![1.3 Structure of a Model 11
0
24
25
40
41
60
65
115
126
208
260
360
400
Present Day
Top Oligocene
Top Lutetian
Top Upper Cretaceous
Top Lower Cretaceous
L.Cret.Unconf.
Top Shublik
Top Lisburne
Top Basement
Base Basement
Kingak
Shublik
Lisburne
Basement
Brookian D
Brookian C
Brookian B
Brookian A
EROSION
EROSION
EROSION
BrFac D
BrFac C
BrFac B
BrFac A
Kingak_Facies
Shublik_Facies
Lisburne_Facies
Basement_Facies
none
none
none
Erosion3
Erosion2
Erosion1
Age
[My]
Facies
Maps
Erosion
Maps
Paleo-Water
Depth Maps
PWD_0
PWD_25
PWD_41
PWD_60
PWD_65
PWD_115
...
...
...
PWD_24
PWD_40
... ... ... ...
Horizon Layer
Fig. 1.4. Excerpt from the age assignment table of the Alaska North Slope model
sediment surface of the example model in Fig. 1.3.d acts as a unconformity
and cuts many layers. A simple approach is to construct the missing erosion
amount for each layer separately and to assume uniform erosion during the
time period of erosion. This is illustrated in Fig. 1.3.e with the virtual horizons
of the Brookian formation above the sediment surface. However, in the consid-
ered model it is further known that there were three main erosion periods and
thus the corresponding erosion maps could be constructed (Fig. 1.3.g.). These
maps together with the virtual Brookian horizons yield the erosion amounts
for each of the layers in the three erosion events.
The above model description would have been sufficient, if the Brookian
formation were eroded after complete deposition. In reality, compressional
deformation in the Tertiary produced a fold–and–trust belt resulting in uplift
and erosion and in a broad shift of the basin depocenters from WSW to ENE,
which lead to mixed erosion and deposition events. A schematic description
is illustrated in Fig. 1.5 which is finally realized in the age assignment table
of Fig. 1.4. Note, that each erosion mentioned in the age assignment table
consists of several layer specific maps with the erosion amounts related to
the respective event. Unfortunately, such a complicated behavior is rather
typical than exceptional. Input building tools often provide sophisticated map
calculators with special features to make the construction of erosion maps
easier. A preliminary simulation result of an ongoing Alaska North Slope study
is shown in Fig. 1.6.
The occurrence of salt diapirs requires paleo-thickness maps for the main
phases of salt doming. The reconstruction of the salt layers is usually based
on geometrical principles, in the simplest case the present day thickness map](https://image.slidesharecdn.com/fundamentalsofbasinandpetroleumsystemsmodeling-230616124321-162f3155/85/Fundamentals-of-Basin-and-Petroleum-Systems-Modeling-pdf-25-320.jpg)
![14 1 Introduction to Basin Modeling
0 0.4 0.8 1.2 2.0 km
Salt Thickness
Salt Domes in the Northern Campos Basin
Selected Thickness Maps of the Autochtonous Salt Layer
2
3
4
5
6
7
Cross-section
Cross-section with Salt
Depth
[km] non-salt sediment
autochtonous salt layer
allochtonous salt bodies
Upper
Cret.
Layer
Lower
Tertiary
Layer
Salt Penetration at 32 My
Salt Penetration at 44 My
Salt Penetration at 55 My
Salt Penetration at 90 My
Upper
Bota
Layer
Albian
Layer
Depositon
117 My
100 My Opening
of Salt
Windows
55 My
Present
Day
Selected Salt Penetration Maps of the Layers above the Autochtonous Salt Layer
Fig. 1.7. Paleo–salt maps: example from the Northern Campos Basin in Brazil
The interplay of paleo-water depth, erosion, salt thickness, and other paleo-
thickness maps finally determines the paleo-geometries and often requires
some experience of the basin modeler to build geological reasonable scenarios.
Boundary Conditions
Boundary conditions need to be defined for the heat, pressure, and fluid flow
analysis through the entire simulated geologic history. The usual boundary
condition data for the heat flow analysis are temperature maps on the sedi-](https://image.slidesharecdn.com/fundamentalsofbasinandpetroleumsystemsmodeling-230616124321-162f3155/85/Fundamentals-of-Basin-and-Petroleum-Systems-Modeling-pdf-28-320.jpg)
![2.2 Terzaghi Type Models 39
pl(h) = g
h
hs
ρb dz onshore
pl(h) = g
hw
0
ρsea dz + g
h
hw
ρb dz offshore
(2.6)
where hs is the sediment surface. The integral over the weight of overburden
sediments can be replaced by a sum of the weights of the single layers with
thicknesses di (i is the layer number), rock densities ρri, and and porosities
φi.
pl(z) = ρseaghw + g
n
i=1
di [ρwφi + ρri (1 − φi)] . (2.7)
For a homogeneous sediment column with a constant rock density ρr equa-
tion (2.7) can further be simplified as follows:
pl(h) = gρseahw + gρr(h − hw) − g(ρr − ρw)
h
hw
φdz , onshore
pl(h) = gρr(h − hs) − g(ρr − ρw)
h
hs
φdz , offshore.
(2.8)
The remaining integral in the above equation is the weight percentage of
water in the overlaying sediment column.
At larger depths, the term (1 − φ) does not significantly change, which
means the curve tends toward a straight line for a unique sediment type. Litho-
static pressure curves for shale and sandstone, for hydrostatic compaction with
compaction parameters of Fig. 2.8, are shown in Fig. 2.4. The term lithostatic
potential ul is used for the lithostatic pressure minus hydrostatic pressure
ul = pl − ph.
Pore Pressure Equation
The pore pressure equation is a one phase fluid flow equation based on the
mass balance of pore water. A flow equation relates driving forces with flow
rates. The driving force for pore water flow is the overpressure gradient. Darcys
law establishes a linear relationship between the discharge velocity v of the
pore fluid and the overpressure gradient ∇u assuming relatively slow flow
for a Newtonian fluid. The proportionality factor is the mobility μ = k/ν,
which is a function of the rock type dependent permeability k and the fluid
dependent viscosity ν.
v = −
k
ν
∇u . (2.9)
This flow equation is an analogy to Fourier’s equation of heat flow, which
similarly relates temperature gradient and heat flux with the thermal con-
ductivity tensor. The permeability tensor is often simplified using only two](https://image.slidesharecdn.com/fundamentalsofbasinandpetroleumsystemsmodeling-230616124321-162f3155/85/Fundamentals-of-Basin-and-Petroleum-Systems-Modeling-pdf-53-320.jpg)
![52 2 Pore Pressure, Compaction and Tectonics
a) b)
0 10 20 30 40 50 60
0
10
20
30
40
50
60
70
Effective Stress in MPa
Porosity
in
%
1
6
6
1
0 1000 2000 3000 4000 5000 6000
0
10
20
30
40
50
60
70
Depth in m
Porosity
in
%
1
6
6
1
1..100% Shale
2..80% Shale; 20% Sandstone
3..60% Shale; 40% Sandstone
4..40% Shale; 60% Sandstone
5..20% Shale; 80% Sandstone
6..100% Sandstone
1..100% Shale
2..80% Shale; 20% Sandstone
3..60% Shale; 40% Sandstone
4..40% Shale; 60% Sandstone
5..20% Shale; 80% Sandstone
6..100% Sandstone
Fig. 2.14. Compaction curves for mixtures of shale and sandstone using arith-
metic averages of all compaction parameters. (a) Athy’s depth model with shale
φ0 = 0.70, k = 0.83 km−1
and sandstone φ0 = 0.41, k = 0.31 km−1
. (b) Schneider
model with shale φ0 = 0.70, φa = 0.35, ka = 0.1916 MPa−1
, kb = 0.0527 MPa−1
and
sandstone φ0 = 0.42, φa = 0.205; ka = 0.0416 MPa−1
, kb = 0.0178 MPa−1
The viscosity of saline water can be estimated from McCain Jr. (1990) in
Danesh (1998) as follows:
ν = νT (0.9994 + 4.0295 × 10−5
P + 3.1062 × 10−9
P2
)
νT = T−a
(109.547 − 8.40564 s + 0.313314 s2
+ 8.72213 × 10−3
s3
)
a = 1.12166 − 2.63951 × 10−2
s + 6.79461 × 10−4
s2
+5.47119 × 10−5
s3
− 1.55586 × 10−6
s4
(2.39)
with ν in mPa s, T in ◦
F, P in psi, s in mass %, and the validity intervals for
νT of 38◦
C T 200◦
C and s 26%, and for ν of 30◦
C T 75◦
C, and
P 100 MPa.
Another formulation was published by Hewlett-Packard (1985) in Mc Der-
mott et al. (2004).
ν = ν0 [1 − 1.87 × 10−3
s0.5
+ 2.18 × 10−4
s2.5
+(T0.5
− 0.0135 T)(2.76 × 10−3
s − 3.44 × 10−4
s1.5
)]
ν0 = 243.18 × 10−7
10247.8/(TK−140)
[1 + 1.0467 × 10−6
P (TK − 305)]
(2.40)
with ν in mPa s, T in ◦
F, TK in K, P in bar, s in %, and the validity interval
0◦
C T 300◦
C, s 25% and P 430 ◦
C. In the published equation P is](https://image.slidesharecdn.com/fundamentalsofbasinandpetroleumsystemsmodeling-230616124321-162f3155/85/Fundamentals-of-Basin-and-Petroleum-Systems-Modeling-pdf-66-320.jpg)
![2.2 Terzaghi Type Models 53
the difference between the real and the saturation pressure, but the latter one
can be neglected for geological conditions.
Both formulations yield similar results for moderate pressures (Fig. 2.15).
The uncertainties of mobilities are controlled more by permeability than vis-
cosity. Hence, the following simplification of equation (2.39) without salinity
and pressure dependence is also proposed here (dotted curve inFig. 2.15.a).
a) b)
0 50 100 150 200 250 300
0
0.5
1
1.5
2
2.5
Temperature in o
C
Viscousity
in
mPa
s
1
4
0 50 100 150 200 250 300
0
0.5
1
1.5
2
2.5
Temperature in o
C
Viscousity
in
mPa
s
1
2
3
4
no pressure, no salinity
1..p = 0.2 MPa/K
2..p = 0.3 MPa/K
3..p = 0.5 MPa/K
4..p = 1.0 MPa/K
Fig. 2.15. Pressure and temperature dependent water viscosity curves assuming
a salinity of 10% after (a) McCain Jr. (1990) in Danesh (1998) and (b) Hewlett-
Packard (1985) in Mc Dermott et al. (2004). The pressure and salinity independent
curve for the simplified equation (2.41) is the dashed curve in (a)
ν[cp] = 109.5 T−1.122
(2.41)
with temperature T in ◦
F. The viscosity of liquid and vapor petroleum is
dependent on its composition. Viscosity ranges for standard oils and gases
and more sophisticated methods for calculating oil and gas viscosities from
compositions are described in (Sec. 5.6.5).
Permeability
Permeability consists of two factors namely rock (intrinsic) permeability and
relative permeability, the latter one is further described in Sec. 6.3. The in-
trinsic permeability k is mainly affected by the pore structure, especially pore
throat diameters and pore connectivity. Hence, it is dependent on the com-
paction state and usually tabulated as a function of porosity (App. A). The
unit of permeability is Darcy (1 D = 0.98692×10−12
m2
), or millidarcy (mD),
but logarithm millidarcy (log mD) is also used, since permeability values
often vary over orders of magnitude with decreasing porosity (1 log mD =
10 mD, 0 log mD = 1 mD, −1 log mD = 0.1 mD, −2 log mD = 0.01 mD, . . .).](https://image.slidesharecdn.com/fundamentalsofbasinandpetroleumsystemsmodeling-230616124321-162f3155/85/Fundamentals-of-Basin-and-Petroleum-Systems-Modeling-pdf-67-320.jpg)
![72 2 Pore Pressure, Compaction and Tectonics
αw = 5 × 10−4
K and rock αr = 3.3 × 10−5
K.
fa = α
∂T
∂t
(2.63)
which yields the following estimation for the total load source:
Fa =
t
α
∂T
∂t
dt ≈ Δφ ≈ 0.07 (2.64)
with an assumed temperature T = 150◦
C in a depth of 5 km. Assuming, the
overburden is not converted to overpressure for early sedimentation, a value
of 0.05 is more realistic. It is one order of magnitude smaller than the effect
from the overburden load. The aquathermal pressure formation depends on
the heating rate and it becomes higher for fast burial.
Mineral Transformations
Some mineral transformations, such as smectite to illite and anhydrite to
gypsum conversion are related to pore fluid volumes changes. The conversion
from smectite to illite occurs in all shales and shaly rocks and is described as
a complex multi–stage process (Pytte and Reynolds, 1989; Swarbrick et al.,
2002). Some of the related processes increase and some decrease the pore
water volume with a general release of bound water into pore space and with
a total increase of the water relative volume up to 5 %. The volume of the
solid matrix is also generally increased, since mainly Na+
ions are exchanged
by K+
ions with a higher ion radius. This process is controlled by temperature
and the availability of K+
ions in the rock matrix. A widely accepted model
was proposed by Pytte and Reynolds (1989) as a fifth–order reaction of the
following type.
∂x
∂t
= −x5
k1 k2 (2.65)
where x is the smectite to illite ratio with an initial value of 0.8, k1 =
74.2 exp(−2490/T[K]) is the chemical activity of potassium to sodium and
k2 = 1.64 × 1021
My−1
exp(−16600/T[K]) is a Arrhenius type temperature
dependence. The equation can be written as a usual unimolecular forward
reaction type (Chap. 4) as follows:
∂x
∂t
= −x5
A e−E/RT
(2.66)
with the activation energy E = 37.9 kcal/mol, the frequency factor A =
1.217 × 1023
My−1
and the gas constant R = 8.31447 Ws/mol/K. The conver-
sion depends on the heating rate, which is controlled by sedimentation rates
(Fig. 2.33). The simple approach is, that the transformation ratio of the re-
action TRs = (x − 0.2)/0.8 is related with a constant factor κ ≈ 0.05 to the
load source fa as follows:](https://image.slidesharecdn.com/fundamentalsofbasinandpetroleumsystemsmodeling-230616124321-162f3155/85/Fundamentals-of-Basin-and-Petroleum-Systems-Modeling-pdf-86-320.jpg)
![2.4 Overpressure Calibration 75
C
∂u
∂t
− ∇ ·
k
ν
· ∇u = C
∂ul
∂t
+ fc(σ
z, T)+
α
∂T
∂t
+ κ
∂TR
∂t
+
1
ρl
−
1
ρk
∂μl
∂t
−
1
ρv
−
1
ρk
∂μv
∂t
.
(2.71)
2.4 Overpressure Calibration
The overall overpressure is mainly determined by mechanical compaction.
Other sources for overpressure, such as chemical compaction or fluid expan-
sion, are often rather localized phenomena and for that reason not included
in this section.
Mechanical compaction, as formulated in (2.13), relates pore water flow
with porosity reduction and overpressure. An overpressure calibration is there-
fore a calibration of compressibility and permeability. It can be performed in
two major steps. The first step deals with the adjustment of rock compress-
ibilities and the second with permeabilities.
Compressibility is introduced via a relationship of effective stress and
porosity in (2.17). Effective stress is defined as the difference σ
= σ − pI.
Relationship (2.17) describes a local property of the rock, and does not con-
tain permeability. If porosity is known, a compaction model such as Athy’s
law or the Schneider model can be fitted to each lithology in the following
way: porosity and pressure value pairs of the same lithology are collected for
different depths and locations. It is possible to calculate the corresponding
overburden from the basin model for these points. Thus, effective stress can
be evaluated and plotted against porosity (Fig. 2.35). Finally, a compaction
model with an effective stress versus porosity formulation, is fitted against
these data points.
Fig. 2.35. Fit of Athy’s law in porosity–effective
stress formulation against a few data values Porosity [%]
Effective
Stress
[MPa]
Calibrated
Not
Calibrated](https://image.slidesharecdn.com/fundamentalsofbasinandpetroleumsystemsmodeling-230616124321-162f3155/85/Fundamentals-of-Basin-and-Petroleum-Systems-Modeling-pdf-89-320.jpg)
![106 3 Heat Flow Analysis
39.8 C 62.5 C
o o
Dz
=
700
m
z
T
q
D
D
= l
DT = 22.7 K
= 48 mW/m
2
Vertical rock thermal conductivities: Shale =1.70
Sandstone =3.95 - 0.01* (T[° ]-20)
Lime
l
l
W/m/K
C W/m/K
stone =3.00 - 0.001*(T[° ]-20)
l C W/m/K
= 1.48 W/m/K
22.7 K / 700 m
x
Heat Flow q = 48 mW/m
2
Fig. 3.2. 1D steady state example of a simplified model of a North Sea well. The
bulk conductivity of the Hordaland Shale (1.48 W/m/K) is geometrically averaged
from rock with λr = 1.70 W/m/K and water with λw = 0.7 W/m/K at a porosity
of 15.6 %
The temperature–versus–depth curve clearly shows intervals of steep and
low increases due to low and high values of bulk thermal conductivities. This
simple steady state argumentation shows that a heat flow analysis which based
on a constant thermal gradient is a very rough approximation.
However, the heat flow from the base upward does not remain constant
because sediments contain radioactive elements like uranium, thorium and
potassium. Radioactivity causes additional heat production, which increases
the heat flow through the sediments. Thus, the surface heat flow is higher
than the basal value by the amount of generated heat. Each of the radioactive
elements generates gamma rays with radiogenic heat production rates Qr
estimated by Rybach (1973) as follows:
Qr = 0.01 ρr (9.52 U + 2.56 Th + 3.48 K) (3.5)
where ρr is the rock density in kg/m3
, U and Th are the concentration of
uranium and thorium in ppm, K is the concentration of potassium in % and
Qr is in μW/m3
. Pore fluids do not contribute to radioactive heat production.
The resulting increase of vertical heat flow Δq in a layer of thickness h due
to a rock heat production rate Qr is as follows:](https://image.slidesharecdn.com/fundamentalsofbasinandpetroleumsystemsmodeling-230616124321-162f3155/85/Fundamentals-of-Basin-and-Petroleum-Systems-Modeling-pdf-119-320.jpg)
![108 3 Heat Flow Analysis
where c and ρ are bulk values, which should be arithmetically mixed from the
pore fluid and rock values corresponding to the actual porosity. The transient
effect is important during deposition and erosion and when thermal boundary
conditions, namely SWI temperatures or basal heat flow, change rapidly.
Deposition results in deeper burial and higher temperatures of underlaying
sediments. The actual sediment temperatures are lower than for the steady
state solution. Heat is absorbed for heating of the layer and heat flow values
decrease towards the surface. In case of constant deposition without com-
paction the heat flow decreases linearly in vertical direction (Fig. 3.4.a and
b). It can indeed be analytically proven, that a constant deposition rate S
yields a constant heat flow gradient after a relatively short time of deposition,
according to (F.24) with
∂q
∂z
≈
qSρc
λ
. (3.8)
The example values used in Fig. 3.4 result in a gradient of
∂q
∂z
[mW/m2
/km] = 3.678 S[km/My] . (3.9)
In the case of erosion the effect is reverse: the sediment temperatures are
higher than the steady state solution and the heat flow increases toward the
surface. The above rule (3.8) can also be applied to estimate the magnitude
of erosion–induced heat change (Fig. 3.4.c). During a hiatus the heat flow
gradually returns to the steady state solution, with a constant heat flow value
in regions without radioactivity.
The transient effect of an instantaneous heat flow change is shown in
Fig. 3.5. Herein, the basal heat flow jumps from 40 mW/m2
to 60 mW/m2
while the near surface heat flow change is delayed for more than 5 Ma. Simi-
lar examples can be calculated analytically (App. F).
A change in SWI temperature also yields transient heat flow curves
(Fig. 3.6). Typical SWI temperature variations yield much lower magnitudes
compared to those caused by basal heat flow changes. Here, SWI or surface
temperatures are average values over a range of 1000 to 10000 years. Sea-
sonal cycles or short time changes are not taken into account. Sudden surface
temperature rises lower the temperature difference between top and base and
yield lower surface heat flow values. This effect occurs in glacial intervals of the
quaternary period as shown in Fig. 3.21. The presented model uses small time
steps of 2000 years for the approximation of the surface temperature variation.
The temperature increases are especially steep and cause downward trends in
the surface heat flow. Generally, near surface heat flow is correlated with the
average surface temperatures of the previous thousands of years.
The heat flow is constant from base to top for 1D solutions without ra-
dioactivity and transient effects. Erosion and radiogenic sediments increases
the surface heat flow, while deposition lowers surface heat flow. A transient
solution of the 1D example model in Fig. 3.2 and Fig. 3.3 is shown in Fig. 3.7.](https://image.slidesharecdn.com/fundamentalsofbasinandpetroleumsystemsmodeling-230616124321-162f3155/85/Fundamentals-of-Basin-and-Petroleum-Systems-Modeling-pdf-121-320.jpg)
![3.3 Thermal Conductivity 111
-200 -150 -100 -50 0
0
5
10
15
20
25
30
Temperature
in
Celsius
1
2
a) b) c)
d) e)
-200 -150 -100 -50 0
0
20
40
60
80
Geological Time [Ma]
Heat
Flow
in
mW/m2
Geological Time [Ma]
Geological Time [Ma]
1..Earth Surface
2..Sediment-Water-Interface (SWI)
Fig. 3.7. 1D transient solution of a simplified model of a well in the North Sea.
(a) Present day heat flow without radioactivity. (b),(c) Present day heat flow and
temperature with radioactivity. (d) Temperatures during burial history and SWI
temperature trend with radioactivity. (e) Heat flow values during burial history and
basal heat flow trend with radioactivity](https://image.slidesharecdn.com/fundamentalsofbasinandpetroleumsystemsmodeling-230616124321-162f3155/85/Fundamentals-of-Basin-and-Petroleum-Systems-Modeling-pdf-124-320.jpg)
![3.4 Specific Heat Capacity 117
The specific heat capacity or specific heat is defined as heat capacity per mass
c = Cp/m.
The unit of specific heat is J/kg/K. The specific heat of a rock sample
is measured by determining the temperature changes and the corresponding
amount of heat entering or leaving the sample. The specific heat capacity is
therefore the storage capacity for heat energy per unit mass. The ratio of
the heat capacity and the thermal conductivity is a measure of the transient
effect.
Heat capacity also controls the magnitude of convection as it determines
how much of the stored heat can be moved together with a moving body. The
specific heat capacity is a volumetric type property (Sec. 8.3). Rock and fluid
component values are mixed arithmetically.
3.4.1 Rock and Mineral Functions
Specific heat capacities depend on temperature. The 20◦
C values c20 for min-
erals and some standard lithotypes are tabulated in App. E.
Waples Model
The temperature dependency of the heat capacity for any mineral, lithology,
or rock value except kerogen and coal can be calculated using the following
equation, which has been adopted from Waples and Waples (2004a) and is
shown in Fig. 3.12.a.
c(T) = c20 (0.953+2.29×10−3
T −2.835×10−6
T2
+1.191×10−9
T3
) (3.21)
where c20 is the heat capacity at 20◦
C and the temperature is given in ◦
C.
Waples and Waples (2004a) also proposed a special function for heat capacity
of kerogen and coal as follows:
c(T)[J/kg/m] = 1214.3 + 6.2657 T − 0.12345 T2
+ 1.7165 × 10−3
T3
− 1.1491 × 10−5
T4
+ 3.5686 × 10−8
T5
− 4.1208 × 10−11
T6
.
(3.22)
Linear Dependency Model
The temperature dependency can be approximated with a simple linear func-
tion.
c(T) = c20 + cT (T − 20◦
C) (3.23)
where c20 is the heat capacity at 20◦
C, cT is the heat capacity increase per
degree temperature. Default lithological values for c20 and cT are tabulated
in App. E for a temperature interval from −20◦
C to 300◦
C.](https://image.slidesharecdn.com/fundamentalsofbasinandpetroleumsystemsmodeling-230616124321-162f3155/85/Fundamentals-of-Basin-and-Petroleum-Systems-Modeling-pdf-130-320.jpg)
![118 3 Heat Flow Analysis
a) b)
0 50 100 150 200 250 300 350
0
500
1000
1500
2000
Temperature in °C
Heat
Capacity
in
J/kg/K
0 50 100 150 200 250 300 350
0
1000
2000
3000
4000
5000
Temperature in °C
Heat
Capacity
in
J/kg/K
Water
Coal
Gas Hydrate Oil
Gas
Ice
Fig. 3.12. Heat capacity for: (a) rocks, (b) coal and fluids
3.4.2 Pore Fluid Functions
Somerton (1992) developed equations for the specific heat capacity of pure
water cw as a function of temperature. For a relatively constant density the
heat capacity decreases linearly to 290◦
C followed by a strong decrease at
higher temperatures according to
ρ cw =
⎧
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎩
4245 − 1.841 T for T 290◦
C
3703 exp[−0.00481 (T − 290)+
2.34 × 10−4
(T − 290)2
] for T 290◦
C
(3.24)
with cw in J/m/K, ρ in kg/m3
and T in ◦
C. The above described temperature
dependent function is shown in Fig. 3.12.b assuming a constant water density
of 1040 kg/m3
. According to Kobranova (1989) the specific heat of saline water
is slightly lower than that of pure water.
Gambill (1957) published the equation below for the specific heat capacity
of oil co as a function of temperature and density:
co = (1684 + 3.389 T)
√
ρ (3.25)
with co in J/m/K, ρ in kg/m3
, and T in ◦
C. It is satisfactory to use a value for
the specific heat capacity of natural gas for all compositions at all tempera-
tures and pressures. A reasonable value is 3250 J/kg/K (Waples and Waples,
2004b). Waples also proposed the following equation for the heat capacity of
gas hydrates chin the same paper.
ch = 2097 + 7.235 T + 0.0199 T2
(3.26)
with ch in J/m/K and T in ◦
C. The specific heat capacity of ice (2115 J/kg/K)
is about half that of liquid water for temperatures around 0 ◦
C. The pore and
bulk values of heat capacity values are mixed arithmetically.](https://image.slidesharecdn.com/fundamentalsofbasinandpetroleumsystemsmodeling-230616124321-162f3155/85/Fundamentals-of-Basin-and-Petroleum-Systems-Modeling-pdf-131-320.jpg)
![3.6 Three Dimensional Heat Flow Equation 119
3.5 Radiogenic Heat
Some minerals and rocks contain traces of the radioactive elements uranium
(U), thorium (Th) and potassium (K) which are additional heat sources (3.5).
Measurements and data catalogs are sparse, but D. Waples developed a data
base for most rocks and minerals based on modeling experience and some
literature data (App. E). These values should be used carefully especially
since the effects on heat flow calculations are tremendous as was discussed
in Sec. 3.2.1. Radiogenic heat values for lithologies are given as rock matrix
values and are converted to bulk values during simulation by multiplying by
(1 − φ). They can be derived from the following data sources:
• uranium, thorium and potassium concentrations from spectral gamma ray
measurements. These values are bulk values with a core sample porosity
φc. One has to use Rybachs law (3.5) and divide by (1 − φc) to get the
corresponding matrix heat flow production rate Qr.
• Gamma Ray API values. Buecker and Rybach (1996) proposed the fol-
lowing law to convert Gamma Ray APIs to matrix heat flow production
values:
Qr[μW/m3
] = 0.0158 (API − 0.8) . (3.27)
Concentrations of radioactive elements are present-day values U0, T0, K0.
Paleo-values are higher corresponding to their half-lives as follows (with t in
My).
U = U0 (1 + 2.77 × 10−4
t − 7.82 × 10−8
t2
+ 4.53 × 10−12
t3
)
Th = Th0 exp(0.00005 t)
K = K0 exp(0.000555 t) .
(3.28)
Uranium consists of two isotopes having different half–lives. The sum of the
two exponential functions is approximated by a third order polynomial. The
time correction is small on geological time scales.
3.6 Three Dimensional Heat Flow Equation
In 1D models, heat flows vertically upward from base to top. In multi–
dimensional problems, heat flow can also laterally divert to follow layers of
high thermal conductivity. The formulation of the multi-dimensional heat flow
problem yields a transport–type differential equation with temperature as
the field variable and heat flow as the corresponding flow variable. The heat
transport equation is based on energy balances, which means that the tem-
perature induced internal energy change in a volume element is equal to the
heat conducted into or out of the volume element plus the heat transferred
by convection plus radiogenic heat production. It is](https://image.slidesharecdn.com/fundamentalsofbasinandpetroleumsystemsmodeling-230616124321-162f3155/85/Fundamentals-of-Basin-and-Petroleum-Systems-Modeling-pdf-132-320.jpg)
![3.8 Crustal Models for Basal Heat Flow Prediction 135
from crustal layer thicknesses. The principle is illustrated for two very simple
two layer (crust and mantle) models in Fig. 3.28: the Airy and the Platt model.
The Airy model supposes a constant density for the entire crust. Thus, the
mountain height or water depth is a simple function of the total crust thick-
ness. Hence, the high mountains are above thick crust and large water depths
suggest a thin crust below. The Platt model presumes a crust of varying den-
sity at the same depth level of the asthenosphere. Thus, the surface elevation
is a function of the crustal density only: the higher the mountain the lighter
the crust below. These examples illustrate the principle of a hydrostatic litho-
sphere: each vertical column of lithosphere is able to move independently of
the adjacent column to balance itself via its own weight.
rc
a) b)
Crust
rm
Mantle
h1
b1
Sea level
Water
h2
b2
Onshore
(rm- )
r = r
c 1 c 1
b h Offshore
(rm- ) - )
r = (r r
c c w
b h
2 2
rw
c)
d)
rm
rc r1
r2
h1
h2
Crust
Mantle
Onshore
(rc- )
r = r
1 b h
1 1 1 Offshore
(rc- ) - )
r = (r r
2 2 2
b h
c w
Water
b1
b2
Fig. 3.28. Isostatic compensations: (a) Airy compensation. (b) Platt compensation.
(c) Hydrostatic isostasy. (d) Flexural isostasy
In reality, there is an influence from the connected areas, which is de-
scribed as flexural compensation, but which is not considered here. However,
the weakness of the above models is the assumption of constant density with
depth, since higher temperatures lower the density by thermal expansion ac-
cording to
ρ(T) = ρ0 [1 − α(T − T0)] (3.37)
with the linear expansion factor α = 3.28 × 10−5
/◦
C (McKenzie, 1978) and
a reference density ρ0 for surface temperature T0 = 20 ◦
C. Another conse-
quence of this equation is that cooling of the lithosphere causes subsidence
and warming leads to uplift.
Uniform Stretching Model
The hydrostatic equation of isostasy applied to the uniform stretching model
yields the following total mass per unit area for a column of water, crust, upper
mantle and asthenosphere above a reference depth in the asthenosphere at any
time (Fig. 3.26).](https://image.slidesharecdn.com/fundamentalsofbasinandpetroleumsystemsmodeling-230616124321-162f3155/85/Fundamentals-of-Basin-and-Petroleum-Systems-Modeling-pdf-148-320.jpg)
![136 3 Heat Flow Analysis
m = ρw hw + ρc0
dc
hw
[1 − αcT(x)]dx + ρm0
dm
dc
[1 − αmT(x)]dx
+ ρm0[1 − αm(Ta − Tswi)] ha
(3.38)
where αc, αm are the thermal expansions of the crust and mantle, and dc, dm
are the depths of base crust and mantle, and hw, ha are the thicknesses of
water and asthenosphere.
The tectonic water depths after instantaneous stretching hw1
and after
inifinite cooling hw2
can be analytically calculated with the assumptions of
no crustal radioactive heat production, a unique stretching factor β for crust
and mantle, equal and constant thermal properties of the crust and mantle.
and an linearly increase of the temperature with depth(Jarvis and McKenzie,
1980).
hw1
=
(hm+hc)[(ρm0−ρc0) hc
hm+hc
(1−αTa
hc
2hm+2hc
)−
αTaρm0
2 ](1− 1
β )
ρm0(1−αTa)−ρw
,
hw2
= (ρm0−ρc0)hc
ρm0(1−αTa)−ρw
1 − 1
β
− αTahc
2hm+2hc
1 − 1
β2
.
(3.39)
The subsidence during stretching is related to an inflow of additional heavy
asthenospheric material, while additional subsidence is caused by the cooling
of the entire column. Again it should be noted, that the above premises, espe-
cially the assumption of instantaneous stretching as seen in equation (3.39) are
drastic simplifications. More comprehensive equations should be used instead.
Such subsidence curves through geological time are illustrated in Fig. 3.29 for
various stretching factors. In Figs. 3.30 and 3.31 radioactive heat production
in the crust is taken into account. The subsidence after stretching hw1
can
become negative (uplift), when crustal stretching is very small compared to
mantle stretching, as shown in the example in Fig. 3.27.a at the margin of the
pure shear model and in Fig. 3.32.b.
The subsidence is much larger, when the weight of the sediments is taken
into account, in contrast to pure water filled basins. Total subsidence, which
is the real subsidence with sediments is different from the tectonic subsidence,
which is the theoretical subsidence for water fill only. The relation between
tectonic subsidence hw and total subsidence ht is as follows:
ht =
ρa − ρw
ρa
hw +
1
ρa
n
i=1
ρsi hsi (3.40)
where ρsi and hsi are the density and thickness of the i-th sediment layer.
Sometimes, a basement is introduced between the upper crust and the
sediments, which is the sediment package before stretching. It has to be de-
termined whether the total and tectonic subsidence is then understood as the
top or bottom basement.](https://image.slidesharecdn.com/fundamentalsofbasinandpetroleumsystemsmodeling-230616124321-162f3155/85/Fundamentals-of-Basin-and-Petroleum-Systems-Modeling-pdf-149-320.jpg)
![142 3 Heat Flow Analysis
a)
b)
0
20
40
60
80
100
120
140
0
2
4
6
8
10
Geological Age in My
Subsidence
in
km
1
2
3
0
20
40
60
80
100
120
140
0
20
40
60
80
100
120
Geological Age in My
Heat
Flow
in
mW/m
2
Tectonic Subsidence Map
Present Day Heat Flow Map
c)
d)
42
40
38
38
36
34
e) f)
1.5
2.0
2.5
3.0
1.0
1.5
2.5
2.0
3.0
2.0
3.0
4.0
5.0
6.0
5.0
3.0
Stretching factor map of the crust Stretching factor map of the mantle
1..Total subsidence
2..Tectonic subsidence
3..Theoretical subsidence
Stretching Cooling
[mW/m2]
[m]
Fig. 3.35. Example from the Northern Campos basin for crustal heat flow analysis
with a rift period of 132 − 113 My, hc = 35 km, hm = 95 km, Qr0 = 2.5μW/m3
,
zh = 7 km: (a) Total and tectonic subsidence from input geometry and calculated
theoretical subsidence after calibration at the location of the map midpoint. (b)
Calculated heat flow at the basin midpoint with the calibrated stretching factors
βc = 3.0, βm = 4.7. (c) Tectonic subsidence map from input geometry. (d) Calcu-
lated present day heat flow map. (e),(f) Maps with calibrated stretching factors for
the crust and mantle](https://image.slidesharecdn.com/fundamentalsofbasinandpetroleumsystemsmodeling-230616124321-162f3155/85/Fundamentals-of-Basin-and-Petroleum-Systems-Modeling-pdf-155-320.jpg)
![4.2 Distributed Reactivity Kinetics 155
Fig. 4.4. Influence of the heating
rate on first order Arrhenius type
reactions. The first peak is related
to geological heating rates 10 K/My,
while the other two peaks are labora-
tory heating rates with 0.1 K/min and
5 K/min
0 100 200 300 400 500 600
0
1
2
3
4
5
6
Temperature in o
C
10 K/Ma A = 10
E = 55 kcal/mol
14
s-1
5 K/min
0.1 K/min
Transformation
/
Heating
Rate
in
%/K
Xi
ki
−→ Y,
∂xi
∂t
= −ki xα
,
∂y
∂t
=
n
i=1
kixα
. (4.3)
Each of the subreactions i can be described with a pair (Ai, Ei), but usually
a unique frequency factor is used, which results in a discrete distribution of
activation energies p(Ei) equal to the initial masses of the reactants x0i =
p(Ei). The transformation ratio is TR = 1 − x = y with x =
i xi. It also
varies between 0 and 1.
Parallel reactions can be used for the decomposition of complex macro–
molecules having a wide range of chemical bond strengths to one type of
cracking product. Herein, the initial masses x0i correspond to the mass of the
reactant related to the cracking of chemical bonds of the corresponding acti-
vation energy levels. However, the chemistry of the decomposition of organic
matter is generally more complex and the use of parallel reactions is just an
empirical formalism for compositional effects of reactants and products.
Arbitrary discrete distributions p(Ei) are mainly used for petroleum and
coal formation. Approximations to continuous Gaussian distributions are also
popular. Gaussian distributions are described with a mean activation energy
μ and a variance σ as follows (Chap. 7):
p(E) =
1
σ
√
2π
exp
−(E − μ)2
2σ2
. (4.4)
Burnham and Braun (1999) reported the usage of two other continuous
distributions: the Gamma distribution
p(E) =
aν
(E − γ)ν−1
Γ(ν)
e−(E − γ)a (4.5)
with parameters ν, a and threshold activation energy γ, and the Weibull
distribution
p(E) =
β
η
E − γ
η
β−1
e− [(E − γ)/η]
β
(4.6)](https://image.slidesharecdn.com/fundamentalsofbasinandpetroleumsystemsmodeling-230616124321-162f3155/85/Fundamentals-of-Basin-and-Petroleum-Systems-Modeling-pdf-168-320.jpg)
![156 4 Petroleum Generation
with the width parameter η, the shape parameter β, and again a threshold
activation energy γ. The mean energy of the Weibull distribution is Ē =
γ + ηΓ(1/β + 1), (Fig. 4.5).
Activation Energy Distribution [kcal/mol]
Probability
80%
10% 10%
50.2659 51.9449
50 51 52 53
0
0.2
0.4
0.6
Activation Energy Distribution [kcal/mol]
Probability
80%
10% 10%
50.6492 53.0349
50 51 52 53 54 55
0
0.1
0.2
0.3
0.4
0.5
Fig. 4.5. Examples of Gamma (left) and Weibull (right) distributed activation
energies. Here it is ν = a = η = β = 2 and γ = 50 kcal/mol for both distributions
Equation (4.3) can be numerically integrated in time for stepwise constant
heating rates, which yields the following mass reduction from time step l to
l + 1 with duration Δt and temperature T(l)
:
x
(l+1)
i = x
(l)
i +
Δtki(T(l)
)
α − 1
for α = 1,
x
(l+1)
i = x
(l)
i
2 − Δtki(T(l)
)
2 + Δtki(T(l))
else .
(4.7)
The extension of the above approach to the formation of multiple prod-
ucts is usually performed by superposition of single kinetic reaction schemes,
which yields sequential and parallel connected super–schemes. However, the
actual processing resembles a bookkeeping of all reactions that are defined in
the source rock according to the organic facies. The computing effort is low
compared to other processes; e.g. heat and fluid flow calculations. It increases
linearly with the number of cells, but the required computer memory can be-
come significant as one value for each of the discrete activation energies of
each mass component has to be stored for each cell.
4.3 Petroleum Generation Kinetics
The total content of organic matter (kerogen and bitumen) is usually given in
terms of the total organic carbon content (TOC) in mass %. It is the ratio of
the mass of all carbon atoms in the organic particles to the total mass of the
rock matrix. Hence, it is a concentration value and one needs the total rock](https://image.slidesharecdn.com/fundamentalsofbasinandpetroleumsystemsmodeling-230616124321-162f3155/85/Fundamentals-of-Basin-and-Petroleum-Systems-Modeling-pdf-169-320.jpg)
![172 4 Petroleum Generation
the kinetics is approximated by superposition of the four reactions with only
one frequency factor. The final distribution of the activation energies is shown
in Fig. 4.19. The reflection value of the Easy–Ro model is then exponentially
correlated with the TR to the interval [0.20%, 4.66%], as follows:
Ro[%] = 0.20
4.66
0.20
TR
. (4.26)
Carr (1999) extended the above model to incorporate overpressure u retarda-
tion, as there is an expansion of the products due to the generation of volatiles.
The pressure dependency is proposed to be included via a modified frequency
factor A(u).
A(u) = Ah e−u/c (4.27)
where Ah = 3.17×1026
My−1
is the original frequency factor from the Easy–Ro
model, defined here as the hydrostatic frequency factor, u is the overpressure,
and c = 590 psi is a scaling factor.
34 36 38 40 42 44 46 48 50 52 54 56 58 60 62 64 66 68 70 72
0
2
4
6
8
10
Activation Energy in kcal/mol
Frequency
in
%
30 31 33 35.1 37 38.9 40.7 42.6 44.5 46.3 48.2 50.1 51.9 53.8 56 58
0
5
10
15
20
25
Activation Energy in kcal/mol
Frequency
in
%
Sweeny Burnham Larter
A = 3.17x1026
My-1
A = 2.36x1022
My-1
Fig. 4.19. Distributions of the activation energies used in the vitrinite models of
Sweeney and Burnham (1990) and Larter (1988)
Larter Model
Larter (1988) utilized quantitative pyrolysis gas chromatographic data of iso-
lated vitrinite kerogen to achieve the concentrations of structurally specific
moieties (alkylphenol precursor) in vitrinite as a function of rank. The kinet-
ics describe the phenol precursor loss from vitrinite.
precursor
ki
−→ phenol . (4.28)
The pyrolysis data are inverted into a parallel decomposition reaction with
normal Gaussian distributed activation energies of mean energy Ē = 186 kJ,
standard deviation σ = 13.0 kJ, and frequency factor A = 2.36 × 1022
My−1
.
An approximated discrete distribution is shown in Fig. 4.19. The phenol yield](https://image.slidesharecdn.com/fundamentalsofbasinandpetroleumsystemsmodeling-230616124321-162f3155/85/Fundamentals-of-Basin-and-Petroleum-Systems-Modeling-pdf-185-320.jpg)
![4.4 Thermal Calibration Parameters 173
is then linearly correlated to the interval [0.45%, 1.60%] with vitrinite re-
flectance.
Ro[%] = 0.45 + 1.15 TR (4.29)
where TR is the transformation rate of the parallel reaction. The model limits
the reflection value to 1.6% since the study is based on chemical reactions
primarily in oil generating zones.
TTI Model
This model was proposed by Waples (1980). It calculates TTI (Time–Temper-
ature–Index) maturity, which is converted to vitrinite reflectance. The defini-
tion of the TTI is based on the assumption that the maturity rate of vitrinite
almost doubles every 10◦
C.
TTI =
n
(Δtn) 2n
(4.30)
where the integer n represents a temperature interval with the following
scheme:
n = −1 for T = [90◦
C, 100◦
C],
n = 0 for T = [100◦
C, 110◦
C],
n = 1 for T = [110◦
C, 120◦
C],
n = 2 for T = [120◦
C, 130◦
C] . . . .
(4.31)
he value Δtn is the time in My spent by the vitrinite in the corresponding
temperature interval. The TTI value is converted to vitrinite reflectance with
empirical functions, e.g. with the following rule by Goff (1983).
Ro[%] = 0.06359 (1444 TTI)0.2012
. (4.32)
Although this model is only rule based, it is still used.
Vitrinite reflectance is the most common thermal maturity parameter and
is generally used for calibration of heat flow history. Herein, paleo-heat flow
values as the most uncertain input parameters are changed according to differ-
ences between calculated and measured vitrinite reflectance parameters. This
requires reliable vitrinite reflectance models and measurements.
Calculated vitrinite reflectance models are shown in Fig. 4.20 for different
sedimentation curves. They differ in the initial (immature) value on the surface
and further at greater depth. The most commonly used model is the Burnham
Sweeney model.
Vitrinite reflectance curves commonly have offsets at erosional discor-
dances (Fig. 4.21) as the uplifted layers have already experienced a heating
period with a corresponding increase in maturity. The offset is typically higher
for larger erosional thickness. Thus, jumps in the vitrinite data at a certain
depth can be used to estimate the erosion thickness. The maturation process](https://image.slidesharecdn.com/fundamentalsofbasinandpetroleumsystemsmodeling-230616124321-162f3155/85/Fundamentals-of-Basin-and-Petroleum-Systems-Modeling-pdf-186-320.jpg)
![4.4 Thermal Calibration Parameters 181
generated with a slightly different rate described by a constant shift in the ac-
tivation energy distribution ΔE of 25 cal/mol. The isotopic ratio R of methane
can then be tracked in petroleum migration and accumulation analysis. The
isotopic fractionation can be used to distinguish between biogenic and ther-
mogenic gases.
4.4.5 Fission–Track Analysis
Fission–tracks are trails of disrupted atoms in the crystal structures of vari-
ous minerals caused by radioactivity. The trails can be etched using acid so
that they become visible under a microscope. For example they are found
in apatites and are almost exclusively generated by the spontaneous fission
of the uranium isotope 238
U. It has a decay constant of 4.92 × 10−18
s−1
=
1.55 × 10−4
Ma−1
, which corresponds to a half life of 4471 Ma. Fission–tracks
show a temperature dependent annealing and are thus used to constrain paleo–
temperature histories. Introductions to fission–track analysis are given in Gal-
lagher et al. (1998), and Beardsmore and Cull (2001).
Track length distributions can be predicted in a forward modeling ap-
proach from temperature histories which can easily be extracted from basin
models. These distributions can be compared with sample data for calibration
purposes. In the case of good data quality, the temperature history can even be
evaluated based on an inversion of forward modeling (Ketcham et al., 2000).
This section deals with a short summary of forward modeling of the track
length distributions only. More details can be found in Green et al. (1986);
Laslett et al. (1987); Duddy et al. (1988); Green et al. (1989a,b); Carlson et al.
(1999), and Donelick et al. (1999).
Track length distributions are usually formulated with the reduced track
length r = l/l0 with l the confined track length during annealing and l0 the
initial track length. Forward modeling of track length reduction through time
is described in Laslett et al. (1987). According to their results it is possible to
fit the relationship between time t in seconds and annealing described by r at
constant temperature T in Kelvin by a “fanning” fission–track Arrhenius fit
of the form
ln t = A(r) + B(r)/T (4.35)
with e.g. A = −28.12,
B(r) = [g(r) + 4.87]/0.000168 (4.36)
and
g(r) = ([(1 − r2.7
)/2.7]0.35
− 1)/0.35 . (4.37)
In the case of time dependent temperature T(t) (4.35) can be differentiated
and becomes
dr
dt
=
1
A(r) + B(r)/T(t)
e−A(r)−B(r)/T (t)
+
B(r)
T2
dT
dt
(4.38)](https://image.slidesharecdn.com/fundamentalsofbasinandpetroleumsystemsmodeling-230616124321-162f3155/85/Fundamentals-of-Basin-and-Petroleum-Systems-Modeling-pdf-194-320.jpg)
![186 4 Petroleum Generation
mads =
mads,maxKLc
1 + KLc
(4.44)
with mads,max as the maximum mass which can be adsorbed in one monolayer
on the kerogen surface and KL = kads/kdes as the ratio of reaction rate con-
stants for adsorption and desorption. Obviously, KL is strongly dependent on
the temperature. It may be possible to model kads and kdes with an Arrhenius
law.
The BET theory is an enhancement of (4.44) for the adsorption of multiple
layers of adsorbates. It becomes
mads =
mads,maxKBc
(1 − c) [1 + (KB − 1)c]
(4.45)
with KB similar to KL and mads,max again as the mass of one monolayer.
The Langmuir equation can easily be extended for multiple types of
molecules which might be adsorbed with the assumption, that each molecule
of each type occupies the same space on the adsorbent. The adsorbed amount
of component i becomes
mads,i =
mads,max,iKL,ici
1 +
k KL,kck
(4.46)
with KL,i the Langmuir factor, mads,max,i the mass of one monolayer, and ci
the concentration of species i.
The mass of one monolayer can be estimated to be
mads,max = g m
2/3
ker (4.47)
with a geometrical factor g. It can be assumed that the kerogen is distributed
randomly in the sediments so that the exponent 2/3 should be replaced by
an exponent from a fractal theory. A value about 1 can be interpreted as an
adsorption potential proportional to the number of kerogen atoms, which is
consistent with the Freundlich equation.
In practice, temperature and adsorption kinetics are often ignored. Ad-
sorbed masses are equal to maximum adsorption amounts if enough adsorbates
are available. The maximum adsorbed mass is assumed to be proportional to
the mass of kerogen, which encompasses the actual generative and the inert
kerogen. Two different basic approaches are commonly in use.
Pepper and Corvi (1995b) proposed adsorption of each species independent
from other species (Fig. 4.31). The following formulation is often used:
mads,max,i = ai mker (4.48)
where ai is the adsorption coefficient and mads,max,i the (maximum) adsorp-
tion amount of component i.](https://image.slidesharecdn.com/fundamentalsofbasinandpetroleumsystemsmodeling-230616124321-162f3155/85/Fundamentals-of-Basin-and-Petroleum-Systems-Modeling-pdf-199-320.jpg)
![192 4 Petroleum Generation
Temperature
API Undegraded
API Degraded
Age [Ma]
Volume
[Mm^3]
Fig. 4.37. API and temperature of accumulated petroleum with and without
biodegradation according to (4.54). The filling history with spilling and partial re-
filling is shown at the bottom right. It started at 25 Ma. The first API values are
available for the following time step at 22 Ma
M
e
t
h
a
n
e
E
t
h
a
n
e
P
r
o
p
a
n
e
i
-
B
u
t
a
n
e
n
-
B
u
t
a
n
e
i
-
P
e
n
t
a
n
e
n
-
P
e
n
t
a
n
e
n
-
H
e
x
a
n
e
C
7
-
1
5
C
1
6
-
2
5
C
2
6
-
3
5
C
3
6
-
4
5
C
5
6
+
C
4
6
-
5
5
Fig. 4.38. Composition of undegraded and degraded oil at present day. Note that
the molar fraction of methane changed from 34% to 61%
eration, and expulsion times already gives a first idea about possible reservoir
charge even if migration is not yet considered in detail. The most important
input data and results for a source rock analysis are illustrated in Fig. 4.39
and Fig. 4.40 for a model of the Northern Campos basin. Herein, three source
rocks are defined in the Bota, Coquina and Lagoe Faia formations, each with
primary and secondary kinetics. The source rock kinetics can be based on
measured sample data or equivalent default kinetics from a database. Kinet-
ics related adsorption factors should also be defined for each HC component.
The kerogen content and quality are defined with TOC and HI maps, which](https://image.slidesharecdn.com/fundamentalsofbasinandpetroleumsystemsmodeling-230616124321-162f3155/85/Fundamentals-of-Basin-and-Petroleum-Systems-Modeling-pdf-205-320.jpg)
![5.5 Flash Calculations 217
The performance of flash calculations is very good when a relatively small
number of components are involved. Flash calculations can even be explicitly
performed during solution of differential equations for each grid cell and in
each time step in basin modeling. By experience, the extra amount of time
compared to a symmetrical black oil model is about 10% of total simulation
time.7
Phase diagrams, such as calculated for Fig. 5.14 from more than 10.000
flash calculations can be generated on a modern PC within a few seconds.
However, flash calculations are also very helpful for the modeling of fluid flow
with symmetrical black oil descriptions, because they can be used for the
consistent construction of needed lookup tables (Sec. 5.3).
5.5.1 Classification of Petroleum
Petroleum is commonly classified as dry gas, wet gas, gas condensate, volatile
oil, and black oil (Danesh, 1998). The main classification parameter is the
gas oil ratio (GOR) (Table 5.1). Examples, which are constructed with flash
calculations are shown in Fig. 5.14. The corresponding compositions and com-
ponent properties are listed in App. K.
Class GOR [m3
/m3
] ◦
API Composition
Dry Gas – – CH4 + other light gas comp. only
Wet Gas 10 000 – mainly CH4 + other light gas comp.
Gas Condensate 570 . . . 10 000 40 . . . 60 12.5 mol % C7+
Volatile Oil 310 . . . 570 40 12.5 . . . 20 mol % C7+
Black Oil 310 45 20 mol % C7+
Table 5.1. Classification of petroleum according to Danesh (1998). The API gravity
refers to liquid after outgassing. Black oil is additionally classified by a relatively
low bubble point pressure
5.5.2 PT–Paths
A spectrum of typical geological PT–paths is introduced in Fig. 5.2. Phase
compositions can quickly be evaluated along PT–paths with flash calculations
(Figs. 5.15, 5.16).
In typical geological environments the critical point can be reached. In
this case strong variations of phase compositions might occur within certain
pressure temperature intervals (Fig. 5.15).
7
The number is highly dependent on the fraction of the grid cells that hold HCs
inside.](https://image.slidesharecdn.com/fundamentalsofbasinandpetroleumsystemsmodeling-230616124321-162f3155/85/Fundamentals-of-Basin-and-Petroleum-Systems-Modeling-pdf-230-320.jpg)
![218 5 Fluid Analysis
Dry Gas
C
Wet Gas
C
Gas Condensate
C
Volatile Oil
C
Black Oil
C
Example GOR ◦
API C7+
[m3
/m3
] [mol %]
Dry Gas – – 0.0
Wet Gas 22 452 (79) ≈ 1.5
Gas Condensate 1 997 52 ≈ 4.0
Volatile Oil 473 48 ≈ 13.0
Black Oil 110 32 ≈ 36.0
Fig. 5.14. Five PT phase diagrams which are calculated with the SRK EOS. The
critical point is marked with a “C”. Compositions and component properties are
listed in App. K
5.6 Property Prediction
5.6.1 Density
Density can directly be calculated from the results of a flash calculation. The
molar volume v is known from (5.15) or (5.16) and the composition from
(5.30). Hence, density can be directly calculated from](https://image.slidesharecdn.com/fundamentalsofbasinandpetroleumsystemsmodeling-230616124321-162f3155/85/Fundamentals-of-Basin-and-Petroleum-Systems-Modeling-pdf-231-320.jpg)
![5.6 Property Prediction 221
they are strongly limited in their pressure and temperature ranges. Moreover,
predetermined grouping and lumping of components restricts their usability.
However, the predictions of empirical formulas can be compared to the results
of the more general EOS and used for the calibration of phase properties.
API Method
The API density prediction is based on an available compositional analysis of
the components up to C6 with appropriate component densities and the mea-
sured density of C7+ at standard conditions. The density ρl at temperature
T and pressure p can be evaluated according to the following formulas:
ρl = ρ1
C(288.706 K)
C(T)
, ρ1 =
i
ziMi
i
ziMi/ρ0,i
, (5.38)
with Mi the molecular weight of component i, zi the molar fractions, and ρ0,i
the dissolved component density at standard conditions according to
Component N2 CO2 H2S C1 C2 C3 i–C4 n–C4 i–C5 n–C5 C6
Density [kg/m3
] 804 809 834 300 356 508 563 584 625 631 664
Furthermore it is
C = A1 + A2 T
r + A3 T
r
2
+ A4 T
r
3
with
Ai = Bi1 + Bi2 p
r + Bi3 p
r
2
+ Bi4 p
r
3
+ Bi5 p
r
4
.
(5.39)
The coefficients Bij are defined through
i j 1 2 3 4 5
1 1.6368 −0.04615 2.1138 × 10−3
−0.7845 × 10−5
−0.6923 × 10−6
2 −1.9693 0.21874 −8.0028 × 10−3
−8.2328 × 10−5
5.2604 × 10−6
3 2.4638 −0.36461 12.8763 × 10−3
14.8059 × 10−5
−8.6895 × 10−6
4 −1.5841 0.25136 −11.3805 × 10−3
9.5672 × 10−5
2.1812 × 10−6
and it is T
r = T/T
c, p
r = p/p
c with T
c =
i
ziTci and p
c =
i
zipci.
Standing–Katz Method
Standing and Katz proposed another set of formulas for density calculations
by known oil composition: The density of a C3+ mixture is
ρC3+
=
C7+
i=C3
ziMi
C7+
i=C3
ziMi/ρ0,i
(5.40)](https://image.slidesharecdn.com/fundamentalsofbasinandpetroleumsystemsmodeling-230616124321-162f3155/85/Fundamentals-of-Basin-and-Petroleum-Systems-Modeling-pdf-234-320.jpg)
![5.6 Property Prediction 223
a = λ en/(T +460)
and b = m(T + 460) + C (5.45)
which are tabulated in (5.2). The C7+ component has exceptional parameters
aC7+
= exp
3.8405985 × 10−3
MC7+
− 9.5638281 × 10−4
MC7+
/SC7+
+ 261.80818/(T + 460)
+ 7.3104464 × 10−6
M2
C7+
+ 10.753517
,
bC7+
= 3.499274 × 10−2
MC7+
− 7.2725403 SC7+
+ 2.232395 × 10−4
(T + 460)
− 1.6322572 × 10−2
MC7+
/SC7+
+ 6.2256545
(5.46)
with SC7+
as the specific gravity of the C7+ component. Finally the density
can be calculated easily from the molar volume, the composition, and the
molecular weights of the components.
Component λ n m × 104
C
C1 ( 70 − 300 ◦
F) 9160.6413 61.893223 3.3162472 0.50874303
C1 (301 − 460 ◦
F) 147.47333 3247.4533 −14.072637 1.8326695
C2 (100 − 249 ◦
F) 46709.537 −404.48844 5.1520981 0.52239654
C2 (250 − 460 ◦
F) 17495.343 34.163551 2.8201736 0.62309877
C3 20247.757 190.24420 2.1586448 0.90832519
i–C4 32204.420 131.63171 3.3862284 1.1013834
n–C4 33016.212 146.15445 2.9021257 1.1168144
C5 37046.234 299.62630 2.1954785 1.4364289
C6 52093.006 254.56097 3.6961858 1.5929406
H2S 13200.0 0 17.900 0.3945
N2 4300.0 2.293 4.490 0.3853
CO2 8166.0 126.00 1.8180 0.3872
Table 5.2. Component parameters of Alani–Kennedy EOS
The different methods for calculation of liquid densities are compared in
Fig. 5.17 for a black oil and a volatile oil. Their compositions and component
properties are listed in App. K. The amount and molar weight of the C7+
fraction, which is needed as an input parameter for the API and the Standing–
Katz methods, are calculated by molar average with the following component
densities:
Component C6−14 C15 C25 C35 C45
Density [kg/m3
] 700 800 850 920 940
The direct density calculation from composition and molecular weight in many
examples, deviates a lot from the results of the other methods. This can be](https://image.slidesharecdn.com/fundamentalsofbasinandpetroleumsystemsmodeling-230616124321-162f3155/85/Fundamentals-of-Basin-and-Petroleum-Systems-Modeling-pdf-236-320.jpg)
![5.6 Property Prediction 227
Black Oil 0.3 MPa/ C
°
1 MPa/ C
°
Volatile Oil 0.3 MPa/ C
°
1 MPa/ C
°
Fig. 5.19. Oil formation volume factor Bo calculated from the liquid phase of a
black oil and of a volatile oil. The liquid phase is calculated with the SRK EOS
along two different PT–paths which both end at standard surface conditions
5.6.5 Viscosity
Next to density, viscosity is an important indicator for phase property char-
acterizations. It varies greatly between different oil types (Table 5.3). Due to
the strong dependency of production recovery factors from viscosity it is of
great interest.
Viscosity in [cP] at
Oil Type 50◦
C 100◦
C 150◦
C
very high viscosity 1000 25 2.5
high viscosity 100 2.5 0.5
medium viscosity 10 0.5 0.25
low viscosity 3 0.35 0.1
very low viscosity 1 0.25 0.05
Table 5.3. Revised viscosity values according to Ungerer et al. (1990)
Viscosity is often modeled as a quantity that is dependent only on pres-
sure, temperature, density, and the amount of dissolved gas (Danesh, 1998).
An overview about empirical correlations is given by Bergman and Sutton
(2007). Models taking detailed compositional effects into account are found
in (Pedersen et al., 1984; Pedersen and Fredenslund, 1987; Zéberg-Mikkelsen,
2001). Obviously, the effect of long–chained compounds has a significant in-
fluence on the viscosity. Hence a good characterization of the fluid heavy end
is even more important for viscosity than for density prediction.
A recent comparison of actual models with measurement data can be found
in Zéberg-Mikkelsen (2001). Advanced theories such as the friction–theory or](https://image.slidesharecdn.com/fundamentalsofbasinandpetroleumsystemsmodeling-230616124321-162f3155/85/Fundamentals-of-Basin-and-Petroleum-Systems-Modeling-pdf-240-320.jpg)
![234 5 Fluid Analysis
Liquid–Vapor Interfacial Tension
The liquid–vapor IFT γl,v is usually determined with the parachor method
γl,v
1/4
= Pγ(ρn,l − ρn,v) (5.83)
with γl,v in mN/m, molar densities ρn,l, ρn,v in mol/cm3
, and “parachor”
constant Pγ. Parachor values are tabulated for components and usually mixed
based on molar fractions (Danesh, 1998). Equation (5.83) then becomes
γl,v
1/4
= ρn,l
i
xiPγ,i − ρn,v
i
yiPγ,i . (5.84)
Petroleum–Water Interfacial Tension
Usually petroleum–water IFT is listed in lookup–tables for simulation. IFT
values are mainly used for determination of capillary pressure for migration
and accumulation in basin modeling (Sec. 6.3.1). Very often rough estimates
such as constant values of γp,w = 42 mN/m are used because uncertainties
in IFT can be neglected compared to uncertainties of knowledge about pore
throat radii.
However, it is found that pressure dependency of IFT is much weaker than
temperature dependency. An improved empirical correlation which reflects
this behavior is
γp,w = 111(ρw − ρp)1.024
(T/Tp,c)−1.25
(5.85)
with the densities in g/cm3
and γp,w in mN/m (Danesh, 1998, Fig. 5.21).
The critical temperature Tp,c of the petroleum phase must be known. Its
calculation with mixing rules for critical parameters such as the Lee–Kesler–
Average (5.6) are reported in the literature.
Fig. 5.21. Petroleum–water interfacial
tension γp,w according to (5.85) for dif-
ferent density contrasts ρw − ρp T/Tp,c
g
p,w
[mN/m]
10 kg/m^3
200 kg/m^3
400 kg/m^3
600 kg/m^3
800
kg/m
^3
1000
kg/m
^3](https://image.slidesharecdn.com/fundamentalsofbasinandpetroleumsystemsmodeling-230616124321-162f3155/85/Fundamentals-of-Basin-and-Petroleum-Systems-Modeling-pdf-247-320.jpg)
![5.7 Calibration of a Fluid Model 239
Molar M Tc pc vc Acentric Volume Shift
Fraction [%] [g/mol] [◦
C] [MPa] [m3
/kmol] Factor [m3
/kmol]
Methane 50 16.043 −82.59 4.599 0.0986 0.0115 0.0007
Ethane 6 30.070 32.17 4.872 0.1455 0.0995 0.0028
Propane 5 44.096 96.68 4.248 0.2000 0.1523 0.0052
n–Butane 3 58.123 151.97 3.796 0.2550 0.2002 0.0080
n–Pentane 2 72.150 196.55 3.370 0.3130 0.2515 0.0122
C6 2 84.000 236.85 3.271 0.3480 0.2510 0.0134
C7+ 32 247.000 500.00 1.700 0.7237 0.5295 0.0778
Table 5.4. Composition of an “Example–Oil”. The corresponding phase diagram
is shown in Fig. 5.23
fore be implemented efficiently because MCMC inversions are based on many
model evaluations, typically 100.000 times and more.
A prototype for the objective χ2
PVT of the MCMC inversion is given by
(7.12) and becomes
χ2
PVT =
API − APIm[pci, Tci, ωi, Mi]
ΔAPI
2
+
GOR − GORm[pci, Tci, ωi]
ΔGOR
2
+
pb − pbm[pci, Tci, ωi]
Δpb
2
+
Bo − Bom[pci, Tci, ωi]
ΔBo
2
.
(5.87)
The index m points out that the related quantities are modeled with flash
calculations and the square brackets indicate a dependency on the set of all
pseudo–components, which are labeled here with the index i. The importance
of each quantity in the inversion is controlled by its Δ factor. Decreasing Δ
increases the importance in the same way as the reduction of a measurement
error increases the importance of the related measurement. Minimization of
(5.87) calibrates the fluid model. The MCMC algorithm is briefly outlined in
Sec. 7.5.6.
An application of this algorithm to the lumped fluid is depicted in Fig. 5.22
with the calibration values taken from the original unlumped results quickly
providing a perfect calibration. Unfortunately, the size of the coexistence area
remains smaller than the one from the original data. Changing the critical
pressure or temperature of the C7+ pseudo–component manually, before per-
forming a new MCMC inversion, leads to a different calibrated end result
with a differently coexistence area. The MCMC finds just one solution out
of multiple possibilities. In such a case it is advantageous to add a Bayesian
term to (5.87) to ensure that the MCMC algorithm does not alter the pseudo–
component parameters too much. The objective becomes now
φPVT = χ2
PVT + φTc
+ φpc
+ φω + φM (5.88)](https://image.slidesharecdn.com/fundamentalsofbasinandpetroleumsystemsmodeling-230616124321-162f3155/85/Fundamentals-of-Basin-and-Petroleum-Systems-Modeling-pdf-252-320.jpg)
![240 5 Fluid Analysis
with e.g.
φTc
=
N
i=1
Tci − Tc0i
ΔT
2
, (5.89)
N as the number of components, and Tc0i the critical value from which the
calibration result should not deviate too much. The formulas for φpc
, φω,
and φM can be constructed analogously. Smaller the Δ factors are chosen as
less variations of the related parameters are allowed. Iterative modification of
the pseudo–component parameters and usage of the MCMC inversion quickly
leads to the result shown in Fig. 5.24.
It must be noted that a calibration via the variation of molecular weight
M, violates mass conservation. This is demonstrated with an extreme case:
a lumped two-component system with API = 29, GOR = 300 m3
/m3
and
pb = 11 MPa and pseudo–component parameters, as in Table 5.5, is calibrated
against API = 30, GOR = 100 m3
/m3
, and pb = 14 MPa. The molar fraction
of both pseudo–components is kept constant. A calibration is easily possible by
varying “Medium Oil” to “Medium Oil A”. However, the variation is dramatic.
The original “Medium Oil” has properties in the range of C5 – C8 on a SCN
– scale, whereas the calibrated component parameters can be found in a wide
region around C15. The molecular weight is more than doubled. Alternatively,
a calibration can also be performed under a constant mass fraction constraint
for each component. Here this is unfortunately not possible via variation of
only the heavy component. However, a calibration of both components yields
quickly new parameters “Dry Gas B” and “Medium Oil B”. The variation of
the pseudo–component parameters is less drastic. Note the variation of the
molar composition.
Molar M Tc pc Acentric
Fraction [%] [g/mol] [◦
C] [MPa] Factor
Dry Gas 50.0 17.943 −75.730 4.850 0.02210
Medium Oil 50.0 101.141 238.770 3.459 0.26060
Medium Oil A 50.0 208.075 292.118 1.768 1.24167
Dry Gas B 34.7 37.498 −90.806 5.298 0.10547
Medium Oil B 65.3 112.313 207.621 2.790 1.07016
Table 5.5. Two component example
5.7.3 Tuning of the Binary Interaction Parameter (BIP)
It is rather unusual in basin modeling to perform a fine–tuning of the BIP
because it is assumed that the modeling of large compositional variations un-
der the strongly varying subsurface conditions of geological processes, are not
significantly improved with fine–tuned parameters, which are often adapted](https://image.slidesharecdn.com/fundamentalsofbasinandpetroleumsystemsmodeling-230616124321-162f3155/85/Fundamentals-of-Basin-and-Petroleum-Systems-Modeling-pdf-253-320.jpg)
![242 5 Fluid Analysis
Pressure
[MPa]
Temperature [ C]
°
Methane + Water
Hydrate + Ice
Hydrate
+
Water
Methane + Ice
Q1
Fig. 5.25. Methane water phase diagram according to (5.90). Methane is meant to
be in the vapor and water in the liquid phase. The quadruple point Q1 is located at
−0.2◦
C and 2.563 MPa (Table 5.6). The dashed line indicates a temperature pressure
PT–path. It starts at 5 MPa, which is equivalent to a water depth of about 500 m, and
continues with a linear temperature gradient of 30 K/km and hydrostatic conditions
of 10 MPa/km. The dash–dotted line depicts a permafrost example with the same
temperature and pressure gradients. For both PT–path examples it is possible that
hydrates form in a limited depth range above the phase separation line (Gas Hydrate
Stability Zone – GHSZ). Note that according to (5.90) the phase boundary between
hydrate and water and the boundary between hydrate and ice are fitted empirically
and independently. This leads to a small artificial discontinuity at the quadruple
point
Q1 Q2
T[◦
C] p[MPa] T[◦
C] p[MPa]
CH4 −0.2 2.563 –
CO2 0 1.256 9.9 4.499
H2S −0.3 0.093 29.6 2.293
Table 5.6. Quadruple points and
parameters for (5.90) from Sloan Jr.
(1998). Note, that the formula for
CO2 is limited to 9.9 ◦
C due to a
second quadruple point Q2
Methane CH4
−25 ◦
C T −0.2 ◦
C −0.2 ◦
C T 25 ◦
C
a b[K] a b[K]
14.717 1886.79 38.980 8533.80
Carbon Dioxide CO2
−25 ◦
C T 0 ◦
C 0 ◦
C T 9.9 ◦
C
a b[K] a b[K]
18.594 3161.41 44.580 10246.28
Hydrogen Sulfide H2S
−25 ◦
C T −0.3 ◦
C −0.3 ◦
C T 25 ◦
C
a b[K] a b[K]
16.560 3270.41 34.828 8266.10](https://image.slidesharecdn.com/fundamentalsofbasinandpetroleumsystemsmodeling-230616124321-162f3155/85/Fundamentals-of-Basin-and-Petroleum-Systems-Modeling-pdf-255-320.jpg)
![260 6 Migration and Accumulation
5.7
5.7
5.7
5.7 5.7
5.7 5.7
5.3
6.1
6.5
4.5
4.9
Regions of varying
oil potential and
critical oil saturation
Region of uniform
oil potential
4.50
4.75
5.00
5.25
5.50
5.75
6.00
6.25
6.50
Oil Potential
in MPa
0 S S
wc co 1
pint
Sw
p
c
0 S S
wc co 1
Sw
1
2
3
4
5
Cell
1 Res. 1 0.1 6.1
2 Res. 1 0.1 5.7
3 Res. 55 0.5 5.7
4 Res. 95 0.9 5.7
5 Res. 96 1.3 5.7
6 Seal 1 1.7 5.7
S p u
o c o
[%] [Mpa] [MPa]
Seal
Reservoir
2
3
4
5
6
6
p
c
pint
pce
pce
90
80
70
60
50
40
30
20
10
0
Oil Saturation
in %
a)
b)
Seal
Reservoir
Fig. 6.6. Reservoir in static equilibrium
A principle scheme of an accumulation in static equilibrium below a seal
is shown in Fig. 6.6. The oil potential at base seal (cell 6) is equal to the oil
potential at the top reservoir (cell 5). Thus, the two corresponding capillary
pressures differ only by the amount of buoyancy pressure between the two
adjacent cells. The capillary pressure in the base of the seal is equal to the
capillary entry pressure, while the same capillary pressure at the top reservoir
location is related to a very high saturation value. The oil potential is constant
in the entire accumulation, which means that the capillary entry pressure
decreases linearly with depth in the reservoir (cells 5 — 2). The buoyancy
decreases by the same amount.](https://image.slidesharecdn.com/fundamentalsofbasinandpetroleumsystemsmodeling-230616124321-162f3155/85/Fundamentals-of-Basin-and-Petroleum-Systems-Modeling-pdf-272-320.jpg)
![6.3 Multi–Phase Darcy Flow 263
Petroleum
Saturation
[%]
Basement
Shale
Shale
Silt
ShaleSilt
Shale
Sandstone
Kitchen Area
Layer 1
Layer 2
Layer 3
Layer 4
Layer 5
Layer 6
Layer 7
Layer 8
Layer 9
Layer 10
Layer 11
Layer 12
Layer 13
Impermeable
Fault
Fig. 6.8. Two dimensional example model with reservoir equilibrium as outlined in
Fig. 6.6. The black vectors indicate oil and the light grey vectors gas flow. Layers 5
and 8 are sandstone reservoirs
A source term qp which describes the generation or loss of phase p (e.g. by
chemical reactions) is added here for the sake of completeness.
Furthermore, compaction is described by dφ/dt = −C dσ
z/dt with σ
z as
the vertical component of the effective stress and C as the matrix compress-
ibility. For simplicity, the equations are here, according to (2.3), only noted for
a coordinate system which moves with the solid rock. The compaction term
becomes
∂φ
∂t
= −C
∂(ul − uw)
∂t
(6.23)
with ul = pl − ph as the depositional or overburden load potential (Chap. 2).
A comprehensive treatment in a spatially fixed coordinate system can e.g. be
found in Luo and Vasseur (1992). Additional factors of form 1/(1−φ) such as
in (2.10), which are also known from Yükler et al. (1979), enter the equations
(App. B).
Equations 6.20, 6.21, and 6.23 can be inserted in (6.22) for the construction
of final three phase flow equations:](https://image.slidesharecdn.com/fundamentalsofbasinandpetroleumsystemsmodeling-230616124321-162f3155/85/Fundamentals-of-Basin-and-Petroleum-Systems-Modeling-pdf-275-320.jpg)
![294 6 Migration and Accumulation
Petroleum
Saturation
[%]
Hybrid
Accumulation
Accumulation
Flow Vectors
Saturation
Flow Vectors
Break Through
Fault
Fault
Oil
Large
Gascap
Vertical Transport
from Source to Carrier
Oil
Oil with
small
Gascap
Flowpath
Kitchen Area
Lateral Transport
in the Carrier
Lateral Transport
in the Carrier
Fig. 6.40. Hybrid and flowpath results of the same model as in Fig. 6.8. The hybrid
model has the same break through path as the flowpath model](https://image.slidesharecdn.com/fundamentalsofbasinandpetroleumsystemsmodeling-230616124321-162f3155/85/Fundamentals-of-Basin-and-Petroleum-Systems-Modeling-pdf-306-320.jpg)
![6.9 Discussion 325
Petroleum
Saturation
[%]
Petroleum
Saturation
[%]
Darcy
Hybrid
Fig. 6.62. Darcy and hybrid runs of the model which is also depicted in Fig. 6.61](https://image.slidesharecdn.com/fundamentalsofbasinandpetroleumsystemsmodeling-230616124321-162f3155/85/Fundamentals-of-Basin-and-Petroleum-Systems-Modeling-pdf-337-320.jpg)
![334 6 Migration and Accumulation
0 Ma
11 Ma
3 Drainage Areas
with
3 Accumulations
1 Drainage Area
with
1 Accumulation
Age [Ma]
Age [Ma]
Age [Ma]
Sum
Volume
[Mm^3]
Volume
[Mm^3]
Volume
[Mm^3]
Age [Ma]
Volume
[Mm^3]
Fig. 6.66. Map view of tracked accumulations. Three accumulations are found at
the present day but only one at 11 Ma. This leads to discontinuities in the tracking
of accumulated volumes of the individual accumulations at 11 Ma. However, the sum
over all drainage areas does not show discontinuities
x x
Drainage Area
Border
Accumulation
Highest Point
x
x
Paleo Event
Spill
Drainage Area
Border (unmerged)
Present Day
Fig. 6.67. A schematic map view of a reservoir with accumulations at a paleo
event and at present day. Both small accumulations belong to the history of the
big accumulation at present day. The spilled amounts at paleo time should not be
accounted for when spilled amounts are tracked through time
field (e.g. a main structure and its satellites) and the full reservoir. Obviously,
it can be expected that wrongly tracked paleo structures become relatively
less important if the number of present day drainage areas increases.25
Er-
roneously tracked paleo structures might only appear at the border of the
tracked field, which becomes less important with the increasing size of the full
area.
25
The areas should not be disjunct.](https://image.slidesharecdn.com/fundamentalsofbasinandpetroleumsystemsmodeling-230616124321-162f3155/85/Fundamentals-of-Basin-and-Petroleum-Systems-Modeling-pdf-346-320.jpg)
![7.5 Metamodels 373
Fig. 7.25. Response surface isolines for
temperature in Celsius in a source rock.
Dependent on heat flow and SWI tem-
perature variations the isolines are lin-
ear as expected
Fig. 7.26. Response surface isolines
for transformation ratio in [%] at same
point as in Fig. 7.25. Negative values
indicate a region where the response
surface method cannot be used
equation
ρc
∂T
∂t
− ∇ · (λ · ∇T) = Q (7.28)
of heat flow. Here T is the temperature, λ are the thermal conductivities, t is
the time, ρ the density, c the specific heat capacity, and Q are external heat
sources. For a unique compilation of heat flow analysis, boundary and initial
conditions must be specified. At the top of the basin usually the temperature
is given, at the sides, a condition of prohibited horizontal heat flow is applied
and at the bottom, basement heat flows are specified:
T = TSWI on top,
∇T = 0 at the sides,
−λ · ∇T = q at the bottom,
T|t=initialtime = T0 at initial time
(7.29)
with TSWI the “Sediment Water Interface” temperature at top of the basin,
q the basement heat flow, and T0 the temperature profile at initial time.
If λ, ρ, and c are assumed to be smooth and weakly temperature–
dependent then (7.28) is almost linear. This property can be utilized:
Firstly, one should take a look at the following boundary value problem:
ρc
∂T̃
∂t
− ∇ · (λ · ∇T̃) = 0 with
T̃ = 0 on top,
∇T̃ = 0 at the sides
−λ · ∇T̃ = q̃ at the bottom and
T|t=initialtime = 0 at initial time.
(7.30)](https://image.slidesharecdn.com/fundamentalsofbasinandpetroleumsystemsmodeling-230616124321-162f3155/85/Fundamentals-of-Basin-and-Petroleum-Systems-Modeling-pdf-385-320.jpg)
![406 A Compaction and Flow Parameter
Depo. Athy Athy Compress. y Schneider Factor
Poro. k k Max. Min. ka kb φ
[%]
1
km
1
MPa
10−7
kPa
10−7
kPa
1
MPa
1
MPa
Limestone
ooid grainstone 35.0 0.01 0.001 2.0 1.0 0.001 0.001 0.18
Waulsort. mound 16.0 0.01 0.001 2.0 1.0 0.001 0.001 0.08
micrite 51.0 0.52 0.04766 850.0 19.8 0.03212 0.07894 0.26
shaly 48.0 0.50 0.04493 686.5 18.8 0.03091 0.07090 0.24
org.–rich
typical 51.0 0.52 0.04939 881.2 20.5 0.03329 0.08185 0.26
1-2% TOC 51.0 0.52 0.04851 865.1 20.1 0.03270 0.08037 0.26
10% TOC 51.0 0.52 0.05365 956.8 22.3 0.03616 0.08890 0.26
chalk, typical 70.0 0.90 0.10546 4613.3 41.6 0.20793 0.05813 0.35
chalk, 95% calcite 70.0 0.90 0.10546 4611.2 41.7 0.20793 0.05813 0.35
chalk, 75% calcite 67.0 0.90 0.10124 3871.1 40.6 0.19000 0.05738 0.34
chalk, 40% calcite 65.0 0.90 0.09870 3453.3 39.9 0.05891 0.18975 0.33
Marl 50.0 0.50 0.04651 778.0 19.8 0.07375 0.03086 0.25
Dolomite
typical 35.0 0.39 0.03084 252.3 11.0 0.04219 0.02344 0.17
org.–lean, sandy 35.0 0.39 0.03119 255.2 11.2 0.04312 0.02344 0.17
org.–lean, silty 35.0 0.39 0.03138 256.7 11.2 0.04188 0.02406 0.17
org.–rich 35.0 0.39 0.03459 283.0 12.4 0.02656 0.04634 0.17
Table A.1. Compaction Parameter: Carbonate Rocks](https://image.slidesharecdn.com/fundamentalsofbasinandpetroleumsystemsmodeling-230616124321-162f3155/85/Fundamentals-of-Basin-and-Petroleum-Systems-Modeling-pdf-418-320.jpg)
![A Compaction and Flow Parameter 407
Depo. Athy Athy Compress. y Schneider Factor
Poro. k k Max. Min. ka kb φ
[%]
1
km
1
MPa
10−7
kPa
10−7
kPa
1
MPa
1
MPa
Biogenic Sediments
Chalk, typical 70.0 0.90 0.10546 4611.2 41.7 5.00000 0.04500 0.35
Chalk, 95% calcite 70.0 0.90 0.10546 4611.2 41.7 0.20793 0.05813 0.35
Chalk, 75% calcite 67.0 0.90 0.10124 3871.1 40.6 0.19000 0.05738 0.34
Chalk, 40% calcite 65.0 0.90 0.09870 3453.3 39.9 0.05891 0.18975 0.33
Coal, pure 76.0 0.43 1.51396 87407.4 564.6 3.41736 0.74043 0.38
Coal, impure 74.0 0.42 0.22597 11692.7 87.9 0.49695 0.11328 0.37
Coal, silty 68.0 0.40 0.13392 5289.1 53.2 0.07308 0.28035 0.34
Clastic Sediments
Sandstone
typical 41.0 0.31 0.02660 274.7 11.5 0.04156 0.01781 0.20
clay–rich 40.0 0.32 0.02661 265.4 11.1 0.04000 0.01812 0.20
clay–poor 42.0 0.30 0.02627 280.5 11.8 0.04047 0.01734 0.21
quartzite
typical 42.0 0.30 0.02726 291.0 12.2 0.04281 0.01781 0.21
very quartz–rich 42.0 0.27 0.02461 252.6 11.8 0.01615 0.03977 0.21
subarkose
typical 41.0 0.28 0.02468 244.8 11.4 0.01631 0.03874 0.20
quartz–rich 42.0 0.28 0.02533 263.4 11.9 0.03984 0.01641 0.21
clay–rich 42.0 0.30 0.02643 282.2 11.8 0.04063 0.01750 0.21
clay–poor 42.0 0.30 0.02627 280.5 11.8 0.04047 0.01734 0.21
dolomite–rich 40.0 0.30 0.02572 249.8 11.2 0.04000 0.01688 0.20
arkose
typical 39.0 0.33 0.02772 267.3 11.2 0.01981 0.04212 0.20
quartz–rich 41.0 0.30 0.02623 267.1 11.6 0.04094 0.01750 0.20
quartz–poor 40.0 0.32 0.02646 263.9 11.1 0.04000 0.01812 0.20
clay–rich 40.0 0.32 0.02661 265.4 11.1 0.04000 0.01812 0.20
clay–poor 39.0 0.32 0.02722 258.9 11.2 0.04063 0.01812 0.20
dolomite–rich 39.0 0.32 0.02658 252.8 11.0 0.03875 0.01812 0.20
wacke 39.0 0.34 0.02773 271.1 10.9 0.01992 0.04192 0.20
Table A.2. Compaction Parameter](https://image.slidesharecdn.com/fundamentalsofbasinandpetroleumsystemsmodeling-230616124321-162f3155/85/Fundamentals-of-Basin-and-Petroleum-Systems-Modeling-pdf-419-320.jpg)
![408 A Compaction and Flow Parameter
Depo. Athy Athy Compress. y Schneider Factor
Poro. k k Max. Min. ka kb φ
[%]
1
km
1
MPa
10−7
kPa
10−7
kPa
1
MPa
1
MPa
Shale
typical 70.0 0.83 0.09613 4032.7 40.3 0.19157 0.05270 0.35
org.–lean
typical 70.0 0.83 0.09613 4032.7 40.3 0.19157 0.05270 0.35
sandy 65.0 0.83 0.08999 3005.2 38.7 0.17479 0.05340 0.33
silty 67.0 0.83 0.09230 3373.4 39.4 0.18394 0.05357 0.34
silic., typical 70.0 0.83 0.09556 4007.5 40.1 0.19043 0.05238 0.35
silic., 95% Opal–CT 80.0 0.83 0.14631 7397.5 37.8 0.33248 0.06983 0.40
black 70.0 0.83 0.10931 4584.3 45.8 0.22348 0.06082 0.35
organic–rich
typical 70.0 0.83 0.10931 4291.1 42.9 0.18549 0.05265 0.30
3% TOC 70.0 0.83 0.10165 4264.1 42.6 0.20782 0.05656 0.35
8% TOC 70.0 0.83 0.10931 4584.3 45.4 0.22348 0.06082 0.35
20% TOC 70.0 0.83 0.12975 5441.2 54.4 0.26527 0.07219 0.35
Siltstone
org.–lean 55.0 0.51 0.04907 1036.1 21.1 0.03165 0.08611 0.28
organic–rich
typical 55.0 0.51 0.04937 1042.2 21.2 0.03185 0.08660 0.28
10% TOC 55.0 0.51 0.05459 1152.6 23.5 0.09119 0.03419 0.28
2-3% TOC 55.0 0.51 0.04966 1048.7 21.4 0.03204 0.08714 0.28
Conglomerate
typical 30.0 0.30 0.02429 142.1 8.8 0.01897 0.03228 0.15
quartzitic 30.0 0.30 0.02429 142.1 8.8 0.01897 0.03228 0.15
Tuff, felsic 60.0 0.35 0.03711 959.1 16.5 0.06961 0.02117 0.3
Tuff, basaltic 60.0 0.35 0.03212 829.3 14.3 0.06000 0.01844 0.3
Table A.3. Compaction Parameter: Clastic Sediments – Part 2](https://image.slidesharecdn.com/fundamentalsofbasinandpetroleumsystemsmodeling-230616124321-162f3155/85/Fundamentals-of-Basin-and-Petroleum-Systems-Modeling-pdf-420-320.jpg)
![A Compaction and Flow Parameter 409
An- Porosity [%] Permeability [log mD]
iso- at Point at Point
tr. 1 2 3 1 2 3
Biogenic Sediments
Chalk, typical 1.5 1.00 25 70 −6.75 −3.10 1.00
Chalk, 95% calcite 1.5 1.00 25 70 −6.75 −3.10 1.00
Chalk, 75% calcite 2.0 1.00 25 67 −6.75 −3.10 1.00
Chalk, 40% calcite 3.0 1.00 25 65 −6.75 −3.10 1.00
Coal, pure 4.0 3.02 25 76 −7.50 −2.60 0.00
Coal, impure 4.0 3.17 25 74 −7.50 −2.60 0.00
Coal, silty 4.0 3.39 25 68 −7.50 −2.60 0.00
Diatomite, clay–poor 3.0 1.00 25 80 −2.55 −1.30 3.00
Diatomite, clay–rich 10.0 1.00 25 75 −3.05 −1.80 2.50
Carbonate Rocks
Limestone
ooid grainstone 1.1 1.00 15 35 −2.44 2.60 3.00
Waulsortian mound 1.1 1.00 12 16 −2.50 3.00 3.35
micrite 1.1 1.00 25 51 −2.20 1.00 1.52
shaly 2.0 1.00 25 48 −1.99 0.72 1.50
organic–rich, typical 2.0 1.00 25 51 −1.99 0.73 1.00
organic–rich, 1-2% TOC 1.2 1.00 25 51 −1.99 0.73 1.00
organic–rich, 10% TOC 2.0 1.00 25 51 −1.99 0.72 1.00
chalk, typical 1.5 1.00 25 70 −6.75 −3.10 1.00
chalk, 95% calcite 1.5 1.00 25 70 −6.75 −3.10 1.00
chalk, 75% calcite 2.0 1.00 25 67 −6.75 −3.10 0.73
chalk, 40% calcite 3.0 1.00 25 65 −6.75 −3.10 0.54
Marl 1.2 1.00 25 50 −5.05 −2.25 −0.78
Dolomite
typical 1.1 1.00 25 35 0.11 2.83 3.92
organic–lean, sandy 1.1 1.00 25 35 0.60 2.98 3.93
organic–lean, silty 1.1 1.00 25 35 0.31 2.88 3.91
organic–rich 1.2 1.00 25 35 −0.37 2.69 3.91
Table A.4. Porosity and Permeability Parameter](https://image.slidesharecdn.com/fundamentalsofbasinandpetroleumsystemsmodeling-230616124321-162f3155/85/Fundamentals-of-Basin-and-Petroleum-Systems-Modeling-pdf-421-320.jpg)
![410 A Compaction and Flow Parameter
An- Porosity [%] Permeability [log mD]
iso- at Point at Point
tr. 1 2 3 1 2 3
Sandstone
typical 5.0 1 25 41 −1.80 3.00 4.33
clay–rich 10.0 1 25 40 −2.80 2.00 3.62
clay–poor 5.0 1 25 42 −1.80 3.00 4.84
quartzite, typical 3.0 1 25 42 −1.80 3.00 4.84
quartzite, very quartz–rich 2.0 1 25 42 −1.80 3.00 4.84
subarkose, typical 3.0 1 25 41 −1.80 3.00 4.73
subarkose, quartz–rich 3.0 1 25 42 −1.80 3.00 4.84
subarkose, clay–rich 10.0 1 25 42 −2.80 2.00 3.84
subarkose, clay–poor 3.0 1 25 42 −1.80 3.00 4.84
subarkose, dolomite–rich 5.0 1 25 40 −2.30 2.50 4.12
arkose, typical 5.0 1 25 39 −1.80 3.00 4.51
arkose, quartz–rich 4.0 1 25 41 −1.80 3.00 4.73
arkose, quartz–poor 7.0 1 25 40 −2.30 2.50 4.12
arkose, clay–rich 10.0 1 25 40 −2.80 2.00 3.62
arkose, clay–poor 4.0 1 25 39 −1.80 3.00 4.51
arkose, dolomite–rich 5.0 1 25 39 −2.30 2.50 4.01
wacke 7.0 1 25 39 −2.80 2.00 3.51
Shale
typical 1.2 1 25 70 −8.52 −3.00 −1.00
org.–lean, typical 1.2 1 25 70 −8.52 −3.00 −1.00
org.–lean, sandy 1.5 1 25 65 −8.42 −2.00 1.00
org.–lean, silty 1.3 1 25 67 −8.47 −2.50 0.50
org.–lean, silic., typical) 1.5 1 25 70 −8.47 −2.00 1.30
org.–lean, silic., 95% Opal-CT 1.1 1 25 80 −8.42 −1.50 −1.00
black 5.0 1 25 70 −8.52 −3.00 −1.00
organic–rich, typical 5.0 1 25 70 −8.52 −3.00 −1.00
organic–rich, 3% TOC 2.0 1 25 70 −8.52 −3.00 −1.00
organic–rich, 8% TOC 5.0 1 25 70 −8.52 −3.00 −1.00
organic–rich, 20% TOC 10.0 1 25 70 −8.52 −3.00 −1.00
Siltstone
organic–lean 10.0 1 25 55 −6.28 −1.00 0.71
organic–rich, typical 10.0 1 25 55 −6.28 −1.00 0.71
organic–rich, 10% TOC 10.0 1 25 55 −6.28 −1.00 0.71
organic–rich, 2-3% TOC 10.0 1 25 55 −6.28 −1.00 0.71
Conglomerate, typical 1.1 1 25 30 −2.80 2.00 4.00
Conglomerate, quartzitic 1.1 1 25 30 −2.80 2.00 4.00
Tuff, felsic 1.3 1 28 60 −1.87 1.75 4.00
Tuff, basaltic 1.3 1 28 60 −1.87 1.75 4.00
Table A.5. Porosity and Permeability Parameter: Clastic Sediments](https://image.slidesharecdn.com/fundamentalsofbasinandpetroleumsystemsmodeling-230616124321-162f3155/85/Fundamentals-of-Basin-and-Petroleum-Systems-Modeling-pdf-422-320.jpg)
![A Compaction and Flow Parameter 411
Entry Pr. [MPa] Parameter
at Porosity a b
1% 25% [MPa]
Biogenic Sediments
Chalk, typical 15.42 5.78 16.78 −0.037
Chalk, 95% calcite 15.42 5.78 16.78 −0.037
Chalk, 75% calcite 15.42 5.78 16.78 −0.037
Chalk, 40% calcite 15.42 5.78 16.78 −0.037
Coal, pure 23.35 5.36 26.14 −0.049
Coal, impure 23.35 5.36 26.16 −0.050
Coal, silty 23.35 5.35 26.19 −0.050
Diatomite, clay–poor 2.94 1.47 1.56 −0.013
Diatomite, clay–rich 4.30 2.15 2.05 −0.013
Carbonate Rocks
Limestone
ooid grainstone 1.42 0.08 1.74 −0.086
Waulsortian mound 1.47 0.06 1.94 −0.120
micrite 1.25 0.26 1.34 −0.032
shaly 1.11 0.32 1.18 −0.027
organic–rich, typical 1.11 0.31 1.18 −0.027
organic–rich, 1-2% TOC 1.11 0.31 1.18 −0.027
organic–rich, 10% TOC 1.11 0.32 1.18 −0.027
chalk, typical 15.42 5.78 16.78 −0.037
chalk, 95% calcite 15.42 5.78 16.78 −0.037
chalk, 75% calcite 15.42 5.78 16.78 −0.037
chalk, 40% calcite 15.42 5.78 16.78 −0.037
Marl 6.03 3.03 6.43 −0.028
Dolomite
typical 0.35 0.06 0.37 −0.027
organic–lean, sandy 0.27 0.06 0.28 −0.024
organic–lean, silty 0.31 0.06 0.33 −0.026
organic–rich 0.45 0.07 0.49 −0.031
Table A.6. Capillary Entry Pressure](https://image.slidesharecdn.com/fundamentalsofbasinandpetroleumsystemsmodeling-230616124321-162f3155/85/Fundamentals-of-Basin-and-Petroleum-Systems-Modeling-pdf-423-320.jpg)
![412 A Compaction and Flow Parameter
Entry Pr. [MPa] Parameter
at Porosity a b
1% 25% [MPa]
Sandstone
typical 1.00 0.06 1.12 −0.048
clay–rich 1.74 0.12 1.94 −0.048
clay–poor 1.00 0.06 1.12 −0.048
quartzite, typical 1.00 0.06 1.12 −0.048
quartzite, very quartz–rich 1.00 0.06 1.12 −0.048
subarkose, typical 1.00 0.06 1.12 −0.048
subarkose, quartz–rich 1.00 0.06 1.12 −0.048
subarkose, clay–rich 1.74 0.12 1.94 −0.048
subarkose, clay–poor 1.00 0.06 1.12 −0.048
subarkose, dolomite–rich 1.32 0.08 1.47 −0.048
arkose, typical 1.00 0.06 1.12 −0.048
arkose, quartz–rich 1.00 0.06 1.12 −0.048
arkose, quartz–poor 1.32 0.08 1.47 −0.048
arkose, clay–rich 1.74 0.12 1.94 −0.048
arkose, clay–poor 1.00 0.06 1.12 −0.048
arkose, dolomite–rich 1.32 0.08 1.47 −0.048
wacke 1.74 0.12 1.94 −0.048
Shale
typical 41.02 5.36 46.58 −0.055
organic–lean, typical 41.02 5.36 46.58 −0.055
organic–lean, sandy 38.82 2.50 45.00 −0.064
organic–lean, silty 39.90 3.66 45.78 −0.060
organic–lean, siliceous, typical 39.90 2.50 46.31 −0.065
organic–lean, siliceous, 95% Opal–CT 38.82 1.71 45.52 −0.069
black 41.02 5.35 46.58 −0.055
organic–rich, typical 41.02 5.35 46.58 −0.055
organic–rich, 3% TOC 41.02 5.35 46.58 −0.055
organic–rich, 8% TOC 41.02 5.35 46.58 −0.055
organic–rich, 20% TOC 41.02 5.35 46.58 −0.055
Siltstone
organic–lean 11.90 1.17 13.43 −0.053
organic–rich, typical 11.90 1.17 13.43 −0.053
organic–rich, 10% TOC 11.90 1.17 13.43 −0.053
organic–rich, 2-3% TOC 11.90 1.17 13.43 −0.053
Conglomerate, typical 1.74 0.12 1.94 −0.048
Conglomerate, quartzitic 1.74 0.12 1.94 −0.048
Table A.7. Capillary Entry Pressure: Clastic Sediments](https://image.slidesharecdn.com/fundamentalsofbasinandpetroleumsystemsmodeling-230616124321-162f3155/85/Fundamentals-of-Basin-and-Petroleum-Systems-Modeling-pdf-424-320.jpg)
![E
Thermal Properties
The following data of this section is provided by Doug Waples. It is based on
a comprehensive literature review and his modeling experience over 20 years.
Vert. An- Sp. Den. U Th K Radio-
Cond. iso- Heat active
tr. Cap. Heat
W
mK
J
kgK
kg
m3 [ppm] [ppm] [%]
μW
m3
Amphibolite 2.40 1.25 1130 2960 0.50 1.60 1.00 0.37
Eclogite 3.55 1.10 750 3400 0.20 0.40 0.20 0.12
Gneiss 2.70 1.40 800 2740 5.00 13.00 3.00 2.50
Marble 2.80 1.02 860 2700 1.00 1.00 0.20 0.34
Quartzite 5.40 1.03 770 2650 0.40 2.20 1.10 0.35
Schist 2.90 1.35 920 2740 2.10 9.70 2.20 1.44
Slate 1.80 1.75 860 2750 3.00 12.00 4.00 2.01
Serpentinite 2.60 1.45 785 2900 0.03 0.07 0.005 0.01
Table E.1. Thermal Properties: Metamorphic Rocks](https://image.slidesharecdn.com/fundamentalsofbasinandpetroleumsystemsmodeling-230616124321-162f3155/85/Fundamentals-of-Basin-and-Petroleum-Systems-Modeling-pdf-432-320.jpg)
![422 E Thermal Properties
Vert. An- Sp. Den. U Th K Radio-
Cond. iso- Heat active
tr. Cap. Heat
W
mK
J
kgK
kg
m3 [ppm] [ppm] [%]
μW
m3
Rock Forming Minerals
Quartz 7.69 1.00 740 2650 0.70 2.00 0.00 0.31
K–Spar 2.40 1.00 675 2550 1.60 5.00 14.05 1.96
Plagioclase 1.85 1.00 740 2680 2.50 1.50 0.00 0.74
Mica 0.60 6.00 770 2860 12.00 12.00 9.40 5.08
Chlorite 1.55 2.65 865 2800 1.00 4.00 0.65 0.62
Mixed–layer Clays 0.85 2.75 835 2780 1.00 4.00 4.00 0.94
Calcite 3.25 1.00 820 2710 0.00 0.00 0.00 0.00
Dolomite 5.30 1.00 870 2870 0.00 0.00 0.00 0.00
Halite, Mineral 6.50 1.00 865 2160 0.00 0.00 0.00 0.00
Anhydrite, Mineral 6.30 1.00 550 2960 0.00 0.00 0.00 0.00
Gypsum, Mineral 1.30 1.15 1070 2310 0.00 0.00 0.00 0.00
Olivine 4.20 1.00 750 3320 0.01 0.01 0.00 0.00
Orthoclase 2.30 1.00 620 2570 0.00 0.00 14.05 1.26
Rock Fragments
Igneous, Granite 2.60 1.15 800 2650 4.00 13.50 3.50 2.25
Volcanic, Basalt 2.10 1.17 790 2870 0.90 2.70 0.80 0.52
Metamorphic,
Schist/Slate 2.35 1.55 890 2745 2.55 10.85 3.10 1.73
Opaques,
Amphibole 2.90 1.00 750 3100 0.00 0.00 0.20 0.02
Other Minerals
Serpentine 2.60 1.00 785 2600 0.00 0.00 0.00 0.00
Siderite 3.00 1.00 700 3940 0.00 0.00 0.00 0.00
Sphalerite 12.70 1.00 460 4090 0.00 0.00 0.00 0.00
Sphene, Titanite 2.35 1.00 840 3520 400.00 350.00 0.00 165.58
Sylvite 6.70 1.00 685 1990 0.00 0.00 52.45 3.63
Talc 5.85 1.00 840 2780 0.00 0.00 0.00 0.00
Topaz 11.25 1.00 790 3200 0.00 0.00 0.00 0.00
Tourmaline 4.25 1.00 540 3100 0.00 0.00 0.00 0.00
Volcanic glass 1.35 1.00 770 2400 5.00 15.00 5.00 2.48
Zeolite 1.80 1.00 1050 2250 0.00 0.00 2.00 0.16
Zircon 5.00 1.00 630 4670 3000 2000 0.00 1573
Table E.2. Thermal Properties: Minerals – Part 1](https://image.slidesharecdn.com/fundamentalsofbasinandpetroleumsystemsmodeling-230616124321-162f3155/85/Fundamentals-of-Basin-and-Petroleum-Systems-Modeling-pdf-433-320.jpg)
![E Thermal Properties 423
Vert. An- Sp. Dens. U Th K Radio-
Cond. iso- Heat active
tr. Cap. Heat
W
mK
J
kgK
kg
m3 [ppm] [ppm] [%]
μW
m3
Other Minerals
Kaolinite 1.25 2.67 950 2590 1.00 4.00 0.50 0.56
Smectite 0.85 2.80 808 2610 0.50 2.00 0.50 0.30
Illite 0.85 2.75 822 2660 1.50 5.00 6.03 1.28
TOC (weight%) 0.28 1.10 1500 1100 100 0.00 0.00 10.47
Amphibole 2.90 1.00 750 3100 0.00 0.00 0.2 0.02
Analcime 1.40 1.00 960 2260 0.00 0.00 0.00 0.00
Anorthite 1.70 1.00 740 2760 0.00 0.00 0.00 0.00
Apatite 1.35 1.00 700 3180 30.00 150.00 0.00 21.29
Aragonite 2.25 1.00 800 2930 0.00 0.00 0.00 0.00
Barite 1.35 1.00 460 4480 0.00 0.00 0.00 0.00
Brucite 12.90 1.00 1260 2370 0.00 0.00 0.00 0.00
Chalcopyrite 8.20 1.00 535 4200 0.00 0.00 0.00 0.00
Chert, Mineral 5.00 1.00 740 2600 0.00 0.00 0.00 0.00
Clinoptilolite 1.40 1.00 1140 2100 0.00 0.00 3.30 0.24
Diopside 4.90 1.00 760 3280 0.00 0.00 0.00 0.00
Epidote 2.80 1.00 770 3590 35.00 275.00 0.00 37.24
Fluorite 9.50 1.00 870 3180 0.00 0.00 0.00 0.00
Forsterite 5.10 1.00 820 3210 0.00 0.00 0.00 0.00
Galena 2.70 1.00 210 7600 0.00 0.00 0.00 0.00
Garnet 3.50 1.00 750 3800 0.00 0.00 0.00 0.00
Gibbsite 2.60 1.00 1170 2440 0.00 0.00 0.00 0.00
Glauconite 1.60 1.00 770 2300 0.00 0.00 5.49 0.44
Goethite 2.70 1.00 825 4270 0.00 0.00 0.00 0.00
Hematite 11.30 1.00 625 5280 0.00 0.00 0.00 0.00
Hornblende 2.80 1.00 710 3080 15.00 25.00 1.75 6.56
Hypersthene 4.40 1.00 800 3450 0.00 0.00 0.00 0.00
Ilmenite 2.40 1.00 700 4790 25.00 0.00 0.00 11.40
Magnesite 5.80 1.00 880 3010 0.00 0.00 0.00 0.00
Magnetite 5.10 1.00 620 5200 15.00 10.00 0.00 8.76
Microcline 2.50 1.00 710 2560 0.00 0.00 14.05 1.25
Opal–CT 1.70 1.00 725 2300 0.00 0.00 0.00 0.00
Polyhalite, Min. 1.55 1.00 700 2780 0.00 0.00 12.97 1.25
Pyrite 3.00 1.00 510 5010 0.00 0.00 0.00 0.00
Pyrophyllite 3.75 1.00 805 2820 0.00 0.00 0.00 0.00
Pyroxene 4.20 1.00 770 3500 20.00 13.00 0.00 7.83
Rutile 5.10 1.00 700 4250 0.00 0.00 0.00 0.00
Table E.3. Thermal Properties: Minerals, Part 2](https://image.slidesharecdn.com/fundamentalsofbasinandpetroleumsystemsmodeling-230616124321-162f3155/85/Fundamentals-of-Basin-and-Petroleum-Systems-Modeling-pdf-434-320.jpg)
![424 E Thermal Properties
Vert. An- Sp. Den. U Th K Radio-
Cond. iso- Heat active
tr. Cap. Heat
W
mK
J
kgK
kg
m3 [ppm] [ppm] [%]
μW
m3
Albitite 2.00 1.00 785 2600 0.02 0.02 0.40 0.04
Andesite 2.70 1.00 820 2650 1.35 2.50 1.40 0.64
Anorthosite 1.70 1.00 750 2800 1.00 3.50 0.45 0.56
Basalt, normal 1.80 1.17 800 2870 0.90 2.70 0.80 0.52
Basalt, weathered 2.10 1.17 790 2870 0.90 2.70 0.80 0.52
Bronzitite 3.80 1.00 760 3450 0.02 0.02 0.05 0.01
Diabase 2.60 1.00 800 2800 0.25 0.90 0.45 0.18
Diorite 2.70 1.00 1140 2900 2.00 8.50 1.10 1.29
Dolerite 2.30 1.00 900 2930 0.40 1.60 0.70 0.30
Dunite 3.80 1.30 720 3310 0.004 0.004 0.01 0.00
Gabbro 2.60 1.00 800 2870 0.25 0.80 0.50 0.18
Granite, 150 My old 2.60 1.15 800 2650 6.50 17.00 5.70 3.32
Granite, 500 My old 2.60 1.15 800 2650 4.50 15.00 4.50 2.57
Granite, 1000 My old 2.60 1.15 800 2650 4.00 13.50 3.50 2.25
Granodiorite 2.60 1.00 730 2720 2.30 9.30 2.80 1.51
Harzburgite 6.90 1.00 760 3300 0.02 0.02 0.05 0.01
Hypersthenite 4.10 1.00 760 3450 0.02 0.02 0.05 0.01
Lamprophyre 2.50 1.00 760 3000 1.20 5.50 4.10 1.19
Lherzolite 3.30 1.00 780 3150 0.015 0.05 0.0006 0.01
Monzonite 2.70 1.00 740 2600 2.70 8.50 4.00 1.60
Monzonite, quartz 2.80 1.00 880 2700 6.50 27.00 3.40 3.86
Nepheline syenite 2.40 1.00 750 2650 9.10 27.00 4.00 4.50
Norite 2.20 1.00 670 2860 0.15 0.50 0.30 0.11
Olivinite 3.20 1.00 800 3450 0.01 0.08 0.05 0.02
Peridotite, typical 4.00 1.00 800 3200 0.025 0.025 0.055 0.02
Peridotite, serpentinized 2.20 1.00 700 3100 0.03 0.03 0.01 0.01
Pyroxenite 3.80 1.00 1000 3220 0.20 0.55 0.75 0.19
Quartz monzonite 2.80 1.00 880 2700 6.50 27.00 3.40 3.86
Rhyolite 2.60 1.20 800 2500 5.80 13.00 3.70 2.53
Syenite, typical 2.60 1.00 800 2760 2.70 15.00 4.70 2.22
Syenite, nepheline 2.40 1.00 750 2650 9.10 27.00 4.00 4.50
Tonalite 2.55 1.00 800 2800 0.00 12.00 2.50 1.10
Ultramafics 3.80 1.00 900 3310 0.001 0.004 0.003 0.00
Table E.4. Thermal Properties: Igneous Rocks](https://image.slidesharecdn.com/fundamentalsofbasinandpetroleumsystemsmodeling-230616124321-162f3155/85/Fundamentals-of-Basin-and-Petroleum-Systems-Modeling-pdf-435-320.jpg)
![E Thermal Properties 425
Vert. An- Sp. Th. Den. U Th K Radio-
Cond. iso- Heat Sort. active
tr. Cap. Fac. Heat
W
mK
J
kgK
f
kg
m3 [ppm] [ppm] [%]
μW
m3
Sandstone
typical 3.95 1.15 855 1.00 2720 1.30 3.50 1.30 0.70
clay–rich 3.35 1.20 860 1.00 2760 1.50 5.10 3.60 1.10
clay–poor 5.95 1.05 820 1.00 2700 0.70 2.30 0.60 0.40
quartzite, typical 6.15 1.08 890 1.00 2640 0.60 1.80 0.90 0.36
quart., very quartz–rich 6.45 1.06 890 1.00 2640 0.50 1.70 0.70 0.30
subarkose, typical 4.55 1.14 870 1.00 2680 1.00 3.30 1.30 0.60
subarkose, quartz–rich 5.05 1.13 880 1.00 2650 0.90 2.70 0.90 0.49
subarkose, clay–rich 3.40 1.40 870 1.00 2690 1.20 3.90 2.30 0.79
subarkose, clay–poor 4.80 1.09 870 1.00 2700 0.70 2.30 0.60 0.40
subarkose, dolomite–rich 4.10 1.07 840 1.00 2710 0.90 2.70 0.90 0.50
arkose, typical 3.20 1.25 845 1.00 2730 3.00 8.00 2.50 1.58
arkose, quartz–rich 4.05 1.20 860 1.00 2690 2.00 6.00 0.90 1.01
arkose, quartz–poor 2.00 1.50 835 1.00 2770 3.00 8.00 2.50 1.60
arkose, clay–rich 2.30 1.45 865 1.00 2760 2.00 7.00 3.60 1.37
arkose, clay–poor 4.00 1.12 835 1.00 2710 2.00 7.00 1.00 1.10
arkose, dolomite–rich 4.35 1.05 850 1.00 2750 1.50 5.10 3.60 1.10
wacke 2.60 1.40 850 1.00 2780 2.50 8.00 2.50 1.47
Shale
typical 1.64 1.60 860 1.38 2700 3.70 12.0 2.70 2.03
organic–lean, typical 1.70 1.55 860 1.35 2700 3.70 12.0 2.70 2.03
organic–lean, sandy 1.84 2.00 860 1.64 2700 2.80 11.0 2.50 1.71
organic–lean, silty 1.77 1.80 860 1.51 2700 3.00 11.0 2.60 1.78
organic–lean
siliceous, typical 1.90 1.17 860 1.00 2710 2.00 4.5 2.00 1.02
siliceous, 95% Opal–CT 1.75 1.16 860 1.00 2330 1.00 1.3 1.00 0.38
black 0.90 3.27 940 2.40 2500 19.00 11.0 2.50 5.44
organic–rich, typical 1.25 2.19 900 1.76 2600 5.00 12.0 2.80 2.29
organic–rich, 3% TOC 1.45 1.84 880 1.54 2610 5.00 12.0 2.80 2.30
organic–rich, 8% TOC 1.20 2.20 900 1.74 2500 10.00 11.0 2.90 3.34
organic–rich, 20% TOC 0.80 3.09 980 2.22 2270 20.00 11.0 2.60 5.17
Siltstone
organic–lean 2.05 1.50 910 1.00 2720 2.00 5.00 1.00 0.96
org.–rich, typical 2.01 1.71 940 1.47 2710 2.00 5.00 1.00 0.96
org.–rich, 10% TOC 1.82 2.64 960 2.03 2550 15.00 6.00 2.00 4.21
org.–rich, 2-3% TOC 2.00 1.76 930 1.52 2700 2.50 6.50 2.00 1.28
Conglomerate, typical 2.30 1.05 820 1.00 2700 1.50 4.00 2.00 0.85
Conglomerate, quartzitic 6.10 1.05 780 1.00 2700 1.00 4.00 1.20 0.65
Tuff, felsic 2.60 1.17 830 1.00 2650 3.00 6.50 3.90 1.56
Tuff, basaltic 1.90 1.17 830 1.00 2900 0.65 2.20 0.85 0.43
Table E.5. Thermal Properties: Sedimentary Rocks, Part 1 – Clastic Sediments](https://image.slidesharecdn.com/fundamentalsofbasinandpetroleumsystemsmodeling-230616124321-162f3155/85/Fundamentals-of-Basin-and-Petroleum-Systems-Modeling-pdf-436-320.jpg)
![426 E Thermal Properties
Vert. An- Sp. Th. Den. U Th K Radio-
Cond. iso- Heat Sort. active
tr. Cap. Fac. Heat
W
mK
J
kgK
f
kg
m3 [ppm] [ppm] [%]
μW
m3
Biogenic Sediments
Chalk, typical 2.90 1.07 850 1.00 2680 1.90 1.40 0.25 0.60
Chalk, 95% calcite 3.00 1.04 830 1.00 2680 1.90 1.40 0.25 0.60
Chalk, 75% calcite 2.65 1.15 840 1.00 2680 1.90 1.40 0.25 0.60
Chalk, 40% calcite 3.20 1.10 860 1.00 2680 1.90 1.40 0.25 0.60
Coal, pure 0.30 1.20 1300 1.03 1600 1.50 3.00 0.55 0.38
Coal, impure 1.00 1.15 1200 1.00 1600 2.00 3.00 0.70 0.47
Coal, silty 1.60 1.30 1100 1.00 1600 2.00 3.00 0.70 0.47
Diatomite, clay–poor 1.60 1.00 790 1.00 2100 1.30 1.00 1.00 0.39
Diatomite, clay–rich 1.50 1.10 800 1.00 2300 3.00 2.60 2.50 1.01
Kerogen 1.00 1.00 6.3 1.00 1100 100.00 0.00 0.00 10.47
Carbonate Rocks
Limestone
ooid grainstone 3.00 1.19 835 1.11 2740 1.00 1.00 0.20 0.35
Waulsortian mound 3.00 1.19 835 1.11 2740 1.00 1.00 0.20 0.35
micrite 3.00 1.19 835 1.11 2740 1.00 1.00 0.20 0.35
shaly 2.30 1.70 850 1.38 2730 2.00 4.00 1.00 0.89
org.–rich, typical 2.00 1.95 845 1.47 2680 5.00 1.50 0.26 1.40
org.–rich, 1-2% TOC 2.63 1.40 840 1.16 2710 2.50 1.70 0.27 0.79
org.–rich, 10% TOC 1.45 2.68 850 1.84 2550 10.00 1.40 0.25 2.54
chalk, typical 2.90 1.07 850 1.00 2680 1.90 1.40 0.25 0.60
chalk, 95% calcite 3.00 1.04 830 1.00 2680 1.90 1.40 0.25 0.60
chalk, 75% calcite 2.65 1.15 840 1.00 2680 1.90 1.40 0.25 0.60
chalk, 40% calcite 3.20 1.10 860 1.00 2680 1.90 1.40 0.25 0.60
Marl 2.00 1.45 850 1.00 2700 2.50 5.00 2.00 1.18
Dolomite
typical 4.20 1.06 860 1.00 2790 0.80 0.60 0.40 0.29
organic–lean, sandy 4.25 1.06 860 1.00 2770 0.90 0.90 0.70 0.37
organic–lean, silty 3.00 1.10 860 1.00 2760 0.90 0.90 0.80 0.38
organic–rich 2.15 2.44 910 1.76 2600 10.00 1.40 0.60 2.62
Table E.6. Thermal Properties: Sedimentary Rocks, Part 2](https://image.slidesharecdn.com/fundamentalsofbasinandpetroleumsystemsmodeling-230616124321-162f3155/85/Fundamentals-of-Basin-and-Petroleum-Systems-Modeling-pdf-437-320.jpg)
![E Thermal Properties 427
Vert. An- Sp. Th. Den. U Th K Radio-
Cond. iso- Heat Sort. active
tr. Cap. Fac. Heat
W
mK
J
kgK
f
kg
m3 [ppm] [ppm] [%]
μW
m3
Chemical Sediments
Polyhalite 1.00 1.00 700 1.00 2780 0.02 0.01 12.90 1.25
Salt 6.50 1.01 860 1.00 2740 0.02 0.01 0.10 0.02
Sylvinite 1.00 1.02 685 1.00 2100 0.02 0.01 20.60 1.51
Anhydrite 6.30 1.05 750 1.00 2970 0.10 0.30 0.40 0.09
Chert 4.80 1.00 890 1.00 2650 1.00 1.00 0.70 0.38
Gypsum 1.50 1.15 1100 1.00 2320 0.08 0.20 0.30 0.05
Halite 6.50 1.01 860 1.00 2200 0.02 0.01 0.10 0.01
Table E.7. Thermal Properties: Sedimentary Rocks, Part 3](https://image.slidesharecdn.com/fundamentalsofbasinandpetroleumsystemsmodeling-230616124321-162f3155/85/Fundamentals-of-Basin-and-Petroleum-Systems-Modeling-pdf-438-320.jpg)
![450 I Component Properties
Name MW Tc Pc vc Acentric Rackett Shift ρ
g
mol
[K] [MPa]
m3
kmol
Fac. ωc ZRa Fac. SE
kg
m3
Oil: Alkanes
n–Hexane 86.177 507.6 3.025 0.371 0.3013 0.26355 −0.01478 663.8
n–Heptane 100.204 540.2 2.740 0.428 0.3495 0.26074 0.02509 688.2
n–Octane 114.231 568.7 2.490 0.486 0.3996 0.25678 0.04808 707.0
n–Nonane 128.258 594.6 2.290 0.544 0.4435 0.25456 0.06799 721.9
n–Decane 142.285 617.7 2.110 0.600 0.4923 0.25074 0.08546 734.2
n–Undecane 156.312 639.0 1.949 0.659 0.5303 0.24990 0.10100 744.5
n–Dodecane 170.338 658.0 1.820 0.716 0.5764 0.24692 0.11500 752.7
n–Tridecane 184.365 675.0 1.680 0.775 0.6174 0.24698 0.12764 761.7
n–Tetradecane 198.392 693.0 1.570 0.830 0.6430 0.24322 0.13923 763.3
n–Pentadecane 212.419 708.0 1.480 0.889 0.6863 0.23030 0.14989 772.2
n–Hexadecane 226.446 723.0 1.400 0.944 0.7174 0.22760 0.15974 777.2
n–Heptadecane 240.473 736.0 1.340 1.000 0.7697 0.23431 0.16890 779.7
n–Octadecane 254.500 747.0 1.270 1.060 0.8114 0.22917 0.17744 782.0
n–Nonadecane 268.527 758.0 1.210 1.120 0.8522 0.21580 0.18545 786.9
n–Eicosane 282.553 768.0 1.160 1.170 0.9069 0.22811 0.19297 792.7
n–Heneicosane 296.580 781.7 1.147 1.198 0.9220 0.20970 0.20007 795.4
n–Docosane 310.610 791.8 1.101 1.253 0.9550 0.20680 0.20678 798.1
n–Tricosane 324.630 801.3 1.059 1.307 0.9890 0.20380 0.21314 800.4
n–Tetracosane 338.680 810.4 1.019 1.362 1.0190 0.20110 0.21919 802.5
Oil: Alkenes
1–Hexene 84.161 504.03 3.140 0.3540 0.2800 0.26600 676.9
1–Heptene 98.188 537.29 2.830 0.4130 0.3310 0.26150 701.5
Oil: Aromatics
Benzene 78.114 562.16 4.898 0.2589 0.2108 0.26967 −0.04870 882.9
Toluene 92.141 591.79 4.109 0.3158 0.2641 0.26390 −0.01450 874.3
Ethylbenzene 106.167 617.17 3.609 0.3738 0.3036 0.26186 0.01397 874.4
o–Xylene 106.167 630.37 3.734 0.3692 0.3127 0.26200 0.01397 884.9
m–Xylene 106.167 617.05 3.541 0.3758 0.3260 0.26200 0.01397 869.4
p–Xylene 106.167 616.26 3.511 0.3791 0.3259 0.28700 0.01397 866.6
Oil: Cycloalkanes
Cyclopentane 70.134 511.76 4.502 0.2583 0.1943 0.26824 −0.11868 760.3
Methylcyclopent. 84.161 532.79 3.784 0.3189 0.2302 0.27040 −0.07227 754.0
Cyclohexane 84.161 553.54 4.075 0.3079 0.2118 0.27286 −0.07227 783.5
Methylcyclohex. 98.188 572.19 3.471 0.3680 0.2350 0.26986 −0.03454 774.8
Ethylcyclopent. 98.188 569.52 3.397 0.3745 0.2715 0.26670 −0.03454 771.2
Ethylcyclohex. 112.215 609.15 3.040 0.4500 0.2455 0.26900 −0.00292 792.1
Oil: Methyl–Alkanes
2 2-Dimethylprop 72.150 433.78 3.199 0.3036 0.1964 0.27570 −0.04350 597.4
2–Methylpentane 86.177 497.50 3.010 0.3664 0.2781 0.26620 −0.01478 657.8
Table I.1. Component parameters – Part 1](https://image.slidesharecdn.com/fundamentalsofbasinandpetroleumsystemsmodeling-230616124321-162f3155/85/Fundamentals-of-Basin-and-Petroleum-Systems-Modeling-pdf-460-320.jpg)
![I Component Properties 451
Name MW Tc Pc vc Acentric Rackett Shift ρ
g
mol
[K] [MPa]
m3
kmol
Fac. ωc ZRa Fac. SE
kg
m3
Oil: Single Carbon Number (SCN)
C6 84 510 3.271 0.348 0.251 0.269 −0.01478 690
C7 96 547 3.071 0.392 0.280 0.266 0.01745 727
C8 107 574 2.877 0.433 0.312 0.263 0.03669 749
C9 121 603 2.665 0.485 0.352 0.260 0.05804 768
C10 134 627 2.481 0.532 0.389 0.256 0.07540 782
C11 147 649 2.310 0.584 0.429 0.253 0.09088 793
C12 161 670 2.165 0.635 0.467 0.250 0.10583 804
C13 175 689 2.054 0.681 0.501 0.247 0.11932 815
C14 190 708 1.953 0.727 0.536 0.244 0.13243 826
C15 206 727 1.853 0.777 0.571 0.240 0.14512 836
C16 222 743 1.752 0.830 0.610 0.237 0.15670 843
C17 237 758 1.679 0.874 0.643 0.234 0.16669 851
C18 251 770 1.614 0.915 0.672 0.232 0.17536 856
C19 263 781 1.559 0.951 0.698 0.229 0.18235 861
C20 275 793 1.495 0.997 0.732 0.226 0.18900 866
C21 291 804 1.446 1.034 0.759 0.224 0.19729 871
C22 300 815 1.393 1.077 0.789 0.221 0.20174 876
C23 312 825 1.356 1.110 0.815 0.219 0.20743 881
C24 324 834 1.314 1.147 0.841 0.217 0.21286 885
C25 337 844 1.263 1.193 0.874 0.214 0.21849 888
C26 349 853 1.230 1.226 0.897 0.212 0.22346 892
C27 360 862 1.200 1.259 0.944 0.200 0.22784 896
C28 372 870 1.164 1.296 0.968 0.198 0.23244 899
C29 382 877 1.140 1.323 0.985 0.196 0.23614 902
C30 394 885 1.107 1.361 1.008 0.194 0.24044 905
C31 404 893 1.085 1.389 1.026 0.193 0.24390 909
C32 415 901 1.060 1.421 1.046 0.191 0.24759 912
C33 426 907 1.039 1.448 1.063 0.189 0.25117 915
C34 437 914 1.013 1.480 1.082 0.188 0.25464 917
C35 445 920 0.998 1.502 1.095 0.187 0.25710 920
C36 456 926 0.974 1.534 1.114 0.185 0.26040 922
C37 464 932 0.964 1.550 1.124 0.184 0.26275 925
C38 475 938 0.941 1.583 1.142 0.182 0.26589 927
C39 484 943 0.927 1.604 1.154 0.181 0.26840 929
C40 495 950 0.905 1.636 1.172 0.180 0.27139 931
C41 502 954 0.896 1.652 1.181 0.179 0.27325 933
C42 512 959 0.877 1.680 1.195 0.178 0.27586 934
C43 521 964 0.864 1.701 1.207 0.177 0.27815 936
C44 531 970 0.844 1.733 1.224 0.175 0.28065 938
C45 539 974 0.835 1.749 1.232 0.174 0.28261 940
Table I.2. Component parameters – Part 2](https://image.slidesharecdn.com/fundamentalsofbasinandpetroleumsystemsmodeling-230616124321-162f3155/85/Fundamentals-of-Basin-and-Petroleum-Systems-Modeling-pdf-461-320.jpg)
![452 I Component Properties
Name MW Tc Pc vc Acentric Rackett
g
mol
[K] [MPa]
m3
kmol
Fac. ωc ZRa
Oil: Boiling Range Groups
Oil–range
Mahakam(waxy crude) 133.440 617.95 2.600 0.5103 0.3795 0.25725
Oil–range
Tuscaloosa(Live Oil) 159.140 657.60 2.334 0.5920 0.4451 0.25137
Oil–range
Smackover(S–rich crude) 184.440 693.37 2.105 0.6773 0.5083 0.24593
Oil BO 196.032 705.54 2.026 0.7084 0.5392 0.24244
Oil–range
Tualag(waxy crude) 201.620 713.24 1.998 0.7237 0.5485 0.24155
Oil: Compound Classes
C6–C14ARO 89.616 588.92 4.321 0.3047 0.2546 0.26788
C6–C14SAT 121.422 587.96 2.518 0.4980 0.4003 0.25812
C5–C14 132.314 595.03 2.305 0.5594 0.4546 0.25367
C6–C14 137.378 605.58 2.215 0.5801 0.4717 0.25242
C15+ 311.950 824.32 1.392 1.0926 0.8079 0.21739
C6–C13ARO 104.484 618.13 3.703 0.3669 0.3077 0.26604
C6–C13SAT 137.936 609.86 2.191 0.5792 0.4730 0.25313
C7+ 209.610 724.56 1.932 0.7551 0.5686 0.23983
C14+ISAT 299.180 813.82 1.442 1.0524 0.7797 0.22017
C14+NSAT 299.180 813.82 1.442 1.0524 0.7797 0.22017
C14+ARO U 299.180 813.82 1.442 1.0524 0.7797 0.22017
C15+ARO 282.553 768.00 1.165 1.1700 0.9069 0.22811
C15+SAT 255.342 747.73 1.309 1.0531 0.8085 0.22382
C14+NSO 402.700 891.20 1.109 1.3697 1.0050 0.19590
NSO 338.680 810.40 1.009 1.3620 1.0190 0.20110
Standard Oils
Volatile Oil 53.135 367.79 4.373 0.1956 0.1267 0.27919
Light Oil 48.439 357.38 4.389 0.1895 0.1238 0.27968
Black Oil 90.072 491.62 3.535 0.3121 0.2423 0.26892
Medium Oil 101.141 511.92 3.459 0.3302 0.2606 0.26722
Heavy Oil 127.492 568.42 3.041 0.4080 0.3290 0.26099
Tuscaloosa T2 Live Oil 57.930 400.71 4.112 0.2241 0.1634 0.27631
Smackover T2–S Live Oil 142.418 604.26 2.669 0.4839 0.3930 0.25608
Tualag T1 High Waxy 167.860 650.49 2.298 0.5886 0.4750 0.24916
Mahakam T3 High Waxy+Aro 102.266 529.78 3.082 0.3768 0.3165 0.26429
Table I.3. Component parameters – Part 3](https://image.slidesharecdn.com/fundamentalsofbasinandpetroleumsystemsmodeling-230616124321-162f3155/85/Fundamentals-of-Basin-and-Petroleum-Systems-Modeling-pdf-462-320.jpg)
![I Component Properties 453
Name MW Tc Pc vc Acentric Rackett Shift ρ
g
mol
[K] [MPa]
m3
kmol
Fac. ωc ZRa Fac. SE
kg
m3
Gas: Alkanes
Methane 16.043 190.56 4.599 0.0986 0.0115 0.28941 −0.15400 300.0
Methane–C13 17.043 190.56 4.599 0.0986 0.0115 0.28941 −0.15400 318.7
Ethane 30.070 305.32 4.872 0.1455 0.0995 0.28128 −0.10020 356.2
Propane 44.096 369.83 4.248 0.2000 0.1523 0.27664 −0.08501 507.0
n–Butane 58.123 425.12 3.796 0.2550 0.2002 0.27331 −0.06413 584.0
n–Pentane 72.150 469.70 3.370 0.3130 0.2515 0.26853 −0.04183 631.1
Gas: Alkenes
Ethylene 28.054 282.36 5.032 0.1291 0.0852 0.28054 500.0
Propadiene 40.065 393.15 5.470 0.1620 0.1596 0.27283 599.7
Propylene 42.081 364.76 4.612 0.1810 0.1424 0.27821 521.0
1–Butene 56.107 419.58 4.020 0.2399 0.1867 0.27351 600.5
cis–2–Butene 56.107 435.58 4.206 0.2340 0.2030 0.27044 628.6
trans–2–Butene 56.107 428.63 4.103 0.2382 0.2182 0.27212 611.2
1 2–Butadiene 54.092 444.00 4.500 0.2190 0.2509 0.26850 657.6
1 3–Butadiene 54.092 425.37 4.330 0.2208 0.1932 0.27130 627.3
1–Pentene 70.134 464.78 3.529 0.2960 0.2329 0.27035 645.8
cis–2–Pentene 70.134 475.93 3.654 0.3021 0.2406 0.26940 659.8
trans–2–Pentene 70.134 475.37 3.654 0.3021 0.2373 0.26970 652.4
Gas: Methyl–Alkanes
i–Butane 58.123 408.14 3.648 0.2627 0.1770 0.27569 −0.07935 562.9
i–Pentane 72.150 460.43 3.381 0.3058 0.2275 0.27060 −0.04350 624.7
Gas: Methyl–Alkenes
2–Methyl–
1–Butene 70.134 465.00 3.400 0.2920 0.2287 0.2705 656.3
3–Methyl–
1–Butene 70.134 450.37 3.516 0.3021 0.2286 0.2705 632.2
2–Methyl–
2–Butene 70.134 471.00 3.400 0.2920 0.2767 0.2663 668.3
Table I.4. Component parameters – Part 4](https://image.slidesharecdn.com/fundamentalsofbasinandpetroleumsystemsmodeling-230616124321-162f3155/85/Fundamentals-of-Basin-and-Petroleum-Systems-Modeling-pdf-463-320.jpg)
![454 I Component Properties
Name MW Tc Pc vc Acentric Rackett ρ
g
mol
[K] [MPa]
m3
kmol
Fac. ωc ZRa
kg
m3
Gas: Compound Classes
C2–C4 47.130 379.57 4.192 0.2120 0.1615 0.27623
C2–C5 53.070 400.97 3.996 0.2360 0.1828 0.27440
C3–C5 62.752 430.72 3.599 0.2721 0.2008 0.27340
Standard Gases
Hydrogen 2.016 33.18 1.313 0.0642 −0.2150 0.31997
Carbon Monoxide 28.010 132.92 3.499 0.0931 0.0663 0.28966
Nitrogen 28.014 126.10 3.394 0.0901 0.0403 0.28971 809.4
Oxygen 31.999 154.58 5.043 0.0734 0.0218 0.28962 1142.1
Hydrogen Sulphide 34.082 373.53 8.963 0.0985 0.0827 0.28476 801.4
Carbon Dioxide 44.010 304.19 7.382 0.0940 0.2276 0.27275 818.0
Sulfur Dioxide 64.065 430.75 7.884 0.1220 0.2451 0.26729 1394.6
Dry Gas 17.943 197.42 4.850 0.0977 0.0221 0.27992
Wet Gas 30.186 272.40 4.801 0.1345 0.0624 0.28493
Water 18.015 647.13 22.055 0.0560 0.3449 1000.0
Table I.5. Component parameters – Part 5
CO2 H2S N2 CO CO2 H2S N2 CO
H2S 0.097 0.099
N2 −0.017 0.000 −0.032 0.000
Methane 0.092 0.000 0.031 0.030 0.093 0.000 0.028 0.032
Ethylene 0.055 0.083 0.086 0.000 0.053 0.085 0.080 0.000
Ethane 0.132 0.000 0.052 −0.023 0.136 0.000 0.041 −0.028
Propylene 0.093 0.000 0.090 0.000 0.094 0.000 0.090 0.000
Propane 0.124 0.088 0.085 0.026 0.129 0.088 0.076 0.016
Isobutane 0.120 0.047 0.103 0.000 0.128 0.051 0.094 0.000
n–Butane 0.133 0.000 0.080 0.000 0.143 0.000 0.070 0.000
Isopentane 0.122 0.000 0.092 0.000 0.131 0.000 0.087 0.000
n–Pentane 0.122 0.063 0.100 0.000 0.131 0.069 0.088 0.000
n–Hexane 0.110 0.000 0.150 0.000 0.118 0.000 0.150 0.000
n–Heptane 0.100 0.000 0.144 0.000 0.110 0.000 0.142 0.000
n–Decane 0.114 0.000 0.000 0.000 0.130 0.000 0.000 0.000
Cyclohexane 0.105 0.000 0.000 0.000 0.129 0.000 0.000 0.000
Benzene 0.077 0.000 0.164 0.000 0.077 0.000 0.153 0.000
Toluene 0.106 0.000 0.000 0.000 0.113 0.000 0.000 0.000
Table I.6. Binary Interaction Parameter (BIP) for the Peng–Robinson (left) and
the Soave–Redlich–Kwong (right) equation of state according to Reid et. al. (1987)](https://image.slidesharecdn.com/fundamentalsofbasinandpetroleumsystemsmodeling-230616124321-162f3155/85/Fundamentals-of-Basin-and-Petroleum-Systems-Modeling-pdf-464-320.jpg)
![K
Compositions and Components for Fig. 5.14
Dry Gas Wet Gas Gas Condensate Volatile Oil Black Oil
Methane 75.0 84.0 79.0 54.0 47.0
Ethane 7.0 8.0 10.0 9.0 6.0
Propane 6.0 3.0 3.0 8.0 4.0
n–Butane 5.0 1.0 2.0 8.0 3.0
n–Pentane 4.0 1.0 1.5 7.0 2.0
n–Hexane 3.0 1.0
C6−14 2.0 1.7 6.0 10.0
C15 1.8 4.0 9.0
C25 0.7 2.0 8.0
C35 0.2 1.5 6.0
C45 0.1 0.5 5.0
Table K.1. Compositions of the examples in Fig. 5.14. All entries are in molar %
M Tc Pc vc Acentric Volume Shift
[g/mol] [◦
C] [MPa] [m3
/kmol] Factor [m3
/kmol]
Methane 16.043 −82.59 4.599 0.0986 0.0115 0.0007
Ethane 30.070 32.17 4.872 0.1455 0.0995 0.0028
Propane 44.096 96.68 4.248 0.2000 0.1523 0.0052
n–Butane 58.123 151.97 3.796 0.2550 0.2002 0.0080
n–Pentane 72.150 196.55 3.370 0.3130 0.2515 0.0122
n–Hexane 86.177 234.45 3.025 0.3710 0.3013 0.0176
C6−14 133.680 292.11 1.999 0.5801 0.3111 0.0389
C15 183.516 456.45 1.723 0.7770 0.5882 0.0724
C25 323.821 573.739 1.128 1.1930 0.8361 0.1821
C35 442.868 638.133 0.886 1.5020 1.1327 0.3356
C45 537.418 699.490 0.737 1.7490 1.1940 0.4761
Table K.2. Component properties of the examples in Fig. 5.14](https://image.slidesharecdn.com/fundamentalsofbasinandpetroleumsystemsmodeling-230616124321-162f3155/85/Fundamentals-of-Basin-and-Petroleum-Systems-Modeling-pdf-469-320.jpg)
![L
An Analytic Solution for the Diffusion
of Methane Through a Cap Rock
A homogeneous cap rock of thickness h is sketched in Fig. L.1. The concen-
tration at bottom is determined by an methane accumulation with a rather
high concentration value cb. At top the methane might quickly migrate away,
which can be described by a rather small concentration ct. The initial con-
centration within the cap rock is given by c0. The mathematical formulation
is
∂
∂t
c(z, t) = D
∂2
∂z2
c(z, t) (L.1)
with boundary conditions
c(z, 0) = c0 and c(0, t) = ct , c(h, t) = cb for t 0 . (L.2)
The problem is similar to App. F.5 and App. F.6. The solution becomes for
t 0
c(z, t) = ct + (cb − ct)
z
h
+
∞
n=1
2 [c0 − ct + (−1)n
(cb − c0)]
sin μnz
μnh
exp(−μ2
nDt)
(L.3)
with μn = nπ/h.
An example solution is shown in Fig. L.1.More than 1 My are needed for
the establishment of a static concentration gradient in the cap rock. Note
that the time dependency scales quadratical with height h due to the form
of exponent μ2
nDt = n2
π2
Dt/h2
. Hence a cap rock with doubled thickness of
200 m needs four times as long and with 10 times thickness of 1 km about 100
times as long for the convergence to the static solution. These are here times
of 4 My and 100 My. Additionally, the time dependency scales linear inversely
with the diffusion coefficient. Realistic values can become D = 10−11
m2
/s,
which is one order of magnitude smaller than in example Fig. L.1 or even yet
smaller (Sec. 6.4).Transient effects decelerate respectively.](https://image.slidesharecdn.com/fundamentalsofbasinandpetroleumsystemsmodeling-230616124321-162f3155/85/Fundamentals-of-Basin-and-Petroleum-Systems-Modeling-pdf-470-320.jpg)
Fundamentals of Basin and Petroleum Systems Modeling.pdf
- 1. Fundamentals of Basin and Petroleum Systems Modeling
- 2. Thomas Hantschel · Armin I. Kauerauf Fundamentals of Basin and Petroleum Systems Modeling 123
- 3. Dr. Thomas Hantschel Integrated Exploration Systems GmbH A Schlumberger Company Ritterstr. 23 52072 Aachen Germany thantschel@slb.com Dr. Armin I. Kauerauf Integrated Exploration Systems GmbH A Schlumberger Company Ritterstr. 23 52072 Aachen Germany akauerauf@slb.com ISBN 978-3-540-72317-2 e-ISBN 978-3-540-72318-9 DOI 10.1007/978-3-540-72318-9 Springer Dordrecht Heidelberg London New York Library of Congress Control Number: Applied for c Springer-Verlag Berlin Heidelberg 2009 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: Bauer, Thomas Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)
- 4. Preface It is with great satisfaction and personal delight that I can write the foreword for this book Fundamentals of Basin and Petroleum Systems Modeling by Thomas Hantschel and Armin Ingo Kauerauf. It is a privilege for us geoscien- tists that two outstanding physicists, with scientific backgrounds in numerical methods of continuum-mechanics and in statistical physics respectively could be won to deeply dive into the numerical simulation of complex geoprocesses. The keen interest in the geosciences of Thomas Hantschel and Armin I. Kauer- auf and their patience with more descriptive oriented geologists, geochemists, sedimentologists and structural geologists made it possible to write this book, a profound and quantitative treatment of the mathematical and physical as- pects of very complex geoprocesses. In addition to their investigative inter- est during their patient dialogue with afore mentioned geological specialists Thomas Hantschel and Armin I. Kauerauf gained a great wealth of practical experience by cooperating closely with the international upstream petroleum industry during their years with the service company IES, Integrated Explo- ration Systems. Their book will be a milestone in the advancement of modern geosciences. The scientific and the practical value of modern geosciences rests to a large degree upon the recognition of the complex interrelationship of individual processes, such as compaction, heat-, fluid- and mass-flow, reaction kinetics etc. and upon the sequential quantification of the entire process chain. The intelligent usage of modern high speed computers made all this possible. Basin modeling was for many years considered as “a niche discipline”, mainly propagated and used by geochemists. What a fundamental error and misunderstanding! The absolute contrary is the truth. Basin modeling in- tegrates practically all geoscientific disciplines, it allows an unprecedented quantitative understanding of entire process chains and it detects quickly inconsistencies or uncertainties in our knowledge base. In short, the basin modeling–approach is a big step forward in modern geosciences. This book is a challenge for academic teachers in the geosciences and likewise for scien- tists and engineers in the petroleum and mining industry. The challenge is to
- 5. VI Preface educate much more than in the past the younger ones among us to be able to walk along the borderline between the exact sciences with a physcial and mathematical background and the classical geosciences and vice versa. In 1984 Prof. Bernard Tissot and I wrote in the Preface of the second edition of our book Petroleum Formation and Occurrence: “It is evident that computer modeling is here to stay, and may very well revolutionize the field. The computer can be used as an experimental tool to test geological ideas and hypotheses whenever it is possible to provide adequate software for normally very complicated geological processes. The enormous advantages offered by computer simulation of geological processes are that no physical or physico- chemical principles are violated and that for the first time the geological time factor, always measured in millions of years rather than in decades, can be handled with high speed computers with large memories. Thus, the age of true quantification in the geosciences has arrived. We believe that this computer- aided, quantitative approach will have an economic and intellectual impact on the petroleum industry, mainly on exploration.” All this indeed is the case now. And even more so, basin modeling enhances and deepens the intelligent interpretation of geological data acquired by geophysical, geological and geo- chemical methods and thus converts static information into dynamic process understanding. I congratulate the two authors for their excellent textbook. I urge the geoscientific community to dig into the wealth of scientific information offered in this book. It will help us to understand and quantify the dynamics occurring in the subsurface. Dietrich Welte
- 6. VII In the late 1970s “Basin Modeling” was introduced as the term describing the quantitative modeling of geological processes in sedimentary basins on geological timescales. At that time basin models found their main application in heat and pore water flow modeling with regard to sediment compaction and temperature controlled chemistry of hydrocarbon generation. Since then geological, chemical, and transport related models have much improved. Basin modeling turned into a complex and integrated framework of many processes, such as multiphase fluid flow for hydrocarbon migration and accumulation, advanced reaction schemes for organic and mineral transformations or com- pressional and extensional tectonics. The term “Basin Modeling” is not only used for the modeling of processes in sediments, but also for the modeling of crustal and mantle heat and mass flow processes to predict the sedimentary basin type and the related tectonic subsidence. We prefer the naming “Crustal Models”’ for this type of analysis. Obviously, processes in the crust are tightly linked to the sedimentary basin and hence integrated basin and crustal models have also been developed. In addition to pure scientific research there has always been a commercial motivation for basin modeling as a means to understand, quantify and predict petroleum repositories. From the start, the petroleum industry has been the main sponsor for the development of basin modeling tools for exploration and resource assessment. Over time, a number of specialized tools and different types of basin modeling simulators have been developed and with them new terminologies have been introduced, such as “Petroleum Systems Modeling”, “Exploration Risk Assessment” or “Prospect and Play Analysis”. We, the authors of this book, are both physicists with a focus on nu- merical modeling and software design. Since 1990 and 1997 respectively, we have developed major parts of various generations of the commercial basin simulation software PetroMod® . Furthermore, we have offered many training courses on the subject of the theory and fundamental principles behind basin modeling. The training courses contain a fair amount of mathematics, physics and chemistry – the basic building blocks of the software tools. A complete simulation of an actual geological basin often displays complex fluid flow and accumulation patterns which are difficult to interpret. We believe that a basic understanding of the theory behind the tools is essential to master the models in detail. Most basin modelers, in scientific research institutions or the petroleum industry, are expert geologists, coming from an entirely different academic domain. They may therefore be unfamiliar with the mathematics and quanti- tative science related to the software. This results in an abundance of excellent literature about basin modeling from the geological point of view but no com- prehensive study regarding mathematics, physics and computer science. The book is intended above all as an introduction to the mathematical and physical backgrounds of basin modeling for geologists and petroleum explo- rationists. Simultaneously, it should also provide (geo)physicists, mathemati- cians and computer scientists with a more in–depth view of the theory behind
- 7. VIII Preface the models. It is a challenge when writing for an interdisciplinary audience to find the balance between the depth and detail of information on the one hand and the various educational backgrounds of the readers on the other. It is not mandatory to understand all of the details to comprehend the basic principles. We hope this book will be useful for all parties. With this work we also wanted to create a handbook offering a broad picture of the topic, including comprehensive lists of default values for most parameters, such as rock and fluid properties and geochemical kinetics. We hope that our compilation will ease the work of many modelers. The book is not intended as an introduction to the geological principles of basin formation nor as a tutorial to practical basin modeling. Case studies have not been included. A second volume focusing on case studies and the practical aspects of the application is planned for the future. Experts in sedimentology, petrology, diagenesis, fault seal analysis, fractur- ing, rock mechanics, numerics, and statistics may find the approach to some topics in this book too simplistic, but we deliberately came to the decision to open the book to a broader interdisciplinary understanding. At the same time we also feel that we present in many instances ideas which could inspire further studies. The main focus has been on numerical models and features. Naturally, there is a tendency to focus on features which we ourselves developed for PetroMod® , but most of the basic models are also applicable for other aca- demic and commercial software programs. Since there are not many publica- tions by other development groups about the fundamentals, theory and pa- rameters of their work, we were often unable to include appropriate references in our discussion. Basin modeling is a multi–disciplinary science. We hope that students, re- searchers and petroleum explorers with very different experiences will benefit from the presented work. Acknowledgments We wish to acknowledge our families who tolerated many long hours on the computer, in the evening and at the weekend. IES, which recently became part of Schlumberger, was very generous in supporting this work. It is hardly conceivable for us to write such a book without the infrastructure, support and friendly atmosphere at IES. We appreciate the IES geologists, who tried to teach us basics about geol- ogy and supplied us continuously with test models and data. Many examples in this volume can be attributed to their work. Special thanks go to the IES software development team, which provided us among others with excellent software for building and analyzing basin models.
- 8. IX We are indebted to Thomas Fuchs, Michael de Lind van Wijngaarden and Michael Fücker for many interesting discussions. Our understanding was again and again improved, in general and in detail. It must be pointed out, that many rock data values and correlations which were not published up to now are originally from Doug Waples. He did a great job of collecting data over many years. Special acknowledgments are due to the patient referees Michael Hertle, Bernd Krooss, Tim Matava, Ken Peters, Øyvind Sylta, René Thomson, Doug Waples, Dietrich Welte, Michael de Lind van Wijngaarden, Bjorn Wygrala and Gareth Yardley. Hopefully, we did not stress them too much. Chris Bendall and Katrin Fraenzel checked spelling and grammar. Thanks for the hard job. Finally we should gratefully mention all colleagues, customers and friends who accompanied us during the last years. We wanted to avoid missing a person, so we abdicated a huge list. Further special acknowledgments from Armin are due Dr. Gerich–Düssel- dorf, Dr. Kasparek and Dr. Schäfer who diagnosed a serious disease in April 2008. Prof. Dr. Autschbach and his team at the RWTH–Klinikum were able to save Armin’s life in an emergency surgery. Many thanks. Unfortunately, this delayed the appearance of the book at least for two additional months and we have a suitable justification for some typos. December, 2008 Thomas Hantschel and Armin I. Kauerauf
- 9. Contents 1 Introduction to Basin Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 History. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Geological Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.3 Structure of a Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.4 Petroleum Systems Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 1.5 Modeling Workflows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 1.6 Structural Restoration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 1.7 Comparison with Reservoir Modeling . . . . . . . . . . . . . . . . . . . . . . 27 1.8 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 2 Pore Pressure, Compaction and Tectonics . . . . . . . . . . . . . . . . . 31 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 2.1.1 Bulk Stresses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 2.1.2 Pore Pressure Formation and Fluid Flow . . . . . . . . . . . . . 33 2.1.3 Compaction and Porosity Reduction . . . . . . . . . . . . . . . . . 35 2.2 Terzaghi Type Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 2.2.1 Basic Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 2.2.2 Mechanical Compaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 2.2.3 Permeability and Viscosity . . . . . . . . . . . . . . . . . . . . . . . . . 51 2.2.4 1D Pressure Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 2.2.5 Pressure Solutions in 2D and 3D . . . . . . . . . . . . . . . . . . . . 59 2.3 Special Processes of Pressure Formation . . . . . . . . . . . . . . . . . . . . 65 2.3.1 Chemical Compaction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 2.3.2 Fluid Expansion Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 2.4 Overpressure Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 2.5 Geomechanical Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 2.6 Stress and Deformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 2.6.1 Failure Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 2.7 Faults . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
- 10. XII Contents 2.8 Paleo–Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 2.8.1 Event–Stepping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 2.8.2 Paleo–Stepping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 2.8.3 Overthrusting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 3 Heat Flow Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 3.2 One Dimensional (1D) Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 3.2.1 Steady State Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 3.2.2 Transient Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 3.3 Thermal Conductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 3.3.1 Rock and Mineral Functions . . . . . . . . . . . . . . . . . . . . . . . . 112 3.3.2 Pore Fluid Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 3.4 Specific Heat Capacity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 3.4.1 Rock and Mineral Functions . . . . . . . . . . . . . . . . . . . . . . . . 117 3.4.2 Pore Fluid Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 3.5 Radiogenic Heat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 3.6 Three Dimensional Heat Flow Equation . . . . . . . . . . . . . . . . . . . . 119 3.6.1 Heat Convection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 3.6.2 Magmatic Intrusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 3.6.3 Permafrost . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 3.7 SWI Temperatures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 3.8 Crustal Models for Basal Heat Flow Prediction . . . . . . . . . . . . . . 129 3.8.1 The Principle of Isostasy . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 3.8.2 Heat Flow Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 3.8.3 Workflow Crustal Preprocessing . . . . . . . . . . . . . . . . . . . . . 139 3.9 Heat Flow Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 3.9.1 Example Workflow for 3D Heat Calibration . . . . . . . . . . . 145 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 4 Petroleum Generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 4.2 Distributed Reactivity Kinetics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 4.3 Petroleum Generation Kinetics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156 4.3.1 Bulk Kinetics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 4.3.2 Oil–Gas Kinetics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 4.3.3 Compositional Kinetics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 4.4 Thermal Calibration Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . 169 4.4.1 Vitrinite Reflectance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 4.4.2 Molecular Biomarkers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176 4.4.3 Tmax Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 4.4.4 Isotopic Fractionation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
- 11. Contents XIII 4.4.5 Fission–Track Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 4.5 Adsorption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 4.6 Biodegradation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188 4.7 Source Rock Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196 5 Fluid Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199 5.2 Water Phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201 5.3 Binary Mixtures and Black Oil Models . . . . . . . . . . . . . . . . . . . . . 203 5.4 Equations of State (EOS) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207 5.4.1 Mixing Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210 5.4.2 Phase Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 5.5 Flash Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 5.5.1 Classification of Petroleum . . . . . . . . . . . . . . . . . . . . . . . . . 217 5.5.2 PT–Paths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217 5.6 Property Prediction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218 5.6.1 Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218 5.6.2 Bubble Point Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224 5.6.3 Gas Oil Ratio (GOR) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225 5.6.4 Oil Formation Volume Factor Bo . . . . . . . . . . . . . . . . . . . . 225 5.6.5 Viscosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227 5.6.6 Interfacial Tension (IFT) . . . . . . . . . . . . . . . . . . . . . . . . . . . 233 5.7 Calibration of a Fluid Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235 5.7.1 Calibration and Fluid Heavy End . . . . . . . . . . . . . . . . . . . 236 5.7.2 Tuning of Pseudo–Component Parameters . . . . . . . . . . . . 237 5.7.3 Tuning of the Binary Interaction Parameter (BIP) . . . . . 240 5.8 Gas Hydrates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243 6 Migration and Accumulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247 6.2 Geological Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248 6.3 Multi–Phase Darcy Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250 6.3.1 Capillary Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254 6.3.2 Pressure at Phase Boundaries . . . . . . . . . . . . . . . . . . . . . . . 259 6.3.3 Three Phase Flow Formulation without Phase Changes 261 6.3.4 Multicomponent Flow Equations with Phase Changes . . 265 6.3.5 Black Oil Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266 6.4 Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267 6.5 Reservoirs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268 6.5.1 Flowpath Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 270 6.5.2 Drainage Area Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272
- 12. XIV Contents 6.5.3 Accumulation Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276 6.5.4 Faults and Small Scale Features . . . . . . . . . . . . . . . . . . . . . 280 6.5.5 Overpressure and Waterflow . . . . . . . . . . . . . . . . . . . . . . . . 282 6.5.6 Non–Ideal Reservoirs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283 6.6 Hybrid Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286 6.6.1 Domain Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287 6.6.2 Break Through. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289 6.6.3 Fault Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292 6.7 Flowpath Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295 6.8 Invasion Percolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297 6.8.1 Physical Background. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298 6.8.2 Percolation on Microscopic Length Scales . . . . . . . . . . . . . 304 6.8.3 Upscaling of Microscopic Percolation . . . . . . . . . . . . . . . . . 306 6.8.4 One Phase Invasion Percolation . . . . . . . . . . . . . . . . . . . . . 309 6.8.5 Two Phase Migration with Displacement . . . . . . . . . . . . . 312 6.8.6 Discretization of Space and Property Assignment . . . . . . 313 6.8.7 Anisotropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317 6.9 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319 6.10 Mass Balances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327 6.10.1 Fundamental Laws of Mass Conservation . . . . . . . . . . . . . 327 6.10.2 The Petroleum System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 330 6.10.3 Reservoir Structures and Accumulations . . . . . . . . . . . . . . 332 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 336 7 Risk Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341 7.2 Monte Carlo Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344 7.2.1 Uncertainty Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . 348 7.2.2 Derived Uncertainty Parameters . . . . . . . . . . . . . . . . . . . . . 351 7.2.3 Latin Hypercube Sampling (LHC) . . . . . . . . . . . . . . . . . . . 352 7.2.4 Uncertainty Correlations . . . . . . . . . . . . . . . . . . . . . . . . . . . 354 7.2.5 Analysis of Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357 7.2.6 Model Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359 7.3 Bayesian Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 360 7.3.1 Prior Information of Derived Parameters . . . . . . . . . . . . . 365 7.3.2 Correlations of Priors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365 7.3.3 Prior Information of Nominal Uncertainties . . . . . . . . . . . 365 7.4 Deterministic Sampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367 7.4.1 Cubical Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 368 7.4.2 Other Deterministic Designs . . . . . . . . . . . . . . . . . . . . . . . . 369
- 13. Contents XV 7.5 Metamodels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 370 7.5.1 Response Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 370 7.5.2 Fast Thermal Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 372 7.5.3 Kriging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375 7.5.4 Neural Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 376 7.5.5 Other Methods for Metamodeling . . . . . . . . . . . . . . . . . . . 376 7.5.6 Calibration with Markov Chain Monte Carlo Series . . . . 376 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 378 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 378 8 Mathematical Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 381 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 381 8.2 Physical Quantities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382 8.3 Mixing Rules and Upscaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384 8.4 Finite Differences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 387 8.5 Finite Element Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 389 8.6 Control Volumes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394 8.7 Solver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 396 8.8 Parallelization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 396 8.9 Local Grid Refinement (LGR). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 399 8.9.1 Tartan Grid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 400 8.9.2 Windowing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 400 8.9.3 Coupled Model in Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 401 8.9.4 Faults. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 402 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404 A Compaction and Flow Parameter . . . . . . . . . . . . . . . . . . . . . . . . . . 405 B Deviation of the Pressure Equation . . . . . . . . . . . . . . . . . . . . . . . . 413 C Analytic Groundwater Flow Solution from Tóth . . . . . . . . . . . 415 D One Dimensional Consolidation Solution from Gibson . . . . . 419 E Thermal Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 421 F Analytic Solutions to Selected Heat Flow Problems . . . . . . . . 429 F.1 Influence of Radiogenic Heat Production on a Steady State Temperature Profile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 430 F.2 Steady State Temperature Profile with a Lateral Basal Heat Flow Jump . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 430 F.4 Steady State Temperature Profile for a Two Block Model . . . . . 433 F.5 Non Steady State Model with Heat Flow Jump. . . . . . . . . . . . . . 434 F.3 Steady State Temperature Profile with SWI Temperature Jump 432
- 14. XVI Contents F.6 Non Steady State Model with SWI Temperature Jump . . . . . . . 436 F.7 An Estimate for the Impact of Continuous Deposition on Heat Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 437 G Petroleum Kinetics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 441 H Biomarker . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445 I Component Properties. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 449 J Methane Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 457 K Compositions and Components for Fig. 5.14 . . . . . . . . . . . . . . . 459 L An Analytic Solution for the Diffusion of Methane Through a Cap Rock . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 461 M Flowpath Bending . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 463 N Unit Conversions and Constants . . . . . . . . . . . . . . . . . . . . . . . . . . . 467 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 469
- 15. 1 Introduction to Basin Modeling 1.1 History Geology and geochemistry in sedimentary basins have been established sci- ences for centuries. Important textbooks, such as Tissot and Welte (1984); Hunt (1996); Gluyas and Swarbrick (2004); Peters et al. (2005); Allen and Allen (2005), summarize the knowledge especially related to petroleum geo- sciences. The first basin modeling computer programs were developed around 1980 (Yükler et al., 1979). The main concept encompassed multi–1D heat flow simulation and subsequent geochemical models to construct petroleum gener- ation and expulsion maps for the evaluation of source rock maturity. One of the key tasks was to calculate and calibrate the temperature history during the evolution of a geological basin. Heat flow calculation is one of the best in- vestigated problems in applied engineering. A formulation and solution of the corresponding differential equations can be easily achieved. Once the paleo– temperatures were known, equations for chemical kinetics could be used to evaluate the cracking rates of petroleum generation. Another important part of the analysis was the prediction of pore fluid pressures. Transport equations for one fluid phase with a special term for the overburden sedimentation rate were used to calculate the compaction of the sediments. The compaction state and related porosity facilitated the determination of bulk thermal conductiv- ities for heat flow calculations. At that time, practical studies were mainly performed as 1D simulations along wells, because the computer capabilities were still limited and multiphase fluid flow for migration and accumulation of petroleum had not been well implemented. Temperature profiles from multi– well analysis were used to calculate petroleum generation with source rock maturity maps over time and the determination of the peak phases of oil and gas expulsion. This concept is still used when data are scarce in early exploration or when the project requires some quick output. From 1990 to 1998 a new generation of basin modeling programs became the standard in the petroleum industry. The most important new feature was T. Hantschel, A.I. Kauerauf, Fundamentals of Basin and Petroleum 1 Systems Modeling, DOI 10.1007/978-3-540-72318-9 1, © Springer-Verlag Berlin Heidelberg 2009
- 16. 2 1 Introduction to Basin Modeling the implementation of refined fluid flow models with three phases: water, liquid petroleum, and gas. In commercial packages, 2D Darcy flow models and map based flowpath analysis were realized (Ungerer et al., 1990; Hermanrud, 1993). Darcy flow models are able to model all relevant processes of flow, accumula- tion, and seal break through. They are based on differential equation systems for the competing fluid phases. However, they are restricted to 2D simula- tors, since they require a high computing and development effort. The map based flowpath technique redistributes pre-calculated expulsion amounts of petroleum along reservoir–seal interfaces within the reservoirs. Accumulation bodies are calculated under correct conservation of the petroleum mass and volume. The approach is based on some crude approximations concerning flow. However, it considers horizontal spilling from one drainage area to the next and simple break through when the column pressure exceeds the seal capabil- ity. Most models under study were first performed in 2D along cross sections because pre-interpreted horizons and faults along 2D seismic lines were read- ily available. Calculated generation and expulsion amounts were again used for the flowpath analysis afterwards. Although 2D Darcy flow models work very well, they were rarely used in practical exploration studies as horizontal petroleum migration in the third dimension can not be neglected. Another important innovation was the implementation of special geological processes such as salt dome tectonics, refined fault behavior, diffusion, cementation, fracturing, and igneous intrusions. In 1998, a new generation of modeling programs were released changing the workflow of most basin modeling studies once again. Many new features were related to petroleum migration and the characteristics of reservoirs. Most programs and tools focused on 3D functions with improved features for model building and increased simulator performance. From that time on, most of the heat and pore pressure calculations were performed in full 3D. This re- quired the interpretation and mapping of a relatively complete set of horizons instead of just the horizons of the reservoirs. Three–phase–Darcy flow mod- els were also made available in 3D. However, high computation efforts were necessary while simplifying the model’s premises to a large degree. Conse- quently the model’s resolution was restricted which often led to unrealistic or oversimplified geometries. Pure Darcy flow models were not applicable in practice. Three alternatives for modeling migration were developed. One was the use of the well established flowpath models, the other two are new devel- opments: hybrid flow simulators and the invasion percolation method. Hybrid fluid flow models use domain decomposition to solve the Darcy flow equations only in areas with low permeabilities and flowpath methods in areas with high permeabilities, resulting in a significant decrease of computing time. In- vasion percolation is another rule based transport technique which focuses on capillary pressure and buoyancy without any permeability controlled flow tim- ing. Another new feature was the implementation of multicomponent resolved petroleum phases and the development of fast thermodynamic PVT (Pres- sure Volume Temperature) controlled fluid analysis based on flash calculation
- 17. 1.2 Geological Processes 3 for these components. Between four and fourteen fluid components (chemical species) are usually taken into consideration, replacing the traditional two component (oil–gas) black oil models. Reservoir composition and petroleum quality prediction were significantly improved. Simultaneously, better com- puter hardware especially PC clusters combined with parallelized simulators, reduced computing times significantly. Furthermore, statistics for calibration, risk analysis for quantification of probability for success or failure and the con- sideration of extensional and compressional tectonics significantly increased the applicability of basin modeling. Integrated exploration workflows, which incorporate basin modeling, became a standard in the industry. 1.2 Geological Processes Basin modeling is dynamic modeling of geological processes in sedimentary basins over geological time spans. A basin model is simulated forward through geological time starting with the sedimentation of the oldest layer until the entire sequence of layers has been deposited and present day is reached. Several geological processes are calculated and updated at each time step (Fig. 1.1). Most important are deposition, compaction, heat flow analysis, petroleum generation, expulsion, phase dissolution, migration, and accumulation. Deposition Layers are created on the upper surface during sedimentation or removed during erosion. It is assumed that the geological events of deposition and hiatus are known. Therefore, paleo times of deposition can be assigned to the layers. The depositional thickness of a new layer is calculated via porosity con- trolled backstripping from present day thickness or imported from structural restoration programs. The overall geometry may also change due to salt move- ment or magmatic intrusions. Estimated backstripping amounts yield calcu- lated present day thicknesses which are not identical with the given present day geometry. The differences facilitate a better estimation of the depositional thicknesses in the next simulation run. This method of organizing multiple for- ward simulations to calibrate against the present day geometry is referred to as optimization procedure. Pressure Calculation and Compaction Pressure calculation is mainly a one–phase water flow problem which is driven by changes of the overburden weight due to sedimentation. Additionally, in- ternal pressure building processes such as gas generation, quartz cementation and mineral conversions can be taken into account. Pore pressure reduction entails compaction and leads to corresponding changes in the geometry of the basin. That is why pressure calculation and compaction have to be performed before heat flow analysis in each time step.
- 18. 4 1 Introduction to Basin Modeling Deposition Pressure Calculation Heat Flow Analysis Petroleum Generation Fluid Analysis Petroleum Migration Reservoir Volumetrics (Sedimentation, Erosion, Salt Doming, Geological Event Assignment) and Compaction and Kinetics of Thermal Calibration Parameters and Adsorption and Expulsion (Phase Compositions) ( Invasion Percolation, Flowpath Analysis) Darcy Flow, Diffusion, Timesteps Migration Timesteps Fig. 1.1. Major geological processes in basin modeling Heat Flow Analysis Temperature calculation is the target of the heat flow analysis. It is a nec- essary prerequisite for the determination of geochemical reaction rates. Heat conduction and convection as well as heat generation by radioactive decay must be taken into consideration. Igneous intrusions require the inclusion of thermal phase transitions in sediments. Thermal boundary conditions with in- flow of heat at the base of the sediments must be formulated. These basal heat flow values are often predicted with crustal models in separate preprocessing programs or are interactively calculated for each geological event. Kinetics of Calibration Parameters It is possible to predict vitrinite reflectance values, the concentration of molec- ular biomarkers and apatite fission tracks with suitable models which are based on Arrhenius type reaction rates and simple conversion equations. These predictions are temperature sensitive and can therefore be compared to mea- sured data so that uncertain thermal input data, such as paleo–heat flow values, can be restricted or even calibrated.
- 19. 1.2 Geological Processes 5 Petroleum Generation The generation of petroleum components from kerogen (primary cracking) and the secondary cracking of the petroleum is usually described with sets of parallel reactions of decomposition kinetics. The number of chemical com- ponents vary between two (oil, gas) and twenty. The cracking schemes can be quite complex when many components and secondary cracking are taken into account. Adsorption models describe the release of hydrocarbons into free pore space of the source rock. Fluid Analysis The generated hydrocarbon amounts are mixtures of chemical components. Fluid flow models deal with fluid phases which are typically liquid, vapor and supercritical or undersaturated phases. Therefore temperature and pressure dependent dissolution of components into the fluid phases is studied during fluid analysis. The two most important fluid models are the rather simple black oil model and the thermodynamically founded multicomponent flash calculations. Fluid phase properties, such as densities and viscosities, are also derived from fluid models. They are essential for accurate migration modeling and reservoir volumetrics. Darcy Flow and Diffusion Darcy flow describes multicomponent three phase flow based on the relative permeability and capillary pressure concept. It can be applied for migration. Migration velocities and accumulation saturations are calculated in one pro- cedure. Special algorithms are used to describe break through and migration across or in faults. Diffusion effects can be evaluated for the transport of light hydrocarbons in the water phase. Flowpath Analysis In carriers lateral petroleum flow occurs instantaneously on geological time- scales. It can be modeled with geometrically constructed flowpaths. Informa- tion about drainage areas and accumulations with compositional information can easily be obtained. Spilling between and merging of drainage areas must be taken into account. Flowpath analysis in combination with Darcy flow in low permeability regions is called the hybrid method. Migration modeling without sophisticated Darcy flow, instead using simplified vertical transport of generated hydrocarbons into carriers, is commonly called flowpath modeling. Invasion Percolation Migration and accumulation can alternatively be modeled with invasion per- colation. This assumes that on geological timescales petroleum moves instan- taneously through the basin driven by buoyancy and capillary pressure. Any time control is neglected and the petroleum volume is subdivided into very small finite amounts. Invasion percolation is very convenient to model in– fault flow. The method is especially efficient for one phase flow with the phase consisting of only a few hydrocarbon components.
- 20. 6 1 Introduction to Basin Modeling Reservoir Volumetrics The column height of an accumulation is balanced by the capillary entry pressure of the corresponding seal. Leakage and break through are therefore important processes reducing the trapped volume. Other processes such as secondary cracking or biodegradation also have a serious impact on the quality and quantity of the accumulated volume. In principle all processes depend on each other. Therefore, at a given time, all these coupled processes must be solved together with the solution of the last time step as the initial condition. For numerical reasons such an approach can be performed implicitly in time and is thus called an implicit scheme. In practice it is found, that the processes can be decoupled, very often to some high order of accuracy. Finally it is possible to solve for all the processes which are shown in Fig. 1.1 in the given order. Extra loops with iterative updates for higher accuracy can easily be performed. Decoupled schemes are often called explicit schemes, especially if the processes itself are treated explicitly in time. For example, migration and accumulation seldom has an important effect on basin wide compaction. Thus migration can often be treated independently. However, a coupling of migration with compaction might arise with pressure updates due to gas generation and subsequent local modification of the geome- try. By re-running the entire simulation with consideration of the gas pressure of the previous run, the modified geometry can in principle be iteratively im- proved until convergence is reached. In practice, it is often found, that only very few iterative runs are necessary. For the implicit scheme, the temporal evolution of the basin must obviously be calculated on the smallest timescale of all involved geological processes. A big advantage of an explicit scheme is the fact, that each explicitly treated process can be solved on its own timescale. On the other hand, time steps of implicitly treated processes can often, for numerical reasons, be longer than time steps of explicitly treated processes. This increases the performance of the implicit scheme, especially when iterative feedback loops have to be taken into account in explicit schemes. In practice, a combination of both schemes is found to be most advantageous. This yields three types of time steps, which are often called events, basic and migration time steps. The outer time loops are identical with geological events. They characterize the period in which one layer has been uniformly deposited or eroded or when a geological hiatus occurred. Thus, the total number of events is almost equal to the number of geological layers and usually ranges between 20 and 50. Events are subdivided into basic time steps with one solution for pressure or compaction and the heat equations. The length of the basic time step depends on deposition or erosion amounts and on the total duration of the event. The total number of time steps usually lies between 200 and 500. The basic time steps are further subdivided into migration steps for an explicitly treated Darcy flow analysis. In one migration time step the transported fluid amount per cell is usually restricted to the pore volume of that cell. Therefore
- 21. 1.2 Geological Processes 7 the total number ranges from 1000 up to 50000 and more and depends on the flow activity and the selected migration modeling method. All time loops for events, basic time steps and migration time steps are commonly managed automatically in most simulators. Mathematical convergence is often ensured by empirical rules for step length calculation. Transport Processes Heat flow, pore pressure and compaction, Darcy flow migration processes, and diffusion are transport processes. They follow a similar scheme of description, derivation, and formulation of the basic equations. The core problem is the interaction of two basic quantities, the state and the flow variable (Table 1.1). The influence of a flow variable acting from any location on any other neigh- boring location is the main part of the mathematical formulation. Modeling of transport problems requires a major computing effort. For example, temperature and heat flow are the corresponding basic vari- ables for heat conduction. Temperature is the state variable and heat flow is the corresponding flow variable. A temperature difference (or gradient) causes a heat flow, and the heat flow decreases the temperature difference. The heat flow is controlled by the thermal conductivity and the temperature response by the heat capacity. State variable Flow variable Flow equation Material property Temperature T Heat flow q q = −λ · grad T Thermal conductivity λ Pressure p Water flow vw vw = − k ν · grad(p − ρgz) Permeability k and viscosity ν Fluid potential up Fluid flow vp vp = − kkrp νp · grad up Relative perm. kkrp and viscosities νp Concentration c Diffusion flux J J = −D grad c Diffusion coeff. D Table 1.1. Fundamental physical transport laws and variables In general, an energy or mass balance can be used to formulate a boundary value problem with appropriate boundary conditions and to calculate the development of both the state and the flow variables through geological time. A solution to the boundary value problem requires in practice a discretization of the basin into cells and the construction and inversion of a large matrix. The matrix elements represent the change of the state variable caused by the flow between two neighboring cells. The number of cells is the number of unknowns. Finally, an inversion of the matrix results in the solution vector, e.g. containing a temperature inside of each cell.
- 22. 8 1 Introduction to Basin Modeling The inversion of transport processes is often the major computing effort in basin modeling (Chap. 8). It depends strongly, almost exponentially, on the number of cells and therefore the resolution. Examples of non-transport processes are fluid analysis, chemical kinetics and accumulation analysis, which depend only linearly on the number of cells if they are separated and explicitly treated. These processes can then be modeled very efficiently. 1.3 Structure of a Model The general analysis of the basin type and the main phases of basin evo- lution precede the construction of the model input data. This encompasses information about plate tectonics, rifting events, location of the basin, and depositional environments through geological time, global climates, paleo– bathymetries, and tectonic events. The model input is summarized in Fig. 1.2, and includes: present day model data with depth horizons, facies maps, fault planes, the age assignment table for the geological event definition, additional data for the description of paleo–geometries, thermal and mechanical bound- ary conditions through geologic time, the property values for lithologies, fluids, and chemical kinetics. 1 - Horizons (Depth/Structure Maps) - Facies Maps - Fault Surfaces 3 - Water Depth Maps - Erosion Maps - Salt Thickness Maps - Paleo Thickness Maps Present Day Model Paleo Geometry 4 - SWI-Temperature Maps - Basal Heat Flow Maps 5 - Facies Definitions - TOC HI Maps - Rock Composition Maps 6 (optional) - Attributes (Cubes, Maps) - Depth Conversion Boundary Conditions Facies Seismic Reference Horizons for 2 Age Assignment Fig. 1.2. Basic elements of model input Present Day Model Data A sedimentary basin is a sequence of geological layers. Each of the layers contains all the particles which have been deposited during a stratigraphic event. A horizon is the interface between two layers (Fig. 1.3) and usually interpreted from a seismic reflection surface. Seismic interpretation maps and lines (in 2D) are usually not extended over the entire model area and have to be inter– and extrapolated and calibrated with well data. The construction of the horizon stacks often requires most of the time for the model building.
- 23. 1.3 Structure of a Model 9 L. Cret. Unconf. Shublik Fm. Basement Present-day Surface Lisburne Gr. Brookian Forsets L. Cret. Unconf. Shublik Fm. Basement Present-day Surface Lisburne Gr. Brookian Forsets L. Cret. Unconf. Shublik Fm. Basement Present-day Surface Lisburne Gr. Brookian Forsets Horizon ( ) Horizon ( ) L.Cret.Unconf. Top Shublik Layer ( ) Kingak Facies ( ) Kingak_Facies a) Horizons, Layer, Facies c) Stratigraphy and Horizons b) Example Facies Map for the Layer deposited between 115 and 110 My Sand Shale Topset Clinoform Bottomset -4000 m (Eroded Brookian) Horizons Sediment Surface 4000 m Brookian Formation up to 5000 m Beaufortian Ellesmerian Reconstructed Horizons f) Example Paleo-Water Depth Map 115 My g) Erosion Maps Erosion 2 40 My Erosion 3 24 My Erosion 1 60 My d) e) meter meter meter meter Fig. 1.3. Present day and paleo–geometry data: example from Alaska North Slope
- 24. 10 1 Introduction to Basin Modeling A complete stack of horizon maps subdivides the space for volumetric prop- erty assignments. Parts of layers with similar sedimentation environments are called geological facies (Fig. 1.8). Facies are related to common property val- ues of geological bodies. They are the main “material types” of the model. Layers can consist of several different facies and the same facies can appear in different layers. The distribution of facies is usually described with one facies map in each layer, based on well data information and sedimentological princi- ples, e.g. clastic rocks are distributed corresponding to relationships between grain size and transport distances, particularly the distance from the coast (Fig. 1.3). In simple cases a layer can be characterized only by one unique fa- cies type, whereas high resolution seismic facies maps allow the construction of very detailed facies maps (Fig. 1.10). Fault planes are constructed from seismic interpretations, well data, and dips, which can also require a lot of effort. Depth horizons, facies maps, and fault planes constitute the present day model. Age Assignment The age assignment or stratigraphic table relates the present day horizons and layers with the geologic age of their deposition and erosion. In layer sequences without erosions, horizons represent all sedimentary particles, which are deposited during the same geological events (Fig. 1.3). If valid for the model, erosion and hiatus events also have to be included in the stratigraphic table. Erosion events require additional maps for the amounts of erosion and have to be combined with the corresponding water-depth for the description of the related uplift of the basin. Stratigraphic diagrams with facies variations (Fig. 1.3) have to be simpli- fied in order to get a relatively low number of model horizons in the range of 10−50. Migrating patterns of facies through time generally require a Wheeler diagram instead of one single simplified age table. However, this feature is rather difficult to implement into a computer program. Paleo-Geometry Data The present day model can be built from measured data, such as seismic and well data. The paleo–model is mainly based on knowledge and princi- ples from historical and regional geology, sedimentology and tectonics, which results in higher degrees of uncertainty. Water depth maps are derived from isostasy considerations of crustal stretching models together with assumptions on global sea level changes. They describe the burial and uplift of the basin. Water depth maps can also be derived from known distributions of sediment facies and vice versa (see e.g. the equivalence of the water–depth and facies map at 115 My in Fig. 1.3.b and f). The construction of the erosion maps is usually more difficult. In the sim- plest case, one layer is partially eroded during one erosional event. The erosion thickness can be re–calculated by decompaction of the present day thickness and subtraction from an assumed relatively uniform depositional map. The
- 25. 1.3 Structure of a Model 11 0 24 25 40 41 60 65 115 126 208 260 360 400 Present Day Top Oligocene Top Lutetian Top Upper Cretaceous Top Lower Cretaceous L.Cret.Unconf. Top Shublik Top Lisburne Top Basement Base Basement Kingak Shublik Lisburne Basement Brookian D Brookian C Brookian B Brookian A EROSION EROSION EROSION BrFac D BrFac C BrFac B BrFac A Kingak_Facies Shublik_Facies Lisburne_Facies Basement_Facies none none none Erosion3 Erosion2 Erosion1 Age [My] Facies Maps Erosion Maps Paleo-Water Depth Maps PWD_0 PWD_25 PWD_41 PWD_60 PWD_65 PWD_115 ... ... ... PWD_24 PWD_40 ... ... ... ... Horizon Layer Fig. 1.4. Excerpt from the age assignment table of the Alaska North Slope model sediment surface of the example model in Fig. 1.3.d acts as a unconformity and cuts many layers. A simple approach is to construct the missing erosion amount for each layer separately and to assume uniform erosion during the time period of erosion. This is illustrated in Fig. 1.3.e with the virtual horizons of the Brookian formation above the sediment surface. However, in the consid- ered model it is further known that there were three main erosion periods and thus the corresponding erosion maps could be constructed (Fig. 1.3.g.). These maps together with the virtual Brookian horizons yield the erosion amounts for each of the layers in the three erosion events. The above model description would have been sufficient, if the Brookian formation were eroded after complete deposition. In reality, compressional deformation in the Tertiary produced a fold–and–trust belt resulting in uplift and erosion and in a broad shift of the basin depocenters from WSW to ENE, which lead to mixed erosion and deposition events. A schematic description is illustrated in Fig. 1.5 which is finally realized in the age assignment table of Fig. 1.4. Note, that each erosion mentioned in the age assignment table consists of several layer specific maps with the erosion amounts related to the respective event. Unfortunately, such a complicated behavior is rather typical than exceptional. Input building tools often provide sophisticated map calculators with special features to make the construction of erosion maps easier. A preliminary simulation result of an ongoing Alaska North Slope study is shown in Fig. 1.6. The occurrence of salt diapirs requires paleo-thickness maps for the main phases of salt doming. The reconstruction of the salt layers is usually based on geometrical principles, in the simplest case the present day thickness map
- 26. 12 1 Introduction to Basin Modeling 97 110 85 65 Deposition until 65 My WSW ENE Erosion A: 65 and 60 My 60 97 110 85 65 WSW ENE Deposited Uplifted Eroded Deposition until present day WSW ENE 97 110 85 65 55 41 25 0 Deposition until 41 My 97 110 85 65 WSW ENE 55 41 Erosion B: 41 and 40 My 40 97 110 85 65 55 41 WSW ENE Deposited Uplifted Eroded Deposition until 25 My WSW ENE 97 110 85 65 55 41 25 Erosion C: 25 and 24 My 24 WSW ENE 97 110 85 65 55 41 25 Deposited Uplifted Eroded Fig. 1.5. Paleo–geometry data: example from the Alaska North Slope
- 27. 1.3 Structure of a Model 13 Celsius Gas from Kingak Oil from Kingak Gas from Shublik Oil from Shublik Gas from Hue Oil from Hue Insitu Liquid Composition Insitu Liquid Composition Surface Composition Liquid Vapor Fig. 1.6. Source rock tracking in Alaska North Slope. The two big visible accumu- lations are the Kuparuk (center) and Prudhoe Bay (right) fields is linearly interpolated to an uniform deposition map. Corrections are made, if the resulting paleo-geometries show unrealistic kinks in the reconstructed base–salt maps. Salt layers can also be reconstructed based on calculated lithostatic pressures or total stresses at the salt boundaries because salt moves along the gradient of the lowest mechanical resistivity. The reconstructed salt thickness maps can be implemented in the input model by two methods: paleo–thicknesses for autochthonic salt layers and penetration maps for al- lochthonous salt bodies as illustrated in Fig. 1.7 for the Jurassic salt layer of the Northern Campos Model. Autochthonous salt maps through geologic times can be simply realized by adjusting the layer thickness in each grid- point. The occurrence and timing of the salt windows is often very important for petroleum migration and pressure development as subsalt fluids and pres- sures are released afterwards. The penetration of shallower sediments by salt and the formation of single allochthonous salt bodies is usually implemented with the replacement of the original sediment facies by the salt facies. Both methods have to be combined with adjustments of the other sediment thicknesses to maintain the mass balance. These correction maps can be added to the input data as paleo- thickness maps during the corresponding events.
- 28. 14 1 Introduction to Basin Modeling 0 0.4 0.8 1.2 2.0 km Salt Thickness Salt Domes in the Northern Campos Basin Selected Thickness Maps of the Autochtonous Salt Layer 2 3 4 5 6 7 Cross-section Cross-section with Salt Depth [km] non-salt sediment autochtonous salt layer allochtonous salt bodies Upper Cret. Layer Lower Tertiary Layer Salt Penetration at 32 My Salt Penetration at 44 My Salt Penetration at 55 My Salt Penetration at 90 My Upper Bota Layer Albian Layer Depositon 117 My 100 My Opening of Salt Windows 55 My Present Day Selected Salt Penetration Maps of the Layers above the Autochtonous Salt Layer Fig. 1.7. Paleo–salt maps: example from the Northern Campos Basin in Brazil The interplay of paleo-water depth, erosion, salt thickness, and other paleo- thickness maps finally determines the paleo-geometries and often requires some experience of the basin modeler to build geological reasonable scenarios. Boundary Conditions Boundary conditions need to be defined for the heat, pressure, and fluid flow analysis through the entire simulated geologic history. The usual boundary condition data for the heat flow analysis are temperature maps on the sedi-
- 29. 1.3 Structure of a Model 15 ment surface or the sediment–water interface and basal heat flow maps for the respective events. The surface temperature maps are collected from general paleo–climate databases. The basal heat-flow maps can be estimated from crustal models and calibrated with thermal calibration parameters, which is explained in more detail in Chap. 3. Specific inner and upper igneous intru- sion temperature maps should be added for magmatic intrusion and extrusion events, respectively. The boundary conditions for the pore pressure and fluid flow analysis are often defined as ideal open (e.g. at sediment surface) and ideal closed (e.g. at base sediment). Exceptions are onshore basins or erosion events, which require the definition of groundwater maps to calculate the groundwater potential as the upper boundary condition for the pore pressure analysis. Herein, the sediment surface could be a good approximation. It is a common method to determine the boundary values through geo- logic history as trend curves at single locations (gridpoints) first and calculate boundary value maps for the geological events by inter– and extrapolation af- terwards. Facies Properties Facies are sediment bodies with common properties. The name facies is widely used in geoscience for all types of properties. Here, the facies is characterized by two sub–group facies types: the rock facies (or lithology) and the organic facies (or organofacies, Fig. 1.8). A classification of lithologies is also shown in Fig. 1.8. It is used for the rock property tables in the appendix. The main rock properties are ther- mal conductivities, heat capacities, radiogenic heat production, permeabili- ties, compressibilities, and capillary entry pressures. Most of them depend on temperature and porosity. Functions for fracturing and cementation are also rock specific properties. A classification of the organic facies is discussed in Chap. 4. The organic facies encompass all kinetic parameters for the generation and cracking of petroleum and the parameters to specify the quantity and quality of organic matter. The kinetic parameters are mainly Arrhenius–type activation energy and frequency data for primary and secondary cracking of hydrocarbon com- ponents. The total organic content (TOC) and the hydrogen index (HI) are usually defined by distribution maps. Furthermore, adsorption parameters are also related to the organic facies type. Fluid properties are either given directly for the different fluid phases or calculated from compositional infor- mation. Fluid phase properties are e.g. densities or viscosities. Typical fluid component properties are critical temperatures, pressures, and specific vol- umes. Seismic Seismic attribute cubes or maps can be used to refine the facies distribution maps in some layers, e.g. the ratio of shear to compressional velocity is cor- related to the average grain size of clastic rock. The conversion of seismic
- 30. 16 1 Introduction to Basin Modeling Facies Lithology (Rock Facies) - Thermal Properties: Conductivity, Heat Capacity, Radiogenic Heat Production - Mechanical Properties: Compressibility - Fluid Flow Properties: Permeabilities, Capillary Pressures Organic Facies - Organic Content: TOC, HI, Kerogen Type - Primary and Secondary Cracking Kinetics: Activation Energy Distributions - Adsorption Coefficients Sedimentary Rocks - Clastic Sediments: Sandstone, Shale, Silt - Chemical Sediments: Salt, Gypsum, Anhydrite - Biogenic Sediments: Chalk, Coal, Kerogen - Carbonate Rocks: Limestone, Marl, Dolomite Metamorphic and Igneous Rocks - Igneous Rocks: Granite, Basalt, Tuff - Metamorphic Rocks: Marble, Gneiss Minerals (for mixing of rock types) - Rock Fragments - Rock Forming Minerals: Quartz, Feldspar, Olivine - Other Minerals: Smectite, Illite Lithology 0.00001 0.0001 0.001 0.01 0.1 1 10 grain size in mm Clay Silt Sand Gravel Classification WENTWORTH FOLK very fine fine medium coarse very coarse granule pebble cobble Micrite Lutite Arenite Rudite Siltite Carbonates Clastic Sediments Clastic Sediments and Carbonates Fig. 1.8. Classification of facies, lithologies with the most important examples and terminology of clastic sediments and carbonates according to grain sizes. The picture is from Bahlburg and Breitkreuz (2004) attributes to a “lithocube” requires a lot of effort and is only available in a few projects. Seismic facies cubes are usually available for the reservoir layers. In Fig. 1.9 and 1.10 two example cases from Australia and the North Sea are shown. Seismic facies cubes and maps are used, respectively. Seismic cubes can be given in two–way–time or depth. They require reference horizons to map the corresponding cells from the seismic to the depth model. The re- sulting facies distribution can be even finer than the major model grid. The
- 31. 1.3 Structure of a Model 17 invasion percolation method, which is used for modeling of migration, works on a sub–gridding of the cells and takes high resolution features into account (Chap. 6). Capillary entry pressures from the finer scale seismic facies control migration and accumulation. Seismic Cube with Facies Attributes 3D Depth Model Color: Capillary Entry Pressure Model Horizons Reference Horizons Seismic Facies Seismic Facies mapped to Lithologies Invasion Percolation on Refined Grid Fig. 1.9. Seismic cube with facies attributes and migration with invasion percola- tion. The attributes are mapped via reference and model horizons to the 3D model. For example, a point which lies at 35 % vertical distance between two reference hori- zons is here assumed to lie on the same relative position between the corresponding model horizons The North Sea petroleum migration example (Fig. 1.10) is mainly re- stricted to two layers only: the upper Jurassic layer, and the overlaying chalk layer. The Jurassic layer contains high organic content shale and sandstone. It
- 32. 18 1 Introduction to Basin Modeling Middle Jurassic Invasion percolation grid with 34 million cells and 250 m resolution in the regional scale and 60 m resolution in the prospect scale Cross section from a 3D Model in the Danish North Sea Salt Upper Jurassic Layer contains Source Rock (Shale) and Reservoir Rock (Sandstone) Cretaceous and Lower Tertiary Layers contain Chalk Reservoirs Sandstones Shales Low velocities Middle Jurassic Facies Map 30 m Resolution Calk Content Map 30 m Resolution and Lower Tertiary Upper Cretaceous High velocities Fig. 1.10. High resolution maps and migration modeling with invasion percolation. The figures are courtesy of MAERSK
- 33. 1.3 Structure of a Model 19 is both, a source and a reservoir layer. The chalk layer also contains petroleum accumulations and it is sealed by a dense overlaying shale. The two seismic attribute maps are applied to the layers without any further subdivision in vertical directions. In this case, the invasion percolation method is especially suitable, as high resolution is important and the migration distances are short. Discretization of a Model A continuum approach is commonly applied for the general description of heat and fluid flow processes on a macroscopic scale. Practical solutions can, on the other hand, only be obtained for discretized models. A mesher generates grids with the cells as the smallest volumetric units of the geological model. The basin or region of interest is assumed to be covered continuously with cells. Every physical or geological quantity such as temperature, pressure, satura- tion, concentration, permeability, thermal conductivity, etc. is well defined in the cell as a single, effective or average value. Furthermore, the value can vary continuously from cell to cell at least within parts of the structure. Each cell is used as a finite element or finite volume within the mathematical solvers. The approach requires that the size of the cell must be small compared to the system being modeled (basin scale) but, at the same time, large compared to the pore scale and grain size. Typical scale sizes are Molecular Scale: 10−9 . . . 10−8 m Pore Scale: 10−6 . . . 10−3 m Bulk Continuum: 10−3 . . . 10−2 m Cells of the Grid: 100 . . . 102 m Basin Scale: 103 . . . 105 m with cells which are much larger than the pore scale and grain sizes and much smaller than the basin scale.1 However, modern simulation programs might contain different grid scales and even different basin scales for the modeling of different geological pro- cesses. Such multigrids are typically created with sampled and refined repre- sentations of a master grid. Optimal methods can then be applied for each geological process. For example, heat flow is often modeled on the full basin scale with grid cells seldom smaller than 100 m, whereas petroleum systems modeling is sometimes restricted to smaller areas of source rock expulsion and active migration pathways with corresponding grid cells, which can become very small. However, sophisticated up- and downscaling functions (e.g. for fractal saturation patterns) may be required. Many quantities can be defined as gridded maps at certain events. Al- ternatively, geological time dependent trend functions are often specified at 1 In finite element simulators, a continuous crossover within a cell is modeled and the bulk continuum scale, rather than the cell size of the grid, must be compared with the basin scale. Finite elements therefore often show an implicitly higher resolution than other cell types.
- 34. 20 1 Introduction to Basin Modeling individual well locations. Maps are then generated for each event by spatial interpolation over the whole model area. In both cases maps are the central objects for the creation of a basin model. Size of a Model A primary target of basin modeling is the assessment of exploration risk by calculation of generated and accumulated petroleum volumes for different ge- ological migration scenarios. Herein, basin to reservoir scale models are used from a total length of hundreds of kilometers down to only a few kilometers (Fig. 1.11). Another study type concerns resource assessments, which cover even more extensive geographical areas such as entire countries (Fig. 1.11). The total amount of oil and gas resources in several layers is estimated. This task often encompasses source rock maturity studies including volumetrics for migration losses with simplified reservoir distributions. Governmental geolog- ical surveys and academic institutes often contribute to such studies. Typical model dimensions and grid data are shown in Fig. 1.11. In prac- tice, there are in general two requirements, a minimum model resolution to approximate the geological structures of interest and a simulation run time of less than 12 hours. This is a “rule of thumb” of the authors: a simulation must to be able to run in one night. Computer performance has significantly increased since the introduction of parallelized simulations on computer clusters. The average number of cells for a complete simulation is 1−2 million cells which corresponds to 200−300 gridpoints in the horizontal directions. Heat, pressure, and Darcy flow com- puting times depend almost exponentially on the number of cells. Doubling the number of gridpoints in one direction often increases the computing effort by one order of magnitude. That is why big improvements in computer per- formance and numerical methods often have only a small effect on the grid resolution. However, computing time is very difficult to estimate as some im- portant controlling parameters, such as the number of hydrocarbon containing cells, average and peak fluid flow rates or the number of migration time steps for good convergence, are not known prior to the special conditions of each simulation. 1.4 Petroleum Systems Modeling A “Petroleum System” is a geologic system that encompasses the hydrocarbon source rocks and all related oil and gas, and which includes all of the geologic elements and processes that are essential if a hydrocarbon accumulation is to exist (Magoon and Dow, 1994). A petroleum systems model is a digital data model of a petroleum sys- tem in which the interrelated processes and their results can be simulated
- 35. 1.4 Petroleum Systems Modeling 21 Exploration Risk Assessments - Northern Campos Basin (Brasil) Hydrocarbon Resource Assessment (Iraq) 140 km 600 km Maps: 20..50 Grids: 500..1000 x 500..1000 Cells: 1 .. 10 Million Cell Size: 2 km ..50 km Timesteps: 200..2000 Processors: 4..10 Analysis: Source Rock Computing Time: 10..30 hours Maps: 20..50 Grids: 100..500 x 100..500 Cells: 0.1 .. 4 Million Cell Size: 100..2000 m Timesteps: 2000..20000 Processors: 1..10 (..20 for Risk Runs) Analysis: Petroleum System Computing Time: 1..12 hours 11 km North East Fig. 1.11. Studies on prospect and regional scales. The figure from Iraq is courtesy of the U.S. Geological Survey and described in Pitman et al. (2003) in order to understand and predict them. It is a preferably 3D representa- tion of geological data in an area of interest, which can range from a single charge or drainage area to an entire basin. A petroleum systems model is dynamic which means that petroleum systems modeling provides a complete and unique record of the generation, migration, accumulation and loss of oil and gas in a petroleum system through geologic time. Petroleum systems modeling includes basic assessments such as:
- 36. 22 1 Introduction to Basin Modeling Fig. 1.12. Simplified petroleum system chart of Alaska North Slope after Magoon et al. (2003) Have hydrocarbons been generated? This includes a full range of services from initial charge risking in frontier areas to regional resource assessments of yet–to–find hydrocarbons. Where were hydrocarbons generated? If hydrocarbons were generated, their locations can be defined quite accurately so that their possible rela- tionships to prospects can be risked. When were hydrocarbons generated? There are many clear examples of where basins, plays, and prospects have failed due to timing problems. For example, when oil and gas was generated early and the structures were created much later. Could hydrocarbons have migrated to my prospect? Modeling of the dy- namic process of generation, expulsion, and migration makes it possible to determine if the oil and gas charge could reach the trap. What are the properties of the hydrocarbons? Modeling of the phase be- havior of the hydrocarbons during migration, accumulation and loss makes it possible to determine oil vs. gas probabilities and even predict properties such as API gravities and GORs.
- 37. 1.4 Petroleum Systems Modeling 23 Petroleum systems modeling can be interpreted as a sub-group of basin models, which model the full hydrocarbon lifecycle. It covers the most sophis- ticated targets of basin modeling. Each source rock develops its own petroleum system. The petroleum sys- tem elements are facies, which contained, transported or sealed the generated petroleum from one source rock. These facies were named according to their function as source rock, carrier rock or seal. All the distributed petroleum of one petroleum system is more or less connected with rest saturation drops, migration stringers and accumulation bodies (Fig. 1.10) and is usually mixed with other petroleum systems from the same basin. The petroleum system chart shows the timing of the petroleum systems elements and allows a first assessment of the process chain (Fig. 1.12). Timing and Migration Risk - relates the changes to the trap ... migration - takes dependencies and processes into account - takes dynamics into account Seal Carrier/ Reservoir Carrier Source Trap Risk for example: - Prospect geometry - Reservoir quality - Seal quality Charge Risk for example: - Source rock quality - Source rock maturity - Generated petroleum Fig. 1.13. Risk factors of petroleum systems modeling A primary target of petroleum systems modeling are hydrocarbon explo- ration risk factors (Figs. 1.13, 1.14). They are the hydrocarbon charge, the reservoir quality, the trap capabilities and the timing relationship between the charge, reservoir, and seal (Fig. 1.13). Exploration risk commissions often evaluate the risk related to charge, reservoir, and seal, separately and subdi- vided into several factors (Fig. 1.14). Obviously, most of these risk factors can be assessed from a well designed basin model with special emphasis to the charge factors. Probability analysis methods (Chap. 7) allow the total risk to be quantified as a result of special uncertainties of the single risk factors and also take into account the timing relationships. Thus, basin modeling com- bined with probability analysis can be used as a decision support system for exploration risk assessment.
- 38. 24 1 Introduction to Basin Modeling Gas Hydrates Processes Cementation Bio- degradation Deposited Lithology Reservoir Quality - Seal CapPress Seismic Velocities Seismic Fault SGR Structure Properties Structure Geometry Interpretation Reservoir Trap Carrier Permeab. Maturity Kinetics incl. CBM Source Quality TOC HI Thermal Properties Heat Transfer Generation Biogenic Gas Migration Carrier Extent Key Risk Factors Carrier CapPress Charge Fig. 1.14. Petroleum Systems Modeling as a Decision Support System 1.5 Modeling Workflows The employment of some geological processes is optional and sometimes mod- eling of pressure or migration is not needed. It also makes sense to completely decouple pressure, temperature, and hydrocarbon fluid flow modeling from each other especially for pressure and temperature calibrations or if several migration scenarios should be tested. The following schemes for source rock analysis, reservoir volumetrics, and migration modeling demonstrate some common workflows (Fig. 1.15). Source Rock Maturation Study This type of study is performed when knowledge about the basin is sparse or when project deadlines are near. Large uncertainties in the data may not allow a sophisticated modeling. Only basic facts are investigated and emphasis is put on small simulation times. In the initial step a model is calibrated for pressure and then again for temperature.2 Both calibrations are performed fully decoupled. Feedback of temperature effects on compaction are not taken into account. This enhances the performance of the procedure drastically. Possible errors are neglected. After the calibration, generated hydrocarbon masses give a first idea about source rock maturity, peak expulsion times and maximum reservoir fillings. 2 It is important to perform the pressure calibration before the heat flow analysis since pressure formation influences the paleo–geometry which can have a signifi- cant effect on temperature history.
- 39. 1.5 Modeling Workflows 25 Pressure Compaction Temperature Vitrinite Reflectance Pressure Compaction Temperature Generation Expulsion PVT, Migration Volumetrics (FP, IP, D, Hyb) Vitrinite Reflectance Pressure Compaction Temperature Generation Expulsion PVT, Migration Volumetrics (FP, IP, D, Hyb) Vitrinite Reflectance a) b) c) Single Simulation Run Multi-Simulation Runs for Risking of Scenarios Multi-Simulation Runs for Calibartion Generation Expulsion PVT Volumetrics (FP or IP) Fig. 1.15. Modeling Workflows for (a) Source Rock Maturation Study. (b) De- coupled Migration Study. (c) Petroleum Migration Study. (FP..Flowpath Modeling, IP..Invasion Percolation, D..Darcy Flow Modeling, Hyb..Hybrid Flow Modeling) Drainage areas of interest in the reservoir are mapped to the source rocks with some simple procedures and the corresponding expulsion amounts are collected for the volumetrics. Flowpath modeling and invasion percolation techniques can be used additionally in a more advanced manner to consider losses, spill and seal break through amounts. Multi–simulation runs are often performed for calibration and inversion. Statistical methods can be used to improve the calibration workflow. His- tograms with generated amounts are often evaluated as functions of uncertain parameters, such as basal heat flow or SWI temperatures. However, this type of modeling is too crude for risking of individual accumulations. Decoupled Migration Study A decoupled migration study is typically performed when multiple migration scenarios are studied. It is often not reasonable to recalculate compaction and temperature for each migration scenario anew because feedback effects between migration and compaction or temperature are usually very small. On the other hand, a lot of simulation time is saved when the pressure and temperature field is not recalculated for each simulation run. Migration and accumulation are performed on the most sophisticated level. They are considered in more detail than in a source rock maturation study. The selection of the migration model depends on the type of the geological migration process, the model input, the available computer soft- and hardware and the output preferences of the user. Very often different migration methods are tested for their performance in a given basin under certain geological conditions. Darcy flow with time control is often applied, especially as a part of the hybrid migration method. For example, a petroleum system very often consists of several sources. The interaction of different reservoir layers can play an important role. Especially the charging of traps can be studied with the
- 40. 26 1 Introduction to Basin Modeling hybrid migration method. A large part of this book deals with explanations and comparisons of the different migration modeling methods. Multi–simulation runs are performed to explore the range of calculated reservoir fillings dependent on unknown input parameters of the petroleum system. Statistical models are often applied to quantify the risk assessment procedure (Chap. 7). Coupled Migration Study The decoupled mode ignores the influences of the petroleum system on tem- perature and pressure, such as gas generation pressure or oil and gas influences on thermal conductivities. Coupled scenarios ensure modeling of the full inter- action. Of course, high resolution 3D models need a lot of computer power for such fully integrated runs, especially when multi–simulation runs are needed for calibration and risk assessment. The calibration of the petroleum system is also part of the procedure when information about known accumulations is available. It cannot be done automatically since there are too many uncertain input parameters which affect the resulting accumulation pattern. Workflows for modeling geological processes are numerous and most peo- ple have their own preferred data and workflows to achieve the desired results. There is no doubt that many of the controlling geological factors involved in these processes are not very well known and difficult to quantify, and that this limits the numerical accuracy of the models. For example, it is still un- clear how short–term thermal events (“heat spikes”) influence the kinetics of petroleum formation, or how significant errors in the heat flow history that result from insufficient knowledge of the intensity and time of erosional phases can be avoided. Additional restrictions are our limited knowledge of factors affecting carbonate diagenesis (early or late diagenetic cementation?), and subsequent inaccurate estimates of thermal conductivities at the respective diagenetic stages. This list can surely be extended. In many cases, it can be assumed that uncertainties resulting from missing knowledge about uncertain processes are often larger than small errors due to a missing feedback effect. More conceptual models with less coupled processes can be understood, cal- ibrated, and studied more easily. For example, due to higher simulation per- formance more uncertain parameters can be varied to assess their influence on the modeling results. 1.6 Structural Restoration Structural restoration deals with the determination of the shape of geological structures at paleo times. Overthrusting and faulting are the main topics. It is often performed with a backstripping approach which is mainly based on the mass and volume balances of rock material. Structural restoration is tightly linked to basin modeling as the shapes of layers and faults are often used as inputs in basin modeling. Optimization
- 41. 1.7 Comparison with Reservoir Modeling 27 procedures for geometry calculations can then be omitted. However, multiple simulation runs cannot be avoided if porosity is to be calibrated. Fully restored geometries of basins at certain events are needed and extensive restorations have to be performed (Chap. 2). Structural modeling, geomechanics, and tectonics incorporate the model- ing of stresses and strains. They are needed when fault properties, fracturing, and lateral effects on compaction are of interest. 1.7 Comparison with Reservoir Modeling A role, similar to that of basin modeling in exploration, has traditionally been played in production by reservoir modeling (Aziz and Settari, 1979). There are many fundamental similarities between reservoir modeling and basin mod- eling, as both technologies are used to model transport processes for hydro- carbon fluid flow in geologic models in order to provide an improved under- standing, so that better predictions of possible results can be made. The scaling of basin and petroleum systems models is however completely different than that of reservoir models, as dynamic geologic processes are considered in basin modeling. Sedimentary basins evolve through geologic time with significant changes in their geometries due to burial subsidence and compaction, uplift, and erosion, and structural complexities. Additionally, the size of sedimentary basins is also orders of magnitude larger than typical field sizes. For example, mega-regional models cover areas the size of the Gulf of Mexico and include the entire sedimentary sequence up to depths of 10 km and more. As a result, pressure and temperature conditions in sedimentary basins vary over a much wider range. Besides this there are some other fundamental differences which are less important from a technical viewpoint. For example, reservoir modeling deals with forecasts of future production.3 The Influence of humans on the results, e.g. due to the injection of steam, play a central role. In contrast basin model- ing is performed for geological times only. Human influences on the basin are obviously of no interest. Likewise an optimization routine, which is not found in reservoir modeling, is necessary for calibration of the present day geome- try. Despite all these differences, basin modeling has benefited greatly from reservoir modeling. For example, fluid analysis was first applied in reservoir modeling and has now evolved to become a sophisticated addition to basin modeling. 3 History matching is similar to calibration in basin modeling. It is performed to improve the quality of future predictions.
- 42. 28 1 Introduction to Basin Modeling 1.8 Outlook Future trends in basin modeling will involve the refinement of the implemen- tation of all the above listed geological processes. As already mentioned there are, for example, developments for the integration of stresses and strains into simulators. This example is an enhancement of compaction and pore pressure prediction. Besides this there are other developments which try to incorporate seismic information more directly into basin models. For example, invasion percola- tion models have a higher resolution than other processes in basin models. The resolution approaches almost the resolution of seismic data. A direct incorporation of seismic data is therefore desired. Seismic data can also be used in general for facies and lithology assign- ment. However, appropriate attribute analyses and upscaling laws must be developed.
- 43. 1.8 Outlook 29 Summary: Basin modeling is dynamic forward modeling of geological pro- cesses in sedimentary basins over geological time spans. It incorporates de- position, pore pressure calculation and compaction, heat flow analysis and temperature determination, the kinetics of calibration parameters such as vitrinite reflectance or biomarkers, modeling of hydrocarbon generation, ad- sorption and expulsion processes, fluid analysis, and finally migration. Transport processes for water (pore pressure and compaction), heat (tem- perature calculation), and petroleum (migration and diffusion) can be for- mulated in terms of flow equations with appropriate conservation equations for mass or energy which finally yield diffusion type differential equations. A sedimentary basin is a sequence of geological layers. Each layer was deposited in a given stratigraphical event and is subdivided into regions of similar facies. A facies type specifies the lithological rock type and the organic facies. The lithology includes quantities such as permeability, compaction parameters, heat capacities, thermal conductivities and so on. The organic facies contain the total organic carbon content (TOC), the hydrogen index (HI) and the specification of the kinetic for petroleum generation. Boundary conditions must also be defined. Basal heat flow can be determined from crustal models for basin evolution. Migration is the most sophisticated process in modeling. Due to its un- certain nature and extensive computing requirements different modeling ap- proaches exist. Hybrid simulators combine the advantages of all approaches. Additionally, a basin model contains special submodels concerning faults and fault properties, cementation, thermal calibration parameters, salt move- ment, intrusions, fluid phase properties, secondary cracking, and so on. Basin models typically cover areas about 10 x 10 km up to 1000 x 1000 km and to a depth of 10 km. They are gridded into volume elements with up to 500 gridpoints in the lateral directions and up to 50 layers. Each volume element contains a constant facies in a bulk continuum approximation. Ap- propriate upscaling of physical properties from core to grid size might be necessary. In practice, different workflows for risk evaluation and calibration exist. Dependent on the quality of the data, the geological processes are modeled decoupled, partially coupled or fully coupled. Source rock maturation studies are typically decoupled and petroleum migration studies are fully coupled. In between, decoupled migration studies are performed for risking, when, for example, the migration pathways are not known and different migration scenarios are tested. Structural restoration yields valuable information about overthrusted layers and faulted geometries. It is an important step for modeling many of the world’s basins. Basin modeling has been performed since about 1980 and became fully three dimensional in respect to all important processes around 1998 when sophisticated 3D-simulators with migration were published.
- 44. 30 1 Introduction to Basin Modeling References P. A. Allen and J. R. Allen. Basin Analysis. Blackwell Publishing, second edition, 2005. K. Aziz and A. Settari. Petroleum Reservoir Simulation. Elsevier, 1979. H. Bahlburg and C Breitkreuz. Grundlagen der Geology. Elsevier GmbH, Muenchen, second edition, 2004. J. Gluyas and R. Swarbrick. Petroleum Geoscience. Blackwell Publishing, 2004. C. Hermanrud. Basin modeling techniques – an overview. Basin Mod- elling: Advances and Applications, pages 1–34. Norwegian Petroleum So- ciety (NPF), Special Publication No. 3, Elsevier, 1993. J. M. Hunt. Petroleum Geochemistry and Geology. W. H. Freeman and Com- pany, New York, second edition, 1996. L. B. Magoon and W. G. Dow. The petroleum system from source to trap. AAPG Memoir, 60, 1994. L. B. Magoon, P. G. Lillis, K. J. Bird, C. Lampe, and K. E. Peters. Alaska North Slope Petroleum Systems: U.S. Geologi- cal Survey open file report. Technical report, 2003. URL geopubs.wr.usgs.gov/open-file/of03-324. K. E. Peters, C. C. Walters, and J. M. Moldowan. The Biomarker Guide, volume 1 and 2. Cambridge University Press, second edition, 2005. J. K. Pitman, D. W. Steinshouver, and M. D. Lewan. A 2 1/2 and 3D mod- eling study of Jurrasic source rock: U.S. Geological Survey open file report. Technical report, 2003. URL pubs.usgs.gov/of/2003/ofr-03-192. B. P. Tissot and D. H. Welte. Petroleum Formation and Occurrence. Springer– Verlag, Berlin, second edition, 1984. P. Ungerer, J. Burrus, B. Doligez, P. Y. Chenet, and F. Bessis. Basin evalu- ation by integrated two–dimensional modeling of heat transfer, fluid flow, hydrocarbon gerneration and migration. AAPG Bulletin, 74:309–335, 1990. M. A. Yükler, C. Cornford, and D. Welte. Simulation of geologic, hydrody- namic, and thermodynamic development of a sediment basin – a quanti- tative approach. In U. von Rad, W. B. F. Ryan, and al., editors, Initial Reports of the Deep Sea Drilling Project, pages 761–771, 1979.
- 45. 2 Pore Pressure, Compaction and Tectonics 2.1 Introduction Most physical transport and related processes depend on both, temperature and pressure. Pressure is one of the fundamental physical values. It is a scalar, which is represented with a single value in each location. The term pressure has only a real meaning for fluids and not solids. In porous media, pressure is often introduced as the pressure within the fluids in the pores, the pore pressure. The equivalent physical entity in solids is the stress tensor, which is a symmetrical 3x3 tensor with six independent values (Sec. 8.2). It can be illustrated with an ellipsoid, whose axes represent the principal stresses in size and direction. Usually, only single components or invariants of the stress tensor are important. Both, rock stress and pore pressure describe the response of the material to an external load. The “average” stress of the porous volume element is called bulk stress. It is therefore a superposition or mixture of pore pressure and rock stress and it has to be in equilibrium with all external loads. The primary pressure and stress causing process is sedimentation with subsidence, which produces overburden load on the subsurface rocks. Stresses and pore pressures generally increase with depth. Rock stresses and fluid pres- sures interact with compaction and porosity reduction. The main mechanisms for compaction are rearrangement of the grains to denser packages and cemen- tation, which are called mechanical and chemical compaction, respectively. In summary, three main ingredients needed to formulate a model for the me- chanics of the porous sediments, are the concepts of bulk stress, pore pressure and compaction. Additional effects, like mineral transformation, aquathermal pressuring, and kerogen cracking or fracturing, should also taken into account. 2.1.1 Bulk Stresses A homogeneous body under a constant load from above deforms horizontally and vertically as shown in Fig. 2.1.a. The vertical stress in each location is T. Hantschel, A.I. Kauerauf, Fundamentals of Basin and Petroleum 31 Systems Modeling, DOI 10.1007/978-3-540-72318-9 2, © Springer-Verlag Berlin Heidelberg 2009
- 46. 32 2 Pore Pressure, Compaction and Tectonics then equal to the top load and the horizontal stress is equal to zero. This stress state is called uniaxial. If the side boundaries of the bodies are fixed (Fig. 2.1.b), the horizontal stress components are compressive as well and equal to a fixed ratio of the top load, namely σh/σv = ν/(1 − ν). The Poisson ratio ν is a material constant and sediments have numbers of 0.1 . . . 0.4, which yield stress ratios of 0.11 . . . 0.67. Exceptions are salt and unconsolidated sands with Poisson ratios close to 0.5. sh a) b) c) d) e) sh sh sh sh sh sh sv sh sv sv sv sv sv sv sv sv sv Fig. 2.1. Vertical and horizontal stresses in a homogeneous solid with overburden (a) load on top with free moving sides; (b) load on top with fixed side boundaries (c) gravity loads with fixed side boundaries; (d) together with additional constant compressions on the side boundaries; (e) together with additional constant tensions on the side boundaries The situation in non–tectonically influenced basins is similar to the fixed solid case (Fig. 2.1.c) with vertical loads increasing approximately linearly with depth. Heterogeneities of geomechanical properties cause different stress ratios and rotation of the main stress axes. Fault planes and salt domes disturb homogeneous trends in stress. Tectonic processes generally add a compressive or tensile stress to the horizontal component (Fig. 2.1.d,e). Extensions of the model to lower horizontal stresses can revert the compressive (positive) stresses into tensile (negative) stresses, while compressive boundaries can in- crease the horizontal stresses so that they exceed the vertical stresses and become the maximum principal stress. The stress state in solid grains is mainly controlled by the overburden load of the considered volume element in the case of negligible tectonic forces and homogeneously layered rocks (Fig. 2.1.c). Then, the ”lithostatic pressure” approach can be used, which describes the three dimensional stress field by one single value, the lithostatic pressure, assuming the following simplifications:
- 47. 2.1 Introduction 33 • The three main stress axes are straight vertically and horizontally directed. This assumption is not valid in heterogeneous layers and salt domes, where the coordinate system of the principal stress components rotates. • The boundaries of the basins are fixed in terms of displacements, the hor- izontal stresses are equal in both directions and the stress ratio (σh/σv) is constant. The elastic properties are isotropic and layerwise homogeneous. It also means that the model has no tectonic stresses due to compressional or extensional forces or displacements. • The vertical stress component is equal to the overburden load. This means that all stresses are conducted straight vertically and will not influence each other. The vertical component is then equal to the overburden weight, the litho- static pressure. It represents the ’pressure’ state of the porous bulk element. 2.1.2 Pore Pressure Formation and Fluid Flow The measurable pressure value in the pore fluid is the pore pressure. It is mainly caused by the overburden weight, but fluid flow together with com- paction can decrease the overburden induced pressure and the resulting pore pressure is usually smaller than the lithostatic pressure. In a non–compactable porous rock, the lithostatic pressure and the pore pressure are both equal to the overburden load. Fluid outflow allows grain rotation to more compact packages, which decreases pore pressure and porosity. Thus, the difference between lithostatic and pore pressure is a measure of compaction. Ideal compaction does not reduce pore pressure to zero. Instead a hydro- static pressure remains, which is equal to the weight of the overlaying water column. Generally, the hydrostatic pressure is defined as the part of the pore pressure which does not contribute to water flow. The hydrostatic zero level can be arbitrarily defined, since only gradients and not absolute values of pressures control pore water flow. The groundwater table is not suitable as a constant reference level, since it varies over basin scale. Instead, the seawater level is used as the hydrostatic zero level. The hydrostatic pressure is then equivalent to the water column weight measured from the seawater level and therefore depends on sea and pore water density. Note that the hydrostatic pressure is not a measurable pressure. It is a theoretical pressure for ideal compactable layers or slow sedimentation. The difference between the pore pressure and the hydrostatic pressure is the overpressure which directly controls water flow (Fig. 2.2). The pore pressure lies usually between hydrostatic and lithostatic pressure, but there are exceptions. It can be lower than the hydrostatic pressure when high uplift and erosion rates act on deep sand layers which are connected to near surface pressure areas along permeable facies. It can also exceed lithostatic pressures when large overpressures are built up by gas generation or highly permeable facies are connected at large depth levels.
- 48. 34 2 Pore Pressure, Compaction and Tectonics Water Mountain Groundwater-Level Sediment-Water-Interface Sea Level Pressure Pressure Depth Depth 0 0 Hydrostatic Pressure Pore Pressure Lithostatic Pressure Hydrostatic Pressure Pore Pressure Lithostatic Pressure Over- pressure Effective Stress Effective Stress Over- pressure Excess Hydraulic Groundwater Potential Offshore Onshore Sea Level Fig. 2.2. Definitions of pressures and stresses. The groundwater level is often as- sumed to match the surface in basin modeling. Then, pore and lithostatic pressure have the same zero level, as shown here One can distinguish between three processes of overpressure build up: over- burden load together with mechanical under–compaction, cementation and overpressuring caused by fluid expansion processes (Osborne and Swarbrick, 1997; Swarbrick et al., 2002). Overburden load induced pore pressure formation due to incomplete sed- iment compaction, as explained above, is the main process for overpressure formation. Here, compaction is the rearrangement of grains to denser pack- ages with a reduction in pore space related to a decrease in pore throats and connectivity of the pore network. This process of grain rotation, crushing and deformation is called mechanical compaction. Compaction is caused by over- burden load. The load acts on the pore fluid and the rock grains according to their compressibilities. Incremental fluid outflow generates a difference be- tween rock stresses and pore pressure, which allows compaction. Compaction in turn changes the ratio between the rock stresses and the fluid pressure, since it decreases the rock and bulk compressibility, enforces further fluid outflow, and decreases the thickness of the solid matrix. The result of this coupled process is always a reduction of overpressure since the outflow from the compacting element is greater then the local increase of overpressure due to the thinning of the solid matrix. This is ensured as the compaction law is formulated in a manner, that relates porosity loss with effective stress in- crease. Finally, mechanical compaction is considered to be an overpressure reducing process (Fig. 2.3.a). The remaining overpressure could be simply in-
- 49. 2.1 Introduction 35 terpreted as a result of incomplete compaction and that is why this process of overpressure formation is called under–compaction. p p 1. Water outflow 2. Decrease of pore pressure 3. Mechanical compaction (grain rotation) vw p 1.Chemical compaction: quartz dissolution , diffusion transport and precipitation 2. Increase of pore pressure 3. Water outflow Q p 1 2 3 Q Q vw 4. Decrease of pore pressure vw vw vw vw Q2 Q1 Q3 a) b) Fig. 2.3. Overpressure and Compaction: (a) mechanical compaction is a result of water outflow, it is always related to decrease in overpressure. (b) Quartz cementa- tion and related compaction transfers lithostatic to pore pressure. It increases pore pressure. Water outflow can partially decrease the overpressure afterwards Another source for overpressure is chemical compaction due to cementa- tion. Cementation occurs in all sandstones and carbonates. It significantly decreases the porosity, and is mainly responsible for porosity reduction at large depths, where mechanical compaction is almost negligible. Cementation is the result of dissolution of quartz from the horizontal contact areas, diffusive transport within the pore water, and precipitation of a silican cement on free quartz surfaces. Quartz dissolution is mainly stress controlled. Temperature affects the diffusion constant and precipitation rate. Chemical compaction in- creases overpressure, since rock stress is transfered from the rock matrix to pore pressure. Cementation also drives fluid outflow and compaction with the generated overpressure as the main driving force (Fig. 2.3.b). The third group of overpressure generating processes encompasses fluid expansion mechanisms: oil and gas generation, oil to gas cracking, aquather- mal expansion and mineral changes such as smectite to illite conversion. In all these processes, mass or the density of the fluids changes and yields fluid pressure increase controlled by fluid compressibility. The overpressure increase due to fluid expansion mechanisms is usually small compared to those related to mechanical and chemical compaction. 2.1.3 Compaction and Porosity Reduction Compaction is the reduction of the sediment bulk volume and is equivalent to volumetric strain v = V/V0, the ratio of a load bearing volume V to the unloaded initial volume V0. The average of the volumetric charge of a specimen is called the mean stress σ̄. Stresses and strains are further explained in the Sec. 2.6. A compaction law relates volumetric strain to mean stress changes with an elastic parameter.
- 50. 36 2 Pore Pressure, Compaction and Tectonics Rock Compaction Pore Space Compaction Elastic Elasticity of the Grains Elasticity of the Skeleton Elasticity of the Pore Fluid Plastic Plasticity of the Grains Rearrangement of the Grains Pressure Dissolution Table 2.1. Compaction related Mechanisms C = − 1 V ∂V ∂σ̄ = ∂v ∂σ̄ . (2.1) Compaction mainly decreases porosity, but also reduces the grain volume. Generally, the rock and pore volumes are reduced with reversible (elastic) and irreversible (plastic) contributions. Some of the special mechanisms acting on microscopic and mesoscopic scales are listed in Table 2.1 after Schneider et al. (1996). The compressibility in equation (2.1) is mainly a property of the grain framework and is called bulk compressibility. In the absence of a pore fluid, it relates the bulk volume decrease with the mean total stress. The presence of a pore water retards compaction, which as a first approximation can be described as the introduction of a mean effective stress σ̄ = σ̄ − p instead of the mean total stress in (2.1) with a reduction of the pore pressure p. Terzaghi (1923) confirmed this thesis experimentally, by proving that increasing the mean total stress or decreasing the pore pressure yields the same amount of compaction. Generally, Terzaghi’s effective stress can also be introduced as a stress tensor σ . σ = σ − p I (2.2) where I is the unit tensor (Chap. 8). In practice, compaction laws on the basis of Terzaghi’s effective stress definition are written in terms of porosity loss versus the vertical component of the effective stress σ z. The usage of porosity change instead of the volumetric bulk strain neglects volume changes of the solid matrix which are small. The restriction to the vertical effective stress means that a fixed ratio between horizontal and vertical stresses is assumed. The corresponding vertical total stress can then be simply approximated by the overburden sediment load pressure pl. ∂φ ∂t = −CT ∂σ z ∂t = −CT ∂(pl − p) ∂t . (2.3) For most rock types, the Terzaghi compressibility CT decreases rapidly during compaction. This type of compaction law is widely used in basin modeling. However, the formulation with only the vertical components of the stress ten- sor fails, when active extensional or compressional tectonics occur. Therefore, an extension of the law is proposed in Sec. 2.8. Biot (1941) worked out a more detailed poro–elastic model for the exten- sion of equation (2.1) for water filled porous rocks, taking into account the
- 51. 2.2 Terzaghi Type Models 37 effect of the rock compressibility Cr, which yields the following compaction law with the Biot compressibility CB. ∂v ∂t = CB ∂σ̄ ∂t with σ = σ − αpI and α = 1 − Cr CB . (2.4) This formulation means, that the retardation of the compaction due to pore pressure drops with lower bulk compressibilities, since the rock compress- ibilities remain almost constant during compaction. Exceptions are mineral transformations or plastic flow of the grains. In unconfined sediments, Cr C (soil mechanical approach) and α ≈ 1, while at large depth Cr ≈ CB φ and α ≈ 1 − φ (rock mechanical approach). The case α = 1 also means, that the effective stress is equal to the Terzaghi’s assumption of negligible rock grain deformations. Note, that these effective stresses are only formal entities and not measurable physical values. 2.2 Terzaghi Type Models Terzaghi type models are based on the simplifications of the lithostatic stress concept. In these models, overpressure formation related to incomplete me- chanical compaction is considered and a fixed relation between porosity re- duction and sediment compaction is assumed. The models have been widely used in 1D–Basin modeling programs since the early 90’s. The assumptions are as follows: • The ”lithostatic pressure” concept is considered taking into account only the vertical component of the stress tensor as the maximum principal stress. The lithostatic pressure is equal to the overburden weight. The horizontal stresses are fixed ratios of the lithostatic pressure. Additional tectonic stresses, due to compressional or extensional forces, are neglected. • Pore pressure formation is caused by overburden load. Fluid flow and compaction determine how the pressure is formed and distributed in the basin. Compaction is related to pore fluid outflow and decreases overpres- sure. One phase fluid flow in a fully saturated rock is considered, which is controlled by permeabilities. Pressure communication within the porous network is assumed. • Mechanical compaction of the pore space takes into account the rearrange- ment of the grains to more compact blocks. All compaction is related to porosity reduction caused by pore fluid outflow. This porosity reduction process is controlled by the Terzaghi’s effective stress value which is equal to the difference of the lithostatic and the pore pressure: σ = σz − p. A relationship between maximum effective stress and porosity is assumed. • Water is treated as incompressible.
- 52. 38 2 Pore Pressure, Compaction and Tectonics 2.2.1 Basic Formulation Hydrostatic and Lithostatic Pressure The hydrostatic pressure ph at depth h is equal to the weight of a pure water column from sea level with the water density ρw. ph(h) = h 0 gρwdz (2.5) with z = 0 at sea level. This yields positive values below and negative values above sea level. The negative hydrostatic pressure at groundwater level is the groundwater potential. In basin modeling, the groundwater level is often assumed to be identical to the sediment surface. The water density varies with changing salinity values, while the depen- dency on temperature and pressure is relatively small and often neglectable. A further simplification is the assumption of two constant densities for seawater ρsea = 1100 kg/m3 and pore water ρw = 1040 kg/m3 . This yields piecewise linear curves for hydrostatic pressure versus depth in sediments below sea water (Fig. 2.4). a) b) -20 0 20 40 60 80 100 120 -1 0 1 2 3 4 Pressure in MPa Depth in km 0 20 40 60 80 100 120 0 1 2 3 4 5 Pressure in MPa Depth in km Hydrostatic Lithostatic Shale Lithostatic Sandstone hw Hydrostatic Lithostatic Shale Lithostatic Sandstone hs Fig. 2.4. Hydrostatic and lithostatic pressure curves for normal compacted rocks with the following properties: sea water density ρsea = 1100 kg/m3 , pore water density ρw = 1040 kg/m3 , shale density ρs = 2700 kg/m3 , sandstone density ρs = 2720 kg/m3 . (a) Offshore with a water depth hw = 1 km. (b) Onshore with a height of hs = 1 km. The lithostatic curves cross each other, since shale starts with a higher initial porosity but compacts faster The lithostatic pressure pl is equivalent to the total load of the overlaying sediments of bulk density ρb and sea water. Lithostatic zero level is the surface onshore and the seawater level offshore.
- 53. 2.2 Terzaghi Type Models 39 pl(h) = g h hs ρb dz onshore pl(h) = g hw 0 ρsea dz + g h hw ρb dz offshore (2.6) where hs is the sediment surface. The integral over the weight of overburden sediments can be replaced by a sum of the weights of the single layers with thicknesses di (i is the layer number), rock densities ρri, and and porosities φi. pl(z) = ρseaghw + g n i=1 di [ρwφi + ρri (1 − φi)] . (2.7) For a homogeneous sediment column with a constant rock density ρr equa- tion (2.7) can further be simplified as follows: pl(h) = gρseahw + gρr(h − hw) − g(ρr − ρw) h hw φdz , onshore pl(h) = gρr(h − hs) − g(ρr − ρw) h hs φdz , offshore. (2.8) The remaining integral in the above equation is the weight percentage of water in the overlaying sediment column. At larger depths, the term (1 − φ) does not significantly change, which means the curve tends toward a straight line for a unique sediment type. Litho- static pressure curves for shale and sandstone, for hydrostatic compaction with compaction parameters of Fig. 2.8, are shown in Fig. 2.4. The term lithostatic potential ul is used for the lithostatic pressure minus hydrostatic pressure ul = pl − ph. Pore Pressure Equation The pore pressure equation is a one phase fluid flow equation based on the mass balance of pore water. A flow equation relates driving forces with flow rates. The driving force for pore water flow is the overpressure gradient. Darcys law establishes a linear relationship between the discharge velocity v of the pore fluid and the overpressure gradient ∇u assuming relatively slow flow for a Newtonian fluid. The proportionality factor is the mobility μ = k/ν, which is a function of the rock type dependent permeability k and the fluid dependent viscosity ν. v = − k ν ∇u . (2.9) This flow equation is an analogy to Fourier’s equation of heat flow, which similarly relates temperature gradient and heat flux with the thermal con- ductivity tensor. The permeability tensor is often simplified using only two
- 54. 40 2 Pore Pressure, Compaction and Tectonics values parallel and perpendicular to the facies layering, named as vertical and horizontal permeabilities. Mass balance requires, that any fluid discharge from a volume element is compensated by change in the contained fluid mass. The internal fluid mass changes when the fluid density or the fluid volume is modified (App. B). ∇ · v = − 1 1 − φ ∂φ ∂t + 1 ρ ∂ρ ∂t . (2.10) Local changes of the fluid densities occur for fluid expansion processes like aquathermal pressuring, mineral transformations or petroleum generation and cracking. Changes in the fluid volume or porosity are related to mechanical and chemical compaction, which are considered as two independent processes. The porosity reduction due to mechanical compaction is formulated with Terzaghi’s compaction law, while chemical compaction induced porosity loss is a temperature and effective stress dependent function fc(T, σ ), as specified later in Sec. 2.3. ∂φ ∂t = −C ∂σ z ∂t − fc(T, σ z) . (2.11) The basic model deals with mechanical compaction only and supposes Terza- ghi’s effective stress definitions, which yield the following pressure equation. − ∇ · k ν · ∇u = − 1 1 − φ ∂φ ∂t = C 1 − φ ∂σ z ∂t = C 1 − φ ∂(ul − u) ∂t . (2.12) Thus, C 1 − φ ∂u ∂t − ∇ · k ν · ∇u = C 1 − φ ∂ul ∂t . (2.13) The equation shows that the overburden load causes overpressure increase and compaction. In the absence of all overpressure generating sources, fluid flow is still admissible, but then the total inflow is equal to the total outflow of each element. Pore water loss is always related to the corresponding overpres- sure discharge and the grain structure reacts instantaneously with mechanical compaction. The two lithological parameters, compressibility and permeability con- trol fluid flow and pressure formation. The bulk compressibility describes the ability of the rock framework to compact and it also controls how overburden influences pore pressure. The bulk compressibility in the pressure equation should not be mixed up with pure grain or fluid compressibility, which is orders of magnitude smaller. The higher the compressibility of the element the higher the pore pressure decrease and the smaller the overpressure for- mation. The permeability controls flow rates, flow paths, and the resulting pore pressure fields. The overpressure in an element cannot decrease if the elements surroundings are impermeable even when the element itself is highly permeable and compressible.
- 55. 2.2 Terzaghi Type Models 41 The permeability can vary by several orders of magnitude, ranging from highly permeable facies (sandstone) to low permeability facies (shale) to al- most impermeable facies (salt). The two end members of almost impermeable and highly permeable facies are handled with special methods, which will be further discussed later in the 2D– and 3D–pressure examples. Boundary values of equation (2.13) are overpressures and water flow veloc- ities as illustrated in Fig. 2.5. The upper boundary condition is zero overpres- sure at the sediment–water–interface offshore and an overpressure equal to the groundwater potential at the sediment surface onshore. The groundwater potential yields topographic driven flow, which is explained in Sec. 2.2.5. Sediment 1 Sediment 2 Sediment 3 No Flow Condition at Base ( ) n×Ñu = 0 Water Salt: u = ul Side Boundary: n×Ñu = 0 Upper Boundary: Surface Groundwater Potential or Sediment Water Interface u = u u = 0 g k , C 2 2 k , C 3 3 k , C 1 1 Salt Boundary: = 0 n×Ñu Fig. 2.5. Boundary value problem for overpressure calculation The lower and side boundaries are no–flow areas, which means the over- pressure gradient along the surface normal n is set to zero n·∇u = 0. They are called closed boundaries. In small (prospect) scale models, special overpres- sures are usually set as side boundary values for some layers. For example, zero overpressure should be set at a permeable layer boundary, if it has a highly permeable connection to a hydrostatic area. To fix an overpressure value as a boundary condition at a certain point, is like injecting or releasing water until the given pressure is achieved. One can also apply a complete pressure array as side boundary values on prospect scale models from precalculated and cal- ibrated basin scale models (Sec. 8.9). Special inner boundary conditions have to be set to impermeable rocks, namely no flow across the boundaries to these areas n · ∇u = 0 and lithostatic pressure within impermeable regions u = ul.
- 56. 42 2 Pore Pressure, Compaction and Tectonics Compaction and Porosity Reduction In the basic model, a simple relationship between mechanical compaction and porosity decrease is considered. Hence, the related porosity change is equivalent to the bulk strain and a function of the Terzaghi’s effective stress. Several relationships between porosity and effective stress have been developed and they are described in the following section. Although the formulations look different, they are similar to exponential relationships of the following type: φ ≈ k1 e−k2 σ z . (2.14) The compaction in each volume element is usually realized with contrac- tion of its vertical edges when only vertical compaction occurs. The relative decrease in any vertical length is equal to a relative decrease in volume. Then, the actual thickness d is calculated using any previous or initial thickness d0 from the present and previous porosities φ, φ0 as follows: z = d d0 = 1 − φ0 1 − φ . (2.15) 2.2.2 Mechanical Compaction Mechanical compaction is almost irreversible. Hence, porosity is maintained when effective stress is decreased due to uplift, erosion, or an overpressure increase. The general porosity-effective stress relationship (2.14) could then still be used, but with the maximum effective stress value instead of the ac- tual effective stress. This is taken into account when the following compaction laws are formulated in terms of effective stresses. Most mechanical compaction functions are porosity–effective stress relationships with decreasing porosity for increasing effective stress. The lithotype dependent functions can be mea- sured through a triaxial compression test. Soil mechanical models use loga- rithmic functions between the void ratio e = φ/(1 − φ) and the effective stress, which yields a similar curve as equation (2.14). e ≈ k1 − k2 log(σ z) . (2.16) The equivalence of the relationships (2.14) and (2.16) is illustrated in Fig. 2.6. The exponential porosity–effective stress has a wide range of linear porosity versus the logarithm effective stress relationship for most lithologies, and it also behaves almost linearly in the high porosity range when trans- formed into the corresponding void ratio diagram. Hence, the pure soil me- chanical formulation should only be applied to an effective stress of 15 MPa or approximately 1 km. Compaction curves generally depend on the stress path, but usually only normal compaction curves, with an uniform increase in overburden, are taken into account. Stress release caused by uplift and erosion shows an elastic
- 57. 2.2 Terzaghi Type Models 43 rebound, usually described with a low incline in the compaction diagram as illustrated in Fig. 2.6. a) b) 1 3 10 30 100 0 10 20 30 40 50 60 70 Effective Stress in MPa Porosity in % 1 3 10 30 100 0 0.5 1 1.5 2 Effective Stress in MPa Void Ratio A A C C B D B D Fig. 2.6. Normal compaction curves (A and D) of a typical shale with an exponen- tial porosity–effective stress relationship. The parameters are given in Fig. 2.8. The paths (B) and (C) represent load removal (erosion and uplift) and reload, respec- tively. (a) The porosity–stress relationship plotted versus the logarithmic stress axis has a wide range of linear behavior. (b) The void ratio–logarithm effective stress plot also behaves linearly for high porosities. The dashed curves represent linear approximations between porositiy and void ratio and the logarithm of the effective stress Relationship between Effective Stress, Equivalent Hydrostatic Depth and Compressibility The following compaction laws relate porosity to either effective stress φ(σ z), frame compressibility φ(C), or equivalent hydrostatic depth φ(ze), which is the depth of the sample with the same porosity and rock type under hydro- static pressure conditions. A formulation in terms of one of these independent variables can always be converted analytically or numerically into either of the others. A compaction law has to encompass all three relations during simulation: • φ(σ z) determines new porosities after the pore pressure equation yields the effective stresses with the calculated new pore pressures. • C(φ) or C(σ z) determines the actual frame compressibilities which are required in the pressure equation. The compressibility is the derivation of the φ(σ z) function after σ z. It defines the slope of the porosity versus the effective stress curve.
- 58. 44 2 Pore Pressure, Compaction and Tectonics • φ(ze) is the theoretical porosity versus depth curve assuming hydrostatic pressures and the deposition of the entire column with the same lithotype. Many log and well data are available in terms of porosity versus equivalent hydrostatic depth rather than for porosity versus effective stress data. They are often used to determine the lithotype dependent parameters in the compaction laws. Effective stress and compressibility based functions can simply be converted to each other by derivation or integration. C(σ z) = − dφ(σ z) dσ z , φ(σ z) = − σ z 0 C(σ)dσ . (2.17) The hydrostatic porosity–depth function can be derived from the com- pressibility and porosity–effective stress equations as follows. For hydrostatic conditions, the effective stress change is equal to the change of the lithostatic minus the hydrostatic pressure dσ z dze = Δρ g (1 − φ) (2.18) where Δρ = ρr − ρw is the difference of the rock and water density. Thus, dφ dze = dφ dσ z dσ z dze = −Δρg(1 − φ)C(φ) . (2.19) The porosity–depth function can be analytically expressed if (1 − φ)C(φ) can be integrated. Analytical porosity–depth functions are very advantageous for the calibration of well data, but some relationships require numerical iter- ation schemes. Athy’s Law formulated with Effective Stress Athy (1930) proposed a simple exponential decrease of porosity with depth for a given rock type described only with an initial porosity φ0 and a compaction parameter k. As already explained above, effective stress rather than total depth should be used in the compaction law. A corresponding simple expo- nential porosity–effective stress function was first proposed by Smith (1971). φ = φ0 e−kσ z . (2.20) The compressibility function C(φ) and the hydrostatic porosity–depth func- tion φ(ze) are according to (2.17) and (2.19) as follows. C(φ) = kφ, (2.21) φ(ze) = φ0 φ0 + (1 − φ0) exp(kΔρ gze) . (2.22)
- 59. 2.2 Terzaghi Type Models 45 The exponential function (2.20) is a straight line in the logarithmic poros- ity versus effective stress diagram with k as the decline angle. Typical com- paction curves for clastic rocks are shown in Fig. 2.7. The previous model can be easily extended with consideration of a non–zero minimum porosity φm. φ = φm + (φ0 − φm) exp(−kσ z), (2.23) C(φ) = k(φ0 − φm) exp(−kσ z) = k(φ − φm), (2.24) φ(ze) = (φ0 − φm) + φm(1 − φ0) exp(k(1 − φm)Δρ gze) (φ0 − φm) + (1 − φ0) exp(k(1 − φm)Δρ gze) . (2.25) This model is frequently used in basin modeling (Giles et al., 1998), although the use of only one compaction parameter does not give a good match with observed data for many rock types. a) b) 0 10 20 30 40 50 60 0 10 20 30 40 50 60 70 Effective Stress in MPa Porosity in % 1 2 3 0 10 20 30 40 50 60 1 10 100 Effecive Stress in MPa Porosity in % 1 2 3 1..Shale 2..Siltstone 3..Sandstone 1..Shale 2..Siltstone 3..Sandstone Fig. 2.7. Porosity versus effective stress curves on (a) logarithmic and (b) linear scale for various lithologies using Athy’s Effective Stress law with the following pa- rameters: shale φ0 = 0.70; k = 0.096 MPa−1 , siltstone φ0 = 0.55; k = 0.049 MPa−1 , sandstone φ0 = 0.41; k = 0.0266 MPa−1 . The minimum porosity is zero Athy’s Law formulated with Hydrostatic Depth A depth related porosity law (Athy, 1930) is used with the introduction of an equivalent hydrostatic depth ze instead of the total depth. φ = φ0 exp(−kze) . (2.26) The advantage of this formulation is, that the compaction parameter k can be easily determined when measured porosity versus equivalent depth data
- 60. 46 2 Pore Pressure, Compaction and Tectonics is available. The compressibility function C(φ) and the hydrostatic porosity– depth function φ(ze) are according to (2.17) and (2.19) as follows. C = k Δρ g φ (1 − φ) , (2.27) σ z(φ) = Δρ g k (φ − φ0 − ln φ φ0 ) . (2.28) The inverse function φ(σ ) can be calculated with the Newton iteration method. The resulting porosity–effective stress curves are generally steeper in the high porosity and shallower in low porosity ranges than the Athy versus effective stress functions. Hence, they are more applicable for most rock types even though they are based on only one compaction parameter. The authors prefer this law as a default for most lithologies. Example compaction curves for clastic rocks and carbonates are illustrated in Fig. 2.8. Schneider Model An extension of Athy’s effective stress law to two exponential terms was pro- posed by Schneider et al. (1996). φ = φ1 + φa exp(−kaσ z) + φb exp(−kbσ z) . (2.29) Different compaction parameters ka, kb for lower and higher porosity ranges are realized with the superposition of two exponential terms. The initial poros- ity is equal to the sum of the three porosity parameters φ1 + φa + φb. Both porosities φa and φb are usually assumed to be half of the initial porosity value φ0. The corresponding compressibility function is as follows: C(σ z) = kaφa exp(−kaσ z) + kbφb exp(−kbσ z) . (2.30) The hydrostatic porosity versus depth function can be obtained, when equation (2.18) is integrated numerically to get σ z(ze), and then φ(ze) is calculated with equation (2.29) afterwards. Numerical integration can also be applied to any other model with a given analytical expression for φ(σ z). The proposed default parameters in App. A yield curves almost identical to those of Athy’s depth model (Fig. 2.8). Compressibility Model Compressibilities are the derivatives of the porosity versus effective stress and are proportional to the slope of the porosity versus effective stress curves. This model assumes an exponential decrease in the compressibilities from the depositional value C0 to a value Cm corresponding to the minimum porosity φm (Fig. 2.9).
- 61. 2.2 Terzaghi Type Models 47 0 10 20 30 40 50 60 0 20 40 60 80 Effective Stress in MPa Porosity in % 3 4 2 1 0 1000 2000 3000 4000 5000 6000 0 20 40 60 80 Depth in m Porosity in % 1 2 3 4 a) b) c) d) 0 1000 2000 3000 4000 5000 6000 0 10 20 30 40 50 60 70 Depth in m Porosity in % 1 2 3 0 10 20 30 40 50 60 0 10 20 30 40 50 60 70 Effective Stress in MPa Porosity in % 1 2 3 1..Coal 3..Limestone 4..Dolomite 1..Coal 2..Chalk 3..Limestone 4..Dolomite 1..Shale 2..Siltstone 3..Sandstone 1..Shale 2..Siltstone 3..Sandstone 2..Chalk Fig. 2.8. Porosity versus hydrostatic depth and effective stress curves for vari- ous lithologies using Athy’s depth law with the following parameters: shale φ0 = 0.70, k = 0.83 km−1 , siltstone φ0 = 0.55, k = 0.34 km−1 , sandstone φ0 = 0.41, k = 0.31 km−1 , coal φ0 = 0.76, k = 0.43 km−1 , chalk φ0 = 0.70, k = 0.90 km−1 , limestone φ0 = 0.51, k = 0.52 km−1 , dolomite φ0 = 0.70, k = 0.39 km−1 log C(φ) = φ0 − φ φ0 − φm log Cm + φ − φm φ0 − φm log C0 . (2.31) This is equivalent to the following expression: C(φ) = α exp(βφ) with ln(α) = φ0lnCm − φmlnC0 φ0 − φm , β = lnC0 − lnCm φ0 − φm . (2.32) Integration of the above exponential function yields the corresponding effective stress correlations.
- 62. 48 2 Pore Pressure, Compaction and Tectonics φ(σ z) = − 1 β ln(αβσ z + exp(−βφ0)) C(σ z) = α αβσ z + exp(−βφ0) . (2.33) Numerical integration of (2.19) can be used to determine the hydrostatic porosity versus depth function φ(ze) from C(σ z). Compressibility models gen- erally decrease too fast for low porosities. Default parameters are shown in Appendix A and the curves for clastic rocks are illustrated in Fig. 2.9. 0 10 20 30 40 50 60 70 1 10 100 1000 Porosity in % Compressibility in GPa -1 1 2 3 a) b) 0 10 20 30 40 50 60 0 10 20 30 40 50 60 70 Effective Stress in MPa Porosity in % 1 2 3 P P 1..Shale 2..Siltstone 3..Sandstone 1..Shale 2..Siltstone 3..Sandstone Fig. 2.9. Compaction curves for various lithologies using the compressibility model with the following parameters: Shale C0 = 403 GPa−1 , Cm = 4.03 GPa−1 , Siltstone C0 = 103 GPa−1 ; Cm = 2.11 GPa−1 , Sandstone C0 = 27.5 GPa−1 , Cm = 1.15 GPa−1 Mudstone Model The following law from soil mechanics is especially applicable for clastic rocks. e = φ 1 − φ = e100 − β log σ z 0.1 MPa . (2.34) The reference void ratio e100 at 0.1 MPa can be considered as an initial void ratio e100 = φ0/(1 − φ0) although this compaction law yields a singularity e → ∞ for the void ratio at zero stress. The two functions φ(σ z) and C(σ z) are as follows. φ(σ z) = σ z − β log(σ z/0.1 MPa) 1 + e100 − β log(σ/0.1 MPa) , (2.35)
- 63. 2.2 Terzaghi Type Models 49 C(σ z) = β σ z(1 + e100 − β log(σ/0.1 MPa)2 . (2.36) The above model does not provide an analytical expression for the porosity- depth function. However, with the compressibility model equation (2.19), it can be integrated numerically. The material dependent constants are the initial void ratio e100 and the compressibility β. They can be related to the (volumetric) clay content r for mudstones with the following relationships proposed by Yang and Aplin (2004) Fig. 2.10. e100(r) = 0.3024 + 1.6867 r + 1.9505 r2 β(r) = 0.0937 + 0.5708 r + 0.8483 r2 . (2.37) a) b) 1 3 10 30 100 0 0.5 1 1.5 2 2.5 3 Effective Stress in MPa Void Ratio 1 6 2 3 4 5 0 10 20 30 40 50 60 0 20 40 60 80 Effective Stress in MPa Porosity in % 1 6 5 4 3 2 1..r= 1 2..r=0.8 3..r=0.6 4..r=0.4 5..r=0.2 6..r=0 1..r= 1 2..r=0.8 3..r=0.6 4..r=0.4 5..r=0.2 6..r=0 Fig. 2.10. Compaction curves for various lithologies using the mudstone model with the clay dependent functions of Yang and Aplin Lauvrak (2007) proposed the EasySoil model with the following correla- tions to sample data for e∗ 100, β∗ and an upscaling to e100, β for natural rocks (Fig. 2.11). e∗ 100(r) = 0.725 − 0.252 r + 2.53 r2 , β∗ (r) = 0.218 − 0.119 r + 1.193 r2 , e100 = e∗ 100 + 0.76 β∗ , β = 1.07 β∗ . (2.38)
- 64. 50 2 Pore Pressure, Compaction and Tectonics a) b) 1 3 10 30 100 0 0.5 1 1.5 2 2.5 3 Effective Stress in MPa Void Ratio 1 6 2 3 4 5 0 10 20 30 40 50 60 0 20 40 60 80 Effective Stress in MPa Porosity in % 1 6 2 3 4 5 1..r= 1 2..r=0.8 3..r=0.6 4..r=0.4 5..r=0.2 6..r=0 1..r= 1 2..r=0.8 3..r=0.6 4..r=0.4 5..r=0.2 6..r=0 Fig. 2.11. Compaction curves for various lithologies using the mudstone model with the clay dependent functions of Lauvraks EasySoil model Compressibilities Bulk compressibilities can be directly derived from the compaction law as pre- viously explained. Example curves for clastic rocks and carbonates are shown in Fig. 2.12 with the parameters of the Athy’s hydrostatic depth model. Other compaction models yield similar curves with the proposed default parameters. a) b) 0 10 20 30 40 50 60 70 0 20 40 60 80 100 120 Porosity in % Compressibility in GPa -1 1 2 3 0 20 40 60 80 0 50 100 150 200 250 Porosity in % Compressibility in GPa -1 1 2 3 4 1..Shale 2..Siltstone 3..Sandstone 1..Coal 2..Chalk 3..Limestone 4..Dolomite Fig. 2.12. Compressibility curves for various lithologies using Athy’s depth law with the parameters of Fig. 2.8
- 65. 2.2 Terzaghi Type Models 51 Comparison of Various Lithologies Although the formulation of the various compaction models look very different from each other, default parameters for most lithologies yield very similar compaction curves. An exception is the mudstone model, which is generally not suitable for approximation of the compaction trend in the lower porosity range. The standard shale curves are shown in Fig. 2.13 for all described models. Compaction parameters are mixed arithmetically. Example curves for shale–sandstone mixtures are shown in Fig. 2.14. Fig. 2.13. Comparison of different compaction laws for shale: The curves for the Schneider and Athy’s depth model are almost identical and plotted in one line. (1) Compressibility model: C0 = 403 GPa−1 , Cm = 4.03 GPa−1 (2) Athy’s effective stress law: φ0 = 0.70, k = 0.096 MPa−1 (3) Schneider model: φ0 = 0.70, φa = 0.35, ka = 0.1916 MPa−1 , kb = 0.0527 MPa−1 and Athy’s hydrostatic depth law: k = 0.83 km−1 (4) Mudstone model: e100 = 1.2889, β = 0.458 0 10 20 30 40 50 60 0 10 20 30 40 50 60 70 Effective Stress in MPa Porosity in % 1 4 3 1..Compressibility Model 2..Athy Stress Model 3..Schneider Model and Athy Depth Model 4..Mudstone Model 2 2.2.3 Permeability and Viscosity The mobility μ is a measure of the ability of a material to transmit fluids. It includes the rock permeability k and the fluid viscosity ν: μ = k/ν. In the Darcy law (2.9), the mobility is the proportional factor between pressure gradients and fluid flow velocities. This applies to slow flowing (Newtonian) fluids. The permeability is mainly affected by the pore structure of the rock, and the viscosity describes the internal friction of the moving phase. This indicates that flow velocity rises with rising permeability and reduces with increasing viscosity. Viscosity Fluid viscosity is a measure of the resistance of the fluid against flow. It is related to the attractive forces between the molecules. Viscosity generally depends on pressure, temperature, and phase composition. The considered viscosity is the dynamic viscosity ν in contrast to the kinematic viscosity ν/ρ. The unit of the dynamic viscosity is Poise (P, 1 Pa s = 0.1 P).
- 66. 52 2 Pore Pressure, Compaction and Tectonics a) b) 0 10 20 30 40 50 60 0 10 20 30 40 50 60 70 Effective Stress in MPa Porosity in % 1 6 6 1 0 1000 2000 3000 4000 5000 6000 0 10 20 30 40 50 60 70 Depth in m Porosity in % 1 6 6 1 1..100% Shale 2..80% Shale; 20% Sandstone 3..60% Shale; 40% Sandstone 4..40% Shale; 60% Sandstone 5..20% Shale; 80% Sandstone 6..100% Sandstone 1..100% Shale 2..80% Shale; 20% Sandstone 3..60% Shale; 40% Sandstone 4..40% Shale; 60% Sandstone 5..20% Shale; 80% Sandstone 6..100% Sandstone Fig. 2.14. Compaction curves for mixtures of shale and sandstone using arith- metic averages of all compaction parameters. (a) Athy’s depth model with shale φ0 = 0.70, k = 0.83 km−1 and sandstone φ0 = 0.41, k = 0.31 km−1 . (b) Schneider model with shale φ0 = 0.70, φa = 0.35, ka = 0.1916 MPa−1 , kb = 0.0527 MPa−1 and sandstone φ0 = 0.42, φa = 0.205; ka = 0.0416 MPa−1 , kb = 0.0178 MPa−1 The viscosity of saline water can be estimated from McCain Jr. (1990) in Danesh (1998) as follows: ν = νT (0.9994 + 4.0295 × 10−5 P + 3.1062 × 10−9 P2 ) νT = T−a (109.547 − 8.40564 s + 0.313314 s2 + 8.72213 × 10−3 s3 ) a = 1.12166 − 2.63951 × 10−2 s + 6.79461 × 10−4 s2 +5.47119 × 10−5 s3 − 1.55586 × 10−6 s4 (2.39) with ν in mPa s, T in ◦ F, P in psi, s in mass %, and the validity intervals for νT of 38◦ C T 200◦ C and s 26%, and for ν of 30◦ C T 75◦ C, and P 100 MPa. Another formulation was published by Hewlett-Packard (1985) in Mc Der- mott et al. (2004). ν = ν0 [1 − 1.87 × 10−3 s0.5 + 2.18 × 10−4 s2.5 +(T0.5 − 0.0135 T)(2.76 × 10−3 s − 3.44 × 10−4 s1.5 )] ν0 = 243.18 × 10−7 10247.8/(TK−140) [1 + 1.0467 × 10−6 P (TK − 305)] (2.40) with ν in mPa s, T in ◦ F, TK in K, P in bar, s in %, and the validity interval 0◦ C T 300◦ C, s 25% and P 430 ◦ C. In the published equation P is
- 67. 2.2 Terzaghi Type Models 53 the difference between the real and the saturation pressure, but the latter one can be neglected for geological conditions. Both formulations yield similar results for moderate pressures (Fig. 2.15). The uncertainties of mobilities are controlled more by permeability than vis- cosity. Hence, the following simplification of equation (2.39) without salinity and pressure dependence is also proposed here (dotted curve inFig. 2.15.a). a) b) 0 50 100 150 200 250 300 0 0.5 1 1.5 2 2.5 Temperature in o C Viscousity in mPa s 1 4 0 50 100 150 200 250 300 0 0.5 1 1.5 2 2.5 Temperature in o C Viscousity in mPa s 1 2 3 4 no pressure, no salinity 1..p = 0.2 MPa/K 2..p = 0.3 MPa/K 3..p = 0.5 MPa/K 4..p = 1.0 MPa/K Fig. 2.15. Pressure and temperature dependent water viscosity curves assuming a salinity of 10% after (a) McCain Jr. (1990) in Danesh (1998) and (b) Hewlett- Packard (1985) in Mc Dermott et al. (2004). The pressure and salinity independent curve for the simplified equation (2.41) is the dashed curve in (a) ν[cp] = 109.5 T−1.122 (2.41) with temperature T in ◦ F. The viscosity of liquid and vapor petroleum is dependent on its composition. Viscosity ranges for standard oils and gases and more sophisticated methods for calculating oil and gas viscosities from compositions are described in (Sec. 5.6.5). Permeability Permeability consists of two factors namely rock (intrinsic) permeability and relative permeability, the latter one is further described in Sec. 6.3. The in- trinsic permeability k is mainly affected by the pore structure, especially pore throat diameters and pore connectivity. Hence, it is dependent on the com- paction state and usually tabulated as a function of porosity (App. A). The unit of permeability is Darcy (1 D = 0.98692×10−12 m2 ), or millidarcy (mD), but logarithm millidarcy (log mD) is also used, since permeability values often vary over orders of magnitude with decreasing porosity (1 log mD = 10 mD, 0 log mD = 1 mD, −1 log mD = 0.1 mD, −2 log mD = 0.01 mD, . . .).
- 68. 54 2 Pore Pressure, Compaction and Tectonics The most commonly used permeability relationship is the Kozeny–Carman relation. A derivation can be drafted from Hagen–Poiseuille’s law for fluid flow through a porous structure, which is approximated by a bundle of tubular parallel capillaries. The flow velocity of a viscous fluid of N parallel tubes of radius r embedded in an rock matrix of bulk area A can be expressed with the Hagen–Poiseuille law as follows. v = N A r4 π 8ν ∇p (2.42) where ∇p is the driving pressure gradient along the tubes. The porosity of the considered tubular bundle is φ = Nπr2 /A, which yields the following fluid velocity. v = r2 φ 8ν ∇p . (2.43) The comparison with Darcy’s Law (2.9) results in a permeability of k = r2 φ/8. The introduction of a tortuosity τ , which is defined as “the length of the path actually followed between two points divided by the apparent path between these two points” (Vidal-Beaudet and Charpentier, 2000) or as “the averaged ratio of path–lengths traveled by a petroleum fluid to the geometrical length of the region of rock considered” (England et al., 1987) yields k = r2 φ 8τ2 . (2.44) It can be estimated with τ = √ 3 for many rocks. This equation can be rewritten to a so called Kozeny–Carman type rela- tionship of the form k = Bφ3 τ2S2 (2.45) with the specific surface area S = N2πr/A and B a geometrical factor (Mavko et al., 1998). From consideration of sphere packing it is possible to estimate S = (3/2)(1 − φ)/d with d as grain size. Furthermore, it is common, to replace the porosity φ with (φ − φc) by assuming that the permeability vanishes be- low a threshold porosity φc where the pores become unconnected (Mavko et al., 1998). However, the following revised Kozeny–Carman relationship has been pro- posed by Ungerer et al. (1990) for practical use in basin modeling. k(φ) = 2 × 1016 κ φ5 S2(1 − φ)2 if φ 0.1 k(φ) = 2 × 1014 κ φ3 S2(1 − φ)2 if φ 0.1 (2.46)
- 69. 2.2 Terzaghi Type Models 55 with specific surface area S in m2 /m3 , κ a lithotype dependent scaling factor and φ a corrected porosity φ = φ − 3.1 × 10−10 S. Example parameters for clastic rocks are given in Table 2.2. Lithology Specific Surface Area Scaling Factor in m2 /m3 Shale 108 0.01 Siltstone 107 0.5 Sandstone 106 10.0 Table 2.2. Kozeny–Carman parameters for various lithologies The Kozeny–Carman type relation (2.46) has two different exponential factors for the high and low porosity range. The permeability decrease for highly porous rocks is mainly caused by the reduction of the pore throat radius, while in the highly compacted rocks, the closure and elimination of pore throats yields a decrease in pore connectivity (coordination number) and a drop in permeability. However, all Kozeny–Carman type models describe the intrinsic perme- ability dependent on porosity, pore size, pore throat radii distribution, and coordination number, which is a measure of the pore connectivities. Further considerations based on more complex geometrical models are e.g. given in Vidal-Beaudet and Charpentier (2000), Doyen (1988). An alternative approach describes the permeability with a piecewise linear function in the log permeability versus porosity diagram. Example curves for many lithologies are tabulated in App. A in terms of three porosity versus log permeability pairs. Some of these curves for clastic rocks and carbonates together with the corresponding Kozeny–Carman curves with the parame- ters from Table 2.2 are illustrated in Fig. 2.16. Salt, granite and basalt are considered as impermeable. Permeabilities are mixed geometrically for homogeneous mixtures or litho- types. In layered mixtures the horizontal values are mixed arithmetically and the vertical values are mixed harmonically. Generally, permeability k is a symmetrical tensor with six independent components. Similar to the thermal conductivity, it is often approximated with only two independent components: the permeability along the geological layer kh and permeability across the geological layer kv with an anisotropy factor ak = kh/kv. Typical anisotropy values are ak = 2 . . . 10 for clastic rocks and ak = 1 . . . 3 for carbonates, they are tabulated for many rock types in App. A. The above permeability curves and tabulated permeability values mean vertical permeabilities and equivalent hand-sample values, since most pub- lished data and in-house databases in oil companies are derived from hand- sample measurements. Basin scale values for horizontal and vertical perme-
- 70. 56 2 Pore Pressure, Compaction and Tectonics 0 20 40 60 80 -8 -6 -4 -2 0 2 4 Porosity in % Permeability in log mD 1 2 3 5 a) b) 0 10 20 30 40 50 60 70 -10 -8 -6 -4 -2 0 2 4 6 Porosity in % Permeability in log mD 1 2 3 1..Sandstone 2..Siltstone 3..Shale 4 1..Dolomite 2..Limestone 3..Marl 4..Chalk 5..Coal Fig. 2.16. Permeability curves for various lithologies with piecewise linear (solid) and Kozeny–Carman (dashed) relationships. The parameters are from Table 2.2 and from the Appendix A. A special curve is proposed for coal abilities are calculated from the hand specimen values multiplied with a hori- zontal and vertical upscaling factor, respectively. The higher values for larger scales are caused by macro-fractures, inhomogeneities and permeable inclu- sions. Upscaling factors to basin scale elements with lengths greater than 50 m are reported for sandstones: 500 (horizontal) and 10 (vertical) (Schulze- Makuch et al., 1999). Based on the authors’ experience, we suggest upscaling factors of 50 (horizontal) and 1 (no upscaling vertical) for all clastic rocks and carbonates, and no upscaling otherwise. Different horizontal to vertical upscaling increases the anisotropy factor in clastic rocks to ak = 100 . . . 500, respectively. The permeability of fractured rock is much higher than that of undisturbed samples and is discussed in Sec. 2.6.1. 2.2.4 1D Pressure Solutions Simplified 1D models can be used to discuss some fundamental processes of overpressure formation and compaction, although 1D solutions are less practi- cal, since most overpressure distributions are strongly influenced by horizontal water flows along highly permeable layers (App. D). In this section, only me- chanical compaction is considered in describing the interaction of overburden due to sedimentation, overpressure formation and compaction. The 1D for- mulation of the general pressure equation (2.13) is as follows. C ∂u ∂t − k ν ∂u ∂z = C ∂ul ∂t . (2.47)
- 71. 2.2 Terzaghi Type Models 57 Pressure curves for a unique rock type deposited with constant sedimenta- tion rates are shown in Fig. 2.17 for shales and siltstones. Shale permeability decreases rapidly during burial, since the log permeability to porosity curve is very steep. Hence, there is only a small sedimentation rate dependent tran- sition zone between the uppermost 1 . . . 3 km and the deeper part, where the pressure gradient is equal to the lithostatic gradient. The transition zone in lower permeable rocks like siltstone occurs over a broader region of sometimes several kilometers. The corresponding porosity curves for homogeneous de- positions are shown in Fig. 2.18. The porosity reduction stops in the deep impermeable blocks, when water outflow is near zero. 0 50 100 150 200 250 0 2 4 6 8 10 Pressure in MPa Depth in km 1 2 3 4 5 a) b) 0 20 40 60 80 100 120 0 1 2 3 4 5 Pressure in MPa Depth in km 1..Hydrostatic 2..Rate 100 m/My 3..Rate 200 m/My 4..Rate 1000 m/My 5..Lithostatic 1..Hydrostatic 2..Rate 50m/My 3..Rate 100 m/My 4..Rate 200 m/My 5..Rate 1000 m/My 6..Lithostatic 1 2 5 6 Fig. 2.17. Sedimentation rate dependent overpressure formation of (a) siltstones and (b) shales with piecewise linear permeability curves of Fig. 2.16 The pressure formation in an alternating sandstone–shale sequence is shown in Fig. 2.19. The pressure gradient in sandstone is equal to the hy- drostatic gradient, while the pressure in the shale layer returns relatively quickly (after 500 m in the example) to almost the level of the pure shale curve. Hence, the increase of pressure in seals could be much higher than the lithostatic gradients. The behavior of an impermeable seal is illustrated in Fig. 2.20. All over- burden load above the seal is added to the pore pressure of all layers below the seal, since no pore water can cross the impermeable seal. This yields an increase to lithostatic pressure in the seal and a constant offset equal to the overburden load of the overpressure below the seal during the time after the sedimentation of the seal. This displacement with additional pressure is marked with the dotted line in Fig. 2.20.b. The pore pressure (solid line) also includes a small pressure exchange within the block below the seal. The following calculation shows reservoir pressure decrease by water flow through a permeable seal (Fig. 2.21). The considered reservoir has a thickness
- 72. 58 2 Pore Pressure, Compaction and Tectonics 0 10 20 30 40 50 60 0 1 2 3 4 5 6 Porosity in % Depth in km 1 4 a) b) 0 10 20 30 40 50 60 70 0 1 2 3 4 5 6 Porosity in % Depth in km 1 2 3 4 1..Hydrostatic 2..Rate 100 m/My 3..Rate 200 m/My 4..Rate 1000 m/My 1..Hydrostatic 2..Rate 50 m/My 3..Rate 100 m/My 4..Rate 200 m/My Fig. 2.18. Sedimentation rate dependent compaction of (a) siltstones and (b) shales with compaction curves of Fig. 2.8 Hydrostatic gradient in sandstone Steep increase in the upper part of the shales Pure shale pressure line Pressure in MPa Fig. 2.19. 1D overpressure formation in an alternating sand–shale sequence hr, a bulk compressibility Cr and an overpressure ur. The overpressure in the seal with a permeability ks and a thickness hs drops to zero at or near the top of the seal. The flow velocity in the seal is according to the Darcy’s law as follows. v = ks ν |∇u| ≈ ks ν ur hs . (2.48) Integration of equation (2.13) over the reservoir area with the assumption of no sedimentation yields the following relationship. v · n dS = C ∂u ∂t dV . (2.49)
- 73. 2.2 Terzaghi Type Models 59 0 5 10 15 20 25 30 35 0 1 2 3 4 5 Overpressure in MPa Depth in km 1 2 3 a) b) Pressure in MPa Lithostatic Pressure Hydrostatic Pressure Pore Pressure Seal 1..Pressure at 20 My (after depostion of salt) 2..Pressure at present 3..Pressure at 20 My plus post-salt overburden Fig. 2.20. 1D overpressure formation below a perfect seal. (a) Present day pressure versus depth curve. (b) Overpressure versus depth curves for the time of seal depo- sition and at present day. The pressure curve at seal deposition (20 My) is shifted to the corresponding present day locations to illustrate, that overpressure increase in all layers below the seal from 20 My to present day is a almost the same. The dashed curve is the 20 My curve plus the overburden load after seal deposition. The difference of the dashed curve and the present day overpressure curve is caused by water exchange in the layers below the seal The outflow of the reservoir is restricted to the reservoir–seal interface with surface Ar. Hence, Ar v = −Cr Vr ∂u ∂t (2.50) with the reservoir volume Vr = Ar hr. Thus, ∂ur ∂t = − ks Crνhshr ur . (2.51) It yields an exponential decrease in the reservoir pressure as follows. ur(t) = u0exp(− t τ ), τ = Crνhshr ks . (2.52) with an initial reservoir pressure u0. The time th = τ ln(2) when half of the overpressure is dropped is controlled by the permeability of the seal, the bulk compressibility of the reservoir, and the reservoir and seal thicknesses. Typical values for th (Fig. 2.22) show, that very low permeabilities are necessary to seal pressure over significant times. 2.2.5 Pressure Solutions in 2D and 3D Most of the effects discussed in the previous sections are also important in multidimensional pressure calculations: the upper part of the basin is in a
- 74. 60 2 Pore Pressure, Compaction and Tectonics Fig. 2.21. Overpressure formation and decrease below a permeable seal. It is assumed, that the overpressure in a permeable seal drops linearly from reservoir pressure to zero. The overpressure in the compartment is constant. The water flow through the seal and the related decrease in reser- voir pressure depend on seal permeability, reser- voir compressibility and the thicknesses of the two layers hs hr Overpressure u Depth z Seal Pressure Compartment Fig. 2.22. Reservoir pressure de- crease times th for various seal per- meabilities ks and hr = 200 m, hs = 200 m, ν = 0.5 mPa s -8 -7 -6 -5 -4 0.01 0.1 1 10 100 Seal Permeability in log(mD) Reservoir Pressure Decrease Time t h in My 1 2 3 1..C 2..C 3..C = 2 GPa =10 GPa -1 -1 -1 r r = 5 GPa r hydrostatic state, pressure increases in impermeable layers, and large over- pressure areas occur below low permeability seals. Additionally, high perme- able layers transmit high water flow rates and yield overpressure equalization in the layer. Even thin high permeable layers affect the multidimensional over- pressure field, especially when they are expanded over long distances or large depths. This is illustrated in Fig. 2.23, where high permeability sand layers of very different depths are well connected to each other and yield almost the same overpressure everywhere in the sands. The calculated difference for the connected layers with a permeability of k = −2.5 × log(mD) = 3.2 × 10−3 mD is about Δu = 0.01 MPa. The pressure difference is higher, when low permeability rocks interrupt the connectivity of the sands. Darcy’s law states, that the assumption of the same overall flow rate results in an increase in the pressure difference by one order of magnitude, when the connected permeability decreases by one order of magnitude (10 mD). This example shows how sensitive the multidimensional pressure solution depends on the connectivity of the highly permeable facies.
- 75. 2.2 Terzaghi Type Models 61 14.55 14.55 14.55 14.55 A B C D MPa 0 10 20 30 40 50 2 3 4 5 6 0 10 20 30 40 50 2 3 4 5 6 Overpressure in MPa Overpressure in MPa Permeable Barrier -6 -5 -4 -3 -2 0.01 0.1 1 10 100 Permeability in log(mD) Pressure Difference to A in MPa B C D a) b) Permeabilities (vertical) Shale k=-1.5 ..-6 log(mD) Shale k=-6 ..-7 log(mD) Sand k = 2 .. 0 log(mD) c) Fig. 2.23. Overpressure equalization along high permeability layers: (a) connected highly permeable sand layers are embedded in a thick shale package and yield almost the same overpressure in the three sandy sub-layers. (b) The pressure difference in the sand layers is almost proportional to the log permeability of the barrier. (c) Pressure solution with highly permeable barriers between the sands Another similar example is shown in Fig. 2.24. Here a permeable layer connects a highly overpressured area below a thick shale block with a shal- low hydrostatic pressure area. This permeable layer is able to discharge the pressure below the shale with resting pressure gradients equivalent to the permeability values of the connecting layer. A 3D example (Fig. 2.25) with a thin permeable layer varying over several kilometers of depth, also shows the pressure equilibration effect along a highly permeable flow avenue. These examples show how the architecture of the sediments in the basin control the pressure distribution. The overpressure equation (2.13) does not deliver a solution in imperme- able facies, such as salt, granite, or basalt, since these permeabilities are equal to zero. The pressure in impermeable structures should be equal to lithostatic pressure, since any fluid inclusion enclosed in an impermeable environment could never drop its pressure due to fluid outflow and must bear the total overburden. Hence, the inner points of impermeable rocks and salt are set as inner boundary conditions with values equal to the lithostatic potential. The overpressure gradient at the top of a salt dome can be a multiple of the lithostatic gradient as the overpressure can increase from a nearly hydrostatic
- 76. 62 2 Pore Pressure, Compaction and Tectonics -2.45 -4.25 +2.00 -7.13 -6.47 -6.01 -2.06 -3.88 Permeabilities (vertical) Shale k=-1.5 ..-7 log(mD) Sand k = 2 .. 0 log(mD) Basement (impermeable) a) b) c) d) Sandstone Permeability 1.0 log(mD) Sandstone Permeability -3.0 log(mD) 0.08 0.07 5.01 17.52 -3 -2 -1 0 1 0.01 0.1 1 10 Permeability in log(mD) Pressure Difference in MPa MPa Fig. 2.24. Overpressure discharge in a highly permeable layer: (a) the highly per- meable layer connects a high pressure area with a hydrostatic exit. (b) The pressure gradient in the sand layer gradually decreases with increasing permeability. (c), (d) 2D–overpressure fields for two different permeabilities of the permeable layer Fig. 2.25. Small overpressure differ- ences along a 100 m thin layer with a permeability of k = 10−2 mD 30.42 MPa 30.69 MPa 30.76 MPa 30.68 MPa regime in the sediments above to a lithostatic regime in the salt layer over a very short distance. Uplift, together with erosion, yields overpressure release, since overburden load is decreased, but the porosity is almost maintained and the decompaction path during uplift is different from the normal compaction line (Fig. 2.6). Hence, the compressibility during uplift is much smaller (or close to zero),
- 77. 2.2 Terzaghi Type Models 63 which results in lower pressure release during erosion when compared with pressure formation during burial. Some of the multidimensional pressure effects are illustrated in the exam- ple calculation of a 2D cross section from the Santos basin offshore Brazil (Fig. 2.26). The pressure is hydrostatic in the shallow area up to the top over- pressure surface in 1 to 3 km depth. The pressure is lithostatic in the imperme- able salt domes. High overpressure occurs below the thick salt domes, which gradually decrease toward the salt window. The overpressure is much lower below the smaller salt bodies. A thick block of low permeable shale layers also causes overpressure formation, while overpressures are almost equilibrated in the highly permeable facies. In the above description, the upper pressure boundary condition at the sediment-water-interface is set to zero overpressure. In areas above sea level, the upper boundary is the groundwater surface and the pressure boundary condition is the groundwater potential, which is equal to the weight of the groundwater column above sea level. The pressure variable u in equation (2.13) is named hydraulic potential instead of overpressure in the terminology of groundwater specialists. Both terms are synonymous. The onshore ground- water level far from the coast is usually only some meters beneath the surface. The topographic surface can be taken as the approximate groundwater sur- face. The groundwater level close to the coast, or in very steep mountains can be significantly decreased, so that the boundary value of the corresponding hydraulic potential must be applied nearer to sea level with a much lower value. An onshore example with a groundwater potential is shown in Fig. 2.27. The model has an aquifer with a depth of 3 km and a water flow towards the hydrostatic zone. The resulting water flow system and overpressure field of the basin is a superposition of three effects: the topographic driven flow near the surface follows the surface profile, the sedimentation controlled overpressure flow is directed out of the thick sediments, and the aquifer layer transports water toward the hydrostatic area. The water flow is much faster and the overpressure is much smaller, when the mountains are less extended as shown in Fig. 2.27.c. The formation of mountains is always related with uplift and erosion, which is accompanied by a decrease in the overpressure potential of the uplifted blocks, since overburden is released during erosion. The overpressure release in rapid uplift and erosional periods below the mountains can be so high, that under-pressures arise in the aquifers, and the water flow can redirect toward the mountains as shown in Fig. 2.27.d. An analytical solution of a linearly varying horizontal groundwater potential is described in App. C.
- 78. 64 2 Pore Pressure, Compaction and Tectonics MPa Lithostatic pressure in salt High overpressure in dense shales Shallow hydrostatic area log(mD) impermeable low moderate high Lithology Permeability (vertical) Overpressure No overpressure gradient in sands Moderat over- pressure below salt windows High overpressure below the salt 50% Carb 50% Marl 35% Silt 35% Shale 30% Carb 70% Sand 30% Shale 50% Sand 50% Shale 10% Sand 80% Shale 10% Carb Sandstone 15% Sand 85% Shale Siltstone 70% Sand 30% Shale Shale 33% Sand 34% Shale 33% Carb 5% Sand 95% Shale Salt Basement Fig. 2.26. Overpressure formation along a 2D cross section in the Santos basin, Brazil. The blue vectors indicate water flow
- 79. 2.3 Special Processes of Pressure Formation 65 -50 0 50 100 150 -2 -1 0 1 2 3 4 Pressure in MPa Depth in km a) b) c) Groundwater related overpressure Decrease of overpressure due to horizontal outflow Overburden related overpressure MPa 0 km 500 km d) 0 km 500 km 0 km 50 km Fig. 2.27. Overpressure formation in an schematic onshore model: (a) 1D– extractions of pressures along a well with an aquifer at 3 km depth. (b) Overpressure formation and water flow vectors at present day. (c) Effect of mountain width, the model length is one tenth compared to model (b). (d) Effect of erosion: uplift is linked with erosion. The water flow vectors in the aquifer change direction from left to right 2.3 Special Processes of Pressure Formation Special processes of pressure formation are quartz cementation (chemical com- paction), aquathermal pressuring, pressure formation due to petroleum gen- eration and cracking, and mineral transformations such as smectite–illite or gypsum–anhydrite. 2.3.1 Chemical Compaction All sandstones and carbonates are cemented during burial. Quantitative de- scriptions of cementation processes are proposed by several authors (Walder- haug, 1996, 2000; Walderhaug et al., 2001; Bjørkum, 1996; Bjørkum and Nade- nau, 1998; Bjørkum et al., 1998, 2001; Schneider et al., 1996; Schneider and
- 80. 66 2 Pore Pressure, Compaction and Tectonics Hay, 2001; Lander and Walderhaug, 1999). Quartz cementation can be re- garded as a three step process: quartz dissolution at grain-grain contacts, transport of the dissolved silica through pore space and precipitation of sil- ica on free quartz grain surfaces (Walderhaug 1996, Figs. 2.28, 2.29). The transport of the solutes is performed via diffusion or pore water flow. All three processes (dissolution, solute transport and precipitation) have different effects on compaction, porosity reduction and pore pressure change. The ce- mentation rate is controlled by the subsurface conditions, the water flow and water chemistry. Subsurface conditions are temperature, total vertical stress, and pore pressure. Water flow rates depend on the permeabilities of adjacent rock, and water chemistry is characterized by the dissolved minerals and the pH–value. It is a common approach to reduce the model to the precipitation process and assume that the other processes always supply enough silica. Fig. 2.28. Principal processes of chem- ical compaction: (A) Pressure dissolu- tion of silica into pore water. (B) Diffu- sion of dissolved silica within the pore water phase. (C) Precipitation of silica at quartz grains A B Pore Space Diffusion Dissolution Precipitation Quarz Grain C a) Initial Volumes d) After Compaction c) After Precipitation Vr Vr Vw Vq Vq Pore Water Rock Matrix V V -V w w q = b) After Dissolution Vr Vw Pore Water Rock Matrix Vr Vw Pore Water Rock Matrix ‘ Pore Water Rock Matrix Fig. 2.29. Schematic volume balance for quartz dissolution and precipitation The volume balance includes changes in the the solid volume (including cement) Vs, the pore fluid volume Vw, the volume of the precipitated cement Vq, and the total volume Vt = Vs + Vw with the porosity φ = Vw/Vt and the cementation ψ = Vq/Vt. Dissolution of silica occurs along the grain contacts. The contact zone is a thin film of adsorbed fluids between the rough surface of quartz grains. The dissolution rate Cd is mainly controlled by the effective stress (pressure dissolution) and is dependent on temperature.
- 81. 2.3 Special Processes of Pressure Formation 67 Cd(σ z, T) = − 1 Vt ∂Vs ∂t . (2.53) During dissolution, the solid rock volume is decreased by the amounts of dissolved quartz, while the pore volume is increased by the same amount. The dissolved silica is transported in water by diffusion and together with water as a separate phase flow. Hence, it depends on the quartz solubility of water, the diffusion rate, the permeabilities and overpressure gradients. The literature distinguishes between an open and closed systems approach, as- suming relatively long and short transport paths (Schneider et al., 1996). The closed systems approach is more important, since quartz is usually precipi- tated near to the location of dissolution. However, the transport of dissolved silica does not influence the porosity, compaction and overpressure. Precipi- tation of silica as cement occurs on the free grain surfaces with preference to pore throats, which decreases permeability significantly. Precipitation rates Cp are usually temperature dependent (Walderhaug, 1996). Cp(T) = 1 Vt ∂Vs ∂t . (2.54) Pure precipitation increases the amount of solid material and reduces the pore space by the same amount. The total balance of quartz dissolution and precipitation is as follows. ∂Vs ∂t = (Cp − Cd) V, ∂ψ ∂t = Cp . (2.55) The total process yields much lower porosities for high effective stresses than pure mechanical compaction would allow. It also increases the pore pressure, since the dissolution of the solid matrix transfers lithostatic pressure to pore pressure. Closed System Approach In the closed system approach, short diffusion tracks are assumed with pre- cipitation near to the locations of dissolution. Hence, the precipitation rate is equal to the dissolution rate and the total solid volume remains constant. The ability to drop the porosity by cementation additionally to mechanical compaction requires a change in the compaction law by either increasing the bulk compressibility or adding an additional term fc as follows. ∂φ ∂t = −C ∂σ ∂t − fc(T, σ ) (2.56) with the Terzaghi’s compressibility C for mechanical compaction. The cementation controlled porosity loss is also realized by accompanied water outflow and it is usually almost equal to the relative volume of the
- 82. 68 2 Pore Pressure, Compaction and Tectonics precipitated cement (fc ≈ ∂ψ/∂t). The measured relative volumes of silican cement ψ are often used to derive empirical rules for the compaction term fc. Empirical laws for the cementation rate are proposed by Walderhaug (2000) and Schneider et al. (1996), named respectively the Walderhaug and Schneider models. The Walderhaug model is a precipitation rate-limited re- action controlled by the temperature and the quartz surface area available for precipitation. Walderhaug argues, that there is usually enough effective stress at large depth to supply enough dissolved quartz and that the effective stress dependency of the chemical compaction can be neglected. He proposed the following relationship with an Arrhenius type temperature dependency: ∂φ ∂t = − Mq ρq 6(1 − fq)fv dq φ φ0 A e−E/RT (2.57) where R is the gas constant with R = 8.31447 Ws/mol/K, fq is the quartz grain coating factor (the fraction of the quartz grain surface that is coated and unsuitable for precipitation), fv is the quartz grain volume fraction when precipitation starts (the fraction of the detrital grains that are quartz), dq is the average quartz grain size, and A and E are the frequency fac- tor and activation energy of the quartz precipitation rate. Fixed param- eters are Mq = 0.06009 kg/mol and ρq = 2650 kg/m3 , the quartz molar mass and density. Default parameters are fq = 5, fv = 1, dq = 0.03 cm, A = 10−11 mol/cm2 /s, and E = 61 kJ/mol. The activation energy is primar- ily used for calibration, when sample data are available. The porosity loss described by the Walderhaug model is shown in Fig. 2.30 for various activa- tion energies with smaller cementation rates for higher activation energies. A viscoplastic type compaction model is proposed by Schneider et al. (1996), who introduced the porosity loss rate proportional to the effective stress σ z, which represents the quartz supply by pressure induced dissolution. The rate is dropped by a viscosity μ, which decreases with higher temperature according to a Arrhenius type dependency. ∂φ ∂t = −(1 − φ) σ z μ , μ = μ0 exp E k 1 T − 1 T0 (2.58) with the reference temperature T0 = 15 ◦ C and viscosity μ0 = 50 GPa/My. The activation energy E can be used for calibration with default values be- tween 16 and 18 kJ/mol. Porosity loss curves for fast and slow sedimentation are shown in Fig. 2.31 for various activation energies. Contrary to the Walder- haug model, porosity rates increase with higher activation energies. The additional term in the compaction law also appears in the revised pressure equation according to (2.11) as follows: C ∂u ∂t − ∇ · k ν · ∇u = C ∂ul ∂t + fc(σ z, T) . (2.59) The increase in pressure is caused by the transfer of rock stress to pore pressure due to the abbreviation of the vertical stress bearing rock elements, by the
- 83. 2.3 Special Processes of Pressure Formation 69 a) b) 0 50 100 150 200 0 5 10 15 20 25 30 35 Temperature in o C Porosity in % 1 2 3 4 0 20 40 60 80 100 120 140 160 0 5 10 15 20 25 30 35 Temperature in o C Porosity in % 1 2 3 4 1..E=61 kJ/mol 2..E=64 kJ/mol 3..E=67 kJ/mol 4..E=70 kJ/mol 1..E=61 kJ/mol 2..E=64 kJ/mol 3..E=67 kJ/mol 4..E=70 kJ/mol Fig. 2.30. Cemented porosity calculated with the Walderhaug model with coating factor fq = 0.5, quartz grain volume fraction f = 1, quartz grain size dq = 0.3 mm, initial porosity φ0 = 41 %, frequency factor A = 10−11 mol/cm2 /s: (a) with high sed- imentation rates S = 1.0 km/My, (b) with low sedimentation rates S = 0.1 km/My a) b) 0 50 100 150 200 250 300 0 2 4 6 8 10 12 14 16 Temperature in o C Porosity in % 1 2 3 4 0 50 100 150 200 250 0 5 10 15 20 25 30 Temperature in o C Porosity in % 1 2 3 4 1..E=16 kJ/mol 2..E=18 kJ/mol 3..E=20 kJ/mol 4..E=22 kJ/mol 1..E=16 kJ/mol 2..E=18 kJ/mol 3..E=20 kJ/mol 4..E=22 kJ/mol Fig. 2.31. Cemented porosity calculated with the Schneider model with a temper- ature gradient dT/dz = 30 ◦ C/km, effective stress gradient dσ z/dz = 10 MPa/km: for sedimentation rates (a) S = 0.1 km/My, (b) S = 1 km/My
- 84. 70 2 Pore Pressure, Compaction and Tectonics pore space reduction due to precipitated cement, and by the permeability decrease due to thinner pore throats. An one dimensional example is shown in Fig. 2.32 with alternating shale and sandstone layers. The Walderhaug and Schneider models are compared with respect to porosity loss and additional overpressure generation. The Walderhaug model generally predicts higher cementation rates than the Schneider model. The difference in porosity loss (Fig. 2.32) is very high when the proposed default values are used. The effects are more similar when higher activation energies, rather than the default values, are used in both models. Cementation of the sandy layers yields lower permeability values which sig- nificantly influence the dewatering of the shale layers below. Additional over- pressures in the sandstones also influence the pressure formation and porosity loss of the overlaying shales. 2.3.2 Fluid Expansion Models Fluid expansions yield fluid density increases and related overpressure for- mation, which can be described with an additional source term fa in the overpressure equation (2.59) as follows: C ∂u ∂t − ∇ · k ν · ∇u = C ∂ul ∂t + fc(σ z, T) + fa(T) . (2.60) The source terms in the above equation are understood as the relative pore fluid volume increase over time. The volumetric formulation is obtained, as in the initial mass balance equation (2.10) all the terms were already divided by the pore water density assuming that the water density variations with depth and time are relatively small on the considered scale. The fluid ex- pansion models are here formulated on volume and not on mass balances, although variable fluid densities are considered. They are easier to implement and overview and the differences to more complex formulations are of minor importance (Luo and Vasseur, 1992, 1996). One can also integrate the source terms over the entire burial history for a deep sediment, for example in 5 km depth, to compare the order magnitudes of the different sources for pressure formation with each other. The total source for overburden load Fo is: Fo = t C ∂ul ∂t dt ≈ C̄ ūl ≈ 0.75 (2.61) where C̄ ≈ 10 GPa−1 is the average bulk compressibility and ūl ≈ 75 MPa is an average total load of a sediment in 5 km depth. Assuming, that in the initial sedimentation phase approximately one third of the overburden was not converted to overpressure, a value of 0.5 is more realistic. The total load source for chemical compaction Fc is equal to the total porosity reduction Δφ by cementation: Fc = t ∂φ ∂t dt ≈ Δφ ≈ 0.15 . (2.62)
- 85. 2.3 Special Processes of Pressure Formation 71 a) b) c) d) 0 10 20 30 40 0 10 20 30 40 50 Geologic Time in My Porosity in % 1 2 3 4 5 0 5 10 15 20 25 30 0 1 2 3 4 5 6 Porosity in % Depth in km No cementation Walderhaug Model Schneider Model Sand Sand Sand 0 20 40 60 80 100 0 1 2 3 4 5 6 Pressure in MPa Depth in km Hydrostatic Lithostatic No cementation Walderhaug Model Schneider Model Sa Sa Sa 0 10 20 30 40 0 10 20 30 40 50 Geologic Time in My Porosity in % 1 2 3 4 1..No cementation 2..E=72.0 kJ/mol 3..E=66.5 kJ/mol 4..E=61.0 kJ/mol (default) 1..No cementation 2..E=16 kJ/mol 3..E=18 kJ/mol 4..E=20 kJ/mol 5..E=22 kJ/mol Schneider Model Walderhaug Model Fig. 2.32. 1D example of three sandstone layers embedded in shale depositions. Porosity loss and overpressure formation is calculated with the Walderhaug and Schneider model for cementation of the sandstone layers and without cementation for comparison. (a) Porosity loss due to mechanical and chemical compaction. The activation energies of the Walderhaug and Schneider model are 61 kJ/mol and 20 kJ/mol. (b) Overpressure formation during cementation. (c), (d) Porosity loss of the lower sandstone layer calculated with the Walderhaug and Schneider model, respectively, for different activation energies This has a significant effect compared to the overburden load, but is locally restricted to sandstone and carbonate layers only. Aquathermal Pressuring Luo and Vasseur (1992) investigated the effect of overpressure formation caused by thermal expansion of the pore water. The additional source term fa in the overpressure equation depends on the isobaric thermal expansion coefficient α = αw − αr, which is the difference between the values for water
- 86. 72 2 Pore Pressure, Compaction and Tectonics αw = 5 × 10−4 K and rock αr = 3.3 × 10−5 K. fa = α ∂T ∂t (2.63) which yields the following estimation for the total load source: Fa = t α ∂T ∂t dt ≈ Δφ ≈ 0.07 (2.64) with an assumed temperature T = 150◦ C in a depth of 5 km. Assuming, the overburden is not converted to overpressure for early sedimentation, a value of 0.05 is more realistic. It is one order of magnitude smaller than the effect from the overburden load. The aquathermal pressure formation depends on the heating rate and it becomes higher for fast burial. Mineral Transformations Some mineral transformations, such as smectite to illite and anhydrite to gypsum conversion are related to pore fluid volumes changes. The conversion from smectite to illite occurs in all shales and shaly rocks and is described as a complex multi–stage process (Pytte and Reynolds, 1989; Swarbrick et al., 2002). Some of the related processes increase and some decrease the pore water volume with a general release of bound water into pore space and with a total increase of the water relative volume up to 5 %. The volume of the solid matrix is also generally increased, since mainly Na+ ions are exchanged by K+ ions with a higher ion radius. This process is controlled by temperature and the availability of K+ ions in the rock matrix. A widely accepted model was proposed by Pytte and Reynolds (1989) as a fifth–order reaction of the following type. ∂x ∂t = −x5 k1 k2 (2.65) where x is the smectite to illite ratio with an initial value of 0.8, k1 = 74.2 exp(−2490/T[K]) is the chemical activity of potassium to sodium and k2 = 1.64 × 1021 My−1 exp(−16600/T[K]) is a Arrhenius type temperature dependence. The equation can be written as a usual unimolecular forward reaction type (Chap. 4) as follows: ∂x ∂t = −x5 A e−E/RT (2.66) with the activation energy E = 37.9 kcal/mol, the frequency factor A = 1.217 × 1023 My−1 and the gas constant R = 8.31447 Ws/mol/K. The conver- sion depends on the heating rate, which is controlled by sedimentation rates (Fig. 2.33). The simple approach is, that the transformation ratio of the re- action TRs = (x − 0.2)/0.8 is related with a constant factor κ ≈ 0.05 to the load source fa as follows:
- 87. 2.3 Special Processes of Pressure Formation 73 fa = κ ∂ ∂t TR . (2.67) Hence, the overpressure formation caused by smectite to illite conversion also depends on the sedimentation rates with a total source of Fa = t κ ∂ ∂t TR dt = κ ≈ 0.05 . (2.68) This effect is one order of magnitude smaller than the overburden load ef- fect. Osborne and Swarbrick (1997) and Swarbrick et al. (2002) proposed the separate consideration of smectite dehydration with the release of interlayer water, but the authors consider this effect to be included in the usual me- chanical compaction model for shale. Fig. 2.33. Smectite–Illite after Pytte and Reynolds (1989) calculated for uni- form sedimentation with different sedi- mentation rates 0 20 40 60 80 100 0 1 2 3 4 Illite to smectite ratio in % Depth in km 1 2 3 1..S = 25 m/My 2..S = 100 m/My 3..S = 1000 m/My Petroleum Generation Pressure Catagenetic processes of organic matter change the relative volumes of kero- gen, liquid, and vapor petroleum (Fig. 2.34). Here, the primary cracking of kerogen and secondary cracking of the heavier petroleum components are taken into account. The controlling parameters are the changes of the petroleum phase masses and volumes, which result from the generated chem- ical components and the PVT controlled dissolution into the two petroleum phases (Chaps. 4, 5). The fluid models also provide corresponding modifica- tions of the phase densities, which control the generation driven overpressure formation. The actual densities (mass per volume) of the generated petroleum components, which are dissolved in liquid and vapor phase, are denoted as μl and μv with the phase densities ρl and ρv, respectively. Then, the load source for primary generation is as follows:
- 88. 74 2 Pore Pressure, Compaction and Tectonics fa = 1 ρl − 1 ρk ∂μl ∂t + 1 ρv − 1 ρk ∂μv ∂t . (2.69) The kerogen density ρk has to be taken into account, since the reduction of kerogen volume during cracking is also a significant value. For the estimation of the magnitude of the load source, all generated petroleum with mass density μp is assumed to be dissolved in one super–critical petroleum phase with density ρp.It follows from (4.8): Fa = t 1 ρp − 1 ρk ∂μp ∂t dt ≈ ρr ρp − ρr ρk TOC0 HI0 (1 − φ) ≈ 0.033 (2.70) with an initial total organic carbon content TOC0 = 5%, an initial hydrogen index HI0 = 500 mgHC/gTOC, a porosity φ = 20%, and densities of ρk = 800 kg/m3 , ρp = 500 kg/m3 , ρr = 2200 kg/m3 . This effect is one order of magnitude smaller than the overburden load effect and it is restricted to source rocks only. Exceptions are coals with TOC values higher than 50 % and coal bed methane production, which can form very high overpressures in place. Gas has a much higher compressibility (100 GPa−1 ) than the porous framework (10 GPa−1 ), which yields a retarded overpressure drop by pore fluid outflow as described in equation (2.52), when high gas saturations occur. This slightly increases the effect of gas generation controlled overpressures. Fig. 2.34. Gas generation from kerogen changes the volumetrics of kerogen and pore fluids Secondary cracking can also change phase volumes and can be described analogously to primary cracking. The resulting overpressure build up is much smaller, especially because coke with higher density is formed as a by–product. The generation of petroleum amounts can be more accurately formulated in the multi–phase fluid flow equations (Chap. 2.9) with source terms for the generated masses. Luo and Vasseur (1996) published a detailed analysis for a two-phase system formulation with similar magnitudes for overpressure build-up as described in equation (2.70). The overpressure equation with all described effects is as follows. Rock Rock H O 2 H O 2 Kerogen Kerogen Gas Gas
- 89. 2.4 Overpressure Calibration 75 C ∂u ∂t − ∇ · k ν · ∇u = C ∂ul ∂t + fc(σ z, T)+ α ∂T ∂t + κ ∂TR ∂t + 1 ρl − 1 ρk ∂μl ∂t − 1 ρv − 1 ρk ∂μv ∂t . (2.71) 2.4 Overpressure Calibration The overall overpressure is mainly determined by mechanical compaction. Other sources for overpressure, such as chemical compaction or fluid expan- sion, are often rather localized phenomena and for that reason not included in this section. Mechanical compaction, as formulated in (2.13), relates pore water flow with porosity reduction and overpressure. An overpressure calibration is there- fore a calibration of compressibility and permeability. It can be performed in two major steps. The first step deals with the adjustment of rock compress- ibilities and the second with permeabilities. Compressibility is introduced via a relationship of effective stress and porosity in (2.17). Effective stress is defined as the difference σ = σ − pI. Relationship (2.17) describes a local property of the rock, and does not con- tain permeability. If porosity is known, a compaction model such as Athy’s law or the Schneider model can be fitted to each lithology in the following way: porosity and pressure value pairs of the same lithology are collected for different depths and locations. It is possible to calculate the corresponding overburden from the basin model for these points. Thus, effective stress can be evaluated and plotted against porosity (Fig. 2.35). Finally, a compaction model with an effective stress versus porosity formulation, is fitted against these data points. Fig. 2.35. Fit of Athy’s law in porosity–effective stress formulation against a few data values Porosity [%] Effective Stress [MPa] Calibrated Not Calibrated
- 90. 76 2 Pore Pressure, Compaction and Tectonics Note that this approach relates effective stress to porosity and not with depth. A porosity depth fit is not achieved until the second major calibration step of permeability adjustment is performed. The second step consists of a permeability adjustment against overpres- sure. This step can be sophisticated because overpressure depends in general non–locally and ambiguously on permeability. For example, a pressure drop due to water outflow can often be modeled by different positions and sizes of the “leak”. However, some general rules of thumb can be stated. Highly imper- meable rocks are not a matter for calibration. If there is no water flow within these rocks, a small change in permeability does not change the overpressure pattern at all. The situation is the other way around for highly permeable sand layers. A change in permeability will not change the overpressure pattern. It remains equilibrated inside these layers. Important for pressure calibration are the layers in which overpressure builds up or is released (Fig. 2.36). Obvi- ously, a permeability variation for these rocks will cause a significant change in the overpressure pattern. Identification of these layers is a key point in over- pressure calibration because it drastically reduces the number of calibration parameters. Fig. 2.36. An example of a pressure cal- ibration by adjustment of the permeabili- ties. Pressure builds up in layer Fm6 and is slightly released in layer Fm5. Hence Fm5 and Fm6 are the key layers in this example. It is possible to calibrate the model by variation of the permeabilities in these layers. Below Fm5 a highly per- meable sandstone is located. It transports some water from this region through a slightly higher permeable window area in Fm5/Fm6, far away from this well. Hence it is necessary to incorporate and adjust the permeabilities in this window area for good calibration Overpressure can, in principal, only be calibrated if the water flow and wa- ter balance is adjusted correctly throughout the entire basin. In practice this leads to a situation where the permeabilities of many layers, lithologies, and rock types must be adjusted simultaneously. Due to long range pressure inter-
- 91. 2.5 Geomechanical Models 77 actions caused by water flow, this is usually very problematic. It is found that it is possible to tackle the problem iteratively. Calibration is first performed on key layers which are connected directly or via other permeable layers to the top of the basin. Water is transported along these pathways out of the basin. Overpressure can be calibrated best if the total amount of water in the basin is adjusted first. According to this picture adjustments of permeabilities in more deeply buried regions will lead to a minor overall adjustment at least on the global water balance.1 On average, the water flow is upward and therefore calibration should usually be performed from the top down and from more to less permeable lithologies. The procedure can be repeated iteratively until convergence is reached. This workflow assumes, that recently no erosion with a reduction of over- burden appeared. Otherwise, under the assumption of non-decompactable rocks, porosity must be fitted against the maximum effective stress. How- ever, the maximum effective stress might not be calculable as necessary paleo overpressures are possibly unknown. This problem can be overcome with ad- ditional overpressure shifts at paleo times, which are also calibrated against the present day overpressure pattern. The whole overpressure calibration pro- cedure including both steps must then iteratively be refined. 2.5 Geomechanical Models The fundamentals of geomechanics would require a full book in itself. Here only the most important equations, which are needed for basin modeling, are mentioned. Detailed descriptions are given e.g. in Fjaer et al. (1992) and Parry (2004). Solids conduct forces through the material and react with deformations. The forces and moments acting on each small volume element are described in terms of stress and the deformations are represented in terms of strains. Most materials respond with linear dependent recoverable strains on small stresses, which is called linear elasticity. In practice, stress–strain relations also have terms of non-linearity, irreversibly (plasticity), rate dependency (viscosity or creep) and yield failure, when certain limits of the stress components are exceeded. The traditional stress–strain concept has been determined for solids. It can be extended to porous media with an introduction of effective stresses, which takes pore pressure into account. The difference between the concepts of rock and soil mechanics is that the first takes into account cement between the grains, while the second refers to unconsolidated rock with loosely connected grain particles. 1 More deeply buried rocks are usually more compacted and therefore less perme- able. This also reduces the capability of water transport and the range of influence on the overpressure pattern.
- 92. 78 2 Pore Pressure, Compaction and Tectonics 2.6 Stress and Deformation The total bulk stress tensor is a superposition of the stress tensors of the grains and the pore pressure of the fluids. Tensors, their principal values and invariants are introduced in Sec. 8.2. The stress tensor σij has normal (i = j) and shear (i = j) components, which act on surfaces perpendicular to the coordinate axes (Fig. 2.37). Compressional normal components are positive. The principal values are denoted as σ1, σ2, and σ3 with σ1 ≥ σ2 ≥ σ3. The boundary vector t = n · σ acting on arbitrary area with the normal n has a normal and a tangential component tn and tt. szz t Normal Stress sn tn A a) x y z szx szy sxx sxz sxy syy syx syz b) x z y n Area A Shear Stress st s1 s3 s2 d) s1 c) s2 s3 s1 s2 s3 tt tn tt Fig. 2.37. Representations of the stress tensor. (a) Normal and shear components, (b) Boundary stresses at an arbitrary area, (c) Principal Stresses, (d) Mohr circles There are two important representations of the three dimensional stress state: the Mohr circles used in rock mechanics and the “p-q” plots used in soil mechanics. The Mohr circle construction is based on principal stresses. An arbitrary 3D stress tensor is pictured with three circles (Fig. 2.37) in a normal- shear stress diagram and the area between the circles represents the boundary stress vector acting on any cut-plane of the volume element. The outer circle is important to analyze and illustrates rock failure. Stress in two dimensions is represented with only one circle. Mohr circles for the special cases of biaxial and isotropic stresses, and pure shear are illustrated in Fig. 2.38.
- 93. 2.6 Stress and Deformation 79 a) c) Shear Effective Stress ’ s t Normal Effective Stress ’ s n b) Deviator Effective Stress q Mean Effective Stress ’ s s1 s3 s3 s1 A..Biaxial Stress (Compression) s1 s3 s1 s3 B..Biaxial Stress (Compression/Extension) C..Isotropic Stress (Pressure) s3 s1 s1 s3 D..Pure Shear s1 s1 s3 s3 szx sxz sxz B D C A A B C D szx Fig. 2.38. Stress characterization of special load cases (a) Principal stresses for biaxial stresses, (b) Mohr circles, (c) soil mechanical p–q plot The “p-q” plot uses two characteristic values, the mean stress σ̄ and the deviatoric stress q.2 The mean stress is an average volumetric (compressional) stress and the deviatoric stress represents an average shear stress as follows: σ̄ = σ1 + σ2 + σ3 q = 1 √ 2 (σ1 − σ2)2 + (σ1 − σ3)2 + (σ2 − σ3)2 1/2 . (2.72) Any three dimensional stress state is a point in the “p-q” plot as illustrated in Fig. 2.38. Any movement, rotation and deformation yields a change in the position of the sample particles, which is described with the displacement u(r). The deformation of a volume element is called strain and can be derived from a given displacement vector as follows: = 1 2 (∇u + (∇u)T ) . (2.73) This equation is only valid for small deformations. In case of large deforma- tions, additional terms with products of ∇u need to be incorporated in the above equation (Zienkiewicz, 1984). The strain tensor ij also has normal 2 The symbol “p” is usually used in soil mechanics terminology instead of σ̄, but the symbol “p” is already used for pore pressure here.
- 94. 80 2 Pore Pressure, Compaction and Tectonics (or direct) (i = j) and shear (or distortion) (i = j) components. The total volumetric deformation v = 3 ¯ is the sum of the principal strains: v = 1 + 2 + 3 . (2.74) The two special deformations of pure and simple shear have no volumetric strain v = 0 (Fig. 2.39). ezz ezz exx exx exx=-ezz exz ezx exz=ezx exz ezx x z x z Shear Strain e1 e3 Direct Strain en es Pure Shear Simple Shear Fig. 2.39. Pure shear and simple shear are special deformations without volumetric strain. They are represented with the same Mohr circle, but they have different orientations of the axes The elasticity tensor E relates stresses and strains linearly σ = E · assuming the linear theory of elasticity. It contains only two elastic parameters for isotropic behavior: the shear modulus G and the Poisson’s ratio ν. σ = 2G + 2Gν 1 − 2ν vI (2.75) where I is the unit tensor. Alternatively, the Young’s modulus E or the bulk modulus K can be used as follows. E = 2G(1 − ν), K = 2G(1 + ν) 3(1 − 2ν) . (2.76) Note, that the inverse of the bulk modulus is the bulk compressibility C = 1/K. The meaning of the elasticity parameters is especially descriptive for uni- axial compression with σx and the two resulting strains x and y, where the elastic properties are E = σx/x, ν = −y/x, and K = σ̄/v. Anisotropy is described with more then two elastic parameters in the elasticity tensor and non-linear elastic behavior with additional terms of higher order strains. The principle of force equilibrium states, that any body force f is compen- sated by the stress tensor. ∇ · σ + ρ f = 0 (2.77) where ρ is the bulk density. This yields the differential equation based bound- ary value problem for the model of linear elasticity (2.75) with the gravity (overburden) forces as follows.
- 95. 2.6 Stress and Deformation 81 GΔu + G 1 − 2ν ∇∇ · u + ρgez = 0 . (2.78) The boundary values are displacements u and boundary stresses t = n · σ. The differential equation is slightly different when large deformations are taken into account (Zienkiewicz, 1984). The above theory of stresses and strains has been developed and proofed for pure solids. Extensions for composite media, as needed for pore fluids and rocks, require very complex models for pore pressure and rock stresses. There are simplified models proposed by Terzaghi on an experimental basis and Biot on theoretical derivations, both are based on the idea of introducing an effective stress σ instead of the total stress σ and using the principal equations of the above concept with some modifications. σ = σ − α p I, with α = 1 − Kb Ks (2.79) where Kb, Ks are the bulk moduli for the bulk framework and the solid rock matrix, respectively. Terzaghi’s effective stress is defined for α = 1, which is a good approximation except at very large depths. The corresponding effec- tive stress based Mohr circles and “p’-q” plots for compacted sediments are illustrated and explained in Fig. 2.40.3 2.6.1 Failure Analysis Elastic material response means, that characteristic stress values like σz or σ̄ increase linearly with the equivalent strain values z, v. The correspond- ing stress-strain plots also show non-elastic behavior. The curves are usually obtained by rock mechanical laboratory measurements, such as drained and undrained uni- and triaxial tests. Mean stress and volumetric strain are the im- portant parameters in basin modeling. Typical curves are shown in Fig. 2.41. Usually, elastic and elasto-plastic regions are distinguished, separated by the yield point, the point of maximum stress and the critical state point. Elastic behavior means no permanent changes. Thus, the rebound curve is identical to the load curve. Beyond the yield point the specimen will not return to the original state, but it still supports increasing loads with yield- ing. Softening begins at the point of maximum stress. Further deformation yield less ability of the specimen to withstand stress. At the critical state, an instable deformation occurs, like rupture or pore collapse. Failure is usually defined at the point of maximum stress, although it is sometimes used for yielding, since the material structure changes. 3 Equivalent to the term “p-q” plot, which was introduced as the mean stress versus deviatoric stress diagram, the “p’-q” plot depicts the mean effective versus the deviatoric effective stress diagram. Note that the deviatoric effective stress is equal to the deviatoric total stress for Terzaghi’s definition of the effective stress (2.2).
- 96. 82 2 Pore Pressure, Compaction and Tectonics a) Shear Effective Stress ’ s t Normal Effective Stress ’ s n Burial b) Shear Effective Stress ’ s t Normal Effective Stress ’ s n Pore Pressure Increase Uplift Pore Pressure Decrease c) Shear Effective Stress ’ s t Normal Effective Stress ’ s n Extensional Tectonics Compressional Tectonics d) Deviator Effective Stress q Mean Effective Stress ’ s n Burial (normal compaction) Pore Pressure Increase Compressional Tectonics Fig. 2.40. Effective stress based Mohr circles and “p’-q” plots for (a) burial and uplift, (b) pore pressure changes, (c) tectonics. (d) Equivalent “p’-q” plot Volumetric Strain en Normal Stress ’ s n Elastic YP CS Brittle Material Volumetric Strain en Normal Stress ’ s n Elastic YP MS CS Elasto-Plastic Hardening Elasto-Plastic Softening Ductile Material Elasto-Plastic Hardening Rebound curves Fig. 2.41. Schematic stress versus strain plot for rocks. Characteristic points are the yield point (YP), the maximum stress point (MS) and the critical state point (CS). Brittle and ductile materials are distinguished by the relative length of the elasto-plastic region. The rebound curve has approximately the same steep angle as the linear elastic curve with a small hysteresis The behavior of a sample is called ductile or brittle, when the elasto- plastic region is large or small. The curve in the elasto-plastic region of the same sample strongly depends on temperature and the speed of deformation. Mohr Type Failure For an arbitrary three dimensional stress state, the failure criterion is a func- tion of all principal stresses, the yield function f:
- 97. 2.6 Stress and Deformation 83 f(σ1, σ2, σ3) = 0 . (2.80) Mohr proposed a function, which depends only on maximum and min- imum principal stresses. This is equivalent to a curve in the Mohr diagram (Fig. 2.42). Failure occurs when the Mohr circle intersects the failure line. The often used Mohr failure curve is a straight line with cohesion C and internal friction μ as offset and steep angle of the line, respectively, which is called a Mohr–Coulomb failure. Effective Normal Stress s’n Pore Collapse Mohr-Coulomb Failure Griffith Failure Cohesion C Tensile Strength T Internal Friction m Shear Stress st 1 2 s1 s2 s3 Fig. 2.42. Stresses and pressures in porous rocks σt = C + μσn . (2.81) Mohr–Coulomb failure initiates plastic flow along a failure plane, which is directed at an angle β = (π + 2μ)/4 between the axes of σ3 and σ1. The Mohr–Coulomb failure criterion is also equivalent to the surface of a hexagonal pyramid in the principal stress space (Fig. 2.42). A Mohr-type failure curve can also used to describe pore collapse, but with a negative friction angle, which is equivalent to the “cap” of the Mohr-Coulomb pyramid. In the extensional region, a Griffith type criterion is usually taken into account. It is derived from a microscopic theory of crack extension in two- dimensional samples. A simple generalization to three dimensional rock sam- ples can be made with a parabolic failure curve in the Mohr diagram (Fig. 2.42), which is equivalent to a parabolic “top” in the principal stress space. Griffith formulated the failure equation for 2D only. The simplest ex- tension to 3D-phenomena is Murell’s extension, where the failure surface, in terms of principal stresses, is also a parabolic surface with a simple pyramid on top. (σ1 − σ3)2 + (σ1 − σ2)2 + (σ2 − σ3)2 = 24 T0 (σ1 + σ2 + σ3) or σ1 = −T0 or σ2 = −T0 or σ3 = −T0 . (2.82) Note, that the maximum tensile strength is the only failure parameter for Griffith failure and it is related to cohesion as C = 12 T0. Griffith failure
- 98. 84 2 Pore Pressure, Compaction and Tectonics can also be extended to compressional regions with the same formula. This is usually used for the description of fracturing as explained in Sec. 2.6.1. Plastic Flow and Critical State The term critical state indicates damage and the inability of the specimen to support stresses. Usually, the same type of failure condition is used as for yielding, but with different material parameters C0, T0 and μ. Plastic flow occurs from the onset of yielding until the critical state is reached. Then, the strain consists of elastic and plastic parts (Fig. 2.43), = e + p . (2.83) The elastic strain is still related to the effective stress tensor with the elastic modules, and the plastic deformations are directed perpendicular to the failure surface in the principal stress space. dp,ij = dλ ∂f ∂σ ij (2.84) where λ is the hardening parameter. In the above equation the plastic flow vector dp,ij is directed perpendicular to the yield surface in the σ–space. In porous media, a constant angle between flow vector and yield surface is usually assumed. In general, the function f can be different from the yield function and it is then called plastic potential. It is very important, to note, that the direction of the plastic flow is controlled by the failure parameters C0, T0 and μ. Fault planes are directed along fixed angles between the minimum and maximum principal stresses. The formulation of the corresponding elasto– plastic boundary problem takes into account the plastic hardening law (2.84) and the yield condition (2.81). Detailed descriptions are given in Zienkiewicz (1984). Fig. 2.43. Failure surfaces for yield- ing and critical state after Fjaer et al. (1992). The process of yielding is equiv- alent to hardening until the critical state is reached and damage occurs ELASTO- PLASTIC ELASTIC σ1 σ2 Initial yield surface Current yield surface Critical state surface The soil mechanical equivalent of the yield surface is the the Roscoe and Hvorslev surface in the “p–q–v” diagram (Fig. 2.44), for normally consolidated
- 99. 2.6 Stress and Deformation 85 and overconsolidated rocks, respectively.4 When the effective stress state in a rock intersects the yield surfaces, further compaction with increasing effective stress occurs along both failure surfaces until the critical state line. Fig. 2.44. Failure surfaces in a soil me- chanics “p’–q–v” diagram. When the compaction controlled effective stress paths hits the failure surface, further compaction follows the failure surfaces until the critical state line (CSL) Roscoe Surface CSL Hvorslev Surface p’ q v Deviator Effective Stress Mean Effective Stress Void Ratio Fracturing Fracturing is another type of failure, which is the formation and growth of microfractures in rocks. Most fracture models are based on the Griffith theory, which defines a brittle failure surface in principal effective stress diagrams. The most common type of fractures are tensile fractures. They are formed when traction exceeds the tensile strength T0. Following the usal conven- tion, traction is negative stress. Hence the maximum traction is equal to the minimum principal effective stress σ 3 and the condition for the initiation of fractures is σ3 − p = T0 . (2.85) Obviously, the above condition is valid when the Mohr circle contacts the failure line on the left side (Fig. 2.42). The Mohr circle moves to the left mainly by overpressuring. The minimum overpressure, which is needed to initiate fracturing for a given stress state, is called the fracturing pressure. The fracturing pressure can be drawn in the pressure–depth space to illustrate the threshold pore pressure for fracturing (Fig. 2.42). It is a very common simplification in basin modeling programs to describe the fracturing condition with a fixed fracturing pressure versus depth curve (Fig. 2.45). The tensile strength differs for the different rock types. Hilgers et al. (2006) reported T0 = 10 MPa for sandstone and a significant smaller value for shale. Thus, the fracturing pressure gradient alternates within a shale–sand sequence. Fracturing increases the rock permeabilities and drops the capillary entry pressures as described with the following relationship. 4 Normally consolidated rocks are under the maximum effective stress, while over- consolidated rocks have a lower effective stress than at maximum burial
- 100. 86 2 Pore Pressure, Compaction and Tectonics Pressure or Stress Depth a) ph s3 p s1 s3 s3 Pressure or Stress Depth b) ph s3 s1 Pressure or Stress Depth c) ph p s1 pf df Fracturing pf T0 0 0 Fig. 2.45. Pressure and stress versus depth diagram for a fracturing model with hy- drostatic pressure ph, pore pressure p, fracturing pressure pf , principal bulk stresses σ1, σ3 and effective stresses σ 1, σ 3. The maximum stress is almost equal to the lithostatic pressure and the minimum stress is assumed to be a fixed fraction of the lithostatic pressure. (a) The difference between the minimum and maximum effective stress increases with depth. (b) The Griffith model defines the fracturing pressure as the required pore pressure to initiate fracturing. (c) Fracturing occurs when the pore pressure exceeds the fracturing pressure log kf = log k + λk p − pf pf , pc = pcf − λc p − pf pf (2.86) where k and kf are the permeability of the unfractured and fractured rock, pc and pcf are the capillary entry pressures of the unfractured and fractured rock, and λk and λc are the fracturing parameters. For clastic rocks, the fracturing parameters of λk=3 log mD and λc = 3 MPa are frequently used. Fractures can partially anneal when the overpressure decreases below the minimum effective stress, so that the tension turns into compression. It is usually not necessary to exceed tensile strength again when fractures are re-opened, that means that the pore pressure has to be equal to the minimum effective stress. In some models, the simplified fracturing condition, that the fracturing pressure is equal to the minimum principal stress, is used. However, the inclusion of multiple closing and opening behaviors requires hysteresis effects with different fracturing pressures and permeabilities. 2.7 Faults Faults occur in most basins with large variations in length, thickness, throws, gouge content and related properties. They play an important role in fluid flow and pressure formation. Faults are initiated in consolidated sediments due to extensional and compressional forces mainly caused by plate tectonics. The process of fault formation and growth can be described and modeled with
- 101. 2.7 Faults 87 kinematic approaches. This is usually not part of basin modeling, instead the fault geometry and main properties at present and paleo–times are given as a predefined input. Geologists distinguish between normal, reverse, transform and strike–slip faults, although most of the faults are mixed mode faults. The fault type depends on stress conditions in the of formation Fig. 2.46. The fault properties can be predicted by structural or fault seal analysis methods. a) Normal fault Reverse fault Hanging wall block Footwall block ψ σ1 σ1 σ1 vertical vertical σ3 vertical σ2 σ2 σ3 b) Transform fault σ2 c) Strike-slip fault d) Mixed mode σ3 Fig. 2.46. Fault types formed under different stress conditions: (a) maximum prin- cipal stress in a vertical direction cause normal and reverse faults. (b) Minimum principal stress in a vertical direction causes transform faults. (c) Medium principal stress in a vertical direction causes strike-slip faults. (d) Most faults in nature are mixed modes. The pictures are from Bahlburg and Breitkreuz (2004) Fault extensions often exceed several hundreds of meters. Fault zones are often in the range of several meters and much smaller than gridcells of basin scale models. Location and orientation of faults are thus geometrically de- scribed with fault planes in 3D–models and lines in 2D–models. Fault lines and planes can be approximated with boundary elements along cell faces and edges in cellular models as illustrated in Fig. 2.47 and Fig. 2.55. The fault planes in 3D–models are constructed from fault lines, which are usually inter- preted from seismic at the surface of horizon maps. Faults can act as preferred migration avenues (in-fault flow) or as hydro- carbon seals which hold column heights of hydrocarbons. The two related flow properties are permeability and capillary entry pressure. Boundary fault
- 102. 88 2 Pore Pressure, Compaction and Tectonics Boundary Element Fault Volume Element Fault Volume Element Fault with Local Grid Refinement Volume Element a) b) Fig. 2.47. 2D–fault models. (a) Fault lines in a 2D–cross section. (b) Fault line approximation with boundary elements, adjacent volume elements, or with locally refined grid cells elements can be used for modeling petroleum migration and accumulation. They act between two volumetric elements with a zero volume. Capillary en- try pressure can be defined for cells with an infinitesimally small volume, but permeabilities cannot assigned to boundary elements without a volume. This yields instantaneous flow for in–fault flow, which is assumed in some petroleum migration models anyway. Permeabilities can not be neglected for pore pressure calculations, if fault gouge material with low permeability causes pressure contrasts and compart- mentalization. Hence, the inclusion of fault permeabilities for pressure mod- eling requires the consideration of a fault volume. The simplest method to work with volumetric fault elements is to define all cells adjacent to the fault plane as fault cells and assign the correspond- ing fault permeabilities. Obviously, fault zones can be overestimated with this
- 103. 2.7 Faults 89 approach, which can yield large errors in the calculated pressures. This is especially problematic for very low permeability faults, which have to be con- tinuously connected to model compartments. Double or triple bands of cells are necessary when topologically regular grids are considered. Irregular grid- line spacing with higher gridline density in the vicinity of faults can be used to lower the effect, but this is usually only applied in 2D–models, since the number of gridcells increases significantly. A good solution to the problem is the introduction of locally refined ele- ments around faults (Figs. 2.47, 8.13), where the real fault width can be taken into account. This method requires significant effort for the development of automatic meshers in 3D, especially for the special cases of layer pinch–outs and dendritic fault segments. An example pressure calculation with locally re- fined elements around faults, fault widths of 10 m and fault permeabilities of 10−6 mD is shown in Fig. 2.56. The pressure is constant within the sandstone compartments and varies in the sandy shales. The fault permeabilities and capillary pressures are mainly determined by the gouge composition. Very thin faults can be handled as neutral or juxta- position faults without any property assignment. The gouge composition is a mixture of the rocks of all layers, which slipped along the location during faulting. An important parameter of the composition is the clay content. Various indicators are proposed (Yielding et al., 1997; Fulljames et al., 1996), such as the shale smear factor or the shale gouge ratio (Fig. 2.48). All of them pay attention to the juxtaposition of sediments between the foot and the hanging wall and depend on fault distance or throw. The shale gouge ratio (SGR) is the volumetric ratio of grains smaller then 100 nm to the larger grains assuming that the value at an actual location is simply the arithmetic average of all the material that slipped since fault movement began. In this approach, it is not considered that different rock types have different supply rates to the gouge. An advantage of the SGR concept is, that the values can be calculated by simple volumetrics of the fault adjacent sediments for each point on the fault surface. Yielding (2002) proposed simple relations to convert SGR values to capil- lary entry pressures and permeabilities. Capillary entry pressures control col- umn heights at sealing faults. They are given as mercury-air values and can be converted to the present petroleum-water system via in-situ interfacial tension values of the compositional dependent petroleum phases. The fault capillary pressures (FCP) are therefore capillary entry pressures for the mercury-air displacement. Yielding found linearly increasing FCP values for SGR larger than a threshold SGR, with different ascent angles but unique minimum SGR of 18% in most of the samples. The average value for the parameter k in the following equation is 50 MPa. pc = k (SGR − 0.18) . (2.87)
- 104. 90 2 Pore Pressure, Compaction and Tectonics a) Throw z1 throw z SGR i å = Distance m n i dist z SF . S = b) Juxtaposition Fault Gouge Fault z2 Shale Smear Factor (SF) Shale Gouge Ratio (SGR) Fig. 2.48. Definition of fault properties after Yielding et al. (1997): (a) juxtaposition and gouge fault, (b) definition of shale smear factor (SF) and shale gouge ratio (SGR) The permeability value controls in-fault flow, once the accumulation is able to break into the fault, but this seems to be less important as the distances are usually short before an exit to more conductive sediments is found. It is obvious, that the fault properties (SGR, FCP) experience large varia- tions through geological time. Thus, they have to be specified or precalculated for several time periods. A common simplification is the introduction of spe- cial fault properties: ideal open (SGR 18%, FCP 0.1 MPa) and ideal closed (SGR 95%, FCP 50 MPa) to define faults as completely open or completely closed or via special FCPs or SGRs as in Fig. 2.57. Diagenetic processes or cataclasis in faults can be described by additional temperature or effective stress dependent corrections of the SGR values. 2.8 Paleo–Models In a basin with low faulting, throw and tectonics, back-stripping of the present day geometry under consideration of decompaction, erosion and paleo– thickness corrections is a good approximation of the paleo–geometries. De- compaction and erosion are typically vertical phenomena, which do not take into account any horizontal movements and changes in the total length of the layer. Horizontal movements of single layers like salt domes are described with paleo–thickness corrections based on rock volume balances which re- sults in layer squeezing and stretching. Complex tectonic events often yield strongly deformed geometries, which usually overstretch the possibilities of backstripping. Complete paleo–geometries are alternatively used as input for the simulation. They are constructed from structural modeling methods be- fore basin modeling is performed. The simulator then jumps from predefined
- 105. 2.8 Paleo–Models 91 paleo–geometry to paleo–geometry in the analysis. It has to identify the new location of each single facies and has to take into account facies movements and deformations. In compressional tectonics, overthrusted layers can mul- tiply be defined along a depth line, which is handled in practice with the implementation of a block concept for the thrust belts. Each block represents a compartment treated as a separate unit which can be moved along and against any other block. 2.8.1 Event–Stepping Backstripping is also called event stepping, since the paleo–geometries are re- constructed from the present day geometry due to given “geological events” with a suitable set of sophisticated rules, which yields topologically similar paleo–models. Decompaction of a layer from present day thickness dp to de- positional thickness d0 is calculated with the assumption of the conservation of the solid matrix volume according to d0 (1 − φ0) = dp (1 − φp) (2.88) with present day and depositional porosities φp and φ0, respectively. The present day porosity is not known prior to analysis, since it depends on the pore pressure development. Hence, the decompaction in the first simulation run can only be made with an estimation of the present day porosities, used as the steady state values for hydrostatic pressure conditions. The forward simulation then yields calculated present day geometry based on pore pres- sure controlled compaction, which usually differs from the given present day geometry (Fig. 2.49). This difference is much smaller in the next simulation run, when the calculated present day porosity can be derived for decompaction instead of the estimated steady state values. This optimization procedure can be applied multiple times, but usually two or three loops yield good results. Modeling of erosion requires the definition of the eroded thicknesses and the erosion ages. Eroded thicknesses can be given with virtual horizons or thicknesses at the time of deposition, at present day or any other geological event (Fig. 2.50). Multiple erosions of one layer and one erosion on multiple layers can also easily be recognized with virtual horizons. The interpretation of eroded thickness is often easier to perform on a backstripped and decom- pacted paleo–geometry. Herein, the porosity at the erosion age has also to be considered for decompaction of overconsolidated rocks. The eroded thickness and the compaction history of the layer before erosion has to be taken into account in the optimization procedure. Horizontal movements of layers like salt can be described with additional thickness maps during doming. The changes are realized by layer stretching and thinning. The additional salt thickness layer should be calculated under the assumption of total volume conservation. The simplest model considers a homogeneous depositional layer with the total volume equal to the total
- 106. 92 2 Pore Pressure, Compaction and Tectonics Present Day Geometry Estimated Calculated Calculated Given Paleo- Geometry Estimated Initial Thickness Calculated Porosity Thickness and Overburden Lithostatic Pressure Estimated Steady State Porosity Paleo- Geometry Fig. 2.49. Backstripping with decompaction is based on estimated present day porosities. The calculated porosities of the forward simulation usually improve back- stripping in the next run Present Day Model h1 h2 h3 h4 h5 h6 Paleo-Model at h3 h1 h2 h3 Paleo-Model at h4 h1 h2 h3 h4 Deposition Erosion d d a) b) c) Fig. 2.50. Definition of erosional thicknesses: (a) with virtual horizons at the present day geometry, (b) with additional thicknesses at the time of sedimentation, (c) with virtual horizons at any geological event volume of the present day salt domes and the definition of the doming ages. A linear interpolation between the initial and the final salt thicknesses can then be realized during doming. The opening of the salt windows should be described with an additional salt map, since the salt windows would otherwise open only during the last time step of the doming (Fig. 2.51). Structural geologists often provide salt maps for various geological events based on kinematic models, which also can be considered during simulation. Additional thickness maps can be used for the thinning of the salt adjacent layers or for doming of other lithologies, e.g. shale. High overburden can also yield reverse domes and single salt pillows as illustrated in Fig. 2.52. This
- 107. 2.8 Paleo–Models 93 Salt Deposition Present Day Model Begin of Salt Doming End of Salt Doming Interpolated Intemediate Geometry Interpolated Intemediate Geometry Opening of Salt Windows Fig. 2.51. A simple geometrical model with a linear interpolation of the salt thick- nesses between the geometries of salt sedimentation, salt window opening and the final doming requires the introduction of several layers to avoid multiple occurrences of one layer along a depth–line. Another method for handling salt intrusions into the overburden layers is to exchange the lithology of the intruded layers with salt. This is recommended when the intruded layers have big gaps in the present geometry. In very complex basins, pre–computed paleo-geometries might be necessary. This is described in the next subsection. Fig. 2.52. Reverse salt domes and salt pillowing require multiple layer defini- tions Base Salt Dome Reverse Salt Dome Single Salt Pillow 2.8.2 Paleo–Stepping The introduction of complete geometrical models for certain paleo–times re- quires the recognition of facies locations together with the corresponding types of movements and deformations during stepping from one paleo–geometry
- 108. 94 2 Pore Pressure, Compaction and Tectonics to the next. A section of a layer can be folded, migrated and/or otherwise stretched so that location and shape in two successive time steps might be very different (Fig. 2.53). A meshing algorithm based on pre-defined gridpoints and sublayers can yield new volumetric cells which are no longer related to the same rock of the previous time step. The consequence is that all bulk rock properties have to be transferred according to the new location. a) b) Uniform Folding Non-Uniform Stretching New Move in from Side Boundary Paleo- Event Present Day Folding Stretching Move in Fig. 2.53. Deformation types of facies during tectonics In most cases, the deformation is uniform stretching or thinning, which can be achieved with linear mapping operations. Any non-uniform deformations have to be specified manually between paleo–geometries. For moving–in layers, the side boundary values can be taken as the values for the previous time step. In the following a method is described, how the pressure and compaction problem is solved, when the compaction has already been predefined via paleo– geometries. Both the pressure and the compaction equations can be solved in the usual way. The change in the overburden load of each layer is calculated from one paleo–geometry to the next. The transient equation for overpressure (2.13) can then be solved with the transformed cell values of the previous time step. The results are a change in the overpressure as well as a reduction in the porosity. The only difference from the usual procedure is, that the porosity change is not converted into the new layer thicknesses, since they are already predefined with given paleo–geometries. Hence, porosity reduction and compaction are decoupled processes here and it is accepted that the volume rock matrix is no longer conserved.
- 109. 2.8 Paleo–Models 95 Backstripping or event–stepping is applied before the first occurrence of a paleo–geometry with the usual method for optimization (Fig. 2.54). Present Day Model Backstripped Paleo-Geometry Given Geometry Event before Predefined Paleo-Models Overburden Lithostatic Pressure Estimated Steady State Porosity Calculated Porosity GivenThickness Estimated Initial Thickness Backstripping First Predefined Paleo-Model Calculated Porosity Calculated Thickness Given Paleo-Geometry Paleo-Stepping Predefined Paleo-Model Given Paleo-Geometry Event-Stepping Fig. 2.54. Decoupling of compaction and porosity calculation during paleo–stepping A difficult problem is the automatic generation of additional paleo–models for time steps between the interpretations. The simplest idea is to use linear interpolations so that the thickness values of each gridpoint are interpolated, but this often yields unsuitable connections in steep faults. Another method is to directly jump to the next paleo–model, and use intermediate time steps for the solution heat and fluid flow equations, but with the same geometry for the whole geological event. The above procedure clearly separates structural reconstruction from for- ward basin modeling analysis by work flow and by data. The advantage of this decoupled link is that it is possible to use advanced special tools for both structural and basin modeling and the functionality of both tools are retained. Due to decoupling of processes information is lost. Feedback between mod- eled processes as well as coupled tools, handling structural and basin modeling together, are principally possible. 2.8.3 Overthrusting Blocks for overthrust belts are introduced to avoid multiple layer occurrences along one depth–line (Fig. 2.58). Each block is then treated like a ”single basin model” with suitable and varying coupling conditions between the block boundaries. The number of blocks can vary during paleosteps, since the break- ing of a so called super block into separate pieces leads to the development of complex block substructures from a homogeneous initial model. A hierarchy
- 110. 96 2 Pore Pressure, Compaction and Tectonics of block heritages has to be specified as a model input. Splitting of a super block into subblocks also has to be taken into account when considering the shift of the fundamental layer values according to their new locations. All block boundaries are faults. They can be treated as neutral (juxtapo- sition), partially or ideally open or closed faults with capillary entry pressures and permeabilities (Sec. 2.7). Compression or extension yields a total section abbreviation or stretching, which yields an increase or decrease in horizontal stress and causes additional or retarded overpressuring and compaction. Then, compaction should be con- trolled by mean effective stress instead of vertical effective stress components with the following modified compaction law, which replaces equation (2.3). ∂φ ∂t = −Cv ∂σ̄ ∂t = −Cv ∂(σ̄ − p) ∂t == −Cv ∂(σ̄ − ph − u) ∂t (2.89) where Cv is the volumetric bulk compressibility, which is related to the Terza- ghi compressibility CT as follows:5 Cv = CT 3(1 − ν) 1 + ν (2.90) which yields a factor of 1.28 . . . 2.45 for Poisson ratios of ν of 0.1 . . . 0.4. The pressure equation (2.13) is modified as follows. Cv 1 − φ ∂u ∂t − ∇ · k ν · ∇u = Cv 1 − φ ∂(σ̄ − ph) ∂t (2.91) where ph is the hydrostatic pressure and σ̄ is the mean total stress. Assuming, that the total stress differs from Terzaghi’s lithostatic pressure assumption only by an additional horizontal stress component the tectonic stress σt, the total mean stress is related with the overburden weight pressure pl as follows: σ̄ = σv + 2 σh = 1 + ν 3(1 − ν) pl + 2σt . (2.92) The additional assumption of a uniform compression σt 0 or extension σt 0 with a constant tectonic stress σt yields a simple extension of the pressure equation and compaction law which includes tectonic processes. 5 In the case without tectonics, it is σh/σv = ν/(1−ν) and with σ̄ = (1/3)(σv +2 σh) it follows that σv/σ̄ = 3(1 − ν)/(1 + ν).
- 111. 2.8 Paleo–Models 97 b) c) a) Fig. 2.55. (a) Fault approximation with boundary elements in 3D. (b) Vertical view with horizontal fault elements. (c) Map view of cutout with fault traces Lithology Shale sandy Sand shaly Dolomite Silt shaly Sand Chalk Marl Shale sandy Salt Basement 14.4 MPa 0 5 10 15 20 MPa 10.4 MPa 6.2 MPa Overpressure Fig. 2.56. Overpressure example with locally refined elements around faults Fault Capillary Entry Pressure (FCP) Shale Gouge Ratio (SGR) 12 MPa 0 MPa 3 MPa 35% 0% 70% Fig. 2.57. Capillary pressures and SGR values on fault planes in a 3D–model A B C D E F G H I Domain decomposition into blocks A-I Compressional tectonics yield mutliple layer occurences Fig. 2.58. Introduction of blocks for compressional tectonics
- 112. 98 2 Pore Pressure, Compaction and Tectonics Summary: Overburden load and tectonic stresses cause rock stresses, fluid pressure formation, and sediment compaction. An external load on a bulk volume element is balanced partially by the rock skeleton and partially by the pore water. Rock stresses and pore water pressure equalize overburden and external tectonic stresses. Many geomechanical processes are formulated with overpressure instead of pore pressure and effective stress instead of total rock stress. Overpres- sure is pore pressure minus hydrostatic pressure, which is the weight of the overlaying pure water column (plus a depth independent shift for zero level adjustment). Effective stress is the total stress minus pore pressure. Water is mobile. Overpressure gradients causes pore fluid flow, which is mainly controlled by the rock permeabilities. This allows for further rock compaction with reordering of grains. The rock becomes more dense and its internal stress rises as overpressure is usually reduced. All basic effects of mechanical compaction and overpressure formation can be modeled quite accurately with a Terzaghi–type approach. It is based on the assumptions that rock grains and water are incompressible and that rock compaction is a function of the vertical effective stress only, which is called lithostatic pressure. Water flow is modeled with Darcy’s law. Overall mass conservation is taken into account. Appropriate conditions for water in– and outflow at model boundaries must be defined. Various models for compaction vs. effective stress are proposed. The main characteristic is a logarithmic dependency of effective stress on porosity. The related compaction or bulk compressibilities functions are well known over a wide porosity range for various lithotypes. Overpressure calibration is a two step process. Firstly, the material pa- rameters of the compaction law must be fitted locally to suitable porosity vs. effective stress relationships. Secondly, permeabilities of relevant layers, which control the overall water flow, must be adjusted. The second step is rather sophisticated and relies on full simulation runs due to possibilities of long range lateral water flows. Besides pure mechanical compaction, pressure effects due to cementa- tion of pore space, aquathermal expansion, mineral transformations and petroleum generation are found as locally significant. Alternatively to a calculation of the geometry from compaction laws (event–stepping) the geometry might be imported from purely structural analysis (paleo–stepping). However, overpressures and effective stresses are simulated in any case with similar algorithms. Faults can be approximated with special volumetric and boundary ele- ments. The main mechanical properties are fault transmissibilities and capil- lary entry pressures, which can be derived from measured or calculated shale gouge ratios. The inclusion into pressure and fluid flow analysis requires so- phisticated numerical models.
- 113. REFERENCES 99 References L. F. Athy. Density, porosity and compaction of sedimentary rocks. American Association of Petroleum Geophysicists Bulletin, (14):1–24, 1930. H. Bahlburg and C Breitkreuz. Grundlagen der Geology. Elsevier GmbH, Muenchen, second edition, 2004. M. A. Biot. General theory of three-dimensional consolidation. Journal of Applied Physics, (12):155–164, 1941. P. A. Bjørkum. How improtant is pressure in causing dissolution of quartz in sandstones. Journal of Sedimentary Research, 66(1):147–154, 1996. P. A. Bjørkum and P. H. Nadenau. Temperature Controlled Porosity/Per- meability Reduction, Fluid Migration, and Petroleum Exploration in Sedi- mentary Basins. APPEA Journal, 38(Part 1):452–464, 1998. P. A. Bjørkum, E. H. Oelkers, P. H. Nadeau, O. Walderhaug, and W. M. Murphy. Porosity Prediction in Quartzose Sandstones as a Function of Time, Temperature, Depth, Stylolite Frequency, and Hydrocarbon Satura- tion. AAPG Bulletin, 82(4):637–648, 1998. P. A. Bjørkum, O. Walderhaug, and P. H. Nadeau. Thermally driven porosity reduction: impact on basin subsidence. In The Petroleum Exploration of Ireland’s Offshore Basins, volume 188 of Special Publication, pages 385– 392. Geological Society of London, 2001. A. Danesh. PVT and Phase Behaviour of Petroleum Reservoir Fluids. Num- ber 47 in Developments in petroleum science. Elsevier, 1998. P. M. Doyen. Permability, Conductivity, and Pore Geometry of Sandstone. Journal of Geophysical Research, 93(B7):7729–7740, 1988. W. A. England, A. S. MacKenzie, D. M. Mann, and T. M. Quigley. The movement and entrapment of petroleum fluids in the subsurface. Journal of the Geological Society, London, 144:327–347, 1987. E. Fjaer, R. M. Holt, P. Horsrud, A. M. Raan, and Risnes R. Petroleum related rock mechanics. Elsevier, 1992. J. R. Fulljames, L. J. J. Zijerveld, R. C. M. W. Franssen, G. M. Ingram, and P. D. Richard. Fault seal processes. In Norwegian Petroleum Society, editor, Hydrocarbon Seals - Importance for Exploration and Production, page 5. Norwegian Petroleum Society, Oslo, 1996. M. R. Giles, L. Indrelid, and D. M. D. James. Compaction – the great un- known in basin modelling. In S. J. Düppenbecker and J. E. Iliffe, editors, Basin Modelling: Practice and Progress, number 141 in Special Publication, pages 15–43. Geological Society of London, 1998. Hewlett-Packard. Petroleum fluids, manual. Technical Report HP-41C, 1985. C. Hilgers, S. Nollet, J. Schönherr, and J. L. Urai. Paleo–overpressure forma- tion and dissipation in reservoir rocks. OIL GAS European Magazine, (2): 68–73, 2006. R. H. Lander and O. Walderhaug. Predicting Porosity through Simulating Sandstone Compaction and Quartz Cementation. AAPG Bulletin, 83(3): 433–449, 1999.
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- 116. 3 Heat Flow Analysis 3.1 Introduction Heat can be transferred by conduction, convection, and radiation in sediments (Beardsmore and Cull, 2001). The sediment–water–interface temperature and the basal heat flow are the main boundary conditions for heat flow analysis in sediments. Magnitude, orientation and distribution of the heat inflow at the base of the sediments are determined by mechanical and thermal processes of the crust and mantle (Allen and Allen, 2005). Two processes result in permanent heat flow from the Earth’s interior to its surface: earth cooling and radiogenic heat production with a ratio of 17% to 83% respectively (Turcotte, 1980). Heat conduction is defined as the transfer of thermal energy by contact according to thermal gradients. It is the primary process in the shallow litho- sphere. The controlling lithological parameter is the thermal conductivity. It usually decreases from solids to liquids to gases. Generally, heat conduction is more effective with higher density. Heat convection is thermal energy transported with the movement of a fluid or solid. In sedimentary basins, it is mainly related to fluid flow of pore water, liquid petroleum and gas. Convection can be more efficient than con- duction when flow rates are high, e.g. in permeable layers or in fractures. It is the dominant thermal transport mechanism in the asthenosphere. Fluid move- ments can either add or remove thermal energy from a sedimentary sequence and can considerably distort conductive heat transfer systems. Solid convec- tion occurs for example during overthrusting and salt doming. It requires very high thrusting rates to be of significance. A special type of combined conduc- tion and convection is advection, which describes, for example, the heating of grains by groundwater flow. Heat radiation is thermal transport via electromagnetic waves usually with wavelengths of 800 nm to 1 mm. The amount of thermal energy is proportional to the fourth power of temperature. Therefore, only heat transfer from very T. Hantschel, A.I. Kauerauf, Fundamentals of Basin and Petroleum 103 Systems Modeling, DOI 10.1007/978-3-540-72318-9 3, © Springer-Verlag Berlin Heidelberg 2009
- 117. 104 3 Heat Flow Analysis hot areas requires attention. It is negligible in sediments, but should be con- sidered in deep parts of the lithosphere or asthenosphere. The heat conductivity law states, that a temperature difference between two locations causes a heat flow q. Its magnitude depends on the thermal conductivity of the material and the distance between these locations. In mathematical notation it becomes q = −λ · ∇T (3.1) with the temperature gradient ∇T and the thermal conductivity tensor λ. The tensor λ is often assumed to have only two independent components: the conductivity along a geological layer λh and the conductivity across a geo- logical layer λv. The heat flow vector at any location is mainly directed along the steepest decrease of temperature from a given location. In the lithosphere, it is mainly caused by the difference between its top and base temperatures: the surface temperature or sediment–water–interface (SWI) temperature at the top and the asthenosphere–lithosphere boundary temperature at its base. Hence, the resulting heat flow is mainly vertically directed when the two boundary surfaces are almost spherical and when the lateral variations of the boundary temperatures are small. The average thermal conductivity and the thickness of the mantle and crustal layer mainly control the heat in–flux into the sediments. This heat flow at the base of the sediments defines the lower boundary condition for the heat flow analysis in the sediments. In practice, heat flow analysis is commonly subdivided into two prob- lems: the consideration of the crustal model to calculate the heat in–flux into the sediments and the temperature calculation in the sediments afterwards (Fig. 3.1). SWI temperature Tswi SWI temperature Tswi Sediments Crust Upper mantle Tb Base lithosphere temperature q a) b) Water Sediments q Base sediment heat flow Fig. 3.1. Boundary value problem for a heat flow analysis (a) of the lithosphere and (b) in the sediments
- 118. 3.2 One Dimensional (1D) Models 105 3.2 One Dimensional (1D) Models In the 1D approach, it is assumed that all heat flow vectors are directed ver- tically. 1D solutions often provide a good estimate for temperatures since the boundary values define radial core to surface aligned paths. They are espe- cially used for well–based calibrations of basal heat flow trends. Exceptions, which cannot be modeled with 1D approaches are local areas of extraordinar- ily high thermal conductivities like salt domes, which bundle heat flow vectors from adjacent areas along highly conductive avenues. 3.2.1 Steady State Models The most simple 1D models are steady state solutions in which all time depen- dent terms such as transient or convection effects are neglected. In the absence of radioactivity, the heat flow q is constant throughout the sediments and the temperature gradient in a layer is higher the lower the thermal conductivity. Multilayer solutions can then be directly derived from the heat flow equation (3.1) with the assumption that the average bulk thermal conductivity λb of a layer sequence is equal to the harmonic average of the corresponding single layer bulk thermal conductivities λi. The temperature controlled boundary value problem of the lithosphere yields the following 1D steady state solution with vertical thermal conductivity λb and thickness hl of the lithosphere, and the corresponding properties of the upper mantle λm, hm, the crust λc, hc and sediments λs, hs. q = λb Tb − Tswi hl , hl λb = hm λm + hc λc + hs λs . (3.2) The advanced solution of the lithosphere problem, taking into account ra- dioactive heat production, transient effects and convection caused by stretch- ing, is discussed in Sec. 3.8. An equivalent steady state solution for the n–layer model of sediments with a base sediment heat flow qbs can be derived from equation (3.1) for the temperature at base of the sediments Tbs and a tem- perature increase ΔTi within a layer i as follows. Tbs = Tswi + n i=1 ΔTi, ΔTi = qbs hi λi . (3.3) The temperatures at all layer boundaries can be calculated from the sur- face temperature down to the base of section. This algorithm is illustrated in Fig. 3.2 for a simple sediment column consisting of only three lithotypes: shale, sandstone and limestone. The bulk conductivity for each layer λi is here approximated by a geometric average of the values for water λw and rock λr with the porosity φ as follows. λi = λr (1−φ) λw φ . (3.4)
- 119. 106 3 Heat Flow Analysis 39.8 C 62.5 C o o Dz = 700 m z T q D D = l DT = 22.7 K = 48 mW/m 2 Vertical rock thermal conductivities: Shale =1.70 Sandstone =3.95 - 0.01* (T[° ]-20) Lime l l W/m/K C W/m/K stone =3.00 - 0.001*(T[° ]-20) l C W/m/K = 1.48 W/m/K 22.7 K / 700 m x Heat Flow q = 48 mW/m 2 Fig. 3.2. 1D steady state example of a simplified model of a North Sea well. The bulk conductivity of the Hordaland Shale (1.48 W/m/K) is geometrically averaged from rock with λr = 1.70 W/m/K and water with λw = 0.7 W/m/K at a porosity of 15.6 % The temperature–versus–depth curve clearly shows intervals of steep and low increases due to low and high values of bulk thermal conductivities. This simple steady state argumentation shows that a heat flow analysis which based on a constant thermal gradient is a very rough approximation. However, the heat flow from the base upward does not remain constant because sediments contain radioactive elements like uranium, thorium and potassium. Radioactivity causes additional heat production, which increases the heat flow through the sediments. Thus, the surface heat flow is higher than the basal value by the amount of generated heat. Each of the radioactive elements generates gamma rays with radiogenic heat production rates Qr estimated by Rybach (1973) as follows: Qr = 0.01 ρr (9.52 U + 2.56 Th + 3.48 K) (3.5) where ρr is the rock density in kg/m3 , U and Th are the concentration of uranium and thorium in ppm, K is the concentration of potassium in % and Qr is in μW/m3 . Pore fluids do not contribute to radioactive heat production. The resulting increase of vertical heat flow Δq in a layer of thickness h due to a rock heat production rate Qr is as follows:
- 120. 3.2 One Dimensional (1D) Models 107 Δq = (1 − φ) h Qr . (3.6) The simple layer sequence example of Fig. 3.3 shows an average increase of the heat flow of about 1 mW/m2 per km sediment, which is a good general estimate. 51.15 51.74 mW/m mW/m 2 2 Dz = 350 m Dq = 0.59 mW/m 2 Radioactive heat production: Shale Sandstone Lime = 2.00 = 0.70 Qr m m W/m W/m 3 3 Q Q r r stone = 0.90 mW/m 3 Dq z = (1- ) = 0.59 mW/m D f Qr = 350 m (1-0.156) 2.00 W/m x x m 3 2 Fig. 3.3. Heat flow increase for the North Sea well with an example calculation in the Hordaland layer. The total increase of 5 mW/m2 from base to top is about 1 mW/m2 per km sediment 3.2.2 Transient Effect A system is in thermal steady state when the heat flow is constant everywhere. Any change of a thermal boundary condition, geometry, properties or temper- atures yields non-steady or transient state. The system will gradually return to a new flow equilibrium, when the new conditions do not change any more. The transition time depends generally on the ratio of the transported heat and the inner thermal energy, which is controlled by the size of the system, and ratio of the heat capacity and the thermal conductivity. The transient 1D temperature distribution along the downward directed z–axis is the solution of the following differential equation: ρc ∂T ∂t − ∂ ∂z λ ∂T ∂z = Qr (3.7)
- 121. 108 3 Heat Flow Analysis where c and ρ are bulk values, which should be arithmetically mixed from the pore fluid and rock values corresponding to the actual porosity. The transient effect is important during deposition and erosion and when thermal boundary conditions, namely SWI temperatures or basal heat flow, change rapidly. Deposition results in deeper burial and higher temperatures of underlaying sediments. The actual sediment temperatures are lower than for the steady state solution. Heat is absorbed for heating of the layer and heat flow values decrease towards the surface. In case of constant deposition without com- paction the heat flow decreases linearly in vertical direction (Fig. 3.4.a and b). It can indeed be analytically proven, that a constant deposition rate S yields a constant heat flow gradient after a relatively short time of deposition, according to (F.24) with ∂q ∂z ≈ qSρc λ . (3.8) The example values used in Fig. 3.4 result in a gradient of ∂q ∂z [mW/m2 /km] = 3.678 S[km/My] . (3.9) In the case of erosion the effect is reverse: the sediment temperatures are higher than the steady state solution and the heat flow increases toward the surface. The above rule (3.8) can also be applied to estimate the magnitude of erosion–induced heat change (Fig. 3.4.c). During a hiatus the heat flow gradually returns to the steady state solution, with a constant heat flow value in regions without radioactivity. The transient effect of an instantaneous heat flow change is shown in Fig. 3.5. Herein, the basal heat flow jumps from 40 mW/m2 to 60 mW/m2 while the near surface heat flow change is delayed for more than 5 Ma. Simi- lar examples can be calculated analytically (App. F). A change in SWI temperature also yields transient heat flow curves (Fig. 3.6). Typical SWI temperature variations yield much lower magnitudes compared to those caused by basal heat flow changes. Here, SWI or surface temperatures are average values over a range of 1000 to 10000 years. Sea- sonal cycles or short time changes are not taken into account. Sudden surface temperature rises lower the temperature difference between top and base and yield lower surface heat flow values. This effect occurs in glacial intervals of the quaternary period as shown in Fig. 3.21. The presented model uses small time steps of 2000 years for the approximation of the surface temperature variation. The temperature increases are especially steep and cause downward trends in the surface heat flow. Generally, near surface heat flow is correlated with the average surface temperatures of the previous thousands of years. The heat flow is constant from base to top for 1D solutions without ra- dioactivity and transient effects. Erosion and radiogenic sediments increases the surface heat flow, while deposition lowers surface heat flow. A transient solution of the 1D example model in Fig. 3.2 and Fig. 3.3 is shown in Fig. 3.7.
- 122. 3.2 One Dimensional (1D) Models 109 c) d) b) Sedimentation Hiatus Erosion a) Heat Flow mW/m 2 62 64 60 58 56 60 54 56 58 60 0 1 2 3 4 5 Heat Flow in mW/m2 Depth in k m 60 61 62 63 64 0 1 2 Heat Flow in mW/m 2 Depth in km 0 200 400 600 800 1000 0 1 2 3 4 Deposition/Erosion Rate in m/My Heat Flow Change in mW/m 2 per km depth 20 My 0 My Fig. 3.4. Transient effect of deposition and erosion: (a) burial history with heat flow overlay for periods of uniform deposition, hiatus and uniform erosion. (b) Heat flow vs. depth after deposition. (c) Rate dependent heat flow change per 1000 m sediment. (d) Heat flow vs. depth after erosion. The example is calculated with a constant basal heat flow. Compaction is neglected Herein, the basal heat flow variation through geological time is the main con- trol on the heat flow in the well. However, the heat flow isolines are not perfectly vertical lines as radioactivity causes a significant slope. Figure 3.7.a shows calculated heat flow values for present day, without radioactivity, to quantify the transient effects. The transient effect of 0.8 mW/m2 is mainly controlled by deposition and it is much smaller than the increase of 6 mW/m2 caused by radioactivity. It is often of interest to evaluate the difference between non–steady and steady–state thermal conditions in a basin. This can be done by solving the heat flow equation twice, once with and once without the transient term. The
- 123. 110 3 Heat Flow Analysis a) b) 0 50 100 150 200 250 0 1 2 3 4 5 Temperature in Celsius Depth in km 1 3 4 30 40 50 60 70 0 1 2 3 4 5 Heat Flow in mW/m2 Depth in km 2 3 4 1 1..initial 2..after 1 My 3..after 2 My 4..after 10 My 2 Fig. 3.5. Transient effect due to a basal heat flow switch a) b) 0 50 100 150 200 250 0 1 2 3 4 5 Temperature in Celsius Depth in km Temperature Depth 4 56 57 58 59 60 61 62 0 1 2 3 4 5 Heat Flow in mW/m2 Depth in km 2 3 1 4 1..initial 2..after 1 My 3..after 2 My 4..after 10 My 1 3 2 Fig. 3.6. Transient effect of SWI temperature change from 15◦ C to 25◦ C difference between the two types of calculated temperature fields is called the “thermal disequilibrium indicator”. 3.3 Thermal Conductivity Thermal conductivity describes the ability of material to transport thermal energy via conduction. For a given temperature difference a good heat conduc- tor induces a high heat flow, or a given heat flow maintains a small tempera- ture difference. Steep temperature gradients occur in layers with low thermal conductivities. The unit for thermal conductivity is W/m/K.
- 124. 3.3 Thermal Conductivity 111 -200 -150 -100 -50 0 0 5 10 15 20 25 30 Temperature in Celsius 1 2 a) b) c) d) e) -200 -150 -100 -50 0 0 20 40 60 80 Geological Time [Ma] Heat Flow in mW/m2 Geological Time [Ma] Geological Time [Ma] 1..Earth Surface 2..Sediment-Water-Interface (SWI) Fig. 3.7. 1D transient solution of a simplified model of a well in the North Sea. (a) Present day heat flow without radioactivity. (b),(c) Present day heat flow and temperature with radioactivity. (d) Temperatures during burial history and SWI temperature trend with radioactivity. (e) Heat flow values during burial history and basal heat flow trend with radioactivity
- 125. 112 3 Heat Flow Analysis The bulk thermal conductivity is controlled by conductivity values of rock and fluid components. Mixing rules for rock and fluid components are gener- ally complex and depend on whether the mixture is homogeneous or layered (Sec. 8.3). Sedimentary rocks are anisotropic with higher horizontal than ver- tical thermal conductivities. Generally, the thermal conductivity λ is a sym- metrical tensor with six independent components. It is often considered to have only two independent components: the conductivity along the geological layer λh and the conductivity across the geological layer λv with an anisotropy factor aλ = λh/λv. 3.3.1 Rock and Mineral Functions Thermal conductivities commonly depend on temperature and vary widely according to the type of rock. Some 20◦ C values of vertical conductivities λ20 v and anisotropy factors a20 λ = λ20 h /λ20 v are given in App. E. Sekiguchi–Waples Model The temperature dependence of matrix conductivity of any mineral, lithology, kerogen or coal can be calculated using the following equations adapted from Sekiguchi (1984) and plotted in Fig. 3.8.a. λi(T) = 358 × (1.0227 λ20 i − 1.882) × 1 T − 0.00068 + 1.84 (3.10) with i = v, h, λ in W/m/K and T in K. 0 50 100 150 200 250 300 350 0 1 2 3 4 5 Temperature in °C Thermal Conduktivity in W/m/K a) b) Temperature in °C Thermal Conductivity in W/m/K Ice: 2.23 W/m/K 0 50 100 150 200 250 300 350 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Water Oil Gas Gas Hydrate Fig. 3.8. Thermal conductivity functions: (a) rocks, (b) fluids Most rocks and minerals do not experience significant changes in their anisotropy factors during compaction. Exceptions are claystones with signif- icant dependency on compaction states. The effect can be described with a
- 126. 3.3 Thermal Conductivity 113 factor f defining the ratio of the depositional vertical conductivity λvd to the vertical conductivity of rock with zero porosity λv0. The latter is an extrapo- lated value for a fully compacted rock. Horizontal conductivities are calculated using the principle of Waples and Tirsgaard (2002) assuming that the mean thermal conductivity value of the clay minerals λm = λv + 2 λh remains con- stant during compaction. It is further assumed that the decrease in vertical conductivity with compaction is exponential as follows: λv(φ) = λv0fφ/φ0 (3.11) where λv0 is the vertical conductivity of an ideally compacted rock with φ = 0, φ0 is the depositional porosity and f is the grain rotation factor with f = 1 for porosity independent anisotropy.1 The principle of constant mean conductivity yields a porosity dependent horizontal conductivity λh(φ) from the horizontal conductivity of an ideally compacted rock λh0 as follows: λh(φ) = λh0 + 1 2 (λv0 − λv(φ)) . (3.12) Thus, the conductivity values of a rock with any porosity and temperature are calculated in two steps. First, the porosity related corrections are made using (3.12) for claystones only and second, the temperature corrections are made using (3.10) for both (vertical and horizontal) values separately. In the following example, the matrix thermal conductivity values of a shale with λ20 v0 = 1.64 W/m/K, aλ = 1.6, φ0 = 70% and f = 1.38 are calculated at φ = 30% and T = 80◦ C: λ20 h0 = 1.60 × 1.64 = 2.624 λ20 vφ = 1.64 × 1.380.3/0.7 = 1.883 λ20 hφ = (1.64 + 2 × 2.624 − 1.883)/2 = 2.520 λv = 358 × (1.0227 × 1.883 − 1.882) × (0.00283 − 0.00068) + 1.84 = 1.876 λh = 358 × (1.0227 × 2.520 − 1.882) × (0.00283 − 0.00068) + 1.84 = 2.402 The complete thermal conductivity versus porosity functions of the above example shale are shown in Fig. 3.9. Linear Dependency Model A simplified alternative model for rock matrix conductivities is based on the assumption of linearly temperature dependent values only. 1 Anisotropy values are often given by the depositional conductivity λvd, the fully compacted conductivity λv0 at φ = 0 and anisotropy factors aλd = λhd/λvd, aλ0 = λh0/λv0. The constant sum of horizontal and vertical conductivities be- comes λvd(1 + 2aλd) = λv0(1 + 2aλ0) and therefore f = (2aλ0 + 1)/(2aλd + 1).
- 127. 114 3 Heat Flow Analysis a) b) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 1 1.5 2 2.5 3 Porosity in Fraction Thermal Conductivity in W/m/K lh(T) lv(T) 0 50 100 150 200 250 300 350 1 1.5 2 2.5 3 Temperature in °C Thermal Conduktivity in W/m/K lv(T) l (T) l l (20 C) (20 C) h o h v o Fig. 3.9. Rock thermal conductivity functions of a typical shale under hydrostatic compaction and a constant temperature gradient of 30◦ C/1000 m with a temperature T = 20 ◦ C at deposition. (a) Porosity dependency (b) Temperature dependency λv(T) = λ20 v + λT (T − 20◦ C) (3.13) where λ20 v is the thermal conductivity at 20◦ C and λT is the conductivity increase per degree temperature. Default values for some lithologies are listed in App. E. Such a linear temperature dependency is often assumed in an interval from −20◦ C to 300◦ C. Rock Mixtures Mineral component based matrix conductivity values are calculated with the geometric average of their component minerals when the minerals are homoge- neously distributed in the rock (Fig. 8.4). The geometrical average is also used for rocks consisting of homogeneously distributed lithological components. In the case of layered structures the horizontal conductivities are calculated with arithmetic averages and the vertical conductivities with harmonic averages. 3.3.2 Pore Fluid Functions Thermal conductivities of fluids are isotropic and depend on temperature (Fig. 3.8). Deming and Chapman (1989) published the following formulas for water: λw = ⎧ ⎨ ⎩ 0.565 − 1.88 × 10−3 T − 7.23 × 10−6 T2 for T 137◦ C 0.602 − 1.31 × 10−3 T − 5.14 × 10−6 T2 for T 137◦ C (3.14) with λw in W/m/K and T in Kelvin. The following equations for conductivities of liquid and vapor petroleum are based on Luo et al. (1994) and a personal communication between Ming Luo and Doug Waples.
- 128. 3.3 Thermal Conductivity 115 λo = 0.2389 − 4.593 × 10−4 T + 2.676 × 10−7 T2 for T 240◦ C λg = −0.0969 + 4.37 × 10−4 T for T 120◦ C λo = λg = 0.075 else (3.15) with λo and λg in W/m/K and T in Kelvin. Most hydrocarbon components approach conductivities of about 0.075 W/m/K at high temperatures (Poling et al., 2001). Similarly, methane conductivities also approach values close to 0.075 W/m/K (Lide, 2006). These results seem intuitively reasonable, since under those conditions both phases are supercritical and similarities of the properties dominate. The thermal conductivity of solid pore substances, such as gas hydrates (clathrates) and ice, are 0.49 W/m/K, and 2.23 W/m/K, respectively (Sloan, 1998). All the above thermal conductivities are shown together in Fig. 3.8.b. Water is a superior thermal conductor, while gas has the lowest conductivity which yields isotherm bending below gas accumulations (Fig. 3.10). 2.0 2.5 2.0 3.0 4.0 km a) b) Labels with Thermal Conductivity (in W/m/K) Isolines of Temperature (in Celsius) 100 o C 110o C 105o C 95 o C 115 o C Isotherm Bending 1.43 1.96 1.63 1.67 2.0 2.5 2.0 3.0 4.0 km Gas Reservoir Seal Depth in km Depth in km Fig. 3.10. Thermal effects of gas accumulations: (a) vertical bulk thermal conduc- tivities, (b) isotherm bending If the pore filling is a mixture of several (fluid or solid) phases then the geometrical average of the phase values is used. For water, liquid petroleum and gas it is λp = (λw)Sw (λo)So (λg)Sg (3.16) where λp is the pore fluid thermal conductivity, λw, λo, λg are the thermal conductivities of water, oil, and gas phases and Sw, So, Sg are water, oil, and gas saturations. The bulk thermal conductivity is obtained by averaging the rock matrix and pore values with the geometrical average λ = λr (1−φ) λp φ . (3.17)
- 129. 116 3 Heat Flow Analysis A better but more complicated mixing rule is proposed by Buntebarth and Schopper (1998) based on mixing rules for spherical voids in a matrix. λ = λr 1 − Eφ 1 + αEφ with E = 1 − Z 1 + αZ and Z = λp λr . (3.18) The authors propose a value of α = 5 for water. This yields the following simplified law: λ = λr + 5λp + φ(λp − λr) λr + 5λp + 5φ(λp − λr) . (3.19) The whole procedure of calculating bulk conductivity values is summarized in Fig. 3.11. Vertical rock values Shale Sandstone Siltstone Compaction correction Temperature correction Vertical rock matrix value mixed with harmonic or geometric average Horizontal rock matrix value mixed with arithmetic or geometric average Vertical bulk value mixed with Buntebarth law or geometric average Horizontal bulk value mixed with Buntebarth law or geometric average Pore value mixed with geometrical averaging Fluid phase values Water Oil Gas Horizontal rock values Shale Sandstone Siltstone Fig. 3.11. Mixing of bulk thermal conductivities. The compaction and temperature corrections are described, e.g. with equations (3.11), (3.12), and (3.10) 3.4 Specific Heat Capacity The heat capacity at constant pressure Cp is the ratio of a small (infinitesimal) amount of heat ΔQ absorbed from a body which increases the temperature by ΔT Cp = ΔQ ΔT p . (3.20)
- 130. 3.4 Specific Heat Capacity 117 The specific heat capacity or specific heat is defined as heat capacity per mass c = Cp/m. The unit of specific heat is J/kg/K. The specific heat of a rock sample is measured by determining the temperature changes and the corresponding amount of heat entering or leaving the sample. The specific heat capacity is therefore the storage capacity for heat energy per unit mass. The ratio of the heat capacity and the thermal conductivity is a measure of the transient effect. Heat capacity also controls the magnitude of convection as it determines how much of the stored heat can be moved together with a moving body. The specific heat capacity is a volumetric type property (Sec. 8.3). Rock and fluid component values are mixed arithmetically. 3.4.1 Rock and Mineral Functions Specific heat capacities depend on temperature. The 20◦ C values c20 for min- erals and some standard lithotypes are tabulated in App. E. Waples Model The temperature dependency of the heat capacity for any mineral, lithology, or rock value except kerogen and coal can be calculated using the following equation, which has been adopted from Waples and Waples (2004a) and is shown in Fig. 3.12.a. c(T) = c20 (0.953+2.29×10−3 T −2.835×10−6 T2 +1.191×10−9 T3 ) (3.21) where c20 is the heat capacity at 20◦ C and the temperature is given in ◦ C. Waples and Waples (2004a) also proposed a special function for heat capacity of kerogen and coal as follows: c(T)[J/kg/m] = 1214.3 + 6.2657 T − 0.12345 T2 + 1.7165 × 10−3 T3 − 1.1491 × 10−5 T4 + 3.5686 × 10−8 T5 − 4.1208 × 10−11 T6 . (3.22) Linear Dependency Model The temperature dependency can be approximated with a simple linear func- tion. c(T) = c20 + cT (T − 20◦ C) (3.23) where c20 is the heat capacity at 20◦ C, cT is the heat capacity increase per degree temperature. Default lithological values for c20 and cT are tabulated in App. E for a temperature interval from −20◦ C to 300◦ C.
- 131. 118 3 Heat Flow Analysis a) b) 0 50 100 150 200 250 300 350 0 500 1000 1500 2000 Temperature in °C Heat Capacity in J/kg/K 0 50 100 150 200 250 300 350 0 1000 2000 3000 4000 5000 Temperature in °C Heat Capacity in J/kg/K Water Coal Gas Hydrate Oil Gas Ice Fig. 3.12. Heat capacity for: (a) rocks, (b) coal and fluids 3.4.2 Pore Fluid Functions Somerton (1992) developed equations for the specific heat capacity of pure water cw as a function of temperature. For a relatively constant density the heat capacity decreases linearly to 290◦ C followed by a strong decrease at higher temperatures according to ρ cw = ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ 4245 − 1.841 T for T 290◦ C 3703 exp[−0.00481 (T − 290)+ 2.34 × 10−4 (T − 290)2 ] for T 290◦ C (3.24) with cw in J/m/K, ρ in kg/m3 and T in ◦ C. The above described temperature dependent function is shown in Fig. 3.12.b assuming a constant water density of 1040 kg/m3 . According to Kobranova (1989) the specific heat of saline water is slightly lower than that of pure water. Gambill (1957) published the equation below for the specific heat capacity of oil co as a function of temperature and density: co = (1684 + 3.389 T) √ ρ (3.25) with co in J/m/K, ρ in kg/m3 , and T in ◦ C. It is satisfactory to use a value for the specific heat capacity of natural gas for all compositions at all tempera- tures and pressures. A reasonable value is 3250 J/kg/K (Waples and Waples, 2004b). Waples also proposed the following equation for the heat capacity of gas hydrates chin the same paper. ch = 2097 + 7.235 T + 0.0199 T2 (3.26) with ch in J/m/K and T in ◦ C. The specific heat capacity of ice (2115 J/kg/K) is about half that of liquid water for temperatures around 0 ◦ C. The pore and bulk values of heat capacity values are mixed arithmetically.
- 132. 3.6 Three Dimensional Heat Flow Equation 119 3.5 Radiogenic Heat Some minerals and rocks contain traces of the radioactive elements uranium (U), thorium (Th) and potassium (K) which are additional heat sources (3.5). Measurements and data catalogs are sparse, but D. Waples developed a data base for most rocks and minerals based on modeling experience and some literature data (App. E). These values should be used carefully especially since the effects on heat flow calculations are tremendous as was discussed in Sec. 3.2.1. Radiogenic heat values for lithologies are given as rock matrix values and are converted to bulk values during simulation by multiplying by (1 − φ). They can be derived from the following data sources: • uranium, thorium and potassium concentrations from spectral gamma ray measurements. These values are bulk values with a core sample porosity φc. One has to use Rybachs law (3.5) and divide by (1 − φc) to get the corresponding matrix heat flow production rate Qr. • Gamma Ray API values. Buecker and Rybach (1996) proposed the fol- lowing law to convert Gamma Ray APIs to matrix heat flow production values: Qr[μW/m3 ] = 0.0158 (API − 0.8) . (3.27) Concentrations of radioactive elements are present-day values U0, T0, K0. Paleo-values are higher corresponding to their half-lives as follows (with t in My). U = U0 (1 + 2.77 × 10−4 t − 7.82 × 10−8 t2 + 4.53 × 10−12 t3 ) Th = Th0 exp(0.00005 t) K = K0 exp(0.000555 t) . (3.28) Uranium consists of two isotopes having different half–lives. The sum of the two exponential functions is approximated by a third order polynomial. The time correction is small on geological time scales. 3.6 Three Dimensional Heat Flow Equation In 1D models, heat flows vertically upward from base to top. In multi– dimensional problems, heat flow can also laterally divert to follow layers of high thermal conductivity. The formulation of the multi-dimensional heat flow problem yields a transport–type differential equation with temperature as the field variable and heat flow as the corresponding flow variable. The heat transport equation is based on energy balances, which means that the tem- perature induced internal energy change in a volume element is equal to the heat conducted into or out of the volume element plus the heat transferred by convection plus radiogenic heat production. It is
- 133. 120 3 Heat Flow Analysis ρc ∂T ∂t − ∇ · λ · ∇T = ρpcp∇ · (vp T) + Qr (3.29) where λ, ρ, c are the bulk thermal conductivity tensor, bulk density, and bulk specific heat capacity, vp, ρp, cp are the pore fluid velocity vectors, density, and specific heat capacity and Qr is the bulk radioactive heat production. The full heat flow problem with boundary conditions is shown in Fig. 3.13. The four main terms in the heat flow equation describe the transient effect, heat conduction, heat convection, and the influence of heat sources, respec- tively. Two material parameters control the magnitude of the effects: thermal conductivity for heat conduction and heat capacity for transient effects and convection. The same temperatures for rock and pore fluid are assumed. For fast moving pore fluids, different temperatures have to be considered instead. Upper boundary: surface temperature or sediment water interface temperature T T s swi Ts Tswi Ts Fluid flow v Igneous Intrusions Tint Sediment 3 Sediment 2 Sediment 1 Lower boundary: basal heat flow q Side boundary: grad = 0 T Water l1 1 , c l2 2 , c l3 3 , c Fig. 3.13. Boundary value problem for the heat flow analysis Temperatures or heat flow values have to be defined at all model bound- aries. Common thermal boundary conditions are surface temperature (on- shore) or sediment–water–interface (SWI) temperature (offshore) at the top of the sediments T = Tswi, basal heat flow at the sediment base q = qb and no heat flow n · ∇T = 0 at the basin sides (Fig. 3.14.a). All boundary condi- tions have to be defined through geological time as paleo SWI temperatures and paleo heat flow trends, respectively. The lower boundary condition can alternatively be defined partially or completely with fixed base temperatures (Fig. 3.14.b and c). A deep isotherm map for the definition of the lower ther-
- 134. 3.6 Three Dimensional Heat Flow Equation 121 mal boundary can also be applied. Then, the model is subdivided into two domains and each of them is separately solved (Fig. 3.14.d). Tb Tswi a) Tswi Tswi Tswi qb Tiso Tb qb qb qb Submodel 1 d) c) b) Submodel 2 Fig. 3.14. Various types of boundary conditions Three dimensional effects are important when large variations of the ther- mal conductivities occur. Salt has much higher conductivities than most other sediments. Thus, salt domes bundle heat flow as shown in Fig. 3.15.a. The preservation of the total energy requires that the heat flow adjacent to the dome has a corresponding lower value. The surface heat flow reflects this effect as well. The corresponding isotherms (Fig. 3.15.b.) bend down at the base of the salt due to higher salt conductivities. They also form a bow upwards at the top of the salt dome due to the higher heat flow values which is a typical 3D-effect and cannot be found in multi–1D models. Calculated temperature and heat flow distributions of a multi–dimensional model are shown in Figs. 3.16 and 3.17. Thermal conductivities of the clastic rocks increase with depth according to the lower content of pore water, while salt domes are zones of high thermal conductivities. Shales correlate with very high radioactive heat sources. Heat flow from base to top generally increases due to radioactive heat production and it is concentrated along the high con- ductive salt domes and causes the bending of the isotherms as previously discussed. The higher surface heat flow above the salt domes can be clearly observed in the 3D–model. In such complex situations multi–1D models fail and show errors of more than 50◦ C (Fig. 3.17). Some multi–dimensional schematic models can be solved analytically and the results can be used for “benchmarking” of simulation programs. The so- lutions of the following problems are given and discussed in the Appendix: in- fluence of radiogenic heat production on steady state temperature (App. F.1),
- 135. 122 3 Heat Flow Analysis Labels with heat flow values (in mW/m ) Isolines of temperature (in Celsius) 2 100 o 120 o 60 o 40 o 80 o 55 55 70 48 48 108 72 Isotherm Bend b) a) meter meter Depth in meter Fig. 3.15. Heat flow through a salt dome: the basal heat flow is 60 mW/m2 along the entire sediment base. The actual heat flow within the sediments increases to 108 mW/m2 within the dome. Note that heat flows near but outside the dome are lower than the basal heat flow influence of lateral basal heat jump on temperature (App. F.2), influence of SWI temperature jump on temperature profiles (App. F.3), the steady state temperature field for a two block model (App. F.4), the transient temper- ature field of a model with basal heat flow jump (App. F.5), the transient temperature field of a model with SWI temperature jump (App. F.6). 3.6.1 Heat Convection Heat convection is related to moving masses, solid and liquid. In sediments, heat convection is mainly caused by water flow. Water velocities are calculated when solving the pressure–compaction equations and so the convection term couples heat and fluid flow calculations. The amount of heat ΔQ transfered between two points with a temperature difference ΔT for a moving water mass mw is ΔQ = cw mw ΔT = cw ρw Vw ΔT (3.30) where cw, ρw and Vw are the specific heat capacity, the density, and trans- ported volume of the water. The corresponding convective heat flow qv of water moving trough a cell with a length l, a slice–plane A, a bulk volume V = A l, and a velocity vw is as follows: qv = ΔQ AΔt = cw ρw φ V ΔT AΔt = cw ρwvw ΔT . (3.31) The above equation yields a convective heat flow value of qv = 0.04 mW/m2 with a water velocity of 1 mm/y, a temperature difference of 1◦ C between the flow boundaries, a porosity φ = 0.3, a heat capacity of cw = 4186 J/kg/K, and a water density ρw = 1035 kg/m3 . Compaction and overpressure driven water velocities are much smaller. Hence compaction driven convection can
- 136. 3.6 Three Dimensional Heat Flow Equation 123 Celsius Low SWI temperature in deep water Base of salt isotherm bow High vertical temperature gradient in a low conductive area Low vertical temperature gradient in a high conductive area mW/m 3 low moderate high very high not radioactive lowly radioactive highly radioactive Basement 20% Sand 40% Shale 40% Carb 5% Sand 80% Shale 15% Carb Shale Marl Salt Sandstone 50% Sand 40% Shale 10% Carb 25% Sand 60% Shale 15% Carb Lithology Radioactive Heat Production Temperature Top of salt isotherm bow W/m/K Thermal Conductivity (vertical) Fig. 3.16. Multi–dimensional heat flow analysis part I, cross-section from Cam- pos Basin, Brazil. The present day temperature distribution shows several multi– dimensional effects like isotherm bending around salt, SWI temperature variation in deep water and temperature gradient variations according to the thermal conduc- tivities
- 137. 124 3 Heat Flow Analysis mW/m 2 Increased surface heat flow above salt domes Low heat flow through shallow water areas Low heat flow next to salt Multi 1D Model Present Day Heat Flow (3D Model) Salt Domes in a 3D Model Very high heat flow through salt bodies Surface Heat Flow in mW/m 2 75 58 37 Celsius Temperatures are very different from 2D/3D solutions in some areas Fig. 3.17. Multi–dimensional heat flow analysis part II, multi–1D temperature model, multi–dimensional heat flow distribution at present day and surface heat flow anomalies above salt domes for a 3D model
- 138. 3.6 Three Dimensional Heat Flow Equation 125 be neglected in the thermal budget. Topographically driven aquifer flow and flow of hot water through high permeable fractures and faults can have higher flow velocities and must for that reason be taken into account. 3.6.2 Magmatic Intrusions Magmatic intrusions can have substantial effects on paleo–temperatures and all thermal calibration parameters. Although the duration of such events is relatively short, extremely high temperatures can trigger rapid chemical re- actions in the adjacent environment. Igneous intrusions are modeled with the magmatic temperature as inner boundary condition at the location and time of the intrusion. In subsequent time steps, the temperature decreases in both the intrusion and the surrounding layers. Then, hot liquid magma crystal- lizes to solid rock. The related crystallization heat is important and has to be taken into account in the heat balance. The principal processes together with some typical values according to Delaney (1988) are shown in Fig. 3.18. Here, it is necessary to switch the lithological properties of the intrusion volume elements twice, at the time of intrusion and again at the time of solidification. The temperature development during cooling in a simple example is shown in Fig. 3.19, where effects on temperature can still be seen 100,000 years after the time of intrusion. Older intrusions can be recognized in vitrinite reflectance peaks in the vicinity of the intrusions. The use of smaller time steps after the time of intrusion is necessary. Time steps of 500 y, 1 000 y, 2 000 y, 5 000 y, 10 000 y, 20 000 y, 50 000 y, 100 000 y yield suitable results. Liquid Magma = 0.7 kg/kcal/K λ = 2.0 W/m/K = 1000 kg/m c r 3 Any Solid Lithology Solid Basalt λ = 1.95 W/m/K = 2750 kg/m c = 0.22 kg/kcal/K r 3 Intrusion Temperature (1000 C) o Solidus Temperature (950 C) Crystallization Heat (700 MJ/m ) o 3 Displacement by Magma Crystallisation Fig. 3.18. Intrusion model and default values from Delaney (1988) 3.6.3 Permafrost Modeling permafrost requires the introduction of permafrost lithologies with ice in the pores instead of water. Furthermore, additional heat sources and sinks for ice solidification and melting have to be taken into account. The trigger parameter for converting a lithology into a permafrost lithology is a temperature of 0.7 ◦ C. Hence, temperatures below permafrost are much lower
- 139. 126 3 Heat Flow Analysis At the time of intrusion After 1000 years After 50000 years After 5000 years 50 C o 50 C o 50 C o 50 C o 100 C o 100 C o 150 C o 150 C o 150 C o 648 C o 1000 C o 180 C o 347 C o 100 C o 100 C o m m m m m m m m Fig. 3.19. Temperature development around an intrusion of size 300 m × 3000 m compared to ice–free periods (Fig. 3.20). The high thermal conductivity of ice λ = 2.33 W/m/K compared to liquid water yields low temperature gradients (Fig. 3.20) and supports the cooling effect. The cooling is further increased by the solidification heat of ice Qs = 335 J/kg, which is removed from the permafrost environments. The specific heat capacity of ice (0.502 J/kg/m) is relatively small compared to water. Modeling the sequence of interglacial periods such as in the Pleistocene, requires the use of very small time steps of about 1000 years to get an appropri- ate solution for the fluctuations in surface temperature and the corresponding surface heat flow (Fig. 3.21). The surface heat flow peaks generally coincide with the steep changes in surface temperatures, which especially occurred during the change from cold to warm periods. Ice loading can also be simulated in permafrost periods. Then, an upper- most layer is introduced with the thermal and mechanical properties of pure ice. This yields special characteristics of pore- and lithostatic pressure curves as shown in Fig. 3.21. 3.7 SWI Temperatures The sediment–water–interface temperature Tswi or bottom–water–tempera- ture is the upper boundary for the heat flow problem. It can be determined with estimated paleo mean surface or air temperatures Ts and corrections for water depths. The annual mean ground surface temperature is primarily ob- tained from mean air temperatures (www.worldclimate.com), which depends
- 140. 3.7 SWI Temperatures 127 Lithology Thermal Conductivity (vertical) Temperature low moderate high very high Permafrost Area Celsius W/m/K Low surface temperatures and low temperature gradients in permafrost areas 0.625 My Present Day High surface temperatures and high temperature gradients in ice-free periods Isotherm bows dueto salt domes Clastic Sediments, Evaporites Marly Shale Shale, Silty Shale, Calc. Shale Sandy Silt Sandstone, Calc. Sandstone Chalk, Evap. Limestone Salt Fig. 3.20. Heat flow analysis in a permafrost area, sample cross–section from the Lower–Saxony Basin, Germany (Grassmann et al., 2005; Delisle et al., 2007). The model has a 200 m thick permafrost layer with very high conductivities at 0.625 My. The resulting temperature gradients differ significantly from the ice free present day temperatures. The thermal conductivities in the salt are generally very high
- 141. 128 3 Heat Flow Analysis Glacier Top-Ice Pressure Depth 0 Hydrostatic Pressure Pore Pressure Lithostatic Pressure Over- pressure Effective Stress Excess Hydraulic Glacier Potential Sea Level 0 0.2 0.4 0.6 0.8 1 35 40 45 50 55 60 65 70 Geological Time in My Heat Flow in mW/m 2 0 0.2 0.4 0.6 0.8 1 -10 -5 0 5 10 15 Geological Time in My Surface Temperature in Celsius Fig. 3.21. Heat and pressure analysis in a Pleistocene sample cross–section of Fig. 3.20. The surface heat flow is low when the surface temperature increases and visa versa. The glacier causes an excess hydraulic potential on the top glacier surface on latitude. Beardsmore and Cull (2001) proposed the following latitude and water depth dependent equation for the present day sediment–water–interface temperature with an error bar of 2 ◦ C. ln(Tswi − Tf ) = a + b ln z , Tf = −1.90 − 7.64 × 10−4 z , a = 4.63 + 8.84 × 10−4 L − 7.24 × 10−4 L2 , b = −0.32 + 1.04 × 10−4 L − 7.08 × 10−5 L2 (3.32) where Tf is the freezing temperature in ◦ C, z is the water depth in m, and L is the latitude in degree. The corresponding SWI temperature versus depth curves are shown in Fig. 3.22. An average air surface temperature history is given in Fig. 3.23 for different latitudes (Wygrala, 1989). Knowledge of the paleo latitude changes through geological time is therefore necessary to be able to derive paleo surface tem- peratures. This is shown in Fig. 3.24 for several continental areas. The derivation of the paleo–SWI temperatures from average surface tem- perature is very difficult to estimate. Wygrala (1989) proposed a decrease of 1.5◦ C per 100 m in shallow water. The temperature in water deeper than 400 m is primarily controlled by the coldest arctic water temperatures Tn, which are presently affected by polar glaciations. A linear interpolation between the fol- lowing three fixed points is a common approximation for a water depth based SWI temperature correction.
- 142. 3.8 Crustal Models for Basal Heat Flow Prediction 129 Fig. 3.22. Present day sediment–water– interface curves dependent on latitude and depth according to equation (3.32) after Beardsmore and Cull (2001) -5 0 5 10 15 20 25 0 1 2 3 4 SWI-Temperature in Celsius Depth in km 1 2 3 4 5 Latitude 1..0° 2..20° 3..40° 4..60° 5..80° Surface temperature through time for Northern Europe at 70 degrees latitude 350 300 250 200 150 100 50 Equator 5.00 6.50 8.00 9.50 11.00 12.50 14.00 15.50 17.00 18.50 20.00 21.50 23.00 24.50 26.00 27.50 29.00 30.50 South North Degree Time in My Temperature in °C Arctic temperature Tn 0 90.00 74.00 58.00 42.00 26.00 10.00 10.00 26.00 42.00 58.00 74.00 90.00 Fig. 3.23. Paleo–surface temperatures Tswi(0 m) = Ts, Tswi(200 m) = Ts − 3◦ C, Tswi(600 m) = Tn . (3.33) The average arctic temperature is about 4 ◦ C at present, but it was much higher in the past (Fig. 3.23). 3.8 Crustal Models for Basal Heat Flow Prediction Crustal models describe the mechanical and thermal processes of plate tecton- ics. In basin modeling, they are used to estimate the basal sediment heat flow as the lower boundary condition in the heat flow analysis (Fig. 3.1). Another
- 143. 130 3 Heat Flow Analysis -350 -300 -250 -200 -150 -100 -50 0 -10 0 10 20 30 40 50 60 Geological Time in My Delta Latitude in Degree -350 -300 -250 -200 -150 -100 -50 0 -40 -20 0 20 40 60 80 Geological Time in My Delta Latitude in Degree Europe North America South Africa South America West Asia East Asia South Asia India Australia Antarctica Mediterranean, North Africa, Arabia Fig. 3.24. Paleo latitude variations of some continental locations interesting result of plate tectonic models is subsidence through time, which can be compared with sedimentation rates and paleo–water depths. The asthenosphere, upper mantle, oceanic crust, lower and upper continen- tal crust, and sediments are usually distinguished based on differences in their chemical compositions and mechanical properties as illustrated in Fig. 3.25. Mechanical behavior is the determining factor to distinguish between the solid lithosphere and the highly viscous (or pseudo–liquid) asthenosphere, which comprises the upper 250 km of the lower mantle. The lithosphere is further divided into the brittle upper crust and the ductile lower crust and upper mantle. Thus, faults are mainly formed in the upper crust during stretching of the lithosphere. The classification between mantle and crust is based on chemical com- position: mantle material mainly consists of mafic silicates, oceanic crust of mafic minerals and feldspar and continental crust of felsic silicates. There is evidence to assume that the entire mantle has a common convection system
- 144. 3.8 Crustal Models for Basal Heat Flow Prediction 131 Upper continental crust Lower continental crust Upper Mantle Oceanic crust Upper mantle Asthenosphere Asthenosphere Lithosphere (solid, heat conduction) Astenosphere (pseudo-liquid, heat convection) Upper Crust (brittle) =2800 kg/m r0 3 Lower Crust (ductile) r0=2800 kg/m 3 Upper Mantle (ductile) r0=3300 kg/m 3 Lower Mantle (pseudo-liquid) r0=3300 kg/m 3 Pseudo-Liquid-Solid Interface, T=1333 C o Sediment-Water Interface T=4 C o Sediments =2200 kg/m r0 3 Moho Base Sediment Basal Heat Flow Sea Water r0=1060 kg/m 3 Crustal Stretching bcrust Mantle Stretching bmantle a) b) Circulation system: oceanic lithosphere and asthenosphere Crustal compression or stretching bcrust Mantle compression or stretching bmantle Fig. 3.25. Crust and mantle layer definitions. All rock densities are temperature dependent. The densities values ρ0 are rock densities at surface conditions with flow rates of about 10 to 20 cm/y. Oceanic crust, upper mantle and the asthenosphere have similar chemical compositions, since they build a closed circuit with constant formation of oceanic crust material at the mid–ocean ridges and destruction of it in the subduction zones. This circulation system also moves the continental crust pieces causing breakup, stretching, compres- sion and overthrusting. The interface between the upper and lower mantle is the solid to pseudo–liquid boundary with a base lithosphere temperature of Ta = 1100 − 1350 ◦ C. (Parsons and Sclater, 1977). The value of Ta = 1333 ◦ C is used in most publications (McKenzie, 1978), which corresponds to three quarters of the pyrolite melting temperature. The postulate of a fixed and well–known temperature at the base of the lithosphere, is an important as- sumption in crustal heat flow models. Sediments are deposited in accommodation spaces as a result of litho- spheric stretching and compression with usually different stretching velocities in the lithospheric layers and differences in deformation types. Crustal and mantle layers further differ in densities depending on their compositions and temperatures. Thus, a change in layer thicknesses, affects the weight of the total lithospheric column leading to subsidence with sedimentation on top or uplift with erosion. The depth of the top asthenosphere temperature isosur-
- 145. 132 3 Heat Flow Analysis face and the thermal conductivities of the lithospheric layers primarily control the upward heat flow. In summary, a coupled model of lithospheric stretching, heat flow, and subsidence is necessary to obtain the base sediment heat flow through geologic time. Plate tectonics yield different stretching, displacement, folding, and sub- duction processes especially on plate margins, which are related to different phases of basin development. Generalized models have been developed for a stable lithosphere in intra–plate locations, subduction zones at convergent margins and extensional rift–drift phases at divergent margins (Beardsmore and Cull, 2001). Models of extensional rift basins are established since they can easily be quantified and because they can be applied to many petroleum provinces. The most thoroughly investigated and applied model is the model of uniform stretching, also known as the McKenzie model. Uniform Stretching Model This famous model was originally proposed by McKenzie (1978) and it is still frequently used with some minor improvements in basin analysis. It is based on two different periods: an initial stretching phase with constant thinning of the crust and upper mantle and a cooling phase with near or full restoration of the original thickness of the lithosphere. Here uniform stretching is not an uniform decrease of the layer thickness, instead it is a geological time–constant and linearly with depth increasing velocity field. Hence, the base of the lithosphere moves vertically at maximum speed, while the top of the lithosphere is fixed (Fig. 3.26). It also means that the speed of the base lithosphere decreases during uplift, which causes a slowdown in the thinning process. The total thinning of the layers is described with a stretching factor β, which is the ratio of the initial to the final thickness. For the formulation of the heat transfer problem, it is more important to know the velocity vector of the moving layer element during stretching. The vertical component of the velocity vz is linearly decreasing from bottom to top. The maximum velocity vm is the velocity of the rock at the base of the lithosphere at the beginning of the stretching. The relationship between the stretching factor β, the stretching time ts and the maximum velocity vm is ts = h0 h0/β h0 z vm dz = h0 log β vm (3.34) where h0 is the initial thickness if the lithosphere. The vertical component of the velocity at any depth vz(z) of the litho- sphere during the entire stretching phase is vz(z) = z h0 vm . (3.35)
- 146. 3.8 Crustal Models for Basal Heat Flow Prediction 133 hc hm Asthenosphere Crust Upper mantle Asthenosphere Upper mantle Crust hc hm hc/bc h h h m c c c /b + - Stretching Cooling Geologic time Depth Convection velocity increases linearly with depth Tswi= 0..20°C Water Tb = 1333°C hc/bc hm/bm hc/bc hc/bc hm/bm h h h m c c c /b + - Fig. 3.26. McKenzie model with subsidence due to hydrostatic and isostatic com- pensation. The stretching velocity is constant in time and increases linearly with depth. This causes thinning of the crust and upper mantle during stretching. The original lithosphere thickness is restored during cooling In this volume, it is usually dealt with two stretching factors for crust βc and upper mantle βm, and multi–1D heat conduction with radioactive heat production in the upper crust is considered.2 The model was originally worked out for symmetrical rifting, but it can also be applied to asymmetrical rifting and even compression, when each ver- tical lithospheric column experiences a phase of uniform thinning, which could be caused by stretching or sliding on detachment faults. Then, the stretching factors vary asymmetrically depending on their location as shown in Fig. 3.27 for the cases of pure shear, simple shear and simple shear–pure shear, respec- tively. In the pure shear model, the vertical compressional deformation is equal to the horizontal extensional deformation as explained in Chap. 2. The simple shear model comprises only shear components and in the considered case, one large low angle detachment fault cutting through the entire lithosphere. Instead of extending along this detachment fault, the upper plate slips along the detachment surface. The simple shear–pure shear model considers simple shear in the crust and pure shear in the mantle. The corresponding stretching factor distributions along the section are mixtures of both end member models. The most realistic models are mixtures of the three stretching types, but some 2 The original McKenzie model deals with one stretching factor only. Multiple stretching factors were introduced by Hellinger and Sclater (1983); Royden and Keen (1980)
- 147. 134 3 Heat Flow Analysis Crust Upper Mantle Lower Mantle (a) Pure Shear Stretching Factor 3 2 1 b-crust b-mantle Stretching Factor 3 2 1 b-crust b-mantle Stretching Factor 3 2 1 b-crust b-mantle (b) Simple Shear (c) Simple Shear-Pure Shear Horizontal Location Horizontal Location Horizontal Location Fig. 3.27. Rift basins, according to Allen and Allen (2005): (a) pure shear, (b) simple shear, (c) simple shear–pure shear can be approximated by the uniform stretching model when the stretching at each location can be described by only two single stretching parameters. 3.8.1 The Principle of Isostasy The principle of hydrostatic isostasy states that the weight of all overbur- den material (lithosphere plus water depth) measured from a reference depth in the asthenosphere is constant. There is gravitational equilibrium between lithosphere and asthenosphere and the elevation depends on the underlying lithosphere column. An increase in lithospheric weight will therefore yield fur- ther subsidence so that the additional weight is compensated by the load of lighter water on top and less heavier asthenosphere at the base. ρwghw + n i=0 ρsighsi + ρcughcu + ρclghcl + ρmghm + ρagha = constant (3.36) with the subscript indexes w, cu, cl, m, a for water, upper crust, lower crust, upper mantle, and asthenosphere, respectively and si for the ith sediment layer. This equation can be used to calculate water depths or mountain heights
- 148. 3.8 Crustal Models for Basal Heat Flow Prediction 135 from crustal layer thicknesses. The principle is illustrated for two very simple two layer (crust and mantle) models in Fig. 3.28: the Airy and the Platt model. The Airy model supposes a constant density for the entire crust. Thus, the mountain height or water depth is a simple function of the total crust thick- ness. Hence, the high mountains are above thick crust and large water depths suggest a thin crust below. The Platt model presumes a crust of varying den- sity at the same depth level of the asthenosphere. Thus, the surface elevation is a function of the crustal density only: the higher the mountain the lighter the crust below. These examples illustrate the principle of a hydrostatic litho- sphere: each vertical column of lithosphere is able to move independently of the adjacent column to balance itself via its own weight. rc a) b) Crust rm Mantle h1 b1 Sea level Water h2 b2 Onshore (rm- ) r = r c 1 c 1 b h Offshore (rm- ) - ) r = (r r c c w b h 2 2 rw c) d) rm rc r1 r2 h1 h2 Crust Mantle Onshore (rc- ) r = r 1 b h 1 1 1 Offshore (rc- ) - ) r = (r r 2 2 2 b h c w Water b1 b2 Fig. 3.28. Isostatic compensations: (a) Airy compensation. (b) Platt compensation. (c) Hydrostatic isostasy. (d) Flexural isostasy In reality, there is an influence from the connected areas, which is de- scribed as flexural compensation, but which is not considered here. However, the weakness of the above models is the assumption of constant density with depth, since higher temperatures lower the density by thermal expansion ac- cording to ρ(T) = ρ0 [1 − α(T − T0)] (3.37) with the linear expansion factor α = 3.28 × 10−5 /◦ C (McKenzie, 1978) and a reference density ρ0 for surface temperature T0 = 20 ◦ C. Another conse- quence of this equation is that cooling of the lithosphere causes subsidence and warming leads to uplift. Uniform Stretching Model The hydrostatic equation of isostasy applied to the uniform stretching model yields the following total mass per unit area for a column of water, crust, upper mantle and asthenosphere above a reference depth in the asthenosphere at any time (Fig. 3.26).
- 149. 136 3 Heat Flow Analysis m = ρw hw + ρc0 dc hw [1 − αcT(x)]dx + ρm0 dm dc [1 − αmT(x)]dx + ρm0[1 − αm(Ta − Tswi)] ha (3.38) where αc, αm are the thermal expansions of the crust and mantle, and dc, dm are the depths of base crust and mantle, and hw, ha are the thicknesses of water and asthenosphere. The tectonic water depths after instantaneous stretching hw1 and after inifinite cooling hw2 can be analytically calculated with the assumptions of no crustal radioactive heat production, a unique stretching factor β for crust and mantle, equal and constant thermal properties of the crust and mantle. and an linearly increase of the temperature with depth(Jarvis and McKenzie, 1980). hw1 = (hm+hc)[(ρm0−ρc0) hc hm+hc (1−αTa hc 2hm+2hc )− αTaρm0 2 ](1− 1 β ) ρm0(1−αTa)−ρw , hw2 = (ρm0−ρc0)hc ρm0(1−αTa)−ρw 1 − 1 β − αTahc 2hm+2hc 1 − 1 β2 . (3.39) The subsidence during stretching is related to an inflow of additional heavy asthenospheric material, while additional subsidence is caused by the cooling of the entire column. Again it should be noted, that the above premises, espe- cially the assumption of instantaneous stretching as seen in equation (3.39) are drastic simplifications. More comprehensive equations should be used instead. Such subsidence curves through geological time are illustrated in Fig. 3.29 for various stretching factors. In Figs. 3.30 and 3.31 radioactive heat production in the crust is taken into account. The subsidence after stretching hw1 can become negative (uplift), when crustal stretching is very small compared to mantle stretching, as shown in the example in Fig. 3.27.a at the margin of the pure shear model and in Fig. 3.32.b. The subsidence is much larger, when the weight of the sediments is taken into account, in contrast to pure water filled basins. Total subsidence, which is the real subsidence with sediments is different from the tectonic subsidence, which is the theoretical subsidence for water fill only. The relation between tectonic subsidence hw and total subsidence ht is as follows: ht = ρa − ρw ρa hw + 1 ρa n i=1 ρsi hsi (3.40) where ρsi and hsi are the density and thickness of the i-th sediment layer. Sometimes, a basement is introduced between the upper crust and the sediments, which is the sediment package before stretching. It has to be de- termined whether the total and tectonic subsidence is then understood as the top or bottom basement.
- 150. 3.8 Crustal Models for Basal Heat Flow Prediction 137 a) b) 0 20 40 60 80 100 0 1 2 3 4 5 Geological Time in My Subsidence in km b=2 3 4 5 6 0 20 40 60 80 100 0 20 40 60 80 100 120 Geological Time in My Heat Flow in mW/m 2 3 4 5 b=6 2 Stretching Cooling Stretching Cooling Fig. 3.29. McKenzie model: heat flow and tectonic subsidence for several stretching factors β = βc = βm with κ = 0.80410−6 m2 /s−1 , Tswi = 0◦ C, Tb = 1333◦ C, hc = 30 km, hm = 95 km and ts = 50 My Fig. 3.30. Effect of constant radioactive heat production in the crust. The model parameters are the same as in Fig. 3.29 with the stretching factor β = 4 0 20 40 60 80 100 0 20 40 60 80 100 120 Geological Time in My Heat Flow in mW/m 2 1 3 4 1..Qr=0 2..Qr=1 mW/m3 3..Qr=2 mW/m3 4..Qr=3 mW/m3 Stretching Cooling 2 The principle of isostasy is universal, it is not restricted to pure shear processes only, e.g. basin subsidence and uplift can also be predicted in cases of underplating or other tectonic processes. 3.8.2 Heat Flow Models The application of the 3D heat flow equation (3.29) to crustal models re- quires assumptions for thermal properties, boundary conditions and convec- tion fields. The upper and lower boundary values are usually the sediment water interface and the top of the asthenosphere with the temperatures Tswi and Ta. Thermal conductivities and heat capacities of the upper crust depend on temperature as well as on rock composition. Radioactive heat production is
- 151. 138 3 Heat Flow Analysis a) b) 0 20 40 60 80 100 40 60 80 100 120 140 Geological Time in My Heat Flow in mW/m 2 2 3 4 5 b=6 0 20 40 60 80 100 0 1 2 3 4 5 6 Geological Time in My Subsidence in km b=2 3 4 5 6 Stretching Cooling Stretching Cooling Fig. 3.31. Effect of exponentially decreasing radioactive heat production in the crust. The model parameters are the same as in Fig. 3.29 with Qr0 = 2.5 μW/m3 and zh = 7 km known in the crust as an exponentially decreasing function with depth (Sclater et al., 1980; Allen and Allen, 2005). It can be expressed with the half–value depth zh which describes the depth at which the concentration of radioactive elements is half the value of the maximum heat production Qr0. Qr(z) = Qr0 2−z/zh . (3.41) The convection term in (3.29) should be used for the moving lithosphere with velocity v and can be expressed with stretching factors for the crust and mantle (3.34). The problem is often approximated with a multi–1D solu- tion, since horizontal crustal facies variations, together with extreme thermal conductivities, are assumed to be rather rare. λ ∂2 T ∂z2 − ρc ∂T ∂t + ρcvz ∂T ∂z + Qr = 0 . (3.42) The vertical velocity field vz(z) can be derived from mechanical models. The stretching velocity increases linearly in the crust and mantle correspond- ing to the stretching factors, according to (3.34) and (3.35). It is λ ∂2 T ∂z2 − ρc ∂T ∂t + ρc log β ts (h0 − z) ∂T ∂z + Qr = 0 (3.43) where h0 is the initial lithoshere thickness, and ts is the stretching time. The thermal diffusivity κ = λ/ρ/c is often given for the crust and mantle instead of the thermal conductivity λ. Note, that here Qr = 0 in the mantle, vz = 0 during cooling, and β is different for the crust and mantle. The assumptions of the McKenzie model allow a solution of the heat flow equation with analytical methods. This is not possible when modifications of
- 152. 3.8 Crustal Models for Basal Heat Flow Prediction 139 the model are taken into account such as the introduction of radioactive heat production rates for the upper crust, different and variable thermal proper- ties, and several stretching factors and stretching phases. Then, numerical integration methods such as finite difference methods can be applied. The upward and downward movement of the highest asthenosphere surface during stretching and cooling yields the typical heat flow peaks for rift basins. Their height and width depends on stretching factors and stretching duration times (Fig. 3.29). Cooling has already an effect during the stretching phase and lowers the heat flow peak in the case of long stretching times. Thus, the maximum peak occurs at instantaneous stretching. Without radioactivity, the heat flow declines to the initial value after in- finite cooling, as the original lithosphere thickness is restored. Radioactive heat production of the crust increases the total heat flow towards the sur- face, but decreases the relative peak height compared to the initial value, since radioactive heat production decreases with thinning of the crustal layer (Fig. 3.30). This usually yields lower present day heat flow than the original values. Radioactive heat production also results in about 20% higher subsi- dence curves (Fig. 3.31) compared to models without radioactive heat pro- duction (Fig. 3.29). Thus radioactive heat production must not be neglected. The upper mantle often has a higher stretching rate than the crust because it is more ductile. This also yields lower heat flow peaks and less subsidence compared to higher crustal thinning (Fig. 3.32). Heating the lithosphere with the highest rates at the beginning also has an uplift effect, which is usually balanced by the subsidence caused by crustal thinning. Fast and high mantle stretching and less crustal stretching allows the uplift effect to overcome sub- sidence, uplifting the basin when stretching starts (Fig. 3.32). Pure mantle stretching always results in uplift during stretching. The above approach can easily be extended to several phases of uniform stretching with multiple pairs of stretching factors for rifting and cooling periods. Each stretching factor applies the actual thickness of the lithosphere when the new stretching period begins, instead of the initial thickness of the lithosphere (Fig. 3.33). The linear velocity versus depth curve is independent of the total ini- tial depth, it only depends on the stretching factor and time as expressed in equation (3.34), but the velocity vz(dc) of the basal crust layer decreases exponentially in time, since it looses speed during uplift. Stretching factors are inconvenient for the description of any non–uniform stretching behavior. It is possible to work with velocity versus geological time functions instead, and the resulting heat flow equations can still be used in the above manner without any changes. 3.8.3 Workflow Crustal Preprocessing Crustal models are useful to predict tectonic subsidence and paleo–heat flow when stretching and cooling behavior occurs. Present or paleo–subsidence
- 153. 140 3 Heat Flow Analysis a) b) c) d) 0 20 40 60 80 100 0 20 40 60 80 100 120 Geological Time in My Heat Flow in mW/m 2 2 3 4 5 b =6 m 0 20 40 60 80 100 0 20 40 60 80 100 120 Geological Time in My Heat Flow in mW/m 2 2 3 4 5 0 20 40 60 80 100 -0.5 0 0.5 1 1.5 2 2.5 3 Geological Time in My Subsidence in km 4 5 6 0 20 40 60 80 100 -2.5 -2 -1.5 -1 -0.5 0 Geological Time in My Subsidence in km 4 5 6 b =6 m b =2 m b =2 m Cooling Stretching Cooling Stretching Cooling Cooling Stretching Stretching 3 3 Fig. 3.32. Effect of different stretching factors for crust βc and mantle βm: (a),(b) small crustal stretching βc = 2, (c),(d) no crustal stretching βc = 0 from input geometry and stratigraphy can be used “inversely” to determine the stretching factors. The corresponding paleo–heat flow maps can then be calculated afterwards. Fig. 3.34 illustrates the workflow for the calculation of paleo–heat flow maps from input geometry with calibrated stretching maps for crust and man- tle using the uniform stretching model as described in the previous sections. It is also illustrated in Fig. 3.35. The workflow starts with the extraction of the total paleo- and present day subsidence maps from the present day input model (Fig. 3.35.c). The corre- sponding back–stripping routine should also consider estimated paleo–water depth maps, decompaction and salt movement. Then, tectonic subsidence is calculated from total subsidence with the replacement of sediments by water (Eq. 3.40, Fig. 3.35.a). The main computing effort is then needed for inverting
- 154. 3.8 Crustal Models for Basal Heat Flow Prediction 141 a) b) 0 20 40 60 80 100 0 1 2 3 4 5 Geological Age in My Stretching Factor b c) d) 0 20 40 60 80 100 0 5 10 15 20 25 30 Geological Age in My Crustal Thickness in km 0 20 40 60 80 100 0 0.2 0.4 0.6 0.8 1 1.2 Geological Age in My Verlocity of Base Crust in m/Ty 0 0.5 1 1.5 0 5 10 15 20 25 30 Velocity in km/My Depth in km 0 0.5 1 1.5 0 5 10 15 20 25 30 Velocity in km/My Depth in km Uniform Stretching Phase I Uniform Stretching Phase II Cooling Phase Cooling Phase Phase I Phase II Fig. 3.33. Example with two uniform stretching periods, the initial crust thickness is 30 km: (a) Definition of the stretching and cooling periods. (b) The velocity of the base of the crust decreases exponentially during uplift. (c) The crustal thickness also decreases exponentially. (d) The time–constant velocity versus depth curve in the crust for the two stretching periods Calculation of total and tectonic subsidence Present day geometry maps Paleo geometry maps Stratigraphy Facies maps for lithology Stretching phase and times Crust and mantle initial geometry maps Crust and mantle thermal properties Inversion of tectonic subsidence into stretching factors Calculation of heat flow maps Paleo and present subsidence maps for events Stretching factor maps Paleo and present heat flow maps Paleo bathymetry maps Paleo bathymetry maps Paleo bathymetry maps Fig. 3.34. Workflow for (crustal) heat flow preprocessor
- 155. 142 3 Heat Flow Analysis a) b) 0 20 40 60 80 100 120 140 0 2 4 6 8 10 Geological Age in My Subsidence in km 1 2 3 0 20 40 60 80 100 120 140 0 20 40 60 80 100 120 Geological Age in My Heat Flow in mW/m 2 Tectonic Subsidence Map Present Day Heat Flow Map c) d) 42 40 38 38 36 34 e) f) 1.5 2.0 2.5 3.0 1.0 1.5 2.5 2.0 3.0 2.0 3.0 4.0 5.0 6.0 5.0 3.0 Stretching factor map of the crust Stretching factor map of the mantle 1..Total subsidence 2..Tectonic subsidence 3..Theoretical subsidence Stretching Cooling [mW/m2] [m] Fig. 3.35. Example from the Northern Campos basin for crustal heat flow analysis with a rift period of 132 − 113 My, hc = 35 km, hm = 95 km, Qr0 = 2.5μW/m3 , zh = 7 km: (a) Total and tectonic subsidence from input geometry and calculated theoretical subsidence after calibration at the location of the map midpoint. (b) Calculated heat flow at the basin midpoint with the calibrated stretching factors βc = 3.0, βm = 4.7. (c) Tectonic subsidence map from input geometry. (d) Calcu- lated present day heat flow map. (e),(f) Maps with calibrated stretching factors for the crust and mantle
- 156. 3.9 Heat Flow Calibration 143 the tectonic subsidence maps into stretching factors, since it is an inversion of the heat and mechanical McKenzie type equations 3.43, 3.38. Usually, unique rifting and cooling times and initial crustal and mantle thicknesses are used for the entire map. The only unknowns in the inversion step are two stretch- ing factors, which are calculated for each grid point in a multi–1D approach, so that the main output are two stretching maps for the crust and mantle (Fig. 3.35.e and f). The mantle map should be smoothed afterwards, if there is reason to assume high ductility. The inversion can be performed for example with the response surface method, which is explained in Chap. 7. For each gridpoint 10 to 100 runs are necessary with the method of nesting intervals, so that there are about one million 1D forward simulation runs. The final stretching maps for crust and mantle can then be used to calculate the paleo and present heat flow maps through time and to recalculate the paleo water depth maps from simulated subsidence values. This workflow can be extended to more than one rifting event or to de- fine and calibrate other unknowns of the model, such as the initial thickness of the crust. Three stretching maps for the upper and lower crust and the mantle can also be used instead of the presented two layer model, or the two stretching maps can be assigned to upper crust and lower crust/upper man- tle, respectively. Gravitational data can also be used for additional calibration parameters, e.g. when the crust geometry directly controls gravity. Another less accurate and much simpler procedure is use of McKenzie’s equilibrium subsidence (3.39) to directly calculate the stretching factor maps from the total present day subsidence map only, and to predict the paleo–heat flow maps and the new water depth maps afterwards. It is obvious that the McKenzie type models yield only rough estimates of the basal heat flow maps through time and the heat flow maps need to be fine tuned with vitrinite reflectance data and bottom hole temperatures afterwards. A decoupling of the procedures in the two steps of crustal prepro- cessing and calibration against the thermal markers, allows a better overview and handling of the individual parts and leads to a better understanding of the respective processes. 3.9 Heat Flow Calibration Heat flow models can be calibrated with measured temperatures from wells and thermal maturity parameters, such as vitrinite reflectance, biomarkers and fission–track annealing data. Thermal maturity parameters are time and temperature dependent. They indicate how long the rock elements remain at certain temperature levels. Thus, single data points of specific thermal markers are often only useful to calibrate a small temperature interval. They cannot be used to specify a total age for the temperature interval. Exceptions are fluid inclusion temperatures, which are often related to paleo–ages.
- 157. 144 3 Heat Flow Analysis The most commonly used parameter is vitrinite reflectance, since it is widely available in most sediment types, covers typical oil and gas maturity ranges and is easy and cheap to measure. The importance and main temper- ature windows of many other thermal markers are compared and explained in more detail in Chap. 4. The most uncertain input parameters are thermal conductivity and paleo- and present basal heat flow values. There are several workflows and techniques developed to change thermal conductivity and heat flow parameters, when thermal maturity data or mea- sured temperatures, differ from a master run. Recently, more emphasis has been put on the calibration of basal heat flow values as thermal conductivi- ties are much better known. The assumption of a rift–type heat flow peak or any other trend can be obtained as a first estimation from crustal models or other knowledge about geological history. Such trends are typically defined for individual locations and usually calibrated against well data such as bottom hole temperatures or vitrinite reflectance values. The heat flow trends can then be simply shifted entirely or stepwise or other corrections like first order shifting or heat flow peak calibration can be performed until the match with the calibration data is satisfying (Fig. 3.36). a) b) c) d) 0 20 40 60 80 100 120 140 0 20 40 60 80 100 120 140 Geological Age in My Heat Flow in mW/m 2 0 20 40 60 80 100 120 140 0 20 40 60 80 100 120 Geologic Time in My Heat Flow in mW/m 2 0 20 40 60 80 100 120 140 0 20 40 60 80 100 120 Geologic Age in My Heat Flow in mW/m 2 0 20 40 60 80 100 120 140 0 20 40 60 80 100 120 Geologic Age in My Heat Flow in mW/m 2 First order calibration Peak height calibration Stepwise constant shifts (two periods) Constant shift (0-order) Fig. 3.36. Methods of heat flow trend calibration: (a) Constant shift. (b) First order calibration. (c) Stepwise constant shifts. (d) Special peak height change Automatic calibration tools can be used, when numerous thermal cali- bration parameters are available. They allow the definition of time intervals with independent shift corrections, peak corrections or the assumption of ad- ditional uncertainties for SWI temperatures or thermal properties. Numerical models used for the automatic calibration or inversion are Monte Carlo sim-
- 158. 3.9 Heat Flow Calibration 145 ulations, response surface modeling or fast approximated forward simulation techniques, all of which are described in the Chap. 7. One typical workflow of an automatic 3D calibration processor is described below. The final values of an automatic calibration must make physical and ge- ological sense, rather than simply providing an acceptable mathematical fit between measured and calculated values. This is achieved with fixed param- eter ranges or the Bayesian approach (Chap. 7). 3.9.1 Example Workflow for 3D Heat Calibration The workflow under discussion is illustrated in Fig. 3.37. A fit to calibration data such as temperature and vitrinite reflectance values should obviously be achieved while conserving the shape of the heat flow trend through time. Here, only a constant shift of the heat flow trends is considered. The procedure can be applied to regions of limited size, typically to small areas of interest around a well or a group of wells (Fig. 3.38). The extension of these areas should be large enough to incorporate lateral heat flow effects as they appear e.g. in the vicinity of salt domes. The temperature evolution inside each area can be fitted with its own heat flow shift. Performing probability runs for each area of interest Estimated heat flow maps, e.g. from crustal analysis Thermal Calibration Data: vitrinite reflection and temperature Calibration of heat flow trend for each area of interest Interpolate and extrapolate heat flow corrections on maps Response surfaces for single well heat flow trends Heat flow correction maps Define small areas of interest around wells of thermal calibration data Calibrated heat flow maps for all events Fig. 3.37. Workflow for heat flow preprocessor A calibration performed in any area of interest is independent of calibra- tions in other areas. Therefore, it can, for example, be performed in a separate “mini–model”.3 In practice many wells with calibration data are available and it is advantageous to run all the areas together in one big 3D simulation. The areas around the wells are restricted to a small size. As smaller these sizes as the faster the simulation. This is important in practice, because multiple runs with varying heat flow must be performed.4 The calibration in each area 3 If lateral heat flow effects are neglected such a “mini–model” becomes a pure 1D–model. 4 When sufficient computer resources are available, it is also possible to run the full 3D model without any areal cut–outs. This would reproduce all 3D thermal
- 159. 146 3 Heat Flow Analysis Fig. 3.38. Example of a heat flow calibration in areas around wells with temperature and vitrinite reflectance data. The size of each area is equivalent to the thickness of the corresponding column. These column thicknesses are defined as rather thin here because lateral heat flow effects are small is performed independently of each other afterwards. Advanced interpolation and extrapolation between different simulation results with, for example, re- sponse surfaces as described in section Sec. 7.5.1, yield fast and accurate results (Figs. 3.39, 3.40). Finally, calibrated heat flow maps can be constructed by spatial inter- polation between the areas with calibrated heat flow shifts. If necessary, an additional smoothing of the interpolated heat flow shift can be performed. The results can be tested in a final simulation run. Crustal heat flow analysis and heat flow calibration can be performed suc- cessively (Fig. 3.41). It is thus possible to construct calibrated and geological meaningful heat flow maps from a basin model and additional geological infor- mation about stretching phases, crustal structure, bottom hole temperatures and vitrinite reflectance data. effects. However, calibrated heat flow shifts below each calibration well, must be assigned to limited areas for further construction of new structural heat flow maps by interpolation. Additionally, these areas should not be too small. Otherwise in the full 3D model, heat might leave these areas laterally due to lateral temperature gradients. This may yield too low temperatures at the well locations and the calibration might fail.
- 160. 3.9 Heat Flow Calibration 147 Fig. 3.39. Shifted heat flow trend after calibration (solid line) and before calibration (dotted line) Fig. 3.40. Temperature and vitrinite reflectance in the same well calibrated against data values with error bars. Dotted lines represent results from a crustal model only. The corresponding heat flow shift is shown in Fig. 3.39. Note that a fit against temperature data alone would yield a slightly lower tem- perature and vitrinite reflectance pro- file. However, due to small error bars vitrinite reflectance values are treated here with higher importance than tem- perature values Crustal heat flow analysis Model stratigraphy and geometry and crustal input Thermal calibration data: vitrinite reflectance and temperature Heat flow calibration Petroleum systems modeling Heat flow maps, especially predicted for the paleo time of stretching Heat flow maps,especially calibrated against short range paleo-time and present day data Fig. 3.41. General workflow, which links crustal modeling, heat flow calibration and petroleum systems modeling
- 161. 148 3 Heat Flow Analysis Summary: Heat flow analysis is based on a detailed balance of thermal energy that is transported via heat flow through sedimentary basins. Heat flow occurs primarily in form of conduction and convection. The driving forces for conduction are temperature differences. Convection is classified by moving fluid or solid phases that carry their inner thermal energy along. Previously to a detailed energy balance it is necessary to specify heat in– and outflow or alternatively the temperature at the boundary of the sedimentary basin. The main direction of heat flow in sedimentary basins is vertically up- wards. It is thus possible to demonstrate basic effects with crude one dimen- sional models. Steady state heat flow constitutes the most simple heat flow pattern. Explicit formulas can be calculated. Radioactive heat production can easily be incorporated. The complexity of the system rises with consid- eration of transient effects which occur during deposition, erosion, and when thermal boundary conditions change. However, some idealized special cases can be solved analytically. The main thermal properties of the rocks are thermal conductivities, rea- diogenic heat production, and heat capacities. Detailed specifications of these properties for various lithologies and fluids over wide temperature ranges are well known. The general formulation of two and three dimensional heat flow problems incorporates heat convection and magmatic intrusions. The quantification of heat flow and temperature boundary conditions is often a major task. SWI temperatures can be derived from paleo climate models. Effects of permafrost require the specification of paleo surface temperatures. Basal heat flow can be calculated from tectonic stretching and thinning of the crust which causes the evolution of the basin. Models for rift basins are mainly worked out as extensions to the famous McKenzie type crustal mod- els. Basic principles are isostasy and crustal heat flow balance. Finally, the amount of stretching can be calibrated against the known subsidence of the sedimentary package. Comprehensive heat flow trends can be constructed. These trends can locally be adapted to known temperature histories from well logs and samples, e.g. bottom hole temperatures and vitrinite reflectance measurements. Sophisticated workflows are worked out for fast and efficient calibration procedures.
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- 164. 4 Petroleum Generation 4.1 Introduction Modeling of geochemical processes encompasses the generation of petroleum and related maturation parameters, such as vitrinite reflectance, molecular biomarkers, and mineral diagenesis. The transformation and maturation of organic matter can be subdivided into three phases: diagenesis, catagenesis and metagenesis (Tissot and Welte, 1984). The term diagenesis is different from that of the rock types. The formation of petroleum and coal with typical depth and temperature intervals is illustrated in Fig. 4.1. During diagenesis, most organic particles in the sediments are transformed by microbiological processes into kerogen with a release of volatiles such as CH4, NH3 and CO2. Petroleum is mainly generated during catagenesis, when kerogen is thermally cracked to heavier and lighter hydrocarbons and NSO (nitrogen, sulfur, oxygen) compounds. The transformation rates depend on the organic matter type and the time–temperature history. Heavier petroleum components are generally generated first and they are then cracked into lighter components at higher temperatures, resulting in a so called ”oil window” be- tween 1 to 3 km depth. Thermogenic hydrocarbon gas is generated at greater depths. The maturation of coal also depends on time and temperature. It is mainly described by the change of one of its constituents, the maceral vitrinite. One measure for vitrinite maturation is the intensity of reflected light at a stan- dardized wavelength (546 nm). The occurrence of vitrinite in many sediments enables its usage as a general thermal history marker. Vitrinite reflection can thus be correlated to the maturation of petroleum (Fig. 4.1). The generation and maturation of HC-components, molecular biomarkers and coal macerals can be quantified by chemical kinetics. Chemical kinetics are formulated using mass balances. It is therefore important to specify and track all chemical reactants of organic matter during the processes of interest. A simple classification of organic matter in sedimentary rocks after Tissot and T. Hantschel, A.I. Kauerauf, Fundamentals of Basin and Petroleum 151 Systems Modeling, DOI 10.1007/978-3-540-72318-9 4, © Springer-Verlag Berlin Heidelberg 2009
- 165. 152 4 Petroleum Generation T C ° Coal Petroleum and natural gas 2 1 3 4 0 Depht (km) Relative output of generated hydrocarbons Biochemical CH4 Geochemical fossils Petroleum Natural gas CH4 Diagenesis Catagenesis Meta- genesis dry gas wet gas oil zone immature zone Vitrinite reflectance (%) 0.5 1.0 2.0 3.0 Coalification degree peat lignite hard coal anthracite 65 200 Fig. 4.1. Evolution of organic matter: Diagenetic, catagenetic and metagenetic processes describe the generation of oil and gas and coalification. The picture is from Bahlburg and Breitkreuz (2004). The processes are compared with the relative intensities of light reflected from the coal maceral vitrinite Welte (1984) is shown in Fig. 4.2. A comprehensive description is given, for example, in Peters et al. (2005). Diffusion controlled and autocatalytic reactions are not taken into account in this volume, instead decomposition and pseudo–unimolecular kinetics are considered, which can be adequately described by sequential and parallel re- actions. This approach is called kinetics of distributed reactivities. Herein, the rate of each reaction is related to an Arrhenius type activation energy and a frequency factor. 4.2 Distributed Reactivity Kinetics The simplest reaction type, which is used for most sequential and parallel reaction schemes, is the unimolecular forward reaction from an initial reactant X of mass x to the product Y of mass y: X k −→ Y, ∂y ∂t = − ∂x ∂t = k xα (4.1) where α is the reaction order, k is the reaction rate and t is the time. In the following, unit masses are considered with x0 = 1, x(t → ∞) = 0 and y = 1−x.
- 166. 4.2 Distributed Reactivity Kinetics 153 Organic Matter in Sedimentary Rock Alginite (Oil-Prone) Exinite (Oil-Prone) Bitumen (soluble in organic solv.) Hydrocarbons (HCs) Non- Hydrocarbons Kerogen (insoluble in organic solv.) Coke Vitrinite (Gas-Prone) Inertinite (non Source) Gas Components Oil Components N , CO 2 2 NSO Component Light HCs Heavy HCs Fig. 4.2. A geochemical fractionation of organic matter The transformation ratio TR is defined as the converted mass fraction of the initial reactant, which is here TR = y. Most geochemical processes are described with first order reactions α = 1. Higher or lower reaction orders are used when the transformation rate dTR/dt has a nonlinear dependency on the reactant’s concentration. The temperature dependency of the reaction rate k is usually described by an Arrhenius law with two parameters, the frequency factor A and the activation energy E: k = A e−E/RT (4.2) with the gas constant R = 8.31447 Ws/mol/K. The frequency (amplitude or pre–exponential) factor represents the frequency at which the molecules will be transformed and the activation energy describes the required threshold energy to initiate the reaction. The Arrhenius law was originally developed as an empirical equation but it is also confirmed from transition theory with a temperature dependent frequency factor (Glasstone et al., 1941; Benson, 1968). The strong temperature dependence yields significant values for the trans- formation rate when a threshold temperature is exceeded (Fig. 4.3). The ac- tivation energies and frequency factors, which are used in the sample plots, are typical for organic matter decomposition with transformation times of millions of years for heating rates of 10 K/My.
- 167. 154 4 Petroleum Generation Laboratory measurements are performed with higher heating rates of 1 K/min, which yield peak transformation at much higher temperatures than in nature (Fig. 4.4). Small uncertainties in laboratory based kinetic parame- ters can result in large effects on predicted transformation rates at geological time scales. Measured transformation ratios versus time can be inverted into (E, A) pairs with a regression line in the “ln k versus 1/T” Arrhenius plot. Advanced inversion methods are necessary for kinetics with multiple parallel and sequential reactions. a) b) c) d) 0 50 100 150 200 250 300 0 1 2 3 4 5 6 7 Temperature in o C 1 2 3 4 0 50 100 150 200 250 300 0 20 40 60 80 100 Temperature in o C Transformation Ratio in % 1 2 4 5 0 50 100 150 200 250 300 0 2 4 6 8 Temperature in o C 1 3 4 5 0 50 100 150 200 250 300 0 20 40 60 80 100 Temperature in o C Transformation Ratio in % A=10 1..E=45 kcal/mol 2..E=50 kcal/mol 3..E=55 kcal/mol 4..E=60 kcal/mol 5..E=65 kcal/mol 14 s-1 3 5 E=55 kcal/mol 1..A=1010 s-1 2..A=1012 s-1 3..A=1014 s-1 4..A=1016 s-1 5..A=1018 s-1 1 2 3 4 5 2 Transformation / Heating Rate in %/K Transformation / Heating Rate in %/K Fig. 4.3. Influence of activation energy and frequency factor on transformation rate (TR) and transformation ratio for first order Arrhenius type reactions with a heating rate of 10 K/My: (a) and (b) fixed frequency ratio and variable activation energy, (c) and (d) fixed activation energy and variable frequency factor Parallel reactions are used for multiple and not interacting reactants Xi of masses xi with initial masses x0i and i x0i = 1 which are converted into a product Y as follows.
- 168. 4.2 Distributed Reactivity Kinetics 155 Fig. 4.4. Influence of the heating rate on first order Arrhenius type reactions. The first peak is related to geological heating rates 10 K/My, while the other two peaks are labora- tory heating rates with 0.1 K/min and 5 K/min 0 100 200 300 400 500 600 0 1 2 3 4 5 6 Temperature in o C 10 K/Ma A = 10 E = 55 kcal/mol 14 s-1 5 K/min 0.1 K/min Transformation / Heating Rate in %/K Xi ki −→ Y, ∂xi ∂t = −ki xα , ∂y ∂t = n i=1 kixα . (4.3) Each of the subreactions i can be described with a pair (Ai, Ei), but usually a unique frequency factor is used, which results in a discrete distribution of activation energies p(Ei) equal to the initial masses of the reactants x0i = p(Ei). The transformation ratio is TR = 1 − x = y with x = i xi. It also varies between 0 and 1. Parallel reactions can be used for the decomposition of complex macro– molecules having a wide range of chemical bond strengths to one type of cracking product. Herein, the initial masses x0i correspond to the mass of the reactant related to the cracking of chemical bonds of the corresponding acti- vation energy levels. However, the chemistry of the decomposition of organic matter is generally more complex and the use of parallel reactions is just an empirical formalism for compositional effects of reactants and products. Arbitrary discrete distributions p(Ei) are mainly used for petroleum and coal formation. Approximations to continuous Gaussian distributions are also popular. Gaussian distributions are described with a mean activation energy μ and a variance σ as follows (Chap. 7): p(E) = 1 σ √ 2π exp −(E − μ)2 2σ2 . (4.4) Burnham and Braun (1999) reported the usage of two other continuous distributions: the Gamma distribution p(E) = aν (E − γ)ν−1 Γ(ν) e−(E − γ)a (4.5) with parameters ν, a and threshold activation energy γ, and the Weibull distribution p(E) = β η E − γ η β−1 e− [(E − γ)/η] β (4.6)
- 169. 156 4 Petroleum Generation with the width parameter η, the shape parameter β, and again a threshold activation energy γ. The mean energy of the Weibull distribution is Ē = γ + ηΓ(1/β + 1), (Fig. 4.5). Activation Energy Distribution [kcal/mol] Probability 80% 10% 10% 50.2659 51.9449 50 51 52 53 0 0.2 0.4 0.6 Activation Energy Distribution [kcal/mol] Probability 80% 10% 10% 50.6492 53.0349 50 51 52 53 54 55 0 0.1 0.2 0.3 0.4 0.5 Fig. 4.5. Examples of Gamma (left) and Weibull (right) distributed activation energies. Here it is ν = a = η = β = 2 and γ = 50 kcal/mol for both distributions Equation (4.3) can be numerically integrated in time for stepwise constant heating rates, which yields the following mass reduction from time step l to l + 1 with duration Δt and temperature T(l) : x (l+1) i = x (l) i + Δtki(T(l) ) α − 1 for α = 1, x (l+1) i = x (l) i 2 − Δtki(T(l) ) 2 + Δtki(T(l)) else . (4.7) The extension of the above approach to the formation of multiple prod- ucts is usually performed by superposition of single kinetic reaction schemes, which yields sequential and parallel connected super–schemes. However, the actual processing resembles a bookkeeping of all reactions that are defined in the source rock according to the organic facies. The computing effort is low compared to other processes; e.g. heat and fluid flow calculations. It increases linearly with the number of cells, but the required computer memory can be- come significant as one value for each of the discrete activation energies of each mass component has to be stored for each cell. 4.3 Petroleum Generation Kinetics The total content of organic matter (kerogen and bitumen) is usually given in terms of the total organic carbon content (TOC) in mass %. It is the ratio of the mass of all carbon atoms in the organic particles to the total mass of the rock matrix. Hence, it is a concentration value and one needs the total rock
- 170. 4.3 Petroleum Generation Kinetics 157 mass of the source rock for conversion into generated and expelled petroleum masses. The generation of petroleum is a decomposition reaction, from heteroge- neous mixtures of kerogen macromolecules to lighter petroleum molecules. Petroleum kinetics are distinguished from cracking types (primary or sec- ondary), kerogen types (I — IV) and the number and type of the generated petroleum components (bulk, oil–gas, compositional kinetics). Kerogen types are chemically classified according to the abundance of the elements carbon (C), hydrogen (H) and oxygen(O). The most common are the H/C and O/C ratios originally used in coal maceral classifications by van Krevelen (1961), which resulted in the definition of three main kerogen types I — III as shown in Fig. 4.6.a after Peters et al. (2005). 0.1 0.2 0.3 0.5 1.0 1.5 I Oil-prone II Oil-prone III Gas-prone 4.0 4.0 2.0 2.0 1.5 1.5 1.0 1.0 0.5 0.5 Thermal Maturation Pathways R (%) o Atomic O/C Atomic H/C I Oil-prone 150 300 450 600 750 900 50 100 150 200 250 II Oil-prone III Gas-prone Oxygen Index (mg CO /g TOC) 2 Hydrogen Index (mg HC/g TOC) a) b) Fig. 4.6. Characterization of kerogen by van–Krevelen diagrams after Peters et al. (2005): (a) according to the abundance of the elements in kerogen in ratios of H/C and O/C, (b) according to the generative amounts of HC and CO2 in Rock–Eval parameters HI (hydrogen index) and OI (oxygen index) from Fig. 4.7, and vitrinite reflectance R0 The paths are idealized. Real samples can be different The kerogen types are weakly linked to depositional environments. Type I is mostly derived from lacustrine algal matter, although some petroleum source rocks deposited in marine settings are also dominated by type I kero- gen. Type II is the most widespread type. It is common in marine sediments, indicating deposition of autochtonous organic material in a reducing environ- ment. Type III with the highest relative oxygen content of the main kerogen types, indicates an origin in terrigenous environments, i.e. from plant organic matter. Type IV kerogen has very low HI values. The corresponding maturity path is close to the x-axis in the van–Krevelen diagram. Most low maturity coals contain type III kerogen, although coals dominated by type I, II or IV kerogen also occur. Detailed descriptions of the occurrence and the genesis of
- 171. 158 4 Petroleum Generation the different kerogen types are given in Tissot and Welte (1984) and Peters et al. (2005). The maturation paths of decreasing H/C and O/C ratios can be related by simple mass balance calculations to the corresponding decrease in the gen- erative masses of HCs and CO2. The generative HC and CO2 masses of a kerogen sample are measured by Rock–Eval pyrolysis in terms of the HI and OI potential, which yields the ”HI–OI” van-Krevelen diagram as shown in Fig. 4.6.b. The Rock–Eval method is an open system pyrolysis. The rock sample con- taining organic matter is heated at approximately 50 K/min and the released masses of HCs and CO2 are measured (Fig. 4.7). The first HC peak of the thermally distilled HCs (S1) corresponds to the residual bitumen. It is the already generated and not yet expelled mass of HCs in the rock sample. The second peak of pyrolytic generated HC amounts (S2) is the total generative mass of HCs, which is related to the hydrogen index (HI), given in mg/gTOC. Hence, the HI multiplied with the TOC and the rock mass is equal to the total generative mass of HCs in the rock. The peak of the pyrolytic generated CO2 is analogically related to the oxygen index (OI) measured in mg/gTOC. Fig. 4.7. Schematic pyrogram from the Rock–Eval pyrolysis: Hydrogen Index HI, Oxygen Index OI, Produc- tion Index PI. The S1 and S2 peaks mainly contain hydrocarbons. They are measured with a flame ionization detector. The S3 peak contains the generated CO2. It is measured with a thermal conductivity detector after the heating is finished at room tem- perature Frequency of generated components Temperature S1 Tmax Pyrolytic generated HCs~HI Thermally distilled HCs~PI Pyrolytic generated CO ~OI 2 S2 S3 % in TOC , mg in S S , gTOC / mg in OI , HI with TOC / 100 S OI , TOC / 100 S HI 3 , 2 3 2 ´ = ´ = The production index PI=S1/(S1+S2) is a measure of cracked kerogen. It varies between 0 and 1. The PI is equal to the transformation ratio, if no petroleum has been expelled out of the sample. Another characteristic value of the Rock–Eval method is the oven tem- perature Tmax at the maximum HC generation rate for S2 (Fig. 4.7). This temperature can be used as a maturity parameter for the kerogen sample. It can be calculated in source rocks and used as a thermal calibration parameter. The classification of kerogen into the three van Krevelen types is not suf- ficient to predict the generated petroleum composition. The type of the gen- erated petroleum generally depends on various factors, such as the biological input, the oxic or anoxic environment, and marine or deltaic facies. Jones
- 172. 4.3 Petroleum Generation Kinetics 159 (1987) introduced the term organic facies for an improved classification of the kerogen types according to the generated type of petroleum. Di Primio and Horsfield (2006) proposed a characterization of organic facies with the generated portions of the HC classes (C1 − C5, C6 − C14 and C15+) and the sulfur content of the source rock (Fig. 4.8). They also proposed a quantitative description of generation kinetics with 14 hydrocarbon components, which al- lows an improved prediction of HC phases with equation–of–state type flash calculations (Chap. 5). The prediction of phase properties, such as densities, viscosities or phase compositions is more reliable with these organic facies based kinetics than with generalized van–Krevelen type kinetics. Fig. 4.8. Characterization of kero- gen by the generated petroleum type: Five organic facies are defined accord- ing to the generative potential for three HC classes (C1 − C5, C6 − C14, C15+). Facies 3 and 5 are further dis- tinguished into a low and a high sul- fur type . P–N–A means paraffinic– naphthenic–aromatic type C -C 1 5 C15+ C -C 6 14 3 1 2 4 5 1..Gas and Condensate 2..P-N-A Oil (High Wax) 3..P-N-A Oil (Low Wax) 4..Paraffinic Oil (Low Wax) 5..Paraffinic Oil (High Wax) 20 40 60 80 80 60 40 20 80 60 40 20 The number of considered petroleum components depends on the available sample data, the type of pyrolysis system, and whether phase properties such as APIs and GORs should be calculated. Many publications describe primary generation kinetics, while secondary cracking from higher into lower molecular weight HCs with coke as a by– product are rather seldom. Many secondary cracking models are only based on methane and coke as products. Some are described as chain reactions each of them with the next lower molecular weight component and coke as products. In the case of two component oil–gas systems, secondary cracking is defined as an oil to gas reaction. 4.3.1 Bulk Kinetics Bulk kinetics focus on kerogen cracking and do not distinguish between several petroleum components. They are described with n parallel reactions (4.3) from kerogen Xi, i = 1, . . . , n to petroleum Y. Herein, the “i–th” parallel reaction corresponds to the chemical bonds, which have to be cracked with the activation energy Ei in the kerogen molecules. The x0i and xi are the initial and actual relative masses of the generative petroleum according to the activation energy Ei with n i=1 xi0 = 1 and n i=1 xi = x, and y as the
- 173. 160 4 Petroleum Generation generated relative petroleum mass. The energy distribution p(Ei) is equal to the initial relative mass distribution of the generative petroleum x0i. Usually, the petroleum potential yp = y HI0 in gHC/kgTOC is used rather than the relative masses of petroleum y to describe the generated petroleum amounts. Some typical bulk kinetics are shown in Fig. 4.9 for type II and III kerogen after Tegelaar and Noble (1994). The petroleum potential is usually calculated in all layers and not only in the source rock to illustrate the de- pendency and sensitivity of the source rock generation by depth. Herein, the same HI0 is assumed for all layers. The transformation ratio TR = 0.5 defines the critical point of generation. Some example curves for type II kinetics at three different sedimentation rates show the dependency of the generated petroleum on sedimentation or heating rates (Fig. 4.10). The depth of the critical point ranges from 4.5 to 7 km. 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 0 10 20 30 40 Activation Energy in kcal/mol Frequency in % 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 0 5 10 15 20 25 30 35 Activation Energy in kcal/mol Frequency in % Type II Formation: Kimmeridge Clay Age: Jurassic Location: UK Basin: North Sea A = 4.92x1027 My-1 HI = 500 mg/gTOC 0 Type III Formation: Mannville Age: Cretaceous Location: Canada Basin: Alberta A = 2.99x1028 My-1 HI = 131 mg/gTOC 0 Fig. 4.9. Example of bulk kinetics from Tegelaar and Noble (1994) for a type II and a type III kinetic model. A unique frequency factor is assumed The relative petroleum masses y can be converted into real masses as follows. mp = TOC0 HI0 V (1 − φ) ρr y (4.8) where TOC0, HI0 are the initial TOC and HI values V is the considered source rock volume, φ is the porosity, and ρr is the rock density. The initial HI value is usually estimated from the kerogen type, while the initial TOC value is often difficult to determine. When measured HI and TOC values from mature source rock samples are available, the initial TOC value can be reconstructed with the following equation (Peters et al., 2005): TOC0 = p HI TOC HI0(1 − TR)(p − TOC) + HI TOC (4.9) where p = 83 % is the percentage of carbon in generated petroleum, TOC is given in % and HI in mgHC/gTOC. Peters et al. (2005) also give an equation to calculate TR from measured Rock–Eval data and estimated initial values: HI, HI0, PI and PI0 ≈ 0.02. A better alternative is to take the TR values from
- 174. 4.3 Petroleum Generation Kinetics 161 0 20 40 60 80 100 120140 0 2 4 6 8 Generated HCs in mg/gTOC Depth in km 1 2 3 CP1 CP2 CP3 0 10 20 30 40 50 60 70 0 2 4 6 8 Generated HC Rate in mg/gTOC/My Depth in km 1 3 0 50 100 150 200 250 300 0 20 40 60 80 Geologic Time in My Generated HC Rate in mg/gTOC/My 1 3 2 0 50 100 150 200 250 300 0 20 40 60 80 100 120 140 Geologic Time in My Generated HCs in mgHC/gTOC 1 2 3 1..S=25 m/My 2..S=100 m/My 3..S=1000 m/My 1..S=25 m/My 2..S=100 m/My 3..S=1000 m/My 2 1..S=25 m/My 2..S=100 m/My 3..S=1000 m/My 1..S=25 m/My 2..S=100 m/My 3..S=1000 m/My Fig. 4.10. Petroleum generation potential and corresponding rates of the type II bulk kinetics of Fig. 4.9. In each diagram three different sedimentation rates are modeled. The critical points (CP) of the curves correspond to TR=0.5 a simulation with arbitrary HI0 and TOC0 as the TR is not dependent on the initial HC mass. 4.3.2 Oil–Gas Kinetics Oil–gas kinetics are two component models. The gas component lumps to- gether the lighter HCs (C1 − C5) and the oil components comprehend all heavier HCs (C6+). Oil and gas are almost equal to the liquid and vapor phases at surface conditions. They are very different from the liquid and vapor phases at in–situ conditions, i.e. during petroleum migration and in reservoirs. A two component — two phase model is well established in Darcy type fluid flow models by the name “Black Oil model” (Secs. 5.3, 6.3.5). The reaction scheme describes the two primary cracking reactions from kerogen X into oil Y and gas Z and secondary cracking from oil to gas, each described by a set of first order parallel reactions: X1i k1i −→ Y, Yi k i −→ Z, X2i k2i −→ Z . (4.10) The relative masses of generative oil and gas for primary cracking are x1i with i = 1, . . . , n1, and x2i with i = 1, . . . , n2, respectively, the initial values x01i, x02i and the number of parallel reactions n1 and n2. The total actual and initial generative masses are x = n1 i=1 x1i + n2 i=2 x2i, x0 = x01 + x02 = n1 i=1 x01i + n2 i=2 x02i = 1 . (4.11)
- 175. 162 4 Petroleum Generation The two frequency distributions of the activation energies p1(Ei), p2(Ei) for primary cracking of oil and gas are normalized to the initial relative masses x01i, x02i. The total relative amount of generated masses of oil and gas are y and z, respectively, and the frequency distribution over the activation energies for secondary cracking is p (Ei) with i = 1, . . . , n . Then, the mass balance of the above kinetics is as follows. ∂x1i ∂t = −k1i x1i, ∂x2i ∂t = −k2i x2i, ∂yi ∂t = p (Ei) n1 j=1 k1j x1j − k i yi, ∂z ∂t = n2 i=1 k2i x2i + R n i=1 k i yi (4.12) where k1i, k2i, k i are the reaction rates for primary kerogen to oil and gas cracking, and secondary oil to gas cracking, respectively. Each generated oil is distributed to portions yi according to the frequency distribution of the activation energy for secondary cracking. Secondary cracking of oil to gas yields a reduction of organic mass with coke as a by–product, as described by the reduction factor, where R = 1 means no coke and R = 0 all coke. R usually ranges between 0.4 and 0.7. The following principal equation results in a reduction factor of R = 16/28 = 0.57. 2CH2 −→ 1C + 1CH4 2 × 14 −→ 12 + 16 . (4.13) In analogy to bulk kinetics, oil and gas potentials yp, zp are defined as yp = y HI0 and zp = z HI0 with the unit mgHC/gTOC. It is also necessary to integrate ratio (or percentage) values to describe how the total HI value is subdivided into the HI for kerogen to oil HIo and the HI for kerogen to gas HIg reactions. Relative masses for oil and gas are also used to define principle zones of active oil and gas generation. Herein, the generated masses of oil and gas are compared to the maximum generative masses for oil yt and gas zt: yt = n1 i=1 x01i, zt = n2 i=1 x02i + R n1 i=1 x01i . (4.14) with Immature: y/yt 0.1 and z/zt 0.1 Oil Generation: y/yt ≥ 0.1 and z/zt 0.1 Gas Generation: z/zt ≥ 0.1 and z/zt 0.9 Overmature: z/zt ≥ 0.9 (4.15) Oil–gas kinetics are mainly used in source rock maturity studies, when petroleum phase properties are of minor importance. Some typical oil–gas
- 176. 4.3 Petroleum Generation Kinetics 163 kinetics for type I, II and III kerogen after Pepper and Corvi (1995a); Pepper and Dodd (1995) are shown in Fig. 4.11. The calculated oil and gas generation potential versus depth and versus geologic time of the three kerogen types are shown in Fig. 4.12. The effect of the sedimentation rate on the generation potential is illustrated in Fig. 4.13. 4.3.3 Compositional Kinetics In compositional kinetics, more than two HC–components are considered. Each of the components can be generated by primary cracking from kero- gen X, which is described by parallel decomposition reactions. The kinetics for secondary cracking is a triangular scheme where each component can be cracked into all lighter ones. In the formulation of the following kinetic schemes and mass balances, chemical components and distributed activation energies are numbered serially with the indices i, j, l = 1, . . . , N and r, s = 1, . . . , n i, re- spectively. The petroleum components Yi are ordered according to their molar masses, starting with the lowest. The complete reactions scheme encompasses the following N primary and N(N − 1)/2 secondary reactions: Xir kir −→ Yi for i = 1, . . . , N, Yi k ijr −→ Yj for i, j = 1, . . . , N, i j . (4.16) The relative generative component masses of primary cracking are denoted as xir in analogy to the oil–gas kinetics and the initial values are denoted as x0ir, which are equal to the frequency of the activation energies for primary cracking pi(Er) = x0ir. This is an array with the dimension of the number of components and discrete activation energies and with N i=1 ni r=1 pi(Er) = 1 (4.17) (Fig. 4.14). In the most general case of secondary cracking, each petroleum component i can be cracked into any other lighter component j with, i j, which can also be described by a frequency array with dimension of the number of lighter components and activation energies p ij(Er) with N j=1 nj r=1 p ij(Er) = 1 (4.18) for each component i. The mass balance of the coupled reaction is for all i = 1, . . . , N and r = 1, . . . , ni ∂xir ∂t = −kir xir (4.19)
- 177. 164 4 Petroleum Generation 40 50 60 70 80 0 5 10 15 20 ActivationEnergy in kcal/mol Frequency in % 40 50 60 70 80 0 5 10 15 20 ActivationEnergy in kcal/mol Frequency in % 40 50 60 70 80 0 5 10 15 20 ActivationEnergy in kcal/mol Frequency in % 40 50 60 70 80 0 5 10 15 20 ActivationEnergy in kcal/mol Frequency in % 40 50 60 70 80 0 5 10 15 20 ActivationEnergy in kcal/mol Frequency in % 40 50 60 70 80 0 5 10 15 20 Activation Energy in kcal/mol Frequency in % Gas Oil 40 50 60 70 80 0 5 10 15 20 Activation Energy in kcal/mol Frequency in % Gas Oil Gas: A=2.17x1018 s-1 s=4.39 kcal/mol x02=17 % Oil: A=8.14x1013 s-1 s=1.98 kcal/mol x01=83 % 40 50 60 70 80 0 10 20 30 40 Activation Energy in kcal/mol Frequency in % Gas Oil Gas: A=2.29x1016 s-1 s=2.41 kcal/mol x02=13 % Oil: A=2.44x1014 s-1 s=0.93 kcal/mol x01=87 % 40 50 60 70 80 0 5 10 15 20 Activation Energy in kcal/mol Frequency in % Gas Oil Gas: A=1.93x1016 s-1 s=2.36 kcal/mol x02=23 % Oil: A=4.97x1014 s-1 s=1.89 kcal/mol x01=77 % 40 50 60 70 80 0 5 10 15 20 Activation Energy in kcal/mol Frequency in % Gas Oil Gas: A=1.93x1016 s-1 s=2.36 kcal/mol x02=44 % Oil: A=1.23x1017 s-1 x01=56 % Primary Cracking Type A (Kerogen IIS) =617 mg/gTOC 0 HI Oil: A=2.13x1013 s-1 Gas: A=3.93x1012 s-1 E=49.4 kcal/mol =2.56 kcal/mol x02=17 % E=49.3. kcal/mol =1.96 kcal/mol x01=83 % s s Secondary Cracking Type A A=1.0x10 E=58.4 kcal/mol s=2.05 kcal/mol 14 -1 s Secondary Cracking Type B A=1.0x1014 s-1 E=58.4 kcal/mol s=2.08 kcal/mol Primary Cracking Type B (Kerogen II) HI0=592 mg/gTOC E=51.4 kcal/mol E=66.6 kcal/mol Primary Cracking Type C (Kerogen I) HI =600 mg/gTOC 0 E=52.9 kcal/mol E=59.8 kcal/mol Secondary Cracking Type C A=1.0x1014 s-1 E=58.4 kcal/mol s=2.08 kcal/mol Primary Cracking Type DE (Kerogen III) HI0=333 mg/gTOC E=65.7 kcal/mol E=54.5 kcal/mol Secondary Cracking Type C/D A=1.0x1014 s-1 E=57.9 kcal/mol =2.60 kcal/mol s Primary Cracking Type F (Kerogen III/IV) HI0=158 mg/gTOC E=65.7 kcal/mol E=61.9 kcal/mol s=1.58 kcal/mol Secondary Cracking Type F A=1.0x10 s 14 s-1 E=57.3 kcal/mol =3.27 kcal/mol Fig. 4.11. Oil–gas kinetics for typical kerogen types I — III after Pepper and Corvi (1995a); Pepper and Dodd (1995). The kinetics are approximated in this volume by discrete distributions. Note, that the frequency curves for gas are added to the oil curves and that oil and gas are separately normalized to their initial kerogen fraction x0i
- 178. 4.3 Petroleum Generation Kinetics 165 0 200 400 600 800 1000 0 2 4 6 8 Generated HCs in mg/gTOC Depth in km 0 20 40 60 80 0 200 400 600 800 1000 Geologic Time in My Generated HCs in mgHC/gTOC Type II Kerogen Type I Kerogen Type I Kerogen Type III Kerogen Type II Kerogen 0 100 200 300 400 500 0 2 4 6 8 Generated HCs in mg/gTOC Depth in km 0 20 40 60 80 0 100 200 300 400 500 Geologic Time in My Generated HCs in mgHC/gTOC 0 20 40 60 80 -40 -30 -20 -10 0 10 20 30 Geologic Time in My Generated/Cracked HC Rate in mg/gTOC/My 1 3 0 40 80 120 160 0 2 4 6 8 Generated HCs in mg/gTOC Depth in km 0 20 40 60 80 0 20 40 60 80 100 120 140 160 Geologic Time in My Generated HCs in mgHC/gTOC -30 -20 -10 0 10 20 30 0 2 4 6 8 Generated/Cracked HC Rate in mg/gTOC/My Depth in km 1 2 3 Type III Kerogen Secondary Gas Primary Gas Oil Secondary Gas Primary Gas Oil Secondary Gas Primary Gas Oil 1..Oil 2..Primary Gas 3..Secondary Gas Secondary Gas Primary Gas Oil 1..Oil 2..Primary Gas 3..Secondary Gas 2 Secondary Gas Primary Gas Oil Secondary Gas Primary Gas Oil Fig. 4.12. Comparison of kerogen type I, II and III kinetics from Fig. 4.11 with a sedimentation rate of 100 m/My. The gas is further subdivided into primary and secondary cracked gas. The generation potential is calculated for all three kinetic models. Generation rates are also shown for type IIB kerogen and for all i = 1, . . . , N and j = 1, . . . , (i − 1) and r = 1, . . . , n ij ∂yijr ∂t = p ij(Er) ni s=1 kis xis + p ij(Er) N l=i+1 Ril n il s=1 k ils yils − k ijr yijr (4.20) where Rij is the reduction factor for the secondary cracking reaction of com- ponent i to j. The total masses of the generative kerogen x and the petroleum components yi are
- 179. 166 4 Petroleum Generation Typ II Kerogen 0 100 200 300 400 500 0 2 4 6 8 Generated HCs in mg/gTOC Depth in km 0 100 200 300 400 500 0 2 4 6 8 Generated HCs in mg/gTOC Depth in km 0 2 4 8 0 100 200 300 400 500 Geologic Time in My Generated HCs in mgHC/gTOC Secondary Gas Primary Gas Oil Secondary Gas Primary Gas Oil Secondary Gas Primary Gas Oil 6 0 100 200 300 400 0 100 200 300 400 500 Generated HCs in mgHC/gTOC Geologic Time in My Secondary Gas Primary Gas Oil Fig. 4.13. Influence of different sedimentation rates (25 m/My and 1000 m/My) on oil and gas potentials of the type IIB kinetics from Fig. 4.11 x = N i=1 ni r=1 xir, yi = i−1 j=1 n ij r=1 yijr . (4.21) The three terms of equation (4.20) correspond to generated masses from kero- gen, generated masses from the heavier petroleum components and cracked masses into lighter petroleum components. The transformation ratio TR = 1−x cannot be directly derived from the actual masses yi, since the mass loss of coke depends on the ratio of primary and secondary cracked components. 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 0 10 20 30 40 50 Activation Energy in kcal/mol Frequency in % 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 0 5 10 15 20 25 Activation Energy in kcal/mol Frequency in % Boghead Coal A = 1.12x1027 My-1 HI = 551 mg/gTOC 0 Kimmeridge Clay A = 1.64x1028 My-1 HI = 369 mg/gTOC 0 C60 C50 C40 C30 C20 C10 C6 nC5 iC5 nC4 iC4 C3 C2 C1 C60 C50 C40 C30 C20 C10 C6 nC5 iC5 nC4 iC4 C3 C2 C1 Fig. 4.14. 14 component kinetics for Kimmeridge Clay after di Primio and Horsfield (2006) A typical simplification of the secondary cracking scheme is that the heav- ier components are only cracked to methane (i = 1) and only with one acti-
- 180. 4.3 Petroleum Generation Kinetics 167 vation energy. Then all n ij = 0 for j = 1 and small j’s and n ij = 1 for great j’s and (4.20) can be simplified as follows. For methane is ∂y1 ∂t = n1 r=1 k1r x1r + N i=2 Ri k i yi, (4.22) and for all other components i 1 is ∂yi ∂t = ni r=1 kir xir − k i yi (4.23) where Ri is the reduction factor for secondary cracking of a component i to methane and k i(Ai, Ei) is the reaction rate for secondary cracking of the component i based on the pair of frequency factor and activation energy. Two kinetic schemes are often used in practice, the 4-component approach using boiling point classes and the 14-component scheme which is especially suitable for predicting phase properties such as GOR, API and saturation pressure. Boiling Point Classes Espitalie et al. (1988) introduced four HC classes C1, C2 − C5, C6 − C15 and C15+ for multi–component kinetics, based on pyrolysis data of the components with similar boiling point properties. The classes have been used with small modifications in many publications and databases to describe primary crack- ing such as Ungerer et al. (1990), Behar et al. (1997) and Vandenbroucke et al. (1999). In all those papers first order parallel reactions are proposed with a unique frequency factor for each reaction. The models for secondary cracking are restricted to methane generation only as described with equation (4.22). In Fig. 4.15, the kinetics from Behar et al. (1997) for typical type II and type III kerogen are shown, based on anhydrous open and closed pyrolysis. Other kinetics are illustrated in App. G. Note, that the lumped C15+ class was split into the three classes, C15+ saturates, C15+ aromatics and NSOs in the original paper. A separation into higher molecular components is shown in Fig. 4.16 after Abu-Ali et al. (1999). In most boiling point kinetics, the activation energy distributions of the components are approximately normal distributed with increasing deviation and a higher amount of gas generation from type I to type III kerogen, similar to the oil–gas kinetics. Compositional Phase Kinetics Di Primio and Horsfield (2006) proposed a 14 component scheme and pub- lished data for eight sample kinetic models described with a classification of organic facies as shown in Fig. 4.8. These samples were measured with
- 181. 168 4 Petroleum Generation 50 52 54 56 58 60 62 64 66 68 70 72 0 5 10 15 20 25 30 Activation Energy in kcal/mol Frequency in % 46 48 50 52 54 56 58 60 62 64 66 68 0 10 20 30 40 50 Activation Energy in kcal/mol Frequency in % Type II (Paris Basin) A = 5.05x1027 My-1 HI0 = 600.3 mg/gTOC C15+ C6-C14 C2-C5 C1 Type III (Mahakam) A = 9.46x1028 My-1 HI = 193.6 mg/gTOC 0 C15+ C6-C14 C2-C5 C1 Fig. 4.15. Boiling point class kinetics according to Behar et al. (1997) for type II and type III samples from the Paris basin and the Mahakam delta 48 50 52 54 56 58 60 62 64 66 68 0 10 20 30 40 50 60 Activation Energy in kcal/mol Frequency in % Secondary Cracking C1 E=68 kcal/mol, A=8.41x10 My , R=0.7 C1 C1 29 -1 E=68 kcal/mol, A=8.41x10 My , R=0.7 E=67 kcal/mol, A=8.41x10 My , R=0.7 C1 E=67 kcal/mol, A=8.41x10 My , R=0.7 C1 E=67 kcal/mol, A=8.41x10 My , R=0.7 29 -1 29 -1 29 -1 29 -1 NSO C15+aro C15+sat C6-C14aro C6-C14sat Abu-Ali (Type II) Qusaiba A = 1.02x1028 My-1 HI = 469 mg/gTOC 0 NSO C15+aro C15+sat C6-C14aro C6-C14sat C3-C5 C2 C1 Fig. 4.16. Boiling point kinetics after Abu-Ali et al. (1999) with an additional split of the heavier petroleum components. Simple secondary cracking reactions were added by the authors based on calibration data from projects in the Gulf of Arabia combined open– and closed–system pyrolysis. A method is also proposed to determine the gas composition from characteristic petroleum properties GOR (gas oil ratio) and saturation pressures.1 The 14 component scheme takes the following classes into account: C1, C2, C3, iC4, nC4, iC5, nC5, nC6, C7 − C15, C16 − C25, C26 − C35, C36 − C45, C46 − C55 and C55+.2 The eight example kinetic models for primary cracking are shown in Fig. 4.14 for Kimmeridge Clay and in App. G for Boghead Coal, Tasmanite Shale, Woodford Shale, Tertiary Coal, Teruel Oil Shale, Toarcian Shale and Brown Limestone. The PVT behavior of a multi-component kinetic is illustrated in Fig. 4.17 and Fig. 4.18 with the Kimmeridge Clay kinetics. In the open source rock ap- proach no secondary cracking is considered. The generated petroleum masses 1 Gas composition influences phase behavior and pyrolysis alone cannot reproduce gas composition in natural rocks. 2 In the following description, the symbols C10, C20, C30, . . . are used for the higher molecular weight components C7 − C15, C16 − C25, C26 − C35, . . ., respectively.
- 182. 4.4 Thermal Calibration Parameters 169 and related maturity increase with depth. The differences in the PVT dia- grams and petroleum compositions with increasing maturity and GORs and API densities (Sec. 5) are relatively small, e.g. the GOR changes from 85 to 112 kg/kg and densities from 29.4◦ API to 29.6◦ API. The phase diagrams are typical for light oil. The effect of secondary cracking is considered in the closed source rock system approach. In the example kinetics the components C10 to C60+ are cracked to methane with a normal activation energy distribution. The gen- erated petroleum type changes from a gas condensate for early generated petroleum to dry gas for late generation with much higher GORs and APIs. 4.4 Thermal Calibration Parameters 4.4.1 Vitrinite Reflectance The most widely used thermal maturation indicator is the reflectance of the vitrinite maceral in coal, coaly particles, or dispersed organic matter. It in- creases as a function of temperature and time from approximately Ro = 0.25% at the peat stage to more than Ro = 4% at the meta-anthracite stage. Vitrinite is a very complex substance and undergoes a complicated series of changes during pyrolysis. The general reaction leads to the assumption that vitrinite is transformed to residual (modified or mature) vitrinite and some condensate. Vitrinite ki −→ Residual Vitrinite + Volatiles (4.24) Three models were proposed in the 1980s by Waples (1980); Larter (1988); Sweeney and Burnham (1990) and they are still very popular. Burnham Sweeney Model The model from Burnham and Sweeney (1989); Sweeney and Burnham (1990) uses distributions of activation energies for each of the four reactions: the elimination of water, carbon dioxide, methane and higher hydrocarbons. Each of the reactions is described as a parallel decomposition reaction with a set of discrete distributed activation energies. vitrinite k1i −→ residual vitrinite + H2O , vitrinite k2i −→ residual vitrinite + CO2 , vitrinite k3i −→ residual vitrinite + CHn , vitrinite k4i −→ residual vitrinite + CH4 . (4.25) The vitrinite reflectance value is then calculated with the four transformation ratios of the reactions. In a simplified version of the model (Easy–Ro model),
- 183. 170 4 Petroleum Generation Phase diagram of petroleum generated at 6 km depth Phase composition of petroleum generated at 6 km depth flashed to surface condition Critical Point API: 29.4 GOR: 85 Liquid Fraction: 53.9 mol % 89.7 mass % 1.16 vol % m /m 3 3 API: 29.6 GOR: 112 Liquid Fraction: 46.9 mol % 87.5 mass % 0.88 vol % m /m 3 3 Phase diagram of petroleum generated at 3 km depth Phase composition of petroleum generated at 3 km depth flashed to surface condition 0 100 200 300 400 0 2 4 6 8 Generated HCs in mg/gTOC Depth in km C1 C2 C3 iC4 nC4 iC5 nC5 C6 C10 C20 C30 C40 C50 C60 Critical Point Fig. 4.17. Generated petroleum components and related phase diagrams for Kim- meridge Clay kinetics without cracking (open source rock system)
- 184. 4.4 Thermal Calibration Parameters 171 Phase diagram of petroleum generated at 4 km depth Phase composition of petroleum generated at 4 km depth flashed to surface condition Pure Vapor Phase Phase diagram of petroleum generated at 5 km depth Phase composition of petroleum generated at 5 km depth flashed to surface condition 0 100 200 300 400 500 600 700 0 2 4 6 8 Generated HCs in mg/gTOC Depth in km C1 C2 C3 iC4 nC4 iC5 nC5 C6 C10 C20 C30 C40 C50 C60 API: 45 GOR: 1060 m /m Liquid Fraction: 9.37 mol % 47.8 mass % 0.094 vol % 3 3 Fig. 4.18. Generated petroleum components and related phase diagrams for Kim- meridge Clay kinetics with cracking (closed source rock system)
- 185. 172 4 Petroleum Generation the kinetics is approximated by superposition of the four reactions with only one frequency factor. The final distribution of the activation energies is shown in Fig. 4.19. The reflection value of the Easy–Ro model is then exponentially correlated with the TR to the interval [0.20%, 4.66%], as follows: Ro[%] = 0.20 4.66 0.20 TR . (4.26) Carr (1999) extended the above model to incorporate overpressure u retarda- tion, as there is an expansion of the products due to the generation of volatiles. The pressure dependency is proposed to be included via a modified frequency factor A(u). A(u) = Ah e−u/c (4.27) where Ah = 3.17×1026 My−1 is the original frequency factor from the Easy–Ro model, defined here as the hydrostatic frequency factor, u is the overpressure, and c = 590 psi is a scaling factor. 34 36 38 40 42 44 46 48 50 52 54 56 58 60 62 64 66 68 70 72 0 2 4 6 8 10 Activation Energy in kcal/mol Frequency in % 30 31 33 35.1 37 38.9 40.7 42.6 44.5 46.3 48.2 50.1 51.9 53.8 56 58 0 5 10 15 20 25 Activation Energy in kcal/mol Frequency in % Sweeny Burnham Larter A = 3.17x1026 My-1 A = 2.36x1022 My-1 Fig. 4.19. Distributions of the activation energies used in the vitrinite models of Sweeney and Burnham (1990) and Larter (1988) Larter Model Larter (1988) utilized quantitative pyrolysis gas chromatographic data of iso- lated vitrinite kerogen to achieve the concentrations of structurally specific moieties (alkylphenol precursor) in vitrinite as a function of rank. The kinet- ics describe the phenol precursor loss from vitrinite. precursor ki −→ phenol . (4.28) The pyrolysis data are inverted into a parallel decomposition reaction with normal Gaussian distributed activation energies of mean energy Ē = 186 kJ, standard deviation σ = 13.0 kJ, and frequency factor A = 2.36 × 1022 My−1 . An approximated discrete distribution is shown in Fig. 4.19. The phenol yield
- 186. 4.4 Thermal Calibration Parameters 173 is then linearly correlated to the interval [0.45%, 1.60%] with vitrinite re- flectance. Ro[%] = 0.45 + 1.15 TR (4.29) where TR is the transformation rate of the parallel reaction. The model limits the reflection value to 1.6% since the study is based on chemical reactions primarily in oil generating zones. TTI Model This model was proposed by Waples (1980). It calculates TTI (Time–Temper- ature–Index) maturity, which is converted to vitrinite reflectance. The defini- tion of the TTI is based on the assumption that the maturity rate of vitrinite almost doubles every 10◦ C. TTI = n (Δtn) 2n (4.30) where the integer n represents a temperature interval with the following scheme: n = −1 for T = [90◦ C, 100◦ C], n = 0 for T = [100◦ C, 110◦ C], n = 1 for T = [110◦ C, 120◦ C], n = 2 for T = [120◦ C, 130◦ C] . . . . (4.31) he value Δtn is the time in My spent by the vitrinite in the corresponding temperature interval. The TTI value is converted to vitrinite reflectance with empirical functions, e.g. with the following rule by Goff (1983). Ro[%] = 0.06359 (1444 TTI)0.2012 . (4.32) Although this model is only rule based, it is still used. Vitrinite reflectance is the most common thermal maturity parameter and is generally used for calibration of heat flow history. Herein, paleo-heat flow values as the most uncertain input parameters are changed according to differ- ences between calculated and measured vitrinite reflectance parameters. This requires reliable vitrinite reflectance models and measurements. Calculated vitrinite reflectance models are shown in Fig. 4.20 for different sedimentation curves. They differ in the initial (immature) value on the surface and further at greater depth. The most commonly used model is the Burnham Sweeney model. Vitrinite reflectance curves commonly have offsets at erosional discor- dances (Fig. 4.21) as the uplifted layers have already experienced a heating period with a corresponding increase in maturity. The offset is typically higher for larger erosional thickness. Thus, jumps in the vitrinite data at a certain depth can be used to estimate the erosion thickness. The maturation process
- 187. 174 4 Petroleum Generation a) b) c) d) 0 0.5 1 1.5 2 2.5 3 0 2 4 6 8 Vitrinite Reflectance Ro in % Depth in km 1 2 3 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 0 2 4 6 8 Vitrinite Reflectance Ro in % Depth in km 1 2 3 0 0.5 1 1.5 2 2.5 3 0 2 4 6 8 Vitrinite Reflectance Ro in % Depth in km 1 2 3 0 0.5 1 1.5 2 2.5 3 0 2 4 6 8 Vitrinite Reflectance Ro in % Depth in km 1 2 3 1..S = 25 m/My 2..S = 100 m/My 3..S = 1000 m/My 1..S = 25 m/My 2..S = 100 m/My 3..S = 1000 m/My 1..S = 25 m/My 2..S = 100 m/My 3..S = 1000 m/My 1..TTI Model 2..Larter Model 3..Burnham/Sweeny Model Fig. 4.20. Comparison of calculated vitrine reflectance curves after Burnham Sweeney (a), Larter (b), and Waples (c) for different sedimentation curves. Figure (d) shows models together with a sedimentation rate of 100 m/My significantly slows during a hiatus and uplift as there is no further temperature increase, but it does not completely stop. Vitrinite reflectance can be generally correlated with source rock matu- ration and the stage of oil and gas generation. The following intervals define empirical relationships for primary oil and gas generation from kerogen of type II and type III: immature (Ro ≤ 0.55 %), early oil(Ro ≤ 0.70 %), main oil (Ro ≤ 1.00 %), late oil (Ro ≤ 1.30 %), wet gas (Ro ≤ 2.00 %), dry gas (Ro ≤ 4.00 %), overmature (Ro ≥ 4.00 %). This correlation is not always reliable as the generation histories of different petroleum kinetics can signifi- cantly differ from each other. In Fig. 4.22 the correlation of the transformation ratio with the vitrinite reflectance value is shown for the eight different multi– component data sets of Fig. 4.14 and App. G from di Primio and Horsfield (2006) assuming an average sedimentation rate of 100 m/My. The differences
- 188. 4.4 Thermal Calibration Parameters 175 0.5 0.6 0.7 0.8 0.9 2.6 2.8 3 3.2 3.4 3.6 Vitrinite Reflectance Ro in % Depth in km 1 2 3 4 1..he=0 (no erosion) 2..he=2 km 3..he=3 km 4..he=4 km a) 0 0.2 0.4 0.6 0.8 1 1.2 0 1 2 3 4 5 Vitrinite Reflectance Ro in % Depth in km b) VR =0.609 VR =0.763 0 0 Fig. 4.21. Calculated vitrinite reflectance curves can have significant offsets at erosion horizons (a). A discordance occurs at 3 km depth with eroded thicknesses of 2, 3, and 4 km, which corresponds to a maximum burial depth of the eroded horizon of 2, 3 and 4 km. In case of 4 km erosion, the offset is approximately the difference of the vitrinite reflectance values between 3 and 4 km of uniformly deposited layers (b). The Burnham Sweeney model is used here and the sedimentation rate is 100 m/My are obvious, e.g. the Ro interval for main oil corresponds to the TR intervals of (70%, 97%) for Brown Limestone and (5%, 72%) for Alaskan Tasmanite. 0 20 40 60 80 100 0 0.5 1 1.5 2 2.5 Transformation Ratio in % Vitrinite Reflectance in % 0 20 40 60 80 100 0 0.5 1 1.5 2 2.5 Transformation Ratio in % Early Oil Main Oil Late Oil Wet Gas Dry Gas Immature Kimmeridge Clay Brown Limestone Tertiary Coal Boghead Coal Alaskan Tasmanite Woodford Shale Toarcian Shale Teruel Oil Shale Fig. 4.22. Correlation of transformation ratio and vitrinite reflectance There are pitfalls for the interpretation of vitrinite reflectance, such as contamination and sampling problems, misidentification of vitrinite, and poor sample preparation, but the main problem is the dependency on kerogen types. CSIRO Australia proposed a more comprehensive method to take into account
- 189. 176 4 Petroleum Generation the fluorescence measurements of multiple macerals (Wilkins et al., 1998). The method is called FAMM (Fluorescence Alteration of Multiple Macerals). The fluorescence intensities of vitrinite, liptinite and inertinite are measured for different maceral fragments. They are plotted on a fluorescence alteration diagram, which can be used to derive an equivalent vitrinite reflectance value. 4.4.2 Molecular Biomarkers The molecular characteristics of kerogen and petroleum during catagenesis are controlled by the types of deposited organisms, the environmental and preservation conditions and the thermal maturity. Gas chromatography and gas chromatography–mass spectrometry show the relative abundance of in- dividual biomarkers during kerogen pyrolysis (Fig. 4.23). Chromatographic fingerprints and correlations can therefore be used to determine the organic facies types, the thermal maturation state and the degree of biodegradation. Peters et al. (2005) worked out comprehensive lists, which link high concentra- tions of biomarkers and relative ratios of special biomarkers to the biological origins and environments (App. H). For example, biomarkers in crude oil samples can be used to distinguished an origin from shale or carbonate source rocks. The frequency of some biomarkers can also be used to determine the deposition age and the thermal maturity (App. H). Examples are the isomer- ization of the C20 chiral carbon in steranes and of C22 chiral carbon in hopanes and the aromatization of steroid hydrocarbons. These reactions occur before and during the early stages of oil formation. They are commonly described as unimolecular equilibrium reactions with forward and backward reaction rates. Reactant kf ;kb ←→ Product, ∂x ∂t = − ∂y ∂t = −kf x + kb y . (4.33) The reaction rate of the forward reaction is calculated from a single com- ponent Arrhenius law with the activation energy E and the frequency factor A. The ratio between forward and backward reactions c = kb/kf is usually in- dependent of temperature and is usually given as an input parameter instead of specifying the backward reaction rate directly. Sample values for A, E and c are experimentally determined and are described by several authors (Sajgo and Lefler, 1986; Mackenzie and McKenzie, 1983) as shown in Table 4.4.2. Fig. 4.24 shows calculated transformation ratios of the isomerization and aromatization reaction. Similar to all Arrhenius type reactions, the transfor- mation ratio is controlled by the temperature history and therefore by the sedimentation rate. Finally, every equilibrium reaction approaches a transfor- mation ratio equal to c/(c + 1). Other molecular biomarker indices are the methylphenanthrene index (MPI) from Radke and Welte (1983), the methy- ladamantane index (MAI), methyldiamantane index (MDI) from Chen et al. (1996), and the trisnorhopane ratio from Peters et al. (2005).
- 190. 4.4 Thermal Calibration Parameters 177 Fig. 4.23. Gas chromatograms E in kcal A in My−1 c Sterane Aromatization Mackenzie and McKenzie, 1983 47.769 5.680 × 1027 0 Rullkoetter and Marzi, 1988 43.324 1.529 × 1024 0 Sajgo and Lefler, 1986 29.044 3.345 × 1016 0 Sterane Isomerization at C20 Mackenzie and McKenzie, 1982 21.496 1.893 × 1011 1.174 Rullkoetter and Marzi, 1988 40.363 1.533 × 1022 1.174 Sajgo and Lefler, 1986 21.878 7.574 × 1010 1.380 Hopane Isomerization at C22 Mackenzie and McKenzie, 1982 21.496 5.050 × 1011 1.564 Rullkoetter and Marzi, 1988 40.124 2.554 × 1022 1.564 Sajgo and Lefler, 1986 20.971 1.104 × 1012 1.326 Table 4.1. Kinetic parameters for biomarker models 4.4.3 Tmax Values The Tmax value is an output of the Rock–Eval pyrogram: the oven temperature at the peak S2 (Fig. 4.7). In the Rock–Eval pyrolysis, the contained petroleum (S1–curve) and the remaining kerogen potential (S2–curve) are measured by heating chips of the rock sample until all remaining kerogen is cracked to hydrocarbons. The remaining kerogen potential xij can be taken from a simulation. The S2–curve of the Rock–Eval plot can then calculated for each matrix xij of the remaining kerogen potential and the calculated Tmax value can be compared to measured Tmax values from Rock–Eval pyrolysis. The equations (4.3) for bulk kinetics, (4.12) for oil–gas kinetics, (4.19), and for multi–component kinetics
- 191. 178 4 Petroleum Generation 0 20 40 60 80 100 0 2 4 6 8 TR in % Depth in km 1 2 3 0 10 20 30 40 50 60 0 2 4 6 8 TR in % Depth in km 1 2 3 0 10 20 30 40 50 60 70 0 2 4 6 8 TR in % Depth in km 1 2 3 1..S = 25 m/My 2..S = 100 m/My 3..S = 1000 m/My 1..S = 25 m/My 2..S = 100 m/My 3..S = 1000 m/My 1..S = 25 m/My 2..S = 100 m/My 3..S = 1000 m/My Hopane Isomerization Sterane Aromatization Sterane Isomerization Fig. 4.24. Calculated molecular biomarker ratios with kinetic parameters of Mackenzie and McKenzie (1983) for the isomerization of hopanes and steranes. The conversion curves are shown for three different sedimentation rates S are used taking a higher heating rate into account. Sweeney and Burnham (1990) proposed to start with 150 ◦ C in time steps of 0.5 ◦ C until the first decrease of the rate occurs. The procedure is more reliable, when the entire S2–curve is calculated, since more complex multi–component kinetics often yield more than one peak. The Rock–Eval curve and Tmax depend on the heating rate, which is here assumed to be 25 ◦ C/min. Example calculations for the Kimmeridge Clay kinetics (Fig. 4.14) are shown in Fig. 4.25 for different maturation levels. Measured Tmax values always increase with higher thermal maturity of the rock sample, since the remaining activation energies are the highest available and higher temperatures are needed to crack the high activation energy bonds. Calculated Tmax values can drop with increased maturity when different fre- quency factors are used for the components such as for the oil–gas kinetics of Pepper and Corvi (1995a); Pepper and Dodd (1995) of Fig. 4.11 (kerogen type IIB, Fig. 4.26). Higher frequency factors yield smaller shifts when upscaled from geological to laboratory heating rates (Fig. 4.4). For example, both pairs of activation energies and frequency factors: (E = 55 kcal/mol, A = 1014 s−1 ), (E = 65 kcal/mol, A = 2 × 1018 s−1 ), yield peak temperatures of 170 ◦ C at a geological heating rate of 10 K/My, but the peak temperatures for labora- tory heating rate of 5 K/min are 460 ◦ C and 420 ◦ C, respectively. This means, that for the considered Pepper kinetics, the oil peak moves to higher tem- peratures than the gas peak when transferring to the laboratory scale. Thus, kinetics with different frequency factors for oil and gas (in the very popular
- 192. 4.4 Thermal Calibration Parameters 179 450 500 550 600 0 2 4 6 8 Tmax in Celsius Depth in km 0 100 200 300 400 0 2 4 6 8 Generated HCs in mg/gTOC Depth in km Gas Oil A B C D 400 450 500 550 600 0 0.5 1 1.5 2 Temperature in Celsius Frequency of generated HCs in mg/s A B C D A..in 3 km depth B..in 4 km depth C..in 5 km depth D..in 6 km depth Fig. 4.25. Calculated Tmax values for the Kimmeridge Clay kinetics of (Fig. 4.14). The left diagram shows the generated masses of gas (C1 . . . C5) and oil (C6 . . . C60+). The middle diagram shows the calculated Rock–Eval S2–curves for samples from four different depths with the corresponding Tmax values. The entire Tmax versus depth curve is shown in the right diagram Pepper kinetics the frequency factors differ by four orders of magnitude) are not suitable for the use of calculated Rock–Eval plots. 0 200 400 600 800 1000 0 2 4 6 8 Generated HCs in mg/gTOC Depth in km A B C D E 440 460 480 500 520 540 0 2 4 6 8 Tmax in Celsius Depth in km 350 400 450 500 550 0 0.5 1 1.5 2 2.5 3 Temperature in Celsius Frequency of generated HCs in mg/s A..in 3 km depth B..in 4 km depth C..in 5 km depth D..in 6 km depth E..in 7 km depth Gas Oil Fig. 4.26. Calculated Tmax values for the oil–gas kerogen type IIB Pepper and Corvi (1995a); Pepper and Dodd (1995) kinetics of Fig. 4.11
- 193. 180 4 Petroleum Generation Calculated Tmax values are very well correlated to the transformation ratios (TR) from the kinetics. In all the multicomponent phase kinetics shown in Fig. 4.14 and App. G, the TR versus Tmax curves are almost independent of the thermal history. The TR versus Tmax curves differ by less than 0.1% for sedimentation rates between 20 and 1000 m/My for one organic facies, but the curves vary for different organic facies (Fig. 4.27). Hence, there is no general correlation between vitrinite reflectance and Tmax. 460 480 500 520 540 0 20 40 60 80 100 Tmax in Celsius TR in % 460 480 500 520 540 0.2 0.4 0.6 0.8 1 1.2 1.4 Tmax in Celsius Vitrinite Reflectance in % Kimmeridge Clay Brown Limestone Tertiary Coal Boghead Coal Kimmeridge Clay Brown Limestone Tertiary Coal Boghead Coal Fig. 4.27. Correlation of vitrinite reflectance and Tmax for various multi–component kinetic models. The kinetic data for the samples are from Fig. 4.14 and App. G The production index PI and the hydrogen index HI can also be derived from the calculated Rock–Eval curve and used for calibration of kinetic pa- rameters. 4.4.4 Isotopic Fractionation Thermally generated gas components become isotopically heavier with in- creasing maturity, as C12 –C12 bonds are easier to break than C13 –C12 bonds. Cramer et al. (2001) analyzed light gases generated from coals by open system pyrolysis, gas chromatography and mass spectrography to measure δ13 values of methane, ethane and propane with increasing maturity. The δ13 value is defined as follows: δ13 = 1000 R R0 − 1 = 88990 m13 m12 − 1000 (4.34) where R is the isotopic ratio, R0 = 0.01124 the standard isotope ratio, and m12 , m13 are the masses of the C12 and C13 molecules, respectively. Cramer et al. (2001) proposed to distinguished between C12 and C13 methane each
- 194. 4.4 Thermal Calibration Parameters 181 generated with a slightly different rate described by a constant shift in the ac- tivation energy distribution ΔE of 25 cal/mol. The isotopic ratio R of methane can then be tracked in petroleum migration and accumulation analysis. The isotopic fractionation can be used to distinguish between biogenic and ther- mogenic gases. 4.4.5 Fission–Track Analysis Fission–tracks are trails of disrupted atoms in the crystal structures of vari- ous minerals caused by radioactivity. The trails can be etched using acid so that they become visible under a microscope. For example they are found in apatites and are almost exclusively generated by the spontaneous fission of the uranium isotope 238 U. It has a decay constant of 4.92 × 10−18 s−1 = 1.55 × 10−4 Ma−1 , which corresponds to a half life of 4471 Ma. Fission–tracks show a temperature dependent annealing and are thus used to constrain paleo– temperature histories. Introductions to fission–track analysis are given in Gal- lagher et al. (1998), and Beardsmore and Cull (2001). Track length distributions can be predicted in a forward modeling ap- proach from temperature histories which can easily be extracted from basin models. These distributions can be compared with sample data for calibration purposes. In the case of good data quality, the temperature history can even be evaluated based on an inversion of forward modeling (Ketcham et al., 2000). This section deals with a short summary of forward modeling of the track length distributions only. More details can be found in Green et al. (1986); Laslett et al. (1987); Duddy et al. (1988); Green et al. (1989a,b); Carlson et al. (1999), and Donelick et al. (1999). Track length distributions are usually formulated with the reduced track length r = l/l0 with l the confined track length during annealing and l0 the initial track length. Forward modeling of track length reduction through time is described in Laslett et al. (1987). According to their results it is possible to fit the relationship between time t in seconds and annealing described by r at constant temperature T in Kelvin by a “fanning” fission–track Arrhenius fit of the form ln t = A(r) + B(r)/T (4.35) with e.g. A = −28.12, B(r) = [g(r) + 4.87]/0.000168 (4.36) and g(r) = ([(1 − r2.7 )/2.7]0.35 − 1)/0.35 . (4.37) In the case of time dependent temperature T(t) (4.35) can be differentiated and becomes dr dt = 1 A(r) + B(r)/T(t) e−A(r)−B(r)/T (t) + B(r) T2 dT dt (4.38)
- 195. 182 4 Petroleum Generation with initial condition r(0) = 1. The rate equation (4.38) can be solved more easily than the implicit formulation (4.35). However, it must be discretized with caution for a numerical solution. The uranium isotope 238 U is constantly producing new fission tracks with an initial track length distribution of Gaussian form and standard deviation σ = 0.0704 μm (Ketcham, 2003). The resulting present day track length dis- tribution is thus a sum of all track length distributions that were generated continuously over time and annealed according to their generation time. It must be taken into account that the transformation from reduced track length distribution r to observable track length density ρ is biased according to Fig. 4.28. Length distributions must be weighted according to this bias before summing them up. Fig. 4.28. Bias between length r and observable density ρ according to Green (1988); Ketcham (2003) This bias must also be considered if pooled fission track ages ta are cal- culated from reduced track length distributions. According to Ketcham et al. (2000) it is ta = 1 ρst ρ(t) dt (4.39) with the integral limits ranging from sedimentation time to present day. The density ρst is given by the ratio of present–day spontaneous mean track length to mean induced track length with a typical value of ρst = 0.893. Examples of forward modeled track length distributions are shown in Fig. 4.29. Temperature histories can now be calibrated against track length distributions found in real samples. Technically, this can be done with a Kolgomorov–Smirnov test which is designed for comparison of two distribu- tions (Press et al., 2002). Finally, with the outlined procedure only basic approaches of fission track modeling are covered. Topics, such as the integration of track projections to the crystallographic c–axis into the forward modeling procedure or more advanced models incorporating important factors such as the Cl–content, are not included here. These topics would exceed the capacity of this volume.
- 196. 4.5 Adsorption 183 ta = 18.6 Ma ta = 9 Ma ta = 8.4 Ma ta = 11.1 Ma Fig. 4.29. Examples of temperature histories, forward modeled apatite fission track length distributions and pooled ages ta. The marked temperature interval between 60 ◦ C and 110 ◦ C is called “Partial Annealing Zone” 4.5 Adsorption Unreacted and inert kerogen can bind the generated petroleum components by adsorption before releasing them into the open pore space of the source rock. The term expulsion is used to specify the amount of petroleum (phases) pass- ing from the source rock to the carrier interface: that means it encompasses all the processes the petroleum molecule is undergoing within the source.3 The main processes and nomenclature for petroleum migration from source 3 Generally, the term expulsion is used differently in the literature, for desorption and/or primary migration.
- 197. 184 4 Petroleum Generation to reservoir are shown in Fig. 4.30 and are considered as a chain of discrete steps. Primary Migration Accumulated Fluid Phases Secondary Migration Source Rock Kerogen Adsorption Container Pore Space Primary Cracking and Adsorption Desorption and Dissolution Secondary Cracking I Secondary Cracking II Secondary Cracking III Carrier Rock Reservoir Rock Solid Organic Particles Migrating Fluid Phases Source Rock Fluid Phases Adsobed Components Dismigration Cap Rock Fig. 4.30. Adsorption and migration processes • Primary Cracking: kerogen is cracked to petroleum components via first order parallel reactions. • Adsorption: the primary generated components are adsorbed within the unreacted kerogen and coke in the source rocks. The adsorbed amounts are not included in the petroleum in the pore space. The amount of solid organic particles and coke controls the maximum adsorption amounts. Ad- sorbed amounts can be further (secondary) cracked following a secondary cracking scheme. The generated coke as a by-product in secondary cracking increases the maximum adsorption amounts. • Desorption: adsorption amounts can be released into the pore space when the generation amounts exceed the maximum adsorption amounts. The maximum adsorption amounts are therefore modeled as a container that the petroleum must fill completely before entering the pore space. • Dissolution: the desorbed components are dissolved in the fluid phases. Further transport in the pore space is now controlled by the phase (instead of the component) properties and handled with models for fluid flow. • Primary Migration: the fluids have to pass through the source rock pore system until their expulsion into the carrier. A critical saturation value (initial oil saturation or endpoints in the capillary entry and relative per- meability curves) is used in Darcy type models to quantify the amount
- 198. 4.5 Adsorption 185 of initial and residual saturation during expulsion (Sec. 6.3). This value therefore acts as a second container for the expulsion of petroleum. In Darcy flow models, the time for the movement through the source net- work is controlled by permeabilities. Secondary cracking is also modeled for petroleum in the free pore space of source rocks. • Secondary Migration: all further petroleum transport in carrier rocks, reservoirs and through seals is modeled by Darcy, flowpath and/or invasion percolation models (Chap. 6). There is controversy in the literature about whether the secondary cracking scheme in reservoirs is different from or similar to that in source rocks. Most basin modeling programs allow the use of three different (adsorption, source and reservoir) secondary cracking schemes. Quantification of the adsorbed masses is the target of different adsorption models. Herein it is assumed that the adsorption capacity is proportional to the amount of available kerogen, which is given by a crackable or reactive and an inert fraction of kerogen mker = mker,reac + mker,inert . (4.40) Initially, before cracking started, a fraction r of mker,0 is assumed to be inert and a fraction 1 − r is reactive. The inert fraction can directly be evaluated to mker,inert,0 = r 1 − r mker,reac,0 . (4.41) Substitution of this result and of mker,reac = mker,reac,0(1 − TR) into (4.40) yields mker = mker,reac,0 1 1 − r − TR . (4.42) with mker,reac,0 = TOC0 V (1 − φ) ρrHI0 where V is the bulk volume, φ the porosity and ρr the rock density. Hydrocarbons can adsorb on the surface of kerogen. The hydrocarbons are called adsorbate and the kerogen adsorbent. This process is typically described by three different approaches (en.wikipedia.org/wiki/Freundlich equation, en. wikipedia.org/wiki/Adsorption): The Freundlich equation is an empirical equation of the form mads = mker KF c1/n (4.43) where c describes the molar concentration of the adsorbate in the solution and with constants KF and n for a given combination of adsorbent and adsorbate at a particular temperature. The Langmuir equation is based on a kinetic approach of adsorbate molecules which adsorb on the surface of the adsorbent. It is further assumed that the adsorbent is uniform, that the adsorbate molecules do not interact and that only one monolayer of adsorbate is formed. This finally yields
- 199. 186 4 Petroleum Generation mads = mads,maxKLc 1 + KLc (4.44) with mads,max as the maximum mass which can be adsorbed in one monolayer on the kerogen surface and KL = kads/kdes as the ratio of reaction rate con- stants for adsorption and desorption. Obviously, KL is strongly dependent on the temperature. It may be possible to model kads and kdes with an Arrhenius law. The BET theory is an enhancement of (4.44) for the adsorption of multiple layers of adsorbates. It becomes mads = mads,maxKBc (1 − c) [1 + (KB − 1)c] (4.45) with KB similar to KL and mads,max again as the mass of one monolayer. The Langmuir equation can easily be extended for multiple types of molecules which might be adsorbed with the assumption, that each molecule of each type occupies the same space on the adsorbent. The adsorbed amount of component i becomes mads,i = mads,max,iKL,ici 1 + k KL,kck (4.46) with KL,i the Langmuir factor, mads,max,i the mass of one monolayer, and ci the concentration of species i. The mass of one monolayer can be estimated to be mads,max = g m 2/3 ker (4.47) with a geometrical factor g. It can be assumed that the kerogen is distributed randomly in the sediments so that the exponent 2/3 should be replaced by an exponent from a fractal theory. A value about 1 can be interpreted as an adsorption potential proportional to the number of kerogen atoms, which is consistent with the Freundlich equation. In practice, temperature and adsorption kinetics are often ignored. Ad- sorbed masses are equal to maximum adsorption amounts if enough adsorbates are available. The maximum adsorbed mass is assumed to be proportional to the mass of kerogen, which encompasses the actual generative and the inert kerogen. Two different basic approaches are commonly in use. Pepper and Corvi (1995b) proposed adsorption of each species independent from other species (Fig. 4.31). The following formulation is often used: mads,max,i = ai mker (4.48) where ai is the adsorption coefficient and mads,max,i the (maximum) adsorp- tion amount of component i.
- 200. 4.5 Adsorption 187 CH4 C15+ C2 -C5 C6 -C14 Maximum Adsorption Masses Secondary Cracking Primary Cracking Reaction Total Organic Carbon (TOC) Generative Kerogen Inert Kerogen Release into the Free Pore Space Fig. 4.31. Adsorption model for independent species Alternative adsorption models exist. Again the total adsorption mass is proportional to the available kerogen mass. The calculation is similar to that of the adsorption model for independent species with only one adsorption coefficient a as mads,max = a mker . (4.49) In a common formulation with interacting component adsorption, the ad- sorption of each component is assumed to be proportional to the relative mass concentration of the adsorbate in the solution. This yields for ms,i as the mass of component i in solution and Δms,i as a small amount of component i which is currently adsorbed Δms,i wims,i = Δms,k wkms,k . (4.50) It is ms,i = m0,i − ma,i with m0,i the total initial petroleum adsorbate mass and ma,i the already adsorbed mass of component i. The factor wi is an extra weight factor which is introduced for different adsorption amounts of different components. Assuming that there are three components available with weights 1:2:3, then the components are adsorbed in the ratio of 1:2:3 of the unadsorbed relative masses, as long as there are enough generated masses available or until the surface is completely covered (Fig. 4.32). Equation 4.50 can be rewritten in differential form and integrated for the total adsorption mass of each component. It becomes ms,i m0,i 1/wi = ms,k m0,k 1/wk . (4.51) If ms,i is known, ms,k can be calculated. Therefore only the dissolved mass ms,i of one component has to be known. Only two cases have to be consid- ered for the full solution. First, i m0,i ≤ mads,max which is trivial, because
- 201. 188 4 Petroleum Generation Maximum Adsorption Mass Secondary Cracking Primary Cracking Reaction Total Organic Carbon (TOC) Generative Kerogen Inert Kerogen Release into the Free Pore Space CH4 C15+ C2 -C5 C6 -C14 Fig. 4.32. Adsorption model for interacting species everything is adsorbed. Second, i m0,i mads,max. For m0 = i m0,i this yields the condition m0 − mads,max = i ms,i = i m0,i ms,k m0,k wi/wk (4.52) for one fixed reference component k. With ms = i ms,i = m0 − mads,max one may alternatively write ms = i m0,ix wi/wk k (4.53) with xk = ms,k/m0,k. The root xk of this equation can easily be found, e.g. with the Newton–Raphson method. Note that the values of the weights are only important in the ratios relative to each other. For comparison with the adsorption model for independent species they can be scaled according to awi = ai so that the total adsorption becomes similar (Fig. 4.33). Very often there is no data available for a differentiation between gener- ated and expelled compositions. The generation kinetic is thus sometimes cal- ibrated against expulsion. For prevention of a compositional error adsorption must in such a case be identical to the composition of the already generated petroleum. However, wrong results may occur when adsorbed amounts are released later. 4.6 Biodegradation In shallow reservoirs microbes can transform or biodegrade crude oil. The resulting oil is biodegraded. Its density and viscosity is increased and its com- position has changed significantly. The process is assumed to occur mainly at
- 202. 4.6 Biodegradation 189 Fig. 4.33. Adsorption model for four independent species on the left and with interaction on the right. The species are adsorbed with a relative weight of 4 : 3 : 2 : 1. The initial masses m0,i are distributed according to these relative weights on the left and according to equal values of one on the right. In a different case with equal initial mass amounts on the left, the adsorption is additionally limited by the available amount of each species. With increasing total adsorbed mass, a bigger part of the remaining species must be adsorbed. However, all cases are becoming the same for small adsorption masses the free oil water contact (OWC). It is further assumed to be mainly depen- dent on the size of this area, the temperature, the composition of the oil, the filling history and the supply of additional nutrients which are also needed by the microbes (Fig. 4.34). When the temperature rises above a threshold of about 80◦ C almost all biodegradation stops. This effect is called paleopas- teurization. Fig. 4.34. Scheme for biodegradation at oil water contact Nutrients Hydrocarbons Biodegradation Fresh Oil The biodegradation process is not fully understood and quantitative ap- proaches are at best approximations. For example, the biodegradation rate of a lumped compound class is often assumed to follow a simple decay law of the form dmi dt = −rir A (4.54)
- 203. 190 4 Petroleum Generation with mi as the mass of components in compound class i, ri the relative degra- dation rate of class i, r the total degradation rate and A the size of the free OWC area (Fig. 4.35). The total degradation rate r can be approximated by r = ⎧ ⎨ ⎩ rmax, T ≤ T1 rmax exp − (T − T1)2 2σ2 , T T1 (4.55) (Blumenstein et al., 2006; Krooss and di Primio, 2007). The parameters T1 and σ describe the form of the temperature dependency of the model. Tem- perature T1 defines the limit above and σ the temperature range over which biodegradation decreases. Biodegradation is assumed to have a maximum rate rmax below T1, which is dependent on the composition of the oil and the indi- vidual degradation environment particularly the supply of nutrients. Typical values are T1 = 50◦ C, σ = 10◦ C and rmax = 100 kg/m2 /My (Fig. 4.36) and relative degradation rates ri around 0.8 for light components up to C7 and down to 0.1 for e.g. C56+ (Table 4.2). Fig. 4.35. Structural (left) and a stratigraphic (right) traps with their free oil water contact (OWC) area. Free OWC areas can easily be calcu- lated to a high degree of accuracy in reservoir analysis based on flowpath models (Chap. 6.5) Hydrocarbons Free OWC Area Free OWC Area HC Fig. 4.36. Biodegradation rate ac- cording to equation (4.55) s T1 Not every chemical component of oil is degradable. Hence each compound class representing a group of components, is often separated into degradable and non degradable fractions. This fraction of biodegradability is usually as- sumed to be about 1.0 for all light components and decreasing for heavier compound classes starting at C6 down to almost 0 for C56+ (Table 4.2). As a product of biodegradation, methane is obviously an exception with degrad- ability 0. The concept of degradability fractions limits the degradation within
- 204. 4.7 Source Rock Analysis 191 each compound class. This ensures that the mass decrease according to equa- tion (4.54) stops at reasonable values. Compound Degradation Degradability Class Rate ri Methane 0.00 0.00 Ethane 0.40 1.00 Propane 1.00 1.00 i–Butane 0.80 1.00 n–Butane 1.00 1.00 i–Pentane 0.70 1.00 n–Pentane 0.80 1.00 n–Hexane 0.80 1.00 C7−15 1.00 0.80 C16−25 1.00 0.60 C26−35 0.80 0.40 C36−45 0.30 0.20 C46−55 0.20 0.10 C56+ 0.10 0.02 Table 4.2. Example of degradation rates and degradability fractions according to Blumenstein et al. (2006); Krooss and di Primio (2007) Biodegradation is assumed to follow an overall reaction scheme such as 4 C16H34 + 30 H2O −→ 49 CH4 + 15 CO2 (4.56) (Zengler et al., 1999; Larter, 2007). The products of biodegradation are pri- marily methane and carbon dioxide. Other products are generated in rather small amounts and are therefore neglected in quantitative modeling although degraded oils with residual nitrogen or sulfur containing products are found. The generated amount of methane and carbon dioxide is estimated by stoi- chiometry and a balance of carbon atoms. An example of biodegradation is shown in Fig. 4.37 and Fig. 4.38. The API decreases by 9◦ API. All components from ethane up to n–hexane are fully degraded and additional methane is produced. The enhancement of heavy components is a relative effect. Biodegradation generally reduces the amount of oil. Lighter components degrade faster and hence the relative amount of heavier components increases. Besides this effect, outgassing is also taken into account. The class C7−15 is exceptional. It is not fully degraded but only slightly reduced in quality. 4.7 Source Rock Analysis The source rock analysis is an important aspect of petroleum system analysis. The information about generated and expelled petroleum masses, peak gen-
- 205. 192 4 Petroleum Generation Temperature API Undegraded API Degraded Age [Ma] Volume [Mm^3] Fig. 4.37. API and temperature of accumulated petroleum with and without biodegradation according to (4.54). The filling history with spilling and partial re- filling is shown at the bottom right. It started at 25 Ma. The first API values are available for the following time step at 22 Ma M e t h a n e E t h a n e P r o p a n e i - B u t a n e n - B u t a n e i - P e n t a n e n - P e n t a n e n - H e x a n e C 7 - 1 5 C 1 6 - 2 5 C 2 6 - 3 5 C 3 6 - 4 5 C 5 6 + C 4 6 - 5 5 Fig. 4.38. Composition of undegraded and degraded oil at present day. Note that the molar fraction of methane changed from 34% to 61% eration, and expulsion times already gives a first idea about possible reservoir charge even if migration is not yet considered in detail. The most important input data and results for a source rock analysis are illustrated in Fig. 4.39 and Fig. 4.40 for a model of the Northern Campos basin. Herein, three source rocks are defined in the Bota, Coquina and Lagoe Faia formations, each with primary and secondary kinetics. The source rock kinetics can be based on measured sample data or equivalent default kinetics from a database. Kinet- ics related adsorption factors should also be defined for each HC component. The kerogen content and quality are defined with TOC and HI maps, which
- 206. 4.7 Source Rock Analysis 193 are specified for all three source rocks. Usually, TOC and HI values from well data are used to construct interpolation maps. 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 0 10 20 30 40 50 Activation Energy in kcal/mol Frequency in % Sorption Factors in mgHC/gKerogen: C1 C2 C3 iC4 nC4 iC5 nC5 C6 C10 C20 C30 C40 C50 C60+ 2 6 6 3 3 3 3 3 3 4 6 8 10 15 Coquina Map: Transformation Ratio in % Coquina Map: Maturity Classes based on TR Coquina Map: Maturity Classes based on R0 Coquina Map: initial TOC in % Coquina Map: initial HI in mg/gHC Coquina Source Rock: 14 component kinetics Northern Campos Basin: Source Rocks Conquinas Kinetics A = 1.12x1027 My-1 HI = 749 mg/gTOC 0 C60 C50 C40 C30 C20 C10 C6 nC5 iC5 nC4 iC4 C3 C2 C1 Source Rocks: Post Salt Marine (Bota Formation) Pre Salt Lacustrine (Coquinas Formation) Pre Salt Lacustrine (Lagoa Faia Formation) Fig. 4.39. Example: Campos Basin, Brazil The calculated transformation ratios at present day and paleo-times show how and where most of the petroleum is generated. These can be converted
- 207. 194 4 Petroleum Generation Mt 0 20 40 60 80 100 120 0 20 40 60 80 100 120 Geological Time in My Petroleum Masses in Gt 1 2 3 Petroleum System Chart Cretaceous L.Crt U.Crt Paleogene Plc. Eocene Oli. Neocene Mioc. 100 my 50 my Generation Critical Moment Expulsion Reservoirs Seals a b a b c d e f a b salt a: Lagoa Feia Formation, b: Coquina Formation, c: Creaceous Play, d:Tertiary Play, e: Cretaceous Seal f: Tertiary Seal Coquina Formation: Generation and Expulsion Conquina Formation: Generation Gt Conquina Formation: Expulsion Gt 1..Convertible kerogen 2..Generated petroleum 3..Expelled petroleum Critical Moment Fig. 4.40. Example: Campos Basin, Brazil into areas of maturation classes for immature, oil and gas generating, and overmature regions. The principal areas of maturation can also be derived from vitrinite reflectance calculations without consideration of the generated masses, the source rock thickness and the TOC and HI values. The petroleum systems chart summarizes the main periods of generation and expulsion for each source rock for comparison with the periods of reser- voir rock deposition and the occurrence of effective seals. These periods are derived from the generation curves through geologic time and specification of the critical moment of the petroleum system. The petroleum system chart
- 208. 4.7 Source Rock Analysis 195 shows whether the petroleum system elements occur in the correct geological sequence and allow accumulation and preservation of petroleum. Summary: Petroleum generation and coal formation mainly proceed during catagenesis. The maturation processes are time and temperature controlled. Petroleum generation models are generally used for source rock analysis. The amounts and the timing of generation and expulsion of the petroleum system are studied. Petroleum is generated from kerogen, which can be subdivided according to the van Krevelen diagram into type I — IV corresponding to its hydrogen and oxygen content. Oil generation is often modeled with two–component kinetics for oil and gas. The kinetic data are derived from Rock–Eval or similar open system pyrolysis methods. The quantity and quality of the organic matter which is available for petroleum generation is described with the total organic content (TOC) and the hydrogen index (HI), respectively. A different and more detailed characterization of the organic matter is based on the generated petroleum product and from five organic facies types. The corresponding data are derived from combined open and closed system pyrolysis. The petroleum composition is usually described using 14- component kinetics, which have to be combined with fluid models to obtain petroleum properties such as density and viscosity. Quantitative analysis of petroleum generation is based on chemical ki- netics for the primary cracking of kerogen and the secondary cracking of petroleum. Chemical kinetics are formulated with mass balances and dis- tributed reactivity kinetics. These kinetic schemes encompass sequential and parallel reactions that are approximated as first order Arrhenius type reac- tions. Herein, the reaction rate depends exponentially on temperature, de- scribed with the activation energy and the frequency factor. The primary cracking of each petroleum component is modeled with an activation energy distribution. Secondary cracking reactions are usually simplified with the cracking of the heavier components into methane and coke, also formulated with an activation energy distribution. The main results of the analysis are the generation potentials and the generated mass of each petroleum compo- nent through geologic time. The most widely used thermal maturation indicator is the reflectance of vitrinite. Other parameters include molecular biomarkers, Tmax values from Rock–Eval pyrolysis and the annealing of fission tracks. Most of the corresponding kinetic models are also based on distributed kinetic reactions. Thermal maturity parameters can be used to calibrate heat flow histories. They also can be correlated to the maturity of petroleum. Special kinetic models are used for the adsorption of petroleum in the source rock and the biodegradation of petroleum in the reservoir.
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- 210. REFERENCES 197 S. Glasstone, K.J. Laidler, and H. Eyring. The theory of rate processes. McGraw-Hill, 1941. J. C. Goff. Hydrocarbon generation and migration from jurassic source rocks in East Shetland Basin and Viking graben of the northern North Sea. J. Geol. Soc. Lond., 140:445–474, 1983. P. F. Green. The relationship between track shortening and fission track age reduction in apatite: Combined influences of inherent instability, annealing anisotropy, length bias and system calibration. Earth and Planetary Science Letters, 89:335–352, 1988. P. F. Green, I. R. Duddy, A. J. W. Gleadow, P.R. Tingate, and G. M. Laslett. Thermal annealing of fission tracks in apatite 1. A qualitative description. Chemical Geology (Isotope Geoscience Section), 59:237–253, 1986. P. F. Green, I. R. Duddy, A. J. W. Gleadow, and J. F. Lovering. Apatite fission–track analysis as a paleotemperature indicator for hydrocarbon ex- ploration. In N. D. Naeser and T. H. McCulloh, editors, Thermal History of Sedimentary Basins, Methods and Case Histories, pages 181–195. Springer– Verlag, 1989a. P. F. Green, I. R. Duddy, G. M. Laslett, A. J. W. Gleadow, and J. F. Lover- ing. Thermal annealing of fission tracks in apatite 1. Quantitative modelling techniques and extension to geological timescales. Chemical Geology (Iso- tope Geoscience Section), 79:155–182, 1989b. R. W. Jones. Organic facies. In Academic Press, editor, Advances in Petroleum Geochemistry, pages 1–90. 1987. R. A. Ketcham. Personal communication, 2003. R. A. Ketcham, R. A. Donelick, and M. B. Donelick. Aftsolve: A program for multi–kinetic modeling of apatite fission–track data. Geological Materials Research, 2, No. 1 (electronic), 2000. B. M. Krooss and R. di Primio. Personal communication, 2007. S. Larter. Bugs, biodegradation and biochemistry of heavy oil. The 23rd International Meeting on Organic Geochemistry, Torquay, England, 2007. S. R. Larter. Some pragmatic perspectives in source rock geochemistry. Ma- rine and Petroleum Geology, 5:194–204, 1988. G. M. Laslett, P. F. Green, I. R. Duddy, and A. J. W. Gleadow. Thermal annealing of fission tracks in apatite 2. A quantitative analysis. Chemical Geology (Isotope Geoscience Section), 65:1–13, 1987. A. S. Mackenzie and D. McKenzie. Isomerization and aromatization of hy- drocarbons in sedimentary basins. Geological Magazine, 120:417–470, 1983. A. S. Pepper and P. J. Corvi. Simple kinetic models of petroleum formation. Part I: oil and gas generation from kerogen. Marine and Petroleum Geology, 12(3):291–319, 1995a. A. S. Pepper and P. J. Corvi. Simple kinetic models of petroleum formation. Part III: Modelling an open system. Marine and Petroleum Geology, 12(4): 417–452, 1995b.
- 211. 198 4 Petroleum Generation A. S. Pepper and T. A. Dodd. Simple kinetic models of petroleum formation. Part II: oil – gas cracking. Marine and Petroleum Geology, 12(3):321–340, 1995. K. E. Peters, C. C. Walters, and J. M. Moldowan. The Biomarker Guide, volume 1 and 2. Cambridge University Press, second edition, 2005. W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery. Numerical Recipes in C++. Cambridge University Press, second edition, 2002. M. Radke and D. H. Welte. The methylphenanthrene index (MPI): A maturity parameter based on aromatic hydrocarbons. In M. Bjoroy et al., editor, Advances in Organic Geochemistry. Proceedings of the 10th International Meeting on Organic Geochemistry, University of Bergen, Norway, 14–18 September 1981, Wiley and Sons, 1983. C. S. Sajgo and J. Lefler. A reaction kinetic approach to the temperature–time history of sedimentary basins. Lecture Notes in Earth Sciences, 5:123–151, 1986. J. J. Sweeney and A. K. Burnham. Evaluation of a simple model of vitrinite reflectance based on chemical kinetics. AAPG Bulletin, 74:1559–1570, 1990. E. W. Tegelaar and R. A. Noble. Kinetics of hydrocarbon generation as a function of the molecular structure of kerogen as revealed by pyrolysis– gas chromatography. Advances in Organic Geochemistry, 22(3–5):543–574, 1994. B. P. Tissot and D. H. Welte. Petroleum Formation and Occurrence. Springer– Verlag, Berlin, second edition, 1984. P. Ungerer, J. Burrus, B. Doligez, P. Y. Chenet, and F. Bessis. Basin evalu- ation by integrated two–dimensional modeling of heat transfer, fluid flow, hydrocarbon gerneration and migration. AAPG Bulletin, 74:309–335, 1990. D. W. van Krevelen. Coal. Typology–Chemistry–Physics–Constitution. Else- vier, 1961. M. Vandenbroucke, F. Behar, and L. J. Rudkiewicz. Kinetic modelling of petroleum formation and cracking: implications from high pressure, high temperature Elgin Field (UK, North Sea). Organic Geochemistry, 30:1105– 1125, 1999. D. W. Waples. Time and temperature in petroleum formation: application of Lopatin’s method to petroleum exploration. AAPG Bulletin, 64:916–926, 1980. R. W. T. Wilkins, C. P. Buckingham, N. Sherwood, Russel N. J., M. Faiz, and Kurusingal. The current status of famm thermal maturaty technique for petroleum exploration in australia. Australian Petroleum Prduction and Exploration Asociation Journal, 38:421–437, 1998. K. Zengler, H. H. Richnow, R. Rosselló-Mora, W. Michaelis, and F. Widdel. Methane formation from long–chain alkanes by anaerobic microorganisms. Nature, 401:266–269, 1999.
- 212. 5 Fluid Analysis 5.1 Introduction On a macroscopic level fluids consist of physically distinct phases which fill empty space in regions with defined boundaries. The amount, composition, and properties of the phases vary with the overall composition of the fluid and external parameters such as pressure, volume, and temperature (PVT). Migration and other sophisticated aspects of basin modeling can only be sim- ulated if the phases and their properties are known. Hence, the subject of this chapter is modeling of phase compositions and properties. On a molecular level fluids can be grouped into compounds with (approx- imately) the same physical properties. These compounds are called compo- nents and are typically “pure” chemical species such as methane and wa- ter or “lumped” species like alkanes with a defined number of carbon atoms e.g. CnH2n+2 (Danesh, 1998). The subject of fluid analysis can be subdivided into three parts, the determination of the coexisting phases, their compositions and their properties. In basin modeling it is usually assumed that a water phase is present. The polar structure of water causes a separation from the non–polar hydrocarbons, and hence, as a result of this, there are at least two distinct phases. An excep- tion are gas hydrates which occur only under special conditions (Sec. 5.8). It is further assumed that other highly polar components, including salts, dissolve almost completely in the water phase. Small amounts of light HCs can also be dissolved in water. However, usually HCs form their own phases. Furthermore it is commonly assumed that the HC phases exist as liquid and vapor or of only one single phase, a supercritical or undersaturated phase. So, in practice, fluid analysis is reduced to HC phase analysis with dissolution of components in one or the other of two possible phases. From Gibbs’ phase rule it can be deduced that the number of phases κ can become κ = N + 2 where N is the (sometimes large) number of components. It should be kept in mind that more than two HC phases may exist (Sengers and Levelt, 2002; Danesh, 1998; Pedersen et al., 1989; Pedersen and Christensen, 2007). T. Hantschel, A.I. Kauerauf, Fundamentals of Basin and Petroleum 199 Systems Modeling, DOI 10.1007/978-3-540-72318-9 5, © Springer-Verlag Berlin Heidelberg 2009
- 213. 200 5 Fluid Analysis From experience it is known that components with higher molecular weights are enriched in the denser liquid phase, whereas lighter components favor the vapor phase. At ambient surface conditions (standard conditions) of 60 ◦ F = 15.556 ◦ C and 1 atm, a rule of thumb can be applied, accord- ing to which components heavier than pentane are found almost entirely in the liquid and the remaining lighter components almost entirely in the vapor phase.1 In petroleum system studies, components found in the liquid phase are referred to as oil components and components found in the vapor as gas components. At surface conditions the compositional analysis is reduced to a distribution of gas components in the vapor phase and oil components in the liquid phase. Properties of the phases can be estimated using empirical methods such as the API method for density predictions (Sec. 5.4). Retaining these phase compositions for arbitrary pressure and temperature conditions, is called the “Fixed Phase Model” (Fig. 5.1). It is obviously not suitable for conditions which differ a lot from surface conditions. Nevertheless it contains concepts that are useful to the following discussion. Fig. 5.1. The fixed phase model at- tributes light components to the vapor and heavy components to the liquid phase. C5 is an intermediate and can often not be classified unambiguously as a light or a heavy component C C C C C 1 2 3 4 5 Liquid Phase Vapor Phase C C C C C C 5 6 7 8 9 10+ The water phase is commonly well separated from the hydrocarbon phases. Methane is an exceptional component which might dissolve in non–negligible parts in the water phase. This must be taken into account prior to further compositional analysis and thus the next section deals with the influence of the water phase. A simple fluid model for limited pressure and temperature ranges is discussed afterwards in Sec. 5.3. Section 5.4 deals more with the theoretical background and advanced equations of state (EOS). This knowledge is the basis for an algorithm for the prediction of phase compositions in Sec. 5.5. The prediction of properties such as density, viscosity or the gas oil ratio (GOR) of fluids and their phases, is the topic of Sec. 5.6. The chapter continues with a discussion about how fluid models are usually integrated, tuned, and calibrated in basin modeling (Sec. 5.7). Finally, the behavior of gas hydrates is briefly introduced in Sec. 5.8. 1 Pure pentane is a liquid at room temperature with a boiling point of 36◦ C. In multicomponent systems it is usually enriched in the liquid but also the vapor phase contains often non–negligible amounts.
- 214. 5.2 Water Phase 201 Especially in this chapter, many examples are plotted for typical pressure temperature paths (PT–paths) in sedimentary basins, which range from cool and overpressured to hot and hydrostatic (Fig. 5.2). The PT–path in a real basin will often not follow a straight line. For example, overpressured regions often start below 3 km depth. Hence steep curves with values above 0.5 MPa/K are often found only above 100◦ C. However, the range between path 1 and 5 in Fig. 5.2 defines an area of possible pressure temperature points in arbitrary sedimentary basins. Fig. 5.2. Typical pressure temperature paths (PT–paths) in geological basins. Path 1 is cool and overpressured, path 2 is overpressured, path 3 is a typical path, path 4 is hydrostatic and path 5 is hot and hydrostatic 0 50 100 150 200 0 50 100 150 Temperature in Celsius 1 2 3 4 5 Pressure in MPa 2 . 5 M P a / K 1 . 0 M P a / K 0.5 MPa/K 0.3 MPa/K 0.1 MPa/K 5.2 Water Phase In a first rough approximation H2O is found almost completely in the water phase only. More complex approaches such as models including H2O as a component partially dissolved in petroleum (Pedersen et al., 1989), are usually not taken into account. The dissolved amount of water is so small that it can be neglected in basin models. It is also commonly assumed that all HCs, except methane, do not dissolve in water. The dissolved amount of methane in water depends approximately on pressure, temperature, and the overall amount of methane, but not on the properties of the HC phases. Therefore, methane dissolution can be treated independently before HC phase analysis. Methane solubility has been studied, among others, by Haas (1978) and Battino (1987). The IUPAC (International Union of Pure and Applied Chem- istry) recommended the smoothed equation ln x = −55.8111 + 7478.84 T + 20.6794 ln T 100 + 0.753158 ln p (5.1)
- 215. 202 5 Fluid Analysis for the molar fraction x of methane in water, where the unit for temperature T is Kelvin and for pressure p is MPa (Nelson and Simmons, 1995). It is drafted in Fig. 5.3. 500 ppm 1000 ppm 1500 ppm 2000 ppm 2500 ppm 3000 ppm 3500 ppm 4000 ppm 4500 ppm 5000 ppm Fig. 5.3. Isolines of mol fraction solubility x for the dissolution of methane in water according to (5.1) In the subsurface methane dissolves in water only in the immediate neigh- borhood of hydrocarbon pathways. Due to the size and resolution of a basin model it is often not possible to resolve pathways accurately. Additionally, methane solubility in seawater—equivalent pore water is reduced by approx- imately 14% (Nelson and Simmons, 1995). Therefore, the average maximum solubility must be lower than given in (5.1). Another component of special interest concerning petroleum systems mod- eling is carbon dioxide. It is often found in concentrations up to 2% (molar) in petroleum (di Primio, 2008). The physical properties of carbon dioxide dif- fer form typical hydrocarbons. It is slightly polar and dissolves well in water. In basin modeling it is for that reason often assumed that it is completely dissolved in the water phase. An empirically determined fraction of carbon dioxide is sometimes kept in the petroleum phases. This approach is rather crude, especially at locations of high petroleum saturation e.g. in accumula- tions where the water content is relatively small and water is not capable of absorbing all the carbon dioxide. Due to a lack of spatial resolution, basin models are generally not feasible for sophisticated modeling of small concentrations of impurities, which might
- 216. 5.3 Binary Mixtures and Black Oil Models 203 dissolve in the water or in a hydrocarbon phase. Alternatively, empirically determined concentrations are commonly assumed for each phase. 5.3 Binary Mixtures and Black Oil Models Binary mixtures are of great interest for modeling since they represent sim- ple systems which show qualitatively the same phase behavior as arbitrary petroleum fluids. Black oil models are binary mixtures which have been ad- justed to quantitatively match the phase behavior of petroleum, at least in limited pressure temperature ranges.2 A starting point for the discussion of binary mixtures is the phase diagram of a one–component system (Fig. 5.4). The vapor–pressure line defines the conditions at which liquid and vapor coexist. Directly above this line pure liquid and below it pure vapor is found. The line ends at the critical point. Property differences such as a density contrast between the liquid and vapor phase, decrease continuously and finally vanish when approaching the critical point. Gibbs’ phase rule states that the degrees of freedom F equal F = N −κ+2. The number of components is given by N and κ is the number of phases. Here it is N = 1 and therefore κ = 3 − F. An area in the pressure–temperature diagram has two degrees of freedom and thus κ = 1 which means that this area can only be represented by one phase. A line has one degree of freedom so on a line two phases can coexist and on a single point, namely the triple point, even three phases may coexist side by side. The whole pressure–temperature plane is subdivided into areas with one phase, lines with two coexisting phases and one triple point. The solid state behind the melting–point line is with the exceptions of permafrost and gas hydrates of no interest in basin modeling. Quotation marks in Fig. 5.4 are used for liquid and vapor, since the defini- tion of liquid and vapor becomes ambiguous in a one phase region. A simulta- neous temperature and pressure change on a path from a point in the “liquid” region to the “vapor” region, which goes around the critical point through the “supercritical” region and without crossing the vapor–pressure line, causes a continuous change from “liquid” to “vapor”. In other words, there is no point with a physical criterion for the distinction between “liquid” and “vapor” and thus the word supercritical is a better match for the description of this one phase area. An extension to two components causes an “opening” of the vapor– pressure line to an area of coexisting phases (Fig. 5.4). With N = 2 the phase rule yields κ = 4 − F which allows for two phases with two degrees of freedom. All isolines of constant composition end in the critical point. The cricondentherm is defined as the pressure temperature point with highest pos- sible temperature for two coexisting phases. 2 “Black Oil Model” should not be mistaken for “Black Oil”, which describes a class of oils with special properties (Sec. 5.5.1).
- 217. 204 5 Fluid Analysis Pressure Temperature Vapor- Critical Point Tripel Point “Liquid” “Vapor” Supercritical M el tin g- Po in t Lin e Solid L i n e Pressure Pressure Temperature Critical Point Liquid Undersaturated Vapor Undersaturated Supercritical 0% 100% Cricondentherm L in e P o i n t Dew Line Point Bubble Fig. 5.4. Schematic pressure–temperature diagram of a one–component system on the left and of a two–component system on the right The one–phase region is often called “undersaturated liquid” or “under- saturated vapor”. This naming convention can easily be understood with the help of Fig. 5.5 and the example of a one–phase system with 15% methane at point P. This system would be “liquid–like” since it is above curve 8, which indicates the outline of the 15%–methane two–phase region. An increase of the methane–content to 30% moves the limiting slope to curve 7, which touches point P. The liquid is capable of doubling its methane content until it is finally saturated and cannot absorb anymore methane. The remaining methane will then form a separate vapor phase. Temperature in C ° Pressure in kPa CH Fraction 1.. 100.0% 2.. 97.5% 3.. 92.5% 4.. 85.2% 5.. 70.0% 6.. 50.0% 7.. 30.0% 8.. 15.0% 9.. 5.0% 10.. 0.0% 4 Fig. 5.5. Two phase areas of binary methane–ethane mixtures (McCain Jr., 1990)
- 218. 5.3 Binary Mixtures and Black Oil Models 205 In the two–phase region each phase will be saturated. Hence, with a further increase of the methane content a methane–saturated vapor phase appears. The dew point line of this touches point P (Fig. 5.5). This phase consists of 70% methane. At point P the dew point line of mixture 5 (70% methane and 30% ethane) intersects with the bubble point line of mixture 7 (30% methane and 70% ethane), indicating that the co-existing vapor and liquid phases contain 70% and 30% methane, respectively. The total amount of both phases can be calculated with simple material balances: Let x1,2 be the molar ethane and methane fractions in the liquid, y1,2 the molar ethane and methane fractions in the vapor and z1,2 the total molar ethane and methane fractions.3 Then x1 + x2 = 1, y1 + y2 = 1 and z1 + z2 = 1 . (5.2) With the total fractions, nl of liquid and nv of vapor, the balance equations can be formulated as nl + nv = 1 and z1 = x1nl + y1nv, z2 = x2nl + y2nv. (5.3) It is easy to calculate nl and nv with the knowledge of x1,2, y1,2 and z1,2 to nl = z1 − y1 x1 − y1 and nv = z2 − x2 y2 − x2 . (5.4) Component one is the heavier ethane and it prefers the liquid phase, so x1 y1. Methane favors the vapor phase, so y2 x2. Thus undersaturated vapor exists if z1 y1 because nl cannot be negative and undersaturated liquid exists if z2 x2. This means that undersaturated liquid exists if the total amount of methane is less than the capability of a saturated liquid to dissolve methane. In the other cases a two–phase system emerges. The result (5.4) can be interpreted graphically (Fig. 5.6) with so called tie lines (McCain Jr., 1990). Qualitatively, binary mixtures show similar properties to multi–component petroleum. A quantitatively better approximation can now be constructed starting with the idea of gas and oil components from the fixed phase model described in the introduction: All gas components are lumped together into one artificial pseudo gas component and all the oil components into one artifi- cial pseudo oil component. This artificial two component system can then be treated as a binary mixture. The two pseudo components are distributed in both phases. All the data needed for this procedure are the bubble and dew point curves for arbitrary compositions.4 A “Black Oil Model” is a binary 3 In this chapter x is used for liquid, y for vapor and z for the overall composition or if none of both is specified. This is not consistent with Chap. 4 where x denotes kerogen, y oil/liquid and z gas/vapor. However, the notation is commonly used in the literature (Danesh, 1998). 4 It is even more efficient to use two lookup tables for x1 and y2.
- 219. 206 5 Fluid Analysis Fig. 5.6. Tie line construction for a bi- nary mixture with overall composition z1,2, saturated liquid with composition x1 and saturated vapor with composition y2: The vapor and liquid phase amounts nv,l can be calculated with the length of the distances 12, 13 and 23 by nv = 12/23 and nl = 13/23, which is the same as (5.4) Pressure Composition 100% 0% z1,2 Liquid Undersaturated L i n e P o i n t D e w Bubble Point Line Vapor Undersaturated 2 1 3 x1,2 y1,2 mixture with no solubility of the heavier component in the vapor or y1 = 0 (Peaceman, 1977). Binary mixtures without this restriction are often referred to as “Symmetrical Black Oil Models” (SBO) (Fig. 5.7). Fig. 5.7. Symmetrical black oil model: Pure components are grouped into light or heavy components and then lumped into artificial gas and oil pseudo com- ponents. A binary mixture is used for the distribution of the pseudo compo- nents into the liquid and vapor phases. “Un-lumping” of the pseudo components yields the amount of each pure compo- nent in any phase C C C C C 1 2 3 4 5 } } Pseudo Gas Component Pseudo Oil Component Grouping Lumping Bubble and Dew Point Lines Liquid Phase Vapor Phase C C C C C 6 7 8 9 10+ The procedure of grouping gas and oil components, the lumping into pseudo components and the determination of a consistent set of dew and bub- ble point curves, are dependent on the overall composition of the petroleum under consideration. Usually, components are grouped following the fixed phase model. All the light components are collected in the gas component and the heavy components in the oil component. The dew and bubble point curve data are often calculated using thermodynamic methods, if they are not known from oil sample analysis. Therefore, properties such as the criti- cal temperature or pressure of pseudo components must be known. Various approaches such as simple molar averaging according to Tc = i ziTci (5.5) or special formulas such as Lee–Kesler–Averaging (Danesh, 1998) according to
- 220. 5.4 Equations of State (EOS) 207 vc = 1 8 ij zizj v 1/3 ci + v 1/3 cj 3 , Tc = 1 8vc ij zizj TciTcj 1/2 v 1/3 ci + v 1/3 cj 3 , Zc = 0.2905 − 0.085 i ziωi (5.6) are commonly in use. Here Tc is the critical temperature of the lumped com- ponent, Tci the critical temperatures of the pure components, zi the molar fraction of each component, vc the critical molar volume, ω the acentric fac- tor and Zc the critical compressibility. The quantities indexed with i are the values of the pure components. The biggest advantage of black oil models is their high performance during simulation. Grouping and lumping are performed independently before solving the differential equations of basin modeling. Phase changes during migration are calculated just by searching dew and bubble point data in lookup tables and the solution of the simple balance Equations (5.2) – (5.4). The disadvan- tages however are limited pressure and temperature ranges and difficulties to determine the dew and bubble point curves of these artificially constructed binary mixtures. 5.4 Equations of State (EOS) Further thermodynamic aspects, which go beyond the scope of a symmetrical black oil model, can only be considered, if more about the properties of the phases are known. The most important property is the density or its inverse, the molar volume. Relationships of pressure and temperature with volume are called “Equations of State”(EOS). The origin of most EOS is the ideal gas equation pv = RT (5.7) with p as the pressure, v the molar volume, R the universal gas constant and T the temperature. The ideal gas equation is known to be a good approximation to the behavior of dilute gases at low pressures (Fig. 5.9). It can be derived from classical statistical mechanics with assumptions of point–like molecules, which move without interaction (Huang, 1987). The ideal gas equation cannot be used for the description of phase transitions. For further considerations it is convenient to introduce the compressibility factor Z which is defined as Z = pv RT . (5.8) The ideal gas equation then becomes5 Z = 1. 5 Note that Z is not directly related to compressibilities such as introduced in (2.1). For the ideal gas equation it is C = −(1/V ) ∂V/∂p = 1/p.
- 221. 208 5 Fluid Analysis A virial expansion can be interpreted as a systematic starting point to derive more realistic EOS. It has the form Z = 1 + A1 v + A2 v2 + A3 v3 + . . . (5.9) with the virial coefficients A1, A2, A3, . . ., which are temperature dependent. The expansion can also be motivated by arguments from statistical mechan- ics (Huang, 1987). An important example is the Benedict–Webb–Rubin EOS (Danesh 1998, App. J). Many other approaches are used for derivation of EOS. Four families are commonly distinguished (Pedersen et al., 1989): • van der Waals (vdW) • Benedict–Webb–Rubin • Reference–fluid equations • Augmented-rigid–body equations The vdW family is well known for its accuracy and simplicity which makes it the method of choice in basic modeling. Although other families are often very precise, they are less general and thus impractical to use. Van der Waals directly improved the ideal gas equation. He took attractive forces between the molecules into account which are present due to induced dipole–dipole interactions and repulsive forces which originate from the finite volume of the molecules (Becker, 1985; Huang, 1987). The vdW equation has the form p = RT v − b − a v2 . (5.10) The parameter b is a measure of volume of molecules and called “co-volume” whereas a describes the intermolecular attraction. With the notation of compressibility (5.10) becomes Z3 − (1 + B)Z2 + AZ − AB = 0 (5.11) with the dimensionless parameter A = ap (RT)2 and B = bp RT . (5.12) Therefore, vdW equations are referred to as cubic EOS. The form of their isotherms is illustrated in Fig. 5.8. The isotherms have up to three volume– roots for given pressure and temperature. For that reason vdW equations have the ability to model phase transitions. The smallest root describes the liquid– like and the largest the vapor–like state. It can be shown that the intermediate root describes a thermodynamically instable state. Two phases coexist only if the pressure has special values, which can be found graphically according to the so called Maxwell equal area rule (Fig. 5.8). The restriction to special values of pressure for two phase states corresponds to Gibbs’ phase rule.
- 222. 5.4 Equations of State (EOS) 209 Fig. 5.8. Pressure–volume diagram with isotherms according to the vdW equa- tion. Below the critical temperature Tc two phases may exist simultaneously. It can be proven thermodynamically that this is only possible for pressures pb, which cut the temperature isolines in a way that the vertical and horizontal hatched areas are equal in size (Becker, 1985). The region of two phases is situ- ated below the dashed line Pressure Volume T T T T T Critical Point c c c Supercritical (One Phase) V liquid V instable V vapor pb Two Phases Above the critical point there exists only one phase. The location of the critical point can be calculated from the conditions ∂p ∂v Tc = 0 and ∂2 p ∂v2 Tc = 0 (5.13) which yield to b = vc/3 and a = 3 pcv2 c with pcvc = (3/8)RTc . (5.14) Although the vdW equation shows all required qualitative features, it still does not have the accuracy needed for practical purposes in basin modeling. Many improvements have been proposed (Danesh, 1998; Pedersen et al., 1989). The best known are the Soave–Redlich–Kwong EOS (SRK) p = RT v − b − a2 v(v + b) or Z3 − Z2 + (A − B − B2 )Z − AB = 0 (5.15) and the Peng–Robinson EOS (PR): p = RT v − b − a2 v(v + b) + b(v − b) or Z3 − (1 − B)Z2 + (A − 2B − 3B2 )Z − AB + B2 + B3 = 0 . (5.16) Accurate methods for the derivation of the parameters a and b from experi- mental data are available with these EOS: a = α ac , α = 1 + m 1 − T/Tc 2 . (5.17)
- 223. 210 5 Fluid Analysis SRK: ac = 0.42747 R2 T2 c /pc , b = 0.08664 RTc/pc , m = 0.480 + 1.574 ω − 0.176 ω2 . PR: ac = 0.457235 R2 T2 c /pc , b = 0.077796 RTc/pc , m = 0.3796 + 1.485 ω − 0.1644 ω2 + 0.01667 ω3 . (5.18) In contrast to the vdW EOS which depends on two parameters a and b, the PR and SRK EOS implicitly depend on three parameters namely pc, Tc, and the dimensionless acentric factor ω which is defined by ω = − log10 pb(T = 0.7 Tc)/pc − 1.0 (5.19) with pb(T) as the pressure on the vapor–pressure line. The acentric factor is zero for spherical molecules and related to their deviation from a spherical shape (Danesh, 1998). PR and SRK EOS are known to be quite accurate for p 100 MPa and 300 K T 500 K. Densities of some light compo- nents, which are calculated with the SRK EOS and the ideal gas equation, are compared in Fig. 5.9. It can be seen, that the results differ enormously at high pressures and temperatures. A comparison of different EOS for methane only is shown in Fig. 5.10. The ideal gas equation is only accurate at low pressures and temperatures. As expected, SRK and PR EOS differ little. The modified Benedict–Webb–Rubin (MBWR) EOS (J.1) is exclusively adjusted to methane behavior (McCarty, 1974). It agrees well with the SRK EOS for temperatures, which are not too high. 5.4.1 Mixing Rules The vdW equations can also be used for the description of multi–component phases with appropriate phase parameters. These phase parameters can be acquired through so called mixing rules which are often arithmetic or geomet- ric averages of the component parameters. Formally, the approach is similar to lumping procedures described in (5.5) and (5.6). Usually, lumping refers to average component and mixing to average phase properties. However, there is no sharp boundary because the properties of lumped components are often identical with the phase properties of the corresponding mixture. The mixing rule for co-volume b is usually b = i zibi (5.20) with bi as the co-volumes of the components and zi as molar fractions of the components in the phase. A reasonable mixing rule for the attractive parameter a is a geometric average with corrective factors kij ≈ 0 which take into account special inter- molecular forces between components i and j. It has the form a = ij zizjaij with aij = aiaj(1 − kij) . (5.21)
- 224. 5.4 Equations of State (EOS) 211 0.1 MPa/°C 0.3 MPa/°C 0.5 MPa/°C 0.1 MPa/°C 0.3 MPa/°C 0.5 MPa/°C 1 MPa/°C SRK SRK SRK 0.1 MPa/°C 0.3 MPa/°C 0.5 MPa/°C 1 MPa/°C 2.5 MPa/°C 1 MPa/°C 2.5 MPa/°C 0.1 MPa/°C 0.3 MPa/°C 0.5 MPa/°C 1 MPa/°C 2.5 MPa/°C 2.5 MPa/°C 0.1 MPa/°C 0.3 MPa/°C 0.5 MPa/°C 1 MPa/°C 2.5 MPa/°C Ideal Gas Ideal Gas 0.1 MPa/°C 0.3 MPa/°C 0.5 MPa/°C 1 MPa/°C 2.5 MPa/°C Ideal Gas Fig. 5.9. Density ρ = Mwp/(RT) of methane, nitrogen, and carbon dioxide accord- ing to (5.7) for different PT–paths with all of them starting at 0 MPa and 0 ◦ C. The left column is calculated with the ideal gas equation (5.7) and the right column with the SRK EOS (5.15) Many sets of these binary interaction parameters (BIP) kij are in use (Danesh, 1998; Reid et al., 1987, App. I). They are sometimes customized as tuning parameters for “fine–adjustment” of EOS. It is sometimes argued that this usage is problematic (Pedersen et al., 1989). 5.4.2 Phase Equilibrium Gibbs’ energy G represents the thermodynamic potential for systems with pressure and temperature as independent variables (Becker, 1985; Huang, 1987). The potential must be minimal for systems in equilibrium. This leads
- 225. 212 5 Fluid Analysis 1 MPa/°C 2.5 MPa/°C 0.1 MPa/°C 0.3 MPa/°C 0.5 MPa/°C 1 MPa/°C 2.5 MPa/°C 0.1 MPa/°C 0.3 MPa/°C 0.5 MPa/°C Fig. 5.10. Methane density variation between ideal gas equation (5.7) and SRK EOS (5.15) on the top left, be- tween SRK and PR EOS (5.16) on the top right and between MBWR (J.1) and SRK EOS on the right for PT–paths as in Fig. 5.9 1 M Pa/°C 2.5 MPa/°C 0.1 MPa/°C 0.3 MPa/°C 0.5 MPa/°C to the conclusion that the chemical potential μ of each component in each phase must be equal, namely μl,i = μv,i . (5.22) The chemical potential change from μ0,i to μi can be calculated for an ideal gas with an isothermal pressure change from p0 to p as μi − μ0,i = RT ln p/p0 (5.23) in one phase. Similar to the definition of the compressibility factor Z in (5.8) a quantity labeled fugacity is commonly defined with reference to the behavior of an ideal gas implicitly with μi − μ0,i = RT ln fi/f0,i (5.24) for non–ideal systems. The fugacity coefficient φi is defined as φi = fi pzi (5.25) which must behave as an ideal gas for low pressures, so φi → 1 for p → 0.It can be evaluated explicitly to ln φi = bi b (Z − 1) − ln(Z − B) − A (δ2 − δ1)B 2 a j zjaij − bi b ln Z + δ2B Z + δ1B (5.26)
- 226. 5.5 Flash Calculations 213 with δ1,2 = 1, 0 for SRK and δ1,2 = 1 ± √ 2 for PR (Danesh, 1998). 5.5 Flash Calculations The straight forward generalization of the material balance Equations (5.2) – (5.3) for a two–phase N–component system are i xi = 1, i yi = 1, i zi = 1 (5.27) and nl + nv = 1, zi = xinl + yinv . (5.28) With the definition (5.25) the equality of the chemical potentials in a two– phase system leads to xiφl,i = yiφv,i (5.29) for given pressure and temperature. Equations (5.27) – (5.29) with the fugacity coefficients as defined in (5.26), form a set of 2N + 2 independent equations6 with corresponding unknowns xi, yi, nl, and nv. The solution of this set of nonlinear equations is referred to as a “flash calculation” (Fig. 5.11). C C C C C C C C C C 1 2 3 4 5 6 7 8 9 10+ } Flash Calculation: - Van der Waals Type Equations of State - Minimization of Gibbs Free Energy Liquid Phase Vapor Phase Fig. 5.11. Flash model which is based on thermodynamics Equations (5.27) are linear in the unknowns and have a simple structure. Thus it is possible to solve these partly with the help of equilibrium ratios, which are defined as Ki = yi/xi. With knowledge of Ki and nv it is then possible to calculate 6 The condition i zi = 1 does not count since it does not contain independent variables. But from 1 = i zi = nl i xi + nv i yi = 1 + nv i (yi − xi) it follows directly i (yi − xi) = 0. So one of the sums i xi = 1 or i yi = 1 is dependent and should also not be counted.
- 227. 214 5 Fluid Analysis xi = zi 1 + (Ki − 1)nv , yi = Kizi 1 + (Ki − 1)nv , and nl = 1 − nv . (5.30) From 0 = i (yi − xi) = i (Ki − 1)zi 1 + (Ki − 1)nv (5.31) however, it is possible to determine nv numerically. The remaining Ki can now be treated iteratively. The complete algorithm is made up of the following sequence: 1 Estimate Ki 2 Calculate nv, nl and xi, yi with (5.31) and (5.30) 3 Calculate al, bl and av, bv with mixing rules 4 Calculate Zl and Zv with (5.15) or (5.16) 5 Calculate φl,i and φv,i with (5.26) 6 Calculate Knew i = φl,i/φv,i 7 If i (1 − Knew i /Ki)2 adjust Ki ← Knew i and go back to step 2 The criterion of convergence in step 7 is given by , which should be a small number such as = 10−12 . Good initial estimates of Ki are very important. Wilson (Danesh, 1998; Pedersen et al., 1989) proposed approximations of equilibrium ratios below p 3.5 MPa with the formula Ki = pci p exp 5.37 (1 + ωi) 1 − Tci T (5.32) which can be used as start values for flash calculations at even higher pres- sures. A failure in convergence indicates a one–phase solution. Additionally, a stability analysis for the explicit evaluation of the number of phases can be performed (Danesh, 1998; Pedersen et al., 1989). However, these methods are very sophisticated and in themselves unstable. Ambiguous cases are rare in real case studies. In case of a one–phase state, two alternative thermodynamic stable roots may exist, according to Fig. 5.8. The root with the smaller Gibbs energy is the searched one. The difference of the Gibbs energy between two roots Zl and Zv can be calculated as Gv − Gl RT = Zv − Zl + ln Zl − B Zv − B − A B(δ2 − δ1) ln Zl + δ1B Zl + δ2B × Zv + δ2B Zv + δ1B . (5.33) Due to limited data availability and system resources in basin modeling the number of components in flash calculations are commonly restricted to about
- 228. 5.5 Flash Calculations 215 14 (Sec. 4.3.3). Thus, it is also necessary to use lumped pseudo components in flash calculations. Often they replace the heavier components. The C15+ pseudo–component, which lumps together all HCs of the fluid heavy end with more than or equal to 15 carbon atoms, is an example of this. Examples of some flash calculations are shown in Fig. 5.12 and Fig. 5.13. Molar Composition Mass Composition Volume Composition Fig. 5.12. Pie charts for phase composition and bar graphs for component distribu- tion to phases. The overall composition is the same as the “Black Oil” of Fig. 5.14. It is listed in App. K. All figures are calculated with the SRK EOS for surface con- ditions of 0.1 MPa and 15 ◦ C. The inner circle of each pie chart refers to vapor, the outer to liquid. The sum of all bars of each bar graph is 100%. All components are found in both phases, although only pentane and butane contribute with significant amounts to both phases
- 229. 216 5 Fluid Analysis 10 C 3 MPa 20 C 6 MPa 30 C 9 MPa 40 C 12 MPa Fig. 5.13. Pie charts and bar graphs for the same composition as in Fig. 5.12. Only molar fractions are displayed. Light components are dissolved in the liquid phase with increasing pressure. The pressure temperature values correspond to a PT–path with 0.3 MPa/◦ C, which starts at 0◦ C. The composition at standard conditions is depicted in Fig. 5.12
- 230. 5.5 Flash Calculations 217 The performance of flash calculations is very good when a relatively small number of components are involved. Flash calculations can even be explicitly performed during solution of differential equations for each grid cell and in each time step in basin modeling. By experience, the extra amount of time compared to a symmetrical black oil model is about 10% of total simulation time.7 Phase diagrams, such as calculated for Fig. 5.14 from more than 10.000 flash calculations can be generated on a modern PC within a few seconds. However, flash calculations are also very helpful for the modeling of fluid flow with symmetrical black oil descriptions, because they can be used for the consistent construction of needed lookup tables (Sec. 5.3). 5.5.1 Classification of Petroleum Petroleum is commonly classified as dry gas, wet gas, gas condensate, volatile oil, and black oil (Danesh, 1998). The main classification parameter is the gas oil ratio (GOR) (Table 5.1). Examples, which are constructed with flash calculations are shown in Fig. 5.14. The corresponding compositions and com- ponent properties are listed in App. K. Class GOR [m3 /m3 ] ◦ API Composition Dry Gas – – CH4 + other light gas comp. only Wet Gas 10 000 – mainly CH4 + other light gas comp. Gas Condensate 570 . . . 10 000 40 . . . 60 12.5 mol % C7+ Volatile Oil 310 . . . 570 40 12.5 . . . 20 mol % C7+ Black Oil 310 45 20 mol % C7+ Table 5.1. Classification of petroleum according to Danesh (1998). The API gravity refers to liquid after outgassing. Black oil is additionally classified by a relatively low bubble point pressure 5.5.2 PT–Paths A spectrum of typical geological PT–paths is introduced in Fig. 5.2. Phase compositions can quickly be evaluated along PT–paths with flash calculations (Figs. 5.15, 5.16). In typical geological environments the critical point can be reached. In this case strong variations of phase compositions might occur within certain pressure temperature intervals (Fig. 5.15). 7 The number is highly dependent on the fraction of the grid cells that hold HCs inside.
- 231. 218 5 Fluid Analysis Dry Gas C Wet Gas C Gas Condensate C Volatile Oil C Black Oil C Example GOR ◦ API C7+ [m3 /m3 ] [mol %] Dry Gas – – 0.0 Wet Gas 22 452 (79) ≈ 1.5 Gas Condensate 1 997 52 ≈ 4.0 Volatile Oil 473 48 ≈ 13.0 Black Oil 110 32 ≈ 36.0 Fig. 5.14. Five PT phase diagrams which are calculated with the SRK EOS. The critical point is marked with a “C”. Compositions and component properties are listed in App. K 5.6 Property Prediction 5.6.1 Density Density can directly be calculated from the results of a flash calculation. The molar volume v is known from (5.15) or (5.16) and the composition from (5.30). Hence, density can be directly calculated from
- 232. 5.6 Property Prediction 219 PT Path 1 2 3 PT Path 1 PT Path 2 3 PT Path 4 Critical Point Fig. 5.15. Pressure–temperature phase diagram of a fluid with a high amount of light components on the left. Three PT–paths are marked. The liquid mass fraction along these paths is plotted on the right. Path 1 corresponds to a temperature gradient of 30 ◦ C/km and 10 MPa/km, which is typical in geological systems. Path 3 has the same pressure gradient but 40 ◦ C/km. The intermediate path 2 cuts almost exactly at the critical point. It can be seen, that the phase compositions vary strongly in the vicinity of the critical point. The highest curvature is found, as expected, on path 2 at the critical point. The three PT–paths 1–3 are passing the critical point quite close. The system, which is shown in Fig. 5.23, has a critical point far away from these PT–paths. For comparison, path 4 is extracted with the same PT conditions as path 1 from this system. The transition from the two–phase to the one–phase region is here rather smooth Fig. 5.16. Liquid mass fraction of the same fluid as in Fig. 5.15 at 300◦ C. The formation of liquid under isother- mal pressure release is called “retro- grade condensation” (Danesh, 1998) 300 C ° ρ = 1 v i Mizi (5.34) with Mi as the molar mass and zi as molar fraction of component i. The molar volume v and the composition zi must be replaced by vl and xi for liquids and by vv, yi for vapor. Volume Shift Although the SRK and PR EOS are quite accurate for the prediction of the number of phases and their composition, density calculations show some sys- tematic deviations. Peneloux (Danesh, 1998; Pedersen et al., 1989) proposed
- 233. 220 5 Fluid Analysis to use a corrected molar volume v instead of v. It can be calculated by a shift parameter c according to v = v − c . (5.35) For SRK, the shift factor c can be estimated from the Rackett compress- ibility factor8 ZRA with c = 0.40768 (0.29441 − ZRA) RTc pc . (5.36) For PR, it is sometimes tabulated in the form of the shift factor SE = c/b according to Jhaveri and Youngeren (Danesh, 1998). The mixing rule for the volume shift is arithmetic similar to co-volume mixing in (5.20). It must be explicitly noted that volume shifts, according to Peneloux or Jhaveri and Youngeren for density corrections, do not affect the compositional results of PT–flash calculations (Danesh, 1998). Hence, volume shift values can be modified independently of compositional changes for density calibration. Volume shifts of lumped pseudo–components are often calibrated with their densities. Liquid Density The density of a liquid at standard conditions is often quantified with ◦ API as the unit of choice in the petroleum industry. This emphasizes the fact that the oil price is mainly determined by the oil density, which represents the most common quality factor. The conversion from the specific gravity So to ◦ API is presented as ◦ API = 141.5 So − 131.5 (5.37) with So in g/cm3 . For example, densities from 1000 . . . 702 kg/m3 are mapped to 10 . . . 70 ◦ API. It should be noted that this definition becomes difficult for density differences: a density uncertainty stemming from a measurement error or an estimated error in a model of, for example, 1% of an 800 kg/m3 oil produces an relative error of almost 4% for the corresponding value of 45 ◦ API. This makes the API gravity a very sensitive parameter. High measurement and simulation accuracy must be reached for common ◦ API accuracy. Alternative to the use of densities, which are directly extracted from EOS, empirical methods such as the API or Standing–Katz methods or the op- timized EOS of Alani–Kennedy, can be employed if phase compositions are known (Pedersen et al., 1989; Danesh, 1998). These formulas often yield very accurate density values but they are usually not interpreted as EOS because 8 The Rackett compressibility factor has been introduced for considerations of crit- ical properties (Danesh, 1998). It is tabulated in App. I for may components.
- 234. 5.6 Property Prediction 221 they are strongly limited in their pressure and temperature ranges. Moreover, predetermined grouping and lumping of components restricts their usability. However, the predictions of empirical formulas can be compared to the results of the more general EOS and used for the calibration of phase properties. API Method The API density prediction is based on an available compositional analysis of the components up to C6 with appropriate component densities and the mea- sured density of C7+ at standard conditions. The density ρl at temperature T and pressure p can be evaluated according to the following formulas: ρl = ρ1 C(288.706 K) C(T) , ρ1 = i ziMi i ziMi/ρ0,i , (5.38) with Mi the molecular weight of component i, zi the molar fractions, and ρ0,i the dissolved component density at standard conditions according to Component N2 CO2 H2S C1 C2 C3 i–C4 n–C4 i–C5 n–C5 C6 Density [kg/m3 ] 804 809 834 300 356 508 563 584 625 631 664 Furthermore it is C = A1 + A2 T r + A3 T r 2 + A4 T r 3 with Ai = Bi1 + Bi2 p r + Bi3 p r 2 + Bi4 p r 3 + Bi5 p r 4 . (5.39) The coefficients Bij are defined through i j 1 2 3 4 5 1 1.6368 −0.04615 2.1138 × 10−3 −0.7845 × 10−5 −0.6923 × 10−6 2 −1.9693 0.21874 −8.0028 × 10−3 −8.2328 × 10−5 5.2604 × 10−6 3 2.4638 −0.36461 12.8763 × 10−3 14.8059 × 10−5 −8.6895 × 10−6 4 −1.5841 0.25136 −11.3805 × 10−3 9.5672 × 10−5 2.1812 × 10−6 and it is T r = T/T c, p r = p/p c with T c = i ziTci and p c = i zipci. Standing–Katz Method Standing and Katz proposed another set of formulas for density calculations by known oil composition: The density of a C3+ mixture is ρC3+ = C7+ i=C3 ziMi C7+ i=C3 ziMi/ρ0,i (5.40)
- 235. 222 5 Fluid Analysis with ρ0,i again as the dissolved component density at standard conditions. Just as for the API method, ρC7+ is a measured density. Methane and ethane are treated separately in ρC2+ = ρC3+ (1 − 0.01386 wC2 − 0.000082 w2 C2 ) +0.379 wC2 + 0.0042 w2 C2 , ρC1+ = ρC2+ (1 − 0.012 wC1 − 0.000158 w2 C1 ) +0.0133 wC1 + 0.00058 w2 C1 (5.41) with wC1 as weight percent of C1 in the total phase and wC2 as weight percent of C2 in C2+. The density ρC1+ is the apparent oil density, which must be corrected for pressure and temperature so that ρl = ρC1+ + Δρp − ΔρT (5.42) with Δρp = 0.167 + 16.181 × 10 −0.0425 ρC1+ p 1000 − 0.01 0.299 + 263 × 10 −0.0603 ρC1+ p 1000 2 , ΔρT = 0.0133 + 152.4(ρC1+ + Δρp )−2.45 (T − 60) − 8.1 × 10−6 − 0.0622 × 10 −0.0764(ρC1+ +Δρp ) (T − 60)2 . (5.43) Herein p is given in psi, T in Fahrenheit, and the densities in lbm/ft3 . The formulas are set up for 40 ρC3+ 60 lbm/ft3 , wC1 16 and wC2 10. For low concentrations, CO2 can be taken into account with a specific gravity of 0.420. Alani–Kennedy EOS The Alani–Kennedy EOS is another alternative for density calculations when the liquid compositions are known. It is an EOS which has been optimized empirically for liquid density calculations only: v3 − R(T + 460) p + b v2 + av p − ab p = 0 , (5.44) with v as molar volume in ft3 /lbmol, T as temperature in Fahrenheit, p as pressure in psi, and R = 10.7335. The goal is to find the smallest root of (5.44). It is possible to calculate the parameters a and b by molar arithmetic mixing from the corresponding pure component values. These pure components are parametrized with parameters λ, n, m, and C by
- 236. 5.6 Property Prediction 223 a = λ en/(T +460) and b = m(T + 460) + C (5.45) which are tabulated in (5.2). The C7+ component has exceptional parameters aC7+ = exp 3.8405985 × 10−3 MC7+ − 9.5638281 × 10−4 MC7+ /SC7+ + 261.80818/(T + 460) + 7.3104464 × 10−6 M2 C7+ + 10.753517 , bC7+ = 3.499274 × 10−2 MC7+ − 7.2725403 SC7+ + 2.232395 × 10−4 (T + 460) − 1.6322572 × 10−2 MC7+ /SC7+ + 6.2256545 (5.46) with SC7+ as the specific gravity of the C7+ component. Finally the density can be calculated easily from the molar volume, the composition, and the molecular weights of the components. Component λ n m × 104 C C1 ( 70 − 300 ◦ F) 9160.6413 61.893223 3.3162472 0.50874303 C1 (301 − 460 ◦ F) 147.47333 3247.4533 −14.072637 1.8326695 C2 (100 − 249 ◦ F) 46709.537 −404.48844 5.1520981 0.52239654 C2 (250 − 460 ◦ F) 17495.343 34.163551 2.8201736 0.62309877 C3 20247.757 190.24420 2.1586448 0.90832519 i–C4 32204.420 131.63171 3.3862284 1.1013834 n–C4 33016.212 146.15445 2.9021257 1.1168144 C5 37046.234 299.62630 2.1954785 1.4364289 C6 52093.006 254.56097 3.6961858 1.5929406 H2S 13200.0 0 17.900 0.3945 N2 4300.0 2.293 4.490 0.3853 CO2 8166.0 126.00 1.8180 0.3872 Table 5.2. Component parameters of Alani–Kennedy EOS The different methods for calculation of liquid densities are compared in Fig. 5.17 for a black oil and a volatile oil. Their compositions and component properties are listed in App. K. The amount and molar weight of the C7+ fraction, which is needed as an input parameter for the API and the Standing– Katz methods, are calculated by molar average with the following component densities: Component C6−14 C15 C25 C35 C45 Density [kg/m3 ] 700 800 850 920 940 The direct density calculation from composition and molecular weight in many examples, deviates a lot from the results of the other methods. This can be
- 237. 224 5 Fluid Analysis corrected by calibrated volume shifts and molecular weights of the lumped pseudo components without affecting the composition. More about calibration can be found in Sec. 5.7. 0.3 MPa/°C 0.3 MPa/°C 1 MPa/°C 1 MPa/°C API Method Direct from SRK Alani-Kennedy Standing-Katz Volatile Oil Volatile Oil Black Oil Black Oil API Method Direct from SRK Standing-Katz Alani-Kennedy Standing-Katz Standing-Katz Alani-Kennedy Alani-Kennedy Direct from SRK Direct from SRK API Method API Method Fig. 5.17. Density of liquid phase calculated with different methods for the same black oil and volatile oil as shown in Fig. 5.14. Pres- sure and temperature follow different PT–paths with 0.3 MPa/◦ C and 1 MPa/◦ C. Both PT– paths start at normal conditions. The density is plotted in units of API and is therefore labeled “Insitu API”. The composition of the liquid is calculated with the SRK EOS API (at standard conditions) Volatile Black Oil Oil Direct from SRK 47.9 31.6 API Method 46.0 35.0 Standing–Katz 45.8 34.9 Alani–Kennedy 40.3 28.4 5.6.2 Bubble Point Pressure The bubble point pressure pb is also often called saturation pressure as can be easily seen from the discussion in Sec. 5.3. It can be read from diagrams such as shown in Fig. 5.23 or it can be calculated more accurately with special flash algorithms: At bubble point conditions it is xi = zi and nl = 1 , nv = 0 . (5.47)
- 238. 5.6 Property Prediction 225 The relation 1 = i yi = i xi φl,i φv,i (5.48) can be deduced from the equality of chemical potentials (5.29). Taking into account the definition (5.25) and the equality of fugacities fv,i = fl,i, Danesh (1998) proposed to iteratively solve for the bubble point pressure pb according to pb,n+1 = pb,n i xi φl,i φv,i . (5.49) An algorithm for a bubble point flash can now be formulated as: 1 Calculate al, bl with mixing rules 2 Estimate bubble point pressure pb 3 Estimate Ki with (5.32) 4 Calculate yi = xiKi and av, bv with mixing rules 5 Calculate Zl and Zv with (5.15) or (5.16) 6 Calculate φl,i and φv,i with (5.26) 7 Calculate Knew i = φl,i/φv,i and pnew b = i xiKnew i 8 If i (1 − Knew i /Ki)2 adjust pb ← pnew b , Ki ← Knew i and go back to step 4 The convergence criterion is, as in Sec. 5.5, again given by a small . 5.6.3 Gas Oil Ratio (GOR) The GOR defines the gas to oil volume fraction of a produced fluid at standard conditions. It is often quantified in m3 /m3 or in SCF/STB and it ranges up to 150. 000 SCF/STB (Danesh, 1998). The GOR is an indicator of the amount of heavy components in the fluid. This is in accordance to the rule of thumb that mostly C6+ form the liquid phase at standard conditions. Inversely it can be interpreted as a measure of light components (especially methane) which are dissolved in the liquid under insitu conditions. Besides density it is the most important parameter for the classification of reservoir petroleum Sec. 5.5.1. Outgassing along a typical PT–path can be visualized very well with the GOR (Fig. 5.18). A black oil and a volatile oil both exsolve gas when reaching the bubble point from hotter and therefore deeper regions. These gases have a very high GOR. They are almost dry. The gas condensate behaves differently. Oil condensates when reaching the dew point. 5.6.4 Oil Formation Volume Factor Bo Another important quantity is the oil formation volume factor Bo, which relates the subsurface insitu liquid volume (plus its dissolved vapor) to the
- 239. 226 5 Fluid Analysis Black Oil Volatile Oil Gas Condensate Insitu Vapor Insitu Liquid Insitu Undersaturated Insitu Supercritical Insitu Vapor Bubble Point Bubble Point Dew Point Fig. 5.18. GOR at standard surface conditions. It is calculated from separated phases at insitu conditions. The composition of the phases is calculated from the black oil, the volatile oil and the gas condensate of Fig. 5.14 with the SRK EOS along a PT–path of 0.3 MPa/◦ C, which ends at standard surface conditions. At the bubble or dew point each supercritical/undersaturated phase separates into liquid and vapor liquid volume at surface conditions (Peaceman, 1977). It therefore defines the shrinkage of petroleum when it is produced. It can be evaluated to Bo = Wl + Wv Wl ρl ρl,i (5.50) with liquid, vapor weights Wl, Wv, and liquid density ρl of a sample at stan- dard conditions and liquid density ρl,i at insitu conditions. It can empirically be determined with knowledge of liquid and vapor densities and the amount of vapor dissolved in the liquid under reservoir conditions. (Danesh, 1998). Obviously, the oil formation volume factor is tightly linked to the methane content in the liquid under insitu conditions. In modeling practice, two flash calculations, one for insitu and one for surface conditions must be performed for its calculation. It can be rewritten as Bo = 1 + nv k ykMk nl k xkMk ρl ρl,i (5.51) with xk, yk, nl, and nv at standard conditions. Two examples of Bo along two different PT–paths are shown in Fig. 5.19. Both quantities, GOR and Bo are often referred to if considerations con- cerning light components especially methane are performed (di Primio and Horsfield, 2006). Methane content is often a key quantity in many geolog- ical processes e.g. the proper description of source rock kinetics, secondary cracking or biodegradation.
- 240. 5.6 Property Prediction 227 Black Oil 0.3 MPa/ C ° 1 MPa/ C ° Volatile Oil 0.3 MPa/ C ° 1 MPa/ C ° Fig. 5.19. Oil formation volume factor Bo calculated from the liquid phase of a black oil and of a volatile oil. The liquid phase is calculated with the SRK EOS along two different PT–paths which both end at standard surface conditions 5.6.5 Viscosity Next to density, viscosity is an important indicator for phase property char- acterizations. It varies greatly between different oil types (Table 5.3). Due to the strong dependency of production recovery factors from viscosity it is of great interest. Viscosity in [cP] at Oil Type 50◦ C 100◦ C 150◦ C very high viscosity 1000 25 2.5 high viscosity 100 2.5 0.5 medium viscosity 10 0.5 0.25 low viscosity 3 0.35 0.1 very low viscosity 1 0.25 0.05 Table 5.3. Revised viscosity values according to Ungerer et al. (1990) Viscosity is often modeled as a quantity that is dependent only on pres- sure, temperature, density, and the amount of dissolved gas (Danesh, 1998). An overview about empirical correlations is given by Bergman and Sutton (2007). Models taking detailed compositional effects into account are found in (Pedersen et al., 1984; Pedersen and Fredenslund, 1987; Zéberg-Mikkelsen, 2001). Obviously, the effect of long–chained compounds has a significant in- fluence on the viscosity. Hence a good characterization of the fluid heavy end is even more important for viscosity than for density prediction. A recent comparison of actual models with measurement data can be found in Zéberg-Mikkelsen (2001). Advanced theories such as the friction–theory or
- 241. 228 5 Fluid Analysis the free–volume model match laboratory data best. But due to a lack of field data and unknown component parameters for the advanced theories, especially for many important heavy end compounds, it is common to follow more direct approaches such as the empirical Lohrenz–Bray–Clark (LBC) model (Lohrenz et al., 1964) or the corresponding states (CS) model (Pedersen et al., 1984; Pedersen and Fredenslund, 1987; Lindeloff et al., 2004) in practice. Lohrenz–Bray–Clark (LBC) Model The starting point of the LBC model are empirical formulas for the viscosity ν0 of low–pressure pure component fluids: ν0 = 34 × 10−5 (T/Tc)0.94 /λ for T/Tc ≤ 1.5 17.78 × 10−5 (4.58 (T/Tc) − 1.67) 5/8 /λ for T/Tc 1.5 . (5.52) Herein, λ is called the viscosity reducing parameter. It has its origin in the kinetic theory of gases and is defined as λ = T1/6 c M−1/2 p−2/3 c . (5.53) For higher pressures an empirical correlation is used. It has the form (ν − ν0) λ + 10−4 1/4 = a0 + a1ρr + a2ρ2 r + a3ρ3 r + a4ρ4 r (5.54) with the reduced density ρr = ρ/ρc and the coefficients a0 = 0.1023, a1 = 0.023364, a2 = 0.058533, a3 = −0.040758, a4 = 0.0093324 . (5.55) The model can be improved with an individual fit of the coefficients (5.55) to known viscosity data sets. Multi–component systems are treated with Herning–Zipperer mixing rules. The low density viscosity is “mixed” according to ν0 = i ziν0i Mi i zi Mi (5.56) and the viscosity reducing parameter following λ = i ziTci 1/6 i ziM −1/2 i −1/2 i zipci −2/3 . (5.57) The mixing rule for the reduced density ρr = ρ/ρc = vc/v with v as the molar volume can be rewritten to a rule for the critical volume vc. It is estimated with
- 242. 5.6 Property Prediction 229 vc = i zivci (5.58) and for the C7+ according to vc,C7+ = 1.3468 + 9.4404 × 10−4 MC7+ −1.72651 SC7+ + 4.4083 × 10−3 MC7+ SC7+ (5.59) with S indicating the specific gravity. With these values and knowledge of the density ρ or the molar volume v, formula (5.54) can be used to estimate the viscosity of a mixture. In the above formulas temperatures are in Kelvin, pressures in atmo- spheres, critical volumes in (5.59) in m3 /kmol and viscosities in mPa s = cP. Viscosities can be evaluated very fast due to the simple nature of the LBC–formulas. Models with lower performance are often not usable in fluid flow simulators. But it must be noted that the LBC–model is based on a polynomial of degree 16 as introduced in (5.54). Polynomials of such a high degree are known to easily become numerically unstable and therefore LBC based models must be evaluated with care. Corresponding States (CS) Model The principle of corresponding states is not only applied to viscosity predic- tions (Danesh, 1998). The central assumption is the observation that many properties behave similarly within the same “relative distance” from the crit- ical point. This observation can be shown best with the vdW EOS in its “reduced” form pr = 8 Tr 3 vr − 1 − 3 v2 r . (5.60) Here, “reduced quantities” are defined as pr = p/pc, Tr = T/Tc, and vr = v/vc. Equation (5.60) describes a universal behavior determined only by the relative distance to the critical point. It is only dependent on quantities which are occurring in reduced form. Following the above scheme the CS model for viscosity prediction must basically consist of two parts: Firstly the viscosity behavior of a well known reference fluid must be quantified to a high degree of accuracy and secondly scaling procedures for mapping of this behavior to the fluid under investigation must be formulated. In the model of Pedersen et al. (1984) methane is chosen as the reference fluid. Its viscosity can be calculated with formulas from Hanley et al. (1975, 1977): νref (ρ, T) = ν0 + ν1(T)ρ + Δν (ρ, T) . (5.61) ν0 is the dilute gas viscosity, which can be calculated with
- 243. 230 5 Fluid Analysis ν0 = 9 i=1 GVi T i−4 3 (5.62) and the coefficients GV1 = −2.090975 × 105 GV2 = 2.647269 × 105 GV3 = −1.472818 × 105 GV4 = 4.716740 × 104 GV5 = −9.491872 × 103 GV6 = 1.219979 × 103 GV7 = −9.627993 × 101 GV8 = 4.274152 GV9 = −8.141531 × 10−2 . (5.63) The term ν1ρ describes a low order density correction with ν1 = 1.696985927 − 0.133372346 1.4 − ln T 168.0 2 . (5.64) For methane of higher density the term Δν becomes more important. It is Δν = exp j1 + j4 T × exp ρ0.5 j2 + j3 T3/2 + θρ0.5 j5 + j6 T + j7 T2 − 1.0 (5.65) with θ = ρ − ρc ρc (5.66) and the coefficients j1 = −10.35060586 j2 = 17.571599671 j3 = −3019.3918656 j4 = 188.73011594 j5 = 0.042903609488 j6 = 145.2902344 j7 = 6127.6818706 . (5.67) Pedersen and Fredenslund (1987); Pedersen et al. (1989) extended (5.61) to temperatures below the freezing point of methane at TF = 91 K which corresponds to reduced temperatures below 0.4. Equation (5.61) becomes now νref (ρ, T) = ν0 + ν1(T)ρ + F+Δν (ρ, T) + F−Δν (ρ, T) (5.68) with an additional term Δν = exp k1 + k4 T × exp ρ0.5 k2 + k3 T3/2 + θρ0.5 k5 + k6 T + k7 T2 − 1.0 , (5.69) the coefficients
- 244. 5.6 Property Prediction 231 k1 = −9.74602 k2 = 18.0834 k3 = −4126.66 k4 = 44.6055 k5 = 0.976544 k6 = 81.8134 k7 = 15649.9 (5.70) and the weight factors F± = 1 ± tanh(T − TF ) 2 (5.71) which ensure a continuous crossover between ν and ν at the freezing tem- perature TF . The viscosity is calculated in units of μP with the density in g/cm3 and the temperature in K. The viscosity can now be calculated as mentioned above by relative scaling to the critical point. The general formula has the form ν = Tc Tc,ref −1/6 pc pc,ref 2/3 M Mref 1/2 α αref νref (pref , Tref ) . (5.72) Obviously, the first three factors of (5.72) represent a scaling with the viscosity reducing parameter, which has already been introduced in (5.57) for the LBC model. Additionally corrective factors α and αref are introduced. These α– factors are analogously used for a correction of the reference pressure and temperature following pref = p pc,ref pc αref α and Tref = T Tc,ref Tc αref α (5.73) and can be calculated from α = 1 + 7.378 × 10−3 ρ1.847 r M0.5173 , αref = 1 + 0.031ρ1.847 r . (5.74) The reduced density ρr which is necessary for the calculation of the α–factors can obviously only be evaluated without the corrections itself: ρr = ρref p pc,ref pc , T Tc,ref Tc ρc,ref . (5.75) The methane reference density without α–corrections in (5.75) or with α– corrections in (5.68) can be calculated with a modified Benedict–Webb–Rubin EOS proposed by McCarty (1974). For the sake of completeness this lengthy formula is listed in App. J. Similar as in the LBC model the critical quantities Tc, pc and the molecular weight M must initially be calculated with mixing rules from pure component
- 245. 232 5 Fluid Analysis values. The mixing of critical quantities follows a procedure similar to the Lee–Kesler rules (5.6):9 Tc = ij zizj TciTcjVcij ij zizjVcij and pc = 8 Tc ij zizjVcij (5.76) with Vcij = ⎡ ⎣ Tci pci 1 3 + Tcj pcj 1 3 ⎤ ⎦ 3 . (5.77) The molecular weight is calculated by M = Mn + 1.304 × 10−4 (M2.303 w − M2.303 n ) (5.78) with the molecular weight M in g/mol and the molar average Mn and mass average Mw defined by Mn = i ziMi and Mw = i ziM2 i i ziMi . (5.79) CS for Heavy Oils The CS model can be extended to heavy oils (Lindeloff et al., 2004). Instead using methane as the reference compound a correlation for heavy oils proposed by Rønningsen (1993) is used. It is for viscosity ν0 under atmospheric pressure conditions log10 ν0 = −0.07995 − 0.01101M − 371.8 T + 6.125M T (5.80) with M = Mn for Mw Mn ≤ 1.5 and M = Mn Mw 1.5Mn 0.5 else. (5.81) The first case describes a stabe oil and the second a live oil.10 The exponent 0.5 and the coefficient 1.5 can be used as tuning parameters. Viscosity ν0 is in mPa s = cP and temperature in K. 9 Here it is assumed that all components have the same critical compressibility Zc! 10 A live oil contains dissolved gas that may be released at surface conditions (www.glossary.oilfield.slb.com). Correspondingly, a stable oil (dead oil) does at surface conditions not contain dissolved gas (anymore). It is in thermodynamic equilibrium.
- 246. 5.6 Property Prediction 233 The correlation is only valid for atmospheric pressure. A pressure correc- tion of the form 1 ν0 ∂ν ∂p = 0.008/atm (5.82) is assumed. The heavy oil correlation is applied for methane reference temperatures Tref 65 K. Below 75 K a smooth crossover from methane as the reference compound can be assumed. This can be modeled with a formula such as (5.71). Other modifications and extensions of the CS theory are reported in the literature. For example n−decane is used as the reference fluid by Dexheimer et al. (2001). Viscosities of two example oils according to LBC and CS theory are shown in Fig. 5.20. They are strongly dependent on composition and the type of com- ponents. A heavy end characterization might be necessary and a calibration of fluid data against the LBC coefficients (5.55) or against properties of lumped components can usually not be avoided. The curves in Fig. 5.20 are therefore only example curves. Black Oil Volatile Oil LBC 0.3 MPa/°C CS 0.3 MPa/°C LBC 1 MPa/°C CS 1 MPa/°C LBC 0.3 MPa/°C CS 0.3 MPa/°C LBC 1 MPa/°C CS 1 MPa/°C Fig. 5.20. Liquid phase viscosities according to LBC and CS theories. The black oil and the volatile oil, which are shown in Fig. 5.14 and which are more precisely de- fined in (App. K), are chosen as examples. Viscosities are calculated for two different PT–paths 5.6.6 Interfacial Tension (IFT) A precise prediction of IFT is very sophisticated. It can only be performed if a detailed analysis of the phase composition and physical behavior is performed down to a molecular level. In basin modeling, IFT prediction is therefore restricted to basic approximations. Interfacial tension is a property which is dependent on two phases. All combinations between different fluid phases must be treated explicitly. Obvi- ously, the correct distribution of the components in the different phases is a prerequisite. This can be gained by flash calculations.
- 247. 234 5 Fluid Analysis Liquid–Vapor Interfacial Tension The liquid–vapor IFT γl,v is usually determined with the parachor method γl,v 1/4 = Pγ(ρn,l − ρn,v) (5.83) with γl,v in mN/m, molar densities ρn,l, ρn,v in mol/cm3 , and “parachor” constant Pγ. Parachor values are tabulated for components and usually mixed based on molar fractions (Danesh, 1998). Equation (5.83) then becomes γl,v 1/4 = ρn,l i xiPγ,i − ρn,v i yiPγ,i . (5.84) Petroleum–Water Interfacial Tension Usually petroleum–water IFT is listed in lookup–tables for simulation. IFT values are mainly used for determination of capillary pressure for migration and accumulation in basin modeling (Sec. 6.3.1). Very often rough estimates such as constant values of γp,w = 42 mN/m are used because uncertainties in IFT can be neglected compared to uncertainties of knowledge about pore throat radii. However, it is found that pressure dependency of IFT is much weaker than temperature dependency. An improved empirical correlation which reflects this behavior is γp,w = 111(ρw − ρp)1.024 (T/Tp,c)−1.25 (5.85) with the densities in g/cm3 and γp,w in mN/m (Danesh, 1998, Fig. 5.21). The critical temperature Tp,c of the petroleum phase must be known. Its calculation with mixing rules for critical parameters such as the Lee–Kesler– Average (5.6) are reported in the literature. Fig. 5.21. Petroleum–water interfacial tension γp,w according to (5.85) for dif- ferent density contrasts ρw − ρp T/Tp,c g p,w [mN/m] 10 kg/m^3 200 kg/m^3 400 kg/m^3 600 kg/m^3 800 kg/m ^3 1000 kg/m ^3
- 248. 5.7 Calibration of a Fluid Model 235 5.7 Calibration of a Fluid Model Besides overpressure and heat flow calibration (Secs. 2.4, 3.9) it is also com- mon to calibrate against fluid data. The type of EOS, the choice of an appro- priate set of binary interaction parameters and the method of grouping and lumping of pseudo components, especially for the heavy end of the fluids, are frequently calibrated against easily available fluid sample data. In addition to the compositional analysis, these are often basic properties such as density, viscosity, gas oil ratio (GOR) at surface conditions and bubble point pressure at surface temperature. Other properties, such as heat capacities, or more de- tailed results from fluid analysis, such as swelling tests, are usually not taken into account for flash model calibrations in basin modeling. Flash calculations are sometimes used for compositional predictions but not for density calculations of liquids. Liquid densities at or near standard conditions can also be calculated with the API method, the Standing–Katz procedure or the optimized EOS of Alani–Kennedy (Sec. 5.6.1) in a post processing approach in case of known composition. These methods have a high degree of accuracy within a limited range of applicability. Evidently, they can only be used for the analysis of final simulation results and not for the entire simulation. Consistency between the more general EOS and the precise empirical formulas can be reached by a calibration of both methods against each other. Typically this can be done by proper selection of the pseudo– components and heavy end analysis (see 5.7.1 and 5.7.2 below). Obviously, this only makes sense if fluid sample data is not available. In basin modeling the overall fluid model calibration is not a matter of fluid analysis only. The choice of a proper set of pseudo–components is not exclusively based on fluid characterization. The set of pseudo–components should be on the one hand as small as possible for high simulation performance and on the other hand sufficient to correctly model the PVT–behavior as well as the basic geochemical processes such as generation kinetics and source rock maturation. Secondary cracking or biodegradation are examples of component specific processes, which should also be taken into account for the choice of components. Methane content is a key quantity in many geological processes. It strongly effects all easily accessible fluid parameters such as the gas oil ratio, the oil formation volume factor or the bubble point pressure. Due to its volatile na- ture, it is often used as a calibration parameter for fluid analysis. However, the methane content is usually calibrated against the source rock type and its associated kinetic (di Primio and Skeie, 2004; di Primio et al., 1998; di Primio and Horsfield, 2006). Fluid model calibration with the methane content there- fore becomes obsolete with the quantification of a source rock kinetic. In general it is assumed that the overall composition is not a matter of calibration in fluid analysis. The stoichiometry of geochemistry and mass con- servation in basin modeling permit the calibration of parameters which change the composition of the simulation results.
- 249. 236 5 Fluid Analysis Many kinetics are formulated for a predetermined set of pure and pseudo– components. They thus specify the components of the model. The number of the components is usually relatively small. Good quantitative information about the geochemistry of a source rock is often rare and a small set is nec- essary or at least very advantageous for a good simulation performance of a large basin model. Hence, the number of HC fluid components seldom exceeds 14.11 However, there are still degrees of freedom for the calibration of the fluid model. 5.7.1 Calibration and Fluid Heavy End The fluids heavy end is often relatively unspecified due to the limited number of pseudo–components given by typical kinetics. Hence, heavy end character- ization parameters can be used as calibration parameters in basin modeling. A rough approximation can be based on a fit of the molecular weight of the heaviest pseudo–component against density or API. The PR, the SRK EOS and the fugacities do not depend explicitly on molecular weights of compo- nents. Thus the composition of the phases does not change under the variation of molecular weights, whereas density varies linearly with its change. Such a fit is straight forward and easy to perform. The lumped heavy end zCn+ with its mass MCn+ is usually known from modeling results or from sample analysis. From detailed analysis of real sam- ples it has been found that heavy HCs are often distributed exponentially in single carbon number (SCN) groups Cn according to zCn ∝ exp(−FMCn ) (5.86) with F as a sample–dependent constant and MCn as the molecular weight of the lumped component Cn consisting of pure components with n carbon atoms (Danesh, 1998). Frequently, fluid PVT–property predictions can be improved by former expansion of heavy pseudo–components according to this scheme.12 Partly grouping and lumping these expanded components with regard to the improved predictions, leads to a new reduced set of pseudo–components, which is much smaller than the expanded one but more complex than the 11 If component tracking is modeled the number of components is sometimes much higher. However, in such cases the number of components of the kinetic of each source rock is still limited to a small number. Because of this and the fact that source rock potentials can be compared better if the “assigned components” are physically the same, it turns out that in general the overall number of physically distinct components is rather small. 12 With the assumption MCn+1 −MCn = const. it is possible to analytically calculate a good initial guess F ≈ 1/(MCn+ −MCn0 ) from (5.86) for a heavy end distribution starting at Cn0 . In practice it is conveniently assumed that zCn = 0 for large n, e.g. n 45.
- 250. 5.7 Calibration of a Fluid Model 237 original set. A workflow of this type is often integrated in the proper selection of the entire component set. The predicted fluids in a basin model are sometimes “re-expanded” for higher accuracy of the final result. However, operations such as heavy end expansions, grouping or lumping during the simulation are usually not per- formed, due to a lack of computing performance. More details about heavy end characterizations can be found in Pedersen et al. (1989) or Pedersen and Christensen (2007). 5.7.2 Tuning of Pseudo–Component Parameters Sometimes systematic errors in the process of pseudo–component definition concerning fluid analysis appear. Lumping rules are often only based on rough semi–empirical formulas such as molar averages. The resulting pseudo– component parameters are not of the expected quality. They describe phys- ical properties of a pure one pseudo–component fluid. But they are multi- component systems themselves and therefore more complex than a pure chem- ical species. It is both theoretically and experimentally difficult to determine the critical quantities of a pseudo–component. Furthermore, lumping heavy end components usually decreases the size of the coexistence area because the most heavy components are removed (Fig. 5.22). Hence, the resulting pseudo–component parameter sets are often only starting points for a cali- bration against real data. It must also be mentioned that EOS such as SRK or PR are themselves based on various assumptions and approximations. A calibration of component parameters individually to SRK or PR EOS en- hance at least the predictive quality of the basin model. Therefore, tuning or calibration of pseudo–components can be very helpful in basin modeling. It is easy to modify pseudo–component parameters so, that some quanti- ties, such as the size of the coexistence area, fit better after lumping (Fig. 5.23). However, all quantities are non-linearly coupled and an improvement such as in (Fig. 5.23) could here only be reached for the price of making the GOR prediction worse. The most important quantities for calibration are API, GOR, bubble point pressure pb and oil formation volume factor Bo (Stainforth, 2004). They should at least be calibrated against all pseudo–component parameters namely crit- ical temperature Tc, pressure pc, acentric factor ω, and molecular weight M. In principle this can be subdivided into two major steps: Firstly Tc, pc, and ω can be fitted against GOR, pb, and Bo as these quantities are independent of the molecular weights of the pseudo–components. Molecular weights enter equations of PVT–analysis only for the calculation of API and densities. As mentioned in the previous section, this relation is linear and it therefore follows that API can easily be fitted by variation of the molecular weights of the pseudo–components. In practice both steps can be performed at once with a Markov chain Monte Carlo (MCMC) inversion algorithm. Flash calculations should there-
- 251. 238 5 Fluid Analysis Fig. 5.22. Coexistence area of a fluid which consists of SCN components rang- ing from methane up to C45. The solid line limits the coexistence area for the same fluid modeled with a lumped heavy end of the form C7−25 and C26−45 and the dashed line refers to further lumping with only one C7+ component Fig. 5.23. Coexistence area of the same fluid as in Fig. 5.22 with one heavy pseudo–component C7+. Here pc and Tc of C7+ were shifted manually from pc = 2.14 MPa and Tc = 415.8 ◦ C to pc = 1.7 MPa and Tc = 500 ◦ C so that the size of the coexistence area again ap- proaches the original size. The composi- tion is listed in Table 5.4. Note that some deviations still exist. Especially the loca- tion of the critical point is different. Ad- ditionally, the fraction of liquid and va- por is altered. GOR changed from 212 to 165 m3 /m3 . API could be adjusted man- ually by variation of MC7+ from 193.3 to 247 g/mol. In summary, C7+ parameters were changed dramatically to achieve a calibration against a coexistence area size and API only Fig. 5.24. Coexistence area of the same fluid as in Fig. 5.22 and Fig. 5.23 with one heavy pseudo–component C7+. Here the pseudo–component parameters are fitted automatically with the MCMC against the results from the unlumped composi- tion. The fitting procedure was repeated under manual change of Tc and pc to achieve a coexistence area of almost the same size as for the original fluid in Fig. 5.22. A Bayesian objective provided a limit of change in respect to M, Tc, and pc of C7+. GOR and API are ex- actly matched with M = 191 g/mol, Tc = 492◦ C and pc = 2.07 MPa
- 252. 5.7 Calibration of a Fluid Model 239 Molar M Tc pc vc Acentric Volume Shift Fraction [%] [g/mol] [◦ C] [MPa] [m3 /kmol] Factor [m3 /kmol] Methane 50 16.043 −82.59 4.599 0.0986 0.0115 0.0007 Ethane 6 30.070 32.17 4.872 0.1455 0.0995 0.0028 Propane 5 44.096 96.68 4.248 0.2000 0.1523 0.0052 n–Butane 3 58.123 151.97 3.796 0.2550 0.2002 0.0080 n–Pentane 2 72.150 196.55 3.370 0.3130 0.2515 0.0122 C6 2 84.000 236.85 3.271 0.3480 0.2510 0.0134 C7+ 32 247.000 500.00 1.700 0.7237 0.5295 0.0778 Table 5.4. Composition of an “Example–Oil”. The corresponding phase diagram is shown in Fig. 5.23 fore be implemented efficiently because MCMC inversions are based on many model evaluations, typically 100.000 times and more. A prototype for the objective χ2 PVT of the MCMC inversion is given by (7.12) and becomes χ2 PVT = API − APIm[pci, Tci, ωi, Mi] ΔAPI 2 + GOR − GORm[pci, Tci, ωi] ΔGOR 2 + pb − pbm[pci, Tci, ωi] Δpb 2 + Bo − Bom[pci, Tci, ωi] ΔBo 2 . (5.87) The index m points out that the related quantities are modeled with flash calculations and the square brackets indicate a dependency on the set of all pseudo–components, which are labeled here with the index i. The importance of each quantity in the inversion is controlled by its Δ factor. Decreasing Δ increases the importance in the same way as the reduction of a measurement error increases the importance of the related measurement. Minimization of (5.87) calibrates the fluid model. The MCMC algorithm is briefly outlined in Sec. 7.5.6. An application of this algorithm to the lumped fluid is depicted in Fig. 5.22 with the calibration values taken from the original unlumped results quickly providing a perfect calibration. Unfortunately, the size of the coexistence area remains smaller than the one from the original data. Changing the critical pressure or temperature of the C7+ pseudo–component manually, before per- forming a new MCMC inversion, leads to a different calibrated end result with a differently coexistence area. The MCMC finds just one solution out of multiple possibilities. In such a case it is advantageous to add a Bayesian term to (5.87) to ensure that the MCMC algorithm does not alter the pseudo– component parameters too much. The objective becomes now φPVT = χ2 PVT + φTc + φpc + φω + φM (5.88)
- 253. 240 5 Fluid Analysis with e.g. φTc = N i=1 Tci − Tc0i ΔT 2 , (5.89) N as the number of components, and Tc0i the critical value from which the calibration result should not deviate too much. The formulas for φpc , φω, and φM can be constructed analogously. Smaller the Δ factors are chosen as less variations of the related parameters are allowed. Iterative modification of the pseudo–component parameters and usage of the MCMC inversion quickly leads to the result shown in Fig. 5.24. It must be noted that a calibration via the variation of molecular weight M, violates mass conservation. This is demonstrated with an extreme case: a lumped two-component system with API = 29, GOR = 300 m3 /m3 and pb = 11 MPa and pseudo–component parameters, as in Table 5.5, is calibrated against API = 30, GOR = 100 m3 /m3 , and pb = 14 MPa. The molar fraction of both pseudo–components is kept constant. A calibration is easily possible by varying “Medium Oil” to “Medium Oil A”. However, the variation is dramatic. The original “Medium Oil” has properties in the range of C5 – C8 on a SCN – scale, whereas the calibrated component parameters can be found in a wide region around C15. The molecular weight is more than doubled. Alternatively, a calibration can also be performed under a constant mass fraction constraint for each component. Here this is unfortunately not possible via variation of only the heavy component. However, a calibration of both components yields quickly new parameters “Dry Gas B” and “Medium Oil B”. The variation of the pseudo–component parameters is less drastic. Note the variation of the molar composition. Molar M Tc pc Acentric Fraction [%] [g/mol] [◦ C] [MPa] Factor Dry Gas 50.0 17.943 −75.730 4.850 0.02210 Medium Oil 50.0 101.141 238.770 3.459 0.26060 Medium Oil A 50.0 208.075 292.118 1.768 1.24167 Dry Gas B 34.7 37.498 −90.806 5.298 0.10547 Medium Oil B 65.3 112.313 207.621 2.790 1.07016 Table 5.5. Two component example 5.7.3 Tuning of the Binary Interaction Parameter (BIP) It is rather unusual in basin modeling to perform a fine–tuning of the BIP because it is assumed that the modeling of large compositional variations un- der the strongly varying subsurface conditions of geological processes, are not significantly improved with fine–tuned parameters, which are often adapted
- 254. 5.8 Gas Hydrates 241 to near surface conditions. Besides this, the usage of the BIP decreases the performance of flash calculations so that the overall simulator might slow down significantly. As for heavy end expansions, BIPs are sometimes tuned for higher quality of the final results. 5.8 Gas Hydrates Clathrate hydrates are solid state phases which consist of water in “icy form” with other chemical species occupying the “ice–cavities”. These other chemi- cal species are often gases under normal conditions such as methane, carbon dioxide or hydrogen sulfide which are small enough to occupy even small sized molecular cavities. Additionally, these species appear frequently enough to form significant hydrate amounts in natural systems. It is therefore common to restrict the modeling to gas hydrates. Very often only methane hydrates are modeled. Methane is generated in each depth level and time scale of a basin model. It is produced in deeply buried gas prone source rocks, by secondary cracking in reservoir rocks, by biodegradation or by direct transformation of deposited organic matter in shallow sequences. It is also highly mobile in re- gards to primary and secondary migration with sufficient water solubility to allow it to be transported by groundwater (Sec. 5.2). Thus methane forms the most important hydrate deposits in natural systems. The PT phase diagram for methane hydrate stability of a pure water methane system is depicted in Fig. 5.25. The phase boundaries are calculated from an empirical equation of the form p = exp(a − b/T) (5.90) with p in kPa (Sloan Jr., 1998). Parameters a and b and values for the quadru- ple points13 are listed in Table 5.6. The influence of other chemical species, such as salt must be taken into account in more realistic systems. Impurities often act as hydrate inhibitors. The formation of gas hydrates has some implications on bulk lithology properties such as a change of thermal conductivity (Fig. 3.8). However, gas hydrate layers are usually thin compared to the extensions of a basin model and for that reason it can be assumed that their overall impact is small. An exception is a drastic reduction of permeability because pore throats are choked by hydrates. A layer which contains hydrates usually acts as a seal for water flow and petroleum migration. 13 Four phases coexist at a quadruple point.
- 255. 242 5 Fluid Analysis Pressure [MPa] Temperature [ C] ° Methane + Water Hydrate + Ice Hydrate + Water Methane + Ice Q1 Fig. 5.25. Methane water phase diagram according to (5.90). Methane is meant to be in the vapor and water in the liquid phase. The quadruple point Q1 is located at −0.2◦ C and 2.563 MPa (Table 5.6). The dashed line indicates a temperature pressure PT–path. It starts at 5 MPa, which is equivalent to a water depth of about 500 m, and continues with a linear temperature gradient of 30 K/km and hydrostatic conditions of 10 MPa/km. The dash–dotted line depicts a permafrost example with the same temperature and pressure gradients. For both PT–path examples it is possible that hydrates form in a limited depth range above the phase separation line (Gas Hydrate Stability Zone – GHSZ). Note that according to (5.90) the phase boundary between hydrate and water and the boundary between hydrate and ice are fitted empirically and independently. This leads to a small artificial discontinuity at the quadruple point Q1 Q2 T[◦ C] p[MPa] T[◦ C] p[MPa] CH4 −0.2 2.563 – CO2 0 1.256 9.9 4.499 H2S −0.3 0.093 29.6 2.293 Table 5.6. Quadruple points and parameters for (5.90) from Sloan Jr. (1998). Note, that the formula for CO2 is limited to 9.9 ◦ C due to a second quadruple point Q2 Methane CH4 −25 ◦ C T −0.2 ◦ C −0.2 ◦ C T 25 ◦ C a b[K] a b[K] 14.717 1886.79 38.980 8533.80 Carbon Dioxide CO2 −25 ◦ C T 0 ◦ C 0 ◦ C T 9.9 ◦ C a b[K] a b[K] 18.594 3161.41 44.580 10246.28 Hydrogen Sulfide H2S −25 ◦ C T −0.3 ◦ C −0.3 ◦ C T 25 ◦ C a b[K] a b[K] 16.560 3270.41 34.828 8266.10
- 256. REFERENCES 243 Summary: Water, liquid HCs and vapor HCs are the fluid phases, which occur most frequently in sedimentary basins. HCs are sometimes only present in one supercritical or undersaturated phase with methane bearing the only HC component which is found in non– negligible amounts in water. H2O does not dissolve in petroleum and sep- arates strongly from the other phases. Therefore, the prediction of phase amounts, compositions and properties is reduced to a consideration of the HCs only. Some basic aspects of phase separation are demonstrated with the highly simplified fixed phase and the symmetrical black oil model. These models are only valid in limited pressure and temperature intervals. Basin wide predictions can only be performed with a more precise specifi- cation of the pressure–volume–temperature (PVT) relationships of the fluids. These relationships are called equations of state (EOS). The most well known EOS are the Soave–Redlich–Kwong and the Peng–Robinson EOS. Phase amounts and compositions can be calculated by minimization of thermo- dynamical potentials, namely the Gibbs free energy, in multi-compositional resolution. The resulting algorithm is called a flash calculation and is well known for its accurate results. The properties of phases can be further studied based on compositional information and the consideration of empirical correlations. The focus is put on density and API predictions, bubble point pressure, gas oil ratio (GOR), the oil formation volume factor (Bo) and the viscosity. However, viscosity predictions are very difficult. Overall fluid compositions are not well known in basin models. Sparsely available source rock data, limited geochemical component resolution and many uncertainties about the details of the geological processes, involved in generation and expulsion, do not allow high compositional resolutions. Be- sides this, memory restrictions in computer simulations also limit the amount of processable components. Hence components are often lumped crudely to- gether. A maximum number of 14 pseudo components is seldom exceeded. However, pseudo components are specified by parameters which have de- grees of freedom for calibration. A procedure for simultaneous calibration of pseudo component parameters against known properties of fluid samples is therefore possible. The predictive quality of the fluid analysis can be im- proved significantly when fluid sample data is available. Finally, the conditions for formation of gas hydrates (Gas Hydrate Sta- bility Zone – GHSZ) are outlined. References R. Battino. Methane, volume 27/28 of Solubility data series. Pergamon Press., 1987.
- 257. 244 5 Fluid Analysis R. Becker. Theorie der Wärme. Springer–Verlag Berlin Heidelberg, 3. Auflage, 1985. D. F. Bergman and R. P. Sutton. A consistent and accurate dead–oil–viscosity method. SPE 110194, 2007. A. Danesh. PVT and Phase Behaviour of Petroleum Reservoir Fluids. Num- ber 47 in Developments in petroleum science. Elsevier, 1998. D. Dexheimer, C. M. Jackson, and M. A. Barrufet. A modification of Peder- sen’s model for saturated crude oil viscosities using standard black oil PVT data. Fluid Phase Equilibria, 183–184:247–257, 2001. R. di Primio. Private communication, 2008. R. di Primio and B. Horsfield. From petroleum–type organofacies to hydro- carbon phase prediction. AAPG Bulletin, 90:1031–1058, 2006. R. di Primio and J. E. Skeie. Development of a compositional kinetic model for hydrocarbon generation and phase equilibria modelling: A case study from Snorre field, Norwegian North Sea. In J. M. Cubitt, W. A. England, and S. Larter, editors, Understanding Petroleum Reservoirs: Towards an Integrated Reservoir Engineering, Special Publication, pages 157–174. Ge- ological Society of London, 2004. R. di Primio, V. Dieckmann, and N. Mills. PVT and phase behaviour analysis in petroleum exploration. Organic Geochemistry, 29:207–222, 1998. J. Haas. An empirical equation with tables of smoothed solubilities of methane in water and aqueous sodium chloride solutions up to 25 weight percent, 360◦ C and 138 MPa. Open–file rep 78–1004, USGS, 1978. H. J. M. Hanley, R. D. McCarty, and W. M. Haynes. Equations for the viscosity and thermal conductivity coefficients of methane. Cryogenics, pages 413–417, 1975. H. J. M. Hanley, W. M. Haynes, and R. D. McCarty. The viscosity and thermal conductivity coefficients for dense gaseous and liquid methane. J. Phys. Chem. Ref. Data, 6:597–609, 1977. K. Huang. Statistical Mechanics. John Wiley Sons, second edition, 1987. N. Lindeloff, K. S. Pedersen, H. P. Rønningsen, and J. Milter. The corre- sponding states viscosity model applied to heavy oil systems. Journal of Canadian Petroleum Technology, 43:47–53, 2004. J. Lohrenz, B. G. Bray, and C. R. Clark. Calculating viscosities of reservoir fluids from their compositions. Journal of Petroleum Technology, pages 1171–1176, 1964. W. D. McCain Jr. The Properties of Petroleum Fluids. Pennwell Books, second edition, 1990. R. D. McCarty. A modified Benedict–Webb–Rubin equation of state for methane using recent experimental data. Cryogenics, pages 276–280, 1974. J. S. Nelson and E. C. Simmons. Diffusion of methane and ethane through the reservoir cap rock: Implications for the timing and duration of catagenesis. AAPG Bulletin, 79:1064–1074, 1995. D. W. Peaceman. Fundamentals of Numerical Reservoir Simulation. Num- ber 6 in Developments in petroleum science. Elsevier, 1977.
- 258. REFERENCES 245 K. S. Pedersen and P. L. Christensen. Phase Behavior of Petroleum Reservoir Fluids. CRC Taylor Francis, 2007. K. S. Pedersen and A. Fredenslund. An improved corresponding states model for the prediction of oil and gas viscosities and thermal conductivities. Chemical Engineering Science, 42:182–186, 1987. K. S. Pedersen, A. Fredenslund, P. L. Christensen, and P. Thomassen. Vis- cosity of crude oils. Chemical Engineering Science, 39:1011–1016, 1984. K. S. Pedersen, Aa. Fredenslund, and P. Thomassen. Properties of Oils and Natural Gases, volume 5 of Contributions in Petroleum Geology Engi- neering. Gulf Publishing Company, 1989. R. C. Reid, J. M. Prausnitz, and B. E. Poling. The Properties of Gases and Liquids. McGraw–Hill Book Company, 4th edition, 1987. H. P. Rønningsen. Prediction of viscosity and surface tension of north sea petroleum fluids by using the average molar weight. Energy and Fuels, 7: 565–573, 1993. J. L. Sengers and A. H. M. Levelt. Diederick Korteweg, pioneer of criticality. Physics Today, pages 47–53, 2002. E. D. Sloan Jr. Clathrate Hydrates of Natural Gases. Marcel Dekker Inc., second edition, 1998. J. G. Stainforth. New insights into reservoir filling and mixing processes. In J. M. Cubitt, W. A. England, and S. Larter, editors, Understanding Petroleum Reservoirs: Towards an Integrated Reservoir Engineering, Spe- cial Publication, pages 115–132. Geological Society of London, 2004. P. Ungerer, J. Burrus, B. Doligez, P. Y. Chenet, and F. Bessis. Basin evalu- ation by integrated two–dimensional modeling of heat transfer, fluid flow, hydrocarbon gerneration and migration. AAPG Bulletin, 74:309–335, 1990. C. K. Zéberg-Mikkelsen. Viscosity Study of Hydrocarbon Fluids at Reservoir Conditions. PhD thesis, Technical University of Denmark, Lyngby, Den- mark, 2001.
- 259. 6 Migration and Accumulation 6.1 Introduction The processes of petroleum migration are still under discussion and not very well understood. Reservoir engineering and production modeling, which are usually based on Darcy type separate phase flow and mass conservation, are successfully applied to model petroleum flow, at least in reservoirs (Peaceman, 1977; Aziz and Settari, 1979; Barenblatt et al., 1990; Dake, 2001). Engineer- ing success and the persuasiveness of the approach justify a transfer of the methodology from reservoirs to petroleum systems and from timescales of years to millions of years. The resulting differential equations constitute a consistent and complex description for modeling migration in porous media. The argumentation is supported by the fact that Darcy’s law has already been applied successfully to model water flow and compaction in sedimentary basins on geological timescales (Chap. 2). The straight–forward extension of the single phase water flow model to include petroleum phases yields the most comprehensive and consistent formulation of multi–phase Darcy flow in one set of coupled differential equations. The separation of water flow for the calculation of compaction as demonstrated in Chap. 2 is an accurate approx- imation. The chapter starts with a short introduction to the geological aspects of migration Sec. 6.2. Fundamental aspects of Darcy flow based migration modeling are described in Sec. 6.3. Therein, a general description of all the driving forces and of the transport parameters is given without a detailed introduction to the mathematics of fluid flow. The complete set of coupled differential equations for fluid flow are formulated in Sections 6.3.3—6.3.5. Petroleum transport via diffusion is assumed to be of lesser importance for migration. However, it is shortly outlined in Sec. 6.4. Darcy flow based differential equations are often too complex to be solved in acceptable times. As a consequence, model resolutions are usually very poor. To overcome these difficulties, methods with higher performance are presented in the following sections. T. Hantschel, A.I. Kauerauf, Fundamentals of Basin and Petroleum 247 Systems Modeling, DOI 10.1007/978-3-540-72318-9 6, © Springer-Verlag Berlin Heidelberg 2009
- 260. 248 6 Migration and Accumulation Firstly, flowpath based reservoir analysis is introduced in Sec. 6.5. A hybrid method, which merges the advantages of this approach and Darcy flow, is presented afterwards in Sec. 6.6. Even faster but more approximative is pure flowpath modeling (Sec. 6.7). Alternatively, overall migration modeling can be performed with an invasion percolation technique, which is described in detail in Sec. 6.8. The different methods are discussed and compared in Sec. 6.9. A detailed analysis and understanding of a petroleum system relies on comprehensive bookkeeping of all the petroleum amounts involved in different geological processes. Generation, expulsion, cracking, migration losses, basin outflow, and accumulation, to name only the most important processes, must be tracked in multi–component resolution for all layers and facies. Rules for bookkeeping are finally discussed in Sec. 6.10. 6.2 Geological Background The aim of this section is merely to summarize some basic knowledge about the processes of petroleum migration. It is known that petroleum amounts which are transported via secondary migration are very small, at least on average over space and geological time. Source rocks generate petroleum on a basin–wide length scale over time inter- vals of millions of years. Flow rates based on average expulsion amounts have values in the range of 8 × 10−15 . . . 8 × 10−14 m3 /m2 /s (England et al., 1987). However, it is also known that migration pathways focus and that migra- tion flow is pulsed in time, see e.g. Haines jumps (Wilkinson, 1984; England et al., 1987; Carruthers, 1998; Sylta, 2004; Dembicki Jr. and Anderson, 1989; Catalan et al., 1992). This implies that localized peak flow rates of moving petroleum might be much higher. The overall direction of migration is known to be vertical due to buoy- ancy. Highly permeable features such as reservoir rocks, or on a smaller scale faults, can act as conduits of petroleum. The resulting transport paths follow these conduits in an upward direction (Sec. 6.5). Structures of low permeabil- ity and high capillary pressure can be invaded under sufficient pressurization of the petroleum phases, due to buoyancy forces associated with accumula- tions formed in traps (Dembicki Jr. and Anderson, 1989; Catalan et al., 1992; England et al., 1987). Residual amounts of hydrocarbons (HC) are immobile and “lost” along migration paths. Density and compositional changes of petroleum phases during migration are important. Symmetrically with decreasing pressure, the liquid phase loses light components and the vapor phase, heavy components. The density varies more with compositional changes than with immediate pressure and temper- ature changes (England et al., 1987). Diffusion effects are negligible (England et al., 1987).
- 261. 6.2 Geological Background 249 This picture of migration seems unique and consistent. However, in a more detailed view, many questions arise. For a first impression a few shall be listed here: It is often assumed that petroleum migrates in disconnected stringers or filaments (Berg, 1975; Tissot and Welte, 1984; Sylta, 2004; Schowalter, 1979). What is their size and form? Do these stringers cover a lot of microscopic pores? Obviously, there is at least no connection in terms of pressure from source rock to reservoir. It is commonly assumed that on a macroscopic scale stringer sizes are rather small. Otherwise overpressuring would lead to break through processes and the petroleum would be able to pass barriers almost everywhere. The size must be known more accurately for a detailed flow de- scription. Do flow pulses occur on a macroscopic or on a microscopic scale in space? Such a question can only be answered if more is known about the stringers. How long do the flow pulses last? What is the velocity of a moving stringer? In the case of fast moving macroscopic stringers, viscosity and internal dis- placement might play a significant role. Obviously, pores might be filled which would not be invaded by a slow, almost static flow. Is the flow pattern rather homogeneous in space or does it show a fractal appearance on a macroscopic scale?1 This question is of great importance, as migration losses can be better estimated if “the petroleum traversed pore volume” is known. In a fractal petroleum migration pattern, only parts of space are traversed. As special effects in a more continuous picture, saturation “shocks” or discontinuities occur on a macroscopic level. Immiscible fluid displacement often occurs at the surface of a sharp saturation boundary (Barenblatt et al., 1990).2 This is well known in production and seems to be contradictory to fractal flow patterns. However, production induced flow rates are much higher than the flow rates during migration. For that reason such effects might not appear during migration. On the other hand, the local flow rate belonging to one moving stringer is definitely higher than average migration rates. Hence a stringer might have sharp boundaries while moving through the rock matrix. Observations originating from laboratory experiments and core samples support the fractal view. Does such fractal behavior originate from macro- scopic large scale variations of rock properties or does it evolve through up- scaling of self–similar fractal structures coming from random pore size varia- tions? These are some basic questions. Surely, the list can be further extended and many fundamental questions concerning migration are still open. 1 Fractal means self–similar under a given magnification. More precise definitions are formulated in the literature, e.g. in Stüwe (2007). 2 The saturation is only rapidly varying and appears to be discontinuous on a macroscopic level.
- 262. 250 6 Migration and Accumulation Finally it must be noted that in basin modeling migration is considered as the flow or movement of HCs in the free pore space. It is not principally dis- tinguished into primary migration inside and secondary or tertiary migration outside of source rocks. 6.3 Multi–Phase Darcy Flow Separate flow of non–mixing phases is commonly assumed to be the dominant transport mechanism for secondary migration. The driving forces for fluid flow are pressure potential differences. The fluid pressure potential is just the pressure reduced by the pressure of a static fluid column with a correspond- ing vertical pressure gradient, which is necessary to balance the weight of the column itself (Fig. 2.2).3 Exceeding amounts of pressure cause flow. The potential up for any phase p is thus defined as up = p − ρpgz (6.1) with ρp the density of fluid phase p, g the gravitational acceleration, and z the depth.4 Darcy’s law states that a potential difference causes a flow according to vp = μp Δup Δl . (6.2) Herein vp is the velocity of flow of phase p and μp its mobility. The symbol Δup indicates a potential difference over a distance Δl in space. The flow direction is from high to low pressure potential. Timing of the fluid flow is included and quantified via the introduction of flow velocities. The mobility in multi–phase fluid systems is usually split into three factors according to μp = kkrp νp . (6.3) The relative permeability krp is introduced additionally in comparison with the single–phase formulas of Sec. 2.2.3. The numerator kkrp is called the effective permeability and k the absolute or intrinsic permeability. Models and approximations for the estimation of absolute permeability are already discussed in Sec. 2.2.3. Effective permeability and viscosity can vary drastically with temperature, pressure, saturation, and porosity. Hence, flow and migration velocities might also show variations over several orders of magnitude (Fig. 6.1). 3 Plus a depth independent shift for zero level adjustment. 4 For the indication of different petroleum phases an “oil–gas” instead of a “liquid– vapor” notation is used in this chapter. The authors prefer the latter but the “oil–gas” notation is standard in the literature.
- 263. 6.3 Multi–Phase Darcy Flow 251 Fig. 6.1. Petroleum flow velocity iso- lines. Assuming a fixed viscosity and a varying permeability or vice versa the time for the fluid to travel a given dis- tance for a fixed pressure difference varies here between one day and thousands of years 0.01 0.1 1 10 100 1000 mPa s Gas Oil Viscosity Permeability in mD 104 103 102 101 10-1 10-2 10-3 10-4 10-5 100 1 Day 1 Year 1000 Years The petroleum saturation defines the fraction of pore space which is used for flow (England et al., 1987). It is thus possible to roughly approximate the relative permeability krp with the petroleum saturation Sp by krp = Sp based on the strongly simplified tube bundle model for fluid flow from Sec. 2.2.3. More generally, the fraction of tubes available for fluid flow is proportional to saturation. Tube radii are assumed to be fixed and randomly distributed within certain limits. At low petroleum saturations, the permeability is deter- mined by the probability of randomly drawing an arbitrary tube with radius r. It is k ∝ r 2 with r as the expectation value of drawing a value r. At full saturation the sum of all tubes is used for flow. The permeabililty is pro- portional to the slice plane area of the tubes in the rock and thus k ∝ r2 . Generally r2 r 2 . Intermediate saturations are described by a continuous crossover. However, such a simplified tube model does not take into account some important effects. Actual path lengths might decrease with increasing saturation. Pathways which emerge due to flow branching and flow through partially filled tubes are also not considered. As saturation increases, the permeability also increases in real porous me- dia. Relative permeability dependencies of a more realistic form are shown in Fig. 6.2. Here, water is considered to be a wetting phase with a low con- tact angle between the rock and the water and petroleum to be generally non-wetting, which implies that the grain surface is covered by at least a thin layer of water. The three saturation end points, namely critical oil Soc, critical gas Sgc and connate water saturation Swc, are threshold values. They distin- guish between initial saturations, which must be overcome to allow flow, and residual saturations, which are immobile. Very important is the critical oil saturation value of a source rock. It controls expulsion because it defines a threshold saturation which has to be overcome before oil starts moving and is expelled.
- 264. 252 6 Migration and Accumulation Generally, critical saturation values are lithology dependent and must be distinguished according to the surrounding phases Yuen et al. (2008). Some approximations are usually made in basin modeling. Critical gas saturations Sgc are usually assumed to be negligibly small, allowing every small gas bubble to be mobile. Critical oil saturations Soc in sandstones, which roughly range from 0.1 . . . 10%, are much smaller than in shales, with values from about 0.5 . . . 50.0%. Specific values depend on proper upscaling. Therefore the typical flow channel width, its density and the microscopic saturation distribution along the channels must, in principle, be correctly estimated and rescaled to the common gridcell dimensions of the basin model. krw krow krog krg Sw Sg kr kr Swc 1-Soc 1-Swc Sgc Fig. 6.2. Relative permeability curves for Swc = 5%, Soc = 3%, and Sgc = 1% according to Table 6.1. The dashed curves are according to (6.6). Note that the krog–curve starts at Sg = 0 and that krg = 0 for Sg 1% Se = 0 Se = 0.5 Se = 1 Quadratic Fit krw, krg 0.0 0.1 0.4 0.4 S2 e krow, krog 1.0 0.3 0.0 1 − 1.8 Se + 0.8 S2 e Table 6.1. Supporting points and quadratic fit of relative permeability curves which are shown in Fig. 6.2 Relative permeability curves of water and gas are assumed to be a function of water and gas saturation Sw and Sg respectively and the relative perme- ability of oil depends on both water and gas saturation in the most common relative permeability models (Aziz and Settari, 1979). Hence it is krw = f(Sw) , krg = f(Sg) (6.4) and
- 265. 6.3 Multi–Phase Darcy Flow 253 kro = krow krog, krow = f(Sw), krog = f(Sg) . (6.5) The flow of one phase is treated here as if the other phases are part of the solid rock matrix. This assumption is not valid if fluid phases interact during flow. Other relations than (6.4) and (6.5) can be found in Aziz and Settari (1979) amongst others. The relative permeability of any fluid is zero below its critical satu- ration, it becomes immobile. Saturations are for often rescaled into nor- malized or effective saturations Se which map the saturation interval be- tween the connate and the critical saturation to an interval of 0 . . . 1 as Swe = (Sw−Swc)/(1−Swc−Soc) for krw and krow, Sgoe = Sg/(1−Swc) for krog and Sge = (Sg − Sgc)/(1 − Swc − Sgc) for krg. Due to the lack of precise data in basin modeling it is common to approximate the overall shape of relative permeability curves with a universal form, which is lithology and phase prop- erty independent. Such general shapes are sometimes modeled with quadratic functions, which are based for example on three points as defined in Table 6.1. This is a crude approximation but it is justified by huge uncertainties in absolute permeabilities, which are often only known to one order of magnitude in accuracy. Small relative permeability uncertainties are of comparatively no consequence, at least in basin modeling. The data base is exceptional for sandstones. Empirical formulas exist such as krw = 0.3 S3 we and krow = 0.85 (1 − Swe)3 (6.6) from Ringrose and Corbett (1994). Other frequently used formulas are the Brooks and Corey equations krw = S(2+3λ)/λ we and krow = k0 row(1 − Swe)2 1 − S(2+λ)/λ we (6.7) with a constant k0 row and a parameter λ which describes a “sorting” of the rock. It can vary between 0 and ∞. A small value indicates a poorly sorted (inhomogeneous) rock (Sylta, 2002a; Ataie-Ashtiani et al., 2002). The relative permeabilities in (6.6) and (6.7) become the same for λ → ∞, k0 rwo = 0.85, and an additional pre-factor of 0.3 in the Corey equation for krw. Equation 6.2 is only a basic formulation of Darcy’s law. Some details have not been mentioned yet. The comprehensive formulation is vp = −μp · ∇up (6.8) with the mobility tensor μp = kkrp/νp. This law is formulated in terms of vectors. The driving force −∇up is a gradient, which points in the direction of the steepest decrease of the potential field up. It is multiplied with a tensor μp ∝ k which describes the anisotropy of the rock permeability so that the resulting flow velocity vp is not necessarily pointing in the same direction as −∇up, (compare with 8.2, 8.3). The gradient −∇up is a mathematical formulation of pressure poten- tial differences over infinitesimally small distances in spatial directions. This
- 266. 254 6 Migration and Accumulation point is of a technical nature and ensures that spatially varying potentials are treated correctly in three dimensions. Darcy’s law is a typical friction law with a friction force proportional to the velocity. Fluids which obey this law are called “Newtonian fluids”. Note that for mechanics without friction, acceleration is proportional to the driving force. In other cases (e.g. if flow velocities are very high) other quantities might be related to friction. The kinetic energy of the displaced fluid, which is proportional to v2 , might quantify the energy loss from friction. Other relationships are possible for such “non–Newtonian” fluids. It is sometimes argued that viscous resistance is so small that it can gener- ally be neglected. This implies that flow–velocity analysis is abandoned, which might be reasonable for many geological situations, especially if long geolog- ical timescales are considered. However, time–control of migrating petroleum is lost. Instead, generation rates become the only time controlling factor in the petroleum systems model. The topic is discussed in more detail in Sec. 6.8. The origin of the driving forces for petroleum, which have not been men- tioned yet, is the topic of the following sections. 6.3.1 Capillary Pressure Interfacial tension occurs at the interface between two adjacent immiscible phases. It rises due to differently sized attraction forces between molecules within one phase and across a boundary to molecules within another phase. The corresponding effect in porous media is named capillary pressure which indicates an additional fluid pressure due to geometry and contact forces. It is commonly described by the term pco for oil–water and pcg for oil–gas systems and can be calculated for an ideal capillary tube of radius r as pc = 2γ r cos θ (6.9) with γ as the interfacial tension and θ the contact angle (Fig. 6.3). In porous media the capillary pressure depends on the pore throat radii r, which are pure rock properties, interfacial tensions, which are pure fluid properties, and contact angles for the specific combination of both. Capillary pressure is usually measured for mercury–air systems in labora- tory experiments. It can be transformed to water–petroleum systems by pcPet = pcHg γPet cos θPet γHg cos θHg (6.10) according to (6.9). The interfacial tension of mercury is γHg = 471 mN/m and the contact angle for a mercury–air system is θHg = 140◦ . Assuming an ideal water wet petroleum–water system with θPet = 0◦ the above formula reduces to pcPet = pcHg γPet 360.8 mN/m . (6.11)
- 267. 6.3 Multi–Phase Darcy Flow 255 R r q Oil / Hg Water / Air pc q Fig. 6.3. Pressure in a capillary Capillary pressure is saturation dependent for macroscopic rock samples because rock contains many pores with varying sizes, which can be invaded at different pressures (Fig. 6.4). Commonly, a capillary pressure threshold pint has to be exceeded to start petroleum saturation. It corresponds to the largest pore throat at the petroleum water contact. Higher pressures are necessary to increase petroleum saturation. The entry pressure pce has to be overcome to saturate the rock up to the residual petroleum saturation, which corresponds to the first connected petroleum path through the sample, and which is not removable anymore due to hysteresis. Residual and critical petroleum satura- tion are commonly approximated with the same value in basin modeling. In the following text they are not distinguished anymore. With further pressure increase smaller throats are overcome, smaller pores are filled and a higher saturation is achieved. Idealized forms of capillary pressure curves are used in basin modeling. Hysteresis, as depicted in Fig. 6.4, is usually not taken into account. The only exception are immobile losses. A porous medium can be interpreted as a bundle of capillary tubes. Ac- cording to (2.44) r ∝ k/φ with permeability k for the radius r of the tubes. This can be inserted into (6.9). The resulting equation can be rearranged and according to Fig. 6.4 interpreted as a function of saturation only which is called the Leverett J–function (Barenblatt et al., 1990) J(Sw) = pc(Sw) k/φ γ cos θ . (6.12) This dimensionless function allows measurement results obtained for the same rock type with different fluids at different porosities and permeabilities to be compared. The values should lie approximately on one saturation dependent J(Sw) – curve. For example, Ringrose and Corbett (1994) proposed J(Sw) ∼ S−2/3 we for sandstones.
- 268. 256 6 Migration and Accumulation 0 S S wc co 1 pint Water Saturation Sw Capillary Pressure p c Im bibition Drainage pce Petroleum Grain pc Fig. 6.4. Illustration of drainage and imbibition curves similar to Aziz and Settari (1979), Schowalter (1979), and Wilkinson (1986) on the left. The capillary pressure pc refers to the minimal pore throat radius which is actually reached by petroleum within the rock on the right. Note that the x–axis on the left side shows water saturation. The nomenclature “drainage” and “imbibition” refers to water drainage and imbibition Due to lack of data, only simplified functions of pc(Sw) are used in mod- eling practice. The simplest functional form is pc(Sw) = pce + p0(1 − Sw) (6.13) with e.g. p0 = 1 MPa. However, such a simple model is unrealistic for high petroleum saturation. Capillary pressure rises drastically when water is drained and the connate water saturation Swc is approached. As with permeability, more accurate formulas for sandstones such as pco = poeS−1/λ we (6.14) with poe as the oil entry pressure are available (Sylta, 2002a). The sorting factor λ is the same as in (6.7). The previously mentioned case of λ → ∞ yields a constant capillary pressure of form pco = poe. This simple model is used in invasion percolation, where the capillary pressure curve is sampled by just one value, the entry pressure (Fig. 6.4). It is not sufficient for Darcy flow models where capillary pressure must increase continuously with petroleum saturation Sp (Fig. 6.6). The most important parameter for all commonly used capillary pressure models is the capillary entry pressure pce. It is usually given by mercury– air values pcHG in lithological data bases or empirical equations and can be converted to oil–water and gas–water values with interfacial tension values of oil–water γo and gas–water γg. The interfacial tension values of oil–water and gas–water are temperature and pressure dependent and described in Chap. 5. In the following discussion, the interfacial tension values of γo = 42 mN/m
- 269. 6.3 Multi–Phase Darcy Flow 257 and γg = 72 mN/m are used to compare reference data for capillary entry pressures, which are reported for different fluid systems. According to equation (6.11), this yields a general relationship between oil–water, gas–water and mercury–air capillary entry pressures as pcOil = pcHG/8.6 , pcGas = pcHG/5.0. (6.15) Due to the dependency of capillary pressure on pore throat radii, it is also strongly dependent on the compaction state, which is usually described by porosity or permeability (App. A). It can quite accurately be estimated for many lithologies with the following models. Permeability dependent models Some authors proposed general exponential relationships between capillary entry pressures and permeabilities, which can be used for all lithotypes of the following type: pce = a(k/k0)b . (6.16) Ibrahim et al. (1970) in Ingram and Urai (1999) derived a = 0.548 MPa, b = −0.33, and k0 = 1 mD from calculations of column heights. Hildenbrand et al. (2004) performed gas break through experiments on pelitic rocks with N2, CO2, and CH4, which yield a = 0.0741 MPa, b = −0.24, k0 = 1 mD for methane–water, which is equivalent to a = 0.37 MPa for mercury–air accord- ing to (6.16). Porosity dependent models The combination of the above relationship pce(k) with a linear relationship between log k and φ yields pce = a 10−bφ (6.17) for capillary entry pressure dependency on porosity. The parameters a and b are lithology dependent as the permeability versus porosity curves are also lithotype specific functions. Typical mercury–air values for various lithologies calculated with the Hildenbrand model from permeability curves of App. A are also tabulated in the appendix. The advantage of the porosity dependent curves is, that they can be fit- ted for each rock type independently of permeability calibrations. The pro- posed equation (6.16) from Hildenbrand and Ibrahim for a typical shale with a piecewise linear permeability curve such as in Fig. 2.16 yields capillary entry pressure curves as shown in Fig. 6.5.a. Application in basin modeling shows, that the Hildenbrand model works well for low porosity shales and good expe- riences are made with the Ibrahim model for high porosity shales. Hantschel and Waples (2007) proposed a linear interpolation between the capillary entry pressure calculated for 1 % porosity from the Hildenbrand model, the capillary entry pressure calculated for 25 % porosity from the Ibrahim model and zero capillary entry pressure for the depositional porosity (Fig. 6.5.a).
- 270. 258 6 Migration and Accumulation This construction of piecewise linear capillary entry pressure versus poros- ity curves can generally be used for other lithotypes. The corresponding capil- lary entry pressure values for 1 % and 25 % porosities are tabulated in App. A and shown in Fig. 6.5.b. The conversion of mercury–air into oil–water and gas–water values using (6.15) yields the curves of Fig. 6.5.c. 0 10 20 30 40 50 60 70 0 10 20 30 40 50 Porosity in % Capillary Entry Pressure in MPa 1 2 3 0 10 20 30 40 50 60 70 0 2 4 6 8 10 Porosity in % Capillary Entry Pressure in MPa 1 2 0 5 10 15 20 25 0 10 20 30 40 50 Porosity in % Capillary Entry Pressure in MPa 1 2 3 4 5 6 a) b) c) d) 0 10 20 30 40 50 60 70 0 0.1 0.2 0.3 0.4 0.5 Porosity in % Column Height in km 1 2 Hildenbrand Krooss Ibrahim Hantschel Waples 1 2 3 1..Shale 2..Coal 3..Chalk 4..Siltstone 5..Marl 6..Limestone 1..Gas 2..Oil 1..Gas-Water 2..Oil-Water Fig. 6.5. Capillary entry pressure curves It must further be noted that basin scale entry pressure values are often calculated from hand specimen values divided by an upscaling factor which lowers the entry pressure. An upscaling factor of 2.56 is proposed for clas- tic rocks and carbonates.5 Upscaling of the piecewise linear capillary entry 5 Assuming an upscaling factor for the permeability of 50 (Sec. 2.2.3), the Hilden- brand parameter b=-0.24 yields an upscaling factor for capillary entry pressure of 50−0.24 = 1/2.56.
- 271. 6.3 Multi–Phase Darcy Flow 259 pressure curve of typical shale yields maximum column heights as shown in Fig. 6.5.d for oil and gas densities of ρo = 600 kg/m3 and ρg = 100 kg/m3 . Note, that all basin modeling input capillary pressure values are usually given as mercury–air reference values and that the conversion into oil–water and gas–water values with the corresponding maximum column heights are calculated during the simulation with the actual reservoir porosities and the temperature and pressure controlled oil and gas interfacial tension and density values. 6.3.2 Pressure at Phase Boundaries In the absence of other forces, petroleum migrates from high to low capillary pressure regions. Obstacles such as cap rocks with high capillary pressures inhibit migration. For example, the higher the capillary pressure contrast be- tween a reservoir and a seal, the higher the buoyant pressure of an accumula- tion must be to allow fluid flow into and through the seal. Another force arises through water flow, more precisely through its pres- sure potential uw, which has been discussed in Chap. 2. In general, any phase might be driven by forces originating at the boundary to other phases. Without capillary pressure changes from water to pretroleum the pressure would be continuous at phase boundaries. A pressure jump of the height of the capillary pressure must be taken into account. With (6.1) this yields uo = uw + (ρw − ρo)gz + pco (6.18) and ug = uw + (ρw − ρg)gz + pco + pcg . (6.19) Petroleum buoyancy in water is directly identified by inserting the last two equations into Darcy’s law (6.2) and looking at the terms with the density contrasts. Buoyant forces are strong enough to cause the rapid migration of oil through sandstones. Migration times can often be neglected on geological timescales and replaced by instantaneous movements as done in the hybrid and flowpath methods (Sec. 6.5). The behavior is quite different for water which experiences no buoyant forces in the surrounding medium and no capillary pressure thresholds. Due to the fact that the overall amount of petroleum in a basin is very small compared to the amount of water, it is often assumed that neither the overall water flow nor compaction is distorted very much by the HCs and that the methods presented in Chap. 2 do not need to be or need to be only slightly corrected. Effects such as fluid expansion (Sec. 2.3.2) are usually incorporated in the water flow equations but not implicitly coupled to migration equations. In principle, a correction based on iterative solutions of water and petroleum flow equations for the improvement of the coupling between both flow equations can be easily performed.
- 272. 260 6 Migration and Accumulation 5.7 5.7 5.7 5.7 5.7 5.7 5.7 5.3 6.1 6.5 4.5 4.9 Regions of varying oil potential and critical oil saturation Region of uniform oil potential 4.50 4.75 5.00 5.25 5.50 5.75 6.00 6.25 6.50 Oil Potential in MPa 0 S S wc co 1 pint Sw p c 0 S S wc co 1 Sw 1 2 3 4 5 Cell 1 Res. 1 0.1 6.1 2 Res. 1 0.1 5.7 3 Res. 55 0.5 5.7 4 Res. 95 0.9 5.7 5 Res. 96 1.3 5.7 6 Seal 1 1.7 5.7 S p u o c o [%] [Mpa] [MPa] Seal Reservoir 2 3 4 5 6 6 p c pint pce pce 90 80 70 60 50 40 30 20 10 0 Oil Saturation in % a) b) Seal Reservoir Fig. 6.6. Reservoir in static equilibrium A principle scheme of an accumulation in static equilibrium below a seal is shown in Fig. 6.6. The oil potential at base seal (cell 6) is equal to the oil potential at the top reservoir (cell 5). Thus, the two corresponding capillary pressures differ only by the amount of buoyancy pressure between the two adjacent cells. The capillary pressure in the base of the seal is equal to the capillary entry pressure, while the same capillary pressure at the top reservoir location is related to a very high saturation value. The oil potential is constant in the entire accumulation, which means that the capillary entry pressure decreases linearly with depth in the reservoir (cells 5 — 2). The buoyancy decreases by the same amount.
- 273. 6.3 Multi–Phase Darcy Flow 261 There is no migration within the oil accumulation and from the reservoir into the seal as there is no gradient of the oil potential. Note that the capillary pressure difference between the base reservoir and base seal cells (2 and 6) differ by the difference of the capillary entry pressure of the reservoir and sealing lithology and that it is equal to the buoyancy of the accumulation. This result is used as the fundamental sealing rule in flowpath and invasion percolation methods. The petroleum potential below the accumulation increases with depth. Due to a missing pressure communication between the immobile petroleum droplets, buoyancy is not varying with depth and the capillary pressure is equal to the capillary entry pressure for all locations saturated with the critical oil saturation. Although there is an oil potential gradient here, no migration occurs, as the relative permeability for critical oil saturation is equal to zero. In summary, the petroleum system around a petroleum accumulation in static equilibrium consists of two types of domains: migration regions with critical oil saturations and accumulation regions with constant oil potentials. Due to the fact, that source rock expulsion rates are usually very small, it can often be assumed, that even an accumulation under continuous feeding with a break through on top is well approximated by a static equilibrium. Other accumulation examples under static equilibrium are explained in Fig. 6.7. However, Darcy flow equations are also capable to model dynamic behavior. An example model is shown in Fig. 6.8. It consists mainly of two sandstone layers and a deeply buried source rock. Expelled petroleum is moving upwards. The permeability is highly anisotropic in the shales and thus the petroleum is moving in an intermediate direction between vertically upward and the dipping angle of the layers. Leaking accumulations are found in the sandstone structures beside the impermeable fault. The capillary curve is still below 50 % saturation, reaching pressure values which allow break through from sandstone into shale below 2000 m depth. The resolution of the grid is too rough for a precise estimate of the column height. Some vectors indicate small amounts from locations where flow traversed in paleo times and residual amounts were captured. With ongoing compaction and reduced porosity, residual saturation values are exceeded and small petroleum amounts started to move again. It can finally be summarized that Darcy type migration modeling can be interpreted as a balance of all external forces, such as capillary pressure, buoyancy, and water pressure at the phase boundary, with a viscous resistance force, where, from the viewpoint of a balanced Newtonian force, each force has a counterpart of the same strength but with opposite direction. 6.3.3 Three Phase Flow Formulation without Phase Changes A starting point for the formulation of separate multiphase flow equations are mass balances. Mainly three immiscible fluid phases, namely water, oil, and gas are found in a basin. The mass fraction for any phase p can be calculated from
- 274. 262 6 Migration and Accumulation Res. Sat. Shale, p = 1 MPa ce Sandstone, p = 0.1 MPa ce Siltstone, p = 0.5 MPa ce a) b) 0 1 2 3 4 Pressure/Potential in MPa Depth Capillary Pressure Oil Potential 0 0.5 1 1.5 2 2.5 3 Pressure/Potential in MPa Depth Capillary Pressure Oil Potential 0 0.5 1 1.5 2 2.5 3 3.5 Pressure/Potential in MPa Depth Capillary Pressure Oil Potenial CS CS c) CS CS d) CS e) 0 0.5 1 1.5 2 2.5 3 3.5 Pressure/Potential in MPa Depth Capillary Pressure Oil Potential 0 1 2 3 4 Pressure/Potential in MPa Depth Capillary Pressure Oil Potenial Oil Residual Saturation Residual Saturation Oil Oil Oil Oil Res. Sat. Res. Sat. Res. Sat. Shale S a n d Sand Shale Shale Sand Silt Sand Shale Shale S a n d S a n d S a n d Silt Fig. 6.7. Capillary pressure and oil potential for accumulations under static equi- librium: (a) With siltstone facies in and below a sandstone reservoir. Note, that the oil–water contact in all silt facies have the same depth. (b) With a sandstone lens in a shale layer at the time when the sandstone has been entirely filled. (c) At the time, when the sandstone pressure is increased until break through. Note, that the accumulation saturation from the first entire filling to break–through increases corresponding to the capillary pressure versus saturation curve. (d) With two dis- connected reservoirs. (e) With two connected reservoirs. The pressure and potential curves are shown for the marked cross–sections (CS) through the accumulations mp = φρpSp (6.20) with ρp as the density, Sp the saturation, and φ as the porosity of the rock. Correspondingly, mass fluxes ṁi are given by ṁp = ρpvp . (6.21) The velocities vp can be calculated with Darcy’s law according to (6.8). The mass balance for each phase p is ∂mp ∂t + ∇ · ṁp = qp . (6.22)
- 275. 6.3 Multi–Phase Darcy Flow 263 Petroleum Saturation [%] Basement Shale Shale Silt ShaleSilt Shale Sandstone Kitchen Area Layer 1 Layer 2 Layer 3 Layer 4 Layer 5 Layer 6 Layer 7 Layer 8 Layer 9 Layer 10 Layer 11 Layer 12 Layer 13 Impermeable Fault Fig. 6.8. Two dimensional example model with reservoir equilibrium as outlined in Fig. 6.6. The black vectors indicate oil and the light grey vectors gas flow. Layers 5 and 8 are sandstone reservoirs A source term qp which describes the generation or loss of phase p (e.g. by chemical reactions) is added here for the sake of completeness. Furthermore, compaction is described by dφ/dt = −C dσ z/dt with σ z as the vertical component of the effective stress and C as the matrix compress- ibility. For simplicity, the equations are here, according to (2.3), only noted for a coordinate system which moves with the solid rock. The compaction term becomes ∂φ ∂t = −C ∂(ul − uw) ∂t (6.23) with ul = pl − ph as the depositional or overburden load potential (Chap. 2). A comprehensive treatment in a spatially fixed coordinate system can e.g. be found in Luo and Vasseur (1992). Additional factors of form 1/(1−φ) such as in (2.10), which are also known from Yükler et al. (1979), enter the equations (App. B). Equations 6.20, 6.21, and 6.23 can be inserted in (6.22) for the construction of final three phase flow equations:
- 276. 264 6 Migration and Accumulation ρwφ 1 − φ ∂Sw ∂t −∇ · ρwμw · ∇uw − ρwSwC 1 − φ ∂(ul − uw) ∂t = qw , ρoφ 1 − φ ∂So ∂t −∇ · ρoμo · ∇uo − ρoSoC 1 − φ ∂(ul − uw) ∂t = qo , ρgφ 1 − φ ∂Sg ∂t −∇ · ρgμg · ∇ug − ρgSgC 1 − φ ∂(ul − uw) ∂t = qg , (6.24) Sw + So + Sg = 1 . (6.25) Equations (6.18) and (6.19) can be rearranged to uo − uw = pco(Sw) + (ρw − ρo)gz , (6.26) ug − uo = pcg(So) + (ρo − ρg)gz . (6.27) They close the system. The unknowns are the three pressure potentials and the three saturations. For simplicity, it is here assumed that the densities are only slowly varying over time. Derivatives concerning ∂ρi/∂t could easily be added to (6.24). The saturation dependency of capillary pressure is noted for clarification. However, the oil saturation dependency of pcg only describes the case where gas is surrounded by oil. The gas capillary pressure pcg is similar to the oil capillary pressure pco which is also dependent on water saturation in the case of So = 0. Generally, pco and pcg depend on all saturations Sw, So, and Sg. There are two principal methods, with many variations, for solving such complicated sets of equations. First of all, they can be solved directly af- ter implicit gridding in time (Sec. 8.4). This scheme is generally called im- plicit and usually involves a huge effort for the construction and inversion of the corresponding matrices. The equation system is solved simultaneously for many variables. Such methods are therefore also called “simultaneous solu- tion” methods (Aziz and Settari, 1979). The “classical” implicit approach is based on a simultaneous solution for uo, Sw, and Sg. Alternatively, it is possible to solve such equation systems partially ex- plicit. Some values (e.g. the saturation values) are kept fixed for a small time step and the system is solved implicitly, e.g. for pressures only. Afterwards, HC saturations are updated explicitly according to the calculated pressure gradients and Darcy’s flow law. The “classical” explicit approach is based on an implicit solution for the water pressure only and an explicit treatment of all other unknowns. Such an approach is also called “implicit pressure and explicit saturation” (IMPES) (Aziz and Settari, 1979). The whole procedure must be repeated multiple times because it works only for small time steps. However, iterations are always necessary for both approaches as there are non–linear dependencies within the equations. Densities depend non– linearly on pressures, capillary pressures and relative permeabilities depend non–linearly on saturations, and so on.
- 277. 6.3 Multi–Phase Darcy Flow 265 6.3.4 Multicomponent Flow Equations with Phase Changes Dissolution and exsolution of components in and out of phases was not consid- ered in the previous section. It can be incorporated explicitly with flash calcu- lations before or after each time step (Chap. 5). For an implicit treatment, it is necessary to switch to a multicomponent formulation with N–components. The mass mip denotes the mass of the component i dissolved in the phase p, while mi and mp are the total component and phase masses at a given location. Thus the conditions N i=1 miw = mw, N i=1 mio = mo, N i=1 mig = mg (6.28) and miw + mio + mig = mi for i = 1, . . . , N (6.29) must be fulfilled. The mass fractions Cip = mip mp (6.30) are PVT–functions and can be calculated under consideration of thermody- namic laws or via lookup tables (Chap. 5). Besides this, N i=1 Cip = 1 . (6.31) The mass of each component inside the rock can now be calculated ac- cording to mi = φ(CiwρwSw + CioρoSo + CigρgSg) (6.32) with φ as the porosity of the rock and ρp the density of phase p. Correspond- ingly, mass fluxes ṁi are given by ṁi = Ciwρwvw + Cioρovo + Cigρgvg (6.33) The phase velocities vp obey Darcy’s law (6.8). The local formulation of mass balance for each component i is given by ∂mi ∂t + ∇ · ṁi = qi (6.34) with newly introduced component sources qi. Insertion of (6.32), (6.33), and (6.23) into (6.34) with consideration of App. B yields
- 278. 266 6 Migration and Accumulation φ 1 − φ Ciwρw ∂Sw ∂t + Cioρo ∂So ∂t + Cigρg ∂Sg ∂t −(ρwSwCiw + ρoSoCio + ρgSgCig) C 1 − φ ∂(ul − uw) ∂t −∇ · Ciwρwμw · ∇uw − ∇ · Cioρoμo · ∇uo −∇ · Cigρgμg · ∇ug = qi (6.35) for all i = 1, . . . , N. The equation system consists of n differential equations for the 3N + 6 unknowns Cip, Sp, and pp. The closing equations are again (6.25), (6.26), and (6.27). Additionally, the 2n PVT–functions defining (6.30) must be specified and (6.28) must be fulfilled. Again it is here assumed that the densities are not or only slowly varying over time. An implicit solution is very difficult to process as a simultaneous solu- tion with so many unknowns yields huge matrices, which cannot be inverted without serious problems. The application of the classical explicit method to the multicomponent model is relatively simple as the equations are implicitly solved for the wa- ter pressure only. Afterwards all of the above multicomponent equations are sequentially evaluated for an incremental mass change of each component. Finally, it must be noted that the densities and the PVT–functions (6.30) are strongly temperature dependent. Temperature is also explicitly time de- pendent as can be seen in Chap. 3. Hence, in a correct formulation, partial derivatives of ρp and Cip must also be taken into account and added to (6.35). However, in the classical explicit approach only water pressure is calculated implicitly. Water density does not vary much. Additionally, water does gener- ally not contain much HC in solution and vice versa HC phases do not absorb much H2O so these derivatives can be neglected for water. Other phases are treated explicitly and thus correctly, when small enough time steps are chosen. Therefore the potential error is expected to be small. 6.3.5 Black Oil Model Two dissolution models are widely used in reservoir engineering and basin modeling. These are the black oil model and the multicomponent model. The black oil model is an approximation which consists of three compo- nents only, namely H2O, and two petroleum components, one lighter gas and one heavier oil component (Sec. 5.3). In the following, the water, oil, and gas components are denoted with 1, 2 and 3 respectively. The water component consists of 100% H2O and the oil component is dissolved in the oil phase only. The light gas component can be dissolved in both petroleum phases, depen- dent on temperature and pressure. Thus, only one independent PVT–function is found in the model. It is the ratio of the gas component in the oil phase, named here as x with C3o = x. Most of the Cip are zero, except
- 279. 6.4 Diffusion 267 C1w = 1, C2o = 1 − x, C3o = x, C3g = 1 . (6.36) The mass balance equations for the three components now yield ρwφ 1 − φ ∂Sw ∂t − ∇ · ρwμw · ∇uw − ρwCSw 1 − φ ∂(ul − uw) ∂t = q1 (1 − x) ρoφ 1 − φ ∂So ∂t − ∇ · (1 − x)ρoμo · ∇uo − (1 − x) ρoCSo 1 − φ ∂(ul − uw) ∂t = q2 ρgφ 1 − φ ∂Sg ∂t + x ρoφ 1 − φ ∂So ∂t − ∇ · ρgμg · ∇ug − ∇ · xρoμo · ∇uo − C(ρgSg + xρoSo) 1 − φ ∂(ul − uw) ∂t = q3 (6.37) instead of (6.24). The above formulation is very similar to the three phase flow formulation without phase changes. In principle, explicit and implicit solution methods can be applied in the same way. Obviously, the incorporation of oil dissolution in the gas phase, as described for the symmetrical black oil model, is straight forward (Sec. 5.3). 6.4 Diffusion Petroleum transport as a diffusion process in aqueous solutions, in molecular or micellar form, is of lesser importance for migration. Exceptions are very poor source rocks, which are not able to build up sufficiently large amounts of fluid for separate phase flow, and diffusion of gas through dense unfractured seals, where leakage is prohibited. Another application of diffusion flow is transport of HC components in accumulations. For example, diffusion may act against gravitational separation and thus components are transported from the core of an accumulation to its oil–water contact, where biodegra- dation might occur. In basin modeling it is commonly assumed that lateral diffusion inside of accumulations over geological times is strong enough to achieve a homogeneous mixing of petroleum from different oil types with in- dividual compositions. However, the grid resolution of typical basin models does usually not allow spatial effects within accumulations to be modeled. Diffusion models for secondary petroleum migration are only related to the transport of methane in aqueous solution. The dissolution of methane in water is PVT controlled and described in Sec. 5.2. The resulting concentration gradients cause a diffusion flux, which can be formulated with Fick’s law applied to porous media and a effective rock diffusion coefficient D according to J = −D∇c (6.38) for a methane concentration c (Krooss, 1992; Krooss et al., 1992a,b).
- 280. 268 6 Migration and Accumulation Diffusion coefficients for different temperatures were measured by Schlömer and Krooss (1997, 2004). Values between 0.018 × 10−10 m2 /s = 56.8 × 10−6 km2 /My and 4.46 × 10−10 m2 /s = 0.014 × km2 /My were found. It is commonly assumed that the temperature dependency of a diffusion coefficient follows an Arrhenius law according to D(T) = D0 exp (−EA/RT) (6.39) with D0 as a frequency type pre-factor, activation energy EA, and the univer- sal gas constant R. Activation energies of about 16 . . . 50 kJ/mol are reported (Krooss, 1992). A diffusion equation of form ∂c ∂t = D∇2 c (6.40) can be derived from (6.38) with consideration of mass conservation. Follow- ing Krooss et al. (1992a), an example solution of the diffusion equation for methane diffusion through a cap rock is presented in App. L. The transport via diffusion is regarded as a two step process comprising the dissolution of methane in water and diffusion within the water phase. Once the methane is dissolved in water, transport via water flow also becomes relevant, especially when high rates of aquifer flow exist. 6.5 Reservoirs Accumulations of hydrocarbons are often found in reservoir rocks with high porosities, which can accommodate large amounts of hydrocarbons. High porosities correlate commonly with high permeabilities. Reservoirs with such properties are called carriers. Inside them HCs can migrate long distances in short periods of time before they finally accumulate in traps. The most accurate formulation of migration physics leads to a complicated set of coupled non–linear partial differential equations (Sec. 6.3), which can only be solved with a huge amount of resources (Aziz and Settari, 1979; Øye, 1999). This effort is commonly taken into account in production related reser- voir modeling. However, these solutions are limited to fixed geometries and to time scales of less than a few years. In contrast, geological timescales and non–rigid geometries caused by compaction are studied in basin modeling. It is practically impossible to directly solve the differential equations with vary- ing geometries. However, long geological timescales in basin modeling permit an alternative approach which needs only a fraction of the resources needed for a direct solution: Due to the high mobility of the HCs in the carriers and the density con- trast with surrounding pore water, the resulting force governing migration is buoyancy (Hubbert, 1953; Sylta, 1993; Lehner et al., 1987). Thus HCs move
- 281. 6.5 Reservoirs 269 primarily upwards in carriers until they reach a barrier. A barrier is a region with high capillary pressure and low permeability such as a sealing rock. Still driven by buoyancy, the HCs migrate just below the seal following the inter- face to the highest point of the carrier where they are trapped and accumulate (Fig. 6.9). The migration can be almost lateral over long distances and hap- pens almost instantaneously compared to other geological processes such as deposition and compaction or transient temperature compensations. Fig. 6.9. Section view of the general scheme for reservoir analysis In total, HC migration in carriers can be modeled by construction and analysis of flowpaths, drainage areas and the calculation of volumetrics for accumulation. This is almost a geometric procedure and thus can be accom- plished much more quickly than the direct solution of coupled partial differ- ential equations.6 Reservoir analysis is based on flowpath evaluation. It is therefore also known as “ray tracing” (Hantschel et al., 2000) and is the topic of Sec. 6.5.1. Flowpath analysis can be used for the construction of drainage areas which collect all the flow of a region in one trap. Drainage area decomposition of carriers and properties of drainage areas are treated in Sec. 6.5.2. Volumetrics for accumulation analysis as one major step of reservoir analysis is the subject of Sec. 6.5.3. The treatment of faults in this picture is described in Sec. 6.5.4. Later, in Sec. 6.5.6 non–ideal reservoirs are discussed. The terminology “flowpath modeling” is in this volume used for basin wide migration and is not restricted to reservoirs only Sec. 6.7. 6 The solution performance of the Darcy flow problem can be enhanced by the for- mulation of two–dimensional Darcy flow based equations for the reservoir (Lehner et al., 1987). a Carrier Source Rock Flowpath Sealing Rock Accumulation
- 282. 270 6 Migration and Accumulation Fig. 6.10. Map view with an example of reservoir analysis. Isolines depict the depth of the sealing interface. Flowpaths and accumulated amounts of liquid HCs are displayed in grey. In the bottom left corner, below the region of 3000 meter depth, a kitchen area expels HCs into the carrier. They migrate as far as the top right of the map. The map has 120 × 120 gridpoints and open borders. Some flowpaths are filtered for better visualization 6.5.1 Flowpath Analysis According to Darcy’s law the velocity v of HC flow is v = − kkr ν · ∇u (6.41) with permeability tensor k, relative permeability kr, viscosity ν, and over- pressure gradient ∇u. With buoyancy as the driving force |∇u| = (ρw −ρp)g. Herein ρw,p are the densities of water and petroleum and g = 9.80665 m/s2 . A low estimation of the velocity in carriers with kkr ≈ 1mD ≈ 10−15 m2 ,
- 283. 6.5 Reservoirs 271 ν ≈ 10−3 Pa s, ρw − ρp ≈ 600 kg/m3 and 1 My ≈ 3 × 1013 s yields v ≈ 6 nm s ≈ 180 km My (6.42) for upward movement (Fig. 6.1). The velocity is reduced if migration follows a seal with a dipping angle α by a factor sin α (Fig. 6.9). With an angle of about ten degrees the estimated velocity is v ≈ 30 km My (6.43) and with one degree the velocity becomes v ≈ 3 km/My. This is a conservative estimate. Velocities of up to 1000 km/My have been reported in the literature (Sylta, 2004). Generally, HCs have the capability to migrate long distances through carriers on geological timescales. This conclusion is the basis for migration modeling in carriers. It implies that migration in carriers continues until a trap is reached. Only HCs which have not entered the carrier during the last geological event may not have reached a trap.7,8 HCs entering a carrier move straight upwards until they reach a seal and then follow the steepest direction upwards below the seal. Flowpaths help to visualize the migration just below the seal (Fig. 6.10). They end at the local heights of the carrier indicating the end of migration. Thus flowpaths can easily be constructed on a map of the reservoir seal interface. Migration mod- eling with flowpaths has an “implicit” high resolution because flowpaths can be constructed from interpolations of mapped gridpoints (Fig. 6.14). Flow- paths passing between two neighboring gridpoints can end in different traps (Fig. 6.11) and therefore even low–resolution maps can show a complicated migration pattern. Residual amounts of petroleum, which indicate that petroleum traversed or is actually traversing, are usually only found in the zone below the sealing interface but not in the bulk reservoir rock (Dembicki Jr. and Anderson, 1989; Schowalter, 1979). This confirms the flowpath concept with migration directly below the seal. Flowpaths are constructed from purely geometric analysis and indicate only the direction of flow. A quantification of immobile losses in this picture is postponed until Sec. 6.5.6. Several flowpaths starting at great distances from each other can end at one local height. Thus all the HCs reaching this trap must be summed up before it is possible to perform an accumulation analysis (Sec. 6.5.2). Finally, for the determination of column heights, volumetrics must be performed in each trap (Sec. 6.5.3). Hence reservoir analysis is a three step process which consists of “flowpath analysis”, “drainage area analysis”, and volumetrics for trapping or alternatively “accumulation analysis”. 7 Except for small amounts of immobile HCs 8 In streamline modeling timing is taken into account on each streamline (Datta- Gupta et al., 2001). This approach is necessary on short production timescales.
- 284. 272 6 Migration and Accumulation Fig. 6.11. Zoomed cutout of Fig. 6.10 with a mesh which contains every second gridline. Flowpaths and accumulations have a very high “implicit” resolution caused by the smooth interpolation of gridded maps 6.5.2 Drainage Area Analysis A carrier can be subdivided into drainage areas or fetch areas. All HCs entering the carrier in the same drainage area migrate to the same trap. HCs entering the carrier at other locations migrate to other traps. Each trap belongs to one drainage area and vice versa. Any point of the carrier belongs to exactly one drainage area. The subdivision into drainage areas is a domain decomposition of the carrier. Drainage areas can be calculated by construction of “possible” flowpaths for each point of a carrier (Fig. 6.12). Migration becomes a purely geometric problem of HC distribution in traps when the drainage areas are known. Prior to anything else it is then necessary to know the maximum trap capacity of a structure. A trap can be filled until the accumulated HCs reach the border of the drainage area. The contact point at the border is called the “spill point” and the maximum possible filling volume is called the pore volume of the “closure” (Fig. 6.13). The shape of an accumulation can be constructed using the following scheme. HCs move to the highest point which has not been occupied yet.
- 285. 6.5 Reservoirs 273 Fig. 6.12. Zoomed cutout of the same map as in Fig. 6.10 with highlighted drainage area borders (thick lines) and “possible flowpaths” perpendicular to some depth isolines. Flowpaths originat- ing at the border of a drainage area might follow the border over wide dis- tances until they bend off towards the interior Fig. 6.13. Zoomed cutout of the same map as in Fig. 6.10 with drainage area borders, closures, thin depth isolines, and dotted spill paths. Spill points are lo- cated at the beginning of the spill paths. This is the location where the closure touches the drainage area border Thus, the meniscus of the accumulation is flat and horizontal. The spill point must be the highest point on the border of the drainage area. It is easy to construct the shape of the closure by cutting the reservoir with a horizontal plane through the spill point. The volume between this meniscus plane and the reservoir seal interface is the closure volume. The entire algorithm of the drainage area, spill point, and closure volume calculation is in practice complicated because the construction of flowpaths is based on the calculation of depth gradients. A gradient is dependent on the local shape of the map and therefore strongly dependent on the interpolation method and on more than one grid point (Fig. 6.14). It is therefore very sensitive to gridding uncertainties at the border of drainage areas because it is supposed to be zero or at least perpendicular to the border. Small variations in interpolation might shift the drainage are border. It might be argued that twisted cells, as in Fig. 6.14, are rather rare and therefore not representative for depth maps. However, due to the fact that
- 286. 274 6 Migration and Accumulation flowpaths are perpendicular to depth isolines9 , it can be assumed that twisted cells, such as in Fig. 6.14 with almost the same value at more than one corner point, occur more frequently at the borders of the drainage areas. The accurate location of these borders is obviously very important for the determination of the spill point. 1000 800 1000 800 1000 800 900 1000 900 1000 900 1000 925 Fig. 6.14. Three different examples of interpolation between four gridpoints. The rectangles are cells in a map view with given depth at the corner points. In the two left examples the values are interpolated linearly within two triangles and in the right example within four triangles. The center point on the right is calculated with an arithmetic average. Depth isolines are dotted. An example of a flowpath entering from left is visualized with arrows. The flowpath is perpendicular to the depth isolines. In the first two examples different flowpaths based on locally different gradients are emerging. It can easily be verified that flowpaths entering the cell on the left side from any position always end–up at the top right corner, whereas the other examples show a drainage area border from the top left to bottom right. The first two examples represent the same scheme differing only in the orientation of the decomposition triangles. The right example does not have a comparable inherent orientation and yields a drainage area border consistent with expectations. Hence, the right interpolation scheme is superior Fortunately, the exact location of the border usually does not contribute significantly to volume calculations because the closure is often far inside the drainage area. Besides the fact that fetched amounts of HCs depend on the size of the drainage area itself,10 there is one further problem. The depth of the spill point is the depth of the highest point at the drainage area border. It strongly influences the closure volume (Fig. 6.15) and therefore well interpreted and gridded maps are needed to allow good volumetrics. Locations of HC injection into a reservoir should be the unique starting points of the flowpaths. This implies two different technical approaches for handling drainage areas. If the injection of HCs is performed at grid point locations then drainage areas can be described by areas of grid points with the border lines in between. Alternatively, if HCs are injected according to a cell picture in the center between the grid points then drainage areas should be treated as collections of cells with the drainage area borders on the grid 9 Only if the map is displayed with an equal horizontal to vertical aspect ratio 10 Below one drainage area, source rock properties and expulsion amounts do not usually vary very much.
- 287. 6.5 Reservoirs 275 Carrier Gridlines Fig. 6.15. Illustration of how slight variations of the spill point depth severely alter the trap capacity. Two possible scenarios of spill point depth are indicated by dashed carrier seal interface lines on the left. The closure volume varies between the dotted lines. Note that the horizontal axis is often stretched by a factor of about 10 in typical basin models and that the volume must be interpreted three dimensionally. Large volumetric errors can arise from poor gridding or low resolution lines. The first approach is easier in practical workflows because everything is handled on the same grid, e.g. data export and import. The second is more natural because non–unique flow directions seem to be less frequent (Fig. 6.16). However, the best results can be achieved if flexible drainage area borders are allowed on sub–gridding level (Fig. 6.17). Fig. 6.16. Non–unique flowpath direction for HCs reaching the seal from below. The left case is a section view with petroleum injected at a grid location and the right case is a map view of a cell similar to the left case of Fig. 6.14, with injection at the cell center. In practice, the flowpath is of- ten simply assumed to follow the steepest gradient 1000 700 900 800 Seal Gridlines ? ? 850 Fig. 6.17. An example of a gridding prob- lem at a spill point location. In this exam- ple the closure is allowed to be extended half a grid distance behind the displayed drainage area border. The values between the gridlines are smoothly interpolated. The closure should not only be restricted to the gridcells, which belong completely to the drainage area. This ensures the best possible calculation of the spill point depth and therefore the best closure amount and trap capacity estimation Spill Point Closure Drainage Area Border Isoline Gridline
- 288. 276 6 Migration and Accumulation Absolutely flat horizontal areas are particularly problematic where flow- paths are not defined and in cases where the location of the drainage area border is undefined (Fig. 6.18). For simplicity it is better not to separate such areas. The volumetrics is not affected anyway. A similar problem occurs if a given drainage area has multiple spill points at the same depth. Nevertheless even such unrealistic cases can be handled consistently. Such problems are academic, as absolutely flat areas or different spill points at exactly the same depth do not exist in real case studies. Carrier Flat Carrier Flat Fig. 6.18. Schematic view of two cases with a flat reservoir seal interface. Obviously, the flat area belongs to the drainage area on the left because all HCs coming from below will sooner or later migrate to the left. In the right case the drainage area border and the spill point might be located at any location on the flat area If a trap is filled up to the spill point, it starts spilling excess amounts of HCs over the border of the drainage area into a neighboring drainage area. All these spilled HCs follow one flowpath, the “spill path”. Spill paths can become main migration pathways because huge amounts of HCs may migrate along only a few paths (Fig. 6.10). Drainage area analysis can become more sophisticated if drainage areas merge: This happens if two areas spill into each other and the total amount of HCs entering these areas exceeds the summed capacity of both. This is often the case when excessive amounts of HCs do not fit into the trap of the area into which they are spilled and then they are spilled back into the area where they originally came from. In such cases both areas merge. A new closure and a new spill point for the merged area have to be calculated (Fig. 6.19). With the continuous generation of HCs over geological time the merging of already merged areas may occur (Fig. 6.20). The performance of drainage area analysis can be improved if small drainage areas or drainage areas with overall negligible closure volumes are directly merged with their larger neighbors (Fig. 6.21). In such cases flowpaths sometimes seem to end inside a merged drainage area but outside of the trap itself (right example of Fig. 6.21). However, the calculation of volumetrics and HC distribution is not affected. 6.5.3 Accumulation Analysis HC accumulations are found as interconnected zones of high porosity and permeability. Therefore a HC pressure arises. It can be determined by cal-
- 289. 6.5 Reservoirs 277 Unmerged Drainage Area Boundary Depth Isoline Merged Drainage Area Boundary Oil Spill Path Fig. 6.19. Continuous trapping of HCs with filling, spilling, and merging of ac- cumulations and drainage areas followed again by spilling in a cutout region from Fig. 6.10 Shale Un-merged Drainage Area Border Depth Isoline Oil Shale Oil Large Sandlens Fig. 6.20. Map view of a reservoir layer with some shale lenses inside. It is sur- rounded by low permeability shales which implies closed boundaries. Oil is accumu- lated into big bodies covering multiple structural drainage areas
- 290. 278 6 Migration and Accumulation Closure Micro- Structure Drainage Area Border Micro- Structure Closure Drainage Area Border Closure of Merged Drainage Areas Seal Direction of Flow Fig. 6.21. Two schematic examples of drainage areas in section views with micro– structures which are merged before accumulation analysis. The trap capacity rises enormously in the left example culation of the column height which acts against a seal. When it exceeds a certain limit, a break through occurs (Fig. 6.22). The limiting pressure is given by the capillary pressure of the sealing rock. The meniscus (or column) height which balances the column pressure with the sealing capillary pressure and which fits the remaining volume into the trap must be found. Fig. 6.22. Pressure acting on top of an accumulation against a seal. The column pressure pcol can be evaluated from the density contrast Δρ = ρw − ρp of water and petroleum and column height h via pcol = Δρgh due to pressure communi- cation in the accumulation Carrier h h Capillary Pressure Column Pressure Column Pressure Seal For all the considerations which were made up to now, perfect compo- nent mixing in an accumulation is assumed. Accumulations are studied over geological timescales in basin modeling. So this assumption is reasonable for accumulations with universal pressure contact but must not be valid in ev- ery case. Processes such as compositional grading are not taken into account because in basin modeling the spatial model resolution is usually not high enough. Accumulations often have a size of only a few gridcells.11 Very often HCs occur in two phases, vapor and liquid (Chap. 5). The migration of these phases can be treated separately. Both will reach the trap on geological timescales. In the trap they will interact: the vapor is lighter and therefore it moves to the top of the accumulation displacing the liquid (Fig. 6.23). Liquid can only occupy space which is not occupied by vapor. In extreme cases the liquid and vapor drainage areas must be distinguished. Liquid areas have merged, while at the same time the vapor still “sees” un– merged areas (Fig. 6.24). 11 An exception are local grid refinements (LGR) around accumulations such as described in Sec. 8.9
- 291. 6.5 Reservoirs 279 Fig. 6.23. Vapor (dark grey) displacing liquid (light grey) in comparison with Fig. 6.10: vapor entered the carrier from the kitchen area below 3200 meter depth from the bottom left corner of the map. The vapor displaced most of the liquid Fig. 6.24. Close up view of a sce- nario with less gas than in Fig. 6.23. Two vapor accumulations (dark grey) are located above one liquid accumula- tion (light grey) in the structure on the right side. Here vapor accumulates along different drainage areas than liquid. The “liquid” drainage area has merged, the “vapor” areas have not
- 292. 280 6 Migration and Accumulation Column pressure calculations for the determination of break throughs must take into account the different densities of vapor and liquid in each phase and the continuity of pressure at the liquid–vapor interface (Fig. 6.25, Watts 1987, Dake 2001). Fig. 6.25. Schematic figure of pressure build–up in an accumulation containing liquid and vapor. The pressure gradient in the vapor is steeper than in the liquid HCs, which again is steeper than in the water Pressure Depth Seal Carrier Vapor-Liquid Contact Liquid-Water Contact pcol Component mixing between the two phases and redistribution of com- ponents to the phases after fluid analysis or recalculation of phase densities in the accumulations are possible but usually not necessary because carriers often lie within a limited pressure–temperature interval. Variations in phase composition or density are thus assumed to be rather small. Huge effects of outgassing inside of reservoirs are expected to appear more during ongoing reservoir uplift than during migration within the reservoir. An exception is long distance lateral migration, e.g. due to spilling (Gussow, 1954). However, outgassing over geological timescales is automatically treated by the hybrid method and flowpath modeling (Secs. 6.6, 6.7). Flash calculations can easily be performed before and after each reservoir analysis or Darcy migration time step within each gridcell which contains hydrocarbons. 6.5.4 Faults and Small Scale Features Faults are often interpreted as almost perfect HC “conduits” or barriers, open or closed faults respectively (Chapman, 1983). Both cases can be combined with reservoir analysis. The reservoir is usually a horizontally oriented layer of smaller thickness. By contrast faults are typically vertically oriented without any thickness and thus modeled as vertical planes inserted into the reservoir. They can be visualized by lines on maps (Fig. 6.26). Open or closed faults are easy to combine with reservoir analysis. Open faults act as the endpoints of flowpaths. They are vertical HC con- duits. HCs reaching open faults are transported out of the reservoir and are therefore no longer the subject of reservoir analysis.
- 293. 6.5 Reservoirs 281 Fig. 6.26. A map with one impermeable and one permeable fault. Vapor is colored dark and liquid light grey. The impermeable fault acts as a barrier whereas the HCs are leaving the carrier at the permeable fault Closed faults act as barriers which cannot be crossed by flowpaths. Drainage area and accumulation analysis are automatically adjusted if flow- paths have been correctly calculated. Very often faults are assumed to have a finite fault capillary pressure (FCP) (Sec. 2.7, Yielding et al. 1997; Clarke et al. 2005a,b, 2006). Due to the fact that column pressure is highest at the top of a petroleum column, it is possible to model faults with lower FCP than the seal capillary pressure by simply replacing the seal capillary pressure by the FCP at the location where the fault crosses the seal (Figs. 6.27, 6.28). Break through amounts at fault locations are assumed to migrate into the fault following its structure upwards.
- 294. 282 6 Migration and Accumulation In a grid which consists of hexahedron type cells such as described in Chap. 8 with the gridpoints at the corners, the sealing capillary pressure value of a gridpoint in the reservoir layer is finally selected out of a maximum of 15 values, namely three possible surrounding juxtaposition cells, four overlaying top seal cells, four possible in–reservoir fault walls and four possible reservoir to seal horizontal fault cells. Fig. 6.27. A map view of a reservoir with accumulations but without any faults or break through (left), with a high FCP fault (center) and a low FCP fault (right). As indicated by the “star” a break through appears on top of the structure in the center and on the highest point of the fault at the reservoir seal interface in the scenario to the right Fig. 6.28. Oil column of height h in equilibrium with fault capillary pressure (FCP). Break through amounts follow the fault upwards. The column pres- sure can be calculated as in Fig. 6.22 h Carrier Gridlines (dotted) Fault Capillary Pressure Seal Fault Lithology variations which affect migration or accumulation such as poros- ity variations, fracturing or facies changes, can easily be modeled with addi- tional porosity maps, capillary pressure maps of the seal or in complicated geometries, with artificial faults which limit the horizontal extension of the reservoir (Fig. 6.34). It is often more challenging to create the maps than modeling the effects. 6.5.5 Overpressure and Waterflow Pressure gradients twist flowpaths and deform HC–water contact planes (Hub- bert, 1953; Hindle, 1997). Reservoir rocks with high permeabilities have the
- 295. 6.5 Reservoirs 283 ability to balance overpressures directly with outflow of water.12 Thus only very small pressure gradients are found in reservoir rocks and therefore water flow can only significantly affect the direction of flowpaths in regions of rela- tively slow migration, e.g. in regions of low reservoir seal tilting. Due to the small size of the pressure gradients and a lack of high precision data describ- ing the geometry of the reservoir, it is often impossible to determine these locations and the pressure gradient to the precision which is necessary for accurate flowpath bending in basin modeling. However, it is usually assumed that the effect is small and does not contribute significantly to the overall picture of migration and accumulation. Additionally, continuous aquifer flow might occur. It can originate on length–scales which are even larger than the arbitrary extensions of a basin model (Hubbert, 1953; Ingebritsen and Sanford, 1998; Freeze and Cherry, 1979). The necessary overpressure data is usually not available but it is gen- erally assumed that it can be approximated by a hydraulic head hw = p/gρw which follows onshore topography (Chap. 2). The dipping angle γ of the hy- draulic head is in this case equivalent to the tilting angle of the surface. Hence, topography driven flow occurs in basins with water flow originating in high (on–shore) mountains. The calculation of bending of the lateral petroleum flow direction below a seal is demonstrated in App. M. For example, the angle ψ between the x– direction and the projection of the petroleum flow direction into the horizontal xy–surface is given by tan ψ = ρw − ρp ρw cos α sin α tan γ (6.44) for dipping angles γ of the hydraulic head in the x–direction and α of the seal in the y–direction (Fig. 6.29). 6.5.6 Non–Ideal Reservoirs Until now, migration losses have been neglected. Two different types of losses are distinguished. Petroleum becomes immobile below the critical saturation. It forms small droplets which are stuck. This phenomenon is described with a relative permeability of exactly zero below the critical oil saturation Soc. Be- sides immobile losses, all petroleum which is not found within accumulations is commonly named “losses of the petroleum system” (PS losses). Some amounts are lost at the basin boundaries, e.g. on top, or are mobile and might reach an accumulation later but they are currently not available for production. However, compared to losses during primary migration or in other regions of low permeability, it can be assumed that losses in reservoirs are mostly small so reservoir analysis without losses is a good approximation in basin modeling (Dembicki Jr. and Anderson, 1989). 12 At least on geological timescales
- 296. 284 6 Migration and Accumulation a g g a y=1° y=5° y=10° y=30° y=80° y=45° y=1° y=5° y=10° y=20° y=80° y=45° y=30° y=20° Fig. 6.29. Bending of a flowpath for an oil with ρ = 750 kg/m3 on the left and a gas with ρ = 150 kg/m3 on the right according to (6.44). The water density is here ρw = 1040 kg/m3 . As expected, the same bending angle ψ at the same tilting α needs less “lateral overpressure” γ for the oil . The curves are symmetric around α = 45◦ . A high tilting α of the seal implies a moderate lateral flow component in cases without lateral overpressure. Thus only a moderate overpressure is necessary to bend the lateral flow component significantly However, losses in reservoirs can be treated via an extra processing step before accumulation analysis. For an estimation of both immobile losses and PS losses, it is necessary to determine the height of the migrating oil stringer below the reservoir seal interface. Let Q be a volume of petroleum which is transported per time unit along a flowpath of thickness H and width w (Fig. 6.30). It has the velocity v = Q/w/T which can also be calculated by (6.41). Hence it becomes T = Qν wkkrΔρg sin α (6.45) for isotropic permeability and α as the dipping angle of the seal. This formula can only be used for rough estimates of thickness H because the relative permeability kr is not known. It is mainly determined by the oil saturation of the migrating oil which is not known very well (Fig. 6.2). Very often it is assumed that the saturation is rather high and therefore the influence of kr is small. a Seal H a Fig. 6.30. Stringer flow of thickness H below seal
- 297. 6.5 Reservoirs 285 Sylta (1991, 2002a) derived formulas for a better estimation of thickness H. The volume flow along a flowpath is given by Q = w H 0 v(h)dh = w H 0 k ν Δρg sin α kr(Sw(h))dh (6.46) with a saturation dependent relative permeability, which is again dependent on the distance h below the seal within the moving stringer. Based on an equilibrium of buoyancy with capillary pressure according to Fig. 6.4, which can be formulated in equations such as (6.14), it is possible to calculate the saturation within the moving oil stringer by inversion of pco(Sw) = Δρg(H − h) cos α + poe (6.47) for distance h. The result can be inserted into (6.46) and numerically solved for total stringer thickness H. It was shown that migration often takes place under low average saturations. This implies that kr becomes small and hence the thickness H large. Finally immobile losses, which increase with thickness H, are significantly higher than initially expected and PS losses are rather small due to the low average saturation (Sylta, 2002a). In (6.46) and (6.47) it is not considered that the flow rate Q changes due to saturation losses along the flow path. For example, this can technically be taken into account by gridding along a flowpath and stepwise reduction of Q. The idea of adjusting saturation and permeability according to a given flow amount can be formulated more generally for reservoirs. Reservoirs are often rather small structures compared to the size of the basin. They are in flow balance with their environment (Fig. 6.31), which is equivalent to an approximation without any time derivative in the Darcy flow equations.13 Thus the Darcy flow equations for oil flow can be reduced from an initial value problem such as (6.24) to a boundary value problem of the form vo = − kkro(Sw) νo · ∇uo, ∇ · vo = 0, uo − uw = pco(Sw) + Δρgz (6.48) with five unknowns uo, Sw, and the three components of vo. Here, for sim- plicity, migration of gas is not taken into account. Oil inflow velocities vo at the bottom of the reservoir are known from source rock expulsion rates. Top capillary pressure and permeability at areas of leakage must be estimated by taking into account leakage flow rates. In principle, sophisticated boundary conditions must be formulated or evaluated by repeated iterative reservoir analysis. However, capillary pressure curves and permeabilities of seals are usually not known very well. Rough approximations are therefore justified. Additionally, a “homogeneous” solution without inflow (and therefore without leakage) but with a finite oil saturation, must also be constructed 13 Compaction is also neglected here because it occurs on longer timescales than petroleum flow in reservoirs.
- 298. 286 6 Migration and Accumulation Accumulation Saturation on Path below Seal Fig. 6.31. A reservoir with overall flow balance. Petroleum saturations are adjusted in such a way that flow rates, capillary pressure and buoyancy are kept locally in balance. Thus total inflow and leakage is also in balance and added in unsaturated regions for the modeling of preserved accumula- tions from previous geological events. An exact numerical solution of these equations is still very complex. High resolution reservoir models must be constructed. Thus a high effort is neces- sary for small improvements in accuracy compared to a more simple flowpath analysis. 6.6 Hybrid Method Multi–phase flow can be described by coupled nonlinear partial differential equations (Sec. 6.3.3). Due to irregular geometries varying through time com- bined with wide ranges of parameters and long geological timescales in basin modeling, it is practically impossible to solve these equations. Based on flow- path analysis, an approximate solution in high permeability reservoir regions can be constructed. On the other hand the flow in low permeability regions is slow, so it can be calculated based on an explicit and smooth evolution of the saturation pattern through time. Putting these two approaches together yields the hybrid method of flow modeling (Hantschel et al., 2000). The flow velocity v can be calculated with Darcy’s law (6.8) or (6.41). It can easily be estimated that it becomes very small in low permeability regions. For example, with k = 10−4 mD and conditions such as in Sec. 6.5.1 it becomes smaller than 20 m/My. Isolines of flow velocity taking permeability and viscosity variations into account are depicted in Fig. 6.1. In low permeability regions and during one geological event, HCs might only move a total of a few gridcells in basin modeling. Thus, it is possible to explicitly solve the differential equations by calculating velocities from pres- sure gradients and then updating HC saturations according to this velocity field. Numerical stability is achieved by introducing small migration time steps. According to the Courant–Friedrichs–Lewy criterion, explicit solutions of dif- ferential equations are stable if the time steps are small enough (Press et al., 2002). The size of the time steps scales inversely with the velocity of flow. Thus, in regions of low velocity, this approach is feasible whereas in highly
- 299. 6.6 Hybrid Method 287 permeable reservoir regions the number of time steps explodes so that an explicit solution becomes impossible. A domain decomposition in low and high permeability regions is the basis for the hybrid method of basin wide flow modeling. A special challenge of the hybrid method are break throughs, which are treated in Sec. 6.6.2, and fault flow, which is described in Sec. 6.6.3. 6.6.1 Domain Decomposition A domain decomposition for the hybrid method is a spatial disjunction of the model into low and high permeability regions. Highly permeable regions are reservoir rocks which act as carriers and containers of accumulations. Obvi- ously, these accumulations are objects of special interest. In most cases only a few layers in a basin model are reservoirs. A much larger part is occu- pied by low permeability rocks. Usually, the source rock belongs to this part (Fig. 6.32). Hence the domain decomposition is a cut–out of reservoirs. One must be careful, because the permeability limit which defines if a gridcell be- longs to a reservoir or not depends on the grid distances and the size of the time steps. In practice is more relevant to ensure that the cut–out does not become disjunkt into small pieces, e.g. when the permeability limit is exceeded in some celles during subsidence. This can be achieve by application of the cut–out condition on a fixed instead of the insitu porosity. A value of 10−2 mD for a porosity of 30% is found as a good default value, which works well in many basins. A model with flow in a layer in which the permeability jumps between regions with low permeability and reservoirs is shown in Fig. 6.33. Reservoirs can have arbitrary outlines due to sometimes complicated facies distributions. In such cases a technical complication arises concerning the flow boundaries in the reservoir. The borders of the low permeability regions act as barriers, which create the possibility of stratigraphic traps, whereas the model boundaries are usually treated as open, which indicates a continuity of the facies and allows outflow into neighboring regions. Thus, a tool for reservoir analysis must be able to allow closed and open boundary conditions in one reservoir (Fig. 6.34). One main advantage of reservoir analysis is rapid processing. For exam- ple, maps of 200 × 200 cells can be processed within seconds on modern PCs. Detailed geometric information on small scales, which is necessary for accu- rate flow modeling, can be retained. Thus it is possible to use a higher grid resolution for the reservoir analysis than for migration in the regions with low permeability (Fig. 6.32). The price to be paid for this multigrid technique are grid transformations, which are more elaborate than initially appears. Besides the reservoir cut–out itself, saturation values in full multicomponent resolu- tion must be transformed between the fine and the coarse grid (Fig. 6.35).
- 300. 288 6 Migration and Accumulation Break Through Flowpath Analysis Darcy Flow Accumulation Fig. 6.32. The principle scheme of domain decomposition into reservoirs and low permeability regions for petroleum flow analysis with the hybrid method. In the cut– out on the right the finite element grid is displayed. The accumulation is calculated on a finer grid Less Permeable Facies Flowpath Darcy Flow Highly Permeable Facies Fig. 6.33. Flow in a layer with facies change from low to high permeability. Darcy flow is indicated by vectors in the low permeability region, whereas flowpaths and accumulation bodies represent the reservoir analysis
- 301. 6.6 Hybrid Method 289 Fig. 6.34. A map view of a reservoir from a study of the Campos basin, offshore Brazil. Isolines indicate the depth in 200 meter intervals from 2000 to 5000 me- ters. Oil accumulations are colored grey. The thickly gridded outlines mark barriers to facies with low permeability. The two reservoir “lenses” are here open for HC migration to the sides, compared with a different scenario shown in Fig. 6.20 6.6.2 Break Through Break through and leaking of accumulations are of special interest. This effect is caused by a column pressure acting against the reservoir seal (Fig. 6.22). If the pressure becomes high enough, a break through occurs. The pressure is highest at the highest point of the accumulation. The break through appears at the point where the contrast between column pressure and capillary pressure is the highest.14 Exeeding amounts are transported into the seal. Advandced 14 In general this need not be the highest point if there are capillary pressure vari- ations in the seal.
- 302. 290 6 Migration and Accumulation Fig. 6.35. Multigrid technique for integrated flow analysis, figure taken from Hantschel et al. (2000) break through models take dynamic leaking into account. The sealing strength decreases during leakage down to 35% of its original value Vassenden et al. (2003). Break throughs can be problematic concerning timing. Break through amounts which cross the interface between both domains cannot easily be split into small portions which are moved in many small time steps, as nec- essary for the framework of explicitly treated Darcy flow. Time control has already been dismissed for reservoir analysis on the highly permeable side of the interface. The use of simplified break through models is quite common for both spatial and timing reasons. The simplest model is a purely vertical break through (Fig. 6.36). Grid cell after grid cell is filled with residual saturation upwardly in a vertical direction according to an average flow direction based on buoyancy (Sec. 6.5). This con- tinues until the break through amount of petroleum is completely distributed. If the model’s top surface is reached, the excess amount is assumed to leave the basin. A special case occurs if another reservoir is reached. This reservoir is then assumed to accommodate the remaining petroleum. Alternatively, a barrier in the form of a highly sealing lithology such as salt might be reached. In this case the procedure is stopped and the excess amount is redistributed into the initial reservoir (reinjection). Lateral migration is assumed not to occur. Such a model might be improved by taking into account permeability or capillary pressure gradients for the determination of the flow direction. Of course, this is only consistent if lateral flow is allowed. Cell by cell is now saturated in decreasing order of capillary entry pressure with an upward ten- dency instead of an exclusively upward flow. Such models are usually called “Invasion Percolation” (IP) models (Wilkinson and Willemsen, 1983; Meakin,
- 303. 6.6 Hybrid Method 291 1991; Carruthers, 1998; Carruthers and Ringrose, 1998). A more detailed dis- cussion is postponed to Sec. 6.8. Shale: low permeability high capillary pressure Sand: high permeability low capillary pressure Salt: impermeable Purely vertical breakthrough ending in shale in sand below salt Breakthrough with lateral migration Fig. 6.36. A schematic section view of a break through from a reservoir with either simplified vertical flow or invasion percolation (IP). The arrow pointing down indicates a case of reservoir reinjection Some additional complications must be taken into account: In cases of reinjection an additional reservoir analysis must be performed. This is necessary because column heights would have changed if even a small amount of petroleum were transported through the seal. Obviously, this reser- voir analysis must be performed artificially with ideal seals. If cells which already contain petroleum are percolated, this petroleum should be added to the excess amounts for consistency. In rare cases of non– residual amounts in the cells, this might have a significant influence on the migration pattern. A break through is commonly assumed to appear on a scale which is smaller than the grid resolution used for the solution of the Darcy flow equa- tions (Fig. 6.45). Even whole accumulations may not be resolved properly on the Darcy flow grid if a higher resolution for the reservoir analysis was chosen. A localized break thought path could become difficult to model in a Darcy flow picture. The effect is commonly attributed by residual saturations. However, it has to be mentioned that local and small scale break throughs are unlikely and that accumulations may leak over a wide range. In an accu- mulation with one break through point, all migration amounts that originate in a possibly huge source rock volume and are continuously feeding the ac- cumulation from a laterally wide drainage area focus on one break through migration path. Such a huge amount cannot be transported through a small lo- calized break through area on top of the accumulation. A “wide” leakage area might be more appropriate (Sylta, 2005, 2004, 2002b; Vassenden et al., 2003). However, a percolation based break through method can easily be adapted to this situation. Grid cells on the break through path are now filled with full saturation instead of residual saturation. The disadvantage of too low spatial resolution in the Darcy flow region does not exist in this picture anymore and a modeling of focused break through paths with residual saturations in
- 304. 292 6 Migration and Accumulation huge grid cells is not necessary anymore. Both break through types, wide area leaking or focused break through paths, can thus be properly modeled. A two dimensional example model with two leaking accumulations is shown in Fig. 6.40. The results of the hybrid and Darcy runs are almost the same (Fig. 6.8). A corresponding invasion percolation run is displayed in Fig. 6.48. 6.6.3 Fault Flow A single fault cell is said to be a locally open fault when the capillary pressure is lower then the value of the adjacent cell and otherwise named locally closed. Closed faults act as seals and open faults as conduits, e.g. as additional flow avenues. This approach allows migration within faults (Fig. 6.37). Following the literature, such cases might be questionable (Allan, 1989; Knipe, 1997). Much literature is related to production with much higher flow rates. A fault which is sealing in production might be open for migration over geological time scales of million of years. However, avoiding migration within faults can in any case easily be achieved with high capillary pressure values. Fig. 6.37. Section view of migration in a fault at the top and across a fault at the bottom according to Chapman (1983) Migration in Fault Migration across Fault Reservoir analysis with faults is discussed in Sec. 6.5.4. An example with ideally open and closed faults is shown in Fig. 6.26. Generally, fault capil- lary pressures (FCP) are compared with capillary pressure values of the seal (Sec. 6.5.4). Excess break through amounts are injected into the fault wall (Fig. 6.38). The Darcy flow method does not allow zero volume elements in flow equa- tions. Thus fault flow must be modeled with thin locally refined volume cells. A realization of this approach is exhausting and yields long computing times. Alternatively, migration in the fault is performed with the same percolation methods as for break through. It is controlled by capillary pressure only. The
- 305. 6.6 Hybrid Method 293 Fig. 6.38. Section view of a break through from a reservoir into a fault system. In comparison with Fig. 6.36 this case only makes sense if lateral flow is allowed Shale: low permeability high capillary pressure Sand: high permeability low capillary pressure Salt: impermeable Breakthrough into fault with lateral migration Fault with FCP lower than CP of Shale only difference is that the fault cells cannot contain significant amounts of petroleum. Fault inflow does not necessarily originate from reservoirs only. It might occur also in pure Darcy flow regions (Fig. 6.39). This is especially impor- tant when permeable faults cross source rocks and significant amounts are transported away along these faults. For that reason an additional “petroleum drainage” or “fault inflow” function is added and called after each Darcy step. High FCP Bulk Cell CP FCP High Saturation Low Saturation Fig. 6.39. Schematic view of flow from bulk cells into faults. According to capillary drainage and imbibition curves (Figs. 6.4, 6.6), only small amounts from highly saturated cells can migrate into a fault in the case of high FCP (left side). In the case of generally lower FCP than bulk cell capillary pressure, the fault might quickly drain the neighboring cells of petroleum (right side). Note that due to the negligible bulk volume of a fault, an inflow is only possible if there is a corresponding outflow at some other location Additionally, fault inflow might also occur from break through paths. How- ever, in practice only one generalized break through and fault flow method is used for all cases. A break through and fault flow routine as described in this and the previous section is an invasion percolation technique. Invasion percolation is capillary– driven flow without any timing, which results in high flow velocities. The approach is especially meaningful for highly permeable cells with shorter flow times than the Darcy time steps, or for small amounts which are quickly redistributed over short distances. Invasion percolation is discussed in more detail in Sec. 6.8.
- 306. 294 6 Migration and Accumulation Petroleum Saturation [%] Hybrid Accumulation Accumulation Flow Vectors Saturation Flow Vectors Break Through Fault Fault Oil Large Gascap Vertical Transport from Source to Carrier Oil Oil with small Gascap Flowpath Kitchen Area Lateral Transport in the Carrier Lateral Transport in the Carrier Fig. 6.40. Hybrid and flowpath results of the same model as in Fig. 6.8. The hybrid model has the same break through path as the flowpath model
- 307. 6.7 Flowpath Modeling 295 6.7 Flowpath Modeling Very often it can be assumed that HC migration through layers of low per- meability is almost vertical in an upward direction. Typical vertical migration distances are at least one order of magnitude smaller than lateral distances in basin modeling. In such cases migration may happen quickly on geological timescales. The basin wide migration pattern can therefore become an almost pure reservoir analysis. The HCs are generated in the source rock and after expulsion they are directly “injected” into the next reservoir above. Losses can roughly be estimated as proportional to the thickness of the rocks which are passed through (Fig. 6.41). Break through and fault flow can be treated in a similar way as in the hybrid method (Fig. 6.42). Models of this type are called “flowpath models”. They are usually not referred to as hybrid, although they are based on a domain decomposition and contain elements of invasion percolation. 5.2 Oil Potential in MPa (Buoyancy plus capillary pressure plus overpressure in water) Capillary Entry Pressure in MPa 5.7 5.1 6.2 5.6 6.0 5.0 6.3 Darcy Flowpath Invasion Percolation 1.0 1.1 1.2 1.0 1.1 1.2 0.8 0.9 1.3 5.5 Fig. 6.41. Section view for comparison of Darcy flow, flowpath modeling, and in- vasion percoalation migration schemes in low permeable regions The advantage of flowpath modeling is that processing is extremely fast. Small migration time steps are completely avoided. The price paid is an ap- proximation which disregards timing and lateral migration in low permeability regions almost completely. The same example model, which was processed with Darcy flow or with the hybrid method, is for comparison also simulated as a pure flowpath model and shown in Fig. 6.40. Migration pathways, size, and location of the accumu- lations are almost the same as in the Darcy or hybrid runs. The composition of the lower accumulation is very different. It contains much more gas from late expulsion of the source. These gas amounts have not yet reached the structure in the Darcy and hybrid models. Flowpath models are very advantageous if the overall basin model is “sim- ple”. In general: heatflow can very often be modeled one–dimensionally in the vertical direction if lateral effects are small. If overpressures do not occur the pressure is hydrostatic and can also be calculated one–dimensionally. In such cases all submodels are very efficient and the complete simulation performance is very high. Very often an speedup of more than one order of magnitude can
- 308. 296 6 Migration and Accumulation Expulsion from Source Leakage Spilling Oil Gas Source Rock Top Carrier Top Carrier Top Carrier Gas Flowpaths Oil Flowpaths Stacked Reservoir System Fig. 6.42. Principle scheme of flowpath modeling: A stacked reservoir system with some accumulations and a source rock is shown in the top figure. The corresponding flowpaths and migration vectors are shown below
- 309. 6.8 Invasion Percolation 297 be reached compared to hybrid models. Simulation runs can be performed in an hour instead of a day. It is even possible to easily integrate simple schemes of source rock down- ward expulsion into the methodology just by splitting up the expelled amounts of HCs by simple formulas into two parts. One is moved vertically upwards and the other downwards to the next reservoir layer. Due to their high efficiency, flowpath models are usually processed as spe- cial scenarios, as first–guess models or as crude approximations, even if the assumptions are poorly fulfilled. However, basic constraints such as an overall HC balance are kept and a good overview of fundamental issues such as the generated amount of HCs or reservoir capacities with consistent distributions of accumulations can be acquired in a fast and efficient manner, unrivaled by other methodologies. The approach is very often justified by good results but care needs to be taken. 6.8 Invasion Percolation Fluid flow in porous media is assumed to be best and most consistently de- scribed by Darcy flow equations (Sec. 6.3), although it was shown in Sec. 6.2 that a lot of unknowns concerning petroleum migration still exist. It is fur- ther widely assumed that the application of Darcy flow for migration modeling might only be crude due to the low grid resolution coming from limited com- puter resources, although the method is physically correct on the macroscopic length scales of rough gridding. Darcy flow is based on microscopic averaging and upscaling of “smooth and continuous” flow to macroscopic length scales. One of the biggest disadvantages is the resulting low spatial resolution, which constricts modeling of migration channels. In its formulation, as in Sec. 6.3, it is also doubtful if it can be used for the modeling of migration based on disconnected stringers with sharp boundaries, fractal distribution, and the possibility to perform almost instantaneous spatial jumps. Although often yielding good results it must be concluded that Darcy flow equations might sometimes not be applicable for migration. Some important technical drawbacks of pure Darcy flow modeling can be overcome with the hy- brid approach (Sec. 6.6). Flowpath modeling is also a well accepted alternative (Sec. 6.7). Unfortunately these improvements are restricted to reservoirs only. Invasion percolation can be interpreted as an extension of such approaches. Invasion percolation has already been introduced in Sec. 6.6.2 for the mod- eling of break through processes, which are often believed to be difficult to model on a “rough Darcy grid”. The technique is commonly used for low permeability seals. Therefore it should also be possible to apply invasion per- colation for migration in rocks with low permeability. Consequently, Darcy flow in hybrid models can in some cases be completely replaced by invasion percolation. This allows processing of higher spatial resolutions. The price is cruder approximations, especially in regard to the timing of migration.
- 310. 298 6 Migration and Accumulation Timing is not of importance in reservoirs (Sec. 6.5). Reservoir analysis is based on buoyancy driven flow and can be modeled from purely geometrical analysis. Thus it should also be possible to use the percolation approach in reservoirs. Finally, invasion percolation can be applied as one migration method for the whole model. Domain decompositions which are necessary for hybrid or flowpath modeling can be avoided. Migration patterns that have been created without domain decomposition and which have been modeled in only one technique are easier to present and to understand. Additionally, there is the benefit that one can easily include complex and detailed geometries such as vertical sandstone channels, which pose the greatest challenge in the hybrid approach (Fig. 6.43). On the other hand some benefits such as the extremely high processing speed of flowpath modeling (Sec. 6.7), especially for multi–phase models with displacement, or the implicitly high resolution of the flowpaths traversing the grid in slightly skewed directions, are lost (Fig. 6.11). A more detailed discussion of the physics of migration with special consid- eration of the applicability and restrictions of the invasion percolation method is presented in 6.8.1. Percolation on microscopic length scales is introduced in Sec. 6.8.2. Upscaling of a microscopic percolation technique to a method usable in basin modeling is treated in Sec. 6.8.3. The complete technique is described in Sects. 6.8.4 and 6.8.5. Gridding, small scale property variations and the incorporation of high resolution data are the subjects of Sec. 6.8.6. 6.8.1 Physical Background The physical background which leads to the idea of developing the rule–based invasion percolation migration method shall now be summarized.15 Three forces are commonly assumed to determine petroleum flow (Hub- bert, 1953). These are buoyancy, which originates from gravity and the density contrast between petroleum and the surrounding water, capillary pressure, which is due to interfacial tension between water and petroleum, and fric- tion of the moving fluid, which is usually described by viscosity and mobility (Sec. 6.3). As mentioned in the previous section, average flow velocities are very small in secondary migration. This leads to the assumption that viscous forces, which are proportional to velocity according to Darcy’s law, might be negligible. Prior to a further analysis, viscous forces are therefore compared with forces originating from interfacial tensions. Pressure drops due to viscous forces over a distance R have, according to Wilkinson (1984), the magnitude Δpvisc ∼ νvR k , (6.49) 15 The term invasion percolation was originally defined in a slightly different con- text (Wilkinson and Willemsen, 1983; Meakin, 1991) but it is also used in basin modeling.
- 311. 6.8 Invasion Percolation 299 Depth Shale Sand Oil Shale Oil Shale Shale Oil Sand Sand Oil Fig. 6.43. Geometries which cannot easily be modeled with layer based domain decomposition. The sand layers are overthrusted in the example on the top left. Such cases are difficult to handle in layer based domain decomposition of hybrid and flowpath models. All sand layer parts on the left and on the right side of the fault must be treated separately. This causes an enormous effort in big models with many overthrusted layers. For example, the determination of the correct order of processing of these layer fragments is not trivial. The example on the top right is even worse for hybrid modeling. The sand object can principally not be separated into horizontally aligned layer parts, which are a prerequisite for a map based flowpath analysis. Accumulations which span over branching sand layers or sand layers with large shale lenses, are shown on the bottom. Such geometries are obviously also problematic in hybrid models. All examples in this figure are calculated with invasion percolation with velocity v, viscosity ν, and permeability k (6.41). The magnitude of capillary pressure differences is given by Δpint ∼ γ R (6.50) with γ being the interfacial tension (Sec. 5.6.6). With this choice the size of R is of the order of a minimal pore throat radius. Comparison of both quantities yields Δpvisc Δpint = C K (6.51) with the capillary number
- 312. 300 6 Migration and Accumulation C = νv γ (6.52) and the dimensionless geometrical factor K = k R2 , (6.53) (Wilkinson, 1984). Both factors can be estimated. The capillary number is very small and has a value of C ∼ 10−6 in reservoir flow for production (Barenblatt et al., 1990). An limit of C 10−10 can be estimated by taking into account the average velocity of a gravity driven flow derived from a Darcy flow model of secondary migration (England et al., 1987). The estimation of the factor K is more complicated. Evaluation of (2.44) yields K ∼ φS/24 ∼ 10−3 for tortuosity τ = √ 3 and φS ∼ 0.05 × 0.5 = 0.025 in agreement with Wilkinson (1984). This estimate can be confirmed with some values based on experience. Example values are grain sizes of 0.1 mm for sandstone with estimated R ∼ 1 μm and a permeability of 10 mD ∼ 10−15 m2 or 1μm grain sizes for clay with estimated R ∼ 10 nm and permeability 10−4 mD ∼ 10−19 m2 (England et al., 1987; Wood, 1990). The examples yield a value of K ∼ 10−3 . . . 10−2 . Finally with C 10−10 it follows from (6.51) Δpvisc Δpint 10−6 . (6.54) A possible conclusion would be that, at least to a certain degree of accuracy, viscous effects can be neglected. This conclusion can be criticized. A macro- scopic estimate of velocity is applied to a microscopic scale of varying capillary pressures. Macroscopic capillary variations such as the Hobson type Δpint ∼ γ 1 Rt − 1 Rb (6.55) are a better choice for the comparison (Berg, 1975). Herein Rt is the smallest throat radius, which is currently filled with petroleum and usually found on top of the stringer, and Rb is the smallest throat belonging to the same stringer which is simultaneously drained from petroleum and usually found at the bottom of the stringer (Fig. 6.44). The values Rt and Rb can be assumed to be of the same order of magnitude. Otherwise significant overpressuring must occur within the stringer. Hence 1/Rt − 1/Rb is smaller than 1/Rt or 1/Rb. The geometric factor now has the form K = k Re 1 Rt − 1 Rb . (6.56) Note that in this picture, according to (6.49), Re describes the extension of a stringer moving through the rock matrix. In the case of macroscopic stringers
- 313. 6.8 Invasion Percolation 301 2Rt 2Rb Grain Water Oil Fig. 6.44. Schematic diagram of a stringer according to Berg (1975); Tissot and Welte (1984) Re R. Finally, K decreases significantly and so the small estimate (6.54) rises drastically. Pathway focusing is an additional effect disturbing the estimate (6.54). It is argued that a typical accumulation is filled with 106 m3 /My (Sylta, 2004, 2002b). This corresponds to a flow rate v = 10−14 m3 /m2 /s for a drainage area of A ∼ 3 km2 belonging to the accumulation. In the case of a break through on top of the accumulation, this flow amount must be transported through the break through area. Otherwise the column height would increase leading to higher pressures, which would finally create additional break through paths. If the break through occurs only at “weak” points in the seal over small distances with, for example, less than 30 m in diameter, this leads to a break through area, which is about 3000 times smaller than the drainage area (Fig. 6.45). Hence the flow rate is about 3000 times higher and thus the estimation (6.54) must be enhanced by this factor.16 Flow pulsing is difficult to estimate. Permeabilities and capillary pressures were introduced with consideration of flow in tubes (Sec. 6.3). Hence a starting point for the study of flow pulses could be the rate of fluid penetration into a thin lengthy capillary, which has been well researched by Washburn (1921). It is a dynamic process which depends on time and penetration length. A further consideration of flow snap–off (e.g. such as described in Vassenden et al. 2003) with periodical or continuous supply of HCs from below, suggests the occurrence of flow pulses. However, flow pulsing indicates time intervals with high flow and time intervals with low flow rates. Correspondingly, the flow velocity during a flow pulse must be higher than the time averaged flow velocity. An enhanced flow velocity increases the capillary number (6.52) and thus the viscosity numerator in (6.54). In summary, a more detailed view of geometry and stringer sizes, the in- corporation of pathway focusing and the consideration of flow pulsing leads 16 It is even argued that such high flow rates cannot occur in break through pro- cesses and therefore leaking must occur on wider areas (Sylta, 2004, 2002b).
- 314. 302 6 Migration and Accumulation Backfilled Oil Residual Saturation Source Rock Depth Fig. 6.45. Schematic section views of an accumulation with break through, which is fed over the whole drainage area from a source rock at the bottom of the picture. Random capillary entry pressure heterogeneities are present in the right but not in the left. Dark grey indicates backfilled oil and light grey residual amounts. All the oil coming from below has to pass a small break through path. Note, that the column height is slightly lower in the right figure due to capillary entry pressure variations in the seal to the conclusion that the limiting factor in (6.54) rises dramatically. Vis- cous effects should generally not be neglected. Even low velocity viscous flow probably explains the existence of flow pulses and migration in disconnected stringers: On migration pathways, microaccumulations form below barriers with en- hanced capillary pressure. The saturation within a microaccumulation rises until the saturation dependent capillary pressure is high enough to overcome the barrier (compare with Fig. 6.6) All capillary pressure gradients are evened out at the barrier. The resulting forces acting on the petroleum are only vis- cous friction and buoyancy. Hence, the minimum bulk velocity of continuous petroleum outflow can be estimated with Darcy’s law. It is v ∼ k ν Δρg 10−13 m s (6.57) with Δρ = 300 kg/m3 , ν = 3 × 10−3 Pa s, and k ≥ 10−19 m2 = 10−4 mD.17 The estimated velocity v is faster than source rock expulsion velocities, which can be estimated with a maximum of 8 × 10−14 m/s (Sec. 6.2, England et al. 1987, Carruthers 1998). Hence snap–off must occur because feeding of the microaccumulation is slower than its outflow. Finally, petroleum migrates in small disconnected blobs which are the stringers. It must be noted, that the migration velocity of such a stringer is not given by (6.57). Porosity φ must be taken into account. The actual velocity of movement would thus become va = v/φ. Additionally, the continuous pen- 17 Velocity reduction due to a (possible) small relative permeability is assumed to balance approximately with velocity raise due to upscaling and anisotropy effects (Sec. 2.2.3).
- 315. 6.8 Invasion Percolation 303 etration of petroleum into pores which are not or are only partially saturated must be considered (Washburn, 1921). As argued previously, stringers might therefore move in pulses with strongly varying velocity. However, knowledge of the details of stringer migration are fortunately not necessary for rough estimates such as in (6.57). Darcy’s law joins timing with viscous effects. Even pure capillary–driven flow must take into account viscous effects (Washburn, 1921). A model based on entirely static capillary effects neither explains dynamic effects such as snap–off nor gives hints about migration velocities (Meakin et al., 2000). A stringer follows a migration path by combination of buoyancy–driven movement in an upward direction and following of the smallest capillary re- sistance given by the widest pore throats. During movement, some petroleum is lost as immobile microscopic droplets and truncated parts. It shrinks until it finally disappears or becomes trapped below a barrier of small throats. A barrier of small throats might occur due to the laws of probability in an envi- ronment with randomly distributed pore throats, if the stringer is very small, or due to a macroscopic variation of the lithology. The trapped stringer stays there until a new stringer from below reaches it. The newly arriving stringer is usually bigger, since its losses are balanced with collected droplets from its predecessor. It did not follow saturated dead ends and moved on the backbone of the migration pattern. Both stringers merge and the movement might con- tinue, depending on the maximum throat width on top of the merged stringer and its height, which determines its buoyancy pressure. Alternatively a mi- croaccumulation may arise. If a big capillary threshold has to be overcome, stringers can continue to merge and a visible accumulation might form. The principle scheme is depicted in Fig. 6.46. Moving stringers do not dissipate since the petroleum at their inner and bottom part may even move faster than on top, which enforces cohesion. In the inner part the saturation is higher and therefore the permeability increases. Extra upward forces act due to interfacial tension at the bottom.18 In total the petroleum in the bottom and inner part is “pushing” against the slower moving top. Sharp stringer boundaries evolve and inner stringer convection might arise eventually. Finally, an overall picture of migration might be a percolation of stringers. The similar behavior of moving droplets, connected strings of blobs, snap–off and disconnected fingers are reported from experiments (Frette et al., 1992; Meakin et al., 1992, 2000; Catalan et al., 1992; Vassenden et al., 2003). The overall estimation of migration velocity is given by an expulsion con- trolled average velocity of a first front of stringers. Oil expelled into existing pathways moves faster and arrives earlier in traps than oil expelled into un- 18 This is expressed by the saturation dependency of relative permeabilities denoted by e.g. the Buckeley–Leverett function (Barenblatt et al., 1990). In production it leads to well defined petroleum water boundaries.
- 316. 304 6 Migration and Accumulation D i r e c t i o n o f M i g r a t i o n Stringer Break- through Capillary Pressure Barrier Microscopic Capillary Pressure Heterogeneities Micro- accumulation Fig. 6.46. Scheme of stringer migration along a migration pathway. Migration chan- nels exist due to inherent properties of the rock matrix such as fractures or faults or they evolve due to microaccumulations, which “planate” or “smooth” the path- ways for the stringers. Microscopic heterogeneities can be passed automotively by stringers saturated pathways. However, the average flow rate remains limited by the expulsion amounts. It has been argued that migration can be described as a percolation of stringers through a network of throats. In the following a new migration algo- rithm is derived from an enhancement and modification of existing microscopic percolation models. 6.8.2 Percolation on Microscopic Length Scales In a simplified view, the space between the sediment grains can be divided into pores and throats connecting the pores. Only the case of water as the wetting phase is considered here, so the entire mineral surface is covered with a thin layer of water. The migration of petroleum is mainly controlled by the throats representing the smallest structures which must be traversed. A throat of size R can only be entered if a capillary pressure barrier has been overcome (6.50). Pore and especially throat sizes are randomly distributed within some limits in a natural rock. Ordinary percolation theory mainly deals with the subject of finding paths through a network of random sized pores and throats assuming that a fluid under pressure p can overcome barriers of capillary
- 317. 6.8 Invasion Percolation 305 pressure pint p. Regular grids are studied analytically and with computer simulations (Fig. 6.47). The gridcells are either called sites, if they indicate pores or bonds, if they are related to pore throats (Stauffer and Aharony, 1994; Nickel and Wilkinson, 1983). The main results are critical pressure values pcrit describing the threshold for the creation of a path through a sample and that the probability P of finding such a percolation path scales according to P ∼ (p − pcrit)β (6.58) for |p − pcrit| 1 with an exponent β = 0.41, which can be determined from simulations (Winter, 1987; Stauffer and Aharony, 1994). The critical pressure can be interpreted as the entry pressure pce, which is necessary to achieve a significant macroscopic fluid penetration. The number N of percolated sites scales analogously as N ∼ LD (6.59) with L as the grid or sample size and D as the exponent of the fractal perco- lation dimension. The fraction of percolated sites is commonly interpreted as saturation. Thus saturation is also expected to follow this scaling law. The exponents are usually dependent on the grid dimension only and not on the specific choice of the grid, which is why they are called univer- sal (Wilkinson and Willemsen, 1983). A value of D ≈ 2.5 has been calculated theoretically (Stauffer and Aharony, 1994) and proved experimentally (Hirsch and Thompson, 1995). Depth Fig. 6.47. Microscopic invasion percolation patterns for three different values of density contrast which increase from left to right. The pictures are from Meakin et al. (2000). Note the periodic boundary conditions
- 318. 306 6 Migration and Accumulation In invasion percolation, percolating paths are typically created starting from one given boundary, and describe a process of invasion and displacement. This alone does not yield significant differences to the “original” percolation method (Wilkinson and Willemsen, 1983; Wilkinson, 1984). Very often wa- ter trapping processes are also simulated. Water trapping might occur if a water cluster is completely surrounded by petroleum, with the consequence that sites belonging to this cluster cannot be invaded anymore (Wilkinson and Willemsen, 1983). Controversially, it is argued that the water can escape through the thin wetting layer covering the grains, giving rise to the assump- tion that trapping phenomena do not exist or that the trapping probability is so small that the effect can be neglected (Wilkinson, 1986; Carruthers, 1998; Frette et al., 1992). Extensions to the method take correlated disorder or buoyancy into ac- count (Meakin, 1991; Meakin et al., 1992). Experiments related to buoyancy– driven invasion percolation have also been successfully performed (Frette et al., 1992; Meakin et al., 2000; Hirsch and Thompson, 1995). 6.8.3 Upscaling of Microscopic Percolation Basin models are so large that migration cannot be modeled based on perco- lation within microscopic pores. Percolation must be simulated with macro- scopic sites. The degree to which pore throat distribution and macroscopic capillary pressure variations affect percolation with macroscopic sites has yet to be investigated. A fundamental question is weather upscaling also yields fractal patterns, or weather it instead yields more continuous saturation pat- terns as commonly assumed for the Darcy flow equations. Fractal means self– similar under a given magnification. This definition includes the possibility of fractal patterns with macroscopic extension. The relationship of buoyant to capillary forces is the most important pa- rameter for the characterization of invasion percolation processes. The dimen- sionless Bond number is defined as B = ΔρgR2 γ (6.60) (Wilkinson, 1984, 1986). With Δρ = 300 kg/m3 , R = 0.01 . . . 1 μm as in Sec. 6.8.1 and interfacial tension γ ∼ 30 mN/m (Danesh, 1998) it can be estimated to have a rather small value of B ∼ 10−11 . . . 10−7 . (6.61) Wilkinson introduced a length scale ξ above which the characteristic behavior of the system alters. It scales as ξ ∼ R/B0.47 with R/B0.47 ≈ 1 . . . 2 mm. For a “stabilizing gradient” system, which means injection at the top and not at the bottom of the system, percolative character is lost above this length scale (Wilkinson, 1986). In the case of a “destabilizing gradient”
- 319. 6.8 Invasion Percolation 307 with injection at the bottom, investigated here, it was shown that the satu- ration pattern is characterized by a “directed random walk” of small “blobs” (Meakin et al., 1992; Frette et al., 1992). The maximum horizontal deviation w of the random walk scales according to w ∼ h1/2 or w h ∼ 1 √ h (6.62) with h as the vertical height. Upscaling from a length scale of ξ = 1 mm to 10 m yields a reduction for w/h of about 10−3/10 = 0.01. Blobs which perform random walks with w/h = 1 on a scale of ξ show horizontal deviations of about 0.1 m in sites of 10 m in size. Hence, it is not necessary to consider pore throat variations when studying capillary pressure variations. In basin modeling invasion percolation cells are huge compared to a mi- croscopic scale and small compared to the basin scale. They are even small compared to “Darcy flow” cells. The same argument which has just been used for upscaling from pore to invasion percolation size can now be used again. Due to the small size of an invasion percolation site, which is usually above seismic resolution, it does not contain macroscopic variations relevant for basin mod- eling. It is therefore a realistic assumption to treat a site as rather smooth, homogeneous, and non–fractal in capillary pressure variation. Hence an inva- sion percolation site has properties similar to a pore in a microscopic percola- tion model. For example, its capillary pressure can be discretized by just one value. This behavior can even be verified for sandstones: non–fractal behavior corresponds to the Corey equations according to Timlin et al. (1999) with λ → ∞. Application of this limit to (6.14) yields a saturation independent capillary pressure as already demonstrated in Sec. 6.3.1. Finally, the capillary pressure curve Fig. 6.4 can be described by one entry pressure value and two saturation values, a residual start– and a connate water end–saturation for each invasion percolation site. When dealing with macroscopic invasion percolation, the capillary pressure should vary randomly between sites. This variation should reflect the impact of random heterogeneity of capillary pressure variation on length scales above the site size. The range of capillary pressure variation, which defines the range of these random variations, must be specified. Again, formula (6.62) yields some hints. It can roughly be estimated down to which length scale ξ noise coming from heterogeneities must be considered. If w ≈ 1 m is the cut–off threshold of the influence of noise in a 10 m spacing grid, which indicates an almost straight line crossing the grid, then it is w/h = 0.1. The corresponding length scale ξ for w/h = 1 becomes ξ = 0.12 10 m = 0.1 m. Roughly estimated, capillary heterogeneity variations shall be considered down to a resolution of 1/100th of the grid spacing. Variations due to smaller structures do not play a significant role. They do not affect the macroscopic appearance of migration pathways. However, site saturations must be upscaled properly based on the frac- tal dimensionality of the microscopic pattern of the blobs. Scaling of the
- 320. 308 6 Migration and Accumulation saturation S with the fractal dimension D ≈ 2.5, provides the behavior S ∼ L2.5 /L3 = L−0.5 , with L as a typical edge length of the considered vol- ume. Extrapolating from experimentally measured saturations of well cores with a diameter of 0.1 m to grid site sizes of 10 m; yields a reduction factor of 100−0.5 = 0.1 for site saturation (Meakin et al., 2000; Hirsch and Thompson, 1995; Stauffer and Aharony, 1994). Hence, residual saturations are expected to be very small on a layer scale (Hirsch and Thompson, 1995). On the other hand micro–accumulations due to macroscopic heterogeneities may also form on all length scales. This must be remembered for the estimation of average residual saturations. An upper limit of the stringer size hmax can be estimated by the height necessary to overcome the percolation pressure pcrit as hmax = pcrit Δρg (6.63) in a microscopic view. The percolation pressure in a cubic lattice with uni- formly distributed capillary pressures between 0 MPa and 1 MPa is pcrit = 0.3116 MPa (Stauffer and Aharony, 1994; Wilkinson and Willemsen, 1983).19 A more realistic distribution would be between 1 MPa and 1.1 MPa. It must be taken into account that filling a new pore with petroleum at the top of the stringer causes another at the bottom to be drained of petroleum. Hence, only a pressure difference such as described by (6.55) needs to be overcome by buoy- ancy (Fig. 6.44). Big pores with up to 1 MPa are filled because the stringer is already located within the rock and spans multiple pores. So only 31.16% of the remaining 0.1 MPa variation must be overcome and with Δρ = 300 kg/m3 the maximum stringer height becomes hmax ≈ 10 m. Analogously, an esti- mated variation of capillary pressures of 1 . . . 2 kPa in sandstone leads to a stringer height of 10 cm (compare with Berg 1975, Schowalter 1979). The phenomenon that pore distributions are usually not uniform, is as- sumed to be unimportant (Meakin et al., 2000). Nevertheless the estimates for the stringer heights are crude. Stringers of such size are obviously con- nected to a lot of pore throats. They have a macroscopic size. An entrapment based on a statistical distribution of small pore throats becomes highly im- probable because in reality, layers are non–random heterogeneous over such macroscopic distances. Variations of capillary pressure in the form of hetero- geneities might occur in all sizes and on all possible length scales above grain size. For example, fractures occurring on or below a macroscopic length scale must also be considered. Finally, it can be estimated that a realistic stringer height is often smaller. It is important that stringers are smaller than the high resolution grid sites of the invasion percolation models for basin modeling. Because of this, a site describes a small ensemble of migrating stringers. Pressure snap–off occurs within a site. Sites are often the smallest objects which are modeled in basin 19 The value refers to site percolation. It is pcrit = 0.2488 MPa for bond percolation.
- 321. 6.8 Invasion Percolation 309 modeling. For that reason snap–off is modeled to occur at site height because stringer heights are directly below the site height in shales and down to one order of magnitude below it in sandstones. This introduces a vertical gridding error with a small error for the calculation of overpressuring. However, it does not influence the saturation significantly because the saturation of the macroscopic sites is chosen independently of pressure. In the case of macroscopic capillary pressure variation, accumulations can form below capillary pressure barriers. This can be mapped by backfilling sites. These sites have “full saturation” and only a residual amount of water. Almost all pores are filled and the sites are pressure connected. The value of “full saturation” can be specified by accounting for the amount of pores which can be reached. In practice, drainage and imbibition curves are considered (Fig. 6.4). They often show a plateau (Schowalter, 1979). For such cases, residual and connate saturations are chosen as limiting values of this plateau. Finally, upscaling can be summarized by comparison of the macroscopic in- vasion percolation processes to well established microscopic approaches: on the macroscopic scale, migration can be treated as a random walk of stringers. The stringers are not connected during migration. Hence the macroscopic picture is equivalent to a microscopic picture of invasion percolation without buoyancy but with a bias of preferred vertical migration direction. The stringers are pressure–connected during backfilling. Here the macroscopic model is equiva- lent to a microscopic description of invasion percolation with buoyancy. The method uses two values of saturation, namely the residual petroleum satura- tion associated with exceeding the capillary entry pressure, and a full satu- ration associated with backfilling. Random capillary pressure variations are based on macroscopic heterogeneities down to a scale of 1/100th of the grid spacing. A big advantage of an invasion percolation approach is that capillary entry pressures can easily be biased by overpressure in the water. Aquifer flow, which causes a petroleum water contact area deformation at the bottom of an accumulation, can easily be taken into account (Fig. 6.52). The biased entry pressure is called the threshold pressure. 6.8.4 One Phase Invasion Percolation A one phase invasion percolation algorithm for migration can now be formu- lated. The space is subdivided into grid sites with a higher resolution than finite elements for Darcy flow analysis. A threshold pressure value is assigned to each site. The value is determined randomly from a distribution which describes the capillary pressure variations due to heterogeneities of the flow unit it belongs to plus the overpressure in the water.20 20 A flow unit is defined here as a region of space with similar flow properties. In its definition it has originally been limited to reservoir rocks (Stolz and Graves, 2005).
- 322. 310 6 Migration and Accumulation Migration starts in the source rock. Migration sites are saturated with residual saturation following a path of decreasing capillary pressure resistance with a preferential direction upwards. The first filling of a site with residual petroleum saturation is called invasion. If no path of decreasing or constant capillary pressure is found, backfilling begins. A column pressure due to backfilling is taken into account and a break through is sought. If a break through point is found, migration continues at this point, otherwise the last site according to the opposite order of invasion is investigated. If a neighboring site with lower threshold pressure exists, mi- gration continues in this direction, otherwise the site is backfilled and the algorithm continues as above. Three non–trivial examples are shown in Figs. 6.49 and 6.51. The first one on the left side of Fig. 6.49 shows a rather complicated example of capillary equilibrium in sand and silt. The liquid–water contact of the upper accumu- lation in the silt is in equilibrium with the contact height in the intermediate sand. The difference of these contact heights is the same as the column height of the lower accumulation. The second example on the right side of Fig. 6.49 demonstrates that it is possible to correctly model completely filled and overpressured sand lenses with invasion percolation. Although the column height in the sandlens is less than the one of the accumulation below it, a break through occurs. The reason for this behavior can be easily understood: The saturation throughout the sand lens reaches the point of high saturation with complete filling. According to the drainage curve Fig. 6.4 pressure rises in the lens until it balances with the shale (see also Fig. 6.6). Finally, a break through occurs. The third example is about expulsion, with the overall primary migration downward due to a vertical overpressure variation (Fig. 6.51). The migration object evolving from one expulsion point is called a stringer path. During the whole procedure a mass and volume balance must be kept. The algorithm stops when feeding amounts at the injection point of the stringer path have been distributed. The biggest disadvantage of the algorithm is that the time necessary for the migration of the petroleum from the source rock expulsion point to the top of the stringer path is not taken into account. Such an estimation would be difficult because the velocity of stringers moving along the stringer paths is difficult to calculate. Darcy flow methods can only be used for the estimation of flow velocities in pressure connected flow regions. Displacement of water and complicated intra–stringer flow patterns can in principle only be evaluated at resolutions which must be higher than the size of the stringers. This is obviously not possible and one must therefore rely on crude average flow velocity estimates such as given by (6.57). Usually, invasion percolation is performed without time control and based on the assumption that migration happens on a faster timescale than generation and expulsion. It must be noted that different stringer paths evolving from different ex- pulsion points might merge if a site is invaded from both. The result is a
- 323. 6.8 Invasion Percolation 311 Kitchen Area Fig. 6.48. Invasion percolation results of the same model as in Figs. 6.8 and 6.40. Light red and green cells are backfilled whereas the dark cells contain only residual amounts of petroleum. The same color scheme is used in Figs. 6.49 and 6.50 Sand Shale Silt Shale Sand Lens Sand Fig. 6.49. A percolation example with accumulations in a shale – sand – silt – sand layering on the left and an example of a sand lens which is completely filled with oil Fig. 6.50. An example of tilted accumulations under constant lateral overpressure in a reservoir below a seal (grey)
- 324. 312 6 Migration and Accumulation Fig. 6.51. An example of down- ward expulsion due to overpres- sure rising in the shales above the reservoir Overpressure- gradient Active Source Rock Reservoir stringer path with two feeding points. This behavior models the process of pathway focusing (Fig. 6.45). Obviously, the order of invasion and backfill- ing is dependent on the order of processing different stringers. This order is not predetermined by the method itself and must be chosen in a geologically meaningful manner. 6.8.5 Two Phase Migration with Displacement Petroleum often occurs in the two phases liquid and vapor. To a first–order approximation, liquid and vapor migration can be treated almost indepen- dently. Each of the phases has its own density contrast and its own pathways. Hence, sites can be traversed by liquid and vapor without interaction. Only sites with backfilling must be treated in a special way. Vapor usually displaces liquid whereas liquid cannot enter fully vapor–saturated sites. An additional complication is pressure build–up in accumulations which contain a gas cap with connection to an oil body. The column pressure on top of the oil must be added to the gas pressure. A simple example of a two phase accumulation under lateral overpressure conditions is shown in Fig. 6.50. Tilting of petroleum water contact areas under lateral water pressure variation is demonstrated. Vapor buoyancy is higher than liquid buoyancy and therefore the vapor–water contact is tilted less than the liquid–water contact. The liquid–vapor contact is not tilted due to the assumption that lateral overpressure does not occur within the accumulation. The petroleum is in static equilibrium and hence it can be assumed that it does not move. Note that a constant lateral water pressure is applied here. This is a first order approximation because water flow and overpressure must be calculated under consideration of layer and accumulation geometry. The latter is an obstacle for the water flow. However, the accumulation forms on top of the reservoir
- 325. 6.8 Invasion Percolation 313 and it can be assumed that the water flow and the overpressure pattern are not disturbed very much. It is known that compositional changes during migration give rise to im- portant effects during migration. Especially the changing pressure and tem- perature conditions during vertical migration affect these processes (England et al., 1987). Phase properties can only be properly determined if the com- position is known (Chap. 5). It is not possible to take compositional changes into account in each percolation cell, due to computer memory restrictions. Computing resources are not sufficient for advanced fluid analysis on each in- dividual tiny site either. However, this problem can be overcome by additional approximations. For example, phase compositions can be modeled with sim- ple symmetrical black oil models. But this approach is not very sophisticated because symmetrical black oil models only work well in restricted pressure and temperature intervals (Sec. 5.3). In an alternative approach phase properties are calculated only after long time steps. After each geological event, all hydrocarbon amounts are trans- formed to the rough finite element grid. On this grid scale fluid analysis can be performed. Afterwards, the high resolution percolation sites can be updated accordingly. During the percolation steps themselves the phase compositions remain fixed. Migration will be limited in range if source rock expulsion is small in one event step, especially if the event step is short. Hence the error in phase composition will also be small. This method allows an iterative refine- ment with decreasing time step length, which can be tested for convergence. Invasion percolation results are shown in Fig. 6.48 for the same exam- ple as modeled with Darcy flow (Fig. 6.8), the hybrid method and flowpath modeling (Fig. 6.40). The results are almost the same as for the Darcy and hybrid calculations. Even the migration pathways from the source to the car- rier are tilted with similar angles as in the Darcy and hybrid cases. Most of the petroleum does not enter the carrier vertically above the source and it can be seen that the deeply buried source is only generating gas. However, due to micro–accumulations and further compaction, which also drives some oil migration from the residual saturations, it is found that some gas is stuck below and some oil is still reaching the carrier layer 8. Hence, the compositions and phases which are found in the lower accumulation of layer 8 are similar to the hybrid and not to the flowpath results. 6.8.6 Discretization of Space and Property Assignment There is a choice between completely regular gridding or a gridding adapted to the geological structures of interest, i.e. the flow units. Regular grids have the advantage of easier algorithms, faster implemen- tation and higher performance in execution. On the other hand, a regular gridding might be insufficient for the modeling of small scale structures, such as thin layers. Even a high resolution model with one billion sites and a
- 326. 314 6 Migration and Accumulation 1000 × 1000 × 1000 grid resolution has, for a maximum depth of 10, 000 me- ters, a vertical grid resolution of 10 meters, which is too poor for modeling migration in thin reservoir rocks. Thus non–regular grids are recommended (Fig. 6.55). This is consistent with arbitrary percolation theory where many results are independent of the type of the grid (Stauffer and Aharony, 1994). A disadvantage of non regular grids is related to the shape of the petroleum water contact below an accumulation. Due to the irregularity of the grid the meniscus is not sampled horizontally but instead follows the structure of the layer. In general, this problem can only be overcome with high resolution models (Fig. 6.52). In models with aquifer flow and lateral pressure gradi- ents, the contact is not horizontal, so even regularly gridded models have this problem.21 Fig. 6.52. Magnified cut–out of a section view. The background pattern depicts the facies. Backfilled irregular sites are marked with regular grey rectangles. Some layers contain water flow and overpressure. Tilted and deformed oil water contacts are visible. Sampling of the column height is difficult to observe due to high resolution Another problem concerning gridding in general, is long distance migration in dipped reservoirs. Gridding directions usually do not follow the dipping 21 Regularly gridded models have the additional problem, that the top surface of an accumulation does not arbitrarily follow seal dipping. Hence a smooth oil–water contact is sometimes achieved for the price of a poorly modeled seal interface.
- 327. 6.8 Invasion Percolation 315 directions. Migrating petroleum “sees” only the neighboring sites and moves to the highest neighbor site if capillary pressure variations are small. It therefore follows the grid direction instead of the direction of the steepest ascent. Fig. 6.53. Section view of two schematic invasion percolation paths in a reservoir rock. No capillary pressure variation is as- signed on the left side. The capillary en- try pressure is modified by overpressure in the water. Obviously, invaded sites fol- low the grid in an upward direction. On the right side a capillary pressure variation in the form of random noise is assigned to each site. Hence, jumps to the left or right might occur and in average the path approximately follows the direction of the steepest ascent Direction of Steepest Ascent Overpressure Isoline The problem does not exist in cases of significant capillary pressure vari- ation. Here the mean migration direction follows the direction of steepest ascent, at the price of smearing out the migration path (Figs. 6.53, 6.54). Therefore a proper amount of heterogeneity is often needed in invasion per- colation models. Depth Fig. 6.54. Invasion percolation pathways below an accumulation according to Fig. 6.53. The figures are calculated without and with heterogeneities, respectively Faults are sometimes of special importance for migration. They are two dimensional on a basin scale and can be modeled by surface pieces adjacent to the “volume sites” of the rock matrix. However, in a microscopic picture they are fully three–dimensional and the fluid flow through them can be modeled with percolation methods in the same way as through rock. Therefore, fault surfaces can easily be integrated in an invasion percolation algorithm simply by treating them as arbitrary sites whose volume, consistent with their surface
- 328. 316 6 Migration and Accumulation description, is zero (Figs. 6.55, 6.56). However, they act as conduits or barriers which means that backfilling might occur. Backfilled amounts are negligible but the pressure rise due to pressure connection is the same as in an ordinary accumulation in a sandstone below a seal. It must be noted that the modeling of faults as surfaces is not possible in a pure Darcy flow framework where all cells must have a finite volume and surfaces appear only as boundaries. Fig. 6.55. Regular 6 × 6 subdivision of a two dimensional non–regular fi- nite element with fault sites at the element boundary as implemented in PetroMod® . A finite element grid is shown in Fig. 8.6 Fig. 6.56. Section view of an example with migration. Migration is colored dark grey, migration through the fault light grey and accumulated petroleum black. The two upper accumulations are in pressure contact along the fault. Due to this pressure contact both up- per accumulations have the same oil water contact height. The pressure at top of the fault originates from the col- umn down to the oil water contact of both upper accumulations meter Depth in meter Fault Fault Migration Pressure Contact Accumulation Invasion percolation can be performed on grids with a higher resolution than temperature calculations or Darcy flow. It is therefore possible to directly incorporate high resolution data such as seismic facies in the population of the capillary pressure field. Seismic inversion also provides porosity and clay content, which are good indicators for the size of capillary entry pressure. An example of direct incorporation of seismic data for invasion percola- tion into a basin model is shown in Fig. 6.57. Seismic velocities have been converted to porosities and permeabilities and subsequently into capillary en- try pressures. Finally, facies maps have been refined on the site resolution
- 329. 6.8 Invasion Percolation 317 scale.22 Calculated gas chimneys from seismic attribute analysis agree with leaking accumulations. Three basic pitfalls must be recognized when seismic data is used. Firstly, seismic data is usually only available for the present day and not for paleo times when migration actually took place. This problem can be theoretically overcome by backstripping the seismic data according to basin evolution. How- ever, an enhancement porosity and therefore an reduction of capillary pressure due to decompaction must be taken into account. This problem can be handled with a well defined basin model. Secondly, interpreted seismic data is needed because flow units must be assigned. The invasion percolation method needs an overall underlying knowl- edge of the rock types for construction of flow units. Up to now this cannot be done automatically. Again, this problem can be handled with a well defined basin model. Facies distributions from basin models can be used for construc- tion of flow units for migration. However, direct incorporation of seismic data is not as easy as it seems at first glance. As a last pitfall it must be mentioned that care needs to be taken with a “direct” usage of seismic data. Seismic data contains noise which is not related to the variation of rock properties but comes from the restrictions of the measurement setup and the physical processes of sound propagation. For example, noise is created from microphones or the dissipation of sonic waves. This contribution of noise to the measured signal must be clearly distinguished from the “pure” signal due to the rock specific distribution of pores, throats and macroscopic heterogeneities. Obviously, this is very difficult. Direct usage of seismic data must be performed very carefully, otherwise there is a danger of miscalculating basic quantities and producing uncertain results. Simple deterministic mapping of noisy seismic data to throat distributions or capillary pressure heterogeneities may be unreliable. 6.8.7 Anisotropy Migration based on invasion percolation, as formulated up to now, does not in- corporate effects of anisotropy. Anisotropy is described in the Darcy equations (Sec. 6.3) with a permeability tensor. Capillary pressure is usually defined as a scalar quantity for the description of porous media. It has a direction inde- pendent nature, which is stated explicitly with the word “pressure”. Hence the introduction of a horizontal and a vertical capillary pressure component appar- ently makes no sense. The only simple possibilities for introducing anisotropy are given by site size anisotropy and by anisotropic variation of the capillary pressure distribution. 22 The regional model of 80 × 100 km has gridcells of 1 km in length. The area of interest is refined with a resolution of 300 m (see LGR in Sec. 8.9). An inva- sion percolation grid with 5 × 5 × 5 sampling according to Fig. 6.55 yields 60 m resolution. The inverted seismic cube has a resolution of 30 m.
- 330. 318 6 Migration and Accumulation Gas Chimney Basin Model Seismic Processing Fig. 6.57. Chimneys which are modeled with IP on the left. Similar structures are observed in processed seismic on the right. Pictures are courtesy of MAERSK Hybrid Invasion Percolation Fig. 6.58. Comparative study of Shengli basin, China Hybrid Invasion Percolation Fig. 6.59. Roncador field in the Campos basin, Brazil (Bartha, 2007)
- 331. 6.9 Discussion 319 The requirements to construct a meaningful anisotropic capillary pressure field are rather high. Without incorporation of heterogeneities migration will be completely vertical or horizontal depending on the buoyancy force, site dimensions and capillary pressure variation. As in Fig. 6.53, heterogeneities must be incorporated to achieve a smooth crossover from these extreme cases. But even backfilling based on a very small enhancement of capillary pressure due to low heterogeneities might lead to a relatively drastic buoyant pressure rise in the case of not very thin sites. Strong vertical capillary variations might be overcome and migration will follow upwards. Therefore, even small varia- tions of capillary pressure in a lateral direction might destroy an anisotropy of preferred horizontal migration pathways. This problem can be overcome technically by introducing anisotropy as a vertical heterogeneity variation at a sub–site resolution. This corresponds to a picture with “sub–layers” crossing the sites. Micro–accumulations build up until preferred sub–layers with reduced threshold in a lateral direction are found. Anisotropy can then be interpreted as a site averaged quantity defining a capillary pressure heterogeneity level, which can be overcome in a lateral di- rection by migration (Fig. 6.60). A continuous crossover from non–anisotropic to very anisotropic can be achieved easily by increasing this anisotropy value. This method only works consistently if heterogeneity (e.g. in the form of ran- dom noise) is assigned in the model. Both, the sub–site and the global hetero- geneity variation are directly and quantitatively related. An anisotropy much smaller than the global heterogeneity does not affect the migration picture and an anisotropy much bigger than the global heterogeneity has the same effect as completely ignoring it in the lateral direction. Therefore, this is a consistent approach and, as expected, the number of micro–accumulations at the site scale will decrease with increasing anisotropy. Obviously, the deter- mination of residual saturation values and general upscaling procedures are affected and become quite complicated in this picture. By visual inspection the authors found 10% capillary pressure heterogene- ity variations with a value of 10% for an anisotropic threshold level as an appropriate default for most lithologies and typical gridcell sizes in basin mod- eling. 6.9 Discussion The Darcy flow model is a well established and physically consistent for- mulation of a transport problem with separate phases considering buoyancy, overpressure, capillary, and viscous forces. It also incorporates water flow and compaction in the most comprehensive formulation. The basic idea is a sum- mation and balance of all the forces acting on the fluids. Multi–phase Darcy flow models are too complex and too inefficient for basin modeling. Alternative methods are therefore presented in this chapter. They rely exclusively on the assumption that overall migration timing can
- 332. 320 6 Migration and Accumulation a) b) c) d) Fig. 6.60. Invasion percolation with capillary pressure heterogeneity variations of 10% and different anisotropy threshold levels: a) 0%, b) 5%, c) 10%, d) 15% be neglected. An application of this approximation in reservoirs, for fault flow and for break through processes yields the hybrid method. Flowpath modeling is a further approximation with even better processing performance and with the main advantages of the hybrid approach, but it is rather vague regarding the overall timing and lateral migration in low permeability regions. Both, flowpath and invasion percolation methods delegate the whole time control to HC generation and expulsion. Migration is treated as if it occurs instantaneously on geological timescales. Hybrid and flowpath models are characterized by reservoir analysis. Accu- mulation bodies with well defined volumes and column heights are calculated. Two phase effects, such as the displacement of liquid by vapor in accumula- tions, are taken very precisely into account. The spatial resolution can usually be chosen to be significantly higher than for heat analysis or compaction calcu- lations of the overall basin model. High spatial resolution allows high precision volumetrics and accurate petroleum–water contact height predictions. Hence, correct calculations of column pressures from accumulations are very easy. A reservoir analysis can be carried out very quickly on modern computers. Data uncertainties can be tested by interactive risking of different migration and accumulation scenarios on most PCs. The biggest problem of hybrid and flowpath models are complex geome- tries such as vertical sandstone channels or permeable faults, which connect several reservoirs (Fig. 6.43). The concept of reservoir analysis as discussed in this chapter is map based and thus essentially a two dimensional concept. Hybrid and flowpath models are limited in complexity of geometry. They work best when the geometry follows a layer cake topology.
- 333. 6.9 Discussion 321 However, without loss of generality the flowpath concept could be extended into the third dimension. Drainage areas become drainage volumes and flow- paths, which are located at the reservoir seal interface, would become complete three dimensional pathways. This approach ensures a more “natural” domain decomposition, which enables a more complex geometry. However, an imple- mentation in the form of a computer program would cause some difficulties. Standard data formats such as maps have to be replaced by more proprietary 3D – formats. Three dimensional data sets, which are obviously less common than simple maps, must be populated. Increasing data amounts and reduced performance confine the options of interactive handling. It is an open question whether the implementation of a full 3D flowpath concept would be worth the large amount of effort required. Besides the complex geometry, some smaller problems exist concerning hybrid and flowpath modeling. The automatic construction of a dataset by domain decomposition is often an expensive task. A large amount of data must be collected for a reservoir sub–model. Among the main components are the mapped reservoir, capillary pressures, porosities, faults, and the component resolved amounts of HCs entering the reservoir. Grid transformations require extra efforts in modeling. Fortunately, a cut out from a fully populated basin model can be automated. A disadvantage comes with parallelization. Reservoir analysis itself is to some degree parallelizable (Bücker et al., 2008). The main challenge is found in parallelization of the domain decomposition and the collection of injection amounts. Other small problems rise due to effects such as HC loss in the carriers, which are only processable with extra efforts or the impact of water flow within a carrier, which is easy to consider for flowpath bending but difficult to integrate into an appropriate prediction of contact area deformation. Invasion percolation offers an alternative migration method, which is based on an interpretation of migration as a movement of separated stringers. It can be performed on a higher resolution grid than commonly used in basin mod- eling for Darcy flow or temperature analysis. Physically, the method does not differ significantly from flowpath based migration. However, its techni- cal implementation is completely different. High resolution seismic data can be directly incorporated and a domain decomposition is not a prerequisite anymore. It is a big advantage of the technique that it is possible to model HCs percolating through a whole basin model. Complicated geometries with strongly varying migration properties can be easily modeled (Fig. 6.43). Ef- fects of laminated, crossbedded, and pervasively faulted strata can be taken into account to a high degree of accuracy (Ringrose and Corbett, 1994). Small scale structures such as faults can be integrated into the algorithm in a “nat- ural” and “intuitive” way. Principally, migration losses and hence migration efficiency can be calculated to a high degree of accuracy (Luo et al., 2007, 2008). Column heights in accumulations are calculated quite accurately even under aquifer flow conditions.
- 334. 322 6 Migration and Accumulation A simple two dimensional example model for comparison of hybrid, flow- path, and invasion percolation is shown in Fig. 6.61. Accumulated amounts are quite similar here. The invasion percolation and flowpath models are almost the same. The hybrid model differs slightly, in that the structure on the top left is not fed by a break through from below. Besides this all accumulations in all examples have the same break through behavior. On the right, in layers 5—8, some migration activity over wider regions can be seen in the hybrid and the IP models. This flow originates from additional break throughs in previous events, which give rise to residual saturations or microaccumulations and further transport of small HC amounts with ongoing compaction. This process is completely ignored by the flowpath model. The composition of the big accumulation within Layer 5 is also depicted in Fig. 6.61. It is very similar in all three models. A major difference in PVT– behavior can be found in the accumulation in Layer 9 beside the fault on the right. The accumulation consists mainly of vapor in the hybrid model, whereas it is almost entirely liquid in the flowpath and IP results. A closer inspection shows that the composition is very similar in all three models. During the last event the geometry changed significantly. Flash calculations are performed more often in the hybrid than in the flowpath and IP models. Thus the phases of the hybrid are already updated whereas the “faster migrating” flowpath and IP models show a systematic error due to time steps being too long. However, pressure and temperature are not far from the critical point. The density difference between vapor and liquid is relatively small and the error is not very big. A Darcy flow model is also calculated for comparison (Fig. 6.62). The prin- cipal migration scenario including the location of the accumulations is very similar. The biggest differences are found for saturation values inside the accu- mulations. Due to the low grid resolution and an improper capillary pressure curve (compare Fig. 6.4), it is not possible to model saturation patterns such as in the hybrid case, which approach almost 100 % petroleum saturation in the center of some accumulations. Another important difference in the Darcy model in Fig. 6.62 is the sat- uration in Layer 9 on the right side, which is not located in a structure or stratigraphic trap. Layer 9 is a highly permeable sandstone. The amounts on the right are stuck due to convergence problems of the explicitly treated Darcy flow. A reduction in the length of the migration time steps reduces this artifact. However, the calculation time of the Darcy flow model is already three times as much as for the hybrid model. The difference in simulation time increases even more for three dimensional models. Finally it must be noted that the composition within the accumulations of the Darcy flow model differs significantly from the other results. The sat- uration never exceeds 40 % inside the gridcells. The corresponding capillary pressure is so high that liquid cannot enter these cells. The vapor displaces all the liquid and finally most of the liquid leaves the model at the sides. Only in
- 335. 6.9 Discussion 323 the Darcy model do all accumulations contain vapor. This is an error which is inherited from low gridding and improper capillary pressure curves. Invasion percolation demands the analysis of multiple migration scenarios if, due to the random assignment of capillary pressure values, several widely distinct saturation patterns evolve. The resulting scenarios are not determin- istic and cannot be reproduced (at least not in detail). Thus, in theory, one specific migration pattern is not representative because major migration paths might be dependent on small scale variations. Further analyses, such as sen- sitivity studies with multiple scenarios, are necessary. A risking procedure should be integrated into the migration analysis (Fig. 6.63). However, this is not a major drawback, as risking should be performed in basin modeling anyway (Chap. 7). High resolution seismic data combined with advanced flow unit interpre- tation enables sophisticated percolation analysis on a high resolution scale. The data, however, must be available and interpreted. Fully populated three dimensional models must be constructed. Possible pitfalls associated with slightly dipped seals or with the interpretation of noise in seismic data com- plicate the workflow (Sec. 6.8.6). Flowpath models might be a better alternative in cases of incomplete data sets or simple geometries such as “layer cake” structures. Generated amounts are vertically transported into the reservoir rocks. The implicit high resolu- tion of the flowpath calculation (Sec. 6.5.1), petroleum water contact height analysis without vertical gridding problems and the high performance of the calculation, especially for multi phase migration, leads often to almost the same results with a simpler and faster workflow. As mentioned before, timing is a big disadvantage of invasion percolation as well as for pure flowpath modeling. The time the hydrocarbons need to migrate from the source rock to the structure is generally not taken into account by the method. An example is shown in Fig. 6.58. Colored planes indicate faults which are partially open for migration. Similar structures are charged in both models. In the IP model, petroleum is transported to the reservoirs more quickly and in larger amounts Another example with severely affected timing is shown in Fig. 6.59. In the Campos basin, salt windows might open at a certain time allowing HCs to pass. Charging and accumulated amounts in the hybrid and in the invasion percolation model are almost the same for the Roncador field. But the two source rocks charge the structures differently in each model and there are huge migration differences in sub salt layers which are not visible in the figure. Elaborated mass balances such as Table 6.2 are here very advantageous. Important information can often be extracted from balance tables more efficiently than by visual interpretation of migration and accumulation scenarios. Mass balances are discussed in general in Sec. 6.10. Additionally, it should not be forgotten that break through and leakage flow rates are usually so high that the basic assumption of low flow rates for invasion percolation is violated. Darcy flow is argued to be the better alternative here (Sylta, 2004, 2002b). However, this problem could principally
- 336. 324 6 Migration and Accumulation Hybrid Closed Fault Closed Fault Accumulation Flow Vectors Invasion Percolation Breakthrough Flowpath Lithology Shale sandy Sand shaly Dolomite Silt shaly Sand Chalk Marl Shale sandy Salt Basement Components in Mass % Components in Mass % Liquid Vapor Big Circle: Liquid Small Circle: Vapor Big Circle: Liquid Small Circle: Vapor Fig. 6.61. Hybrid, flowpath, and IP runs of a simplified North Sea example model. Layer 10 and 12 are source rocks. Break through pathways are not displayed in the hybrid plot. The resolution of this figure is unfortunately too poor to clearly distinguish between full and residually saturated IP cells. Flowpaths in the reservoir layers of the hybrid and the flowpath model are graphically converted here to flow vectors for clearer presentation
- 337. 6.9 Discussion 325 Petroleum Saturation [%] Petroleum Saturation [%] Darcy Hybrid Fig. 6.62. Darcy and hybrid runs of the model which is also depicted in Fig. 6.61
- 338. 326 6 Migration and Accumulation Depth Fig. 6.63. Invasion percolation scenarios with two different realizations of hetero- geneity. The top left structure is filled in the left example and the top right structure in the right example Petroleum System Hybrid IP Generated 60.0 60.0 Expelled 54.0 54.0 HC in Reservoirs 2.0 2.0 Losses 52.0 52.0 Losses in Detail Hybrid IP Migration 29.5 0.5 Secondary Cracking 3.0 0.0 Outflow Top 7.0 2.5 Outflow Sides 12.5 49.0 Table 6.2. The masses of the petroleum Systems in Gtons for the two versions of the Campos model which are shown in Fig. 6.59. The main differences are due to different migration scenarios in the shale between source and salt. In the IP model most of the mass is transported to the border and lost at the basin sides whereas petroleum moves slowly in the hybrid case. The mass balance is additionally affected by secondary cracking in the hybrid model. The petroleum is kept long enough in hot regions just above the source rock. Sub salt migration timing is important in this study (Bartha, 2007) be solved: break through flow rates could be estimated and additional break through points could be calculated if necessary. Explicitly treated Darcy flow and invasion percolation are not so different from their technical approach. Both are grid cell (site) based and both modify the saturation in the grid cells according to physical conditions in a range containing neighboring cells. The central differences are that invasion perco- lation usually works on a higher spatial grid resolution but at a much lower resolution in time. The explicit Darcy method performs many small migra- tion time steps to achieve a picture of continuously moving saturation under time control, which is neglected in invasion percolation. Saturation is modeled with continuous values in Darcy flow whereas discrete fillings, namely resid- ually and fully saturated, are used in invasion percolation. The implications of the saturation discretization are expected to be rather small, because the smearing out of the saturation values of invasion percolation in the low res- olution Darcy cells also yields (almost) continuous pictures. However, a big difference is found in column height calculations, which cannot be performed accurately within a low resolution Darcy grid. All approaches have advantages and disadvantages. Neither Darcy flow, hybrid approach, flowpath modeling nor invasion percolation is superior in
- 339. 6.10 Mass Balances 327 general. All known migration models are based on a number of assumptions, simplifications and approximations. There is no modeling method which fully covers the whole spectrum of all migration related effects to a high degree of accuracy. However, depending on the specific geological environment and the availability of data in a concrete case study, one of the methods is often more suitable than the others. Better modeling results can be achieved just by selection of the most appropriate method. It must also be mentioned that the comparison of different methods for the same study yields at least an idea of result variations and uncertainties. 6.10 Mass Balances An accurate analysis of a petroleum system is equivalent to the detailed quan- tification of all the petroleum involved in each geological process. Petroleum amounts must be therefore separately quantified and tracked for the processes of deposition, generation, cracking, adsorption, phase separation, dissolution, migration, and accumulation. The large number of processes, layers, phases, and components involved, make this topic a challenge. A systematic approach is necessary. 6.10.1 Fundamental Laws of Mass Conservation A sediment with kerogen within it has at time t a potential to generate a mass MP (t) of petroleum. The potential might increase by additional deposition of sediment with mass generation potential MSed(t) and decrease via erosion by MEro(t). The amount of kerogen available for generation before the start of generation is MP (t) = MSed(t) − MEro(t). Generated HC mass is quantified by MG(t). Hence, the potential through time is finally given by MP (t) = MSed(t) − MEro(t) − MG(t) . (6.64) The mass amounts in balance equations such as (6.64) are cumulative over and uniquely dependent on time t. Amount differences ΔM(t, t ) = M(t) − M(t ) for any time interval, which starts at time t and ends at time t, can be calculated just as differences between the values at the two different time points. Hence ΔMP (t, t ) becomes ΔMP (t, t ) = ΔMSed(t, t ) − ΔMEro(t, t ) − ΔMG(t, t ) . (6.65) It is not called cumulative anymore. Equation 6.65 has the same structure for mass differences ΔM as its cumulative counterpart (6.64) for total masses M. Thus all the following cumulative balance equations can also be interpreted as non–cumulative. The time dependence is, for the simplicity of the expressions, not noted anymore in this section.
- 340. 328 6 Migration and Accumulation HC mass amounts MHC, which are found in a sediment, are given by the sum of liquid and vapor petroleum ML,V , HCs which are dissolved in water MW and HCs which are adsorbed MAd: MHC = ML + MV + MW + MAd . (6.66) Generated mass and current HC mass differ by secondary cracked and by migrated amounts only. The amount of the secondary cracking product coke with mass MC can be calculated as the difference of HC mass MD which is destroyed by cracking and newly generated (transformed) mass MT by MC = MD − MT . Migrated masses are usually distinguished by in– and outflow MIn/Out across the border of the object under consideration. Hence the continuity equation can be written as MHC + MC + MOut = MG + MIn (6.67) or ML + MV + MW + MAd + MD + MOut = MG + MIn + MT . (6.68) The last equation contains on both sides only terms with positive values, typical for bookkeeping. The left side defines where the HCs are found and the right from where they come (assets and liabilities). The last two equations (6.67) and (6.68) are obviously valid for any geo- logical object, especially any layer or facies at each time. For example, there is usually no generation in a reservoir r, hence it is MG,r = 0. The equations are also separately valid for each HC component. Hence each term is usually labeled with additional indices such as MV,l,i, which indicate the mass of com- ponent i found in the vapor phase inside of layer l. The index i is skipped in this section. It makes the equations rather ugly and it is not necessary for the argumentation. Sums of all components yield total masses and formulas with skipped component indices can also be interpreted as total mass balances. Tracking should be performed for each finite element cell in modeling prac- tice. Any geological object in a basin model, which is important for HC balance considerations (e.g. any layer l), can be constructed by sums over its cells. It is therefore possible to calculate any balance for any geological object in post processing steps. A difference formulation such as (6.65) between two succeeding time steps with infinitesimal duration and infinitesimally small volumes for (6.67) yields a continuity equation such as (6.34) for mass balance. A rearrangement of (6.67) in difference form such as (6.65) and division by Δt = t − t yields ΔMHC Δt + Δ(MOut − MIn) Δt = Δ(MG − MC) Δt (6.69) and makes the relationship with (6.34) more obvious. Existing HCs constitute the transient term and in– and outflow the flow term. Generation is formulated with source terms on the right side of the Darcy flow equations.
- 341. 6.10 Mass Balances 329 Equation 6.68 contains the important quantities which are necessary for an understanding and a detailed analysis of petroleum systems whereas (6.69) demonstrates the relationship to the flow equations. Losses from the border of the whole model also need to be tracked to ensure the completeness of the bookkeeping. The masses which leave the model are usually quantified and collected under the term model outflow MMO. Some- times it is additionally distinguished between outflow at the top and outflow at the sides with MMO,l = MMO top,l + MMO sides,l. Any outflow from a geo- logical object is an inflow into another geological object except the amounts which leave the basin. Hence it is l MOut,l − MIn,l = l MMO,l . (6.70) The summation index l must cover all geological objects with in– and outflow, e.g. all layers and faults. Conducting faults are obviously of special interest for migration. In basin modeling resolution they have a two-dimensional character which implies that all quantities which are found in equations (6.64) — (6.68) except MIn/Out are zero. Additionally ΔMFlow,F = l(F ) ΔMFlow,l = 0 (6.71) with the sum over all layers l (and other faults) which touch the fault F. Biodegradation is for convenience often included in the secondary cracking scheme. It usually cannot be mixed up with secondary cracking because it is found only in shallow sediments at temperatures too cold for cracking. Similarly, petroleum dissolved in hydrate phases is usually added to the water balance. However, an extension of the balance scheme (6.68) with a separate incorporation of these masses is straight forward. Separate tracking of MOut and MIn is not recommended in modeling as demonstrated in Fig. 6.64. Due to gridding artifacts neither MOut nor MIn can be determined accurately. Only the difference ΔMFlow = MOut − MIn is reliable. A similar problem rises with flowpath modeling and invasion perco- lation. Petroleum which traverses a facies in one event is not tracked as in– and outflow. For example, in flowpath models, the petroleum is transported vertically from source to reservoir without considering vertical migration path- ways in detail (Sec. 6.7). An in– and outflow tracking is elaborate and time consuming especially in models with complex facies distributions and sublay- ering. Due to random capillary pressure variations and high resolution cells, similar problems as in Fig. 6.64 arise in invasion percolation. A stringer path might enter and leave a facies multiple times over short geological distances. This leads to large and meaningless in– and outflow amounts. Again only the difference is useful. All amounts should be tracked in units of mass. Insitu volume is inappro- priate because it changes with pressure and temperature (Chap. 5). For an
- 342. 330 6 Migration and Accumulation Facies A Facies B Facies B Facies A F l o w Flow Fig. 6.64. Schematic map view of flow along a facies boundary on the left. A correctly modeled flow may follow the gridlines in a model on the right. Only the average flow direction is the same as on the left side. Each facies has multiple in– and outflows. Obviously, the vanishing difference of in– and outflow is correctly conserved. Models with many time steps such as explicitly treated Darcy flow models might erroneously cause enormous amounts of in– and outflow impression of volumetrics it is sometimes advisable to transform all masses with one fixed density to volumes. Such a transformation conserves all bal- ance and conservation laws. Nevertheless, vapor volumes are sometimes trans- formed separately with another density. Conservation laws such as (6.67) are not valid in such a case and must be treated with caution. 6.10.2 The Petroleum System All HC masses which are involved in all the important geological processes of petroleum geology in a sedimentary basin, are tracked when all the quantities of (6.68) the generation potentials and the basin outflow are known all over the basin. It is now possible to evaluate all characteristic numbers for petroleum systems analysis. However, it is first necessary to introduce the expulsion masses ME and HC losses MLoss of a petroleum system. In basin modeling expulsion masses of a source s are defined by ME,s = MG,s − MC,s − MHC,s = ΔMFlow,s . (6.72) Classically, expulsion is defined slightly differently in geology. Expulsion is only related to the petroleum which is generated in the expelling source rock. The balance according to (6.72) adds masses of hydrocarbons which were gen- erated in another source rock, migrated into the considered source and again left it.23 Classically defined expulsion amounts can only be calculated with high effort in basin models with migration and multiple source rocks. The situation is shown in Fig. 6.65. First of all, each component must be tracked according to the source rock where it was generated. But source rock tracking alone is not sufficient because free pore space in source B might be occupied 23 According to (6.72) even “non–source” rocks might expell if ΔMFlow,l 0.
- 343. 6.10 Mass Balances 331 by petroleum of source A. An interaction of such a type causes an additional amount of expulsion from source B, which would not occur if source A did not exist. This additional amount is usually not be taken into account in the defi- nition of classical expulsion. It is defined as a property of the considered source alone. Furthermore, classical expulsion is only defined for the first time when petroleum leaves the source. Inflow back into its generating source rock (e.g. after downward expulsion) followed again by outflow must also not taken into account. This causes additional complications according to Fig. 6.64. Finally, it must be stated that classical expulsion can not be directly calculated from or within a basin model with secondary migration.24 However, it can easily be calculated from special simulation runs which model primary migration only without performing any further migration. These extra simulations can in both cases be performed very quickly if temperature and pressure results are reused from runs with migration instead of recalculating them every time (Sec. 1.3). Fig. 6.65. Stacked source system. Expelled amounts of the deeper source A migrate into source B and mix with petroleum inside of source B. Free pore space in source B is occu- pied by petroleum which is generated in source A. Expulsion amounts of source B are therefore difficult to determine Source B Source A HC losses of a petroleum system (PS) are now defined by the expelled masses minus the amount found in the reservoirs as MLoss,PS = s(PS) ME,s − r(PS) MHC,r . (6.73) The first sum is over all sources s and the second over all reservoirs r be- longing to the petroleum system. It does not matter, if a classically defined expulsion amount or the definition of (6.72) is used in (6.73). The flow balance is invariant because inflow into a source rock is counted negatively according to (6.72). HC losses can alternatively be evaluated with MLoss,PS = l(PS) MMO,l + l(PS) MC,l + l(PS) MHC,l . (6.74) 24 Classical expulsion is often not present as an overlay for visualization in model viewers. Source rock leakage, which is defined as the outflow at the upside or downside of a source rock, is commonly available as a compensation.
- 344. 332 6 Migration and Accumulation The first sum represents the flow out of the basin. It is over all the layers of the petroleum system including the sources and the reservoirs. The second sum describes the amount of losses by cracking. The index l runs over all layers exclusive of all source rocks. The last sum describes the amount of HCs in the petroleum system which is neither located within the source nor is found within a reservoir. The index l therefore runs over all layers exclusive of all source rocks and reservoirs. Finally, a petroleum system is commonly characterized by the masses found in its reservoirs MHC,PS = r(PS) MHC,r , (6.75) the losses MLoss,PS, which can be calculated from (6.73) or (6.74), the expelled masses ME,PS = s(PS) ME,s , (6.76) and its generation potential given by MP,PS = s(PS) MP,s . (6.77) Again, the sum over s(PS) indicates a sum over all sources and r(PS) over all reservoirs of the petroleum system. A requirement for the evaluation of characteristic numbers of a petroleum system is knowledge of its encompassing elements (Sec. 1.4). This is not a triv- ial problem, neither in reality nor in models with multiple source rocks and a complicated geometry. Two supplementary methods are commonly used. Firstly, flowpaths, drainage areas, flow vectors and cells with residual satu- rations, which indicate that petroleum has passed through them, are tracked just by visualization with model viewers. This is a very fast but also rather crude approach and often does not reveal all affiliations. Compaction and varying geometry makes the mapping of flowpaths over time a sophisticated task. Secondly, source rock tracking is applied. Components which are gener- ated in one source are separated during the whole simulation from identical components generated within other source rocks. This method allows a close tracking but unfortunately a lot of computer memory is needed if many source rocks are present in the model. Another disadvantage is the rather low spatial resolution, especially if the source rocks are wide. However, this method has the additional benefit, that petroleum can be accurately associated with its origin, even if it is mixed from different sources. The best results are achieved with a combination of both methods. 6.10.3 Reservoir Structures and Accumulations Individual structures in reservoirs and accumulations are another important class of geological objects for which mass balances are commonly formulated.
- 345. 6.10 Mass Balances 333 Outflow out of a single structure is often subdivided into spilling, fault flow and leakage or break through MOut = MSpill + MF + MBT . (6.78) Usually, only one of these three outflow types exists. Generation and absorp- tion usually do also not exist in reservoirs and hence (6.68) becomes ML + MV + MW + MC + MF + MSpill + MBT = MIn . (6.79) Sometimes inflow is also separated into spilled amounts coming from other accumulations and amounts caught over the whole drainage area from below Min = MSpillIn + MDA. A severe problem arises with compaction and tectonic movements. Struc- tures do not permanently exist over the whole lifecycle of a basin. They mostly develop ages after deposition and they may move laterally with an overall tec- tonic shift or vanish during further compaction. The tracking of such variable geological objects over several geological events is sometimes impossible and technically very difficult to automate. Similar problems arise with accumula- tions which merge over several traps with proceeding source rock expulsion (Fig. 6.19). Generally, it is only possible to formulate non–cumulative balances such as (6.65), e.g. for one geological event only. A tracking algorithm for traps and accumulations can be based on a present day drainage area subdivision of a reservoir. Each structure of a pre- vious event has a highest point which is often located in its center. If this point is found in the present day drainage area it is added to its time track. This rule has several consequences. Firstly, each structure found in paleo time is related uniquely to a present day structure. This implies that in sums over multiple structures no paleo time structure is counted twice and that no struc- ture is skipped if a sum over all structures is performed. Secondly, no paleo time structure might be found. In such a case, a gap is found in time ex- tractions. Finally, more than one paleo time structure may be related to one present day structure. In such a case all the amounts of liquid, vapor, break through etc. must be summed up over this paleo set of structures Fig. 6.66. An exception are spilled amounts (Fig. 6.67). It must be checked if they spill out of the tracked set to which they belong. Internal spills obviously need not be counted. The summed spill amount of the set outflow is called net spill. The total paleo outflow MOut according to (6.78) might not be provided by one term alone. For example, one of the related paleo structures might have a break through and another might spill. The system of summing a group of many substructures has a big advan- tage. It can be easily extended to a group of present day structures or drainage areas which can be summed together with the same rules. A group of present day structures is associated with a group of structures at each paleo event, which is just the union of all sets belonging to each drainage area. This allows an almost continuous crossover from one present day structure to a larger
- 346. 334 6 Migration and Accumulation 0 Ma 11 Ma 3 Drainage Areas with 3 Accumulations 1 Drainage Area with 1 Accumulation Age [Ma] Age [Ma] Age [Ma] Sum Volume [Mm^3] Volume [Mm^3] Volume [Mm^3] Age [Ma] Volume [Mm^3] Fig. 6.66. Map view of tracked accumulations. Three accumulations are found at the present day but only one at 11 Ma. This leads to discontinuities in the tracking of accumulated volumes of the individual accumulations at 11 Ma. However, the sum over all drainage areas does not show discontinuities x x Drainage Area Border Accumulation Highest Point x x Paleo Event Spill Drainage Area Border (unmerged) Present Day Fig. 6.67. A schematic map view of a reservoir with accumulations at a paleo event and at present day. Both small accumulations belong to the history of the big accumulation at present day. The spilled amounts at paleo time should not be accounted for when spilled amounts are tracked through time field (e.g. a main structure and its satellites) and the full reservoir. Obviously, it can be expected that wrongly tracked paleo structures become relatively less important if the number of present day drainage areas increases.25 Er- roneously tracked paleo structures might only appear at the border of the tracked field, which becomes less important with the increasing size of the full area. 25 The areas should not be disjunct.
- 347. 6.10 Mass Balances 335 Summary: The most comprehensive formulation of fluid flow in porous media is given by Darcy flow. It is based on a balance of all forces which act on the fluid. These include capillary pressures which rise due to interfacial tensions in rock pores and throats, friction which is described by petroleum viscosity and rock permeability, buoyancy of the petroleum in water, and additional external forces, e.g. due to aquifer flow. The differential equations with appropriate boundary conditions can be formulated. They allow for petroleum mass conservation. Practically, Darcy flow can only be modeled on a coarse grid resolution. However, migration in carriers and reservoirs can be approximated very accurately with flowpath (ray tracing) methods. Due to high permeability, migration occurs instantaneously on geological timescales. Hence, migration and accumulation can be modeled geometrically including the flowpaths, drainage areas, and volumetrics of the structures. In rocks with low permeability petroleum migrates slowly, which allows for a solution of Darcy flow based differential equations. Further more, a domain decomposition of a sedimentary basin into high and low permeability rocks can be performed. Highly permeable regions can be modeled with flowpath based reservoir analysis and low permeability regions with Darcy flow. This approach is called the hybrid method. Vertical migration between source rock and reservoir occurs very quickly on geological timescales when the vertical distance is short. Darcy based flow can often be approximated by direct injection of expelled petroleum into the reservoirs above the source rocks. Reservoir analyses can be performed with the hybrid method. This highly efficient method is called flowpath modeling. Time control of migration is completely determined by expulsion. Obviously, flowpath modeling is not applicable everywhere. Darcy flow can be criticized because it is based on continuous flow and macroscopic length scales. It might not be suitable when migration is in- terpreted as a non–uniform movement of microscopic disjunct petroleum stringers along preferred migration pathways. Upscaling considerations give rise to an alternative invasion percolation method, which is in many phys- ical aspects very similar to flowpath modeling because time control of the migrating petroleum is again completely neglected. Domain decompositions can be avoided and migration can be performed in complex geometries on high resolution grids. Seismic data can be incorporated without upscaling. Each of these methods have advantages and disadvantages. Darcy flow is the most comprehensive method but practically unusable. The hybrid method is very accurate and can be well applied in practice. Flowpath mod- eling is the most efficient method and often a very good approach when data is sparse. Invasion percolation works best on complex geometries such as multi–faulted regions and when high resolution seismic data is incorporated. A systematic assembly of mass tracking in all geological processes of petroleum generation and migration can be incorporated.
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- 353. 7 Risk Analysis 7.1 Introduction In previous chapters two assumptions were made about data needed for suc- cessful simulation runs. It was first proposed that necessary data is completely available and second that it is good quality. So it was implicitly concluded that each model is unique. In practice, this is usually not the case. Data sets have gaps and the data values often have wide error bars. These uncertainties lead to the following three types of questions: 1. What is the impact of uncertainties in the input data on the model? What is the chance or the probability of having special scenarios? How large is the risk or the probability of failure? Is the simulation result stable or does a slight variation of some input parameters cause a completely different result? How sensitive is the relationship between a given parameter variation and the resulting model variation or how do the error bars of the input data map to the error bars of the results? 2. What are the important dependencies in our model? Not every uncertainty of an input parameter has an impact on each un- certainty of the simulation result values. Which parameter influences which result? How strong are the different influences? Do they have a special form? Studying these questions is especially necessary for the understanding of the model and the processes it contains. Understanding is again necessary if con- clusions are to be drawn, which go beyond a plain collection of results. 3. Which set of input data leads to agreement when considering additional comparison data? Very often additional calibration data are available which cannot be used directly for the modeling but can be compared to simulation results. Is it possible to reduce the uncertainty in the input data by excluding models related to simulation results which are not matching the calibration data? In the literature, procedures treating this problem are often listed under the keywords “inversion” or “calibration”. T. Hantschel, A.I. Kauerauf, Fundamentals of Basin and Petroleum 341 Systems Modeling, DOI 10.1007/978-3-540-72318-9 7, © Springer-Verlag Berlin Heidelberg 2009
- 354. 342 7 Risk Analysis This chapter deals with these three topics under the headings “Risking”, “Understanding”, and “Calibration”. The classical approach of tackling such problems would be to perform sce- nario runs. Starting with a first best guess model, which is commonly named “reference model” or “master run” (Fig. 7.1), model parameters are modi- fied manually and scenario runs are performed according to the knowledge, speculations, expectations, and understanding of the modeler (Fig. 7.2). l l l l Geometry Lithologies Boundary Conditions ... Input Values: Simulation Result One Input Set One Simulation One Model Fig. 7.1. Result of one – deterministic – 3D simulation. It is implicitly assumed, that uncertainties in input data do not exist An example, with an uncertain temperature history caused by unknowns in heat flow and thermal conductivity could have this typical form: A high heat flow scenario and a low heat flow scenario are simulated, whereas other uncertain input parameters such as thermal conductivities are held at fixed values. The high heat flow scenario is found to be realistic by looking for example at the calibration data or through the experience of the modeler. Next, a high thermal conductivity scenario and a low conductivity scenario are modeled with a fixed high heat flow. High thermal conductivity matches the calibration data best. So, a high heat flow combined with high thermal conductivity is found to be the most realistic scenario. Two main problems arise with such an approach: The procedure of variation and selection of the input parameters is not systematic. There is a possibility of overseeing other realistic scenarios e.g.: in this example, low heat flow with low thermal conductivity. With a higher amount of uncertain parameters such mistakes can become normal. The choice of “probable” scenarios is not very well quantified: A sensitivity analysis of how precise the heat flow has to be known to match the result is not performed, a quantification of the reduction of uncertainty is missing and the repercussions of the variation of other parameters, simultaneously with the heat flow, is omitted completely. The reliability of risk results gained by scenario runs is primarily depen- dent on the knowledge of the involved modelers. Scenarios are usually not
- 355. 7.1 Introduction 343 l l l l Geometry Lithologies Boundary Conditions ... Input Values: Simulation Several Input Sets Several Simulations Several Models l l l l Geometry Lithologies Boundary Conditions ... Modified Input Values: Simulation l l l l Geometry Lithologies Boundary Conditions ... Again Modified Input Values: Simulation . . . . . . . . . Result Result 2 Result 3 Fig. 7.2. Approach with “Scenario Runs”. The first run is usually called the “Master Run” or “Reference Model” performed systematically and the discussion of the results is qualitative but not quantitative. The main goal of this chapter is to describe a systematic approach to deal with these issues. A more concrete formulation of the tasks involved with the three topics are: • Risking: Calculation of probabilities, confidence intervals and error bars. • Understanding: Calculation and analysis of correlations. • Calibration: Calculation of the probability of how good a model fits calibration data and search for the best fitting model. All topics contain words such as “probability” or “correlation” which are related to the language of stochastics and statistics. It is possible to treat all three topics simultaneously with a stochastic method such as a “Monte Carlo Simulation”. This has the big advantage that expensive and time consuming simulation runs can be reused for the analysis of three distinct topics.
- 356. 344 7 Risk Analysis An introduction into probabilistic methods of applied basin modeling can be found in Thomsen (1998). Other approaches are usually less general and restricted in applicability or assumptions. Nevertheless, the efficiency can be significantly raised by studying limited problems or tasks with different meth- ods. Highly specialized methods of inversion, for interpolation and extrapola- tion of simulation results are discussed in later sections. 7.2 Monte Carlo Simulation The starting point for a Monte Carlo simulation is a reference model and a list of uncertainties belonging to the data. The reference model is based on a parameter set within the limits of these uncertainties, which typically represents a best first guess. Additionally, a quantification of the uncertainties must be known. The most precise quantification is a probability distribution (Fig. 7.3) which defines the probability of a data value to be exact.1 Fig. 7.3. Examples of normally and log. normally distributed uncertainties Very often the distribution is not known but only some more general state- ments about the type and size of the uncertainties. It is usually not difficult and also not critical to construct a distribution from this knowledge. This is discussed in Sec. 7.2.1 and typical examples are demonstrated. One important point, which must be mentioned, is that the uncertain model parameters should be independent. In Sec. 7.2.3 this is discussed in more detail. With this setup the “Monte Carlo Workflow” is straightforward: A set of random numbers according to the distributions is drawn and a simulation run with this parameter set is performed. This procedure is repeated while the results are collected (Fig. 7.4). Output parameters are collected, visualized, and analyzed with statistical tools such as histograms. 1 It more precisely defines a probability density with probabilities of values to be within certain intervals.
- 357. 7.2 Monte Carlo Simulation 345 Draw random numbers according to uncertainty definitions Simulation run Collect data for histograms Enough runs? Finish Start yes no Fig. 7.4. Flow chart for Monte Carlo simulation runs Multiple Simulation Runs Fig. 7.5. Monte Carlo simulation with histograms of accumulated petroleum It will now be shown that the main topics “risking”, “understanding”, and “calibration” can be solved with the Monte Carlo simulation approach: Risking Confidence intervals related to risking can directly be read off from result histograms. They define the probability to find a result within a given interval. E.g. it is possible to formulate statements such as “With 80% probability the accumulated liquid petroleum amount is between 1623 and 1628 million cubic meters” (Fig. 7.6).
- 358. 346 7 Risk Analysis Fig. 7.6. Histogram of liquid accumula- tions: “With 80% probability the accu- mulated liquid petroleum amount is be- tween 1623 and 1628 million cubic me- ters” Fig. 7.7. Decision analysis with tree: The expected value EV for gains or losses of the “drill branch” is the aver- age EV (drill) = 0.2×(−6)+0.5×(−2)+ 0.3 × 10 = $ 0.8 MM Other valuable characteristics of a histogram are the modus, which defines the location of the most probable result, or the average. The concept of cal- culating expectation values, such as the average, is extremely important in economics: For example complex decision procedures in companies are often analyzed with decision trees (Fig. 7.7). These trees are based on the evident statistical law, that the optimal decision strategy is found by following the branches with the highest expectation values, which can be calculated from averages. A measure for the width of a histogram is the standard deviation. This quantity can be set in relation to the less precisely defined error bar. To- gether, average and standard deviation are often used as “value with error bar” (Fig. 7.8). Big standard deviations of resulting histograms indicate high uncertainties and give rise to the conclusion that the master run is not repre- sentative and therefore not probable. Fig. 7.8. Gauss distribution with mean μ = 0 and standard deviation σ = 2. The standard deviation can be interpreted as the size of an error bar. About 68% of numbers drawn from this distribution will be inside the range of the error bar
- 359. 7.2 Monte Carlo Simulation 347 The analysis of the result widths as functions of the uncertainty widths is called “sensitivity analysis”. More precisely, it can be represented by the relation between standard deviations of uncertainty and result parameters. A result is highly sensitive/unsensitive to an uncertainty if its error bar is large/small compared to the error bar of the uncertainty parameter. Therefore, sensitivity analysis can be a guiding tool for the understanding of a system. Understanding A very important problem arises with the question of where future efforts concerning the reduction of uncertainties should be spent. A reduction of un- certainties can be achieved by further data acquisition, which can be very costly. Sensitivity analysis directly leads to the parameters, which are of im- portance (Fig. 7.9). So, an expensive collection of unnecessary data could be avoided. Fig. 7.9. Tornado diagram depicting the influence of some uncertainties on the porosity at a defined location in a well. Spearman rank order correlation coeffi- cients (Press et al., 2002) are plotted as bars. As expected, the permeability shift (highlighted) (anti–)correlates mostly with the porosity Understanding can be improved by searching for dependencies, e.g. via cross plots (Fig. 7.10). Correlations can be visualized and with the help of correlation coefficients quantified. In case of strong correlations, it is possible to interpolate the results and for forecasting purposes state formulas of de- pendency. For expensive simulation runs this is very valuable. Generalizations of such techniques are discussed in Sec. 7.5. Calibration It is obvious that calibration could be performed with Monte Carlo simula- tions in a simple way by just looking at the model that best fits the calibration data. The investigation of uncertainty space is performed by sampling the un- certainties according to their probability distributions. Random combinations of parameters are used for the Monte Carlo models. This method ensures a global sampling of the space of uncertainty. The risk of missing regions with good calibration, becomes small with a high number of Monte Carlo runs.
- 360. 348 7 Risk Analysis Fig. 7.10. Cross plot of temperature against heat flow shift. A correlation is visible and a linear interpolation might be performed The Monte Carlo method is not intelligent in a way that it searches for models with good calibration. Search algorithms would be more efficient but obviously cannot be performed simultaneously in combination with risking and understanding. Additionally, these algorithms often search in local re- gions of space so it could be that they end up with an erroneous calibration. Therefore, a global investigation of the uncertainty space such as with a Monte Carlo analysis has to be performed as a first step before beginning with a search algorithm. Global stability and the prevention of extra simulation runs are often of higher importance compared to high–quality calibration. How- ever, in Sec. 7.5 more sophisticated calibration methods, which combine the advantages of both approaches are discussed. 7.2.1 Uncertainty Distributions Uncertainty distributions must be specified for Monte Carlo simulations. The properties of some well known distributions are now discussed with regard to their usage in Monte Carlo simulations. Normal Distribution The normal or Gauss distribution p(x) = 1 σ √ 2π exp −(x − μ)2 2σ2 (7.1) with mean μ and standard deviation σ is the most widely used distribution in science (Fig. 7.3). Assume that a quantity X is measured independently N times with the values x1 . . . xN . Following the central limit theorem of statistics, the average x = 1 N N i=1 xi (7.2) is Gauss distributed for N → ∞ with2 2 In practice N 7 is enough for high numerical accuracy.
- 361. 7.2 Monte Carlo Simulation 349 μ = 1 N N i=1 μi and σ2 = 1 N N i=1 σ2 i . (7.3) Here, μi and σi are the means and the standard deviations of the proba- bility distributions for each measurement i. They are often the same for all i. Parameters which are used in large scale basin models are often provided as upscaled averages of higher resolution data or averages of multiple mea- surements. Assuming independency, it is often possible to assign a Gauss distributed uncertainty to such a parameter. Logarithmic Normal Distribution This distribution is also called lognormal or lognorm distribution and has the form p(x) = 1 σx √ 2π exp − (ln x − μ)2 2σ2 for x 0 and p(x) = 0 else , (7.4) compare with Fig. 7.3. It has similar properties to the normal distribution. If a quantity Y is normally distributed then X = exp Y is lognormally distributed. The central limit theorem for the arithmetic average of some Yi becomes a geometric average for the related Xi namely x = N ! i=1 x 1/N i . (7.5) The equations for μ and σ stay the same as in (7.3) but it should be remem- bered that μ and σ are not the mean and standard deviation of the lognormal distribution, they are only the mean and standard deviation of the related normal distribution. A lognormal distribution is of special interest to “scale quantities”, which by definition cannot be negative. The logarithm of a scale quantity can be calculated every time and the distribution is zero for negative values. Many physical quantities especially material properties such as thermal conductivi- ties are limited to positive values. And the calculation of averages e.g. upscal- ing, is often performed with geometrical averaging (Chap. 8). The lognorm distribution can be a proper choice for the description of uncertainties related to such quantities. Uniform Distribution The uniform distribution (Fig. 7.11) is defined by p(x) = 1 b − a for a ≤ x ≤ b and p(x) = 0 else (7.6)
- 362. 350 7 Risk Analysis with a b. The uniform distribution is a good choice if nothing except some limiting statements can be made about the form of the uncertainty. The two discontinuities of the distribution are often the subject of criticism: They have sharp edges and are therefore argued to be in contradiction to the assumption of ignorance about the “tails” of the distribution. Additionally, it often seems unreasonable that the central inner parts of an uncertainty have the same probabilistic weight as the more outer parts. Fig. 7.11. Examples of uniform and triangular distributions Triangular Distribution The triangular distribution (Fig. 7.11) does not have the principal problems, which come with the uniform distribution. It is given by p(x) = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 2(x − a) (c − a)(b − a) for a x ≤ b, 2(c − x) (c − a)(c − b) for b x c, 0 else (7.7) with a b c. The distribution is zero at a and c and its median is located at b. A triangular uncertainty distribution can be directly constructed when the uncertainty limits and also the most probable value are known. It can also be used as an “easy to use” approximation to normal and lognorm distributions (Lerche, 1997; Thomsen, 1998). Other distributions, such as exponential or beta distributions, are more sophisticated alternatives to uncertainty descriptions (Figs. 7.12, 4.5, Rinne 1997). They are only used under special circumstances.
- 363. 7.2 Monte Carlo Simulation 351 Fig. 7.12. Examples of exponential and beta distributed uncertainties Nominal Distributions Sometimes it is necessary to assign uncertainties to discrete parameters (Fig. 7.13). In the most general case these parameters are without order re- lation, which signifies that there are no “less than” or “bigger than” defined between them. Then they are called nominal parameters. Typical examples are lithologies or kinetic type assignments as lithologies and kinetics are usu- ally specified by a large number of parameters. Therefore uncertain nominal parameters often imply strong result variations. An interpretation of results derived from nominal uncertainties can be difficult especially if nominal and continuous uncertainties are mixed in the same sequence of risk runs. This should therefore be avoided. Fig. 7.13. Discrete distributed lithologies 7.2.2 Derived Uncertainty Parameters An uncertainty is described as a distribution of one number, e.g. the thermal conductivity of shale. But very often it is associated with more than only one number. For example a heat flow uncertainty is related to the complete basal heat flow, which is space and time dependent and cannot be described with one number only. However, the basal heat flow of the master run can be shifted, tilted, twisted, etc.. A restriction to special forms of variation which can be described with one number only allows the assignment of an uncertainty to this “derived parameter”.
- 364. 352 7 Risk Analysis It is well known from mathematics that arbitrary variations can often be decomposed into infinite series of orthogonal functions which would yield infinite uncertainty parameters. In practice, one is thus restricted to the most important variations, which are often defined by the first terms of such series. The simplest form of basal heat flow variations is a value shift in a defined time interval (Fig. 7.14). Therefore, it is possible to assign an uncertainty distribution to a shift of the whole basal heat flow. 50 mW/m^2 45 mW/m^2 55 mW/m^2 40 mW/m^2 60 mW/m^2 55 mW/m^2 65 mW/m^2 50 mW/m^2 Shift of 10 mW/m^2 Fig. 7.14. Shift of (gridded) basal heat flow map Complex structural uncertainties can easily be risked with the prize of restriction to special forms of variation (Fig. 3.36). 7.2.3 Latin Hypercube Sampling (LHC) Arbitrary random sampling of the uncertainty distributions has some draw- backs. Clusters of drawn numbers can occur (Fig. 7.15) and low probability tails of distributions are often not sampled, although they might contribute significantly to the analysis (e.g. calculation of moments) especially if they have a wide range. The statistics (e.g. estimating a mean with the average over a set of random numbers) becomes increasingly better, as less clusters exist and the smoother the sampling is. x xx x x x xx x x x Fig. 7.15. Clustering of random numbers in one and two dimensions. The variables x and y are uncertainty parameters e.g. a heat flow and a SWI temperature shift x x x xx x x x x x y Latin hypercube sampling is a technique which helps to avoid clustering and samples low probability tails without affecting basic statistics. It consists
- 365. 7.2 Monte Carlo Simulation 353 of primarily two parts: The first part is an improved drawing algorithm in the series drawn from one distribution. The second part refers to the “hyper- cube” and deals with multiple drawings in the multi–dimensional uncertainty “hyperspace”. Latin Hypercube Sampling in One Dimension The interval within which the uncertainty parameter is defined can be divided into intervals of the same cumulative probability which are called “strips” (Fig. 7.16). Drawing a random number is now performed in two steps: First, a strip is selected. Then, a random number is drawn according to the probability distribution in this strip (Fig. 7.17). It is not allowed to use a strip again until all others have been selected for drawing. The best efficiency is obviously achieved if the number of drawings equals the number of strips or is a small integer multiple. x Cumulative Prob . - - - - - - - - - - 0.2 0.4 0.6 0.8 1.0 - - - - - - - - - - Fig. 7.16. Segmentation into ten equal probable intervals LHC sampling is three times more efficient for the calculation of basic statistical quantities such as means or confidence intervals (Newendorp and Schuyler, 2000). When considering the huge size of simulation efforts for big basin models, this is a good deal for the price. On the other hand it is easy to see, that this method does not reproduce auto–correlations between successive drawings. In practical implementations, the additional effort for the calculation of the strips and for the bookkeeping of used and unused strips has also to be taken into account. Latin Hypercube Sampling in Multiple Dimensions In more than one dimension, each distribution is segmented into the same number of strips. When drawing the random numbers, it is necessary to avoid correlations between the selection of the strips. Therefore the strips must be selected randomly too. The final result is a subdivision of uncertainty space into equal probable hypercubes (Fig. 7.17). In the case that the number of drawings equals the number of strips, each cube contains only one drawn number at maximum.
- 366. 354 7 Risk Analysis In two dimensions each column or row contains one drawn number and in N dimensions each N − 1 subspace contains exactly one number. x x x x x x x x x x x - - - - - - - - - - Fig. 7.17. Latin hypercube segmenta- tion with random numbers in one and two dimensions. Compare with Fig. 7.15 x x x x x x x x x x y - - - - - - - - - - - - - - - - - - Again, bookkeeping of strips has to be performed but the advantages are the same as in the one–dimensional case. LHC is a very efficient method for global sampling of the uncertainty space. 7.2.4 Uncertainty Correlations Up to now, it was assumed that the uncertainty parameters were independent. The opposite of independence is dependency. This does not need to be further discussed, as a dependent uncertainty parameter can obviously be eliminated from the list of uncertainties and treated like a simulation result. Besides these two extremes, the region of correlation exists where specified combinations of the drawn numbers are favored above others. An example could be the thermal conductivity of two layers which are known to have similar lithologies but it is not known what they are. So, for heat flow analysis a modeler would prefer to study combinations of similar conductivities. A complete joint probability distribution, which defines the probability for all combinations of all values of the uncertainty parameters, would be the most thorough description. However, data and theoretical foundations of cer- tain joint probability distribution forms usually do not exist. In practice it is sufficient to deal with correlation coefficients which are used to link marginal distributions. The rest of the joint probability distribution remains unspeci- fied. Nevertheless, drawing random numbers of correlated distributions is prob- lematic enough. Explicit formulas exist for correlated Gauss distributions. The simplest case are two correlated Gauss distributions which have the following form (Beyer et al., 1999) p(x) = 1 2π |Σ| e− 1 2 xT Σ−1 x . (7.8) with two variables xT = (x1, x2). The correlation is defined by the covariance matrix
- 367. 7.2 Monte Carlo Simulation 355 Σ = σ2 1 ρσ1σ2 ρσ1σ2 σ2 2 (7.9) which is symmetric and positive definite (Fahrmeir and Hamerle, 1984). Here σi = x2 i are the variances and ρ = x1x2 /σ1σ2 is the correlation coefficient with −1 ≤ ρ ≤ 1. Without loss of generality σ1 = σ2 = 1 is further assumed. The correlation matrix can be Cholesky decomposed (Beyer et al., 1999; Press et al., 2002) into Σ = AT A with A = 1 ρ 0 1 − ρ2 . (7.10) Thus x∗ = Ax = (x1 + ρx2, 1 − ρ2x2)T is Gauss distributed without corre- lation. Obviously, this can easily be generalized to higher dimensions. Correlations of arbitrary marginal distributions can be forced with numer- ical methods. At least three different algorithms are known to exist (Miller, 1998). In the case of many distributions with many correlations, these algo- rithms become computationally very expensive. Especially, if one parameter is correlated multiple times with other parameters, these methods are not affordable anymore. Another disadvantage of these algorithms is their incom- patibility with latin hypercube sampling. Abdication of LHC sampling reduces the performance significantly. A new approximative method to get correlated random numbers is now described: All random numbers can be drawn before performing the risk runs if the total number of simulation runs is known at the beginning and correla- tions are ignored. These random numbers can be sorted afterwards with the following algorithm: An uncertainty parameter is randomly selected and after that two random numbers out of its sequence are randomly selected again. These numbers are swapped, if the resulting covariance matrix approximates the target covariance matrix more closely than before. This procedure can be repeated until a high degree of accuracy is reached. The sum of the squared deviations of all correlation coefficients can be taken as a measure for the total deviation. It is clear that this permutation procedure certainly does not lead to a sufficient approximation and never to the exact reproduction of all correlation coefficients if the number of simulations is small.3 But experience has shown that an almost exact numerical agreement can be reached very fast on modern computers in practical relevant examples. Even for about twenty runs with a few correlated parameters a good numerical agreement could be achieved. 3 An extension of this method with some acceptance/rejection probability of a number swap would transfer it into a “Markov chain Monte Carlo” (MCMC) al- gorithm. It can be proved that such an algorithm finds the optimal approximation over longer time intervals. Due to rejection the MCMC algorithm shows generally a poorer performance. By experience, the authors found the MCMC algorithm here not necessary. MCMC algorithms in general will be discussed in more detail in Sec. 7.5.
- 368. 356 7 Risk Analysis Another big advantage of this procedure is its compatibility with latin hypercube sampling. Hypercubes and therefore higher performance coming from a lower number of necessary risk runs can be conserved. An example of a correlation matrix constructed with this permutation method is given here: the matrix which links some marginal probability dis- tributions of uniform, triangular, normal, and lognorm form is defined as ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 1 0 1 0 0.2 1 0 0.3 0 1 −0.1 0 0 0 1 −0.7 0 −0.1 0.1 0 1 0 −0.2 0 0 0 0 1 −0.6 0.2 0 0 0 0 −0.5 1 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ . Because of symmetry, only the lower triangular part of the matrix is shown here. For 20 random numbers, which correspond to 20 simulation runs, the fol- lowing approximation could be achieved with a maximum deviation of 0.0321 of any correlation value ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 1 −0.0085 1 −0.0092 0.2088 1 0.0131 0.3004 −0.0008 1 −0.0978 −0.0060 −0.0005 0.0057 1 −0.6705 −0.0063 −0.0973 0.1091 0.0006 1 0.0178 −0.1956 0.0008 −0.0033 −0.0015 0.0172 1 −0.5679 0.1957 −0.0102 −0.0078 0.0024 0.0310 −0.4865 1 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ and for 100 runs ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 1 −0.0009 1 0.0004 0.2000 1 0.0000 0.2998 0.0002 1 −0.1000 −0.0002 0.0002 0.0005 1 −0.6938 0.0000 −0.0998 0.0999 −0.0002 1 0.0031 −0.1997 0.0001 0.0000 0.0016 −0.0023 1 −0.5928 0.1992 0.0005 0.0000 0.0011 0.0051 −0.4965 1 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ with a maximum deviation of 0.0072. The maximum relative error of any correlation coefficient with an absolute value bigger than 0.1 is then less than 2%. This error is usually far beyond the accuracy of the knowledge of the correlation coefficients.
- 369. 7.2 Monte Carlo Simulation 357 7.2.5 Analysis of Results Many good textbooks are available about probability theory and statistics, e.g. Beyer et al. (1999) or Spiegel and Stephens (1999). Due to some specific problems associated with basin modeling some subjects are reviewed here. An introduction to statistical analysis has already been given in Sec. 7.2. Histograms and cross plots are used to visualize the data while numbers, such as average values, can be calculated for further analysis. Statements can be quantified, e.g. with percentiles. A histogram is a binned approximation of a probability distribution. The width of a bin should neither be too narrow nor too wide because the visualization would become meaningless or group- ing errors would become an issue (Spiegel and Stephens, 1999). Especially the calculation of percentiles for risking directly from histograms yields only “gridded values” (Fig. 7.18). In this case it is often better to use a linear inter- polated form of cumulative probability. It is more precise for the calculation (e.g. percentiles) and has a smoother visualization (Fig. 7.18) but the data have to be available in raw form. This corresponds to a higher allocation of resources in a computer implementation. Nevertheless, the binning width of a well sampled histogram indicates the statistical error of extracted quantities such as mean values or percentiles. Analysis (Fig. 7.18) shows that at least about 100 data values and therefore 100 risk runs are necessary for statistics with an acceptable relative error of around a few percent. Fig. 7.18. Histogram with cumulative frequency of 100 drawn temperature values on the left and linear interpolated cumulative probability based on the same data on the right Correlation coefficients are usually calculated for the analysis of possible dependencies. One should not forget that correlation is necessary but not sufficient for dependency. So, finding dependencies is not only part of analysis but also interpretation. Unfortunately, dependencies can have a variety of forms. Standard approaches of statistics test only for special forms. Most commonly used is the Pearson correlation coefficient. It is a measure for the deviation of a cross plot from a straight line. If its correlation value is 1
- 370. 358 7 Risk Analysis then the cross plot fits perfectly to a straight line, which is positively inclined, and if it is −1 to a line, which is negatively inclined. If the correlation is 0 then there is no similarity to a straight line at all. Intermediate values indicate an approach to a straight line which becomes better with increasing absolute values (Fig. 7.19). A straight line is the most important form of dependency but it is also a strong restriction to a special form. The Spearman rank order correlation coefficient is more general. It models the order of data points and is a measure of the deviation to an arbitrary monotonic increasing or decreasing correlation. Even a little more general but nearly the same is “Kendall’s tau”. It relies more on relative ordering and less on ranks. Some example cross plots with different correlation coefficients are shown in Fig. 7.19. Pearson: 1 Spearman: 1 Kendall: 1 Pearson: 0.98 Spearman: 1.00 Kendall: 1.00 Pearson: 0.83 Spearman: 1.00 Kendall: 1.00 x x x y y y y x Pearson: -1 Spearman: -1 Kendall: -1 Pearson: 0 Spearman: 0 Kendall: Not Defined y x x x Pearson: 0.383 Spearman: 0.015 Kendall: -0.104 y Pearson: 0.99 Spearman: 0.99 Kendall: 0.92 Pearson: 0.04 Spearman: 0.05 Kendall: 0.04 y Pearson: 0.69 Spearman: 0.68 Kendall: 0.49 y y x x x Fig. 7.19. Some examples of cross plots and their correlation coefficients Spearman’s rank order coefficient ranges also from −1 to 1 but it is overall known to be more robust than Pearson’s correlation coefficient (Press et al., 2002). Commonly, it is used for tornado diagrams where lists of correlations are ranked and visualized (Fig. 7.9). The existence of correlation can generally not be determined with a cor- relation coefficient alone. It only describes the strength of the correlation of a
- 371. 7.2 Monte Carlo Simulation 359 specific data set. For example a small dataset can be randomly correlated. The most extreme case are two points which fall on a straight line every time. But fortunately it is often possible to estimate significance levels for the existence of correlation. For a more detailed discussion see Press et al. (2002). Nominal distributions must be treated differently. Because of missing order relations a correlation cannot be defined properly anymore. Instead, associ- ations are calculated. Typical association values are Cramer’s V or the con- tingency coefficient C. Their interpretation can be complicated. Alternative measures of association exist and are based on entropy (Press et al., 2002). The values of association reach from 0 to 1 from no– to full– association. They cannot be negative just like their continuous counterparts, which must be kept in mind when plotted e.g. in tornado diagrams. Similar to continuous correlations, significance levels can also be estimated for associations. If a uncertain nominal parameter is associated with a result distribution of the corresponding model, the result distribution must obviously be available in discretized, e.g. binned, form. 7.2.6 Model Data Basin models are usually very large in size and contain a vast amount of data. It is not possible to store all the data of each Monte Carlo run completely. Only selected and restricted amounts of data can be handled and therefore not all statistical methods can be used for analysis every time. The calculation of the average is exceptional. It can be obtained just by adding the results of each run and finally, after the last run, by dividing through the number of runs. Hence a storage of all results of all runs can be avoided. Variance can be treated in a similar way. More precisely, average and variance of a quantity need only resources of the same size necessary for the storage of the corresponding result values of two simulation runs. It is therefore possible to store them for all quantities of interest even for grid based spatial overlays on huge three dimensional models. Sophisticated statistical analysis can only be performed if the complete data sets or at least histograms are available. They are usually collected only at some special points or for quantities of special interest. In basin modeling it is common to collect all the results of the different simulation runs at all well locations with logging information because calibration against these mea- surements might be performed. Additionally, the sizes of petroleum accumu- lations and column heights as primary targets of petroleum systems modeling are tracked over all risk runs. It is common to define “risk points” which are special points of interest, where additionally all the data of all runs is collected. These are usually points in source rocks which are of interest for maturation and expulsion timing or points located at faults which can e.g. be important for petroleum migration. If the spatial density of these points, with full risk data, is high and the intermediate behavior of the fields belonging to the stored values smooth, then
- 372. 360 7 Risk Analysis it is possible to interpolate the data through space for full reconstruction of each risk run. Especially, in combination with predictive methods for risking, which are discussed later in Sec. 7.5, these methods can be used to estimate full modeling results without performing the accordant expensive risk run. Such forecasted data sets can be generated very fast and used for further more sophisticated statistical analyses. Besides the risk points it is also common to explicitly track hydrocar- bon mass amounts related to layers, facies, faults or individual structures such as introduced in Sec. 6.10. Especially the characteristic HC masses of a petroleum system are subjected to statistical analysis (Sec. 6.10.2). Analyses of individual reservoir structures and accumulations raise similar problems for identification and tracking of a structure in different risk runs as in different events (Sec. 6.10.3). It can be solved in the same manner as in Sec. 6.10.3 just by treating a risk run similarly as a paleo event. Again, the same structure and accumulation tracking problems arise but are assumed to decrease with an increasing number of grouped drainage areas. 7.3 Bayesian Approach Calibration can be non–unique or numerically unstable dependent on the available data. A bayesian approach for generalized calibration is presented in this section. It can be read almost independently from the rest of this chapter and can also be skipped if calibration topics are not of special interest. In the following, it is assumed that N calibration data values dT = (d1, . . . , dN ) are available. They are measured values and have an error, so they can be described by di ±Δi. Further, it is assumed that M uncertainties xk exist. Performing a simulation run with fixed values xT = (x1, . . . , xM ) yields a model with simulation results fi(x) as model data, which can be compared to the calibration data. In arbitrary calibrations it is possible to calculate the probability of how calibration data fit a given model. Under the assumption of small error bars and a statistical independency of measurements belonging to the data points, it is postulated that the measurement values are normally distributed. So, the probability of how well a model fits the calibration data is given by p(d|x) ∝ N ! i=1 exp − 1 2 di − fi(x) Δi 2 . (7.11) Calibration does now imply a search of the values x, which fit the cali- bration data best. An obviously good criterion for the best fit is looking for the highest probability, which is called the “Maximum Likelihood” method in statistics (Beyer et al., 1999). Caused by the minus sign in the exponent of (7.11) it is equivalent to the search for the minimum of
- 373. 7.3 Bayesian Approach 361 χ2 = N i=1 di − fi(x) Δi 2 (7.12) which is the classic chi–square formula for fitting models to data. Its interpre- tation is easy and can be visualized for simple cases: Equation (7.12) becomes a linear regression of a straight line with the assumption of M = 2, a simple “simulator” fi = x1ri + x2 and data values measured at some locations r namely di = dri (Fig. 7.20). Fig. 7.20. Regression for the fit of a straight line through a “cloud” of measure- ment data. The inverse of the error bar size determines the weight of each point r f = x r + x 1 2 f Some problems can arise with (7.12) in practice. Often it occurs that a calibration is not unique (Fig. 7.21). In such a case the practitioner would choose a value somewhere out of the middle or at the highest probability of the according uncertainty distribution of the heat flow. Fig. 7.21. Example with M = 5, N = 19, and objective χ2 plotted against heat flow shift, which is known to be the most sensitive parameter. The calibration of heat flow is not unique in the range of −5 . . . 3 mW/m2
- 374. 362 7 Risk Analysis Even worse, occasionally a calibration is numerically unstable or yields completely unrealistic results caused by insufficient data points combined with some outliers. Such awkward effects can already be found in mathematically very simple situations: Without loss of generality σi = 1 is assumed for the following discussion. Further a simple “linear simulator” with fi = M j=1 Rijxj or f = R · x (7.13) is studied. The matrix R describes this simple “linear simulator”. It is easy to show that minimizing (7.12) directly leads to Rx = d (7.14) which is a set of linear algebraic equations with x as unknowns. This leads directly to the following statements: 1. The inverse R−1 does not exist in general, especially if M N which denotes that a unique calibration is not possible. 2. If R−1 exists it could be numerically unstable (Press et al., 2002). 3. If a solution is found, it is not ensured to be physically or geologically meaningful. The first statement means that calibration data could be insufficient for cali- bration. For example present day temperature data alone is never sufficient for paleo–heat flow calibration. The second statement expresses that calibration data might be inconsistent leading to possibly different calibration scenarios and the third states that calibration might be optimal outside of the allowed parameter range of the model, e.g. a negative thermal conductivity. The problem now, is how to get rid of these possibly awkward calibration behaviors and introduce a method which is at least as good as the workflow of the practitioner. A possible solution could be a so called “Singular Value Decomposition” of the matrix R (Press et al., 2002). This is a projection on parameters which can be calibrated numerically stable with the available data. The rest of the uncertainty parameters are ignored. This method has two drawbacks: First of all it is only well defined for linear problems such as the “linear simulator”. A generalization to non–linear problems would be very complicated if possible at all. Second, there is still a problem with the parameters which cannot be calibrated. Which value should they have? Regularization is another attempt which can be tried. Instead of minimiz- ing χ2 it is proposed to minimize χ2 + λ ( xT x ) (7.15) with a number λ which has to be selected properly. It is easy to see that at least the first and second statement are solved with this method because (7.14) changes to
- 375. 7.3 Bayesian Approach 363 ( RT R + λ1 ) x = RT d which are regularized normal equations of (7.14).4 But which value should be taken for λ? A few ideas can be found in (Press et al., 2002) but in general the problem remains unsolved. All three problems are of principle nature and it is necessary to go back to the basics of the probability definition (7.11). It is written in conditional form stating a probability of calibration data fitting a given model. Instead it is possible to evaluate the probability of models fitting given calibration data following Bayes law p(x|d) = p(x) p(d|x) p(d) (7.16) which leads down to the roots of probability theory and logic (Jaynes, 2003; Robert, 2001). The term on the left side is called the “posterior”, the classical probability (7.11) is situated in the nominator and called the “likelihood” and the first term p(x) on the right side the “prior”. The term in the denominator does not play a central role, it is for normalization only. It is possible to evaluate (7.16) similar to (7.11) under the assumption that all distributions have Gaussian form. One yields a minimization rule for the objective function Φ with Φ = N i=1 di − fi(x) Δi 2 + M i=1 xi − μi σi 2 (7.17) where μi and σi are means and standard deviations of the uncertainty distri- butions. The first term on the right side is the “classical” χ2 followed by an additional term. It is derived from the uncertainty distributions and implies that the knowledge for the definition of their shape has the same value as the knowledge about error bars of calibration data and should be taken into account with the same weight for calibration. The knowledge entering the def- inition of the uncertainty distributions is therefore called “prior information” and the distributions often just “priors”. In case of the linear simulator, the objective Φ = N i=1 di − M j=1 Rijxj Δi 2 + M i=1 xi − μi σi 2 must be minimized. This formula has basically the same form as (7.15). In case of μi = 0 and σi = σ for all i they are the same with λ = 1/σ2 .5 It is 4 This is only half of the truth because it is known that normal equations usually have worse numerical properties than their “non–normal” counterparts. Depend- ing on the numeric value of λ, the stability can still be a problem. 5 This relation yields some additional hints to how parameters such as λ should be chosen in regularization problems.
- 376. 364 7 Risk Analysis easy to see that the first and second statements concerning the existence and numerical stability of R−1 vanish with the usage of (7.17). The term associated with the prior tries to move the calibration in the direction of the μi where the center of the distribution is located. The most extreme case would be if there were no given calibration data. Then (7.17) would lead to xi = μi. In general, one can assume that the priors are defined for physically and geologically meaningful parameter ranges. The attraction of the parameters into this region by the prior therefore ensures meaningful solutions and solves for the problem of the third statement. This behavior automates the procedure of the practitioner. On the other hand if either a huge amount of data or qualitatively very good calibration data is available the prior term can be neglected and cali- bration approaches the classical χ2 method. In the intermediate region both terms balance Φ in the same way as different data points balance pure χ2 calibration. The discussion is the same for the nonlinear case (7.17) and therefore it is expected that the prior term removes the problems associated with all three statements in almost all cases. The formula (7.17) provides the very simple interpretation that an un- certainty parameter is used for calibration in exactly the same manner as a calibration data point. For example, if definitions of uncertainty distributions are deduced from measurements, there is no reason why they should not be used for calibration in the same way as calibration data. A calibration with the classical χ2 takes only calibration data into account, whereas calibration with the objective Φ calibrates the whole model including parameter uncertainties as well as calibration data uncertainties. The important point about a Bayesian approach for calibration is the definition of the prior distribution. If it is derived from measurements with error bars, everything is o.k. But very often priors are defined just through the experience of the modeler. So, e.g. the basement heat flow is simply known not to be below 20 mW/m 2 and never to be above 140 mW/m 2 . With the definition of a prior such knowledge is taken quantitatively into account and must now withstand critical considerations. An iterative refinement of uncertainties as feedback of risk results is not allowed in the Bayesian approach because independency of all calibration data values must be ensured. This is completely different to the classic approach where calibration error bars are usually mapped to uncertainty parameter ranges. These ranges are afterwards often taken as “obvious” limits for un- certainty distribution definitions. Distributions constructed in such a way are not allowed to be used as priors in objective functions. Nevertheless, they are often a good choice for Monte Carlo simulations in general.
- 377. 7.3 Bayesian Approach 365 7.3.1 Prior Information of Derived Parameters It is sometimes problematic to use the Bayesian approach with uncertain derived parameters. For example the shift of a whole basement heat flow in a huge basin model is taken into account with the same weight as e.g. one measured bottom hole temperature value by (7.17). The weights are only given by the size of the uncertainty but one shift of a basement heat flow, shifts many grid values. Its prior knowledge is not based on experience alone but form a variety of argumentations, consistency arguments, indirect observations, etc. So the prior information which enters the calibration is obviously larger than assumed by (7.17). For that reason it should be possible to increase the weight of such a parameter. There is no fixed rule for how much the weight should be increased. If the relative uncertainties of the calibration data and the model parameter are of the same size, then the prior term should be multiplied by approximately the number of correlated calibration data points. If it would be much less, the prior term would not affect the calibration and if it would be much bigger, the calibration data would not show significant contributions. In balance, the prior information is believed to be as important as the calibration data itself, which is a reasonable starting point in many cases. In the example which is shown in Figs. 7.21 and 7.22 the uncertainty in the heat flow shift could be reduced by about two thirds by using the Bayesian approach. 7.3.2 Correlations of Priors Correlations of priors can be directly taken into account in the Bayesian ap- proach. Instead of using only the diagonal elements of the covariance matrix in (7.17) the whole matrix Σ which is explicitly written down in the two dimensional case in (7.9) must be used: Φ = (d − f(x))T C−1 (d − f(x)) + (x − μ)T Σ−1 (x − μ) . (7.18) Here a matrix notation with Cij = Δiδik with δik = 1 for i = j and δik = 0 else was chosen. 7.3.3 Prior Information of Nominal Uncertainties Nominal distributions (Fig. 7.13) need a special treatment in the Bayesian framework. For example, one continuous uncertainty of the objective is Φ = χ2 + Φpc (7.19) with the continuous prior term Φpc = x − μ σ 2 (7.20)
- 378. 366 7 Risk Analysis Fig. 7.22. The same example as Fig. 7.21 but with Bayesian objective Φ plotted against the heat flow shift. The calibration is almost unique with a shift in the range of −2 . . . 1 mW/m2 . An extra prior weight of 19 was assumed for the heat flow shift and further rise narrows the range and moves it continuously into the direction of the master run without shift but for nominal distributions a mean μ or a variance σ does not exist by definition. In Sec. 7.2.5 association was used instead of correlation. Variance is an “auto–correlation” so it is obvious to try Φpn = n i=1 (Ni − ni)2 ni (7.21) with n defined as the number of bins of the distribution, ni = Npi with N as the number of samples, pk the probability of the bin k and Ni the number of samples in bin i. It is N i=1 pi = 1. Equation (7.21) is known to follow χ2 statistics as well as its continuous counterpart Φpc (Press et al., 2002). The prior must be calculated for one run so N = 1 and Ni = δik with k as the bin of the drawn sample. Evaluation of (7.21) yields Φpn = 1 pk − 1 . (7.22) This is a reasonable choice because the objective Φpn is decreasing with in- creasing pk similar as Φpc with σ2 . With rising uncertainty the prior becomes less important. In practice, it is possible to add a constant which does not influence the minimization procedure of the objective and use
- 379. 7.4 Deterministic Sampling 367 Φpn = 1 pk − 1 pm (7.23) with m as the index of the bin of the master run instead of (7.22). This has the advantage that Φpn = 0 for the master run. If x = μ is chosen in the continuous case then it is analogously Φpc = 0 and Φ = χ2 for the master run. 7.4 Deterministic Sampling In the previous section it was shown that the Monte Carlo method is very general. Many topics such as risking, calibration and understanding could be treated simultaneously by just analyzing the results of one Monte Carlo simulation. Occasionally, one is interested only in special questions which are not related to the topic of general risking. In these cases a random and global sampling of the space of uncertainty is often not necessary anymore and it is possible to avoid expensive simulation runs. The most extreme cases are special algorithms for highly specialized ques- tions, e.g. one is only interested in classical calibration. This can be seen as a minimization problem of one χ2 function. Special algorithms exist to optimize such a minimization (Press et al., 2002). Expensive simulations are avoided and high numerical accuracy is achieved. However, such algorithms have some serious drawbacks. First, in basin modeling high numerical accuracy is usu- ally not needed because of many uncertainties. Second, these algorithms are so specialized that expensive simulations performed for a minimization of χ2 are not reusable for a minimization of Φ, which is also often an issue. Other disadvantages are technical in nature (e.g. bad parallelization prop- erties) because many sophisticated algorithms are of primarily sequential na- ture, e.g. following a gradient downhill to the minimum. Thus one is looking for methods, which are more efficient than arbitrary Monte Carlo simulations, for the price of losing generality and which are less special than sophisticated algorithms with high numerical accuracy. Ob- viously, the targets of interest must be specified exactly before starting to search for appropriate methods. Risking is the part of Monte Carlo simulations which is most dependent on the random structure of sampling caused by the nature of probabilities. Thus one has to dispense with risking in its general form. On the other hand, one does not need to dispense with “simple” risking such as the calculation of minimum and maximum scenarios. Meaningful targets are hence “simple” risking, calibration, and “simple” understanding as far as understanding can be found without the calculation of statistical quantities. Other targets, which will be treated more explicitly in Sec. 7.5, are inter– and extrapolation techniques between different simulations for forecasting re- sults. Abdication of risking is not necessary anymore. It can be studied with forecasted models.
- 380. 368 7 Risk Analysis 7.4.1 Cubical Design The most simple uncertainty sampling design, which fulfills the conditions of the previous discussion, is simply sampling all combinations of minimum and maximum choices of all uncertainty parameters. In uncertainty space this has the form of a (hyper)cube (Fig. 7.23) and is therefore called cubical design.6 Fig. 7.23. Example of cubical design in uncertainty space with three param- eters. The bold circles depict the param- eter choices for the simulation runs Heat Flow Erosion SWI Temperat ure Cubical sampling can be used for “simple” risking, calibration, “simple” understanding, and forecasting (Sec. 7.5): the topic of “simple” risking is solved under the assumption of “non–pathological” behavior of the simulator. In such a case, minimum and maximum values of uncertainty parameters would map to minimum and maximum simulation results and thus to result ranges similar to error bars. It is clear that complicated processes such as migration cannot be treated this way. Calibration is performed analogously. Cubical design samples the uncer- tainty space regularly and so simulation results are easy to interpolate for good calibration. Understanding is improved because at least all extreme combinations are simulated. Again under the assumption of smoothness it is possible to cal- culate interaction effects out of the results (Montgomery, 2001). In general, all results can be inter– and extrapolated, e.g. linearly, which is forecasting. More about this in Sec. 7.5.1. Nevertheless, cubical design has the serious drawback that the number of simulation runs to be performed increases exponentially with the number of uncertain parameters. For that reason one is often forced to omit certain combinations. There exists a whole theory of “Design of Experiments (DOE)” treating problems such as this (Montgomery, 2001). Keywords are “fractional factorial design” for omitting special combinations of uncertainty parameters or “screening”. Screening is important because it tries to find the important 6 It is exactly cubical if the units of the uncertainties are chosen so that the mini- mum and maximum values have the same numerical value for all parameters.
- 381. 7.4 Deterministic Sampling 369 and sensitive parameters. The other parameters could be omitted, which dras- tically reduces the number of combinations. “Blocking” is another important method, which omits sensitive parameters for the better recognition of the effects of the less important parameters. Other designs such as pure cubi- cal designs are proposed, too, e.g. cubical face centered, which obviously can be very valuable. But the theory was created for engineering needs based on real experiments and not on deterministic reproducible computer simulations. Thus only basic ideas such as screening or blocking can be transferred. Simulation runs for the cubical design can be performed in parallel. The number of parallel runs is restricted to one cube. Multiple cubes themselves are evaluated sequentially if their design is iteratively refined as proposed in Montgomery (2001). Thus simulation runs can to some degree be performed in parallel but not in such a general way as for arbitrary Monte Carlo simulations. Cubical designs are very valuable for fast uncertainty analyses especially if the number of uncertainties is small. 7.4.2 Other Deterministic Designs A method that is similar to latin hypercube sampling is Sobol’ sequence of “quasi–random” numbers (Press et al., 2002). It guarantees a smoother and more homogeneous sampling than pure “pseudo–random” sampling.7 The sampling is smoothly refined by increasing numbers in the sequence and it is not influenced by extra parameters such as strip widths. Sobol’ sequence generates quasi–random numbers, which are calculated in any case with a deterministic algorithm, whereas LHC sampling is based on pseudo–random numbers.8 It can therefore be expected, that LHC sampling generates random numbers with better statistical properties.9 Additionally, an implementation of Sobol’ sequence, such as in Press et al. (2002), does not allow an arbitrary number of independent and different sampling realizations, which can be achieved easily for LHC sampling by different initialization of the random number generator. Independently created samplings can thus usually not be merged to one large with better statistical properties. In practical work this is e.g. a drawback for parallel processing or merging of different risk scenarios. Workarounds, such as sequential precalculation of the random numbers, must be performed (Bücker et al., 2008). 7 “Pseudo–random” numbers are deterministic numbers which are generated in a way that they pass statistical tests for random numbers. Therefore they are random in practice. “Quasi–random” numbers only appear to be “random”. 8 Obviously, it is even possible to combine LHC sampling with real random num- bers. 9 Better statistical properties are here defined as a larger number of passed statis- tical tests.
- 382. 370 7 Risk Analysis 7.5 Metamodels Interpolation and extrapolation of results between different simulations is fore- casting. A method which forecasts all important results is called a “meta- model” or a “surrogate” (Simpson et al., 1997). A metamodel can be used for everything that can be done with the corresponding “real model”, e.g. risking. Metamodels are very important in basin modeling because of the high simulation effort, especially the long simulation times of 3D basin models. In contrast, forecasting is usually very fast, often by a factor of more than a million. Forecasting does usually not produce the exact results but only approxi- mations. Thus metamodels are often restricted in their applicability. Highly non–linear effects such as hydrocarbon spilling can usually not be forecasted with metamodels. An overview of common metamodeling methods is given in the following. After this the usage of metamodels for calibration is discussed. Based on the high performance of response surface based metamodels, it is shown that Markov chain Monte Carlo algorithms can be applied. 7.5.1 Response Surfaces The usage of response surfaces for interpolation and extrapolation is very popular in many fields of science. Very good textbooks are available (My- ers and Montgomery, 2002; Box and Draper, 1987; Khuri and Cronell, 1996; Montgomery, 2001). The method is also becoming popular in basin modeling (Wendebourg, 2003). Response surfaces are low order multivariate polygons which are fitted with least squares regression techniques to the simulation results. Thus, for example a model f(x1, x2) with two uncertainty parameters, is typically approximated by f ≈ b0 + b1x1 + b2x2 + b11x2 1 + b22x2 2 + b12x1x2 (7.24) with bi and bik calculated from a least squares fit (Fig. 7.24). Fig. 7.24. Illustration of a response sur- face with two uncertainty parameters x1 and x2. Crosses indicate simulation re- sults for given xi e.g. temperatures for given heat flow and bulk conductivity values. Generally, they do not match the response surface exactly f x x x x x2 x1
- 383. 7.5 Metamodels 371 Therefore response surfaces are ideal for approximating smooth and con- tinuous dependencies. Discontinuities and oscillations cannot be reproduced. It is common to introduce a short hand notation for quadratic terms so that (7.24) becomes f ≈ b0 + b1x1 + b2x2 + b3x3 + b4x4 + b5x5 with x3 = x2x2, b3 = b22, . . . . In general there are k parameters xj with j = 1, . . . , k. The number k is determined by the number of uncertainties M and it is k = M for linear response surfaces or k = M(M + 3)/2 for approximations including quadratic terms. For the fit, some data values yi = fi(xi1, . . . , xik) of already performed simulations are needed. So finally a vector bT = (b0, . . . , bk) for optimization of approximation yi ≈ b0 + b1xi1 + b2xi2 + . . . + bkxik or in vector notation y ≈ Xb with Xij = xi,j−1 for j 1 and Xi1 = 1 is searched. A least squares fit results in minimization of (y − Xb)2 and evaluation yields 10 b = (XT X)−1 XT y . (7.25) A measure of goodness σg of this approximation can be evaluated by σ2 g = (y − Xb)2 N . (7.26) This simply denotes the quality of a fit by summing up the quadratic devia- tions and dividing through the number of points.11 A safer alternative, which takes outliers into account, can be defined by the maximal deviation σg = max i yi − k Xikbk . (7.27) Design forms of cubical type are very often used as the “natural” sam- pling procedure for the creation of response surfaces (Myers and Montgomery, 2002; Montgomery, 2001). Under the assumption of smooth behavior of the approximated model, it is obvious that cubical design is an effective sampling 10 Equation (7.25) is known to be highly unstable and badly conditioned in many practical examples. More robust for the solution of b is a decomposition of X into singular values and direct solution of y = Xb in appropriate subspaces (Press et al., 2002). 11 At first glance this seems to be in contradiction to unbiased estimators of variance such as σ2 = (y−Xb)2 /(N −k−1) and defined in Myers and Montgomery (2002). But this formula is linked tightly to the variation of b under random variation of X due to measurement uncertainties. This is a completely different objective.
- 384. 372 7 Risk Analysis strategy as all minimum–maximum combinations are studied. Additionally, the number of unknowns to be determined for a quadratic response surface k + 1 = M(M + 3)/2 + 1 almost matches, in cases of small numbers of uncer- tainties, M the number of simulations 2M + 1 which have to be performed:12 M 1 2 3 4 5 . . . k + 1 3 6 10 15 21 . . . 2M + 1 3 5 9 17 33 . . . Thus only an optimal small number of simulations have to be performed and good matches at the points of simulation itself, leading to small σg, are enforced. In Figs. 7.25 and 7.26 two typical diagrams of response surface models are shown. The formulas describing these isolines can easily be extracted and used for further studies. A high value of the coefficient of a cross term, e.g. x1x2, indicates an interaction between the impact of the corresponding uncertainty parameters. This is information which may help to better understand a model. In Fig. 7.26 negative values for the transformation ratio appear. This is due to the polygonal form of the method. It can therefore not be used in these regions. Generally, it often occurs that simulation results vary faster than can be approximated with simple polygons. In such cases response surface modeling is often performed only in limited regions with adapted sampling of the uncertainty space (Montgomery, 2001). Another example, which could not be treated in general by response sur- faces, are pressure calibrations via variation of permeabilities, especially if permeability is expressed in logarithmic units. Pore pressure is restricted to the lower limit by hydrostatic pressure and to lithostatic pressure at the up- per limit but a polynomial fit of response surface type is generally unlimited for infinitely increasing or decreasing uncertainty parameters. However, in a limited region of permeability variations a smooth behavior of pore pressure with good fitting response surfaces can be obtained. The creation of a metamodel is mainly determined by the solution of a linear set of equations of dimension (k+1)×(k+1) for each result point which is modeled. In case of calibration purposes this number is given by the number of calibration points N. Thus, the creation of a response surface metamodel is performed within seconds on modern computers because in practice mostly M 10 and N 1000. The calculation of response surface metamodel results is an evaluation of simple polygons and this is extremely fast. In most computer applications this appears to be almost instantaneous. 7.5.2 Fast Thermal Simulation Fast thermal simulation is a special method of fast heat flow analysis (Nielsen, 2001). It is based on the approximative linear form of the partial differential 12 All combinations plus master run.
- 385. 7.5 Metamodels 373 Fig. 7.25. Response surface isolines for temperature in Celsius in a source rock. Dependent on heat flow and SWI tem- perature variations the isolines are lin- ear as expected Fig. 7.26. Response surface isolines for transformation ratio in [%] at same point as in Fig. 7.25. Negative values indicate a region where the response surface method cannot be used equation ρc ∂T ∂t − ∇ · (λ · ∇T) = Q (7.28) of heat flow. Here T is the temperature, λ are the thermal conductivities, t is the time, ρ the density, c the specific heat capacity, and Q are external heat sources. For a unique compilation of heat flow analysis, boundary and initial conditions must be specified. At the top of the basin usually the temperature is given, at the sides, a condition of prohibited horizontal heat flow is applied and at the bottom, basement heat flows are specified: T = TSWI on top, ∇T = 0 at the sides, −λ · ∇T = q at the bottom, T|t=initialtime = T0 at initial time (7.29) with TSWI the “Sediment Water Interface” temperature at top of the basin, q the basement heat flow, and T0 the temperature profile at initial time. If λ, ρ, and c are assumed to be smooth and weakly temperature– dependent then (7.28) is almost linear. This property can be utilized: Firstly, one should take a look at the following boundary value problem: ρc ∂T̃ ∂t − ∇ · (λ · ∇T̃) = 0 with T̃ = 0 on top, ∇T̃ = 0 at the sides −λ · ∇T̃ = q̃ at the bottom and T|t=initialtime = 0 at initial time. (7.30)
- 386. 374 7 Risk Analysis A temperature profile T +xqT̃ with T as the solution of (7.28) with boundary and initial conditions (7.29) and T̃, a solution of (7.30) is a solution of (7.28) with boundary and initial conditions such as (7.29), which must only be mod- ified at the bottom by −λ · ∇T = q + xqq̃. Herein, xq is just an arbitrary number, which can be interpreted as a derived uncertainty parameter for a variation of form q̃. The important point is that with only two solutions T and T̃ one can construct multiple solutions T + xqT̃ for heat flow variations q + xqq̃ for any value xq just by linear combination. The form of xqq̃ defines the space of possible heat flow variations. Obviously, multiple variations xq,iq̃i can be combined just by summing up the solutions xq,iT̃i. Hence it is possible to quickly create flexible variations of the original heat flow and temperature pattern. It has been proposed by Nielsen (2001) to vary the heat flow below each of the four model corners.13 Each paleo–heat flow map is calculated by inter- polation with two dimensional form functions analogously to (8.22). A corner variation with its shape function states one variation q̃i with i = 1, . . . , 4 for all four corners. The sum of all four corner variations describes tilting and twisting variations of the original heat flow distribution. The method should not be applied in cases when mismatches to the measured data require heat flow shifts very differently in many locations, e.g. from well to well. For such cases methods as described in Sec. 3.9 are more advantageous. Additionally it must be noted, that q̃ can vary independently to q through time. Hence heat flow variations in time can be incorporated. For example, it is possible to shift the heat flow of each of the four corner points linearely with time. The set T and T̃ can be interpreted as a metamodel for forecasting mani- fold heat flow histories. A response surface which is created as an interpolation of two different solutions T1 and T2 of (7.28) with the boundary conditions (7.29) once with bottom heat flow q and the other with q+q̃, yields almost the same results as the fast thermal simulation caused by the linearity of the differential equation system. The main difference between both methods comes from the fact, that the differential equation is usually not exactly linear. Parameters such as the thermal conductivity are typically weak but non–linearly temperature dependent. The response surface can then be interpreted rather as a “secant approach”, whereas the fast thermal simulation is following more a “tangent” (Fig. 7.27). Response surfaces can incorporate smooth non–linearities to some degree of accuracy with their quadratic terms (7.24). On the other hand, as they are caused by the linear regression, they do not need to match the simulation results from which they were created exactly, whereas fast thermal 13 It is assumed here that the model has a rectangular base area. Generally, it does not matter if the corner points are not inside of the model.
- 387. 7.5 Metamodels 375 simulations reproduce, at least in a region of small variation, the original models exactly. Fig. 7.27. Illustration of the differences between response surface and fast ther- mal simulations. The curvature of tem- perature, which comes from the nonlin- earities is exaggerated for demonstra- tion. Due to quadratic terms the re- sponse surface is able to approximate non–linearities whereas the fast thermal simulation approximation is restricted to a straight “tangential” line. The re- sponse surface does not in general need to match the temperature exactly at any point whereas the fast thermal simula- tion matches at T = T1 Simulated Temperature Response Surface Fast Thermal Approximation T2 T =T1 T q q ~ ~ Heat Flow Temperature The calculation of T and T̃ needs about the same effort. The evaluation of a forecast is just the evaluation of T +xqT̃ and thus can be performed almost instantaneously. Thus the effort and the needed resources for the creation of the metamodel as well as the evaluation performance of the fast thermal simulation and the response surface are almost the same. Fast thermal simulations are not limited to heat flow variations only but can also be applied to thermal conductivity variations. This is achieved by introducing a derived uncertainty parameter xλ for the variation of thermal conductivity according to λ + xλλ̂ with a λ̂ describing the form of the varia- tion. The solution T̂ of the differential equation ∇ · (λ · ∇T̂) = −∇ · (λ̂ · ∇T) (7.31) with boundary conditions such as (7.29) but with q = 0 can be added to the solution T in the same manner as T̃ to construct valid heat flow histories for thermal conductivity variations. Here, the fast thermal simulation is restricted to “small” variations in conductivity because quadratic terms are neglected in the deviation of (7.31). 7.5.3 Kriging Kriging is another method for interpolation and extrapolation in multi dimen- sional spaces. It is based on the minimization of statistical correlations and derived as the best linear unbiased estimator. Originally it was developed for spatial inter– and extrapolation only but it can also be applied to abstract uncertainty spaces. Various different methods of kriging exist. To the authors
- 388. 376 7 Risk Analysis knowledge it has not been applied up to now in any case as a metamodel in basin modeling and thus we refer only to the literature of geostatistics (Davis, 2002). However, it can be expected that kriging might yield good results in many cases especially when simple functions such as (7.24) are not appropriate at all for the description of the model or process of interest. 7.5.4 Neural Networks Neural Networks can be interpreted as metamodels. Neural Networks must be trained. They learn. Three classes of learning are usually distinguished (Zell, 1997): • supervised learning • reinforcement learning • unsupervised learning Supervised learning is based on comparison with correct results. These results are simulation results in basin modeling. Supervised learning is usually the fastest way of learning. Nevertheless many expensive simulation runs must be performed for this way of learning. Reinforcement learning is based on reduced feedback. The network is taught only with information about the correctness of its output but not the correct result itself. Therefore reinforced learning networks need even more training than supervised learning networks. Although the amount of feedback data is small it, too, must be available. This means that many expensive simulation runs must be performed for this method. Unsupervised learning is performed without feedback. The network should learn by classifications in its own right e.g. by “self organization”. Caused by the complexity of a typical basin model, it is expected that unsupervised learning neural networks will be improper for result predictions. The high effort for learning leads to the conjecture that neural networks are not the best alternative for basin metamodeling. 7.5.5 Other Methods for Metamodeling Methods such as rule based expert systems or decision trees are obviously limited in their applicability for forecasting. Under special circumstances they can be interpreted as metamodels but not in general. Other special techniques are based on analysis in frequency space. Due to the complex geometry in geology, these methods cannot usually be applied to basin modeling. 7.5.6 Calibration with Markov Chain Monte Carlo Series The Markov chain Monte Carlo (MCMC) method is designed for the sampling of multi–variate probability distributions (Neal, 1993; Besag, 2000).
- 389. 7.5 Metamodels 377 Calibration is a search of high probability regions where data fits the model which is not trivial in high dimensional spaces. So MCMC can be misused to find the regions of calibration. Additionally, the sampling allows the error bars of measurement values to be mapped to uncertainty distributions.14 The whole subject of search and mapping is called “inversion” and thus MCMC is also a method for inversion. Several different algorithms for MCMC exist, which can be shown to be directly related (Neal, 1993; Besag, 2000). This section is restricted to the classical Metropolis algorithm. It basically works as follows: According to a distribution, with some special properties which are of no interest here, random jumps are performed in uncertainty space. If the probability density of the distribution, which should be sampled increases, the jump is accepted. If the density decreases it can be rejected or accepted by a special criterion with a random level of acceptance. This ensures that MCMC also samples low probability regions but it focuses primarily on the highly probable regions. In Fig. 7.28 such a MCMC “random walk” is illustrated. Sampling can only be performed on a small fraction of jumps typically every 100th or less to ensure independency of the samples. Fig. 7.28. Illustration of Markov chain Monte Carlo sampling with a random walk. Isolines indicate the probability density in the x1 – x2 uncertainty diagram. The “ran- dom walk” is attracted by the region of high probability x2 x1 Obviously, it is clear that MCMC is not very efficient in sampling because most of the jumps which are model results are ignored for sampling, some of the jumps are rejected and additionally, the random walks can become very long before reaching their high probability calibration targets. Proper MCMC sampling can become a delicate choice of the start point and jump width. Therefore, in basin modeling MCMC is only usable with fast metamodels such as response surfaces or fast thermal simulations. On the other hand, in theory MCMC guarantees to find the regions of interest. 14 These distributions are not allowed to be used in a Bayesian approach, see Sec. 7.3.
- 390. 378 7 Risk Analysis Summary: Models are usually constructed on the basis of uncertain data. These uncertainties cause additional tasks during comprehensive model anal- ysis. Firstly, modeled results must be classified according to their probability. For example, confidence intervals of special output scenarios should be spec- ified or even more concrete a risk of failure must be quantified. Secondly, the behavior of model results with the variation of uncertain parameters should be understood. Which parameter affects which part of the result? Finally, uncertainties should be reduced by comparison with additional calibration data. The three tasks are “risking”, “understanding”, and “calibration”. Obviously, all three tasks can be studied with multiple simulation runs. Uncertain parameters must therefore be varied according to their range of uncertainty. Monte Carlo simulations are an effective method of treating all three tasks simultaneously. Multiple simulation runs with randomly drawn uncertainty parameters, according to their probability of occurrence, are per- formed. Another advantage of the approach is the possibility of unrestricted parallel processing. This is especially valuable because simulation runs are often very time consuming and therefore expensive. The method can fur- ther be optimized with latin hypercube sampling, which avoids clustering of parameter combinations. A model can be calibrated in two different ways, with and without consid- eration of information which describes data uncertainties of the model, e.g. limits or ranges of an uncertain input parameter. This “prior” information is taken into account in the Bayesian approach. Ambiguous and geologically meaningless calibrations can be avoided with this approach. Simulation runs are very time consuming. Response surfaces are a method for fast interpolation between simulation results. Other methods for rapid result prediction, such as the fast thermal simulation, are also discussed. Particularly with regard to heat flow problems, response surfaces and fast thermal simulations can be used efficiently for calibration. A very robust algorithm concerning inversion is the Markov Chain Monte Carlo (MCMC) sampling, which in principal guarantees the best possible calibration due to random jumps in the uncertainty space. References J. Besag. Markov Chain Monte Carlo for Statistical Inference. Working paper, Center for Statistics and the Social Sciences, University of Washington, 2000. O. Beyer, H. Hackel, V. Pieper, and J. Tiedge. Wahrscheilichkeitsrechnung und Statistik. B. G. Teubner Stuttgart Leipzig, 8th edition, 1999. G. E. P. Box and N. R. Draper. Empirical Model–Building and Response Surfaces. John Wiley Sons, Inc., 1987.
- 391. REFERENCES 379 H. M. Bücker, A. I. Kauerauf, and A. Rasch. A smooth transition from serial to parallel processing in the industrial petroleum system modeling package petromod. Computers Geosciences, 34:1473–1479, 2008. J. C. Davis. Statistics and Data Analysis in Geology. John Wiley Sons, 3rd edition, 2002. L. Fahrmeir and A. Hamerle, editors. Multivariate statistische Verfahren. Walter de Gruyter Co., 1984. E. T. Jaynes. Probability Theory: The Logic of Science. Cambridge University Press, 2003. A. I. Khuri and J. A. Cronell. Response Surfaces: Designs and Analyses. Marcel Dekker, Inc., second edition, 1996. I. Lerche. Geological Risk and Uncertainty in Oil Exploration. Academic Press, 1997. J. O. Miller. Bivar: A program for generating correlated random numbers. Behavior Research Methods, Instruments Computers, 30:720–723, 1998. D. C. Montgomery. Design and Analysis of Experiments. John Wiley Sons, Inc., 5th edition, 2001. R. H. Myers and D. C. Montgomery. Response Surface Methodology: Process and Product Optimization Using Designed Experiments. John Wiley Sons, Inc., second edition, 2002. R. M. Neal. Probabilistic Inference Using Markov Chain Monte Carlo Meth- ods. Technical Report CRG–TR–93–1, Department of Computer Science, University of Toronto, 1993. P. Newendorp and J. Schuyler. Decision Analysis for Petroleum Exploration. Planning Press, second edition, 2000. Søren Nielsen. Århus University, Denmark, Private communication, 2001. W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery. Numerical Recipes in C++. Cambridge University Press, second edition, 2002. H. Rinne. Taschenbuch der Statistik. Verlag Harri Deutsch, second edition, 1997. C. P. Robert. The Bayesian Choice. Springer–Verlag New York, Inc., second edition, 2001. T. W. Simpson, J. D. Peplinski, P. N. Koch, and J. K. Allen. On the use of statistics in design and the implications for deterministic computer ex- periments. In Proceedings of DETC. ASME Design Engineering Technical Conferences, 1997. M. R. Spiegel and L. J. Stephens. Schaum’s Outline of Theory and Problems of Statistics. The McGraw–Hill Companies Inc., New York, 1999. R. O. Thomsen. Aspects of applied basin modelling: sensitivity analysis and scientific risk. In S. J. Düppenbecker and J. E. Iliffe, editors, Basin Mod- elling: Practice and Progress, number 141 in Special Publication, pages 209–221. Geological Society of London, 1998. J. Wendebourg. Uncertainty of petroleum generation using methods of experi- mental design and response surface modeling: Application to the Gippsland
- 392. 380 7 Risk Analysis Basin, Australia. In S. Düppenbecker and R. Marzi, editors, Multidimen- sional Basin Modeling, volume 7 of AAPG/Datapages Discovery Series, pages 295–307, 2003. A. Zell. Simulation neuronaler Netze. R. Oldenburg Verlag München Wien, 1997.
- 393. 8 Mathematical Methods 8.1 Introduction Basin modeling is a framework of adapted geological and physical models. An overall implementation contains a wide range of algorithms and methods each of them appropriate for each “submodel”. A detailed discussion of all ap- proaches goes beyond the scope of this volume. Many algorithms are standard in other fields, such as statistics or computer science. In these cases additional information can be found in the cited literature. An excellent general overview over numerical methods is given by Press et al. (2002). Adapted or special al- gorithms of basin modeling, which are important for understanding and which are not too comprehensive, are outlined together with the fundamental theory in the previous chapters. From the viewpoint of numerics differential equations constitute the largest class of specific problems in basin modeling. They are fundamental for basin modeling, complex and costly to solve. Therefore they play a central role in all respects. Temperature and pressure fields, which are here denoted with u, are modeled with parabolic diffusion equations in the form of ∂tu − ∇ · λ · ∇u = q . (8.1) Hence this chapter focuses on the methods necessary for the solution of such equations in basin modeling. Mathematical equations, such as (8.1), are formulated in terms of phys- ical quantities, such as temperature or stress. These quantities have some important basic properties, which are shortly summarized in Sec. 8.2. Very often physical quantities are bulk values. Macroscopic averages between differ- ent rock and fluid types must be specified. Typical examples are presented in Sec. 8.3. Multi–dimensional differential equations with complicated geometries are usually solved with the finite element method whereas finite differences are often used in special cases of approximately one–dimensional problems, such as simplified crustal layer models. Finite differences are more elementary T. Hantschel, A.I. Kauerauf, Fundamentals of Basin and Petroleum 381 Systems Modeling, DOI 10.1007/978-3-540-72318-9 8, © Springer-Verlag Berlin Heidelberg 2009
- 394. 382 8 Mathematical Methods and therefore discussed first in Sec. 8.4. Afterwards Sec. 8.5 deals with the finite element method. Control volumes must be mentioned because they are sometimes an alternative especially for flow and pressure modeling. They are discussed in Sec. 8.6. All these methods map the differential equations to a system of linear algebraic equations. Necessary for its solution are solvers, which are briefly summarized in Sec. 8.7. High performance modeling can be obtained with parallelization and is discussed in the Sec. 8.8. Finally, some approaches of local grid refinement (LGR) are discussed in Sec. 8.9. 8.2 Physical Quantities Many physical quantities are functions in space and geological time, e.g. tem- perature T(x, t), pressure p(x, t) or flow velocity v(x, t). They are scalars, vectors or tensors (Fig. 8.1). Scalars are undirected, that means they are rep- resented by just one single value, such as temperature or pressure at a given location x, t in space and time. A vector has both, size and direction. Thus three independent numbers are necessary to describe a vector at a given loca- tion, e.g. a vertical and two horizontal components for a water flow velocity v = ⎛ ⎝ vx vy vz ⎞ ⎠ = (vx, vy, vz)T . (8.2) The index T indicates here a transposition of rows and columns. 40 C o 30 C o 25 C o 60 C o 40 C o 30 C o 80 C o 60 C o 40 C o s Scalar Vector Tensor T v a) b) Fig. 8.1. (a) Temperature, velocity and stress are examples for scalars, vectors, and tensors. (b) A field for temperature with temperature gradients A tensor σ has a different size in each direction and is therefore char- acterized by three vectors. It consists of normal components σii and shear components σij for i = j: σ = ⎛ ⎝ σij ⎞ ⎠ = ⎛ ⎝ σxx σyx σzx σxy σyy σzy σxz σyz σzz ⎞ ⎠ . (8.3)
- 395. 8.2 Physical Quantities 383 Physical quantities, such as the stress tensor, are usually described by sym- metrical tensors with σij = σji. The number of independent values of a sym- metrical tensor is six. These six values can be represented by three orthogonal main directions and corresponding sizes. For example, the tensorial character of a physical quantity can be described with normal and tangential vectors acting at the surface of a volume element. There always exist three perpen- dicular planes, where these vectors are oriented in normal direction of the corresponding surfaces. These surfaces normals are the main directions of the tensor and characterize the tensor similar as the direction characterizes a vec- tor. Corresponding sizes of a tensor are called principal values or eigenvalues λk and can be calculated as the solutions of σij − λkδij = 0 (8.4) with the unit tensor δij = 1 for i = j and δij = 0 else. The principal values λk are usually sorted according to size and renamed to σk with σ1 σ2 σ3. The main directions of many tensors are parallel and perpendicular to the layer directions in geological models. Principal values in horizontal direction are usually of same size and thus the two different principal values are often named σh and σv. For example, thermal conductivities and permeabilities are tensors. In principle they contain direction dependent components, although they are determined in practice by only two values along and across the layering. The terms horizontal and vertical conductivities are used although the values rep- resent layer directions and not fixed space axes. The anisotropy factor is the ratio between the horizontal and the vertical component. Other characteristics of a tensor are the invariants I1 = σxx + σyy + σzz I2 = −σxxσyy − σxxσzz − σyyσzz + σ2 xy + σ2 xz + σ2 yz I3 = σxxσyyσzz + 2σxyσxzσyz − σxxσyz − σyyσxz − σzzσyz , (8.5) which are independent of the choice of the coordinate system. The first in- variant I1 represents the total value of the normal components. The average value of the normal components is defined as σ̄ = I1/3. Any tensor can be represented as the sum of a pure normal and a deviatoric part sij according to σij = σ̄ δij + sij . (8.6) The invariants of the deviatoric tensor are often called J1, J2, J3 with J1 = 0. Deviatoric invariants of the stress tensor are important for the determination of failure criteria, since these are related to distortion. The value change per distance of a quantity into a specified direction (over an infinitesimal distance) is called its derivative. A gradient is a vector with partial derivatives of a scalar quantity in x, y, and z – direction as components.
- 396. 384 8 Mathematical Methods It points into the steepest upward direction (Fig. 8.1) and can be calculated as grad T = ∇T = ∂T ∂x , ∂T ∂y , ∂T ∂z T . (8.7) The (scalar) size of a gradient can be calculated as a derivative in steepest upward direction. For example, the main direction of temperature change is approximately vertically downward. A temperature change per distance into downward direction is therefore often called the vertical temperature gradient. Note, that flow vectors, such as heat or fluid flow, point typically in down- ward and not in upward direction of the corresponding temperature or pres- sure field. They are thus proportional to the negative gradient. 8.3 Mixing Rules and Upscaling Most bulk property values can approximately be derived from single min- eral and fluid component values, such as defined in App. A, E, I, by suitable mixing and upscaling rules. Such component based properties are thermal con- ductivities, heat capacities or densities. The mixed rock matrix and pore fluid values are usually calculated first. The corresponding properties of the miner- als, lithologies, organic compounds, and the fluid phase values are taken into account, respectively. Then the bulk value is derived from the two averages of the rock and pore fluid (Fig. 8.2). Gas Oil Water Minerals Lithologies Rocks Pore Fluid Bulk Value Rock Value Fluid Phases e.g. Limestone, Clay Content Mixing Mixing Upscaling Upscaled Bulk Value Mixing Fig. 8.2. Mixing and Upscaling Other properties such as permeabilities, compressibilities, and capillary entry pressures are rather defined as porosity dependent functions instead of component based mixtures. Fluid properties need not to be taken into account. However, mixing rules are still needed when the rock matrix consists of various parts of different lithologies. Generally, mixing and upscaling can become very complex and dependent on details of the specific system and quantities which are mixed. For example, thermal conductivities should be mixed with a sophisticated procedure which incorporates the Bunterbarth formula, the principle of constant mean con- ductivity, and the variation of anisotropy with porosity (Chap. 3). However,
- 397. 8.3 Mixing Rules and Upscaling 385 there are three basic methods for mixing and upscaling, namely arithmetic, harmonic, and geometric averaging, which are often used when detailed in- formation is missing. If necessary, the accuracy can often be improved by additional or modified rules for specific parameters or special arrangements of the components. The arithmetic average of a property value λ is the sum of the component property values λi weighted with the mass or volume fraction pi according to λ = n i=1 pi λi with n i=1 pi = 1 . (8.8) The geometric mean averages the order of magnitude. Therefore, it is also the arithmetic average of the log–values with λ = n ! i=1 (λi)pi or log λ = n i=1 pi log λi . (8.9) The harmonic mean averages the inverse values according to 1 λ = n i=1 pi λi . (8.10) A modified rule is the square–root mean which arithmetically averages the square root of the component values √ λ = n i=1 pi λi (8.11) and increases the importance of small fractions pi. This rule is recommended for thermal conductivities (Beardsmore and Cull, 2001). Generally, it can mathematically be proven that arithmetic averages are larger than geometric averages which are again larger than harmonic averages (Aigner and Ziegler, 2004). The square–root mean is usually found between arithmetic and geometric mean. The difference between the mean values in- creases with larger variance of the single values (Fig. 8.3). The selection of the appropriate averaging method is less important for property values which are spread over a small interval. In case of fluid phase mixing the fraction values of the mixing equation are saturations, in case of rock mixing they are mineral fractions, and when pore and rock values are mixed together they are porosity φ and rock volume fraction 1 − φ. In table Table 8.1 typical mixing methods for different quantities are listed. The quantities are often distinguished between extensive, conductiv- ity or transport parameters, and critical or threshold properties. Sometimes the list is extended for intensive quantities. Each of them usually requires different methods for mixing and upscaling.
- 398. 386 8 Mathematical Methods 0 0.2 0.4 0.6 0.8 1 1 10 Component Fraction Mean 1..Arithmetic 2..Square Root 3..Geometric 4..Harmonic 0 0.2 0.4 0.6 0.8 1 1 2 Component Fraction Mean 0 0.2 0.4 0.6 0.8 1 1 1.1 Component Fraction Mean a) b) c) 2 1 3 4 Fig. 8.3. Mean values of two components versus fractions. The first one component value is equal to 1 while the second value is 10, 2, and 1.1 respectively Example Mixing Extensive Density A Quantity Heat Capacity Porosity Clay Content Radioactive Heat Specific Surface Area Transport Thermal Conductivity Fluids: G Parameter Permeability Rocks: GAH, S Bulk: GAH Threshold Critical Oil Saturation A Property Critical Gas Saturation A Connate Water Saturation A Capillary Entry Pressure G or M Fracture Limit A Compressibility A or M Table 8.1. Arithmetic (A), geometric (G), harmonic (H), and maximum based (M) or special (S) mixing rules, such as square root mixing, are used for different variables and mixture types. GAH indicates that the geometrical average is used for a homogeneous structure and the arithmetic and harmonic average is used for vertical and horizontal components of a layered structure Extensive properties affect the bulk value simply proportional to their abundance and should obviously be mixed with arithmetic averages. Exam- ples are heat capacity, density, and porosity. Exceptions are for example com- pressibility values for vertical compaction when the less compactable rock components consist of vertically well connected columns. Then compressibil- ity behaves similar as a threshold property and the minimum compressibility value should be used. Transport parameters, such as thermal conductivity, are calculated with different mixture and upscaling rules dependent whether and how the compo-
- 399. 8.4 Finite Differences 387 nents have been layered or arranged (Fig. 8.4). In case of straight layering the mixed parameter is equal to the arithmetic mean for the flow along and to the harmonic mean for the flow perpendicular to the layering direction. For mixed domains the mean is somewhere between arithmetic and harmonic mean. Usu- ally the geometric average is taken. In case of porosity dependent permeability curves each curve point has to be mixed separately. Kozeny–Carman relations are based on special parameters (Sec. 2.2.3). R1 R2 R2 R1 a) Layered Along Flow c) Layered Across Flow b) Not Layered Arithmetic Geometric Harmonic Fig. 8.4. Averaging types for conductivity values. The corresponding resistance R in an electrical circuit is calculated in the case a) from parallel and in the case c) from sequential placed resistances R1 and R2 Threshold properties such as the capillary entry pressure often require special considerations for mixing and upscaling based e.g. on the flow and saturation pattern. Mixing and upscaling are discussed in the chapters where the properties are introduced. Due to fractal flow patterns, upscaling must often be treated differently to mixing with special methods. Without known correlations, fluid properties of mixtures of fluids often do not behave proportional to the abundance of their components. These quantities are often more of intensive nature and are therefore often mixed with the geometric mean. An exception are fluid densities, which are mixed arithmetically. However, phase properties can be derived from the chemical components using the properties of the pure substances. This is especially needed for fluid analysis with the corresponding flash calculations (Sec. 5). Special rules such as Lee–Kesseler mixing (5.6) have a proven track record. 8.4 Finite Differences The numerical solution of a partial differential equation can only be specified and evaluated for a limited set of discrete points. Hence the first step is the
- 400. 388 8 Mathematical Methods discretization of space and time. For the finite difference method the gridding is usually chosen in such a way that expressions containing derivatives are approximated by ∂u ∂x ≈ Δu Δx (8.12) with Δu, Δx determining the difference between u and x at adjacent grid- points. An appropriate discretization fulfills the conditions of orthogonal grid directions, which ensures the independency of directions and “small” distances Δ in space and time in consistency with approximation (8.12). Sometimes higher order terms in Δ are taken into account by following a Taylor expan- sion. Dy Dx u1,2 u1,1 u2,1 x0 x1 y1 y0 Dy Dx ui ui,r ui,b ui,l ui,t Control Volumes Finite Differences Fig. 8.5. Cutouts of two–dimensional finite difference and control volume grids A typical two dimensional finite difference grid is depicted in Fig. 8.5. Here, the discretization in space for the diffusion term of (8.1) with constant λ works out to be ∂2 u ∂x2 + ∂2 u ∂y2 ≈ ui+1,k − 2ui,k + ui−1,k (Δx)2 + ui,k+1 − 2ui,k + ui,k−1 (Δy)2 (8.13) with ui,k as the solution at the gridpoint at xi and yk. The discretization in time can be performed independently with the same approach (8.12) and yields un+1 − un Δt = tn+1 tn dt(∇ · λ · ∇u + q) = η(∇ · λ · ∇un+1 + qn+1 ) + (1 − η)(∇ · λ · ∇un + qn ) Δt (8.14) with Δt = tn+1 − tn , discrete time steps tn , solutions un at these time steps, and 0 ≤ η ≤ 1 according to the midpoint rule of integration. The integral can now be approximated with predefined values of η. The choice η = 0 is called the “explicit scheme”. It has the advantage that un+1 can be evaluated
- 401. 8.5 Finite Element Method 389 through time from un explicitly without the inversion of a system of linear equations. But it can be shown that explicit schemes are only stable for very small Δt (Patankar, 1980; Press et al., 2002). The choice η = 1 is called the “fully implicit scheme”, which can be shown to be unconditionally stable. But it has the disadvantage that a system of linear equations has to be inverted for the calculation of un+1 from un . Additionally, it is only accurate to first order in Δt. Alternatively it is possible to choose η = 1/2, which is called “Crank–Nicolson scheme”. This is also stable with higher accuracy. But it is more complicated and it might generate physically unrealistic solutions, which have to be corrected (Patankar, 1980; Press et al., 2002). The accuracy which can be reached with an adapted Crank–Nicolson scheme goes beyond the common accuracy of basin models. Crank–Nicolson schemes conserve small scale features in the solution of differential equations. Due to upscaling approximations in model building it can be assumed that these features cannot be modeled correctly anyway. Therefore it is the au- thors’ opinion that the fully implicit scheme is sufficient in most cases of basin modeling. The finite differences method is always used for the discretization in time but rarely in space. It has big disadvantages concerning irregular spatial grids and discontinuities because the fields u, λ and q are assumed to vary smoothly from grid point to grid point. For example it is problematic to mimic perme- ability jumps over orders of magnitude at sharp edges of layer boundaries even if they are following exactly the grid directions. For that reason more sophisticated methods, such as finite elements or control volumes, must be used. 8.5 Finite Element Method The finite element (FE) method was first developed for mechanical engi- neering purposes. Many good textbooks are available, e.g. Schwarz (1991) or Zienkiewicz (1984). Unlike the finite difference method the finite elements method is not re- stricted to rectangular grid cells only. The grid cells are now called finite elements or just elements and the gridpoints belonging to one element nodes. Finite elements are very flexible for gridding irregular geometries. They can for example easily be used for compaction processes with changing layer thick- nesses. In Fig. 8.6 a typical finite element grid is depicted. In finite element approximation fields, such as pressure or temperature, are approximated by mathematical rather simple but unique and continuous differentiable functions inside the elements. These so called “form” or “shape functions” usually obey some simplified continuity conditions at the element interfaces and it is necessary that they are zero at all nodes except the one which they are related to. In practice low order multi–variate polygons, typi- cally up to quadratic order, are widely used.
- 402. 390 8 Mathematical Methods Fig. 8.6. Zoomed cutout of a two dimensional finite element grid. The annotation on the right side shows layer names In total the finite element approach can be written as u(x) ≈ n k=1 ukNk(x) (8.15) with n as the total number of gridpoints and x = (x, y, z)T .1 It is Nk(x) = 0 outside of the elements containing the node k. Inside Nk(x) can be written as a sum over these elements Nk(x) = e(k) Ne k (x) . (8.16) The functions Ne i are the shape functions. Substitution of (8.16) in (8.15) and changing the order of summation yields u(x) ≈ e pe i=1 uiNe i (x) . (8.17) The first sum is over all elements e, pe is the number of nodes of element e and i a numbering of nodes inside of each element. The shape functions 1 The discretization of time is omitted for simplicity of the description. It can be performed the way outlined in the last section.
- 403. 8.5 Finite Element Method 391 must obey Ne i = 0 only in element e and Ne i (xj, yj, zj) = 1 for i = j and Ne i (xj, yj, zj) = 0 for i = j. This ensures that for an evaluation of u at a given location x only the grid values ui of the element surrounding x are needed. The uk must now be specified so, that (8.15) and (8.17) become numer- ically good approximations. Galerkin proposed the following method: The approximation (8.15) is used in the original differential Equation (8.1). Then the equation is multiplied from the left side with Nk and integrated over the whole volume Ω of the boundary value problem. The approximation (8.15) is not continuously differentiable. Hence second derivatives of (8.1) cannot be evaluated but this problem can be bypassed. Under consideration of bound- ary values it is possible to evaluate the integral with the method of partial integration and Gauss’ integral formula. The resulting equations contain only first order derivatives (Schwarz, 1991). In total, a linear set of equations for the uk is generated. In practice it is common not to evaluate integrals which have been created by multiplication of (8.1) with Nk but with the shape functions Ne k . Then integrals of type pe i=1 Ω uiNe k (∂tNe i − ∇ · λe · ∇Ne i − qe Ne i ) dV (8.18) must be evaluated. The sum over the elements was already been skipped be- cause the functions Ne i and Ne k are zero outside of element e. The conductivity coefficients λ and the source term q are assumed to be constant in element e with values λe and qe . A partial integration yields now expressions in the form of pe i=1 ∂tui * Ω Ne k Ne i dV + ui * Ω ∇Ne i · λe · ∇Ne k dV −te * ∂Ω Ne k Ne i dS − qe * Ω Ne k Ne i dV (8.19) Here ∂Ω is the border of the volume Ω. The value te represents a Neumann boundary condition of form n · λ · ∇u = t (8.20) at the border of element e with normal vector n. If there is no Neumann boundary condition the term can be skipped. Alternatively there might be a Dirichlet boundary condition with given v of form u = v at the border of element e, which can easily be represented by setting ui = ve i at location i. According to (8.15) – (8.17) expressions of form (8.19) can be summed up and set to zero. So it is possible to construct a set of linear equations of form n i=1 (Akiui + ∂tui) = fk . (8.21)
- 404. 392 8 Mathematical Methods with fk containing all the terms with te , ve , and qe and matrix elements Aki containing the terms coming from the integrals over the derivatives of the shape functions. a b a b x z y x h z 1 2 3 4 5 6 7 8 Fig. 8.7. Finite element with rectangular x-y projection on the left and cubical finite element in a ξ–η–ζ coordinate system on the right The specifications of element types and appropriate shape functions used in basin modeling have not been completed for now. Sedimentary basins usu- ally have a lateral extension which is at least one order of magnitude larger than their thicknesses. Most quantities, such as temperature or pressure, vary strongly in depth but only smoothly in lateral directions. Thus for numerical reasons gridding in depth should not be mixed with gridding in lateral di- rections.2 This argumentation implies that gridpoints should be arranged in columns of vertical direction. The finite element method allows variable dis- tances of the gridpoints in depth direction. Hence the most simple choice are cuboid elements of hexahedron type with the nodes at the corners determining layer interfaces Fig. 8.7. In practice data is often available in form of regularly gridded depth maps. So hexahedrons, which are rectangular in top to bottom view, are not only optimal from a numerical point of view but also for the availability of data. The shape functions for a cube can explicitly be denoted. It is common to use a notation with “normalized coordinates” ξ = (ξ, η, ζ)T instead of x = (x, y, z)T in space. Hence for a cube with nodes at the corner points at ξ = ±1, η = ±1, and ζ = ±1 a set of simple shape functions is given by 2 Small variations are mapped in form of small contributions to the matrix ele- ments and large effects in form of large contributions. Small contributions have numerically a more stable impact if they are separated from large contributions. In many cases this can be achieved by separation of directions.
- 405. 8.5 Finite Element Method 393 Ne 1 = 1 8 (1 + ξ)(1 − η)(1 − ζ), Ne 2 = 1 8 (1 + ξ)(1 + η)(1 − ζ), Ne 3 = 1 8 (1 − ξ)(1 + η)(1 − ζ), Ne 4 = 1 8 (1 − ξ)(1 − η)(1 − ζ), Ne 5 = 1 8 (1 + ξ)(1 − η)(1 + ζ), Ne 6 = 1 8 (1 + ξ)(1 + η)(1 + ζ), Ne 7 = 1 8 (1 − ξ)(1 + η)(1 + ζ), Ne 8 = 1 8 (1 − ξ)(1 − η)(1 + ζ) (8.22) with a node numbering according to Fig. 8.7.These shape functions are linear in ξ, η, and ζ at the edges and therefore called “tri–linear”. The shape func- tions of a general hexahedron can only be specified implicitly. Assuming that the corner points are located at the positions xi = (xi, yi, zi)T it is possible to state a coordinate transformation from x to ξ in such a way, that (8.22) can be used for the evaluation of the shape functions: x = 8 i=1 Ne i (ξ)xi . (8.23) Note, that the shape functions are used again for the transformation. It is easy to show that for an element with rectangular projection in the x–y plane and corner points at xi = aξi/2 and yi = bηi/2 the transformation simplifies to x = a 2 ξ, y = b 2 η, z = 8 i=1 Ne i (ξ, η, ζ)zi . (8.24) It is possible now to explicitly invert (8.24). With knowledge of ξ and η from x and y one yields a linear equation for ζ. Hence shape functions can also be explicitly specified but the expressions become lengthy. In a computer program an inversion can be processed almost effortlessly. The calculation of integrals, which are occurring in expressions such as (8.19), can now be drastically simplified. Using the normalized coordinates for integration the Jacobi–matrix J simplifies to J = ∂(x, y, z) ∂(ξ, η, ζ) = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ a 2 0 ∂z ∂ξ 0 b 2 ∂z ∂η 0 0 ∂z ∂ζ ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ (8.25) and the determinant of the Jacobi–matrix becomes det(J) = ab 4 ∂z ∂ζ . (8.26) The volume of the finite element can for example be calculated analytically and becomes
- 406. 394 8 Mathematical Methods V e = ab 4 (z5 − z1 + z6 − z2 + z7 − z3 + z8 − z4) . (8.27) The restriction to rectangular projectable elements ensures that it is possi- ble to evaluate integrals of type (8.19) numerically exact, which in general increases the overall numerical stability. Other types of elements can be used, e.g. hexahedrons with additional nodes (Schwarz, 1991; Zienkiewicz, 1984). More complicated elements give rise to more complicated shape functions with higher order polygons. Higher order does not in general increase accuracy but it ensures higher effort and less numerical stability especially if small elements are located next to larger ones. In basin modeling such effects can occur if e.g. thin layers are modeled. So it is the authors’ opinion that hexahedrons with shape functions of type (8.22) are a good choice and high accuracy should be achieved with high grid resolution. 8.6 Control Volumes In contrast to finite differences or the finite element method where the field u of (8.1) is approximated following a rather technical argumentation control volumes (CV) were developed with the aim to use physical argumentations directly for the construction of the discretized set of equations. Mass or energy conservation with respect to the source term is the basic principle but other points, such as appropriate averages for permeabilities or conductivities, are also taken into account. A very good introduction is given in the textbook of Patankar (1980). The time discretization is usually performed analog to the finite differences following the argumentation of Sec. 8.4. The CV formulation starts with a discussion of a small discrete volume around a grid point. Following mass or energy conservation the total flux across the surface of this volume must equal the generated or absorbed amount inside the volume. Hence the gridpoints are usually located inside the cell of discretization but not at its interface such as typically within the finite element method. For simplicity reasons the method is outlined here only for two dimensions, steady state and isotropic conditions. The generalization up to three dimen- sions and inclusion of transient as well as non–isotropic effects is straightfor- ward. The integration of (8.1) over a volume according to Fig. 8.5 yields λ̄i,r(ui,r − ui)Δy Δx − λ̄i,l(ui − ui,l)Δy Δx + λ̄i,t(ui,t − ui)Δx Δy − λ̄i,b(ui − ui,b)Δx Δy + q̄iΔxΔy = 0 . (8.28)
- 407. 8.6 Control Volumes 395 Here ui is the approximation of u and q̄i the average of q inside the control volume i. The quantities referenced by the expressions λ̄i,k are the “conduc- tivities” between the control volume i and the neighboring volume in the direction k with k as r, l, t, or b indicating “right”, “left”, “top” or “bottom”. The first four terms describe fluxes across the surfaces of the control volume. In case of constant conductivity λi inside of a control volume but variation across the surfaces it can be shown with a simple physical argumentation (Patankar, 1980) that the λ̄i,k must be approximated by a harmonic average such as 1 λ̄i,r = 1 2 1 λi + 1 λi,r . (8.29) The terms of (8.28) can finally be rearranged to aiui = k ai,kui,k + bi (8.30) with the sum over all neighbors k. It is bi = q̄iΔxΔy and the coefficients ai, ai,k can easily be extracted from (8.28) as ai = (λ̄i,r + λ̄i,l) Δy Δx + (λ̄i,t + λ̄i,b) Δx Δy (8.31) and ai,r = λ̄i,r Δy Δx , ai,l = λ̄i,l Δy Δx , ai,t = λ̄i,t Δx Δy , ai,b = λ̄i,b Δx Δy . (8.32) They are always positive and due to the linear nature of the differential equa- tion (8.1) it is ai = k ai,k . (8.33) The inclusion of transient terms leads to a modification of (8.30), which is similar to (8.21). Care must be taken with generalizations of these scheme, especially if non–linearities are introduced (Patankar, 1980). Typically, this might occur with parameters, such as conductivities, capacities or the source term, if they are dependent on the field u itself. A straightforward generalization of the scheme to non–regular cells is not possible. Obviously, the fluxes in x and y direction in (8.28) are independent. This is not the case for non–orthogonal directions of arbitrary cells. In Øye (1999) this problem is solved by subdivision of control volumes into finite elements for the calculation of fluxes through the surfaces. Unfortunately, this can only be done with huge effort. Maybe it would be possible to find an easier description if rectangular projectable volumes, such as in the previous section, were used.
- 408. 396 8 Mathematical Methods A comparison between finite differences, control volumes, and finite ele- ments extends the scope of this chapter. Some detailed arguments can e.g. be found in Gray (1984); Marsal (1976). 8.7 Solver The finite element and the control volume method both lead to a linear set of equations, namely (8.21) and (8.30), which still must be solved. Many textbooks, such as Press et al. (2002), or software packages, such as LAPACK or LINPACK (www.netlib.org), are available providing methods or programs for the solution of linear sets of equations. Unfortunately the scope of the topic goes beyond the possibilities of this volume. Nevertheless a few facts should be listed. Solutions of linear equation systems are the most time consuming parts of computer simulations in basin modeling. Both cases, the FE and the CV, yield symmetric and sparse equation sys- tems. The standard approach for the solution are conjugate gradients (Press et al., 2002). A very good description of the method can be found in Shewchuk (1994). Backsubstitution methods are recommended in cases of one dimen- sional simulations along a well (Press et al., 2002; Patankar, 1980). Because of irregularities in the gridding and wide variations over short distances of parameters such as permeability, it becomes very difficult to use multigrid methods, which are known to have a higher performance in many cases than conjugate gradients. Iterative methods are an alternative if good estimates for the solution are available. This can be the case if time steps are small or steady state condi- tions have been reached and the solution of the previous time step is used as the initial step for the actual iteration. Unfortunately, sedimentation, com- paction and subsequently changing geometries over geological times prevent this approach in many cases.3 Conjugate gradients often work better if appropriate preconditioning is used (Shewchuk, 1994). Even a simple diagonal preconditioning yields a sig- nificant improvement of performance. More advanced methods, such as in- complete Cholesky preconditioning, are also common. 8.8 Parallelization In practice two main computer architectures concerning parallelization must be distinguished. First, separate computers each of them consisting of a pro- cessor and memory, which communicate via network. Networks of arbitrary computers are principally not limited to overall computing power and mem- ory and they are relatively cheap if standard PC’s are used. The alternative 3 Strictly spoken the conjugate gradient method can be interpreted as non–iterative (Shewchuk, 1994). Nevertheless, its main calculation steps are called iterations.
- 409. 8.8 Parallelization 397 architecture is a computer with multiple processors, all of them with access to one shared memory. This has the great advantage that sending and receiving data is not necessary. However, such computers are expensive and do usually have limited capabilities for memory and the number of processors. Parallelization is complicated from a technical viewpoint The usage of ad- ditional software tools is necessary. Parallel processing on networks is often implemented using the Message Passing Interface (MPI), which is the stan- dard in numerics and scientific computing (Gropp et al., 1999; Snir et al., 1998; Gropp et al., 1998). To complete the list, alternatives, such as CORBA (Common Object Request Broker Architecture) or RPC (Remote Procedure Call), are only mentioned with some citations (Linnhoff-Popien, 1998; Redlich, 1996; Fischer and Müller, 1996). Shared memory parallelization is usually im- plemented with threads. The handling of threads can be improved with tools, such as the OpenMP, which is an advanced software tool for the usage of threads in scientific computing (Chandra et al., 2001). As has been mentioned in the previous section the most time–intensive operations are solving linear equations. Thus the best starting point is a paral- lelization of the conjugate gradient method. This is usually done the following way: The model is cut into pieces, each processor computes one piece. Each piece is extended at its boundary by the neighboring and following gridpoints belonging to other pieces. So at each boundary an overlap region is created, which is processed by at least two processors. Data transfer is only necessary for the gridpoints of the overlap region. After each iteration of the conjugate gradient method, the data of the overlap region must be updated between the processors. The amount of data is small and network parallelization can be used if a piece is big compared to its overlap region. In practice the model is cut into slices so that data exchange occurs only between two processors (Fig. 8.9). This again reduces data transfer. Caused by rectangular gridding in lateral direction (Sec. 8.5) the best choice are slices, which are cut vertically in lateral x or y direction. When most of the simulation time is used for the sim- ulation of differential equations, the speedup scales almost linearly with the number of processors, when the number of processors is low (Fig. 8.8).4 If the number of processors becomes high the speedup converges to a constant value because an overhead of data communication rises. In case of hybrid models there is an important part of time used for reservoir analysis, which cannot be parallelized with the network parallelization so the speedup is low in the start. On the other hand the highly explicit treated Darcy flow part is an ef- ficiently parallelizable part of the program, which still yields speedups where pressure and temperature have already converged. Generally, the speedup for parallelization of migration is unique and can be totally different in different case studies. 4 Speedup is the rate of processing times. Linear speedup is found when it doubles with the doubling of processors and triples with tripling of processors.
- 410. 398 8 Mathematical Methods Unfortunately, this approach limits the total number of processors to one fifth of the number of gridpoints perpendicular to the slicing. Theoretically, it would be possible to cut the slices again vertically in the other horizontal direction into sub pieces but this is commonly not done. 1 2 3 4 5 6 7 8 0 1 2 3 4 5 # Processors Speedup Hybrid migration Press. Temp. Fig. 8.8. Example speedup of network parallelized model with hybrid migration (Chap. 6.6) or with 3D pressure–temperature calculation only Fig. 8.9. Illustration of model decomposition into slices for parallelization. The overlap regions are clipped by the dashed lines Caused by the time discretization (8.14) a linear set of equations must be solved for each time step. It must be noted that the performance of parallel computing can be enhanced, if each processor keeps as much data as pos- sible between the time steps to avoid unnecessary communication. With the exception of overlap regions an overall disjunctive data distribution to the pro- cessors has the additional benefit of better balance of the computer memory.
- 411. 8.9 Local Grid Refinement (LGR) 399 This also manifests itself in a better performance. Usually, the effect is negligi- ble but in extreme cases the total speedup can become “superlinear” because different types of computer memory from processor cache to swap space with extremely different performances exist and smaller amounts of data are often stored in faster accessible memory locations. In principle a disjunctive data model is not a major issue, but in practice this needs a lot of detailed work. Besides the solution of linear sets of equations, reservoir analysis is the second most time consuming part in basin modeling (Chap. 6.5). The hydro- carbons can move large distances almost laterally before they are trapped. Usually, the reservoir analysis is map based. Hence a parallelization founded on vertically cut slices is not possible. Injected amounts can reach the map from almost everywhere in the model and might be redistributed almost every- where again, especially if conductive faults are taken into account. The amount of transfered data is high if multi–component descriptions of the hydrocarbon phases are necessary. Thus it is not possible to use network parallelization. Additionally, it is impossible to parallel process the reservoirs because hydro- carbons which leave one carrier usually enter another one. So in general, the reservoirs are strongly interacting. They must be processed sequentially from bottom to top, from source to trap. A possible parallelization can only be performed inside each reservoir analysis. Here the most time consuming part is the volumetrics. A thread parallelization of volumetrics in the carriers is possible (Bücker et al., 2008). But speedups as with the solution of differential equations by parallelization of the conjugate gradient algorithm are not to be expected. 8.9 Local Grid Refinement (LGR) Commonly, two different reasons for LGR are found in basin modeling. Firstly, the overall model resolution is generally too high for processing. In this case, a central area of interest is typically calculated in full resolution whereas the outer rim of the model is sampled. This approach allows for feasible comput- ing times and high resolution results. Secondly, high resolution data is only available for a prospect or field, which is small compared to the extension of a meaningful basin model. Obviously, the high resolution data can be inte- grated via LGR into the basin model. Principally, processing speed can be optimized if a discretization is adapted rigorously to the availability of data and the areas of interest. It was argued contrarily in Sec. 8.5 that discretization in basin modeling should be performed on a basis of stacked maps with vertical columns of gridpoints and rectangular base areas in horizontal alignment. The availability and the handling of data in model builders and viewers, which is restricted almost exclusively to rectangular gridded maps, complies very well with this requirement. A slight improvement of common style regular gridding is called the tartan grid and introduced in Sec. 8.9.1. Compromises between slight
- 412. 400 8 Mathematical Methods improvements and a more rigorous local grid refinement are discussed in the following subsections. 8.9.1 Tartan Grid A tartan grid is shown in Fig. 8.10. It consists of a regular rectangular grid with arbitrary distances in x and y–direction. The central area of interest is here gridded with high resolution. A tartan grid is compatible with the requirements of Sec. 8.5. Such models can be parallelized without any limi- tations in the same manner as described in Sec. 8.8. Obviously, the number of gridpoints is not optimal. Grid points are lying on grid lines and hence additional gridpoints in an area of interest lead also to additional gridpoints outside of the area of interest. Unnecessary gridpoints reduce the performance. x y Fig. 8.10. Map view of a tartan grid. The polygon outlines an area of interest which is gridded in higher resolution 8.9.2 Windowing A low resolution model, which is covering wide areas around a small high resolution model, can be used to calculate boundary conditions (e.g. for heat flow analysis or pressure prediction) at the interface to the small scale high resolution model (Fig. 8.11). Unrealistic “no–flow” Neumann conditions at model sides, such as shown in Fig. 2.5 or Fig. 3.13, are hence shifted away from the area of interest. This approach is often used for heat flow and temperature calculations and sometimes for pressure prediction. A specification of a Dirichlet boundary condition for an explicitly treated migration model is obviously problematic. Catching of all amounts which enter the area of the small model is possible. These amounts can be injected into the small scale model. However, migration becomes complicated when HCs leave the small model but not the big low resolution area. At a later time they
- 413. 8.9 Local Grid Refinement (LGR) 401 may reenter the area of the small model. A strong coupling in space and time with iterations between both models might be necessary. The whole procedure becomes technically very complicated, when accumulations are located at the boundary between both models. Alternatively, the big low scale model can be modeled without taking into account any effects of the smaller one. Petroleum amounts which are found in the area of the high resolution model are redistributed according to the high resolution geometry with some final migration steps. Again, both models must be strongly coupled when migration proceeds differently through the high resolution than through the low resolution model. With coupling this approach becomes like a two grid version of a multigrid method. A big advantage is the possibility of using different migration modeling methods in both models even if they are coupled. For example, a hybrid method with fast long distance migration might be used for the low resolu- tion model and a detailed invasion percolation in the region with the high resolution data. Additionally it must be noted that such an approach is much easier to implement because the model boundary between the low and high resolution area is not of special interest. Generally, a big advantage of windowing is its compatibility with parallel processing as outlined in Sec. 8.8. Node numbering and slicing remain in each model the same. Especially uncoupled models can be build and calculated with high performance. x y Fig. 8.11. Map view of a high resolution grid, which belongs to a small model, and a low resolution grid of a large model, which covers a wide area around the small model 8.9.3 Coupled Model in Model In the windowing approach the lowly resolved large area model is simulated before the highly resolved small area model. This might lead to principal
- 414. 402 8 Mathematical Methods problems as outlined in the previous section. Alternatively, both models can be coupled more tightly and simulated consistently so that feedback of each model is taken into account immediately by the other one. For example, both models must be adapted in a way that flow conditions at any location on the interface between the models for all times are consistently fulfilled on both sides. Technically, this can be achieved by special coupling of both models to one large overall model via extra grid cells, extra coupling conditions and/or by iterative modeling where flow amounts at the model interface are stepwise refined until convergence is reached (Fig. 8.12). Obviously, this approach needs much more computing resources than win- dowing. Additionally, parallelization becomes more complicated. Hanging Node Extra Grid Cell Fig. 8.12. Crossover from a low to a high resolution grid with extra grid cells on the left side and “hanging nodes” without extra grid cells on the right side. Crossovers with hanging nodes are usually treated with extra coupling conditions 8.9.4 Faults Faults are very thin compared to common grid resolutions. Concerning heat flow or migration they can often be modeled as ideal surfaces (Secs. 6.5.4, 6.6.3, 6.8.6). Contrarily, pressure prediction makes it necessary to take into account volumetric properties of faults, e.g. which determine water flow conductivity (Fig. 2.56). The grid is for that reason often improved locally around faults (Figs. 2.47, 8.13). Practically, this can easily be achieved if the faults are following the surfaces and edges of existing cells of the “volume–grid”. A small error in fault location due to gridding is less important than not correctly taking into account water outflow from an overpressured region.
- 415. 8.9 Local Grid Refinement (LGR) 403 Fig. 8.13. Fault of constant thickness, which is gridded with extra cells (grey). It is following sur- faces and edges of the overall “volume–grid” Vertical Grid Line Fault Horizon Thickness Summary: Basin modeling makes extensive use of mathematical and nu- merical methods. The solution of differential equations of diffusion type are the largest group of frequently occurring numerical tasks. The temporal evolution is often modeled with finite differences. Explicit and implicit schemes are discussed. The spatial behavior of the differential equations is treated with finite elements or control volumes. Both methods are shortly outlined in theory. Regularly gridded maps are the most common data source for basin mod- eling. Hexahedrons are for that reason the basic three dimensional spatial building blocks of a model. The finite element method is presented in more detail for hexahedrons with form functions of the most simple tri–linear type. Some typically appearing integrals can be solved analytically for this case. The numerical solution of the resulting linear equations is summarized. Focus is put on a discussion of computational parallelization. An overall model slicing and separate processing of these slices on different computers is described. This method yields a linear speedup for a low number of pro- cessors. However, it is not applicable for flowpath and invasion percolation based migration. Other approaches must be found, e.g. parallelization based on shared memory access. Local grid refinement (LGR) allows processing of models with high reso- lution data, which are usually not feasible due to computer performance and memory restrictions. Small scale field or prospect data can be incorporated into basin models. Special areas of interest can be processed in high reso- lution. LGR is outlined for a continuous crossover from regularly gridded models over the tartan grid and windowing to coupled models in models. Spatially improved fault handling is also described. Mixing rules for upscaling to bulk values are discussed. Some rules of thumb for mixing and upscaling are given, when more detailed imforma- tion is missing. The most simple approaches are arithmetic, geometric, and harmonic averages or maximum based upscaling.
- 416. 404 8 Mathematical Methods References M. Aigner and G. M. Ziegler. Das BUCH der Beweise. Springer, second edition, 2004. G. R. Beardsmore and J. P. Cull. Crustal Heat Flow. Cambridge University Press, 2001. H. M. Bücker, A. I. Kauerauf, and A. Rasch. A smooth transition from serial to parallel processing in the industrial petroleum system modeling package petromod. Computers Geosciences, 34:1473–1479, 2008. R. Chandra, L. Dagum, D. Maydan, J. McDonald, and R. Menon. Parallel Programming in OpenMP. Morgan Kaufmann Publishers, San Francisco, CA, 2001. S. Fischer and W. Müller. Netzwerprogrammierung unter Linux und Unix. UNIX easy. Carl Hanser Verlag München Wien, 1996. W. G. Gray. Comparision of finite difference and finite element methods. In J. Bear and M. Y. Corapcioglu, editors, Fundamentals of Transport Phe- nomena in Porous Media, number 82 in NATO ASI Series, E: Applied Science. Martinus Nijhoff, 1984. W. Gropp, S. Huss-Lederman, A. Lumsdaine, E. Lusk, B. Nitzberg, W. Saphir, and M. Snir. MPI – The Complete Reference, Volume 2, The MPI Exten- sions. Scientific and Engineering Computation. The MIT Press, 1998. W. Gropp, E. Lusk, and A. Skejellum. Using MPI – Portable Parallel Pro- gramming with the Message–Passing Interface. Scientific and Engineering Computation. The MIT Press, 2 edition, 1999. C. Linnhoff-Popien. CORBA–Kommunikation und Management. Springer, 1998. D. Marsal. Die numerische Lösung partieller Differentialgleichungen in Wis- senschaft und Technik. Bibliographisches Institut, 1976. G. Å. Øye. An Object–Oriented Parallel Implementation of Local Grid Re- finement and Domain Decomposition in a Simulator for Secondary Oil Mi- gration. PhD thesis, University of Bergen, 1999. S. V. Patankar. Heat Transfer and Fluid Flow. Hemisphere Publishing Cor- poration, 1980. W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery. Numerical Recipes in C++. Cambridge University Press, second edition, 2002. J.-P. Redlich. CORBA 2.0. Addsion–Wesley Publishing Company, 1996. H. R. Schwarz. Methode der finiten Elemente. B. G. Teubner, Stuttgart, 1991. J. R. Shewchuk. An introduction to the conjugate gradient method without the agonizing pain. Lecture, School of Computer Science, Carnegie Mellon University, Pittsburgh, PA 15213, 1994. M. Snir, S. Otto, S. Huss-Lederman, D. Walker, and J. Dongarra. MPI – The Complete Reference, Volume 1, The MPI Core. Scientific and Engineering Computation. The MIT Press, second edition, 1998. O. C. Zienkiewicz. Methode der finiten Elemente. Carl Hanser, second edition, 1984.
- 417. A Compaction and Flow Parameter The data for depositional porosities and Athy’s compaction parameters vs. depth are worked out by Doug Waples. It is based on a comprehensive litera- ture review and his modeling experience over 20 years. The other compaction parameters are due to IES experience and fit to the curves from Doug Waples. The permeability values are also provided from Doug Waples. The capillary entry pressures are derived from permeabilities with the Hildenbrand equa- tions from Chap. 6.
- 418. 406 A Compaction and Flow Parameter Depo. Athy Athy Compress. y Schneider Factor Poro. k k Max. Min. ka kb φ [%] 1 km 1 MPa 10−7 kPa 10−7 kPa 1 MPa 1 MPa Limestone ooid grainstone 35.0 0.01 0.001 2.0 1.0 0.001 0.001 0.18 Waulsort. mound 16.0 0.01 0.001 2.0 1.0 0.001 0.001 0.08 micrite 51.0 0.52 0.04766 850.0 19.8 0.03212 0.07894 0.26 shaly 48.0 0.50 0.04493 686.5 18.8 0.03091 0.07090 0.24 org.–rich typical 51.0 0.52 0.04939 881.2 20.5 0.03329 0.08185 0.26 1-2% TOC 51.0 0.52 0.04851 865.1 20.1 0.03270 0.08037 0.26 10% TOC 51.0 0.52 0.05365 956.8 22.3 0.03616 0.08890 0.26 chalk, typical 70.0 0.90 0.10546 4613.3 41.6 0.20793 0.05813 0.35 chalk, 95% calcite 70.0 0.90 0.10546 4611.2 41.7 0.20793 0.05813 0.35 chalk, 75% calcite 67.0 0.90 0.10124 3871.1 40.6 0.19000 0.05738 0.34 chalk, 40% calcite 65.0 0.90 0.09870 3453.3 39.9 0.05891 0.18975 0.33 Marl 50.0 0.50 0.04651 778.0 19.8 0.07375 0.03086 0.25 Dolomite typical 35.0 0.39 0.03084 252.3 11.0 0.04219 0.02344 0.17 org.–lean, sandy 35.0 0.39 0.03119 255.2 11.2 0.04312 0.02344 0.17 org.–lean, silty 35.0 0.39 0.03138 256.7 11.2 0.04188 0.02406 0.17 org.–rich 35.0 0.39 0.03459 283.0 12.4 0.02656 0.04634 0.17 Table A.1. Compaction Parameter: Carbonate Rocks
- 419. A Compaction and Flow Parameter 407 Depo. Athy Athy Compress. y Schneider Factor Poro. k k Max. Min. ka kb φ [%] 1 km 1 MPa 10−7 kPa 10−7 kPa 1 MPa 1 MPa Biogenic Sediments Chalk, typical 70.0 0.90 0.10546 4611.2 41.7 5.00000 0.04500 0.35 Chalk, 95% calcite 70.0 0.90 0.10546 4611.2 41.7 0.20793 0.05813 0.35 Chalk, 75% calcite 67.0 0.90 0.10124 3871.1 40.6 0.19000 0.05738 0.34 Chalk, 40% calcite 65.0 0.90 0.09870 3453.3 39.9 0.05891 0.18975 0.33 Coal, pure 76.0 0.43 1.51396 87407.4 564.6 3.41736 0.74043 0.38 Coal, impure 74.0 0.42 0.22597 11692.7 87.9 0.49695 0.11328 0.37 Coal, silty 68.0 0.40 0.13392 5289.1 53.2 0.07308 0.28035 0.34 Clastic Sediments Sandstone typical 41.0 0.31 0.02660 274.7 11.5 0.04156 0.01781 0.20 clay–rich 40.0 0.32 0.02661 265.4 11.1 0.04000 0.01812 0.20 clay–poor 42.0 0.30 0.02627 280.5 11.8 0.04047 0.01734 0.21 quartzite typical 42.0 0.30 0.02726 291.0 12.2 0.04281 0.01781 0.21 very quartz–rich 42.0 0.27 0.02461 252.6 11.8 0.01615 0.03977 0.21 subarkose typical 41.0 0.28 0.02468 244.8 11.4 0.01631 0.03874 0.20 quartz–rich 42.0 0.28 0.02533 263.4 11.9 0.03984 0.01641 0.21 clay–rich 42.0 0.30 0.02643 282.2 11.8 0.04063 0.01750 0.21 clay–poor 42.0 0.30 0.02627 280.5 11.8 0.04047 0.01734 0.21 dolomite–rich 40.0 0.30 0.02572 249.8 11.2 0.04000 0.01688 0.20 arkose typical 39.0 0.33 0.02772 267.3 11.2 0.01981 0.04212 0.20 quartz–rich 41.0 0.30 0.02623 267.1 11.6 0.04094 0.01750 0.20 quartz–poor 40.0 0.32 0.02646 263.9 11.1 0.04000 0.01812 0.20 clay–rich 40.0 0.32 0.02661 265.4 11.1 0.04000 0.01812 0.20 clay–poor 39.0 0.32 0.02722 258.9 11.2 0.04063 0.01812 0.20 dolomite–rich 39.0 0.32 0.02658 252.8 11.0 0.03875 0.01812 0.20 wacke 39.0 0.34 0.02773 271.1 10.9 0.01992 0.04192 0.20 Table A.2. Compaction Parameter
- 420. 408 A Compaction and Flow Parameter Depo. Athy Athy Compress. y Schneider Factor Poro. k k Max. Min. ka kb φ [%] 1 km 1 MPa 10−7 kPa 10−7 kPa 1 MPa 1 MPa Shale typical 70.0 0.83 0.09613 4032.7 40.3 0.19157 0.05270 0.35 org.–lean typical 70.0 0.83 0.09613 4032.7 40.3 0.19157 0.05270 0.35 sandy 65.0 0.83 0.08999 3005.2 38.7 0.17479 0.05340 0.33 silty 67.0 0.83 0.09230 3373.4 39.4 0.18394 0.05357 0.34 silic., typical 70.0 0.83 0.09556 4007.5 40.1 0.19043 0.05238 0.35 silic., 95% Opal–CT 80.0 0.83 0.14631 7397.5 37.8 0.33248 0.06983 0.40 black 70.0 0.83 0.10931 4584.3 45.8 0.22348 0.06082 0.35 organic–rich typical 70.0 0.83 0.10931 4291.1 42.9 0.18549 0.05265 0.30 3% TOC 70.0 0.83 0.10165 4264.1 42.6 0.20782 0.05656 0.35 8% TOC 70.0 0.83 0.10931 4584.3 45.4 0.22348 0.06082 0.35 20% TOC 70.0 0.83 0.12975 5441.2 54.4 0.26527 0.07219 0.35 Siltstone org.–lean 55.0 0.51 0.04907 1036.1 21.1 0.03165 0.08611 0.28 organic–rich typical 55.0 0.51 0.04937 1042.2 21.2 0.03185 0.08660 0.28 10% TOC 55.0 0.51 0.05459 1152.6 23.5 0.09119 0.03419 0.28 2-3% TOC 55.0 0.51 0.04966 1048.7 21.4 0.03204 0.08714 0.28 Conglomerate typical 30.0 0.30 0.02429 142.1 8.8 0.01897 0.03228 0.15 quartzitic 30.0 0.30 0.02429 142.1 8.8 0.01897 0.03228 0.15 Tuff, felsic 60.0 0.35 0.03711 959.1 16.5 0.06961 0.02117 0.3 Tuff, basaltic 60.0 0.35 0.03212 829.3 14.3 0.06000 0.01844 0.3 Table A.3. Compaction Parameter: Clastic Sediments – Part 2
- 421. A Compaction and Flow Parameter 409 An- Porosity [%] Permeability [log mD] iso- at Point at Point tr. 1 2 3 1 2 3 Biogenic Sediments Chalk, typical 1.5 1.00 25 70 −6.75 −3.10 1.00 Chalk, 95% calcite 1.5 1.00 25 70 −6.75 −3.10 1.00 Chalk, 75% calcite 2.0 1.00 25 67 −6.75 −3.10 1.00 Chalk, 40% calcite 3.0 1.00 25 65 −6.75 −3.10 1.00 Coal, pure 4.0 3.02 25 76 −7.50 −2.60 0.00 Coal, impure 4.0 3.17 25 74 −7.50 −2.60 0.00 Coal, silty 4.0 3.39 25 68 −7.50 −2.60 0.00 Diatomite, clay–poor 3.0 1.00 25 80 −2.55 −1.30 3.00 Diatomite, clay–rich 10.0 1.00 25 75 −3.05 −1.80 2.50 Carbonate Rocks Limestone ooid grainstone 1.1 1.00 15 35 −2.44 2.60 3.00 Waulsortian mound 1.1 1.00 12 16 −2.50 3.00 3.35 micrite 1.1 1.00 25 51 −2.20 1.00 1.52 shaly 2.0 1.00 25 48 −1.99 0.72 1.50 organic–rich, typical 2.0 1.00 25 51 −1.99 0.73 1.00 organic–rich, 1-2% TOC 1.2 1.00 25 51 −1.99 0.73 1.00 organic–rich, 10% TOC 2.0 1.00 25 51 −1.99 0.72 1.00 chalk, typical 1.5 1.00 25 70 −6.75 −3.10 1.00 chalk, 95% calcite 1.5 1.00 25 70 −6.75 −3.10 1.00 chalk, 75% calcite 2.0 1.00 25 67 −6.75 −3.10 0.73 chalk, 40% calcite 3.0 1.00 25 65 −6.75 −3.10 0.54 Marl 1.2 1.00 25 50 −5.05 −2.25 −0.78 Dolomite typical 1.1 1.00 25 35 0.11 2.83 3.92 organic–lean, sandy 1.1 1.00 25 35 0.60 2.98 3.93 organic–lean, silty 1.1 1.00 25 35 0.31 2.88 3.91 organic–rich 1.2 1.00 25 35 −0.37 2.69 3.91 Table A.4. Porosity and Permeability Parameter
- 422. 410 A Compaction and Flow Parameter An- Porosity [%] Permeability [log mD] iso- at Point at Point tr. 1 2 3 1 2 3 Sandstone typical 5.0 1 25 41 −1.80 3.00 4.33 clay–rich 10.0 1 25 40 −2.80 2.00 3.62 clay–poor 5.0 1 25 42 −1.80 3.00 4.84 quartzite, typical 3.0 1 25 42 −1.80 3.00 4.84 quartzite, very quartz–rich 2.0 1 25 42 −1.80 3.00 4.84 subarkose, typical 3.0 1 25 41 −1.80 3.00 4.73 subarkose, quartz–rich 3.0 1 25 42 −1.80 3.00 4.84 subarkose, clay–rich 10.0 1 25 42 −2.80 2.00 3.84 subarkose, clay–poor 3.0 1 25 42 −1.80 3.00 4.84 subarkose, dolomite–rich 5.0 1 25 40 −2.30 2.50 4.12 arkose, typical 5.0 1 25 39 −1.80 3.00 4.51 arkose, quartz–rich 4.0 1 25 41 −1.80 3.00 4.73 arkose, quartz–poor 7.0 1 25 40 −2.30 2.50 4.12 arkose, clay–rich 10.0 1 25 40 −2.80 2.00 3.62 arkose, clay–poor 4.0 1 25 39 −1.80 3.00 4.51 arkose, dolomite–rich 5.0 1 25 39 −2.30 2.50 4.01 wacke 7.0 1 25 39 −2.80 2.00 3.51 Shale typical 1.2 1 25 70 −8.52 −3.00 −1.00 org.–lean, typical 1.2 1 25 70 −8.52 −3.00 −1.00 org.–lean, sandy 1.5 1 25 65 −8.42 −2.00 1.00 org.–lean, silty 1.3 1 25 67 −8.47 −2.50 0.50 org.–lean, silic., typical) 1.5 1 25 70 −8.47 −2.00 1.30 org.–lean, silic., 95% Opal-CT 1.1 1 25 80 −8.42 −1.50 −1.00 black 5.0 1 25 70 −8.52 −3.00 −1.00 organic–rich, typical 5.0 1 25 70 −8.52 −3.00 −1.00 organic–rich, 3% TOC 2.0 1 25 70 −8.52 −3.00 −1.00 organic–rich, 8% TOC 5.0 1 25 70 −8.52 −3.00 −1.00 organic–rich, 20% TOC 10.0 1 25 70 −8.52 −3.00 −1.00 Siltstone organic–lean 10.0 1 25 55 −6.28 −1.00 0.71 organic–rich, typical 10.0 1 25 55 −6.28 −1.00 0.71 organic–rich, 10% TOC 10.0 1 25 55 −6.28 −1.00 0.71 organic–rich, 2-3% TOC 10.0 1 25 55 −6.28 −1.00 0.71 Conglomerate, typical 1.1 1 25 30 −2.80 2.00 4.00 Conglomerate, quartzitic 1.1 1 25 30 −2.80 2.00 4.00 Tuff, felsic 1.3 1 28 60 −1.87 1.75 4.00 Tuff, basaltic 1.3 1 28 60 −1.87 1.75 4.00 Table A.5. Porosity and Permeability Parameter: Clastic Sediments
- 423. A Compaction and Flow Parameter 411 Entry Pr. [MPa] Parameter at Porosity a b 1% 25% [MPa] Biogenic Sediments Chalk, typical 15.42 5.78 16.78 −0.037 Chalk, 95% calcite 15.42 5.78 16.78 −0.037 Chalk, 75% calcite 15.42 5.78 16.78 −0.037 Chalk, 40% calcite 15.42 5.78 16.78 −0.037 Coal, pure 23.35 5.36 26.14 −0.049 Coal, impure 23.35 5.36 26.16 −0.050 Coal, silty 23.35 5.35 26.19 −0.050 Diatomite, clay–poor 2.94 1.47 1.56 −0.013 Diatomite, clay–rich 4.30 2.15 2.05 −0.013 Carbonate Rocks Limestone ooid grainstone 1.42 0.08 1.74 −0.086 Waulsortian mound 1.47 0.06 1.94 −0.120 micrite 1.25 0.26 1.34 −0.032 shaly 1.11 0.32 1.18 −0.027 organic–rich, typical 1.11 0.31 1.18 −0.027 organic–rich, 1-2% TOC 1.11 0.31 1.18 −0.027 organic–rich, 10% TOC 1.11 0.32 1.18 −0.027 chalk, typical 15.42 5.78 16.78 −0.037 chalk, 95% calcite 15.42 5.78 16.78 −0.037 chalk, 75% calcite 15.42 5.78 16.78 −0.037 chalk, 40% calcite 15.42 5.78 16.78 −0.037 Marl 6.03 3.03 6.43 −0.028 Dolomite typical 0.35 0.06 0.37 −0.027 organic–lean, sandy 0.27 0.06 0.28 −0.024 organic–lean, silty 0.31 0.06 0.33 −0.026 organic–rich 0.45 0.07 0.49 −0.031 Table A.6. Capillary Entry Pressure
- 424. 412 A Compaction and Flow Parameter Entry Pr. [MPa] Parameter at Porosity a b 1% 25% [MPa] Sandstone typical 1.00 0.06 1.12 −0.048 clay–rich 1.74 0.12 1.94 −0.048 clay–poor 1.00 0.06 1.12 −0.048 quartzite, typical 1.00 0.06 1.12 −0.048 quartzite, very quartz–rich 1.00 0.06 1.12 −0.048 subarkose, typical 1.00 0.06 1.12 −0.048 subarkose, quartz–rich 1.00 0.06 1.12 −0.048 subarkose, clay–rich 1.74 0.12 1.94 −0.048 subarkose, clay–poor 1.00 0.06 1.12 −0.048 subarkose, dolomite–rich 1.32 0.08 1.47 −0.048 arkose, typical 1.00 0.06 1.12 −0.048 arkose, quartz–rich 1.00 0.06 1.12 −0.048 arkose, quartz–poor 1.32 0.08 1.47 −0.048 arkose, clay–rich 1.74 0.12 1.94 −0.048 arkose, clay–poor 1.00 0.06 1.12 −0.048 arkose, dolomite–rich 1.32 0.08 1.47 −0.048 wacke 1.74 0.12 1.94 −0.048 Shale typical 41.02 5.36 46.58 −0.055 organic–lean, typical 41.02 5.36 46.58 −0.055 organic–lean, sandy 38.82 2.50 45.00 −0.064 organic–lean, silty 39.90 3.66 45.78 −0.060 organic–lean, siliceous, typical 39.90 2.50 46.31 −0.065 organic–lean, siliceous, 95% Opal–CT 38.82 1.71 45.52 −0.069 black 41.02 5.35 46.58 −0.055 organic–rich, typical 41.02 5.35 46.58 −0.055 organic–rich, 3% TOC 41.02 5.35 46.58 −0.055 organic–rich, 8% TOC 41.02 5.35 46.58 −0.055 organic–rich, 20% TOC 41.02 5.35 46.58 −0.055 Siltstone organic–lean 11.90 1.17 13.43 −0.053 organic–rich, typical 11.90 1.17 13.43 −0.053 organic–rich, 10% TOC 11.90 1.17 13.43 −0.053 organic–rich, 2-3% TOC 11.90 1.17 13.43 −0.053 Conglomerate, typical 1.74 0.12 1.94 −0.048 Conglomerate, quartzitic 1.74 0.12 1.94 −0.048 Table A.7. Capillary Entry Pressure: Clastic Sediments
- 425. B Deviation of the Pressure Equation The deviation of (2.13) follows the appendix in the article from Luo and Vasseur (1992). The conservation of solid rock material in a fixed coordinate system can be described by a continuity equation of form ∂(ρs(1 − φ)) ∂t + ∇ · (ρs(1 − φ)vs) = 0 (B.1) with the porosity φ, the density ρs of the solid grains and the velocity vs of the compacting and thus moving solid. Note that vs is not a bulk velocity. It is the “real” volocity of the moving rock. Under the assumption that the grain density keeps constant it becomes ∇ · vs = 1 1 − φ dφ dt (B.2) with d dt = ∂ ∂t + vs · ∇ . (B.3) The continuity equation for a fluid of density ρ which is moving with velocity v is ∂(ρφ) ∂t + ∇ · (ρφv) = 0 . (B.4) Again v is the “real” and not the bulk velocity. Darcy’s law states φ(v − vs) = − k μ · ∇u (B.5) with the permeability k, mobility μ, and overpressure potential u = p − ph. The term v−vs is the relative velocity of the fluid in the rock and φ(v−vs) is the relative bulk velocity. Equations (B.5) and (B.2) can be inserted in (B.4). Rearranging yields
- 426. 414 B Deviation of the Pressure Equation ρ 1 − φ dφ dt = ∇ · k μ · ∇u − φ dρ dt . (B.6) According to the compaction law dφ dt = −C dσ z dt (B.7) it becomes with σ z = ul − u ρ 1 − φ du dt − ∇ · k μ · ∇u = ρ 1 − φ dul dt − φ dρ dt . (B.8) A transition to a moving coordinate system, which moves with the solid rock, yields a substitution of d/dt with ∂/∂t and finally to equation (2.13). The result differs only by some 1/(1−φ) factors from the expected result. A direct usage of (2.10) in the moving coordinate system yields the same result. X. Luo and G. Vasseur. Contributions of compaction and aquathermal pressuring to geopressure and the influence of environmental conditions. AAPG Bulletin, 76(10):1550–1559, 1992.
- 427. C Analytic Groundwater Flow Solution from Tóth The steady state groundwater flow for an idealized block as shown in Fig. C.1 can be calculated analytically. The boundary condition on top is given by a hydraulic head of form h0(1 − x/w). At bottom and at the sides a no–flow condition is assumed. All properties such as permeability k or water density ρw within the block are constant and without anisotropy. The pressure equation reduces for the potential u hence to ∂2 ∂x2 u(x, z) + ∂2 ∂z2 u(x, z) = 0 (C.1) with the boundary conditions u(x, 0) = ρwgh0(1 − x w ) for 0 x w ∂ ∂z u(x, z) z=h = 0 for 0 x w ∂ ∂x u(x, z) x=0 = ∂ ∂x u(x, z) x=w = 0 for 0 z h . (C.2) z h x w h0 Fig. C.1. Groundwater flow model according to Tóth (1962) The analytical solution has the form
- 428. 416 C Analytic Groundwater Flow Solution from Tóth u = ρwgh0 2 1 + 8 ∞ n=0 cosh(μn(h − z)) cos(μnx) μ2 nw2 cosh(μnh) (C.3) with μn = (2n+1)π/w. An example is shown in Figs. C.2 and C.3. The water density has been chosen as ρw = 1019 kg/m3 = 1000.0/0.980665 kg/m3 so that u(0, 0) = 10 MPa. The model has also been calculated numerically as a benchmark with finite elements on a regular grid with 100 × 20 cells. Errors were below 0.01 MPa. The differential equation and the solution are formally very similar to those of the heat flow example of Sec. F.8. Obviously, a heat flow example with a linearly varying SWI temperature can easily be constructed from (C.1), (C.2) and (C.3). J. Tóth. A theory of groundwater motion in small drainage basins in cen- tral Alberta, Canada. Journal of Geophysical Research, 67:4375–4387, 1962 MPa Fig. C.2. Pressure potential for groundwater flow according to Tóth. The arrows indicate the direction of water flow. They are only perpendicular to isolines at equal horizontal to vertical aspect ratio (Fig. C.3). Here it is w = 10 km, h = 1 km and ρw = 1019 kg/m3
- 429. C Analytic Groundwater Flow Solution from Tóth 417 MPa Fig. C.3. Cutout from Fig. C.2 with almost the same horizontal to vertical aspect ratio
- 430. D One Dimensional Consolidation Solution from Gibson Gibson (1958) found a closed expression for the compaction problem μ C ∂2 u ∂x2 = ∂u ∂t − Δρg dh dt (D.1) with overpressure u, x = h(t) − z, depth z, and the boundary conditions u(h, t) = 0, ∂ ∂x u(x, t) x=0 = 0 (D.2) for all t and height h(t) = S t (Fig. D.1). The constant factor S determines the velocity with which the layer increases in thickness. Density contrast Δρ = ρr − ρw, rock density ρr, water density ρw, mobility μ, and compress- ibility C are assumed to be constant in space and time. The assumption of constant compressibility is only valid for shallow depths and therefore called consolidation. It has been found as u Δρgh = 1 − 4 √ πK3 e−x2 K/4 ∞ 0 ξ tanh ξ cosh x ξ e−ξ2 /K dξ (D.3) with x = x/h, K = kh, and k = SC/μ. It is drafted in Fig. D.2. u=0 h(t)=S t m/C 0 = ¶ ¶ x u x z Drg Fig. D.1. Illustration of (D.1) and (D.2)
- 431. 420 D One Dimensional Consolidation Solution from Gibson z/h u/ gh Dr kh =4 0.25 1 64 16 Fig. D.2. Example curves according to (D.3) for depth z = h − x The factor K can roughly be estimated as K = 0.04 for a sandstone with C = 0.05/MPa, μ = 10−16 m2 /0.5 mPa s, a sedimentation rate of S = 1 km/Ma and a height h = 5 km. Hence a sandstone will never build up pressure on its own. When the permeability decreases about one or two orders of magnitude and K is approaching K ≈ 1, pressure build up might start. However, (D.3) can be expanded for small K. The exponential function in the integrand of (D.3) is decaying much faster for small K than other terms rise. Hence the tanh– and the cosh–terms can be Taylor–expanded up to second order. The resulting expression can be integrated analytically and finally yields u Δρgh ≈ K 4 (1 − 2x2 ). (D.4) This shows again that practically no overpressure is existing for K → 0. Gibson found additional closed expressions of related problems e.g. with drainage boundary condition u = 0 at x = 0 or h(t) ∼ √ t. R. E. Gibson. The progress of consolidation in a clay layer increasing in thickness with time. Géotechnique, 8:171–182, 1958
- 432. E Thermal Properties The following data of this section is provided by Doug Waples. It is based on a comprehensive literature review and his modeling experience over 20 years. Vert. An- Sp. Den. U Th K Radio- Cond. iso- Heat active tr. Cap. Heat W mK J kgK kg m3 [ppm] [ppm] [%] μW m3 Amphibolite 2.40 1.25 1130 2960 0.50 1.60 1.00 0.37 Eclogite 3.55 1.10 750 3400 0.20 0.40 0.20 0.12 Gneiss 2.70 1.40 800 2740 5.00 13.00 3.00 2.50 Marble 2.80 1.02 860 2700 1.00 1.00 0.20 0.34 Quartzite 5.40 1.03 770 2650 0.40 2.20 1.10 0.35 Schist 2.90 1.35 920 2740 2.10 9.70 2.20 1.44 Slate 1.80 1.75 860 2750 3.00 12.00 4.00 2.01 Serpentinite 2.60 1.45 785 2900 0.03 0.07 0.005 0.01 Table E.1. Thermal Properties: Metamorphic Rocks
- 433. 422 E Thermal Properties Vert. An- Sp. Den. U Th K Radio- Cond. iso- Heat active tr. Cap. Heat W mK J kgK kg m3 [ppm] [ppm] [%] μW m3 Rock Forming Minerals Quartz 7.69 1.00 740 2650 0.70 2.00 0.00 0.31 K–Spar 2.40 1.00 675 2550 1.60 5.00 14.05 1.96 Plagioclase 1.85 1.00 740 2680 2.50 1.50 0.00 0.74 Mica 0.60 6.00 770 2860 12.00 12.00 9.40 5.08 Chlorite 1.55 2.65 865 2800 1.00 4.00 0.65 0.62 Mixed–layer Clays 0.85 2.75 835 2780 1.00 4.00 4.00 0.94 Calcite 3.25 1.00 820 2710 0.00 0.00 0.00 0.00 Dolomite 5.30 1.00 870 2870 0.00 0.00 0.00 0.00 Halite, Mineral 6.50 1.00 865 2160 0.00 0.00 0.00 0.00 Anhydrite, Mineral 6.30 1.00 550 2960 0.00 0.00 0.00 0.00 Gypsum, Mineral 1.30 1.15 1070 2310 0.00 0.00 0.00 0.00 Olivine 4.20 1.00 750 3320 0.01 0.01 0.00 0.00 Orthoclase 2.30 1.00 620 2570 0.00 0.00 14.05 1.26 Rock Fragments Igneous, Granite 2.60 1.15 800 2650 4.00 13.50 3.50 2.25 Volcanic, Basalt 2.10 1.17 790 2870 0.90 2.70 0.80 0.52 Metamorphic, Schist/Slate 2.35 1.55 890 2745 2.55 10.85 3.10 1.73 Opaques, Amphibole 2.90 1.00 750 3100 0.00 0.00 0.20 0.02 Other Minerals Serpentine 2.60 1.00 785 2600 0.00 0.00 0.00 0.00 Siderite 3.00 1.00 700 3940 0.00 0.00 0.00 0.00 Sphalerite 12.70 1.00 460 4090 0.00 0.00 0.00 0.00 Sphene, Titanite 2.35 1.00 840 3520 400.00 350.00 0.00 165.58 Sylvite 6.70 1.00 685 1990 0.00 0.00 52.45 3.63 Talc 5.85 1.00 840 2780 0.00 0.00 0.00 0.00 Topaz 11.25 1.00 790 3200 0.00 0.00 0.00 0.00 Tourmaline 4.25 1.00 540 3100 0.00 0.00 0.00 0.00 Volcanic glass 1.35 1.00 770 2400 5.00 15.00 5.00 2.48 Zeolite 1.80 1.00 1050 2250 0.00 0.00 2.00 0.16 Zircon 5.00 1.00 630 4670 3000 2000 0.00 1573 Table E.2. Thermal Properties: Minerals – Part 1
- 434. E Thermal Properties 423 Vert. An- Sp. Dens. U Th K Radio- Cond. iso- Heat active tr. Cap. Heat W mK J kgK kg m3 [ppm] [ppm] [%] μW m3 Other Minerals Kaolinite 1.25 2.67 950 2590 1.00 4.00 0.50 0.56 Smectite 0.85 2.80 808 2610 0.50 2.00 0.50 0.30 Illite 0.85 2.75 822 2660 1.50 5.00 6.03 1.28 TOC (weight%) 0.28 1.10 1500 1100 100 0.00 0.00 10.47 Amphibole 2.90 1.00 750 3100 0.00 0.00 0.2 0.02 Analcime 1.40 1.00 960 2260 0.00 0.00 0.00 0.00 Anorthite 1.70 1.00 740 2760 0.00 0.00 0.00 0.00 Apatite 1.35 1.00 700 3180 30.00 150.00 0.00 21.29 Aragonite 2.25 1.00 800 2930 0.00 0.00 0.00 0.00 Barite 1.35 1.00 460 4480 0.00 0.00 0.00 0.00 Brucite 12.90 1.00 1260 2370 0.00 0.00 0.00 0.00 Chalcopyrite 8.20 1.00 535 4200 0.00 0.00 0.00 0.00 Chert, Mineral 5.00 1.00 740 2600 0.00 0.00 0.00 0.00 Clinoptilolite 1.40 1.00 1140 2100 0.00 0.00 3.30 0.24 Diopside 4.90 1.00 760 3280 0.00 0.00 0.00 0.00 Epidote 2.80 1.00 770 3590 35.00 275.00 0.00 37.24 Fluorite 9.50 1.00 870 3180 0.00 0.00 0.00 0.00 Forsterite 5.10 1.00 820 3210 0.00 0.00 0.00 0.00 Galena 2.70 1.00 210 7600 0.00 0.00 0.00 0.00 Garnet 3.50 1.00 750 3800 0.00 0.00 0.00 0.00 Gibbsite 2.60 1.00 1170 2440 0.00 0.00 0.00 0.00 Glauconite 1.60 1.00 770 2300 0.00 0.00 5.49 0.44 Goethite 2.70 1.00 825 4270 0.00 0.00 0.00 0.00 Hematite 11.30 1.00 625 5280 0.00 0.00 0.00 0.00 Hornblende 2.80 1.00 710 3080 15.00 25.00 1.75 6.56 Hypersthene 4.40 1.00 800 3450 0.00 0.00 0.00 0.00 Ilmenite 2.40 1.00 700 4790 25.00 0.00 0.00 11.40 Magnesite 5.80 1.00 880 3010 0.00 0.00 0.00 0.00 Magnetite 5.10 1.00 620 5200 15.00 10.00 0.00 8.76 Microcline 2.50 1.00 710 2560 0.00 0.00 14.05 1.25 Opal–CT 1.70 1.00 725 2300 0.00 0.00 0.00 0.00 Polyhalite, Min. 1.55 1.00 700 2780 0.00 0.00 12.97 1.25 Pyrite 3.00 1.00 510 5010 0.00 0.00 0.00 0.00 Pyrophyllite 3.75 1.00 805 2820 0.00 0.00 0.00 0.00 Pyroxene 4.20 1.00 770 3500 20.00 13.00 0.00 7.83 Rutile 5.10 1.00 700 4250 0.00 0.00 0.00 0.00 Table E.3. Thermal Properties: Minerals, Part 2
- 435. 424 E Thermal Properties Vert. An- Sp. Den. U Th K Radio- Cond. iso- Heat active tr. Cap. Heat W mK J kgK kg m3 [ppm] [ppm] [%] μW m3 Albitite 2.00 1.00 785 2600 0.02 0.02 0.40 0.04 Andesite 2.70 1.00 820 2650 1.35 2.50 1.40 0.64 Anorthosite 1.70 1.00 750 2800 1.00 3.50 0.45 0.56 Basalt, normal 1.80 1.17 800 2870 0.90 2.70 0.80 0.52 Basalt, weathered 2.10 1.17 790 2870 0.90 2.70 0.80 0.52 Bronzitite 3.80 1.00 760 3450 0.02 0.02 0.05 0.01 Diabase 2.60 1.00 800 2800 0.25 0.90 0.45 0.18 Diorite 2.70 1.00 1140 2900 2.00 8.50 1.10 1.29 Dolerite 2.30 1.00 900 2930 0.40 1.60 0.70 0.30 Dunite 3.80 1.30 720 3310 0.004 0.004 0.01 0.00 Gabbro 2.60 1.00 800 2870 0.25 0.80 0.50 0.18 Granite, 150 My old 2.60 1.15 800 2650 6.50 17.00 5.70 3.32 Granite, 500 My old 2.60 1.15 800 2650 4.50 15.00 4.50 2.57 Granite, 1000 My old 2.60 1.15 800 2650 4.00 13.50 3.50 2.25 Granodiorite 2.60 1.00 730 2720 2.30 9.30 2.80 1.51 Harzburgite 6.90 1.00 760 3300 0.02 0.02 0.05 0.01 Hypersthenite 4.10 1.00 760 3450 0.02 0.02 0.05 0.01 Lamprophyre 2.50 1.00 760 3000 1.20 5.50 4.10 1.19 Lherzolite 3.30 1.00 780 3150 0.015 0.05 0.0006 0.01 Monzonite 2.70 1.00 740 2600 2.70 8.50 4.00 1.60 Monzonite, quartz 2.80 1.00 880 2700 6.50 27.00 3.40 3.86 Nepheline syenite 2.40 1.00 750 2650 9.10 27.00 4.00 4.50 Norite 2.20 1.00 670 2860 0.15 0.50 0.30 0.11 Olivinite 3.20 1.00 800 3450 0.01 0.08 0.05 0.02 Peridotite, typical 4.00 1.00 800 3200 0.025 0.025 0.055 0.02 Peridotite, serpentinized 2.20 1.00 700 3100 0.03 0.03 0.01 0.01 Pyroxenite 3.80 1.00 1000 3220 0.20 0.55 0.75 0.19 Quartz monzonite 2.80 1.00 880 2700 6.50 27.00 3.40 3.86 Rhyolite 2.60 1.20 800 2500 5.80 13.00 3.70 2.53 Syenite, typical 2.60 1.00 800 2760 2.70 15.00 4.70 2.22 Syenite, nepheline 2.40 1.00 750 2650 9.10 27.00 4.00 4.50 Tonalite 2.55 1.00 800 2800 0.00 12.00 2.50 1.10 Ultramafics 3.80 1.00 900 3310 0.001 0.004 0.003 0.00 Table E.4. Thermal Properties: Igneous Rocks
- 436. E Thermal Properties 425 Vert. An- Sp. Th. Den. U Th K Radio- Cond. iso- Heat Sort. active tr. Cap. Fac. Heat W mK J kgK f kg m3 [ppm] [ppm] [%] μW m3 Sandstone typical 3.95 1.15 855 1.00 2720 1.30 3.50 1.30 0.70 clay–rich 3.35 1.20 860 1.00 2760 1.50 5.10 3.60 1.10 clay–poor 5.95 1.05 820 1.00 2700 0.70 2.30 0.60 0.40 quartzite, typical 6.15 1.08 890 1.00 2640 0.60 1.80 0.90 0.36 quart., very quartz–rich 6.45 1.06 890 1.00 2640 0.50 1.70 0.70 0.30 subarkose, typical 4.55 1.14 870 1.00 2680 1.00 3.30 1.30 0.60 subarkose, quartz–rich 5.05 1.13 880 1.00 2650 0.90 2.70 0.90 0.49 subarkose, clay–rich 3.40 1.40 870 1.00 2690 1.20 3.90 2.30 0.79 subarkose, clay–poor 4.80 1.09 870 1.00 2700 0.70 2.30 0.60 0.40 subarkose, dolomite–rich 4.10 1.07 840 1.00 2710 0.90 2.70 0.90 0.50 arkose, typical 3.20 1.25 845 1.00 2730 3.00 8.00 2.50 1.58 arkose, quartz–rich 4.05 1.20 860 1.00 2690 2.00 6.00 0.90 1.01 arkose, quartz–poor 2.00 1.50 835 1.00 2770 3.00 8.00 2.50 1.60 arkose, clay–rich 2.30 1.45 865 1.00 2760 2.00 7.00 3.60 1.37 arkose, clay–poor 4.00 1.12 835 1.00 2710 2.00 7.00 1.00 1.10 arkose, dolomite–rich 4.35 1.05 850 1.00 2750 1.50 5.10 3.60 1.10 wacke 2.60 1.40 850 1.00 2780 2.50 8.00 2.50 1.47 Shale typical 1.64 1.60 860 1.38 2700 3.70 12.0 2.70 2.03 organic–lean, typical 1.70 1.55 860 1.35 2700 3.70 12.0 2.70 2.03 organic–lean, sandy 1.84 2.00 860 1.64 2700 2.80 11.0 2.50 1.71 organic–lean, silty 1.77 1.80 860 1.51 2700 3.00 11.0 2.60 1.78 organic–lean siliceous, typical 1.90 1.17 860 1.00 2710 2.00 4.5 2.00 1.02 siliceous, 95% Opal–CT 1.75 1.16 860 1.00 2330 1.00 1.3 1.00 0.38 black 0.90 3.27 940 2.40 2500 19.00 11.0 2.50 5.44 organic–rich, typical 1.25 2.19 900 1.76 2600 5.00 12.0 2.80 2.29 organic–rich, 3% TOC 1.45 1.84 880 1.54 2610 5.00 12.0 2.80 2.30 organic–rich, 8% TOC 1.20 2.20 900 1.74 2500 10.00 11.0 2.90 3.34 organic–rich, 20% TOC 0.80 3.09 980 2.22 2270 20.00 11.0 2.60 5.17 Siltstone organic–lean 2.05 1.50 910 1.00 2720 2.00 5.00 1.00 0.96 org.–rich, typical 2.01 1.71 940 1.47 2710 2.00 5.00 1.00 0.96 org.–rich, 10% TOC 1.82 2.64 960 2.03 2550 15.00 6.00 2.00 4.21 org.–rich, 2-3% TOC 2.00 1.76 930 1.52 2700 2.50 6.50 2.00 1.28 Conglomerate, typical 2.30 1.05 820 1.00 2700 1.50 4.00 2.00 0.85 Conglomerate, quartzitic 6.10 1.05 780 1.00 2700 1.00 4.00 1.20 0.65 Tuff, felsic 2.60 1.17 830 1.00 2650 3.00 6.50 3.90 1.56 Tuff, basaltic 1.90 1.17 830 1.00 2900 0.65 2.20 0.85 0.43 Table E.5. Thermal Properties: Sedimentary Rocks, Part 1 – Clastic Sediments
- 437. 426 E Thermal Properties Vert. An- Sp. Th. Den. U Th K Radio- Cond. iso- Heat Sort. active tr. Cap. Fac. Heat W mK J kgK f kg m3 [ppm] [ppm] [%] μW m3 Biogenic Sediments Chalk, typical 2.90 1.07 850 1.00 2680 1.90 1.40 0.25 0.60 Chalk, 95% calcite 3.00 1.04 830 1.00 2680 1.90 1.40 0.25 0.60 Chalk, 75% calcite 2.65 1.15 840 1.00 2680 1.90 1.40 0.25 0.60 Chalk, 40% calcite 3.20 1.10 860 1.00 2680 1.90 1.40 0.25 0.60 Coal, pure 0.30 1.20 1300 1.03 1600 1.50 3.00 0.55 0.38 Coal, impure 1.00 1.15 1200 1.00 1600 2.00 3.00 0.70 0.47 Coal, silty 1.60 1.30 1100 1.00 1600 2.00 3.00 0.70 0.47 Diatomite, clay–poor 1.60 1.00 790 1.00 2100 1.30 1.00 1.00 0.39 Diatomite, clay–rich 1.50 1.10 800 1.00 2300 3.00 2.60 2.50 1.01 Kerogen 1.00 1.00 6.3 1.00 1100 100.00 0.00 0.00 10.47 Carbonate Rocks Limestone ooid grainstone 3.00 1.19 835 1.11 2740 1.00 1.00 0.20 0.35 Waulsortian mound 3.00 1.19 835 1.11 2740 1.00 1.00 0.20 0.35 micrite 3.00 1.19 835 1.11 2740 1.00 1.00 0.20 0.35 shaly 2.30 1.70 850 1.38 2730 2.00 4.00 1.00 0.89 org.–rich, typical 2.00 1.95 845 1.47 2680 5.00 1.50 0.26 1.40 org.–rich, 1-2% TOC 2.63 1.40 840 1.16 2710 2.50 1.70 0.27 0.79 org.–rich, 10% TOC 1.45 2.68 850 1.84 2550 10.00 1.40 0.25 2.54 chalk, typical 2.90 1.07 850 1.00 2680 1.90 1.40 0.25 0.60 chalk, 95% calcite 3.00 1.04 830 1.00 2680 1.90 1.40 0.25 0.60 chalk, 75% calcite 2.65 1.15 840 1.00 2680 1.90 1.40 0.25 0.60 chalk, 40% calcite 3.20 1.10 860 1.00 2680 1.90 1.40 0.25 0.60 Marl 2.00 1.45 850 1.00 2700 2.50 5.00 2.00 1.18 Dolomite typical 4.20 1.06 860 1.00 2790 0.80 0.60 0.40 0.29 organic–lean, sandy 4.25 1.06 860 1.00 2770 0.90 0.90 0.70 0.37 organic–lean, silty 3.00 1.10 860 1.00 2760 0.90 0.90 0.80 0.38 organic–rich 2.15 2.44 910 1.76 2600 10.00 1.40 0.60 2.62 Table E.6. Thermal Properties: Sedimentary Rocks, Part 2
- 438. E Thermal Properties 427 Vert. An- Sp. Th. Den. U Th K Radio- Cond. iso- Heat Sort. active tr. Cap. Fac. Heat W mK J kgK f kg m3 [ppm] [ppm] [%] μW m3 Chemical Sediments Polyhalite 1.00 1.00 700 1.00 2780 0.02 0.01 12.90 1.25 Salt 6.50 1.01 860 1.00 2740 0.02 0.01 0.10 0.02 Sylvinite 1.00 1.02 685 1.00 2100 0.02 0.01 20.60 1.51 Anhydrite 6.30 1.05 750 1.00 2970 0.10 0.30 0.40 0.09 Chert 4.80 1.00 890 1.00 2650 1.00 1.00 0.70 0.38 Gypsum 1.50 1.15 1100 1.00 2320 0.08 0.20 0.30 0.05 Halite 6.50 1.01 860 1.00 2200 0.02 0.01 0.10 0.01 Table E.7. Thermal Properties: Sedimentary Rocks, Part 3
- 439. F Analytic Solutions to Selected Heat Flow Problems Heat flow and temperature distributions in sedimentary basins are usually formulated in terms of differential equations of diffusion type with boundary conditions T = Tswi at the sediment water interface, ∇T = 0 at the sides and −λ∇T = q below basement. The first boundary condition is obvious. The second describes a condition which does not allow lateral heat flow at the sides. This approximation can be justified if the lateral extension of the model is large compared to its thickness. The last condition is not obvious from scratch. The basal heat flow is not well known and it does not seem natural to choose the heat flow as the important parameter describing the boundary condition. Temperature seems to be more obvious but is usually not known either. However, it is often assumed that the temperature distribution is in or near steady state and that the heat flow boundary condition is independent of the thickness of the basin. It is therefore robust under variations of the stratigraphic interpretation, whereas the temperature in the basement varies strongly with its thickness and structure. Following an argumentation which includes crust and mantle in the heat flow analysis, such as the McKenzie stretching models, it is possible to estimate the heat flow by q = Ta − Ts h λ (F.1) with Ta as the temperature of the asthenosphere, Ts the temperature below the sediments, h the thickness of the basin, and λ its average thermal con- ductivity (Sec. 3.8). An approach with inclusion of the sediments into (F.1) yields only minor corrections to the overall temperature difference in the nu- merator, thickness in the denominator and overall thermal conductivity. Thus numerous uncertain parameters such as the thermal conductivity of crust and mantle and their thicknesses are mapped into one key quantity, namely the basal heat flow. A few heat flow examples that can be solved analytically are presented in the following. These examples can be used for basic understanding and for validating numerical algorithms.
- 440. 430 F Analytic Solutions to Selected Heat Flow Problems F.1 Influence of Radiogenic Heat Production on a Steady State Temperature Profile q Ts Q l z h Fig. F.1. Model with radioactive heat production Figure F.1 exhibits a simple one layer model. The major difference from the steady state models of Sec. 3.2.1 is the inclusion of radioactive heat pro- duction. The differential equation becomes λ d2 dz2 T = −Q (F.2) with constant thermal conductivity λ and radiogenic heat production Q. The boundary conditions are T = Ts at top surface and λT (h) = q with q as the basal heat flow. The solution can be easily evaluated to T = Ts + q λ z − Q λ z(z − 2h) 2 . (F.3) Some example curves are shown in Fig. F.2 Thus, in steady state situations, a constant heat flow q is related to a linear temperature rise. A constant heat production Q introduces a parabolic dependence of temperature in depth. The heat flow is linearly depth dependent and becomes λT = q − Q (z − h). Note that the heat flow is defined here by q = −qez with ez pointing downward and q positive for flow upwards. F.2 Steady State Temperature Profile with a Lateral Basal Heat Flow Jump A model with a lateral basal heat flow jump from q1 to q2 is studied. It is exhibited in Fig. F.3, does not contain radioactive heat production and can be formulated as ∂2 ∂x2 T(x, z) + ∂2 ∂z2 T(x, z) = 0 (F.4) with the boundary conditions
- 441. F.2 Steady State Temperature Profile with a Lateral Basal Heat Flow Jump 431 Fig. F.2. Some temperature profiles with varying radiogenic heat production. Here Ts = 0◦ C, λ = 2 W/m/K and q = 50 mW/m2 q Ts l z h q 1 2 x w w Fig. F.3. Model with lateral heat flow jump T(x, 0) = Ts for 0 x 2w ∂ ∂z T(x, z) z=h = q1 λ for 0 x w ∂ ∂z T(x, z) z=h = q2 λ for w x 2w ∂ ∂x T(x, z) x=0 = ∂ ∂x T(x, z) x=2w = 0 for 0 z h . (F.5) The analytical solution can be obtained via a separation technique. It is T(x, z) = Ts + q1 + q2 2λ z + q2 − q1 λ ∞ n=1 (−1)n sinh(μnz) cos(μnx) wμ2 n cosh(μnh) (F.6) with μn = (n − 1/2)π/w. An example is shown in Fig. F.4.
- 442. 432 F Analytic Solutions to Selected Heat Flow Problems Fig. F.4. Temperatures at given depth according to (F.6) with w = 50 km, h = 10 km, Ts = 0◦ C, λ = 2 W/m/K, q1 = 50 mW/m2 and q2 = 100 mW/m2 F.3 Steady State Temperature Profile with SWI Temperature Jump q T 1 l z h 2 x w w T Fig. F.5. Model with SWI temperature jump This model is similar to Fig. F.3. It differs in that there is a jump in SWI temperature instead of basal heat flow as shown in Fig. F.5. Thus only the first three boundary conditions of (F.5) must be modified to T(x, 0) = T1 for 0 x w T(x, 0) = T2 for w x 2w ∂ ∂z T(x, z) z=h = q λ for 0 x 2w . (F.7)
- 443. F.4 Steady State Temperature Profile for a Two Block Model 433 The solution is T(x, z) = q λ z+ T1 + T2 2 +(T2 −T1) ∞ n=1 (−1)n cosh(μn(z − h)) cos(μnx) μnw cosh(μnh) (F.8) again with μn = (n − 1/2)π/w. An example is shown in Fig. F.6. Fig. F.6. Temperatures at given depth according to (F.8) with w = 50 km, h = 10 km, T1 = 0◦ C, T2 = 20◦ C, λ = 2 W/m/K and q = 50 mW/m2 F.4 Steady State Temperature Profile for a Two Block Model q Ts l z h q 1 2 x w w l Fig. F.7. Two block model with jump in thermal conductivity λ
- 444. 434 F Analytic Solutions to Selected Heat Flow Problems The model consists of two equal sized rectangular blocks with different thermal conductivities. It is depicted in Fig. F.7 and its mathematical formu- lation is given by T(x, 0) = Ts for −w x w ∂ ∂z T(x, z) z=h = q λ1 for 0 x w ∂ ∂z T(x, z) z=h = q λ2 for −w x 0 ∂ ∂x T(x, z) x=−w = ∂ ∂x T(x, z) x=w = 0 for 0 z h T(0+, z) = T(0−, z) for 0 z h λ1 ∂ ∂x T(x, z) x=0+ = λ1 ∂ ∂x T(x, z) x=0− for 0 z h . (F.9) Explicitly one obtains T(x, z) = Ts + q λ1 z + q λ2 λ2 − λ1 λ1 + λ2 × ∞ n=−∞ (−1)n hμ2 n sin(μnz) (cosh(μnx) − tanh(μnw) sinh(μnx)) (F.10) for 0 x w and T(x, z) = Ts + q λ2 z − q λ1 λ2 − λ1 λ1 + λ2 × ∞ n=−∞ (−1)n hμ2 n sin(μnz) (cosh(μnx) + tanh(μnw) sinh(μnx)) (F.11) for −w x 0 with μn = (n + 1/2)π/h. An example is shown in Fig. F.8. F.5 Non Steady State Model with Heat Flow Jump The model consists of a layer, which is extended to infinity in the horizontal direction. In the vertical direction the temperature is set to Ts at the upper side. In addition the layer is exposed to a constant heat flow at its bottom side. At time t = 0 the heat flow is switched to another value. The heat flow equation becomes ρc ∂ ∂t T(z, t) = λ ∂2 ∂z2 T(z, t) . (F.12)
- 445. F.5 Non Steady State Model with Heat Flow Jump 435 Fig. F.8. Temperatures at given depth according to (F.10) and (F.11) with w = 50 km, h = 10 km, TS = 0◦ C, λ1 = 2 W/m/K, λ2 = 4 W/m/K, and q = 50 mW/m2 The density ρ, the specific heat capacity c, and the thermal conductivity are assumed constant. The boundary conditions are T(0, t) = Ts for all t ∂ ∂z T(z, t) z=h = q1 λ for t 0 ∂ ∂z T(z, t) z=h = q2 λ for t 0 . (F.13) The temperature should be in steady state for t 0: T(z, t) = Ts + q1 λ z . (F.14) For t 0 it becomes in explicit form T(z, t) = Ts + q2 λ z + q1 − q2 λ ∞ n=−∞ (−1)n hμ2 n sin(μnz) exp(−μ2 nλt/ρc) (F.15) with μn = (n + 1/2)π/h. An example is shown in Fig. F.9.
- 446. 436 F Analytic Solutions to Selected Heat Flow Problems Fig. F.9. Temperatures at given depth according to (F.15) with h = 10 km, Ts = 0◦ C, q1 = 50 mW/m2 , q2 = 100 mW/m2 , λ = 2 W/m/K, ρ = 2700 kg/m3 , and c = 860 J/kg/K F.6 Non Steady State Model with SWI Temperature Jump The model is almost the same as in Sec. F.5. Only the boundary conditions (F.13) are different: ∂ ∂z T(z, t) z=h = q λ for all t T(0, t) = T1 for t 0 T(0, t) = T2 for t 0 . (F.16) Again temperature is in steady state for t 0: T(z, t) = T1 + q λ z . (F.17) For t 0 it is T(z, t) = T2 + q λ z + (T2 − T1) ∞ n=−∞ (−1)n μnh sin(μn(z − h)) exp(−μ2 nλt/ρc) (F.18) with μn = (n + 1/2)π/h.
- 447. F.7 An Estimate for the Impact of Continuous Deposition on Heat Flow 437 Fig. F.10. Temperatures at given depth according to (F.18) with h = 10 km, T1 = 0◦ C, T2 = 20◦ C, q = 50 mW/m3 , λ = 2 W/m/K, ρ = 2700 kg/m3 , and c = 860 J/kg/K F.7 An Estimate for the Impact of Continuous Deposition on Heat Flow An example of continuous deposition, a hiatus, and a following erosion with its impact on heat flow is shown in Fig. 3.4 and discussed in Sec. 3.2.2. A derivation of (3.8) is given below. For a formulation of the heat flow problem with continuous deposition as shown in Fig. 3.4 it must be considered that the bulk rock including its captured heat subsides with velocity v = S which is here equal to the sedi- mentation rate S. Equation (3.7) must therefore be enhanced by an additional term proportional to ρ c v∂T/∂z similar as in (3.29) or (3.42). It becomes ∂T ∂t + S ∂T ∂z − λ ρ c ∂2 T ∂z2 = 0 (F.19) with boundary conditions T(0, t) = 0, λ ∂ ∂z T(z, t) z=h = q (F.20) for all t and h(t) = S t (Fig. F.11). Transient heat flow effects are often negligible in geological systems. Ten- tatively, the ∂T/∂t term is discarded here. The boundary value problem can now easily be solved and the temperature becomes
- 448. 438 F Analytic Solutions to Selected Heat Flow Problems T=0 h(t)=S t q l rc Fig. F.11. Illustration of (F.19) and (F.20) T = q λk e−kSt ( ekz − 1 ) (F.21) with k = Sρc/λ. The factor k is independent of time t and therefore it follows directly ∂T ∂t = −kS T . (F.22) An example value of k can be calculated for a typical shale with 10 % porosity, a heat capacity of 4000 J/kg/K for water and a sedimentation rate of S = 1 km/Ma as k = Sρc λ = 1000 m 2700 kg/m3 3.1536 × 1013s (0.9 × 860 + 0.1 × 4000) J kg K 1 1.64 J/s/m/K = 0.0613/km . (F.23) This is a rather small value and the factor kS is here with kS = 0.0613/Ma also very small. Finally it can be assumed that the transient term ∂T/∂t is commonly small compared with other terms in (F.19).1 Formula (F.21) is thus a good approximation to a solution of (F.19) if the sedimentation rate S is not too large.2 It is further possible to expand the approximation (F.21) for small K = kh which yields for the heat flow λ ∂T ∂z = q − Δq with Δq = qkx = qSρc λ x (F.24) 1 More evidence can be achieved with following argument: a small geological tem- perature gradient is 30 ◦ C/km. The first term of (F.19) can be estimated with kST and the second with S 30 ◦ C/km. The first term can hence in example Fig. 3.4 be neglected if T 30/0.0613 ≈ 500◦ C. 2 The pre-factor kS in (F.22) is proportional to S2 whereas the spatial variation is proportional to S for small S according to (F.21) or (F.24). Hence the transient term decays much faster than the spatial terms for a small decreasing S.
- 449. F.7 An Estimate for the Impact of Continuous Deposition on Heat Flow 439 and x = h − z which is the height above the basement. For a typical shale it is hence Δq = 0.0613 qS̄x̄ with S̄ and x̄ as dimensionless numbers. It is S̄ in km/Ma and x̄ in km. It follows with q = 60 mW/m2 for an example such as in Fig. 3.4 Δq = 3.68 S̄x̄ mW/m2 . This approximation is only valid for K = 0.0613 S̄h̄ 1 again with h̄ as h in units of km. This implies roughly S̄h̄ 10 or Sh 10 km2 /Ma. However, a closed expression for a solution without approximation can also be achieved by first transforming the differential equation from depth z to the height x = h − z over the basement and introduction of the function u = T − q(x − h)/λ. It can easily be shown, that the differential equation and the boundary conditions for u(x, t) are identical to the consolidation problem from Gibson in App. D. Obviously, the parameter Δρg must be substituted by −q/λ, the other parameter names are already chosen identical. Additionally, the solution can be expanded for small K = kh as already demonstrated in (D.4) which yields for K → 0 the same result as (F.24). A few example curves of the full solution for (F.19) and (F.20) are plotted in Fig. F.12 and the approximation (F.21) in Fig. F.13. The error is below 0.3% for the case kh = 0.001 and below 8% for kh = 0.1. Approximation (F.21) can be improved for larger kh if k is replaced by k according to Fig. F.14. The value of k is calculated from the exact solution at z = h. It must finally be noted that (F.21) does not incorporate transient effects which arise due to improper initial conditions. Such effects might need a few million years to decay, which can be seen very clearly in Fig. 3.4 for the case of erosion.
- 450. 440 F Analytic Solutions to Selected Heat Flow Problems T /qh l z/h kh = 0.001 0.1 0.5 1 2 5 Fig. F.12. Example curves of the solution for (F.19) and (F.20) T /qh l z/h kh = 0.001 0.1 0.5 1 2 5 Fig. F.13. Approximation according to (F.21) z/h T /qh l kh = 1 kh = 2 kh = 5 Fig. F.14. Approximation (dashed) according to (F.21) with a newly adapted k . Here it is found k h = 0.678 for kh = 1, k h = 1.103 for kh = 2, and k h = 1.941 for kh = 5
- 451. G Petroleum Kinetics 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 0 5 10 15 20 25 30 35 Activation Energy in kcal/mol Frequency in % 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 0 10 20 30 40 50 60 70 Activation Energy in kcal/mol Frequency in % 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 0 20 40 60 80 100 Activation Energy in kcal/mol Frequency in % 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 0 20 40 60 80 100 Activation Energy in kcal/mol Frequency in % 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 0 10 20 30 40 Activation Energy in kcal/mol Frequency in % 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 0 10 20 30 40 50 60 Activation Energy in kcal/mol Frequency in % Type I Formation: Mae Sot Age: Miocene Location: Thailand Basin: Mae Sot A = 1.42x1026 My-1 HI = 704 mg/gTOC 0 Type I Formation: Green River Shale Age: Eocene Location: Utah Basin: Uinta A = 5.58x1027 My-1 HI = 867 mg/gTOC 0 Type I Formation: Tasmanites Age: Cretaceous Location: Alaska Basin: North Slope A = 3.78x1029 My-1 HI = 941 mg/gTOC 0 Type II Formation: Bakken Age: Devionian-Mississipian Location: North Dakota Basin: Williston A = 5.61x1027 My-1 HI = 439 mg/gTOC 0 Type II Formation: Barnett Shale Age: Mississipian Location: Texas Basin: Forth Worth A = 9.15x1027 My-1 HI = 381 mg/gTOC 0 Type II Formation: La Luna Age: Cretaceous Location: Venezuela Basin: Maracaibo A = 2.43x1027 My-1 HI = 661 mg/gTOC 0 Fig. G.1. Bulk kinetics after Tegelaar and Noble (1994)
- 452. 442 G Petroleum Kinetics 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 0 2 4 6 8 10 12 14 Activation Energy in kcal/mol Frequency in % 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 0 20 40 60 80 100 Activation Energy in kcal/mol Frequency in % 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 0 10 20 30 40 Activation Energy in kcal/mol Frequency in % 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 0 10 20 30 40 50 60 70 Activation Energy in kcal/mol Frequency in % Type II Formation: Monterey Age: Miocene Location: California Basin: Ventura A = 3.07x1027 My-1 HI = 531 mg/gTOC 0 Type II Formation: Pematang Age: Eocene Location: Indonesia Basin: Central Sumatra A = 2.06x1028 My-1 HI = 399 mg/gTOC 0 A = 3.82x1028 My-1 HI = 455 mg/gTOC 0 Type II Formation: Woodford Age: Devonian-Mississipian Location: Oklahoma Basin: Ardmore A = 5.42x1026 My HI = 1001 mg/gTOC 0 Type I, Sulfur rich Formation: Ribesalbes Age: Miocene Location: Spain -1 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 0 10 20 30 40 50 60 Activation Energy in kcal/mol Frequency in % Type I Formation: Talang Akar (Resin) Age: Oligocene Location: Indonesia Basin: Ardjuna A = 2.0x1025 My-1 HI0 = 843 mg/gTOC Fig. G.2. Bulk kinetics after Tegelaar and Noble (1994) continued
- 453. G Petroleum Kinetics 443 46 48 50 52 54 56 58 60 62 64 66 68 0 20 40 60 80 Activation Energy in kcal/mol Frequency in % 50 52 54 56 58 60 62 64 66 68 70 72 74 0 10 20 30 40 Activation Energy in kcal/mol Frequency in % Type II (North Sea) A = 4.73x1027 My-1 HI = 390 mg/gTOC 0 C15+ C6-C14 C2-C5 C1 Type III (North Sea) A = 1.73x1028 My-1 HI = 274 mg/gTOC 0 C15+ C6-C14 C2-C5 C1 Fig. G.3. Multicomponent kinetics after Ungerer (1990) 46 48 50 52 54 56 58 60 62 64 66 0 10 20 30 40 50 Activation Energy in kcal/mol Frequency in % 54 56 58 60 62 64 66 68 70 72 74 0 5 10 15 20 25 30 35 Activation Energy in kcal/mol Frequency in % A = 5.05x1027 My-1 HI = 578 mg/gTOC 0 Vandenbroucke (Type II) North Sea C14+NSO C14+ARO U C14+NSAT C14+ISAT C6-C13 ARO C6-C13 SAT C3-C5 Ethane Methane A = 9.78x1028 My-1 HI = 232 mg/gTOC 0 Vandenbroucke (Type III) North Sea C14+NSO C14+ARO U C14+NSAT C14+ISAT C6-C13 ARO C6-C13 SAT C3-C5 Ethane Methane Fig. G.4. Multicomponent kinetics after Vandenbroucke et al. (1999)
- 454. 444 G Petroleum Kinetics 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 0 20 40 60 80 100 Activation Energy in kcal/mol Frequency in % 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 0 5 10 15 20 25 Activation Energy in kcal/mol Frequency in % 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 0 5 10 15 20 25 Activation Energy in kcal/mol Frequency in % 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 0 10 20 30 40 Activation Energy in kcal/mol Frequency in % 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 0 5 10 15 20 25 30 35 Activation Energy in kcal/mol Frequency in % 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 0 100 200 300 400 Activation Energy in kcal/mol Frequency in % Woodford Shale Alaskan Tasmanite Teruel Oil Shale Toarcian Shale Brown Limestone A = 8.111x1026 My-1 HI = 440 mg/gTOC 0 C60 C50 C40 C30 C20 C10 C6 nC5 iC5 nC4 iC4 C3 C2 C1 A = 1.35x1027 My-1 HI = 749 mg/gTOC 0 C60 C50 C40 C30 C20 C10 C6 nC5 iC5 nC4 iC4 C3 C2 C1 A = 9.767x1027 My-1 HI = 226 mg/gTOC 0 C60 C50 C40 C30 C20 C10 C6 nC5 iC5 nC4 iC4 C3 C2 C1 A = 5.74x1026 My-1 HI = 755 mg/gTOC 0 C60 C50 C40 C30 C20 C10 C6 nC5 iC5 nC4 iC4 C3 C2 C1 Tertiary Coal A = 1.908x1026 My-1 HI = 610 mg/gTOC 0 C60 C50 C40 C30 C20 C10 C6 nC5 iC5 nC4 iC4 C3 C2 C1 A = 5.538x1025 My-1 HI = 788 mg/gTOC 0 C60 C50 C40 C30 C20 C10 C6 nC5 iC5 nC4 iC4 C3 C2 C1 Fig. G.5. 14 component kinetics after di Primio and Horsfield (2006)
- 455. H Biomarker With much help from K. E. Peters the following tables are compiled from K. E. Peters, C. C. Walters, and J. M. Moldowan. The Biomarker Guide, volume 1 and 2. Cambridge University Press, second edition, 2005. Compound Biological Origin Environment 2-Methyldocosane Bacteria? Hypersaline Mid-chain Cyanobacteria Hot springs, marine monomethylalkanes Pristane/phytane (low) Phototrophs, archaea Anoxic, high salinity PMI (PME)∗ Archaea, methanogens Hypersaline, anoxic and methanotrophs Crocetane Archaea, methanotrophs? Methane seeps? C25 HBI∗∗ Diatoms Marine and lacustrine Squalane Archaea Hypersaline? Botryococcane, Green algae, Lacustrine-brackish-saline polymethylsqualanes Botryococcus Table H.1. Acyclic biomarkers as indicators of biological input or depositional environment (assumes high concentration of component) ∗ PMI: 2,6,10,15,19-pentamethylicosane (current IUPAC nomenclature), previously spelled pentamethyleicosane (PME). ∗∗ C25 HBI: 2,6,10,14-tetramethyl-7-(3-methylpentyl)-pentadecane.
- 456. 446 H Biomarker Compound Biological Origin Environment Saturates C25 − C34 macrocyclic Green algae, Lacustrine-brackish alkanes Botryococcus β-carotane Cyanobacteria, algae Arid, hypersaline Phyllocladanes Conifers Terrigenous C19 − C30 tricyclic terpanes Tasmanites? Marine, high latitude C30 24-n-propylcholestanes Chrysophyte algae Marine (4-desmethyl) Pregnane, homopregnane Unknown Hypersaline Diasteranes Algae/higher plants Clay-rich rocks Dinosteranes Dinoflagellates Marine, Triassic or younger 28,30-bisnorhopane Bacteria Anoxic marine, upwelling? Bicadinanes Higher plants Terrigenous 23,28-bisnorlupanes Higher plants Terrigenous Gammacerane Tetrahymanol in ciliates Stratified water, feeding on bacteria sulfate-reducing, hypersaline (low sterols) 18α-oleanane Cretaceous or younger, Paralic higher plants Hexahydro-benzohopanes Bacteria Anoxic carbonate-anhydrite Aromatics Benzothiophenes, Unknown Carbonate/evaporite alkyldibenzo-thiophenes Methyl n-pristanyl, Chlorobiaceae, methy i-butyl maleimides, anaerobic green sulfur Photic zone anoxia isorenieratane bacteria Trimethyl chromans∗ Phytoplankton Saline photic zone? Table H.2. Cyclic biomarkers as indicators of biological input or depositional en- vironment (assumes high concentration of component) ∗ Trimethyl chromans: 2-methyl-2-(4,8,12-trimethyl-tridecyl)-chromans.
- 457. H Biomarker 447 Characteristics Shales Carbonates Non-biomarker Parameters API, Gravity Medium-High Low-Medium Sulfur, wt.% Variable High (marine) Thiophenic sulfur Low High Biomarker Parameters Pristane/phytane High (≥ 1) Low (≤ 1) Steranes/17α-hopanes High Low Diasteranes/steranes High Low C24 Tetra-/C26 tricyclic diterpanes Low-Medium Medium-High C29/C30 Hopane Low High ( 1) C35 Homohopane index Low High Hexahydrobenzohopanes and benzohopanes Low High Dia/(Reg + Dia) MA-steroids∗ High Low Ts/(Ts + Tm)∗∗ High Low C29 MA-steroids Low High Table H.3. Some characteristics of petroleum from carbonate versus shale source rocks ∗ Monoaromatic-steroid ratio ∗∗ Tm: C27 17α-trisnorhopane, Ts: C27 18α-trisnorhopane II
- 458. 448 H Biomarker Property Marine Terrigenous Lacustrine Sulfur (wt.%) High (anoxic) Low Low Pristane/phytane 2 3 ∼ 1 − 3 C27 − C29 steranes High C28 High C29 High C27 C30 24-n-propylcholestane Low Low or absent Absent Steranes/hopanes High Low Low Bicyclic sesquiterpanes Low High Low Tricyclic diterpanes Low High High Tetracyclic diterpanes Low High Low Lupanes, bisnorlupanes Low High Low 28,30-bisnorhopane High (anoxic) Low Low Oleananes Low or absent High Low β-carotane Absent Absent High (arid) Botryococcane Absent Absent High (brackish) V/(V + Ni) High (anoxic) Low or absent Low or absent Table H.4. Generalized geochemical properties∗ differ between nonbiodegraded crude oils from marine, terrigenous, or lacustrine source-rock organic matter (mod- ified from Peters and Moldowan, 1993) ∗ Quoted properties encompass most samples, but exceptions occur. For example, many nearshore oxic marine environments resulted in source rocks that generated oils with low sulfur and some very high sulfur oils originated from source rocks deposited in hypersaline lacustrine settings. The terms marine, terrigenous, and lacustrine can be misleading. “Marine oil” might refer to: (1) oil produced from marine reservoir rock, (2) oil generated from source rock deposited under marine conditions, or (3) oil derived from marine organic matter in the source rock. The table refers to provenance of the organic matter (3).
- 459. I Component Properties The following tables with component properties are compiled from: A. Danesh. PVT and Phase Behaviour of Petroleum Reservoir Fluids. Number 47 in Developments in petroleum science. Elsevier, 1998, R. C. Reid, J. M. Prausnitz, and B. E. Poling. The Properties of Gases and Liquids. McGraw–Hill Book Company, 4th edition, 1987, IES PetroMod® , Petroleum Systems Modeling Software, Release 10.0, 2007, URL ies.de. Cells are blanked if there was no data available. The definition of the listed quantities can be found in Chapter 5. Note that some values, such as the molar weight of C7+ or C15+, are obviously not representative for all fluids. This are example values which must be adapted.
- 460. 450 I Component Properties Name MW Tc Pc vc Acentric Rackett Shift ρ g mol [K] [MPa] m3 kmol Fac. ωc ZRa Fac. SE kg m3 Oil: Alkanes n–Hexane 86.177 507.6 3.025 0.371 0.3013 0.26355 −0.01478 663.8 n–Heptane 100.204 540.2 2.740 0.428 0.3495 0.26074 0.02509 688.2 n–Octane 114.231 568.7 2.490 0.486 0.3996 0.25678 0.04808 707.0 n–Nonane 128.258 594.6 2.290 0.544 0.4435 0.25456 0.06799 721.9 n–Decane 142.285 617.7 2.110 0.600 0.4923 0.25074 0.08546 734.2 n–Undecane 156.312 639.0 1.949 0.659 0.5303 0.24990 0.10100 744.5 n–Dodecane 170.338 658.0 1.820 0.716 0.5764 0.24692 0.11500 752.7 n–Tridecane 184.365 675.0 1.680 0.775 0.6174 0.24698 0.12764 761.7 n–Tetradecane 198.392 693.0 1.570 0.830 0.6430 0.24322 0.13923 763.3 n–Pentadecane 212.419 708.0 1.480 0.889 0.6863 0.23030 0.14989 772.2 n–Hexadecane 226.446 723.0 1.400 0.944 0.7174 0.22760 0.15974 777.2 n–Heptadecane 240.473 736.0 1.340 1.000 0.7697 0.23431 0.16890 779.7 n–Octadecane 254.500 747.0 1.270 1.060 0.8114 0.22917 0.17744 782.0 n–Nonadecane 268.527 758.0 1.210 1.120 0.8522 0.21580 0.18545 786.9 n–Eicosane 282.553 768.0 1.160 1.170 0.9069 0.22811 0.19297 792.7 n–Heneicosane 296.580 781.7 1.147 1.198 0.9220 0.20970 0.20007 795.4 n–Docosane 310.610 791.8 1.101 1.253 0.9550 0.20680 0.20678 798.1 n–Tricosane 324.630 801.3 1.059 1.307 0.9890 0.20380 0.21314 800.4 n–Tetracosane 338.680 810.4 1.019 1.362 1.0190 0.20110 0.21919 802.5 Oil: Alkenes 1–Hexene 84.161 504.03 3.140 0.3540 0.2800 0.26600 676.9 1–Heptene 98.188 537.29 2.830 0.4130 0.3310 0.26150 701.5 Oil: Aromatics Benzene 78.114 562.16 4.898 0.2589 0.2108 0.26967 −0.04870 882.9 Toluene 92.141 591.79 4.109 0.3158 0.2641 0.26390 −0.01450 874.3 Ethylbenzene 106.167 617.17 3.609 0.3738 0.3036 0.26186 0.01397 874.4 o–Xylene 106.167 630.37 3.734 0.3692 0.3127 0.26200 0.01397 884.9 m–Xylene 106.167 617.05 3.541 0.3758 0.3260 0.26200 0.01397 869.4 p–Xylene 106.167 616.26 3.511 0.3791 0.3259 0.28700 0.01397 866.6 Oil: Cycloalkanes Cyclopentane 70.134 511.76 4.502 0.2583 0.1943 0.26824 −0.11868 760.3 Methylcyclopent. 84.161 532.79 3.784 0.3189 0.2302 0.27040 −0.07227 754.0 Cyclohexane 84.161 553.54 4.075 0.3079 0.2118 0.27286 −0.07227 783.5 Methylcyclohex. 98.188 572.19 3.471 0.3680 0.2350 0.26986 −0.03454 774.8 Ethylcyclopent. 98.188 569.52 3.397 0.3745 0.2715 0.26670 −0.03454 771.2 Ethylcyclohex. 112.215 609.15 3.040 0.4500 0.2455 0.26900 −0.00292 792.1 Oil: Methyl–Alkanes 2 2-Dimethylprop 72.150 433.78 3.199 0.3036 0.1964 0.27570 −0.04350 597.4 2–Methylpentane 86.177 497.50 3.010 0.3664 0.2781 0.26620 −0.01478 657.8 Table I.1. Component parameters – Part 1
- 461. I Component Properties 451 Name MW Tc Pc vc Acentric Rackett Shift ρ g mol [K] [MPa] m3 kmol Fac. ωc ZRa Fac. SE kg m3 Oil: Single Carbon Number (SCN) C6 84 510 3.271 0.348 0.251 0.269 −0.01478 690 C7 96 547 3.071 0.392 0.280 0.266 0.01745 727 C8 107 574 2.877 0.433 0.312 0.263 0.03669 749 C9 121 603 2.665 0.485 0.352 0.260 0.05804 768 C10 134 627 2.481 0.532 0.389 0.256 0.07540 782 C11 147 649 2.310 0.584 0.429 0.253 0.09088 793 C12 161 670 2.165 0.635 0.467 0.250 0.10583 804 C13 175 689 2.054 0.681 0.501 0.247 0.11932 815 C14 190 708 1.953 0.727 0.536 0.244 0.13243 826 C15 206 727 1.853 0.777 0.571 0.240 0.14512 836 C16 222 743 1.752 0.830 0.610 0.237 0.15670 843 C17 237 758 1.679 0.874 0.643 0.234 0.16669 851 C18 251 770 1.614 0.915 0.672 0.232 0.17536 856 C19 263 781 1.559 0.951 0.698 0.229 0.18235 861 C20 275 793 1.495 0.997 0.732 0.226 0.18900 866 C21 291 804 1.446 1.034 0.759 0.224 0.19729 871 C22 300 815 1.393 1.077 0.789 0.221 0.20174 876 C23 312 825 1.356 1.110 0.815 0.219 0.20743 881 C24 324 834 1.314 1.147 0.841 0.217 0.21286 885 C25 337 844 1.263 1.193 0.874 0.214 0.21849 888 C26 349 853 1.230 1.226 0.897 0.212 0.22346 892 C27 360 862 1.200 1.259 0.944 0.200 0.22784 896 C28 372 870 1.164 1.296 0.968 0.198 0.23244 899 C29 382 877 1.140 1.323 0.985 0.196 0.23614 902 C30 394 885 1.107 1.361 1.008 0.194 0.24044 905 C31 404 893 1.085 1.389 1.026 0.193 0.24390 909 C32 415 901 1.060 1.421 1.046 0.191 0.24759 912 C33 426 907 1.039 1.448 1.063 0.189 0.25117 915 C34 437 914 1.013 1.480 1.082 0.188 0.25464 917 C35 445 920 0.998 1.502 1.095 0.187 0.25710 920 C36 456 926 0.974 1.534 1.114 0.185 0.26040 922 C37 464 932 0.964 1.550 1.124 0.184 0.26275 925 C38 475 938 0.941 1.583 1.142 0.182 0.26589 927 C39 484 943 0.927 1.604 1.154 0.181 0.26840 929 C40 495 950 0.905 1.636 1.172 0.180 0.27139 931 C41 502 954 0.896 1.652 1.181 0.179 0.27325 933 C42 512 959 0.877 1.680 1.195 0.178 0.27586 934 C43 521 964 0.864 1.701 1.207 0.177 0.27815 936 C44 531 970 0.844 1.733 1.224 0.175 0.28065 938 C45 539 974 0.835 1.749 1.232 0.174 0.28261 940 Table I.2. Component parameters – Part 2
- 462. 452 I Component Properties Name MW Tc Pc vc Acentric Rackett g mol [K] [MPa] m3 kmol Fac. ωc ZRa Oil: Boiling Range Groups Oil–range Mahakam(waxy crude) 133.440 617.95 2.600 0.5103 0.3795 0.25725 Oil–range Tuscaloosa(Live Oil) 159.140 657.60 2.334 0.5920 0.4451 0.25137 Oil–range Smackover(S–rich crude) 184.440 693.37 2.105 0.6773 0.5083 0.24593 Oil BO 196.032 705.54 2.026 0.7084 0.5392 0.24244 Oil–range Tualag(waxy crude) 201.620 713.24 1.998 0.7237 0.5485 0.24155 Oil: Compound Classes C6–C14ARO 89.616 588.92 4.321 0.3047 0.2546 0.26788 C6–C14SAT 121.422 587.96 2.518 0.4980 0.4003 0.25812 C5–C14 132.314 595.03 2.305 0.5594 0.4546 0.25367 C6–C14 137.378 605.58 2.215 0.5801 0.4717 0.25242 C15+ 311.950 824.32 1.392 1.0926 0.8079 0.21739 C6–C13ARO 104.484 618.13 3.703 0.3669 0.3077 0.26604 C6–C13SAT 137.936 609.86 2.191 0.5792 0.4730 0.25313 C7+ 209.610 724.56 1.932 0.7551 0.5686 0.23983 C14+ISAT 299.180 813.82 1.442 1.0524 0.7797 0.22017 C14+NSAT 299.180 813.82 1.442 1.0524 0.7797 0.22017 C14+ARO U 299.180 813.82 1.442 1.0524 0.7797 0.22017 C15+ARO 282.553 768.00 1.165 1.1700 0.9069 0.22811 C15+SAT 255.342 747.73 1.309 1.0531 0.8085 0.22382 C14+NSO 402.700 891.20 1.109 1.3697 1.0050 0.19590 NSO 338.680 810.40 1.009 1.3620 1.0190 0.20110 Standard Oils Volatile Oil 53.135 367.79 4.373 0.1956 0.1267 0.27919 Light Oil 48.439 357.38 4.389 0.1895 0.1238 0.27968 Black Oil 90.072 491.62 3.535 0.3121 0.2423 0.26892 Medium Oil 101.141 511.92 3.459 0.3302 0.2606 0.26722 Heavy Oil 127.492 568.42 3.041 0.4080 0.3290 0.26099 Tuscaloosa T2 Live Oil 57.930 400.71 4.112 0.2241 0.1634 0.27631 Smackover T2–S Live Oil 142.418 604.26 2.669 0.4839 0.3930 0.25608 Tualag T1 High Waxy 167.860 650.49 2.298 0.5886 0.4750 0.24916 Mahakam T3 High Waxy+Aro 102.266 529.78 3.082 0.3768 0.3165 0.26429 Table I.3. Component parameters – Part 3
- 463. I Component Properties 453 Name MW Tc Pc vc Acentric Rackett Shift ρ g mol [K] [MPa] m3 kmol Fac. ωc ZRa Fac. SE kg m3 Gas: Alkanes Methane 16.043 190.56 4.599 0.0986 0.0115 0.28941 −0.15400 300.0 Methane–C13 17.043 190.56 4.599 0.0986 0.0115 0.28941 −0.15400 318.7 Ethane 30.070 305.32 4.872 0.1455 0.0995 0.28128 −0.10020 356.2 Propane 44.096 369.83 4.248 0.2000 0.1523 0.27664 −0.08501 507.0 n–Butane 58.123 425.12 3.796 0.2550 0.2002 0.27331 −0.06413 584.0 n–Pentane 72.150 469.70 3.370 0.3130 0.2515 0.26853 −0.04183 631.1 Gas: Alkenes Ethylene 28.054 282.36 5.032 0.1291 0.0852 0.28054 500.0 Propadiene 40.065 393.15 5.470 0.1620 0.1596 0.27283 599.7 Propylene 42.081 364.76 4.612 0.1810 0.1424 0.27821 521.0 1–Butene 56.107 419.58 4.020 0.2399 0.1867 0.27351 600.5 cis–2–Butene 56.107 435.58 4.206 0.2340 0.2030 0.27044 628.6 trans–2–Butene 56.107 428.63 4.103 0.2382 0.2182 0.27212 611.2 1 2–Butadiene 54.092 444.00 4.500 0.2190 0.2509 0.26850 657.6 1 3–Butadiene 54.092 425.37 4.330 0.2208 0.1932 0.27130 627.3 1–Pentene 70.134 464.78 3.529 0.2960 0.2329 0.27035 645.8 cis–2–Pentene 70.134 475.93 3.654 0.3021 0.2406 0.26940 659.8 trans–2–Pentene 70.134 475.37 3.654 0.3021 0.2373 0.26970 652.4 Gas: Methyl–Alkanes i–Butane 58.123 408.14 3.648 0.2627 0.1770 0.27569 −0.07935 562.9 i–Pentane 72.150 460.43 3.381 0.3058 0.2275 0.27060 −0.04350 624.7 Gas: Methyl–Alkenes 2–Methyl– 1–Butene 70.134 465.00 3.400 0.2920 0.2287 0.2705 656.3 3–Methyl– 1–Butene 70.134 450.37 3.516 0.3021 0.2286 0.2705 632.2 2–Methyl– 2–Butene 70.134 471.00 3.400 0.2920 0.2767 0.2663 668.3 Table I.4. Component parameters – Part 4
- 464. 454 I Component Properties Name MW Tc Pc vc Acentric Rackett ρ g mol [K] [MPa] m3 kmol Fac. ωc ZRa kg m3 Gas: Compound Classes C2–C4 47.130 379.57 4.192 0.2120 0.1615 0.27623 C2–C5 53.070 400.97 3.996 0.2360 0.1828 0.27440 C3–C5 62.752 430.72 3.599 0.2721 0.2008 0.27340 Standard Gases Hydrogen 2.016 33.18 1.313 0.0642 −0.2150 0.31997 Carbon Monoxide 28.010 132.92 3.499 0.0931 0.0663 0.28966 Nitrogen 28.014 126.10 3.394 0.0901 0.0403 0.28971 809.4 Oxygen 31.999 154.58 5.043 0.0734 0.0218 0.28962 1142.1 Hydrogen Sulphide 34.082 373.53 8.963 0.0985 0.0827 0.28476 801.4 Carbon Dioxide 44.010 304.19 7.382 0.0940 0.2276 0.27275 818.0 Sulfur Dioxide 64.065 430.75 7.884 0.1220 0.2451 0.26729 1394.6 Dry Gas 17.943 197.42 4.850 0.0977 0.0221 0.27992 Wet Gas 30.186 272.40 4.801 0.1345 0.0624 0.28493 Water 18.015 647.13 22.055 0.0560 0.3449 1000.0 Table I.5. Component parameters – Part 5 CO2 H2S N2 CO CO2 H2S N2 CO H2S 0.097 0.099 N2 −0.017 0.000 −0.032 0.000 Methane 0.092 0.000 0.031 0.030 0.093 0.000 0.028 0.032 Ethylene 0.055 0.083 0.086 0.000 0.053 0.085 0.080 0.000 Ethane 0.132 0.000 0.052 −0.023 0.136 0.000 0.041 −0.028 Propylene 0.093 0.000 0.090 0.000 0.094 0.000 0.090 0.000 Propane 0.124 0.088 0.085 0.026 0.129 0.088 0.076 0.016 Isobutane 0.120 0.047 0.103 0.000 0.128 0.051 0.094 0.000 n–Butane 0.133 0.000 0.080 0.000 0.143 0.000 0.070 0.000 Isopentane 0.122 0.000 0.092 0.000 0.131 0.000 0.087 0.000 n–Pentane 0.122 0.063 0.100 0.000 0.131 0.069 0.088 0.000 n–Hexane 0.110 0.000 0.150 0.000 0.118 0.000 0.150 0.000 n–Heptane 0.100 0.000 0.144 0.000 0.110 0.000 0.142 0.000 n–Decane 0.114 0.000 0.000 0.000 0.130 0.000 0.000 0.000 Cyclohexane 0.105 0.000 0.000 0.000 0.129 0.000 0.000 0.000 Benzene 0.077 0.000 0.164 0.000 0.077 0.000 0.153 0.000 Toluene 0.106 0.000 0.000 0.000 0.113 0.000 0.000 0.000 Table I.6. Binary Interaction Parameter (BIP) for the Peng–Robinson (left) and the Soave–Redlich–Kwong (right) equation of state according to Reid et. al. (1987)
- 465. I Component Properties 455 N2 CO2 C1 Ethyl. C2 Propyl. C3 iC4 nC4 N2 0.0000 CO2 0.0000 0.0000 C1 0.0311 0.1070 0.0000 Ethylene 0.0500 0.1200 0.0215 0.0000 C2 0.0515 0.1322 0.0026 0.0089 0.0000 Propylene 0.0600 0.1300 0.0330 0.0000 0.0890 0.0000 C3 0.0852 0.1241 0.0140 0.0100 0.0011 0.0100 0.0000 iC4 0.1000 0.1400 0.0256 0.0200 −0.0067 0.0080 −0.0078 0.0000 nC4 0.0711 0.1333 0.0133 0.0200 0.0096 0.0080 0.0033 0.0000 0.0000 iC5 0.1000 0.1400 −0.0056 0.0250 0.0080 0.0080 0.0111 −0.0040 0.0170 Neopent. 0.1000 0.1400 −0.0056 0.0250 0.0080 0.0080 0.0111 −0.0040 0.0170 nC5 0.1000 0.1400 0.0236 0.0250 0.0078 0.0100 0.0120 0.0020 0.0170 nC6 0.1496 0.1450 0.0422 0.0300 0.0140 0.0110 0.0267 0.0240 0.0174 Methyl- cyc.pent. 0.1500 0.1450 0.0450 0.0310 0.0141 0.0120 0.0270 0.0242 0.0180 Cyc.hex. 0.1500 0.1450 0.0450 0.0310 0.0141 0.0120 0.0270 0.0242 0.0180 nC7 0.1441 0.1450 0.0352 0.0300 0.0150 0.0140 0.0560 0.0250 0.0190 Methyl- cyclohex. 0.1500 0.1450 0.0450 0.0300 0.0160 0.0150 0.0580 0.0250 0.0200 Toluene 0.1700 0.1800 0.0600 0.0400 0.0200 0.0210 0.0600 0.0300 0.0110 o–Xylene 0.1500 0.1400 0.0470 0.0300 0.0160 0.0150 0.0590 0.0260 0.0120 nC8 0.1500 0.1400 0.0470 0.0300 0.0160 0.0150 0.0590 0.0260 0.0120 nC9 0.1550 0.0145 0.0474 0.0400 0.0190 0.0200 0.0070 0.0060 0.0100 nC10– nC14 0.1550 0.0145 0.0500 0.0450 0.0300 0.0250 0.0200 0.0100 0.0010 nC15– nC19 0.1550 0.0145 0.0600 0.0500 0.0400 0.0300 0.0250 0.0150 0.0010 nC20– nC24 0.1550 0.0145 0.0700 0.0600 0.0500 0.0350 0.0300 0.0200 0.0015 Table I.7. Binary Interaction Parameter (BIP) for the Peng–Robinson equation of state according to Danesh (1998)
- 466. 456 I Component Properties N2 CO2 C1 Ethyl. C2 Propyl. C3 iC4 nC4 N2 0.0000 CO2 0.0000 0.0000 C1 0.0278 0.1107 0.0000 Ethyl. 0.0300 0.1000 0.0189 0.0000 C2 0.0407 0.1363 −0.0078 0.0026 0.0000 Propyl. 0.0800 0.1000 0.0289 0.0000 0.0200 0.0000 C3 0.0763 0.1000 0.0080 0.0080 −0.0220 0.0033 0.0000 iC4 0.0944 0.1000 0.0241 0.0900 −0.0010 −0.0144 −0.0100 0.0000 nC4 0.0700 0.1000 0.0056 0.1000 0.0067 0.0000 0.0000 0.0000 0.0000 iC5 0.0867 0.1000 −0.0078 0.0120 0.0050 0.0000 0.0078 0.0000 0.0000 Neopent. 0.0870 0.1000 −0.0078 0.0120 0.0050 0.0000 0.0078 0.0000 0.0000 nC5 0.0878 0.1000 0.0019 0.0120 0.0056 0.0050 0.0230 −0.030 0.0204 nC6 0.1400 0.1000 0.0374 0.0140 −0.0156 0.0050 −0.0022 0.0000 −0.0111 Methyl- cyc.pent. 0.1400 0.1000 0.0400 0.0140 0.0330 0.0050 0.0030 0.0000 0.0000 Cyc.hex. 0.1400 0.1000 0.0333 0.0150 0.0230 0.0050 0.0030 0.0005 0.0000 nC7 0.1422 0.1000 0.0307 0.0144 0.0411 0.0100 0.0044 0.0005 0.0000 Methyl- cyc.hex. 0.1450 0.1000 0.0500 0.0150 0.0230 0.0100 0.0050 0.0005 0.0000 Toluene 0.1500 0.1000 0.0978 0.0300 0.0900 0.0300 0.0300 0.0200 0.0100 o–Xyl. 0.1500 0.1000 0.1000 0.0250 0.0500 0.0300 0.0300 0.0200 0.0100 nC8 0.1500 0.1000 0.0448 0.0200 0.0170 0.0100 0.0040 0.0015 0.0000 nC9 0.1500 0.1000 0.0448 0.0200 0.0170 0.0100 0.0040 0.0015 0.0000 nC10– nC14 0.1500 0.1000 0.0550 0.0300 0.0200 0.0150 0.0040 0.0020 0.0010 nC15– nC19 0.1500 0.1000 0.0600 0.0400 0.0350 0.0250 0.0005 0.0025 0.0010 nC20– nC24 0.1500 0.1000 0.0650 0.0450 0.0400 0.0300 0.0010 0.0050 0.0015 Table I.8. Binary Interaction Parameter (BIP) for the Soave–Redlich–Kwong equa- tion of state according to Danesh (1998)
- 467. J Methane Density The Modified Benedict–Webb–Rubin (MBWR) EOS has the form P = ρRT + ρ2 (N1T + N2T1/2 + N3 + N4/T + N5/T2 ) +ρ3 (N6T + N7 + N8/T + N9/T2 ) + ρ4 (N10T + N11 + N12/T) +ρ5 N13 + ρ6 (N14/T + N15/T2 ) + ρ7 N16/T +ρ8 (N17/T + N18/T2 ) + ρ9 N19/T2 +ρ3 (N20/T2 + N21/T3 )e−γρ2 + ρ5 (N22/T2 + N23/T4 )e−γρ2 +ρ7 (N24/T2 + N25/T3 )e−γρ2 + ρ9 (N26/T2 + N27/T4 )e−γρ2 +ρ11 (N28/T2 + N29/T3 )e−γρ2 +ρ13 (N30/T2 + N31/T3 + N32/T4 )e−γρ2 (J.1) with γ = 0.0096, R = 0.08205616 atm/mol/K, temperature in K, pressure in atm, density ρ in mol/l and
- 468. 458 J Methane Density N1 = −1.8439486666 × 10−2 N2 = 1.0510162064 N3 = −1.6057820303 × 10 N4 = 8.4844027562 × 102 N5 = −4.2738409106 × 104 N6 = 7.6565285254 × 10−4 N7 = −4.8360724197 × 10−1 N8 = 8.5195473835 × 10 N9 = −1.6607434721 × 104 N10 = −3.7521074532 × 10−5 N11 = 2.8616309259 × 10−2 N12 = −2.8685285973 N13 = 1.1906973942 × 10−4 N14 = −8.5315715699 × 10−3 N15 = 3.8365063841 N16 = 2.4986828379 × 10−5 N17 = 5.7974531455 × 10−6 N18 = −7.1648329297 × 10−3 N19 = 1.2577853784 × 10−4 N20 = 2.2240102466 × 104 N21 = −1.4800512328 × 106 N22 = 5.0498054887 × 10 N23 = 1.6428375992 × 106 N24 = 2.1325387196 × 10−1 N25 = 3.7791273422 × 10 N26 = −1.1857016815 × 10−5 N27 = −3.1630780767 × 10 N28 = −4.1006782941 × 10−6 N29 = 1.4870043284 × 10−3 N30 = 3.1512261532 × 10−9 N31 = −2.1670774745 × 10−6 N32 = 2.4000551079 × 10−5 . (J.2) For the selection of the appropriate root it is often very helpful to solve a SRK or PR EOS firstly. This can be performed analytically and the solution can then be used as a start value for the numerical solution of the MBWR EOS.
- 469. K Compositions and Components for Fig. 5.14 Dry Gas Wet Gas Gas Condensate Volatile Oil Black Oil Methane 75.0 84.0 79.0 54.0 47.0 Ethane 7.0 8.0 10.0 9.0 6.0 Propane 6.0 3.0 3.0 8.0 4.0 n–Butane 5.0 1.0 2.0 8.0 3.0 n–Pentane 4.0 1.0 1.5 7.0 2.0 n–Hexane 3.0 1.0 C6−14 2.0 1.7 6.0 10.0 C15 1.8 4.0 9.0 C25 0.7 2.0 8.0 C35 0.2 1.5 6.0 C45 0.1 0.5 5.0 Table K.1. Compositions of the examples in Fig. 5.14. All entries are in molar % M Tc Pc vc Acentric Volume Shift [g/mol] [◦ C] [MPa] [m3 /kmol] Factor [m3 /kmol] Methane 16.043 −82.59 4.599 0.0986 0.0115 0.0007 Ethane 30.070 32.17 4.872 0.1455 0.0995 0.0028 Propane 44.096 96.68 4.248 0.2000 0.1523 0.0052 n–Butane 58.123 151.97 3.796 0.2550 0.2002 0.0080 n–Pentane 72.150 196.55 3.370 0.3130 0.2515 0.0122 n–Hexane 86.177 234.45 3.025 0.3710 0.3013 0.0176 C6−14 133.680 292.11 1.999 0.5801 0.3111 0.0389 C15 183.516 456.45 1.723 0.7770 0.5882 0.0724 C25 323.821 573.739 1.128 1.1930 0.8361 0.1821 C35 442.868 638.133 0.886 1.5020 1.1327 0.3356 C45 537.418 699.490 0.737 1.7490 1.1940 0.4761 Table K.2. Component properties of the examples in Fig. 5.14
- 470. L An Analytic Solution for the Diffusion of Methane Through a Cap Rock A homogeneous cap rock of thickness h is sketched in Fig. L.1. The concen- tration at bottom is determined by an methane accumulation with a rather high concentration value cb. At top the methane might quickly migrate away, which can be described by a rather small concentration ct. The initial con- centration within the cap rock is given by c0. The mathematical formulation is ∂ ∂t c(z, t) = D ∂2 ∂z2 c(z, t) (L.1) with boundary conditions c(z, 0) = c0 and c(0, t) = ct , c(h, t) = cb for t 0 . (L.2) The problem is similar to App. F.5 and App. F.6. The solution becomes for t 0 c(z, t) = ct + (cb − ct) z h + ∞ n=1 2 [c0 − ct + (−1)n (cb − c0)] sin μnz μnh exp(−μ2 nDt) (L.3) with μn = nπ/h. An example solution is shown in Fig. L.1.More than 1 My are needed for the establishment of a static concentration gradient in the cap rock. Note that the time dependency scales quadratical with height h due to the form of exponent μ2 nDt = n2 π2 Dt/h2 . Hence a cap rock with doubled thickness of 200 m needs four times as long and with 10 times thickness of 1 km about 100 times as long for the convergence to the static solution. These are here times of 4 My and 100 My. Additionally, the time dependency scales linear inversely with the diffusion coefficient. Realistic values can become D = 10−11 m2 /s, which is one order of magnitude smaller than in example Fig. L.1 or even yet smaller (Sec. 6.4).Transient effects decelerate respectively.
- 471. 462 L An Analytic Solution for the Diffusion of Methane Through a Cap Rock ct D z h cb c0 Fig. L.1. Model for diffusion through a cap rock of thickness h and example curves of solution (L.3) for ct = 0, c0 = 0, h = 100 m, and a rather big diffusion co- efficient D = 10−10 m2 /s The cumulative volume Q which is flown through a horizontal plane of unit size at depth z is given by Q = t 0 D ∂ ∂z c(z, t )dt (L.4) and can be calculated at top z = 0 or bottom z = h, here again for ct = 0, c0 = 0, as Qt hcb = Dt h2 − 1 6 − 2 ∞ n=1 (−1)n μ2 nh2 exp(−μ2 nDt) Qb hcb = Dt h2 + 1 3 − 2 ∞ n=1 1 μ2 nh2 exp(−μ2 nDt) . (L.5) The figure which is based on the same values as Fig. L.1 is shown in Fig. L.2.The vertical difference ΔQ = Qb − Qt between the volume which passed the bottom and the top boundary defines the methane volume in the cap rock. In the static limit for t → ∞ it approaches the expected value ΔQ = hcb/2. Fig. L.2. Example curves for diffusion amounts according to (L.5). The curve on top quantifies the diffusion amounts on bottom and vice versa
- 472. M Flowpath Bending The angle ψ which indicates the direction of petroleum flow in a reservoir below a planar seal with dipping angle α in the y–direction and lateral water flow with angle β to the x–axis is sketched in Fig. M.1. The normal vector on the seal is given by ns = − sin α ey + cos α ez. The vectors ex,y,z are unit vectors in the x, y,, and z–directions. The petroleum potential up according to (6.18) is up = uw − Δρgz (M.1) with Δρ = ρw − ρp and ρw,p the densities of water and petroleum. a b y x y z ns Fig. M.1. Seal with dipping angle α Without a seal the flow direction of the petroleum is vp ∝ −∇up. Directly below the seal the petroleum cannot move in the direction of ns. The part of the flow which points in the direction of ns must be subtracted and the condition vp · ns = 0 must be ensured. Hence the final direction of flow is vp ∝ −∇up + (∇up · ns)ns . (M.2) A lateral water flow in direction
- 473. 464 M Flowpath Bending vw ∝ cos β ex + sin β ey (M.3) exists. The hydraulic head has a dipping angle of γ which leads to uw = −ρwg tan γ(cos β x + sin β y). Insertion into (M.1) and calculation of the gra- dient yields − ∇up ∝ ρw tan γ(cos β ex + sin β ey) + Δρez . (M.4) It must be noted that the lateral pressure gradient can become so large that the petroleum flow points away from the seal into the reservoir. Such a flow exists when according to Fig. M.2 tan α Δρ ρw tan γ sin β (M.5) and need not to be considered further. Equation (M.4) can now be combined with (M.2) to calculate vp and afterwards tan ψ = vpy vpx = tan β cos2 α + Δρ ρw cos α sin α tan γ cos β . (M.6) A lateral overpressure which is orthogonal to the dipping direction of the seal is equivalent to the special case β = 0 which leads to (6.44). y z a Seal Dr r g b wtan cos Fig. M.2. Lateral waterflow which has to be overcome to release petroleum from the seal The result (M.6) can be checked for some special cases. For γ → 0 it is vpy/vpx → ∞ which correctly yields ψ → 90◦ , the “non–bending” case. For very small α (M.6) can be developed into a Taylor–series and it becomes tan ψ → tan β + Δρ ρw α tan γ cos β → tan β for α → 0 . (M.7) For a flat untilted surface, the flow follows the lateral overpressure direction. The impact of overpressure on flowpath bending is maximized.
- 474. M Flowpath Bending 465 y z a Seal Dr r g wtan Fig. M.3. Lateral waterflow which has to be overcome to reverse the lateral y–flow direction A case which is not covered by (M.6) is β = −90◦ which can easily be cal- culated with Fig. M.3. The lateral flow reverses its direction into the negative y–axis if tan γ Δρ tan α/ρw. A problem rises from (M.6) for Δρ = 0 which results in tan ψ = tan β cos2 α. This is at a first glance unexpected. Without buoyancy the petroleum flow should follow the lateral overpressure with ψ = β. However, water flow in the lateral direction passes through the seal Fig. M.4. This is not allowed for the petroleum. The flow has to follow the barrier. Seal Waterflow Reservoir Fig. M.4. Water flow at seal according to (M.3) A slightly more elegant formula for flowpath bending can be obtained if the water flow is also assumed to follow the dipping of the seal. This behavior can be obtained by the constrains vw · ns = 0 , vw · ex ∝ cos β , vw · ey ∝ sin β . (M.8) The water potential becomes uw = − ρwg tan γ 1 + sin2 β tan2 α (cos β x + sin β y + sin β tan α z) (M.9) when these conditions are fulfilled. The same straight forward calculation as for (M.6) yields now tan ψ = tan β + Δρ ρw 1 + sin2 β tan2 α cos α sin α tan γ cos β . (M.10)
- 475. 466 M Flowpath Bending This result is very similar to (M.6). All special cases are identical except for Δρ = 0 which now yields the more intuitive result ψ = β. Orthogonal water flow with β = 0 is exactly the same (Fig. 6.29). A petroleum release as shown in Fig. M.2 can obviously not occur with a water flow parallel to the barrier. The inversion of the flow direction analogously as in Fig. M.3 for β = −90◦ yields now the condition tan γ Δρ sin α/ρw.
- 476. N Unit Conversions and Constants Quantity Unit Conversion Unit Distance/Depth foot (ft) 0.3048 meter (m) yard (yd) 0.9144 mile (mi) 1609.344 Time million years (My,Ma) 3.1536 × 1013 second (s) Pressure psi 6.89475 × 103 Pascal (Pa) bar 105 atmosphere (atm) 1.01325 × 105 Temperature Celsius (◦ C) ◦ C = K − 273.15 Kelvin (K) Fahrenheit (◦ F) ◦ F = 1.8 ◦ C + 32 Heatflow Unit HFU 0.041868 W/m2 Density g/cm 3 1000 kg/m 3 Mudweight pounds per gallon (ppg) 119.83 ◦ API kg/m 3 = 141500/(131.5 + ◦ API) Mass ton (t) 1000 kilogram (kg) pound (lb) 0.45359237 Force pound force (lbf) 4.448221615 Newton (N) dyne (dyn) 10−5 Volume barrel (bbl,bo) 0.158987 m3 cubic foot (cf) 0.02831685 gallon US (gal) 0.003785412 acre–foot (af) 1233.482 liter (l) 0.001 barrel of oil equivalent gas (boe) ∗ 1068.647751 Energy calorie (cal) 4.1868 Joule (J) Permeability Darcy (D) 0.98692 × 10−12 m2 Viscosity Poise (P) 0.1 Pa s Gas Oil Ratio GOR (SCF/STB) 0.178137 m3 /m3 ∗ Approximately 6000 cubic feet of natural gas are equivalent to one barrel of crude oil (chevron.com/investor/annual/2006/glossary.asp, moneyterms.co.uk/boe).
- 477. 468 N Unit Conversions and Constants Common Unit Prefixes: tera, trillion (T,t): 1012 giga, billion (G,B): 109 mega, million (M): 106 kilo (k): 103 hecto (h): 102 deci (d): 10−1 centi (c): 10−2 milli (m): 10−3 micro (μ): 10−6 Standard acceleration due to gravity: g = 9.80665 m/s 2 Universal gas constant: R = 8.31447 Pa m3 /K/mol
- 478. Index Accumulation analysis, 269, 271, 276–282 break through, 289 leakage, 291 Acentric factor, 207, 210 Activation energy, 15, 68, 72, 152–191, 268, 441–458 Adsorption models, 5, 183–188 Advection, 103 Alani–Kennedy EOS, see Equations of state (EOS) Anisotropy, 383 elasticity, 80 Percolation, 319 percolation, 253, 317 permeability, 55–56 thermal conductivity, 112–113 API, 220 gamma ray, 119 method, 200, 221, 235 Aquathermal pressuring, 71–72 Aquifer flow, 268, 283 Arrhenius law, 4, 15, 68, 72, 152, 153, 176, 186, 268 fanning, 181 Artificial component, see Pseudo component Association, 359, 366 Asthenosphere, 104, 130, 132, 134–137 Athy’s depth model, 45–46, 50, 52 Athy’s effective stress model, 44–45, 51, 75 Auto–correlation, 353, 366 Average arithmetic, 108, 114, 117, 118, 385–387 geometric, 105, 114, 115, 385–387 harmonic, 105, 114, 385–387, 395 square–root, 385 Backstripping, 3, 90, 91, 95, 317 Bayes law, 363 Bayesian statistic, 239, 360–367 BET theory, 186 Beta distribution, 350 Binary interaction parameter (BIP), 211, 240 Binary mixture, 203–207 Binning, 357 Biodegradation, 188–191, 329 Biomarker, 4, 151, 176, 445 methyladamantane index (MAI), 176 methyldiamantane index (MDI), 176 methylphenanthrene index (MPI), 176 trisnorhopane ratio, 176 Biot compressibility, 37 Bitumen, 158 Black oil, 203, 217, 223–225, 227, 233 Black oil model, 3, 5, 161, 203–207, 266–267 Block concept for thrust belts, 96 Blocking, 369 Boiling point classes, 167 Bond number, 306 Boundary condition, 14, 391, 400 crustal model, 129, 137
- 479. 470 Index Dirichlet, 391, 400 heat flow, 103, 121, 107–122 igneous intrusion, 125 inner, 41, 61, 125 Neumann, 391, 400 pressure, 41, 41, 61, 63 Boundary value problem heat flow analysis, 104, 119 overpressure calculation, 41 Break through, 2, 278–280, 289–291, 297, 301, 323, 333 Brittle, 82, 130 Bubble point, 205 calculation, 224 Bulk kinetic, 157, 159–161, 162 Bulk value, 384 Buoyancy, 2, 259, 268, 285, 290, 298, 302, 306, 308 Butane, 215 Calibration, 3, 341, 360, 362 fluid Model, 235 heat flow, 143, 146 Markov chain Monte Carlo, 376–377 overpressure, 75 Capillary entry pressure, 255, 257 Capillary number, 299 Capillary pressure, 254–259, 278 heterogeneity, 307, 307, 308, 309, 315, 317, 319 mercury–air, 254 water–petroleum, 254 Carrier, 268–286, 287, 321 Catagenesis, 73, 151, 176 Central limit theorem, 348 Chemical kinetics, 441–458 Chemical potential, 212, 225 Chi–square χ2 , 361–367 Chimney, 317 Cholesky decomposition, 355 Clathrate hydrate, see Gas hydrate Closed system approach, see Com- paction, chemical Closure, see Drainage area, closure Co-volume, 208, 210 Coal bed methane, 74 Column height, 278 Column pressure, 2, 278, 281, 310, 320 Compaction, 3, 31–99, 108, 112, 247, 257, 259, 263, 269, 319 chemical, 31, 35, 40, 65–70 mechanical, 31, 34, 34, 35, 37, 40, 42, 56, 75 shale, 51 shale sandstone mixture, 52 Component, 199 oil, gas, 200, 205 Component mixing, see Mixing components Component tracking, 236 Compositional grading, 278 Compositional kinetic, 157, 163–167 Compositional phase kinetic, 167–169 Compressibility gas, 74 volumetric, 96 Compressibility factor Z, 207, 212 critical Zc, 207, 232 Rackett ZRA, 220 Compressibility model, 46–48, 51 Conduction, see Heat, conduction Conductivity tensor, 39, 104, 112 Confidence interval, 345 Contact height, 320 Control volumes, 382, 394–396 Convection, 103, 105, 117, 119, 122–125, 131, 138 Correlation, 347 Correlation of priors, 365 Corresponding states principle, 229 Courant–Friedrichs–Lewy criterion, 286 Covariance matrix, 354 Cracking, 328 primary, 15, 73, 157–186 secondary, 15, 73, 74, 157–186, 329 Cramer’s V , 359 Cricondentherm, 203 Critical moment, 194 Critical point, 203, 209 transformation ratio (TR), 160 Critical pressure, 305 Critical state, 81, 84–85 Critical state point, 81 Cross plot, 347 Crust, 105, 129–143 lower, 130
- 480. Index 471 upper, 130 Crustal model, 104 Cubical design, 368–369, 371 Cumulative probability, 357 Darcy flow, 2, 5, 20, 247, 250–268, 270, 297, 316, 319–327, 397 Dead oil, 232 Decision tree, 346 Density contrast, 278 Deposition, 3 Derived uncertainty parameter, 351–352, 365, 374 Design of experiments (DOE), 368 Desorption, 184 Deterministic sampling, 367 Dew point, 205 Diagenesis, 26, 151 Differential equation, 381 Diffusion, 5, 247, 248, 267–268 coefficient, 268 Diffusion equation, 381 Discrete distribution, 351, 359 Discretization, 388 space, 388 time, 388 Displacement, 278, 306, 310, 312, 320 Domain decomposition, 2, 272, 287, 298, 321 Drainage, 256 Drainage area, 2, 5, 269–282 analysis, 272–276 closure, 272 liquid/vapor, 278 merging, 276 spill path, 276 spill point, 272 Dry gas, 217 Ductile, 82, 130, 139, 143 Earth cooling, 103 EasySoil model, 49 Effective saturation, 253 Eigenvalue, 383 Elasticity tensor, 80 Entropy, 359 Equations of state (EOS), 207–241 Alani–Kennedy, 220–223, 235 ideal gas, 207 modified Benedict–Webb–Rubin (MBWR), 210, 457 Peng–Robinson (PR), 209 Soave–Redlich–Kwong (SRK), 209 van der Waals (vdW), 208 Equilibrium ratio, 213–214 Wilson, 214 Equivalent hydrostatic depth, 43, 45 Error bar, 346 Event, 3, 6 Expectation value, 346 Explicit scheme, see Scheme explicit Exponential distribution, 350 Expulsion, 1, 183, 184, 248, 251, 285, 295, 302, 304, 310, 313, 320, 330 downward, 297, 310 Facies, 10 organic, 15, 156, 159, 159, 169, 176, 180, 195 Failure Griffith, 83 Mohr–Coulomb, 83 Murrell, 83 Fast thermal simulation, 145, 372–375 Fault, 86–90 closed, 281, 292 conducting, 280, 292, 329 open, 280, 292, 323 Fault capillary pressure (FCP), 89, 281, 292 Feeding point, 312 Fetch area, see Drainage area Ficks’s law, 267 Finite differences, 381, 387–389, 396 Finite elements, 19, 381, 389–394, 395 form function, 389 Galerkin method, 391 grid, 389 hexahedron, 392 Jacobi–matrix, 393 node, 389 shape function, 389–394 Fission–track, 4, 143, 181–182 Fixed phase model, 200 Flash calculation, 3, 5, 159, 213–241, 387 performance, 217 PT–path, 217
- 481. 472 Index stability, 214 volume shift, 220 Flexural compensation, 135 Flow pulsing, 301 Flow unit, 309, 313, 317 Flowpath analysis, 2, 5, 248, 269–286 bending, 283, 321, 463 modeling, 5, 248, 269, 295–297, 297, 298, 319–327 Fluid analysis, 3, 5, 27, 280, 313, 387, 199–387 Fluid expansion, 34, 35, 40, 75, 259, 70–259 Fluid heavy end, 215, 227, 236 Fluid inclusion, 143 Fractal flow pattern, 249 Fractal saturation pattern, 306 Fractional factorial design, 368 Fracturing pressure, 85 Frequency factor, 15, 68, 72, 152–179, 268, 441–458 Freundlich equation, 185 Fugacity, 212 Gamma distribution, 155 Gas condensate, 217 Gas hydrate, 115, 199, 203, 241 stability zone (GHSZ), 242, 243 Gas oil ratio (GOR), 200, 217, 225, 235, 243 Gauss distribution, 155, 172, 348 Gibbs’ energy, 211, 214 phase rule, 199, 203, 208 Glacial periods, 126 Gradient, 383 Groundwater potential, 38, 41, 63 Grouping of components, 206, 237 Hagen–Poiseuille law, 54 Hanging nodes, 402 Heat conduction, 103 convection, 103, 122–125 crystallization, 125 radiation, 103 radiogenic, 103, 106, 119, 133–136 solidification, 126 Heat flow, 103–149 basal, 4, 103, 105, 106, 108, 129–143 boundary condition, 103, 121, 107–122 calibration, 143–146 Heavy end characterization, 236 Herning–Zipperer mixing, 228 Heterogeneity, see Capillary pressure heterogeneity Histogram, 344, 357, 359 Horizon, 8 Hybrid method, 2, 5, 248, 295, 286–297, 297, 319–327 break through, 287 domain decomposition, 287, 287 fault flow, 287, 292 grid transformation, 287 Hydraulic head, 283 Hydrogen index (HI), 15, 74, 158, 158, 180, 195 Hysteresis, 255 Ideal gas chemical potential, 212 equation, 207–208 Imbibition, 256 Implicit scheme, see Scheme implicit In/Outflow, 328, 329, 330 Interfacial tension (IFT), 233–234, 254, 299 Intermolecular attraction, 208 Intrusion, see Magmatic intrusion Invasion percolation (IP), 2, 5, 248, 291, 295, 297–327 aquifer flow, 309, 314 backfilling, 309, 310 break through, 310 one phase, 309–312 pathway focusing, 312 two phases, 312–313 Inversion, 143, 239, 341, 377 Isostasy, 134–135 Isotopic fractionation, 180–181 Joint probability distribution, 354 Kendall’s tau, 358 Kerogen density, 74
- 482. Index 473 Kerogen type, 157–158, 159, 160, 163, 165, 167 Kozeny–Carman relation, 54, 56 Kriging, 375 Langmuir equation, 185 Latin hypercube sampling (LHC), 352 Least squares fit, 370–371 Lee–Kesler–Averaging, 206, 232, 234 Leverett J–function, 255 Likelihood, 363 Live oil, 232 Local grid refinement (LGR), 399–402 Logarithmic normal distribution, 349 Losses, 321 border, 329 immobile, 255, 271, 283 petroleum system (PS), 283, 331 Lumping, 189, 199, 205, 237 Magmatic intrusion, 125 Mantle, 129–143 lower, 130 upper, 105, 130, 133 Marginal distribution, 354 Markov chain Monte Carlo, 237, 355, 376–377 Mass balance, 39, 261, 265, 267, 323–334 Master run, 144, 342 Maximum likelihood, 360 Maximum stress point, 81 Maxwell’s equal area rule, 208 McKenzie model, see Uniform stretching model Melting–point line, 203 Meniscus, 273 Metagenesis, 151 Metamodel, 372, 374, 376, 370–377 Methane, 229, 235, 241, 267 density, 457 freezing point, 230 solubility in water, 201 Methane–ethane mixture, 204 Micro–accumulation, 308 Migration carrier, 271 losses, 283 primary, 185, 250 secondary, 185, 248, 250, 267 tertiary, 250 Mineral transformations, 72 Mixing compaction parameter, 51 components, 210–211, 278, 280 density, 108 heat capacity, 108, 118 permeabilities, 55 rock and fluid, 117 rocks, 114 Mixing rules, 384 Mobility, 39, 51, 250, 253, 268 Modified Benedict–Webb–Rubin (MBWR) equation, see Equations of state (EOS) Modulus bulk, 80 shear, 80 Young, 80 Modus, 346 Mohr circle, 78, 78, 79–82 Molar averaging, 206 Molecular weight mass average, 232 molar average, 232 Monte Carlo simulation, 343–360, 364 Mudstone model, 48–49, 51 Multicomponent flow equations, 265 Multigrid, 401 hybrid method, 287 solver, 396 Net spill, 333 Neural network, 376 Nominal distribution, 351, 359 Normal distribution, 348 Oil formation volume factor Bo, 226, 243 Oil water contact (OWC), 189 Oil window, 151 Oil–gas kinetic, 157, 161–163, 167 Open system approach, see Compaction, chemical Optimization procedure, 3, 91, 95 Organofacies, see Facies organic Outlier, 362 Overburden, 263 Overpressure, 282
- 483. 474 Index calibration, 75 Oxygen index (OI), 158, 158 Paleo–model, 90–96 Paleopasteurization, 189 Parachor method, 234 Parallelization, 20, 321, 382, 396–399 conjugate gradient, 397 message passing interface (MPI), 397 network, 396–399 OpenMP, 397 overlap region, 397 shared memory, 397–399 thread, 397 Pathway focusing, 301 Pearson correlation coefficient, 357 Peng–Robinson (PR) equation, see Equations of state (EOS) Pentane, 200, 215 Percentiles, 357 Percolation anisotropy, 253, 317–319 bond, 305 fault, 315 ordinary theory, 305 seismic data, 316 site, 305 universal exponent, 305 water trapping, 306 Permafrost, 125–126 Permeability absolute, 250 anisotropy, 55, 56 effective, 250 intrinsic, 53, 55, 250 relative, 53, 250, 283 rock, 53 tensor, 40, 55 Petroleum generation, 151–195 Petroleum generation potential, 161, 327 Petroleum system (PS), 20–23, 191, 327, 329–332 losses, 283 Petroleum systems chart, 22, 194 Phase, 199 coexistence area, 203–205, 208 coexistence line, 203 compounds, 199 degrees of freedom, 203 diagram, 203 equilibrium, 211 fluid, 199 properties, 199 supercritical, 5, 199, 203 transition, 207, 208 undersaturated, 5, 199, 204 water, 199, 201–202 Poisson ratio, 32, 80 Posterior, 363 Precipitation, 35, 65–70 Pressure communication, 278 Pressure potential, 250 Pressure temperature path, see PT–path Pressure volume temperature (PVT), 199 Principal value, 78, 383 Prior, 363 Probability distribution, 344 Production index (PI), 158, 158, 180 Pseudo component, 205 Pseudo–random number, 369 PT–path, 201, 211, 212, 216–217, 224–227, 233, 242 Pure shear model, 133 PVT–analysis, 243 Quadruple point, 241–242 Quarterny period, 108 Quartz cementation, see Compaction, chemical Quasi–random number, 369 Radiation, see Heat, radiation Radioactivity, see Heat radioactive Random walk, 307, 309, 377 Ray tracing, see Flowpath analysis Reduction factor, 162, 165, 167 Reference model, see Master run Regression, 361 Regularization, 362 Reinjection, 290 Reservoir analysis, 248, 269–297, 399 Reservoir modeling, 27, 268 Resolution, 2, 247, 275, 278–332 implicit, 271, 298, 323
- 484. Index 475 Response surface, 143, 145, 146, 370–372 Risk analysis, 3, 323, 341–378 Rock mechanics, 77–78 Rock–Eval pyrolysis, 157, 158, 177–180 Roscoe and Hvorslev surface, 84 Salinity, 38, 53 Saturation connate, water, 251, 256 critical, gas, 251 critical, oil, 251, 283 residual, 255, 271, 319, 326 Saturation discontinuity, 249 Saturation pressure, see Bubble point Scalar, 31, 382 Scale quantity, 349 Scenario run, 342 Scheme, 389 Crank–Nicholson, 389 explicit, 6, 264–267, 286, 322, 326, 389 implicit, 6, 264–267, 389 Schneider chemical compaction model, 68 mechanical compaction model, 46, 51, 52 Screening, 368 Sediment–water–interface (SWI) temperature, 108, 120, 126–129, 144, 373, 432, 436 Sedimentary basin, 8 Seismic attribute analysis, 317 interpretation, 8, 317 inversion, 316 noise, 317 Sensitivity analysis, 347 Separate phase flow, 247, 250, 261 Shale gouge ratio (SGR), 89 Shale smear factor, 89 Shape function, see Finite elements shape function Significance level, 359 Simple shear model, 133 Simple shear–pure shear model, 133 Single carbon number (SCN), 236 Singular value decomposition, 362 Slicing, 397, 401 Smectite dehydration, 73 Smectite–illite conversion, 35, 72 Soave–Redlich–Kwong (SRK) equation, see Equations of state (EOS) Sobol’ sequence, 369 Soil mechanics, 48, 77, 78, 84 Solver, 382, 396 backsubstitution, 396 conjugate gradients, 396 multigrid, 396 preconditioning, 396 Source rock kinetic, 236 Spearman rank order coefficient, 347, 358 Speedup flowpath modeling, 297 linear, 397 parallel processing, 397 Spill path, see Drainage area, spill path Spill point, see Drainage area, spill point, see Drainage area Spill point Spilling, 333 Stable oil, 232 Standard conditions, 200, 216, 220–222, 225, 226, 235 Standard deviation, 346 Standard isotope ratio, 180 Standing–Katz method, 220–222, 235 Steady state, 105–107, 108, 109, 396 Strain tensor, 80 Stratigraphical event, 8 Streamlines, 271 Stress biaxial, 78 deviatoric, 79 principal, 31, 32, 37, 78, 86, 87 tectonic, 96 tensor, 31, 36, 37, 78, 80 uniaxial, 32 Stretching, 105, 129–143 Stringer, 249, 302, 303 cohesion, 303 path, 310 size, 308, 308 snap–off, 302, 308 Study coupled, 25, 26
- 485. 476 Index decoupled, 25 source rock maturation, 24 Subsidence tectonic, 136, 139, 143 total, 136, 140 Supercritical phase, see Phase supercritical Surface conditions, see Standard conditions Surrogate, 370 Symmetrical black oil (SBO) model, 206, 217, 267, 313 T–max, 158, 177–180 Tartan grid, 400 Tectonic, 2 crustal model, 129 plate, 132 subsidence, 136 Tensor, 31, 36, 78, 80, 104, 112, 253, 270, 317, 382 deviatoric, 383 invariant, 78, 383 Terzaghi compressibility, 36 Thermal diffusivity, 138 Thermal disequilibrium indicator, 110 Thermal expansion coefficient, 72 Thermodynamic equilibrium, 211 potential, 211 Threshold pressure, 309 Thrust belts, 91, 95 Tie line, 205 Time step, 6, 286 Topographic driven flow, 41, 63, 283 Tornado diagram, 347, 358 Tortuosity, 54 Total organic carbon content (TOC), 15, 74, 156, 195 Tracking accumulation, 333, 360 source rock, 332 Transformation ratio (TR), 72, 153, 155, 158, 160, 166, 169, 176, 193, 372 critical point, 160 Tri–linear shape function, 393 Triangular distribution, 350 Triple point, 203 Uncertainty, 341, 348–352 Undersaturated phase, see Phase undersaturated Uniform distribution, 349 Uniform stretching model, 132–134, 135–137, 139, 140 Universal gas constant, 207, 468 Upscaling, 19, 252, 258, 297, 298, 306, 319, 349, 384, 385 permeability, 56 residual saturation, 308 Van der Waals equation, see Equations of state (EOS) Van–Krevelen diagram, 157, 157 Vapor–pressure line, 203 Variance, 359 Vector, 382 Virial expansion, 208 Viscosity, 39, 51 chemical compaction, 68 corresponding states (CS) model, 229–233 heavy oil, 232 Lohrenz–Bray–Clark (LBC) model, 228–229 oil, 227–233 water, 51–53 Vitrinite reflectance, 4, 151, 180, 194, 169–194 TTI model, 173 Burnham Sweeney model, 169–172 Easy–Ro model, 172 Larter model, 172–173 Void ratio, 42 Volatile oil, 217, 223–225, 227, 233 Volume shift Jhaveri and Youngeren, 220 Peneloux, 220 Volumetrics, 2, 5, 269–282, 399 Weibull distribution, 155 Wet gas, 217 Windowing, 400–401 Yield point, 81 Yielding, 81, 84–85 Zero level hydrostatic, 33 lithostatic, 38 pressure, 34