Q913 rfp w3 lec 9

Reservoir Fluid Properties Course (1st Ed.)

1.
2.
3.
4.

General Notes about EoS
Ideal Gas EoS
Compressibility Factor
Van Der Waals EoS

2013 H. AlamiNia

Reservoir Fluid Properties Course: Advanced EoS and C7+ Characterization

2

1. Cubic EoS:
A. SRK EoS
B. PR EoS
C. Other Cubic EoS

2. Non Cubic EoS
3. EoS for Mixtures
4. Hydrocarbons
A. Components
B. Mixtures
C. Heavy Oil

2013 H. AlamiNia


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Developments of
Cubic Equations of State
The van der Waals equation is seen to qualitatively
describe the pure-component phase behavior at
temperatures above, equal to, and below the
critical temperature.
Later developments of cubic equations of state
have primarily served to improve the quantitative
predictions of either vapor pressure or phase
properties.
In addition, much effort has been used to extend
the application area of cubic equations of state
from pure components to mixtures.
2013 H. AlamiNia


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Redlich and Kwong Equation
The equation of Redlich and Kwong (1949) is, by
many, considered the first modern equation of state
and takes the form
𝑹𝑻
𝑷=
−
𝑽− 𝒃

𝒂

𝑻𝑽 𝑽 + 𝒃

By comparing this equation with the van der Waals
equation, it is seen that the attractive term has a
more complicated temperature dependence.
This temperature modification serves to improve the
vapor pressure predictions.

The parameters a and b are found by imposing the
critical point criteria.
2013 H. AlamiNia


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The Soave–Redlich–Kwong (SRK)
Equation
Soave (1972) found the pure-component vapor
pressures calculated from the Redlich–Kwong (RK)
equation to be somewhat inaccurate.
He suggested replacing the term in the RK equation
by a more general temperature dependent term, a
(T), giving an equation of state of the form:
𝑹𝑻
𝒂 𝑻
𝑷=
−
𝑽− 𝒃
𝑽 𝑽+ 𝒃

This equation is usually referred to as the Soave-Redlich-Kwong
or just SRK equation.

2013 H. AlamiNia


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Terms in SRK Equation
0.08664𝑹𝑻 𝒄
𝒃=
𝑷𝒄

0.42747𝑹2 𝑻2
𝒄
𝒂 𝑻 = 𝒂𝒄 𝜶 𝑻 , 𝒂𝒄 =
𝑷𝒄

2

𝜶 𝑻 =

1+ 𝒎 1−

𝑻
𝑻𝒄

, 𝒎 = 0.48 + 1.574𝝎 − 0.176𝝎2

ω is the acentric factor
2013 H. AlamiNia


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SRK Equation in Terms of Z
With the classical Soave temperature dependence,
α (T) = 1 at the critical temperature, where a (T)
therefore becomes equal to ac.
Recalling that the compressibility factor Z is defined
as Z= (PV)/ (RT), SRK Equation (P=RT/ (V-b)-a (T)/ (V
(V+b))) may be rewritten in terms of Z:
𝒁3 − 𝒁2 + (𝑨 − 𝑩 + 𝑩2 )𝒁 − 𝑨𝑩 = 0
𝒂 𝑻 𝑷
𝒃𝑷
𝑨= 2 2 , 𝑩=
𝑹 𝑻
𝑹𝑻

With the SRK equation, the compressibility factor
of a pure component at its critical point will always
be equal to 0.333.
2013 H. AlamiNia


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Peng–Robinson (Pr) Equation
The liquid-phase densities predicted using the SRK
equation are in general too low.
Peng and Robinson (1976) traced this deficiency to
the fact that the SRK equation predicts the pure
component critical compressibility factor to be
0.333. The critical compressibility factors are
generally of the order 0.25 to 0.29, i.e., somewhat
lower than simulated using the SRK equation.

2013 H. AlamiNia


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PR Formulation
Peng and Robinson suggested an equation of the
form:
𝑷=

𝑹𝑻
𝒂 𝑻
−
𝑽− 𝒃
𝑽 𝑽 + 𝒃 + 𝒃(𝑽 − 𝒃)
0.07780𝑹𝑻 𝒄
𝒃=
𝑷𝒄

𝑹2 𝑻2
𝒄
𝒂 𝑻 = 𝒂 𝒄 𝜶 𝑻 , 𝒂 𝒄 = 0.45724
𝑷𝒄
2

𝜶 𝑻 =

2013 H. AlamiNia

1+ 𝒎 1−

𝑻
𝑻𝒄

, 𝒎 = 0.37464 + 1.54226𝝎 − 0.26992𝝎2


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Other Cubic Equations of State
The increasing popularity of cubic equations of
state in the 1970s and 1980s inspired
thermodynamics research groups to propose
alternatives to the SRK and PR equations. Many of
these equations have the general form:
𝑹𝑻
𝒂 𝑻
𝑷=
−
𝑽 + 𝜹1
𝑽 + 𝜹2 𝑽 + 𝜹3

The Equation offers the opportunity to include three
different volumetric correction parameters, δ1, δ2, and
δ3.

2013 H. AlamiNia


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Which EoS to Use?
In the petroleum industry, it is important with
some kind of industrial standards to enable
different companies working on the same project to
produce consistent calculation results.
PR seems to be the preferred choice in North America.
Europe generally prefers SRK, while
The rest of the world is more divided between the two
equations of state.

2013 H. AlamiNia


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Other Equations of State
Much exploration activity is currently directed
towards deep reservoirs at high temperature and
high pressure.
The ability of the classical cubic equations of state
to represent the molecular interactions at such
conditions has often been questioned.
More sophisticated equations of state have been
proposed, some of which include terms to account
for the strong repulsive forces acting at high
pressures.
2013 H. AlamiNia


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Other Equations of State (Cont.)
There is little evidence that any of these equations
should be more suited for representing the PVT
properties of petroleum reservoir fluids at elevated
pressures and temperatures than a conventional
cubic equation of state.
 When it comes to simulating hydrocarbon liquid–
liquid split as, for example, oil–asphaltene
equilibria, more advanced equations of state, for
example, the PC-SAFT equation may be needed.

2013 H. AlamiNia


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Differences between EoS for a Pure
Component and Mixtures
Typically, a model for a pure component physical
property contains parameters that are constant or
temperature-dependent and found either by fitting
to data or by CSP.
Thus, the EoS models for pure gases and liquids
express the relationship among the variables P, V,
and T.

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Differences between EoS for a Pure
Component and Mixtures (Cont.)
To describe mixture properties, it is necessary to
include composition dependence which adds
considerable richness to the behavior, and thus
complicates modeling.
Therefore, a mixture equation of state (EoS) is an
algebraic relation between P, V, T, and {y}, where {y}
is the set of n-1 independent mole fractions of the
mixture’s n components.

2013 H. AlamiNia


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Challenges to EoS Models for Mixtures
Composition Dependence of Liquid Partial
Properties,
Multiphase Equilibria,
The Critical Region and High Pressures

2013 H. AlamiNia


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Composition Dependence of
Liquid Partial Properties
The composition dependence of the properties of
liquid mixtures is fundamentally different from that
of a vapor or gas.
The strongest effect on gaseous fluids is caused by
changes in system density from changes in
pressure; composition effects are usually of
secondary importance, especially when mixing is at
constant volume.
Except at high pressures, vapors are not dissimilar
to ideal gases and deviations from ideal mixing are
small.
2013 H. AlamiNia


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Applications of the Equation of State
The Determination of the Equilibrium Ratios
The system temperature T, the system pressure p, and
the overall composition of the mixture zi must be
known.

Determination of the Dew-Point Pressure
A saturated vapor exists for a given temperature at the
pressure at which an infinitesimal amount of liquid first
appears (Pd).

Determination of the Bubble-Point Pressure
The pressure at which the first bubble of gas is formed

Determination of the Mixture Critical Properties
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C7+ Characterization
Naturally occurring oil or gas condensate mixtures
may contain thousands of different components.
 Such high numbers are impractical in flash calculations.

 Some components must be lumped together and
represented as pseudocomponents.
C 7 + characterization consists of representing the
hydrocarbons with seven and more carbon atoms (the
heptane plus or C 7 + fraction) as a convenient number
of pseudocomponents and to find the needed equation
of state parameters (T c, P c, and ω) for each of these
pseudocomponents.
2013 H. AlamiNia


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The Characterization
(or Lumping) Problem

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Classes of Components
The components contained in oil and gas
condensate mixtures can be divided into three
classes:
Defined components:
The defined components contained in petroleum reservoir
fluids are N 2 , CO 2 , H 2 S, C 1 , C 2 , C 3 , iC 4 , nC 4 , iC 5 , nC 5
, and C 6

C 7 + fractions:
Each C 7 + fraction contains hydrocarbons with boiling points
within a given temperature interval.

Plus fraction:
 The plus fraction consists of the components that are too
heavy to be separated into individual C 7 + fractions.
2013 H. AlamiNia


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Sample of Components

Molar Composition of North Sea
Gas Condensate

2013 H. AlamiNia


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Mixing of Multiple Fluids
There is often a need to mix a number of reservoir
fluid compositions into one.
This is, for example, the case when multiple fluids are let
to the same process plant.

When representing the mixed stream, one may
either work
With a weaved composition where the
pseudocomponents of each stream are retained or
With a truly mixed composition.

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Weaved Composition

The two compositions have initially
been characterized individually. For
both fluids, the C7+ fraction is
represented using three
pseudocomponents. As is seen, the
pseudocomponent properties differ
between the two fluids.

2013 H. AlamiNia


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The Molar Amounts of
the Weaved Composition
In the weaved composition, the molar amounts of
the defined components have been obtained as a
simple average of the molar concentrations of
these compounds in each individual composition.
The weaved composition contains all the
pseudocomponents found in each of the two
compositions.

2013 H. AlamiNia


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Mixing
It is recommended to carry out the mixing before
lumping into pseudocomponents. Say NFLUID different
fluids are to be mixed, the properties of carbon number
fraction i of the mixed fluid are found from
𝒎𝒊𝒙
𝑻 𝒄𝒊 =

𝒋

𝑵𝑭𝑳𝑼𝑰𝑫
𝒋=1

𝒋

𝑭𝒓𝒂𝒄 𝒋 𝒛 𝒊 𝑻 𝒄𝒊

𝒋=1

𝑭𝒓𝒂𝒄

𝒎𝒊𝒙
𝝎 𝒄𝒊

=

𝒎𝒊𝒙
, 𝑷 𝒄𝒊 =

𝒋
𝒋 𝒛𝒊
𝒋=1

𝑭𝒓𝒂𝒄 𝒋

𝒋=1

𝑭𝒓𝒂𝒄

𝒋

𝒛 𝒊 𝒎𝒊𝒙 =

𝑭𝒓𝒂𝒄 𝒋 𝒛 𝒊 , 𝑴 𝒊 𝒎𝒊𝒙 =
𝒋=1

𝒋=1

𝒋

𝒋=1
𝒋 𝒋
𝒛 𝒊 𝝎 𝒄𝒊
,
𝒋
𝒋 𝒛𝒊

𝒋=1

𝒋

𝑭𝒓𝒂𝒄 𝒋 𝒛 𝒊 𝑷 𝒄𝒊

𝒋

𝒋
𝒛𝒊

𝒋

𝑭𝒓𝒂𝒄 𝒋 𝒛 𝒊 𝑴 𝒊

𝒋=1


,

𝒋
𝒛𝒊

,

In these equations, zij is the molar fraction of carbon number
fraction i in the j-th composition to be mixed. Frac (j) is the mole
fraction of the j-th composition of the total mixture.
2013 H. AlamiNia


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Heavy Oil Composition Simulation
PVT simulations on heavy oil mixtures have
traditionally been carried out using black oil
correlations expressing the fluid properties in terms
of easily measurable quantities such as API oil
gravity, gas gravity, and gas/oil ratio.
With the application of secondary recovery
techniques such as gas injection and thermal
stimulation it has become more interesting also for
heavy reservoir oils to make compositional
equation-of-state-based simulations.
2013 H. AlamiNia


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Heavy Oil Compositions
A heavy oil is one of a high density at standard
conditions. Crude oils are essentially mixtures of
paraffinic (P), naphthenic (N), and aromatic (A)
compounds.
The densities of aromatics are higher than those of
naphthenes and paraffins of the same molecular
weight.
This is consistent with chemical analyses showing that
heavy oil mixtures are rich in aromatic compounds.
 The term heavy oil may be used for oil mixtures of an
API gravity below 30.
2013 H. AlamiNia


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Wax Precipitation
The majority of the C10+ aromatics present in
crude oil mixture will be components containing
one or more aromatic ring structures with paraffinic
side branches.
The melting temperature of that type of
compounds is low as compared to that of normal
and slightly branched paraffins of approximately the
same molecular weight.
For this reason, wax precipitation is unlikely to take place
from a heavy oil mixture.

2013 H. AlamiNia


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Viscosity of Heavy Oil Mixtures
Since high-molecular-weight compounds may be
kept in solution in the oil at low temperatures, the
viscosity of heavy oil mixtures can be very high
indeed at production conditions and even at
reservoir conditions.
Gas injection is often applied to heavy oil
reservoirs. If the gas is dissolved in the oil, it will
lower the oil viscosity and facilitate production and
possibly also enhance the recovery rate.

2013 H. AlamiNia


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1. Pedersen, K.S., Christensen, P.L., and Azeem,
S.J. (2006). Phase behavior of petroleum
reservoir fluids (CRC Press). Ch4 & Ch5.
2. Poling, B.E., Prausnitz, J.M., John Paul, O., and
Reid, R.C. (2001). The properties of gases and
liquids (McGraw-Hill New York). Ch1 & Ch4 &
Ch5 & Ch8.
3. Tarek, A. (1989). Hydrocarbon Phase Behavior
(Gulf Publishing Company, Houston). Ch3.

2013 H. AlamiNia


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1. Phase Equilibrium Calculations
2. Tc, Pc, and ω Calculation
3. K-Factor & Delumping

2013 H. AlamiNia


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Q913 rfp w3 lec 9

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Q913 rfp w3 lec 9