2014 a method for evaluation of water flooding performance in fractured reservoirs

A method for evaluation of water flooding performance in
fractured reservoirs
Shaohua Gu a,b,n,1
, Yuetian Liu a
, Zhangxin Chen b
, Cuiyu Ma a
a
MOE Key Laboratory of Petroleum Engineering, China University of Petroleum, Beijing 102249, China
b
Department of Chemical and Petroleum Engineering, University of Calgary, Calgary, Alberta, Canada T2N 1N4
a r t i c l e i n f o
Article history:
Received 19 December 2013
Accepted 3 June 2014
Available online 17 June 2014
Keywords:
Water flooding
Fractured reservoir
Dual-porosity
Imbibition
Relative permeability
a b s t r a c t
A mathematical model is developed for evaluation of water flooding performance in a highly fractured
reservoir. The model transforms a dual-porosity medium into an equivalent single porosity medium by
using a pseudo relative permeability method to normalize the relative permeability. This approach
allows both fractures and matrix to have permeability, porosity, endpoint saturation, and endpoint
relative permeability by themselves. Imbibition is also taken into account by modifying Chen's equation.
Some effects, including imbibition and recovery rates are investigated. The investigation shows that
imbibition can determine the potential of a fractured reservoir and a low recovery rate can improve the
water flooding situation in terms of retarding water breakthrough and controlling the rise of water cut.
A new chart composed by water cut vs. recovery curves is protracted to estimate the ultimate water-
flooding recovery rate. The water flooding performance of two reservoirs is evaluated. Compared with
numerical simulation method, the error of these two cases are not more than 2%, which proved that this
method is reliable. Both lab test data and field data are applied to a further discussion of the
characteristics of water flooding performance in fractured reservoirs. On comparison with the classical
method, such as Tong's method and the X-plot method, the reason why the new method is more suitable
to fractured reservoirs is addressed by a theoretical analysis. An appropriate application of this method
can help the reservoir engineer to optimize the reservoir management with low costs and high
efficiency.
& 2014 Elsevier B.V. All rights reserved.
1. Introduction
Experiences from oil recovery around the globe have shown
distinct water flooding performance in fractured reservoirs than in
conventional reservoirs. In most cases, the recovery usually begins
with a high production rate in an early stage and then declines
dramatically once water breaks through due to a rapid rise in
water cut, especially in some high yield wells. Moreover, the
geological complexity is also a barrier for accurate estimation of
the water flooding performance and the potential of a fractured
reservoir. Furthermore, as everyone knows, it is significant to
perform reservoir management and investment decision.
For interpretation of water flooding performance in fractured
reservoirs, many research papers have been published. Currently
used methods can be classified as two categories: reservoir
simulation and a reservoir performance analysis. The reservoir
simulation methods consist of numerical simulation and physical
simulation. Models of dual-porosity (Barenblatt et al., 1960) and
shape factors (Warren and Root, 1963; Kazemi et al., 1976) are
widely used in numerical simulation of the fractured reservoirs.
But one of the main problems is that these models are over-
simplified to meet the demand of computing. Another problem is
that history matching is a subjective process. That is, various
results may be obtained on the basis of the same data. Because of
more tunable parameters in a dual-porosity model, more probable
choices may be made by reservoir engineers. Some new technol-
ogies, such as a discrete fracture network (DFN) model and
unstructured grids (Hoteit and Firoozabadi, 2008a, 2008b; Huang
et al., 2011), can characterize a fracture network more accurately.
However, technical limitation on information collection of in-situ
fractures and enormous amount of computing are impediments to
their application. Actually, the physical simulation (Yuetian et al.,
2013) provides an objective way to present the water flooding
performance in fractured reservoirs, but high costs and low
efficiency are bottleneck problems.
Compared with the reservoir simulation methods, the reservoir
performance analysis methods are easy, fast and cheap tools, which
are composed of analytical models, empirical models and semi-
empirical models. But these types of methods need more field data
Contents lists available at ScienceDirect
journal homepage: www.elsevier.com/locate/petrol
Journal of Petroleum Science and Engineering
http://dx.doi.org/10.1016/j.petrol.2014.06.002
0920-4105/& 2014 Elsevier B.V. All rights reserved.
n
Corresponding author. Tel.: þ86 10 89732260.
E-mail address: cc0012@126.com (S. Gu).
1
Visiting scholar of University of Calgary.
Journal of Petroleum Science and Engineering 120 (2014) 130–140

and recovery experience to develop, and the predicting results also
need more checks with field production. The theory of Buckley and
Leverett (1942) and the Welge (1952) equation were first proposed
to explain the phenomena of two-phase flow in reservoirs. Accord-
ing to experiments of Efros (1958), a relationship between oil cut,
oil viscosity and outflow end water saturation in a process of water–
oil displacement was obtained. Timmerman (1971) found a rela-
tionship between cumulative oil production and an oil–water ratio
(i.e., (1fw)/fw) by field data, which was from a water flooding
reservoir in Illinois. Tong (1978, 1988) studied statistical data from
more than 20 water flooding reservoirs around the globe and drew
a chart for engineers to evaluate the water flooding performance.
Chen (1985) deduced some water displacement curve(WDC) meth-
ods by using the theory of Buckley–Leverett, the Welge equation
and the relation found by Efros, and the results were consistent
with Tong's survey. As more advances in the technology of reservoir
water flooding evaluation are made, more types of reservoirs have
been put into consideration by researchers. El-khatib (2001, 2012)
applied the Buckley–Leverett displacement theory to study water
flooding in non-communicating stratified reservoirs and in inclined
communicating stratified reservoirs. Yang (2009) proposed a new
diagnostic analysis method for water flooding performance in
conventional reservoirs.
In fact, many lessons and much experience have already been
learned from hundreds of fractured reservoirs (Allan and Sun,
2003; Sun and Sloan, 2003) during past many years (Dang et al.,
2011). Many researchers have published many mathematical
models to interpret multi-phase flow in fractured medium, such
as the De Swaan (1978) model, the Kazemi analytical model (1992)
and the Civan (1998) model. However, the existing problems of
evaluating water flooding performance in fractured reservoirs
have not been figured out properly. One of the critical problems
is how to deal with oil–water flow in a dual-porosity medium.
Another issue is how to detect the influence of imbibition on the
in-situ flow and the performance of oil wells. This paper aims to
solve the above mentioned problems. First, a model is proposed
for water–oil flow in a matrix-fracture medium by using the
method of pseudo relative permeability curves. Then Chen's model
(1982) is modified for calculation of the water breakthrough time
and water saturation at the breakthrough time. A chart is com-
posed for water-flooding evaluation by estimation of the ultimate
recovery factor. Then the water flooding performance in two
fractured reservoirs is evaluated. Compared with the classical
method, such as Tong's chart and X-plot method (1978), some
analyses are conducted and influential factors are discussed.
2. Mathematical model
2.1. Assumptions and definitions
A well group consists of one injector and one producer in a
highly fractured reservoir, and the Kazemi modeling concept
(1976) is used, as shown in Fig. 1. The additional assumptions
are given as follows: the flow is linear, isothermal, and incom-
pressible, and it obeys Darcy's law; in a dual-porosity model,
fracture and matrix have its own irreducible water saturation,
permeability, porosity and relativity permeability; the water–oil
displacement in this case is non-piston-like; finally, the reservoir is
water-wet and the imbibition effect is taken into account.
2.2. Pseudo relative permeability
Hearn (1971) used the pseudo relative permeability method to
simulate a stratified reservoir by water flooding, which means that
the reservoir is divided into many layers. Babadagli and Ershaghi
(1993) introduced this method into the dual porosity concept and
proposed the effective fracture relative permeability (EFRP)
method to reduce the model to a single porosity fracture network
model. In the stratified reservoir, each layer has its own thickness,
porosity, initial water saturation, and residual oil saturation.
Similarly, in a fractured reservoir, either fractures or matrix has
Nomenclature
A coefficient, dimensionless
B coefficient, dimensionless
b fracture aperture [L], m
fw water cut, dimensionless
f/ wf the derivative of water cut of fracture, dimensionless
h formation thickness [L], m
kf, kff conventional/intrinsic fracture permeability [L]2
, μm
km matrix permeability [L]2
, μm
kT total permeability [L]2
, μm
krof, krom, kroT oil relative permeability in fracture/matrix/total,
dimensionless
krwf, krwm, krwT water relative permeability in fracture/matrix/
total, dimensionless
L length [L], m
P1–P27 coefficient, dimensionless
Qo cumulative oil, dimensionless
qimb imbibition rate, dimensionless
qwf, qwm, qwT fracture/matrix/total flow rate [L]2
[T]1
, m2
/s
R recovery factor of OOIP, dimensionless
R0
ultimate recovery factor, dimensionless
Rn
recovery in normalized range, dimensionless
Rf, Rm, RT fracture/matrix/total recovery factor of OOIP,
dimensionless
Rf', Rm
' , RT
' fracture/matrix/total ultimate recovery factor,
dimensionless
Swf, Swm, SwT water saturation of fracture/matrix/total, dimen-
sionless
Sof, Som, SoT oil saturation in fracture/matrix/total, dimen-
sionless
Sorf, Sorm, SorT residual oil saturation in fracture/matrix/total,
dimensionless
Swif, Swim, SwiT initial water saturation in fracture/matrix/total,
dimensionless
Sn
wef , Sn
wem, Sn
weT fracture/matrix/total water saturation at out-
flow end in normalized range, dimensionless
SA
wf , SA
wm, SA
wT average water saturation in fracture/matrix/total,
dimensionless
SnA
wT fracture average water saturation in normalized range,
dimensionless
SnA
wBT water saturation at breakthrough time in normalized
range, dimensionless
t time [T], s
tB water breakthrough time [T], s
Vwf, Vwm, VwT fracture/matrix/total water volume [L]3
, m3
W recovery rate [L] [T]1
, m/s
X length [L], m
μo, μw oil viscosity [M][L]1
[T], Pa s
ϕf, ϕm fracture/matrix porosity, dimensionless
λ imbibition index, dimensionless
S. Gu et al. / Journal of Petroleum Science and Engineering 120 (2014) 130–140 131

its own properties, so they can be regarded as two different
“layers”. The pseudo relative permeability method is introduced
to transform a dual-porosity medium into an equivalent single
porosity medium, as displayed in Fig. 2. This process can simplify
the calculation by reducing the number of equations and para-
meters. Actually, the end points of saturations of both matrix and
fractures are not the same value. Therefore, the movable satura-
tion ranges (from Sor to 1Swi) of the two media, are totally
different from each other. In the process of calculation of pseudo
relative permeability, normalization is a necessary procedure for
eliminating the effect of the end points. The normalization process
aims to transform various original saturation ranges to the normal-
ized range from zero to one, which enables the end points of
matrix and fractures to be the same value, as demonstrated in
Fig. 2(a) and (b). The equation is given as follows:
Sn
w ¼
Sw Swi
1Swi Sor
ð1Þ
By the normalization process, all saturations are transformed to
the normalized range, and then the process for pseudo relative
permeability begins. The relative permeability can be tested and
calculated by the Welge–JBN method (Johnson et al., 1959) and the
saturation used in calculation is the water saturation at the
outflow end Swe; therefore, the water relative permeability can
be written as krw(Swe). The pseudo relative permeability of water in
the normalized range is (see derivation in Appendix A)
krwT ðSn
weT Þ ¼
ðkf f =kmÞUðϕf =ϕmÞUkrwf ðSn
wef ÞþkrwmðSn
wemÞ
ðkf f =kmÞUðϕf =ϕmÞþ1
ð2Þ
Similarly, the pseudo relative permeability of oil in the normal-
ized range is
kroT ðSn
weT Þ ¼
ðkf f =kmÞUðϕf =ϕmÞUkrof ðSn
wef ÞþkromðSn
wemÞ
ð3Þ
The fracture relative permeability curves seem X-shaped, as
displayed in Fig. 2(c). They can be written as follows:
krwf ðSn
wef Þ ¼ Sn
wef ð4Þ
krof ðSn
wef Þ ¼ 1Sn
wef ð5Þ
Fig. 1. Model of water flooding in water-wet fractured media and imbibition process.
Fig. 2. Model of water flooding in water-wet fractured medium: (a) matrix relative permeability curves in original range; (b) matrix relative permeability curves in
normalized range; (c) fracture relative permeability curves in normalized range and (d) total relative permeability curve in normalized range.
S. Gu et al. / Journal of Petroleum Science and Engineering 120 (2014) 130–140
132

The matrix relative permeability curves in the normalized
range (Fig. 2(b)) are transformed from relative permeability curve
in the original range (Fig. 2(a)). Here the oil/water relative
permeability ratio can be presented in a polynomial fitting form,
and why the polynomial form becomes the choice will be
explained in Section 4
kromðSn
wemÞ ¼ p1þp2USn
wem þp3USn2
wem þp4USn3
wem þp5USn4
wem
p6USn5
wem þp7USn6
wem þp8USn7
wem þp9USn8
wem ð6Þ
krwmðSn
wemÞ ¼ p10þp11USn
wem þp12USn2
wem þp13USn3
wem þp14USn4
wem
þp15USn5
wem þp16USn6
wem þp17USn7
wem þp18USn8
wem ð7Þ
According to Eqs. (6) and (7), the total relative permeability can
be written as
krwT ðSn
weT Þ
kroT ðSn
weT Þ
¼
1
ðp19þp20USn
weT þp21USn2
weT þp22USn3
weT þp23USn4
weT
þp24USn5
weT þp25USn6
weT þp26USn7
weT þp27USn8
weT Þ
ð8Þ
where the coefficients P19–P27 can be determined by fitting. The
total relative permeability can be referred to as Fig. 2(d). The total
water saturation is(see derivation in Appendix A)
SwT ¼
ðϕf =ϕmÞUSwf þSwm
ðϕf =ϕmÞþ1
ð9Þ
Similarly, the total oil saturation SoT, the total residual oil
saturation SorT and the total initial water SwiT saturation are as
follows, respectively:
SoT ¼
ðϕf =ϕmÞUSof þSom
ðϕf =ϕmÞþ1
ð10Þ
SorT ¼
ðϕf =ϕmÞUSorf þSorm
ðϕf =ϕmÞþ1
ð11Þ
SwiT ¼
ðϕf =ϕmÞUSwif þSwim
ðϕf =ϕmÞþ1
: ð12Þ
2.3. Flow in fracture medium with imbibition
Another key problem in this case is how to detect the effect of
exchange between fracture and matrix. Production from the
matrix blocks can be associated with various physical mechanisms
including oil expansion, capillary imbibition, gravity imbibition,
diffusion and viscous displacement. In water flooding reservoir, oil
expansion is not significant role. Diffusion is not an obvious
phenomenon. When fracture permeability is far higher than
matrix permeability, viscous displacement is negligible as well.
And the main mechanism in production from matrix to fracture is
imbibition. Aronofsky et al. (1958) proposed an empirical model of
imbibition correlated with the oil recovery factor and ultimate
recovery factor, which is
Rm ¼ R0
mð1e λt
Þ ð13Þ
where λ is an imbibition index, which determines of the conver-
gence rate to the ultimate recovery factor. In fact, it illustrates the
magnitude of imbibition, and the unit is [1/s]. Although λ is an
empirical parameter, it includes physical meaning. Kazemi et al.
(1992) use an equation to characterize this parameter. And λ
can also be obtained by spontaneous imbibition test or history
matching. By using the Duhamel principle, Chen and Liu (1982)
deduced a new dynamic imbibition model by using the Aronofsky
model with respect to dynamic water saturation in the fracture
system. Terez and Firoozabadi (1999) used the same model in their
research to interpret the experimental result. However, Chen's
equation has an error leading to an obvious calculation mistake,
which will be discussed in Section 4. Thus the model needs
correction, and Chen's model can be modified as (see derivation
in Appendix B)
qimbðx; y; z; tBÞ ¼ ð1SwimÞϕmR0
mλ Swf ðx; y; z; tBÞλ
Z tB
0
Swf ðx; y; z; τÞe λðtB τÞ
dτ

ð14Þ
Suppose that there is a horizontal, linear, water-wet, naturally
fractured oil-bearing formation of length L, as Fig. 3. The initial
water saturation distributions of the matrix and fracture are Swm(x,
0)¼Swim and Swf(x, 0)¼0, respectively. Water has been injected
into the inlet end (x¼0) since t¼0. The dimensionless parameters
are introduced, such as x ¼ x=L,t ¼ λt, WðtÞ ¼ WðtÞ=lλ and
qimb ¼ qimb=λ. The equations of dimensionless flow in the fractured
porous medium can be written as follows:
WðtÞf
0
wf ðSwf Þ
∂Swf
∂t
þϕf
∂Swf
∂t
þð1SwimÞϕmR0
m Swf
R t
0 Swf ðx; τÞeðt τÞ
dτ
h i
¼ 0
Swf ðx; 0Þ ¼ Swif ðxÞ
Swf ð0; tÞ ¼ 1
8

:
ϕm
dSwm
dt
þqimb ¼ 0
qimb ¼ ð1SwimÞϕmR0
m
R t
0 Swf ðx; τÞeðt τÞ
dτSwf
h i
Swmðx; 0Þ ¼ SwimðxÞ
8

:
ð15Þ
where the derivative of water cut f
0
wf ðSwf Þ can be written as
f
0
wf ðSwf Þ ¼
μo=μw
½ððμo=μwÞ1ÞSwf þ12
ð16Þ
A program is crafted to solve the two-phase flow equation for
numerical analysis. The data applied in numerical calculation can
be referred in Table 1. Some dynamic parameters can be deter-
mined through this computing, including the fracture water
saturation Swf and the matrix water saturation Swm at different
times, as shown in Figs. 4 and 5.
According to the computing results, the average water satura-
tion in fractures SA
wf , the average water saturation in matrix SA
wm,
the water saturation at the outflow end of fractures Swef, and the
Fig. 3. Imbibition process during water flooding.

water saturation at the outflow end of matrix Swem can be
obtained. Here the average means that the values of saturation
distributed in the range of dimensionless length from 0 to 1 are
averaged. Once the above parameters are determined, the average
total water saturation in the normalized range can be obtained
according to Eqs. (1) and (9)–(12), which is
SnA
wT ¼
SA
wT SwiT
1SwiT SorT
¼
ððϕf =ϕmÞUðSA
wf þSA
wmÞ=ðϕf =ϕmÞþ1ÞSwiT
1SwiT SorT
ð17Þ
Similarly, the total water saturation at the outflow end in the
normalized range Sn
weT is
Sn
weT ¼
ððϕf =ϕmÞUSwef þSwem=ðϕf =ϕmÞþ1ÞSwiT
1SwiT SorT
ð18Þ
Then the curve of Sn
weT vs. SnA
wT is plotted in Fig. 6. Fig. 6 shows
that Sn
weT is zero before water breaks through, and the total water
saturation at water breakthrough time in the normalized range is
Sn
wBT , as displayed in Fig. 6. Since the water breaks through, Sn
weT
and SnA
wT have a linear relationship. Compared with the calculation
data, the Welge equation data shows a non-linear relationship
since water breaks through. According to the calculation result of
the Sn
weT vs. SnA
wT curve shown in Fig. 6, an approximate equation
can be established as follows:
Sn
weT ¼ 0; SnA
wT rSn
wBT
Sn
weT ¼
SnA
wT Sn
wBT
1 Sn
wBT
; Sn
wBT rSnA
wT r1
8

:
: ð19Þ
2.4. Fractional oil recovery and water cut
The total recovery factor in the matrix-fracture system is
RT ¼
ϕf ð1Swif ÞRf þϕmð1SwimÞRm
ϕf ð1Swif Þþϕmð1SwimÞ
ð20Þ
From Fig. 4, we can clearly see that the recovery factor in
fractures Rf rises to 1 when at time t¼0.05. It means that Rf rises to
Table 1
Parameters of dimensionless model and corresponding
values.
Parameter, label Value
Ultimate recovery factor, R0
0.1
Matrix porosity, ϕm 0.15
Fracture porosity, ϕf 0.01
Initial fracture water saturation, Swif 0
Initial matrix water saturation, Swim 0.2
Oil–water viscosity ratio, μo/μw 10
Flow rate, W(t) 1
Fig. 4. Water saturation in fractures at different time. The dash line is Chen's (1982)
calculation, and the solid line is our calculation result.
Fig. 5. Water saturation in matrix at different time: (a) the solid line is our
calculation result and (b) the dash line is Chen's (1982) calculation result.
Fig. 6. Water saturation at out flow end in normalized range vs. average water
saturation in total by numerical calculation and Welge equation.
134

1 in a short time. In this case, it can be seen as Rf ¼1. Therefore the
total ultimate recovery factor is
R0
T ¼
ϕf ð1Swif Þþϕmð1SwimÞR0
m
ϕf ð1Swif Þþϕmð1SwimÞ
ð21Þ
According to the Buckley–Leverett theory and Eqs. (11) and
(12), the water cut is
f wðSn
weT Þ ¼
1
1þðμo=μwÞUðkrwðSn
weT Þ=kroðSn
weT ÞÞ
¼
1
1þðμo=μwÞUð1=ðp19þp20USn
weT þp21USn2
weT þp22USn3
weT þp23USn4
weT
þp24USn5
weT þp25USn6
weT þp26USn7
weT þp27USn8
weT ÞÞ
ð22Þ
Then, applying the numerical solution to Eqs. (15) and (16), a
curve of fw vs. Sn
weT can be obtained. Here the curve can be named
the water flooding characteristic curve, since this curve can
characterize the water flooding performance of a reservoir. From
Eq. (19), the relationship of Sn
weT vs. SnA
wT is known, and then the
relationship of fw vs. SnA
wT can also be obtained. However, we need
to investigate the relationship of fw vs. RT. The value of oil recovery
factor RT in total can be acquired by the following equation:
RT ¼
SA
wT SwiT
1SwiT
ð23Þ
In addition, the ultimate recovery factor in total is
R0
T ¼
1SwiT SorT
1SwiT
ð24Þ
According to Eqs. (17), (23) and (24), we can know that
RT
R0
T
¼
SA
wT SwiT
1SwiT SorT
¼ SnA
wT ð25Þ
On the basis of Eqs. (19), (22) and (25), the equation of fw vs. RT
can be established
Finally, according to Eq. (26), the relationship between fw and
RT with different RT
' is obtained.
3. Computational procedure and application
3.1. Evaluation chart plotting process
The computational process has three steps
(1) The determination of a water flooding characteristic curve is
necessary. Here we provide a series of methods. The first one is
to normalize the existing relative permeability curves of a core
sample, and then the parameters P19–P27 can be determined
by fitting. However, this method needs sufficient data.
Research shows that the number of relative permeability
curves should be more than twenty, which will be addressed
in Section 4. Otherwise, the result by insufficient data may not
be reliable. If the relative permeability curves are deficient, the
production data from other reservoirs of the same type can
still be used to generate the water flooding characteristic curve
by regression. Another feasible method is to select one classic
water flooding characteristic curve which can represent the
water flooding performance in such type of reservoirs. Once
the water flooding characteristic curve is confirmed, Eq. (22)
can be used for fitting to determine the coefficients P19–P27.
(2) By substituting different R0
values into Eq. (21), we have the
corresponding Rm
' values. Then substituting the corresponding
Rm
' value into Eq. (15), the corresponding value Sn
wBT can be
obtained.
(3) Substituting the parameters P19–P27, R0
and the corresponding
value Sn
wBT into Eq. (26), a series of curves of fw vs. RT with R0
can be obtained, and the evaluation chart is plotted in Fig. 9.
3.2. The application of field case evaluation
There are two fractured basement reservoirs in an early stage of
water-flooding, shen625 and Biantai, located in Damintun Basin,
northeastern China. Three other mature water flood reservoirs of
the same type are nearby, named Jingbei, Jinganbu and Dong-
shengbu, as shown in Fig. 7, and the properties of these reservoirs
are as Table 2. Qitai (2000) summarized four frequently-used
water flooding characteristic curves, including the Sazpnov curve,
Cipachev curve, Maksimov curve and Nazalov curve, as demon-
strated in Fig. 8. The data of the three mature waterflood
reservoirs can be plotted in the same Figure. From the Fig. 8, we
observe that the Maksimov curve is the most approaching curve to
the curves of field data. So it can be selected as the representative
curve of this type of reservoirs. Then the parameters can be
determined by using Eq. (22) for fitting, which are as follows:
P11 3.702939, P12 20.2722, P13 39.83089, P14 20.7128, P15
29.1592, P16 24.3778, P17 34.81122, P18 49.2859, P19
16.70984.
In the second step, different values of the ultimate recovery
factor RT are selected for calculation, which are 0.05, 0.1, 0.15, 0.2,
RT
R0
T
rSn
wBT ; f wðRT Þ ¼ 0
Sn
wBT rRT
R0
T
r1; f wðRT Þ ¼ 1
1þ
μo
μw
U 1
p19þp20U
RT =R0
T
ð ÞSn
wBT
1Sn
wBT

þp21U
RT =R0
T
ð ÞSn
wBT
1Sn
wBT
2
þp22U
RT =R0
T
ð Þ Sn
wBT
1 Sn
wBT

þp23U
RT =R0
T
ð ÞSn
wBT
1 Sn
wBT
4

þp24U
RT =R0
T
ð ÞSn
wBT
1 Sn
wBT
5
þp25U
RT =R0
T
ð Þ Sn
wBT
1 Sn
wBT
6
þp26U
RT =R0
T
ð Þ Sn
wBT
1 Sn
wBT
7
þp27U
RT =R0
T
ð ÞSn
wBT
1 Sn
wBT
8

8

:
ð26Þ
Fig. 7. The location of the five fractured reservoirs in Damintun Basin, north-
eastern China.

0.25, 0.3, 0.35, 0.4, 0.45 and 0.5. The range ability of recovery rate
W(t) of the three reservoirs is from 0.018 PV/year to 0.022 PV/year,
where the average value 0.02 PV/year is selected. Then the
corresponding value Sn
wBT of the different ultimate recovery factor
can be determined. After the third step, the evaluation chart can
be obtained in Fig. 9.
The Shen625 oil reservoir is in a middle stage of recovery, but
its water cut rises dramatically in recent time, and the fw vs. R
curve approaches the R0
¼0.2 curve. It indicates that field-
development strategies need to be changed badly. Some measure-
ments, including water-shutoff, reducing the choke size, new
perforation, and even adjustment in well pattern, need to be taken
for water-cut control. Otherwise, if the current field-development
strategies are not changed, the ultimate recovery factor will
merely be around 20%, which is not a desired result. The situation
of the Biantai oil reservoir is similar to the case of Shen625, which
can achieve the ultimate recovery factor around 23%, as demon-
strated in Fig. 9. As a contrast, we also use Petrel and Eclipse for
reservoir geomodelling and simulation; the dash lines in Fig. 9 are
the numerical predicting outcomes: 19.7% for shen625, 23.12% for
Biantai. The outcomes of numerical simulation do not appear
much different from those of our method. However, the cost is
totally different, because reservoir simulation is a labor-intensive
and time-consuming work.
4. Discussions
4.1. Imbibition model
According to Eq. (14) and Fig. 9, it can be known that the
recovery rate of a matrix-fracture reservoir depends on the
ultimate recovery factor of matrix. That is, the higher the matrix
recovery factor is, the stronger the imbibition is, and the more
slowly the water cut rises. In Chen's calculation, the parameter Rm
'
(ultimate recovery factor of matrix) is set to 0.1, but the variation
range of water saturation in matrix is from Swm ¼0.2 to Swm ¼0.88
at different time, as shown in Fig. 5(b). As a result, the ultimate
recovery factor of matrix Rm
' can reach 0.85, far beyond the
precondition Rm
' ¼0.1, which means that model is not self-
consistent. It can be seen that Chen and Liu (1982) miscalculated
the imbibition rate, which leads to over-flow of oil from matrix
and the numerical results mismatch the pre-condition. However,
our computing result presents that the variation range of water
saturation in matrix is from 0.2 to 0.28, so Rm
' cannot exceed
0.1. Hence a correct calculation of the imbibition rate is very
significant.
4.2. Effect of injection rate and fracture distribution
From the result of calculation using different recovery rates, it
can be found from Fig. 10 that the curves of different injection
rates show obvious distinction in total water saturation at break-
through time in the normalized range Sn
wBT , as Fig. 10 shows.
For an analysis of the post water breakthrough stage, the
relative permeability curves of two fractured core samples with
the same properties but different fracture distribution are tested,
as Fig. 11 shows. Then the fw vs. Sw plot are calculated by the
relative permeability curve on the basis of Eq. (22), as displayed in
Fig. 12. Sample (a) has joint fractures which connect the two ends
of core samples, while sample (b) has two disconnected fractures.
In the test, the differential pressure between the inflow end and
the outflow end is 0.1 MPa. Because of the different permeability
by fracture connectivity, the flow rates of two samples are
different. The flow rate of sample (a) is 0.04 PV/min, and that of
sample (b) is 0.13 PV/min.
From Fig. 12, the water cut of sample (b) rises dramatically at
first and then the curve approaches the Sazpnov curve, while the
fw vs. Sw curve approaches the Maksimov curve and those of the
other three fractured reservoirs located in Damintun Basin, as
Fig. 8 shows. From the above observation, we draw the conclusion
Fig. 9. Evaluation of two fractured reservoirs by our chart: (a) the solid curve is
field data and (b) the dash curve is numerical simulation result.
Fig. 10. Water saturation at breakthrough time with different recovery rates.
Fig. 8. The water flooding characteristic curves and field data.
136

that the higher the recovery rate is, the faster the water cut rise
since water break through. To check this conclusion, field data
from another three fractured reservoirs are plotted, which are
Renqiu reservoir (Qitai, 2000), Yanling reservoir and Casablanca
reservoir(Allan and Sun, 2003; Sun and Sloan, 2003), respectively.
The annual oil recovery rate of Yanling is 8% per year, that of
Renqiu is 1.6%, and that of Casablanca is 1.2%. As shown in Fig. 12,
the trend of field data also proves the conclusion.
From the above analysis, it can be known that the fracture
distribution plays a significant role in water performance as well.
However, the fracture distribution is uncertain everywhere. Then
we calculate the average curve calculated by 20 relative perme-
ability curves of the three mature waterflood reservoirs. As Fig. 12
shows, the average curve is close to the Maksimov curve, which
can represent the type of the reservoirs. That is why we recom-
mend the number of relative permeability curves should be
more than twenty, because only one or two sample cores cannot
represent the features of the whole reservoir, such as the different
presentations of sample (a) and sample (b) in the relative
permeability test.
4.3. Comparison with Tong's method
Most of the classical methods, such as Tong's method (1988),
are based on the Buckley–Leverett theory, together with the
assumption of single porosity and no capillary pressure. The
Fig. 11. Image and properties of core samples.
Fig. 12. The water flooding characteristic curves, field data and lab test data.
Fig. 13. Comparison of evaluation method: (a) the solid line is our chart (b) and the
dash line is Tong's chart.

empirical formula of Tong's method is as follows:
lg
f w
1f w
¼ 7:5 ðRR0
Þþ1:69 ð27Þ
Our method is based on a dual-porosity model and takes the
imbibition into consideration. The effect of imbibition is mainly
reflected in the following aspects: in most matrix-fracture reser-
voirs without an aquifer, connate water in fractures cannot be a
continuous phase. If water can be a continuous phase, water flows
more easily so the capillary pressure can suck the water into
matrix until no free flowing water is left in the fractures. There-
fore, when reservoir recovery begins, oil in the fracture medium is
produced first, and water cut is zero during that time. Tong's
chart (Fig. 13) shows that the connate water is an important factor
to the ultimately recovery rate. From his plot of statistics from 24
single porosity reservoirs, the initial water cut of these reservoirs
is symptomatic of connate water saturation. The higher the initial
water cut is, the higher the connate water saturation is. In
addition, the higher initial water cut also leads to a lower recovery
rate. So it is an apparent distinction of water performance between
matrix-fracture reservoirs and single porosity reservoirs. It is
also the reason why Tong's method is not suitable for water-wet
fractured reservoirs.
4.4. Comparison with X-plot approach
Ershaghi and Omorigie (1978) and Ershaghi and Abdassah
(1984) developed the X-plot waterflood-analysis method on the
basis of a semi-log linear relative permeability ratio for inter-
mediate saturation values
krwðSweÞ
kroðSweÞ
¼ AeBSwe
ð28Þ
It is also the main assumption that the plot of log (krw/kro) vs. Sw
is a straight line, as shown in Fig. 14. The real data of log (krw/kro)
vs. Sw
n
of matrix and fractures both have a straight line section.
Actually the log (krw/kro) vs. Sw lines of matrix and fractures in a
semi-log plot are not straight once water saturation in the
normalized range is close to 0, which are corresponding to the
early stage. That is why this method cannot work until the water
cut reaches 50%. Compared with the matrix, the real data of log
(krw/kro) vs. Sw
n
of the fractures has a shorter straight line. If the
straight line assumption is used to predict the ultimate recovery
factor of the fractures, the deviation of the fracture curve (krwf/krof)
is larger, as marked in Fig. 14. That is, the more the fractures are in
the reservoir, the shorter the straight line is for the curve of log
(krw/kro) vs. Sw
n
of the total matrix-fracture system. It illustrates
that the methods on the basis of the assumption of Eq. (28) are no
longer suitable to a highly fractured reservoir, due to the mismatch
of real data by the exponential form. Because of good fitting, the
polynomial form becomes the choice, as Fig. 14 shows.
5. Conclusions
(1) To develop the new method for evaluation of the water flooding
performance in fractured reservoirs, some unique features of
water-wet matrix-fracture reservoirs must be taken into con-
sideration, such as the imbibition process and dual-porosity.
These features will lead to an obvious distinction in the fractured
reservoir water flooding performance. In addition, the recovery
rate also has some effects on water flooding performance.
(2) To study a matrix-fracture reservoir, a dual-porosity model is a
common method. But numerous operations and parameters
make the model hard to be used in simulation directly. The
pseudo relative permeability and saturation average can be a
solution to this problem.
(3) In application of the modified imbibition model, the imbibi-
tion flow rate can be related to the ultimate recovery factor. It
provides a way to evaluate water flooding performance and
estimate the potential of a reservoir by using the ultimate
recovery factor. A different ultimate recovery factor yields a
different water cut curve in a matrix-fracture reservoir. Thus
the data of water cut with the recovery rate can be used for
judging how much the ultimate recovery factor can finally be.
(4) From the comparison with numerical simulation, our method
is a faster and easier tool which can provide reliable results.
Compared with the classical methods, such as Tong's method
and the X-plot method, our method takes more unique
features of water-wet fractured reservoirs. So it is more
suitable to the fractured reservoirs.
Acknowledgments
The authors are grateful for financial support from National Sci-
ence and Technology Major Project (Grant No. 2011ZX05009-004-001)
Table 2
The main properties of the five fractured reservoirs in Damintun Basin
Reservoir Lithology OOIP
( 109
kg)
Average permeability
(103
μm)
Reservoir medium
depth (m)
Past producing
time (yr)
Producing oil
in total ( 109
kg)
Dongshegnbu Metamorphic 15.1 98.7 2840 24 3.9
Jinganbu Sandstone Metamorphic 10.5 68.5 2903 12 1.3
Jingbei Carbonate 32.9 162 2725 23 7.1
Shen625 Sandstone Metamorphic 13.5 36.3 3430 7 1.4
Biantai Sandstone Metamorphic 18.1 99.8 1975 11 1.89
Fig. 14. Krof/Krom vs. Sn
w curve of matrix and fracture and their fitting model.
138

of China, The National Natural Science Foundation (Grant No.
51374222) of China and China Scholarship Council.
Appendix A. Pseudo relative permeability and normalized
saturation
In the dual porosity model (as Fig. 1), the total permeability kT
is (Van Golf-Racht, 1982)
kT ¼ kf þkm ¼ kf f
b
h
þkm ¼ kf f ϕf þkm ðA:1Þ
where the total means the saturation in the whole porous medium
system including both matrix and fractures. The water flow rate in
total is
qwT ¼
kT UkrwT ðSn
weT ÞUh
μw UL
dP
dx

ðA:2Þ
The water flow rate in the fracture system is
qwf ¼
kf Ukrwf ðSn
wef ÞUh
μw UL
dP
dx

¼
kf f Ukrwf ðSn
wef ÞUb
μw UL
dP
dx

ðA:3Þ
The water flow rate in the matrix system is
qwm ¼
km UkrwmðSn
wemÞUðhbÞ
μw UL
dP
dx

ðA:4Þ
According to the mass balance, we have the total water flow
rate
qwT ¼ qwf þqwm ðA:5Þ
Thus, the pseudo relative permeability of water in the normal-
ized range is
krwT ðSn
weT Þ ¼
ðkf f =kmÞUðϕf =ϕmÞUkrwf ðSn
wef ÞþkrwmðSn
wemÞ
ðA:6Þ
Because of the incompressibility assumption, the volume con-
servation equation is
VwT ¼ Vwf þVwm ðA:7Þ
Then it can be expanded as follows:
VT Uðϕm þϕf ÞUSwT ¼ VT Uϕf USwf þVT Uϕm USwm ðA:8Þ
Thus the total water saturation is obtained
SwT ¼
ðϕf =ϕmÞUSwf þSwm
ðϕf =ϕmÞþ1
ðA:9Þ
Appendix B. Imbibition flow model
We have Aronofsky's model
Rm ¼ R0
mð1e λt
Þ ðB:1Þ
According to Eq. (B.1), the dimensionless cumulative oil pro-
duction at time t is
QoðtÞ ¼ ð1SwimÞϕmR0
mð1e λt
Þ ðB:2Þ
With respect to the effect of variation of saturation in fractures,
the imbibition rate is
qimbðtÞ ¼
dQoðtÞ
dt
¼ ð1SwimÞϕmR0
mλe λt
ðB:3Þ
where the matrix blocks can be divided into many cells, such as
cell A and cell B, as displayed in Fig. 3.
Once the injected water enters into the fracture-matrix med-
ium, the imbibition process begins. During the water flooding
process, each cell has its own imbibition flow once the water
contacts it. Hence the imbibition flow rate in fractures at time t1 is
qimbðt1Þ ¼ ð1SwimÞϕmR0
mλSwf ðtoÞeλðt1 toÞ
ðB:4Þ
After a period of time Δt, the newly injected water touches cell
A. Meanwhile, the water that contacted cell A now flows to the
fracture area near beside cell B. Cell B obeys the same rule as cell A,
but water becomes less and less because some water has been
imbibed into cell A. Hence the imbibition flow rate in the fractures
at time t2 is
qimbðt2Þ ¼ ð1SwimÞϕmR0
mλSwf ðtoÞeλðt2 toÞ
þð1SwimÞϕmR0
mλ½Swf ðt1ÞSwf ðtoÞeλðt2 t1Þ
ðB:5Þ
Then the water injection continues until the water breaks
through, and the time at water breakthrough is tB. We assume
that the time from 0 to tB is divided into n sections. The flow rate
at the water breakthrough time tB is
qimbðtBÞ ¼ ð1SwimÞϕmR0
mλSwf ðtoÞeλðtB toÞ
þð1SwimÞϕmR0
mλ½Swf ðt1ÞSwf ðtoÞeλðtB t1Þ
þð1SwimÞϕmR0
mλ½Swf ðt2ÞSwf ðt1Þe λðtB t2Þ
:::
þð1SwimÞϕmR0
mλ½Swf ðtn1ÞSwf ðtn 2ÞeλðtB tn 1Þ
ðB:6Þ
Eq. (B.6) can also be written as follows:
qimbðtBÞ
¼ ð1SwimÞϕmR0
mλ ∑
n 2
i ¼ 0
Swf ðtiÞ
d
dt
eλðtB tÞ

t ¼ ti þ θΔti
Δti þSwf ðtn 1ÞeλðtB tn 1Þ
( )
ðB:7Þ
where t0¼0, Δti ¼ti þ 1 ti, and 0rθr1. If n-1, Δti-0 and ti is
replaced by the characteristic time τ, then our modification model
is
qimbðx; y; z; tBÞ
¼ ð1SwimÞϕmR0
mλ Swf ðx; y; z; tBÞλ
Z tB
0
Swf ðx; y; z; τÞe λðtB τÞ
dτ

ðB:8Þ
Appendix C. Supplementary information
Supplementary data associated with this article can be found in
the online version at http://dx.doi.org/10.1016/j.petrol.2014.06.002.
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