mirzaei-paiaman and masihi 2014
TRANSCRIPT
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DOI: 10.1002/ente.201300155
Scaling of Recovery by Cocurrent Spontaneous Imbibitionin Fractured Petroleum Reservoirs
Abouzar Mirzaei-Paiaman*[a, b] and Mohsen Masihi[a]
Introduction
Spontaneous imbibition of an aqueous phase (i.e., the wet-
ting phase, WP) into matrix blocks arising from capillary
forces is an important mechanism to displace the nonwetting
phase (NWP) from fractured reservoirs.[1] This can be either
countercurrent or cocurrent displacement. In countercurrent
spontaneous imbibition (COUCSI) all the permeable faces
of a rock saturated with NWP are brought into contact with
a WP and the WP flows in the opposite direction to the ex-
pelled NWP.[2,3] However, in cocurrent spontaneous imbibi-
tion (COCSI) only a portion of the permeable surfaces is in
contact with a WP and, while keeping the remaining permea-
ble surfaces covered by the NWP, both phases flow in thesame direction.
In a typical COCSI experiment, matrix surfaces covered
by WP and NWP are kept constant during the process.[4–14] In
this case, level of the WP in the fracture system is kept con-
stant at a certain level and there is no viscous force in the
fracture system. We use the same COCSI case in this study.
There are also other cases in which the recovery per-
formance of a matrix block or a stack of matrix blocks under
advancing WP level in the fracture system are studied.[15–20]
Despite the fact that the COCSI can take place in frac-
tured reservoir,[19,21,22] most attention has been received on
scale-up of COUCSI.[23–25] Use of the scaled-up results of
COUCSI experiments leads to pessimistic forecasts regarding
the rate of recovery and final recovery.[10,18,26] It has been em-
phasized that COCSI is more efficient in terms of both final
recovery and displacement rate than COUCSI.[7,9–11,18–20,26,27]
Due to the significant differences between the recovery per-
formances in these processes, the corresponding scaling
equations cannot interchangeably be used. Several studies
show that the scaling equations developed for the COUCSI
process fail to scale up the COCSI data. [8,10,28–30]
A few scaling equations have been proposed for recovery
prediction of COCSI.[31–33] They have been applied to one-di-
mensional displacement as no characteristic length has been
defined for the COCSI. These scaling equations do not incor-
porate all of the factors influencing the process; consequent-
ly, they cannot properly describe the process. The scaling
equation proposed by Rapoport[31] was developed by using
the inspectional analysis of the main governing equations,
making some simplifying assumptions; including that the
prototype WP/NWP viscosity ratio must be duplicated in the
model tests, initial fluid saturations in the prototype must be
duplicated in the model tests, the relative permeability func-
tions must be the same for both the model and the proto-
type, and the capillary pressure functions for the both the
model and the prototype must be related by direct propor-
tionality. Several studies have attempted to develop the scal-ing equation of Rapoport (e.g., Ref. [34, 35]) to COUCSI
data only. The scaling equations of Li[32] and Bourbiaux[33]
were derived based on a restricting approximate solution to
the main governing equations. They take the assumption of
piston-like displacement which is valid only for a few particu-
lar cases. However, according to Mirzaei-Paiaman and
Masihi,[25] the development of these two scaling equations is
not consistent with common scaling practices.
The main purpose of this study is to present universal scal-
ing equations for one-dimensional COCSI based on the
recent finding of Schmid[36] who notices that the analytical
solution to unidirectional displacement given by McWhorter
and Sunada[37] applies to COCSI with no artificial boundary
conditions. We consider the consistency between the devel-
opment of the new scaling equations and common practices
as was considered for COUCSI in Mirzaei-Paiaman and
Masihi.[25] These new scaling equations are rewritten in terms
[a] Dr. A. Mirzaei-Paiaman, Dr. M. Masihi
Department of Chemical and Petroleum Engineering
Sharif University of Technology
P.O. Box 11365-9465, Azadi Ave., Tehran (Iran)
E-mail: [email protected]
[b] Dr. A. Mirzaei-Paiaman
Department of Petroleum Engineering
NISOC, Ahvaz (Iran)
Cocurrent spontaneous imbibition (COCSI) of an aqueous
phase into matrix blocks arising from capillary forces is an
important mechanism for petroleum recovery from fractured
petroleum reservoirs. In this work, the analytical solution to
the COCSI is used to develop the appropriate scaling equa-
tions. In particular, the backflow production of the nonwet-
ting phase at the inlet face is considered. The resulting scal-
ing equations incorporate all factors that influence the pro-
cess and are found in terms of the Darcy number (Da) and
capillary number, (Ca). The proposed scaling equations are
validated against the published experimental data from the
literature.
166 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim Energy Technol. 2014, 2, 166–175
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of two physically meaningful dimensionless numbers: the
Darcy number (Da) and the capillary number (Ca). Then the
new scaling equations are validated by using experimental
data from the literature. Furthermore, we simplified the scal-
ing equations, which can be useful in some certain applica-
tions.
The Analytical Solution
The partial differential equation for one-dimensional hori-
zontal displacement of two immiscible, incompressible fluids
can be expressed by:[37]
uw ¼ f Swð Þ ut D Swð Þ@ Sw@ x
ð1Þ
This equation in combination with the material balance con-
dition gives:[37]
@ Sw@ t
¼ ut@ f Swð Þ@ x
þ @ @ x
D Swð Þ@ Sw@ x
ð2Þ
in which the subscript w denotes the WP, f is the porosity, S
is the saturation, t is the time, ut(=uw+unw) is the total volu-
metric flux (ut=0 for countercurrent flow and u t>0 for uni-
directional displacement), the subscript nw denotes the
NWP, x is the spatial coordinate, f (Sw) is the fractional flow
in the absence of capillary pressure defined as
f Sw
ð Þ ¼
krw mnwk
rw
mnw þ
krnw
mw ð
3
Þand D(Sw) is the capillary diffusion function defined as,
D Swð Þ ¼ f Swð Þkkrnw mnwdP cdSw
ð4Þ
in which kr is the relative permeability, m is the dynamic vis-
cosity, k is the absolute permeability, and P c is the capillary
pressure.
The appropriate initial and boundary conditions can be de-
fined as,[37]
Sw x; 0ð Þ ¼ Swi ð5ÞSw þ1; t ð Þ ¼ Swi ð6Þ
uw 0; t ð Þ ¼ At 1=2 ð7Þ
The initial condition in Equation (5) states that at time
zero the porous medium is at initial WP saturation, Swi. The
medium is assumed as a semi-infinite host at Swi at the far
boundary for all times [Eq. (6)]. Equation (7) is an imposed
inlet boundary condition defined by McWhorter and
Sunada[37] to solve the problem given in Equation (2).
McWhorter and Sunada[37] define the positive parameter A
as,
A ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2 1 f iRð ÞZ
Sw;BC
Swi
Sw Swið ÞD Swð ÞF Swð Þ f n Swð Þ dSw
s ð8Þ
in which f i is the WP fraction of the total efflux, R ¼ utuw 0;t ð Þ
is the ratio of the total volumetric flux to the volumetric flux
of the WP at the inlet (R=0) for countercurrent flow, andR=1 for unidirectional displacement, Sw,BC is the saturation
of WP at the inlet open boundary, f n is the normalized f , and
F is the fractional flow function in the presence of capillary
effects.[37]
f n ¼ f f ið ÞR
1 f iR ð9Þ
F Swð Þ ¼ 1 R Sw;BC
Sw
bSwð ÞD bð ÞF bð Þ f n bð Þ d b
R Sw;BC
Swi
SwSwið ÞD Swð ÞF Swð Þ f n Swð Þ dSw
ð10Þ
The exact solution to Eqation (2) is given implicitly by
McWhorter and Sunada[37] as:
x Sw; t ð Þ ¼ 2 A 1 f iRð Þ
F 0 Swð Þt 1=2 ð11Þ
in which F ’ is the derivate of F with respect to Sw defined as:
F 0 Swð Þ ¼R Sw;BC
Sw
D bð ÞF bð Þ f n bð Þ d bR Sw;BC
Swi
SwSwið ÞD Swð ÞF Swð Þ f n Swð Þ dSw
ð12Þ
The solution expressed in Equation (11) can be used byfirst prescribing Sw,BC and calculating F (Sw) from Equa-
tion (10), and then finally computing A from Equation (8).
However, the use of Equation (10) is indirect, as it has the
form of an implicit functional equation, from which F (Sw)
has to be extracted. Therefore the computation of F (Sw)
from the integral Eq. (10) is performed by an iterative proce-
dure. A convenient first trial is F (Sw)=1 as suggested by
McWhorter and Sunada.[37]
By combining Equation (1) with the definition of R and
the self-similar variable l ¼ xt 1=2, the volumetric flux of theWP at the inlet boundary can be written as: [36]
uw 0; t ð Þ ¼ D Sw;BC
1 f Sw;BC
R
dSwd l l¼0
j t 1=2 ¼ At 1=2 ð13Þ
Using this equation, Schmid[36] noticed that the inlet boun-
dary condition imposed by McWhorter and Sunada[37] to
solve the original problem is redundant for the case of both
COUCSI and COCSI and the solution given in Equa-
tion (11) describes the standard situation found in the labora-
tory experiments. Schmid and Geiger[23,24] and Mirzaei-Paia-
man and Masihi[25] use this finding to develop the appropri-
ate scaling equations for COUCSI cases. Mirzaei-Paiaman
et al.[38]
used the analytical solution for the COUCSI case to
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propose a mathematically based index for characterizing the
wettability of reservoir rocks. In an another modeling ap-
proach, Cai et al.[13,14] proposed analytical expressions for
characterizing a spontaneous cocurrent imbibition process of
a wetting fluid into gas-saturated porous media based on the
fractal character of the porous media. In their work, [13,14] the
mass of imbibed liquid is expressed as a function of the frac-
tal dimensions for pores and for tortuous capillaries, the min-
imum and maximum hydraulic diameters of pores, and the
ratio for minimum to maximum hydraulic diameters as well
as the porosity, fluid properties, and the fluid–solid interac-
tion.
Application of the Analytic Solution to COCSIProcess
McWhorter and Sunada[37] considered a special case of uni-
directional displacement that could be realized in laboratory
by using a semipermeable membrane that is permeable to
only the WP (i.e., R=1). However in practice if a NWP is in-itially present and is being displaced by a WP, depending on
the magnitude of the forced injection rate, three flow re-
gimes may occur in a linear flow domain.[39,40] If there is no
forced injection at the inlet boundary, there will be counter-
current flow (with respect to the NWP) at the inlet face im-
plying R
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common scaling practices should be considered. In practice,
to scale imbibition data, recovery curves are normalized by
the reference volume which can either be the ultimate recov-
ery, pore volume, or initial the NWP in place. If the recovery
in the scaling practice is normalized by the ultimate recovery,
then the appropriate dimensionless time would be in the
form of
t D ¼ 2 AF 0 Swið Þ 1 f iRð Þ
L t 1=2 ð23Þ
If normalizing recovery to pore volume V p is required, the
corresponding dimensionless time t D;V p should be defined as,
t D;V p ¼ 2 A
Lt 1=2 ð24Þ
Similarly, for the case of normalizing the recovery to the
initial NWP in place(V i), the corresponding dimensionless
time t D;V i should be defined as,
t D;V i ¼ 2 A
L 1 Swið Þ t 1=2 ð25Þ
Moreover, these scaling equations can easily be extended
to other wetting conditions by using the method proposed by
Schmid and Geiger.[24]
The above scaling equations can also be generalized by
presenting them in terms dimensionless numbers. Inserting
Equation (7) into Equation (19) yields:
Q
Q1¼ 2 A
2F 0 Swið Þ 1 f iRð ÞLuw
ð26Þ
Equation (20) gives the relationship between the Darcy ve-
locity uw and the linear velocity vw as :
uw ¼ vw
F 0 Swið Þ 1 f iRð Þ t 1=2 ð27Þ
Capillary pressure can be related to the Leverett J function
J (Sw) as:[42]
P c Swð Þ ¼ s ffiffiffi
k
r J Swð Þ ð28Þ
Inserting Equations (8), (12), and (27) into Equation (26)
and combining with Equations (4) and (28) yields:
Q
Q1¼
ffiffik
q L
s
vw mw
R Sw;BCSwi
f Swð Þ mw mnw krnwdJ
dSw
F Swð Þ f n Swð Þ dSw 2R Sw;BC
Swi
SwiSwð Þ f Swð Þ mw mnwkrnwdJ
dSw
F Swð Þ f n Swð Þ dSw
ð29Þ
Similarly, using the relationship between different volumes,
we can write:
Q
V p¼
ffiffik
q L
s
vw mw
R Swi
Sw;BC
f Swð Þ mw mnw krnwdJ
dSw
F Swð Þ f n Swð Þ d b
1 f iRð Þð30Þ
Q
V i¼
ffiffik
q L
s
vw mw
R Swi
Sw;BC
f Swð Þ mw mnw krnwdJ
dSw
F Swð Þ f n Swð Þ d b
1 Swið Þ 1 f iRð Þð31Þ
The generalized Darcy number (Da) defined as the ratio
of two characteristic lengths (of the pore and domain)[25,43]
can be written as :
Da ¼ kL2
ð32Þ
The generalized capillary number Ca representing the rela-
tive effect of viscous forces versus interfacial tension can fur-
ther be written as:
CaQ1 ¼ vw mw
s R Sw;BCSwi SwiSwð Þ f Swð Þ
mw mnw
krnwdJ
dSw
F Swð Þ
f n Swð Þ
dSwR Sw;BCSwi
f Swð Þ mw mnwkrnwdJ
dSw
F Swð Þ f n Swð Þ dSw 2 ð33Þ
CaV p ¼ vw mw
s
1 f iRð ÞR Swi
Sw;BC
f Swð Þ mw mnw krnwdJ
dSw
F Swð Þ f n Swð Þ d bð34Þ
CaV i ¼ vw mw
s
1 Swið Þ 1 f iRð ÞR Swi
Sw;BC
f Swð Þ mw mnw krnwdJ
dSw
F Swð Þ f n Swð Þ d bð35Þ
The right-hand sides in Equations (29), (30), and (31) can
be rewritten as Da1=2
Ca implying that during COCSI, the recov-
ery is controlled by the Darcy and Capillary numbers.
Validation of the New Scaling Equations
A comprehensive survey of the related literature was per-
formed and eight sets of strongly “water-wet” experimental
data from Hamon and Vidal,[6] Bourbiaux and Kalaydjian,[7]
Akin et al.,[8] and Standnes[10] were found to fulfill the re-
quirements of this study. Hamon and Vidal[6] performed four
experiments on a synthetic porous medium by using water
(WP) and a purified refined oil (NWP). In the experiment
by Bourbiaux and Kalaydjian,[7] the porous medium was
sandstone and brine and Soltrol 130 were used (WP andNWP, respectively). In the liquid–liquid and gas–liquid ex-
periments performed by Akin et al.[8] the porous medium
was diatomite with an initial zero saturation of the WP. In
both liquid–liquid and gas–liquid experiments, the WP fluid
was water and the NWP fluids were n-Decane and air, re-
spectively. In the experiment by Standnes[10] chalk was the
porous medium, and water (WP) and n-Decane (NWP) were
used with an established initial WP saturation of zero. These
experiments are summarized in Table 1.
The ability of the new equations to scale the recovery ex-
periments is compared to the scaling equations of Ma
et al.,[34]
Mason et al.,[35]
and Li.[32]
An appropriate scaling
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equation should reduce the observed scatter in the recovery
curves (Figure 1) to an acceptable limit. To use the scaling
equations of Ma et al.[34] and Mason et al.[35] only the com-
monly measured core and fluid property information are
needed, as summarized in Table 1. However, to use the scal-ing equation of Li[32] the measured endpoint relative permea-
bility and capillary pressure data at initial WP saturation are
needed. Comprehensive studies reporting all required
COCSI recovery data and capillary pressure and relative per-
meability information are very rare in the literature. In this
work we therefore assume that for “water-wet” materials,
the relative permeability and capillary pressure information
from a certain rock type is representative for a given materi-
al. The relative permeability and capillary pressure data re-
ported by Graue et al.[44] and Schembre and Kovscek[45] are
used. However to use the new scaling equations, the relative
permeability and capillary pressure data over the entire satu-ration range are needed (i.e., kr and P c vs. Sw data are
needed). Use of the aforementioned approach to run the
model of Li[32] may not provide reliable estimates for F ’(Swi)
and A. As the analytical solution presented by McWhorter
and Sunada,[37] after consideration of backflow production of
NWP, describes exactly the COCSI, numerical values of
F ’(Swi) and A can be obtained by fitting the analytical solu-
tion to experiments by using Equations (19) and (21) or
Equations (19) and (22). Based on Equation (21), a plot of Q
V p
versus 2
L t 1=2 gives a straight line with slope A that can be
computed by using a regression analysis. The numerical
value of A can also be obtained by using plot of Q
V iversus
2L 1Swið Þ t
1=2 and computing the slope of the resulting straightline [Eq. (22)]. After determination of A, the numeral value
of F ’(Swi) can be computed by plotting Q
Q1 versus 2 A 1 f iRð Þ
L t 1=2
and determining slope of the resulting straight line
[Eq. (19)]. In the slope analyses, there may be some observa-
tions deviating from linearity, particularly at early times,
which besides the experimental reading errors can be related
to heterogeneities at the pore level (i.e., pore shape and
pore-level roughness) and/or some geometrical effects that
result in the violation of the one-dimensional flow assump-
tions such that the spatial gradient of the capillary pressure
was not linear.[8,46] Buoyancy effects are assumed negligible
for the strongly “water-wet” small-size rock samples. There is
also some deviation from line-
arity at very late times when
the imbibition front reaches the
far boundary. Obviously, the
analytical solution considered
in this study is not valid any-
more when the WP front con-
tacts the far boundary. Data
points related to the nonlinear
portions should therefore all be
excluded from the regression
analyses. The procedure used to
determine the numeral values
Table 1. Summary of data for experiments performed by Hamon and Vidal[6] (HV1, HV2, HV3, and HV4), Bour-
biaux and Kalaydjian[7] (BK1), Akin et al.[8] (AK1 and AK2), and Standnes[10] (ST1).
Experiment S wi L
[cm]
k [m2] f mw[mPas]
mnw[mPas]
s [mNm1] Endpoint
k rw
End
point
k rnw
Maximum
P c [kPa]
F ’(S wi) A [m ffiffi
sp ]
HV1 0.35 9.7 1.8 10-15 0.27 1 11.5 49 0.14 0.98 480.1 4.7 9.5 10-6
HV2 0.35 19.9 2 10-15
0.28 1 11.5 49 0.12 0.97 461.2 5.2 1.7 10-5
HV3 0.36 49.8 2 10-15 0.28 1 11.5 49 0.10 0.95 458.5 5.7 2.0 10-5
HV4 0.36 84.8 1.6 10-15 0.27 1 11.5 49 0.12 0.95 512.8 5.4 1.7 10-5
BK1 0.39 29 1.37 10-13 0.23 1.2 1.5 35 0.06 0.47 12.5 4.3 2.6 10-5
AK1 0 9.5 7 10-15 0.69 1 0.0182 72 0.09 1.00 428.6 1.0 4.8 10-4
AK2 0 9.5 7 10-15 0.69 1 0.84 51.4 0.15 1.00 200.4 1.6 4.6 10-5
ST1 0 5.0 2 10-15 0.43 1 0.95 46 0.30 1.00 541.5 1.4 5.4 10-5
Figure 1. Recovery in terms of different reference volumes: a) ultimate recov-
ery, b) pore volume, and c) initial NWP in place versus time for the experi-
ments collected from the literature.
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of A and F ’(Swi) is shown in Figure 2 and the summarized re-
sults are given in Table 1.
As the scaling equations of Ma et al.[34] and Mason et al.[35]
are based on the application of general principles of dimen-
sional analysis by Rapoport,[31] rather than by a mathematical
treatment, establishing an exact relationship between the ver-tical and horizontal axes in scaling plots is not possible.[25]
Thus, in scaling plots, the vertical axis may be recovery nor-
malized by any of the reference volumes. Figure 3 and
Figure 4 show, respectively, the performance of the scaling
equations of Ma et al.[34] and Mason et al.[35] after normaliz-
ing the recovery data by the ultimate recovery, pore volume,
or initial oil/gas in place, which is plotted against these scal-
ing equations. These figures show that these equations
cannot scale recovery curves as there is significant scatter in
all plots. There may be several reasons for the poor scaling
performance of the scaling equations of Ma et al.[34] and
Mason et al..[35]
These were derived based on the application
of the general principles of di-
mensional analysis by Rapo-
port.[31] This approach causes
the effects of some of the fac-
tors influencing the process be
systematically neglected. How-
ever the main reason may be
that these scaling equations
have not been derived in a con-
sistent way according to
common scaling practices.[25]
The ability of the scaling
equation from Li[32] to plot the
recovery data normalized by
different reference volumes is
shown in Figure 5 versus this
scaling equation. The scaling
result is not satisfactory and
there still exists a nontrivial
scatter in scaling plots. Oneshould note that this scaling
equation has been specifically
proposed for COCSI. There
may be several reasons for such
poor quality of the scaling rela-
tion. This scaling equation has
been derived based upon a re-
stricting approximate solution
to the main governing equation.
Furthermore the development
of this scaling equation is not
consistent with common scalingpractices as highlighted by Mir-
zaei-Paiaman and Masihi.[25] In
Figure 5, the experiments re-
ported by Hamon and Vidal[6]
are not shown because the scal-
ing equation of Li[32] predicts
negative values if the viscosity
of the NWP (11.5 mPas, in these experiments) becomes large
compared to viscosity of WP (1 mPas, in these experiments).
The ability of the new scaling equations to scale the ex-
periments is shown in Figure 6, in which the recovery data
normalized by different reference volumes are plotted
against the appropriate scaling equations. Depending on thetype of normalization on the y-axis, the corresponding scaling
equation should be put on the x-axis. In each plot the curve
given by the analytical solution is also included. The ability
of the new equations to scale the experiments is much better
than the existing scaling equations. However, for some ex-
periments, some scatter around the analytical solution curve
is noticeable which may be due to the quality of the reported
experimental data. To be able to compare among the scaling
equations (Figures 1, 3, 4, 5, 6, and 7) we use the same
8 cycles on their horizontal axes.
Figure 2. The procedure used to compute the numeral values of A and F 0ðSwiÞ from the experimental data: a) plotof
Q
V pversus 2L t
1=2 gives a straight line with slope A that can be computed by using a regression analysis, b) the nu-
meral value of F 0ðSwiÞ can be computed by plotting QQ1 versus 2 A 1 f i Rð Þ
L t 1=2 and determining slope of the resulting
straight line.
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Derivation of a Simple Scaling Equation for “Water-Wet” Systems
As the presented scaling equations incorporate the capillary
pressure and relative permeability parameters, they are ap-
plicable to systems with a wide range of wetting conditions.
However, from a practical perspective, this information is
generally not easily known, so there is still need for some
utility in simpler scaling groups. By simple scaling groups we
mean scaling equations that do not incorporate capillary
pressure and relative permeability information and are
mostly applicable to the systems with the same wettability
conditions (e.g., all “water-wet”). These equations usually
contain only simple-to-measure rock and fluid parameters
such as porosity, absolute permeability, interfacial tension,
geometrical dimensions, and the wetting and nonwetting
phase viscosities. With the exception of scaling purposes,
there exists significant interest in using simple equations,
mainly in comparative studies, for the objective of studying
the effect of aging time on the wettability alteration,[47,48] the
effect of water adsorption and resulting microfractures in or-
ganic shale rocks,[49,50] or the spontaneous imbibition charac-
teristics of different porous media.[8,12]
Figure 3. Recovery in terms of different reference volumes: a) ultimate recov-
ery, b) pore volume, and c) initial NWP in place versus the scaling equation
of Ma et al.[34]
Figure 4. Recovery in terms of different reference volumes: a) ultimate recov-
ery, b) pore volume, and c) initial NWP in place versus the scaling equation
of Mason et al.[35]
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Inserting Equations (3), (4), (8), (12), and (28) into the di-
mensionless time equations brings them into the new formsas:
t D ¼ t D;V p ¼ t D;V i ¼ G
ffiffiffiffiffiffiffiffiffiffiffiffi2s
ffiffik
q mnwL
2
v uut t 1=2
ð36Þ
in which G, a group of variables, differs (Table 2).
Our objective is to extract a simple scaling equation from
Equation (36). We note that G is a function of wettability, in-
itial wetting phase saturation, fluid viscosity, and pore struc-
ture. Therefore the exclusion of G from the scaling equation
[Eq. (36)] makes this equation ¼ ffiffiffiffiffiffiffiffiffiffi2s ffiffik
p mnwL2
r t
1=2 ! independent
of wettability and thus applicable to “water-wet” systems.
However, some parameters in the G variable are dependent
on fluid viscosity, initial wetting phase saturation, and pore
structure, and their overall effect should be considered, even
after the exclusion of G. In the case of the fluid viscosity de-
pendence, this can be performed by replacing mnw in the sim-
plified equation by an appropriate argument, called the vis-
cosity group. To do this, several forms of the viscosity group
were used and the scaling performance of the resulting sim-
plified scaling equations was checked. The best scaling group
was mnw+ ffiffiffiffiffiffiffiffiffiffiffiffi mw mnwp
. Therefore the resulting simplified scaling
equation can be written as:
Figure 5. Recovery in terms of different reference volumes: a) ultimate recov-
ery, b) pore volume, and c) initial NWP in place versus the scaling equation
of Li.[32]Figure 6. Recovery in terms of different reference volumes: a) ultimate recov-
ery, b) pore volume, and c) initial NWP in place versus the corresponding
new scaling equations.
Energy Technol. 2014, 2, 166– 175 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim 173
Recovery in Fractured Petroleum Reservoirs
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8/18/2019 Mirzaei-Paiaman and Masihi 2014
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t D;simplified ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2s ffiffik
q
mnw þ ffiffiffiffiffiffiffiffiffiffiffiffi mw mnwp L2v uut t 1=2 ð37ÞWe should note that because this simplified scaling equa-
tion contains the ffiffi
k
q term, the dependence on the pore struc-
ture may be assumed to be relaxed. However, the initial wet-
ting phase saturation dependence cannot be removed. There-
fore the simplified scaling equation presented above may not
present very good scaling performance for systems with dif-
ferent initial saturations of the wetting phase. As this scaling
equation is a simplified form of the general scaling equations,
the vertical axis in scaling plots can be recovery normalized
by any reference volume. The ability of this equation to scaledata is shown in Figure 7, which shows an improvement as
compared to Figure 3, Figure 4, and Figure 5. Moreover,
Figure 7 (in comparison to Figure 6) shows acceptable accu-
racy for the new simplified equation as compared to the new
general scaling equations.
Conclusions
The following conclusions can be drawn from this work:
* As the analytical solution to unidirectional displacement
given by McWhorter and Sunada[37] applies to COCSI,[36]
suitable scaling equations for one-dimensional COCSI canbe found by using this solution.
* Backflow production of the NWP at the inlet open boun-
dary is inherent to COCSI and its contribution to the pro-
cess should be taken into account when using the analyti-
cal solution given by McWhorter and Sunada[37] and when
presenting new scaling equations.
* The strategy to account for the contribution of the back-
flow production by assuming R¼6 1 avoids the occurrenceof the possible instabilities in computing the integrals in
the McWhorter and Sunada[37] solution.
* The new scaling equations presented in this study are uni-
versal, incorporating all factors influencing the process, as
the exact analytical solution to the problem without any
assumption are used.
* Consistency between the development of the new scaling
equations and common practices should be considered to
obtain reliable results.
* The new scaling equations can be rewritten in terms of
two physically meaningful dimensionless numbers, Da1/2/
Ca (Da : Darcy number, Ca : capillary number).
* The ability of the new equations to scale the experiments
was found to be much better than the existing scaling
equations.
Figure 7. Recovery in terms of different reference volumes: a) ultimate recov-
ery, b) pore volume, and c) initial NWP in place versus the new simplified
scaling equation.
Table 2. Expressions for G for different dimensionless time equations.
Dimensionless time Corresponding expression for G
tD
G ¼ R Sw;BC
Swi
f Swð Þkrnw dJ dSwF Swð Þ f n Swð Þ dSw
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiR Sw;BCSwi
Swi Swð Þ f Swð Þkrnw dJ dSwF Swð Þ f n Swð Þ dSw
q t D;V p
G ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiR Sw;BC
Swi
Swi Swð Þ f Swð Þkrnw dJ dSwF Swð Þ f n Swð Þ dSw1 f i Rð Þ2
s
t D;V i
G ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiR Sw;BC
Swi
Swi Swð Þ f Swð Þkrnw dJ dSwF Swð Þ f n Swð Þ dSw
1Swið Þ2 1 f i Rð Þ2
s
174 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim Energy Technol. 2014, 2, 166–175
A. Mirzaei-Paiaman and M. Masihi
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8/18/2019 Mirzaei-Paiaman and Masihi 2014
10/10
* For the literature scaling equations derived based on the
application of the general principles of dimensional analy-
sis, establishing an exact relationship between the vertical
and horizontal axes in scaling plots is not possible. Thus,
in scaling plots the vertical axis may be recovery normal-
ized by any of the reference volumes.
* The scaling equations developed for COUCSI fail to scale
up COCSI data.
* The former scaling equations proposed for COCSI, de-
rived based on assumption of piston-like displacement,
yield poor scaling results.
* The general scaling equations presented in this study can
be used to present a simple scaling equation for “water-
wet” systems. The ability of the new simplified scaling
equation to scale “water-wet” data was found to be better
than other existing scaling equations and also acceptable
in comparison to new general scaling equations.
Acknowledgements
The authors thank Sharif University of Technology and the
Research and Technology Departments of NIOC and NISOC
for permission to publish this paper.
Keywords: fluid dynamics · fractured reservoirs · petroleum ·
scaling equations · wetting phase
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Received: October 25, 2013
Revised: December 5, 2013
Published online on February 10, 2014
Energy Technol. 2014, 2, 166– 175 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim 175
Recovery in Fractured Petroleum Reservoirs
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