Mirzaei-Paiaman and Masihi 2014

8/18/2019 Mirzaei-Paiaman and Masihi 2014

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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

Energy Technol. 2014, 2, 166– 175 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim 167

Recovery in Fractured Petroleum Reservoirs


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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


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


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


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


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.


A. Mirzaei-Paiaman and M. Masihi


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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]


ery, b) pore volume, and c) initial NWP in place versus the scaling equation

of Ma et al.[34]



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:



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.




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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.


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




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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