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Three-Phase Hydraulic Conductances inAngular Capillaries

Ahmed Al-Futaisi, SPE, Sultan Qaboos U., and Tad W. Patzek, SPE, U.C. Berkeley/Lawrence BerkeleyNatl. Laboratory

SummaryIn this paper, we extend to three fluid phases a prior finite-elementstudy of hydraulic conductance of two-phase creeping flow inangular capillaries. Previously, we obtained analytic expressionsfor the hydraulic conductance of water in corner filaments. Here,we present the results of a large numerical study with a high-resolution finite-element method that solves the three-phase creep-ing flow approximation of the Navier-Stokes equation. Using theprojection-pursuit regression approach, we provide simple analyticexpressions for the hydraulic conductance of an intermediate layerof oil sandwiched between water in the corners of the capillary andgas in the center. Our correlations are derived for the oil layersbounded by the concave or convex interfaces that are rigid or allowperfect slip. Therefore, our correlations are applicable to drainage,spontaneous imbibition, and forced imbibition with maximum fea-sible hysteresis of each contact angle, oil/water (O/W), and gas/oil(G/O). These correlations should be useful in pore-network calcu-lations of three-phase relative permeabilities of spreading oils.Finally, we compare our results with the existing correlations byZhou et al.18 and Hui and Blunt,19 who assumed thin-film flowwith an effective film thickness proportional to the ratio of theflow area to the length of the no-flow boundary. On average, ourcorrelations are two to four times closer to the numerical resultsthan the corresponding correlations by Zhou et al. and Hui andBlunt.

IntroductionBecause direct measurement of flow of three immiscible fluids isvery difficult, the pore-scale models of three-fluid systems1–5 haveblossomed. One of the more important advancements in such mod-els was the approximation of single-pore geometries as angularcapillaries with square or arbitrary triangular cross sections. Al-though real pores are not exactly square or triangular, this approxi-mation allows one to capture the flow of water in the pore cornersand the flow of oil and gas in the pore center. As illustrated in Fig.1a, when three fluids are moving in a single angular capillary, themost wetting fluid (water or Fluid 1) resides in the corner and themost nonwetting fluid (gas or Fluid 3) fills the center. The thirdfluid (oil or Fluid 2) forms an intermediate layer sandwiched be-tween the other two fluids. In some cases of large contact anglesand positive spreading coefficients, we may find more than onesandwiched layer (Fig. 1b). These intermediate layers are a fewmicrometers thick and have been observed in micromodel experi-ments.6–9 It is drainage through these oil layers that is responsiblefor the high oil recoveries seen experimentally.10–12 Although itwas initially thought that only spreading oils could form suchlayers in angular pores, it has been theoretically predicted andexperimentally verified that nonspreading oil can also form inter-mediate layers in the crevices of the pore space.6,9,13 Therefore,the formation of sandwiched layers is not only related to the posi-tive spreading coefficient, but also depends on the curvaturesof the O/W and G/O interfaces, the corner geometry, and thecontact angles.3

Creeping flow of oil in these intermediate layers is the subjectof this paper. In particular, we study the hydraulic conductances ofoil flow in stable fluid layers of different sizes and geometries. Weprovide simple and accurate correlations for these conductances byrelating them to the interface geometry, fluid contact angles, andpore geometry. The proposed correlations should be useful in porenetwork calculations of three-phase relative permeabilities. Asimilar approach has been successful in single-phase14 and two-phase15 flow in angular capillaries. Assuming that equilibriumO/W and G/O interfaces are stable in creeping flow,16,17 one cansolve the Navier-Stokes equation in the intermediate oil layer,given its fixed boundaries. From the solution of the Navier-Stokesequation, the average flow velocity of fluid i, ⟨vi⟩, is calculated,and its hydraulic conductance, gi, is estimated from a linear rela-tion between the volumetric flow rate in the layer, Qi, and thegradient of the total driving force per unit area, �i:

Qi = �vi�Ai = gi�i , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (1)

where Ai�the layer cross-sectional area.Zhou et al.18 presented an approximate analytical solution for

oil flow along a layer sandwiched between water and air in angularcapillaries, and derived expressions for flow resistance and, there-fore, hydraulic conductance with no-slip and perfect-slip condi-tions at the interfaces. Zhou et al. derived their expressions byassuming thin-film flow with an effective film thickness propor-tional to the ratio of the flow area and the length of the no-flowboundary. Their expressions were derived for the zero O/W con-tact angle, and are limited to the oil flow in the layers bounded byconcave menisci, as in drainage, but not in forced imbibition. Fornonzero O/W contact angles and convex interfaces, Hui andBlunt19,20 proposed a modified version of the Zhou et al. expres-sions. Later in this paper, we will discuss in some detail relativeaccuracy of all these expressions.

Our objectives are twofold: First, to develop a numerical algo-rithm that solves the velocity distribution and, therefore, hydraulicconductance in three-phase flow with various geometries and in-terface boundary conditions; and, second, to provide simple andaccurate correlations for the hydraulic conductances of the inter-mediate layers of oil sandwiched between water and gas. Thepaper is organized as follows. First, we calculate the various geo-metrical descriptors of the oil layer, such as its area, perimeter, andshape factor. Second, we present the mathematical formulation ofthe boundary value problems in creeping flow. Third, we describethe finite-element approximations of these problems, and discussthe numerical results. Finally, we correlate the layer hydraulicconductance with the layer geometry, the pore corner geometry,and the O/W and G/O contact angles.

Layer Geometry and StabilityBefore simulating fluid flow in a sandwiched layer, we study itsgeometry and stability. As shown in Fig. 2, the layer formation ofdepends on five parameters: corner half-angle ß, O/W contactangle �21, G/O contact angle �32, O/W meniscus-apex distance b1,and G/O meniscus-apex distance b2. Zhou et al.18 and Hui andBlunt19 define their hydraulic conductance expressions using theradius of meniscus curvature instead of the meniscus-apex dis-tance. The radius of curvature and the meniscus-apex distance arerelated by Eq. 2, where r1 and r2 are the radii of curvature of theO/W and G/O interfaces, respectively. We select the meniscus-

Copyright © 2003 Society of Petroleum Engineers

This paper (SPE 86889) was revised for publication from paper SPE 75193, prepared forpresentation at the 2002 SPE/DOE Improved Oil Recovery Symposium, Tulsa, 13–17 April.Original manuscript received for review 30 April 2002. Revised manuscript received 30 May2003. Manuscript peer approved 18 June 2003.

252 September 2003 SPE Journal

apex distances in order to avoid the infinite meniscus radius as theinterface becomes flat:

b1 = r1

cos��21 + ��

sin��, b2 = r2

cos��32 + ��

sin��. . . . . . . . . . . . . . . . . . (2)

The G/O interface is always concave (i.e., �32 + ß < �/2). For �32

+ ß � �/2, the layer becomes unstable and cannot form. On theother hand, the O/W interface can be concave (�21 + ß < �/2), flat(�21 + ß=�/2), or convex (�21 + ß > �/2) (see Fig. 2). We shallnow define the dimensionless meniscus-apex distances b1 and b2

by scaling the spatial coordinates with the G/O meniscus-apexdistance, b2. As a result, the dimensionless distance b2 is 1 and thedimensionless distance b1 is b1/b2. The three relevant descriptorsof the sandwiched layer geometry are its dimensionless area, cir-cumference, and shape factor. First, however, we define the threeconstants that will be used repeatedly in the calculations.

E0ij =

�

2− �ij − �

E1ij =

cos��ij + ��

sin�

E2ij =

cos��ij + ��

sin�cos�ij

i = 1, 2, 3, j � i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3)

The layer dimensionless cross-sectional area, AL, is definedas follows:

AL = ��E2

32 − E032�

�E132�2

− �b1

b2�2

sin � cos � if �21 + � = � � 2

�E232 − E0

32�

�E132�2

− �b1

b2�2 �E2

21 − E021�

�E121�2

otherwise.

. . . . (4)

The layer dimensionless circumference, PL, is defined as

PL = �2�1 −b1

b2� + L21 + L32�, . . . . . . . . . . . . . . . . . . . . . . . . . (5)

where L21�the dimensionless length of the O/W interface calcu-lated with

L21 = � 2b1

b2sin � if �21 + � = � � 2

2b1

b2

E021

E121

otherwise,

. . . . . . . . . . . . . . . . . . . (6)

and, similarly, the dimensionless length of the G/O interface, L32,is calculated by

L32 = 2E0

32

E132

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (7)

The layer normalized shape factor, GL, is defined as the ratio of thelayer dimensionless area and square of the layer dimensionlesscircumference as follows:

GL =AL

�PL�2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (8)

The layer actual area is calculated as AL =(b2)2 AL, and the actualperimeter is given by PL=b2 PL. The actual and normalized shapefactors are equal. Finally, the layer cannot exist, because the twomenisci touch, if

�b1

b2�

1

cos2��1 −

cos�

E132

+cos� sin�32

E132 � if �21 + � = � � 2

b1

b2� min�1,�E1

21

E132

cos�32 − sin�

cos�21 − sin�� otherwise

. . . . (9)

Mathematical FormulationAssuming steady-state, creeping isothermal flow of Newtonian,incompressible, and constant viscosity fluids, the combined con-tinuity and Navier-Stokes equations that describe the flow reduceto an elliptic Poisson equation are as follows:

�2v1 = −≡i

�i�x1, x2� ∈ i, . . . . . . . . . . . . . . . . . . . . . . . . . . . . (10)

where i�1,2,3 denotes water, oil, and gas, respectively; vi � theith fluid velocity; �i � the ith fluid viscosity, x1 and x2 � thespatial coordinates across the capillary (Fig. 3), and ��the gra-dient of the total driving force per unit area defined as

�i = −�pi + �i f, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (11)

where�pi�the pressure gradient in fluid i, �i�the ith fluid den-sity, and f�the body force per unit mass. We also assume that theinterfacial tensions are constant, and the buoyancy forces are neg-ligible in the capillary (i.e., both Bond numbers are much lessthan one).

Using a scaling scheme similar to that implemented in Ref. 15,we scale the spatial coordinates with the G/O meniscus-apex dis-tance, b2, and the fluid velocities with the oil viscosity, �2, and therespective gradient of the force, �2, driving oil flow:

xj =xj

b2j = 1, 2, 3

vi =vi �2

b22 �2

j = 1, 2, 3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (12)

After scaling, the dimensionless form of Eq. 10 is

�i�2vi = −1

�1 =�1

�2, �2 = 1, �3 =

�3

�2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (13)

Although Eqs. 10 and 11 are applicable to any of the three fluids,we focus here on the oil in the sandwiched layer. Our formulationis incomplete without specifying the boundary conditions for thislayer. As shown in Fig. 3, we need to impose boundary conditionsalong the duct walls, �s, the O/W interface, �21, and the G/Ointerface, �32.

Fig. 1—Different configurations of (1) water, (2) oil, and (3) gas in a single triangular pore.

253September 2003 SPE Journal

We impose a no-slip boundary condition along the duct wallsas follows:

vi = 0 on �s. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (14)

We consider two different boundary conditions along the O/W andG/O interfaces. The first boundary condition requires infinite sur-face shear viscosity of the interface, which becomes a surfactant-laden rigid wall with a no-slip condition,

vi = 0 on �21 or �32. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (15)

The second boundary condition describes a perfect-slip conditionwith zero surface shear viscosity of the interface. This conditiontranslates mathematically to

�vi ni = 0 on �21 or �32, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (16)

where ni�the unit outward normal vector along the interface.These two interface boundary conditions result in four possibleconfigurations of boundary conditions:

1. BC-1: No-slip O/W interface and no-slip G/O interface.2. BC-2: No-slip O/W interface and perfect-slip G/O interface.3. BC-3: Perfect-slip O/W interface and no-slip G/O interface.4. BC-4: Perfect-slip O/W interface and perfect-slip G/O interface.

Another possible interface boundary condition assumes continuityof velocity and shear stress along the interface:

�v2 n2 = ��v1 n1 on �21

�v2 n2 = �3�v3 n3 on �32. . . . . . . . . . . . . . . . . . . . . . . . . . . (17)

This last boundary condition is not considered here. Finally, thereis a no-flow boundary condition along the lines of symmetry (seethe following discussion).

Numerical ApproximationWe solve the boundary value problem (Eqs. 13 through 16) nu-merically using the finite-element method (FEM). The FEM solu-tion was obtained using the MATLAB Partial Differential Equa-tion (PDE) Toolbox (MathWorks, Natick, Massachusetts).21 TheToolbox provides a powerful and flexible environment for thestudy and solution of partial differential equations in two spacedimensions and time. We now present examples of mesh genera-tion, solution visualization, and convergence.

Mesh Generation. Accuracy of the FEM solution depends on howwell the computational domain is discretized. A good numericalsolution requires a mesh that follows the shape of the duct wallsand the curvatures of the interfaces. The PDE toolbox has manypowerful built-in functions that enable the user to generate thefinite-element meshes with the required properties. In our simula-tions, we have used the built-in MATLAB function initmesh toinitiate the first discretization of the domain. Then, the initial meshis refined five more times using the function refinemesh. In eachrefinement, each triangle in the mesh is divided into four newtriangles. Because of the layer symmetry with respect to the cor-ner-angle bisector, the FEM analysis is performed on 1⁄2 of thedomain. Depending on the layer geometry, the maximum numberof elements generated for a layer (half-domain) in our simulationswas 109,824, whereas the minimum number of elements was5,376. The average number of elements of all the cases consideredin this analysis was 20,446. In Fig. 4, we present examples of layermeshes for two geometrical configurations. The number of ele-ments shown in the plot is just for illustration; the meshes used inthe numerical analysis were much finer.

Dimensionless Velocity Profile. Once we have generated a meshand specified appropriate boundary conditions, we solve Eq. 13 inorder to calculate the profile of the dimensionless velocity in thelayer. In Fig. 5, we show the dimensionless velocity profiles in aspecific layer for the four boundary conditions: BC-1, BC-2, BC-3,and BC-4. The layer dimensionless area, AL, the circumference,PL, and the shape factor, GL, are 0.1901, 3.7913, and 0.0132,respectively. Benefitting from the symmetry of the layer with re-spect to the corner-angle bisector, the FEM analysis is performedusing 16,896 elements on 1⁄2 of the layer; the second half mirrorsthe first one. From Fig. 5, one can see that the interface boundaryconditions influence significantly the velocity profile and, there-fore, the average velocity and hydraulic conductance of the layer.The average dimensionless velocity in the layer, <vL>, is calcu-lated as follows:

�vL� =�

N

k = 1vkAk

�N

k = 1Ak

, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (18)

where N�the total number of elements in the layer, Ak�the ele-ment k dimensionless area, and vk�the element k dimensionless

Fig. 3—Spatial coordinates of the sandwiched layer: The flowis along the duct in the x3-direction and the velocity distributionis calculated in the x1 and x2 directions. �s denotes the ductwall, whereas �21 and �32 denote the O/W and G/O interfaces,respectively.

Fig. 2—Sandwiched layers with concave, flat, or convex O/W interface.


velocity calculated at the center of the element. The average di-mensionless velocities in the layers shown in Fig. 5 are 0.0016,0.0053, 0.0027, and 0.0163, respectively, with the boundary con-ditions BC-1, BC-2, BC-3, and BC-4.

The dimensionless hydraulic conductance, gL, of the layer is

gL = 2�k=1

N

vkAk. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (19)

The factor of 2 compensates for the calculations on 1⁄2 of the layer.For the layers in Fig. 5, the dimensionless hydraulic conductancesare 0.000302, 0.0010, 0.000513, and 0.0031, respectively, for BC-1, BC-2, BC-3, and BC-4.

Noticing that AL�(b2)2 AL, and using Eqs. 1 and 12, we canrelate the layer actual hydraulic conductance, gL, to the dimen-sionless conductance, gL, as follows:

gL =�b2�

4

�2gL. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (20)

Convergence. To test the convergence of our FEM solution, wecalculate the ratio of the hydraulic conductances, gL

N1 and gLN2,

calculated with the number of finite elements N1 and N2�4N1.This ratio approaches 1 when the solution is converging. In Fig. 6,we show an example of this calculation for BC-2: no-slip O/Winterface and perfect-slip G/O interface. Similar results were ob-tained for the other three boundary conditions. From Fig. 5, onecan see that with a few thousands of elements, about 6,000, thenumerical solution converges. Similar observation holds for simu-lations with the other boundary conditions.

ResultsTo derive a universal curve, which approximates the hydraulicconductance of a sandwiched layer, intensive simulations of dif-ferent layer geometries are required, set by the different combina-tions of ß, �21, �32, and b1/b2. In this analysis we have generated17,167 stable layer geometries with arbitrary values of ��5 to85°, �21�0 to 170°, �32�0 to 80°, and b1/b2�0.1 to 0.9. For eachgenerated layer, the dimensionless hydraulic conductance, gL, iscalculated for the four boundary conditions shown in Fig. 5.

In this section, we investigate the e.ects of the layer dimen-sionless parameters ß, �21, �32, and b1/b2 on its dimensionlesshydraulic conductance, gL. All four boundary conditions presented

in Fig. 5 exhibit similar behavior; therefore, we restrict our de-scription to a single boundary configuration, BC-2: The no-slipO/W interface and the perfect slip G/O interface. The same con-clusions can be applied to the other three boundary conditions.

Fig. 7 relates the logarithm of the dimensionless hydraulicconductance to the corner half-angle �, the O/W contact angle �21,and the G/O contact angle �32 for a fixed ratio b1/b2. For smallcorner half-angles, ß < 10°, the variation of the hydraulic conduc-tance with �21 and �32 is relatively minor. However, as the cornerhalf angle increases, � < 10°, the hydraulic conductance variesconsiderably with �21 and �32. For a fixed ß, increasing �21 de-creases the layer hydraulic conductance, whereas increasing �32

increases it. Also, both contact angles, �21 and �32, affect thehydraulic conductance considerably. The effect of the ratio b1/b2

on the layer hydraulic conductance is shown in Fig. 8. One maynotice that increasing b1/b2 results in a large variation of the hy-draulic conductance. A smaller variation is observed when b1/b2 isless than 0.4.

As mentioned in the mathematical formulation of the problem,we are not considering here the momentum transfer boundary con-dition that assumes continuity of velocity and shear stress alongthe interfaces (Eq. 17). However, because this boundary conditionis somewhere between the no-slip and perfect-slip ones, it isworthwhile to compare the hydraulic conductances obtained withthe latter two boundary conditions by fixing some parameters: Weset the boundary condition and the contact angle at the G/O inter-face to no-slip and 0°, respectively. With the same layer geometry,the problem is solved twice, first with the no-slip O/W interface(BC-1), and second with the perfect-slip O/W interface (BC-3).The dimensionless hydraulic conductances for the two cases areplotted in Fig. 9. The type of BC has a strong influence on thehydraulic conductance. For small values of b1/b2 (i.e., for the twointerfaces of the sandwiched layer far from each other), the valuesof the two conductances are similar, and their ratio is close to 1.However, as the corner half-angle � increases, this ratio becomessmaller, reflecting the clear difference between the two BCs. Forthe extreme cases of large b1/b2 (the interfaces very close to eachother) and large �, the conductance ratio can be less than 0.3. Thissimple comparison illustrates the importance of considering thethird boundary condition (Eq. 17), because the real flow is perhapsneither perfect-slip nor no-slip.

Figs. 7 through 9 demonstrate that the different combinations ofß, �21, �32, and b1/b2, and the boundary conditions may result in a

Fig. 5—Velocity distribution in a sandwiched layer (ß=60°, �21=90°, �32=10°, and b1/b2=0.25) with different interface boundaryconditions: BC-1, BC-2, BC-3, and BC-4, respectively.

Fig. 4—Layer mesh generation for two geometrical configurations: (a) ß=24°, �21=0°, �32=0°, and b1/b2=0.25; and (b) ß=60°, �21=80°,�32=10°, and b1/b2=0.3.


huge variation of the layer hydraulic conductance. They also fore-shadow the diffculty of creating an easy-to-use relationship amongthese parameters and the layer hydraulic conductance. As ouranalysis shows, there is no straightforward relation between thelayer conductance and its geometrical features, such as area, pe-rimeter, or shape factor. Therefore, a rigorous statistical procedureis required to derive “best” correlations for the layer hydraulicconductance.

Universal Curves for the LayerHydraulic ConductanceHere we attempt to derive the universal curves that approximatethe hydraulic conductance of the sandwiched layer. As shownabove, regular statistical procedures may not describe adequatelythe nonlinear interactions among the dimensionless dependent

variable Y�(gL) and any of the layer dimensionless predic-tor variables:

x = ��,�21,�32,b1�b2,PL,AL,L21,L32,GL�.

Therefore, we choose projection-pursuit regression to obtain theuniversal curves for the layer hydraulic conductances. Projection-pursuit regression22,23 is a computer-intensive procedure that ap-plies an additive model to the projected variables:

��Y � = �0 + �j=1

M

�j fj ��k X� + �, . . . . . . . . . . . . . . . . . . . . . . . . . (21)

where �(Y)�any transformation of the dependent variable Y,�0�the mean of �(Y), �j and �k are the constants calculated by the

Fig. 7—The logarithm of the dimensionless hydraulic conduc-tance vs. corner half-angles � and contact angles �21 and �32 forb1/b2=0.8, and with BC-2.

Fig. 8—The logarithm of the dimensionless hydraulic conduc-tance vs. the b1/b2 and contact angles �21 and �32 for �=15°, andwith BC-2.

Fig. 6—Convergence of the FEM solution for eight different layers with BC-2: No-slip O/W interface and perfect-slip G/O interface.


regression model, fj are the functions fitted by the model, M�adimension chosen by the user, and ��the deviation of the fittedvalues from the corresponding true ones. Thus projection-pursuitregression uses an additive model on the predictor variables, whichare formed by “projecting” matrix X in M carefully chosen direc-tions. The “pursuit” part indicates that an optimization technique isused to find “good” direction vectors, �k. The statistical languageS-PLUS24,25 is used to implement the projection-pursuit regres-sion. Using projection-pursuit regression, the dimensionless hy-draulic conductance in the layer is given by

ln�gL� = �0 + �1f1�z�, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (22)

where �0, �1, z, and f1(z) depend on the interfaces boundary con-ditions as follows.

BC-1: No-Slip O/W Interface and No-Slip G/OInterface.

�0 = −7.9998

�1 = 1.7474

z = 0.0797ln�� + 0.5540ln�GL�

+ 0.7698ln�AL� − 0.1494�32 − 0.2679b1

b2

f1�z� = 1.3062z + 4.9465. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (23)

BC-2: No-Slip O/W Interface and Perfect-Slip G/OInterface.

�0 = −7.2153

�1 = 1.8026

z = 0.3227ln�� + 0.5948ln�GL�

+ 0.7141ln�AL� − 0.1516�32 − 0.0959b1

b2

f1�z� = 0.0008541z2 + 4.9495. . . . . . . . . . . . . . . . . . . . . . . . . . . (24)

BC-3: Perfect-Slip O/W Interface and No-Slip G/OInterface.

�0 = −7.5325

�1 = 1.6435

z = 0.1367ln�� + 0.4339ln�GL�

+ 0.7444ln�AL� − 0.1740�32 − 0.4568b1

b2

f1�z� = 1.5380z + 4.7172. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (25)

BC-4: Perfect-Slip O/W Interface and Perfect-SlipG/O Interface.

�0 = −6.4543

�1 = 1.7558

z = 0.4405ln�� + 0.14981ln�GL�

+ 0.3587ln�AL� − 0.7446L21 − 0.3169L32

f1�z� = 0.0409z5 + 0.4377z4 + 1.7096z3.

+ 3.0028z2 + 4.1682z + 3.8056. . . . . . . . . . . . . . . . . . . . . . . . (26)

In Figs. 10, 12, 14, and 16, we plot the deviations of the hydraulicconductances calculated with the projection-pursuit regressionfrom the corresponding FEM solutions, for the four boundary con-ditions, respectively. The relative error is calculated as follows:

Relative Error =gL − gL

FE

gLFE

� 100. . . . . . . . . . . . . . . . . . . . . . . (27)

where gLFE�the hydraulic conductance calculated by the finite-

element model. The mean absolute relative errors of these ap-proximations are 9.5, 9.8, 13.9, and 18.0% for BC-1, BC-2, BC-3,and BC-4, respectively. The actual conductance is calculated withEq. 20.

The most widely used expressions for the hydraulic conduc-tance of the sandwiched layers were derived by Zhou et al.18

Fig. 9—The ratio of the dimensionless hydraulic conductance of BC-1 to BC-3 for different layer geometries.


Zhou’s layer dimensionless hydraulic conductance for �32 + ß <�/2 and �21�0° is given by

gL =�

�

� = Ac2�3

2�1 − sin��2

� ��2cos�32 − �1 − cot��1 − �3�Ro2�3

� = 12sin2��1 − �3�2��2cos�32 − �1�

2

� ��2 + f1�1 − cot��1 − f2�3�Ro�2, . . . . . . . . . . . . . . . . . . . (28)

where the constants �1, �2, �3, Ac, Ro are as follows:

�1 =�

2− � − �32

�2 = cot�cos�32 − sin�32 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (29)

�3 = ��

2− ��tan�,

Ac = � sin�

cos��32 + ��2cos�32 � �cot�cos�32 − sin�32�

+ �32 + � −�

2, . . . . (30)

Ro =b1

b2

cos��32 + ��

cos��21 + ��. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (31)

The quantity f is used to indicate the boundary condition at theFluid/Fluid interface. A value of f�1 represents a no-slip bound-ary, while a value of f�0 is a perfect-slip boundary. f1 is theboundary condition at G/O interface and f2 is the boundary con-dition at the O/W interface. For nonzero O/W contact angles, or forconvex interfaces, Hui and Blunt19,20 proposed the followingmodification to Eq. 28:

gL =Ao

3�32�1 − sin��2tan�

12Ac sin2��1 − �3��1 + f1�3 − �1 − f2�3��Aw

Ac

�2. . . . . (32)

where Aw and Ao are calculated by

Aw = �b1

b2

sin�

cos��21 + ��2

cos�21 � �cot�cos�21 − sin�21�

+ �21 + � −�

2

Ao = Ac − Aw. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (33)

Deviations of the hydraulic conductances calculated with Eqs. 28and 32 from the corresponding FEM solutions with the fourboundary conditions in this analysis are shown in Figs. 11, 13, 15,and 17. The points with light colors were generated with the Zhouet al. approximation (Eq. 28) for the zero O/W contact angle,

Fig. 11—Relative deviation of Zhou et al.18 and Hui and Blunt19

flow conductance from the corresponding FEM solution for BC-1 (the mean absolute error is 22.2%).

Fig. 12—Relative deviation of projection-pursuit regressionflow conductance from the corresponding FEM solution forBC-2 (the mean absolute error is 9.8%).


flow conductance from the corresponding FEM solution forBC-2 (the mean absolute error is 24.1%).



whereas the dark-color points belong to the Hui and Blunt approxi-mation (Eq. 32) for nonzero O/W contact angles. The mean abso-lute relative errors of these approximations are 22.2, 24.1, 28.0,and 64.9% for BC-1, BC-2, BC-3, and BC-4, respectively. Thus,the projection-pursuit regression method gives results better thanthe expressions proposed by Zhou et al. and Hui and Blunt.

DiscussionThe hydraulic conductance of an intermediate layer of oil (see Fig.1), sandwiched between the corner water and the central water orgas, has a rather complex dependance on the combination of theboundary conditions at both interfaces, the contact angles, �21 and�32, and the layer thickness at the wall, b2–b1, that in turn dependson the respective capillary pressures, the corner half-angle, �, andthe contact angles. When a combination of the corner half-angleand the contact angles causes the layer surfaces to be either con-cave or flat and concave (see Fig. 2), the layer hydraulic conduc-tance is relatively high. When the inner surface of the layer be-comes convex, �21 + � > �/2, the layer conductance rapidly de-creases. The diminishing layer thickness at the wall, b2–b1 → 0,which corresponds to b2/b1 → 1, causes the layer conductance todecrease exponentially, especially when both contact angles go tozero and the corner half-angle increases (see Fig. 7), or when theinner contact angle �12 goes to 180° (see Fig. 8). The componentsof the independent variable z in Eq. 22 and their scaling followedthose in Ref. 15. In particular, the inverse of the layer thickness atthe wall b1/b2 enters z with the negative sign: The thinner the layer,the more negative the logarithm of the hydraulic conductance.

The oil hydraulic conductance at low oil saturations is a com-plex function of the system geometry and rock wettability. Its

dependence on the system and displacement processes is dis-cussed elsewhere.26,27

ConclusionsA numerical approach has been developed to obtain the hydraulicconductance of an intermediate fluid layer sandwiched betweentwo other fluids. This approach has led to the simple and accu-rate correlations of the layer hydraulic conductance with the vari-ous layer geometry descriptors, and with four different bound-ary conditions:

1. Expressions for calculating the layer perimeter, area, shapefactor, and stability have been derived from the corner half-angle�, the interface contact angles �21 and �32, and the meniscus-apexdistances b1 and b2.

2. A standard finite-element method has been used to solvenumerically Poisson’s equation for creeping, isothermal flow in anintermediate fluid layer formed in a corner of a polygonal capil-lary. Each flow domain was discretized with a mesh that followedthe shape of the capillary walls and the curvatures of the interfaces.

3. The layer hydraulic conductance has been calculated for foursets of boundary conditions: BC-1 (no-slip water/oil interface andno-slip oil/gas interface), BC-2 (no-slip water/oil interface andperfect-slip oil/gas interface), BC-3 (perfect-slip water/oil inter-face and no-slip oil/gas interface), and BC-4 (perfect-slip water/oilinterface and perfect-slip oil/gas interface).

4. Simple comparison between the hydraulic conductance val-ues for perfect-slip and no-slip boundary conditions indicates thatit might be essential to consider the momentum transfer boundarycondition that assumes continuity of velocity and shear stressalong the interfaces.


flow conductance from the corresponding FEM solution forBC-3 G/O interface (the mean absolute error is 28.0%).



flow conductance from the corresponding FEM solution forBC-4 (the mean absolute error is 64.9%).



5. Projection-pursuit regression has been used to obtain theuniversal correlations of the layer hydraulic conductance with therelevant descriptors of the layer geometry. These expressions havebeen listed for each boundary condition described in Conclusion 3.The hydraulic conductance expressions proposed here have beencompared with the expressions proposed by Zhou et al.18 and Huiand Blunt.19 We have compared the relative deviation of the con-ductance calculated with each method from the corresponding fi-nite-element solution. We have shown that the mean absolute rela-tive errors of the projection-pursuit regression expressions are 9.5,9.8, 13.9, and 18.0% for the four boundary conditions, respec-tively. The corresponding mean absolute errors of the Zhou et al.and Hui and Blunt expressions are 22.2, 24.1, 28.0, and 64.9%.Thus, the expressions proposed here are better than those proposedby Zhou et al. and Hui and Blunt by a factor of 2 to 4.

6. Overall, our correlations of the layer hydraulic conductancesare simple and can be used with confidence in the computationallyintensive two-phase and three-phase pore-network simulations ofdrainage and imbibition with contact-line pinning and contact-angle hysteresis.

NomenclatureA � cross-sectional area, L2

A � dimensionless areab1 � oil/water meniscus-apex distance, Lb2 � gas/oil meniscus-apex distance, LEi � geometrical constantf � body force per unit mass, FM-1

fj � jth statistical approximation functiong � hydraulic conductance, L6F-1T-1

g � dimensionless hydraulic conductanceG � shape factorG � normalized shape factor

L21 � dimensionless length of O/W interfaceL32 � dimensionless length of G/O interface

n � unit outward normal vectorP � circumference, LP � dimensionless circumferencer1 � radius of curvature of O/W interface, Lr2 � radius of curvature of O/W interface, LQ � volumetric flow rate L3T-1

<v> � average flow velocity, LT-1

� � velocity, LT-1

X � vector of independent variablesY � dependent variablez � statistical fit variable� � statistical fit constant� � corner half-angle�i � ith boundary of flow domain� � approximation error

�21 � O/W contact angle�32 � G/O contact angle

� � transformation of dependent variable� � statistical fit constant� � viscosity, FL-2T� � gradient of driving force per unit area, FL-3

� � density, ML-3

�� pressure gradient, FL-3

AcknowledgmentsThe first author was sponsored by a fellowship from the SultanQaboos U., the Sultanate of Oman. Partial support was provided bythe U.S. DOE ORT Partnership under Contract DE-ACO3-76FS0098 to the Lawrence Berkeley Natl. Laboratory. Partial sup-port was also provided by gifts from ChevronTexaco and PhillipsPetroleum to UC Oil, Berkeley. Extensive help with the projec-tion-pursuit method has been provided by Dr. David Brillinger,Statistics Dept., U.C. Berkeley.

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Ahmed Al-Futaisi is currently an assistant professor of civil andarchitectural engineering at the Sultan Qaboos U. in Oman.e-mail: [email protected]. His research interests in-volve pore-level modeling of multiphase flow in porous media,and groundwater management. He holds a PhD degree incivil and environmental engineering from U.C. Berkeley. Tad W.Patzek is a professor of geoengineering at the Civil and Envi-ronmental Engineering Dept., U.C. Berkeley, and is Mr. Al-Futaisi’s thesis advisor. e-mail: [email protected]. Hisresearch interests involve multiphase flow in porous media at avariety of scales, evolution of rock microstructure, mechanicalproperties of fragile rocks, and reservoir management andcontrol. He is a co-author of more than 100 papers.


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