Directory UMM :Data Elmu:jurnal:A:Advances In Water Resources:Vol24.Issue3-4.2001:

Pore-morphology-based simulation of drainage in totally wetting

porous media

Markus Hilpert

*

_{, Cass T. Miller}

Center for Advanced Study of the Environment, Department of Environmental Sciences and Engineering, University of North Carolina, CB 7400, 104 Rosenau Hall, Chapel Hill, NC 27599-7400, USA

Received 11 December 1999; received in revised form 29 May 2000; accepted 31 August 2000

Abstract

We develop and analyze a novel, quasi-static, pore-scale approach for modeling drainage in a porous medium system. The approach uses: (1) a synthetic, non-overlapping packing of a set of spheres, (2) a discrete representation of the sphere packing, and (3) concepts from pore morphology and local pore-scale physics to simulate the drainage process. The grain-size distribution and porosity of two well-characterized porous media were used as input into the drainage simulator, and the simulated results showed good agreement with experimental observations. We further comment on the use of this simulator for determining the size of a representative elementary volume needed to characterize the drainage process. Ó _{2001 Elsevier Science Ltd. All rights reserved.}

Keywords:Capillarity; Morphology; Network model; Percolation

1. Introduction

Many engineering applications, such as groundwater remediation and enhanced oil recovery (EOR), involve multiphase ¯uid ¯ow in porous media. Though most of these applications use a continuum approach to describe the ¯ow processes, pore-scale modeling provides an important means to improve our understanding of the underlying physical processes and to determine macro-scale constitutive relationships, such as the capillary pressure±saturation relation.

Due to the natural pore space's complex morphology, ¯uid ¯ow has been modeled at the pore scale primarily using network models that rely upon idealized rep-resentations of the pore morphology. For example, a common approach is to represent the pore morphology by spheres that are connected by cylindrical throats (e.g. [14]). The most dicult task when using network models as a quantitative predictive tool is identifying and specifying coordination numbers and size distributions for pore bodies and pore throats [3]. Some noteworthy contributions, which derive the network structure from a pore-morphological analysis of detailed

three-dimen-sional pore geometries, can be found in [1,2,11,19,26]. But there are also pore-scale modeling approaches, which work with digital representations of the pore morphology. Lattice-gas and lattice-Boltzmann [22] have been increasingly used over the last two decades. While these approaches better represent the porous medium morphology and ¯ow process than idealized network models, they are computationally much more demanding [17]. Hazlett [9] recently suggested an ap-proach for simulating quasi-static two-phase ¯ow based upon a size and connectivity analysis of the digital pore space. This heuristic approach is computationally very attractive and yields reasonable agreement with exper-imental data [5] but has not been widely used.

The overall goal of this work is to develop an ecient and accurate method of simulating drainage of a wetting phase in a real porous medium system. The speci®c objectives are: (1) to develop an approach to link easily quanti®able macroscopic measures of a porous medium to an accurate mapping of the pore morphology; (2) to develop a methodology for using detailed information of pore morphology as a direct input into a drainage sim-ulator without idealizing the morphology; (3) to com-pare simulations based upon the developed method with experimental data from well-characterized systems and with Hazlett's method; and (4) to utilize the developed method to investigate certain aspects of drainage, such

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*

Corresponding author.

E-mail addresses: Markus_Hilpert@unc.edu (M. Hilpert), Casey_Miller@unc.edu (C.T. Miller).

as the representative elementary volume needed to characterize the process.

2. Background

We ®rst introduce some concepts from mathematical morphology, which we will use to formulate our simu-lation approach. The textbooks of Matheron [15] and Serra [24] provide introductions to mathematical mor-phology that may be of interest to the reader; we present only a few fundamental notions.

The morphological erosion E_{of a set Xby a}

struc-turing element Bis the locus of the centers~rof the B~r, which are included inX:

E_{B…X† ˆ}n_~r:B~rXo_{: …1†}

E_{,B, andXare subsets of the underlying space (e.g.,}R2 orR3_{). The subscript~rdenotes the translate ofBby the} vector~r. Another way of writing Eq. (1) is

E_{B…X† ˆX B; …2†}

where stands for the Minkowski subtraction andBfor the re¯ected set of B with respect to the origin,

Bˆ fÿ~r:~r2Bg. A distinction betweenBandBis only necessary for non-symmetric B. The actual choice of B

depends on the speci®c application, but spheres are a common choice. Note that, unlike real spheres, their digital representations are not necessarily symmetric [8]. Fig. 1(a) shows a two-dimensional set X and a

sym-metric two-dimensional sphere S (circle). Fig. 1(b) shows the erosion ofXbySˆ_S.

The morphological dilationD_{ofXbyBis the locus}

of the centers of the B~r, which hitX:

D_{B…X† ˆ}n_{~r:B~r\X 6ˆ ;}o_{; …3†}

which can also be written as

D_{B…X† ˆX B; …4†}

where stands for the Minkowski addition. Fig. 1(c) shows the dilation of X, which was shown in Fig. 1(a), bySˆS.

The mathematical openingO_{ofXbyBis the domain}

swept out by all the translates of Bthat are included in

X:

O_{B…X† ˆ […B~r:B~rX†: …5†}

The opening can be re-expressed by

O_{B…X† ˆ …X B† B: …6†}

Fig. 1(d) shows the opening of X, which was shown in Fig. 1(a), by SˆS.

The openingO_{is closely related to the morphological}

grain-size distribution ifXrepresents the solid phase of a porous medium. This leads to the mathematical con-cept of a granulometry, which seeks to capture the result of a sieve analysis for granular media. A granulometry is essentially an opening with variable size structuring el-ements, kB, where k is a positive real number, i.e., the structuring elements are self-similar. The opening of X

by kS is assumed to be that part of X that would be

hc capillary pressure head

L domain length

M mass of the percolating NWP cluster

N number of spheres

P pore space

pc _{capillary pressure}

R1 principal radius of curvature

R2 principal radius of curvature

S sphere

sn _{NWP saturation}

sn

p NWP saturation at percolation

point

sw _{WP saturation}

X set inR2 _orR3

Vt toroidal volume

Z coordination number of the solid phase

U porosity

q density

c interfacial tension

h contact angle

Mathematical operations

C _{that component of a set that is connected to}

withheld by a sieve with a mesh whose size equals the size ofk>0. Thus, one expects that a sieve with a small mesh size retains more of a given medium than a sieve with a larger mesh size, i.e.,

k06k)O_kBO_k0_B: …7†

This requirement only holds true if B is convex [15]. Then, the cumulative morphological grain-size distri-bution

f…k† ˆVol‰OkB…X†Š

Vol‰XŠ ; …8†

is a monotonically decreasing function of k, which quanti®es the grain size. Vol stands for the volume of its

argument, being a subset of the underlying space. IfXis a pore space, then f…k† represents the morphological pore-size distribution.

For a digital representation, where the pore space is represented by voxels on a cubic lattice, the structuring elementsBare naturally given by digital representations as well. There are many ways of de®ning digital spheres with integer diameter D as structuring elements. One possibility is

S…D† ˆ f~r2N3_:…~rÿ~c†2_6D2_{=4g; …9†}

where the center point of the sphere is ciˆD=2‡1=2 for iˆ1;2;3. Fig. 2 shows these symmetric digital spheres forDˆ1 to 4. The shape ofSis not self-similar

and not convex. Hence,D06Dgenerally does not imply

O_S…D†O_S…D0_†. For example, it would not hold true for a cubic cavity, the walls of which contain cross-shaped holes with extension 3 voxels. Then, structuring spheres of diameter 3 would penetrate into the holes but not structuring spheres of diameter 2. Other examples can be easily constructed.

In order to alleviate the violation of Eq. (7), Glantz [8], who investigated the percolation of solid particles in digital pore spaces, suggested the use of structuring el-ements S0 _{with the property S}0_{D† S}0_{D‡1†. The}

modi®ed structuring elements S0_{, being de®ned by} S0_{1† ˆS…1† and S}0_{n† ˆS…n† [S…nÿ1† possess this}

property. The use of theS0weakens but does not prevent the violation of axiom (7) [8]. Fig. 3 shows these digital spheres forDˆ1 to 4.

3. Approach

The physical model is a digital porous medium with the bottom connected to a non-wetting phase (NWP) reservoir and the top connected to a wetting phase (WP) reservoir. This is a model of a tempe cell, which is commonly used to determine capillary pressure±satura-tion relapressure±satura-tions. Our algorithm for modeling quasi-static primary drainage works as follows:

1. At the beginning, the porous medium is saturated with WP. NWP exists only in its reservoir. The capil-lary pressurepc _{is zero.}

2. Then,pc is increased incrementally. The diameterDs

of a sphere is calculated from Laplace's equation for a spherical interface,

pcˆ4c=Ds; …10†

wherecis the interfacial tension.

3. The sphere serves as a probe. It makes new pore space accessible to the NWP, if it can be moved with-out intersecting with the solid phase from a location that is totally NWP-saturated to a neighboring loca-tion that might be partially WP-saturated. All probe

locations that are totally included in the NWP-®lled portion of the pore space are tested. This step is re-peated until an equilibrium state is reached.

4. The NWP saturation sn _{is computed.}

5. The algorithm proceeds with step (2).

The methods imply a vanishing contact angle,hˆ0°

and thus a totally wetting system. The approach also assumes that there is no trapped or irreducible WP. This requires either (1) that WP be connected through the edges of the pore space; (2) WP ®lms; or (3) a groove system on the solid surface. Our algorithm does not resolve WP ®lm ¯ow, and the groove system can only be modeled for an unrealistically ®ne resolution of the pore space. There is another slight inconsistency in our ap-proach: a throat that is invaded from two sides by NWP (connected to its reservoir) does not get ®lled totally with NWP if the two menisci just touch each other as would happen in reality [20]. But we expect the error in

sn introduced by this shortcoming to be only of minor importance because of the small volume of the throats. We use a pore-morphological framework in order to express sn _{formally during primary drainage. First, we}

determine E_{S…D†…P† ˆP S…D†, i.e., the erosion of the}

pore space by a sphere of diameter D, which corre-sponds to the capillary pressure given by Eq. (10). We call C_{‰XŠ the part of X that is connected to the NWP}

reservoir. We then identify C_‰E_{S…D†…P†Š S…D† with the}

NWP-®lled portion ofP. The NWP saturation becomes

sn…D† ˆ

VolhCh_{P S…D†}i_S…D†i

Vol‰PŠ : …11†

Similar to Eq. (7) for the morphological grain-size distribution, we want sn_D†6sn_D0_{† if D}0_{6D. For}

reasons already discussed, this requirement is often but not always ful®lled if digital spheres are used as struc-turing elements. But the use of the modi®ed spheres S0

instead of the Sshould alleviate violations of it. For illustrative purposes, we applied the pore-mor-phology-based drainage algorithm to a two-dimensional pore space P. Fig. 4(a) shows E_{S…P†. Fig. 4(b) then} Fig. 2. Digital spheresS…D†forDˆ1±4.

showsC_‰E_{S…P†Š, which is the component of}E_{S…P†that is}

connected to the NWP reservoir. Finally Fig. 4(c) shows

C_‰E_{S…P†Š S, which represents the NWP-®lled portion}

ofP. The capillary pressurepc _{is inversely proportional}

to the diameter of the circle S that was used as the structuring element for the morphological operation.

We implemented a Fortran90 code for pore-mor-phology-based drainage in three-dimensional, digital porous media. The centers of the solid phase voxels are located at subsets of Z3 _{f…1=2;1=2;1=2†g. Then,} possible centers of the structuring elements are Z3 _{f…1=2;1=2;1=2†g and} Z3 _{for odd and even}

di-ameters, respectively. Connected components of digital sets were determined based upon the six nearest neighbors. Other centers for the structuring elements were not considered; this ensured that the structuring elements extended to the restricting solid phase. Fig. 5 illustrates the simulations for a three-dimensional digital porous medium with 353 _{voxels. Fig. 5(a)}

shows the porous medium. Figs. 5(b) and (c) show the NWP for sphere diameters Dˆ6 and Dˆ4 voxels, respectively.

One can see from Eq. (11) or from Fig. 4 that one point on the drainage curve is obtained in a three-step approach, which involves an erosion, a connected component analysis, and a dilation. Both the erosion and the dilation involve boolean operations, and the values of these operations at any point in space can be calculated independently. The connected component analysis can either be performed recursively with bool-ean operations or in a ®xed number of steps with op-erations on integer variables [13,21]. Thus, our approach is, like lattice-Boltzmann, highly suitable for parallel computer platforms. But our algorithm requires only a small and ®xed number of operations on boolean or integer variables in order to determine a point on the primary drainage curve, whereas lattice-Boltzmann

methods need many time steps, involving real number operations, in order to achieve convergence. The com-putations of this work, for example, were performed on a SGI Origin 2000 using only one processor. The sim-ulations on the largest domain considered (8003_voxels)

required 2 GB of system memory.

4. Porous medium systems

We simulated random sphere packings, which follow the grain-size statistics of experimental porous media, using the sphere-packing computer code developed by Yang et al. [28]. For the porous media, we used a uni-form glass bead packing, labeled as GB1b, and a less uniform sand, labeled as C-109. See Table 1 for the porous medium properties. For both porous media, measured primary drainage curves were available, which were inferred from equilibrium tetrachloroethylene± water distributions in long vertical columns [10]. The surface tension was‰36:230:21Šdyn/cm. Although not being an input parameter for the simulator, we also

re-port the densities of the ¯uids, which were

‰1:6130:002Šg=cm3 _{for tetrachloroethylene and} ‰0:99780:002Šg=cm3_{for water. The contact angle was}

assumed to be zero,hˆ0°_{. The grain-size statistics were}

obtained using image analysis [4]. We assumed the dis-tribution functions to be lognormal.

Table 2 shows the properties of the simulated packings that represent the GB1b and C-109 porous media. Both packings contain approximately 10,000 spheres. The output of the sphere packing code (centers and diameters of the spheres, coordination number of the grains) was then used to generate digital represen-tations of the porous media. We generated discretiza-tions with 2003_{, 400}3 _{and 800}3 _voxels.

5. Simulation results

5.1. Discretization e€ects

Fig. 6 shows the primary drainage curves for dis-cretization containing 2003_{, 400}3 _{and 800}3 _{voxels for}

both the simulated GB1b and C-109 systems described in Table 2. In all plots presented, the capillary pressures

pc _{are expressed as heights of a corresponding water}

column,hc_ˆpc_=…q

wg†, wherehcis the capillary pressure

head,qw the density of water, andgˆ9:81 m=s2 is the

gravitational acceleration. We used the modi®ed spheres

S0 as structuring elements. The di€erences in the WP saturation sw _{for the same h}c _{values due to di€erent}

spatial discretization can be quite high in the ¯at parts of the primary drainage curves. The di€erences may be on the order of 0.3 for the C-109 system, for example. Discretization e€ects occur because both the digital structuring elements and the digital pore space are not

self-similar as the resolution changes. The overall shape and entry pressure of the primary drainage curve, however, are not a€ected that much by the discretiza-tion. The data density increases with the resoludiscretiza-tion.

The mean throat radius in a random packing of uniform spheres is given by hRti ˆ0:21Dg [18]. From that, one can estimate the entry pressure head

hc_{ˆ2c=…0:21Dgq}

wg†. We tested the validity of this

equation for non-uniform sphere packings by assuming that Dg stands for the arithmetic mean grain diameter hDgi. For the GB1b and C-109 systems, we estimated NWP entry athc_{ˆ31 cm andh}c_{ˆ15 cm, respectively,}

which is in acceptable agreement with both experimental and simulated data.

In order to investigate the in¯uence of the structuring element on the primary drainage curve, we also used the symmetric structuring elementsS…D†for the simulations in the GB1b medium. Fig. 7 shows hc _{versus the}

di€er-ence in sw between the simulation with the modi®ed

Fig. 5. Three-dimensional illustration: (a) solid phase; (b) NWP forDˆ6 voxels; (c) NWP forDˆ4 voxels.

Table 1

Properties of the two experimental multiphase systems

GB1b C-109

Arithmetic mean grain diameterDg(mm) 0.1156a 0.24

Geometric mean grain diameterDg(mm) 0.115 0.2182a

Arithmetic standard deviation ofDg(mm) 0.0121 0.11

PorosityU 0:3560:002 0:3460:002

a

spheresS0_{D†and theS…D†,s}w_‰S0_{Š ÿs}w_{‰SŠ. This di€erence}

is always positive, althoughSandS0_{are not convex. For}

our digital porous media, the conditionS…D† S0_D†is

sucient to ensure that the smallerSmakes more space accessible to NWP than the larger S0. Large values of

sw_‰S0_{Š ÿs}w_{‰SŠ only occur after NWP breakthrough; for}

smallsn_{values, the porous medium does not experience}

the di€erence between S and S0, because they look identical from the top and enter the porous medium

from below. Interestingly, the coarsest discretization with 2003_{voxels yields the smallests}w_{di€erence, because}

at NWP breakthrough (hc_{ˆ63 cm), the diameter of the}

structuring element is Dsˆ2 voxels, for whichSandS0

are identical. Overall, the structuring element does not have too much impact on the simulated primary drain-age curve. The ®nal choice of the structuring element is somewhat arbitrary and not crucial for our application. Eventually, we followed Glantz [8] and used the modi-®ed structuring elements S0 for the remaining simula-tions.

5.2. WP at very low saturations

The open circles in Fig. 8 show again the simulation results for the discretization with 8003 _{voxels. The}

largestpc _{corresponds to a diameterDsˆ1 voxel of the}

sphereS0_{. The simulation always predicts the minimum}

WP saturation to be zero, because it does not account for WP trapping, except if residual pore space exists between the grains, which is not the case for our digital porous media.

Assuming that there is no WP trapping, the simula-tion overpredicts sw _{for lows}w _{values, where all WP is}

pendular. This is because Eq. (10) for spherical inter-faces does not hold true for pendular WP. The general form of the Young±Laplace equation

0 0.2 0.4 0.6 0.8 1

Fig. 6. In¯uence of spatial resolution on primary drainage curves: (a) GB1b system; (b) C-109 system. Table 2

Sphere-packing realizations for the GB1b and C-109 porous media

GB1b C-109

Arithmetic mean grain diameterDg(mm) 0.1149 0.2397

Geometric mean grain diameterDg(mm) 0.1155 0.2198

Arithmetic standard deviation ofDg(mm) 0.0116 0.1024

Domain lengthL(mm) 2.35 5.78

PorosityU 0.356 0.345

Number of spheresN 9532 10,666

Coordination numberZ 5.95 5.99

pcˆc 1

must be used, whereR1andR2 are the principal radii of curvature. For pendular WP, one of the principal radii is negative, the other positive. Pendular WP saturation predicted by Eq. (10) is larger than the one predicted by the correct Eq. (12).

Signi®cant error is introduced by assuming implicitly a spherical interface when calculatingpc_{. To show this,}

we estimated the primary drainage curve in the range of small WP saturations, where all WP is pendular, only from the sphere pack parameters, namely the mean grain size hDgi and the coordination number Z. As-suming that a WP ring between two solid spheres with diameterDghas the shape of a torus, its volume is given byVtˆ2p‰fj2_ÿf2_jcotjÿR2

1jj‡fR21ÿDgf2=2Šwhere

sinjˆDg=…Dg‡2R1†, jˆ …Dg=2‡R1†cosj, and f ˆ R1sinj [16]. See Fig. 9 for the geometry. The negative principal radius of curvature,R2, is given by

R2ˆR1ÿ

Assuming that Z of the simulated sphere packing corresponds to that of the experimental system and that all grains have the same diameterhDgi, the WP satura-tion is then given bysw_ˆNZVt=…2L3_{U†[16], which varies}

only with respect toR1for a given porous media system. The capillary pressure can then be calculated from Eq. (12) and (13). Note that the torus approximation neglects the variation in R2 when calculating pc_{. The}

solid lines in Fig. 8 show the resulting primary drainage curves. If we falsely assume that pc_{ˆ2c=R1, we obtain}

the dashed lines, which overpredictsw_{signi®cantly. But}

then the match between simulated and estimated data is good for low sw values, because both methods rely on the use of Laplace's equation for a spherical interface.

As a comparison between the solid lines and the ®lled circles in Fig. 8 shows, the experimental sw _{values for}

both porous media systems are larger in the range of high pc _{than the ones predicted above, where we}

as-sumed all WP to be pendular and the absence of WP trapping. These observations suggest the existence of WP other than pendular ± for example WP ®lms ± or WP trapping. We believe that a WP ®lm was likely to exist because the experiments started out with the WP-saturated, water-wet porous medium [23]. This fact and the surface roughness, which was likely greater for the sand grains of the C-109 medium than for the smoother glass beads of the GB1b system, exclude the occurrence of trapped WP [6].

The failure of the simulation in thepc_{calculation for}

pendular WP can be compensated by rescaling the pc

axis. If one assumes again that all WP is pendular and that all grains have the same diameter hDgi, one can estimate the negative principal radius of curvature R2, which is given by Eq. (13), and substitute this expression into Eq. (12). The positive radius of curvature is given

Fig. 9. Geometry of pendular WP.

0 0.2 0.4 0.6 0.8 1

O) _{estimate for saddleshaped interface}

estimate for spherical interface experiment

O) _{estimate for saddleshaped interface}

estimate for spherical interface experiment simulation corrected simulation

(a) (b)

by the radius of the structuring element,R1ˆDs=2. The corrected capillary pressure then becomes

pc0ˆ2c

The triangles in Fig. 8 show the modi®ed simulated primary drainage curves. As expected there is a good agreement between the corrected simulation and the estimate based upon both the sphere packing parameters and the general form of Laplace's equation for low sw

values, as a comparison between the triangles and the solid lines shows. We do not suggest the rescaling of the

pc _{axis as a means to obtain better simulation results,}

since there is no sharp and identi®able transition from pendular WP saturation to the one with spherical menisci.

5.3. Domain size e€ects

We used the discretizations with 4003 _{voxels and cut}

them into non-overlapping, rectangular, equal-sized subdomains by generating four subdomains with 200200400 voxels, 32 with 100100200 voxels, 256 with 5050100 voxels, and 2048 with 2525 50 voxels. The subdomains were rectangular for reasons concerning percolation theory. We averaged the simu-lation results of the subdomains with equal size.

Fig. 10 shows the primary drainage curve for the various subdomain sizes for both the GB1b and C-109 systems. For increasing domain size, the shoulder of the primary drainage curve at the entry point becomes sharper, consistent with the pore-network model simu-lations and experiments of Larson and Morrow [12]. The volume of NWP, which tries to invade the domain per area of the NWP reservoir, is independent of the domain size, resulting in smaller sn _{values for larger}

domains.

Next, we sought to determine the fractal dimension of the percolating NWP cluster, which represents the NWP

that ®rst extends to the boundary opposed to the NWP reservoir when increasing pc _{incrementally. An object}

possesses the spatial dimensiondif its massMscales as

M ˆLd_{, where L is the domain length. We used the} number of NWP-occupied voxels as a measure for M. The object is said to be fractal if d is a non-integer. Following Wilkinson and Willemsen [27], who investi-gated the fractal properties of the percolating cluster formed by an invasion percolation process, we used the rectangular domains of sizeLL2Land determined the percolating cluster over the central region with size

LLL. This geometry suppresses boundary e€ects resulting from the NWP reservoir and the sample outlet. The outlet causes a boundary e€ect because the NWP can advance opposite to the external capillary pressure. Contrary to the investigations of [27], we did not have periodic boundary conditions on the sides because the sphere packing was not periodic. Fig. 11 shows a double-logarithmic plots ofMversusLincluding linear regression curves. The slope of these curves equals the fractal dimension of the percolating cluster. We ob-tained d ˆ2:840:09 for the GB1a system, and

d ˆ2:890:12 for the C-109 system. We also deter-mined the local fractal dimension, dˆd log…M†=

d log…L†[25], and evaluated this expression only based on the results of the two largest subdomains. We ob-tainedd ˆ3:05 for both the GB1b and C-109 systems. The local fractal dimension is larger than thedobtained by use of all four data points, likely because of boundary e€ects: for example, the length of the smallest investi-gated subdomain with 252550 voxels amounts to only very few sphere diameters. Thus, boundary e€ects dominate and reduce d [7]. Hence, the local fractal di-mension provides a more faithful estimate for d, but error bounds are not provided by our analysis. As

d ˆ3:05 is not smaller than the spatial dimension 3, there is no evidence that the percolating NWP cluster is a fractal object.

A local fractal dimension dˆ3:05 for both porous media systems suggests that the fractal dimension of the

0 0.2 0.4 0.6 0.8 1

percolating NWP cluster predicted by the morphology-based approach is larger thand ˆ2:52, the value for an invasion percolation process in three-dimensional net-works [27]. This is not too surprising because the two approaches di€er in some important respects. In a bond invasion percolation model, the NWP invasion starts from one side of the lattice. Then, the largest radius throat on the NWP-WP interface is searched and in-vaded by NWP. This whole process is repeated until no change in the ¯uid distribution occurs. For the case of the no WP trapping rule, the entire network is eventu-ally NWP saturated. Many pore-network model for-mulations (e.g. [29]) are based upon the invasion percolation concept. In our algorithm, we increase the capillary pressurepc_{(decrease the size of the structuring}

element) in steps. We then check in the entire pore space as to whether the structuring element can access new pore space. Then, the entire accessible pore space is in-vaded by NWP simultaneously, not just the largest radius throat, as for the case of invasion percolation. Many quasi-static pore-network models are based on the assumption that the entire accessible pore space is invaded simultaneously (e.g. [14]). The morphological approach may yield a larger fractal dimension than in-vasion percolation theory, because the inin-vasion of all accessible throats at once has more the character of a macroscopic piston ¯ow. The fact that the throat sizes are discretized increases this surge when increasing pc_.

Invasion percolation generates more fractal patterns, because ®ngers are more likely to develop when NWP invades the throats one by one.

Invasion percolation mimics the dynamics of the displacement process. It assumes that throats with the least resistance (largest radius) tend to be invaded ®rst. This assumption is reasonable, but examples can be easily constructed where a simultaneous invasion into all accessible throats yields better results. Both invasion percolation and the morphological approach must

be seen as approximations of the actual dynamic processes.

Whereas we observed that sn _{increases with L for}

large pc_{, Zhou and Stenby [29] observed the opposite}

using an invasion-percolation approach: their pore-net-work model included a trapping mechanism for WP, which we believe was not important for our porous medium systems.

5.4. Randomness of pore space

Fig. 12 shows the standard deviation of the WP sat-uration,Dsw_{, versush}c_{for the four sizes of the simulated}

subdomain. Clearly, smaller subdomain sizes generate a larger scatter in sw. The C-109 system has a larger Dsw

value than the GB1b system because of the larger vari-ability of the grain sizes. Fig. 13 shows the maximum

standard deviation of the WP saturation, max

fpc

:Dsw_{g, and the standard deviation of the porosity,}

DU, versus the height of the subdomain. For both the GB1b and the C-109 system, max fpc

:Dsw_{g is larger}

than DU. This suggests that a REV with respect to the porosity is smaller than that with respect to the primary drainage curve. The randomness of the pore space has a negligible impact on the primary drainage curve if one considers domains with 200200400 voxels, which corresponds approximately to 2500 spheres.

5.5. Comparison to Hazlett's method

Hazlett [9] suggested a model for quasi-static, two-phase ¯ow in porous media, the drainage part of which is similar to our approach. In the language of pore-morphology, Hazlett assumes that

The deviation from our Eq. (11) seems minor but turns out to be crucial. We illustrate the di€erence in a two-dimensional example. Fig. 14 shows two grains. The NWP reservoir is on the left side. We assume thatpc_and

thus the circle diameter Ds is chosen such that the menisci formed by the openingO_{just touch the vertical}

symmetry axis. Hazlett's approach would assume that the entire opening O_ˆE_{S…D†…P† S…D† represents}

NWP. Clearly, this is not true since the meniscus on the left side still has to pass the throat. The meniscus does not know about the hypothetical NWP on the right side, to which it can connect. We implemented Hazlett's ap-proach and simulated the drainage curve for the GB1b medium. The simulated pc _{in the horizontal portion of}

the primary drainage curve underpredicted the exper-imentalpc_{on the order of 20%. This phenomenon is not}

a discretization e€ect.

Coles et al. [5] used Hazlett's model to simulate capillary pressure-saturation curves in sandstone. The three-dimensional pore space was obtained by computed microtomography (CMT). Coles et al. obtained only an

acceptable agreement between experimental and simu-lated data. Their simulation overpredicted pc _unlike

ours. Uncertainties in interfacial tensionc, contact angle h, and the CMT data seem to outweigh the negative bias introduced by the approximations used to represent drainage events.

NWP Reser

v

oir

Fig. 14. Drainage simulation in a two-dimensional pore space. The solid phase is black. Hazlett's method assumes that the opening O (shown in gray) represents NWP, becauseOis completely connected to the NWP reservoir. This is wrong, because the left meniscus cannot pass the throat.

Fig. 13. Standard deviation of the porosity,DU, and maximum standard deviation of WP saturation, maxfpc

:Dsw_{g, versus the height 2Lof the} rectangular subdomain: (a) GB1b system; (b) C-109 system.

0 0.1 0.2 0.3 0.4

Fig. 12.hc_{versus standard deviation of the WP saturation,}

6. Discussion and summary

We suggested a pore-morphology-based approach for modeling quasi-static drainage. We compared our sim-ulations to experimental data and found very good agreement between simulated and experimental results in the horizontal part of the primary drainage curves. We attribute deviations at the entry point to size e€ects and mismatches in the minimum WP saturation to WP ®lms and grooves on the solid surface, neither of ac-counted for. The inappropriate use of Laplace's equa-tion for spherical interfaces overpredictssw _{in the range}

of highpc_{. Lattice-Boltzmann methods do not have this}

shortcoming, but they are computationally much more expensive.

We obtained the digital representation of exper-imental porous media by using a sphere-packing algorithm [28], which only needs the grain-size distri-bution and the porosity as input parameters. The very good agreement between simulated and experimental data testi®es to the ability of the sphere-packing al-gorithm [28] used to model natural unconsolidated porous media accurately, for which a grain-size distri-bution can be readily obtained. Further, the numerical simulation of sphere packings is much less expensive than computed microtomography, and thus much larger porous medium domains can be investigated.

The suggested simulator yields good predictions, if the menisci are spherical, which is the case for sphere packings, unconsolidated porous media, and sand stones in the range of high WP saturations. Other me-dia, such as fractured rocks and clays, will likely yield worse predictions.

The morphology-based approach is currently re-stricted to quasi-static drainage. But in conjunction with the ability to simulate natural porous media, it bears an important application: compared to lattice-Boltzmann simulations, the model is computationally very inex-pensive, because it requires less memory and much less CPU time and thus allows the simulation of much larger domains. Thus, many random realizations of statisti-cally identical pore spaces can be investigated and the random error of primary drainage curves estimated. This information can then be used to perform lattice-Boltzmann simulations wisely, which also can be used to model imbibition and dynamic displacement processes.

Acknowledgements

This work was supported by the the National Science Foundation (NSF) grant EAR-9901660, the National Institute of Environmental Health Sciences (NIEHS) grant 5 P42 ES05948-02, and the Department of Energy (DOE) grant DE-FG07-96ER14703. The authors ac-knowledge the insightful comments and suggestions of

William G. Gray. The authors are also grateful to Hans-Jorg Vogel, Martin Blunt, and one anonymous reviewer for their valuable comments.

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