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

Advances in Water Resources 24 (2001) 243±255
www.elsevier.com/locate/advwatres

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 macroscale 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 representations 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*

Corresponding author.

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

(M.

Hilpert),

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 approach 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 experimental 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 simulator without idealizing the morphology; (3) to compare 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

0309-1708/01/$ - see front matter Ó 2001 Elsevier Science Ltd. All rights reserved.
PII: S 0 3 0 9 - 1 7 0 8 ( 0 0 ) 0 0 0 5 6 - 7

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M. Hilpert, C.T. Miller / Advances in Water Resources 24 (2001) 243±255

sn
snp

Notation
NWP
REV
WP

non-wetting phase
representative elementary volume
wetting phase

sw
X
Vt
Z
U
q
c
h

Variables

B
structuring element
D
diameter
grain diameter
Dg
sphere diameter
Ds
d
fractal dimension
g
gravitational acceleration
capillary pressure head
hc
L
domain length
M
mass of the percolating NWP cluster
N
number of spheres

P
pore space
capillary pressure
pc
principal radius of curvature
R1
principal radius of curvature
R2
S
sphere

Mathematical operations
C
that component of a set that is connected to
the NWP reservoir
D
morphological dilation
E
morphological erosion
O

morphological opening
Vol
volume
D
standard deviation

Minkowski addition

Minkowski subtraction

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 simulation approach. The textbooks of Matheron [15] and
Serra [24] provide introductions to mathematical morphology that may be of interest to the reader; we present
only a few fundamental notions.
The morphological erosion E of a set X by a structuring element B is the locus of the centers ~
r of the B~r ,

which are included in X:
n
o
r : B~r  X :
…1†
EB …X † ˆ ~
E, B, and X are subsets of the underlying space (e.g., R2
or R3 ). The subscript ~
r denotes the translate of B by the
vector ~
r. Another way of writing Eq. (1) is


EB …X † ˆ X B;

NWP saturation
NWP saturation at percolation
point
WP saturation
set in R2 or R3

toroidal volume
coordination number of the solid phase
porosity
density
interfacial tension
contact angle

…2†

where stands for the Minkowski subtraction and B for
the re¯ected set of B with respect to the origin,
B ˆ fÿ~
r :~
r 2 Bg. A distinction between B and B is 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 of X by S ˆ S.
The morphological dilation D of X by B is the locus
of the centers of the B~r , which hit X:
n
o
r : B~r \ X 6ˆ ; ;
…3†
DB …X † ˆ ~
which can also be written as

DB …X † ˆ X  B;

…4†

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

by S ˆ S.

The mathematical opening O of X by B is the domain
swept out by all the translates of B that are included in
X:
OB …X † ˆ […B~r : B~r  X †:

…5†

The opening can be re-expressed by
  B:
OB …X † ˆ …X B†

…6†

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

Fig. 1(a), by S ˆ S.
The opening O is closely related to the morphological
grain-size distribution if X represents the solid phase of
a porous medium. This leads to the mathematical concept 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 elements, 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

M. Hilpert, C.T. Miller / Advances in Water Resources 24 (2001) 243±255

245

Fig. 1. Morphological operations in two dimensions: (a) set X and sphere S; (b) erosion E of X by S; (c) dilation D of X by S; (d) opening O of
X by S.

withheld by a sieve with a mesh whose size equals the
size of k > 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.,
k0 6 k ) OkB  Ok0 B :

…7†

This requirement only holds true if B is convex [15].
Then, the cumulative morphological grain-size distribution
f …k† ˆ

Vol‰OkB …X †Š
;
Vol‰X Š

argument, being a subset of the underlying space. If X is
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
elements B are 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
2

…8†

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

S…D† ˆ f~
r 2 N 3 : …~
r ÿ~
c† 6 D2 =4g;

…9†

where the center point of the sphere is ci ˆ D=2 ‡ 1=2
for i ˆ 1; 2; 3. Fig. 2 shows these symmetric digital
spheres for D ˆ 1 to 4. The shape of S is not self-similar

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M. Hilpert, C.T. Miller / Advances in Water Resources 24 (2001) 243±255

Fig. 2. Digital spheres S…D† for D ˆ 1±4.

Fig. 3. Modi®ed digital spheres S 0 …D† for D ˆ 1±4 and S 0 …D†  S 0 …D ‡ 1†.

and not convex. Hence, D0 6 D generally does not imply
OS…D†  OS…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 elements S 0 with the property S 0 …D†  S 0 …D ‡ 1†. The
modi®ed structuring elements S 0 , being de®ned by
S 0 …1† ˆ S…1† and S 0 …n† ˆ S…n† [ S…n ÿ 1† possess this
property. The use of the S 0 weakens but does not prevent
the violation of axiom (7) [8]. Fig. 3 shows these digital
spheres for D ˆ 1 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±saturation relations. 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 capillary pressure pc is zero.
2. Then, pc is increased incrementally. The diameter Ds
of a sphere is calculated from Laplace's equation
for a spherical interface,
pc ˆ 4c=Ds ;

…10†

where c is the interfacial tension.
3. The sphere serves as a probe. It makes new pore
space accessible to the NWP, if it can be moved without intersecting with the solid phase from a location
that is totally NWP-saturated to a neighboring location 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 repeated 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, h ˆ 0°
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 approach: 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

i.e., the erosion of the
determine ES…D† …P † ˆ P S…D†,
pore space by a sphere of diameter D, which corresponds to the capillary pressure given by Eq. (10). We
call C‰X Š the part of X that is connected to the NWP
reservoir. We then identify C‰ES…D† …P †Š  S…D† with the
NWP-®lled portion of P. The NWP saturation becomes
h h
i
i

Vol C P S…D†
 S…D†
:
…11†
sn …D† ˆ
Vol‰P Š
Similar to Eq. (7) for the morphological grain-size
distribution, we want sn …D† 6 sn …D0 † if D0 6 D. For
reasons already discussed, this requirement is often but
not always ful®lled if digital spheres are used as structuring elements. But the use of the modi®ed spheres S 0
instead of the S should alleviate violations of it.
For illustrative purposes, we applied the pore-morphology-based drainage algorithm to a two-dimensional
pore space P. Fig. 4(a) shows ES …P †. Fig. 4(b) then

M. Hilpert, C.T. Miller / Advances in Water Resources 24 (2001) 243±255

247

Fig. 4. Two-dimensional illustration of drainage simulation in a two-dimensional pore space P. The solid phase is black. In red are shown (a) ES …P †,
the erosion E of P by the circle S; (b) C‰ES …P †Š, which is the component of ES …P † that is connected to the NWP reservoir; and (c) C‰ES …P †Š  S, which
represents the NWP-®lled portion of P.

shows C‰ES …P †Š, which is the component of ES …P † that is
connected to the NWP reservoir. Finally Fig. 4(c) shows
C‰ES …P †Š  S, which represents the NWP-®lled portion
of P. The capillary pressure pc 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-morphology-based drainage in three-dimensional, digital
porous media. The centers of the solid phase voxels are
located at subsets of Z 3  f…1=2; 1=2; 1=2†g. Then,
possible centers of the structuring elements are
Z 3  f…1=2; 1=2; 1=2†g and Z 3 for odd and even diameters, 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 D ˆ 6 and D ˆ 4 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 boolean operations or in a ®xed number of steps with operations 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 computations of this work, for example, were performed on
a SGI Origin 2000 using only one processor. The simulations 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 uniform 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 ‰36:23  0:21Š dyn/cm. Although not
being an input parameter for the simulator, we also report the densities of the ¯uids, which were
‰1:613  0:002Š g=cm3 for tetrachloroethylene and
‰0:9978  0:002Š g=cm3 for water. The contact angle was
assumed to be zero, h ˆ 0°. The grain-size statistics were
obtained using image analysis [4]. We assumed the distribution 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 representations of the porous media. We generated discretizations with 2003 , 4003 and 8003 voxels.

248

M. Hilpert, C.T. Miller / Advances in Water Resources 24 (2001) 243±255

Fig. 5. Three-dimensional illustration: (a) solid phase; (b) NWP for D ˆ 6 voxels; (c) NWP for D ˆ 4 voxels.

Table 1
Properties of the two experimental multiphase systems
Arithmetic mean grain diameter Dg (mm)
Geometric mean grain diameter Dg (mm)
Arithmetic standard deviation of Dg (mm)
Porosity U
a

GB1b

C-109

0.1156a
0.115
0.0121
0:356  0:002

0.24
0.2182a
0.11
0:346  0:002

Based on the assumption of a lognormal grain-size distribution but not part of the experimental data set.

5. Simulation results
5.1. Discretization e€ects
Fig. 6 shows the primary drainage curves for discretization containing 2003 , 4003 and 8003 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 ˆ pc =…qw g†, where hc is the capillary pressure
head, qw the density of water, and g ˆ 9:81 m=s2 is the
gravitational acceleration. We used the modi®ed spheres
S 0 as structuring elements. The di€erences in the WP
saturation sw for the same hc 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 discretization. The data density increases with the resolution.
The mean throat radius in a random packing of
uniform spheres is given by hRt i ˆ 0:21Dg [18]. From
that, one can estimate the entry pressure head
hc ˆ 2c=…0:21Dg qw g†. We tested the validity of this
equation for non-uniform sphere packings by assuming
that Dg stands for the arithmetic mean grain diameter
hDg i. For the GB1b and C-109 systems, we estimated
NWP entry at hc ˆ 31 cm and hc ˆ 15 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 elements S…D† for the simulations
in the GB1b medium. Fig. 7 shows hc versus the di€erence in sw between the simulation with the modi®ed

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M. Hilpert, C.T. Miller / Advances in Water Resources 24 (2001) 243±255
Table 2
Sphere-packing realizations for the GB1b and C-109 porous media

120
100
80
60
40
20

(a)

0
0

0.2

0.4

0.6

w

0.8

Saturation s

0.1149
0.1155
0.0116
2.35
0.356
9532
5.95

0.2397
0.2198
0.1024
5.78
0.345
10,666
5.99

60

experiment
2003 voxel
4003 voxel
3
800 voxel

2

experiment
3
200 voxel
4003 voxel
3
800 voxel

C-109

Capillary pressure head (cm H O)

140

2

Capillary pressure head (cm H O)

Arithmetic mean grain diameter Dg (mm)
Geometric mean grain diameter Dg (mm)
Arithmetic standard deviation of Dg (mm)
Domain length L (mm)
Porosity U
Number of spheres N
Coordination number Z

GB1b

1

50
40
30
20
10

(b)
0
0

0.2

0.4

0.6

Saturation sw

0.8

1

140

from below. Interestingly, the coarsest discretization
with 2003 voxels yields the smallest sw di€erence, because
at NWP breakthrough (hc ˆ 63 cm), the diameter of the
structuring element is Ds ˆ 2 voxels, for which S and S 0
are identical. Overall, the structuring element does not
have too much impact on the simulated primary drainage 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 S 0 for the remaining simulations.

2003 voxel
4003 voxel
8003 voxel

2

Capillary pressure head (cm H O)

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

120
100
80
60
40
20
0
0

0.025

w

0.05

s [S’] s

w

0.075

0.1

[S]

Fig. 7. hc versus sw ‰S 0 Š ÿ sw ‰SŠ, the di€erence in sw between the simulation with the modi®ed spheres S 0 …D† and the symmetric spheres S…D†
for the GB1b system.

spheres S 0 …D† and the S…D†, sw ‰S 0 Š ÿ sw ‰SŠ. This di€erence
is always positive, although S and S 0 are not convex. For
our digital porous media, the condition S…D†  S 0 …D† is
sucient to ensure that the smaller S makes more space
accessible to NWP than the larger S 0 . Large values of
sw ‰S 0 Š ÿ sw ‰SŠ only occur after NWP breakthrough; for
small sn values, the porous medium does not experience
the di€erence between S and S 0 , because they look
identical from the top and enter the porous medium

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
largest pc corresponds to a diameter Ds ˆ 1 voxel of the
sphere S 0 . 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 simulation overpredicts sw for low sw values, where all WP is
pendular. This is because Eq. (10) for spherical interfaces does not hold true for pendular WP. The general
form of the Young±Laplace equation

250

400

estimate for saddleshaped interface
estimate for spherical interface
experiment
simulation
corrected simulation

300

200

100

0
0

0.2

0.4

0.6

w

0.8

250

2

500

Capillary pressure head (cm H O)

Capillary pressure head (cm H2O)

M. Hilpert, C.T. Miller / Advances in Water Resources 24 (2001) 243±255

1

200

estimate for saddleshaped interface
estimate for spherical interface
experiment
simulation
corrected simulation

150

100

50

0
0

0.2

0.4

0.6

w

0.8

1

Saturation s

Saturation s
(a)

(b)

Fig. 8. Solid line ± pendular WP saturation based on the coordination number of the simulated packing and the Laplace equation for saddle-shaped
interfaces. Dashed line ± pendular WP saturation based on the coordination number of the simulated packing and the Laplace equation for spherical
interfaces. Closed circles ± experiment; open circles ± simulation; (a) GB1b system; (b) C-109 system.


1
1
p ˆc
‡
;
R1 R 2
c



…12†

must be used, where R1 and R2 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 calculating pc . 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 hDg i and the coordination number Z. Assuming that a WP ring between two solid spheres with
diameter Dg has the shape of a torus, its volume is given
by Vt ˆ 2p‰ fj2 ÿ f 2 j cot j ÿ R21 jj ‡ fR21 ÿ Dg f 2 =2Š where
sin j ˆ Dg =…Dg ‡ 2R1 †, j ˆ …Dg =2 ‡ R1 † cos j, and f ˆ
R1 sin j [16]. See Fig. 9 for the geometry. The negative
principal radius of curvature, R2 , is given by
q
R2 ˆ R1 ÿ R21 ‡ Dg R1 :
…13†

Fig. 9. Geometry of pendular WP.

Assuming that Z of the simulated sphere packing
corresponds to that of the experimental system and that
all grains have the same diameter hDg i, the WP saturation is then given by sw ˆ NZVt =…2L3 U† [16], which varies
only with respect to R1 for 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 ˆ 2c=R1 , we obtain
the dashed lines, which overpredict sw 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 assumed 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 WPsaturated, 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 the pc 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 hDg i, 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

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M. Hilpert, C.T. Miller / Advances in Water Resources 24 (2001) 243±255

by the radius of the structuring element, R1 ˆ Ds =2. The
corrected capillary pressure then becomes
"
#
2c
1
0
p :
1‡
pc ˆ
…14†
Ds
1 ÿ 1 ‡ 2hDg i=Ds

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

Capillary pressure head (cm H2O)

We used the discretizations with 4003 voxels and cut
them into non-overlapping, rectangular, equal-sized
subdomains by generating four subdomains with
200  200  400 voxels, 32 with 100  100  200 voxels,
256 with 50  50  100 voxels, and 2048 with 25  25 
50 voxels. The subdomains were rectangular for reasons
concerning percolation theory. We averaged the simulation 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 simulations 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
140

experiment
25 x 25 x 50 voxels
50 x 50 x 100 voxels
100 x 100 x 200 voxels
200 x 200 x 400 voxels

120
100
80
60
40
20

(a)
0
0

0.2

0.4

0.6

w

Saturation s

0.8

1

Capillary pressure head (cm H2O)

that ®rst extends to the boundary opposed to the NWP
reservoir when increasing pc incrementally. An object
possesses the spatial dimension d if its mass M scales as
M ˆ Ld , 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 investigated the fractal properties of the percolating cluster
formed by an invasion percolation process, we used the
rectangular domains of size L  L  2L and determined
the percolating cluster over the central region with size
L  L  L. 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 of M versus L including linear
regression curves. The slope of these curves equals the
fractal dimension of the percolating cluster. We obtained d ˆ 2:84  0:09 for the GB1a system, and
d ˆ 2:89  0:12 for the C-109 system. We also determined the local fractal dimension, d ˆ d log …M†=
d log …L† [25], and evaluated this expression only based
on the results of the two largest subdomains. We obtained d ˆ 3:05 for both the GB1b and C-109 systems.
The local fractal dimension is larger than the d obtained
by use of all four data points, likely because of boundary
e€ects: for example, the length of the smallest investigated subdomain with 25  25  50 voxels amounts to
only very few sphere diameters. Thus, boundary e€ects
dominate and reduce d [7]. Hence, the local fractal dimension provides a more faithful estimate for d, but
error bounds are not provided by our analysis. As
d ˆ 3:05 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 d ˆ 3:05 for both porous
media systems suggests that the fractal dimension of the

60

experiment
25 x 25 x 50 voxels
50 x 50 x 100 voxels
100 x 100 x 200 voxels
200 x 200 x 400 voxels

50
40
30
20
10

(b)
0
0

0.2

0.4

0.6

Saturation sw

0.8

1

Fig. 10. In¯uence of domain size on primary drainage curve keeping the spatial resolution constant: (a) GB1b system; (b) C-109 system.

252

M. Hilpert, C.T. Miller / Advances in Water Resources 24 (2001) 243±255
7

6.5

6

6

5.5

5.5

log10 (M)

log

10

(M)

6.5

7

(a)

5
4.5

5
4.5

4

4

3.5

3.5

3
1

1.5

log

10

(L)

2

2.5

(b)

3
1

1.5

log10 (L)

2

2.5

Fig. 11. Mass M of the percolating cluster versus the domain length L. The fractal dimension d is given by the slope of the curve. (a) GB1b system.
(b) C-109 system.

percolating NWP cluster predicted by the morphologybased approach is larger than d ˆ 2:52, the value for an
invasion percolation process in three-dimensional networks [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 invaded 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 eventually NWP saturated. Many pore-network model formulations (e.g. [29]) are based upon the invasion
percolation concept. In our algorithm, we increase the
capillary pressure pc (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 invaded 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 invasion percolation theory, because the invasion 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-network 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 saturation, Dsw , versus hc 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 variability 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 200  200  400 voxels, which
corresponds approximately to 2500 spheres.
5.5. Comparison to Hazlett's method
Hazlett [9] suggested a model for quasi-static, twophase ¯ow in porous media, the drainage part of which
is similar to our approach. In the language of poremorphology, Hazlett assumes that
h hh
i
ii

Vol C P S…D†
 S…D†
:
…15†
sn …D† ˆ
Vol‰P Š

253

Capillary pressure head (cm H2O)

Capillary pressure head (cm H O)

M. Hilpert, C.T. Miller / Advances in Water Resources 24 (2001) 243±255
120

250

2

25 x 25 x 50 voxels
50 x 50 x 100 voxels
100 x 100 x 200 voxels
200 x 200 x 400 voxels

200

25 x 25 x 50 voxels
50 x 50 x 100 voxels
100 x 100 x 200 voxels
200 x 200 x 400 voxels

100
80

150

60

100

40

50

20

(a)
0
0

0.1

0.2

∆ sw

0.3

0.4

(b)
0
0

0.1

0.2

0.3

∆ sw

0.4

Fig. 12. hc versus standard deviation of the WP saturation, Dsw : (a) GB1b system; (b) C-109 system.

0.4

0.4
w

w

max { ∆ s }
∆Φ

0.3

max { ∆ s }

0.2

0.2

0.1

0.1

0
0
(a)

100

200

300

∆Φ

0.3

400

Domain height (voxel)

0
0
(b)

100

200

300

400

Domain height (voxel)

Fig. 13. Standard deviation of the porosity, DU, and maximum standard deviation of WP saturation, maxfpc : Dsw g, versus the height 2L of the
rectangular subdomain: (a) GB1b system; (b) C-109 system.

acceptable agreement between experimental and simulated data. Their simulation overpredicted pc unlike
ours. Uncertainties in interfacial tension c, contact angle
h, and the CMT data seem to outweigh the negative bias
introduced by the approximations used to represent
drainage events.

NWP Reservoir

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 that pc and
thus the circle diameter Ds is chosen such that the
menisci formed by the opening O just touch the vertical
symmetry axis. Hazlett's approach would assume that
the entire opening O ˆ ES…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 approach and simulated the drainage curve for the GB1b
medium. The simulated pc in the horizontal portion of
the primary drainage curve underpredicted the experimental pc 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

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, because O is completely connected to
the NWP reservoir. This is wrong, because the left meniscus cannot
pass the throat.

254

M. Hilpert, C.T. Miller / Advances in Water Resources 24 (2001) 243±255

6. Discussion and summary
We suggested a pore-morphology-based approach for
modeling quasi-static drainage. We compared our simulations 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 accounted for. The inappropriate use of Laplace's equation for spherical interfaces overpredicts sw in the range
of high pc . Lattice-Boltzmann methods do not have this
shortcoming, but they are computationally much more
expensive.
We obtained the digital representation of experimental porous media by using a sphere-packing
algorithm [28], which only needs the grain-size distribution and the porosity as input parameters. The very
good agreement between simulated and experimental
data testi®es to the ability of the sphere-packing algorithm [28] used to model natural unconsolidated
porous media accurately, for which a grain-size distribution 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 media, such as fractured rocks and clays, will likely yield
worse predictions.
The morphology-based approach is currently restricted 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 inexpensive, because it requires less memory and much less
CPU time and thus allows the simulation of much larger
domains. Thus, many random realizations of statistically identical pore spaces can be investigated and the
random error of primary drainage curves estimated.
This information can then be used to perform latticeBoltzmann 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 acknowledge the insightful comments and suggestions of

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