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Advances in Water Resources 24 (2001) 257±277
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Pore network modelling of two-phase ¯ow in porous rock: the e€ect
of correlated heterogeneity
Mark A. Knackstedt a,b, Adrian P. Sheppard a,b, Muhammad Sahimi c,*
a

Department of Applied Mathematics, Research School of Physical Science and Engineering, Australian National University,
Canberra ACT 0200, Australia
b
Australian Petroleum Cooperative Research Centre, University of New South Wales, Sydney NSW 2052, Australia
c
Department of Chemical Engineering, University of Southern California, Los Angeles CA 90089-1211, USA
Received 2 November 1999; received in revised form 19 July 2000; accepted 31 August 2000

Abstract
Using large scale computer simulations and pore network models of porous rock, we investigate the e€ect of correlated heterogeneity on two-phase ¯ow through porous media. First, we review and discuss the experimental evidence for correlated heterogeneity. We then employ the invasion percolation model of two-phase ¯ow in porous media to study the e€ect of correlated
heterogeneity on rate-controlled mercury porosimetry, the breakthrough and residual saturations, and the size distribution of
clusters of trapped ¯uids that are formed during invasion of a porous medium by a ¯uid. For all the cases we compare the results
with those for random (uncorrelated) systems, and show that the simulation results are consistent with the experimental data only if

the heterogeneity of the pore space is correlated. In addition, we also describe a highly ecient algorithm for simulation of twophase ¯ow and invasion percolation that makes it possible to consider very large networks. Ó 2001 Elsevier Science Ltd. All rights
reserved.

1. Introduction
Multiphase ¯ow phenomena in porous media are
relevant to many problems of great scienti®c and industrial importance, ranging from extraction of oil, gas
and geothermal energy from underground reservoirs, to
transport of contaminants in soils and aquifers, and ink
imbibition in a printing paper. Aside from the classical
continuum models of such phenomena (for reviews see,
for example, [39,42]), discrete or pore network models
have been used to represent disordered porous media,
and detailed simulations have been carried out in order
to understand two- and three-phase ¯ow in such media.
To interpret the simulations' results, the concepts of
percolation theory (see, for example, [39,40,52]) have
been employed to model slow ¯ow of ¯uids through the
pore space. These models include both random bond or
site percolation [2,3,9,10,15,19,38] and invasion percolation (IP) [6,57], and have provided considerable insight into the physics of multiphase ¯ow in disordered
porous media. In particular, IP, which was introduced

*

Corresponding author.
E-mail address: moe@iran.usc.edu (M. Sahimi).

for describing the evolution of the interface between an
invading and a defending ¯uid in a porous medium, has
provided deeper understanding of such phenomena.
In most previous applications of percolation theory
and pore network models to modelling of multiphase
¯ow in porous media, correlations in the spatial disorder
have either been neglected, or have been assumed to
have a limited extent [1,5,15,33]. However, it has recently been suggested that long-range correlations are
likely to exist in many porous sedimentary formations,
both at the pore [18] and ®eld scales ([12,22,26,29,30,
32,46] for a review see, for example, [43]). This has
motivated studies of percolation in pore networks with
long-range correlations [8,24,27,45,47]. Results of these
studies indicated that correlations have a signi®cant effect on many important characteristics of such systems.

For example, one ®nds [24] that, with the correlations
present, the percolation threshold can no longer be de®ned uniquely but depends on the rule that de®nes when
and how a cluster is sample-spanning. However, these
papers considered only the e€ect of correlations on the
percolation properties, and did not address the corresponding e€ects on ¯uid clusters' con®gurations and
other important properties of multiphase ¯ow in pore
network models of porous media. There have also been

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M.A. Knackstedt et al. / Advances in Water Resources 24 (2001) 257±277

very limited studies of the e€ect of correlations on the
characteristics of IP, the model which is perhaps most
appropriate for investigation of capillary-dominated
displacements in porous media [25,34,56]. Even then,
these studies were limited to two-dimensional (2D)

networks, and considered only ¯uids' saturations up to
the breakthrough, i.e., the point at which the invading
¯uid becomes sample-spanning for the ®rst time. One
major impediment to the study of two-phase displacements in realistic network models of porous media, and
investigating the e€ect of the correlations, has been the
very high computational costs associated with modelling
IP with ¯uid trapping. Fluid trapping occurs when the
defending ¯uid is incompressible, and portions of it are
surrounded by the invading ¯uid. This phenomenon
limited the previous studies to small 2D networks with
random heterogeneity ± networks too small to study the
e€ect of correlated heterogeneity.
In this paper, we describe experimental X-ray computed tomography (CT) measurements at the mm and
10 lm scales that indicate the presence of extended
correlated heterogeneity at the pore scale. We then use
an IP model to simulate rate-controlled mercury injection experiments in models of porous media with both
uncorrelated and correlated disorder to demonstrate
that introduction of the correlations has a marked e€ect
on the nature of the capillary pressure curve, which is an
important characteristic of any porous medium. Using

the IP model, we also show that it is possible to account
for the behaviour of the experimental data for ¯uid ¯ow
in sedimentary rocks only by including correlated heterogeneity. We also use the IP model to investigate the
e€ect of correlated heterogeneity on capillary-dominated displacements in porous media. In particular, it is
shown that the residual saturations, i.e., the ¯uids' saturations when they become disconnected, are strongly
sensitive to the degree of the correlations, and are substantially lower than those in random networks. The
correlations also have a strong e€ect on the distribution
of the trapped ¯uid's clusters.
The plan of our paper is as follows. In Section 2, we
review experimental data which indicate the existence of
correlated heterogeneity in rock samples. In Section 3,
we describe the simulation methods and computer generation of pore network models with correlated heterogeneity. In Section 4, we present the results of simulation
of a modi®ed IP for rate-controlled mercury injection
experiments in models of heterogeneous porous media
with uncorrelated and correlated disorder, and compare
the results with the experimental data. In Section 5, we
describe the e€ect of correlated heterogeneity on the
breakthrough and residual phase saturations. Similar to
random percolation, IP also leads to formation of ¯uid
clusters with fractal properties, and hence we present

and discuss the fractal properties of the IP clusters at
breakthrough, values of the residual saturations and the

size distribution of residual ¯uid residing in the trapped
clusters, all as functions of the extent of the correlated
heterogeneity. The Section 6 of the paper discusses the
implications of these results for interpreting multiphase
¯ow data for sedimentary rock.

2. Experimental evidence for correlated heterogeneity
Characterizing the pore space of complex porous
media requires the ability to examine the microstructure
of the pore space. Until very recently, direct measurements of the pore-space characteristics had been largely
restricted to the stereological study of thin sections
[7,35]. However, thin sectioning requires a considerable
amount of time to polish, slice, and digitize the sample.
Modern imaging techniques now allow scientists and
engineers to observe extremely complex material morphologies in 3D in a minimal amount of time. In particular, X-ray CT is a non-destructive technique for
visualising features in the interior of opaque solid objects and for resolving information on their 3D geometries. Conventional CT can be used to obtain the
porosity map of a piece of sedimentary rock at length

scales down to a millimetre [13]. High-resolution CT [51]
has made possible the measurement of geometric properties at length scales as small as a few microns.
We have obtained millimeter-scale CT images of
Berea sandstone in our laboratory [49] (see Fig. 1(a)).
Heterogeneity in the porosity distribution is evident
from visual inspection. A semivariogram analysis of the
porosity distribution reveals a spatial correlation in the
porosity distribution with a cuto€ length scale of about
3 mm. The variance in this porosity distribution can be
described by a fractional Brownian motion (fBm) [12]
with a cuto€ length `c and with a Hurst exponent
H ' 0:5 (see below for a description of a fBm). We have
also found [49] that more heterogeneous sandstones
exhibit correlated heterogeneity over a more extended
range [O(1 cm)] and stronger correlations with H ' 0:95
(Fig. 1(b)). Other data on carbonate rocks reveal spatial
correlations in porosity on the order of 5 mm [58]. Although the correlation length measured from these CT
images pertains to porosity (pore clustering) rather than
a direct measure of correlation in the pore size, we incorporate this correlation directly into our network
model. This assumption is consistent with the data recently obtained from micro-CT imaging.

Micro-X-ray CT image facilities can now provide
10243 voxel images of porous materials at a voxel resolution of less than 6 lm [51]. We have obtained a
512  512  666 image of a crossbedded sandstone at
10 lm resolution via micro-CT imaging. In Fig. 2, we
compare two consecutive series of six sections of the
crossbedded sandstone. The two series of images are
separated by less than 1 mm. One can see a large change

M.A. Knackstedt et al. / Advances in Water Resources 24 (2001) 257±277

259

Fig. 1. (a) Grey-scale image showing the porosity distribution in a sandstone at 1 mm pixel resolution. Note the clustering of the high and low
porosity is visually evident, implying correlations in porosity on the scale of several mm. (b) Semi-variogram analysis of the porosity distribution in a
heterogeneous limestone. Here correlations extend out to the cm scale ± the scale of the core plug.

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M.A. Knackstedt et al. / Advances in Water Resources 24 (2001) 257±277

Fig. 2. Comparison of two sets of six consecutive slices of a crossbedded sandstone at 10 lm spacing. In the ®rst set the porosity is less than 10%,
while the second set which is less than 1 mm away, the porosity is larger than 15%.

multiphase ¯ow in network models of porous media in
the presence of correlated heterogeneity.

3. Numerical simulation
The very high computational cost involved with network models has previously limited studies of multiphase ¯ow to networks too small for investigating the
e€ect of spatial correlation, especially if the extent of the
correlations is large. This limitation has now been relaxed by development by Sheppard et al. [50], of a highly
ecient algorithm for simulating IP, which we now
describe brie¯y.
Fig. 3. Variance in the porosity distribution showing power-law (fBm)
behaviour.

3.1. Simulation of invasion percolation

in the porosity of the material with such a small change
in depth ± pore sizes, throat sizes and other geometric
properties of the rock di€er signi®cantly despite the

images being only two grain diameters apart. We show
in Fig. 3 a trace of 660 values of the porosity measured
at a separation of 10 lm. A preliminary statistical
analysis of the data indicates that the description of
correlated heterogeneity used to describe rock properties
at the meter scale through borehole analysis [12,21,
23,29,31,32] may also describe the properties at the pore
scale. Direct measurement of pore and throat sizes, the
correlations between them, and also between neighbouring pore volumes have been made on the crossbedded sandstone and on four samples of Fontainbleau
sandstone [20,55]. The results indicate that there is a
strong correlation between the volume of a throat and
the average volume of nodal pores to which they are
connected.
Such direct measurements of pore-scale structure
bring into question the common assumption that rock
properties at the pore scale are randomly distributed
and that the concepts of random percolation (RP) can
be used for modelling ¯ow behaviour at these length
scales. The results also indicate the need to study

Consider the IP model in 3D. IP simulations begin by
assigning a random number to each site on the network
from an arbitrary distribution [57]. Initially, the network
is ®lled with the defending ¯uid and the invading ¯uid
occupies one face of the network. At each step of the
simulation the site with the largest value on the interface
between the invading and defending ¯uids is occupied
by the defender. Two main variants of IP have been
studied: In the ®rst, compressible IP, the defending ¯uid
is compressible and the invading ¯uid can potentially
invade any region on the interface occupied by the defending ¯uid. In the second, trapping IP (TIP), the defending ¯uid is incompressible and can be trapped when
a portion (cluster) of it is surrounded by the invading
¯uid.
The ¯uids' compressibility is, however, only one of
several factors that a€ect the evolution of the system as
the invading ¯uid advances in the porous medium. In
particular, one must also take into account the ability of
the ¯uids to wet the internal surface of the medium
[4,39]. The process by which a wetting ¯uid is drawn
spontaneously into a porous medium is called imbibition, while forcing of a nonwetting ¯uid into the pore
space is called drainage. We model the porous medium
as a network of pores or sites connected by throats or

M.A. Knackstedt et al. / Advances in Water Resources 24 (2001) 257±277

bonds which have smaller radii than the pores. In IP, the
potential displacement events are ranked according to
the capillary pressure threshold that must be exceeded
before a given event takes place. During imbibition, the
invading ¯uid is drawn ®rst into the smallest constrictions, for which the capillary pressure is large and negative, and it goes last into the widest pores.
Displacement events are therefore ranked in terms of the
largest opening that the invading ¯uid must travel
through, since it is from these larger capillaries that it is
most dicult to displace the defender. Imbibition is
therefore a site IP [4,39] and, in contrast, drainage in
which the invader has most diculty with the smallest
constrictions, is a bond IP.
The new IP algorithm [50] allows rapid simulation of
site and bond IP. In the conventional algorithms [39,57]
the search for the trapped regions is done after every
invasion event using a Hoshen±Kopelman [14] algorithm, which traverses the entire network, labels all
the connected regions, and then only those sites that are
connected to the outlet face are considered as potential
invasion sites. A second sweep of the network is then
done to determine which of the potential sites is to be
invaded in the next time step. Thus, each invasion event
demands O…N † calculations, where N is the number of
sites in the network, and hence an entire network demands O…N 2 † time. This is highly inecient for two
reasons. First, after each invasion event only a small
local change is made to the interface; implementing the
global Hoshen±Kopelman search is unnecessary. Secondly, it is wasteful to traverse the entire network at
each time step to ®nd the most favorable site (or bond)
on the interface since the interface is largely static.
The ®rst problem is tackled by searching the neighbours of each newly invaded site (bond) to check for
trapping. This is ruled out in almost all instances. If
trapping is possible, then several simultaneous breadth
®rst `forest-®re' searches are used to update the cluster
labelling as necessary. This restricts the changes to the
most local region possible. Since each site (bond) can be
invaded or trapped at most once during an invasion, this
part of the algorithm scales as O…N †. The second
problem is solved by storing the sites on the ¯uid±¯uid
interface in a list, sorted according to the capillary
pressure threshold (or the sites' sizes) needed to invade
them. This list is implemented using a balanced binary
search tree, so that insertion and deletion operations on
the list can be performed in O…log n† time, where n is the
list size. The sites that are designated as trapped using
the procedures described above are removed from the
invasion list. Each site (bond) is added and removed
from the interface list at most once, hence limiting the
computational e€ort of this part of the algorithm to
O…N log n†. Thus, the execution time for N sites is
dominated (for large N) by list manipulation and scales
at worst as O…N log N †. Since our method searches

261

cluster volumes rather than perimeters, and incorporates
local checking to minimize cluster searching, it is equally
e€ective in both 2D and 3D.
In addition to the new algorithm for simulating IP, a
new optimized algorithm [17,50] has been developed to
identify the minimal path length, the sites comprising
both the elastic backbone [11], i.e., the set of the sites
that lie on the union of all the shortest paths between
two widely separated points, and the usual transport
backbone, i.e., the multiply connected part of the sample-spanning cluster (SSC) that supports ¯ow and
transport in the network (the rest of the SSC is composed of dead-end sites or bonds), so that the backbone
search and computations do not a€ect the overall execution time of the algorithm. Complete details of the
algorithm, which can be used for arbitrary networks, are
given elsewhere [17,50].
3.2. Generation of correlated pore networks
As discussed above, heterogeneity in geological formations exists at all length scales. Such correlations are
often described by a fBm or a related stochastic process
that induces long-range correlations in the system. A
percolation model of ¯ow in porous media in which the
long-range correlations were generated by a fBm was
®rst proposed by Sahimi [41]. The motivation for his
model was provided by the work of Hewett [12] who
analyzed the permeability distributions and porosity
logs of heterogeneous rock formations at large length
scales (of the order of hundreds of meters), and showed
that the porosity logs in the direction perpendicular to
the bedding follow the statistics of fractional Gaussian
noise (fGn) which is, roughly speaking, the numerical
derivative of fBm, while those parallel to the bedding
follow fBm. In addition, there is convincing evidence
that the permeability distributions of many oil reservoirs [26,29,39,46] and aquifers [28] can be described by
fBm.
If the pore size distribution of a network of pores
contains long-range correlations that can be described by
a fBm, then the variance of the pore size is given by
h‰r…x† ÿ r…x0 †Š2 i ˆ C0 jx ÿ x0 j2H ;

…1†

where C0 is a constant, and x and x0 are two points in
the pore space. The type and extent of the correlations
can be tuned by varying the Hurst exponent H. For
H > 1=2 the correlations are positive, while H < 1=2
produces negative correlations in the increments of the
property values; H ˆ 1=2 corresponds to the random
case in which the increments in the property values are
uncorrelated. A fBm has not been used or tested for
representing the correlations at the pore scale, and experimental evidence, such as Fig. 1, indicates that at this
scale the correlated heterogeneity does not extend to the
entire pore space. We therefore introduce a cuto€ length

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M.A. Knackstedt et al. / Advances in Water Resources 24 (2001) 257±277

scale `c such that for jx ÿ x0 j < `c the correlations are
described by an fBm described by Eq. (1), while for
2
2H
jx ÿ x0 j > `c one has h‰r…x† ÿ r…x0 †Š i ˆ C0 j`c j . The
introduction of the cuto€ length scale `c allows us to
choose an appropriate length scale for correlations at
the pore scale.

4. Simulation of rate-controlled mercury injection experiments
To demonstrate the e€ect of correlated heterogeneity
at the pore scale we use a modi®ed IP model to simulate
rate-controlled mercury injection experiments in porous
materials displaying both correlated and uncorrelated
disorder and compare the results with experimental data
for sedimentary rocks. Rate-controlled mercury injection experiments provide far more information on the
statistical nature of pore structure than conventional
porosimetry [59]. Fluid intrusion under conditions of
constant-rate injection leads to a sequence of jumps in
the capillary pressure which are associated with regions
of low capillarity. While the envelope of the curve is the
classic pressure-controlled curve, the invasion into regions of low capillarity adds discrete jumps onto this
envelope. In the experiments of Yuan and Swanson [59],
mercury injection into a sample was done by a steppingmotor-driven positive displacement pump. This method

gives a volume-controlled measurement of the capillary
pressure Pc which is monitored as a dependent variable.
The particular sequence of alternate reversible and
spontaneous changes is determined by the structure of
the porous medium and the saturation history. An understanding of this relationship is essential to converting
Pc ¯uctuations into pore-structure information. In
Fig. 4(a), we show an example of a capillary pressure
curve obtained in our laboratory for Berea Sandstone
under rate-controlled conditions [49]. The detailed geometry of the jumps in the capillary pressure curve over
di€erent saturation ranges is shown in Figs. 4(b)±(d).
This process is naturally mapped onto the IP model.
Such a model of capillary pressure has previously been
used to model the constant-pressure curve alone [39,42].
We model constant-volume porosimetry in both random
and correlated networks. The conventional IP algorithm
requires minimal modi®cations to realistically mimic a
capillary pressure experiment. In the conventional IP
algorithm one considers invasion from one face of the
network, with the defending ¯uid exiting from the opposite face. In mercury porosimetry the geometry of the
displacement is di€erent. The core is placed in a cell and
the mercury completely surrounds the sample. To mimic
this process we allow the invader to enter the pore space
from all sides. The volume of a porous sample studied
by constant-volume porosimetry is of the order of 1 cm3
which, assuming a rock with a grain size of about

Fig. 4. Experimental constant-volume porosimetry curves for Berea sandstone. (a) Over large saturation range; (b)±(d) detailed curves over di€erent
saturation ranges.

M.A. Knackstedt et al. / Advances in Water Resources 24 (2001) 257±277

' 100 lm, implies a porous medium with up to 1 million
individual grains/pores. The simulations were therefore
performed on networks of comparable size (1283 ). The
statistical data were based on a minimum of 1000
realizations. When comparing the correlated and
uncorrelated systems, the pore throat distribution is the
same. Choosing the throat radii from the same distribution ensures that any di€erences in the simulated
curves are due solely to the presence of the correlations.
In Fig. 5, we show the e€ect of altering the
boundary condition on the constant-volume capillary
pressure curve for a correlated network. When injection comes from one side only, the capillary pressure
curve is often punctuated with extremely large drops at
small to intermediate pressures. The e€ect on the
conventional capillary pressure curve is even more
dramatic. These features are not observed in the experiments. When we modify the IP algorithm to allow
the correct condition of invasion from all sides, the
large drops in the pressure are no longer produced.
Fig. 6 shows the simulated rate-controlled capillary
pressure curves for correlated and uncorrelated systems. Qualitatively, the curves are distinctly di€erent.
The uncorrelated curves show a higher frequency of
jumps in capillary pressure and the jumps have a
consistent baseline over the whole saturation range. In
contrast, the porosimetry curve for the correlated networks exhibits a lower frequency of jumps, is characterised by a more gradual rise in the envelope of the
curve, and the baseline of the jumps in the capillary
pressure steadily increases with pressure. A comparison
between Fig. 6 and the experimental data of Fig. 4
shows that the correlated systems give a better qualitative match, while the uncorrelated case displays no
resemblance to the experimental data. This qualitative
agreement between the data and the simulated capillary
pressure curve points to the existence of correlated
heterogeneity in Berea sandstone.

263

To evaluate the appropriate length scale `c of the
correlations we consider a quantitative measure used
by Yuan and Swanson [59], and Swanson [53] to
characterise the porous rocks, which is the size distribution of regions of low capillarity over di€erent pressure ranges. The regions of low capillarity measured by
constant-volume porosimetry can range in volume from
1±1000n` ± from a single pore volume to hundreds of
pore volumes. At low saturations numerous jumps in the
capillary pressure curve of various sizes are noted. At
higher saturations the number of jumps into regions of
low capillarity are less frequent (compare Figs. 4(b)
and (d)), although large regions of low capillarity are
still invaded at high saturations; see Fig. 4(d). We have
measured the size distribution of low capillarity regions
in several Berea sandstone samples in our laboratory.
We use this measure to obtain a quantitative prediction
of the extent of the length scale `c of the correlated
heterogeneity. At lower saturations di€erences between
the predicted size distributions for varying `c are dicult
to discern. At higher saturations di€erences between the
models become more evident. In the uncorrelated case
(`c ˆ 1), for saturations above 60% no regions of low
capillarity are evident (see Fig. 6(a)). This disagrees with
the experimental data shown in Fig. 4. We plot the size
distribution of low-capillarity regions in Fig. 7 for
models with varying `c and compare them with the experimental data. It is clear from this ®gure that the best
®t to the experimental data is consistent with an `c of
about 10 or more pore lengths.
More direct evidence for the presence of correlation
at the pore scale comes from the experimental work of
Swanson [53]. He presented micrographs of the spatial
distribution of a nonwetting phase in a range of reservoir rocks including Berea sandstone, and showed that
appreciable portions of the rock are still not invaded by
the nonwetting phase at low to moderate nonwetting
phase saturations. A micrograph of Berea sandstone at

Fig. 5. E€ect of the modi®cation of the IP algorithm on a constant-volume and conventional porosimetry curve. In this case we consider identical
simple-cubic samples of size 1283 . (a) Invasion from one side; note the large downward jump in the capillary pressure due to the inlet e€ect. The
constant-pressure envelope is therefore very ¯at. (b) Invasion from all the six sides.

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M.A. Knackstedt et al. / Advances in Water Resources 24 (2001) 257±277

Fig. 6. Volume-controlled capillary pressure curves for uncorrelated [(a) and (c)] and correlated [(b) and (d)] fBm networks, `c ! 1. (a) and (b) give
the curves for the full saturation range, while (c) and (d) give those for a small range of saturation. The signature of the curves is distinct in both cases.
(a) and (c) give no resemblance to the data in Fig. 4.

22% saturation showed large unswept regions of more
than 2 mm in extent. Assuming a grain size of 100 lm,
uninvaded regions of this extent would contain thousands of pores. The experiments of Swanson showed,
however, that at saturations higher than 50% the extent
of the uninvaded regions is signi®cantly smaller than
observed at lower saturations.
We have visualised the distribution of the nonwetting
¯uid during drainage and found that the experimental
observations of Swanson can be accounted for if the
pore space is correlated with a cuto€ length `c of approximately 10 pores. We show in Figs. 8(a)±(c) the
results of simulation of a displacement in uncorrelated
and correlated networks at 25% saturation. The morphology of the displacement on the uncorrelated network spans much of the network and has invaded most
of the pore space. No large unswept regions are evident.
In the two correlated cases, however, large regions of
the pore space remain untouched by the invading ¯uid,
in agreement with the observations of Swanson. In
Figs. 8(d)±(f) the results of simulation of a displacement
in uncorrelated and correlated networks at 75% saturation is shown. In our simulation with the fBm network
with no cuto€ length scale (`c ! 1), Fig. 8(f), the re-

gions of the network uninvaded by the nonwetting ¯uid
remain large. The observations of Swanson are consistent with the simulation in both cases for a cuto€ of
about 10 pores; see Figs. 8(b) and (e).
Let us point out that, had we assigned e€ective sizes
to both sites and bonds of the network, i.e., a site-bond
IP [44], the minima on the Pc could have been considerably lower, as they would have represented interfaces
in the pores. Toledo et al. [54], did consider such a
possibility, and investigated rate-controlled mercury
injection in a network of pores and throats. In addition,
they also simulated the retraction process when the
pressure is decreased. However, they considered injection from only one face of the network, as a result of
which we cannot make a direct comparison between
their work and ours.

5. Implications of correlated heterogeneity for two-phase
¯ow in porous media
Having veri®ed experimentally that extended correlated heterogeneity exists at the pore scale even in
the most homogeneous sandstones, we now consider its

M.A. Knackstedt et al. / Advances in Water Resources 24 (2001) 257±277

265

Fig. 7. Size distribution of the low capillarity regions over the saturation range from 60±80%. N is the number of low capillarity jumps measured and
M is the size (number of pores) of the jumps.

e€ect on two-phase capillary-dominated displacement
processes. In this context we use the TIP model: the
defending ¯uid is incompressible and is trapped when
surrounded by the invading ¯uid. We consider the e€ect
of extended correlated heterogeneity on the breakthrough and the residual saturation, and also on the
con®gurations of the invaded regions and the size distribution of the trapped regions of the displaced ¯uid.
At the breakthrough threshold we consider the scaling
of the threshold with the linear size of the sample, the
shortest path and the backbone of the sample-spanning
cluster of the invading ¯uid for various correlation
lengths. The length of the minimal path, i.e., the length
of the minimum path between two sites (pores) on a
¯uid cluster separated by a Euclidean (straight line) path
of length r, is related to an important problem in
multiphase ¯ow, namely, the prediction of the time to
breakthrough of a ¯uid injected at one point and the
subsequent decay in the production of the defending
¯uid at the outlet [16]. The backbone, i.e., the multiply
connected part of the SSC, describes the conducting or

¯owing path through the rock and is directly relevant to
important macroscopic properties, such as the relative
permeability and formation resistivity. We describe the
e€ect of correlated heterogeneity on the residual
threshold values, along with the variability in their
measurement and the size distribution of the trapped
¯uid. The latter has important implications for tertiary
displacements in oil recovery in which a third ¯uid phase
is injected to further reduce the residual saturation.
We ®rst illustrate the e€ect of correlated heterogeneity on the structure of the ¯uid cluster at breakthrough. Fig. 9 shows examples of the clusters'
con®gurations in 2D site TIP at the breakthrough
threshold in an uncorrelated network and also for correlated networks for three values of the Hurst exponent
H. For the correlated networks we show two cases: one
for a cuto€ length in the correlations of `c ˆ 8 and a
second case for which `c ˆ 1, i.e., the extent of the
correlations is as large as the linear size L of the network. In the correlated case, the clusters have a more
compact structure, and as H increases their compactness

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Fig. 8. A number of slices through a 3D 643 simulation, illustrating the distribution of the nonwetting ¯uid in the network after 25% [(a)±(c)] and
75% [(d)±(f)] saturation for `c ˆ 1 (a and d), `c ˆ 8 (b and e), and `c ! 1 (c and f).

also increases. For H ˆ 0:9 the SSC and its backbone
are completely compact, with very small trapped clusters
in their interior. However, when a cuto€ length scale
`c < L is introduced in the network, the clusters' shapes
change drastically. While at length scale ` < `c the
clusters are still compact, for ` > `c they no longer have
a compact structure. Instead, they are fractal objects,
i.e., their mass M (the number of invaded sites in the
cluster) scales with the length scale ` as

the minimal path deviates only slightly from unity. The
same qualitative changes observed in 2D are also evident in 3D displacements. The numerical results for the
various fractal dimensions are discussed below.

M  `Df

SI ˆ ALDf ÿd ;

…2†

with fractal dimensions Df that are strictly less than 2,
the Euclidean dimension of the system. Interestingly,
although the existence of the cuto€ length scale thickens
the invading front, local trapping still occurs while the
displacing ¯uid is advancing. We also show in the same
®gures the minimal paths. For H > 1=2 the minimal
path is not unique: while one can ®x its length, one ®nds
many such paths with the same length, which is why the
set of all the minimal paths with a ®xed length is a thick
band (see Figs. 9(g) and (h)). For H ˆ 0:5 the SSC and
its backbone appear to have begun taking on a noncompact shape, with the sizes of the trapped clusters
becoming much larger than those for H ˆ 0:9 case. If we
introduce the cuto€ length scale `c ˆ 8, then the trapped
clusters become even larger, and for ` > `c the clusters
are again fractal structures. For H ˆ 0:2 the SSC and its
backbone are fractal objects, with or without the cuto€
length scale `c , although the fractal dimension Dmin of

5.1. Breakthrough saturation
In IP models, the saturation of the injected ¯uid at
breakthrough is described by
…3†

where A is a constant, d the Euclidean dimension, and
Df is the fractal dimension of the invading ¯uid's cluster.
The most accurate method of estimating a fractal dimension such as Df is based on studying the local fractal
dimension ([17,36,37,48,50]) and the approach to its
asymptotic value as M, the mass of the cluster, becomes
very large. For example, for the SSC the local fractal
dimension Df …M† is de®ned as
Df …M† 

d ln M
;
d ln Rg

…4†

where Rg is the radius of gyration of the cluster. A
similar equation holds for other fractal dimensions, such
as Db , the fractal dimension of the backbone. According
to ®nite-size scaling theory, Df …M† converges to its
asymptotic value for large M as
jDf ÿ Df …M†j  M ÿx ;

…5†

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267

Fig. 9. Typical cluster con®gurations for site TIP in 2D. The results are for, from top to bottom, random, H ˆ 0:2, 0.5 and 0.9. For the correlated
grids, the ®gures on the left show the results for a cuto€ length scale `c ˆ 8, while those on the right show the clusters for `c ˆ 1. The light
background gray is the sample-spanning cluster, the dark gray is its backbone, and the black area shows the minimal paths.

where x is a priori unknown correction-to-scaling exponent, and thus it must be estimated from the data.
Combining Eqs. (4) and (5) yields a di€erential equation
the solution of which is given by [17,50]
c1 ‡ Df M x ˆ c2 LxDf ;

…6†

where c1 and c2 are constants. We then ®t the data (for
the mass M versus the length scale L) to Eq. (6) to estimate both Df and x simultaneously. By following this
process we avoid the statistical pitfalls of the two-stage
process used by others [36,37,48] in which the data are

®rst divided into various bins, Df …M† are estimated by
numerical di€erentiation, and then x is varied until Eq.
(5) provides the `best' straight line ®t of the data when
Df …M† is plotted versus M ÿx . In addition, this method
enables us to obtain reliable estimates for the con®dence
intervals of the model parameters.
Values of Df for the SSC in site IP (SIP) and bond IP
(BIP) are identical and agree with the most accurate
estimates for random percolation (RP), see Table 1. In
BIP the invading ¯uid invades the most favorable bond
available at its interface with the defending ¯uid. Values

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M.A. Knackstedt et al. / Advances in Water Resources 24 (2001) 257±277

Table 1
The most accurate estimates of various fractal dimensions for IP in 2D
and 3D, and their comparison with those of RP [50]
Model

Dmin

Db

2D
NTIP
Site TIP
Bond TIP
RP

1:1293  0:0010
1:203  0:001
1:2170  0:0007
1:1307  0:0004

1:6422  0:0040
1:217  0:020
1:217  0:0008
1:6432  0:0008

3D
Site NTIP
Site TIP
Bond TIP
RP

1:3697  0:0005
1:3697  0:0005
1:458  0:008
1:374  0:004

1:868  0:010
1:861  0:005
1:458  0:008
1:87  0:03

of Df for uncorrelated systems are given in Table 1.
Results on correlated networks show that for H < 0:5
the fractal dimension Df of the (sample-spanning) invading ¯uid's cluster is dependent on H, such that it
increases with increasing H. For H > 0:5, however, the
cluster at breakthrough is compact (Df ˆ d), i.e., the
breakthrough saturation SI is a constant. For the case of
pore networks in which we introduce a cuto€ length
scale `c , we observe a crossover from the fractal behaviour associated with uncorrelated networks for
length scales `  `c to the H-dependent Df …H † for
` < `c . For example, for H < 0:5 and ` < `c , all the
fractal dimensions depend on H, while for `  `c they
are the same as those of the random IP.
5.2. Backbone, loopless backbone and minimal path
As mentioned above, an important property of a
percolation system is its backbone. While for RP the
backbone contains closed loops of pores and throats of
all sizes, it has been shown [44] that in bond TIP the
backbone is loopless and is in the form of a long strand,
while, similar to RP, the backbone of site TIP contains
closed loops of the invaded sites and bonds. Similar to
the backbone, important di€erences exist between the
structure of the minimal paths of RP and IP, and also
between those of correlated and random IP. In this
section we discuss these di€erences and point out their
implications for two-phase ¯ow in porous media.
5.2.1. Random site and bond invasion percolation
Unlike the fractal dimension of the SSC, which does
not depend on whether one is considering a drainage or
imbibition process (bond or site TIP), important di€erences arise in the structure of the transport pathways in
the two processes. Our simulations indicate that, as a
consequence of whether one considers bond or site TIP,
strong di€erences exist between the backbone and the
minimal path structures. For bond TIP the backbone
coincides with the minimal path [44], indicating that in
this case the minimal path is more tortuous than in the

other two cases. The di€erences are con®rmed quantitatively if we evaluate the fractal dimensions; see
Table 1. For site TIP the value of Dmin is in agreement
with that of RP, while for bond TIP the value of Dmin is
di€erent from that of RP. These results demonstrate
explicitly that the structure of the ¯ow and transport
paths, and hence their fractal dimensions, for bond TIP
are distinct from those of RP: While the SSC has a
fractal dimension Df consistent with RP, the fractal dimensions associated with its transport paths are not the
same as those of RP.
5.2.2. Correlated invasion percolation
Similar to the SSC in site TIP we ®nd that for
H > 1=2 the backbone of the SSC is compact (Db ˆ 3).
In Fig. 10 we show the dependence on H of the fractal
dimensions of the 3D SSC, the backbone and the minimal path for site TIP. These results indicate that in 3D
and for H < 1=2 the SSC and its backbone are fractal
with fractal dimensions that are nearly identical, and
that the minimal path is not fractal for any H, and hence
Dmin ˆ 1.
The results for bond TIP are very di€erent from those
for site TIP. Fig. 11 presents the con®gurations of the
SSC, its backbone, and the minimal paths for bond TIP
for the same values of the Hurst exponents H as those in
Fig. 9. It is clear that the con®gurations of the transport
paths of the SSC clusters in the two models are completely di€erent. In particular, the backbone of bond
TIP does not contain any closed loops and is in the form
of a long strand, which is in striking contrast with the
backbone of site TIP which is compact for H > 1=2 and
is a fractal object for H < 1=2. However, although the
backbone of bond TIP is loopless and looks like a long

Fig. 10. Dependence of the various fractal dimensions on H for TIP.
The results are for the site sample-spanning cluster (stars), site backbone (diamonds), backbone of bond TIP (squares), and site minimal
paths (triangles).

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269

Fig. 11. Cluster con®gurations showing the SSC and backbone/minimal path in black for bond (loopless) TIP in 2D for the same H values shown in
Fig. 9. The minimal path for site TIP is shown in grey in the ®gures to give a direct comparison.

strand, our analysis indicates that its fractal dimension
D`b is always greater than one for any value of H. Fig. 10
also shows the results for the fractal dimension D`b of
the backbone of bond TIP.
As discussed above, introducing a cuto€ length scale
`c causes a crossover from a value of the fractal dimension for length scales `  `c , that corresponds to
that of TIP without any correlations, to a compact
cluster for H > 0:5 or to a H ±dependent fractal dimension for H < 0:5 for ` < `c . This crossover has been
observed by Knackstedt et al., [17] in 2D simulations
where one could span over two orders of magnitude in
L. The same behaviour can be expected to be observed
for 3D systems.
5.3. Residual saturations
Values of the residual saturation Sr for site TIP have
been obtained for uncorrelated and correlated models
and are given in Fig. 12 for di€erent H and `c over a
range of L. The spread in the data is due to variation in
the pore size distribution for correlated networks and
not because of insucient numerical sampling. To understand the e€ect of ®nite size of the networks on
scaling of the residual saturations, the data for the uncorrelated networks were ®tted to the relationship
Sr …L† ˆ Sr …1† ‡ cLÿa

…7†

from which we found a ˆ 1=m ˆ 1:14  0:02, where m is
the critical exponent of correlation length np . This is in
good agreement with the critical exponent m for random
percolation, m ' 0:88. The ®nite-size scaling relationship
also allows one to predict the residual saturation for an
in®nite system; we obtain, Sr …L ! 1† ' 0:3402
0:0003. The ®nite-size scaling behaviour for the correlated networks was also evaluated at length scales up to
L ˆ 128, an example of which is shown in Fig. 13 for
H ˆ 0:8. The asymptotic values of Sr …L ! 1† are given
in Table 2 along with the corresponding values of the

scaling exponent a. Once again, these values depend
on H.
From the results for the residual saturations we make
the following observations. First, introduction of the
correlations leads to a large reduction in the observed
residual saturation. The value of the residual saturation
is smaller for large H and generally decreases with increasing `c . This is consistent with the structure of the
¯uid clusters and their dependence on H and `c , which
was discussed above. Recall that as H or `c increases, the
invading ¯uid's cluster becomes more compact, resulting
in better displacement and sweep of the defending ¯uid,
and hence reducing its residual saturation. However, the
residual saturation can exhibit a minimal value for ®nite
`c , beyond which it increases slightly. The small increase
of the residual saturation at larger cuto€ length scales
may be due to the possibility of trapping very large regions of the defending ¯uid at larger `c . Remarkably, the
reduction in Sr is signi®cant even for correlations at
small length scales. For example, for a network with
only a nearest-neighbour correlation, `c ˆ 2, and
H ˆ 0:8 the residual saturation drops from 0:34 to 0:26,
a reduction of over 20%. Small-scale correlations clearly
have a profound e€ect on resultant residual saturations
even at large scales.
5.4. Variability in measurements
Each realization of the IP gives a numerically di€erent result. It has been a common practice to make many
realizations and examine the results that are averaged
over all the realizations. However, laboratory core
measurements are necessarily performed on only a small
number of samples, so it is of interest to consider the
variability between realizations, which is of physical
signi®cance because it provides an indication of the
scatter which can be expected to occur in laboratory
measurements.

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M.A. Knackstedt et al. / Advances in Water Resources 24 (2001) 257±277

Fig. 12. Residual saturations for correlated networks for a range of `c . (a) H ˆ 0:2; (b) H ˆ 0:5; (c) H ˆ 0:8.

For RP, percolation theory predicts that the variance
in the threshold scales with the length scale L according
to
r…L† / Lÿb

…8†

where for RP, b ˆ 1=m, and m is critical exponent of the
percolation correlation length mentioned above. We

®nd that Eq. (8) holds for TIP in an uncorrelated network. In Table 3 we report the scaling exponent b of
Eq. (8) for the correlated networks with ®nite cuto€
length scales `c . Most values are close to but slightly
larger than 1=m ' 1:14 for RP.
As seen in Fig. 12 the standard deviation of the residual saturations for fully correlated networks (`c ˆ 1)

M.A. Knackstedt et al. / Advances in Water Resources 24 (2001) 257±277

271

Fig. 12 (continued).

Fig. 13. Finite size scaling of the residual saturation for a correlated network with H ˆ 0:5. The upper curve is for an uncorrelated network. Curves
for `c ˆ 2, `c ˆ 4, `c ˆ 8, `c ˆ 16, and `c ˆ 1 follow from upper left to bottom right. The values of the asymptotic saturations and the scaling
exponent a are given in Table 2.

is independent of L. In Fig. 14 we show individual realisations for the fully correlated networks, illustrating
the wide variation in the observed residuals even at large

L. The data also show the large skewness in the data to
higher values of the saturations. Clearly, for correlated
systems the distribution of the thresholds deviates

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M.A. Knackstedt et al. / Advances in Water Resources 24 (2001) 257±277

Table 2
Residual saturations Sr and variability for di€erent correlated networks with various `c a
H ˆ 0:2
`c ˆ 2
`c ˆ 4
`c ˆ 8
`c ˆ 16
fBm

H ˆ 0:5

H ˆ 0:8

Sr …1†

a

Sr …1†

a

Sr …1†

a

0:278  0:0003
0:257  0:0005
0:248  0:0008
0:245  0:002
0:250  0:03

0.85
0.62
0.50
0.32
0.22

0:271  0:0003
0:240  0:0006
0:225  0:0012
0:222  0:0035
0:223  0:065

0.93
0.65
0.55
0.36
0.18

0:262  0:0003
0:219  0:0007
0:128  0:001
0:158  0:008
0:180  0:094

0.86
0.78
0.58
0.25
0.08

a

Numerical predictions are given along with the value of exponent a in Eq. (7). For comparison, the value of Sr for a random network is 0.340 with
a ˆ 1:14.

Table 3
Exponent b (Eq. (8)) describing scaling of the standard deviation of the
residual saturation with linear network size L
`c
`c
`c
`c

ˆ2
ˆ4
ˆ8
ˆ 16

H ˆ 0:2

H ˆ 0:5

H ˆ 0:8

1.25
1.26
1.24
1.12

1.36
1.34
1.30
1.30

1.20
1.36
1.31
1.07

correlations `c . As Eq. (8) indicates, the variance of the
residual saturations decreases quickly with L since
b > 1. From this result we expect that variances in the
measured residuals for L=`c > 10 to be small, i.e.,
measurements of the residuals should be made on
sample sizes that are at least 10 times larger than the
extent of correlation.
5.5. Size distribution of the clusters of trapped ¯uids

strongly from a Gaussian [24], the expected distribution
for random systems. Moreover, the individual realizations show explicitly that measurements on a small
number of samples on a correlated pore network will
lead to poor estimation of the residual saturation. This
result highlights the need to experimentally measure on
core sizes L that are larger than the length scale of the

The cluster size distribution of the trapped ¯uid is
also of great interest in the study of immiscible displacement processes. We have studied the size distribution of the trapped ¯uid's clusters at the
residual saturation and found that the distribution is
s