Pore network modelling of two-phase ¯ow in porous rock: the eect
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 eect of correlated heter-ogeneity on two-phase ¯ow through porous media. First, we review and discuss the experimental evidence for correlated hetero-geneity. We then employ the invasion percolation model of two-phase ¯ow in porous media to study the eect 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 dataonlyif the heterogeneity of the pore space is correlated. In addition, we also describe a highly ecient algorithm for simulation of two-phase ¯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 in-dustrial 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 perco-lation (IP) [6,57], and have provided considerable in-sight into the physics of multiphase ¯ow in disordered porous media. In particular, IP, which was introduced
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 re-cently 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 ef-fect 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 eect of correlations on the percolation properties, and did not address the corre-sponding eects 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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*
Corresponding author.
E-mail address:moe@iran.usc.edu (M. Sahimi).
very limited studies of the eect 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 displace-ments in realistic network models of porous media, and investigating the eect 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 eect of correlated heterogeneity.
In this paper, we describe experimental X-ray com-puted 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 injec-tion experiments in models of porous media with both uncorrelated and correlated disorder to demonstrate that introduction of the correlations has a marked eect 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 het-erogeneity. We also use the IP model to investigate the eect of correlated heterogeneity on capillary-domi-nated displacements in porous media. In particular, it is shown that the residual saturations, i.e., the ¯uids' sat-urations when they become disconnected, are strongly sensitive to the degree of the correlations, and are sub-stantially lower than those in random networks. The correlations also have a strong eect 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 gen-eration of pore network models with correlated hetero-geneity. 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 eect 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 measure-ments 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 mor-phologies in 3D in a minimal amount of time. In par-ticular, X-ray CT is a non-destructive technique for visualising features in the interior of opaque solid ob-jects and for resolving information on their 3D geome-tries. 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 prop-erties 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 withH '0:95 (Fig. 1(b)). Other data on carbonate rocks reveal spatial correlations in porosity on the order of 5 mm [58]. Al-though 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 in-corporate this correlation directly into our network model. This assumption is consistent with the data re-cently obtained from micro-CT imaging.
Micro-X-ray CT image facilities can now provide 10243 voxel images of porous materials at a voxel
res-olution of less than 6 lm [51]. We have obtained a
512512666 image of a crossbedded sandstone at 10 lm resolution via micro-CT imaging. In Fig. 2, we
in the porosity of the material with such a small change in depth ± pore sizes, throat sizes and other geometric properties of the rock dier 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 neigh-bouring pore volumes have been made on the cross-bedded 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
multiphase ¯ow in network models of porous media in the presence of correlated heterogeneity.
3. Numerical simulation
The very high computational cost involved with net-work models has previously limited studies of multi-phase ¯ow to networks too small for investigating the eect of spatial correlation, especially if the extent of the correlations is large. This limitation has now been re-laxed by development by Sheppard et al. [50], of a highly ecient algorithm for simulating IP, which we now describe brie¯y.
3.1. Simulation of invasion percolation
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 fending ¯uid. In the second, trapping IP (TIP), the de-fending ¯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 aect 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 imbi-bition, 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
Fig. 2. Comparison of two sets of six consecutive slices of a crossbedded sandstone at 10lm 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%.
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 constric-tions, for which the capillary pressure is large and neg-ative, 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 dicult to displace the defender. Imbibition is therefore a site IP [4,39] and, in contrast, drainage in which the invader has most diculty with the smallest constrictions, is abondIP.
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] al-gorithm, 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 Nis the number of sites in the network, and hence an entire network de-mands O N2 time. This is highly inecient 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. Sec-ondly, 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 neigh-bours 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 logntime, wherenis 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 eort of this part of the algorithm to O Nlogn. Thus, the execution time for N sites is dominated (for largeN) by list manipulation and scales at worst as O NlogN. Since our method searches
cluster volumes rather than perimeters, and incorporates local checking to minimize cluster searching, it is equally eective 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 sam-ple-spanning cluster (SSC) that supports ¯ow and transport in the network (the rest of the SSC is com-posed of dead-end sites or bonds), so that the backbone search and computations do not aect the overall exe-cution 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 for-mations 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 reser-voirs [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
hr x ÿr x02i C0jxÿx0j2H; 1
where C0is a constant, andx andx0are two points in
the pore space. The type and extent of the correlations can be tuned by varying the Hurst exponent H. For
scale `c such that forjxÿx0j< `c the correlations are
described by an fBm described by Eq. (1), while for
jxÿx0j> `c one has hr x ÿr x0 2
i C0j`cj 2H
. 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 exper-iments
To demonstrate the eect 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 injec-tion experiments provide far more informainjec-tion 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 pressucontrolled curve, the invasion into re-gions 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 stepping-motor-driven positive displacement pump. This method
gives a volume-controlled measurement of the capillary pressurePc 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 un-derstanding 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 ge-ometry of the jumps in the capillary pressure curve over dierent 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-pressurecurve alone [39,42]. We modelconstant-volumeporosimetry 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 op-posite face. In mercury porosimetry the geometry of the displacement is dierent. 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 allsides. 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
'100lm, 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 distri-bution ensures that any dierences in the simulated curves are due solely to the presence of the correlations. In Fig. 5, we show the eect of altering the boundary condition on the constant-volume capillary pressure curve for a correlated network. When injec-tion comes from one side only, the capillary pressure curve is often punctuated with extremely large drops at small to intermediate pressures. The eect on the conventional capillary pressure curve is even more dramatic. These features are not observed in the ex-periments. 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 sys-tems. Qualitatively, the curves are distinctly dierent. 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 net-works exhibits a lower frequency of jumps, is charac-terised 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 quali-tative 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.
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 distri-bution of regions of low capillarity over dierent pres-sure ranges. The regions of low capillarity meapres-sured 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 dierences between the predicted size distributions for varying`care dicult
to discern. At higher saturations dierences between the models become more evident. In the uncorrelated case (`c 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
ex-perimental 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 reser-voir 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
22% saturation showed large unswept regions of more than 2 mm in extent. Assuming a grain size of 100lm,
uninvaded regions of this extent would contain thou-sands 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
ap-proximately 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 mor-phology of the displacement on the uncorrelated net-work spans much of the netnet-work 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% satu-ration 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 consis-tent 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 eective sizes toboth sites and bonds of the network, i.e., a site-bond IP [44], the minima on the Pc could have been
consid-erably 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 injec-tion 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 corre-lated heterogeneity exists at the pore scale even in the most homogeneous sandstones, we now consider its
eect 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 eect of extended correlated heterogeneity on the break-through and the residual saturation, and also on the con®gurations of the invaded regions and the size dis-tribution 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 eect 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 eect of correlated hetero-geneity on the structure of the ¯uid cluster at break-through. Fig. 9 shows examples of the clusters' con®gurations in 2D site TIP at the breakthrough threshold in an uncorrelated network and also for cor-related 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 `c8 and a
second case for which `c 1, i.e., the extent of the
correlations is as large as the linear size L of the net-work. In the correlated case, the clusters have a more compact structure, and asHincreases their compactness
also increases. For H0:9 the SSC and its backbone are completely compact, with very small trapped clusters in their interior. However, when a cuto length scale
`c<Lis introduced in the network, the clusters' shapes
change drastically. While at length scale ` < `c the
clusters are still compact, for` > `cthey 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
M `Df 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)). ForH0:5 the SSC and its backbone appear to have begun taking on a non-compact shape, with the sizes of the trapped clusters becoming much larger than those forH0:9 case. If we introduce the cuto length scale`c8, then the trapped
clusters become even larger, and for ` > `c the clusters
are again fractal structures. ForH0:2 the SSC and its backbone are fractal objects, with or without the cuto length scale `c, although the fractal dimension Dmin of
the minimal path deviates only slightly from unity. The same qualitative changes observed in 2D are also evi-dent in 3D displacements. The numerical results for the various fractal dimensions are discussed below.
5.1. Breakthrough saturation
In IP models, the saturation of the injected ¯uid at breakthrough is described by
SI ALDfÿd; 3
where Ais 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 di-mension such asDfis based on studying thelocalfractal
dimension ([17,36,37,48,50]) and the approach to its asymptotic value asM, the mass of the cluster, becomes very large. For example, for the SSC the local fractal dimensionDf Mis de®ned as
Df M dlnM dlnRg
; 4
where Rg is the radius of gyration of the cluster. A
similar equation holds for other fractal dimensions, such asDb, the fractal dimension of the backbone. According
to ®nite-size scaling theory, Df M converges to its
asymptotic value for large Mas
jDfÿDf Mj Mÿ
x
; 5
where x is a priori unknown correction-to-scaling ex-ponent, and thus it must be estimated from the data. Combining Eqs. (4) and (5) yields a dierential equation the solution of which is given by [17,50]
c1DfMxc2LxDf; 6
wherec1 andc2 are constants. We then ®t the data (for
the mass Mversus the length scaleL) to Eq. (6) to es-timate bothDf 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 dierentiation, and thenxis varied until Eq. (5) provides the `best' straight line ®t of the data when
Df Mis plotted versus Mÿx. In addition, this method
enables us to obtain reliable estimates for the con®dence intervals of the model parameters.
Values ofDf 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
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)
in-vading ¯uid's cluster is dependent on H, such that it increases with increasing H. ForH >0:5, however, the cluster at breakthrough is compact (Df d), i.e., the
breakthrough saturationSIis a constant. For the case of
pore networks in which we introduce a cuto length scale `c, we observe a crossover from the fractal
be-haviour 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 dierences 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 dierences 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 dier-ences 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 dierences exist between the backbone and the minimal path structures. For bond TIP the backbone
coincideswith the minimal path [44], indicating that in this case the minimal path is more tortuous than in the
other two cases. The dierences are con®rmed quanti-tatively 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 ofDminis
dierent 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
di-mensions 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 (Db3).
In Fig. 10 we show the dependence on Hof the fractal dimensions of the 3D SSC, the backbone and the min-imal 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 isnotfractal forany H, and hence
Dmin1.
The results for bond TIP are very dierent 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 exponentsHas those in Fig. 9. It is clear that the con®gurations of the transport paths of the SSC clusters in the two models are com-pletely dierent. 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 forH >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
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
strand, our analysis indicates that its fractal dimension
D`bis always greater than one foranyvalue ofH. 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
di-mension for length scales ``c, that corresponds to
that of TIP without any correlations, to a compact cluster forH>0:5 or to aH±dependent fractal dimen-sion 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 Srfor site TIP have
been obtained for uncorrelated and correlated models and are given in Fig. 12 for dierent H and `c over a
range ofL. The spread in the data is due to variation in the pore size distribution for correlated networks and not because of insucient numerical sampling. To un-derstand the eect of ®nite size of the networks on scaling of the residual saturations, the data for the un-correlated networks were ®tted to the relationship
Sr L Sr 1 cLÿa 7
from which we founda1=m1:140:02, wheremis the critical exponent of correlation lengthnp. This is in good agreement with the critical exponentmfor 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 corre-lated networks was also evaluated at length scales up to
L128, an example of which is shown in Fig. 13 for
H0:8. The asymptotic values ofSr L! 1are 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 in-creasing `c. This is consistent with the structure of the
¯uid clusters and their dependence onH and `c, which
was discussed above. Recall that asHor`cincreases, 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 re-gions 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, `c2, and H 0:8 the residual saturation drops from 0:34 to 0:26, a reduction of over 20%. Small-scale correlations clearly have a profound eect on resultant residual saturations even at large scales.
5.4. Variability in measurements
Each realization of the IP gives a numerically dier-ent 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.
For RP, percolation theory predicts that the variance in the threshold scales with the length scaleLaccording to
r L /Lÿb 8
where for RP,b1=m, andmis critical exponent of the percolation correlation length mentioned above. We
®nd that Eq. (8) holds for TIP in an uncorrelated net-work. 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 re-sidual saturations for fully correlated networks (`c 1)
is independent ofL. In Fig. 14 we show individual re-alisations 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
Fig. 12 (continued).
strongly from a Gaussian [24], the expected distribution for random systems. Moreover, the individual realiza-tions 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 sizesL that are larger than the length scale of the
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
The cluster size distribution of the trapped ¯uid is also of great interest in the study of immiscible dis-placement processes. We have studied the size distribu-tion of the trapped ¯uid's clusters at the residual saturation and found that the distribution is strongly aected by the type and extent of correlation.
Table 2
Residual saturationsSrand variability for dierent correlated networks with various`ca
H0:2 H0:5 H0:8
Sr 1 a Sr 1 a Sr 1 a
`c2 0:2780:0003 0.85 0:2710:0003 0.93 0:2620:0003 0.86
`c4 0:2570:0005 0.62 0:2400:0006 0.65 0:2190:0007 0.78
`c8 0:2480:0008 0.50 0:2250:0012 0.55 0:1280:001 0.58
`c16 0:2450:002 0.32 0:2220:0035 0.36 0:1580:008 0.25
fBm 0:2500:03 0.22 0:2230:065 0.18 0:1800:094 0.08 aNumerical predictions are given along with the value of exponentain Eq. (7). For comparison, the value ofS
rfor a random network is 0.340 with
a1:14.
Fig. 14. Individual variability in the measured residual phase saturation for a network with long-range correlations. A comparison to an uncor-related network is shown. The fully coruncor-related networks exhibit a wide variation in the observed residuals even at largeL. The data is also skewed showing the poor ®t to a Gaussian distribution.
Table 3
Exponentb(Eq. (8)) describing scaling of the standard deviation of the residual saturation with linear network sizeL
H0:2 H0:5 H0:8
`c2 1.25 1.36 1.20
`c4 1.26 1.34 1.36
`c8 1.24 1.30 1.31
In Fig. 15 we show the number of trapped clusters of sizes for a correlated network for dierent H and dif-ferent values of the cuto length`c. We plot ns s, the
number of clusters of size s, where s is simply the number of invaded sites. From percolation theory one expects ns sto follow the following scaling law:
Fig. 15. (a) Size distribution of the trapped clusters forH0:5 as a function of`c. They-axis is the log of the cumulative cluster size distribution logN sversus logs, whereNs s
P
ns s /sÿs; 9
where s'2:18 for RP. A more accurate way of measuring the cluster size statistics is by investigating
Ns s Ps0>ss0ns0, the average total number of clusters
with a size s0greater than a given sizes. In general one expects to have
Fig. 15 (continued).
Ns s /s2ÿs: 10
If there are no long-range correlations in the system, percolation theory predicts that the exponent s is universal. Since the 3D spanning cluster for TIP with no long-range correlations has the same fractal dimension as that of RP, and because sd=Df1,
we expect the uncorrelated networks and the those with a ®nite `c to show this scaling behaviour. This is
seen in Fig. 15. For pore networks with long-range correlationsDf is nonuniversal and depends on H. We
therefore obtain nonuniversal values of s that depend on H for `c! 1.
We can make some qualitative observations on the size of the trapped clusters as a function of `c. For
small `c there is little eect on the distribution of the
trapped ¯uid. At intermediate and larger values of `c
we observe a higher proportion of trapped sites (pores) lie in larger trapped clusters. For `c! 1 and large H
one single trapped cluster dominates over 30% of the residual phase. The dynamics of the trapping is also strongly dependent on the presence of correlated heterogeneity. We present in Fig. 16 the total propor-tion of the trapped clusters as a funcpropor-tion of the saturation. We see very dierent dynamics when com-paring random and correlated networks. In the random systems, trapping occursonly near the end of the ¯ood ± when over 80% of the total invading ¯uid is present, less than 5% of the defending ¯uid is trapped. In contrast, when we introduce, for example, a cuto
`c16, over 30% of the defending phase is trapped at
80% invader saturation.
6. Discusssion
Our experimental results suggest that correlated het-erogeneity exists down to the pore scale even in a rock like Berea sandstone, which is generally considered to be homogeneous and to exhibit no signi®cant correlations in its pore size distribution. Moreover, direct compari-son of experiments on Berea sandstone and simulation of porosimetry on correlated pore networks provide compelling evidence that correlationsdopersist beyond one or two pore lengths, and are quite extended. Most of the previous applications of percolation theory and pore network simulations to two-phase displacements in po-rous media have assumed that the spatial disorder is uncorrelated. Our results suggest that the use of random percolation concepts to derive the pore size distribution from mercury porosimetry, without considering such extended spatial correlations, may neglect an essential aspect of the physics of sedimentary rocks and hence yield misleading results.
We further illustrated this eect by considering the eect of correlated heterogeneity on
capillary-dominated two-phase ¯ow properties. Small scale correlations have a profound eect on the structure of the ¯uid clusters at breakthrough and at the residual saturation. Introduction of correlated heterogeneity leads to lower residual phase saturations than those observed in random pore networks, and has a strong eect on the resultant distribution of clusters of the trapped ¯uid.
These results highlight the need to incorporate re-alistic descriptions of pore-scale heterogeneity in the scale up of the residual saturation measurements. The very high computational eort involved with TIP sim-ulations had limited previous studies to small networks with random distributions of heterogeneities ± networks too small to study the eect of long-range correlations. Having developed a highly ecient simulator for TIP, we can now examine scaling behavior of all the prop-erties of interest, from the pore to meter scales. Future work will consider the scale-up behavior for rocks. Such studies have the potential to in¯uence the manner in which the oil industry carries out laboratory measure-ments and the procedures used to relate these mea-surements to the log and reservoir scales. Even modest improvements in our understanding of these areas would signi®cantly reduce the risk associated with new oil and gas developments and groundwater remediation strategies.
Acknowledgements
We would like to thank our collaborators on this project: Val Pinczewski, Tim Senden and Rob Sok. Micro-CT imaging was performed by Richard Ketcham at the University of Texas High-Resolution CT Facility. Work at USC was supported in part by the Petroleum Research Fund, administered by the American Chemi-cal Society. MAK thanks the Australian Research Council for support and the ANU Supercomputer Fa-cility and the High Performance Computing Unit at the University of Queenland for generous allocations of computer time.
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