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Pore network modelling of two-phase ¯ow in porous rock: the e�ectof 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, Australiab 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 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 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 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 arerelevant to many problems of great scienti®c and in-dustrial importance, ranging from extraction of oil, gasand geothermal energy from underground reservoirs, totransport of contaminants in soils and aquifers, and inkimbibition in a printing paper. Aside from the classicalcontinuum models of such phenomena (for reviews see,for example, [39,42]), discrete or pore network modelshave been used to represent disordered porous media,and detailed simulations have been carried out in orderto understand two- and three-phase ¯ow in such media.To interpret the simulations' results, the concepts ofpercolation theory (see, for example, [39,40,52]) havebeen employed to model slow ¯ow of ¯uids through thepore space. These models include both random bond orsite 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 disorderedporous media. In particular, IP, which was introduced

for describing the evolution of the interface between aninvading and a defending ¯uid in a porous medium, hasprovided deeper understanding of such phenomena.

In most previous applications of percolation theoryand pore network models to modelling of multiphase¯ow in porous media, correlations in the spatial disorderhave either been neglected, or have been assumed tohave a limited extent [1,5,15,33]. However, it has re-cently been suggested that long-range correlations arelikely 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 hasmotivated studies of percolation in pore networks withlong-range correlations [8,24,27,45,47]. Results of thesestudies indicated that correlations have a signi®cant ef-fect on many important characteristics of such systems.For example, one ®nds [24] that, with the correlationspresent, the percolation threshold can no longer be de-®ned uniquely but depends on the rule that de®nes whenand how a cluster is sample-spanning. However, thesepapers considered only the e�ect of correlations on thepercolation properties, and did not address the corre-sponding e�ects on ¯uid clusters' con®gurations andother important properties of multiphase ¯ow in porenetwork models of porous media. There have also been

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Advances in Water Resources 24 (2001) 257±277

* Corresponding author.

E-mail address: [email protected] (M. Sahimi).

0309-1708/01/$ - see front matter Ó 2001 Elsevier Science Ltd. All rights reserved.

PII: S 0 3 0 9 - 1 7 0 8 ( 0 0 ) 0 0 0 5 7 - 9

very limited studies of the e�ect of correlations on thecharacteristics of IP, the model which is perhaps mostappropriate for investigation of capillary-dominateddisplacements in porous media [25,34,56]. Even then,these studies were limited to two-dimensional (2D)networks, and considered only ¯uids' saturations up tothe breakthrough, i.e., the point at which the invading¯uid becomes sample-spanning for the ®rst time. Onemajor impediment to the study of two-phase displace-ments in realistic network models of porous media, andinvestigating the e�ect of the correlations, has been thevery high computational costs associated with modellingIP with ¯uid trapping. Fluid trapping occurs when thedefending ¯uid is incompressible, and portions of it aresurrounded by the invading ¯uid. This phenomenonlimited the previous studies to small 2D networks withrandom heterogeneity ± networks too small to study thee�ect of correlated heterogeneity.

In this paper, we describe experimental X-ray com-puted tomography (CT) measurements at the mm and10 lm scales that indicate the presence of extendedcorrelated heterogeneity at the pore scale. We then usean IP model to simulate rate-controlled mercury injec-tion experiments in models of porous media with bothuncorrelated and correlated disorder to demonstratethat introduction of the correlations has a marked e�ecton the nature of the capillary pressure curve, which is animportant characteristic of any porous medium. Usingthe IP model, we also show that it is possible to accountfor the behaviour of the experimental data for ¯uid ¯owin sedimentary rocks only by including correlated het-erogeneity. We also use the IP model to investigate thee�ect of correlated heterogeneity on capillary-domi-nated displacements in porous media. In particular, it isshown that the residual saturations, i.e., the ¯uids' sat-urations when they become disconnected, are stronglysensitive to the degree of the correlations, and are sub-stantially lower than those in random networks. Thecorrelations also have a strong e�ect on the distributionof the trapped ¯uid's clusters.

The plan of our paper is as follows. In Section 2, wereview experimental data which indicate the existence ofcorrelated 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 simulationof a modi®ed IP for rate-controlled mercury injectionexperiments in models of heterogeneous porous mediawith uncorrelated and correlated disorder, and comparethe results with the experimental data. In Section 5, wedescribe the e�ect of correlated heterogeneity on thebreakthrough and residual phase saturations. Similar torandom percolation, IP also leads to formation of ¯uidclusters with fractal properties, and hence we presentand discuss the fractal properties of the IP clusters atbreakthrough, values of the residual saturations and the

size distribution of residual ¯uid residing in the trappedclusters, all as functions of the extent of the correlatedheterogeneity. The Section 6 of the paper discusses theimplications of these results for interpreting multiphase¯ow data for sedimentary rock.

2. Experimental evidence for correlated heterogeneity

Characterizing the pore space of complex porousmedia requires the ability to examine the microstructureof the pore space. Until very recently, direct measure-ments of the pore-space characteristics had been largelyrestricted to the stereological study of thin sections[7,35]. However, thin sectioning requires a considerableamount of time to polish, slice, and digitize the sample.Modern imaging techniques now allow scientists andengineers 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 forvisualising 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 theporosity map of a piece of sedimentary rock at lengthscales 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 ofBerea sandstone in our laboratory [49] (see Fig. 1(a)).Heterogeneity in the porosity distribution is evidentfrom visual inspection. A semivariogram analysis of theporosity distribution reveals a spatial correlation in theporosity distribution with a cuto� length scale of about3 mm. The variance in this porosity distribution can bedescribed by a fractional Brownian motion (fBm) [12]with a cuto� length `c and with a Hurst exponentH ' 0:5 (see below for a description of a fBm). We havealso found [49] that more heterogeneous sandstonesexhibit correlated heterogeneity over a more extendedrange [O(1 cm)] and stronger correlations with H ' 0:95(Fig. 1(b)). Other data on carbonate rocks reveal spatialcorrelations in porosity on the order of 5 mm [58]. Al-though the correlation length measured from these CTimages pertains to porosity (pore clustering) rather thana direct measure of correlation in the pore size, we in-corporate this correlation directly into our networkmodel. This assumption is consistent with the data re-cently obtained from micro-CT imaging.

Micro-X-ray CT image facilities can now provide10243 voxel images of porous materials at a voxel res-olution of less than 6 lm [51]. We have obtained a512� 512� 666 image of a crossbedded sandstone at10 lm resolution via micro-CT imaging. In Fig. 2, wecompare two consecutive series of six sections of thecrossbedded sandstone. The two series of images areseparated by less than 1 mm. One can see a large change

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

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.

M.A. Knackstedt et al. / Advances in Water Resources 24 (2001) 257±277 259

in the porosity of the material with such a small changein depth ± pore sizes, throat sizes and other geometricproperties of the rock di�er signi®cantly despite theimages being only two grain diameters apart. We showin Fig. 3 a trace of 660 values of the porosity measuredat a separation of 10 lm. A preliminary statisticalanalysis of the data indicates that the description ofcorrelated heterogeneity used to describe rock propertiesat the meter scale through borehole analysis [12,21,23,29,31,32] may also describe the properties at the porescale. Direct measurement of pore and throat sizes, thecorrelations between them, and also between neigh-bouring pore volumes have been made on the cross-bedded sandstone and on four samples of Fontainbleausandstone [20,55]. The results indicate that there is astrong correlation between the volume of a throat andthe average volume of nodal pores to which they areconnected.

Such direct measurements of pore-scale structurebring into question the common assumption that rockproperties at the pore scale are randomly distributedand that the concepts of random percolation (RP) canbe used for modelling ¯ow behaviour at these lengthscales. The results also indicate the need to study

multiphase ¯ow in network models of porous media inthe 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 thee�ect of spatial correlation, especially if the extent of thecorrelations is large. This limitation has now been re-laxed by development by Sheppard et al. [50], of a highlye�cient algorithm for simulating IP, which we nowdescribe brie¯y.

3.1. Simulation of invasion percolation

Consider the IP model in 3D. IP simulations begin byassigning a random number to each site on the networkfrom an arbitrary distribution [57]. Initially, the networkis ®lled with the defending ¯uid and the invading ¯uidoccupies one face of the network. At each step of thesimulation the site with the largest value on the interfacebetween the invading and defending ¯uids is occupiedby the defender. Two main variants of IP have beenstudied: In the ®rst, compressible IP, the defending ¯uidis compressible and the invading ¯uid can potentiallyinvade any region on the interface occupied by the de-fending ¯uid. In the second, trapping IP (TIP), the de-fending ¯uid is incompressible and can be trapped whena portion (cluster) of it is surrounded by the invading¯uid.

The ¯uids' compressibility is, however, only one ofseveral factors that a�ect the evolution of the system asthe invading ¯uid advances in the porous medium. Inparticular, one must also take into account the ability ofthe ¯uids to wet the internal surface of the medium[4,39]. The process by which a wetting ¯uid is drawnspontaneously into a porous medium is called imbi-bition, while forcing of a nonwetting ¯uid into the porespace is called drainage. We model the porous mediumas 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 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%.

Fig. 3. Variance in the porosity distribution showing power-law (fBm)

behaviour.


bonds which have smaller radii than the pores. In IP, thepotential displacement events are ranked according tothe capillary pressure threshold that must be exceededbefore a given event takes place. During imbibition, theinvading ¯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 thelargest opening that the invading ¯uid must travelthrough, since it is from these larger capillaries that it ismost di�cult to displace the defender. Imbibition istherefore a site IP [4,39] and, in contrast, drainage inwhich the invader has most di�culty with the smallestconstrictions, is a bond IP.

The new IP algorithm [50] allows rapid simulation ofsite and bond IP. In the conventional algorithms [39,57]the search for the trapped regions is done after everyinvasion event using a Hoshen±Kopelman [14] al-gorithm, which traverses the entire network, labels allthe connected regions, and then only those sites that areconnected to the outlet face are considered as potentialinvasion sites. A second sweep of the network is thendone to determine which of the potential sites is to beinvaded in the next time step. Thus, each invasion eventdemands O�N� calculations, where N is the number ofsites in the network, and hence an entire network de-mands O�N 2� time. This is highly ine�cient for tworeasons. First, after each invasion event only a smalllocal change is made to the interface; implementing theglobal Hoshen±Kopelman search is unnecessary. Sec-ondly, it is wasteful to traverse the entire network ateach 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 fortrapping. This is ruled out in almost all instances. Iftrapping is possible, then several simultaneous breadth®rst `forest-®re' searches are used to update the clusterlabelling as necessary. This restricts the changes to themost local region possible. Since each site (bond) can beinvaded or trapped at most once during an invasion, thispart of the algorithm scales as O�N�. The secondproblem is solved by storing the sites on the ¯uid±¯uidinterface in a list, sorted according to the capillarypressure threshold (or the sites' sizes) needed to invadethem. This list is implemented using a balanced binarysearch tree, so that insertion and deletion operations onthe list can be performed in O�log n� time, where n is thelist size. The sites that are designated as trapped usingthe procedures described above are removed from theinvasion list. Each site (bond) is added and removedfrom the interface list at most once, hence limiting thecomputational e�ort of this part of the algorithm toO�N log n�. Thus, the execution time for N sites isdominated (for large N) by list manipulation and scalesat worst as O�N log N�. Since our method searches

cluster volumes rather than perimeters, and incorporateslocal checking to minimize cluster searching, it is equallye�ective in both 2D and 3D.

In addition to the new algorithm for simulating IP, anew optimized algorithm [17,50] has been developed toidentify the minimal path length, the sites comprisingboth the elastic backbone [11], i.e., the set of the sitesthat lie on the union of all the shortest paths betweentwo widely separated points, and the usual transportbackbone, i.e., the multiply connected part of the sam-ple-spanning cluster (SSC) that supports ¯ow andtransport in the network (the rest of the SSC is com-posed of dead-end sites or bonds), so that the backbonesearch and computations do not a�ect the overall exe-cution time of the algorithm. Complete details of thealgorithm, which can be used for arbitrary networks, aregiven 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 areoften described by a fBm or a related stochastic processthat induces long-range correlations in the system. Apercolation model of ¯ow in porous media in which thelong-range correlations were generated by a fBm was®rst proposed by Sahimi [41]. The motivation for hismodel was provided by the work of Hewett [12] whoanalyzed the permeability distributions and porositylogs of heterogeneous rock formations at large lengthscales (of the order of hundreds of meters), and showedthat the porosity logs in the direction perpendicular tothe bedding follow the statistics of fractional Gaussiannoise (fGn) which is, roughly speaking, the numericalderivative of fBm, while those parallel to the beddingfollow fBm. In addition, there is convincing evidencethat the permeability distributions of many oil reser-voirs [26,29,39,46] and aquifers [28] can be described byfBm.

If the pore size distribution of a network of porescontains long-range correlations that can be described bya fBm, then the variance of the pore size is given by

h�r�x� ÿ r�x0��2i � C0jxÿ x0j2H ; �1�where C0 is a constant, and x and x0 are two points inthe pore space. The type and extent of the correlationscan be tuned by varying the Hurst exponent H. ForH > 1=2 the correlations are positive, while H < 1=2produces negative correlations in the increments of theproperty values; H � 1=2 corresponds to the randomcase in which the increments in the property values areuncorrelated. A fBm has not been used or tested forrepresenting the correlations at the pore scale, and ex-perimental evidence, such as Fig. 1, indicates that at thisscale the correlated heterogeneity does not extend to theentire pore space. We therefore introduce a cuto� length


scale `c such that for jxÿ x0j < `c the correlations aredescribed by an fBm described by Eq. (1), while forjxÿ x0j > `c one has h�r�x� ÿ r�x0��2i � C0j`cj2H

. Theintroduction of the cuto� length scale `c allows us tochoose an appropriate length scale for correlations atthe pore scale.

4. Simulation of rate-controlled mercury injection exper-

iments

To demonstrate the e�ect of correlated heterogeneityat the pore scale we use a modi®ed IP model to simulaterate-controlled mercury injection experiments in porousmaterials displaying both correlated and uncorrelateddisorder and compare the results with experimental datafor sedimentary rocks. Rate-controlled mercury injec-tion experiments provide far more information on thestatistical nature of pore structure than conventionalporosimetry [59]. Fluid intrusion under conditions ofconstant-rate injection leads to a sequence of jumps inthe capillary pressure which are associated with regionsof low capillarity. While the envelope of the curve is theclassic pressure-controlled curve, the invasion into re-gions of low capillarity adds discrete jumps onto thisenvelope. 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 capillarypressure Pc which is monitored as a dependent variable.The particular sequence of alternate reversible andspontaneous changes is determined by the structure ofthe porous medium and the saturation history. An un-derstanding of this relationship is essential to convertingPc ¯uctuations into pore-structure information. InFig. 4(a), we show an example of a capillary pressurecurve obtained in our laboratory for Berea Sandstoneunder rate-controlled conditions [49]. The detailed ge-ometry of the jumps in the capillary pressure curve overdi�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 beenused to model the constant-pressure curve alone [39,42].We model constant-volume porosimetry in both randomand correlated networks. The conventional IP algorithmrequires minimal modi®cations to realistically mimic acapillary pressure experiment. In the conventional IPalgorithm one considers invasion from one face of thenetwork, with the defending ¯uid exiting from the op-posite face. In mercury porosimetry the geometry of thedisplacement is di�erent. The core is placed in a cell andthe mercury completely surrounds the sample. To mimicthis process we allow the invader to enter the pore spacefrom all sides. The volume of a porous sample studiedby 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.


' 100 lm, implies a porous medium with up to 1 millionindividual grains/pores. The simulations were thereforeperformed on networks of comparable size (1283). Thestatistical data were based on a minimum of 1000realizations. When comparing the correlated anduncorrelated systems, the pore throat distribution is thesame. Choosing the throat radii from the same distri-bution ensures that any di�erences in the simulatedcurves are due solely to the presence of the correlations.

In Fig. 5, we show the e�ect of altering theboundary condition on the constant-volume capillarypressure curve for a correlated network. When injec-tion comes from one side only, the capillary pressurecurve is often punctuated with extremely large drops atsmall to intermediate pressures. The e�ect on theconventional capillary pressure curve is even moredramatic. These features are not observed in the ex-periments. When we modify the IP algorithm to allowthe correct condition of invasion from all sides, thelarge drops in the pressure are no longer produced.Fig. 6 shows the simulated rate-controlled capillarypressure curves for correlated and uncorrelated sys-tems. Qualitatively, the curves are distinctly di�erent.The uncorrelated curves show a higher frequency ofjumps in capillary pressure and the jumps have aconsistent baseline over the whole saturation range. Incontrast, 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 thecurve, and the baseline of the jumps in the capillarypressure steadily increases with pressure. A comparisonbetween Fig. 6 and the experimental data of Fig. 4shows that the correlated systems give a better quali-tative match, while the uncorrelated case displays noresemblance to the experimental data. This qualitativeagreement between the data and the simulated capillarypressure curve points to the existence of correlatedheterogeneity in Berea sandstone.

To evaluate the appropriate length scale `c of thecorrelations we consider a quantitative measure usedby Yuan and Swanson [59], and Swanson [53] tocharacterise the porous rocks, which is the size distri-bution of regions of low capillarity over di�erent pres-sure ranges. The regions of low capillarity measured byconstant-volume porosimetry can range in volume from1±1000n` ± from a single pore volume to hundreds ofpore volumes. At low saturations numerous jumps in thecapillary pressure curve of various sizes are noted. Athigher saturations the number of jumps into regions oflow capillarity are less frequent (compare Figs. 4(b)and (d)), although large regions of low capillarity arestill invaded at high saturations; see Fig. 4(d). We havemeasured the size distribution of low capillarity regionsin several Berea sandstone samples in our laboratory.We use this measure to obtain a quantitative predictionof the extent of the length scale `c of the correlatedheterogeneity. At lower saturations di�erences betweenthe predicted size distributions for varying `c are di�cultto discern. At higher saturations di�erences between themodels become more evident. In the uncorrelated case(`c � 1), for saturations above 60% no regions of lowcapillarity are evident (see Fig. 6(a)). This disagrees withthe experimental data shown in Fig. 4. We plot the sizedistribution of low-capillarity regions in Fig. 7 formodels 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 ofabout 10 or more pore lengths.

More direct evidence for the presence of correlationat the pore scale comes from the experimental work ofSwanson [53]. He presented micrographs of the spatialdistribution of a nonwetting phase in a range of reser-voir rocks including Berea sandstone, and showed thatappreciable portions of the rock are still not invaded bythe nonwetting phase at low to moderate nonwettingphase 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.


22% saturation showed large unswept regions of morethan 2 mm in extent. Assuming a grain size of 100 lm,uninvaded regions of this extent would contain thou-sands of pores. The experiments of Swanson showed,however, that at saturations higher than 50% the extentof the uninvaded regions is signi®cantly smaller thanobserved at lower saturations.

We have visualised the distribution of the nonwetting¯uid during drainage and found that the experimentalobservations of Swanson can be accounted for if thepore space is correlated with a cuto� length `c of ap-proximately 10 pores. We show in Figs. 8(a)±(c) theresults of simulation of a displacement in uncorrelatedand correlated networks at 25% saturation. The mor-phology of the displacement on the uncorrelated net-work spans much of the network and has invaded mostof the pore space. No large unswept regions are evident.In the two correlated cases, however, large regions ofthe pore space remain untouched by the invading ¯uid,in agreement with the observations of Swanson. InFigs. 8(d)±(f) the results of simulation of a displacementin uncorrelated and correlated networks at 75% satu-ration is shown. In our simulation with the fBm networkwith no cuto� length scale (`c !1), Fig. 8(f), the re-

gions of the network uninvaded by the nonwetting ¯uidremain large. The observations of Swanson are consis-tent with the simulation in both cases for a cuto� ofabout 10 pores; see Figs. 8(b) and (e).

Let us point out that, had we assigned e�ective sizesto both sites and bonds of the network, i.e., a site-bondIP [44], the minima on the Pc could have been consid-erably lower, as they would have represented interfacesin the pores. Toledo et al. [54], did consider such apossibility, and investigated rate-controlled mercuryinjection in a network of pores and throats. In addition,they also simulated the retraction process when thepressure is decreased. However, they considered injec-tion from only one face of the network, as a result ofwhich we cannot make a direct comparison betweentheir 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 inthe most homogeneous sandstones, we now consider its

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.


e�ect on two-phase capillary-dominated displacementprocesses. In this context we use the TIP model: thedefending ¯uid is incompressible and is trapped whensurrounded by the invading ¯uid. We consider the e�ectof extended correlated heterogeneity on the break-through and the residual saturation, and also on thecon®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 scalingof the threshold with the linear size of the sample, theshortest path and the backbone of the sample-spanningcluster of the invading ¯uid for various correlationlengths. The length of the minimal path, i.e., the lengthof the minimum path between two sites (pores) on a¯uid cluster separated by a Euclidean (straight line) pathof length r, is related to an important problem inmultiphase ¯ow, namely, the prediction of the time tobreakthrough of a ¯uid injected at one point and thesubsequent decay in the production of the defending¯uid at the outlet [16]. The backbone, i.e., the multiplyconnected part of the SSC, describes the conducting or

¯owing path through the rock and is directly relevant toimportant macroscopic properties, such as the relativepermeability and formation resistivity. We describe thee�ect of correlated heterogeneity on the residualthreshold values, along with the variability in theirmeasurement and the size distribution of the trapped¯uid. The latter has important implications for tertiarydisplacements in oil recovery in which a third ¯uid phaseis injected to further reduce the residual saturation.

We ®rst illustrate the e�ect 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 breakthroughthreshold in an uncorrelated network and also for cor-related networks for three values of the Hurst exponentH. For the correlated networks we show two cases: onefor a cuto� length in the correlations of `c � 8 and asecond case for which `c � 1, i.e., the extent of thecorrelations is as large as the linear size L of the net-work. In the correlated case, the clusters have a morecompact structure, and as H increases their compactness

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.


also increases. For H � 0:9 the SSC and its backboneare completely compact, with very small trapped clustersin their interior. However, when a cuto� length scale`c < L is introduced in the network, the clusters' shapeschange drastically. While at length scale ` < `c theclusters are still compact, for ` > `c they no longer havea compact structure. Instead, they are fractal objects,i.e., their mass M (the number of invaded sites in thecluster) 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 thickensthe invading front, local trapping still occurs while thedisplacing ¯uid is advancing. We also show in the same®gures the minimal paths. For H > 1=2 the minimalpath is not unique: while one can ®x its length, one ®ndsmany such paths with the same length, which is why theset of all the minimal paths with a ®xed length is a thickband (see Figs. 9(g) and (h)). For H � 0:5 the SSC andits backbone appear to have begun taking on a non-compact shape, with the sizes of the trapped clustersbecoming much larger than those for H � 0:9 case. If weintroduce the cuto� length scale `c � 8, then the trappedclusters become even larger, and for ` > `c the clustersare again fractal structures. For H � 0:2 the SSC and itsbackbone 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. Thesame qualitative changes observed in 2D are also evi-dent in 3D displacements. The numerical results for thevarious fractal dimensions are discussed below.

5.1. Breakthrough saturation

In IP models, the saturation of the injected ¯uid atbreakthrough is described by

SI � ALDfÿd ; �3�where A is a constant, d the Euclidean dimension, andDf is the fractal dimension of the invading ¯uid's cluster.The most accurate method of estimating a fractal di-mension such as Df is based on studying the local fractaldimension ([17,36,37,48,50]) and the approach to itsasymptotic value as M, the mass of the cluster, becomesvery large. For example, for the SSC the local fractaldimension Df�M� is de®ned as

Df�M� � d ln Md ln Rg

; �4�

where Rg is the radius of gyration of the cluster. Asimilar equation holds for other fractal dimensions, suchas Db, the fractal dimension of the backbone. Accordingto ®nite-size scaling theory, Df�M� converges to itsasymptotic value for large M as

jDf ÿ Df�M�j � Mÿx; �5�

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 � 1 (a and d), `c � 8 (b and e), and `c !1 (c and f).


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 di�erential equationthe solution of which is given by [17,50]

c1 � DfMx � c2LxDf ; �6�where c1 and c2 are constants. We then ®t the data (forthe mass M versus the length scale L) to Eq. (6) to es-timate both Df and x simultaneously. By following thisprocess we avoid the statistical pitfalls of the two-stageprocess used by others [36,37,48] in which the data are

®rst divided into various bins, Df�M� are estimated bynumerical di�erentiation, and then x is varied until Eq.(5) provides the `best' straight line ®t of the data whenDf�M� is plotted versus Mÿx. In addition, this methodenables us to obtain reliable estimates for the con®denceintervals 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 accurateestimates for random percolation (RP), see Table 1. InBIP the invading ¯uid invades the most favorable bondavailable at its interface with the defending ¯uid. Values

Fig. 9. Typical cluster con®gurations for site TIP in 2D. The results are for, from top to bottom, random, H � 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 � 8, while those on the right show the clusters for `c � 1. The light

background gray is the sample-spanning cluster, the dark gray is its backbone, and the black area shows the minimal paths.


of Df for uncorrelated systems are given in Table 1.Results on correlated networks show that for H < 0:5the fractal dimension Df of the (sample-spanning) in-vading ¯uid's cluster is dependent on H, such that itincreases with increasing H. For H > 0:5, however, thecluster at breakthrough is compact (Df � d), i.e., thebreakthrough saturation SI is a constant. For the case ofpore networks in which we introduce a cuto� lengthscale `c, we observe a crossover from the fractal be-haviour associated with uncorrelated networks forlength scales `� `c to the H-dependent Df�H� for` < `c. For example, for H < 0:5 and ` < `c, all thefractal dimensions depend on H, while for `� `c theyare the same as those of the random IP.

5.2. Backbone, loopless backbone and minimal path

As mentioned above, an important property of apercolation system is its backbone. While for RP thebackbone contains closed loops of pores and throats ofall sizes, it has been shown [44] that in bond TIP thebackbone is loopless and is in the form of a long strand,while, similar to RP, the backbone of site TIP containsclosed loops of the invaded sites and bonds. Similar tothe backbone, important di�erences exist between thestructure of the minimal paths of RP and IP, and alsobetween those of correlated and random IP. In thissection we discuss these di�erences and point out theirimplications for two-phase ¯ow in porous media.

5.2.1. Random site and bond invasion percolationUnlike the fractal dimension of the SSC, which does

not depend on whether one is considering a drainage orimbibition process (bond or site TIP), important di�er-ences arise in the structure of the transport pathways inthe two processes. Our simulations indicate that, as aconsequence of whether one considers bond or site TIP,strong di�erences exist between the backbone and theminimal path structures. For bond TIP the backbonecoincides with the minimal path [44], indicating that inthis case the minimal path is more tortuous than in the

other two cases. The di�erences are con®rmed quanti-tatively if we evaluate the fractal dimensions; seeTable 1. For site TIP the value of Dmin is in agreementwith that of RP, while for bond TIP the value of Dmin isdi�erent from that of RP. These results demonstrateexplicitly that the structure of the ¯ow and transportpaths, and hence their fractal dimensions, for bond TIPare distinct from those of RP: While the SSC has afractal dimension Df consistent with RP, the fractal di-mensions associated with its transport paths are not thesame as those of RP.

5.2.2. Correlated invasion percolationSimilar to the SSC in site TIP we ®nd that for

H > 1=2 the backbone of the SSC is compact (Db � 3).In Fig. 10 we show the dependence on H of the fractaldimensions of the 3D SSC, the backbone and the min-imal path for site TIP. These results indicate that in 3Dand for H < 1=2 the SSC and its backbone are fractalwith fractal dimensions that are nearly identical, andthat the minimal path is not fractal for any H, and henceDmin � 1.

The results for bond TIP are very di�erent from thosefor site TIP. Fig. 11 presents the con®gurations of theSSC, its backbone, and the minimal paths for bond TIPfor the same values of the Hurst exponents H as those inFig. 9. It is clear that the con®gurations of the transportpaths of the SSC clusters in the two models are com-pletely di�erent. In particular, the backbone of bondTIP does not contain any closed loops and is in the formof a long strand, which is in striking contrast with thebackbone of site TIP which is compact for H > 1=2 andis a fractal object for H < 1=2. However, although thebackbone 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

2D

NTIP 1:1293� 0:0010 1:6422� 0:0040

Site TIP 1:203� 0:001 1:217� 0:020

Bond TIP 1:2170� 0:0007 1:217� 0:0008

RP 1:1307� 0:0004 1:6432� 0:0008

3D

Site NTIP 1:3697� 0:0005 1:868� 0:010

Site TIP 1:3697� 0:0005 1:861� 0:005

Bond TIP 1:458� 0:008 1:458� 0:008

RP 1:374� 0:004 1:87� 0:03

Fig. 10. Dependence of the various fractal dimensions on H for TIP.

The results are for the site sample-spanning cluster (stars), site back-

bone (diamonds), backbone of bond TIP (squares), and site minimal

paths (triangles).


strand, our analysis indicates that its fractal dimensionD`b is always greater than one for any value of H. Fig. 10also shows the results for the fractal dimension D`b ofthe 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 tothat of TIP without any correlations, to a compactcluster for H > 0:5 or to a H±dependent fractal dimen-sion for H < 0:5 for ` < `c. This crossover has beenobserved by Knackstedt et al., [17] in 2D simulationswhere one could span over two orders of magnitude inL. The same behaviour can be expected to be observedfor 3D systems.

5.3. Residual saturations

Values of the residual saturation Sr for site TIP havebeen obtained for uncorrelated and correlated modelsand are given in Fig. 12 for di�erent H and `c over arange of L. The spread in the data is due to variation inthe pore size distribution for correlated networks andnot because of insu�cient numerical sampling. To un-derstand the e�ect of ®nite size of the networks onscaling 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 found a � 1=m � 1:14� 0:02, where m isthe critical exponent of correlation length np. This is ingood agreement with the critical exponent m for randompercolation, m ' 0:88. The ®nite-size scaling relationshipalso allows one to predict the residual saturation for anin®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 toL � 128, an example of which is shown in Fig. 13 forH � 0:8. The asymptotic values of Sr�L!1� are givenin Table 2 along with the corresponding values of the

scaling exponent a. Once again, these values dependon H.

From the results for the residual saturations we makethe following observations. First, introduction of thecorrelations leads to a large reduction in the observedresidual saturation. The value of the residual saturationis smaller for large H and generally decreases with in-creasing `c. This is consistent with the structure of the¯uid clusters and their dependence on H and `c, whichwas discussed above. Recall that as H or `c increases, theinvading ¯uid's cluster becomes more compact, resultingin better displacement and sweep of the defending ¯uid,and hence reducing its residual saturation. However, theresidual saturation can exhibit a minimal value for ®nite`c, beyond which it increases slightly. The small increaseof the residual saturation at larger cuto� length scalesmay be due to the possibility of trapping very large re-gions of the defending ¯uid at larger `c. Remarkably, thereduction in Sr is signi®cant even for correlations atsmall length scales. For example, for a network withonly a nearest-neighbour correlation, `c � 2, andH � 0:8 the residual saturation drops from 0:34 to 0:26,a reduction of over 20%. Small-scale correlations clearlyhave a profound e�ect on resultant residual saturationseven at large scales.

5.4. Variability in measurements

Each realization of the IP gives a numerically di�er-ent result. It has been a common practice to make manyrealizations and examine the results that are averagedover all the realizations. However, laboratory coremeasurements are necessarily performed on only a smallnumber of samples, so it is of interest to consider thevariability between realizations, which is of physicalsigni®cance because it provides an indication of thescatter which can be expected to occur in laboratorymeasurements.

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.


For RP, percolation theory predicts that the variancein the threshold scales with the length scale L accordingto

r�L� / Lÿb �8�where for RP, b � 1=m, and m is critical exponent of thepercolation 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 ofEq. (8) for the correlated networks with ®nite cuto�length scales `c. Most values are close to but slightlylarger 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)

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


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

L. The data also show the large skewness in the data tohigher values of the saturations. Clearly, for correlatedsystems the distribution of the thresholds deviates

Fig. 12 (continued).

Fig. 13. Finite size scaling of the residual saturation for a correlated network with H � 0:5. The upper curve is for an uncorrelated network. Curves

for `c � 2, `c � 4, `c � 8, `c � 16, and `c � 1 follow from upper left to bottom right. The values of the asymptotic saturations and the scaling

exponent a are given in Table 2.


strongly from a Gaussian [24], the expected distributionfor random systems. Moreover, the individual realiza-tions show explicitly that measurements on a smallnumber of samples on a correlated pore network willlead to poor estimation of the residual saturation. Thisresult highlights the need to experimentally measure oncore sizes L that are larger than the length scale of the

correlations `c. As Eq. (8) indicates, the variance of theresidual saturations decreases quickly with L sinceb > 1. From this result we expect that variances in themeasured residuals for L=`c > 10 to be small, i.e.,measurements of the residuals should be made onsample sizes that are at least 10 times larger than theextent of correlation.

5.5. Size distribution of the clusters of trapped ¯uids

The cluster size distribution of the trapped ¯uid isalso 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 theresidual saturation and found that the distribution isstrongly a�ected by the type and extent of correlation.

Table 2

Residual saturations Sr and variability for di�erent correlated networks with various `ca

H � 0:2 H � 0:5 H � 0:8

Sr�1� a Sr�1� a Sr�1� a

`c � 2 0:278� 0:0003 0.85 0:271� 0:0003 0.93 0:262� 0:0003 0.86

`c � 4 0:257� 0:0005 0.62 0:240� 0:0006 0.65 0:219� 0:0007 0.78

`c � 8 0:248� 0:0008 0.50 0:225� 0:0012 0.55 0:128� 0:001 0.58

`c � 16 0:245� 0:002 0.32 0:222� 0:0035 0.36 0:158� 0:008 0.25

fBm 0:250� 0:03 0.22 0:223� 0:065 0.18 0:180� 0:094 0.08a 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 � 1:14.

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 correlated networks exhibit a wide variation in the observed residuals even at large L. The data is also skewed

showing the poor ®t to a Gaussian distribution.

Table 3

Exponent b (Eq. (8)) describing scaling of the standard deviation of the

residual saturation with linear network size L

H � 0:2 H � 0:5 H � 0:8

`c � 2 1.25 1.36 1.20

`c � 4 1.26 1.34 1.36

`c � 8 1.24 1.30 1.31

`c � 16 1.12 1.30 1.07


In Fig. 15 we show the number of trapped clusters ofsize s for a correlated network for di�erent 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 thenumber of invaded sites. From percolation theory oneexpects ns�s� to follow the following scaling law:

Fig. 15. (a) Size distribution of the trapped clusters for H � 0:5 as a function of `c. The y-axis is the log of the cumulative cluster size distribution

log N�s� versus log s, where Ns�s� �P

s>s0 sns, the average total number of clusters with a size greater than a given size s0. In the uncorrelated case and

also for ®nite cuto� length scales `c we see that the scaling predicted for RP holds. For the case of in®nite correlations, the scaling behaviour di�ers

strongly. Note that for the fBm ®eld one trapped cluster has a huge proportion of the trapped ¯uid with more than half of it being within a few

trapped clusters. However, the proportion of small trapped clusters is also large. (b) Same as (a), but for H � 0:8. (c) Same as (a), but for H � 0:2.


ns�s� / sÿs; �9�where s ' 2:18 for RP. A more accurate way ofmeasuring the cluster size statistics is by investigating

Ns�s� �P

s0>s s0ns0 , the average total number of clusterswith a size s0 greater than a given size s. In general oneexpects to have

Fig. 15 (continued).

Fig. 16. Proportion of the trapped ¯uid as a function of total invading ¯uid saturation. In the random system all the trapping occurs at the end of the

invasion. In contrast, for ®nite `c much of the defending ¯uid is trapped at earlier stages of the invasion.


Ns�s� / s2ÿs: �10�

If there are no long-range correlations in the system,percolation theory predicts that the exponent s isuniversal. Since the 3D spanning cluster for TIP withno long-range correlations has the same fractaldimension as that of RP, and because s � d=Df � 1,we expect the uncorrelated networks and the thosewith a ®nite `c to show this scaling behaviour. This isseen in Fig. 15. For pore networks with long-rangecorrelations Df is nonuniversal and depends on H. Wetherefore obtain nonuniversal values of s that dependon H for `c !1.

We can make some qualitative observations on thesize of the trapped clusters as a function of `c. Forsmall `c there is little e�ect on the distribution of thetrapped ¯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 Hone single trapped cluster dominates over 30% of theresidual phase. The dynamics of the trapping is alsostrongly dependent on the presence of correlatedheterogeneity. We present in Fig. 16 the total propor-tion of the trapped clusters as a function of thesaturation. We see very di�erent dynamics when com-paring random and correlated networks. In the randomsystems, trapping occurs only 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. Incontrast, when we introduce, for example, a cuto�`c � 16, over 30% of the defending phase is trapped at80% invader saturation.

6. Discusssion

Our experimental results suggest that correlated het-erogeneity exists down to the pore scale even in a rocklike Berea sandstone, which is generally considered to behomogeneous and to exhibit no signi®cant correlationsin its pore size distribution. Moreover, direct compari-son of experiments on Berea sandstone and simulationof porosimetry on correlated pore networks providecompelling evidence that correlations do persist beyondone or two pore lengths, and are quite extended. Most ofthe previous applications of percolation theory and porenetwork simulations to two-phase displacements in po-rous media have assumed that the spatial disorder isuncorrelated. Our results suggest that the use of randompercolation concepts to derive the pore size distributionfrom mercury porosimetry, without considering suchextended spatial correlations, may neglect an essentialaspect of the physics of sedimentary rocks and henceyield misleading results.

We further illustrated this e�ect by consideringthe e�ect of correlated heterogeneity on capillary-

dominated two-phase ¯ow properties. Small scalecorrelations have a profound e�ect on the structure ofthe ¯uid clusters at breakthrough and at the residualsaturation. Introduction of correlated heterogeneityleads to lower residual phase saturations than thoseobserved in random pore networks, and has a stronge�ect on the resultant distribution of clusters of thetrapped ¯uid.

These results highlight the need to incorporate re-alistic descriptions of pore-scale heterogeneity in thescale up of the residual saturation measurements. Thevery high computational e�ort involved with TIP sim-ulations had limited previous studies to small networkswith random distributions of heterogeneities ± networkstoo small to study the e�ect of long-range correlations.Having developed a highly e�cient simulator for TIP,we can now examine scaling behavior of all the prop-erties of interest, from the pore to meter scales. Futurework will consider the scale-up behavior for rocks. Suchstudies have the potential to in¯uence the manner inwhich the oil industry carries out laboratory measure-ments and the procedures used to relate these mea-surements to the log and reservoir scales. Even modestimprovements in our understanding of these areaswould signi®cantly reduce the risk associated with newoil and gas developments and groundwater remediationstrategies.

Acknowledgements

We would like to thank our collaborators on thisproject: Val Pinczewski, Tim Senden and Rob Sok.Micro-CT imaging was performed by Richard Ketchamat the University of Texas High-Resolution CT Facility.Work at USC was supported in part by the PetroleumResearch Fund, administered by the American Chemi-cal Society. MAK thanks the Australian ResearchCouncil for support and the ANU Supercomputer Fa-cility and the High Performance Computing Unit at theUniversity of Queenland for generous allocations ofcomputer time.

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