PHYSICAL REVIEW E FEBRUARY 1997VOLUME 55, NUMBER 2

Invasion percolation on self-affine topographies

G. Wagner, P. Meakin, J. Feder, and T. Jo”ssangDepartment of Physics, University of Oslo, Box 1048, Blindern, 0316 Oslo 3, Norway

~Received 12 August 1996!

Invasion percolation~IP! with trapping was studied on two-dimensional substrates with a correlated distri-bution of invasion thresholds. The correlations were induced by using the heights of~211!-dimensionalself-affine rough surfaces with Hurst exponents in the range 0,H,1 to assign the threshold values. Theresulting IP clusters consist of ‘‘blobs’’ with sizes up to the entire cluster size that are connected by fine‘‘threads.’’ The fractal dimensionalityDH of the IP clusters is dominated by the blobs. The blob size distri-bution is is related toH andDH . @S1063-651X~97!11401-5#

PACS number~s!: 47.55.Mh, 47.60.1i, 05.40.1j

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I. INTRODUCTION

The invasion percolation~IP! algorithm @1–3# is remark-ably successful in describing the slow immiscible displament of a wetting fluid by a nonwetting fluid in a poroumedia@4–10#. In slow displacements, viscous forces canneglected, and the process is governed by the capilforces. In equilibrium, the pressure of the nonwetting flumust exceed the pressure of the wetting fluid by an amopc , the capillary pressure, to sustain the curvature ofinterface. When a nonwetting fluid is injected into a pofilled with wetting fluid, the capillary pressure must be ovecome. For a circular throat of radiusR and for the interfacialtensionG acting between the two fluids,pc522Gcosu/R,whereu is the contact angle. The contact angle denotesangle at which the interface between the two phases mthe solid surface of the matrix. Thus the nonwetting flupreferentially invades the pores with the largest throats.

The displacement process can be mapped on the IP min a straightforward manner. In site IP, the porous mediumrepresented by a lattice of sites. Each sitei is assigned arandom numberr i , and represents a pore with the diame1/r i . Initially, all sites are occupied by the wetting ‘‘defender’’ fluid. An injection site is chosen and filled with thnonwetting ‘‘invader’’ fluid. The algorithm consists of repeating the following three steps:~1! Identify all defendersites that are adjacent to the invaded sites.~2! Among theseperimeter sites, find the one with the lowest numberr i . ~3!Fill this site with an invader fluid. The process is terminatwhen the edge of the lattice is reached by the cluster formby the invader fluid.

In two dimensions, the incompressibility of the two fluidmust be taken into account by using a ‘‘trapping rule’’@3#.When the invader fluid has surrounded a region of defenfluid, the defender fluid cannot be displaced, and is trappGrowth of the IP cluster must take place at unoccupiedrimeter sites that are not trapped, i.e., a path consistingsteps between nearest-neighbor unoccupied sites to theside of the cluster must exist. The trapping rule appearchange the universality class of the model.

In standard IP, the thresholds are distributed uniformlythe unit interval without any spatial correlation in their vaues, representing a homogeneous porous medium. Howthe pore sizes in geological fields clearly are strongly co

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lated on large scales, and there are indications that corrtions also exist on scales down to the pore scale@11#. Moti-vated by this observation, the IP model with and withotrapping was studied using a multifractal distributionthresholds@12,13#. In these studies the fractal dimensionaliof the IP cluster was reported to be little or not at all affectby the spatial correlations. In contrast, changes in thecluster growth pattern were observed in simulations ussite-bond lattices in which correlations were induced byconstraint that a bond size be less than or equal to the sizthe smaller site to which it was connected@14#.

In the present work, IP with trapping in two dimensionwas studied using substrates with a different kind of corlated disorder. For each sitei at the position (xi ,yi) in atwo-dimensional lattice ofL3L sites, the threshold valueassigned to the site was given by thez value of a roughsurfacez(xi ,yi) with the same extensionL3L in the x-yplane. The substrate obtained in this way may be usedmodel the slow displacement of a wetting fluid in a fractuby a nonwetting fluid. The three-dimensional fracture is reresented by the two-dimensional lattice such that each lasite corresponds to a region of the fracture plane, andthresholdr i assigned to the site corresponds to the aperof the fracture.

For a perfectly nonwetting fluid (u5180°) to advanceand displace a wetting fluid in a fracture region of infiniextension and of aperturea, the capillary pressure

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must be overcome. The nonwetting fluid tends to invafracture regions with wide apertures, and does not displthe wetting fluid from narrow aperture regions. In the quastatic case, the displacement process is governed entirelthe fracture geometry.

Field measurements of natural rock surfaces indicatfractal character@15–18#. Fresh brittle fractures of differentypes of rock were shown to generate self-affine rough sfaces @19#. An isotropic self-affine surfacez(x,y) remainsstatistically invariant to the scaling transformationx→lx,y→ly, andz→lHz. The roughness exponent or Hurst eponentH lies in the range 0,H,1. ForH50.5, a ‘‘verti-cal’’ cross section of the surface has the same statist

1698 © 1997 The American Physical Society

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55 1699INVASION PERCOLATION ON SELF-AFFINE TOPOGRAPHIES

properties as a Brownian process. ForH.0.5, the cross section is persistent~segments leading in the positive or thnegativez direction are likely to be followed by segmenleading into the samez direction!, and forH,0.5, the sur-face is antipersistent. ForHÞ0.5, the cross section is chaacterized by a process called fractional Brownian mot@20#. A simple model of a fracture aperture fielda(x,y) isprovided by taking the difference of two self-affine surfacz1(x,y) andz2(x,y) ~with the same amplitude and roughneexponent! representing the fracture boundaries,

a~x,y!5z2~x,y!2z1~x,y!. ~2!

As long as the surfaces do not overlap@a(x,y).0#, the in-vasion thresholdsr i are well defined after discretizing thaperture fielda(x,y) on a lattice of sites. The aperture fiealso forms a self-affine surface with the same roughnessponent as the two surfaces@21#. In the present work, aperturfields were modeled by thez-value fieldsz(x,y) of singleself-affine surfaces:

a~x,y!5z~x,y!. ~3!

Invasion percolation using this type of substrates was astudied recently by Patersonet al. @22# and Du, Satik, andYortsos@23#. Percolation on self-affine topographies was aplied by Sahimi@24# to model transport phenomena in heerogeneous media. Invasion percolation in three dimensiusing heterogeneous substrates characterized by fractBrownian motion, was studied by Paterson and Painter@25#.

II. SIMULATION

Periodic self-affine surfacesz(xi ,yi) were generated onsquare lattice using a random midpoint displacement arithm with random successive addition@26,27#. The Hurstexponent characterizing the surfaces was measured usinheight difference correlation function

D~r !5^uz~xi ,yi !2z~xi1r x ,yi1r y!u2& rx21r

y25r2. ~4!

For self-affine surfaces with positive Hurst exponentH, thecorrelation function scales asD(r );r 2H @26#. The invasionthreshold fieldsr (xi ,yi)5z(xi ,yi) were obtained from thesubstrates. All sites on the lattice were filled with the wettifluid, and the central site of the lattice was filled with thnonwetting fluid. The IP simulation was carried out by leting the cluster of nonwetting fluid grow stepwise and invaperimeter sites with the smallest threshold.

Figure 1 shows a series of IP clusters obtained inmanner. For low values ofH (0,H,0.5), the clusterswere reminiscent of ordinary IP clusters grown on a lattwith uncorrelated invasion thresholds. Numerous regionsdefender fluids became trapped, and the distribution ofsizes of the regions followed a power law.

For higher values ofH (0.5,H,1), the clusters had adisordered shape, and could be described in terms‘‘blobs’’ of different sizes, connected by thin ‘‘threads.Compared to the case of lowH, a lesser amount of defendefluid became trapped. The IP cluster size distributions wskew, and extended over a large interval. UsingH'0.85, thelargest clusters covered more than 50% of the substr

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while the smallest ones covered less than 0.005% whenedge of the substrate was reached. In contrast, the sizetribution of IP clusters grown on uncorrelated substratesapproximately Gaussian.

A compact blob was formed when the growing IP clustreached a local threshold minimumM 1 on the substrate, andneighboring sites with higher thresholds were invaded susequently~see Fig. 2!. When the cluster reached a siteSseparatingM1 from a second local minimumM2, a thread ofinvaded sites was formed as the growth of the IP clusfollowed a path of steepest descent on the threshold surfRepeating the cycle, a new blob developed when the regaroundM2 was filled out, incorporating an increasing fraction of the thread. Adjacent blobs coalescenced if no furthlocal threshold minima were found. Blobs could thus acqua sizeb of the order of the entire cluster size.

III. CLUSTER STRUCTURE AND DIMENSIONALITY

Figure 3 shows, on a log-log scale, the mean numbS(R) of cluster sitess counted in a circle of radiusR around

FIG. 1. Clusters of nonwetting fluid~black! obtained from simu-lations of IP with trapping using self-affine substrates of siL5256, at the stage when the edge of the substrate was reacThe grey shade indicates the invasion thresholds used, with brshades corresponding to high thresholds. For comparison, thedom number generator seed was kept constant, and the Hurst enent was varied fromH50.13 ~a!, H50.23 ~b!, H50.39 ~c!,H50.55 ~d!, H50.73 ~e! to H50.89 ~f!, respectively. The arrowindicates the injection site.

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1700 55G. WAGNER. P. MEAKIN, J. FEDER, AND T. JO”SSANG

the center of mass of an IP cluster. The mean size~mass! Sis defined by the ratio of the second moment of the sdistribution N(s,R) to the first moment,S5(s2N(s,R)/(sN(s,R). The clusters grew on the correlated substrates using periodic boundary conditions. If a cter reached the edge of the substrate, it could reentersubstrate from the opposite edge. Each simulation wasminated when a cluster attempted to intersect itself. Foself-similar fractal of fractal dimensionalityD,

S~R!;RD. ~5!

In an intermediate range 1,R!L, the measured mean sizfollows a power lawS(R);RD. As a check,D was deter-mined for IP clusters grown on uncorrelated substrateslinear least-squares fit yieldedDuc51.8260.01, consistentwith earlier studies of IP@2,3,28,29,13#. Using correlated

FIG. 2. Illustration of the IP cluster growth model. The invasithreshold of each sitei is given by the heightz(xi ,yi) of a roughsurface. A blob is formed as the cluster~shaded sites! fills out aregion around the local threshold minimumM1. When the clusterreaches the siteS, a thread leading to a second local minimuM2 is formed. The region aroundM2 is then filled.

FIG. 3. The dependence of the mean number of sitesS(R) in acircle of radiusR around the center of gravity of IP clusters, onlog-log scale.R22S(R) was measured for IP with trapping on uncorrelated substrates~circles! and on correlated substrates wiH'0.31 ~squares!, H'0.47 ~diamonds!, andH'0.85 ~triangles!,respectively.R22S(R) is also plotted for IP without trapping oncorrelated substrates withH'0.47 ~filled diamonds!. The solidlines represent linear least-squares fits for distances in the r2<R<64. The substrate size was 5123512 sites.

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substrates, linear least-squares fits yieldedD0.3151.8960.01,D0.4751.9160.00, andD0.8551.9460.01, forH'0.31,H'0.47, andH'0.85, respectively. The errors indcate the standard deviation of the fit parameter. Figurindicates that the IP clusters became more dense as the Hexponent increased.

The two-point density correlation functionC(r ) is fre-quently used to characterize fractal structures. This quanis defined as

C~r !5^r~r0!r~r01r !& ur u5r

^r~r0!r~r0!&, ~6!

where 0<r(r )<1 is the cluster mass density and the avaging is over the occupied originsr0, orientations of thespace vectorr , and a large sample of clusters. For a sesimilar fractal of dimensionD on a two-dimensional substrate, C(r ) is expected to have the formC(r );r 22Df (r /r c), wherer c is a distance characteristic othe overall cluster extension. The cutoff functionf (x) hasthe form f (x)51 for x!1, and decreases faster than apower ofx with increasingx for x@1. Figure 4 shows a ploof C(r ), using different values of the Hurst exponentH andthe same sample of clusters that was used to measureS(R)~Fig. 3!. The correlation function obtained from simulationon uncorrelated substrates shows a decay consistentC(r );r 22Duc for small r . A linear least-square fit toC(r )yielded Duc51.8060.01. Turning to correlated substratelinear least-squares fits toC(r ) ~for r!L) yielded dimen-sions of D0.3151.8860.01, D0.4751.8960.01, andD0.8551.9560.01, respectively, forH'0.31,H'0.47, andH'0.85, respectively. These values are consistent withresults presented in Fig. 3 and confirm the systematiccrease of the fractal dimensionality of the IP clusters withdegree of spatial correlations.ge

FIG. 4. Log-log plot of the density-density correlation functioC(r ) versusr , on a log-log scale.C(r ) was measured for IP withtrapping on uncorrelated substrates~circles! and on correlated substrates withH'0.31 ~squares!, H'0.47 ~diamonds!, andH'0.85~triangles!, respectively. Also plotted isC(r ) for IP without trap-ping on correlated substrates withH'0.47 ~filled diamonds!. Thesolid lines represent linear least-square fits to the curves for thvalues of r where log„C(r )… appears to be a linear function olog(r). The substrate size was 5123512 sites.

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55 1701INVASION PERCOLATION ON SELF-AFFINE TOPOGRAPHIES

The IP cluster structure was studied by measuring thnumber~per lattice site! n(b) of blobs of size~mass! b ob-tained by removing the connecting ‘‘threads.’’ For each sitof an IP cluster, it was determined if the removal of the sitfragmented the cluster. In this case, the site was marked athread site. When all sites had been tested, the thread swere removed. The remaining sites defined the blobs. Figu5 shows the distributionn(b) measured on substrates of2563256 sites counted at the stage when the growing cluters reached the edge of the substrate. At this stage, clussites were removed if the removal implied fragmentation othe IP cluster. The remaining sites defined the blobs. ThdistributionsnL,H(b) of the number of blobs with sizes in therangeb to b1db (db→0) in on a substrate of sizeL with aHurst exponentH may be described by the scaling form

nL,H~b!;L2Hb2t f ~b/LDH!, ~7!

where f (x) is a scaling function that decreases faster thaany power of x for x@1. Here the cutoff blob sizebc;LDH was assumed to be equal to the IP cluster size. Texponentt is given by the size distribution of regions in theself-affine surfaces with a heightz(x,y) less than a ‘‘hori-zontal’’ cut parallel to the thex-y plane at an arbitrary heightz0. For self-affine surfaces with 0,H,1, the linear exten-sionr of such regions scales asN(r );r H23 @20,30#. Makinguse of Eq.~5!, the size distribution of blobs with fractaldimensionalityDH filling out a random sample of regionswith z(x,y),z0 is found to be N(b);b2t, witht5(21DH2H)/DH . TheL dependence in the scaling formEq. ~7! is obtained from the requirement that the first moment m (1)5*bn(b)db of the blob size distribution scalewith the system size asm (1);LDH22. Figure 6 shows an

FIG. 5. Log-log plot of the numbern(b) of ‘‘blobs’’ per latticesite of sizeb obtained after removing single-connecting cluster sitefrom IP clusters~with trapping!, using a log-log scale. The distri-butions were averaged over a sample of clusters in bins of logaritmically increasing size. The IP clusters were grown on uncorrelatesubstrates~circles! and on correlated substates withH'0.31~squares!, H'0.47 ~diamonds!, and H'0.85 ~triangles!, respec-tively. The substrate size was 2563256 sites. Each simulation wasterminated when the IP cluster reached the substrate edge.

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attempt to collapse several blob size distributionsnL,H(b)corresponding to different values ofL and H on a singlecurve f (x), using Eq.~7! and the fractal dimensionalitieDH found from the scaling behavior of the mean cluster siFrom the figure, the scaling functionf (x) is found to be apower law with an exponent of approximately 0.33 fx!1.

IV. DISCUSSION

It was recently shown that the percolation transition@31#on a correlated substrate of the type used here is nevercal for H.0 @32,33#. The percolation exponentnH ~charac-terizing the divergence of the correlation length at the perlation threshold! becomes infinite for substrates wittopographies given by thez-value field of self-affine surfacewith positive Hurst exponent. On such a substrate, the pcolation threshold may be interpreted as the minimal threold heightzc up to which the underlying self-affine surfacmust be ‘‘flooded’’ ~representing the invasion of the corrsponding substrate sites! in order to obtain a spanning clusteof flooded regions that are connected to each other. Ifrelations between critical exponents known from percolattheory carry over, the fractal dimensionality of the infinipercolation cluster at the percolation threshold must be eqto the substrate dimensiond52. Du, Satik, and Yortsos@23#measured the density of percolation clusters at the critthreshold on self-affine topographies withH50.5, and re-ported no dependence on the size of the lattice, implyincluster dimension of 2. The same result was found earlierSchmittbuhl, Vilotte, and Roux@32#.

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FIG. 6. Attempt to collapse the blob size distributionsn(b)measured for IP with trapping onto a single curve by usingscaling form given in Eq.~7!. For low values of the argumenb/LDH, the scaling functionf (b/LDH) is linear with a slope of ap-proximately 0.33~straight solid line!. The IP clusters were grownon correlated substrates withH'0.31 ~squares!, H'0.47 ~dia-monds!, andH'0.85 ~triangles!, respectively. The substrates weof 64364 sites ~solid lines!, 1283128 sites ~dotted lines!,2563256 sites ~dashed lines!, and 5123512 sites ~dot-dashedlines!, respectively. Each simulation was terminated when thecluster reached the substrate edge.

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1702 55G. WAGNER. P. MEAKIN, J. FEDER, AND T. JO”SSANG

There are strong indications that IP without trapping gerates a percolation cluster that is equivalent to the infipercolation cluster@34,35#. IP without trapping thus may beexpected to lead to IP clusters with the fractal dimensionaDH

! 52. Figures 3 and 4 show attempts to determine the frtal dimensionality for IP without trapping on correlated sustrates withH'0.47. From the measurement ofS(R) in Fig.3 and C(r ) in Fig. 4, D0.47

! 51.9560.01 andD0.47

! 51.9560.01 were obtained, respectively. These resudo not permit a definite conclusion since finite-size effemay reduce the effective fractal dimensionality from 2.0the measured values ofDH

! '1.95. The IP clusters studied ithese measurements had a size between approxim50 000 and 150 000 sites. This finding may be compawith the fractal codimensionalitya522DH

! 50.0360.01measured forH50.5 in IP without trapping by Patersoet al. @22#.

Isichenko@30# analyzed the percolation problem on seaffine topographies, and predicted that the perimeter ofinfinite percolation cluster is fractal, with a fractal dimesionality DH

P5(1023H)/7 (0,H,1). The trapping ruleonly applies to sites in the interior of the IP clusters, suthat the structure of the cluster perimeters is not affectedthe rule. IP with trapping may thus lead to IP cluster perieters that are equivalent to the perimeters of infinite perction clusters.DH

P was measured by box counting the perimeters of some of the IP clusters and was found toconsistent with the theoretical prediction.

For IP with trapping, the results presented here indicthat the cluster dimensionalityDH depends on the Hurst exponentH characterizing the threshold correlations of the uderlying substrate. Trapping is not a local rule, but affethe growth of the cluster on a global scale, leading todeviation ofDH fromDH

! A similar difference is well knownfor IP on uncorrelated substrates in two dimensions@3#. Inthree dimensions, trapping occurs rarely, and no such gleffect of the trapping rule on the cluster dimensionalshould be expected.

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The effect of the trapping rule on two-dimensional sustrates is related to the dynamics of the IP process. Thenamics may be expected to be quite different from thenamics of standard IP@29#. For example, the pair correlatiofunctionP(r ), giving the probability that IP cluster sites thaare invaded in two subsequent steps are separated by atancer , decays approximately asP(r );r22 on uncorrelatedsubstrates, forr below a cutoff length. On correlated substrates usingH50.5, a much slower decay was foun@P(r );r2g with g'0.9#. The cluster growth proceede‘‘more smoothly,’’ with prominent ‘‘bursts’’ occurring lessfrequently than in standard IP. For large values ofH, trap-ping of large regions was rare and the deviation ofDH fromDH

! became small.For low values ofH, the fractal dimensionalityDH of the

IP clusters appears to be close toDuc, characterizing IP withtrapping on uncorrelated substrates. This result is consiswith the findings of Meakin@13# mentioned in Sec. I. Themultifractal substrates studied in the cited work were genated recursively and are equivalent to the self-affine ‘‘hiarchical’’ substrates used by Schmittbuhl, Vilotte, and Ro@32# in theH→0 limit.

In summary, IP with trapping on substrates with a sptially correlated threshold distribution resulting from thmapping of a self-affine surface has been studied. The fradimensionality of the IP clusters appears to be a tunablerameter, depending on the Hurst exponentH characterizingthe threshold correlations.

ACKNOWLEDGMENTS

We gratefully acknowledge support by VISTA, a researcooperation between the Norwegian Academy of Scieand Letters and Den norske stats oljeselskap a.s.~STATOIL!and by NFR, the Research Council of Norway. The wopresented received support from NFR and from the Institfor Computer Applications at the University of Stuttgathrough a grant of computing time.

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