Flow and Transport in Porous Media and Fractured Rock (From Classical Methods to Modern Approaches)...

179

7Models of Porous Media

Introduction

In Chapters 4–6, we learned how various properties of porous media – fracturedand unfractured – are measured, analyzed and interpreted. The natural questionthat comes to mind is: how do we model porous media? Any realistic modelingof flow and transport phenomena in a disordered porous medium must include,as the first ingredient, a realistic model of the porous medium itself. However,any model that we may use should depend on the type of the porous media thatwe wish to study and compute its various properties as well as the computationallimitations that we may have. In fact, while many fundamental models for naturalporous media, for example sandstones and carbonate rock, have been developed,the present computational capabilities do not allow their routine use in the simula-tion of complex phenomena, such as, the immiscible and miscibile displacementprocesses. Thus, we still must utilize models that are simple enough for use incomputer simulation of various flow and transport phenomena with reasonablecomputation, yet contain the essential features of the porous medium of interest.

In this chapter, we describe and discuss various models of unfractured porousmedia. Models of fractures, fracture networks, and fractured porous media will bedescribed in Chapter 8. We consider models in which one may incorporate threefundamental scales of heterogeneities described in Chapter 1, namely, microscop-ic (pore level), macroscopic (core plug level) and megascopic (field-scale) hetero-geneities. Porous media with the first two types of heterogeneities constitute whatwe refer to as the laboratory-scale porous media, while those with the third type ofheterogeneities represent large-scale porous media, for example, oil reservoirs andgroundwater aquifers.

7.1Models of Porous Media

Pore space models are needed for estimating the flow and transport properties andother important dynamical features of porous media. The simplest of such prop-

Flow and Transport in Porous Media and Fractured Rock, Second Edition. Muhammad Sahimi.© 2011 WILEY-VCH Verlag GmbH & Co. KGaA. Published 2011 by WILEY-VCH Verlag GmbH & Co. KGaA.

180 7 Models of Porous Media

erties are perhaps the effective permeability Ke and electrical conductivity ge of afluid-saturated porous medium. The simplest property of a pore space is its poros-ity φ and, therefore, an obvious goal for many years was to find a relationshipbetween φ and Ke, the existence of which had seemed so obvious that in the earlyliterature on flow of oil through reservoir rock, no distinction had been made be-tween Ke and φ; it had been assumed that Ke and φ are proportional. Later on,many empirical correlations between Ke and φ were suggested, the best-known ofwhich was perhaps that due to Rose (1945) who proposed that

Ke φm0

(7.1)

where m0 is some undetermined constant. Equation (7.1) is similar to the well-known Archie’s law (Archie, 1942) for the electrical conductivity of a fluid-saturatedporous medium given by

ge D gfφm , (7.2)

where gf is the fluid’s conductivity. In general though, there cannot be any generaland exact relationship between Ke and φ because, obviously, two porous media thathave the same porosity may have very different effective permeabilities since howthe porosity is distributed in a material is crucial to its effective properties. Such anobvious example prompted Cloud (1941) to conclude that there is no sensible relation

between porosity and permeability.

Over the years, many models of porous media have been developed, most ofwhich have been motivated by a certain phenomenon. However, often while a mod-el could be used to study a particular phenomenon and predict some of its proper-ties, it was not general enough to be useful for studying other types of phenomena.In addition, such models often contained parameters that were either defined veryvaguely, or had no physical meaning whatsoever. The sole purpose of such param-eters was to make the models’ predictions agree with the experimental data for aparticular phenomenon.

7.1.1One-Dimensional Models

In this class of models, the pore space is envisioned to be made of a bundle of paral-lel capillary tubes (pores), or a collection of tubes in series. The radius of the tubescan be the same for all, or it can be selected from a pore size distribution. The tubescan be all cylindrical, or may have converging-diverging segments. Sometimes, thetubes are deformed in order to give them a local tortuosity, or they may be giv-en periodic constrictions. None of such models can take into account the effect ofthe interconnectivity of the pores, the existence of closed loops of interconnectedpores, and so on. As a result, many predictions of such models are grossly in error.Scheidegger (1974) and van Brakel (1975) give lucid discussions of such models.

7.1 Models of Porous Media 181

7.1.2Spatially-Periodic Models

The spatially-periodic models of porous media models have been described byNitsche and Brenner (1989) and Adler (1992). In this class of models, the porespace is represented by a periodic structure, the unit cell of which can be a capil-lary periodic network or some other geometrical element. An example is shown inFigure 7.1. A spatially-periodic model is characterized by an associated lattice thatcontains the translational symmetries of the porous medium for which the mod-el is intended. Due to its periodic structure, the lattice is of infinite extent and isgenerated from any one lattice point by discrete displacements of the form

R D i1e1 C i2e2 C i3e3 , (7.3)

where I D (i1, i2, i3) is a triplet of integers, and fe1, e2, e3g is a triad of the basiclattice vectors. The triad is not unique because by applying any unimodular 3 3matrix with integer entries to the basis fe1, e2, e3g, one obtains another equally validbasis. It is often convenient to use cells that are parallelepipeds,that is, built on agiven choice of basic lattice vectors. Given the flexibility that one is afforded withthe choice of the unit cells, their shape is often ambiguously defined.

Spatially-periodic models are also characterized by two length scales. One is themicroscopic length scale lm of the lattice, defined as, lm D max[dmin(r)], wheredmin(r) is the distance between the point at r and the nearest lattice points. For ex-ample, for a cubic lattice of lattice constant a, lm D p

3a/2. The second length scaleL is one over which the averages of the physical fields of interest, for example, thepressure or concentration field, vary in a reasonable (relatively smooth) manner.This length scale is typically of the order of the linear size of the porous medi-um for which the model in intended. For a porous medium to be macroscopicallyhomogeneous, one must have L lm.

Figure 7.1 A spatially-periodic model of disordered porous media.


The simplest spatially-periodic lattice model consists of a two-dimensional (2D)array of circular (infinitely-long) cylinders which represents a relatively simplemodel of unconsolidated porous media if the cylinders are not allowed to overlap.Despite its simplicity, no rigorous results for flow and transport processes in such amodel were obtained until Sangani and Acrivos (1982) analyzed square and hexag-onal arrays of circular cylinders, determined their permeability, and discussed theapplication of the results to heat transfer in unconsolidated porous media. Later,Larson and Higdon (1987) considered flow in the same periodic lattices in boththe axial and transverse directions. However, it was, in fact, Hasimoto (1959) whoderived the first results for the effective permeability of 3D periodic lattices ofspheres – a reasonable model of unconsolidated porous media and packings ofparticles – in the limit of small volume fraction of the spheres. The first set ofresults for the full range of the spheres’ volume fraction was derived by Zick andHomsy (1982) and Sangani and Acrivos (1982). Their results will be described inChapter 9.

The analysis of flow and transport processes in such models is a relatively sim-ple problem when numerical or analytical calculations are confined to a unit cell.In principle, the unit cell may have an arbitrary shape, but if one is to analyze adisordered unit cell (see Figure 7.1) of arbitrary shape, the analysis would be no eas-ier than that of non-periodic models of porous media described below. In a sense,spatially-periodic models represent a type of mean-field approximation to the trulydisordered media because they do not contain any real heterogeneities and attemptto mimic the properties of the disordered media in some average way.

In some cases, the predicted effective properties do come close to those of somereal disordered media. In fact, over 50 years ago, Philip (1957) stated that, “[the]particular case of flow through a cubical lattice of uniform spheres . . . appearscapable of providing information on permeability-geometry relations.” This state-ment turned out to be true in the case of the models studied by Hasimoto (1959),Zick and Homsy (1982), and Sangani and Acrivos (1982). Brenner (1980), Carbonelland Whitaker (1983), and Eidsath et al. (1983) studied hydrodynamic dispersion inspatially-periodic models (see Chapter 11), and found a qualitative agreement be-tween some of their results and the experimental data of Gunn and Pryce (1969).Ryan et al. (1980) showed that the predicted effective reaction rate of a spatially-periodic model provides a useful estimate for some highly unconsolidated porousmedia, such as, packed beds. We should, however, point out that the main reasonfor the agreement between the predictions and the experimental data in all of suchstudies is that the geometry of the models closely resembled that of the experi-mental system. For example, Gunn and Pryce (1969) performed their dispersionexperiments in a spatially-periodic porous medium.

Nitsche and Brenner (1989, p. 244) argued that, “[while] any given model of sam-ple porous rock cannot generally be expected to possess perfect geometrical order,this does not mean that a spatially-periodic model is not useful for understandingthe fundamentals of a penetrant fluid flow through its interstices.” Nonetheless,the usefulness of such models for predicting the effective flow and transport prop-erties of, as well as numerical simulation of various phenomena in disordered and

7.1 Models of Porous Media 183

consolidated porous media, is very limited. Nitsche and Brenner (1989) and Adler(1992) provide an extensive list of references for spatially-periodic models.

What are the main shortcomings of spatially-periodic models of porous media?

1. Regular arrays of spheres (or other particles or inclusions with regular shapes)are limited to relatively low maximum volume fractions of the spheres whichare significantly below the solid (matrix) volume fraction of many real porousmedia.

2. Flow in regular lattices of isolated spheres occurs around the spheres, insteadof through the narrow pores found in real porous media.

3. The spatially-periodic models may be useful for unconsolidated porous mediain which the solid phase does not form a sample-spanning percolation cluster,whereas in consolidated porous media, for example, sandstone, both the solidand the fluid phases are macroscopically connected.

While the effect of solid volume fraction may not be very important for estimatingthe single-phase permeability of a porous medium if the heterogeneities are notbroadly distributed, it is important to other flow phenomena in porous media, suchas, two-phase flow, hydrodynamic dispersion, and so on, and even to single-phaseflow if the medium is highly disordered. Moreover, for heat transfer in porousmedia (e.g., geothermal reservoirs), the effect of heat conduction through the solidmatrix is important and, clearly, heat conduction through a sample-spanning solidmatrix is completely different from that in isolated solid inclusions.To extend the applicability of the spatially-periodic models to consolidated porousmedia, Larson and Higdon (1989) developed an interesting extension. Beginningwith a regular (spatially-periodic) lattice of spheres, they allowed the sphere radii toincrease beyond the point of touching in order to form overlapping spheres. Clear-ly, the solid fraction in this model can be anywhere between the original fractionbefore the growth of the spheres was started, and unity. Different lattices result indifferent pore shapes and sizes. Such a model is similar to the grain consolidationmodel of Roberts and Schwartz (1985) described below, except that Roberts andSchwartz mostly used a random distribution of spheres, whereas Larson and Hig-don used only a spatially-periodic lattice as the starting point. The advantage of themodel by Larson and Higdon is that it is amenable to certain analytical and semian-alytical calculations and, at the same time, mimics certain features of consolidatedporous media.

7.1.3Bethe Lattice Models

Next to spatially-periodic models are branching network models. These are noth-ing but the Bethe lattices of a given coordination number Z (see Chapter 3) thathave been used routinely in the statistical mechanics literature for investigatingcritical phenomena in the mean-field approximation. As far as their applicability to


modeling porous media is concerned, branching networks suffer from two majorshortcomings.

First, although they contain interconnected bonds (pores) that may mimic thatin a pore space, they lack closed loops of bonds, which are a major element of thetopology of any real pore space, particularly natural porous media.

Second, for a Bethe lattice of coordination number Z, the ratio of the number ofsites on the external surface of the network and the total number of sites is (Z 2)/(Z 1) (Ziman, 1979), which takes on finite values for any Z ¤ 2, whereas forlarge 3D networks, the ratio is very small. Thus, surface effects may strongly affectany property of a Bethe lattice and sometimes lead to anomalous phenomena, forexample, those described by Hughes and Sahimi (1982) who investigated diffusionin Bethe lattices. The advantage of the Bethe lattices is that it is often possibleto derive exact analytical formulae for the properties of interest, and sometimes,surprisingly, the predictions of such formulae agree well with those of 3D systems.Examples include diffusion and conduction in disordered Bethe lattices that willbe described in Chapter 10.

Liao and Scheidegger (1969) and Torelli and Scheidegger (1972) were the firstto use the Bethe lattices for modeling transport in porous media. They studiedhydrodynamic dispersion in a porous medium modeled by a Bethe lattice of a givencoordination number. In particular, Torelli and Scheidegger showed that such amodel is fairly successful in predicting the dependence of the dispersion coefficienton the average flow velocity (see Chapter 11). Others have also used the Bethelattices to model transport and reactions in porous catalysts (for a review see Sahimiet al., 1990).

7.1.4Pore Network Models

Chapter 3 provided the theoretical foundations for pore network models, and de-scribed the way they are developed. At the same time, a pore network model ofporous media is intuitively appealing because it is clear that a fluid’s paths in aporous medium branch out and, later on, join one another. A pore network modelof a pore space also provides precise meaning to the concept of the pore size distri-bution. The network that results from the mapping usually has a random topology,and its coordination number varies from node to node. Thus, random networkssuch as the Voronoi network (see Chapter 3) have also been used in the literatureas models of porous media (see, for example, Jerauld et al., 1984b,d; Sahimi andTsotsis, 1997; Rajabbeigi et al., 2009a; Rajabbeigi et al., 2009b).

The development of pore network models has been underway for severaldecades. Bjerrum and Manegold (1927) used a random network made of ran-domly distributed points in space connected to one another by cylindrical tubes inorder to study transport in porous media. However, the computational limitationsof that era severely limited their ability for carrying out any extensive computa-tions. Extensive analytical calculations with such models were first carried out byde Josselin de Jong (1958) and Saffman (1959) in the context of hydrodynamic

7.2 Continuum Models 185

dispersion in porous media that will be described in Chapter 11. Nevertheless, inorder to make their analytical calculations tractable, they both had to make certainassumptions. As pointed out in Chapter 3, the first application of network modelsto modeling two-phase flow in porous media was pioneered by Fatt (1956a,b,c). Heused various 2D networks of bonds representing the pore throats. The radius of thebonds was selected from a probability density function, representing the pore sizedistribution of the medium. No volume was attributed to the nodes. The lengthof each bond was assumed to be proportional to the inverse of its radius. We willcome back to this problem in Chapter 15.

Later, Rose (1957) and Dodd and Keil (1959) used pore network models to studyimmiscible displacement processes in porous media. We already mentioned inChapter 4 the work of Ksenzhek (1963) who used a pore network model to pre-dict the capillary pressure curves for porous media. Thus, although in the physicsliterature two seminal papers of Kirkpatrick (1971, 1973) are generally credited forpopularizing the use of networks of interconnected bonds for modeling transportin disordered media, the earlier pioneering works had already used such modelsto study transport processes in disordered porous media. Note that computer sim-ulations of Jerauld et al. (1984b,d) showed that as long as the average coordinationnumber of a topologically-random network is very close to, or identical with, the co-ordination number of a regular network, the effective flow and transport propertiesof the two networks are nearly equal.

The pore network models described so far are, in some sense, mathematical mod-

els that are used in computer simulations of flow phenomena in porous media.Another class of pore network models consists of a physical network – the man-made and transparent networks of pore bodies and pore throats. Such models havebeen developed for flow visualization studies, and have been particularly usefulfor gaining a deeper understanding of the displacement of one fluid by another.The first of such models – usually referred to as micromodels – was constructedby Chatenever and Calhoun (1952), who made bead packs from single layers ofglass and Lucite beads, and studied the displacement of oil with brine. Mattax andKyte (1961) made the first etched glass network to study displacement processes inporous media, and Davis and Jones (1968) significantly improved their techniqueby introducing photoetching techniques. Bonnet and Lenormand (1977) developeda resin technique for controlling the geometry of the network. Currently, etchedglass and molded resin are routinely used for constructing most micromodels.Lenormand (1990) and Buckley (1991) reviewed various techniques of construct-ing such physical networks.

7.2Continuum Models

Generally speaking, there are at least three different classes of continuum modelsof porous media.


1. Models that are made of a distribution of inclusions, such as circles, ellipses,cylinders, spheres, or ellipsoids, in an otherwise uniform background. The in-clusions may or may not overlap. If they do not, the model represents an uncon-solidated porous medium. If they do overlap, the model may be a reasonablerepresentation of consolidated porous media.

2. Model that are based on tessellating the space into regular or random polygonsor polyhedra and designating at random (or based on a correlation function)some of the polygons or polyhedra as representing the fractures, with the restrepresenting the pore space. This class of models will be described in the nextchapter, where we will describe models of fracture networks.

3. Those in which one distributes sticks of a given aspect ratio or plates of giv-en extents. Such models have been used for representing fibrous materials –a special type of porous materials (such as printing paper) – with the sticksrepresenting the fibers. Alternatively, the sticks or plates may be thought of asthe fractures in a porous medium. We will return to such models in the nextchapter.

The main attractive feature of such models is that with the appropriate choice of theparameters (to be described below), they may represent many real heterogeneousporous media. Their main disadvantage is the complexities that are involved in thestudy of transport processes through them. Thus, although the effective transportproperties of such continuum models can be and have been computed using avariety of techniques, they are still too complex for routine use in the investigationof many important phenomena in disordered porous media.

7.2.1Packing of Spheres

Consider a statistical distribution of N identical d-dimensional spheres of radiusR, which we refer to as phase 2, distributed in an otherwise uniform backgroundthat represents phase 1 of a two-phase disordered material. The system’s total vol-ume is Ω . The model is not as simple or restricted as it may seem. For example,by allowing the particles to overlap and cluster, one can generate a wide varietyof models with complex microstructures. Its 2D version, that is, a system of disksor, equivalently, a system of infinitely-long cylinders distributed in the matrix canbe utilized for modeling fiber-reinforced materials and thin porous films. An ex-ample is shown in Figure 7.2, where the disks’ sizes are distributed according toa uniform probability density function. The 3D version can be used for modelingunconsolidated porous media (for example, packed beds of particles) if the spheresdo not overlap, and consolidated porous media if they do.

If the spheres are not allowed to overlap, then one obtains what is popularlyreferred to as a fully-impenetrable or hard-particle model. In addition to modelingof a packing of particles, the model has been used in the study of a variety phe-nomena, for example, powders, cell membranes (Cornell et al., 1981), thin films(Quickenden and Tan, 1974), particulate composites, colloidal dispersions (Russel


Figure 7.2 A 2D continuum model of porous media represented by a random distribution ofdisks in a uniform background.

et al., 1989), and granular materials such as powders. The model becomes quitegeneral if the spheres are allowed to overlap (Weissberg, 1963). The intersection ofthe spheres does not have to represent a true physical entity, but can only be a wayof generating a heterogeneous porous medium with a certain microstructure. Anexample is the penetrable-concentric shell model or the cherry-pit model (Torquato,1984) in which each d-dimensional sphere of diameter 2R is composed of a hardimpenetrable core of diameter 2λR , encompassed by a perfectly penetrable shell ofthickness (1 λ)R . An example is shown in Figure 7.3.

The limits λ D 0 and one correspond, respectively, to the cases of fully-penetrablesphere model, also called the Swiss-cheese model (see Chapter 3), and the totally-impenetrable sphere model – a packed bed of particles. Thus, varying λ betweenzero and one allows one to tune the connectivity of the particles (i.e., the number of

Figure 7.3 A 2D example of the cherry-pit model in which a shell of thickness (1 λ)R of thedisks is fully penetrable (after Torquato, 1991).


particles that are in touch with a given particle) and obtain a wide variety of modelswith the desired morphology. In the fully-penetrable sphere model (when λ D 0),there is no correlation between the particles’ positions. If φ2 is the volume fractionof the particles, then the particle phase becomes sample-spanning (i.e., percolating)for φ2c ' 0.67 and 0.29, for d D 2 and 3, respectively. Note that, as pointed out inChapter 3, no bicontinuous structure exists in 2D and, therefore, when the particlephase is sample-spanning, the pore space is not. On the other hand, for d D 3, thesystem can be bicontinuous. In particular, both the particle phase and the pores aresample spanning for 0.3 φ2 0.97.

A related model is the so-called equilibrium hard-sphere model in which the par-ticles do not interact if the interparticle separation is larger than the sphere diam-eter, but there is an infinite repulsive force between them if the distance is lessthan or equal to the sphere diameter. An important property of this model is thatthe impenetrability constraint does not uniquely specify the statistical ensemble.That is, the system can be either in thermal equilibrium, or in one of the infinitelymany non-equilibrium states. A fundamental difference between the equilibrium2D hard-sphere model and its 3D counterpart should be noted. The difference isdue to the fact that in 2D, the densest global packing is consistent with the densestlocal packing. The maximum number of d-dimensional spheres that can be packedin such a way that each sphere touches the others is d C 1. This implies that thecoordination number of the 3D pore network that results from mapping the chan-nels between the touching spheres onto a lattice is exactly four. The d-dimensionalpolyhedron that results by taking the spheres’ centers as vertices is a simplex result-ing in line segments, equilateral triangles, and regular tetrahedra for d D 1, 2, and3, respectively. Whereas identical simplices can fill 1D and 2D systems with over-lapping, they cannot do so in 3D, that is, one cannot fill a 3D system with identicalnon-overlapping regular tetrahedra. Due to this inconsistency between the localpacking rules and the global packing constraints, the 3D systems are said to begeometrically frustrated.

7.2.2Particle Distribution and Correlation Functions

A dispersion of spheres is characterized by a number of fundamental microstruc-tural properties, including several n-point correlation and distribution functions(Torquato, 2002; Sahimi, 2003a). The most important of such distribution, corre-lation functions, and microstructural properties are as follows. In the discussionsbelow, when we speak of porous media, we mean those that are represented by thedispersion of spheres.

1. The most fundamental distribution function is n(r n), where n(rn)d r1

d r2 . . . d r n is the probability of simultaneously finding a particle centered involume element around d r1, another particle centered in volume elementaround d r2, and so on, where r n represents the set fr1 . . . r ng. For statisti-cally homogeneous media, the probability density n(rn) only depends upon


the relative distances r12, . . . , r1n , where r1i D r i r1. Thus, for example,2(r1, r2) D 2(r12). Note also that 1(r1) D p , where p is the density of theparticles

2. An important probability distribution function is Sn(x n), defined as the prob-ability of simultaneously finding n points at positions x1, x2, . . . , x n in one ofthe phases (pore or particle). Similarly, S

(i)n (x n) is the probability that the n

points at x1, x2, . . . , x n are in phase i (pore or particle). If the porous medi-um is statistically homogeneous, then S

(i)1 is simply the volume fraction φ i

of phase i. Moreover, S2(r1, r2) D S2(r12). For statistically homogeneous (orstrictly stationary) porous media, the joint probability distributions are trans-lationally invariant so that S

(i)n (x n) D S

(i)n (x n C y ). If a porous medium pos-

sesses phase-inversion symmetry (between the pore and particle phases), thenS

(1)n (x n I φ1, φ2) D S

(2)n (x n I φ2, φ1).

3. The point/q-particle distribution functions G(i)n (x1I r q) characterize the proba-

bility of finding a point at x1 in phase i (pore or particle) and a configuration ofq spheres with their centers at r n , respectively, with n D q C 1.

4. We may also define surface–surface, surface–matrix, and surface–void correla-tion functions Fss(x1, x2), Fsm(x1, x2), and Fsv (x1, x2), associated with findinga point on the interface between the two phases (i.e., on the pores’ walls) andanother point in the pore phase, or on the interface, respectively. Other valu-able information may be gained from a surface–particle correlation functionFs p that yields the correlations associated with finding a point on the parti-cle–pore interface and the center of a sphere in some volume element. Thefunctions Fss and Fsv can be obtained from any plane cut through an isotropicmedium. For homogeneous and isotropic media, such functions only dependon jrj D jx2 x1j. For homogeneous porous media, as jrj ! 1, one hasFsv (r) ! φ1, and Fss(r) ! 2 where is the specific surface area (surfacearea per unit volume), and φ1 D φ the porosity.

5. Two important properties of disordered porous media are the exclusion proba-bilities EV (r) and EP (r). The former is the probability of finding a region ΩV –taken to be a d-dimensional spherical cavity of radius r centered at some arbi-trary point – to be empty of any particle centers. Physically, EV (r) is the expectedfraction of space available to a test sphere of radius r R inserted into the sys-tem. EP (r), on the other hand, is the probability of finding a region ΩP – takento be a d-dimensional spherical cavity of radius r centered at some arbitraryparticle center – to be empty of other particle centers.

6. The void nearest-neighbor probability density functions HV (r) and HP (r) are alsouseful quantities. HV (r)dr is the probability that, at an arbitrary point in thesystem, the center of the nearest particle lies at a distance in the interval (r, r Cdr), whereas HP (r)dr is the probability that, at an arbitrary particle center inthe system, the center of the nearest particle lies at a distance in the interval(r, r C dr).


The definitions of the void nearest-neighbor and exclusion probabilities implythat

Ei (r) D 1 rZ

0

Hi (x )dx , i D V, P (7.4)

or, equivalently, Hi (r) D @Ei/@r . A mean nearest-neighbor distance h`P i isalso defined by

h`P i D1Z0

r H(r)dr D 2R C1Z0

EP (r)dr , (7.5)

which is an important characteristic of a packing of spheres. Utilizing h`P i,one can precisely define the random-close packing fraction φcp in the fully-impenetrable spheres model. Specifically, φcp is defined as the maximum ofpacking fraction φ2 over all the homogeneous and isotropic ensembles of thepackings at which h`P i D 2R , where R is the spheres’ radius. If the spheresare polydispersed with a mean radius hRi, then one may define φcp in a sim-ilar manner by h`P i D 2hRi. From a practical view point, φcp is the packingfraction at which randomly-arranged hard (impenetrable) spheres cannot becompressed any more if a hydrostatic pressure is applied to the packing. Atφcp, the exclusion and the void nearest-neighbor probabilities are both zero forr > 2R . The numerical estimates for φcp, as mentioned earlier, are φcp ' 0.82and 0.64 for d D 2 and 3, respectively. Note that φcp is different from the closest

packing fraction that are π/(2p

3) ' 0.907 and π/(3p

2) ' 0.74 for d D 2 and 3,respectively.

7. Two other important characteristics are the lineal-path function L(i)p (z) (Lu and

Torquato, 1992a,b) and the chord-length distribution function L(i)c (z) (Torquato

and Lu, 1993). L(i)p (z) is the probability that a line segment of length z is entire-

ly in phase i (pore or particle) when randomly thrown into the sample. For 3Dmedia, L

(i)p (z) is equivalent to the area fraction of phase i (pore or particle), mea-

sured from the projected image onto a plane, a highly important problem instereology (Underwood, 1970). Moreover, L

(i)p (0) D φ i and L

(i)p (1) D 0. L

(i)p (z)

contains information on the microscopic connectivity of a porous medium and,thus, it is a useful quantity. In stochastic geometry, the quantity φ i [1 L

(i)p (z)]

is sometimes referred to as the linear contact distribution function (Stoyan et al.,1995).L

(i)c (z)dz, on the other hand, is the probability of finding a chord of length

between z and z C dz in phase i (pore or particle). Thus, L(i)c (z) is actually a

probability density. Chords represents distributions of lengths between inter-sections of lines with the interface between the phases, that is, with the pores’walls. Chord-length distributions are relevant to transport processes in hetero-geneous porous media, for example, diffusion, radiative heat transfer (Ho andStrieder, 1979; Tassopoulos and Rosner, 1992), and flow and conduction (Krohn


and Thompson, 1986). The quantities L(i)p (z) and L

(i)c (z) are, in fact, closely re-

lated (see below).8. A very general distribution function for characterizing the microstructure of

a heterogeneous medium of the type we study here is Gn(x p I r q), which isthe n-point distribution function that characterizes the correlations associatedwith finding p particles centered at positions x p D fx1, . . . , x p g and q particlescentered at positions r q D fr1, . . . , r qg, with n D p C q. Clearly, for q D 0, wehave Gn(x n I ;) D Sn(x n), where ; denotes the null set, and in the limit p D 0,Gn(;I r q) D n(rn). Moreover, if p D 1 and the radius of the p-particles is takento be zero, then Gn(x1I r q) D G (1)(x1I r q), as defined in item 3.

9. A most general n-point distribution function Hn(x mI x pmI r q) is defined(Torquato, 1986) to be the correlation function associated with finding m pointswith positions x m on certain surfaces within the medium, p m with posi-tions x pm in certain spaces exterior to the spheres, and q sphere centers withpositions r q with n D p C q, in a statistically heterogeneous porous medium ofN identical d-dimensional spheres. Most of the n-point correlation and distri-bution functions that were described above represent some limiting cases ofHn(x mI x pmI r q). By way of its construction, the function Hn , unlike the lessgeneral function Gn , also contain interfacial information about the availablesurfaces associated with the first m test particles. Note that in the cherry-pitmodel, Hn(x m I x pmI r q) is identically zero for certain r q . In particular, for anyvalue of the impenetrability λ (see above), we have Hn(x mI x pmI r q) D 0 ifjr i r j j < 2λR , where R is the radius of the spheres.

The functions defined above are not all independent and are, in fact, related toone another. Suppose that the d-dimensional spheres are spatially distributed ac-cording to a specific N-particle probability density PN (r N ). Then, n(rn), which issometimes called the n-particle generic probability density, is given by

n (r n) D N !(N n)!

ZPN

r N

d rnC1 d r N . (7.6)

Many of correlation and distribution functions defined above are related to the n-point distribution function Hn :

Sn (x n) D limai !R

Hn (;I x nI ;) , 8 i , (7.7)

Gn(x1I r q) D lima1!R

Hn (;I x1I r q) , (7.8)

Fss (x1I x2) D limai !R

H2 (x1, x2I ;, ;) , 8 i, (7.9)

HV (r) D H1 (x1I ;, ;) , EV (r) D H1 (;I x1I ;) . (7.10)

One can also show that EP (r) D H2(;I x1I r1)/1(r1), as jx1 r1j ! 0, from whichthe relation HP (r) D @EP /@r is obtained.As for the lineal-path function, we focus on the quantity L p (z) D L

(1)p for a two-

phase (pores and particles) heterogeneous materials of interest in this book. The


key idea is that L p (z), which is a type of exclusion probability function, yields theprobability of inserting a test particle – a line of length z – into the system which isequal to the probability of finding an exclusion zone ΩE(z) between a line of lengthz and a sphere of radius R. The region ΩE is a d-dimensional spherocylinder ofcylindrical length z and radius R with hemispherical caps of radius R on either endof the cylinder. Therefore, if we define the exclusion indicator function m(yI z) by

m(y I z) D

1 , y 2 ΩE(z)0 , otherwise

(7.11)

then, L p (z) is given by (Lu and Torquato, 1992a,b)

L p (z) D 1 C1X

kD1

(1)k

k!

Zk

r k

kYj D1

mx r j I z

d r j . (7.12)

(Lu and Torquato, 1993a,b) showed that for statistically isotropic two-phase materi-als of arbitrary microgeometry, the chord-length distribution function is related tothe lineal-path function L p (z) by

L c(z) D `C

φ1

d2L p (z)dz2 , (7.13)

where `C is the average of L c(z),

`C D1Z0

zL c(z)dz . (7.14)

Lu and Torquato (1991) generalized Eqs. (7.7)–(7.14) to a polydispersed medium inwhich the spheres’ radii follow a probability density function.

7.2.3The n-Particle Probability Density

Given the n-particle probability density n , one can compute the correlation func-tion Hn , from which most other properties follow. The advantage of this formula-tion is that the function n has been studied in great detail in the context of thestatistical mechanics of liquids (see, for example, Hansen and McDonald, 1986)and, therefore, the extensive results obtained in such studies can be immediatelyemployed for modeling porous media of the type we are describing here.

In the fully-penetrable sphere model – the Swiss-cheese model that representsthe limit λ D 0 of the impenetrability parameter λ – with a particle density (num-ber of particles per unit volume) p , there is no spatial correlation between theparticles. Therefore, one has the exact relation

n (r n) D np , 8 n . (7.15)


In this case,

Sn (r n) D expp Vn

, (7.16)

where Vn is the union volume of the n spheres (see below). Determining n(rn) forother types of the dispersion of spheres is considerably more difficult. For exam-ple, for fully-impenetrable spheres, the impenetrability condition cannot by itselfuniquely determine the ensemble, and one must supply more information. Onemust, for example, state that the spheres are distributed in the most random fash-ion that, together with the impenetrability condition, determines the state of thesystem.

Let us define

Θ (x ) D

1 , r > 00 , r < 0

(7.17)

and v i2(rI R , R), the intersection volume of two identical d-dimensional spheres of

radius R with their centers separated by a distance r, which is given by

v i2(rI R , R) D 2R2

"cos1

r

2R

r

2R

1 r2

4R2

12#

Θ (2R r) , d D 2

(7.18)

v i2(rI R , R) D 4πR3

3

1 3r

4RC r3

16R3

Θ (2R r) , d D 3 . (7.19)

Determining the intersection volume of three or more spheres is non-trivial, espe-cially if the spheres’ radii are not the same (Torquato, 2002; Sahimi, 2003a).

7.2.4Distribution of Equal-Size Particles

Consider a system of identical particles of arbitrary shapes with number densityp . We define a dimensionless density η D p v , where v is the volume of a particlethat, for example, for a d-dimensional sphere of radius R is given by

v (R) D πd2 R d

Γ

1 C d2

, (7.20)

where Γ (x ) is the gamma function. For fully-impenetrable particles, the reduceddensity η is exactly the particle volume fraction φ2, that is,

η D φ2 D 1 φ1 D 1 φ , (7.21)

where φ is the porosity. The equality does not hold if the particles can overlap. Inparticular, for the penetrable-concentric shell model, one has

η(λ) φ2(λ) , (7.22)

where λ is the impenetrability index defined above so that the equality applies whenλ D 1.


7.2.4.1 Fully-Penetrable SpheresFor the fully-penetrable spheres model which represents the limit λ D 0, the firsttwo functions S1 and S2 are given by

Sn D8<:

φ D φ1 D 1 φ2 D exp(η) , n D 1 ,

expη

V2(rI R)v

, n D 2 ,

(7.23)

where V2 is the union volume of two spheres to be defined shortly. Given thatfor this model, the n-particle probability density function is given by Eq. (7.15),calculation of Hn(x m I x pmI r q) is straightforward. In particular, if we let m Dq D 0, then Hn(x n), the probability of inserting n spheres of radii a1, . . . , an intoa system of N spheres of radius R at positions x1, . . . , x n , that is, inserting theparticles into the available space of the medium, is given by

Hn (x n) D expp Vn (x nI a1, . . . , an)

. (7.24)

Here, Vn(x n I a1, . . . , an) is the union volume of nd-dimensional spheres of radiia1, . . . , an , centered at x1, . . . , x n , respectively. In the limit that ai ! R for 8 i ,one recovers Eq. (7.16). Note that the union volume V2(r12I a1, a2) of two spheresis given by V2(r12I a1, a2) D v (a1) C v (a2) v i

2(r12I a1, a2), and

V3(r12, r13, r23I a1, a2, a3) Dv (a1) C v (a2) C v (a3) v i2(r12I a1, a2)

v i2(r13I a1, a3) v i

2(r23I a2, a3)

C v i3(r12, r13, r23I a1, a2, a3) , (7.25)

with ri j D jx i x j j, and v being the volume of one sphere given by Eq. (7.20).Other properties of the model can also be determined. The model predicts that

the specific surface area is given by

D lima1!R

H1 (x1I ;, ;) D p φ1dv

dRD ηφ1d

RD ηφd

R, (7.26)

where v and φ are given by Eqs. (7.20) and (7.23), respectively. Equation (7.41) mustbe compared with the corresponding results for the impenetrable sphere model:s D dv/dR . Since there are no correlations in the model, there is no differencebetween the void and particle nearest-neighbor functions, HP (r) D HV (r) D H(r).One can then show that,

H(r) D p

dv (r)dr

exp[p v (r)] , (7.27)

where v (r) is given by Eq. (7.20). Using Eqs. (7.4) and (7.27), we then obtain

E(r) D expη

vd(x )vd(1)

, (7.28)

where

vd (x ) D 12d

πd2 x d

Γ

1 C d2

. (7.29)


One can also derive the lineal-path function L p (z) D L(1)p (z) for this model. Us-

ing Eqs. (7.12) and (7.15), it is not difficult to see that

L p (z) D expp vE(z)

, (7.30)

where vE(z) is the d-dimensional volume of the exclusion zone ΩE defined byEq. (7.11), which in d dimensions is given by

vE(z) D πd2 R d

Γ

1 C d2

C πd1

2 R d1

Γh1 C d1

2

i z . (7.31)

Once L p (z) is known, the chord-length distribution L c(z) can be immediately com-puted using Eq. (7.13). Utilizing Eq. (7.23), we can rewrite the results for L p (z) interms of the volume fraction of the pores, that is, the porosity φ D φ1 D 1 φ2,with the results being

L p (x ) D

φ1C(4/π)x , d D 2 ,φ1C(3/2)x , d D 3 ,

(7.32)

where x D z/(2R). Equations (7.32) indicate that with increasing z, the lineal-pathfunction decreases with the porosity since it becomes increasingly more difficult toinsert a line segment of length z in the pore space.

7.2.4.2 Fully-Impenetrable SpheresUnlike the case of fully-penetrable spheres, the fully-impenetrable sphere modelis much more difficult to analyze because there are significant correlations in thesystem that are imposed by the impenetrability condition. Despite this difficulty,significant progress has been made that has been described by Torquato (2002)and Sahimi (2003a).

In general, one must have EP (r) D 1 and, HP (r) D 0 for 0 r < 2R . It canalso be shown that EV (r) D 1 p v (r) and HV (r) D p s for 0 r R , where v

is the volume of a cavity of radius r, and s is the specific surface area. In general,the functions HP (r) and HV (r) are not truncated series, but are expressed as aninfinite series. For d D 2, an accurate approximation to HP (x ) is given by Torquatoet al. (1990)

HP (x ) D 4η(2r η)(1 η)2

exp 4η

(1 η)2[x2 1 C η(x 1)]

, x > 1 ,

(7.33)

where η D p v is the dimensionless density (volume fraction of the particles)defined above, and x D r/(2R). For d D 3, we have

HP (x ) D 24η

f1 C f2x C f3x3exp

˚η24 f1(x 1) C 12 f2

x2 1

C 8 f3x3 1

, x > 1 , (7.34)


with

f1 D 12

η2 f (η) , f2 D 12

η(3 C η) f (η) , f3 D (1 C η) f (η) , (7.35)

where f (η) D (1 η)3. Given Eqs. (7.34) and (7.35) for HP (r) and HV (r), thecorresponding equations for EP (r) and EV (r) can be immediately determined us-ing Eq. (7.4). Accurate approximations for L p (z) are given by (Lu and Torquato,1992a,b)

L p (z) D (1 η) exp dηz

R(1 η)

, (7.36)

where D π and four for d D 2 and three, respectively. Finally, it can be shownthat for any ergodic ensemble of isotropic packings of identical d-dimensional hardspheres, the mean nearest-neighbor distance h`P i, defined by Eq. (7.5), is boundedfrom above: h`P i 1 C (2d ηd)1.

7.2.4.3 Interpenetrable SpheresUsing computer simulations, Lee and Torquato (1988) calculated the pore volumefraction (porosity) φ D φ1 in the penetrable concentric shell model as a functionof the parameter λ. The following approximate, but very accurate, formulae for φwere derived by Rikvold and Stell (1985) that are in excellent agreement with thenumerical results of Lee and Torquato (1988):

φ(η, λ) D

1 λd η

exp

" (1 λd )η

(1 λd η)d

#Ψd (η, λ) , (7.37)

where Ψd (η, λ) is a d-dependent function given by

Ψ2(η, λ) D exp λ2η2(1 λ)2

(1 λ2η)2

, (7.38)

Ψ3(η, λ) D exp 3λ3 η2

2(1 λ3η)3 (2 3λ C λ3 3λ4 η C 6λ5 η 3λ6 η)

.

(7.39)

Note that ηλd represents the hard-core volume fraction in d dimensions.

7.2.5Distribution of Polydispersed Spheres

For porous media applications, a more realistic version of a model of dispersion ofspheres is one in which the radii of the spheres are distributed according to a nor-malized probability density function f (R). Polydispersivity leads to more flexibilityin the model and, hence, can be exploited for a variety of purposes.


7.2.5.1 Fully-Penetrable SpheresFor this class of models, the volume fraction of the pores, that is, the porosity, canbe obtained from a modification of Eq. (7.23) (Chiew and Glandt, 1984),

S1 D φ1 D φ D expp hv (R)i , (7.40)

and the specific surface s area is given by

D p

@hv (R)i@R

expp hv (R)i . (7.41)

Stell and Rikvold (1987) showed that

Sn (x n) D expp hVn (x nI R , . . . , R)i , (7.42)

where Vn is the union volume of n spheres of radius R defined and discussedearlier.

The lineal-path function has also been determined for this class of models. In or-der to compute this function, the exclusion indicator function defined by Eq. (7.11)must be generalized for a polydispersed system. In this case, one defines an exclu-sion region indicator function by

m j (x I z) D

1 , x 2 ΩE(z, R j )0 , otherwise ,

(7.43)

where R j is the radius of the jth sphere. The lineal-path function L p (z) D L(i)p (z)

is then given by (Lu and Torquato, 1993a,b),

L p (z) D 1C1X

kD1

(1)k

k!

ZdR1 . . . dRk f (Rk ) k

r k I R1, . . . , Rk

kYj D1

m j (x I z)d r j ,

(7.44)

which is a generalization of Eq. (7.12) to polydispersed systems. Then, usingEqs. (7.15) and (7.44), one obtains

L p (z) D expp hvE(z, R)i , (7.45)

where vE(z, R) is given by Eq. (7.31). Combining Eqs. (7.23), (7.31) and (7.45), andproceeding in the same manner as that for the equal-sized particles, we obtain theanalogs of Eqs. (7.32) for polydispersed systems:

L p (z) D

8<:

φ1C

2hRizπhR2i , d D 2 ,

φ1C

3hR2iz

4hR3i , d D 3 ,(7.46)

where φ is the porosity.


7.2.5.2 Fully-Impenetrable SpheresFully-impenetrable sphere models have many intriguing properties, many of whichare not well-understood yet (Torquato, 2002; Sahimi, 2003a). Very little is alsoknown about the packing characteristics of polydisperse hard spheres. For exam-ple, even if we consider one of simplest of such polydisperse packings, namely,a binary mixture of hard spheres of arbitrary radii R1 and R2, its largest achiev-able packing fraction is not known. Despite such difficulties, some progress hasbeen made which is now summarized. For polydisperse impenetrable spheres, thefollowing results are known:

S1 D φ1 D φ D 1 p hv (R)i , (7.47)

where φ is the porosity, and the specific surface area s is given by

D p

dhv (R)idR

D dηhR d1ihR di . (7.48)

Accurate approximations for the lineal-path function L p (z) have also been derivedfor this class of models, with the result for a d-dimensional system being

L p (z) D φ exp

" Γ

1 C 1

2 d

ηhR d1iπ

d2 φhR di

#. (7.49)

7.2.6Simulation of Packings of Spheres

What is the most efficient method for computer simulation of a packing of spher-ical particles? If the spheres are allowed to overlap, then the simulation is ratherstraightforward. The problem is more difficult if one wishes to model the fully-impenetrable spheres model, or to take into account the effect of the hard coresthat the spheres may have (represented by the impenetrability parameter λ).

The classical method of computer generation of a packing of spherical particlesin the fully-impenetrable spheres model is that of Visscher and Bolsterli (1972). Intheir algorithm, a sphere is dropped into the simulation box from the top. If theparticle hits the “floor”, it stops. If it hits another particle, say p1, it rolls down onp1 until it hits another particle p2. Then, it rolls on both p1 and p2 until it hitsparticle p3. If its contact with p1, p2, and p3 is stable, it stops. Otherwise, it rollson the double contacts, and so on. Since the particle’s motion is always downward,the effect of the gravity is automatically taken into account.

As a simple method that takes into account the effect of the hard core of sphericalparticles, consider, for example, a system of d-dimensional spheres. The numberN of the particles and the volume Ω of the system are fixed. The particles are ini-tially placed in a cubical cell (with volume Ω D Ld ) on the sites of a regular lattice,for example, the face-centered or body-centered lattice. No hard-core overlap is as-sumed initially. The particles are then moved randomly by a small distance to newpositions. The new positions are either accepted or rejected according to whether

7.3 Models Based on Diagenesis of Porous Media 199

or not the hard cores overlap. One usually uses periodic boundary conditions thatmeans that, if a particle exits from an external face of the system, an identical par-ticle enters the system from the opposite face of the system. This type of boundarycondition eliminates the boundary effects, hence allowing one to simulate an es-sentially infinitely-large packing. After the particles have been moved a sufficientlylarge number of times, the system reaches equilibrium and its configuration nolonger changes.

7.3Models Based on Diagenesis of Porous Media

A review of the literature indicates that there have been a few attempts to modelthe diagenetic processes that have given rise to the present rock, and in particularoil reservoirs. Among such attempts the work of Roberts and Schwartz (1985) isnotable. They developed a geometrical model for sandstones – usually referred toas the grain-consolidation model – that mimics many features of the natural ones.In their model the initial porous medium (before the diagenetic processes began)is a dense pack of randomly-distributed spherical grains of radius R. The modelwas originally suggested by Bernal (1959, 1960); see also Alben et al. (1976) forstudying of liquids. In the model of Roberts and Schwartz (1985) the coordinatesof the spheres’ centers followed the Bernal distribution. Figure 7.4 shows the initialdense packing of the grain with the volume fraction of the spheres being 0.636 and,therefore, an initial porosity of 0.364.

Figure 7.4 Two stages of the grain-consolidation model with porosities φ D 0.36 (a) andφ D 0.03 (b). Dark areas represent the pores (after Schwartz and Kimminau, 1987).


The radii of the spheres are then allowed to increase simultaneously, as a result ofwhich the porous medium’s porosity and effective permeability both decrease. Thegrowth of the particles is continued until a desired porosity is reached. The perco-lation threshold of the porous medium, that is, the critical porosity below whichno fluid can flow macroscopically in the porous medium, was determined to beφc ' 0.03˙0.004. The porous medium at the critical porosity is also shown in Fig-ure 7.4. Such a low critical porosity implies that the grain-consolidation model cangenerate models of porous media in which the porosity can vary over more thanone order of magnitude. The model shown in Figure 7.4 also has striking resem-blance to natural sandstone, shown in Figure 7.5. Given that the porosity of sand-stone is typically less than 0.4, the grain-consolidation model generates reasonablyaccurate representation of such natural porous media. The model was further stud-ied by Schwartz and Kimminau (1987) and by Schwartz et al. (1989). In particular,the latter group utilized an initial dense packing of nonspherical grains in order togenerate a model of anisotropic or stratified porous media.

An advantage of the grain-consolidation model is its flexibility. If, instead of theBernal distribution, one begins with a simple-cubic lattice of spherical grains andutilizes the algorithm for the growth of the particles, then the critical porosity ofthe medium will be 0.349, which is close to that of the random sphere packing. Onthe other hand, if one begins with a body-centered cubic lattice of spherical grains,the critical porosity will be 0.0055, one order of magnitude smaller than what canbe achieved with the Bernal distribution.

Figure 7.5 Cemented Devonian sandstone from Illinois. Compare this with Figure 7.4b (cour-tesy of Dr. L.M. Schwartz).

7.4 Reconstruction of Porous Media 201

Moreover, the sedimentation and diagenetic processes that give rise to many sed-imentary rocks, such as sandstones, tend to favor a distribution of particles thatare roughly equal in size (Pittmann, 1984). Thus, from this perspective, the grain-consolidation model is much more efficient than a model in which the porosity isreduced by adding additional spherical particles with smaller and smaller sizes, inorder to progressively fill the pore space of the original packing. In fact, to obtain acomparable porosity range by adding smaller particles, one must use spheres withradii that vary over many orders of magnitude. Even then, the final result has verylittle resemblance to most naturally occurring granular porous media.

While the grain-consolidation model is intended for sandstones, similar and per-haps more advanced models for other types of rock are also needed. Biswal et al.

(2007) developed a multiscale model for the diagenesis of carbonate rock. Carbon-ate porous media are characterized by a wide pore size distribution, 2–3 ordersof magnitude variations in their effective permeability (at the same porosity) and,perhaps most importantly, the dependence of their properties on the resolution. Inaddition, their morphology contains extended correlations. The model by Biswal et

al. (2007) captures all the three characteristics. It uses 2D sections of the sampleporous medium, and is based on assuming that the rock is random, but with a cor-related sequence of points that are decorated with crystallites which are convex sets,for example, spheres or polyhedra. All the properties can be computed at arbitraryresolution, which offers a grain advantage over other models.

7.4Reconstruction of Porous Media

An emerging method for developing a model of a porous medium is by reconstruc-

tion: Given a set of data for some properties of a porous medium, one tries to devel-op a model that reproduces the data most accurately by somehow minimizing thedifferences between the calculated and target data. The idea was first developed byJoshi (1974) and later by Quiblier (1984). It consisted of determining the first twomoments – the porosity and two-point correlation function (see Section 4.17.3) –from 2D thin section images of the sample. Then, stochastic 3D samples are gener-ated such that they match the measured statistical properties of the sample. Adler et

al. (1990, 1992) used such a technique to reconstruct Fontainebleau sandstone. Thecomputed effective permeability and electrical conductivity were in fair agreementwith the data, but also consistently smaller than them. As Øren and Bakke (2002)pointed out, the discrepancy was due to the fact that the percolation threshold ofthe sandstone is around 0.02–0.03, whereas the reconstructed stochastic modelshad a percolation threshold of around 0.1. Higher-order statistics were used by Ok-abe and Blunt (2005) in reconstruction of porous media. Keehm et al. (2004) useddata from thin sections to carry out a 3D reconstruction. They then used the latticeBoltzmann approach (see Chapters 9–12) to compute the effective permeability,which was found to be in agreement with the measured data.


One method of reconstruction is based on a stochastic optimization method bywhich the sum of the squared differences between the computed and measureddata is minimized. Several optimization methods are currently available (Sahimiand Hamzehpour, 2010). Notable among them are the gradient-based optimizers(GBOs), the genetic algorithm (GA), and the simulated-annealing (SA) method.The GBOs are deterministic methods, whereas the SA and GA are both stochastictechniques. Among other disadvantages of the GBOs (see, for example, Deschampset al., 1998) are the fact that they might get trapped in the local minima of the en-ergy function, and might also produce incorrect results for complex energy func-tions of the type given by Eq. (7.51) below. They also require computation of thederivatives of the functions, thus making them expensive as well as accurate initialguesses for the parameters to be optimized because, otherwise, the optimizationtechnique might produce different results for different initial guesses.

Thus, the SA method (Kirkpatrick et al., 1983) and the GA (Reeves and Rowe,2003) have been utilized frequently in the reconstruction methods. Let us definean “energy” E by

E DnX

j D1

f

O j

fS j

2 , (7.50)

where O j and S j represent the observed (or measured) and the correspondingsimulated (or calculated) properties of the porous medium, respectively, with n

being the number of data points. If there are more than one set of data for distinct

properties of the porous medium, the energy E is generalized to

E DmX

iD1

Wi Ei , (7.51)

where Ei is the total energy for the data set i, defined by Eq. (7.50), and Wi is thecorresponding weight, as two distinct set of data for the same porous medium donot have the same weight or significance.

To initiate the reconstruction process, one begins with an initial guess for thestructure of the porous medium and uses the SA method or the GA or any othersuitable technique. For example, in the SA approach, two randomly-selected pointsof the model are then interchanged. For example, if the model is in the form of apore network, two pores are selected at random and their radii are interchanged.The new energy E 0 of the system and the energy difference, ∆E D E 0 E , arethen computed. The interchange is then accepted with a probability p (∆E ). Then,according to the Metropolis algorithm,

p (∆E ) D8<:

1 , ∆E 0 ,

exp

∆E

T

, ∆E > 0 ,

(7.52)

where T is a fictitious temperature.Statistical mechanics of thermal systems provides the theoretical foundation for

such an algorithm. We know that if a system is heated up to a high temperature T

7.4 Reconstruction of Porous Media 203

and then slowly cooled down to absolute zero, the equilibrium state of the systemwill be its ground state. The cooling, usually called the annealing schedule, is select-ed to be sufficiently slow so as to allow the system to reach its true equilibriumstate rather than getting trapped in a local energy minimum (a metastable state),and attain its true global minimum energy. At each annealing step i, the system isallowed to evolve long enough to “thermalize” at T(i). We then lower T according toa prescribed annealing schedule. The cooling continues until the true ground stateof the system is reached within some acceptable error, that is, when E is deemed tobe small enough. This concludes the reconstruction process.

Such a procedure was utilized by Yeong and Torquato (1998a,b), Cule and Torqua-to (1999), Manwart and Hilfer (1999), and Sheehan and Torquato (2001) for recon-structing various models of random media. Manwart et al. (2000) used the methodto reconstruct sandstones, given some data for the porous medium. Talukdar andTorsaeter (2002) and Talukdar et al. (2002a,b) used the SA method to reconstructa variety of porous media, including chalk. Levitz (1998) reviewed methods of off-lattice reconstruction of porous media.

Another reconstruction method is based on mimicking the actual processes thathave given rise to the present natural porous media, and in particular, sedimentaryrock. We already described one such reconstruction method, namely, the grain-consolidation model. Bakke and Øren (1997) developed a more advanced versionof the method by incorporating grain size distribution and other petrographicaldata obtained from 2D thin sections. Their reconstruction model attempts to mim-ic three fundamental processes that contribute to the formation of sandstones,namely, sedimentation, compaction, and diagenesis. Øren et al. (1998) applied themethod to reconstructing Bentheimer sandstone, Øren and Bakke (2002) did thesame for Fontainebleau sandstone, while Øren and Bakke (2003) reconstructedthe Bera sandstone. In all cases, they successfully predicted the flow and trans-port properties, including the drainage and imbibition relative permeabilities (seeChapter 15) to two-phase flow in the porous medium.

To model sedimentation, the measured grain size distribution was used. Grainswere picked at random from their size distribution and dropped into a predefinedbox using a sequential deposition algorithm that simulated the successive deposi-tion of the individual grains in a gravitational field. Each grain falls under its ownweight and, after hitting the surface, tries to find the most stable position to settlein.

The compaction process, which plays an important role in decreasing the poros-ity, was modeled as a linear process in which the vertical ordinate of every sandgrain was shifted downwards according to

z D z0 (1 c) , (7.53)

where z0 is the initial position. Here, 0 c 1 is a compaction parameter thatcontrols the amount of compaction.

A diagenetic process was modeled based on the work of Schwartz and co-workerswho developed the grain consolidation model described earlier. Thus, quartz ce-


ment growth of particles with initial radius R0 was modeled according to

R(r) D R0 C min[al(r)γ , l(r)] . (7.54)

Here, l(r) is the distance between the surface of the original spherical grains andthe surface of its Voronoi polyhedron (see Chapter 3 for the relation between thepacking of spherical particles and the Voronoi polyhedra) along the direction r . Theconstant a controls the amount of cement growth and, hence, the porosity, and γcontrols the direction of growth. For γ > 0, the growth of the quartz cement inthe direction of large l(r) is favored (i.e., pore bodies shrink), whereas the oppositeis true for γ < 0, that is, in the direction of the pore throats, which means thatthey are blocked off with the cement growth. The growth increases the tortuosi-ty and leads to nonzero percolation threshold for the porosity, which was alreadymentioned when we described the grain consolidation model.

Two other ingredients were included in the reconstruction method. One was thelocal porosity distribution, first suggested by Hilfer (1991a, 1992); see also Hilfer(1996) and Biswal et al. (1998, 1999), and described briefly in Chapter 4. The essen-tial idea is to measure geometric observables within a finite subset of a porousmedium and then to prepare their histograms or distributions. Then, the localporosity distribution represents the probability of finding the local porosity in theinterval (φ ∆φ, φ C ∆φ), which is similar to spatial distribution of the porosity(and other characteristics) of the FS porous media described in Chapter 5, exceptthat the size of the cells within which the porosity is measured could be quite smalland, therefore, applicable to even small-scale samples.

To incorporate the effect of the connectivity in their reconstruction, Øren andBakke used the concept of local percolation probabilities introduced by Hilfer(1991a, 1996). The local percolation probability Pi is the fraction of measurementcells of sidelength L and local porosity φ that percolates in the direction i.

Hilfer and Manwart (2001) computed the effective permeability and conductiv-ity of several reconstructed models of a Fontainebleau sandstone, and comparedthe results with the data. They reported that although the geometrical propertiesof the models were similar, the flow and transport properties were not. Only thereconstruction model of Øren and Bakke described earlier provided accurate esti-mates of the flow and transport properties. Hilfer and Manwart (2001) stated thatthe reason for the discrepancies between the various models is apparently due tothe truncation of the correlations in the reconstruction process. A more detailedanalysis of Fontainebleau sandstone and comparison with reconstruction modelswas made by Manwart et al. (2002) for several models. The conclusion was that thereconstruction model that takes into account the diagenesis of porous media alsoproduce accurate predictions for their flow and transport properties. Most recently,Latief et al. (2010) developed a model of Fontainebleau sandstone using 2D images,which represents an extension of the model of Biswal et al. (2007) briefly describedearlier. See also Thovert et al. (2001) for an earlier reconstruction model.

Another comparison between reconstruction models of Berea sandstone and theactual data was made by Biswal and Hilfer (1999). They computed the local porositydistributions of the model and compared the results with the data.

7.5 Models of Field-Scale Porous Media 205

7.5Models of Field-Scale Porous Media

At the next level of complexity are field-scale (FS) porous media. The discussion inChapter 4 should have made it clear that small- or laboratory-scale porous mediacan be analyzed in great detail, and reasonable understanding of their structure andproperties can be obtained. This is not, however, the case with the FS porous media.As described in Chapter 6, experimental data for the important properties of suchporous media are usually grossly incomplete, their flow properties, such as theeffective permeability, vary greatly over distinct length scales by at least a few ordersof magnitude, and the relations between such spatially-varying properties and thevolume or the relevant length scales are often unknown. Such complexities makethe transition from models of small porous media to the FS type very difficult.

Even if detailed information about the variability of all the quantities of interest atthe FS is available, we still may not be able to obtain a reasonably complete under-standing, and meaningful average properties of the porous medium at large lengthscales. Due to such reasons, modeling of the FS porous media usually involvesstochastic techniques. The reason for using stochastic techniques is that althoughthe FS porous media are, in principle, intrinsically deterministic due to the rea-sons just discussed, one often must think of them in stochastic terms and describetheir properties in terms of statistical quantities because there are never enoughdata to reduce the uncertainty to negligible levels. By the same token, a continu-um approach to fluid flow and transport in FS porous media based on the classicalequations of flow and transport is often unsuccessful because such equations canprovide information only about quantities that vary in a deterministic manner. Onecan, of course, modify the continuum approach by developing stochastic equations

of flow and transport in which one or more variables, such as the permeability,porosity, and the fluid velocity vary stochastically. We will describe such approach-es in Chapter 11, where we study transport and dispersion in porous media. Here,it suffices to point out that two major problems restrict the usefulness of stochasticequations of flow and transport. One is that the appropriate form of the governingstochastic equations is not often obvious. The second problem is that even if oneknows and uses the appropriate stochastic equations, taking into account the effectof long-range correlations that often exist in the FS porous media (see Chapter 5)is very difficult.

Due to such difficulties, stochastic discrete models have been developed for de-scribing the FS porous media. Such models represent, in some sense, an extensionof the network models for small-scale porous media, except that in such models, ablock of the network represents a portion of the porous medium at a length scaleover which information is available. Moreover, it is relatively straightforward to in-clude the effect of long-range correlations in such discrete models. A comprehen-sive discussion of such models would be too long to be given here. The interestedreader should consult Haldorsen et al. (1988). Here, we briefly describe four ba-sic approaches to modeling of FS porous media. Most of such approaches model2D porous media, as the thickness of the FS porous media in which fluid flow and


transport take place is usually small compared with their length or width and, thus,such porous media are essentially 2D system. It is, however, not difficult to extendsuch approaches to 3D FS porous media.

7.5.1Random Hydraulic Conductivity Models

In this approach, the 2D porous medium is represented by a rectangle which isdiscretized into many smaller rectangular blocks that are supposed to represent aportion of the medium that is homogeneous on the scale of the block’s size. To eachblock, a randomly-selected hydraulic conductivity is assigned. This type of modelwas pioneered by Warren and Skiba (1964) and Heller (1972). In both studies, it wasassumed that there are no correlations between the conductivities of the blocks.Schwartz (1977) modified the model by inserting blocks of lower conductivities inan otherwise homogeneous 2D region. One can also accommodate a non-randomspatial structure by controlling the density and mode of aggregation of the insertedblocks. In principle, the blocks do not have to be rectangular.

Smith and Freeze (1979) and Smith and Schwartz (1980, 1981a,b) modified thisbasic model by including correlations between the blocks’ hydraulic conductivitiesthat usually exist in the FS porous media (see Chapter 5). In their model, it wasassumed that the spatial variations of the hydraulic conductivities are described bya statistically-homogeneous stochastic process. The spatial structure of the conduc-tivity field was represented by a first-order nearest-neighbor stochastic process. Itwas assumed that the hydraulic conductivity gb of the blocks is log-normally dis-tributed but, of course, any other distribution may also be used. If Y D log gb ,then the first-order nearest-neighbor stochastic process implies that Yi j , the ran-dom variable for the block whose center’s coordinates are (i, j ), is given by

Yi j D αx (Yi1, j C YiC1, j ) C αz (Yi, j 1 C Yi, j C1) C i j , (7.55)

where αx and αz are, respectively, autoregressive parameters that express the de-gree of spatial dependence of Yi j on its two neighboring values in the x and z

directions, and i j is a normal random variable uncorrelated with all the otheri j . If αx D αz , then the medium has a statistically-isotropic covariance struc-ture. Otherwise, the porous medium is anisotropic and the covariance between theconductivity values is dependent upon the orientation. The random variables i j

are distributed according to a normal distribution with a zero mean and a givenvariance.

7.5.2Fractal Models

The models developed by Smith and Freeze (1979) and Smith and Schwartz (1980,1981a,b) that contained short-range correlations between the hydraulic conductivi-ties of the blocks were significantly generalized by many others in which the perme-ability and porosity of the blocks followed fractal statistics and, therefore, contained


correlations with an extent that is comparable with the linear size of the system(see Chapter 5). As described in Chapter 5, this type of model was first suggestedby Hewett (1986) who showed that the vertical porosity logs may be representedaccurately by a fractional Gaussian noise (FGN), while the lateral distribution ofthe porosity may follow the statistics of a fractional Brownian motion (FBM). Thetwo stochastic distributions induce long-range correlations in the property values,characterized by a Hurst exponents 0 < H < 1 (see Chapter 5). A FBM with withH > 0.5 induces positive or persistent correlations, while one with H < 0.5 gen-erates negative or anti-persistent correlations. The limit H D 1 represents a totallysmooth distribution.

7.5.3Multifractal Models

Multifractal models were suggested by Meakin (1987) and Lenormand et al. (1990)as a tool for generating highly heterogeneous porous media with long-range corre-lations. Consider a 2D system, for example, a square grid, and a probability p thatcan be related at the end of the construction of the model to a measure, such as, thepermeability or porosity, and is distributed uniformly in the interval (1 a, 1 C a)with 0 a 1. In the first step of constructing the model, a value p11 is se-lected at random and is attributed to all the pixels of the initial square. The firstdichotomy is then carried out to make four squares of size 2n1 2,n1 and fourvalues, p21, p22, p23, and p24, are selected at random and attributed to each of thefour squares. The same procedure is repeated n times. At the end of the process,each pixel is characterized by n values of the probability p. A new measure P is thendefined that is the product of the n random values of p.

The main property of this process is that similar to any fractal distribution, itintroduces correlations between pixels at all scales. A lower cut off can also beintroduced into the model by considering m steps of dichotomy, where m < n

and n is the dimension of the pattern. One first makes 2nm 2nm independentmultifractal patterns of size m0, and then computes 2nm 2nm independentproducts pmC1 pmC2 . . . pn . The pixel values of each multifractal are thenmultiplied by the products in order to obtain products of order n. This procedurecan be further generalized to anisotropic media by considering two probabilities px

and p y for the x- and y-axes, distributing px and p y uniformly in (1 ax , 1 C ax )and (1 ay , 1 C ay ), and identifying p by the product px p y .

For simulating flow in a FS porous medium, the measure p may be thought of asthe effective permeability of a portion of the porous medium. Given a pattern and apermeability distribution, flow and displacement processes in such a porous medi-um can be simulated and investigated. Mukhopadhyay and Sahimi (2000) used themultifractal model to study flow in the FS porous media.


7.5.4Reconstruction Methods

Just as one may develop a model of a small-scale porous medium by a reconstruc-tion method that was described in Section 7.4, one may utilize the same for devel-oping a model for a FS porous medium. The difference is that a FS porous mediummay require a much larger grid and, therefore, determining the global minimumof the energy function defined by Eq. (7.50) or (7.51) is a much more difficult task.

The simulated annealing (SA) method has been used in the past for reconstruct-ing models of the FS porous media, for example, oil reservoirs. For example, Pandaand Lake (1993) used the SA method together with a parallel computational strate-gy in order to reconstruct a model of an oil reservoir by optimizing its permeabilityfield. Quenes and Saad (1993) also used the SA method for oil reservoir character-ization, utilizing a parallel computational strategy, as did Savioli et al. (1996) andSaccomano et al. (2001). Sen et al. (1995) compared the performance of the SA andGA methods for stochastic reservoir modeling. Both methods were able to pro-duce major features of the reservoir’s permeability distribution, and generated theappropriate pressure profiles.

The main disadvantage of the SA method is that it cannot be used easily inmassively-parallel computations as efficiently as the GA can be. Although algo-rithms have been developed for this purpose (see, for example, Bevilacqua, 2002;Ye and Lin, 2006), they have yet to be tested for the type of large and complex opti-mization problems that one must tackle for reconstructing a FS porous medium.

The GA (see, for example, Reeves and Rowe, 2003) has also been utilized inthe reconstruction of the FS porous media. For example, Romero et al. (2000) andRomero and Carter (2001) used a GA to develop a model of an oil reservoir byoptimizing the performance of a flow simulator that computed the pressures andwater flow rates at a number of extraction wells in the reservoir. Sanchez et al.

(2007) utilized the GA to reconstruct a model of a landfill, which is a large-scaleporous medium with the additional complexity that, in addition to flow and trans-port, biodegradation and reactions with nonlinear kinetics also occur there.

One important advantage of the GA over other optimization techniques is thatit is not sensitive to the initial guesses that one utilizes for the parameters that areto be determined and optimized, for example, the spatial distributions of the per-meabilities and porosities. Thus, even an arbitrary initial guess may be employedby the GA. The GA ensures that practically the entire search domain is tested foroptimality and, therefore, the probability of finding the global minimum of theobjective function can, in principle, approach one.

Another advantage that the GAs have over many deterministic optimization tech-niques is that there is no need for calculating any derivative of the energy function,which can be computationally difficult and expensive if the function is complex.One disadvantage of the GAs is that its use can be computationally expensive ifcomputing the energy function is time consuming, so much so that some optimiza-tion problems might require many weeks or even months for completion. Howev-er, the GAs spend up to 99% of the total computation time to evaluate the energy


function. Due to this fact, utilizing high-performance computers and massively-parallel computational strategies alleviates the problem.

Since the reconstruction method based on the SA algorithm was already de-scribed in Section 7.4, it may be useful to also describe the GA. The descriptionand discussion of the GA that we present below follow those given by Sanchez et

al. (2007).

7.5.4.1 The Genetic Algorithm for ReconstructionAny optimization problem that utilizes a GA involves four steps:

1. the selection process for generating solutions;2. the design of the “genome” to constrain the variables that define a possible so-

lution, and the generation of the “phenotype” which, in the present problem, isthe model of transport and reaction that we use;

3. the crossover and mutation operations that are used for generating new solutionsand approaching the true optimal one, and

4. eliticism, which selects those solutions that eventually lead to the global mini-mum of the energy function.

Each time we carry out all four sets of operations, we have completed one generation

of the computations. Let us briefly describe the various steps that are involved inusing the GAs.Selection: In Darwin’s theory, species that can adapt to their environment are se-lected to produce the next generation of the offsprings. In reconstruction of a FSporous medium, each specie is the set of all the parameters that we identify for de-termining their optimal values, given some data for one or more properties of themedium. Such parameters usually include the spatial distributions of the perme-abilities and porosities. Selection is the method by which the species are chosen toproduce the offsprings – the updated species for the next generation. Except for thefirst generation (the first trial species), selection of the species in the GAs is basedon the evaluation of the energy function defined by Eq. (7.50) or (7.51). The specieswith a smaller energy function possesses a greater probability of producing one ormore offsprings for the next generation. The set of all the possible species that weexamine during the optimization process is called the population. One must specifythe population’s size at the beginning of the optimization process.

As mentioned above, the GAs are not sensitive to the initial guesses (the initialpopulation of the species) for the parameters to be optimized. Therefore, the firstpopulation of the species may be generated randomly in the ranges over whichthe parameters to be optimized are expected to vary. Of course, if one has someadditional insights into the spatial distributions of the parameters to be optimized,they are also used for generating more accurate initial species and their population.

Using the population of the species, we use a model for numerical simulationof the phenomenon in the porous medium for which there are some data. Forexample, if the data are the time-dependent pressures at one or more production


wells in an oil reservoir, we solve the equation for the pressure transient analysisin order to compute the time-dependent pressures at the production wells. Then,the next species is selected based on the evaluation of the energy function. Thereare four common selection techniques used in the GAs. We describe one of themthat is called the binary tournament selection, which is as follows.

We set a priori the probability ps of selection. The higher ps, the more likely it isthat the GA will eventually yield the true optimal solution. One selects a value ofps that is high enough to ensure a reasonably accurate solution, but not too highto make the computations prohibitively expensive. One then picks at random twoof the species from the population. A random number r, uniformly distributed in(0,1), is generated. If r < ps, one selects the species that resulted in a smallerenergy. Otherwise, the one with a larger energy is picked. Another pair of speciesis then selected from the population and the same procedure is repeated. Eachpair of the selected species constitute a mating pair. Note that since the species arechosen at random, any species may be selected numerous times, which is why thepopulation of the selected species generally results in a set with lower energies.Repeating this procedure a large number of times finally yields a set of matingpairs, the size of which is half the population’s size. Each mating pair can generatetwo offsprings during the crossover step. This completes the selection process.

Crossover is achieved when a mating pair produces two offsprings. To do so,the values of the parameters to be optimized are first converted to binary num-bers involving zero and one. Similar to the selection, there are several types ofcrossover methods that one may utilize. One may, for example, utilize uniform

crossover, since it has been shown to encourage the exploration of a robust searchdomain or population. For each parameter value in its binary representation, theGA selects a zero and one at random and exchanges their places in the param-eter’s binary representation. The procedure is then repeated a number of timesto generate a new binary representation of that parameter of the offspring. Forexample, suppose that two of the parameters to be optimized are represented by01110011010 and 10101100101. Then, one generates, by a uniform crossover, theoffsprings 01110100101 and 10101011010. This procedure is repeated for all theparameters in all the mating pairs, resulting at the end in the new species or off-springs (which explains why using a large population of the species is very useful).The crossover step is then completed.

Mutation: After the offsprings are generated, it is possible that they exhibit somekind of “defect” which exhibits itself by resulting in an energy of larger numericalvalue. Such a defect in the GAs is known as mutation. The mutation probability isselected from a Gaussian distribution with zero mean and a given variance (e.g.,0.1). All the binary representations of the parameters are converted back to realnumbers, hence generating the new population of the species. Numerical simula-tion of the phenomenon for which the data are available is then carried out usingthe new population of the offsprings or species, and the energies are computedfor all the species based on the newly-calculated properties for which the data areavailable. This completes one generation of the computations. Then, the entireprocedure, consisting of selection, crossover, mutation, and eliticism (see below),


is repeated again, resulting in the second generation. Enough generations are pro-duced until the true optimal solution (in the sense that the minimum of the energyfor a species is its true global minimum, not a local one) is obtained.

Eliticism: In order to speed up the computations and ensure that the true globalminimum of the energy will eventually be reached, one also uses eliticism that, forevery generation, selects those species with the energies of the lowest values andinserts them into the next generation.

Let us point out that the efficiency and performance of any GA depend on the nu-merical values of its parameters, ranging from the population size, to the selection,crossover, and mutation probabilities. This is a virtue of any GA since, by selectingproper (optimal) values of such parameters, the method’s efficiency can be greatlyincreased, while lowering its computational cost.

7.5.4.2 Reconstruction Based on Flow and Seismic DataHamzehpour and Sahimi (2006b) and Hamzehpour et al. (2007) developed a re-construction method for the FS porous media that utilized three distinct types ofdata.

1. Flow data that provide insight into the connectivity of the permeable regions of aporous medium. Hamzehpour et al. (2007) utilized data for the time-dependentfluid pressure at a production well.

2. Seismic data that provide insight into large-scale heterogeneities of a porousmedium. Hamzehpour et al. (2007) utilized the arrival times of seismic wavesat particular points in a porous medium.

3. Porosity logs (or any other type of log) that provide information about the local

heterogeneities of a porous medium.

Using the SA method for minimizing the energy of the system, Hamzehpour andSahimi (2006b) and Hamzehpour et al. (2007) showed that their reconstructionmethod produces models of the FS porous media that provide accurate informationfor those properties of porous media for which no data are available. In particular,Hamzehpour et al. (2007) showed that the reconstructed model provides accuratepredictions for two-phase flow in porous media, which is usually very difficult todo, without using any data for the two-phase flow.

The crucial aspect of the reconstruction model of Hamzehpour et al. (2007) wasthat they used fluid flow data. Such data are of fundamental importance to not onlysingle-phase flows, but, more importantly, to two-phase flows. Without such dataand given that it is very difficult to obtain other types of data that provide informa-tion on the connectivity of high-permeability regions of a FS porous medium, noreconstruction method can be expected to result in realistic models of porous me-dia that can provide accurate predictions for flow and transport processes in suchmedia.

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