Hunt a percolation theory for flow in porous media (lnp 674 2005)(isbn 3540261109)(212s)

Thenovelty in this course is the combined use of both scaling and critical pathapplications of percolation theory to realistic models of porous media; us-ing this combination it is possi

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Allen G Hunt

Percolation Theory

for Flow in Porous Media

ABC

Allen G Hunt

Department of Physics and Geology

Wright State University

Dayton, OH 45431

U.S.A

Email: allenghunt@msn.com

Allen G Hunt, Percolation Theory for Flow in Porous Media,

Lect Notes Phys 674 (Springer, Berlin Heidelberg 2005), DOI 10.1007/b136727

Library of Congress Control Number: 2005930812

ISSN 0075-8450

ISBN-10 3-540-26110-9 Springer Berlin Heidelberg New York

ISBN-13 978-3-540-26110-0 Springer Berlin Heidelberg New York

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Though a sledge hammer may be wonderful for breaking rock, it is a poorchoice for driving a tack into a picture frame There is a fundamental, thoughoften subtle, connection between a tool and the application When Newtonand Leibniz developed the Calculus they created a tool of unprecedentedpower The standard continuum approach has served admirably in the de-scription of fluid behavior in porous media: the conservation of mass andlinear response to energy gradients fit conveniently, and are solid foundationsupon which to build But to solve these equations we must characterize theup-scaled behavior of the medium at the continuum level The nearly univer-sal approach has been to conceive the medium as a bundle of capillary tubes.Some authors made the tubes porous, so they could fill and drain throughtheir walls; others “broke and reconnected” them so each tube had a range ofdiameters along its length In the end it must be admitted that the marriage

of tool (capillary tube bundles) and task (to derive the constitutive relationsfor porous media) has not yet proven to be entirely satisfactory Lacking inthese conceptual models is a framework to describe the fluid-connected net-works within the medium which evolve as functions of grain size distribution,porosity, saturation, and contact angle This is fundamentally a geometryproblem: how to concisely describe the particular nature of this evolving,sparse, dendritic, space-filling network

Recognizing this basic problem, the community flocked to the fractal els as they became better understood in the 1990s But fractals alone werenot enough, as the real problem was to understand not the geometry of themedium, but the geometry of the fluids within the medium, and moreover,

mod-to correctly identify the geometry of the locations that control the flow

I met Allen Hunt in the late 1990s, and over coffee he described his ideasabout critical path analysis for the development of constitutive relationshipsfor unsaturated conductivity I was immediately sold: it was transparent thatthe geometric model (with the equally important framework for mathemat-ical analysis) was ideally suited to the problem at hand Since resistance toflow is a function of the fourth power of the pore aperture, clearly the keywas to systematize the determination of the “weak link” to compute overallresistance to flow Paths that had breaks were irrelevant; and paths that con-tained very small pores provided negligible contribution The permeability

should be proportional to the fourth power of the radius of the smallest pore

in the connected path which has the largest small pore Read that sentencetwice: we are looking for the path of least resistance, and that path’s re-sistance will be a function of the smallest pore in that path Allen had thetool to identify this path as a function of fluid content A very useful, ap-propriately sized, hammer had arrived for our nail Over the following yearsAllen’s work showed the power of using the right tool: he could explain therelationship between the geometry of the medium and liquid content versuspermeability, residual fluid content, electrical resistance, diffusion of solutes,and even the thorny issues of the scale of a representative elementary unit.Critical path analysis is not a panacea, but due to the focus on the control-ling geometric features, it provides a remarkably concise parameterization offluid–medium relationships based on physically measurable properties thataccurately predict many of the basic ensemble properties

A fundamental problem in having these results be broadly understood andadopted is sociological Consider how much time we spend learning calculus

to solve the occasional differential equation Critical path analysis requirescalculus, but also understanding of the mathematics of fractals, and the geo-metric strategy of percolation theory When Allen started his remarkablyproductive march into flow through porous media he deftly employed thesetools that none of our community had mastered There is a natural inertia

to any discipline since re-tooling requires major investments of time Fromthis perspective I have long encouraged Allen to help the community makeuse of this essential set of tools by providing a primer on their application toflow though porous media In this book Allen has once again moved forwardstrategically, and with great energy He has provided an accessible tutorialthat not only provides the bridge for the hydrologist to these new tools, butalso the physicist a window into the specialized considerations of flow throughnatural porous media

Learning new mathematical constructs is much like learning a new guage There is a great deal of investment, and the early effort has few re-wards Ultimately, however, without language there is no communication.Without mathematics, there is no quantitative prediction If understandingthe behavior of liquids in porous media is central to your work, I urge you

lan-to make the investment in learning this material By way of this book Allenprovides a direct and efficient avenue in this venture Your investment will

be well beyond repaid

April, 2005

The focus of research in porous media is largely on phenomena How do youexplain fingering? What causes preferential flow? What “causes” the scaleeffect on the hydraulic conductivity? Why can the incorporation of 5% of hy-drophobic particles into soil make the soil water repellent? Where do long tails

in dispersion come from? These are merely a few examples of a very long list ofquestions addressed The approach to “solving” problems is frequently to (1)take standard differential equations such as the advection–diffusion equationfor solute transport, or Richards’ equation for water transport; (2) substituteresults for what are called constitutive relations such as the hydraulic con-

ductivity, K, molecular diffusion constants, or air permeability as functions

of saturation, and pressure-saturation curves, including hysteresis, etc.; (3)apply various models for the variability and the spatial correlations of thesequantities at some scale; and (4) solve the differential equations numericallyaccording to prescribed initial and/or boundary conditions In spite of contin-uing improvement in numerical results, this avenue of research has not led tothe hoped-for increase in understanding In fact there has been considerablespeculation regarding the reliability of the fundamental differential equations(with some preferring fractional derivatives in the advection–diffusion equa-tion, and some authors questioning the validity of Richards’ equation) whileothers have doubted whether the hydraulic conductivity can be defined atdifferent scales

Although other quite different approaches have thus been taken, let usconsider these “constitutive” relations The constitutive relationships usedtraditionally are often preferred because (1) they generate well-behaved func-tions and make numerical treatments easier; (2) they are flexible This kind

of rationale for using a particular input to a differential equation is not likely

to yield the most informative solution The most serious problem associatedwith traditional constitutive relations is that researchers use such concepts asconnectivity and tortuosity (defined in percolation theory) as means to ad-just theory to experimental results But the appropriate spatial “averaging”scheme is inextricably connected to the evaluation of connectivity In fact,when percolation theory is used in the form of critical path analysis, it is notthe spatial “average” of flow properties which is relevant, but the most resis-tive elements on the most conductive paths, i.e the dominant resistances on

the paths of least resistance An additional problem is that usual constitutiverelations often cover simultaneous moisture regimes in which the representedphysics is not equilibrium, and thus time-dependent, as well as those mois-ture regimes where the dominant physics is equilibrium, so that they must

be overprescribed (while still not describing temporal effects) Finally, therehas been no progress in making the distributions and spatial correlations of,

e.g K, consistent with its values at the core scale, because there is no

sys-tematic treatment of the connectivity of the optimally conducting regions

of the system This book shows a framework that can be used to develop aself-consistent and accurate approach to predict these constitutive relation-ships, their variability, spatial correlations and size dependences, allowing use

of standard differential equations in their continuum framework (and, it ishoped, at all spatial scales)

Although applications of percolation theory have been reviewed in theporous media communities (e.g Sahimi, 1993; Sahimi and Yortsos, 1990)(in fact, percolation theory was invented for treating flow in porous media,Broadbent and Hammersley, 1957) it tends to be regarded as of limited ap-plicability to real systems This is partly a result of these summaries them-selves, which state for example that “Results from percolation theory arebased on systems near the percolation threshold and the proximity of realporous rocks to the threshold and the validity of the critical relationshipsaway from the threshold are matters of question,” (Berkowitz and Balberg,1993) However, it is well-known that percolation theory provides the mostaccurate theoretical results for conduction also, in strongly disordered sys-tems far above the percolation threshold (using critical path analysis) Thenovelty in this course is the combined use of both scaling and critical pathapplications of percolation theory to realistic models of porous media; us-ing this combination it is possible to address porous media under generalconditions, whether near the percolation threshold or not

This book will show how to use percolation theory and critical path sis to find a consistent and accurate description of the saturation dependence

analy-of basic flow properties (hydraulic conductivity, air permeability), the cal conductivity, solute and gas diffusion, as well as the pressure–saturationrelationships, including hysteresis and non-equilibrium effects Using suchconstitutive relationships, results of individual experiments can be predictedand more complex phenomena can be understood Within the framework ofthe cluster statistics of percolation theory it is shown how to calculate the

electri-distributions and correlations of K Using such techniques it becomes easy

to understand some of the phenomena listed above, such as the “scale” effect

on K, as well.

This work does not exist in a vacuum In the 1980s physicists and leum engineers addressed basic problems by searching for examples of scalingthat could be explained by percolation theory, such as Archie’s law (Archie,1942) for the electrical conductivity, or invasion percolation for wetting front

petro-behavior, hysteresis, etc or by using the new fractal models for porous media.The impetus for further research along these lines has dwindled, however, andeven the basic understanding of hysteresis in wetting and drainage developed

in the 1980s is lacking today, at least if one inquires into the usual literature

In addition, the summaries of the work done during that time suggest thatthe percolation theoretical treatments are not flexible enough for Archie’s law(predict universal exponents), or rely on non-universal exponents from con-tinuum percolation theory without a verifiable way to link those exponentswith the medium and make specific predictions An identifiable problem hasbeen the inability of researchers to separate connectivity effects from pore-size effects This limitation is addressed here by applying percolation scalingand critical path analysis simultaneously While there may have been addi-tional problems in the literature of the 1980s (further discussed here in theChapter on hysteresis), it is still not clear to me why this (to me fruitful)line of research was largely abandoned in the 1990s This book represents anattempt to get percolation theory for porous media back “on track.”

It is interesting that many topics dealt with as a matter of course byhydrologists, but in a rather inexact manner, are explicitly treated in perco-lation theory Some examples are:

1 upscaling the hydraulic conductivity = calculating the conductivity frommicroscopic variability,

2 air entrapment = lack of percolation of the air phase,

3 residual water, oil residuals = critical moisture content for percolation,sum of cluster numbers,

4 grain supported medium = percolation of the solid phase;

5 Representative Elementary Volume = the cube of the correlation length

of percolation theory,

6 tortuosity = tortuosity,

7 flow channeling = critical path

These concepts and quantities are not, in general, treatable as tion functions or parameters in percolation theory because their dependencesare prescribed Note that in a rigorous perspective for disordered systems,

optimiza-however, one does not “upscale” K The difficulty here is already contained

within the language; what is important are the optimal conducting paths,not the conductivities of certain regions of space The conductivity of thesystem as a whole is written in terms of the rate-limiting conductances onthe optimal paths and the frequency of occurrence of such paths Definingthe conductivity of the system as a whole in terms of the conductivities ofits components is already a tacit assumption of homogeneous transport Fur-ther, some elementary rigorous results of percolation theory are profoundlyrelevant to understanding flow in porous media In two-dimensional systems

it is not possible for even two phases to percolate simultaneously (in a supported medium there is no flow or diffusion!), while in three dimensions a

grain-number of phases can percolate simultaneously As percolation thresholds areapproached, such physical quantities as the correlation length diverge, andthese divergences cause systematic dependences of flow and transport prop-erties on system size that can only be analyzed through finite-size scaling.Thus it seems unlikely that treatments not based on percolation theory can

be logically generalized from 2D to 3D

I should mention that a book with a similar title, “Percolation Modelsfor Transport in Porous Media,” by Selyakov and Kadet (1996) also notedthat percolation theory could have relevance further from the percolationthreshold, but overlooked the existing literature on critical path analysis, andnever mentioned fractal models of the media, thereby missing the importance

of continuum percolation as well As a consequence, these authors did notadvance in the same direction as this present course

The organization of this book is as follows The purpose of Chap 1 is toprovide the kind of introduction to percolation theory for hydrologists which(1) gives all the necessary basic results to solve the problems presented later;and which (2) with some effort on the part of the reader, can lead to arelatively solid foundation in understanding of the theory The purpose ofChap 2 is to give physicists an introduction to the hydrological science liter-ature, terminology, experiments and associated uncertainties, and finally atleast a summary of the general understanding of the community This generalunderstanding should not be neglected as, even in the absence of quantita-tive theories, some important concepts have been developed and tested Thusthese lecture notes are intended to bridge the gap between practicing hydrol-ogists and applied physicists, as well as demonstrate the possibilities to solveadditional problems, using summaries of the background material in the firsttwo chapters Subsequent chapters give examples of critical path analysis forconcrete system models Chap 3; treat the “constitutive relationships for un-saturated flow,” including a derivation of Archie’s law Chap 4; hysteresis,non-equilibrium properties and the critical volume fraction for percolation

Chap 5; applications of dimensional analysis and apparent scale effects on K

Chap 6; spatial correlations and the variability of the hydraulic conductivityChap 7; and multiscale heterogeneity Chap 8

I wish to thank several people for their help in my education in hydrologyand soil physics, in particular: Todd Skaggs, whose simulation results haveappeared in previous articles and also in this book; John Selker, who showed

me the usefulness of the Rieu and Sposito model for the pore space; GlendonGee, who helped me understand experimental conditions and obtain datafrom the Hanford site; Eugene Freeman for providing additional Hanfordsite data; Bill Herkelrath, again for data; Toby Ewing, whose simulationsfor diffusion were invaluable; Tim Ellsworth for showing me the relevance ofthe experiments of Per Moldrup; Per Moldrup for giving me permission torepublish his figures; Max Hu for providing me with his diffusion data; and

Sally Logsdon for her data on soil structure; Alfred Huebler for giving me aforum among physicists to discuss these ideas I also thank my wife, BeatrixKarthaus-Hunt, for her support.

April, 2005

1 Percolation Theory 1

1.1 What is Percolation? 1

1.2 Qualitative Descriptions 2

1.3 What are the Basic Variables? 4

1.4 What is Scale Invariance and Why is it so Important? 6

1.5 The Relationship of Scale Invariance and Renormalization, and the Relationship of the Renormalization Group to Percolation Theory 7

1.6 Cluster Statistics of Percolation Theory 8

1.7 Calculation of the Critical Site Percolation Probability for the Two-dimensional Triangular Lattice and of the Critical Exponent for the Correlation Length in Two Dimensions 10

1.8 Mean-field Treatment of the Probability of being Connected to the Infinite Cluster 13

1.9 Value of p c for Bond Percolation on the Square Lattice 15

1.10 Estimations of p c for Bond Percolation on the Triangular and Honeycomb Lattices 15

1.11 Summary of Values of p c 17

1.12 More General Relationships for p c 18

1.13 Derivation of One-Dimensional Cluster Statistics and Discussion of Fractal Dimensionality 20

1.14 Argument for Dimensionally-Dependent Scaling Law, Implications for Critical Exponent, τ , and Applications to Critical Exponents 21

1.15 Exponents for Transport Properties 23

1.16 Finite-size Scaling and Fractal Characteristics 26

1.17 Summary of Derived Values of Critical Exponents 26

1.18 Critical Path Analysis 27

Problems 31

2 Porous Media Primer for Physicists 33

2.1 Background: Previous Studies 33

2.2 Relevant Soil Physics 35

2.2.1 Porosity and Moisture Content 35

2.2.2 Classification of the Pore Space 36

2.2.3 Network vs Fractal Models 39

2.2.4 Soil Morphology 40

2.2.5 Representative Elementary Volume and General Guidance for “Upscaling” 43

2.2.6 Porosity and Fractal Media 44

2.3 Soil Water Potential and Water Retention 49

2.4 Hysteresis and Time Dependences in Pressure-Saturation Relationships 54

2.5 Hydraulic and Transport Properties 58

2.6 Some Notes on Experimental Procedures 65

Problems 66

3 Specific Examples of Critical Path Analysis 67

3.1 r-Percolation 67

3.2 rE-percolation (Variable-Range Hopping) 74

3.3 Saturated Hydraulic Conductivity 79

3.4 Unsaturated Hydraulic Conductivity 83

Problems 86

4 Basic Constitutive Relations for Unsaturated Media 89

4.1 Hydraulic and Electrical Conductivities of Porous Media: Universal vs Non-universal Exponents in Continuum Percolation Theory 90

4.2 Air Permeability 107

4.3 Solute and Gas Diffusion 108

4.4 Electrical Conductivity for θ < θ t 118

Problems 119

5 Pressure Saturation Curves and the Critical Volume Fraction for Percolation 121

5.1 Structural Hysteresis 121

5.2 Hydraulic Conductivity Limited Equilibration and Dry-end Deviations from Fractal Scaling 126

5.3 Analysis of Water-Retention Curves in Terms of the Critical Moisture Content for Percolation 131

5.4 Wet-end Deviations from Fractal Scaling of Water-Retention Curves and Discussion of the Critical Volume Fraction for Percolation 135

5.5 General Formulation for Equilibrium and Analogy to Ideal Glass Transition 140

5.6 Oil Residuals 142

Problems 143

6 Applications of the Correlation Length 145

6.1 Effects of Dimensional Cross-overs on Conductivity 146

6.2 Comparison with Field Data 152

6.3 Effects of Hydrophobicity on Water Uptake of Porous Media 154

Problems 155

7 Applications of the Cluster Statistics 157

7.1 Spatial Statistics and Variability of K from Cluster Statistics of Percolation Theory 157

7.2 Cluster Statistics Treatment of Non-equidimensional Volumes and Anisotropy 163

7.3 Semi-Variograms and Cross-Covariance 168

Problems 171

8 Effects of Multi-Scale Heterogeneity 173

8.1 Soil Structure 173

8.2 Variable Moisture Content 180

8.3 A Simplified Hierarchical Problem 182

Problems 186

Summary 187

References 189

1.1 What is Percolation?

Percolation describes properties related to the connectivity of large numbers

of objects which individually have some spatial extent, and for which theirspatial relationships are relevant and statistically prescribed

Percolation theory comes in three basic varieties: bond, site, and uum We will consider all three varieties

contin-A simple bond percolation problem can be represented by a window screen

which maps out a square grid (lattice) Consider cutting a fraction p of the elements of this grid at random At some critical fraction p ≡ p c, (which willturn out to be 0.5), the window screen will lose its connectedness and fallapart Percolation theory addresses directly the question, “at what fraction

of cut bonds does the screen fall apart,” and related questions, such as, “what

is the largest hole that can be found in the screen if some fraction of bondsless than the critical fraction is cut,” and what the structure of such holes

is Percolation theory also readily answers the questions of what the cal conductivity of such an incompletely connected network of (conducting)bonds is, or what the diffusion constant of a network of the same structurecomposed of water-filled tubes would be

electri-A simple site percolation problem can be represented by the random placement of metallic and plastic balls in a very large container If two metalballs are nearest neighbors (touch each other) a current could pass from one

em-to the other If the number of metallic balls is high enough em-to reach a criticaldensity, a continuous connected path through metal can be established Thispath will conduct electricity The larger the fraction of metallic balls, thebetter connected the path will be and the larger the electrical conductivity.Percolation theory generates the electrical conductivity as a function of thefraction of the balls made of metal Site percolation problems can also bedefined on grids

A continuum percolation problem that had received attention already inthe 1970’s is a network of sintered glass and metallic particles Such networkshave analogues in the xerox industry If the detailed structure is known,percolation theory can predict the electrical conductivity of these systems aswell The continuum percolation problem that we will be most interested inhere is that of water flowing in variably saturated porous media

A.G Hunt: Percolation Theory for Flow in Porous Media, Lect Notes Phys 674, 1–31 (2005) www.springerlink.com
Springer-Verlag Berlin Heidelberg 2005c

In all of these variants of percolation theory it will be seen that the values

of p cvary widely from system to system, but that such relationships as thosethat give the size of the largest hole in the screen, or the conductivity, as afunction of the difference between the fraction of cut bonds and the critical

value, p c, are universal, Stauffer (1979); Stauffer and Aharony (1994) Hereuniversal means that the property is independent of the details of the system

and depends only on the dimensionality, d, of the medium The ultimate goal

of this book is to demonstrate how percolation theory can be used to solvepractical problems of transport and related properties of porous media Ithas been hoped that the universal behavior near the percolation thresholdcould be used to guide understanding of real physical systems (for exam-ple, Berkowitz and Balberg, 1993; Sahimi, 1993; Sahimi and Yortsos, 1990)

It has, however, also been pointed out more than once that it is not clearhow close real systems are to the percolation threshold Thus it is important

to emphasize at the outset that this book will explain the use of percolationtheory to calculate transport properties not merely near the percolation tran-sition, but also far from it The two methods have important differences Farfrom the percolation transition it will be non-universal aspects of percolation

theory, i.e the value of p c, which, together with the statistical characteristics

of the medium, tend to dominate, while near the percolation transition it isthe universal aspects that dominate This perspective will be seen to be farmore useful than a restriction to either case by itself, and it will be shown ul-timately to allow calculation of all the transport properties of porous media,

as well as their variability and the structure of their spatial correlations.This first chapter is devoted to the development of basic methods and con-cepts from percolation theory with some concentration on those subjects mostrelevant to the applications The material here is drawn from many sources,but most importantly from Stauffer (1979), Sahimi (1983) and Stauffer andAharony (1994) The second chapter will serve as an introduction to prob-lems and terminology of porous media Subsequent chapters will detail theapplications

1.2 Qualitative Descriptions

Consider again a square grid of points and connect line segments between

nearest neighbor points “at random.” For very small values of p these ments will only connect pairs of nearest neighbor sites As p increases more

seg-pairs will connect and gradually clusters of interconnected sites will appear

As p nears p c many of these clusters will become large, and their internalstructure will begin to be very complex The perimeter (the number of con-nected sites with boundaries on the exterior of the cluster) has two contribu-tions: one proportional to the volume (Kunz and Souillard, 1978), the second

plays a role of a surface area and is proportional to the volume to the 1-1/d

power, where d is the dimensionality (Stauffer, 1979) The radii of large

con-nected clusters is not given in terms of their volume by usual relationshipsvalid for Euclidean objects, and the density profile reflects a reduction inprobability with increasing distance that an arbitrary site is on the cluster

In fact the clusters are fractal objects, without scale reference (except in the

small scale limit when the scale of the grid becomes visible) As p passes p c

the largest interconnected cluster just reaches infinite size (and the largest

region without connected bonds just shrinks to finite size) For p > p c in thevicinity of percolation a very large number of sites connected to the infinitecluster are located on what are called “dead ends.” Dead ends are connected

to the infinite cluster at only one point If current were to flow only throughbonds that connect one side of the system to the other, these dead ends would

be avoided If the dead ends are “pruned” from the cluster, what remains iscalled the “backbone.” Flow of current would be restricted to the “backbone”cluster There are also a large number of loops, or cycles Figure1.1 shows

the “infinite” cluster for p > p c on a square lattice Figure 1.2 shows thebackbone cluster

The backbone cluster has itself been described using the following terms,

“links,” “nodes,” and “blobs.” A pictorial definition of these terms is given inFig.1.3 The characteristic separation of nodes, or the length of a link will be

Fig 1.1 A finite size sample of bond percolation on a square lattice above the

percolation threshold

Fig 1.2 The same system and realization of Fig.1.1, but for which the dead endshave been removed from the infinite cluster to form the “backbone” cluster Notethe existence of many closed loops (Figure from Todd Skaggs, unpublished)

equivalent to the correlation length, defined in (1.1) below The discussion ofthe exponent for the vanishing of the conductivity (Skal and Shklovskii, 1975)

is based on a picture of the percolation cluster, which is essentially equivalent

to Fig.1.3 Note that considerable work on non-linear effects on the electricalconductivity, as well as the usefulness of effective-medium theoretical descrip-tions is based on this kind of a pictorial concept This literature will not bediscussed here, and if interested, the reader should consult Shklovskii andEfros (1984) or Pollak (1987) and the references therein

1.3 What are the Basic Variables?

The most fundamental variable is p, which for the bond percolation problem

is defined to be the fraction of (cut) bonds in the above screen problem, or,equivalently the fraction of bonds emplaced on a background without bonds

In site percolation p stands normally for the fraction of, e.g., the metallic

balls mentioned above It can also stand for the number of lattice (grid) sites

Fig 1.3 Definition of links nodes and blobs (after Junqiao Wu,

can percolate simultaneously In continuum percolation, p can stand for a fractional volume The most important value of p is p c, the critical value at

which percolation occurs In an infinitely large system, p c is precisely defined

– larger values of p guarantee “percolation,” or the existence of an infinitely

large cluster of interconnected sites (bonds or volume) Smaller values of

p guarantee that percolation does not occur For finite sized systems this

transition is spread out over a range of values of p By spread out is meant

that percolation is observed in some finite-sized systems with a given value

of p, but not others.

The next quantities are all functions of the difference, p − p c P is defined

to be the fraction of connected bonds (or painted sites), which is connected

to the infinite cluster (if an infinite cluster exists, i.e., for p > p c ) χ, known

as the correlation length, is defined to be the linear dimension of the largest

cluster for p < p c , the largest hole for p > p c n s is defined to be the volume

concentration of clusters of sites or bonds with s interconnected elements, and

is a function of s and of the difference p − p c How to obtain such quantitiesand how to use them to calculate realistic and often very accurate values oftransport coefficients of disordered porous media is the point of this book

1.4 What is Scale Invariance

and Why is it so Important?

The most important single aspect of percolation theory, which allows most

of the theoretical development is that the correlation length diverges in the

limit p → p c At p = p c, the largest cluster of interconnected bonds, sites,

or volume, just reaches infinite size Starting from p > p c, the divergence ofthe correlation length implies that the largest hole in the infinite connected

cluster just reaches infinite size and there is symmetry between p > p c and

is no finite length scale left in the problem In reference to the qualitativediscussion of Sect 1.2, this scale invariance shows up also in the shapesand internal structure of the clusters, and is represented through the fractaldimension The fractal dimension of large clusters will be discussed further

later The divergence of the correlation length is described by a power, ν,

whose value will be discussed later,

What does the lack of a finite length scale at the percolation thresholdimply? It requires the use of functions of powers Powers may appear to have

a scale, i.e., under some circumstances the relationship of a conductivity to a

length scale, x, can, simply by dimensional analysis be shown to have a form

where the choice of the power –(d − 1) is not intended to be anything beyond

illustrative at present, K is a conductivity, K0 is a particular value of the

conductivity, x is a length, and x0is a particular value of x In contrast to an exponential function, x0 in this case need not identify a particular “scale,”though if used in a judicious fashion it may imply a boundary of the validity

of the scale invariance (the existence of a lower limit of the validity of invariance in site and bond percolation problems is clearly required by thefinite dimensions of the underlying lattice) On the other hand an exponential

scale-function must also have an argument x/x0 The difference, however, is that

in the power law case (1.2) for d = 3 a system that is twice as large as a

given system will always have one fourth the conductivity, regardless of theconductivity of the given system In the case of the exponential function, theratio of the conductivities of systems of size ratio 2 will also depend on theactual value of the conductivity of the smaller system, and thus of the ratio

of the size of the smaller system to x0; this is in no sense a scale-independentrelationship

The lack of any length scale, known as scale invariance, also implies therelevance of fractal analysis, or self-similarity Especially self-similarity is im-portant because it allows the application of the mathematical techniques of

renormalization Application of renormalization will permit us to calculatesome important quantities of percolation theory and employ analogies fromthe existing framework of the theory of critical phenomena Note that fordisordered systems that are far above the percolation threshold it is alwayspossible to define some variable, which defines a subset of the system, which

is at the percolation threshold If this variable is defined in such a way as

to relate to local transport coefficients, then it will be possible to identifythe chief contribution to the transport properties of the medium Then onehas the interesting result that for disordered systems of nearly any structuretransport is dominated by connecting paths near the percolation threshold,and the fractal characteristics of percolation will be relevant to transporteven in media, which seem to have no resemblance to fractals The basis forthis application, called critical path analysis, is described in the last section

of this chapter

The fact that percolation variables behave as power laws in p − p c, as

in (1.1) means that they must either diverge or vanish at p − p c, depending

on whether the power is negative or positive The term singular behavior,however, refers to either divergences or zeroes

1.5 The Relationship of Scale Invariance

and Renormalization, and the Relationship

of the Renormalization Group to Percolation Theory

Renormalization is a rather complex mathematical procedure, which sponds, in real space, to a relatively simple physical operation This operation

corre-is a kind of “coarse-graining,” which accompanies the drawing back of theobserver to a greater distance If the system has true scale-invariance, i.e., isright at the percolation threshold, it will be impossible to detect a change

in the appearance of the system as the scale of observation is increased Asystem with an infinite correlation length looks the same at all length scales

If the system is merely near the percolation threshold, however, and the relation length is finite, then drawing back to a greater distance will makethe correlation length look smaller Eventually the distance of the observerwill be larger than the correlation length This (relative) diminution of thecorrelation length means that at new (and larger) length scales the systemmust appear as though it were further from the percolation threshold Thus

cor-it must also be possible to redefine p simultaneously to be enough closer to

p c so that the appearance of the system does not change, a general conceptwhich underlies the assertion that it is possible to define ‘scaling’ variables

In the operation of renormalization, systems, which are right at tion, remain right at percolation, but if they are not right at percolation theytend to move away from percolation In an abstract space of variables repeti-tion of the renormalization operation leads to completely different trajecto-ries The trajectory produced by repeated applications of renormalization to

percola-a system percola-at the percolpercola-ation threshold is but percola-a point, since no chpercola-anges cpercola-an beobserved The trajectory produced by such repeated applications of renor-malization to a system, which is not at percolation, however, will always be

away from the percolation threshold In fact, if the starting state has p > p c,

the trajectory will always be towards p = 1, while if the starting state has

either zero or one, renormalization will not affect p Thus p = 0, p = p c, and

p = 1 are all “fixed points” of the renormalization procedure, but p = 0 and

p = 1 are “trivial” fixed points Clearly a system with either no bonds and

no clusters or all bonds attached will look the same at all length scales.This same behavior is observed at second order phase transitions, forwhich the correlation length also diverges While the language and under-standing of phase transitions has become more complex since the study ofpercolation theory commenced, the percolation transition does qualify as asecond order phase transition in the traditional definition and the theoreti-cal development for such critical phenomena can be adopted for percolationtheory

1.6 Cluster Statistics of Percolation Theory

Perhaps the most elegant means to summarize the theory of percolation is touse the scaling theory of percolation clusters Stauffer (1979) In principle onecan formulate percolation theory relatively completely in terms of its clusterstatistics, while these statistics also allow easy analogies to other phase tran-sitions The purpose of this section is not to provide such a detailed overview;for that the reader is referred to Stauffer’s review (1979) What will be dis-cussed here is sufficient to demonstrate the redundancy of percolation theoryand provide more unfamiliar readers with multiple bases for understandingand application

The cluster statistics of percolation define the concentration, n s, of

clus-ters of volume (number of sites) s as a function of p Clearly as p approaches

p c the number of large clusters increases rapidly In fact, right at p c, since

there is no length scale, there can be no volume scale either, and n s must

follow a power law in s So at percolation n smust obey,

n s (p = p c)∝ s −τ (1.3)

Here τ is an exponent, whose value will be discussed later How does n s

depend on p? For p = p c a length scale exists, and its value is the

corre-lation length, χ Suppose that one increases the “observation” length scale (reducing χ) and simultaneously takes the system closer to p c (increasing χ);

then one could recover the original configuration if these operations cancelperfectly Recovering the original configuration means that the system looks

the same Stauffer (1979) states “We assume that the ratio [ ] n (p)/n (p )

and similar ratios of other cluster properties are a function of the ratio s/s χ.”

Here s χ is a typical cluster volume at p, which, since cluster sizes follow a power law, is proportional to the limiting (or largest) cluster volume at p Since the linear dimension of the largest clusters with s = s χ is χ, which diverges as p → p c, the limiting cluster volume must also diverge in the limit

p → p c For convenience the power of (p −p c) which restricts the largest ter volume is assumed to be−1/σ The power must be negative in order that

clus-the largest cluster size can diverge at p = p c Thus (s/s χ)σ = s σ (p − p c)≡ z

becomes a scaling variable, and

n s (p)

n s (p c) = f [s

σ (p − p c)] (1.4)

The value of σ is not known, a priori This equation may be called

semi-empirical in that it was designed: 1) to accommodate results of simulations,

which revealed that the cluster numbers, n s, decay according to a power law

(with value τ ) right at critical percolation, 2) to allow simultaneous rescaling

of s and p in such a way that the system looks the same (as long as the product (p −p c )s σ remains the same) The values of τ and σ are expected to be found

from simulations or renormalization procedures Note that substitution of

the function f was uncertain for a long time, with various approximations

proposed In the Stauffer review it was pointed out that a Gaussian form for

The approximation made here, to omit z0, comes partly from the fact that

z0must have some dependence on the system investigated and thus cannot be

universal Nevertheless the fact that a z0exists makes the cluster statistics, inprinciple, asymmetric about the percolation transition For bond percolation

on a square lattice in 2− D, however, there is perfect symmetry between

connected and unconnected bonds and the existence of a term z0would implythat extrema for the clusters of interconnected bonds would occur at different

values of p than for clusters of unconnected bonds However, it must be kept

in mind that the neglect of this detail cold lead to small discrepancies The

form of (1.5) makes it apparent that for p = p c the cluster statistics decay

as a power law only up to a certain maximum s = s χ, which is proportional

above, since clusters of larger volume rapidly become extremely rare.One can use (1.5) in many ways As a first basic application let us use(1.5) to find out for p > p c what fraction of sites is connected to the infinitecluster Stauffer’s argument, which has been amply verified, is that it is the

“singular” behavior of cluster sums, which gives the percolation quantities of

interest Singular means the lowest non-analytic power The sum of sn sfrom

1 to infinity must give the product of p and the total number of sites If we

choose the lowest non-analytic power, we will develop information about thefraction of sites connected to the infinite cluster A sum over a power law

distribution, which is truncated at a maximum s value will be dominated by the largest s allowed The exponential function will essentially truncate the sum, or integral, over sn s at a value of s proportional to (p − p c)−1/σ Thisprocedure gives the following result,

 (p−p c)−1 σ

1

The first term has no dependence on the variable p − p c and is related

to the total number of occupied sites (p) (as p approaches p c from below,

there is no upper limit on the integral, and the integral must give p) The

second term gives the singular behavior, and so describes the behavior of

the fraction of sites, P , connected to the infinite cluster More explicitly, if

the concentration of sites on the finite clusters plus the sites on the infinite

cluster must equal p, then the fraction on the infinite cluster must cancel the

second term of (1.6) Thus we have,

The exponent β is customarily used for the critical behavior of P If

the cluster statistics of percolation theory are known accurately it becomes

possible to calculate β directly Note that P must vanish at p = p c and

therefore β ≥ 0 In systems of practical interest (two and three dimensional

systems) 0≤ β ≤ 1 However, we will only be able to calculate accurately two

values of β, 0 in one dimension and 1 in six dimensions or higher (or on Bethe

lattices) Though we only calculate a few values of the critical exponents ofpercolation theory, summaries of values given elsewhere are also provided.The next few sections provide some “hands-on” calculations from perco-lation theory, which allow someone unfamiliar with the theory to reproducesome of the simpler calculations for him/her self The experience should makethe theory more accessible for people from the porous media communitiesand, I hope, provide some satisfaction in the process

1.7 Calculation of the Critical Site Percolation

Probability for the Two-dimensional Triangular Lattice and of the Critical Exponent for the Correlation Length

in Two Dimensions

While the following discussion is expanded from Stauffer (1979), his sourcewas Reynolds et al (1977) Consider the image of the triangular lattice in

Fig.1.4 The circles represent sites Each site can potentially be connected to

six nearest neighbors Imagine coloring in a fraction p of the sites Whenever

two colored sites are nearest neighbors they can be considered to connect (as

in the case of metallic balls, which could conduct electricity between them ifthey were in contact) If a colored site is neighbor to an uncolored site, or twouncolored sites are neighbors, then no connection is made A renormalizationprocess can be developed, which constructs a new lattice out of “supersites,”which replace groupings of three sites as shown A replacement of three sites

by a single site must involve a rescaling of the length, or distance betweensites, by the factor 31/2 That result can be checked directly in Fig.1.4 byexamining the geometry of the system; the line separating the new sites formsthe bisector of the vertex of the triangles, as shown, thus developing 30-60-

90 triangles The sides of these triangles are in the ratios 1, 31/2, 2 Theratio of the separation of sites in the renormalized lattice to the old lattice is

2/(2 × 3 1/2)

Fig 1.4 A small portion of a site percolation problem on a triangular lattice.

The circles are sites Supersites of a real-space renormalization are located at thecenters of the triangles shown The line drawn in is an aid to measuring distancesusing 30-60-90 right triangles

Now consider how p changes with such a rescaling of the lattice The

approximation that is used here has been called “majority rule.” If either two

or three sites on the original lattice are colored in, the new site is colored in

Clearly, if all three sites of the original lattice were colored in, a connectioncould be made across the triangle in any direction, while if two sites arecolored in, often a connection across the triangle can still be made, thoughnot in an arbitrary direction If one or zero sites are colored in on the originallattice, the new site is not colored in, because no connection across the trianglecan be made, and in most cases such a triangle will interrupt the continuity

of paths constructed across other nearby triangles The new probability, p’,

of coloring in a site is thus constructed from the old probability, p The

above conceptualization is a very reasonable assumption, and nearly precise

It basically means that if you can get across a given “supersite” from oneside to the other, presenting potential connections to new sites on both sides,

it should be colored in Otherwise, it should not be Mathematically this can

be represented as,

p  = p3+ 3p2(1− p) (1.8)The justification for this result is that the probability that all three sites

are colored in independently, each with probability p, is p3 The probability

that two particular sites are colored in and the third is not is p2(1−p) There

are three possible locations for the site, which is not colored in, justifying

the factor 3 In the case that p  = p, the new lattice has precisely the same appearance and statistics as the old, and p  = p ≡ p c If the substitution

p  = p is made in the above equation, it is possible to rewrite the equation

(obviously if all sites or no sites are initially colored in, this condition will

persist) The root p = 1/2 represents p c

The divergence of the correlation length must be according to a powerlaw as discussed The only reasonable form for this relationship is,

scale, take p slightly different, e.g., larger than p c , p − p c = δ, where δ  1,

and find the behavior of p  − p c as a function of δ This kind of procedure

is known as “linearization,” because it will define only the lowest order

vari-ability in p  Using (1.8) without equating p and p  (because if p > p c , p  > p)

write p  as,

p  − p c = (p c + δ)3+ 3 (p c + δ)2(1− p c − δ) − p c (1.12)

and expand the result to first order in δ The result (you should verify this)

is that p  − p c = (3/2)δ But since p − p c = δ by definition, one finds that,

 1

31/2

log3δ

2

 = 1.355 (1.13)

Note that the value of p cfor the site percolation problem on the triangular

lattice is precisely 1/2, while the value of ν is 4/3 = 1.333 and the estimate

ratio of (1/3)(1+(5/2)δ), which is slightly larger than 1/3 (1.14) gives a first

estimate of the power σ in the cluster statistics, s max ∝ (p − p c)−1/σ,

be based on the argument provided, since that argument does not produce

a consistent power, independent of the value of p It will turn out that the correct value of 1/σ is 91/36 = 2.53, or about 7% different from the estimate.

While this difference is not large, it is critical We will revisit this questionafter a discussion of some of the relationships between critical exponents

1.8 Mean-field Treatment of the Probability

of being Connected to the Infinite Cluster

Consider a “mean-field” treatment of the bond percolation problem on a

lat-tice with coordination number (number of nearest neighbors), z In mean-field

treatments all sites are regarded as equivalent While all sites were equivalentbefore the bonds were actually assigned, this equivalence is lost afterwards,and this is a reason why mean-field treatments can fail Nevertheless, a mean-field treatment does illuminate some important concepts, and we can applythese further

Assume that an infinite cluster of connected sites exists Define the

prob-ability that some particular site is connected to the infinite cluster as P

Then the probability that it is not connected is 1− P The probability that

the site is connected to one of its nearest neighbors, chosen arbitrarily, is p.

According to the mean-field hypothesis, the probability that that neighbor

site is connected to the infinite cluster is assumed to have the same value P

The probability that the given site is connected to the infinite cluster over

this particular nearest neighbor is pP, where the product is used because of the independence of the bond probability and the probability P The proba-

bility that it is not connected to the infinite site over this particular nearestneighbor is 1− pP The probability that it is not connected to the infinite

cluster over any of its nearest neighbors is thus (1− pP ) z Thus we musthave,

(1.16)which states that the probability that a site is not connected to the infinitecluster is equal to the probability that it is not connected over any one of itsnearest neighbor sites, and that the probability that each of those neighborsites is not connected to the infinite cluster is identical Equation (1.16) can

be rewritten as,

(1− P )1

= 1− pP (1.17)

Note that for p < 1/z, this equation has only one solution, namely, P = 0.

If the probability that any arbitrary site is connected to the infinite cluster

is 0, there must not be an infinite cluster If an infinite cluster does not exist,the system must be below the percolation threshold This indicates that to

lowest order 1/z is the percolation threshold We expand (1.17) in the variable

1 + 1

2(−P )2

1

z


1



= 1−

1

the mean-field treatment directly predicts p c = 1/z as can be seen from the numerator Note the implication that p c = 1/z has no dependence on d This

is incorrect However, we are going to assume that the result that p c ∝ 1/z is

correct and that the proportionality constant may depend on dimensionality.For our purposes this is the most important result of the application of the

mean-field treatment to find P and we will use it next to deduce some further values of p c.

mean-field result It can be derived for a Bethe lattice (an infinite-dimensionalconstruction discussed in Sect.1.12) Thus, the “classical” exponents are as-sumed valid in systems of large dimensions In problem 4 the scaling rela-

tionship dν = 2 − α is derived from the cluster statistics This relationship is

satisfied by the classical, or mean-field, exponents (Table 1.2) only for d = 6.

Therefore it was concluded that mean-field theories of percolation apply fordimensionality greater than or equal to 6 (Toulouse, 1974), while the scal-ing formulation discussed here holds for dimensionality less than or equal

to 6 This makes d = 6 the cross-over between classical and hyperscaling treatments, a result which forms the basis of the so-called ε-expansion for renormalization, where ε = 6 − d I will not discuss the ε-expansion further

except to note that it has been of great use in calculating critical exponentsfor percolation theory

1.9 Value of pc for Bond Percolation

on the Square Lattice

Consider Fig 1.5 The solid squares form a square lattice Imagine that a

fraction p of the bonds (light lines) have been filled in at random as shown.

Next construct the square lattice denoted by the open squares, which areplaced at the centers of the individual squares formed by four neighboring

solid squares Imagine that a total of q of the potential bonds on this

lat-tice are connected (heavy lines) Each of these square latlat-tices has the same

coordination number, z = 4 Further, every potential bond on each lattice

intersects (blocks) exactly one bond on the other lattice, which we can call a

dual lattice Thus p for the first lattice is precisely 1 − q for the other lattice

and p + q = 1 Given the two-dimensional nature of the lattices, however, it

is not possible that both can “percolate” simultaneously Either one latticepercolates or the other does Given the identical natures of the two lattices,

however, it does not make sense for p c > q c , or for p c < q c The only

alter-native is to choose p c = q c= 1/2 The fact that the square lattice is its owndual lattice means that its percolation probability must be 1/2 The product

of zp c for this lattice is 2

1.10 Estimations of pc for Bond Percolation

on the Triangular and Honeycomb Lattices

Consider Fig.1.6 It includes a honeycomb lattice of solid squares (z = 3) and

a triangular lattice of open squares (z = 6), which are fully complementary

Fig 1.5 A square lattice (open squares and thick bonds) and its dual lattice (solid

squares and thin bonds), which is also a square lattice Note that the bonds of thedual lattice percolate

(or each other’s duals), as were the two square lattices above Thus everybond that is connected on the triangular lattice would “break” a bond onthe honeycomb lattice and vice-versa This means that if no bonds from one

lattice are allowed to cross bonds from the other one, the bond probability p

on the triangular lattice is 1−q, with q the bond probability on the honeycomb

lattice and p = 1 − q The light lines represent bonds on the triangular

lattice, while the heavy lines represent bonds on the honeycomb lattice As

is generally true in two dimensions, either the triangular lattice percolates,

or the honeycomb lattice percolates, but not both simultaneously In the

figure the triangular lattice percolates The result p + q = 1 together with the exclusionary result on the two percolation probabilities implies that p c+

q c = 1 But the result from Sect 1.8 that p c ∝ 1/z implies that q c = 2p c

a good approximation) Simultaneous solution of these two equations yields

p c = 1/3, q c = 2/3 Note that these estimates for p c are both consistent

with the relationship zp c = 2, but the exact results are p c = 0.3473, q c =

0.6527, for which values zp c = 2.08 and 1.96, making this product only an

approximate invariant Vyssotsky et al (1961) suggested that the product

zp should take on the values d/(d − 1), for d ≥ 2, and such an approximate

Fig 1.6 A triangular lattice (open squares and thin bonds) and its dual lattice

(solid squares and thick bonds), which is a honeycomb lattice Here the bonds ofthe triangular lattice percolate

invariant as this can be quite useful if a system, for which p c is not knownand cannot be readily calculated, is encountered

1.11 Summary of Values of pc

The known results for p c are summarized in the table The four cases, forwhich simple approximations are known and repeated above are noted These

four cases correspond also to the only exact values of p c known

All the estimated bond p c values given are exactly consistent with theVyssotsky et al (1961) relationship, though, when compared with the most

accurate determinations of p cthat relationship gives values that are accurateonly to within about 4%

Lattice Type z p cbond Estimated p c zp c p csite (est.)

1.12 More General Relationships for pc

Some relationships for p c are mentioned, which may help guide estimations

in more complex, but more realistic models Bethe lattices may well be ofrelevance to discuss the permeability of such objects as root systems of plants,for example In any case Bethe lattices are essentially non-overlapping tree

structures, with each bond the starting point for z − 1 additional bonds The

critical bond fraction for coordination number z is p c = 1/(z − 1) This value

is easy to understand Consider such a tree structure with a site on a branch

of nth order, which is connected to a site on the n − 1st branch There are

is connected is less than 1/(z − 1), the average number sites on the n + 1st

branch that this particular site is connected to is less than 1, and repetition ofthis process an infinite number of times is guaranteed not to yield an infinitenumber of interconnected sites If the probability that a link is connected

is larger than 1/(z − 1) the average number of sites on the n + 1st branchconnected to the given site is greater than 1, and connection to infinite size

is possible Note that in the limit z → ∞, p c → 1/z, and the result zp c= 1

is established The Vyssotsky relationship yields zp c = 1 in the limit d → ∞,

and Bethe lattices can be regarded as infinite dimensional The relevance ofBethe lattices to non-living porous media is, however, limited, particularly ascritical exponents in Bethe lattices are also integral, rather than non-integral.The lattice structures mentioned so far by no means exhaust the types in-vestigated, and the Vyssotsky relationship is valid only for bond percolation

Galam and Mauger (1998) have developed a more general relationship for p c

of the following form, p c = p0[(d − 1)(q − 1)] −a d b For regular lattices, q = z, the coordination number For non-periodic tilings, q is an effective value of

z The relationship is considered to be valid for anisotropic lattice with

non-equivalent nearest neighbors, non-Bravais lattices with two atom unit cellsand quasi-crystals The biggest strength of the relationship, however, may

be that it can be applied to both site and bond percolation problems In the

former case, b = 0, while in the latter b = a The biggest weakness is probably that the known systems fall into two classes, each with different values of p0

and a The first class includes 2-D triangle, square and honeycomb lattices with a = 0.3601 and p = 0.8889 for site percolation and a = 6897 and

p0= 0.6558 for bond percolation Two dimensional Kagome and all

(hyper)-cubic lattices in 3≤ d ≤ 6 constitute the second class with a = 0.6160 and

p0 = 1.2868 for site and a = 0.9346 and p0 = 0.7541 for bond percolation, respectively But in order to use these results to predict p c, one must know towhich class of lattice a particular system belongs Nevertheless it is important

here to provide guidance for prediction of p c in new system geometries.Finally we come to the result of Scher and Zallen (1970) for the criticalvolume fraction for continuum percolation Such results have the potential

to be of great use in percolation problems in porous media Scher and Zallen(1970) found that for regular lattices the critical occupied volume fraction,

V c = p c f (1.20)

where p c is the critical bond fraction, and f is the filling factor (the fractional

volume covered) of a lattice when each site of the lattice is occupied by asphere in such a way that two nearest neighbor impenetrable spheres touchone another at one point For a simple cubic lattice the value of this product

is 0.163, and in fact the value of this product for all the lattices consideredscarcely differed from 0.17 Note that an analogous model (with different

shaped objects) could have direct relevance to porous media with f replaced

by the porosity, and there is indeed evidence for the validity of (1.20) in thiscontext

Shante and Kirkpatrick (1971) generalized this idea to overlapping spheres,

and showed that the average number, B c , of bonds per site at p c (equal to

the product of zp c) is related to the corresponding critical volume fractionby,

Note that the choice of B c = 1.5 for three dimensions yields V c = 0.17.

This result is generalized to an arbitrary continuum of spheres by choosing

B c to be the limiting value of p c z in the limit z → ∞ Values of V c on theorder of 0.17 have often been suggested to be relevant to real media Balberg(1987) has developed these ideas further, finding,

cannot be located without overlapping For spheres this ratio is (4/3)πr3/

(4/3)π(2r)3 = 1/8, in agreement with the result of Shante and Kirkpatrick

(1971)

For results for the critical volume fractions for percolation for a number

of anisotropic shapes one can also consult the following web page http://ciks.cbt.nist.gov/∼garbocz/paper59/node12.html#SECTION00050000000000000000

(geometrical percolation threshold of overlapping ellipsoids, Garboczi et al.,(1995) These values may be of considerable use in geologic applications, atleast to guide conceptualization In particular, the critical volume fraction forpercolation has a strong tendency to diminish for increasing shape anisotropy.

1.13 Derivation of One-Dimensional Cluster Statistics and Discussion of Fractal Dimensionality

In one dimension the critical bond fraction for percolation, p c = 1 Thisresult is necessary since any break in the chain of elements will prevent theformation of a cluster of infinite size that spreads from negative to positiveinfinity The following discussion follows Stauffer (1979) in its general content

The probability of finding a cluster of s interconnected bonds, all in a row, is

n s = p s(1− p)2

(1.23)

In the case that p c − p  1, this result can be rewritten to lowest order

n s = s −2[1− (p c − p)] s

s2(p c − p)2

= s −2exp [−s (p c − p)] s2(p c − p)2

(1.24)The cluster statistics of percolation theory can always be written in thefollowing form,

n s = s −τ f [s σ (p − p c)] (1.25)Equation (1.24) and (1.25) show that for one dimensional systems τ = 2 and σ = 1 Also, one sees that there is a cut-off in the occurrence of clusters for sizes s > smax ≈ 1/(p c − p) meaning that σ = 1 In 1 − D the length

of a cluster of size s is s (times the fundamental bond length), so that the linear dimension of the largest cluster for p < p c is also (p c − p) −1 This

result then defines the correlation length and yields the value ν = 1 Note that the fact that smax ∝ (p c − p) 1/σ and that χ ∝ (p c − p) −ν implies the

result that in 1− D smax ∝χ 1/σν = χ1 Any time the total “volume” (s)

of an object is proportional to its linear dimension (χ) to an exponent, the

implication is that the “dimensionality” of the object is that exponent As a

consequence, the combination of exponents, 1/σν has become known as the fractal dimensionality, d f, of percolation clusters, i.e., of large clusters near

the percolation threshold Note that for d = 1, d f = 1 as well, and in onedimension large clusters near the percolation threshold do not (cannot) havethe rough “surface” associated with fractal objects However, in systems of

larger dimensionalities, d f typically turns out to be less than d Other values

of critical exponents are given in Table 1.2

1.14 Argument for Dimensionally-Dependent Scaling

Law, Implications for Critical Exponent, τ ,

and Applications to Critical Exponents

The dimensionally-dependent scaling law relating various critical exponentsfrom percolation theory can be derived by considering the cluster statistics

of percolation We will need to rewrite the cluster statistics of percolationtheory in terms of the linear extent of such clusters We know that the volume

of a cluster of linear extent N is equal to s = N 1/σν, on account of thefractal dimensionality of the clusters We have derived the form of the clusterstatistics in one-dimension, and it has been demonstrated that this form

is appropriate in larger dimensions too Stauffer (1979) Further, the invariance of the system right at percolation requires that the cluster statisticsfollow a power-law decay, so we could certainly write,

If one integrates this probability density function over a range of values

from, say, N0to 2N0, which in a power-law discretization scheme (appropriatefor self-similar media) would represent one size class, one obtains,

P (N0) = N −

τ −1 σν

The next argument is that typically one cluster of linear dimension N0should be found in a volume of linear dimension N0in order to be 1) consistentwith the idea of percolation, i.e., that one can expect percolation to occur inany size system, all the way to infinite size, 2) consistent with the concepts ofself-similarity, i.e., that all such volumes look alike Thus the concentration

of clusters of size N0 is proportional to N0−d so that the product of N d and

N0−d= 1 The implication is that,

d

This is one of the fundamental scaling relationships of percolation theory

It relates the fractal dimensionality, 1/σν, to the Euclidean dimensionality.

It is also straightforward to derive the ratio of the number of connectedsites of a large cluster and the total number of occupied sites in the volumespanned by that cluster This result is,

Note that this result is the same as that derived directly from the cluster

statistics Some conclusions can be made on the basis of the definition of n s

itself Consider the integral,

 ∞

1

sn s ds = s2−τ | ∞

which represents the total number of connected sites This integral diverges

unless τ > 2 Although in dimensions d > 1 this is strictly an inequality, in

d = 1 the value τ = 2 is allowed The reason for this is that the percolation

probability is identically 1 At p = p c, however, the concentration of clusters

of linear size s cannot really follow a power law; all sites are connected and the only cluster is infinite in extent Right at p c in one dimension the structure

of the infinite cluster must be Euclidean Equation (1.23), from which n s

was derived, yields fundamentally different results in the limit p → 1 and for

p = 1 In fact, a sudden increase in P from 0 to 1 over an infinitesimally

small increase in p is consistent with a value of β = 0 Note that β = 0 is also

obtained by application of the scaling (1.32) or (1.34) consistent with d = d f

or τ = 2.

Consider now our rough results in two dimensions, namely ν = 1.355 and

dimensionality is the same as the Euclidean dimensionality, then, by (1.33)

just as in one dimension, β = 0 Then (1.27) gives τ = 2 as well In fact, however, we saw that σ > 1/2ν (and it turns out that β = 0.14, not 0) While this difference is not great, and d f = 1.9 for d = 2 (only slightly

smaller than 2), the difference is obviously very important So, while the

approximate renormalization procedure to find ν appeared at least in 1979

to generate some hope that the value was accurate, in fact the result for the

exponent σ is sufficient to cast serious doubt on the accuracy of the procedure

generally

1.15 Exponents for Transport Properties

Consider the site percolation problem discussed above and stipulate for plicity that all the metallic balls are of the same size and composition Allow

sim-them to be placed on a simple cubic lattice We have not calculated p c for

this lattice but numerical simulations give the result p c= 0.3116 Thus, in aninfinite lattice, if fewer than 31.16% of the balls emplaced are conducting and

the remainder are insulators, the system will not conduct at all If p > p c,the system will conduct Clearly the conductivity of the system must follow

a functional form, which vanishes (rather than diverge) at p = p c = 0.3116.

The result of percolation theory is that the functional form must be a powerlaw (and the arguments given here justify that), so that what we need to beable to do is predict the exponent

The important aspects of this problem treated by percolation theory arethe connectivity and the tortuosity of the conducting paths Discussions ofthis topic have occupied a great deal of literature but, as will be seen, the orig-inal discussion of Skal and Shklovskii (1975) generates predictions that, forthe properties of porous media at least, appear to be accurate The following

is consistent with the general results of that work

The electrical conductivity of a system is defined as the ratio of the currentper unit area and the applied electrical field If this ratio is independent ofthe field (as is normally the case at small field strengths), the system obeysOhm’s law The current per unit area in the present case involves the currentper path and the number of connected paths per unit area The simplestassumption is that the current for each connected path is identical Then thenumber of connected paths per unit cross-sectional area (in three-dimensions)

is proportional to,

Since in three dimensions, ν = 0.88, the lowest order estimate of the conductivity is that it should vanish as the 2ν = 1.76 power of the differ- ence, p − p c This suggestion is actually almost correct But it turns out thatthe structure of large clusters near the percolation threshold, and by exten-sion also the infinite cluster just above the percolation threshold, is fractalfor distances below the correlation length (which of course diverges right

at percolation) This fractality produces a tortuosity as well Although notdiscussed to this point, it is also known from simulations that the distance

along a connected path, Λ, over a separation equal to the correlation length

is actually longer than the correlation length Λ diverges at the percolation

threshold according to

Thus, assuming that the resistance of the current-carrying path is justthe sum of the resistances of all the metal balls encountered, this resistanceper unit system length must actually increase as the percolation threshold is

approached, and the increase must be given by the ratio of Λ to χ This ratio

is proportional to (p − p c)ν−1 = (p − p c)−0.12 Such an increase in resistanceproduces an alteration of the results for the conductivity to,

Here the first contribution to the exponent is essentially a result of theconnectivity, or separation of the paths along which current can flow, whilethe second contribution is due to the tortuosity of these paths The combined

exponent is thus the sum of two contributions, 1.76 + 0.12 = 1.88 In two

dimensions the argument given here does not work, since there is no way todefine a tortuosity in the same way as in the argument leading up to (1.38)

The reason for this is that in two dimensions the exponent of Λ appears to

be smaller in magnitude than that of χ One can simply use the exponent for the correlation length, ν = 1.33, as a first approximation to t Work of Derrida and Vannimenus (1983) however, shows that the value of t in two dimensions is 1.28 (Jerauld et al (1984) find t = 1.27, while Normand and Herrmann (1990), find t = 1.30) While it is not possible to derive t = 1.28

rather than 1.33 with the limited theoretical techniques developed here, thisvalue is so important to later discussions that it is brought up now In onedimension, the conductivity is either zero (if there are any non-conducting

elements at all), or is a finite value, implying t = 0 But t is, in general,

non-universal for one-dimensional systems If there is a variation in the conductionproperties of the individual elements (not all resistance values identical), the

result p c = 1 implies that the total resistance may be dominated by theresistance of the most resistive element in one dimensional systems Notethat although the concept of conductivity and the discussion of the value of

t were introduced using the example of electrical conduction, the arguments

are perfectly general, and the results could be applied to, e.g., the hydraulicconductivity or to air flow as well

Berkowitz and Balberg (1992) in fact explicitly demonstrated that models

of hydraulic conduction yield (1.38) for the hydraulic conductivity near thepercolation threshold, and found values of the exponents compatible with

found results compatible with non-universal exponents Sen et al (1985); Feng

et al (1986) in certain 3− D systems.

One can also use the Einstein relationship Sahimi (1993) between

diffu-sion, D, and conductivity, σ,

where n is the number of charge carriers, which is normally assumed to be

given by the fraction of sites connected to the infinite cluster, to find

Interestingly enough, as we will find, although other relationships givenhere are verified, (1.40) appears to give inaccurate predictions for solute

diffusion in porous media, although it appears to be fairly close for gas fusion Although these results may not yet be completely understood, themain discrepancy appears to be due to the ability of solutes to diffuse overthin water films present in otherwise dry pores, while there is ordinarily noequivalent possibility for gases to diffuse through water It is curious that

dif-a simple effective-medium theoreticdif-al result yields D ∝ (p − p c)1, which isexactly what is observed, although almost certainly for the wrong reasons

On account of this coincidence, however, and because of the rather close respondence between effective-medium and percolation theories, the essence

cor-of this derivation is repeated here

The lowest order effective medium approximation for the mean diffusivity,

D m, can be obtained via physical arguments Kirkpatrick (1971, 1973) or vialattice Green functions Sahimi (1983) as Keffer et al (1996),

where D b is a very small value and D0is relatively large, and for which these

authors define f ≡ D b /D0 Note that, in an unusual choice, these authors

chose to use the symbol p for the low diffusion elements! The solution of

(1.41) using (1.42) for f (D) is,

12

which would seem to yield p c = 1− 2/z and a critical exponent of 1 But

given that these authors exchanged the roles of p and 1 − p, the actual result

obtained for p c is zp c = 2, which would be in agreement with the results

of percolation theory except that the constant, 2, is more appropriate fortwo-dimensional, rather than the three-dimensional configurations consid-ered Note also that the critical exponent of 1 is unaffected by the transpo-

sition of p and 1 − p.