applications of percolation theory to porous media with ...directory.umm.ac.id/data...

Applications of percolation theory to porous media with distributedlocal conductances

A.G. Hunt

Paci®c Northwest National Laboratory, Atmospheric Sciences and Global Change Resources, Richland, WA 99352, USA

Received 1 December 1999; received in revised form 25 May 2000; accepted 31 August 2000

Abstract

Critical path analysis and percolation theory are known to predict accurately dc and low frequency ac electrical conductivity in

strongly heterogeneous solids, and have some applications in statistics. Much of this work is dedicated to review the current state of

these theories. Application to heterogeneous porous media has been slower, though the concept of percolation was invented in that

context. The de®nition of the critical path is that path which traverses an in®nitely large system, with no breaks, which has the lowest

possible value of the largest resistance on the path. This resistance is called the rate-limiting, or critical, element, Rc. Mathematical

schemes are known for calculating Rc in many cases, but this application is not the focus here. The condition under which critical

path analysis and percolation theory are superior to other theories is when heterogeneities are so strong, that transport is largely

controlled by a few rate-limiting transitions, and the entire potential ®eld governing the transport is in¯uenced by these individual

processes. This is the limit of heterogeneous, deterministic transport, characterized by reproducibility (repeatability). This work goes

on to show the issues in which progress with this theoretical approach has been slow (in particular, the relationship between a critical

rate, or conductance, and the characteristic conductivity), and what progress is being made towards solving them. It describes

applications to saturated and unsaturated ¯ows, some of which are new. The state of knowledge regarding application of cluster

statistics of percolation theory to ®nd spatial variability and correlations in the hydraulic conductivity is summarized. Relationships

between electrical and hydraulic conductivities are explored. Here, as for the relationship between saturated and unsaturated ¯ows,

the approach described includes new applications of existing concepts. The speci®c case of power-law distributions of pore sizes, a

kind of ``random'' fractal soil is discussed (critical path analysis would not be preferred for calculating the hydraulic conductivity of

a regular fractal). Ó 2001 Elsevier Science Ltd. All rights reserved.

1. Introduction

Mere existence of heterogeneities in a medium is fre-quently considered grounds for preferring stochastic todeterministic transport theories. In part, this assumptionseems to follow from a notion that deterministic theoriescannot treat statistical variability. But if such heteroge-neities can be accurately described in a statistical sense,e.g., at the pore scale, then percolation theory can beapplied to generate both mean values and the variabilityof transport properties in a given volume with size muchlarger than the pore scale. Its interpretation as deter-ministic in nature does not imply that percolation theorywill tell which volume has a particular conductivity. But,once one considers the possibility of deterministic disor-der, it becomes easier to incorporate certain non-sto-

chastic tendencies into a general conceptual framework.For example, in a deterministic, but heterogeneous po-rous medium, there is no surprise that preferential ¯owpaths are followed repeatedly. Similarly, the conclusion[1], ``At high [scaled variance], owing to ¯ow localization,extreme values of [the pressure drop squared] occurred atdeterministic positions. The ¯ow pattern is so stronglycontrolled by these huge values that a stochastic de-scription becomes inadequate,'' should be immediatelyrecognized as an obvious possibility in real porous media.

Percolation theoretical applications were given in thephysics literature in the 1970s. Interestingly, Seager andPike [2] as well as Kirkpatrick [3], who, like [1] had donenumerical simulations on transport in heterogeneousmedia, came to the conclusion that percolation theoryperformed best of known approaches when disorder was(relatively) high, while e�ective-medium theories weresuperior when disorder was low. The crossover in ap-plicability was at a critical resistance, which involved a

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Advances in Water Resources 24 (2001) 279±307

E-mail address: [email protected] (A.G. Hunt).

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

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

List of symbols

a localization radius (electronic wave-func-tions)

b typical site separation (solid state)c geometrical constant at pore scalec0 typical separation of pores in random

mediumd Euclidean spatial dimensiondf fractal dimension of percolation clustersdA surface area elementdV volume elementf fraction of resistors in largest resistance

classg generalized conductancegs explicit reference to saturated critical

conductancege electrical conductancege

c critical value of electrical conductancegh hydraulic conductancegh

c critical value of hydraulic conductanceh capillary pressurei denotes site or porej denotes site or porek Boltzmann constantke constant relating to electrical conductancekh constant relating to hydraulic conduc-

tancel separation of critical (hydraulic) resist-

ancesl0 unit pore separation in networkm exponent in van Genuchten functionn ionic concentration in groundwaterns volume concentration of clusters with s

sitesnN volume concentration of clusters of length

Nn0 most likely ionic concentrationp bond probabilityq 1 or 2 (exponent on cluster statistics)qe electrical chargeqr ratio of pores in successive pore classespc critical value of bond probabilityr radius of a pore throat or constrictionrc critical value of pore throatsrs dimension of cluster with s elementsr0 most likely pore radiusr0 smallest pore radiusrij distance from site i to site jrm largest pore throat radiusr> largest pore ®lled with ¯uids no. of sites on a clustersn variance of ln(n)sr variance of log(r)

w0 rate prefactorwij local transition probabilities per unit timex system sizeA normalization constantA0 normalization constantAi local pore surface areaCK�h� hydraulic conductivity covariance at sep-

aration hC pore aspect ratio (when constant)C�d� dimensionally dependent cluster statistics

constantD fractal dimensionality of pore spaceDr fractal dimensionality of volume occupied

by solidE electric ®eldEi(x) exponential integral of xEij energy associated with electron hopping

from i to jEC ¯uid electrical conductivityF electrical formation factorH�x� Heaviside step function of argument xJ constant in hydraulic conductivity distri-

butionK hydraulic conductivityKs explicit reference saturated hydraulic

conductivityK/ constant proportional to solid volume of

soilK�1� hydraulic conductivity of unbounded

systemL separation of steady-state current-carry-

ing pathsN times l is rs� linear dimension of clusterP pressureQ ¯ow rate (volume per unit time)R generalized resistanceRe electrical resistanceRh hydraulic resistanceS relative saturationS�k� characteristic function of wave number kT temperatureV volumeVi pore volumeW �r� distribution of pore radiiW �x;K� hydraulic conductivity distribution at

scale xZ local coordination number at pore scalea volume fractionac critical volume fractionb ratio of system size to largest pore sizebe combination of constants related to rdc

bh combination of constants related to Kv correlation length associated with perco-

lation

280 A.G. Hunt / Advances in Water Resources 24 (2001) 279±307

factor from random distributions of about exp [10],implying distribution widths of 4±5 orders of magnitude.Since e�ective-medium theories are stochastic in ¯avor[4] (a single equation is used to represent any arbitrarypoint in the medium, but includes a representation of allthe variability of that medium) the conclusion regardingthe relative applicabilities of percolation and e�ective-medium theories is compatible with the conclusion [1]``The [square of the pressure gradient] ®eld had a rathersimple random structure at low to moderate [relativevariance] and a stochastic description was an attractiveoption in this case. [...]''. Finally, my contention is thatthe results summarized above are general enough tohold reasonably well in real rocks (in particular they donot seem to depend on the dimensionality of the net-works nor on the speci®c pore radius distributions usedhere).'' The fact that [1] mention rocks as the mediumunder consideration, should not dissuade others fromimagining that the conclusions apply equally to soils. Imake no qualitative distinction here between soil androck, although, due to the greater cementation of thelatter, one should normally expect smaller pores to bethe rule.

Percolation theory is a theoretical framework thatallows an investigator to quantify connections of vol-umes, areas or line segments when arranged at ``ran-dom'', [5,6] When such line segments stand fortransport, e.g., between neighboring pores, or betweenneighboring electronic states (more or less localized ondi�erent sites), the statistics of their connectivity revealinformation about the rate-limiting electrical or hy-draulic conductance of large systems. It has been ob-served that the chief problem with geostatisticalformulations regarding the hydraulic conductivity is alack of information regarding the connections betweenhigher conducting regions [7]. But the fact that perco-lation theory keeps track of connections makes it alogical choice for addressing spatial correlations. Thuspercolation theory has the strength of quantifying con-

nections and emphasizing on heterogeneity. The presentwork considers such e�ects at length scales and underconditions for which pore-scale variability is the relevantheterogeneity. There is, in principle, no size limit onapplicability of percolation theory. But practical issuesmay constrain the most valuable applications of perco-lation theory to the pore scale. This is because it is likelythat the criteria for selection of ``stochastic'' versus``deterministic'' methods are a�ected by loss of detailedinformation (which may accompany change in lengthscales). Combined with the possibility that heterogene-ities in transport at large length scales could have asmaller magnitude than at small length scales, it ispossible that stochastic theories tend to become moresuitable with increasing length scale, and it becomesdi�cult to make a generalized prediction regarding thechoice of an optimal theoretical approach at arbitrarylength scales. Further research in this direction is es-sential.

Applying percolation theory to usual network mod-els, when pore separations are all equal and the coor-dination number is consistent across a lattice is easy.Application to more complicated systems is also poss-ible as long as transport between points i and j is con-sidered limited by the narrowest portion of a connecting``throat'' or ``neck''. The local coordination number canbe random, and constant aspect ratios of the pores maybe considered. In the case of such complications theproper formulation is based on continuum percolation,but the general concepts involved do not change.

1.1. A short history of the hydraulic conductivity insaturated soils

The beginning of the following discussion is mainlyfrom Bernabe and Bruderer [1] (hereafter referred to asBB) who note that formulations of the saturated hy-draulic conductivity in porous media have historicallyutilized expressions of the form

v0 prefactor of correlation length/ porosityk tortuosity parameterl ¯uid viscosity (water in this case)le mobility of charges in groundwaterm � 0:88 critical exponent of correlation length

(3D)m0 constant frequency (attempt frequency)rdc DC conductivityh water contenthsat water content at saturationhr residual water contentr�x� AC electrical conductivity

r � 0:45 critical exponent of percolation theory(3D)

s � 2:2 critical exponent of percolation theory(3D)

nij random variable associated with i±jtransition

x frequency of an applied (electric) ®eldx as superscript, a power relating ge and

gh

CK�h� conductivity semi-variogram at separa-tion h

K length scaleX units of electrical resistance �X�

A.G. Hunt / Advances in Water Resources 24 (2001) 279±307 281

K � r2

cF�1:1�

with r a length related to pore geometry, c � 8 for cy-lindrical pores, and F the ``electrical formation factor'',which gives the ratio of the ¯uid bulk conductivity to therock conductivity (excluding surface conduction). BBclearly show that the evolution of understanding of ¯owin porous media is tied to the conceptual evolution of rin Eq. (1.1). We will see that not all factors of r, whichenter Eq. (1.1), explicitly, or implicitly, need be identical.

In Kozeny [8] and Carman [9] model, based on theconcept of bundles of tubes, r � 2hVii=hAii, where Vi is alocal pore volume, Ai, the average pore surface area, andthe brackets denote a volume average.

A more recent treatment [10] considers a relationshipbetween the electrical and hydraulic conductivities togenerate a di�erent length scale in place of r in the ex-pression for K,

K � 2

RE2 dVRE2 dA

�1:2�

with E the electric ®eld, and E2 essentially the energydensity of the electric ®eld, which can be related todissipation.

In [11] it is argued that e�ective-medium treatmentsmust also yield a K in the form of Eq. (1.1), since everylink between pores has a ¯ow, Qij / r4Pt=l0, where Pt,the pressure di�erence, is linear in the distance betweenpores, l0.

The Katz and Thompson [12] treatment of criticalpath analysis (originally from [13,14]) yields

K � r2c

cF�1:3�

but with c � 56:5 (later amended downwards [15,16]).Here rc is the critical pore radius, de®ned by the con-dition that rc is the largest value of r, for which an in-terconnected path may be found from one side of asystem to the other, on which no radius smaller than rc isencountered. The particular value of rc is system-dependent, but may be calculated analytically, or deter-mined for each particular system depending on theshapes of pores, the distribution of pore radii, and theconnectivity of the pores. While critical path analysiscan be used to ®nd rc, the determination of rc is notsu�cient to ®nd the hydraulic conductivity, as will beshown in this review. It will be seen that application ofcritical path analysis does not always lead to an ex-pression with only one length scale, as appears to beimplied in Eqs. (1.1)±(1.3).

A recent application of critical path analysis to boththe electric and hydraulic conductivities by Friedmanand Seaton [17] (hereafter referred to as FS) led to theexpression,

ghc �

p8ll0

r4c �1:4�

for the critical value, ghc , of the hydraulic conductance,

and

gec � EC

pr2c

l0

�1:5�

for the critical value, gec of the electrical conductance. In

these expressions, l0 is the length of the critical (and all)pores, EC the intrinsic electrical conductivity of the ¯uid(water with whatever ions it may contain), and l is thedynamic viscosity of water. First, note that the hydraulicconductance involves r4

c ; conversion to the hydraulicconductivity may, but need not always, yield a pro-portionality to r2

c . From Eqs. (1.4) and (1.5), FS concludedthat the ratio of the hydraulic and electrical conductiv-ities should be proportional to the square of the criticalpore radius, in accordance with the conclusions of BB.This conclusion should be independent of the methodused to calculate the conductivity from the conductance,as noted by both BB and FS.

Eq. (1.6) gives the hydraulic conductivity of a randomfractal soil obtained by critical path analysis. This resultincludes estimates of the length scales necessary fortransforming an expression for a hydraulic conductanceto a conductivity (derived in the steps up to Eq. (3.43),Hunt and Selker, 2000, in review),

K � p8Cl

� �l

L2

� �r3

m 1� ÿ ac�3= 3ÿDr� �

� p8Cl

� �r2

m 1� ÿ ac�4= 3ÿDr� � �1:6�

and is given in terms of the largest pore radius in thesystem, rm, as well as a constant C, which is a uniformaspect ratio, and ac, which is the critical volume fractionfor percolation, l the separation of critical rate-limitingpore throats, and L is the separation of the main water-carrying paths. The factor �1ÿ ac�4=�3ÿDr� can, if thefractal dimensionality, Dr, is near 3, be very muchsmaller than 1, and an e�ective radius much smallerthan the maximum r, although the result formally pre-serves the proportionality of the hydraulic conductivityto the square of a particular pore (throat) diameter.

BB compared several results for K, including paralleltubes [8,9] the model of Johnson and Schwartz [10], astochastic model [18], and the Katz and Thompsonmodel [12] (KT) but not Eqs. (1.4) and (1.6)) with sim-ulations. They conclude that the KT model provides thebest description of trends of the hydraulic conductivitywith width of the pore distribution. This result should,by itself, be su�cient motivation to pursue the bestmethod for calculating K consistent with percolationtheory. But it is really only the beginning.


1.2. Problems in existing analyses

Several problems in current analyses will be dis-cussed. These include the conversion from a critical rateto a system conductivity, as well as various schemes todescribe resistance distributions that characterize sim-pli®ed versions of a complex network.

2. The basis of critical path analysis and tests of its

validity

Percolation theory and critical path analysis can beapplied in any system in which transport is stronglyheterogeneous. Examples include electrical conductivityof disordered solids, hydraulic and electrical conduc-tivities of rocks and soils, and the viscosity and electricalconductivity of super-cooled liquids. Actually percola-tion theory was originally devised for applications inporous media [19]. Linear transport theories are fairlywell established, although some debate still exists. Whilenon-linear transport theories have been constructed [20]even in relatively well-characterized systems in solid-state physics nothing approaching consensus has beenreached as to their validity.

The simplest application of critical path analysis is toa network model of ¯ow in porous media under satu-rated conditions. Allow each bond to represent a porethroat with a radius selected at random from a distri-bution W �r�. Then the critical radius, rc, is de®ned byZ 1

rc

W �r�dr � pc �2:1�

with 0 < pc < 1. Stated in English, if a fraction, pc, ofthe bonds of a network is chosen at random and con-nected, they must produce an interconnected path ofin®nite length. The implication here is that it must bepossible to ®nd a path through the network which nevertraverses a pore of radius smaller than rc. If this rc isunusually small compared with the other rs on the path,the pressure drop across rc will be very large (comparethe BB quote in the ®rst paragraph). The value of pc,and hence rc, depends mainly on the coordinationnumber, Z, and the dimensionality, d. Many values of pc

are catalogued (e.g., [21]) others can be estimated using[22]

Zpc � dd ÿ 1

: �2:2�

While ®nding the critical resistance is a big step incalculating the conductivity, it is but the ®rst. As itturns out, it is not su�cient for determination of thepaths on which the water ¯ows, and even after thesepaths have been found, it is still necessary to calculatetheir total resistance and how many of them there

are. While no disagreement exists up to Eq. (2.2),di�erent approaches begin to diverge immediatelythereafter.

Critical path analysis generalizes the following ob-servation; the equivalent resistance of a 10 X and a106 X resistance con®gured in parallel is nearly 10 X.The equivalent resistance of a 106 X and a 10 X re-sistance con®gured in series is essentially 106 X. Theargument is then extended to paths through a mediumfor which local resistance values are spread out over avery wide range. Imagine reconstructing the pore spaceof a porous medium by adding individual pores, one byone, in descending order of size. A series of subnetworksincluding more and more pores is derived from theoriginal network. The ®rst such subnetwork containinga cluster of pores connected throughout the network iscalled the critical subnetwork. Any other path throughthe system, if chosen at random, will include pores withmuch smaller radii; such a parallel path has much higherresistance, Rh, and may be ignored (since Rh / rÿ4, apore of half the width carries 1/16 the ¯ow). Thus largerresistances are treated as open circuits. On the otherhand, larger pores on the critical path have resistancesso much smaller, that they may be ignored, and aretherefore replaced by short circuits. Thus this treatmentof critical path analysis (CPA) originally from [23] re-places the entire distribution of resistance values bythree: open circuits (nearly) critical resistances, andshorts. Although this sounds oversimpli®ed, it is themost complex version available. The nearly critical valueof R is then treated as an optimization parameter, inspirit with the tendency for charge or water to ®nd theoptimal conducting path.

The second version of CPA is due to B�ottger andBryksin [24], (hereafter called BOBR), and is of partic-ular relevance since it was chosen as the basis for the KTapproach to porous media. The di�erence is in thesimpli®cation of the network. BOBR employ two classesof resistances to describe the full range of variability.Thus a system with continuously distributed localresistances is represented in the same way as an insula-tor±conductor composite. Resistances larger than anarbitrary value, R, are treated as in®nitely large, allsmaller resistances given the value R. The cuto� R ischosen to maximize the conductivity. Such an algorithmis intended to represent the tendency of water, as well aselectricity, to follow the path of least resistance. But thistreatment tends to overestimate the resistance of thecurrent-carrying paths because it overcounts the numberof large resistances. The problem is di�erent in the ACconduction because the method introduces a bias in thecounting of the resistances vis-�a-vis capacitances of in-dividual portions of the network. This bias overwhelmsthe overestimation of the resistance by displacing theirin¯uence to a lower frequency ± where conductionshould be much more di�cult.


In one-dimensional (1D) solid-state systems wheretransport is by electronic tunneling between localizedsites, nearest neighbor transition rates, wij, are

wij � m expbÿrij=ac; �2:3�where rij is the site separation, m a constant with di-mensions of inverse seconds, and a is a fundamentallength scale. The mean separation of the sites is b. Thedc conductivity of a chain of sites of length L is knownto be proportional to L1ÿ2b=a, and is zero in the limitL!1. So the ac conductivity of an in®nite chainvanishes in the limit of zero frequency, x, of the appliedelectric ®eld. The correct dependence [25],

r�x� / x�1ÿa=2b�=�1�a=2b�: �2:4�The BOBR treatment yields

r�x� / x1ÿa=b: �2:5�Both functions are positive powers of x, and satisfy therequirement that the conductivity vanishes at zero fre-quency. But the ratio of the BOBR expression to thecorrect result, as a function of frequency ``goes like'',

x�ÿa2=2b2�=�1�a=2b�; �2:6�which, in the limit of zero frequency, is in®nite. Ofcourse, if the (percolation) limit a=b! 0 is taken atarbitrary frequency, the two results are identical, but forany ®nite b (site separation), the BOBR result isseriously too large. Thus, under extreme cases, theBOBR formulation can lead to spectacular overestima-tion of the conductivity. The formulation of [23], how-ever, was later shown [26], to yield the correctexpression, Eq. (2.4). That the BOBR treatment couldlead to an underestimation [12] of the hydraulic con-ductivity by a factor 2 (as argued in [15,16]) is thus notsurprising. The strength of the Friedman and Pollak [23]version (henceforth called FP) is that it simultaneouslyexplains the large failure of the BOBR treatment in one-dimensional hopping systems, and its smaller problemsin the saturated hydraulic conductivity.

The second uncertainty involves length scales.Treating the smaller resistances on the critical path asshorts allows its resistance to be written as proportionalto the inverse of the separation, l, of the critical re-sistances on this path, the conductance proportional tol. Then the critical conductance can be converted to acharacteristic conductivity value if the separation ofcontributing paths, L, is known as well. A fairly goodexpression for l is obtained by using the typical sepa-ration of critical resistance values in the bulk sample;although slightly better calculations exist [27], they arefar more di�cult. Using the simplest expression,

l � c0

R Rc�eRc

RcÿRc=e W �R�dRR Rc

0W �R�dR

" #ÿ1=d

�2:7�

with c0 the product of a numerical constant of orderunity and a fundamental pore length. Because the valueof c0 is not well constrained, uncertainty exists in com-parison with experiment and simulation. Now,

rdc � lRcLdÿ1

: �2:8�

The evaluation of L requires re-examination of thechoice of R � Rc. But calculation of the conductivityrequires relation of L to R. It has often been assumedthat L is related to the correlation length from percola-tion theory. This correlation length is unrelated to cor-relations in the positions of resistances of a given size,but is a representation of how large clusters of resistancescan get (by random association) if the concentration ofresistors is anywhere near the critical value. The reasonwhy the choice R � Rc must be re-evaluated is that L�Rc�is always in®nite, and some resistance other than Rc mustbe chosen for Eq. (2.8), otherwise the conductivity isidentically zero. FP developed an optimization schemefor choosing this resistance, and using this optimizationscheme it is possible to reconcile apparently confusingpieces of information. This optimization scheme is dis-cussed in detail in the next section.

Relating R to the correlation length is quite di�erentin cases where R is an exponential function of randomvariables (based on geometry of the pore space), andwhen it is a power of a random variable, such as inPoiseuille ¯ow, where Q / r4. In the exponential case,L / ln�R=Rc��ÿ�dÿ1�m�; while in the power-law case,L / �Rÿ Rc��ÿ�dÿ1�m�

, where m � 0:9 is a critical exponentfrom percolation theory and d is the dimensionality ofthe system. Because of this di�erence, only in the formercase is the structure of the current-carrying paths tor-tuous and describable in terms of concepts of percola-tion theory, while in the latter, appropriate for Poiseuille¯ow, the structure of the current-carrying paths is un-related to percolation. The di�erence between expo-nential and power-law cases is exempli®ed in Figs. 1and 2.

In the present context, I mention again the work ofLe Doussal [16], who starts from an equation similar toEq. (2.8),

rdc � K0gc gcP �gc�� y ; �2:9�

where K0 is a constant, gc the critical conductance, and Pis a function of gc which involves L. Le Doussal [16]asserts that y � �d ÿ 2�m depends only on dimensional-ity. If L and l in Eq. (2.8) were identical, then one couldsubstitute L�dÿ2� for L�dÿ1� in the denominator of Eq.(2.8), and our expressions would be identical to thispoint. Indeed two-dimensional simulations yield r � gc,which has been interpreted [6,21], to mean that l and Lare equal. But, since the optimal and critical values of gare di�erent, this conclusion does not follow. Besides,


experimental results [28] require that L and l besystematically di�erent [29]. A greater di�erence is notedsubsequently. [16] considers exclusively the distributionof ln(g), apparently on the basis of his statement``A useful model to study is g � g0 exp�kx� with a ®xeddistribution, D�x� of x. Then one has gcP �gc� �D�xc�=k� 1, if k is large.'' Such an argument is familiarfrom solid-state physics (and is indeed largely a re-statement of the FP formulation, but with di�erent ex-ponents), where exponential functions of randomvariables are the rule, but does not work in saturated¯ow, if Poiseuille ¯ow is envisioned, and a result for thehydraulic conductivity in terms of a power of a critical

pore radius is sought. In the case where R is an expo-nential function of random variables, then L is related toa logarithm of the conductance, in accord with [16]. Butin the case where R is a power of a random variable (asin pore throats using Poiseuille ¯ow), then L is a powerof Rÿ Rc, and not a logarithmic function. In this case, itis shown here, both theoretically, and numerically, thatwhile the critical conductance is still relevant to system-wide transport, the critical network with tortuous pathsis not. BB came to the same conclusion. Thus, themethod of [16] is internally inconsistent, by virtue of hisrelying on methods appropriate for resistances, whichare exponential functions of random variables. Other

Fig. 1. Computer generated 2D random resistor network with exponential dependences of resistance values on random variables, Rij � R0 exp�nij�.All R's with R < Rmax are shown as bonds, and those with Rmax=e2 < R < Rmax are shown in bold. In (a), Rmax � Rc=3:8; v, the size of the largest

cluster, is about 15 bond lengths, l, the typical separation of R's within a factor e2 of Rmax, is drawn as about 5 bond lengths. In (b) Rmax is chosen

equal to Rc: v is in®nite. l should again be about 5 bond lengths, but in this ®gure l is about 10. In (c), Rmax is chosen as 3:8 Rc � Ropt, the optimal

value for dc conduction. l is again about 5 bond lengths, while v � L is about 20 bond lengths. The ®gure shows that at optimal conduction, the

current path is still rather tortuous, l is a slowly varying function of Rmax, while v is strongly varying ([36]; Hunt and Skaggs, 2000, submitted).


authors, e.g. [20], have also used a similar formulationwith respect to the exponent (proportional to �Dÿ 2�m),but with a power-law dependence, as here.

3. General calculations using CPA

In this section, the structure of the calculation ofgeneral dc transport using percolation theory in theform of critical path analysis is discussed. The generalstructure of such a calculation does not depend directlyon the transport property involved, whether electricalconduction or ¯uid ¯ow, although it may depend on thedetails of the local conductances.

Percolation theory is based on the geometry of con-nectivity [5]. If some number of objects of given size andshape are distributed in a volume of some particularsize, what is the probability that at least one path can befound across the volume which never contacts theseobjects (or which never loses contact)? The result iseither one or zero in the limit of in®nite size [5] Thecrossover from one to zero occurs at a well-de®nedconcentration. As a consequence, in the limit of in®nitesystem size the dc conductivity of ``nominally homo-geneous'' systems can be accurately calculated using crit-ical path analysis [2]. By nominally homogeneous (forporous media) I mean systems with the same bulkproperties, such as bulk density, distributions of particle

Fig. 2. Analogous to Fig. 1, except that resistances are now power laws in the pore radius, compatible with Poiseuille ¯ow. In (a), Rmax � Rc=2, and l

turns out to be about 2, while v is about 5. In (b), Rmax � Rc, l is again about 2, while v is larger than the system. In (c), Rmax � 2Rc, l is about 1.5, and

v � L is about 5. This ®gure demonstrates that the current-carrying path is not tortuous, and that L is a much smaller value, on the order of the pore

separation. l is still a slowly varying function of R ([36]; and Hunt and Skaggs, 2000, in review).


sizes, composition, organic content, ionic concentration,etc. Even when these properties show neither random,nor systematic variability, the local variation in poresize, and for unsaturated systems, variation in moisturecontent, can be so large as to make the systems stronglyheterogeneous from the perspective of transport or ¯ow.When the size of the regions with a given suite of bulkproperties is in some sense small (the particular con-ditions will be clari®ed in the derivations) ®nite-sizedcorrections must be included.

A simpli®ed problem [21], which illustrates theconcept of percolation theory, is that of a square (two-dimensional) lattice, on which bonds between sites canbe connected at random with some probability, p. Forp values less than (greater than) pc � 0:5 no path (atleast one path) can be found which connects places atin®nite separation. In a system in which the bondscorrespond to conductances distributed continuouslyover some wide range, one can arbitrarily regard allconductances greater than some arbitrary value, g, asbeing connected bonds. For some critical value ofg � gc, then, the set of all conductances g > gc pro-duces an in®nitely long connected path. That particularvalue of g � gc is then of great signi®cance for themacroscopic (large-scale) conductivity. gc turns out alsoto be of signi®cance for the statistical variability of theconductivity on smaller length scales, as well as thetransient response of the system, both of which can beexpressed in terms of the characteristic (or critical)conductance.

Quantities such as pc and hence gc are highly depen-dent on many system parameters, such as the distribu-tion of g, mean local coordination numbers, and theshape of regions associated with g [21]. gc is referred toas system-speci®c, or non-universal. Other properties,such as, for a given value of �p ÿ pc�=pc, the clusterstatistics (number of clusters of a given number of el-ements per unit volume), are the same in a wide varietyof cases, provided the system is near critical percolation,i.e., p and pc are of similar magnitude to each other [5].Properties, which do not depend on the value of pc, canbe termed universal (or quasi-universal), because theydo not depend on the geometrical shapes of the indi-vidual objects, although they do depend on spatial di-mension [5]. They also appear to be the same whetherthe individual bonds between sites are geometricallyordered or not [5]. Physical results which involve thedependence of cluster numbers on �p ÿ pc�=pc includethe scale-dependence of the distribution of hydraulicconductivity values [27], the correction in the meanconductivity of ®nite size systems to the in®nite systemconductivity [27], and the spatial dependence of thesemi-variogram [30]. These and other properties, such asthe relationship between the electrical and hydraulicconductivities, can be expressed in terms of gc, and de-pend on the form of the dependence of local conductiv-

ities, but not their values or distributions. These a�ectonly gc. In the systems considered here, with continu-ously distributed values of g, it is always possible toisolate a subsystem with g near gc, for which the clusterstatistics near percolation are relevant. On the otherhand, systems composed of individual volumes witheither very large or essentially zero conductances (suchas fractured impermeable rock) may or may not meetconditions for the relevance of percolation theory.

Functional forms of local resistances in solid-statephysics applications: Critical path analysis applied tosolid-state conduction problems has always started withthe assumption that transport on the microscopic scaleinvolves mechanisms whose rates, wij / Rÿ1, dependexponentially on random variables. These mechanismsinclude:1. Particle hopping over a barrier (from i to j),

wij � w0 exp�ÿEij=kT �.2. Tunneling through barriers, wij � w0 exp�ÿ2rij=a�;

wij � w0 exp�ÿEij=kT ÿ 2rij=a�.In the above applications, a is the localization length, Erandom energies, k the Boltzmann constant, T thetemperature, and r are hopping or tunneling length. Thestandard of applicability of percolation theory (com-pared with e�ective medium theories) has been ex-pressed in terms of the spread of local conductancevalues (greater than, ca. four orders of magnitude [2,4]),not in terms of the functional form of the local con-ductances on random variables. Nevertheless it ispossible that the same criterion does not apply in caseswhere the local conductances are not exponential func-tions of random variables. As in [17] (hereafter referredto as FS), however, we will proceed under the assump-tion that percolation theory and critical path analysisare applicable regardless of the particular form of thedistribution, provided the spread of values is su�cientlylarge. This assumption is in accord with BB who foundthat the particular form of the distribution of pore sizesdid not a�ect the applicability of percolation theorycompared with stochastic methods.

Functional form of local conductances in porous media:In unsaturated soils or rocks, transport of water (hy-draulic conductivity) has been given variously as expo-nentially dependent on the moisture content [31,32] oras power-law in form [33,34]. In saturated soils, thehydraulic conductance, gh, may be treated as a powerlaw, gh / r4, if viscous ¯ow between neighboring poresis considered an example of Poiseuille ¯ow (e.g., FSin their treatment using critical path analysis). FS alsouse for an electrical conductance, ge / r2. Later in thiswork, it is shown that under speci®c conditions, unsat-urated ¯ow may lead to an exponential form ofconductance. Thus it is important to allow for eitherpower-law, or exponential, functions of random vari-ables when developing critical path analysis, and I givethe general results for both.


3.1. Optimization of Eq. (2.8) for the DC conductivity

Whether R � R0 exp nij, where nij is a random variablerelated to pore geometry, and R0 is a fundamental pre-factor with units of (hydraulic) resistance, or whetherR � R0�n�n, the procedure to ®nd the critical value of R,is to ®nd the critical value of n, and then insert it into theappropriate one of these two relationships. In either caseone hasZ nc

0

W �n�dn � pc: �3:1�

The separation of the current-carrying paths inEq. (2.8), however, involves the correlation length. Thecorrelation length is expressed in terms of p ÿ pc. Howdoes one proceed in relating p ÿ pc to resistance values(hydraulic, or otherwise)? One must start by writing thesame equation for an arbitrary p and corresponding n,Z n

0

W �n0�dn0 � p �3:2�

p is then an arbitrary fraction of the bonds. While pc

describes the quantile of the distribution (measuredfrom the most highly conducting bond), which generatescritical percolation, the only stipulation that we mustmake about p is that it is not be too di�erent from pc.For p ÿ pc � 1, cluster statistics of percolation de®nethe number of clusters of a given size which are formed.In the present context, this means the number of clusterswith no resistor exceeding the value that corresponds top through the random variable n. These statistics alsogenerate the density of such clusters, the tortuosity ofthe chief conducting path, called the backbone cluster,and the linear dimension of the clusters. The largestavailable cluster is de®ned by the correlation length, v,which diverges at critical percolation according to

v � v0jp ÿ pcjÿm �3:3�with m � 0:88 and v0 related to the typical bond, or re-sistor length (pore length, l0, on a network). ForR < Rc; v is the size of the largest cluster of intercon-nected resistances with largest resistance R and is ®nite.For R > Rc; v is the size of the largest region with noresistance greater than R, but for which the R's are notshorted out by equal-sized or larger clusters with smallerR's, and is also ®nite. For R P Rc; v is also the typicalseparation of paths, which could carry current. Whenthe optimum value of this separation is found, it is calledL, as above. The divergence is the reason why, if thesubnetwork employed to ®nd the conductivity wascomprised only of resistors smaller than or equal to Rc,the calculated conductivity would be zero. The separa-tion of current-carrying paths would be equal to thelinear dimension of the critical cluster, and substitutionof L � 1 yields zero conductivity. This result is relatedto the one obtained for metal-insulator composites that

the conductivity vanishes in the in®nite system limitbelow the metal percolation threshold [21].

Using p ÿ pc � 1, a relationship of nc to pc in integralform means that p ÿ pc, which appears in the correlationlength, can be written as �nÿ nc�=nc. But if n is pro-portional to the natural logarithm of R (the ®rst caseabove),

jp ÿ pcj � ln�R=Rc�ln�Rc=R0�� 3:4�

l is now the typical separation of the largest resistances,rather than Rc, but Eq. (2.7) demonstrates that lvariesonly weakly with R so that Eq. (2.7) is still used tocalculate l. Thus, resubstitution into Eq. (2.8) leads to

rdc�R� � lR

ln�R=Rc�2m: �3:5�

Optimization of Eq. (3.5) with respect to R yields

Ropt � Rc exp�2m�; �3:6�so that L � v0�2m�2m

. Formulation of the problem interms of the conductance leads to the same answer. Thecalculation is self-consistent, conduction occurringalong tortuous paths through approximately fractalclusters, and

rdc � l

Rcv0 exp�2m��2m�2m : �3:7�

For critical path analysis to be valid, the result for Rmust either be very close to or at least clearly related toRc. Here the correlation length, L, and the resistance, R,are expressed in terms of their critical values and interms of percolation statistics, respectively, guaranteeingself-consistency.

Use of Eq. (3.7) appears to imply that the conduc-tivity could not be increased beyond the optimum valueby including more resistances (and therefore additionalpaths). In fact, including additional resistance valuescannot reduce the conductivity, and the optimization isassumed to denote a crossover to a regime where addinglarger resistances does not materially increase the sys-tem-wide response. When R is an exponential functionof random variables, it is sensitive to system parameters,so the correlation length does not vary rapidly with R.Thus, the optimal value of R is not so close to the criticalvalue (exp�1:8� � 6 times larger), but the structure of theconducting paths is essentially that at critical percola-tion, complex and tortuous.

What happens when R � R0�n�n? Now linearizationyields p ÿ pc / �Rÿ Rc�=nRc. Substitution into Eq. (2.8)leads to,

rdv � ljRÿ Rcj2m=Rv20: �3:8�

Optimization yields a minimum at R < Rc this time,outside the range of validity of the expression. Thus forR > Rc, Eq. (3.8) predicts that the conductivity is a


monotonically increasing function of R. The result doesnot invalidate the application of critical path analysis,nor does it invalidate the result for p ÿ pc. But it doesmean that L in this case cannot be represented in termsof critical exponents of percolation theory, since theconditions for using this representation of L have beenviolated. The solution is to recognize that for relativelysmall increases in R > Rc, the correlation length has di-minished so much that percolation statistics no longerapply, and the separation of current-carrying paths issimilar to the separation of actual paths. Once the sep-aration is reduced to such a value, it can scarcely bereduced any further, and there is no point in increasingR any further. Since this has happened with a very smallincrease in R, R is pinned extremely close to Rc. Thisresult therefore implies that the current may be domi-nated by paths with resistance values very close to Rc,but the structure of the current paths is nothing like thatnear percolation, thus not particularly tortuous. Nu-merical solution of Kircho�'s laws (Section 4) revealsthat typical values of L in this case are about 10 (in unitsof fundamental pore separations). BB also noted, that``even when highly localized, [for very large disorder] the¯ow is not truly restricted to the critical path as de®nedby CPA.'' (The power-law case can also be formulatedin terms of the conductance; here a result is obtainedwhich is not absurd, but since the two answers di�er, theimplied value of the correlation length is outside therange of validity of the percolation-theoretical result.)The two cases, exponential vs. power functions of ran-dom variables, are contrasted in Figs. 1 and 2, re-spectively. In Fig. 1, the current-carrying path is seen tobe tortuous, but not in Fig. 2.

If the empirically determined exponential relationshipbetween the hydraulic conductivity and the moisturecontent [31] is relatively accurate, (an issue to which Ireturn) then these results imply that steady-state ¯ow inunsaturated soils should be more tortuous than in sat-urated soils. The ®nding of non-tortuous ¯ow paths forthe power-law dependence is not so di�erent from tra-ditional treatments of saturated ¯ow, like Kozeny±Carman, in which a number of parallel tubes (which donot communicate with each other) with di�ering hy-draulic conductivities are envisioned. From the ®guresas well as the optimization, L, the separation of thecurrent-carrying paths, is a low multiple of v0, the fun-damental resistor, or pore, length. From Fig. 2, l alsoappears to be a small multiple of the pore length. Ifprecision of the hydraulic conductivity better than towithin a factor of two or three is sought, these estimateswill have to be improved.

3.2. The Friedman±Seaton network

FS calculate the critical electrical and hydraulicconductances for a medium represented as a regular

network. While all throat lengths are thus equal, thethroat radii are assumed widely distributed. The par-ticular distribution chosen determines the value of thecritical conductance, but is not relevant to the argu-ments relating the critical conductance to the systemconductivity.

In this analysis, contributions to the electrical con-ductivity due to sorption of charge on clay particles arenot treated, although inclusion of such complexity ispossible, in principle. FS consider possibilities of eithercylindrical-shaped, or slit-shaped pores. In the lattercase, the power of the random variable r in each con-ductance is reduced by one (and replaced by a uniformvalue w), but this re®nement, of value, is peripheral here.

In Poiseuille ¯ow, each bond of length l0 and radius r,has hydraulic conductance,

gh � p8l

r4

l0

� khr4 �3:9�

with l the viscosity of water, and kh � p=8ll0 a con-venient way to represent all the factors which areconstant.

FS assume that the ionic concentration is also aconstant. Then the electrical conductance for the throatjoining two pores is equal to

ge � pECr2

l0

; �3:10�

where EC is the intrinsic electrical conductivity. Forlater use, we note that EC can be represented as theproduct,

EC � lenqe; �3:11�where le is the mobility of the charges qe, present involume concentration n. Using Eq. (3.10), one can re-write the individual electrical conductances as follows:

ge � pECr2

l0

� pnleqer2

l0

� kenr2; �3:12�

where ke incorporates all constant parameters, thusemphasizing that both n and r are, in principle, randomvariables.

The critical percolation condition for the hydraulicconductivity isZ 1

rc

W �r�dr � pc; �3:13�

where pc is system and dimensionally dependent. Thecritical hydraulic conductance, gh

c is proportional to r4c ,

and, in case, as FS, n is assumed uniform, the criticalelectrical conductance, ge

c, is proportional to r2c . FS use a

log-normal distribution of pore sizes, r,

W r� � � 1

rsr

��2pp exp

�ÿ ln�r� ÿ ln�r0��

2p

sr

� ��: �3:14�

It will be useful also to consider ion concentrations,which are log-normally distributed,


W �n� � 1

nsn

��2pp exp

�ÿ ln�n� ÿ ln�n0��

2p

sn

� ��3:15�

with

n0 � exp�hln�n�i�; r0 � exp�hln�r�i�;sn � Var�ln�n��; sr � Var�ln�r��: �3:16�

This conjecture cannot be con®rmed, although evidencefrom observation, [35] suggests that such a distributionmust be broad, and may well be skewed. Here I onlymean to imply that any spread of values is far morelikely than a uniform concentration; assuming the samedistribution as for r makes calculations easier and moretransparent.

Upon substitution of W �r� into Eq. (3.13), one ob-tains

pc � 1��2pp

Z 1

bh

exp�ÿ t2

�dt � 1

2erfc�bh� �3:17�

with

bh � lnbghc=khr4

0c4��2p

sr

: �3:18�

The solution of Eq. (3.17) de®nes the critical hydraulicconductance in terms of the inverse complementary er-ror function, erfcÿ1(x), of a combination of constantsincluding the percolation probability, pc.

There is no material di�erence in the FS calculationfor the electrical conductivity, and

pc � 1

2p

Z 1

beh

exp�ÿt2

�dt � 1

2erfc�be� �3:19�

with

be � lnbge=kenr20c

2��2p

sr

�3:20�

results. Thus we have (taking for the moment, as FS,n � n0)

ghc � khr4

0 expb4��2p

sr erfcÿ1�2pc�c �3:21�and

gec � ken0r2

0 expb2��2p

sr erfcÿ1�2pc�c: �3:22�The quotient of these two expressions is

ghc

gec

� kh

n0ke

� �r2

0 exp 2��2p

sr erfcÿ1 2pc� �h i

; �3:23�

which is easily seen to be proportional to r2c . Eqs. (3.21)

and (3.22) also show that the critical hydraulic con-ductance is proportional to the square of the criticalelectrical conductance,

ghc / ge

cr2c / ge

c

� �2: �3:24�

FS present evidence that the ®rst formulation is valid,while BB demonstrate the second. The ®rst formulationhas the advantage that the viscosity of water does not

enter the proportionality constant, in contrast to thesecond. FS stop here and do not calculate the electricalor hydraulic conductivities, but note that each is pro-portional, respectively, to the conductances derivedabove, and therefore the same relationship for the con-ductivities holds as for the conductances. However, FSdo note that due to the di�erent dependences of Re andRh on r the characteristic resistance on the critical, orpercolating paths, will be a functional of the distributionof resistance values, di�erent for the hydraulic andelectrical conductivities. Although this comment is cor-rect, since the characteristic resistance is just an integralover the R's from the smallest to a value somewhatlarger than Rc, the ratio of the two R's involves a nu-merical factor very nearly 1. The present procedure ofde®ning a characteristic length between maximal resis-tance yields similar results and allows the distinctionbetween tortuous and non-tortuous paths to be madebased on the functional form of the resistance values.

3.2.1. A modi®cation of the Friedman±Seaton procedurefor calculating ge

The treatment of FS implicitly assumes a uniformconcentration of ions in the groundwater. This as-sumption is not justi®able, but not much is known aboutthe true distribution of ionic concentrations, particularlyon the pore scale. Furthermore, in soils the electricalconductivity can be in¯uenced by the presence of clayminerals. In any case, evidence suggests that a widevariability may be present, particularly in the vadosezone. In Florida aquifers with carbonates and inter-bedded clays [35], ionic concentrations at probe scalescan vary by as much as orders of magnitude. In fact,mean concentrations of particular ions (in ca. 4 l ofwater) can vary by a factor of 7 over vertical distances assmall as 80 cm [35]. Since the given values are alreadyaveraged over a large area of space, it is obvious that themicroscopic variability in charge distribution can bemuch larger. Given such variability, it is possible that alog-normal distribution for pore-scale ionic concentra-tions is also appropriate. Using log-normal distributionsfor both pore throat radii and ionic concentrations it ispossible to extend the FS results for electrical conduc-tivity to a two-variable percolation problem in whichboth are treated on an equal footing. It is also useful topoint out how complications in CPA are dealt with. Thefollowing was given in preliminary form [36] (and morefully in Hunt and Skaggs, 2000, in review). The perco-lation condition reads

pc �Z 1

0

drZ 1

gec=ker2

W �r�W �n�dn; �3:25�

where the lower limit on n prevents inclusion of con-ductances smaller than the implicit critical value, forwhich Eq. (3.25) is then solved. With appropriatechange of variables, the log-normal distribution can be


converted to a normal distribution, the double integralto a single integral [37],

pc � 1

2p

Z 1

be

exp�ÿ t2

�dt �3:26�

with

be � ln�gc=ken0r20�

�2s2n � 8s2

r �1=2: �3:27�

Consequently one can write down for gec,

gec � ken0r2

0 exp��8s2

r � 2s2n

qerfcÿ1�2pc�

j k: �3:28�

Now the relationship between the critical electrical andhydraulic conductances is not so simple as in FS. In fact,

gh

ge� �x �kh

ke� �xr4ÿ2x

0

nx0

�3:29�

with

x � 4sr

s2n � 4s2

r

ÿ �1=2: �3:30�

Note that for sn � 0; x � 2, and the FS relations arerecovered, but otherwise x < 2, and can be less than 1.Although the form of this relationship depends on thedistribution chosen for n, complexity will always in-crease and usual inferences regarding ionic concentra-tion from the electrical conductivity will be incorrect.Consider sn=sr � 1, just a small modi®cation of the SRresult,

ge � ken0r20

� exp 2erfcÿ1 2pc� ��2p

sr

h iexp erfcÿ1 2pc� �

��2p

s2n

4sr

" #�3:31�

and solution of this equation for n0 leads to

n0 � ge

ker20

exphÿ2erfcÿ1 2pc� �

��2p

sr

i� exp

ÿ ��2p

erfcÿ1 2pc� �s2n

4sr

" #: �3:32�

Thus, as usually presumed, the ionic concentration isproportional to the electrical conductivity. However, theproportionality constant includes both the variability inthe pore sizes, and the variability in the ionic concentra-tion. Eq. (3.32) shows that neglect of either of these e�ectscan lead to overestimates (underestimates) of the ionicconcentration if 2pc < 1 (if 2pc > 1). In three dimensions,pc is typically less than 1=2 [21] and overestimationswould be expected. Further, it shows that, when ionicvariability is relatively small, changes in pore size vari-ability can be interpreted as changes in ionic concentra-tion, if e�ects of the ®rst exponential factor are neglected.

For later use, note that it is possible to calculatecritical values of the following combinations of powers,p and m, of n and r, respectively,

nprm� �c � rm0 np

0 exp��2p

erfcÿ1�2pc� p2s2n

�� m2s2

r

�1=2�:

�3:33�Note again that in 3D pc < 0:5 [21], and both ge and gh

are increasing functions of disorder. In 2D, however, pc

may be smaller than 0.5 [21], and both conductancesmay be either increasing, or decreasing functions ofdisorder for log-normal distributions.

3.3. Applications to fractal porous media

The question regarding what statistics to use for poresize variability appears to be unresolved. Some investi-gators, such as FS, choose log-normal distributions, andBB used log-uniform, while others [38±42] choosepower-laws associated with fractal geometry. Insofar asself-similarity in soil or rock particles and aggregation isconcerned, strict applicability of fractal statistics musthave bounds. If soils are neither rocky nor particularlycohesive, this upper bound will likely be in the milli-meter size range. Appeal to fractal fracture for genera-tion of power-law statistics of pore sizes suggests thatsoils produced mainly by physical weathering are morelikely to produce fractal geometries than those producedin sedimentary environments by, e.g., ¯uvial deposition.Whether such systematic variability can be veri®ed is notclear, but in any case, some fairly impressive evidencehas been accumulated that fractal soils have a place inmodel development [43].

Here I use a single distribution to cover a portion ofthe range from tens of microns to tens of centimeters; inindividual cases variations over 1±3 orders of magnitudeare expected to be the rule. Sharp cuto�s at both ends ofthe distribution are employed.

It has been pointed out [41] that Brooks and Corey[34] already reported results for the water content, h interms of its saturated value hsat in a form amenable tointerpretation in terms of fractal soil structures,

h h� � � hsat

hhmin

� �D0ÿ3

; �3:34�

where h is the capillary pressure, and hmin is the air entrypressure. This expression was derived in [38] using afractal geometry. Of course, at that time [34] regarded D0

as an empirical parameter. Further [43], soils (and es-pecially rocks) frequently exhibit systematic trends indensity with sample size, indicating fractal structure.Thus considerable motivation exists for pursuing a soilstructure, and associated transport model, based on afractal perspective.

Soil structure calculations with such a distributionhave been given in [39,40], and their results are adopted


without much discussion, except insofar as it isnecessary to ®nd a distribution of pore sizes and toclarify notational di�erences. The ®rst paper by Rieuand Sposito [39], will henceforth be referred to as RS. Ido not envision regular structures, but imagine a ran-dom soil structure, described nevertheless by the sameparameters and distributions given in RS. Thus theirresults for density and porosity as functions of systemsize are still assumed to hold, as well as the number ofpores at any given size. The structure of percolation-theoretical calculations requires disorder; however, suchstructural disorder also allows calculations of the vari-ability of transport properties. Thus one can derivespatial variability of transport properties within thecontext of deterministic soil structure using a deter-ministic theory of transport. In the process, trends in themean values of the hydraulic conductivity and the den-sity can be derived in accordance with observation, aswell as with simulations, [41] for example, while incor-porating variability.

3.3.1. Soil structure backgroundUsing Eqs. (17), (18), and (20) of RS, it is possible to

®nd the relative number of pores in the ith pore class(using qr > 1 as a size ratio of successive pore classes, r0

as the smallest pore radius, and rm as the largest)

Wi � A0 qÿDr

ÿ �i � A0 qir

ÿ �ÿD; �3:35�

where D is the fractal dimensionality and A0 is a con-stant. The reason for the exchange in the order of i andD is to facilitate representation of the pore radius, r, interms of qi

r.Allowing r to take on a continuous range of values,

W �r� � Arÿ1ÿD �3:36�with A another constant. The reduction in the power by1 is necessary to allow an integral over a ®nite range of r(e.g., one size class) to yield rÿD, corresponding to thediscrete case. We now, as in RS, consider an incom-pletely fragmented porous medium with both the grainsand the pores fractally distributed. In this case thefractional power Dr must be substituted for D in thepore volume distribution. SetR rm

r0dr=r� �r3rÿDrR rm

r0dr=r� �r3rÿDr � K/

� / �3:37�

with / the porosity. The term K/ in the denominatorarises from the solid volume. Constants in the distri-bution, such as A, have been absorbed into K/. Solutionof Eq. (3.37) yields, for K/,

K/ � r3ÿDrm 1ÿ /� �

3ÿ Dr�3:38�

on application of the identity (from RS)/ � 1ÿ �r0=rm�3ÿDr in both numerator and denominator.

3.3.2. Saturated hydraulic conductivityApplication of Darcy's Law at the pore scale, as an

integral over ¯uid velocities, allows possible geometricalcomplications. The safest application is to pore throats,the constrictions between pores. The scaling relation-ships derived in RS are not necessarily consistent withsuch an application, although RS did present results forhydraulic properties (with some similarities to thepresent solution; see the full reference, Hunt and Selker,2000, in review). It would be consistent with the conceptof self-similarity to assume that the constrictionsbetween pores follow the same distribution as inEq. (3.36), making application of CPA consistent withthe framework of RS.

In the present problem, it was assumed (RS) that thescaling of pore sizes a�ects all dimensions similarly, i.e.,the pore aspect ratio is independent of pore size. But it isnot correct to use critical rate analysis for bond perco-lation [17] when the bond lengths vary over orders ofmagnitude, since the bonds will not ®t on a regularnetwork. The appropriate generalization is continuumpercolation (e�ectively a percolation of open volume)[6]. Continuum percolation is de®ned in terms of frac-tional volume; when the fractional volume is greaterthan some particular value, the individual volumesconnect. In such analysis generally, the possibility must,in principle, be considered that the pore space itself doesnot connect. In the present application we assume that itdoes. This is consistent with the assumption of incom-plete fragmentation by RS. Critical path analysis in theform of volume percolation, however, ®nds the smallestpore radius, rc, necessary to complete such an inter-connected path. In the present case we can ®nd thatradius as follows:R rm

rcr2ÿDr drR rm

r0r2ÿDr dr � r3ÿDr

m3ÿDr

� ��1ÿ /�

� ac: �3:39�

The left-hand side of the equation is fraction of the totalvolume in pores with radius larger than the criticalradius, and the right-hand side is the critical volumefraction. The solution of this equation is

rc � rm 1� ÿ ac�1=�3ÿDr�: �3:40�Note that the coincidence

ac � / �3:41�leads to rc � r0. In this particular case, the porositycorresponds exactly to the minimum volume fractionrequired to construct a connected path of open volume,so that any such path will have to traverse the smallestpore available. Normally ac < / by constraints associ-ated with the derivation of the pore radius statistics. Butthe implication that the hydraulic conductivity vanishesif and when ac > / is consistent with general conceptswhich imply that an interconnected volume of pore


space is required for ¯ow. An aspect not treated in thiswork, is the determination of ac as a function of ge-ometry, which normally requires intensive simulations.Nevertheless for practical applications, ac has often beentaken to be about 15% [6].

Assume that the pores have a cross-sectional area r2,and a length (C)(r). The above results are not changed.We can write for the critical value of the hydraulicconductance,

ghc �

p8Cl

� �r3

c

� p8Cl

� �r3

m 1� ÿ ac�3= 3ÿDr� �H /� ÿ ac�; �3:42�

where l is the viscosity of water. The Heaviside stepfunction, H, of argument /ÿ ac is included only to in-dicate that, according to the assumption of a minimumpore size, r0, the hydraulic conductivity must vanishwhen the integration requires smaller pores to produce aconnected path of pore volumes. However, this factor issubsequently dropped for two reasons. First, a rigidlower cuto� in pore radii probably does not exist. Sec-ond, note that the factor with the critical volume frac-tion, ac, is less than 1; consequently when the fractionaldimensionality, Dr, approaches 3, the critical hydraulicconductance approaches 0. If Dr ! 3; / ! 0, which isa guarantee that the hydraulic conductivity vanish. Sowith the choice of a hydraulic conductivity, which van-ishes at a ®nite porosity (theoretically superior) and one,which vanishes only when the porosity is zero, I pick forsimplicity the latter. But this choice should be treatedwith caution.

The critical value of the hydraulic conductance helpsto generate the critical value of the hydraulic conduc-tivity. If it is known how many paths per unit area canbe found which are characterized by the critical con-ductance, as well as the separation of the controlling(critical) conductances on these paths, the hydraulicconductivity of an in®nite system can be constructed.We de®ne the separation of the paths with gc to be L,and the separation of the gc's on these paths to be l. Ithas been shown that for Poiseuille ¯ow in a non-fractalnetwork, both l and L are small multiples of the typicalpore separation [36]. Here no typical pore separationexists by de®nition. But the same calculation schemeapplied in previous cases [26], can also be applied here

for l (Eq. (2.7)) and yields l � rm�1ÿ ac�1=�3ÿDr�. Byarguments [36], L is given in terms of the distributionof all pores, and is, in this case also approximately rm.Such an assumption is consistent with constraints offractal geometry, where it is assumed that the ¯ow pathis biased to including the largest pores. K�1�, de®nedto be the hydraulic conductivity of an in®nitely largesystem is

K 1� � � p8Cl

� �l

L2

� �r3

m 1� ÿ ac�3= 3ÿDr� �

� p8Cl

� �r2

m 1� ÿ ac�4= 3ÿDr� �: �3:43�

3.3.3. Unsaturated hydraulic conductivityIf complications due to di�erences in imbibition and

drainage are neglected (apparent, but not overwhelmingin the simulation of Perrier et al. (1995) with whichI ultimately make comparison), assumption of equili-bration between di�erent pores by ®lm ¯ow appearsreasonable [44,45]. Then the simplest compatibleassumption for the e�ects of drying is that the largestpores empty ®rst and that all smaller pores are still full.As an initial step, de®ne the relative saturation, S, interms of the largest pore still ®lled with water, r>.

S �R r>

r0r2ÿDr drR rm

r0r2ÿDr dr

� r3ÿDr> ÿ r3ÿDr

0

r3ÿDrm ÿ r3ÿDr

0

: �3:44�

Note that the term in the denominator involving thevolume of the solid soil is absent, since S is de®ned interms of the pore space only. Solving for r>,

r> � Sr3ÿDrm

� � 1� ÿ S�r3ÿDr0

�1= 3ÿDr� �: �3:45�

To ®nd the critical value of the radius, however, con-tinuum percolation is again applied, including the termin the denominator representing the contribution of thesolid volume. Also, it is necessary to include as the upperlimit in the integral in the numerator, the largest ®lledvolume, since larger volumes no longer contribute to thehydraulic conductivity.R r>

rcr2ÿDr drR rm

r0r2ÿDr dr � r3ÿDr

m =3ÿ Dr 1ÿ /� � � ac: �3:46�

Using again, / � 1ÿ �r0=rm�3ÿDr, this result can be sim-

pli®ed as follows:

r3ÿDr> ÿ r3ÿDr

c

r3ÿDrm

� ac: �3:47�

Further manipulation generates

r3ÿDrc � Sr3ÿDr

m � 1� ÿ S�r3ÿDr0 ÿ acr3ÿDr

m : �3:48�Introducing gs � gc (from Eq. (3.42)) in the saturatedcase, and g as gc in the unsaturated case, this equationcan be rewritten in the following form:

g � gs 1

�ÿ /

1ÿ S1ÿ ac

�3= 3ÿDr� �: �3:49�

On the premise that a ratio involving l and L is onlyweakly dependent on the moisture content, I write in a®rst approximation,

K � Ks 1

�ÿ /

1ÿ S1ÿ ac

�3= 3ÿDr� �: �3:50�


Eq. (3.50) is compared with a result from Kravchenkoand Zhang [46] (KZ) who also used the RS model, butthen applied the pore-size distribution model of Burdine[47] to ®nd for the unsaturated hydraulic conductivity,

K�S� � KSS�5ÿDr=3ÿDr��1 �3:51�in which they set the tortuosity parameter, k, equal to 1.It is interesting to compare exponents; the KZ exponentis �8ÿ 2Dr�=�3ÿ Dr�, which for typical values of Dr,about 2.6±2.8 (according to the data of [39,40]), gives arange 2:8=�3ÿ Dr�±2:4=�3ÿ Dr�, very similar to thepower on Eq. (3.50) from the present analysis. Eq. (3.51)was obtained from Eq. (12) of KZ by setting the residualmoisture content equal to 0, as they were able to do. KZpresent comparison of Eq. (3.51) with the data, whichdemonstrate a satisfactory ®t for a wide variety of cases,from silt loams to loamy sands. KZ obtain Dr from soilparticle-size plots, as in RS.

Although Eq. (3.50) is incomplete, further calcula-tions are really estimations, so Eq. (3.50), is comparedalready with results from simulations of [41] in Fig. 3.The results of [41] were for a construction, which turnedout to have fractal dimensionality of 2.86. For a simpleDr of 2.875 �3ÿ Dr � 1=8� and a ratio of largest poreradius to smallest pore radius of 100, typical of thevalues considered in [41], the porosity is found to be0.44. We use a typical value [6] of ac � 0:15. The valuesextracted from the Perrier graph are, where possible,intermediate in value between their imbibition anddrainage curves. It appears that Eq. (3.50) captures themain physics. Further, note that the drainage curvestend to follow more closely Eq. (3.50), but are broken o�at higher relative saturations than the imbibition curves;thus at low S, only imbibition curves survive, producingthe majority of the discrepancy.

The theoretical results are also compared with twoHanford site soils [48]. Hanford 1 soil (H1) is fromMcGee Ranch, and has a porosity of at least 40%.Hanford 2 has a lower porosity, ca. 30%. In the resultsof RS the porosity is a function of Dr, and the ratio ofminimum to maximum pore size, r0=rm, If a soil hasfractal characteristics, it is logical to assume that the

ratio of minimum to maximum pore sizes is equal to theratio of smallest to largest particle sizes. The particle sizedistributions for H1 and H2 are given in [48], and re-produced in Table 1, from which r0=rm is determined.Thus using values for Dr obtained from experiment bythe RS equations, and the derivation for K=Ks givenhere, one can sometimes obtain predictions for the un-saturated hydraulic conductivity from particle-size dis-tributions while avoiding use of adjustable parameters.The results are shown in Figs. 4 and 5. The quality of the®t in Fig. 5 is degraded at low S because the particle-sizedistribution was only fractal over about three sizeclasses; lower size classes had to be lumped together toget a range of 2 orders of magnitude.

Eq. (3.50) is somewhat related to van-Genuchten [33]parameterization of the unsaturated hydraulic conduc-tivity,

K � KsS 1� ÿ 1

ÿ ÿ S1=m�m�2

; �3:52�where

S � hs ÿ hhs ÿ hr

�3:53�

with the subscripts s and r referring to saturated andresidual values of the moisture content.

A question that arises is, what are the relative roles ofthe percolation structure of the conductivity calculationsand the fractal structure of the soil in obtainingEq. (3.50)? This question is addressed by taking the limitDr ! 3, in Section 3.3.4, or by using the same soil de-scription with a di�erent means of calculating the hy-draulic conductivity, as in KZ above, or by pickinganother soil description as a test, done next.

In few cases for a pore distribution can one deriveclosed-form results, but one other possibility is an ex-ponential pore distribution,

W �r� � 1

Rexp

r0 ÿ rr

h ir0 < r;

� 0 r < r0: �3:54�Two soluble cases exist. In one, the pore lengths are allthe same. In the second, the pore lengths are pro-

Fig. 3. Graphical representation of Eq. (3.50), derived for K as a fraction of Ks plotted against relative saturation, S, for a fractal soil. Parameters

are, Dr � 2:875; rm=r0 � 100, yielding a porosity of 44%. Comparison is made with results from the simulation by Perrier et al. [41],

Dr � 2:86; rm=r0 � 100. Hunt and Selker, 2000, in review.


portional to the diameters. This case is treated identi-cally to that above using continuum percolation. In ei-ther case I neglect the possible e�ects of the lengths l andL. In the second case,

K � KS 1

�ÿ 1ÿ S

1ÿ ac=/

�3=4

: �3:55�

In the ®rst case, however, it is appropriate to use bondpercolation. Then, one ®nds,

K � KS 1

�ÿ 1ÿ S

ac ln 1=ac� ��4

: �3:56�

Note that in Eq. (3.56) the porosity does not appear,since the guaranteed connectivity makes the actual porevolume irrelevant. The main lesson here is that similarforms for the relationship between the saturated andunsaturated values of the critical hydraulic conductanceare found using percolation for a wide variety of as-sumed soil structures.

One ®nal point involves the value of S for which theunsaturated hydraulic conductivity vanishes. Using thegiven formulas, one can ®nd, S � 1ÿ �1ÿ ac�=/ (Eq.(3.50)), S � 0 (Eq. (3.52)), and S � ac=/ (Eq. (3.55). Eq.(3.56) has no simple interpretation. In Eq. (3.55), ifthe porosity is smaller than the critical volume fraction,than the saturated hydraulic conductivity vanishes. Theunsaturated hydraulic conductivity then vanishes ifthe saturated fraction of the volume is smaller than the

critical volume fraction. Eq. (3.50) was derived con-sistently with Eq. (3.42), where the same condition onthe critical volume fraction and the porosity was drop-ped. There, the condition was generated from a ®niteminimum pore size.

3.3.4. Fractal soils in the limit Dr ! 3Using / � 1ÿ �r0=rm�3ÿDr , in the limit Dr ! 3 Eq.

(3.50) becomes

K � Ks exp

�ÿ 3 ln

rm

r0

� �1ÿ S1ÿ ac

� ��: �3:57�

Eq. (3.57) uses the fundamental de®nition ofe � 2:718 . . .. The derivation of Eq. (3.57) in the limit ofnon-fractal soils gives possible theoretical grounds forusing an exponential form for the parameterization ofthe unsaturated hydraulic conductivity. Of course, forDr identically 3, Ks� 0 from Eq. (3.42).

3.3.5. Possible application of CPA up ``scales''If one now admits the possibility that the moisture

content can vary over length scales much smaller than asample, then it is possible to apply CPA a second time.In this case variability of S in Eq. (3.57) can lead to arather large variability in Ks, meaning that the criticalvalue of Ks is obtained by ®nding the critical value of S.For convenience, and because it is likely to be reason-able, the distribution of moisture contents in the soil istaken to be Gaussian,

Fig. 4. Comparison of the prediction of Eq. (3.50) with the McGee Ranch soil at the Hanford, DOE site. Axes are the same as in Fig. 3 (from Hunt

and Selker, 2000, in review).

Fig. 5. Comparison of the Eq. (3.50) with the North Caisson soil at the Hanford, DOE site. Axes are the same as in the two previous ®gures (from

Hunt and Selker, 2000, in review).


W �S� � 1��2pp

rs

exp

"ÿ S ÿ Sm��

2p

rs

� �2#: �3:58�

Under this condition, the relationship between the val-ues of the unsaturated and saturated hydraulic con-ductivities can be written in terms of the mean, Sm, andvariance, rs of the moisture content distribution asfollows:

K � Ks exp

�ÿ3 ln

rm

r0

� �1ÿ Sm ÿ crs

1ÿ ac

� ��: �3:59�

This result (approximate because Dr cannot be 3) is anexplicit derivation of an exponential dependence of the(unsaturated) hydraulic conductivity on the relativesaturation. Eq. (3.59) may be important for the discus-sion of the variability of the hydraulic conductivity inunsaturated soils with steady-state ¯ow [31], who foundthat under such conditions, a log-normal distribution ofK is measured (derived in Section 4).

4. Finite-size calculations and distributions of the hy-

draulic conductivity

Systems of ®nite size must be treated di�erently thanin®nite systems. While it may be possible to treat ®nite-size corrections using scaling theory, it is easiest to usethe cluster statistics of percolation theory. Moreover,these statistics can be adapted to give concrete expres-sions for cross-covariance, semi-variograms and otherformulations involving spatial correlations as well as thedistributions of the hydraulic conductivity themselves.

4.1. General results

Cluster statistics of percolation theory are given [5] interms of the volume concentration, ns, of clusters of agiven size at a bond (or site) probability, p, relative tothe critical value, pc,

ns � C�d�sÿs exp fÿ sr p�� ÿ pc��qg; �4:1�

where s refers to the number of sites on the cluster, ameasure of the cluster volume (or area in two dimen-sions). C�d� is a numerical factor of order unity, butdepends on dimensionality, d. In 3D C�d� is 1.6, andhenceforth neglected. The value of q is 1 or 2, dependingon the method used to obtain it; while theory appears torequire q � 1, Monte Carlo simulations appear to re-quire q � 2 [5]. Results for both values will be presented.When it materially eases calculations (allows integrals tobe expressed in closed form) I choose q � 1. The expo-nents r � 0:4 and s � 2:2 are critical exponents frompercolation theory. A number of such exponents existfor di�erent aspects of percolation theory, and the in-terested reader is referred to Stau�er [5] review for moredetailed discussion. Scaling relations between the expo-

nents make only two of them independent [5]. Suchexponents and scaling relations were ®rst derived in thetheory of phase transitions. In order to use Eq. (4.1), smust be related to the size, or linear dimension, N, of thecluster (in units of l, the typical separations of the largestresistors), while p ÿ pc must be related to the largestresistance on the cluster. The relationship of s to N [49],is managed using three relationships, the ®rst of which is

rs � srm; �4:2�where rs is the linear dimension of a cluster of s sites (atpercolation), and the combination of exponents 1=rm isknown as the fractal dimensionality for percolationclusters, df : df is completely unrelated to Dr in the pre-vious section, other than the fact that each must be lessthan d. The second relationship is

Nl � rs; �4:3�i.e., the characteristic separation between maximallyvalued resistors and the number of such resistors in apath is the length of the path. The third relationship is

N 1=rm=s � f ; �4:4�where f is the fraction of resistances represented by thelargest values, Rc here. This calculation has been per-formed for both fundamental cases, when the local re-sistances are exponential functions, or power functionsof random variables. The result for the volume con-centration, nN , of clusters of length N is, in the formercase,

nN � Nÿ4lÿ3 exp

(ÿ Nl

L

� �1=m

ln R=Rc� �jj" #q)

: �4:5�

Also, when the tortuosity of the backbone cluster isaccounted for, m is replaced by 1 [27]. Since tortuosity isonly relevant when local resistances are exponentialfunctions of random variables, for the saturated hy-draulic conductivity it is probably appropriate to keepthe power m. On the other hand, m � 0:88 � 1 anyway,and the simpli®cation that arises from substituting 1 isgreat, so henceforth, m is replaced by 1.

In the case of power-law functions, however, we ®nd

nN � Nÿ4 exp

�ÿ Nl

L

� �J

Rÿ Rc

Rc

� �� q�; �4:6�

where J is a constant that depends on the system con-sidered. This particular form will be very useful in dis-cussions of the saturated hydraulic conductivity. Due tothe linearization procedure, which relates p ÿ pc toRÿ Rc, the same results are obtained for the conduc-tance, g, electrical or hydraulic. It should be noted thatthe above result appears to allow the possibility ofnegative values of R, because the derived statistics areGaussian in form, but with a width comparable to themean value. In fact, when Ris reduced much below Rc,the cluster statistics no longer apply. As a consequence,


the true statistics are di�erent for very small values of R,and there is no contradiction. But the exact results in thelimit R! 0 and in the limit R!1 are not known, anddepend on the particulars of the distribution of localconductance values. In these cases, some form of ex-treme value statistics is likely to apply. Thus, while inthis case, percolation theory demands symmetry nearthe peak of the distribution, physical conditions, and theproportionality of the width of the ``Gaussian'' distri-bution to its peak value, require that away from thepeak the distribution be skewed so as to avoid thepossibility of negative resistances.

With the volume concentration of clusters of lengthN, and maximal resistance value R, how can one ®nd theprobability density for measuring a particular value ofthe resistance? First, as in [27] observe that any time thesystem is located entirely within a cluster of maximal Rdi�erent from Rc, that R, rather than Rc, will bemeasured as the characteristic resistance. Thus thequantity needed for the statistics of the hydraulic con-ductivity is the fractional volume occupied by clusterslarger than the system, and with any particular value ofR. This is accomplished by noting that the total volumeoccupied by clusters of linear dimension N and maximalresistance value R, is proportional to [27]

Nl� �3nN �R� � 1

Nlexp

�ÿ Nl

Lln

gc

g

� �� q�; �4:7�

where the quotient gc=g was substituted for R=Rc. Whenq � 1, an absolute value is required to generate sym-metry about gc. But when q � 2, symmetry is guaranteedfor all formulations with either g or R. In the case whenR has a power-law dependence on random variables [27]

Nl� �3nN �R� � 1

Nlexp

�ÿ Nl

LJ

g ÿ gc

g

� �� q�: �4:8�

The probability that a volume of linear dimension x islocated entirely within a cluster of maximal resistance Ris proportional to the total volume occupied by suchclusters with N > x=l [27]

W �x;K� /Z 1

x=l

dNN

exp

�ÿ Nl

LJ ln

KKc

� �� q�; �4:9�

W �x;K� /Z 1

x=l

dNN

exp

�ÿ Nl

LJ

K ÿ Kc

Kc

� �� q�:

�4:10�In Eqs. (4.9) and (4.10), the substitution of K for g isalso trivial; the length scales necessary for the transfor-mation appear in both numerator and denominator anddivide out.

For q � 1, and in the case of exponential functions ofrandom variables, the integral is immediately evaluatedto be

W �x;K� � L=x� � K=Kc� �x=L

ln Kc=K� � K < Kc; �4:11�

W �x;K� � L=x� � Kc=K� �x=L

ln K=Kc� � K > Kc: �4:12�

These results are power-laws; in case q � 2 is chosen, anexponential integral results. In the limit of large x, theexponential integral is approximated by a log-normaldistribution, which, in limited ranges of K, is similar to apower-law, and also highly skewed. Of course theseexpressions are relative, not absolute, probabilities.Since the power increases with increasing x, the distri-bution of conductivity values narrows with increasingsystem size.

The preferred form of the cluster statistics from thestandpoint of generating agreement with Monte Carlowork is the Gaussian [5]. In this case, the lowest orderapproximation of the distribution of conductivity valuesis log-normal [27]. It is probably signi®cant that [31] e.g.,observed log-normal distributions of the hydraulicconductivity under steady-state conditions in unsatu-rated soils. His observation that the values of the hy-draulic conductivity were exponential functions of themoisture content would seem to validate the assump-tions necessary for obtaining the log-normal distribu-tion. Further, the derivation of Eq. (3.58) shows that it ispossible to state the conditions, under which, an expo-nential dependence of the hydraulic conductivity on themean moisture content of a soil can be obtained; stronglocal variance of moisture content, as well as a limit of afractal dimensionality approaching 3. However, somecaution is appropriate for the reason that the exponen-tial-dependence of the unsaturated hydraulic conduc-tivity on the relative saturation is not believed to begenerally appropriate. Nevertheless, the present deriva-tion has shown, from ®rst principles, how it is possibleto derive the results observed by Nielsen [31], and howsteps of the derivation appear to have been veri®ed byexperiment.

In the other extreme, when local conductances arepowers of a random variable, the resulting distributionis again an exponential integral function, but in the limitof large x, it may be approximated as a Gaussian. Theseresults will not be displayed explicitly here, as they arerepeated in the next two sections.

Evaluation of the mean conductivity from Eq. (4.12)is mostly straightforward, and the result is [27]

K�x�h i �R1

0W x;K� �K dKR1

0W x;K� �dK

� K 1� � 1

"� L

x

� �2

� Lx

� �3#; �4:13�

where K�1� is the hydraulic conductivity of an in®nitesystem as found in CPA. In 2D, the powers 2 and 3 are


replaced by 1 and 2, respectively. The third term arisesin either case, whether the local conductances, arepower-law or exponential functions of random vari-ables, but the second term results only when the localconductances are exponential functions of randomvariables. However, the estimation of the integral for thecase of the power-law form, that the second, and not thethird term, would survive [27] was incorrect [50]. In fact,by comparison with results for the low frequency-dependent conductivity [51] it is seen that the third termis always present, and simulations presented here agree.

Eq. (4.13) is compared with the numerical solutionsof Kircho�'s Laws on a two-dimensional (2D) grid ofsize x (for the case when the individual gs are expo-nential functions of random variables) in Fig. 6 (from[50] and Hunt and Skaggs, unpublished). Eq. (4.13) re-produces the simulated results very well. In Fig. 7 theresults when the local conductances are a power-lawfunction of random variables are given. Here only the

third term, proportional to 1=x2, is present. Two othersimulations of each case, but with di�ering local con-ductance distribution widths were also performed, butdid not provide qualitatively di�erent results. In all sixcases, the ®tted value of K�1� was less than 3% di�erentfrom the calculated value of gc, in agreement with resultsof [6]; in the three exponential cases, L was always 30%smaller than calculated, but in the three power cases, Lwas 50% smaller than calculated, and typically less than10. In this case, as noted in Section 2, results for L couldnot be justi®ed because the calculated value depends onwhether the optimization of the conductivity is per-formed with respect to the conductance or the re-sistance, in contrast to the case with exponentialfunctions of random variables. The ®tted values for thepower-law cases are in accord with the deduction that Lis a small multiple of the fundamental pore separation.But the calculated value of l is consistently a factor oftwo or more too small, probably due to the fact thatmany of the largest resistors on the path can be avoided.Although this topic has been addressed [52], no analyt-ical results for the appropriate separation of maximalresistors were given for the present use.

These results are interesting in that they can be in-terpreted to provide support for the case that L and l areequal. In fact, the near coincidence of the in®nite systemconductivity and the critical conductance does not implythe equality of L and l because K�1� involves the op-timal conductance, and this value is not equal to thecritical conductance (a ratio of t to exp�m�).

The reduction of a mean hydraulic conductivity withincreasing size is interpreted thus. Large clusters ofcritical resistances contribute additionally to the con-ductivity in ®nite systems even when they cannot reachboth sides of an in®nite system. The reason is that atcritical resistance �p � pc� the cluster numbers, ns, decayaccording to a power of the cluster size, so these clusters

Fig. 6. Comparison of predicted ®nite-size e�ects on the mean conductivity with numerical simulation. An exponential dependence of the local

conductance on random variables was chosen. The calculated value of the critical conductance was 1.0, the ®t for the in®nite system conductivity was

0.98. The calculated L was 14.76, the ®t 11.34 (in units of fundamental pore separations), and is indicated on the graph. In the simulations, it was

shown that the skewness of the distribution increases rapidly when the system size diminishes below 11, and this is the reason why the ®t deteriorated

rapidly. The calculated Kc is 2% larger than the observed conductivity, while the calculated L is about 30% larger. A crossover from a regime of an

xÿ2 dependence to an xÿ1 dependence occurs with increasing system size. Relationships between results of calculation and simulation were repeated

for di�erent distribution widths (not shown) [50].

Fig. 7. The same as Fig. 6, except that local conductances are powers

of a random variable. Here, only the term proportional to the system

size to the negative 2 power is evident. The theoretical value of L

proved to be approximately a factor 2 too large, but the calculated

value of the critical conductance was still within 2% of the in®nite

system conductivity. The important di�erence is that there is no regime

with an xÿ2 dependence [50].


continue to contribute out to essentially unlimited size.Non-critical resistance values contribute also. In the log-normal case, the skew in the distribution means that aregion of conductivity, say, 2Kc is equally likely to occuras one with Kc=2. The ensemble average over such adistribution must be larger than the conductivity at thepeak. Nevertheless, such clusters become less commonwith increasing size, so that the additive contribution isreduced. The present results are also in general agree-ment with numerical work of Paleologos et al. [53] who®nd an approximate power-law decay to an in®nitesystem hydraulic conductivity in an ensemble averageover heterogeneities under the same conditions. Further,both results imply that the corrections increase in sizeand are important over larger length scales with in-creasing disorder.

4.2. Finite size e�ects in fractal soils

The variability of the hydraulic conductivity for®nite-sized cubical systems of linear dimension x isexpressed in terms of cluster statistics of percolation

theory also when continuum percolation is applied [5]�q � 2�,ns � 1:6sÿs expbÿ srja� ÿ acj�2c �4:14�but in terms of the volume fraction, a and the criticalvolume fraction, ac. a can be any volume fraction, whichis not greatly di�erent from ac. As in [27] as long asx > rm, the largest pore radius, Eq. (4.14) can be trans-formed to (using linearizations of all di�erences, aÿ ac,etc.)

W g� � /Z 1

x=l

dNN

� exp

(ÿ g ÿ gc

gc

� �NlL

� �3ÿ Dr

3

� �1�

�ÿ ac�

�2):

�4:15�The factor Nl/L inside the exponent arises from thefactor sr. The product of the exponential portion of thisintegral and Nÿ4lÿ3, from cluster statistics, representsthe volume concentration of clusters of interconnectedconductances all greater than or equal to g of linear

Fig. 8. Schematic plot in n, r plane (ionic concentration� n, r� pore radius) of the unbiased probability densities for measuring particular values of

n and r, together with the condition relating the measured electrical conductivity to the product of n and r2. Factorization of the product nr2 allows

representation of the allowed phase space as the curve, r / nÿ1=2. The conditional probability of measuring a given value of the hydraulic con-

ductivity is obtained by integrating over the appropriate n density along this curve [36].


dimension Nl [27] from [5]. A factor �Nl�3 is the volumeof the cluster; the product represents the fractionalvolume ®lled by such clusters, proportional to theprobability that an experimental volume will ``cut'' sucha cluster. The product Nÿ4lÿ3�Nl�3 � 1=N . The reasonfor this choice is that such clusters can provide alternatepaths for the movement of water, described by a dif-ferent rate-limiting conductance. Approximate evalua-tion of Eq. (4.15) for large x using �K ÿ Kc�=Kc ��g ÿ gc�=gc yields

W �K� / 1

x

� exp

(ÿ K ÿKc

Kc

� �xrm

� �3ÿDr

3

� �1�

�ÿ ac�

�2):

�4:16�

Eq. (4.16) is just the lowest order approximation to anexponential integral, [27]. Finite-size corrections for anensemble-mean conductivity (from [27]) yield

K x� �h i � K 1� � 1

"� L

x

� �3#

� p8Cl

� �r2

m 1� ÿ ac�4= 3ÿDr� �1

�� rm

x

� �3�

�4:17�

(note that the �rm=x�2 term is absent as discussed inSection 4.1). The complementary problem of how tocalculate the distribution of conductivity values and ®-nite-size corrections to the mean when x < rm, is some-what more di�cult. Calculations are simpli®ed,however, if one considers instead a system, which is anagglomeration of individual systems of size x, repeatedin a statistical sense. Here it will be assumed that thelargest pore has a size x=b, which means that the upperlimits on the integrals must be changed from rm to x=b.R x=b

rcr2ÿDr drR x=b

r0r2ÿDr dr � x=b� �

3ÿDr

3ÿDr1ÿ /� �

� ac: �4:18�

Here the porosity, /, is also understood to be replacedwith a porosity appropriate to a system in which poreslarger than x=b are excluded. This is accomplished bysubstituting x=b for rm in the expression for the porosityas well. As a consequence, x=b has been substituted forrm everywhere in the equation and the result for rc canalso be obtained from Eq. (3.40) by substituting x=b forrm,

rc � xb

� �1� ÿ ac�1= 3ÿDr� �

: �4:19�

Now write down the mean conductivity for a system ofsize x,

K x� � � p8Cl

� �l

L2

� �xb

� �3

1� ÿ ac�4= 3ÿDr� �1

"� L

x

� �3#;

� p8Cl

� �xb

� �2

1� ÿ ac�4= 3ÿDr� �1

"� 1

b

� �3#�4:20�

using again L � x=b, and in analogy with above,l � �x=b��1ÿ ac�1=�3ÿDr�

. In this case, ®nite size correc-tions to the conductivity change only the numericalconstant, independent of size. This result should be nosurprise; the soil is scale invariant in the size ranger0 < x < rm. As smaller system sizes are envisioned, thescale of the chief conducting regions changes identically,and the system, in its ®nite size, does not change itsfundamental appearance or statistical variation. Thecombination of Eq. (4.20), an increasing mean hydraulicconductivity for small system sizes, and Eq. (4.17), adiminishing hydraulic conductivity at large sizes, impliesthat the mean hydraulic conductivity can pass through amaximum at an intermediate size. The same tendencyexists in the width of the distribution of K.

The volume distribution of hydraulic conductivity isdirectly written down,

W �K� / 1

xexp

(ÿ K ÿ Kc

Kc

� �x

x=b

� ��

� 3ÿ Dr

3

� �1� ÿ ac�

�2)

� 1

xexp

8<:ÿ K ÿ Kc� �bp=8Cl� � x=b� �2 1ÿ ac� �4= 3ÿDr� �

!"

� 3ÿ Dr

3

� �1� ÿ ac�

#29=;: �4:21�

The ®rst expression is the same as Eq. (4.16) for W �K�(with the substitution rm ! x=b�, but the second, withexplicit representation of Kc, has an advantage. Itdemonstrates that the width of the distribution of Kvalues increases linearly with increasing system size untilthe size exceeds that of the largest pore, at which pointthe result in Eq. (4.16) demonstrates that the widthdiminishes as xÿ1. The dependence of the width of theconductivity distribution on system size is identical tothat of the mean value, making measurements of a nearzero K equally possible at any size (up to rm). Thischaracteristic is shared by log-normal distributions.However, use of percolation cluster statistics in the limitK ! 0 is not appropriate, as it is far from percolation.Such limitations are probably of greater importance inthe next section.


5. Spatial correlations

The calculation of cross-covariance and semi-vario-grams can also be accomplished with the cluster statis-tics of percolation theory [30]. The de®nition of thesemi-variogram is given in [54] or [55], but a practicalway of estimating the semi-variogram is based on

CK�h� � 1

2N�h�XN�h�j�1

K xj

ÿ �� ÿ K xj

ÿ � h��2 �5:1�

[56]. For stationary random functions, the semi-vario-gram is related to the covariance, CK�h� [7,54,55],

CK�h� � CK�0� ÿ CK�h�: �5:2�The calculation of the variogram follows directly fromcluster statistics and the (unnormalized) probabilitydensity W �x;K�. Typically the variogram may be fac-tored into a spatial dependence and the variance, so thatat large distances the variogram is coincident with thevariance [56]. The distance over which the principlevariation occurs is called the range [56]. Consider that itis known that the conductivity of a given volume is K.This fact implies that the volume intersects a cluster ofsize at least x with minimum conductance g that corre-sponds to K. The probability, that the conductivity isalso K in a volume of the same size a distance h away isthen obtained from the probability that the cluster wasat least x� h in size, given the fact that it was at leastx; P �x� h : x;K�, otherwise the conductivities at the twosites are completely uncorrelated. This is a conditionalprobability, and is given by the quotient of the prob-ability that the cluster is of size at least x� h and theprobability that the cluster was of size at least x. Theresult for P�x� h : x;K� is then directly related toW �x;K� from Section 4,

P x� � h; x;K� �R1

x�h� �=ldNN exp ÿ Nl

L

ÿ �ln Rc

R

ÿ �� q� R1x� �=l

dNN exp ÿ Nl

L

ÿ �ln Rc

R

ÿ �� q� �R1

x�h� �=l I dNR1x=l I dN

; �5:3�

where I is the integrand. Here the expression valid forlocal conductances which are exponential functions ofrandom variables has been chosen in order to be speci®c,and because in the case q � 1, the integrals are el-ementary (if the logarithm is neglected). But the vario-gram is a measure of the lack of correlation, and istherefore proportional to 1ÿ P�x� h : x;K�,

1ÿ P �x� h; x;K� �R1l

x=l I dN ÿ R1x�h� �=l I dNR1x=l I dN

� 1ÿ xx� h

� �exp

�ÿ x

l

� �ln

KKc

� �� ; �5:4�

where the second equality follows when q � 1 is chosen.The spatial dependence of the variogram is now ob-tained as an integration over the distribution of K valueson cubes of size x,

CK h� � / 1ÿZ 1

0

xx� h

� exp

�ÿ x

l

� �ln

RRc

� �� W 0 K; x� �dK; �5:5�

where

W 0 x;K� � � W x;K� �R10

W x;K 0� �dK 0�5:6�

is the normalized probability density that a measure-ment of the conductivity of a volume x lies within dK ofK. Note that expressions of Eqs. (5.4) and (5.5) in termsof the critical resistance are appropriate only for re-sistances with values not too di�erent from critical;percolation theory does not yield the (extremely small)probabilities of ®nding paths with resistances greatlydi�erent from Rc. As a consequence, asymptotic valuesassociated with Eq. (5.6) are not accurate. IntegratingEq. (5.6) leads to an exponential integral, which wasapproximated using Ei�ÿx� � �1=x� exp�ÿx� (valid forlarge x) and generated [30],

CK h� � / 1ÿ xx� h

� �2

: �5:7�

In Eq. (5.7) an additional term in the denominator, ÿL2,was dropped [30] for x and h both large compared to L,otherwise the result would have been

CK h� � / 1ÿ x� �2x� h� �2 ÿ L2

" #: �5:8�

Eq. (5.8) or (Eq. (5.7)) can be analyzed using Bochner'stheorem [57], which says that for a stationary conduc-tivity the Fourier transform of the correlation function(proportional to the negative of the second term) mustexist and be positive de®nite. If the hydraulic conduc-tivity is stationary, it is possible to determine fromEq. (5.2) that the correlation function correspondingto Eq. (5.7) is proportional to

CK h� � / x2

x� h� �2" #

: �5:9�

The Fourier transform, or characteristic function, S�k�of Eq. (5.9) is

S k� � � 4px2

k

� �x

d

dx

�� 1

�ci�kx� sin kx� ÿ cos�kx� si�kx��

�5:10�with ci�x� the cosine integral, and si�x� the sine integralof argument x. The oscillations present in this charac-teristic function, Eq. (5.10), are not acceptable [57], and


Eq. (5.7) (or Eq. (5.8)) for the semi-variogram can onlybe an approximation. h2 � �any length�2 in the denom-inator would remove the inconsistency, but the existenceof the ÿL2 term in Eq. (5.8) merely makes the resultmore complex. The existence of the sum of two squareswould change the slope of the correlation function tozero at the origin, and it is likely that the sharp variationof the derived variogram (discontinuous slope at theorigin) is a factor in the generation of the oscillations.Eq. (5.7) was derived using approximations valid forlarge h, but it is not yet clear whether removal of theseapproximations will remove the problem. The existenceof the factor 1=k does not [58] contradict the assumptionof stationarity, but such a functional form does indicatelong-range correlations. The same proportionality to kÿ1

would obtain also in the case that the denominator ofEq. (5.7) satis®ed Bochner's theorem according to thealternate form above (the sum of two squares). In eithercase, the long-range correlations derive from the slowdecay of the correlation function at large h, for whichthe approximation of the exponential integral is accu-rate. Nevertheless, kÿ1 probably represents a boundaryon the range of possible behaviors; the strength of thedivergence for small k would be reduced by using theGaussian approximation to the cluster statistics (morenearly in accord with experiment and Monte Carlosimulations [5]). In any case long-range correlations areknown from time-series to cause di�culty in obtainingthe variance; classical point estimates of the standarddeviation of the mean of a large number of measure-ments may not converge [58,59].

The other cases, using, e.g., a Gaussian form forcluster statistics, or the appropriate form for local con-ductances, which are powers of random variables, arenot given here, as they are not simply representable interms of elementary functions. But the identical proce-dure can be followed. The variance of K is readily ob-tained from the distribution as proportional to Kc�L=x�2,so that, using Eq. (5.7) (acknowledging theoreticalproblems for small h),

CK h� � � Kc

Lx

� �2

1

"ÿ x

x� h

� �2#: �5:11�

The height of the sill in the variogram (for h� x)diminishes with increased system size, x, in accord withthe reduction in the variance. The length scale, x,(proportional to the range) is derived solely from themeasurement, and not from the medium. That this couldresult from cases with large heterogeneity was noted[56]. In the limit of small h, the variogram derived here isproportional to h, a condition of strong heterogeneitynoted in [60] (in contrast to the case that the slope of thevariogram is 0 [60] an indication of ``a highly regularspatial variability''). But this interpretation requires twocomments; one is that the strong heterogeneity meant is

not a property of a large variance, but of spatial cor-relations. The second is the possibility that the sharpchange in slope of the variogram is what led to the vi-olation of Bochner's theorem.

The cross-covariance can also be formally writtenwithin the framework of cluster statistics, but due to thelength of the expression, the reader is referred to theoriginal work [30].

6. A probabilistic relationship between the electrical and

hydraulic conductivities

A problem of some experimental interest is thededuction of the hydraulic conductivity throughmeasurements of the electrical conductivity. Here wecan give an approximate expression for the probabilitythat a particular volume is characterized by a hydraulicconductivity if the electrical conductivity is measured,and if the ionic concentration as well as its variabilityare known. We must ®rst write down the probabilities ofmeasuring given values of the hydraulic and electricalconductivities, given that the sample size is some ®nitevalue, x. For in®nite systems, the results of Section 3.2apply.

6.1. Finite-size calculations of the hydraulic and electricalconductivities for the Friedman and Seaton model

As pointed out, the cluster statistics of percolationtheory give a framework for calculating statistical vari-ability of the conductivity, hydraulic or otherwise. Thebasic formula to use describes the number of clusters, ns,per unit volume, Eq. (4.1). It is required only to relatep ÿ pc to g ÿ gc, and s to the linear dimension of thecluster. In [49], it is shown that sÿs ds � Nÿ4 dN , while sr

in the exponent is replaced by N 1=m. The critical expo-nent, m, has a value approximately 0.88. Although it hasbeen shown in [27] that 1=m in the exponent is to bereplaced by 1 when the tortuosity of current-carryingpaths on large clusters is accounted for, for power-lawdependence of resistance and hydraulic resistance suchtortuosity was shown above to be irrelevant. Therefore,the exponent 1=m is only approximately equal to 1 in thiscase.

The following discussion is taken largely from [36]and Hunt and Skaggs (2000, in review). Expansion of�p ÿ pc�=pc in terms of g ÿ gc (from Eq. (3.28)) for ge

yields

p ÿ pc

pc

� ÿ ��2=p

ps2

n � 4s2r

ÿ �1=2

exp ÿ b2=2ÿ �

1� erf b=��2pÿ � g ÿ gc

gc

: �6:1�

Henceforth the ratio with the exponential and errorfunctions of b is ignored.


Expression of the cluster statistics in terms of ge andN yields,

nN � 1

N 4exp

"ÿ N 2

L=l� �2 !

2=ps2

n � 4s2r

� �ge ÿ ge

c

gec

� �2#:

�6:2�Here, L is the typical separation of current-carryingpaths, while l is the characteristic separation of thelargest resistors on such paths. Above it was shown thatthese lengths are similar. L is normally larger than l [49],and cannot be smaller than l [23].

The probability that a given cubic volume V � x3

intersects a cluster bigger than or equal to V withcharacteristic conductance g is proportional to the in-tegral of the product of N 3 with the cluster numberunder the constraint N > V 1=3=l. The above clusternumbers are not normalized, so that representation interms of a probability requires a normalization integralover all R. Then the next integral is proportional to theprobability that a given volume, x3, will have a givenconductance, g,

W �ge; x� /Z 1

V 1=3=l

dNN

� exp

"ÿ N 2

L=l� �2 !

2=ps2

n � 4s2r

� �g ÿ gc

gc

� �2#:

�6:3�Here the probability of ®nding a ge other than the typ-ical value is given in terms of exponential integrals. Byanalogy, we can construct from Eq. (3.18) the prob-ability of measuring a particular value of the hydraulicconductivity.

W �gh; x� /Z 1

V 1=3=l

dNN

� exp

"ÿ N 2

L=l� �2 !

2=p16s2

r

� �gh ÿ gh

c

gc

� �2#:

�6:4�In principle, L and l can have di�erent values here, butwe will take them to be the same as for the electricalconductivity. One can also calculate the probability thata path can be found traversing a cube which does notcontain any pore concentration smaller than n,

W �n; x� �Z 1

x=l

dNN

� exp

"ÿ N 2=

L=l� �2=m !

2=ps2

n

� �nÿ nc

nc

� �2#:

�6:5�This integral is not the probability that a given volume x3

has a particular ionic concentration. Such a probabilitywould be expressed as a convolution over log-normal

distributions for individual pores. If paths across ®nite-sized systems are selected on the basis of their electricalconductivity, however, Eq. (6.5) does not give theprobability that the smallest concentration on such apath is n either. When electrical conduction is involved,the e�ective width of the distribution of ionic concen-trations is increased, since some sites with low concen-trations but high pore radii are included in conductingpaths. But the increase in width is less than linear in theindividual widths in n and r, with the result that removalof the variability in r leads to a smaller e�ective width inn, and a reduction in an e�ective nc. The requiredprobability can be written in the form:

W �n; x� /Z 1

x=l

dNN

� exp

"ÿ Nl

L

� �2=p��

s2n � 4s2

r

p ÿ 2sr

� �2

!nÿ n0

n0

� �2#;

�6:6�where

n0 � n0 exp��2p

s2n

ÿ�j� 4s2

r

�1=2 ÿ 2��2p

sr

�erfcÿ1 2pc� �

k:

�6:7�Eq. (6.6) is a conditional probability; the probabilitythat a given minimum value of n is found provided it ison the conducting path, times the probability that agiven critical r is found on that path, must, on averageequal the probability that the path has electrical con-ductance nr2. Then the product of n0r2, where n0 is un-derstood to be the smallest n on a conducting path, andr2 is the smallest value of r2, yields, by construction, theproper dependence of the hydraulic conductivity onsystem parameters (Eq. (6.3)). For sn=sr � 1; n0 ! n0,and W �n; x� approaches d�nÿ n0� for all x. Each of theintegrals Eqs. (6.3)±(6.6) is expressed in terms of theexponential integral function,

Ei�ÿz� �Z 1

z

dtt

exp � ÿ t� �6:8�

with the general form given by

W �g; x� � ÿEi

"ÿ g ÿ gc

gc

� �2

f sn; sr� � xL

� �2=m#

�6:9�

with f �sn; sr� a function of the variances of n and r, andwhich di�ers somewhat from case to case. In each case,however, wider distributions of n or r at the pore scalelead to smaller values of f �sn; sr�; and a wider compositedistribution. In the limit of large values of its argument,ÿEi�ÿx� � �1=x� exp�ÿx�; but it diverges logarithmi-cally in the limit x! 0. Normalizing integrals (6.3)±(6.6)is accomplished by dividing by an integral over gc; gh,or n, respectively. The logarithmic divergence does notlead to problems in the normalization, only in the


representation, if the large argument approximation ischosen.

Because of the factor f �sn; sr�, the widths of the dis-tributions given by integrals (6.3)±(6.6) can, for verysmall system sizes, exceed the mean values. For quan-tities such as the hydraulic and electrical conductivities,which must be positive, such a result is rejected ontheoretical grounds. There is no real problem here.Percolation theory is not valid in the limit of smallsystems; for the statistics to be valid, large clusters mustbe important, and such clusters are only relevant in largesystems (in very small systems, the statistics of shortsde®ne the response [e.g., 61]). Because of the factor,f �sn; sr�, however, the widths of these distributions aresigni®cant even for fairly large systems, as long as theoriginal distributions in n and r are wide. TakingL=l > 1 and sr � 4 yields for example a width of gh

values in excess of 5000 ghc at x�L, far out of the range

of acceptable values. Only for x > 100L is the conditionsatis®ed that the width of the distribution is less than themean. From Section 3, L will usually be on the order ofseveral individual resistor lengths (pore separations).Thus, the statistics developed here would be applicablefor systems perhaps 500 pore separations on a side andup, with the variability in transport properties becominginsigni®cant only when the system reaches 5000 poreseparations on a side. In the case of sands or loams,several thousand pore separations could be meters,while for purely clay-sized particles, perhaps onlymillimeters. For larger system size, additional variationsin the statistics of density, particle-sizes, etc. can con-tinue to introduce variability into the local transportcoe�cients, making continued use of percolation theory/critical path analysis possible.

6.2. Construction of the probabilistic relationship

In terms of Eq. (6.3), the measurement of a given ge

determines the smallest value, �nr2�min of the product ofnr2, necessary to traverse the volume, and consequentlycontains information regarding r2, and thus r4. The mostimportant complication is that the operation of taking aminimum, or percolating, value of a product of twodi�erent variables is not equal to the product of theirtwo minimum values, as illustrated by Eq. (3.33). Butrepresenting a quantity in terms of conditional prob-abilities requires taking a product of two expressions.The values of n useful for constructing a critical path forthe electrical conductivity are distributed according to thenormal distribution, as in Eq. (6.5), but with a meann � n0, given in Eq. (6.6). Note that in the limitsn ! 0; n0 ! n0, but in the limit sn=sr !1; n0 ! nc,clearly correct limits. We will make the approximation,�kenr2�min � ken0minr2

min, in analogy with Eq. (6.6), onaccount of using n0. Thus paths for which rmin is un-usually small must be compensated, on the average, by

having larger nmin. Then we can use Eq. (6.6) for thedistribution of n values together with the constraintkenminr2

min � ge.It is then possible to represent the prob-lem graphically in the n; r plane, as in Fig. 8. Thus theapproximation consists of factoring the product nr2 onlyin the condition, not in the individual distributions.

In Fig. 8, shading is used to indicate which portion ofthis plane the system is likely to be found in, the vari-ation in the shading schematically corresponding to thedistributions derived above. Knowing the value of theelectrical conductivity restricts the system to a smallportion of the plane, changing the probability that theminimum value of r4 (yielding the hydraulic conductiv-ity) necessary to traverse the system, is a given value.Setting knr2 equal to ge, restricts the system to liesomewhere on the curve, r / nÿ1=2. Where the curveknr2 � ge intersects the heaviest shading corresponds tothe most likely value of the hydraulic conductivity to bemeasured. Relaxation of the factorization approxima-tion would require substituting a distribution for thecurve. For the visually oriented, it would be equivalentto smearing out the curve into a ``fuzzy'' area. In anycase, the probability desired is a conditional probability,and is given as a ratio of two generalized areas. In ourparticular case, the appropriate conditional probabilityis

W �r; x : ge�

�

R10

dnd nÿ ge=kr2� � nge

� �Ei ÿ x

L

ÿ �2 2=p��s2n�4s2

r

pÿ2sr

� �2

!nÿn0��

2p

n0

� �2

" #R1

0dgh

R10

dnd nÿ ge=kr2� � nge

� �Ei ÿ x

L

ÿ �2 2=p��s2

n�4s2r

pÿ2sr

� �2

!nÿn0��

2p

n0

� �2

" #

�1

k�gh=kh�1=2

� �Ei ÿ x

L

ÿ �2 2=p��s2

n�4s2r

pÿ2sr

� �2

!ge�kh�1=2=ke�gh�1=2ÿn0

n0

� �2

" #R1

0dgh 1

ke gh=kh� �1=2

� �Ei ÿ x

L

ÿ �2 2=p��s2

n�4s2r

pÿ2sr

� �2

!ge�kh�1=2=ke�gh�1=2ÿn0

n0

� �2

" # :

�6:10�The Dirac delta function, d�nÿ ge=kr2�, whereby n, theabove quotient is implied, restricts integration to thecurve representing the measured electrical conductivity;the process of obtaining this form from d�ge ÿ knr2�generates the factor, n=ge (from �dge=dn�ÿ1

evaluated atn � ge=kr2. The argument of the exponential integralbecomes proportional to

xL

� �2 ge�kh�1=2=ke�gh�1=2

n oÿ n0

n0

0@ 1A2

� xL

� �2

�ge�kh�1=2

=ke�gh�1=2n o� �

ÿ n0 exp��2p

erfÿ1 2pc ÿ 1� � ��S2

n � 4S2r

p ÿ 2Sr

ÿ �� n0 exp

��2p

erfÿ1 2pc ÿ 1� � ��S2

n � 4S2r

p ÿ 2Sr

ÿ �� 0@ 1A2

:

�6:11�

In the limit x!1, the exponential integral functionbecomes a Dirac delta-function of the argument,


ge�kh�1=2=ke�gh�1=2 ÿ n0

n0

!; �6:12�

which from Eqs. (3.28) and (3.29) yields

W �gh� � d ghÿ ÿ gh

c

�� d gh

�ÿ ge� �x kh

ke� �xr4ÿ2x

0

nx0

�: �6:13�

While Eqs. (6.10) and (6.11) are correct within thecontext of percolation theory, they are not directlyuseful. The factor sr, required to construct the quantityn0, is certainly not known a priori. If it were, one couldwrite down gh without ever measuring ge. Since n0 is atransport related quantity, there is no way to measure itother than by appeal to transport measurements,speci®cally the hydraulic and electrical conductivities.Thus we are left with the question of whether estima-tions of n0 can be useful, or with a second possible ap-plication. I have no suggestions for estimating n0, but dohave a possible suggestion for an alternate use of theresults derived here.

A possible alternate application is to use the datafrom both the electrical and hydraulic conductivities todetermine the ionic concentration. If both the mean andvariance of ge and gh can be determined, the mean andvariance of both the ionic concentration and pore sizedistributions can be found. However, it would benecessary to perform such measurements at severallength scales, since none of the equations derived is afunction of only the mean or only the variance of eitherr, n, or a product such as nr2. Clearly, for composi-tionally homogeneous soils, the variability of transportcoe�cients disappears on length scales long comparedwith the sill of the variogram. Measurements of thehydraulic and electrical conductivities at such lengthscales would allow comparison with Eqs. (6.3) and (6.4),in which n0; r0; sn, and sr all appear. But measurementof the electrical and hydraulic conductivities at shortenough length scales to observe the conditional distri-bution, Eq. (6.10), will allow a determination of n0 aswell. This additional information will allow a separationof the quantities sn and sr, as they appear in n0 in adi�erent combination than in ge or gh.

7. Conclusions and possible future research directions

While many applications of percolation theory toporous media are known, [43], its application in asso-ciation with critical path analysis to systems with dis-tributed local conductances is still in its infancy. Despitethe early stage of development, it appears likely thatsuch applications will help to solve a number of fun-damental, hitherto unsolved, problems. The weakestpoint in critical path analysis is still the lack of a uni®ed

structure for calculating the separation of paths con-tributing the most to the dc conduction properties,called L. This length is reasonably well de®ned whenlocal conductances are exponential functions of randomvariables, but less so in the case of powers of randomvariables. Naturally, the latter is of clear relevance toporous media, whereas the ®rst is of more doubtfulrelevance (apart from the results of [31] which are notuniversally acknowledged). One of the two strongestpoints is that CPA is clearly a deterministic theory, butas a subset of percolation theory, it is also associatedwith a means, namely cluster statistics of percolationtheory, to generate distributions of the conductivity andspatial correlations. The second is that in tests oftheoretical calculations of the hydraulic conductivity inheterogeneous porous media it usually turns out supe-rior, at least when the heterogeneity is strong. This resulthas been interpreted (as I see it, correctly) by BB [1] asdue to the controlling in¯uence of the largest, or critical,resistance values on the critical path, for which the po-tential drop is so large that it in¯uences the entire net-work (or system if a natural system is discussed). Ofcourse, as I pointed out, [27] such a large potential dropacross a single element can easily produce non-lineare�ects on such blocking, or critical resistances. As aconsequence, non-linear e�ects can occur at smallervalues of the pressure head than one might have other-wise expected.

So much for general comments. Where do we go fromhere (assuming that CPA and percolation theory willhave a large impact)?1. Eliminate the uncertainty in the calculation of L (as

above) in order to get an accurate value of the dc con-ductivity.

2. Apply to geological complexity.3. Apply to cases with anisotropy (if ¯ow at an acute

angle to layering is envisioned, analogies to the Halle�ect in electrical conduction may be exploited).

4. Find numerical values for both l and L so that thestatistics of variability in K can be more quanti®ablytested.

5. When necessary apply cluster statistics far from per-colation [5].

6. Find an expression, analytical or numerically ob-tained, for the semi-variogram, which does not con-tradict Bochner's theorem.

7. Try to verify whether there is a well-de®ned cross-over in the respective abilities of stochastic anddeterministic calculation schemes to reproduce simu-lations and experiments, and whether this crossover,as suggested, occurs at a consistent value of theheterogeneity, [4]. Investigate the possibility that thisvalue is dependent on how well the system is charac-terized.

8. When the conditions of applicability of percolationtheory become more rigidly de®ned, begin to establish


a database of solved problems based on real soils inorder to aid in inverse modeling.

9. Explore di�erences between imbibition and drainagein establishment of the relationship between K�S�and Ks. Here there exists considerable literature re-garding the role of connectivity in establishing thehysteretic e�ects; application of percolation theorywithin the framework of CPA should be a reasonablesynthesis (see Table 1).

Acknowledgements

I am grateful to Todd Skaggs, who showed me thework of Friedman and Seaton and worked out many ofthe details of the generalization of that problem to in-clude variability in ionic concentration, as well as per-forming the numerical simulations for the ®nite-sizee�ects on the hydraulic conductivity given here. Dr.Skaggs is responsible for all the numerical work pre-sented. I am grateful to John Selker, who showed me thepossibilities inherent in the Rieu and Sposito model.This study was supported by the US Department ofEnergy Climate Change Prediction Program, which ispart of the DOE Biological and Environmental Re-search Program. The Paci®c Northwest National Lab-oratory is operated for the DOE by Battelle MemorialInstitute under contract DE-AC06-76RLO 1830.

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Table 1

Two representative samples of McGee Ranch soil

Particle size (lm) % Less than (by weight) hVolume incrementi2:5� 102 93.5 96.7 4.9%

1:1� 102 77.8 87.8 12.3%

0:75� 102 63.8 76.2 12.8%

0:53� 102 55.0 63.8 10.6%

0:27� 102 41.0 40.6 18.6%

0:15� 102 32.5 31.6 8.8%

0:089� 102 25.5 23.1 7.2%

0:063� 102 22.5 18.1 4%


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