Applications of percolation theory to porous media with distributed
local 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 calculatingRcin 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 arenew. 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 to deterministic transport theories. In part, this assumption seems to follow from a notion that deterministic theories cannot treat statistical variability. Butifsuch heteroge-neities can be accurately described in a statistical sense, e.g., at the pore scale, then percolation theory can be applied to generate both mean values and the variability of transport properties in a given volume with size much larger than the pore scale. Its interpretation as deter-ministic in nature does not imply that percolation theory will 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 ¯ow paths are followed repeatedly. Similarly, the conclusion [1], ``At high [scaled variance], owing to ¯ow localization, extreme values of [the pressure drop squared] occurred at deterministic positions. The ¯ow pattern is so strongly controlled by these huge values that a stochastic de-scription becomes inadequate,'' should be immediately recognized as an obvious possibility in real porous media. Percolation theoretical applications were given in the physics literature in the 1970s. Interestingly, Seager and Pike [2] as well as Kirkpatrick [3], who, like [1] had done numerical simulations on transport in heterogeneous media, came to the conclusion that percolation theory performed best of known approaches when disorder was (relatively) high, while eective-medium theories were superior when disorder was low. The crossover in ap-plicability was at a critical resistance, which involved a
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E-mail address:allen.hunt@pnl.gov (A.G. Hunt).
List of symbols
a localization radius (electronic wave-func-tions)
b typical site separation (solid state) c geometrical constant at pore scale
c0 typical separation of pores in random medium
d Euclidean spatial dimension
df fractal dimension of percolation clusters dA surface area element
dV volume element
f fraction of resistors in largest resistance class
g generalized conductance
gs explicit reference to saturated critical conductance
ge electrical conductance ge
c critical value of electrical conductance gh hydraulic conductance
gh
c critical value of hydraulic conductance
h capillary pressure i denotes site or pore j denotes site or pore k Boltzmann constant
ke constant relating to electrical conductance kh constant relating to hydraulic
conduc-tance
l separation of critical (hydraulic) resist-ances
l0 unit pore separation in network m exponent in van Genuchten function n ionic concentration in groundwater
ns volume concentration of clusters with s
sites
nN volume concentration of clusters of length
N
n0 most likely ionic concentration p bond probability
q 1 or 2 (exponent on cluster statistics)
qe electrical charge
qr ratio of pores in successive pore classes
pc critical value of bond probability r radius of a pore throat or constriction
rc critical value of pore throats
rs dimension of cluster withselements
r0 most likely pore radius r0 smallest pore radius
rij distance from siteito sitej rm largest pore throat radius r> largest pore ®lled with ¯uid
s no. of sites on a cluster
sn variance of ln(n) sr variance of log(r)
w0 rate prefactor
wij local transition probabilities per unit time
x system size
A normalization constant
A0 normalization constant
Ai local pore surface area
CK h hydraulic conductivity covariance at
sep-arationh
C pore aspect ratio (when constant)
C d dimensionally dependent cluster statistics constant
D fractal dimensionality of pore space
Dr fractal dimensionality of volume occupied
by solid E electric ®eld
Ei(x) exponential integral ofx
Eij energy associated with electron hopping
fromito j
EC ¯uid electrical conductivity F electrical formation factor
H x Heaviside step function of argumentx J constant in hydraulic conductivity
distri-bution
K hydraulic conductivity
Ks explicit reference saturated hydraulic conductivity
K/ constant proportional to solid volume of
soil
K 1 hydraulic conductivity of unbounded system
L separation of steady-state current-carry-ing paths
N timeslisrslinear dimension of cluster P pressure
Q ¯ow rate (volume per unit time) R generalized resistance
Re electrical resistance Rh hydraulic resistance
S relative saturation
S k characteristic function of wave numberk T temperature
V volume
Vi pore volume
W r distribution of pore radii
W x;K hydraulic conductivity distribution at scalex
Z local coordination number at pore scale a volume fraction
ac critical volume fraction
b ratio of system size to largest pore size be combination of constants related tordc
bh combination of constants related toK v correlation length associated with
factor from random distributions of about exp [10], implying distribution widths of 4±5 orders of magnitude. Since eective-medium theories are stochastic in ¯avor [4] (a single equation is used to represent any arbitrary point in the medium, but includes a representation of all the variability of that medium) the conclusion regarding the relative applicabilities of percolation and eective-medium theories is compatible with the conclusion [1] ``The [square of the pressure gradient] ®eld had a rather simple random structure at low to moderate [relative variance] and a stochastic description was an attractive option in this case. [...]''. Finally, my contention is that the results summarized above are general enough to hold reasonably well in real rocks (in particular they do not seem to depend on the dimensionality of the net-works nor on the speci®c pore radius distributions used here).'' The fact that [1] mention rocks as the medium under consideration, should not dissuade others from imagining that the conclusions apply equally to soils. I make no qualitative distinction here between soil and rock, although, due to the greater cementation of the latter, one should normally expect smaller pores to be the rule.
Percolation theory is a theoretical framework that allows an investigator to quantify connections of vol-umes, areas or line segments when arranged at ``ran-dom'', [5,6] When such line segments stand for transport, e.g., between neighboring pores, or between neighboring electronic states (more or less localized on dierent sites), the statistics of their connectivity reveal information about the rate-limiting electrical or hy-draulic conductance of large systems. It has been ob-served that the chief problem with geostatistical formulations regarding the hydraulic conductivity is a lack of information regarding the connections between higher conducting regions [7]. But the fact that perco-lation theory keeps track of connections makes it a logical choice for addressing spatial correlations. Thus percolation theory has the strength of quantifying
con-nections and emphasizing on heterogeneity. The present work considers such eects at length scales and under conditions for which pore-scale variability is the relevant heterogeneity. There is, in principle, no size limit on applicability of percolation theory. But practical issues may constrain the most valuable applications of perco-lation theory to the pore scale. This is because it is likely that the criteria for selection of ``stochastic'' versus ``deterministic'' methods are aected by loss of detailed information (which may accompany change in length scales). Combined with the possibility that heterogene-ities in transport at large length scales could have a smaller magnitude than at small length scales, it is possible that stochastic theories tend to become more suitable with increasing length scale, and it becomes dicult to make a generalized prediction regarding the choice of an optimal theoretical approach at arbitrary length 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 andj is con-sidered limited by the narrowest portion of a connecting ``throat'' or ``neck''. The local coordination number can be random, and constant aspect ratios of the pores may be considered. In the case of such complications the proper formulation is based on continuum percolation, but the general concepts involved do not change.
1.1. A short history of the hydraulic conductivity in saturated soils
The beginning of the following discussion is mainly from Bernabe and Bruderer [1] (hereafter referred to as BB) who note that formulations of the saturated hy-draulic conductivity in porous media have historically utilized expressions of the form
v0 prefactor of correlation length
/ porosity
k tortuosity parameter
l ¯uid viscosity (water in this case) le mobility of charges in groundwater
m0:88 critical exponent of correlation length (3D)
m0 constant frequency (attempt frequency)
rdc DC conductivity
h water content
hsat water content at saturation
hr residual water content
r x AC electrical conductivity
r0:45 critical exponent of percolation theory (3D)
s2:2 critical exponent of percolation theory (3D)
nij random variable associated with i±j
transition
x frequency of an applied (electric) ®eld x as superscript, a power relating ge and
gh
CK h conductivity semi-variogram at
separa-tionh K length scale
K r
2
cF 1:1
with ra length related to pore geometry,c8 for cy-lindrical pores, andFthe ``electrical formation factor'', which gives the ratio of the ¯uid bulk conductivity to the rock conductivity (excluding surface conduction). BB clearly show that the evolution of understanding of ¯ow in porous media is tied to the conceptual evolution ofr in Eq. (1.1). We will see that not all factors ofr, which enter Eq. (1.1), explicitly, or implicitly, need be identical. In Kozeny [8] and Carman [9] model, based on the concept of bundles of tubes,r2hVii=hAii, whereViis a
local pore volume,Ai, the average pore surface area, and
the brackets denote a volume average.
A more recent treatment [10] considers a relationship between the electrical and hydraulic conductivities to generate a dierent length scale in place ofr in the ex-pression forK,
K2
R
E2dV
R
E2dA 1:2
with E the electric ®eld, and E2 essentially the energy
density of the electric ®eld, which can be related to dissipation.
In [11] it is argued that eective-medium treatments must also yield aKin the form of Eq. (1.1), since every link between pores has a ¯ow, Qij/r4Pt=l0, where Pt,
the pressure dierence, is linear in the distance between pores,l0.
The Katz and Thompson [12] treatment of critical path analysis(originally from [13,14]) yields
K r
2 c
cF 1:3
but with c56:5 (later amended downwards [15,16]). Here rc is the critical pore radius, de®ned by the con-dition thatrc is the largest value ofr, for which an in-terconnected path may be found from one side of a system to the other, on which no radius smaller thanrcis encountered. The particular value of rc is system-dependent, but may be calculated analytically, or deter-mined for each particular system depending on the shapes of pores, the distribution of pore radii, and the connectivity of the pores. While critical path analysis can be used to ®nd rc, the determination of rc is not sucient to ®nd the hydraulic conductivity, as will be shown in this review. It will be seen that application of critical path analysis does not always lead to an ex-pression with only one length scale, as appears to be implied in Eqs. (1.1)±(1.3).
A recent application of critical path analysis to both the electric and hydraulic conductivities by Friedman and Seaton [17] (hereafter referred to as FS) led to the expression,
ghc p
8ll0r 4
c 1:4
for the critical value, gh
c, of the hydraulic conductance,
and
gecEC
prc2
l0 1:5
for the critical value,ge
cof the electrical conductance. In
these expressions,l0is the length of the critical (and all) pores, EC the intrinsic electrical conductivity of the ¯uid (water with whatever ions it may contain), andlis the dynamic viscosity of water. First, note that the hydraulic conductance involves r4
c; conversion to the hydraulic
conductivity may, but need not always, yield a pro-portionality tor2
c. From Eqs. (1.4) and (1.5), FS concluded
that the ratio of the hydraulic and electrical conductiv-ities should be proportional to the square of the critical pore radius, in accordance with the conclusions of BB. This conclusion should be independent of the method used to calculate the conductivity from the conductance, as noted by both BB and FS.
Eq. (1.6) gives the hydraulic conductivity of a random fractal soil obtained by critical path analysis. This result includes estimates of the length scales necessary for transforming an expression for a hydraulic conductance to a conductivity (derived in the steps up to Eq. (3.43), Hunt and Selker, 2000, in review),
K p
8Cl
l L2
rm3 ÿ1 ac 3= 3ÿDr
p
8Cl
r2m ÿ1 ac 4= 3ÿDr
1:6
and is given in terms of the largest pore radius in the system, rm, as well as a constant C, which is a uniform
aspect ratio, andac, which is the critical volume fraction
for percolation,l the separation of critical rate-limiting pore throats, andLis the separation of the main water-carrying paths. The factor 1ÿac4= 3ÿDr can, if the
fractal dimensionality, Dr, is near 3, be very much
smaller than 1, and an eective radius much smaller than the maximum r, although the result formally pre-serves the proportionality of the hydraulic conductivity to the square of a particular pore (throat) diameter.
1.2. Problems in existing analyses
Several problems in current analyses will be dis-cussed. These include the conversion from a critical rate to a system conductivity, as well as various schemes to describe 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 be applied in any system in which transport is strongly heterogeneous. Examples include electrical conductivity of disordered solids, hydraulic and electrical conduc-tivities of rocks and soils, and the viscosity and electrical conductivity of super-cooled liquids. Actually percola-tion theory was originally devised for applicapercola-tions in porous media [19]. Linear transport theories are fairly well established, although some debate still exists. While non-linear transport theories have been constructed [20] even in relatively well-characterized systems in solid-state physics nothing approaching consensus has been reached as to their validity.
The simplest application of critical path analysis is to a network model of ¯ow in porous media under satu-rated conditions. Allow each bond to represent a pore throat with a radius selected at random from a distri-butionW r. Then the critical radius,rc, is de®ned by
Z 1
rc
W rdrpc 2:1
with 0<pc<1. Stated in English, if a fraction, pc, of the bonds of a network is chosen at random and con-nected, they must produce an interconnected path of in®nite length. The implication here is that it must be possible to ®nd a path through the network which never traverses a pore of radius smaller than rc. If this rc is unusually small compared with the otherrs on the path, the pressure drop acrossrc will be very large (compare the BB quote in the ®rst paragraph). The value of pc, and hence rc, depends mainly on the coordination number,Z, and the dimensionality,d. Many values ofpc
are catalogued (e.g., [21]) others can be estimated using [22]
Zpc d
dÿ1: 2:2
While ®nding the critical resistance is a big step in calculating the conductivity, it is but the ®rst. As it turns out, it is not sucient for determination of the paths on which the water ¯ows, and even after these paths have been found, it is still necessary to calculate their total resistance and how many of them there
are. While no disagreement exists up to Eq. (2.2), dierent approaches begin to diverge immediately thereafter.
Critical path analysis generalizes the following ob-servation; the equivalent resistance of a 10 X and a 106 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. The
argument is then extended to paths through a medium for which local resistance values are spread out over a very wide range. Imagine reconstructing the pore space of a porous medium by adding individual pores, one by one, in descending order of size. A series of subnetworks including more and more pores is derived from the original network. The ®rst such subnetwork containing a cluster of pores connected throughout the network is called the critical subnetwork. Any other path through the system, if chosen at random, will include pores with much smaller radii; such a parallel path has much higher resistance, Rh, and may be ignored (since Rh /rÿ4, a
pore of half the width carries 1/16 the ¯ow). Thus larger resistances are treated as open circuits. On the other hand, larger pores on the critical path have resistances so much smaller, that they may be ignored, and are therefore replaced byshort circuits. Thus this treatment of critical path analysis (CPA) originally from [23] re-places the entire distribution of resistance values by three: open circuits (nearly) critical resistances, and shorts. Although this sounds oversimpli®ed, it is the most complex version available. Thenearlycritical value of R is then treated as an optimization parameter, in spirit with the tendency for charge or water to ®nd the optimal conducting path.
In one-dimensional (1D) solid-state systems where transport is by electronic tunneling between localized sites, nearest neighbor transition rates,wij, are
wijmexpbÿrij=ac; 2:3
where rij is the site separation, m a constant with
di-mensions of inverse seconds, and a is a fundamental length scale. The mean separation of the sites isb. The dc conductivity of a chain of sites of lengthLis known to be proportional to L1ÿ2b=a, and is zero in the limit L! 1. So the ac conductivity of an in®nite chain vanishes in the limit of zero frequency,x, of the applied electric ®eld. The correct dependence [25],
r x /x 1ÿa=2b= 1a=2b: 2:4
The BOBR treatment yields
r x /x1ÿa=b: 2:5
Both functions are positive powers ofx, and satisfy the requirement that the conductivity vanishes at zero fre-quency. But the ratio of the BOBR expression to the correct result, as a function of frequency ``goes like'',
x ÿa2=2b2= 1a=2b; 2:6
which, in the limit of zero frequency, is in®nite. Of course, if the (percolation) limit a=b!0 is taken at arbitrary frequency, the two results are identical, but for any ®nite b (site separation), the BOBR result is seriously too large. Thus, under extreme cases, the BOBR formulation can lead to spectacular overestima-tion of the conductivity. The formulaoverestima-tion of [23], how-ever, was later shown [26], to yield the correct expression, Eq. (2.4). That the BOBR treatment could lead to an underestimation [12] of the hydraulic con-ductivity by a factor 2 (as argued in [15,16]) is thus not surprising. The strength of the Friedman and Pollak [23] version (henceforth called FP) is that it simultaneously explains the large failure of the BOBR treatment in one-dimensional hopping systems, and its smaller problems in the saturated hydraulic conductivity.
The second uncertainty involves length scales. Treating the smaller resistances on the critical path as shorts allows its resistance to be written as proportional to the inverse of the separation, l, of the critical re-sistances on this path, the conductance proportional to l. Then the critical conductance can be converted to a characteristic conductivity value if the separation of contributing paths, L, is known as well. A fairly good expression 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 are far more dicult. Using the simplest expression,
lc0
RRceRc
RcÿRc=eW RdR
RRc
0 W RdR
" #ÿ1=d
2:7
with c0 the product of a numerical constant of order unity and a fundamental pore length. Because the value of c0 is not well constrained, uncertainty exists in com-parison with experiment and simulation. Now,
rdc l
RcLdÿ1: 2:8
The evaluation of L requires re-examination of the choice of RRc. But calculation of the conductivity requires relation of L to R. It has often been assumed thatLis related to the correlation length from percola-tion theory. This correlapercola-tion length is unrelated to cor-relations in the positions of resistances of a given size, but is a representation of how large clusters of resistances can get (by random association) if the concentration of resistors is anywhere near the critical value. The reason why the choiceRRcmust be re-evaluated is thatL Rc
is always in®nite, and some resistance other thanRcmust be chosen for Eq. (2.8), otherwise the conductivity is identically zero. FP developed an optimization scheme for choosing this resistance, and using this optimization scheme it is possible to reconcile apparently confusing pieces of information. This optimization scheme is dis-cussed in detail in the next section.
RelatingRto the correlation length is quite dierent in cases where Ris an exponential function of random variables (based on geometry of the pore space), and when it is a power of a random variable, such as in Poiseuille ¯ow, where Q/r4. In the exponential case, L/ln R=Rcÿ dÿ1m; while in the power-law case,
L/ RÿRcÿ dÿ1m, wherem0:9 is a critical exponent from percolation theory and d is the dimensionality of the system. Because of this dierence, only in the former case 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 dierence between expo-nential and power-law cases is exempli®ed in Figs. 1 and 2.
In the present context, I mention again the work of Le Doussal [16], who starts from an equation similar to Eq. (2.8),
rdcK0gcgcP gc y
; 2:9
whereK0is a constant,gcthe critical conductance, andP is a function of gc which involves L. Le Doussal [16] asserts that y dÿ2mdepends only on dimensional-ity. IfLandlin Eq. (2.8) were identical, then one could substitute L dÿ2 for L dÿ1 in the denominator of Eq.
experimental results [28] require that L and l be systematically dierent [29]. A greater dierence is noted subsequently. [16] considers exclusively the distribution of ln(g), apparently on the basis of his statement ``A useful model to study isgg0exp kxwith a ®xed distribution, D x of x. Then one has gcP gc
D xc=k1, ifkis large.'' Such an argument is familiar from solid-state physics (and is indeed largely a re-statement of the FP formulation, but with dierent ex-ponents), where exponential functions of random variables are the rule, but does not work in saturated ¯ow, if Poiseuille ¯ow is envisioned, and a result for the hydraulic conductivity in terms of a powerof a critical
pore radius is sought. In the case where Ris an expo-nential function of random variables, thenL isrelated to a logarithm of the conductance, in accord with [16]. But in the case whereRis a power of a random variable (as in pore throats using Poiseuille ¯ow), thenLis a power ofRÿRc, and not a logarithmic function. In this case, it is shown here, both theoretically, and numerically, that while the critical conductance is still relevant to system-wide transport, the critical network with tortuous paths is not. BB came to the same conclusion. Thus, the method of [16] is internally inconsistent, by virtue of his relying on methods appropriate for resistances, which are exponential functions of random variables. Other
Fig. 1. Computer generated 2D random resistor network with exponential dependences of resistance values on random variables,RijR0exp nij.
AllR's withR<Rmaxare shown as bonds, and those withRmax=e2<R<Rmaxare shown in bold. In (a),RmaxRc=3:8; v, the size of the largest
cluster, is about 15 bond lengths,l, the typical separation ofR's within a factor e2ofR
max, is drawn as about 5 bond lengths. In (b)Rmaxis chosen
equal toRc:vis in®nite.lshould again be about 5 bond lengths, but in this ®gurelis about 10. In (c),Rmaxis chosen as 3:8RcRopt, the optimal
authors, e.g. [20], have also used a similar formulation with respect to the exponent (proportional to Dÿ2m), but with a power-law dependence, as here.
3. General calculations using CPA
In this section, the structure of the calculation of general dc transport using percolation theory in the form of critical path analysis is discussed. The general structure of such a calculation does not depend directly on the transport property involved, whether electrical conduction or ¯uid ¯ow, although it may depend on the details of the local conductances.
Percolation theory is based on the geometry of con-nectivity [5]. If some number of objects of given size and shape are distributed in a volume of some particular size, what is the probability that at least one path can be found across the volume which never contacts these objects (or which never loses contact)? The result is either one or zero in the limit of in®nite size [5] The crossover from one to zero occurs at a well-de®ned concentration. As a consequence, in the limit of in®nite system size the dc conductivity of ``nominally homo-geneous'' systems can be accurately calculated using crit-ical path analysis [2]. By nominally homogeneous (for porous media) I mean systems with the same bulk properties, 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),RmaxRc=2, andl
sizes, composition, organic content, ionic concentration, etc. Even when these properties show neither random, nor systematic variability, the local variation in pore size, and for unsaturated systems, variation in moisture content, can be so large as to make the systems strongly heterogeneous from the perspective of transport or ¯ow. When the size of the regions with a given suite of bulk properties is in some sense small (the particular con-ditions will be clari®ed in the derivations) ®nite-sized corrections must be included.
A simpli®ed problem [21], which illustrates the concept of percolation theory, is that of a square (two-dimensional) lattice, on which bonds between sites can be connected at random with some probability,p. For p values less than (greater than) pc0:5 no path (at least one path) can be found which connects places at in®nite separation. In a system in which the bonds correspond to conductances distributed continuously over some wide range, one can arbitrarily regard all conductances greater than some arbitrary value, g, as being connected bonds. For some critical value of
ggc, then, the set of all conductances g>gc pro-duces an in®nitely long connected path. That particular value of ggc is then of great signi®cance for the macroscopic (large-scale) conductivity.gc turns out also to be of signi®cance for the statistical variability of the conductivity on smaller length scales, as well as the transient response of the system, both of which can be expressed in terms of the characteristic (or critical) conductance.
Quantities such as pc and hencegc are highly depen-dent on many system parameters, such as the distribu-tion of g, mean local coordination numbers, and the shape of regions associated withg[21].gc is referred to as system-speci®c, or non-universal. Other properties, such as, for a given value of pÿpc=pc, the cluster statistics (number of clusters of a given number of el-ements per unit volume), are the same in a wide variety of cases, provided the system is near critical percolation, i.e.,p andpcare of similar magnitude to each other [5]. Properties, which do not depend on the value ofpc, can be termed universal (or quasi-universal), because they do not depend on the geometrical shapes of the invidual objects, although they do depend on spatial di-mension [5]. They also appear to be the same whether the individual bonds between sites are geometrically ordered or not [5]. Physical results which involve the dependence of cluster numbers on pÿpc=pc include the scale-dependence of the distribution of hydraulic conductivity values [27], the correction in the mean conductivity of ®nite size systems to the in®nite system conductivity [27], and the spatial dependence of the semi-variogram [30]. These and other properties, such as the relationship between the electrical and hydraulic conductivities, can be expressed in terms ofgc, and de-pend on the form of thedependenceof local
conductiv-ities, but not their values or distributions. These aect only gc. In the systems considered here, with continu-ously distributed values of g, it is always possible to isolate a subsystem withgneargc, for which the cluster statistics near percolation are relevant. On the other hand, systems composed of individual volumes with either very large or essentially zero conductances (such as fractured impermeable rock) may or may not meet conditions for the relevance of percolation theory.
Functional forms of local resistances in solid-state physics applications: Critical path analysis applied to solid-state conduction problems has always started with the assumption that transport on the microscopic scale involves mechanisms whose rates, wij /Rÿ1, depend
exponentially on random variables. These mechanisms include:
1. Particle hopping over a barrier (from i to j),
wij w0expÿEij=kT.
2. Tunneling through barriers, wijw0expÿ2rij=a; wijw0expÿEij=kT ÿ2rij=a.
In the above applications,ais the localization length,E random energies, k the Boltzmann constant, T the temperature, andrare hopping or tunneling length. The standard of applicability of percolation theory (com-pared with eective medium theories) has been ex-pressed in terms of the spread of local conductance values (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 is possible that the same criterion does not apply in cases where the local conductances are not exponential func-tions of random variables. As in [17] (hereafter referred to as FS), however, we will proceed under the assump-tion that percolaassump-tion theory and critical path analysis are applicable regardless of the particular form of the distribution, provided the spread of values is suciently large. This assumption is in accord with BB who found that the particular form of the distribution of pore sizes did not aect the applicability of percolation theory compared 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] or as power-law in form [33,34]. In saturated soils, the hydraulic conductance, gh, may be treated as a power
law, gh/r4, if viscous ¯ow between neighboring pores
is considered an example of Poiseuille ¯ow (e.g., FS in their treatment using critical path analysis). FS also use for an electrical conductance, ge/r2. Later in this
3.1. Optimization of Eq. (2.8) for the DC conductivity
WhetherRR0expnij, wherenijis a random variable
related to pore geometry, andR0 is a fundamental pre-factor with units of (hydraulic) resistance, or whether
RR0 nn, the procedure to ®nd the critical value ofR, is to ®nd the critical value ofn, and then insert it into the appropriate one of these two relationships. In either case one has
Z nc
0
W ndnpc: 3:1
The separation of the current-carrying paths in Eq. (2.8), however, involves the correlation length. The correlation length is expressed in terms of pÿpc. How does one proceed in relatingpÿpc to resistance values (hydraulic, or otherwise)? One must start by writing the same equation for an arbitrarypand correspondingn,
Z n
0
W n0dn0p 3:2
p is then an arbitrary fraction of the bonds. While pc
describes the quantile of the distribution (measured from the most highly conducting bond), which generates critical percolation, the only stipulation that we must make about p is that it is not be too dierent frompc. For pÿpc1, cluster statistics of percolation de®ne the number of clusters of a given size which are formed. In the present context, this means the number of clusters with no resistor exceeding the value that corresponds to p through the random variable n. These statistics also generate the density of such clusters, the tortuosity of the chief conducting path, called the backbone cluster, and the linear dimension of the clusters. The largest available cluster is de®ned by the correlation length,v, which diverges at critical percolation according to vv0jpÿpcjÿ
m
3:3
withm0:88 and v0 related to the typical bond, or
re-sistor length (pore length, l0, on a network). For
R<Rc; v is the size of the largest cluster of intercon-nected resistances with largest resistanceRand is ®nite. For R>Rc; v is the size of the largest region with no resistance greater thanR, but for which theR's are not shorted out by equal-sized or larger clusters with smaller R's, and is also ®nite. ForRPRc; v is also the typical separation of paths, which could carry current. When the optimum value of this separation is found, it is called L, as above. The divergence is the reason why, if the subnetwork employed to ®nd the conductivity was comprised only of resistors smaller than or equal toRc, the calculated conductivity would be zero. The separa-tion of current-carrying paths would be equal to the linear dimension of the critical cluster, and substitution ofL 1yields zero conductivity. This result is related to the one obtained for metal-insulator composites that
the conductivity vanishes in the in®nite system limit below the metal percolation threshold [21].
Usingpÿpc 1, a relationship ofnctopcin integral
form means thatpÿpc, which appears in the correlation length, can be written as nÿnc=nc. But if n is
pro-portional to the natural logarithm of R (the ®rst case above),
jpÿpcj lnln R=Rc Rc=R0
3:4
lis now the typical separation of the largest resistances, rather than Rc, but Eq. (2.7) demonstrates that lvaries only weakly with R so that Eq. (2.7) is still used to calculatel. Thus, resubstitution into Eq. (2.8) leads to
rdc R l
Rln R=Rc 2m
: 3:5
Optimization of Eq. (3.5) with respect toRyields
RoptRc exp 2m; 3:6
so that Lv0 2m 2m
. Formulation of the problem in terms of the conductance leads to the same answer. The calculation is self-consistent, conduction occurring along tortuous paths through approximately fractal clusters, and
rdc
l Rcv0exp 2m 2m
2m: 3:7
For critical path analysis to be valid, the result for R must either be very close to or at least clearly related to
Rc. Here the correlation length,L, and the resistance,R, are expressed in terms of their critical values and in terms of percolation statistics, respectively, guaranteeing self-consistency.
Use of Eq. (3.7) appears to imply that the conduc-tivity could not be increased beyond the optimum value by including more resistances (and therefore additional paths). In fact, including additional resistance values cannot reduce the conductivity, and the optimization is assumed to denote a crossover to a regime where adding larger resistances does not materially increase the sys-tem-wide response. When Ris an exponential function of random variables, it is sensitive to system parameters, so the correlation length does not vary rapidly with R. Thus, the optimal value ofRis not so close to the critical value (exp 1:8 6 times larger), but the structure of the conducting paths is essentially that at critical percola-tion, complex and tortuous.
What happens when RR0 nn? Now linearization yieldspÿpc/ RÿRc=nRc. Substitution into Eq. (2.8) leads to,
rdvljRÿRcj 2m
=Rv2
0: 3:8
Optimization yields a minimum at R<Rc this time, outside the range of validity of the expression. Thus for
monotonically increasing function ofR. The result does not invalidate the application of critical path analysis, nor does it invalidate the result for pÿpc. But it does mean thatLin this case cannot be represented in terms of critical exponents of percolation theory, since the conditions for using this representation ofLhave been violated. The solution is to recognize that for relatively small increases inR>Rc, the correlation length has di-minished so much that percolation statistics no longer apply, and the separation of current-carrying paths is similar to the separation of actual paths. Once the sep-aration is reduced to such a value, it can scarcely be reduced any further, and there is no point in increasing Rany further. Since this has happened with a very small increase in R, R is pinned extremely close to Rc. This result 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 that near percolation, thus not particularly tortuous. Nu-merical solution of Kircho's laws (Section 4) reveals that typical values ofLin this case are about 10 (in units of 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®ned by CPA.'' (The power-law case can also be formulated in terms of the conductance; here a result is obtained which is not absurd, but since the two answers dier, the implied value of the correlation length is outside the range 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 to be tortuous, but not in Fig. 2.
If the empirically determined exponential relationship between the hydraulic conductivity and the moisture content [31] is relatively accurate, (an issue to which I return) then these results imply that steady-state ¯ow in unsaturated soils should be more tortuous than in sat-urated soils. The ®nding of non-tortuous ¯ow paths for the power-law dependence is not so dierent from tra-ditional treatments of saturated ¯ow, like Kozeny± Carman, in which a number of parallel tubes (which do not communicate with each other) with diering hy-draulic conductivities are envisioned. From the ®gures as well as the optimization, L, the separation of the current-carrying paths, is a low multiple ofv0, the
fun-damental resistor, or pore, length. From Fig. 2, l also appears to be a small multiple of the pore length. If precision of the hydraulic conductivity better than to within a factor of two or three is sought, these estimates will have to be improved.
3.2. The Friedman±Seaton network
FS calculate the critical electrical and hydraulic conductances for a medium represented as a regular
network. While all throat lengths are thus equal, the throat radii are assumed widely distributed. The par-ticular distribution chosen determines the value of the critical conductance, but is not relevant to the argu-ments relating the critical conductance to the system conductivity.
In this analysis, contributions to the electrical con-ductivity due to sorption of charge on clay particles are not treated, although inclusion of such complexity is possible, in principle. FS consider possibilities of either cylindrical-shaped, or slit-shaped pores. In the latter case, the power of the random variable r in each con-ductance is reduced by one (and replaced by a uniform valuew), but this re®nement, of value, is peripheral here. In Poiseuille ¯ow, each bond of lengthl0and radiusr, has hydraulic conductance,
gh p
8l
r4
l0k
h
r4 3:9
with l the viscosity of water, and khp=8ll0 a
con-venient way to represent all the factors which are constant.
FS assume that the ionic concentration is also a constant. Then the electrical conductance for the throat joining two pores is equal to
gepECr
2
l0 ; 3:10
where EC is the intrinsic electrical conductivity. For later use, we note that EC can be represented as the product,
EClenqe; 3:11
where le is the mobility of the charges qe, present in
volume concentration n. Using Eq. (3.10), one can re-write the individual electrical conductances as follows:
gepECr
2
l0
pnleqer2
l0 k
e
nr2; 3:12
where ke incorporates all constant parameters, thus
emphasizing that bothnandrare, in principle, random variables.
The critical percolation condition for the hydraulic conductivity is
Z 1
rc
W rdrpc; 3:13
where pc is system and dimensionally dependent. The critical hydraulic conductance, gh
c is proportional tor4c,
and, in case, as FS, n is assumed uniform, the critical electrical conductance,ge
c, is proportional torc2. FS use a
log-normal distribution of pore sizes,r,
W r 1
rsr
2p
p exp
ÿ ln r ÿln r0
2
p
sr
: 3:14
W n 1
This conjecture cannot be con®rmed, although evidence from observation, [35] suggests that such a distribution must be broad, and may well be skewed. Here I only mean to imply that any spread of values is far more likely than a uniform concentration; assuming the same distribution as forrmakes calculations easier and more transparent.
The solution of Eq. (3.17) de®nes the critical hydraulic conductance in terms of the inverse complementary er-ror function, erfcÿ1(x), of a combination of constants including the percolation probability,pc.
There is no material dierence in the FS calculation for the electrical conductivity, and
pc21
results. Thus we have (taking for the moment, as FS,
nn0) The quotient of these two expressions is
gh
which is easily seen to be proportional tor2
c. Eqs. (3.21)
and (3.22) also show that the critical hydraulic con-ductance is proportional to the square of the critical electrical conductance,
FS present evidence that the ®rst formulation is valid, while BB demonstrate the second. The ®rst formulation has the advantage that the viscosity of water does not
enter the proportionality constant, in contrast to the second. FS stop here and do not calculate the electrical or hydraulic conductivities, but note that each is pro-portional, respectively, to the conductances derived above, and therefore the same relationship for the con-ductivities holds as for the conductances. However, FS do note that due to the dierent dependences ofRe and Rh on r the characteristic resistance on the critical, or
percolating paths, will be a functional of the distribution of resistance values, dierent for the hydraulic and electrical conductivities. Although this comment is cor-rect, since the characteristic resistance is just an integral over the R's from the smallest to a value somewhat larger than Rc, the ratio of the two R's involves a nu-merical factor very nearly 1. The present procedure of de®ning a characteristic length between maximal resis-tance yields similar results and allows the distinction between tortuous and non-tortuous paths to be made based on the functional form of the resistance values.
3.2.1. A modi®cation of the Friedman±Seaton procedure for calculatingge
The treatment of FS implicitly assumes a uniform concentration of ions in the groundwater. This as-sumption is not justi®able, but not much is known about the true distribution of ionic concentrations, particularly on the pore scale. Furthermore, in soils the electrical conductivity can be in¯uenced by the presence of clay minerals. In any case, evidence suggests that a wide variability may be present, particularly in the vadose zone. In Florida aquifers with carbonates and inter-bedded clays [35], ionic concentrations at probe scales can vary by as much as orders of magnitude. In fact, mean concentrations of particular ions (in ca. 4 l of water) can vary by a factor of 7 over vertical distances as small as 80 cm [35]. Since the given values are already averaged over a large area of space, it is obvious that the microscopic variability in charge distribution can be much larger. Given such variability, it is possible that a log-normal distribution for pore-scale ionic concentra-tions is also appropriate. Using log-normal distribuconcentra-tions for both pore throat radii and ionic concentrations it is possible to extend the FS results for electrical conduc-tivity to a two-variable percolation problem in which both are treated on an equal footing. It is also useful to point out how complications in CPA are dealt with. The following was given in preliminary form [36] (and more fully in Hunt and Skaggs, 2000, in review). The perco-lation condition reads
converted to a normal distribution, the double integral to a single integral [37],
pc 1
Consequently one can write down forge c,
Now the relationship between the critical electrical and hydraulic conductances is not so simple as in FS. In fact,
gh
recovered, but otherwisex<2, and can be less than 1. Although the form of this relationship depends on the distribution chosen for n, complexity will always in-crease and usual inferences regarding ionic concentra-tion from the electrical conductivity will be incorrect. Considersn=sr1, just a small modi®cation of the SR
and solution of this equation forn0leads to
n0 g
Thus, as usually presumed, the ionic concentration is proportional to the electrical conductivity. However, the proportionality constant includes both the variability in the pore sizes, and the variability in the ionic concentra-tion. Eq. (3.32) shows that neglect of either of these eects can lead to overestimates (underestimates) of the ionic concentration if 2pc<1 (if 2pc>1). In three dimensions,
pc is typically less than 1=2 [21] and overestimations would be expected. Further, it shows that, when ionic variability is relatively small, changes in pore size vari-ability can be interpreted as changes in ionic concentra-tion, if eects of the ®rst exponential factor are neglected.
For later use, note that it is possible to calculate critical values of the following combinations of powers, p andm, of nandr, respectively,
nprm
are increasing functions of disorder. In 2D, however,pc
may be smaller than 0.5 [21], and both conductances may be either increasing, or decreasing functions of disorder for log-normal distributions.
3.3. Applications to fractal porous media
The question regarding what statistics to use for pore size variability appears to be unresolved. Some investi-gators, such as FS, choose log-normal distributions, and BB used log-uniform, while others [38±42] choose power-laws associated with fractal geometry. Insofar as self-similarity in soil or rock particles and aggregation is concerned, strict applicability of fractal statistics must have bounds. If soils are neither rocky nor particularly cohesive, 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 that soils produced mainly by physical weathering are more likely to produce fractal geometries than those produced in sedimentary environments by, e.g., ¯uvial deposition. Whether such systematic variability can be veri®ed is not clear, but in any case, some fairly impressive evidence has been accumulated that fractal soils have a place in model development [43].
Here I use a single distribution to cover aportionof the range from tens of microns to tens of centimeters; in individual cases variations over 1±3 orders of magnitude are expected to be the rule. Sharp cutos at both ends of the distribution are employed.
It has been pointed out [41] that Brooks and Corey [34] already reported results for the water content, h in terms of its saturated value hsat in a form amenable to
interpretation in terms of fractal soil structures,
h h hsat h hmin
D0ÿ3
; 3:34
wherehis the capillary pressure, andhminis the air entry pressure. This expression was derived in [38] using a fractal geometry. Of course, at that time [34] regardedD0
as an empirical parameter. Further [43], soils (and es-pecially rocks) frequently exhibit systematic trends in density with sample size, indicating fractal structure. Thus considerable motivation exists for pursuing a soil structure, and associated transport model, based on a fractal perspective.
without much discussion, except insofar as it is necessary to ®nd a distribution of pore sizes and to clarify notational dierences. The ®rst paper by Rieu and Sposito [39], will henceforth be referred to as RS. I do not envision regular structures, but imagine a ran-dom soil structure, described nevertheless by the same parameters and distributions given in RS. Thus their results for density and porosity as functions of system size are still assumed to hold, as well as the number of pores at any given size. The structure of percolation-theoretical calculations requires disorder; however, such structural disorder also allows calculations of the vari-ability of transport properties. Thus one can derive spatial variability of transport properties within the context of deterministic soil structure using a deter-ministic theory of transport. In the process, trends in the mean values of the hydraulic conductivity and the den-sity can be derived in accordance with observation, as well as with simulations, [41] for example, while incor-porating variability.
3.3.1. Soil structure background
Using Eqs. (17), (18), and (20) of RS, it is possible to ®nd the relative number of pores in the ith pore class (usingqr>1 as a size ratio of successive pore classes,r0
as the smallest pore radius, andrm as the largest)
WiA0 qÿrD ÿ i
A0ÿqirÿD; 3:35
where D is the fractal dimensionality and A0 is a con-stant. The reason for the exchange in the order ofiand Dis to facilitate representation of the pore radius,r, in terms ofqi
r.
Allowing rto take on a continuous range of values,
W r Arÿ1ÿD 3:36
withAanother constant. The reduction in the power by 1 is necessary to allow an integral over a ®nite range ofr (e.g., one size class) to yield rÿD, corresponding to the
discrete case. We now, as in RS, consider an incom-pletely fragmented porous medium with both the grains and the pores fractally distributed. In this case the fractional power Dr must be substituted for D in the
pore volume distribution. Set
Rrm r0 dr=rr
3rÿDr
Rrm r0 dr=rr
3rÿDrK /
/ 3:37
with / the porosity. The term K/ in the denominator
arises from the solid volume. Constants in the distri-bution, such asA, have been absorbed intoK/. Solution
of Eq. (3.37) yields, forK/,
K/
r3ÿDr
m 1ÿ/
3ÿDr
3:38
on application of the identity (from RS) /1ÿ r0=rm
3ÿDrin both numerator and denominator.
3.3.2. Saturated hydraulic conductivity
Application of Darcy's Law at the pore scale, as an integral over ¯uid velocities, allows possible geometrical complications. The safest application is to pore throats, the constrictions between pores. The scaling relation-ships derived in RS are not necessarily consistent with such an application, although RS did present results for hydraulic properties (with some similarities to the present solution; see the full reference, Hunt and Selker, 2000, in review). It would be consistent with the concept of self-similarity to assume that the constrictions between pores follow the same distribution as in Eq. (3.36), making application of CPA consistent with the framework of RS.
In the present problem, it was assumed (RS) that the scaling of pore sizes aects all dimensions similarly, i.e., the pore aspect ratio is independent of pore size. But it is not correct to use critical rate analysis for bond perco-lation [17] when the bond lengths vary over orders of magnitude, since the bonds will not ®t on a regular network. The appropriate generalization is continuum percolation (eectively a percolation of open volume) [6]. Continuum percolation is de®ned in terms of frac-tional volume; when the fracfrac-tional volume is greater than some particular value, the individual volumes connect. In such analysis generally, the possibility must, in principle, be considered that the pore space itself does not connect. In the present application we assume that it does. This is consistent with the assumption of incom-plete fragmentation by RS. Critical path analysis in the form of volume percolation, however, ®nds the smallest pore radius, rc, necessary to complete such an inter-connected path. In the present case we can ®nd that radius as follows:
Rrm rc r
2ÿDrdr
Rrm r0 r
2ÿDrdr r 3ÿDr m 3ÿDr
1ÿ/
ac: 3:39
The left-hand side of the equation is fraction of the total volume in pores with radius larger than the critical radius, and the right-hand side is the critical volume fraction. The solution of this equation is
rcrm ÿ1 ac 1= 3ÿDr:
3:40
Note that the coincidence
ac/ 3:41
leads to rcr0. In this particular case, the porosity corresponds exactly to the minimum volume fraction required to construct a connected path of open volume, so that any such path will have to traverse the smallest pore available. Normally ac </ by constraints
associ-ated with the derivation of the pore radius statistics. But the implication that the hydraulic conductivity vanishes if and when ac>/ is consistent with general concepts
space is required for ¯ow. An aspect not treated in this work, is the determination of ac as a function of
ge-ometry, which normally requires intensive simulations. Nevertheless for practical applications,achas often been
taken to be about 15% [6].
Assume that the pores have a cross-sectional arear2,
and a length (C)(r). The above results are not changed. We can write for the critical value of the hydraulic conductance,
where l is the viscosity of water. The Heaviside step function,H, of argument/ÿac is included only to
in-dicate that, according to the assumption of a minimum pore size, r0, the hydraulic conductivity must vanish when the integration requires smaller pores to produce a connected path of pore volumes. However, this factor is subsequently dropped for two reasons. First, a rigid lower 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 fractional
dimensionality,Dr, approaches 3, the critical hydraulic
conductance approaches 0. IfDr ! 3; / ! 0, which is
a guarantee that the hydraulic conductivity vanish. So with 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 for simplicity the latter. But this choice should be treated with caution.
The critical value of the hydraulic conductance helps to generate the critical value of the hydraulic conduc-tivity. If it is known how many paths per unit area can be found which are characterized by the critical con-ductance, as well as the separation of the controlling (critical) conductances on these paths, the hydraulic conductivity of an in®nite system can be constructed. We de®ne the separation of the paths with gc to beL, and the separation of the gc's on these paths to bel. It has been shown that for Poiseuille ¯ow in a non-fractal network, bothlandLare small multiples of the typical pore separation [36]. Here no typical pore separation exists by de®nition. But the same calculation scheme applied in previous cases [26], can also be applied here for l (Eq. (2.7)) and yields lrm 1ÿac
1= 3ÿDr
. By arguments [36], L is given in terms of the distribution of all pores, and is, in this case also approximately rm.
Such an assumption is consistent with constraints of fractal geometry, where it is assumed that the ¯ow path is biased to including the largest pores. K 1, de®ned to be the hydraulic conductivity of an in®nitely large system is
3.3.3. Unsaturated hydraulic conductivity
If complications due to dierences in imbibition and drainage are neglected (apparent, but not overwhelming in the simulation of Perrier et al. (1995) with which I ultimately make comparison), assumption of equili-bration between dierent pores by ®lm ¯ow appears reasonable [44,45]. Then the simplest compatible assumption for the eects of drying is that the largest pores empty ®rst and that all smaller pores are still full. As an initial step, de®ne the relative saturation, S, in terms of the largest pore still ®lled with water,r>.
S
Note that the term in the denominator involving the volume of the solid soil is absent, since Sis de®ned in terms of the pore space only. Solving forr>,
r> Sr3mÿDr
To ®nd the critical value of the radius, however, con-tinuum percolation is again applied, including the term in the denominator representing the contribution of the solid volume. Also, it is necessary to include as the upper limit in the integral in the numerator, the largest ®lled volume, since larger volumes no longer contribute to the hydraulic conductivity.
, this result can be sim-pli®ed as follows:
Introducing gsgc (from Eq. (3.42)) in the saturated case, andg as gc in the unsaturated case, this equation can be rewritten in the following form:
ggs 1
Eq. (3.50) is compared with a result from Kravchenko and Zhang [46] (KZ) who also used the RS model, but then applied the pore-size distribution model of Burdine [47] to ®nd for the unsaturated hydraulic conductivity,
K S KSS 5ÿDr=3ÿDr1 3:51
in which they set the tortuosity parameter,k, equal to 1. It is interesting to compare exponents; the KZ exponent is 8ÿ2Dr= 3ÿDr, which for typical values of Dr,
about 2.6±2.8 (according to the data of [39,40]), gives a range 2:8= 3ÿDr±2:4= 3ÿDr, very similar to the
power on Eq. (3.50) from the present analysis. Eq. (3.51) was obtained from Eq. (12) of KZ by setting the residual moisture content equal to 0, as they were able to do. KZ present comparison of Eq. (3.51) with the data, which demonstrate a satisfactory ®t for a wide variety of cases, from silt loams to loamy sands. KZ obtainDr from soil
particle-size plots, as in RS.
Although Eq. (3.50) is incomplete, further calcula-tions are really estimacalcula-tions, so Eq. (3.50), is compared already with results from simulations of [41] in Fig. 3. The results of [41] were for a construction, which turned out to have fractal dimensionality of 2.86. For a simple
Dr of 2.875 3ÿDr1=8 and a ratio of largest pore
radius to smallest pore radius of 100, typical of the values considered in [41], the porosity is found to be 0.44. We use a typical value [6] ofac0:15. The values
extracted from the Perrier graph are, where possible, intermediate in value between their imbibition and drainage curves. It appears that Eq. (3.50) captures the main physics. Further, note that the drainage curves tend to follow more closely Eq. (3.50), but are broken o at higher relative saturations than the imbibition curves; thus at lowS, only imbibition curves survive, producing the majority of the discrepancy.
The theoretical results are also compared with two Hanford site soils [48]. Hanford 1 soil (H1) is from McGee Ranch, and has a porosity of at least 40%. Hanford 2 has a lower porosity, ca. 30%. In the results of RS the porosity is a function ofDr, and the ratio of
minimum to maximum pore size, r0=rm, If a soil has
fractal characteristics, it is logical to assume that the
ratio of minimum to maximum pore sizes is equal to the ratio of smallest to largest particle sizes. The particle size distributions 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 by
the RS equations, and the derivation for K=Ks given here, 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 lowSbecause the particle-size distribution was only fractal over about three size classes; lower size classes had to be lumped together to get 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 KsS1
ÿÿ1
ÿS1=mm2
; 3:52
where
S hsÿh
hsÿhr
3:53
with the subscripts s and r referring to saturated and residual values of the moisture content.
A question that arises is, what are the relative roles of the percolation structure of the conductivity calculations and the fractal structure of the soil in obtaining Eq. (3.50)? This question is addressed by taking the limit
Dr!3, in Section 3.3.4, or by using the same soil
de-scription with a dierent means of calculating the hy-draulic conductivity, as in KZ above, or by picking another soil description as a test, done next.
In few cases for a pore distribution can one derive closed-form results, but one other possibility is an ex-ponential pore distribution,
W r 1
Rexp
r0ÿr r
h i
r0<r;
0 r<r0: 3:54
Two soluble cases exist. In one, the pore lengths are all the same. In the second, the pore lengths are
pro-Fig. 3. Graphical representation of Eq. (3.50), derived forKas a fraction ofKsplotted against relative saturation,S, for a fractal soil. Parameters
are, Dr2:875; rm=r0100, yielding a porosity of 44%. Comparison is made with results from the simulation by Perrier et al. [41],
portional to the diameters. This case is treated identi-cally to that above using continuum percolation. In ei-ther case I neglect the possible eects of the lengthsland L. In the second case,
KKS 1
ÿ 1ÿS
1ÿac=/
3=4
: 3:55
In the ®rst case, however, it is appropriate to use bond percolation. Then, one ®nds,
KKS 1
ÿ 1ÿS
acln 1 =ac
4
: 3:56
Note that in Eq. (3.56) the porosity does not appear, since the guaranteed connectivity makes the actual pore volume irrelevant. The main lesson here is that similar forms for the relationship between the saturated and unsaturated values of the critical hydraulic conductance are found using percolation for a wide variety of as-sumed soil structures.
One ®nal point involves the value ofSfor which the unsaturated hydraulic conductivity vanishes. Using the given formulas, one can ®nd, S 1ÿ 1ÿac=/ (Eq.
(3.50)),S 0 (Eq. (3.52)), andS ac=/(Eq. (3.55). Eq.
(3.56) has no simple interpretation. In Eq. (3.55), if the porosity is smaller than the critical volume fraction, than the saturated hydraulic conductivity vanishes. The unsaturated hydraulic conductivity then vanishes if the 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 on the critical volume fraction and the porosity was drop-ped. There, the condition was generated from a ®nite minimum pore size.
3.3.4. Fractal soils in the limit Dr!3
Using /1ÿ r0=rm
3ÿDr, in the limit
Dr!3 Eq.
(3.50) becomes
K Ksexp
ÿ3 ln rm
r0
1
ÿS
1ÿac
: 3:57
Eq. (3.57) uses the fundamental de®nition of
e2:718. . .. The derivation of Eq. (3.57) in the limit of non-fractal soils gives possible theoretical grounds for using an exponential form for the parameterization of the unsaturated hydraulic conductivity. Of course, for
Dr identically 3,Ks0 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 a sample, then it is possible to apply CPA a second time. In this case variability of S in Eq. (3.57) can lead to a rather large variability in Ks, meaning that the critical value ofKsis obtained by ®nding the critical value ofS. For convenience, and because it is likely to be reason-able, the distribution of moisture contents in the soil is taken 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).
