Directory UMM :Data Elmu:jurnal:A:Advances In Water Resources:Vol24.Issue3-4.2001:

Advances in Water Resources 24 (2001) 279±307
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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 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 frequently 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. But if such heterogeneities 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 deterministic in nature does not imply that percolation theory
will tell which volume has a particular conductivity. But,
once one considers the possibility of deterministic disorder, it becomes easier to incorporate certain non-sto-

E-mail address: allen.hunt@pnl.gov (A.G. Hunt).

chastic tendencies into a general conceptual framework.
For example, in a deterministic, but heterogeneous porous 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 description 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 e€ective-medium theories were
superior when disorder was low. The crossover in applicability was at a critical resistance, which involved a

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

280

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

List of symbols
a
b
c
c0
d
df
dA
dV
f
g
gs
ge
gce
gh

gch
h
i
j
k
ke
kh
l
l0
m
n
ns
nN
n0
p
q
qe
qr
pc
r

rc
rs
r0
r0
rij
rm
r>
s
sn
sr

localization radius (electronic wave-functions)
typical site separation (solid state)
geometrical constant at pore scale
typical separation of pores in random
medium
Euclidean spatial dimension
fractal dimension of percolation clusters
surface area element
volume element

fraction of resistors in largest resistance
class
generalized conductance
explicit reference to saturated critical
conductance
electrical conductance
critical value of electrical conductance
hydraulic conductance
critical value of hydraulic conductance
capillary pressure
denotes site or pore
denotes site or pore
Boltzmann constant
constant relating to electrical conductance
constant relating to hydraulic conductance
separation of critical (hydraulic) resistances
unit pore separation in network
exponent in van Genuchten function
ionic concentration in groundwater
volume concentration of clusters with s

sites
volume concentration of clusters of length
N
most likely ionic concentration
bond probability
1 or 2 (exponent on cluster statistics)
electrical charge
ratio of pores in successive pore classes
critical value of bond probability
radius of a pore throat or constriction
critical value of pore throats
dimension of cluster with s elements
most likely pore radius
smallest pore radius
distance from site i to site j
largest pore throat radius
largest pore ®lled with ¯uid
no. of sites on a cluster
variance of ln(n)
variance of log(r)

w0
wij
x
A
A0
Ai
CK …h†
C
C…d†
D
Dr
E
Ei(x)
Eij
EC
F
H …x†
J
K

Ks
K/
K…1†
L
N
P
Q
R
Re
Rh
S
S…k†
T
V
Vi
W …r†
W …x; K†
Z
a
ac

b
be
bh
v

rate prefactor
local transition probabilities per unit time
system size
normalization constant
normalization constant
local pore surface area
hydraulic conductivity covariance at separation h
pore aspect ratio (when constant)
dimensionally dependent cluster statistics
constant
fractal dimensionality of pore space
fractal dimensionality of volume occupied
by solid
electric ®eld
exponential integral of x

energy associated with electron hopping
from i to j
¯uid electrical conductivity
electrical formation factor
Heaviside step function of argument x
constant in hydraulic conductivity distribution
hydraulic conductivity
explicit reference saturated hydraulic
conductivity
constant proportional to solid volume of
soil
hydraulic conductivity of unbounded
system
separation of steady-state current-carrying paths
times l is rs ˆ linear dimension of cluster
pressure
¯ow rate (volume per unit time)
generalized resistance
electrical resistance
hydraulic resistance
relative saturation
characteristic function of wave number k
temperature
volume
pore volume
distribution of pore radii
hydraulic conductivity distribution at
scale x
local coordination number at pore scale
volume fraction
critical volume fraction
ratio of system size to largest pore size
combination of constants related to rdc
combination of constants related to K
correlation length associated with percolation

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

v0
/
k
l
le
m  0:88
m0
rdc
h
hsat
hr
r…x†

prefactor of correlation length
porosity
tortuosity parameter
¯uid viscosity (water in this case)
mobility of charges in groundwater
critical exponent of correlation length
(3D)
constant frequency (attempt frequency)
DC conductivity
water content
water content at saturation
residual water content
AC electrical conductivity

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 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 e€ectivemedium 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 networks 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 volumes, areas or line segments when arranged at ``random'', [5,6] When such line segments stand for
transport, e.g., between neighboring pores, or between
neighboring electronic states (more or less localized on
di€erent sites), the statistics of their connectivity reveal
information about the rate-limiting electrical or hydraulic conductance of large systems. It has been observed 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 percolation theory keeps track of connections makes it a
logical choice for addressing spatial correlations. Thus
percolation theory has the strength of quantifying con-

r  0:45
s  2:2
nij
x
x
CK …h†
K
X

281

critical exponent of percolation theory
(3D)
critical exponent of percolation theory
(3D)
random variable associated with i±j
transition
frequency of an applied (electric) ®eld
as superscript, a power relating ge and
gh
conductivity semi-variogram at separation h
length scale
units of electrical resistance …X†

nections and emphasizing on heterogeneity. The present
work considers such e€ects 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 percolation theory to the pore scale. This is because it is likely
that the criteria for selection of ``stochastic'' versus
``deterministic'' methods are a€ected by loss of detailed
information (which may accompany change in length
scales). Combined with the possibility that heterogeneities 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
dicult to make a generalized prediction regarding the
choice of an optimal theoretical approach at arbitrary
length scales. Further research in this direction is essential.
Applying percolation theory to usual network models, when pore separations are all equal and the coordination number is consistent across a lattice is easy.
Application to more complicated systems is also possible as long as transport between points i and j is considered 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 hydraulic conductivity in porous media have historically
utilized expressions of the form

282

Kˆ

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

r2
cF

…1:1†

with r a length related to pore geometry, c ˆ 8 for cylindrical pores, and F the ``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 of r
in Eq. (1.1). We will see that not all factors of r, 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, r ˆ 2hVi i=hAi i, where Vi is 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 di€erent length scale in place of r in the expression for K,
R 2
E dV
…1:2†
Kˆ2R 2
E dA

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 e€ective-medium treatments
must also yield a K in the form of Eq. (1.1), since every
link between pores has a ¯ow, Qij / r4 Pt =l0 , where Pt ,
the pressure di€erence, is linear in the distance between
pores, l0 .
The Katz and Thompson [12] treatment of critical
path analysis (originally from [13,14]) yields
r2
Kˆ c
cF

…1:3†

but with c ˆ 56:5 (later amended downwards [15,16]).
Here rc is the critical pore radius, de®ned by the condition that rc is the largest value of r, for which an interconnected path may be found from one side of a
system to the other, on which no radius smaller than rc is
encountered. The particular value of rc is systemdependent, but may be calculated analytically, or determined 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
sucient 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 expression 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,

gch ˆ

p 4
r
8ll0 c

…1:4†

for the critical value, gch , of the hydraulic conductance,
and
gce ˆ EC

prc2
l0

…1:5†

for the critical value, gce 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 the
dynamic viscosity of water. First, note that the hydraulic
conductance involves rc4 ; conversion to the hydraulic
conductivity may, but need not always, yield a proportionality to rc2 . From Eqs. (1.4) and (1.5), FS concluded
that the ratio of the hydraulic and electrical conductivities 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),


 
p
l 3
3=…3ÿDr †
r ‰ 1 ÿ ac Š
Kˆ
8Cl L2 m


p
4=…3ÿDr †

r2 ‰1 ÿ ac Š
8Cl m

…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, and ac , which is the critical volume fraction
for percolation, l the separation of critical rate-limiting
pore throats, and L is the separation of the main watercarrying paths. The factor ‰1 ÿ ac Š4=…3ÿDr † can, if the
fractal dimensionality, Dr , is near 3, be very much
smaller than 1, and an e€ective radius much smaller
than the maximum r, although the result formally preserves the proportionality of the hydraulic conductivity
to the square of a particular pore (throat) diameter.
BB compared several results for K, including parallel
tubes [8,9] the model of Johnson and Schwartz [10], a
stochastic model [18], and the Katz and Thompson
model [12] (KT) but not Eqs. (1.4) and (1.6)) with simulations. They conclude that the KT model provides the
best description of trends of the hydraulic conductivity
with width of the pore distribution. This result should,
by itself, be sucient motivation to pursue the best
method for calculating K consistent with percolation
theory. But it is really only the beginning.

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

1.2. Problems in existing analyses
Several problems in current analyses will be discussed. These include the conversion from a critical rate
to a system conductivity, as well as various schemes to
describe resistance distributions that characterize simpli®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 conductivities of rocks and soils, and the viscosity and electrical
conductivity of super-cooled liquids. Actually percolation theory was originally devised for applications 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 solidstate 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 saturated conditions. Allow each bond to represent a pore
throat with a radius selected at random from a distribution W …r†. Then the critical radius, rc , is de®ned by
Z 1
W …r† dr ˆ pc
…2:1†
rc

with 0 < pc < 1. Stated in English, if a fraction, pc , of
the bonds of a network is chosen at random and connected, 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 other rs on the path,
the pressure drop across rc 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 of pc
are catalogued (e.g., [21]) others can be estimated using
[22]
Zpc 

d
:
d ÿ1

…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 sucient 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

283

are. While no disagreement exists up to Eq. (2.2),
di€erent approaches begin to diverge immediately
thereafter.
Critical path analysis generalizes the following observation; 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 resistance 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 by short circuits. Thus this treatment
of critical path analysis (CPA) originally from [23] replaces 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. The nearly critical 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.
The second version of CPA is due to B
ottger and
Bryksin [24], (hereafter called BOBR), and is of particular relevance since it was chosen as the basis for the KT
approach to porous media. The di€erence is in the
simpli®cation of the network. BOBR employ two classes
of resistances to describe the full range of variability.
Thus a system with continuously distributed local
resistances is represented in the same way as an insulator±conductor composite. Resistances larger than an
arbitrary value, R, are treated as in®nitely large, all
smaller resistances given the value R. The cuto€ R is
chosen to maximize the conductivity. Such an algorithm
is intended to represent the tendency of water, as well as
electricity, to follow the path of least resistance. But this
treatment tends to overestimate the resistance of the
current-carrying paths because it overcounts the number
of large resistances. The problem is di€erent in the AC
conduction because the method introduces a bias in the
counting of the resistances vis-a-vis capacitances of individual portions of the network. This bias overwhelms
the overestimation of the resistance by displacing their
in¯uence to a lower frequency ± where conduction
should be much more dicult.

284

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

In one-dimensional (1D) solid-state systems where
transport is by electronic tunneling between localized
sites, nearest neighbor transition rates, wij , are
wij ˆ m expbÿrij =ac;

…2:3†

where rij is the site separation, m a constant with dimensions of inverse seconds, and a is a fundamental
length scale. The mean separation of the sites is b. The
dc conductivity of a chain of sites of length L is 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†=…1‡a=2b† :

…2:4†

The BOBR treatment yields
r…x† / x1ÿa=b :

…2:5†

Both functions are positive powers of x, and satisfy the
requirement that the conductivity vanishes at zero frequency. But the ratio of the BOBR expression to the
correct result, as a function of frequency ``goes like'',
x…ÿa

2 =2b2 †=…1‡a=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 overestimation of the conductivity. The formulation of [23], however, 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 conductivity 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 onedimensional 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 resistances 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 separation of critical resistance values in the bulk sample;
although slightly better calculations exist [27], they are
far more dicult. Using the simplest expression,

l ˆ c0

" R Rc ‡eRc

Rc ÿRc =e
R Rc
W
0

W …R† dR
…R† dR

#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 comparison with experiment and simulation. Now,
rdc ˆ

l
:
Rc Ldÿ1

…2:8†

The evaluation of L requires re-examination of the
choice of R ˆ Rc . But calculation of the conductivity
requires relation of L to R. It has often been assumed
that L is related to the correlation length from percolation theory. This correlation length is unrelated to correlations 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 choice R ˆ Rc must be re-evaluated is that L…Rc †
is always in®nite, and some resistance other than Rc must
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 discussed in detail in the next section.
Relating R to the correlation length is quite di€erent
in cases where R is 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=Rc †‰ÿ…dÿ1†mŠ ; while in the power-law case,
L / …R ÿ Rc †‰ÿ…dÿ1†mŠ , where m  0:9 is a critical exponent
from percolation theory and d is the dimensionality of
the system. Because of this di€erence, only in the former
case is the structure of the current-carrying paths tortuous and describable in terms of concepts of percolation theory, while in the latter, appropriate for Poiseuille
¯ow, the structure of the current-carrying paths is unrelated to percolation. The di€erence between exponential 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),
rdc  K0 gc ‰ gc P …gc †Šy ;

…2:9†

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

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

285

Fig. 1. Computer generated 2D random resistor network with exponential dependences of resistance values on random variables, Rij ˆ 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 ˆ 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 ˆ 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).

experimental results [28] require that L and l be
systematically di€erent [29]. A greater di€erence is noted
subsequently. [16] considers exclusively the distribution
of ln(g), apparently on the basis of his statement
``A useful model to study is g ˆ g0 exp…kx† with a ®xed
distribution, D…x† of x. Then one has gc P …gc † ˆ
D…xc †=k  1, if k is large.'' Such an argument is familiar
from solid-state physics (and is indeed largely a restatement of the FP formulation, but with di€erent exponents), 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 power of a critical

pore radius is sought. In the case where R is an exponential function of random variables, then L is related to
a logarithm of the conductance, in accord with [16]. But
in the case where R is a power of a random variable (as
in pore throats using Poiseuille ¯ow), then L is a power
of R ÿ Rc , 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 systemwide 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

286

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

Fig. 2. Analogous to Fig. 1, except that resistances are now power laws in the pore radius, compatible with Poiseuille ¯ow. In (a), Rmax ˆ Rc =2, and l
turns out to be about 2, while v is about 5. In (b), Rmax ˆ Rc , l is again about 2, while v is larger than the system. In (c), Rmax ˆ 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).

authors, e.g. [20], have also used a similar formulation
with respect to the exponent (proportional to …D ÿ 2†m),
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 connectivity [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 homogeneous'' systems can be accurately calculated using critical path analysis [2]. By nominally homogeneous (for
porous media) I mean systems with the same bulk
properties, such as bulk density, distributions of particle

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

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 conditions 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 (twodimensional) lattice, on which bonds between sites can
be connected at random with some probability, p. For
p values less than (greater than) pc ˆ 0:5 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
g ˆ gc , then, the set of all conductances g > gc produces an in®nitely long connected path. That particular
value of g ˆ gc 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 hence gc are highly dependent on many system parameters, such as the distribution of g, mean local coordination numbers, and the
shape of regions associated with g [21]. gc is referred to
as system-speci®c, or non-universal. Other properties,
such as, for a given value of …p ÿ pc †=pc , the cluster
statistics (number of clusters of a given number of elements per unit volume), are the same in a wide variety
of 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 , can
be termed universal (or quasi-universal), because they
do not depend on the geometrical shapes of the individual objects, although they do depend on spatial dimension [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 …p ÿ pc †=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 of gc , and depend on the form of the dependence of local conductiv-

287

ities, but not their values or distributions. These a€ect
only gc . In the systems considered here, with continuously distributed values of g, it is always possible to
isolate a subsystem with g near gc , 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 ˆ w0 exp‰ÿEij =kT Š.
2. Tunneling through barriers, wij ˆ w0 exp‰ÿ2rij =aŠ;
wij ˆ w0 exp‰ÿEij =kT ÿ 2rij =aŠ.
In the above applications, a is the localization length, E
random energies, k the Boltzmann constant, T the
temperature, and r are hopping or tunneling length. The
standard of applicability of percolation theory (compared with e€ective medium theories) has been expressed 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 conductances on random variables. Nevertheless it is
possible that the same criterion does not apply in cases
where the local conductances are not exponential functions of random variables. As in [17] (hereafter referred
to as FS), however, we will proceed under the assumption that percolation theory and critical path analysis
are applicable regardless of the particular form of the
distribution, provided the spread of values is suciently
large. This assumption is in accord with BB who found
that the particular form of the distribution of pore sizes
did not a€ect 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 (hydraulic conductivity) has been given variously as exponentially 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
work, it is shown that under speci®c conditions, unsaturated ¯ow may lead to an exponential form of
conductance. Thus it is important to allow for either
power-law, or exponential, functions of random variables when developing critical path analysis, and I give
the general results for both.

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

3.1. Optimization of Eq. (2.8) for the DC conductivity
Whether R ˆ R0 exp nij , where nij is a random variable
related to pore geometry, and R0 is a fundamental prefactor with units of (hydraulic) resistance, or whether
R ˆ 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 the
appropriate one of these two relationships. In either case
one has
Z nc
W …n† dn ˆ pc :
…3:1†
0

The separation of the current-carrying paths in
Eq. (2.8), however, involves the correlation length. The
correlation length is expressed in terms of p ÿ pc . How
does one proceed in relating p ÿ pc to resistance values
(hydraulic, or otherwise)? One must start by writing the
same equation for an arbitrary p and corresponding n,
Z n
W …n0 † dn0 ˆ p
…3:2†
0

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 di€erent from pc .
For p ÿ pc  1, 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
v ˆ v0 jp ÿ pc jÿm

…3:3†

with m  0:88 and v0 related to the typical bond, or resistor length (pore length, l0 , on a network). For
R < Rc ; v is the size of the largest cluster of interconnected resistances with largest resistance R and is ®nite.
For R > Rc ; v is the size of the largest region with no
resistance greater than R, but for which the R's are not
shorted out by equal-sized or larger clusters with smaller
R's, and is also ®nite. For R P Rc ; 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 to Rc ,
the calculated conductivity would be zero. The separation of current-carrying paths would be equal to the
linear dimension of the critical cluster, and substitution
of L ˆ 1 yields 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].
Using p ÿ pc  1, a relationship of nc to pc in integral
form means that p ÿ pc , which appears in the correlation
length, can be written as …n ÿ nc †=nc . But if n is proportional to the natural logarithm of R (the ®rst case
above),


 ln…R=Rc † 

…3:4†
jp ÿ pc j ˆ 
ln…Rc =R0 † 
l is 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
calculate l. Thus, resubstitution into Eq. (2.8) leads to
rdc …R† ˆ

l
2m
ln…R=Rc † :
R

…3:5†

Optimization of Eq. (3.5) with respect to R yields
Ropt ˆ Rc exp…2m†;

…3:6†

2m

so that L ˆ v0 …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 ˆ

l
Rc v0 exp…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 conductivity 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 system-wide response. When R is 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 of R is 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 percolation, complex and tortuous.
What happens when R ˆ R0 …n†n ? Now linearization
yields p ÿ pc / …R ÿ Rc †=nRc . Substitution into Eq. (2.8)
leads to,
rdv ˆ ljR ÿ Rc j2m =Rv20 :

…3:8†

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

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

monotonically increasing function of R. The result does
not invalidate the application of critical path analysis,
nor does it invalidate the result for p ÿ pc . But it does
mean that L in this case cannot be represented in terms
of critical exponents of percolation theory, since the
conditions for using this representation of L have been
violated. The solution is to recognize that for relatively
small increases in R > Rc , the correlation length has diminished 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 separation is reduced to such a value, it can scarcely be
reduced any further, and there is no point in increasing
R any 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 dominated 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. Numerical solution of Kircho€'s laws (Section 4) reveals
that typical values of L in 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 di€er, 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 random variables, are contrasted in Figs. 1 and 2, respectively. 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 saturated soils. The ®nding of non-tortuous ¯ow paths for
the power-law dependence is not so di€erent from traditional treatments of saturated ¯ow, like Kozeny±
Carman, in which a number of parallel tubes (which do
not communicate with each other) with di€ering hydraulic conductivities are envisioned. From the ®gures
as well as the optimization, L, the separation of the
current-carrying paths, is a low multiple of v0 , the fundamental 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

289

network. While all throat lengths are thus equal, the
throat radii are assumed widely distributed. The particular distribution chosen determines the value of the
critical conductance, but is not relevant to the arguments relating the critical conductance to the system
conductivity.
In this analysis, contributions to the electrical conductivity 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 conductance is reduced by one (and replaced by a uniform
value 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 ˆ

p r4
 k h r4
8l l0

…3:9†

with l the viscosity of water, and k h ˆ p=8ll0 a convenient 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
ge ˆ

pECr2
;
l0

…3:10†

where EC is the intrinsic electrical conductivity. For
later use, we note that EC can be represented as the
product,
EC ˆ le nqe ;

…3:11†

where le is the mobility of the charges qe , present in
volume concentration n. Using Eq. (3.10), one can rewrite the individual electrical conductances as follows:
ge ˆ

pECr2 pnle qe r2
ˆ
 k e nr2 ;
l0
l0

…3:12†

where k e incorporates all constant parameters, thus
emphasizing that both n and r are, in principle, random
variables.
The critical percolation condition for the hydraulic
conductivity is
Z 1
W …r† dr ˆ pc ;
…3:13†
rc

where pc is system and dimensionally dependent. The
critical hydraulic conductance, gch is proportional to rc4 ,
and, in case, as FS, n is assumed uniform, the critical
electrical conductance, gce , is proportional to rc2 . FS use a
log-normal distribution of pore sizes, r,

 
1
ln…r† ÿ ln…r0 †
p
:
…3:14†
W …r† ˆ p exp ÿ
rsr 2p
2sr
It will be useful also to consider ion concentrations,
which are log-normally distributed,

290

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

W …n† ˆ
with


 
1
ln…n† ÿ ln…n0 †
p exp ÿ
p
nsn 2p
2s n

n0 ˆ exp‰hln…n†iŠ;
sn ˆ Var…ln…n††;

r0 ˆ exp‰hln…r†iŠ;

sr ˆ Var…ln…r††:

…3:15†

…3:16†

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 for r makes calculations easier and more
transparent.
Upon substitution of W …r† into Eq. (3.13), one obtains
Z 1


1
1
…3:17†
exp ÿ t2 dt ˆ erfc…bh †
pc ˆ p
h
2
2p b

enter the