percolative transport in fractal porous media

$: Percolative transport in fractal porous media$
Chaos, Solitons and Fractals 19 (2004) 309–325

www.elsevier.com/locate/chaos

Percolative transport in fractal porous media

A.G. Hunt

Cooperative Institute for Research in the Environmental Sciences, University of Colorado, Boulder, CO 80309-0216, USA

Abstract

Application of continuum percolation theory to a fractal pore space model yields results for the constitutive

relationships for unsaturated flow in agreement with experiment. This application also unites understanding in that

the same dry end moisture content, ht ¼ 0:039SA0:52vol as a function of the surface area to volume ratio, is shown to be

associated with the deviation of experimental water retention from fractal scaling as well as with the vanishing of

the diffusion constant. Substituted into a critical path analysis (based on continuum percolation theory) for the de-

pendence of the unsaturated hydraulic conductivity, KðhÞ, on moisture, the same value of ht produces excellent

agreement with experimental data (y ¼ 1:0015x� 0:0065, R2 ¼ 0:96), with y experiment, x theory and using no ad-

justable parameters. Though critical path analysis is based on percolation theory, the result obtained for KðhÞ is

more closely tied to the fractal characteristics of the medium, and the dependence is referred to as a fractal scaling of

the hydraulic conductivity. In all three properties, the interpretation of ht is the same; it represents the mini-

mum value for which a continuous interconnected path of capillary flow is possible, making it the critical

volume fraction for percolation. This identification means that the low moisture content deviation from fractal

predictions in hðhÞ does not conflict with fractal models of the pore space, as the deviation is due to dynamics rather

than to structure. Critical path analysis does not yield percolation scaling, in which K vanishes as a power of ðh� htÞ.However, it is shown here that the data for KðhÞ and hðhÞ are consistent with an interpretation in which the

fractal scaling of K at large moisture contents crosses over to a percolation scaling at a moisture content slightly

above ht.� 2003 Elsevier Ltd. All rights reserved.

1. Introduction

The characterization of observed hydraulic properties of natural porous media is a difficult problem. Since mea-

surements are time consuming and expensive, many theoretical predictions for hydraulic properties have been devel-

oped from particle-size data (PSD) [1–6], or indirectly using textural data [7], to generate the PSD. The most important

properties (needed for a variety of purposes) are the hydraulic conductivity, K, as a function of saturation (or pressure)

and the water retention, h, as a function of pressure, referred to as hydraulic head, h. It is typically presumed that water

retention characteristics are properties of the geometry alone, but this is true only in equilibrium. If equilibrium is not

maintained, then hðhÞ depends on past values of KðhÞ and the time in a complex way, and thus so does KðhÞ. Ifequilibrium conditions are guaranteed, hðhÞ can be calculated more easily than KðhÞ, which calculation appears to

present a dilemma even in equilibrium. The dilemma relates to the representation of the complexity of porous media as

well as to the mathematical means chosen to calculate the response.

It seems clear that the computing power necessary to apply the Navier–Stokes (NS) equations at the pore-scale, even

for a lab-sized sample, will not be available for decades. Moreover it is unclear whether three-dimensional imaging

(from, e.g., synchrotron X-ray radiation [8,9], neutron radiography [10,11], laser etching [12,13], or simply injection and

dissolution) will provide a commensurately accurate basis for such numerical modeling, at least for the natural vari-

ability of soils, from sands through silt-loams to clays. But there is no consensus regarding the choice of alternatives to

direct flow simulations of NS (on an accurately mapped medium). In any case representations of the pore space can

0960-0779/04/$ - see front matter � 2003 Elsevier Ltd. All rights reserved.

doi:10.1016/S0960-0779(03)00044-4

310 A.G. Hunt / Chaos, Solitons and Fractals 19 (2004) 309–325

influence the theoretical choices to calculate K, while defects of either can degrade the accuracy of the predicted hy-

draulic properties.

Consider first the pore space. Two common choices are regular network models [14–16], and fractal representations

[17–25]. The network models range from simple networks of tubes to more complex variations, which distinguish

between pores and pore-throats (constrictions between pores). While often accepted as an arbitrary distinction, the

difference between pores and pore-throats can be topologically quantified using dual graph theory [26,27], which

proceeds in a similar manner to the construction of the primitive cell of the reciprocal lattice in solid-state physics. The

variety of fractal models is a bit overwhelming [28], including systems, which have fractal pore space, fractal particle

space, both, or various combinations including fractal surfaces. Even a simple network of tubes can be modeled to have

fractal characteristics if the distribution of tube radii conforms to a power law [29]. The particular model used here is an

abstraction of the [17] model and, in fact, generates identical expressions to theirs for both the porosity and the water

retention curve, though quite different physics is used to determine K. The conceptual alterations consist of taking a

continuous range of pore sizes instead of partitioning into discrete pore-size ranges, and no explicit consideration of

how the geometry of the system actually appears, nor how the particles actually transmit force. The basis for this neglect

lies in percolation theory, which, in 3D allows simultaneous percolation of both particle and pore space over a wide

range of porosities (the same is not true in 2D, where either solid space or pore space must percolate, but not both

simultaneously).

The most frequently used means of calculating K in porous media is based on the concept of ‘‘capillary bundles’’

[30,31], a method equivalent to finding a mean conductivity. Theoretical arguments demonstrate the unsuitability of

mean transport coefficients, 1 and controlled tests expose their failures [34], typically missing correct values by many

orders of magnitude [35–37]. Alternatives have been given in terms of stochastic theories [38–41], which make general

claims regarding the relevance of a geometric mean conductivity via the Matheron conjecture [42]. These generaliza-

tions are over-sold, and neither the concept of any kind of mean, nor of an operation over conductivities, is valid, with

the appropriate quantities to consider being conductances [43]. This is because the conductivity of a heterogeneous

medium does not relate to the flow per unit area of the paths with the dominant response, but rather to the flow on those

paths and their areal density, i.e. the frequency of their occurrence in the medium [44–46]. Failure to recognize this

difference constitutes a tacit assumption that the medium is homogenous. As a consequence of the relevance of per-

colation theory to flow, in some cases these flow paths may be fractal even if the medium is not (for general discussions

see, e.g. [45]). In particular, when the moisture content diminishes to a value near the percolation threshold, capillary

flow paths are organized in a fractal structure.

For systems with large spreads in local values of a conductance, the best means for calculating a system-wide re-

sponse has been shown repeatedly [34–36,47], to be percolation theory, originally developed [48], to treat flow in porous

media. Instead of being thwarted by large variability, the accuracy of percolation theory can only be improved by

increases in local variability; thus the problem of dealing with heterogeneity becomes a non-problem. The other very

commonly cited obstacle for theoretical understanding is the unknown connectivity of the more highly conductive

regions of the medium. The purpose of percolation theory is to quantify such connectivity [45,49]; thus avoidance of

percolation theory means neglect of the problem generally admitted to be the greatest obstacle to understanding.

The most difficult task is the systematic and quantitative determination of the variability of connectivity from system

to system. This has proved difficult at geologic scales. But the unambiguous prediction of pore-scale hydraulic prop-

erties became possible when the critical volume fraction for percolation was identified as a function of surface area to

volume ratio, SAvol, from empirical studies on solute diffusion [50]. This identification was made possible through the

conceptual power of percolation theory.

Percolation theory generally gives the largest value of the smallest (rate limiting) conductance required to construct a

path of interconnected conductances that spans an infinite system [51,52]. It can also for a finite system find the

variability in the minimum conductance required to generate such a finite-length path [53]. In order to map this rate-

limiting conductance to a value for K, additional effort is required, e.g. [45,46,54] and involves some uncertainty in

general, although for ratios of conductivities, which appear in the constitutive relations, ratios of limiting conductances

seem to suffice [54], at least as long as the moisture content defining the critical volume fraction for percolation is not

approached too closely.

1 By guaranteeing connectivity of tubes in each size range, this, and all related methods, weight the tubes with the largest radii the

highest. In fact, however, as well known in other fields of transport in disordered systems, e.g. [32,33] and articles referenced therein,

and demonstrated again explicitly in porous media [34] the most important elements are at or near critical percolation, with larger

elements effectively acting short circuits and rapidly diminishing in importance with increasing conductivity, and smaller elements

effectively acting as open circuits, and having no influence at all on the conductivity.

A.G. Hunt / Chaos, Solitons and Fractals 19 (2004) 309–325 311

Even if it is decided that the important application of percolation theory to transport is through use of critical path

analysis, a major question that remains is in the form of percolation theory to use, bond percolation or continuum

percolation. This question depends on the model of the pore-space used. If a network model is considered, bond

percolation is the natural choice. If a random fractal model is considered, then continuum percolation appears to be

more appropriate [54]. Random fractal models appear to be the most suitable models. In this case, the choice of

continuum percolation is motivated by the fact that soil particle size distributions are typically power laws over about

one and a half orders of magnitude; if self-similarity is invoked (in accord with, e.g., fractal fragmentation models [55])

then pore lengths will vary over a similar range. Such a large variation of pore lengths makes it difficult to construct a

regular network with properties consistent from site to site. Without an explicit network representation, application of

bond percolation is non-trivial. Application of continuum percolation theory to K is based on the concept that in every

system, some subset of the pore space is guaranteed to be connected, and this fraction has been theoretically estimated,

as well as empirically determined.

Taken together, random fractal models and continuum percolation theory appear to give a consistent description of

all three results from basic soil physics, the water retention curve, the hydraulic conductivity, and solute diffusion. This

is the primary reason I have for preferring a random fractal model of porous media, even though different reasons have

been given by others. These include results from imaging [56,57], etc., typically of rocks or ‘‘fractured soils’’ systematic

variations of density with volume [58], derivations of fractal characteristics compatible with observation from fractal

fragmentation models [24,55], and observed power laws in particle-size distributions [18,19,24,59,60] water retention

characteristics [25,60] (compatible with the often used [61], parametrization) and unsaturated K [62]. It is worth noting,

however, that the constitutive relationships are not quite power laws, even if the pore-size distribution is.

2. Pore-space model

As indicated, many fractal models exist, but I choose the [17], model (RS) as being preferable. This choice was

motivated by the success and simplicity of the RS model, though it can be extended to account for different fractal

dimensionalities in different ranges of pore size. The RS model is for a truncated, random fractal, i.e. the pore space is

self-similar over only a specific range of pore sizes, and the topology of the connections is not spatially correlated. The

mathematics of the fractal model itself is yet simpler in the more abstract case of a continuous distribution of pore sizes

considered here and in previous works. I take the fundamental relationship to be the RS result for the porosity,

/ ¼ 1� r0rm

� �3�D

ð1Þ

In this expression, D is the (volume) fractal dimensionality of the pore space, while r0ðrmÞ is the smallest (largest) radius,

over which the fractal description holds. The simple representation of Eq. (1) demonstrates that D tends to 3 with

vanishing porosity, or a large ratio of rm=r0 Though typical values of D are about 2.8, or slightly larger [20], this simply

means that typical values of / are around 0.4 and of rm=r0 � 50.

The attraction of this particular formulation is chiefly in its dependence on parameters that are relatively easily

obtained from experiment. If one assumes, as is customary, that in any given soil, the maximum pore size is proportional

to the maximum particle size [4,63], the related assumption that the minimum pore size is proportional to the minimum

particle size is a natural extension [64]. Then the ratio of r0=rm should be the same for both particles and pores. As a result

it is possible to plot up (log) cumulative particle volume vs. (log) particle size, and determine r0 and rm from the bounds

on the validity of the fractal description (endpoints of the straight line fit), and this was done for the McGee Ranch soil

[64]. Repetition of this exercise on 11 soil samples [64], yielded a mean value of D ¼ 2:8299 with standard deviation,

0.012. The prediction of KðhÞ made at that time, however, utilized an approximate value of ht which satisfied the in-

equality, ht 6 0:15, but in [65], it became possible to deduce the value of ht for this soil, and the prediction of KðhÞ will berevisited here. PSDs for the 11 soil samples are shown in the composite in Fig. 1. No consistent trends of D with soil

texture have been found, nor should they be expected, unless such trends in either porosity or rm=r0 are expected.It seems often to be the case that the fractal dimensionality varies with particle size class [24,64,65], thus D may be

different for the sand size range than for clay, or silt. Note that Eq. (1) is then applied to each size range with distinct Dseparately, but that in each case, / is replaced by the associated partial porosity, /i. The values of i, an integer index,

need not be tied to the classical size ranges, silt, sand, and clay. The fact that D tends to 3 for vanishing porosity means

that any size range, which is associated with a small partial porosity will tend to have 3� D � 1. Thus in clayey soils,

the sandy size range may be associated with D near 3, while D for the clay range may be nearer 2.7 [24], whereas in

sandy soils these tendencies appear to be reversed [64,65].

Fig. 1. Experimental data for the PSDs of 11 samples of the McGee Ranch Soil, US Department of Energy, Hanford Site (data from

Rockhold et al., 1988) [72].


Consistent with a lack of dependence of Eq. (1) on pore shape, it is possible to consider [65] that the pore and particle

spaces are effectively ‘‘dual’’, where the solid fractal dimensionality, Ds is defined by Eq. (1) but with the substitution

/ ! 1� /, i.e.,

/ ¼ r0rm

� �3�Ds

ð2Þ

The probability distribution function for pore radii is defined as W ðrÞ, and the probability that a pore has radius

between r and r þ dr is [64]

W ðrÞdr ¼ 3� Dr3�Dm

� �r�1�D dr ð3Þ

An analogous expression for particle sizes results from comparison with Eq. (2). An integral from r0 to rm of

W ðrÞV ðrÞ may be verified to yield Eq. (1) for pore volumes V ðrÞ ¼ r3 Explicit incorporation of the geometry into the

pore volume changes the overall normalization constant from ð3� DÞ=r3�Dm to some other constant value, if the result,

Eq. (1) is required to remain the same.

Explicit representations of the air entry pressure, or the saturated hydraulic conductivity, require explicit consid-

eration of the pore geometry, but since this geometry is assumed scale independent, the same geometrical factors appear

in both K and its saturated value, thus canceling in the ratio. A similar argument applies to the water retention. The

pores with the smallest radius will also have the greatest curvature, guaranteeing perfect correlation between pore size

and water retention and allowing all pores larger than a given size to be considered empty of water, and all smaller sizes,

full. Thus the geometrical factor relating h to pore radius r relates also the air entry value, hA and the maximum pore

size, rm and will cancel in scaling formulations of hðhA=hÞ. The result for the water retention curve has been derived to

be [17,64]

S ¼ 1� 1

/

� �1

"� hA

h

� �3�D#� hA

h

� �3�D

ð4Þ

implying that extension to a continuous pore-size distribution does not lose any of the fundamental simplicity of their

model. The predictions of water retention from this or closely related models have been verified independently at least

three times [25,60,65], and were reported in a fourth case [59], to be superior to predictions from, e.g., the [18,19],

treatment.

It is of course an important result that at both low and high moisture contents water retention characteristics

frequently deviate from the fractal scaling of Eq. (4). These deviations are often incorporated into phenomenologies

[66], which attempt to match experiment over the entire range of h. Such an outlook appears to be in error, and not


merely on account of its underestimation of the relevance of fractal descriptions. At the high h end, this divergence is,

for finer soils, in some cases, due to small concentrations of sand, and correspondingly rare larger pores. In other cases

suggestions have been made regarding the possible effects of soil aggregation producing larger pores for reasons, which

cannot be related to the particle-size data. These particular physical reasons seem quite reasonable, and have in some

cases been explicitly verified; an example is given in Fig. 2a. The more interesting deviation from the present perspective

is that which occurs at low moisture contents.

The deviation from fractal scaling of hðhÞ at low moisture contents appears to be related solely to problems with

equilibration, as analysis of equilibration of a soil column on a pressure plate can show [65,67]. If this is true, as it

appears to be, then this deviation says nothing about the actual distribution of pore sizes, which are filled with water, at

these low moisture contents, but rather it says more about the magnitude of KðhÞ. This can be seen through elementary

statistical analysis of the moisture contents of a collection of soils, at which the deviation from fractal scaling in hðhÞoccurs [65] and is discussed below as well.

Moldrup et al. [50], analyzed a large selection of soils and found that the ratios of solute diffusion constants in

porous media, Dpm to their values in water, Dw are best represented as,

Fig. 2

culatio

predict

Dpm

Dw

¼ 1:1hðh� htÞ ð5Þ

with ht a threshold moisture content equal to,

ht ¼ 0:039SA0:52vol ð6Þ

The statistical correlation of Eq. (6), with R2 ¼ 0:99, is high enough to use for predicting KðhÞ, but not high enough

for theoretical inference. Values of ht ranged from about 0.05 for coarse sands to about 0.18 for clayey soils. Hunt and

Ewing [68] demonstrated that Eq. (5) is a direct consequence of the applicability of percolation theory. Eq. (6) can be

used to check whether the moisture content, at which fractal scaling of water retention breaks down, is compatible with

the moisture content at which solute diffusion vanishes.

A suite of 43 Hanford site soils were investigated [65]. The data from these soils were obtained by various groups,

such as [69], and compiled by [59,70,71], etc. These soils were taken from sites characterized as Injection Test Site (ITS),

Volatile Organic Carbon (VOC), Field Lysimeter Test Facility (FLTF) and so forth because of the similarities of the

soils at the various sites. The FLTF soils are mainly loams, the ITS soils coarse sands, and the VOC soils coarse sands

and gravels with some finer materials sprinkled in. The PSD was used to generate D and thus hðhÞ, and hA was used as

an adjustable parameter to optimize the fit. This amounts to generating slope and (usually slight) curvature from ex-

periment, but the vertical position is then used as a fit parameter. The points at which the experimental and theoretical

. Experimental data for the water-retention curves for seven soils (from Freeman, 1995) and comparison with theoretical cal-

ns from assumed fractal geometry. Arrows denote the moisture contents, hd at which the experimental curves deviate from the

ed curves.


results diverged were then plotted against SA0:52vol . SAvol was not given for these soils, so it had to be estimated from the

following result,

Thr

e sho

ldM

oist

ure

Fig. 3

denote

of its l

SAvol /R rmr0

r2r�1�Ds drR rmr0

r3r�1�Ds dr¼ 3� Ds

2� Ds

� �1

rm

rmr0

� �Ds�2

� 1

1� /

264

375 ð7Þ

Given the dependence of SAvol on the fractal dimensionality, the porosity, and the ratio of maximum to minimum pore

radius, it can be shown that the critical volume fraction in this formulation is a function of the same variables as in that

of [21]. Because [50], report SAvol in terms of volume, rather than the more customary mass, Eq. (7) must be multiplied

by the factor 1� / in order to be used in Eq. (6). A sample of water retention curves with determination of hd is shownin Fig. 2. The regression on Eq. (6) using Eq. (7) for SAvol is shown in Fig. 3. The full data for the regression are in Table

1, and are taken from [59,70,71]. The correlation, with R2 ¼ 0:83, means that it is probable that the same moisture

content governs both the vanishing of the diffusion constant and the deviation from fractal scaling in the water re-

tention curve. A second regression was performed [65], this time using the inverse geometric mean radius as a proxy for

SAvol. In the latter case, R2 ¼ 0:879. Thus it is now possible to find an expression for ht for arbitrary soils, if SAvol can be

calculated, meaning that the hydraulic conductivity can be predicted without use of unknown parameters.

I consider the McGee Ranch soil from above. When SAvol (or the inverse geometric mean radius) is inserted into

these regressions, ht is predicted to be 0.103 and 0.114, respectively. The McGee Ranch soil is a silt loam, and Moldrup

et al., 2001, asserted that typical values of ht for soils of this texture were approximately 0.12–0.13. Taking the mean of

the two regressed values yields 0.01085. Data for KðhÞ for the McGee Ranch soils were generated for five different

depths in unsteady drainage experiments [72], the theoretical predictions from Eq. (8). In order to compare theory with

experiment, 5 of the 11 soil samples had to be chosen. All were assigned the mean value of ht (the dependence of K on htis much less sensitive than the dependence on D). A different value of D was chosen for each depth. These values were

selected from the measured D values so as to optimize the correspondence between theory and experiment, but subject

to the conditions that the mean and the standard deviation of D for the five soils be as close as possible to the values for

the entire suite of samples. The mean of the five chosen soils was 2.8302 (compare 2.8299), while the standard deviation

was 0.11 (compared with 0.12). The visual comparison of experimental and theoretical values is then given in Fig. 4. A

regression of theoretical on experimental values is shown in Fig. 5. The regression yields y ¼ 1:0015x� 0:0065, withR2 ¼ 0:9613. The implication is that experiment and theory agree within 0.2%, when all parameters are given by ex-

periment, and with 96% of the variability explained by the variation in D. This is an overstatement of the agreement

between experiment and theory, of course. Neither regression for finding ht is precise enough to guarantee such pre-

cision, while there is no evidence that the mean of the two regressed values of ht is precisely the correct value. In

y = 0.2679x + 0.0586R2 = 0.8325

0

0.02

0.04

0.06

0.1

0.12

0.14

0.16

0.18

0.2

0.22

0.24

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

A[SAvol]^1/2

VOC

FLTF

ITS

USE

y=0.382x

0.08

Correlation of Threshold Moisture and Surface Area

. Plot of observed hd vs. calculated SA0:52vol (with unknown proportionality constant A). The FLTF, VOC, and ITS soils are each

d by an oval. The straight line fit through the origin is an approximate minimum value of hd for a given SA0:52vol and is, on account

ack of a y-intercept, compatible with the Moldrup relation, Eq. (4), for the threshold moisture content for solute diffusion.

Table 1

Summary of results of fractal analysis for 43 soils

Soil D Ds / cm3/cm3 hdacm3/cm3 Texture KSb (cm/s) hA (cm) rm (m) hArm r0 (m) h3AKS hmax (cm)

VOC 3-0647 2.773 2.8 0.515 0.134 Loamy sand 0.0002 85 89 7565 3.16 122.83 900

VOC 3-0649 2.823 2.898 0.539 <0.12 Loam ngc 530 30 15,900 0.07 None

VOC 3-0650 2.863 2.917 0.624 0.37d Sandy loam 2.6E)07 51.5 1000 51,500 3.3 0.0355 501

VOC 3-0651 2.857e 2.87 0.374 0.126 Loamy sand 0.0094 25 631 15,775 0.32 146.88 850

VOC 3-0652 2.878 2.56 0.352 0.12 Sand 0.00037 58 316 18,328 8.9 72.191 501

VOC 3-0653 2.9e 2.874 0.419 0.12 Sandy loam 5.8E)06 55 446.7 24,568.5 0.45 0.965 7000

VOC 3-0654 2.931 2.916 0.466 <0.18 Sandy gravel 0.00027 40 8912 356,480 1 17.28 <6310

VOC 3-0654-2 2.849 0.419 0.11 Sandy gravel 0.0136 3 2238 6714 1 0.3672 100–1000

VOC 3-0655 2.927 0.4 <0.15 Silty, sandy gravel 0.000158 13 5012 65,156 0.631 0.3471 <3162

VOC 3-0657 2.955 0.359 <0.1 Gravelly sand 0.0136 30 5012 150,360 0.631 367.2 <3162

ERDF 4-1011 2.871 2.816 0.44 0.135 Loamy sand 0.00001 56 126 7056 1.4125 1.7562 1100

ERDF 4-0644 2.906 2.81 0.38 0.11 Loamy sand 5.7E)06 100.5 33.1 3326.55 2 1000

B8814-135 2.891 2.727 0.356 0.145 Silty sand 1.36E)06 135 91.2 12,312 1.86 2511

B8814-130B 2.886 2.682 0.329 0.14 Loamy sand 4.1E)07 46 150 6900 4.53 0.0399 420

FLTF D02-10 2.778 2.776 0.496 0.205 Silt loam 0.00012 100 11 1100 0.48 120 600

FLTF D02-16 2.718 2.71 0.496 0.2 Silt loam 0.00012 150 9 1350 0.8 405 600

FLTF D04-04 2.806 2.804 0.496 0.203 Silt loam 0.00012 100 12.6 1260 0.355 120 700

FLTF D04-10 2.778 2.773 0.496 0.198 Loam 0.00024 100 11 1100 0.5 240 600

FLTF D05-03 2.737 2.735 0.496 0.213 Loam 0.00029 130 7 910 0.525 637.13 1000

FLTF D07-04 2.796 2.791 0.496 0.198 Silt loam 0.00012 98 9.3 911.4 0.4 112.94 600

FLTF D09-05 2.8 2.83 0.496 0.19 Loam 0.00029 72 11.2 806.4 0.79 108.24 562

FLTF D10-04 2.775 2.769 0.496 0.215 Silt loam 0.00012 90 8.32 748.8 0.4 87.48 510

FLTF D11-06 2.803 2.798 0.496 0.195 Silt loam 0.00012 76 11.22 852.72 0.347 52.677 510

FLTF D11-08 2.802 2.797 0.496 0.22 Silt loam 0.00012 80 10 800 0.316 61.44 510

ITS 1-1417 2.919 2.876 0.566 0.088 Sand 0.00014 35 280 9800 2.8 6.0025 2000

ITS 1-1418 2.953 2.762 0.313 0.08 Gravelly sand 0.00014 2 2818 5636 21.3 0.0011 700

ITS 2-1417 2.9 2.719 0.328 0.037 Sand 0.00014 20 250 5000 3.16 1.12 700

ITS 2-1637 2.932 2.708 0.313 0.078 Sand 0.0042 11 1122 12,342 21 5.5902 100

ITS 2-1639 2.951 2.654 0.239 0.058 Sand 0.0012 5 2000 10,000 32 0.15 100

ITS 2-2225 2.844 2.548 0.322 0.075 Sand 0.0055 15 490 7350 40 18.563 200

ITS 2-2226 2.925 2.573 0.229 0.065 Sand 0.015 7 1995 13,965 63.1 5.145 100

ITS 2-2227 2.919 2.666 0.271 0.065 Sand 0.0087 5.4 1995 10,773 39.8 1.3699 200

ITS 2-2228 2.904 2.376 0.212 0.05 Sand 0.021 10 1905 19,050 158.5 21 100

ITS 2-2229 2.902 2.465 0.234 0.075 Sand 0.0064 11 1202 13,222 79.43 8.5184 89

ITS 2-2230 2.853 2.8 0.447 0.11 Sand 0.00023 40 200 8000 3.56 14.72 1000

ITS 2-2231 2.905 2.716 0.318 0.13 Gravelly sand 0.0075 70 851 59,570 20 2572.5 1995

ITS 2-2232 2.88 2.508 0.272 0.08 Sand 0.041 14 1000 14,000 70 112.5 250

ITS 2-2233 2.9 2.492 0.243 0.08 Sand 0.017 11 1202 13,222 74.1 22.627 100

ITS 2-2234 2.81 <2 0.224 0.039 Sand 0.021 80 2089 167,120 831 10,752 3000

USE MW10-45 2.859 2.634 0.34 0.08 Sand 0.00531 15.2 851 12,935.2 44.68 18.648 306

USE MW10-86 2.764 2.569 0.397 0.09 Sand 0.0197 20 489 9780 57 157.6 306

USE MW10-165 2.8299 2.511 0.324 0.075 Sand 0.00663 22 501 11,022 50.1 70.596 306

A.G.Hunt/Chaos,Solito

nsandFracta

ls19(2004)309–325

315

Table 1 (continued)

Soil D Ds / cm3/cm3 hdacm3/cm3 Texture KSb (cm/s) hA (cm) rm (m) hArm r0 (m) h3AKS hmax (cm)

218 W-5-0005 2.894 2.765 0.366 0.12 Sandy loam 0.000067 35 141.2 4942 1.95 2.8726 1000

NCf 2.806 0.09 Sand 0.02 5 1000 5000 5.2 2.5 38

McGee 2.830 0.15 Silt loam 0.001 45 114 5130 3.25 91.125 330

a In the entry for hd the symbol <x means no deviation was noted above x.bKS values are from Khaleel and Freeman (1995).c ng means not given in Freeman, 1995 or Khaleel and Freeman, 1995.d The deviation of VOC 3-0650 from theory at h ¼ 0:37 is not understood.e implies significant uncertainty in the result.f North Caisson (NC) and McGee Ranch soils from Rockhold et al., 1988.

316

A.G.Hunt/Chaos,Solito

nsandFracta

ls19(2004)309–325

-6.5

-6

-5.5

-5

-4.5

-4

-3.5

-3

0.05 0.15 0.25 0.35 0.45

Fig. 4. Predicted and observed values of KðhÞ for the McGee Ranch soil, using the distribution of D values from Fig. 1 and the mean of

the regressed values of ht from Fig. 3 (and a similar regression on the inverse geometric mean radius).

Fig. 5. Result of regression of predicted values and observed values of KðhÞ from Fig. 4.


addition, the experimental precision is considerably less than the implied precision of the agreement. But the com-

parison is certainly encouraging, as the predicted parameter values represent logical choices.

The two regressions for ht each yielded intercepts of roughly 0.06, and R2 ¼ 0:83 and 0.879, respectively. If the

correlation between hd and ht were perfect, the intercept should be zero. Also, both regressions yield values of R2, which

are much less than 0.99, the value of R2 reported by [50] for the correlation between ht and 0:039SA0:52vol . The calculation

of SAvol in terms of the PSD was admittedly simplified, and the determination of hd was less accurate than that of ht butthere was a more important factor in the reduction of R2. The correlation between hd and ht was shown [65] to be

degraded by the fact that a number of soils in this dataset were almost certainly out of equilibrium at moisture contents

significantly higher than what would be predicted by the percolation condition, and for reasons other than a lack of

percolation of capillary flow paths. These other soils appeared to have hydraulic conductivity values smaller than

5� 10�8 cm/s, which for the geometry given [73] was inadequate to achieve equilibration on the time scales of the

experiment [65,67].

There were several reasons why such low K values occurred in some soils, but one of the most common reasons [65]

was that they had anomalously low values of KS . In essence, this means that the proportionality constant giving the

pore radii in terms of the particle sizes was unusually low, consistent with highly compacted, or very coarse soils. Other

reasons involve the cross-over from a lower value of D to a higher value with diminishing pore sizes, but further

discussion of this point is beyond the scope of the present paper. In any case it was possible from theory to predict [65]

which soils should have such low values of K. For example, for the 15 ITS soils, three of the four soils with the lowest KS

values had 3 of the 4 largest values of hd relative to ht. Altogether the bulk of the soils with unexpectedly large hd and

thus low KðhtÞ were from the VOC site, as shown in Fig. 6. Removal of these soils and the four with low KðhtÞ from the

ITS soils from the correlation, Fig. 7, results in an increase in R2 to 0.94, but little change in the intercept.

The systematic deviation from experiment due to the existence of an intercept on the order of 0.05 presents an

interesting problem in interpretation. The persistence of this intercept was not understood in the discussion of [65], but

may be very important for the interpretations of the present paper.

Fig. 6. Predicted values of KðhÞ, using experimentally determined values of threshold moisture content, air-entry pressure, and the first

experimental head value at, or below, the deviation of hðhÞ from the fractal prediction. While the FLTF loams all have predicted Kvalues above the minimum value for equilibration, roughly half of the sands, or gravelly soils in the VOC and about one-third of those

in the ITS regions, are expected to have K values too low to equilibrate. Note that the ‘‘others’’ category denotes ERDF and B8814

loamy sands.

Correlation of Threshold Moisture and Surface Area

y = 0.2848x + 0.0511R2 = 0.9497

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

0.22

0.24

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

A[SAvol]^1/2

Thr

esho

ldM

oist

ure

VOC

FLTF

ITS

USE

y=0.382x

Fig. 7. Repeat of Fig. 3, but with those soils removed, for which equilibration was predicted to be difficult owing ultimately to reasons

not directly related to the percolation transition, such as very low (10�5 cm/s) values of the hydraulic conductivity under saturated

conditions.


The fundamental relevance of percolation theory to the solute diffusion result, Eq. (5), is shown not merely by the

fact that Eq. (5) can be derived through application of percolation theory [68] (based on results of [74]) but already by

the universality of the power of the function h� ht namely 1. This power is independent of the pore size distribution of

the porous medium [50] and has an explicit dependence neither on D nor / (though ht may depend on /). Such universal

powers have never been reported for the hydraulic conductivity. On the contrary, the slopes of logðKÞ vs. logðhÞ plotsare inversely correlated with the slope of logðhÞ vs. logðhÞ graphs [75] and this slope (¼ 3� D) can be anywhere from

near 0 to near 1. Nevertheless, observation of the percolation scaling result for diffusion, Eq. (5), suggests the possibility


of finding such a relationship for K as well. The next section examines the possibility that percolation scaling is found

only in a narrow range of h values near ht.

3. Fractal scaling vs. Percolation scaling of the unsaturated hydraulic conductivity

The uncertainty regarding the predicted behavior of K can be examined relative to the question regarding the most

important role of percolation theory in capillary flow. Some authors apply percolation scaling [76–79] to variably

saturated porous media, K / ðh� htÞt where for universal percolation scaling, 16 t6 2 [80], but where observation (and

theory, for the Swiss Cheese model, [81]) has suggested that this universality need not hold in continuum percolation,

(with values of t greater than 2, for example [80]). Because K is in fact correlated with the PSD and pore-size distri-

butions through its correlation with the water retention curve, e.g. [75] many authors (just a few being, [82–84] (based

on [85]); [66]) has chosen to calculate a hydraulic conductivity related to some mean value of the hydraulic conduc-

tivities associated with tubes whose radii correspond to the distribution of pore sizes inferred from the PSD. However,

methods to calculate transport coefficients in disordered media, which are based on mean values of local coefficients, are

inferior to critical path analysis, also based on percolation theory. Given two methods to calculate KðhÞ based on

percolation theory, how does one choose between them?

Advocates of percolation scaling results of the hydraulic conductivity would assert that the structure of the con-

nections of the liquid water phase govern the hydraulic conductivity near the percolation threshold. Critical path

analysis, on the other hand, is based on the assumption that the dependence on h of the smallest value of the pore

radius, r, through which water must flow, determines the scaling behavior of the hydraulic conductivity. As will be

shown here, one should expect that for values of h near enough to ht the percolation scaling should indeed become the

more important limiting factor to the hydraulic conductivity. First, however, the basic arguments for the two analytical

predictions will be given.

Critical path analysis finds the rate-limiting conductance by using percolation theory [46,47,51,52,54,86]. The vol-

ume enclosed by all the water-filled pores with radii greater than a critical radius, rc is taken as the critical volume

fraction for percolation, ht and rc then defines the radius of the smallest pore, through which water must flow if it is to

travel a large distance. At saturation, all the pores are filled with water, and the largest pore radius contributing to the

critical volume fraction is rm. At lower water contents, the largest pore, which can contribute to the critical volume

fraction is smaller than rm and is given by the same traditional methods applied to fractal porous media, which gen-

erated the fractal water retention curves above. The conductance of the pore with radius rc is proportional to r4c=l,where l is the length of this pore. But l / rc by the assumption of self-similarity required for a fractal structure, making

the critical conductance proportional to r3c . The scaling of r3c i.e. its value at arbitrary saturation divided by its value for

S ¼ 1, gives the scaling of KðSÞ=KS .

Critical path analysis [54] generates the following expression for KðSÞ (the basis for its derivation is given below),

K ¼ KS 1

�� /

1� S1� ht

�3=ð3�DÞ

ð8Þ

where ht is the critical volume fraction for percolation, and S is the relative saturation. ht is system-dependent, but is, by

definition the water remaining when the network of capillary flow paths breaks up into disconnected regions.

To understand Eq. (8) consider first the fully saturated case. Then the condition that rc be the smallest pore radius

encountered on a percolation path is expressed,

ht ¼3� Dr3�Dm

Z rm

rc

dr r2�D; rc ¼ rmð1� acÞ1=ð3�DÞ ð9Þ

Eq. (9) sets the total volume fraction found in pores with radii r > rc equal to a critical volume fraction for percolation

of that volume.

The next input to Eq. (8), the effects of partial saturation, is generated using the premise that the porous medium can

adjust to removal of water by evacuating all pores with radii larger than some equilibrium value, which we call, r>. Thusno hysteresis is considered here. The relative saturation is the quotient of the pore space in pores with r < r> and the

total pore space,

S ¼ 1

/

� �3� Dr3�Dm

� �Z r>

r0

dr r2�D ¼ 1

/½r3�D

> � r3�D0 �

r3�Dm

ð10Þ

When r> ¼ rm, Eq. (10) yields S ¼ 1. Geometric factors relating to the shapes of pores have been left out here and

throughout; for relationships between the saturated and unsaturated conductivity (or the fractal scaling of the water


retention) such relationships are irrelevant. As long as the soil is characterized by a single fractal dimension, the same

geometric factor describing pore shapes is generated at any pore size, and the relationship of K to KS , is unaffected.

Next, the percolation condition relating the smallest (or critical) pore size needed to be traversed, to the critical volume

fraction when the largest pore filled with water has r ¼ r> is,

ht ¼3� Dr3�Dm

� �Z r>

rc

dr r2�D ¼ r>rm

� �3�D

� rcrm

� �3�D

ð11Þ

Eq. (11) has the same form as Eq. (9), but the upper limit has been reduced from rm to r> producing a related

reduction in rc, consistent with the effects of partial saturation. Together, Eq. (9), Eqs. (10) and (11) allow rc for un-saturated conditions to be expressed in terms of rc for saturated conditions, Eq. (8). This procedure is identical to that

of [54], even though the normalization is expressed differently. The hydraulic conductivity of the medium is controlled

by the hydraulic conductance of the rate-limiting pore-throat [54], which is proportional to the cube of the critical radius

(since for fractal geometries with self-similarity, the length of a pore of radius rc must also be proportional to rc), andEq. (8) is obtained by cubing the relationship between the two values of the critical radius (see [54], for a description of

possible complicating length factors [87] for a complementary discussion based on statistics, or [32], for additional

complications regarding the distribution of rate-limiting resistance values on the dominant paths).

Percolation scaling of the hydraulic conductivity, on the other hand, is based primarily on the fact that the sepa-

ration, L, of the paths carrying the water dramatically increases as the moisture content approaches its critical value.

L, the correlation length, is known to depend on the moisture content as follows ([68], based on [45,88]).

L ¼ ðh� htÞ�m ð12Þ

where m ¼ 0:88 is a critical exponent from percolation theory [88]. Since the hydraulic conductivity of the medium is

given in terms of the flow on the water carrying paths divided by the cross-sectional area, one might expect that the

hydraulic conductivity would be inversely proportional to L2. While this argument is na€ııve, its result that K be pro-

portional to ðh� htÞ1:76 is in agreement with the prediction that K / ðh� htÞt and its value of t is within the allowed

range of t values quoted above [76,80]. This prediction can be written more explicitly,

K ¼ K0ðh� htÞt ð13Þ

with K0 an appropriate scaling value of the hydraulic conductivity. Note that [78], has also emphasized that the scaling

exponent of the conductivity shows evidence of not being universal for continuum percolation theory; this point will

be revisited later in this section.

Note that far from the critical volume fraction, L is a relatively slowly varying function of the moisture content, and

the fractal scaling of rc may be more rapidly varying, whereas the divergence of L at the critical moisture content should

overpower the influence of the fractal scaling of rc at values of h very near ht. A composite dependence of KðSÞ couldthen be envisioned, where the fractal scaling dominates until h is very nearly equal to ht and the percolation scaling

takes over in the vicinity of ht. The easiest way to treat the problem is to assume that Eq. (8) holds until the L-de-pendence becomes very strong, and a cross-over to percolation scaling (Eq. (13)) occurs at a moisture content h1, whichis defined by continuity of both K and its derivative. The general procedure is thus equivalent to accounting for, in each

range of h, that portion of the physics, which produces the most rapid change in KðhÞ, and treating the remaining

physics as constant. More complicated theoretical treatments can be constructed, however. The first of the two con-

ditions may be written as follows (where h may be anticipated to equal h1):

KS1� /þ h� ht

1� ht

� �3=ð3�DÞ

¼ K0

h� hth1 � ht

� �t

ð14Þ

Evaluated at h ¼ h1 Eq. (14) gives

K0 ¼ KS1� /þ h1 � ht

1� ht

� �3=ð3�DÞ

ð15Þ

The continuity of dK=dh at h ¼ h1 requires that

tð1� /þ h1 � htÞ ¼3

3� Dðh1 � htÞ ð16Þ

Solution of Eq. (16) for h1 in terms of ht, using the average fractal dimensionality of the Hanford soils, D ¼ 2:857, theiraverage porosity, 0.394 (from [65], with values reprinted in the Table) and t ¼ 1:76 yields

h1 ¼ ht þ 0:055 ð17Þ


Note for comparison that t ¼ 2 would yield a different numerical factor, 0.064. An exponent closer to 2 is favored, for

example, if the diffusion constant vanishes linearly in h� ht (P. Sen, personal communication, 2002). The cross-over

moisture content thus is expected to exceed the critical moisture content by about 0.06, almost the same difference as noted

between hd and ht and suggesting the equivalence of h1 and hd. The implication is that the percolation scaling of L governs

K in a range of moisture contents closely tied to the critical volume fraction for percolation, and causes a rapid diminution

of K and associated rapid increase in equilibration times over what would be expected from the fractal scaling.

Consider that, without the guidance from the above discussion, an experimentalist would not anticipate the need to

increase suddenly the equilibration time with reduction in moisture content below h1, and would thereby generate water

retention curves that are out of equilibrium for all lower moisture contents. But because of the excess water in the in-

completely drained (non-equilibrium) system, the measured hydraulic conductivity values will also exceed the predicted

values, as discussed in [65], and due to the approximate self-consistency of the experimental procedure and the exper-

imental results, the lack of equilibration and the inappropriate measurements of K may go completely unsuspected.

Note that incomplete description of the cross-over might lead to the conclusion that the percolation scaling exponent

is non-universal, as the scaling exponent in the fractal scaling regime is highly non-universal (varying from as little as

about 5 to as much as several hundred), and the argument of the linear function is the same, except for the addition of a

constant term. Thus it is possible that the effects of fractal scaling make the percolation scaling result appear non-

universal.

Finally, it is interesting to note that in one limit, the fractal scaling is identical to percolation scaling but with non-

universal exponents. In particular, if rm=r0 ! 1 then the porosity, /, behaves as, / ! 1, and the Eq. (16) yields

h1 ! ht, whereas the argument of Eq. (8) reduces to h� ht. The exponent remains 3=ð3� DÞ, so Eq. (8) has the same

form as the percolation scaling, but with an exponent guaranteed to be larger than 2, since D > 2. In this regard it

should also be mentioned that D < 3 is strictly required for the porosity to approach 1. Thus in the limit that the

porosity approaches 1, the explicit percolation scaling regime disappears, but the fractal scaling becomes indistin-

guishable from non-universal percolation scaling.

4. Analogies to the glass transition

Some important similarities between the drying of porous media and the cooling of viscous liquids past the glass

transition exist. The observed glass transition is a ‘‘kinetic’’ transition [89] in which thermal energy is removed from the

system too fast for the system to distribute the remaining thermal energy in accordance with statistical equilibrium. This

result is a consequence of the rapid diminution of transport coefficients with diminishing temperature, e.g., over a range

of temperatures less than 100 K, it is not too unusual for the viscosity to rise by 12 orders of magnitude. In such a case,

a cooling rate of a few degrees per minute can increase the relaxation time from seconds to hours over a few minutes

period and a few degrees temperature change, and the system simply falls out of equilibrium. What is perhaps inter-

esting for the analogy have been the many attempts to find a phase-transition underlying the kinetic glass transition.

These attempts have involved the search for a quantity, which either vanishes (a transport coefficient) or diverges (a

length scale, or the viscosity) at a finite temperature. Such a quantity has never been discovered. The analogy in the

present case could be a search for a hydraulic conductivity, which extrapolates to zero at finite moisture content. What

does the present analysis of experimental data say about this possibility?

The fractal scaling relationship for the hydraulic conductivity does not vanish at the critical volume fraction for

percolation, but its use for smaller moisture contents appears to be inappropriate. The reason why the fractal scaling

result for KðhtÞ does not vanish is that it essentially gives the value of the conductance of the smallest pore in terms of

the conductance of the critical pore for percolation under saturated conditions. As pointed out, however, the separation

of flow paths must diverge in the limit h ! ht. This means that the hydraulic conductivity due to capillary flow will

indeed vanish in this limit. But this vanishing does not have anything to do with the length scale associated with the

minimum pore size, rather it is a result of the divergence of a length scale defining the separation of paths.

Interestingly, the evidence appears to suggest that just in the vicinity at which one would expect to be able to see the

change in behavior, the investigated systems tend to fall out of equilibrium and the measured hydraulic conductivity

curves have no relation to the equilibrium dependencies. It seems unlikely that this is coincidence. Clearly, as soon as

the critical behavior associated with a vanishing hydraulic conductivity sets on, the increasingly rapid diminution of the

hydraulic conductivity tends to throw the system out of equilibrium. Since it is possible to express the moisture content

at which this cross-over in behavior occurs in terms of the critical moisture content for percolation, the moisture content

at which the loss of equilibrium takes place becomes pinned to the moisture content defining the percolation transition.

The mechanism of this pinning is the result that KðhÞ in that range of moisture contents is a power of h� ht. This isprecisely the kind of evidence, which has been sought in the glass transition for decades, but never found.


The second interesting aspect of the present behavior is the discovery of systems, for which non-equilibrium con-

ditions set on at moisture contents, which appear to have no relationship with percolation. In these systems, the hy-

draulic conductivity falls below values required for equilibration on experimental time scales for reasons other than an

approach to percolation. The most common alternate cause for the low value of KðhÞ is that the value of KS is very low.

The value of KS is not closely correlated with the value of the moisture content at which capillary flow ceases to

percolate, since KS can be small simply because all pores are small, e.g., because the ratio of pore sizes to particle sizes is

much smaller than the frequently assumed (e.g. [63]) value of �30%. The dependence of KðhÞ is in that case not a scaling

function of h� ht at any accessible moisture content because at values of hmuch higher than htKðhÞ drops to values well

below 10�8 cm/s, and the time scale for experimental equilibration can exceed months [65,67].

The attractive aspect of porous media is then that the two possibilities that have both been suggested for the glass

transition can be observed and distinguished. In one case, systems fall out of equilibrium at a moisture content pinned

to a value associated with a (percolation) phase transition; in the second case systems fall out of equilibrium at moisture

contents well above this percolation phase transition. In neither case is the evidence for the phase transition typically

seen directly in the hydraulic conductivity, but in one case the phase transition may be deduced from the deviation from

the fractal scaling of the water retention curve and its relationship to diffusion. Insofar as this correlation between the

percolation phase transition and the loss of equilibrium is expressed in the water retention characteristics, it seems to be

traceable to the scaling form of the hydraulic conductivity in the vicinity of the critical volume fraction for percolation.

5. Conclusions

In previous studies by several authors it has been confirmed that fractal scaling can predict water retention char-

acteristics of natural porous media from the PSD. In a prior study of 43 soils from the US Department of Energy

Hanford Site, it was shown [65] possible to relate the dry-end moisture content, hd, at which experimental water re-

tention curves deviated from the predicted fractal scaling to the moisture content, ht, at which solute diffusion vanished

[50]. This result suggested that deviations from fractal scaling were due to a lack of phase continuity, or non-percolation

of the liquid phase. The result, however, that hd was consistently larger than ht, by a term roughly 0.06, was not un-

derstood. Further, it had also been noted [68] that the result for the diffusion constant obeyed percolation scaling [50],

whereas the result for the hydraulic conductivity appeared to obey fractal scaling [64]. These two puzzles were likely

resolved in the present study, which examined the relevance of fractal vs. percolation scaling in the hydraulic con-

ductivity. Due to the percolation transition at ht it can be shown that in a small range of moisture contents near ht thepercolation scaling results for K / ðh� htÞt, t near 2, should dominate the fractal scaling. This range of moisture

contents turned out to be roughly 0.06 for the same soils whose water retention characteristics were investigated above.

The close correspondence of the two results is not likely to be a coincidence. It is thus concluded that an onset of the

percolation-scaling regime was likely associated with a rapid drop in K, and a cross-over to a non-equilibrium response

in the water retention characteristics. This percolation regime was noted to be systematically reduced in width with

increase in porosity towards 1, but in the limit of porosity 1 the fractal-scaling regime was transformed into a non-

universal percolation-scaling regime. The lack of equilibration in the percolation scaling range of moisture contents

makes it difficult to verify the percolation scaling directly in K, but a simpler dependence of the diffusion constant, linear

in h� ht allowed [50] an accurate extrapolation to h ¼ ht and an identification of the defining role of percolation theory

in solute diffusion. The fractal scaling is a property of the structure of the medium, whereas the percolation scaling is a

result of the topology of the liquid phase in that medium near the point at which the liquid phase loses continuity. Thus

it may be no surprise that as the volume fraction of the solid phase becomes vanishingly small, the percolation-scaling

regime disappears, but the fractal scaling becomes identical to a (non-universal) percolation scaling. Defining these

limits on percolation scaling not only helps explain data relating water retention and diffusion, but helps interpret the

roles that the random structure of the porous medium, and the random connections of the fluid phase inside the

structure, play in the hydraulic conductivity. Defining the theoretical bounds on the fractal scaling in the water re-

tention and hydraulic conductivity serves rather to strengthen the experimental support for the interpretation that

natural media are fractal, than to weaken it.

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