Download - Basic Transport Properties in Natural Porous Media · Basic Transport Properties in Natural Porous Media Continuum Percolation Theory and Fractal Model A. G. HUNT Received September

Basic Transport Properties in NaturalPorous Media

Continuum Percolation Theory and Fractal Model

A. G. HUNT

Received September 9, 2004; revision accepted December 7, 2004; accepted December 7, 2004

1. INTRODUCTION

P orous media include many man-made as well as natural materials.In fact, all solid substances are po-

rous either to some degree, or at somelength scale. But the most familiar nat-ural media, which are porous enoughfor significant amounts of water or air

to flow through, are soils and rocks.

This article tells how it has finally been possible to predict

the flow of air and water (and several other fundamental

transport properties) in such natural porous media.

It has not been easy for scientists to agree on the appro-

priate description of such media. These porous media are

disordered on many scales, all the way from the individual

grains or pores, through nominally homogenous laboratory-

sized samples and then field or plot sizes to formation sizes

and larger, altogether over more than 12 orders of magni-

tude of size. Fractal, multifractal, Gaussian, and log-normal

models have all been proposed, probably in every scale

range. The verification of a consistent theoretical framework

for calculation of transport properties, at least at some

scales, has the potential to eliminate much confusion re-

garding both the appropriate theoretical techniques to use

as well as the appropriate model to choose.

The present description has two the-oretical inputs. The first is that power-law distributions of pore sizes best de-scribe the pore space and that suchdistributions are consistent with theself-similarity associated with fractalmodels [1–5]. The second is that theappropriate theoretical description offlow is based on percolation theory

[6 –9]. The choice to apply continuum percolation theory,rather than the site or bond variations, is required by thechoice of the fractal model. But the limitations of the fractalmodel are then predicted by continuum percolation theory.

Using continuum percolation theory on the probabilisticfractal model, it is possible to predict the ratio of the unsat-urated to the saturated hydraulic conductivities [10, 11], thepressure-saturation relationships [12, 13] (including hyster-etic aspects [14]), solute and gas diffusion relationships [15]that are observed in experiment [16 –21], the observed [22,23] air permeability [24] and the known [2] electrical con-ductivity relationships [25]. All parameters are consistentfrom one property to another. The values of the parametersthat describe the hydraulic properties can be predicted [11,12] from physical measurements [16, 19, 20] (or approxi-

Correspondence to: Allen G. Hunt. E-mail: [email protected]. G. Hunt is at the Department of Physics and Departmentof Geology, Wright State University, Dayton, OH 45435.

The first question asked bypercolation theorists is

essentially, can particles or gasmolecules fit through the porespace and arrive at the other

side of the system ….

22 C O M P L E X I T Y © 2005 Wiley Periodicals, Inc., Vol. 10, No. 3DOI 10.1002/cplx.20067

mated from calculations [26]) and are not fit parameters.The understanding that is developed is universal.

2. FRACTAL PORE-SPACE MODELIn porous media fractal models have been used to describethe solid volume, the pore volume, or the interface betweenthe two. Fractal models of pore space were developed ini-tially in the mid-1980s [1, 2] and used in the petroleumphysics and engineering communities. Some applicationswere to the pressure dependence of the saturation, some tohydraulic or electrical conductivity. Turcotte [3] proposed afractal fragmentation model, which identified a physicalbasis for the existence of fractal soils in the scale invarianceof the fragmentation of soil particles. Fragmentation can beviewed as the chief mechanism of physical weathering. Hereit will not be necessary to account for the present complex-ity in fractal modeling. In particular, it will be necessary todiscuss only two different fractal dimensionalities [11, 12].Ds, describes the solid space; Dp, describes the pore space.No particular geometrical assumptions need to be made.The values of Dp and Ds are related by symmetry [12], i.e.,particle quantities are obtained from pore quantities bysubstitution everywhere in an equation of �31 � �.

The particular model used is of a probabilistic, trun-cated, continuous (rather than discrete) fractal. Model char-acteristics are, however, defined so that the porosity andwater retention functions are identical to those of the dis-crete and explicit fractal model of Rieu and Sposito [4](called hereafter the RS model). Moreover, integration overthe continuous pore size distribution between qr and r,where q � 1 is an arbitrary factor, yields the contribution tothe porosity from each size class obtained by RS. Thus thepresent model is just a continuous version of RS.

The distribution of pore sizes is defined by the followingprobability density function [11]:

W�r� �3 � Dp

rm3�Dp

r�1�Dp r0 � r � rm. (1)

The power law distribution of pore sizes is bounded by amaximum radius, rm, and truncated at the minimum radius,r0. Equation (1), as written, is compatible with a volume, r3,for a pore of radius r. The result for the total porosityderived from equation (1) is [11]

� �3 � Dp

rm3�Dp �

r0

rm

r3r�1�Dpdr � 1 � � r0

rm� 3�Dp

, (2)

exactly as in RS. If a particular geometry for the pore shapeis envisioned, it is possible to change the normalizationfactor to maintain the result for the porosity, and also main-tain the correspondence to RS.

Using the substitutions, �31 � �, and Dp3Ds the resultis obtained [12]:

� � � r0

rm� 3�Ds

. (3)

As a reminder, in equation (2) r0 and rm refer explicitly tothe minimum and maximum pore sizes, and in equation (3)to the minimum and maximum particle sizes, respectively.However, I make the assumption, made long before [27–29]that pore and particle radii are proportional to each other.Thus it must also be assumed that the ratio of r0 to rm is thesame for both the pores and the particles. Equations (2) and(3) also imply directly a property pointed out in RS, thattypically Dp � Ds. But the basis for this relationship is thatusually � � 0.5. If � � 0.5, the relationship is reversed. Notethat in the fractal treatment of [2] and [30] the porosity isgiven by equation (3), but using Dp as the fractal dimen-sionality!

It is possible, aside from such complications as individ-ual particle geometry, (including, especially the tendency ofclay particles to be flat, i.e., two dimensional) to develop anexpression for the surface area to volume ratio, A/V, of themedium, assuming that it is related to the ratio of thesurface area (r2) of all the particles to the volume (r3) of allthe particles [12],

A/V ��r0

rm r2r�1�Dsdr

�r0

rm r3r�1�Dsdr� � 3 � Ds

2 � Ds� 1

rm� �

rm

r0� Ds�2

� 1

1 � �� .

(4)

Equation (4) must be multiplied by 1 � � to give a surface areaper unit volume of the porous medium [12]. Though equation(4) is used only for later comparisons, it is given now becauseof its connection with the power-law pore size distribution andits absence of explicit connection with flow.

3. BASIC RELEVANCE OF PERCOLATION

3.1. The Upscaling ProblemIn heterogeneous or disordered media, the difficulty oftransporting mass or energy may vary greatly from place toplace. Thus the hydraulic conductivity may be a strongfunction of the position. The porous media communitiesinvestigate what they call “upscaling” for flow and trans-port. In physics terms this would be formulated as “findingan effective transport coefficient in terms of its microscopic(or local) variability.”

In the hydrology and soil physics literature the mostcommon suggestion is that the upscaled K will lie some-where between the harmonic and the arithmetic mean ofthe individual K values. The basis of this statement is in two

© 2005 Wiley Periodicals, Inc. C O M P L E X I T Y 23

obvious extreme results. One is that a collection of resistors(electrical or hydraulic) arranged in series has an equivalentresistance equal to the sum of all the resistances (the con-ductivity given by the harmonic mean conductivity, as longas all the resistances are of equal length). On the other hand,a collection of the same resistances configured in parallelhas an equivalent conductance equal to the sum of all theconductances (the conductivity given by the arithmeticmean conductivity).

The question is: what fraction of the distribution of re-sistance values in a real medium should be considered asbeing configured in series and what remaining fractionshould be considered as in parallel?

This question has been addressed for the electrical conduc-tivity [7, 8] of disordered solids, and the unambiguous answerobtained [31, 32] that the appropriate configuration is de-scribed by percolation theory. What percolation theory statesis that all resistances smaller than or equal to some criticalvalue must be considered to be in series with each other, butin parallel with the remainder of the distribution. The quantileof this critical resistance in the distribution of resistance valuesis equal to the percolation probability. The same method [33]was later validated [34] in network simulations of the saturatedhydraulic conductivity. Because Ohm’sLaw for electrical conduction and Dar-cy’s Law for fluid flow are mathemati-cally identical, this statement of topolog-ical and analytical equivalence is notsurprising. Percolation theory also per-formed better than stochastic (perturba-

tion) approaches for geologic scale up-

scaling problems [35]. As will be seen here all basic transport

phenomena will be described by some combination of ideas

from percolation theory. The combination will vary according

to the property.

3.2. Percolation PerspectiveThe first question asked by percolation theorists is essen-

tially, can particles or gas molecules fit through the pore

space and arrive at the other side of the system [6]? The

answer is based on whether the pore space is continuously

connected. In case the system is below the percolation

threshold and the pore space is not continuously con-

nected, the answer is no; if above the percolation threshold,

one can also ask how the transport property varies with the

proximity of the system to the percolation threshold (i.e.,

how much gas, water, or solute arrives in a given time

frame). This perspective of percolation theory presumes its

usefulness only for systems near the percolation threshold

[35, 36]. This perspective is too narrow, and has contributed

to an underestimation of the value of percolation theory,

particularly in porous media. In order to develop a full

understanding of transport in porous media it is necessary

to be able to apply percolation theory to systems both nearand far from the percolation threshold.

As a basis to consider both uses of percolation theoryconsider an infinite square grid in 2D (like a very largewindow screen). Imagine cutting single wires (individualbonds) at random. What fraction, p, of bonds do you have tobreak before the structure, or screen, falls apart? That valueof p is called pc, and in this particular case pc � 0.5. Theprobability, P, that an infinite continuous path of brokenbonds exists, is zero for p � pc, but 1 for p � pc. Similarly,emplacement of a fraction, pc, of the wires into their screenpositions will just construct an infinitely large connectedwire screen. This screen will have lots of holes, in fact holesof all sizes (meaning a fractal structure) but it will be con-tinuous. If, in such a system, all the local grid connectionsare identical in size, then the electrical conductivity, �, ofthe screen can be accurately predicted as a function of pusing scaling concepts of percolation theory. Because theconducting portion of the medium near the percolationthreshold is self-similar, � must vanish according to a pow-er-law in (p � pc). The result is, � � (p � pc)t with t � 1.88(1.27) in 3D (2D) [9, 37]. Although pc turns out to depend onthe particulars of the mesh geometry, i.e., whether it is

square, triangular, or hexagonal, thevalue of t turns out to be independentof nearly everything except the dimen-sionality of the system. If all the wires ofthe grid are present, but of very differ-ent radii, however, the optimal meansto calculate the electrical conductivityin terms of the variability of the wires

uses a more abstract application of percolation theory.Consider now that a large number of resistances with a

wide range of values are connected between random pairsof nearest-neighbor sites on such a regular grid. Such a gridcan also be used to represent an idealized problem in the(unsaturated) hydraulic conductivity and is known as anetwork model [38]. In particular one can imagine connect-ing cylindrical tubes with a wide range of tube radii betweenthe sites. Calculating the conductivity of the electrical ana-logue is somewhat more straightforward because in a realsituation, the right-angle junctions of different radius water-filled tubes could contribute to flow nonlinearities, but aslong as such complications are ignored, the fundamentalidea is identical for both cases. In this case pc is a moreimportant quantity than t above. If one knows the pc for anarbitrary regular grid representing a network model with aspread of pore sizes and thus connecting conductances, onecan find the upscaled K [39] or electrical conductivity. Val-ues of pc are tabulated for many specific cases [40].

Percolation theory used in the form of critical path anal-ysis (CPA) quantifies the characteristic resistance of the“path of least resistance” [40, 41], and the “critical,” or“bottleneck” resistances on this path control the entire field

Percolation theory in the form ofcritical path analysis quantifiesthe characteristic resistance of

the path of least resistance.

24 C O M P L E X I T Y © 2005 Wiley Periodicals, Inc.

of potential drops [34]. Consider a problem of a log uniformdistribution of resistance values with a distribution width 10orders of magnitude, e.g., from 100 to 1010 in arbitrary units.Place each resistor at random on a square lattice. Hydrolo-gists intuitively suspect that the conductivity of this ar-rangement is given in terms of the middle value, 105, orgeometric mean resistance [42] and this is correct becausepc � 0.5. What this means is that, for the square lattice, onecan put 1/2 the resistors into grid positions at random, andit will be possible to find an interconnected path of theseresistances, which is guaranteed to reach infinite size (in aninfinite system). If one has chosen that half of the resistancedistribution with the smallest resistances, one has mini-mized the total resistance. In summary, a continuous pathfor current flow can just be found across an infinite squarelattice, which completely avoids resistances larger than themedian value.

For a square grid, the value of pc � 0.5 is obtainedprecisely by the approximate formula, Zpc d/(d � 1), withd the Euclidean dimensionality [43] and Z the number ofnearest neighbors for a given site. In two dimensions thenZpc 2. While pc on the 2D square lattice is 1/2, on thetriangular lattice, with Z � 6, pc � 0.347 1/3, and on thehoneycomb lattice, with Z � 3, pc � 0.653 2/3.

Thus if the same resistors as above are placed on atriangular lattice, the dominant resistance value out of thedistribution is now 103.47. For the triangular lattice, thecurrent never has to utilize the slowest roughly 2/3 of theconnections. And if the same resistors are placed on ahoneycomb lattice, the dominant resistance is 106.53. Forthe honeycomb lattice, the current can only avoid the slow-est roughly 1/3 of the connections. The corresponding val-ues of the conductivity are spread out over more than threeorders of magnitude, depending on the local coordinationnumber, Z, all in two dimensions! Such spreads can be evengreater in 3D. (Note that this description of CPA ignorescomplications from the divergence of the correlation lengthat p � pc. In order to develop an expression for the conduc-tivity for any of these grids, one must take such complica-tions into account, but for treating ratios of conductivityvalues such complications can typically be neglected).

We will see that, by examining experimental relation-ships for solute diffusion, the value of the critical volumefraction for percolation can be predicted for a wide range ofsoils from soil physical quantities to within a few percent.Thus, in the identification of a controlling resistance (in ourproblem, a bottleneck pore) we can also find its radiuswithin a few percent.

3.3. Application of Percolation Theoretical Concepts toProbabilistic Fractal ModelsThis application of percolation theory is to the unsaturatedhydraulic conductivity of random fractal systems. As arguedin [10] one should not apply a network model to probabi-

listic fractal systems. If the system is truly self-similar, and if

particle (and thus pore) radii vary over orders of magnitude,

then the lengths of these pores must also be assumed to

vary over orders of magnitude. It is impossible to construct

a regular network with spatially invariant values of the co-

ordination number under such circumstances. So, instead

of applying percolation theory to the bonds of a network, I

apply it on a continuum. This means that instead of using a

bond probability, p, I use a fractional volume, such as the

moisture content, �, to represent the quantity, which can

take on a critical value defining the onset of percolation. For

� � �t an infinitely large cluster of interconnected, water-

filled pores exists. The drawback of using continuum per-

colation theory is that it is not immediately clear what �t

should be. Experiment [20] however, has delivered �t for us

(together with some theoretical interpretation [26]. The ex-

periments were performed in the vicinity of the percolation

threshold at low moisture contents.

To apply CPA to calculating K of a probabilistic fractal

the appropriate choice is to set the volume of all the water-

filled pore space with radius larger than some radius, rc,

equal to the critical volume fraction for percolation, �t.

From the concept of percolation theory this volume must be

interconnecting, meaning that water can flow through a

continuously connected path of water-filled pores through

the entire medium without ever traversing a pore with ra-

dius less than rc. This makes rc effectively a bottleneck to

flow. The dependence of rc on � ultimately gives K(�).

For a given pressure difference Poiseuille’s law for flow

in a tube (at low Reynolds number) yields a flux propor-

tional to the fourth power of the radius, r, and inversely

proportional to the length, l. Since, in a self-similar me-

dium, l � r, the flux, and therefore the hydraulic conduc-

tance, gh, of a pore of radius r, is proportional to r3. For a

right circular cylinder the geometric (numerical) factors

are known. The same general structure of the result will

apply to an arbitrary shape, although the numerical fac-

tors are not, in general known. But if the fractal descrip-

tion is valid, that numerical factor is the same (or nearly

the same) for all pores in that medium and is thus inde-

pendent of the value of rc. As a consequence one can

derive the ratio of K to its saturated value, KS, without

concern for the geometry of the individual pores. Such

geometry is important for KS, but KS appears best calcu-

lated by existing CPA methods [33].

Accordingly, rc under saturated conditions is given

through [11],

3 � Dp

rm3�Dp �

rc

rm

r3r�1�Dpdr � � t (5)


with �t as yet unknown in value, but taken as the minimummoisture content for which an interconnected network ofwater-filled pores can be found.

If the porous medium is not saturated, but in equilib-rium, the largest pores will be air-filled. It will be a topiclater in this article to consider how this equilibration may ormay not occur. Nevertheless, under equilibrium conditions,there will be a maximum-sized water-filled pore becausethere is a unique relationship between pore curvature andpore size in a fractal medium. The radius, r� of this largestwater-filled pore is found from a condition on the satura-tion, S �/� [11],

1�

3 � Dp

rm3�Dp �

r0

r�

r2�Dpdr � S. (6)

Since the minimum moisture content, �t, required for per-colation of water-filled pores appears to be independent of� (justified later), then we can repeat the process for findinga bottleneck pore radius, but now for unsaturated condi-tions [11],

3 � Dp

rm3�Dp �

rc�s�

r�

r2�Dpdr � � t. (7)

Simultaneous solution of equations (5), (6), and (7) yields[11],

rc�S� � rc�S � 1�� 1 � �1 � S1 � � t

� 1/�3�Dp�

. (8)

Given that the hydraulic conductance of each pore is pro-portional to the cube of its radius, the ratio of the limitingconductance at S to its value at S � 1 is thus,

gc�S� � gc�S � 1�� 1 � �1 � S1 � � t

� 3/�3�Dp�

. (9)

The ratio of the limiting conductance of the network isassumed to define the ratio of the hydraulic conductivity,K(S), and its saturated value, KS, leaving

K�S� � KS� 1 � �1 � S1 � � t

� 1/�3�Dp�

. (10)

Equation (10) may be rewritten as follows,

K�S� � KS� �1 � �� t�

1 � � t� 3/�3�Dp�

. (11)

Note that K(S) is a power of (1 � �) � (� � �t), and as aconsequence does not vanish at the percolation threshold,� � �t. In fact, equation (11) implies that the hydraulicconductivity is governed by the smallest pore in the systemwhen the moisture content equals �t. But K must trendcontinuously to zero in the limit p3pc (here �3�t) as (� �

�t)t. So the question of how these two different dependences

of K(�) are to be handled must be addressed. Note that thefact that water is a wetting fluid allows water films to becontinuously connected even when water-filled pores donot form a continuously connected network. Since the con-ductivity of such water films is generally orders of magni-tude less than that of capillary flow, we can neglect thiscomplication under most circumstances.

As seen from equation (A.2), when � approaches �t fromabove, the separation, , of paths of interconnected, water-filled pore-space, along which water can flow, must diverge.This divergence requires that the hydraulic conductivityvanish, independent of the size pore that limits the rate offlow along such paths. Further, (also from the Appendix) theactual distance, �, that water flows between pores sepa-rated by is (� � �t)

�1. The hydraulic conductivity is definedin terms of the total flow crossing a plane divided by thecross-sectional area of the plane. If the flow path separationdiverges according to � (� � �t)

�0.88, then the hydraulicconductivity in 3D must vanish according to �2, or

K � �� t�2�0.88� � �� t�

t (12)

with t � 2 � 1.76 (or t � � 1.35 in 2D). But if thetortuosity implicit in proportionality (A.3) is taken into ac-count, the value of t in 3D should be t � 1.76 � (1 � 0.88) �

1.88 [9] because the resistance per unit sample length wouldbe increased by (� � �t)

0.12. Note that the tortuosity, with a6% contribution, is not the chief input into t. As pointed outby Stauffer [44], however, such an argument breaks down in2D, since there �/ � 1. 2D media were treated in [37]where it was shown that t � 1.27.

Equation (12) must be appropriate in some range of �

near �t. Thus, equation (11), which describes the depen-dence of K on � due to the diminishing size of the bottleneckpore, must give way to equation (12) at some �x related to �t.�x can be determined [45] by setting the two dependences ofK(�), as well as their derivatives equal to each other at � � �x.Physically, this means that for any �, the appropriate resultto choose has the larger value of dK/d�, i.e., that the ap-propriate form to choose is the one most sensitive tochanges in moisture content. Equivalently, the less sensitivedependence is set equal to a constant. This is the bestapproach, since the result for the correlation length isstrictly valid only in the region, in which it varies rapidly.For short-hand reference, equation (12) will be referred toas percolation scaling of K, whereas equation (10) is referred


to as fractal scaling. Even though both ultimately derivefrom percolation theory, in equation (10) it is the fractalcharacteristics, which make the dominant impact on K. Insoil, treatments based solely on percolation scaling (e.g.,[46, 47]) will not be likely to be relevant for a large enoughrange of moisture contents to be directly verifiable, and thepercolation scaling regime will be mostly inferred indirectly,as will be seen.

The result from the above analysis is [45]

�x � � t � �t�1 � ��

33 � Dp

� t� . (13)

For Hanford soils, t � 1.88 and typical values of � � 0.4 andDp � 2.81 leads to �x � �t 0.08, only about 22% of therange of accessible moisture contents. Figure 1 demon-strates an example of this cross-over for the values of Dp, �,and �t from the McGee Ranch soil as well as the experimen-tally measured values of K. Here Dp was obtained fromequation (2), the particle size data, and the porosity,whereas �t was obtained using the regression described insection 7. An important physical consequence of equation

(12) is the implication that K(�) begins to drop rapidly withdiminishing � as � approaches �t. An analogous result is alsoobtained from effective-medium approximations on a net-work (Robert Ewing, personal communication). Note thatthe experimental results from unsteady drainage field ex-periments cut off at the point where K(�) begins to dropnoticeably below the fractal scaling prediction (at � � 0.15).An analogue of this behavior is seen in laboratory results forwater-retention curves [17, 18, 48] and is discussed in sec-tion 7. In Figure 2 the predicted and observed hydraulicconductivity from a more complex soil are shown. Note thatagain no adjustable parameters were used.

An important implication of equations (11), (12), and(13) is the following. Consider the limit r030, i.e., thatfractal fragmentation has proceeded indefinitely. Fromequation (2) it is seen that �31. If � � 1, equation (11)yields,

K � KS� � � � t

� � � t� 3/�3�Dp�

. (14)

Further equation (13) yields

�x � � t. (15)

Thus, the percolation scaling regime disappears, while thefractal scaling regime develops a dependence on the mois-ture content, which is equivalent to the percolation scaling

FIGURE 1

Comparison of experimental data (E) for K(�) for the McGee Ranchsoil with predictions from equations (10) and (12). The ranges ofmoisture content where equations (10) and (12) are valid are denotedby bold lines, whereas the ranges of moisture contents where they arenot valid are denoted by light lines. The mean Dp � 2.832 was usedfor equation (10). The standard deviation, 0.011, for Dp was used toconstrain predicted data between the sequences of dashes. �t �0.1085, predicted from comparison of equation (4) with the thresholdfor solute diffusion (from [22]), is denoted by the vertical line, whereasthe cross-over in validity from equation (10) to equation (12) occurs at�x � 0.174. t � 1.88. The values of Dp, its standard deviation, and�t are from [45]. The lowest moisture contents reached in theunsteady drainage experiment for the McGee Ranch soil were ca.0.15, the moisture content at which solutions from equation (10) andequation (12) begin to diverge.

FIGURE 2

Predicted and observed hydraulic conductivity for the North CaissonHanford site soil. Although the agreement here is imperfect in com-parison with the McGee Ranch soil, the prediction is made for amultimodal distribution of pore sizes without using adjustable param-eters. The cause for the discrepancy is probably a slight overestima-tion of the porosity in the mode of the distribution corresponding to thelarger pores (and a slight underestimation of that in the smaller pores,where K is underestimated).


regime, but with an exponent, which is related to the spe-cific characteristics of the fractal structure, and is thus non-universal. Also, it is interesting that the argument of equa-tion (14) is the same as that traditionally used in soil physicsphenomenologies such as [49], or [50] (as long as the resid-ual moisture is interpreted to be �t), which therefore be-come accurate in the limit �31.

4. AIR PERMEABILITY AND ELECTRICAL CONDUCTIVITYThe use of the term permeability in porous media usuallyimplies a flow property (rather than a magnetic property).For the permeability corresponding to K, this property is thehydraulic conductivity multiplied by the viscosity (normal-ized to the density). The concept of permeability is meant toisolate the characteristics of the fluid from the propertydescribing transport and relate the constitutive relationsolely to the properties of the medium. Since water is awetting fluid and air is not, however, this isolation is onlypartially successful in developing an equivalence. Thus, al-though the air permeability, ka, in completely dry porousmedia is equal to the (water) permeability in completelysaturated porous media, at arbitrary saturation there is nosimple relationship between the two properties. At arbitrarysaturation the air tends to occupy the largest pores of themedium and the water the smallest.

Consider, in contrast to the drying of an initially satu-rated medium the wetting of an initially dry medium. Ini-tially the critical (bottleneck) pore radius for air flow is giventhrough equation (5). As water enters the smallest pores, thecritical path for airflow is unaffected, since it avoids all ofthe smaller pores anyway. In fact, as long as the watercontent is not high enough for water to enter pores of sizerc � rm(1 � �t)

1/(3-D) or larger, it is possible to find a path ofair-filled pores with all radii larger than or equal to rc. Thismeans that critical path analysis reveals no effects at all ofchanging saturation on the air permeability, ka! This doesnot imply that ka is independent of S, however. What itmeans is that, in the competition between percolation-scaling and fractal-scaling for the dominant effect on ka,percolation-scaling wins at all values of the saturation. Thuska can be written as follows:

ka � k0� � � � t

� � � t� t

(16)

with t � 1.88 (1.27) in 3D (2D), � the air-filled porosity, �t itscritical value for percolation, and k0 the value of ka in acompletely dry medium. Compare the prediction of equa-tion (16) with experiments on 2D materials in Figure 3 usingone adjustable parameter, �t � 0.16. Furthermore, for asuite of experiments on 3D volcanic tuffs, Moldrup et al. [22]report that ka obeys a result like equation (16), but with t �

1.89 0.54. Here the 0.54 means that ca. 2/3 of the

measured systems exhibited t values between 1.35 and 2.43.The difference between the Moldrup experiments andequation (16) is that the power-law dependence works bet-ter if the air permeability is scaled to a value that is closer tosaturation than its value under completely dry conditions,not surprising given that the percolation theoretical de-scription is considered to be strictly valid only in the neigh-borhood of percolation. The continued increase in ka withfurther drying is due to topological effects such as containedin the treatment of the correlation length of percolationtheory, but there is no universal treatment for such effectsthat would be guaranteed to be accurate. In the 2D casedescribed, however, percolation scaling of ka was accurateeven far from percolation.

The electrical conductivity of completely saturated po-rous media (particularly rocks, such as sandstones) obeys“Archie’s law,”

� � �m (17)

with reported values of m ranging from 1.5 to 2.5. But inmore controlled 3D experiments [51] m � 1.86 0.19. CanArchie’s law also be explained in the present context? In Ref.51 it was speculated that Archie’s law might be a conse-quence of percolation, but the authors then rejected the

FIGURE 3

Experiment vs. theory for the relative gas-phase permeability ofhelium as a function of the fraction of helium-filled pore space, Vs/Vt.The empty circles are single-phase and the filled circles are multi-component samples. The diamonds are the prediction from equation(16), which compares strictly with the multicomponent samples.


possibility, partly because of the potential effects of therange of pore sizes in rocks. This question was also raised in[36], namely that Archie’s law might be explained if thecritical volume fraction for percolation were very small, butthey then provided no answer. We have here a consistentmeans to take the variability in pore sizes into account,however, for both K and �. It is interesting to note that,although some investigators have reported that KS couldalso be represented as a power of the porosity (and simpleaveraging procedures can develop such a dependence), [51]reports that the data for KS can under no circumstances berepresented as a simple power law in �. So it is importantthat the framework described here does generate Archie’slaw, but no similar relationship for K at saturation.

The electrical conductivity of a cylindrical wire or tube of(homogeneously) conducting fluid is independent of its ra-dius or length. Thus the current in a pore of radius r andlength l with a potential difference �V is proportional to r2/l.In fractal media, l � r, so that the electrical conductance ofa pore of radius r is proportional to r. The topology ofconnection of the pore space is independent of whether theproperty under consideration is theelectrical or the hydraulic conductivity,so rc is identical for both, but whereasgc

h � r3 (the superscript h refers to hy-draulic), gc

e � r. Repeating the analysisthat led to equation (13) for the cross-over in the dominance from fractal topercolation scaling yields

�x � � t ��1 � ��t

13 � D

� t. (18)

Now, if the same values for �t � 0.04, t � 1.88, D � 2.81, and� � 0.4 are substituted into equation (18) as into equation(13), one finds that the cross-over moisture content is 0.37for the electrical conductivity rather than 0.12 for the hy-draulic conductivity. Instead of fractal scaling (for the hy-draulic conductivity), in the case of the electrical conduc-tivity it is percolation scaling, which dominates over nearlythe entire range of water contents. Thus the saturationdependence of the electrical conductivity, in contrast to thatof the hydraulic conductivity, may to a good approximationbe written in the form of percolation scaling all the way tofull saturation,

� � �� t�t. (19)

As it is shown in section (7) that for media with insignificantclay content �t � �, with a numerical constant indepen-dent of porosity, equation (19) may be rewritten as

� � �1 � � t� t (20)

in the form of equation (20), Archie’s law. In order to de-velop Archie’s law it is not necessary for to be small,though comparisons with experiment suggest that its valueis between 0.1 and 0.16 (accounting for �t � 0.04 if � � 0.4above). Reference 52 performed 2D simulations and con-cluded that m � 1.28 0.07, in very good agreement witht � 1.27 from percolation theory. The results of [51] and [53],m � 1.86 0.19 for 3D systems are also in agreement withthe percolation theoretical prediction t � 1.88. Note that theresult for rc � rm(1 � �t)

1/(3-D) at full saturation is notconsistent with representation as a power law in �, so thatwhenever fractal scaling of a transport property is appropri-ate for a wide range of moisture contents, as it is for K, it isnot possible to represent the value of that property at sat-uration as a power law in �. This result is in accord with theconclusions of [51, 53] and the much stronger dependenceof the critical hydraulic conductance than the critical elec-trical conductance on the pore radius is responsible for amajor difference in the properties of the electrical and hy-draulic conductivities.

Note that it may be possible that at very low porosities,such as found in rocks with low connec-tivity, the hydraulic conductivity mayalso be written as a power law of theporosity in the same form as the elec-trical conductivity. The above analysisuses characteristic parameters fromsoils and may not be appropriate for allporous media. This question is as yetunresolved.

5. PRESSURE-SATURATION CURVESThe fractal scaling of water-retention characteristics is given

briefly because this discussion yields results, equation (23),

identical to those already given in RS. Afterwards the dis-

cussion is extended to include hysteresis.

The water retention characteristics of a medium specify

its water content as a function of the tension at the air-water

interface for experimental procedures, which produce

drainage.

I will use the Young-Laplace relationship, h � A/r, be-

tween the pore radius and the tension, h, chiefly as a math-

ematical transformation. Here A is a number, which de-

pends also on pore geometry in relating curvature to radius.

A is required only to calculate a characteristic pressure, or

scale factor, because in self-similar, fractal, media A can be

considered to be independent of pore size. Equation (6)

may then be rewritten as

S �1�

3 � Dp

rm3�Dp �

r0

A/h

r2�Dpdr. (21)

The major hysteresis associatedwith wetting and drying is

directly related to percolationtheory through the concept of

accessibility (the fraction of sitesor volume connected to the

infinite cluster).


One can then use A/hA, hA an “air entry pressure,” to rep-resent the maximum pore size, and write,

1 �1�

3 � Dp

rm3�Dp �

r0

A/hA

r2�Dpdr. (22)

Equations (21) and (22) may be combined to yield thefollowing scaling expression for the equilibrium water-re-tention curve,

S � 1 � � 1�� 1 � �hA

h � 3�D� . (23)

Although hA represents the tension at which air would enterthe largest pore, if air could reach the largest pore, it doesnot conceptually represent the tension at which air shouldbegin to enter the medium [13]. If effects of edges or of soilstructure are ignored, then air cannot enter the system untila higher tension is reached, known as the bubbling pres-sure, for which the air allowable pore space percolates. Thispressure may be found by setting the water content fromequation (23) equal to the porosity less the critical volumefraction for percolation of air, which yields h � hA. Becausethese pressures may be quite low, especially in coarse mediawith large pores, the finite height of a typical experimentalcolumn under the effects of gravity can introduce non-negligible effects due to the vertical variability of the pres-sure.

How else do the concepts of percolation theory have animpact on the water retention characteristics of porousmedia? As water is extracted from the medium, the hydrau-lic conductivity crosses over to percolation scaling at �1. Atthis moisture content the hydraulic conductivity begins arapid drop, and to the degree that a particular drainageexperiment relies on capillary flow processes to reduce �,the time scales for these processes may increase greatly. Ifexperimental time scales are not adjusted accordingly, theexperimental system may “fall out of equilibrium.” Thistransition to a nonequilibrium condition occurs at a mois-ture content pinned to the percolation transition, makingtypical porous media analogues to the ideal glass transition.Thus one expects, under many experimental circumstances,deviations from the fractal scaling of water retention curvesat both the wet and dry ends; the former is due to the lackof percolation of the air phase, and the latter is due to thelack of percolation of the water phase. But beyond effects onthe water retention curve, percolation theory has funda-mental effects on the difference between the pressure-sat-uration relationships of porous media for wetting and dry-ing. These differences are referred to as hysteresis and arealso of fundamental physical interest.

There are various potential inputs to hysteresis betweenimbibition and drainage. Two such inputs that are consid-ered in Hunt [14] are discussed here. The first is that duringdrainage water must be extracted through a pore neck,whereas during imbibition water must be able to fill anentire pore (see articles by Lenhard and coauthors, e.g., [54,55]). In a self-similar (fractal) medium the ratio of pore neckto pore body radius must be independent of pore size,meaning that this geometric ratio will show up in a ratio ofcharacteristic pressures between imbibition and drainage.Lenhard and coworkers above have treated the problem ofthe different values of the tension for which a given pore canempty and fill with water and have concluded from exper-imental evidence that a typical ratio of these two pressuresis about 2. This means that a ratio of characteristic pressuresfor drainage, hA, and imbibition, call it, hB, in a fractalnetwork should also be roughly 2. Because of the inverserelationship between h and r, and because during drainagewater must be extracted through a throat, but during imbi-bition the water must fill the entire pore body, the experi-mental evidence suggests that the ratio between typicalpore body and pore throat radii is also a factor 2. Thesecond input is accessibility.

Here and subsequently I will make a distinction betweenthe words “allowable” and “accessible.” Allowable will beused to designate pores, which, at a given pressure, couldcontain water (or air). Use of the term accessible means thata continuous path of allowable pore-space to the given porefrom a water (or air) supply exists. In the case of drainagefrom full saturation, water will occupy all allowable porespace. But in imbibition, excluding effects of vapor phasetransport or film flow, [56, 57] water will occupy only thatpore-space, which is both allowable and accessible. Thus, inorder to find the water content of a fractal porous mediumupon imbibition we need only take equation (23), substitutefor hA a different characteristic pressure, hB, and multiply byan accessibility factor from percolation theory, equation(A.1). Equation (A.1), however, must be normalized to 1 atsome moisture content. A convenient and justifiable choiceis to require equation (A.1) to give 1 at � � � [14]. Althoughvirtually all water-allowable pore space is probably accessi-ble already at somewhat lower values of the moisture con-tent than � � �, in the absence of detailed simulations, anyother choice would be arbitrary. The accessibility factorshould be expressed in terms of the tension. The result is

� � � 3 � D

rm3�D � �

r0

A/h

r2�Ddr� �A/h�3�D � �A/hc�3�D

�A/hB�3�D � �A/hc�3�D� �

.

(24)

Here hc is defined as the minimum value of the matricpotential, such that the water-allowable pore space actually


percolates, or hc � A/rc. For a system, in which it waspossible to find the value of �t from experiment, Figure 4[21], it was then possible [24] to compare equation (24) withexperiment using a single adjustable parameter, hB. Theresult is shown in Figure 5 and compared with the resultwhen the factor describing the restriction on accessibility isneglected. The value hB � 5.6 cm turned out to be ca. 0.5hA,

with hA � 11.2 cm, as expected from the cited works ofLenhard [53, 54].

Although the lower value of hB sets the appropriate ver-tical scale on the imbibition curve, the accessibility factorflattens the curve, making it compatible with experiment.

The other complication alluded to, a lack of equilibrationdue to small values of the hydraulic conductivity, is exem-plified in Figures 6 –9. In previous works, the moisture con-tents, �d, at which deviations from fractal scaling predictionset on at the dry end, were compared with �t [equation (13)].

FIGURE 4

Drying curves from [21] for varying fractions of hydrophobic particlecontent. The data shown as the largest diamonds are for zerohydrophobic particle content, i.e., a hydrophilic soil. The criticalvolume fraction occurs at about S � 12%, or, since � � 0.4, at � �0.05.

FIGURE 5

The hydrophilic wetting curve from Bauters [21]. “Theory” is aprediction from equation (3), using a value of the characteristicpressure as 5.6 cm instead of the air entry pressure of 11.2 cm fromthe drying curve. “Hysteresis” is the prediction from equation (24),which accounts for the effects of the accessibility from the infinite,“percolation” network. Note that the arrow demonstrates that thedeviation from prediction at the wet end occurs at an air volumefraction equal to the critical volume fraction for percolation, a sign ofair “entrapment.”

FIGURE 6

The water-retention characteristics of the Injection Test Site 2-1418soil [19, 48]. The solid diamonds are the experimental results. Theopen squares are the theoretical predictions using the fractal scaling[from equation (23)], the known porosity, the fractal dimensionalityfrom the particle-size data, and the air entry pressure as an adjustableparameter. The open circles are the predictions from assuming in-complete equilibration. �t � 0.04 and �1 � 0.06 are not adjustableparameters.

FIGURE 7

The water-retention characteristics of additional Hanford Site soils[19]. See legend to Figure 6 for the meaning of the symbols. Theclassification of this soil by Freeman is ITS 2-2230.


A study of over 40 US Department of Energy Hanford sitesoils was conducted for this purpose [11]; the comparisonmade was statistical. The result was found that �d �x.Later, the following model was developed. Consider that anestimation of an equilibration time based on extrapolationof a K, which seems to be following a known curve, wouldunderestimate the amount of time required for water toflow out of a soil by a factor 100, if K were overestimated bythat value. All subsequent measurements would be out ofequilibrium. The procedure we adopted was simply to allow

only the fraction of water to flow out of a soil that is given by

the ratio of the real value of K for a given moisture content

to the presumed value at that content. After updating the

moisture content, the new moisture content is used to cal-

culate a new K ratio, and the process is repeated. The

procedure to take the system out of equilibrium was initi-

ated at the moisture content nearest �x (because the perco-

lation scaling sets on there). �x was calculated from equa-

tion (13) for known Dp and �. The results were optimized for

the 13 cases investigated if the presumed K was calculated

for two steps behind the real K, so this was equivalent to

using a single adjustable parameter. The results are given in

Figures 6 –9. Such an updating procedure is not accurate in

the high saturation limit, so that end of the curve should be

ignored. The overall comparison with experiment is striking.

Note that the upward curvature of the water retention

curve at low moisture contents has been used as an argu-

ment against the validity of fractal treatments of porous

media. Now it is seen that this curvature is well-predicted

using fractal parameters from physical measurements (den-

sity and particle size, which then yield the fractal dimen-

sionality, and hydraulic conductivity).

6. SOLUTE AND GAS DIFFUSION: THE CRITICALMOISTURE CONTENT FOR PERCOLATIONExperimental results for solute diffusion are often reported

in the form, Dpm/(Dw�), where Dpm is the diffusion constant

of a given solute measured in a porous medium, and Dw is

FIGURE 8

The water-retention characteristics of additional Hanford Site soils[19]. See legend to Figure 6 for the meaning of the symbols. Theclassification of this soil by Freeman is USMW 10-86.

FIGURE 9

The water-retention characteristics of additional Hanford Site soils [19]. See legend to Figure 6 for the meaning of the symbols. The classification of thissoil by Freeman is ITS 2-1418.


the diffusion constant of the same solute measured in water.This ratio has been found to vanish as [20]

Dpm

Dw�� 1.1�� t�. (25)

The value, �t, at which the solute diffusion vanishes, variessystematically with soil texture. The relationship found by[20], is

� t � 0.039�A/V�0.52 (26)

with A/V the surface area-to-volume ratio of the medium.This threshold moisture content �t may be identified withthe critical volume fraction for percolation (see section 7 fora more detailed discussion of �t). Equation (26) gives a verygood way to estimate �t if the surface area-to-volume ratioof a medium is known. To a very good approximation, �t isindependent of moisture; otherwise the experimental re-sults for Dpm/(Dw�) could not be the straight lines observed.Later it was shown [26] that the dependence on surface areaof �t was consistent with absorption of water by clay min-erals. For soils with little clay content the following relation-ship was proposed:

� t � �. (27)

Equation (27) with 0.1 was consistent with experiment[12, 26] for soils with no clay content. Although equation(26) appeared to be accurate for Danish soils (with consid-erable clay content), in the general case probably a sum ofsuch terms is correct [26]. Most of the materials investigatedin [51], however, were sandstones, several of which werewind-deposited, and it is unlikely that such materials shouldhave significant clay content. The justification for equation(27) was simple [26] and this issue is discussed further insection (7).

In diffusion simulations on a cubic lattice with variableporosity (which was used later in [15], to simulate variablemoisture content) [58] considered the “tortuosity factor,” �,defined as the ratio of Dpm/(Dw�). They demonstrated thatin the “scaling region” (near percolation) � scaled with sys-tem length, x, as

� � x�1.11 (28)

as long as x � . For larger system sizes, � was constant. Thisresult implies that

� � �1.11. (29)

Thus for smaller system sizes the fractal structure deter-mined the diffusion properties, whereas for larger system

sizes, the diffusion proceeded in Euclidean space (Fick’slaw) beyond a length scale, defined by the correlationlength, but with the limiting value defined by the smallestvalue reached before that cross-over. At the percolationthreshold the size effects continue to infinitely large systemsand the diffusion constant vanishes.

Take equation (29) and make the usual substitution ofp3�, substitute in the expression for �, � for �, and thefollowing result is obtained:

Dpm

Dw�� t�

�1.11��0.88� �� t�. (30)

Here the approximate sign indicates that the power(1.11)(0.88) is so close to one that the difference is insignif-icant and that the numerical constant was not obtained.

Some comment is required for the choice of substituting �

for �. One could argue that the appropriate substitution wouldbe �(� � �t)

0.4, reflecting the fact that some water-filled poresare not directly connected to the infinite cluster of water-filledpores. Reference 15 argued that the accessibility factor f [equa-tion (A.1)] was not appropriate. The reason is that for � � �t

diffusion along thin films (presumably slower than throughfilled pores) can link up the disconnected clusters with theinfinite cluster and, according to a result from [14], the totaldistance of diffusion over thin films is microscopic (insignifi-cant number of pore lengths) above the percolation threshold.One could consider the possibility that diffusion along thinwater films could also for � � �t provide pathways for diffu-sion, invalidating equation (30). But, for � � �t the distanceover thin films that diffusing particles would have to explore,very rapidly becomes macroscopic rather than microscopic, asshown in [14]. The macroscopic length scales, over which suchslower diffusion must occur, makes solute diffusion negligiblysmall for � � �t.

Reference 15 also applied equation (29) to gas diffusion.The argument they presented must be slightly modified inorder to be correct.

If, as is typical, both gas diffusion and solute diffusion aremeasured along the primary drainage curve, there are issuesrelated to hysteresis to consider. In particular, with drying,a fractional volume of water �t will be left in the medium,but this water will not be useful for solute diffusion, becauseof its lack of connectivity. During drainage, air can enter themedium only along the edges until the matric potentialreaches a value that is low enough (corresponding to the“bubbling pressure”) that the air-allowable pore-space per-colates. Thus the very presence of air in the medium (exceptfor pores near the edges) implies the percolation of the airphase. As a consequence, excluding edge effects, gas diffu-sion during water drainage does not vanish at a finite vol-ume of air, though in imbibition it presumably would (theremaining, disconnected air volume termed “entrapped


air”). Thus to a first approximation, gas diffusion along adrainage curve vanishes only at � � 0.

Gas effectively diffuses only through air-filled pores,since gas diffusion through water is inherently much slowerthan through air, in addition to requiring solution and ex-solution, processes which are slow for typical test gases.Thus air-filled pores, which are not connected to the infinitecluster (of air-filled pores), do not contribute to the relevantporosity, and the relevant air-accessible porosity must bereduced from � by the factor (�/�)0.4, as long as there are anyair-filled pores, which are not connected to the infinitecluster. During drying, however, by assumption all air-filledpores are accessible either to the exterior of the system (orclusters of sites accessible to the exterior) or to the infinitecluster. Thus, in order to use the factor (�/�)0.4, it musteither be assumed that the measurements were made dur-ing wetting or that the pores accessible to the edges of thesystem during drying are the finite-sized clusters of pores.In either case, �t may not be assumed to be identically zeroif the factor (�/�)0.4 is used. If, however, �t is a small fractionof the porosity, then it may still, to a first approximation, beneglected. The evidence from the related property of airpermeability [22] is that �t may be as small as 0.02.

In [15] it was guaranteed that the fraction of air-filled sitesthat is interconnected is unity when � � �. Using equation (29)as the starting point for gas diffusion [15] found

Dpm

Da��

��0.4 � �1;

Dpm

Da� �1.4

�

�0.4 . (30)

The experimental result [20] had a power of 2.5 on the factor� and a power of 1 on the factor �, very nearly the resultderived. Note, however, that the critical volume fraction forair percolation must be very small for this result to beaccurate; the experimental value of �t � 0.02 for the airpermeability [22] is small enough to make this approxima-tion reasonable.

In summary, experimental result equation (25) confirmsthe relevance of percolation theory to solute diffusion. Thethreshold moisture content, equation (23), for solute diffu-sion is identified with the critical volume fraction for per-colation. As will be seen, experimental result, equation (26),confirms the validity of the fractal structure. Note that thereare essentially two differences between gas and solute dif-fusion. In solute diffusion, the critical volume for waterpercolation is fairly large, and diffusion along thin films canalleviate some of the connectivity problems above the per-colation threshold. In gas diffusion the critical volume frac-tion for air percolation must be quite small, and connectiv-ity problems above the percolation threshold are alreadyimportant. The fact that, on the average, �t � �t is signifi-cant. The cause is likely due to the effects of clay minerals,

which absorb water, but not air. But at least for coarser mediawith little or no clay content, we should expect �t �t.

7. THE CRITICAL VOLUME FRACTION FOR PERCOLATIONThe critical volume fraction for percolation can be calcu-lated for very simple models. Consider the model of a po-rous medium described by a network of intersecting iden-tical tubes. In this model the porosity is given by the ratio ofthe tube volume to the total volume occupied by the net-work. If the network is square, then pc � 0.5. In such a case,however, the critical volume fraction for percolation is 0.5�.In fact any such network, whether triangular, hexagonal,cubic (or other) must satisfy �t � pc�. This is an expressionof the impact of correlations in the pore space on the criticalmoisture content for percolation. If the tubes are all madesmaller, but the structure is the same (from, e.g., compac-tion), then the critical volume fraction for percolation mustalways be some consistent fraction of the pore volumefraction. This is not meant to be a proof that such a pro-portionality is always correct in natural media, but it is anargument for investigating the possibility.

On the other hand it is also well known that clay mineralsabsorb a great deal of water onto both internal and externalsurfaces. So, regardless of whether there is sufficient waterto generate a percolating network of filled pores, if a me-dium has significant clay content it will, under most condi-tions of humidity, contain a considerable amount of water.

In a series of analyses [12, 26] the moisture contents, �d, atwhich deviations from fractal scaling of Hanford site waterretention curves set on were compared first with (A/V)0.52 � �t,where A/V was calculated from equation (4) and �t was fromthe solute diffusion experiments of [20], and then was com-pared with theoretical estimates of �. For the reasons given insection 6 we would expect that �d would be slightly above �t, inthe vicinity of �1. In fact the difference between �d and �t foundin this regression was 0.06, very close to �1 � �t. R2 � 0.8 wasfound for this regression. So the deviation from fractal scalingwas clearly related to percolation theory (this same regressionwas then later used to find �t for the McGee Ranch and NorthCaisson soils). The theoretical comparisons involved �d andthe sum of � and a theoretical estimation of the water con-tent bound on clay minerals. Here values of R2 exceeded 0.9, ifthe value of chosen was small, i.e., 0.1 � � 0.16. Theestimation of the water bound on clay minerals again usedequation (4) and was found to have a functional form of(A/V)3-Dp, making it possible to identify tentatively the Mold-rup threshold moisture content with water bound on clayminerals, as long as Dp 2.5. This latter water content wasassumed not to participate in capillary flow, but to add to thetotal water content in the medium. Thus equation (26) pro-vides evidence that the fractal description of natural porousmedia is appropriate.

Note that the Moldrup experiments on solute and gas dif-fusion imply that the critical moisture content for percolation


is considerably larger than the critical air content. But in Ref.

13 it was found that the wet-end deviation of Hanford pres-

sure saturation curves from fractal scaling set on an air con-

tent, which was nearly �t though slightly smaller. This puzzle

is likely the result of the difference in clay contents between

the two suites of soils; the arid soils of the Hanford site do not

contain nearly as much clay as the Danish soils in the Mold-

rup et al. [20, 22] studies. But the Hanford site soils contain

enough examples with clay that some overlap between the

Hanford and the Danish soils exists.

It is also possible to estimate directly the effects of

incomplete equilibration on the observed water retention

characteristics. It can be hypothesized that the change in K

below �1 from the cross-over to percolation scaling would

not be anticipated by an experimenter. Then in a given

step of an experiment, only as much water would drain as

given by the product of the ratio of equation (12) to equa-

tion (11) and the equilibrium drainage. An algorithmic

procedure can then be developed, which updates the wa-

ter content according to the incomplete equilibration,

then recalculates K by both equation (12) and equation

(11) according to the actual water content and repeats the

procedure. An example of this kind of comparison is

shown for a particular Hanford site soil in Figure 7.

The conclusion of these articles and of this section is thatin order for there to be sufficient water in a medium to allowpercolation of capillary flow paths, the water content musthave a contribution proportional to the porosity, and, ifthere is significant clay content in the medium, also a termdescribing the adsorption of water onto clay surfaces. Butair percolation can be guaranteed with only the first term.

8. CONCLUSIONSOnly two inputs taken together explain the fundamentalphysics of transport in porous media. One is that mostnatural media are continuous, truncated, probabilistic frac-tals, the second is that transport is best treated within theframework of continuum percolation theory. Far from thepercolation threshold that framework produces critical pathanalysis, which traces the value of an upscaled transportcoefficient to the dependence of a rate-limiting local trans-port coefficient on, e.g., saturation. The transport is domi-nated by the properties of the medium. Near the threshold,the transport coefficient behaves according to the scalingproperties of percolation theory and is universal. The dif-ference between different transport properties is that thecross-over between the different regimes occurs at differentsaturations: For the hydraulic conductivity, this cross-overis near the percolation threshold, for the electrical conduc-

TABLE 1

Properties of Porous Media Obtained from Percolation Theory

Property 2D 3D

Critical volume fraction (minimal clay content) �3D case Vc 0.1�

Critical volume fraction (large clay content) Vc � 0.039�AV�

0.52

Unsaturated hydraulic conductivity (fractal scaling) K � KS�1 � � � �� t�

1 � �t�3/�3�Dp�

Unsaturated hydraulic conductivity (percolation scaling) K � K0(� � �t)1.27 K � K0(� � �t)

1.88

Cross-over moisture content (hydraulic conductivity)�x � �t �

�1 � ��1.883

3 � Dp� 1.88

Air permeability ka � ka�� t

� � �t�1.27

ka � ka�� t

� � �t�1.88

Electrical conductivity (fractal scaling) � � �0�1 � � � �� t�

1 � �t�1/�3�Dp�

Electrical conductivity (percolation scaling) � � �0(� � �t)1.27 � � �0(� � �t)

1.88

Cross-over moisture content (electrical conductivity)�x � �t �

�1 � ��1.881

3 � Dp� 1.88

Solute diffusionDpm

Dw� �� t�

Gas diffusionDpm

Dg

�2.4

�


tivity and solute (gas) diffusion, it is near saturation (zeromoisture content), and for the air permeability, no suchcross-over takes place. Consequently, in some transportphenomena, percolation scaling can be directly observed,whereas in others it can only be inferred. The major hyster-esis associated with wetting and drying is directly related topercolation theory through the concept of accessibility (thefraction of sites or volume connected to the infinite cluster).Limitations of the fractal description of pressure-saturationcurves are required by basic percolation theoretical consid-erations regarding air phase continuity near saturation andwater-phase continuity at the dry end.

A summary of the derived properties is given in Table 1.

APPENDIX: USEFUL RESULTS FROM PERCOLATION THEORYThe analogy p3�, pc3�t is used repeatedly in the followingsince the critical behavior of the topology of percolationtheory is universal [44] and depends only on the dimensionof the system. Although for p � pc, (� � �t) it is known thatan infinitely large cluster of interconnected bonds (water-filled pore space) will appear, not all bonds (allowably wet-ted pore space) are connected to the infinite cluster ofbonds (of connected water-allowable pore space). In fact,many finite clusters of bonds (allowably wetted pore space)exist as well. The fraction, f, of all the bonds (allowable porespace) connected to the infinite cluster is [44]

f � �p � pc�� t�

�. (A.1)

Here � is a “critical exponent” from percolation theory withvalue 0.4 in three dimensions. Though proportionality (A.1)is accurate only in the vicinity of �t [44], in order to use it formaking testable predictions, it must be normalized. Clearlyf � 1 at � � �, and this was the choice for normalization [14].The factor f will be relevant for the hysteresis betweenwetting and drying and also for air diffusion on the primarydrying curve (or solute diffusion on the wetting curve).

Another quantity that will be important is the size, , of the

largest finite clusters for p � pc. This size is also the size of the

largest holes (bondless regions) as well as the characteristic

separation between connected portions of the infinite cluster.

This important length scale is called the correlation length,

with � 0.88 (1.27) in 3D (2D) another critical exponent

from percolation theory [9, 44, 37]. It is important that the

right hand side of proportionality (A.2) is written as an

absolute value, consistent with the statement that gives

the size of the largest clusters of bonds for p � pc and that

of the largest cluster of holes (or bonds not connected with

the infinite cluster) for p � pc. An additional aspect that will

be important when discussing the diffusion results is that

represents an upper bound on the length scale, over which

the medium “behaves” like a fractal [40]. At larger scales, a

medium behaves as though it possessed a Euclidean geom-

etry. Right at pc there is no length scale, at which the system

obeys Euclidean geometry.

A second important length scale is the actual distance

along the connected path for the largest clusters of con-

nected bonds, �, which behaves like [44]:

� � �p � pc��1 � �� t�

�1. (A.3)

Because in three dimensions the ratio �/ diverges like (p �

pc)�0.12 as p3pc, the path along what is called the backbone

cluster in percolation theory is tortuous—a product of the

fractal structure of the cluster [44]. The exponent, 0.12,

above describes the tortuosity of this path. This means that

it is possible for flow and dispersing solutes in a nonfractal

medium to follow fractal paths. Equations (A.2) and (A.3)

will be relevant for K(S), and equation (A.1) for hysteresis.

Numerical prefactors of equations (A.2) and (A.3) are nei-

ther known nor required.

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