Published in Vadose Zone Journal 4:134-138 (2005)
© 2005 Soil Science Society of America
677 S. Segoe Rd., Madison, WI 53711 USA
ORIGINAL RESEARCH
Continuum Percolation Theory for Saturation Dependence of Air Permeability
A. G. Hunt*
Department of Physics, Wright State University, 3640 Colonel Glenn Highway, Dayton, OH 45435
* Corresponding author (allen.hunt{at}wright.edu)
Received 18 June 2004.
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ABSTRACT
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Continuum percolation theory has recently been used to find the saturation, S, dependence of the hydraulic conductivity, K(S), of probabilistic fractal porous media. Analysis of K(S) in conjunction with solute diffusion revealed the presence of a critical volume fraction, 
t, for percolation in natural porous media. For moisture contents within a few percent of t, K(S) depends on the moisture content as a power of – t. At higher moisture contents, K(S) is determined through critical path analysis, which uses continuum percolation theory to find the dependence of a bottleneck (flow-limiting) pore radius on S. The physics near t is thus dominated by connectivity and tortuosity issues, but far from t by the variations in the radius of a bottleneck pore. Here it is demonstrated that the bottleneck pore radius for air permeability, ka, does not change as a function of saturation. Using the same scaling for the air permeability in the vicinity of the percolation of the air phase as proposed for the hydraulic conductivity in the vicinity of the percolation of the water phase yields results for ka in accordance with experimental data.
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INTRODUCTION
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ON ACCOUNT OF THEIR relevance to climate and subsurface modeling, it is important to be able to predict the hydraulic properties of unsaturated porous media. To calculate the air permeability, hydraulic conductivity, and solute and gas diffusion constants of unsaturated porous media, one must know the geometry and topology of the pore space. Often the only information available is soil texture. While knowledge of soil texture is clearly inadequate to calculate this suite of properties, it is possible to make accurate predictions of scaling formulations of, for example, the ratio of the unsaturated/saturated hydraulic conductivity without complete information regarding the pore space. The best framework chosen for this approach appears to be to apply continuum percolation theory (Hunt, 2001; Hunt and Gee, 2002ab; Hunt and Ewing, 2003; Hunt, 2004a, 2004b, 2004c) to random fractal models (Rieu and Sposito, 1991) of the pore space. While it turns out that the fractal model of the pore space has important implications for the unsaturated hydraulic conductivity, the other properties appear to be understood completely using only the framework of percolation theory (Hunt and Gee, 2002a; Hunt and Ewing, 2003; Hunt, 2004c). Even far from the percolation threshold, percolation theory has already been demonstrated on several occasions to provide the most accurate theoretical treatment of transport properties in disordered solids (Seager and Pike, 1974; Kirkpatrick, 1973) as well as in flow in porous media (Katz and Thompson, 1986; Shah and Yortsos, 1996; Bernabe and Bruderer, 1998). In fact, percolation theory was originally developed to treat problems of flow in porous media (Broadbent and Hammersley, 1957). In previous publications I have treated the unsaturated hydraulic conductivity (Hunt, 2001, 2004a; Hunt and Gee, 2002a) as well as solute and gas diffusion (Hunt and Ewing, 2003). Other studies have showed the relevance of the percolation transition to pressure–saturation curves (Hunt and Gee, 2002b, 2003) including hysteresis (Hunt 2004c). The present work addresses air permeability. Understanding air permeability is greatly facilitated by discussing it in conjunction with the hydraulic conductivity.
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THEORETICAL BACKGROUND: PERCOLATION THEORY
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Experimental results for the permeability are consistent with universal expressions for conduction developed from percolation theory, as long as the critical volume fraction for percolation, which is system dependent, is small. Thus, the discussion of percolation theory will provide the necessary background to describe scaling of transport coefficients, while the discussion of critical volume fractions is postponed to include the experimental context.
Percolation theory describes the connectivity of elements of some physical extent that are placed in a space with some random component (Stauffer and Aharony, 1994). It will not be necessary to define what is meant by randomness in the present context; it will be assumed that natural porous media are sufficiently randomly arranged. The derived results are then compared with experimental data. An extremely simple choice to introduce percolation quantities is a two-dimensional square grid, which has a fraction p of nearest neighbor gridpoints connected at random by "bonds." If the grid is infinitely large, for p < 0.5 
pc no cluster of sites interconnected by bonds will reach infinite size. For p 
pc, the largest cluster of sites with interconnected bonds will reach infinite size. This statement serves as a definition of pc; however, since the value of pc is system dependent, the definition does not necessarily provide a general value.
The correlation length, 
, is the single most important quantity defined in percolation theory. The physical interpretation of is that it represents the distance across the largest connected cluster for p < pc, and across the largest interconnected void for p > pc. It is known (Stauffer, 1979) that must diverge according to a power law at p = pc; the value of the power is known in both two-dimensional and three-dimensional systems. In particular,
 | [1] |
with values v = 1.35 (two-dimensional) and v = 0.88 (three-dimensional) (Stauffer, 1979). Here 0 is a constant, which is proportional to the grid spacing. Equation [1] is only precise for p near pc, a condition which can be expressed approximately as |p – pc|/pc << 1. Because defines the largest voids for p > pc, it also describes the largest expected separations of interconnected paths for p > pc. This particular feature is important for the scaling of hydraulic properties.
While the largest clusters of interconnected sites have the physical dimensions of the correlation length for p > pc, a path along the connected bonds from one side of the cluster to the other (in three dimensions) must have a length at least equal to 
|p – pc|–1 (Stauffer, 1979). Near p = pc the ratio / thus diverges as |p – pc|–0.12, and the distance along an interconnected path from one side of a cluster to the other is infinitely longer than the straight line distance from one side to the other. The property that produces the extra distance is referred to as "tortuosity."
Percolation theory has been applied to problems where the relevant variable is the occupied site fraction (e.g., diseased trees in an orchard), a bond fraction (e.g., random resistor networks), or volume fraction (e.g., volumetric moisture in a porous medium). The third variant of percolation theory is known as "continuum" percolation theory. Thus, in continuum percolation theory, the largest region of interconnected volume of a given type (such as pore space) just reaches infinite size at the critical volume fraction for the pore space.
Solute diffusion in porous media has been shown (Moldrup et al., 2001) to vanish linearly in the quantity, – t. This result has been shown to be expected from an application of percolation theory (Hunt and Ewing, 2003) to simulations (Ewing and Horton, 2003). The same result from simulations (Ewing and Horton, 2003) could be adapted (Hunt and Ewing, 2003) to explain gas diffusion. Moldrup et al. (2001) provided an empirical relationship for t that was highly significant (with an R2 = 0.99).
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CALCULATION OF AIR PERMEABILITY
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While the actual calculation of the scaling of the air permeability presented here is extremely simple, some additional material is presented for completeness and understanding. This additional material is a summary of what has been published before, however, so it is made as brief as possible.
The air permeability, ka, at zero moisture content should equal the permeability at saturation, and the corresponding ratio of the air conductivity to the hydraulic conductivity should be the ratio of the inverse of their viscosities.
Consider the water retention properties of a porous medium. As a medium dries, water exits first from the largest pores. In a truncated continuous random fractal medium, where the pore volume probability density function is given by (Hunt and Gee, 2002a), W
dr = 
dr, r0 
r rm, this condition can be written as
 | [2] |
This expression equals 1 when the radius of the largest pore filled with water, r>, is equal to the largest pore in the system, rm.
Critical path analysis (Ambegaokar et al., 1971; Pollak, 1972) uses the critical value, pc, of p to find the "bottleneck" (or critical) pore radius for flow (Hunt, 2001; Hunt and Gee, 2002a). This critical resistance defines the most resistive element to flow on the most conductive path. Since the critical radius is defined in terms of pc, the separation (equal to the correlation length) of such paths is infinite. Later modifications of critical path analysis optimized the conductivity by choosing more resistive, but more frequently occurring paths. Such an optimization is required if one is seeking the most accurate calculation of, for example, the saturated value of the hydraulic conductivity (such as the treatment of Katz and Thompson, 1986). However, the purpose of the present investigation is to find the scaling of the air permeability at arbitrary saturation to its value at zero moisture content. What is sought is any change in the bottleneck radius with increasing moisture content due to a change in the saturation. The topological aspects of percolation theory can, in principle, be treated separately, as they obviously become important when the critical air fraction for percolation is approached. Since it will turn out that the bottleneck radius for airflow does not change with saturation, this sort of isolation of geometrical and topological effects works perfectly for the air permeability; the only effects present are the topological ones associated with the connectivity and the tortuosity.
Because of the difficulty treating fractal media with a wide range of pore sizes using any network form (Fatt, 1956) of percolation theory, it was found more suitable to use continuum percolation theory (Hunt, 2001). To make a concrete prediction, the critical volume fraction for percolation, t, must be known. This value can be compared with the residual moisture content, r (van Genuchten, 1980), but, in contrast to r, it is unambiguously defined through a series of experiments on solute diffusion (Moldrup et al., 2001).
Experimental information also shows t to be independent of saturation, at least to a very good approximation. This inference depended on the result that the plots (Moldrup et al., 2001) of the ratio, Dpm/Dw, of solute diffusion constants in the porous medium, Dpm, to their values in water, Dw, were straight lines with intercept t. If t were not a constant, the experimental plots would develop a systematic curvature. In these experiments it was found that the solute diffusion constant vanished at a threshold moisture content, t, which depended on the specific surface area, A/V, as follows:
 | [3] |
While Hunt (2004b) showed that such a dependence on the specific surface area could be generated from bound water (on the surface of clay minerals), which did not participate in flow, it was argued that there should also be a contribution proportional to the porosity. The contribution proportional to the porosity was deduced by considering a simplified model. For a regular network of tubes of identical radius and length, it is easy to see that the critical volume fraction is equal to the product of the critical bond fraction and the porosity; the critical bond fraction for a simple cubic network is, for example, 0.249. The value of the dry-end moisture content where deviations from fractal scaling commence was compared with the porosity (Hunt, 2004b), and it was found that in the absence of clay minerals (for coarse porous media)
 | [4] |
where 
is the porosity and 
is an unknown proportionality constant. appeared to be constrained to lie between 1/10 and 1/6 (Hunt, 2004b). While this fraction was rather small compared with typical values of the critical bond probability (even in three dimensions), spatial correlations in the pore space can have the effect of reducing the critical volume fraction for percolation.
Critical path analysis applies percolation theory in the following way. Since any combination of pores, whose total volume is equal to t = , connects to form an infinitely long path of interconnected pore volume, one can at any saturation construct a path that spans the entire system, but is composed only of an arbitrary subset of the water-filled pores. Choosing this subset to include the largest pores in the system defines a collection of connected flow paths, which have the largest possible value of the flow-limiting pore size. This pore size is called the "critical," or "bottleneck" pore. Using the above expression for r>, it is possible to find the radius of the bottleneck pore at any saturation from the following equation (Hunt and Gee, 2002b):
 | [5] |
If = , the bottleneck pore radius is rc = rm(1 – t)1/(3–D). Note that using the assumption (and experimental verification) that t is independent of S, rc(S) must diminish when r> does, that is, with diminishing saturation. Using Eq. [4] it was possible (Hunt and Gee, 2002a) to express K(S)/KS as
 | [6] |
It can be shown (Hunt, 2004a) that this expression for K(S) does not vanish at = t, as K(S) must (Golden et al., 1998). The reason why K must vanish is connected with the path density and tortuosity of the capillary flow paths, which are defined through concepts from percolation theory (Hunt, 2004a). But Eq. [6] yields a K value at = t appropriate for a value of the bottleneck radius corresponding to the smallest pore, r0, in the system. At a value of = 1, with
 | [7] |
the correct formulation for K(S) crosses over to (Hunt, 2004a)
 | [8] |
with t = 1.76 + 0.12 in three dimensions. The first contribution to t comes from the fact that water can only flow along certain paths and that the separations of these paths scale with the correlation length. Thus, the water flux (flow per unit area per unit time) passing through a plane, with unit normal parallel to the mean pressure gradient, must scale inversely with the square of the correlation length from percolation theory. The second contribution to t comes from the tortuosity of the path; its total resistance is increased by a factor / over the resistance of a path of length equal to the linear dimension of the largest cluster of water-filled pores. Thus, the two contributions to t can be identified with "connectivity" and "tortuosity," respectively. This representation of the value of t is inconsistent in two dimensions (Stauffer, 1979), since / < 1, whereas direct representation as K –1 would lead to t = 1.35. Nevertheless Derrida and Vannimenus (1982) determined t = 1.27 (i.e., somewhat smaller).
Let us use 
for the air-filled porosity and t for its critical value for percolation. The assumption here will be that for coarse soils, in which bound water films on the solid phase are expected to be of minimal importance, the critical volume fractions for percolation of air and water will be identical. The largest air-filled pore always has radius rm, regardless of saturation (except for S = 1). Application of critical path analysis to the air permeability then yields rc = rm(1 – t)1/(3–D) for all > t = . Thus, there is no effect from changing saturation on the bottleneck radius for air flow. Although a large fraction at least of natural media are truncated fractals, this argument does not depend on whether the medium is fractal. However, as 
t, the path density and the tortuosity of the paths, along which air can flow, must both change rapidly according to the framework of percolation theory. The expression for the air permeability must follow Eq. [8] exactly for K():
 | [9] |
where k0 is a prefactor, and t = 1.88 (three-dimensional) and t = 1.27 (two-dimensional). t has no other dependence than on dimensionality.
Equation [9] is the central result of this paper. Making value for t equal to t is considered to be accurate only for coarse porous media (without significant clay content and accompanying bound surface water) (Hunt, 2004b). Probably the best estimate of is 0.10, but it may also be as high as 0.16 (Hunt, 2004b). In this expression k0 cannot be determined as accurately as the corresponding prefactor for the case of the hydraulic conductivity. The reason is that this result for the air permeability, like Eq. [8] for K(S), is precise only for a small range of air contents near percolation. Equation [8] for K(S) in contrast could be connected with Eq. [6] by requiring continuity of both K and dK/d at 1. Here there is no expression for ka() over a similar range of moisture contents. Just because critical path analysis does not discover a dependence, that does not mean that it is clear that ka() is constant everywhere except for ( – t)/t << 1. Nevertheless, experimental data bear out the general results of critical path analysis and percolation scaling (Eq. [9]) presented here. While it is somewhat risky, one can rewrite Eq. [9] in the following approximate form so as to normalize it to the S = 0 value of ka,
 | [10] |
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COMPARISON WITH EXPERIMENT
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Comparison is made with experimental data from Moldrup et al. (2003) and Steriotis et al. (1997). Only a brief summary of the results of Moldrup et al. (2003) is given here. In particular their experiments showed that the air permeability vanishes as a power, u, of the air-filled porosity, or as a power of the difference between the air-filled porosity and a small, but finite fraction. This fraction was reported to be roughly 0.02, which is slightly smaller than a typical value of /10 for their experimental systems (undisturbed ash soils), but not greatly different. An average of their u values turned out to be 1.89 ± 0.54. These authors appear to have found a dependence of u on what would correspond here to D, that is, 1 + 0.05/(3 – D) with typical values of 1/(3 – D) approximately 17.8 (i.e., u = 1 + 17.8/20 = 1.89). Although they found a significant value of R2 for this correlation, the dependence (a factor of 1/20) was extremely weak, and it is possible that providing a theoretical basis for a value of u independent of D is sufficient motivation to reject such a correlation.
An important complication is that Moldrup et al. (2003) found that k0 could best be estimated as the value of the air permeability at 100 cm tension, roughly an interfacial tension at field capacity. Then it would be necessary to rewrite Eq. [9] as follows:
 | [11] |
From percolation theory, however, there is no reason to assign any significance to any particular value of the interfacial tension. The important scaling variable is the moisture content, meaning that a better choice for the prefactor would probably be the air permeability for equal to some fixed fraction of the porosity, .
Steriotis et al. (1997) performed experiments of the permeability of He gas in partially saturated porous media. They constructed porous membranes with typical pore sizes tens of nanometers (maximum size 70 nm) and a thickness of roughly 35 µm. The samples were annular cylinders of 6-mm outer diameter and length on the order of 10 mm. Thus, while there were on the order of 1000 pores in the radial direction, the length of the medium was typically on the order of 300 times its thickness. The point of the discussion is that as the percolation threshold is approached, the correlation length from percolation theory tends to diverge, and can for a fairly significant range of He-filled porosity exceed the radial direction without exceeding the vertical direction. Under such conditions, gas flow is actually two-dimensional. Figure 1
compares the experimental results of Steriotis et al. (1997) with the prediction from Eq. [10] with t = 1.27. Note that only one adjustable parameter corresponding to the critical air fraction for percolation was used for this comparison (i.e., t = 0.14).
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Fig. 1. Comparison of experiment with theory for the relative gas-phase permeability of He as a function of the fraction of He-filled pore space, Vs/Vt. The open circles are single-phase and the solid circles are multicomponent samples. The diamonds represent the prediction from Eq. [10].
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Theory is seen to describe the experimental results quite accurately. The present analysis is appropriate even though no evidence is presented that the medium constructed by Steriotis et al. (1997) is fractal.
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CONCLUSIONS
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Continuum percolation theory, which gives all other hydraulic properties in agreement with experimental data, also accurately describes experimental results for air permeability. These results in three-dimensional systems consist of (i) a minimal dependence of ka on saturation S with increasing S until a fairly high saturation is reached and (ii) then a rapid decline according to the power 1.88 of either the air-filled porosity, or, as seems more likely from theory, the difference between the air-filled porosity and a critical value of the air-filled porosity. That critical value is a small fraction of the total porosity, at least for coarse (clay-free) porous media. In two dimensions theory predicts the same form of the air permeability, but with a smaller exponent (1.27 instead of 1.88). This prediction is apparently verified. Note that experiments are also in general accord with the tendency of critical percolation probabilities to be smaller in three dimensions (0.02 observed by Moldrup et al., 2003) than in two dimensions (0.15 observed by Steriotis et al., 1997).
Although the properties of the medium are very important for the dependence of the hydraulic conductivity on saturation, they are almost irrelevant for the dependences of solute and gas diffusion on saturation, and as can now be seen, also for the air permeability. The expressions for air permeability, solute diffusion, and gas diffusion all vanish as powers of either – t, or – t.
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ACKNOWLEDGMENTS
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My sincere thanks to Dr. R.P. Ewing for directing me to the data of Steriotis et al. (1997).
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ABSTRACT
INTRODUCTION
