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Department of Physics, Wright State Univ., 3640 Colonel Glenn Hwy., Dayton, OH 45435
* Corresponding author (allen.hunt{at}wright.edu)
Received 1 March 2004.
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ABSTRACT |
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 TOP ABSTRACT INTRODUCTIONSUMMARY AND CONCLUSIONSREFERENCES |
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Abbreviations: CPA, critical path analysis
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INTRODUCTION |
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TOPABSTRACTINTRODUCTIONSUMMARY AND CONCLUSIONSREFERENCES |
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The van Genuchten parameterization for the unsaturated hydraulic conductivity, K(
), in terms of the saturated hydraulic conductivity, KS, has the following form (van Genuchten et al., 1991),
 | [1] |
is the relative saturation,
 | [2] |
r and s are the minimum and maximum moisture contents reached under reasonable experimental conditions. s is usually less than the porosity, 
, because typical maximum moisture contents achieved in field experiments (van Genuchten et al., 1991) are 5 to 10% less than . On the other hand, it is also difficult to dry soils completely, except under extreme conditions, and some water usually remains. r is thus larger than zero. But r is not a very well-defined parameter in practice. Quoting Luckner et al. (1989) by way of van Genuchten et al. (1991): "The residual water content, r, specifies the maximum amount of water in a soil that will not contribute to liquid flow because of blockage from the flow paths or strong adsorption onto the solid phase." In usual practice, however, no attempt is made to evaluate either of these interpretations and r is treated simply as a fitting parameter.
A fundamental feature of Eq. [1] and [2] and most other common treatments of the unsaturated hydraulic conductivity (e.g., Mualem, 1976) is the separation of the relevant physics into two basic parts. One part is a treatment of the pore network as a regular object, such as a bundle of capillary tubes, and the second is the recognition, a posteriori, that this is an oversimplification, which is then corrected by accounting for the nonregularity of the real medium. Treating the pore network as a regular object allows various constructions (Kozeny, 1927; Carman [1956] and approaches based thereon) that calculate a mean conductivity of a collection of straight tubes, while the correction takes into account that the tubes are not straight and not all of them reach from one side of the system to the other. This physics is generally reflected in the division of Eq. [1] and similar equations into two factors, with one factor representing the averaging over the flow in the tubes and the second factor the "connectivity" or "tortuosity" or a combination of both. Specifically, the connectivity is represented as the 1/2 factor in Eq. [1].
The dependence of the moisture content on the soil water potential in the van Genuchten scheme (van Genuchten et al., 1991) is
 | [3] |
Substituting Eq. [2] into Eq. [1] gives
 | [4] |
Substituting Eq. [3] into Eq. [1] gives
 | [5] |
Commonly (van Genuchten et al., 1991), but not always, the relationship m = 1 – 1/n is applied. While in percolation theory there exists a kind of complementary power law relationship—a power, 3 – D, on the ratio of hA to h and a power, 3/(3 – D), overall (see Eq. [14])—no direct correspondence can be made with percolation theory in the case of Eq. [5] to provide justification for the relationship m = 1 – 1/n.
Critical path analysis (Ambegaokar et al., 1971; Pollak, 1972), abbreviated here as CPA, uses percolation theory (Stauffer and Aharony, 1994) to find the characteristic "bottleneck" pore or resistance in a disordered network. Such analysis can, in principle, be applied to any pore-size distribution. However, considerable evidence (not discussed in detail here) indicates that a large proportion of natural porous media are best modeled as truncated random fractals such as the Rieu and Sposito (1991) model. When applied to such models, CPA yields results similar to the van Genuchten parametrization. The result of using CPA to calculate K()/KS for a power law distribution of pore sizes, consistent with the Rieu and Sposito (1991) truncated fractal model, is (Hunt, 2001, 2004a)
 | [6] |
This result was based on the behavior of a bottleneck pore radius as a function of moisture content, and the fact that for fractal geometries the conductance of that bottleneck pore is proportional to the cube of its radius. Here Dp is the fractal dimensionality of the pore space and t is well defined as the minimum moisture content for the percolation of water-filled pore space. The range of moisture contents within which Eq. [6] is valid, is clearly restricted to > t, though its range of validity actually turns out to be somewhat more restricted (Hunt, 2004a). Due to the coincidence of this water content with the water content at which solute diffusion vanishes (Hunt and Gee, 2002b), there is at least a reliable phenomenological result (Moldrup et al., 2001) for t:
 | [7] |
t can be obtained numerically. The value of K()/KS can be predicted as a function of moisture content if t is known, Dp can be inferred from the particle-size distribution, and is found from the bulk and particle densities. Thus hydraulic properties can be predicted from physical properties.
It is interesting to reconsider the description of the physics behind r. In the analysis that leads to a theoretical estimate of t, contributions from both "water blocked from the flow paths" and "water absorbed to solid surfaces" are considered. The latter factor seems essential to the known dependence of t on the specific surface area.
Focusing exclusively on the bottleneck pore radius (Eq. [6]) ignores other characteristics of the critical paths, which in the limit 
t must be more important than the moisture dependence of this bottleneck radius. These aspects include, as typically assumed, tortuosity and connectivity. These aspects relate directly to solute diffusion, as explained below. Fortunately percolation theory provides a consistent means to estimate these other characteristics as well, at least for moisture contents not far removed from t. The mean distance between critical paths, called the path separation or correlation length, 
, diverges as ( – t)–
, where = 0.88 in three-dimensional systems (Stauffer and Aharony, 1994). When – t > 0, describes the largest clusters of interconnected nonconducting (air-filled) sites, and thus the distance separating water-conducting paths. Recall that in Eq. [6] there was no consideration of the fact that the separation of water-conducting paths must diverge in the limit t. Taking this length divergence into account requires that the hydraulic conductivity vanish in the limit t according to a power, x, of – t; that is,
 | [8] |
describes the path separation, then 2 is the relevant area. Dividing any expression for the net flow per connected path by 2 leads to x = 2 = 2(0.88) = 1.76. As for a tortuosity correction, it can be shown (unpublished data, 2004) that this increases x by 0.12. The reason for this is that the divergence (from percolation theory) of the length, 
, along the backbone cluster at percolation is governed by a critical exponent with value unity, while the size, , of the largest cluster at percolation (the infinite cluster) diverges according to the exponent 0.88 above. Thus the path along the backbone of the infinite cluster at percolation is tortuous in the sense that its length is an infinitely large factor longer than the cluster is itself, and the tortuosity thus behaves as / or, ( – t)0.88–1 = ( – t)–0.12.
The combination of these considerations led to the expression for K at low moisture contents in the vicinity of percolation:
 | [9] |
The value x = 1.88 does not depend on the specific characteristics of the medium, but only on the dimensionality of the space in which the medium is embedded. Power laws with such widely applicable values of the power are often termed universal.
Note that Hunt and Ewing (2003) showed that solute diffusion in unsaturated porous media should vanish at the same moisture content, t, at which Eq. [9] predicts that K should vanish. The result predicted for the ratio of the diffusion constant of a solute in porous media, Dpm, divided by the diffusion constant of the same solute in water, Dw, is
 | [10] |
Note that the left side of Eq. [10] is also typically referred to as the (inverse) of a "tortuosity" factor, but this definition of tortuosity is quite different from that defined in percolation theory, even though both diverge in the limit t. Equation [10] is identical (except for a factor 1.1) to the experimental relationship observed by Moldrup et al. (2001). It should be noted that it would be inconsistent to use Eq. [6] for the hydraulic conductivity and Eq. [10] for solute diffusion; this is the fundamental reason why it was necessary to propose Eq. [9] to describe the behavior of K in the vicinity of t.
It may be tempting to take Eq. [9] and simply multiply it by Eq. [7], thus generating a result similar to Eq. [3] in structure. This would be inaccurate over the majority of the range of moisture contents, however, since the scaling formulation of Eq. [9] for the hydraulic conductivity can only be valid (Stauffer and Aharony, 1994) in the neighborhood of t. It is probably equally valid, though somewhat clumsy in application, simply to multiply Eq. [6] by Eq. [9] when < 1, defined in Eq. [11], though such a procedure would require a revision of the derivation of Eq. [11]. The best, though indirect, experimental estimate of 1 for a suite of 43 Hanford soils (Hunt and Gee, 2002b) was about t + 0.06. This value for 1 was obtained through visual comparison of the (dry end) moisture content at which water-retention curves deviated from fractal scaling with t and then regressing the observed suite of data for 1 on the predicted values of t. Using mean values of Dp and from the same suite of soils (in Eq. [11] below) the same result for 1 was obtained (Hunt, 2004a). The interpretation was that the relatively sudden onset of a rapid diminution of K predicted by a crossover in behavior from Eq. [6] to Eq. [9] led to equilibration problems with the ceramic plate determination of the water-retention characteristics. In these studies the threshold moisture content, t, took on values from a few percent up to >15% and were, at least, well approximated by Eq. [7].
The moisture content at which Eq. [6] should be replaced by proportionality Eq. [9] was found by requiring both K and dK/d to be continuous at that value of . By this procedure the crossover 
1 is found to be
 | [11] |
Equation [11], together with Eq. [6] and [9], allows estimation of K() across the entire range of moisture contents, t < < , which correspond roughly to r < < s.
There is an independent test of the suite of results, Eq. [6], [9], and [11] from percolation theory. It is known that for continuum percolation problems the hydraulic (or electrical) conductivity may obey the kind of scaling relationship shown in Eq. [9] only with a nonuniversal power if the value of t is >2 (Feng et al., 1987; Golden, 1990, 1997). Nonuniversal is used here to describe values of a power, which depend sensitively on the characteristics of a medium, such as the fractal dimensionality, Dp. Note that the proposed universal power in Eq. [9] is x = 1.88, and so x < 2, as required.
Consider the value of the power x = 1.88. Mualem and Dagan (1978) summarized "statistical" models of the unsaturated hydraulic conductivity by Childs and Collis-George (1950), Millington and Quirk (1961), Sharma (1966), Kunze et al. (1968), and Burdine (1953). For those results, which incorporated an important tortuosity correction (excluding Childs and Collis-George [1950]) values of 1 < x < 2 were obtained. Mualem (1976) himself prefers values of x = 1/2, x = 1, or x = 4/3. The statistical advantages cited by Mualem (1976) of x = 1/2 may have contributed to the van Genuchten choice of x = 1/2. Also the argument of Eq. [9] and the argument of the first factor, 1/2, of Eq. [1] are proportional to each other. Thus the typical means to predict effects of tortuosity/connectivity actually predict essentially the correct functional form as well as a power that is not too different from that obtained by more rigorous theory.
In the context of the next discussion it is shown that the same suite of equations (Eq. [6], [9], and [11]) can also yield a nonuniversal power >2. The framework of this derivation is a discussion of the porosity and the water-retention curves as derived from the Rieu and Sposito random fractal model (Rieu and Sposito, 1991).
The porosity from the Rieu and Sposito model has the form
 | [12] |
 | [13] |
Using Eq. [6] and [13] together allows K to be expressed as a function of tension, h,
 | [14] |
Equation [14] reduces to K(h) = KS(hA/h)3 in the limit t 0, expressing the proportionality of K to a bottleneck pore conductance, and the relevance of the power 3 in Poiseuille flow for pores assumed to satisfy the condition that their length is proportional to their radius (self-similarity of fractal media).
Consider Eq. [12] in the limit, r0 0. In this case, 1. Substitution of 1 into Eq. [6], [9], and [11] (Hunt, 2004a) leads to the results that the argument of Eq. [6] becomes equal to the argument of Eq. [9], but the range of validity of Eq. [9] shrinks to nothing. In this particular case Eq. [6] can be rewritten as
 | [15] |
Here the power, 3/(3 – Dp), is restricted to be greater than 3 as long as the fractal dimensionality of the pore space is greater than 2. If a looser constraint, Dp > 1, is applied, however, the result is that the nonuniversal power must be >2. Because of the explicit dependence on Dp, this power is nonuniversal, and the required result that powers >2 be nonuniversal is obtained.
An important feature of Eq. [14] is that the argument is essentially the relative saturation function, Eq. [2], since the unity term in the denominator is just the porosity, and porosity is also the theoretical maximum volume wetness. Thus it is shown here that the argument of the van Genuchten parameterization becomes precisely correct in the limit of a true, rather than truncated, fractal medium. While it has long been recognized that the Brooks and Corey (1964) water-retention relationship was consistent with a truncated fractal model (e.g., Eq. [14]), the (admittedly more tenuous) connection between the van Genuchten parameterization and true fractal media has not been recognized before.
Setting an arithmetic average of the conductivities of all the elements of a system equal to the upscaled conductivity is well known to correspond to a physical model in which all the components are configured in parallel. For example, the bundle of capillary tubes model (Kozeny, 1927; Carman, 1956) has each tube connecting opposite sides of the system in parallel with all the other tubes. Configuring all the components in parallel results in a system, whose critical volume fraction is zero, because each individual element of the system connects to both ends independently of all other elements. But if t = 0, then consistency requires that r = 0. This follows from the vanishing of K due to capillary flow processes (Eq. [9]) at = t. So the van Genuchten model contradicts itself: by allowing both zero and nonzero values for the residual moisture content, its elements are simultaneously parallel and not parallel.
Recognition of the above theoretical inconsistency allows a further test to be made. Consider Eq. [1] in the limit that r = 0. If the parameter m is set to 1, it is seen that the argument of Eq. [1] becomes proportional to that of Eq. [6] for any value of the porosity (the factor in the denominator is not reproduced in Eq. [6]. Thus, under the restriction m = 1, imposition of the condition r = 0 forces the argument of the second factor in the van Genuchten function to be compatible with that of Eq. [6]. But this means that the flexibility of the van Genuchten parameterization comes at the expense of logical consistency in the explicit assumption (through the averaging procedure) that the tubes are in parallel and, implicitly (r 
0) that they are not.
It should be noted that the water-retention function from the Rieu and Sposito (1991) model, as for the Brooks and Corey (1964) phenomenology, predicts the saturation in terms of a power, 3 – Dp, of the ratio, hA/h, whereas Eq. [6] involves the power, 3/(3 – Dp). If one then represents K as a function of h (Eq. [14]), the dependence contains two powers that are inversely related. Such a result is expected generally because of the inverse relationships of K and on h. [The critical volume fraction in CPA, and K in other schemes, as well as (h) are all related to integrals over the pore size distribution.] In the van Genuchten parameterization of K(h) such inverse powers, m and 1/(1 – m), are also present. But no obvious correspondence between these inversely related powers and those from the percolation treatment can be drawn. Further, m and 1/m are already present in the K() formulation, for which percolation theory does not predict two inverse powers, though 3/(3 – D) diverges and approaches zero if D 3. Such a presence of inverse powers in the van Genuchten parameterization allows the conductivity as a function of moisture content to develop a sigmoidal shape, becoming nearly vertical as saturation is approached. However, consider that the near-saturation regime may be dominated either by structure (Nimmo, 1997) or, in coarse soils, simply by the difficulty of wetting the soil completely. I claim that adjusting the formulation of the conductivity over the entire range of saturation to capture a behavior at the end of the range of saturations leads to problems with interpretation in the middle range of saturation. An example of such potential difficulties is included later.
For other reasons as well it is not a theoretical advantage to describe the hydraulic conductivity and the pressure–saturation curves over the entire range of saturations using the same phenomenology, in spite of the expedience of being able to do so. Complications at the wet end due to presence or absence of structure are well known. But the critical path analysis success has also elucidated some of the problems at the dry end. In particular, those processes (film flow, water vapor transport) responsible for conduction at < t are different from capillary flow through a continuous interconnected (percolating) network. It has also already been shown (Hunt and Gee, 2002b) that typical experimental results for water retention in this range of moisture contents are not equilibrium values, and carry an explicit time dependence (Hunt, 2004b). This means that for < t, the physics of the hydraulic conductivity and of pressure–saturation curves is very different from that at > t. Using the same phenomenological expression to link these different regimes may be computationally expedient, but it is not well founded. Linking time-dependent and time-independent physics in one function estimated for time-independent conditions must carry serious risks to accuracy when used to predict conditions in the time-dependent range. For completeness and simplicity, the van Genuchten and percolation results are summarized Table 1. A correspondence between the hydraulic conductivity of the van Genuchten and the percolation formulations is shown in Fig. 1
for representative examples of the relevant parameters.
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SUMMARY AND CONCLUSIONS |
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TOPABSTRACTINTRODUCTIONSUMMARY AND CONCLUSIONSREFERENCES |
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REFERENCES |
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TOPABSTRACTINTRODUCTIONSUMMARY AND CONCLUSIONSREFERENCES |
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h–0.4 for large h.