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DIVISION S-1—NOTES

ON THE VANISHING OF SOLUTE DIFFUSION IN POROUS MEDIA AT A THRESHOLD MOISTURE CONTENT

^a Cooperative Institute for Research in the Environmental Sciences, Univ. of Colorado, Boulder, CO 80309-0216
^b Dep. of Agronomy, Iowa State Univ., Ames, IA 50011-1010

^* Corresponding author (ewing{at}iastate.edu).

	ABSTRACT

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It was recently shown that the solute diffusion coefficient tends to zero at some threshold water content. We examine this phenomenon in light of recent simulation results from a network diffusion model. The experimental and simulation results agree, inviting an explanation from continuum percolation theory.

	INTRODUCTION

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A RECENT PUBLICATION by Moldrup et al. (2001) demonstrated the existence of a predictive relationship between the ratio of surface area A (m²) as measured with BET-N₂, to volume V (cm³), of a porous medium and the threshold water content, _t, (m³ m^-3) at which solute diffusion in that medium vanishes (i.e., becomes immeasurably small):

[1]

This threshold water content arises in the following ratio of the diffusion coefficient of a solute in a porous medium (D_pm) with water content , and the diffusion coefficient of the same solute in water, D_w:

[2]

The ( - _t) term, a difference of fractional volumes, turns out to be the inverse of an important length scale from percolation theory applied on a continuum. In this Note we demonstrate that the proportionality, Eq. [2], can be readily derived in terms of critical exponents of percolation theory.

	Theory

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We first give some definitions from percolation theory (Stauffer and Aharony, 1994; Berkowitz and Ewing, 1998). These are written initially in terms of three-dimensional bond percolation on a lattice, and then for continuum percolation in the case that the moisture content is the relevant volume fraction. Define 0 < p_c < 1 to be the critical value of the fraction of connected bonds. When a fraction p < p_c of the bonds of an infinite lattice are connected at random, the largest cluster of interconnected bonds has a linear dimension L = L₀ |p - p_c|^-, with the prefactor L₀ on the order of an individual bond length, and = 0.88 being a critical exponent of percolation theory. When p = p_c, it is possible to find a sample-spanning (infinite) path across a lattice of arbitrary size using connected bonds only. The so-called infinite cluster forming that path is fractal, and the path is tortuous. If a fraction p > p_c of bonds is connected, then the largest (infinite) cluster is multiply connected, and the typical separation of both paths and nodes is also L = L₀ |p - p_c|^-. On the other hand, the tortuous distance along the percolation path between nodes, , is somewhat larger than L and is given by = |p - p_c|^-1. Finally, not all connected bonds connect to the infinite cluster itself. Only a fraction P = (p - p_c)^ß (with ß = 0.4 in three dimensions) are actually part of the infinite cluster. Thus P = 0 at p = p_c. Note that the form of these results is expected to be the same in continuum as in bond percolation.

In the case of continuum percolation, the relative volume fraction, V, and its critical value for percolation, V_c, play the roles of p and p_c. These volume fractions can be related to fractional moisture contents. In recent treatments of the unsaturated hydraulic conductivity, K(), in terms of critical path analysis (Hunt and Gee, 2002a), it was shown that K() depends on a critical moisture content, _c, for percolation. Specifically, for < _c it is not possible to construct an infinite pathway along which capillary flow occurs, so (in soils terminology) once capillary continuity is broken, water movement across the sample must occur partially (or primarily, depending on the value of the moisture content) through the much slower mechanism of film flow. Similarly, in Hunt and Gee (2002b), this critical value of the moisture content for capillary flow represents a minimum volumetric moisture content at which equilibration occurs by capillary flow, and thus the lower limit of soil moisture contents at which water retention can reasonably be measured at equilibrium. The numerical value of _c for flow was shown (Hunt and Gee, 2002b) to be nearly equal to that of Moldrup et al.'s (2001) threshold moisture content _t at which solute diffusion vanishes. In this contribution we will use continuum percolation theory, with the moisture content playing the role of p (or V) and the threshold moisture content _c the role of p_c (or V_c), to derive Eq. [2] within a proportionality constant.

The effective diffusion coefficient, D_pm, in saturated porous media is often given in terms of the aqueous diffusivity, D_w (Hillel, 1980; Ewing and Horton, 2003):

[3]

where is tortuosity and is porosity. The empirical diffusive tortuosity was shown (Eq. [4] of Ewing and Horton, 2003) to scale as

[4]

where the equality added here restates the fundamental tenet of percolation theory (Stauffer and Aharony, 1994) that scaling results are universal for all variations of percolation theory, and that the values of the powers are functions only of the dimensionality of the medium. Substitution of from Eq. [4] into Eq. [3] yields

[5]

where the equality is written in accord with the supposition that for unsaturated conditions the moisture content, , will play the role of the porosity, , in the prefactor. Equation [5] is very close to the experimentally observed relationship, Eq. [2]. Uncertainties in the exponents given in Ewing and Horton (2003) (the scaling of to L was expected to yield the exponent 1.19 instead of 1.11) allow for a possible spread of values of the exponent on - _c of from 0.98 to 1.05, the mean of which values differs from observation (Moldrup et al.'s [2001] exponent) by only 1.5%. Thus for all intents and purposes, the exponent of Eq. [5] is indistinguishable from that of Eq. [2], the argument - _c is identical, and there is no reason to assume that the prefactor is greatly different.

The generalization, , for incomplete saturation, is somewhat subtle. In theoretical constructs, which do not account for diffusion over water films, it would be necessary to replace with an accessible moisture content, P = ( - _c)^ß , denoting that fraction of the water-filled porosity that is part of the infinite cluster of water-filled pores (all interconnected by water-filled pores). This important potential complication will be seen to be relevant for gas diffusion, but not for diffusion through water. The reason it is not relevant for water is based on the fact that those water filled pores which are not themselves part of the infinite cluster, can always be connected to it over films. The total distance over films (in the connections to the infinite cluster) can be shown to be microscopic (i.e., a few pore lengths; Hunt, 2003) and slowly varying in for > _c, while this distance jumps rapidly to macroscopic (on the order of the sample size) for < _c. Thus if diffusion can proceed through films then the entire water-filled volume is approximately equally accessible for all > _c.

To underscore the validity of Eq. [5] and our substitution we now consider gas diffusion. It is worth noting here that both solute and gas diffusion experiments are typically conducted along a primary drainage curve. During drainage, air cannot enter the system in significant volume until the air-accessible pore space percolates, that is, reaches the bubbling pressure. Thus the very presence of air in the medium (excluding pores near the edges) implies the percolation of the air phase. As a consequence, gas diffusion during drainage does not vanish at a finite volume of air, though in imbibition it would (the remaining, disconnected air volume termed entrapped air). Rather, gas diffusion vanishes when air accessible porosity = 0. As a practical consideration, gas effectively diffuses only through the air-filled pores, since gas diffusion through water is inherently much slower than through air, in addition to requiring solution and exsolution, processes that are slow for typical test gases. Thus air-filled pores, which are not connected to the infinite cluster (of air-filled pores) do not contribute to the relevant porosity, and the relevant air-accessible porosity must be reduced from by the factor (/)^0.4. Here we guarantee that the fraction of air-filled sites that is interconnected is unity when = . In analogy to Eq. [5] we thus arrive at

[6]

which is essentially the observed result (Moldrup et al., 2001):

[7]

The higher exponent, 2.4 in Eq. [6] rather than essentially 1 as in Eq. [5], appears to result from (i) the asymmetry between air entry and water drainage along a primary drainage curve (additional power of 1), and (ii) from the fact that water is wetting, leaving films which allow diffusion relatively efficiently when compared with the efficiency of air diffusion through water-filled pores (additional power of 0.4).

The present note is based on the presumption that this minimum water content represents the break-up of the network for capillary flow into disconnected regions when the moisture content is too low to permit percolation on a continuum. Of course this condition would necessitate the vanishing of any solute diffusion as well (except insofar as such could proceed over macroscopic distances through water films—presumably orders of magnitude more slowly). As a consequence, and through comparison of experimental values (Hunt and Gee, 2002b), identical values for threshold moisture may be concluded to govern both flow and solute diffusion.

	Summary

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1. A published derivation of diffusion scaling consistent with percolation on a lattice is used (consistent with continuum percolation theory) to explain the form of the effective diffusion of solutes in porous media compared with their diffusion in water.
   
2. This same published derivation is also consistent with experimental results for gas diffusion.
   
3. The percolation scaling result for diffusion facilitates the interpretation of the difference between solute and gas diffusion in terms of physical constraints, providing additional confidence in the results for solute diffusion.
   
4. The critical volume fraction, which was previously shown to govern the flow equation and the minimum moisture content at which water retention curves represent equilibrium conditions, is here shown to be identical to the threshold moisture content for solute diffusion. Concepts from (continuum) percolation theory are seen to both enhance and unify our understanding of two-phase flow and transport in porous media.
   

Received for publication August 20, 2002.

	REFERENCES

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• Berkowitz, B., and R.P. Ewing. 1998. Percolation theory and network modeling applications in soil physics. Surv. Geophys. 19:23–72.
• Ewing, R.P., and R. Horton. 2003. Scaling in diffusive transport. p. 49–61. In Y. Pachepsky et al.(ed.) Scaling methods in soil physics. CRC Press, Boca Raton, FL.
• Hillel, D. 1980. Fundamentals of soil physics. Academic Press, NY.
• Hunt, A.G. 2003. Continuum percolation theory for water retention and hydraulic conductivity of fractal soils: Extension to non-equilibrium. Adv. Water Resour. (In review).
• Hunt, A.G., and G.W. Gee. 2002a. Applications of critical path analysis to fractal porous media: Comparison with examples from the DOE Hanford Site. Adv. Water Resour. 25:129–146.
• Hunt, A.G., and G.W. Gee. 2002b. Water retention of fractal soils using continuum percolation theory: Tests of Hanford Site soils. Vadose Zone J. 1:252–260.[Abstract/Free Full Text]
• Moldrup, P., T. Oleson, T. Komatsu, P. Schjoning, and D.E. Rolston. 2001. Tortuosity, diffusivity, and permeability in the soil liquid and gaseous phases. Soil Sci. Soc. Am. J. 65:613–623.[Abstract/Free Full Text]
• Stauffer, D., and A. Aharony. 1994. Introduction to percolation theory (revised 2nd ed.), Taylor and Francis, London.

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