Published in Vadose Zone Journal 4:145-150 (2005)
© 2005 Soil Science Society of America
677 S. Segoe Rd., Madison, WI 53711 USA
ORIGINAL RESEARCH
Hydraulic Conductivity Limited Equilibration
Effect on Water Retention Characteristics
A. G. Hunta,* and
T. E. Skinnerb
a Departments of Physics and Geology, Wright State University, Dayton, OH 45435
b Department of Physics, Wright State University, Dayton, OH 45435
* Corresponding author (allen.hunt{at}wright.edu)
Received 25 June 2004.
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ABSTRACT
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The effects of the percolation phase transition on equilibration of porous media during drainage are shown to set on at moisture contents, 
, somewhat larger than the critical moisture content for percolation, t. An algorithm is developed, which yields the typical upward curvature of log[h()] curves at low moisture contents, where h is the hydraulic head, as well as a flattening of the curve of log[K(h)] for large h, where K is the hydraulic conductivity. Within certain approximations the procedure can be validated through solution of the associated differential equation. The procedure was tested against seven drainage curves from the USDOE Hanford site and found to be predictive. However, theoretical questions remain regarding, among other things, the implicit assumption that the interfacial tension measured is an equilibrium value, even though the moisture content is not. The motivation of the present research is to reduce ambiguities in the interpretation of dry-end pressure–saturation curves.
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INTRODUCTION
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THE PURPOSE OF THIS STUDY is to give a concrete prediction of nonequilibrium water retention characteristics of natural porous media under a specific set of conditions. These conditions are as follows:- The medium can be represented as a continuous, truncated, probabilistic fractal (Hunt, 2001; Hunt and Gee, 2002a).
- Critical path analysis (Ambegaokar et al., 1971; Pollak, 1972; Katz and Thompson, 1986) based on continuum percolation theory gives (Hunt and Gee, 2002a) the dependence of the hydraulic conductivity, K, on moisture content, , except for a small range of moisture contents in the vicinity of the critical volume fraction for percolation, t (Hunt, 2004a).
- In the limit
t, the dependence of the hydraulic conductivity on moisture content crosses over to percolation-based scaling (Hunt, 2004a).
The unexpected change in the dependence of K() is hypothesized to lead researchers to underestimate the time for equilibration, resulting in inadequate drainage. The inadequate drainage leads to larger moisture contents for a given hydraulic head, h, and a larger value of K(h) (Hunt and Gee, 2002b) in sandy soils from the USDOE Hanford site. This hypothesis is in general accord with Topp et al. (1967), who found that when comparing water retention data obtained by equilibrium, steady-state, and transient methods, more water was retained in a sand at a given matric potential for the transient flow case than for the static equilibrium and steady-state cases.
Our work concerns only nonequilibrium effects on drainage curves; the fundamental hysteresis between drainage and imbibition has already been treated (Hunt, 2004c) in percolation theory. Although the present work is restricted to calculations for a fractal soil model, there is no requirement to make such a restriction, and extensions of this work are intended to take into account more general models of porous media in the forward modeling mode. If a general forward modeling problem can be solved, the next step will be to use inverse modeling to recover an equilibrium water retention curve from arbitrary experimental data.
How prevalent and how important are the effects of the lack of equilibration on, for example, predictions of water retention? We checked for the conditions of Fig. 5.53 (p. 578) of Carter (1993) by numerical simulations. Equilibration of a 3-cm soil core starting at 0.5 MPa (5 bars) being taken to 1.5 MPa (15 bars) on an infinitely permeable porous plate will take at least 1000 h, but often more than 10000 h. From this perspective we question the notion of equilibrium, and suggest that in the future investigators should document the actual pressure potential of the core, at a minimum at the top and bottom of the core, and ideally in the midpoint. This can be quickly done with a psychrometer.
It is not our purpose to advocate that the particular scenario leading to nonequilibrium results we discuss here must be relevant to every system; methods of investigation that avoid capillary flow at low saturation should certainly be free of the particular defects discussed here. Nevertheless, we point out that field drainage experiments cannot so easily avoid such complications. We also do not mean to imply that fractal models for the pore space are necessarily accurate down to the lowest moisture contents reached. Fractal models for the contributions of surface films to the water retention curve were suggested already in the 1980s (de Gennes, 1985), for example. Nor is it our intent necessarily to contest the comments of Hassanizadeh and Gray (1993), "Burial of dynamic effects into relative permeability and capillary pressure hysteresis is shown to be an unsatisfactory theoretical construct for modeling the actual processes occurring in two-phase flow." But in our view, the ability to predict the observed forms of water retention curves under certain conditions is a worthy goal. In particular, we make comparisons with experiments (Freeman, 1995) for a number of USDOE Hanford site soils. A second valuable result is that we are able to infer the relevance of continuum percolation theory under conditions where it is difficult to detect the percolation threshold directly. Finally, we can support the contention that it is necessary to exercise caution in the evaluation of water retention curves at low moisture contents, at least if the measurements were obtained using ceramic plates.
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SUMMARY OF EXISTING THEORETICAL RESULTS
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The random fractal model used here and in previous publications of the author is that of Rieu and Sposito (1991). The porosity, 
, is known to have the following dependence on fractal dimensionality, D, and on the minimum and maximum pore radii for which the fractal description holds (r0 and rm, respectively):
 | [1] |
The equilibrium water retention characteristics of such a medium, in the absence of any complications due to phase continuity of either the water or air phases and neglecting contributions of thin surface films of water, is given by (Rieu and Sposito, 1991; Hunt and Gee, 2002a)
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In Eq. [2] S 
/ is saturation, and hA is the air entry value of the pressure, h. Note for later use that the equilibrium reduction, 
, in water constant on successive measurements, hi and hi+1, should then be
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Critical path analysis applied to a power-law distribution of pore sizes consistent with the Rieu and Sposito (1991) model yields the following expression (Hunt and Gee, 2002a) for the ratio of the unsaturated to the saturated hydraulic conductivity:
 | [4] |
where t is the critical volume fraction for percolation, KS the saturated hydraulic conductivity, and K() the value of K at moisture content, . An empirical relationship for t exists (Moldrup et al., 2001), but the theoretical explanation of this expression is not complete (Hunt, 2004b). Equation [4] does not vanish in the limit t, however, as it must (Golden et al., 1998), and as solute diffusion is observed and predicted to do (Moldrup et al., 2001; Hunt and Ewing, 2003). Percolation scaling requires the following result for the hydraulic conductivity in this limit (Hunt, 2004a):
 | [5] |
It has been argued (Hunt, 2004a) that the exponent t = 1.88 and, similarly, that the air permeability must vanish (Hunt, 2004d) according to this power of 
– t, where is the air-filled porosity, and t is its critical value for percolation. This value of t has been confirmed (Hunt, 2004d) by comparison with experimental data (Moldrup et al., 2003), so that there is reason for confidence that the value is accurate for K as well. Under these circumstances, demanding continuity of both the hydraulic conductivity and its derivative with respect to the moisture content yielded (Hunt, 2004a)
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for t 
1, where
 | [7] |
The predicted hydraulic conductivity as a function of moisture content is shown in Fig. 1
for characteristic system parameters, = 0.44, D = 2.84, and t = 0.107. Figure 1 also shows the extension of Eq. [3] for < 1. This is important for the following considerations of nonequilibrium water retention.
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NONEQUILIBRIUM PROPERTIES
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Several possible causes for nonequilibrium results for pressure–saturation curves have been discussed in the present context of continuum percolation theory (Hunt, 2004b). Those dealing with the differences between imbibition and drainage are not discussed here. Two main possible explanations for nonequilibrium drainage results were discussed in that reference. One was that some of the pores that permit the presence of water at an initial hydraulic head are not connected by allowed pathways to the infinite, percolating, cluster of "permissible" pore space, and thus may remain filled even after the hydraulic head is increased to a value incompatible with the presence of water in these pores. This was proposed by Wildenschild and Hopmans (1999) as well: "We suggest that the rate-dependent variation of our experiments is due to 1) entrapped water occupying dead-end pore space; we speculate that the amount of water increases with increasing flow rate, and 2) initial drainage of the lower portion of the sample, thereby preventing further drainage of the upper portion of the sample." Hunt (2004c) proposed that this first possibility was not relevant to usual field experiments because it predicted greater hysteresis in K as a function of saturation than as a function of hydraulic head and because it predicted that nonequilibration would lead to a lower K value for a given moisture content than otherwise observed. It was also demonstrated that a relatively short time scale could allow this "entrapped" water to be transported over microscopic distances by film flow to the connected, infinite cluster. However, in the fast flow experiments described by Wildenschild and Hopmans (1999), this time scale might not be reached. Furthermore, Wildenschild and Hopmans (1999) observed, "the unsaturated hydraulic conductivity decreases with increasing flow rate for a specific water content for the Lincoln soil. In some cases, this difference is close to two orders of magnitude." Thus the experiments of Wildenschild and Hopmans (1999) appear to confirm the analysis of Hunt (2004c).
Also the second possibility mentioned by Wildenschild and Hopmans (1999) is related to the concept here (and the second possibility discussed in Hunt, 2004c), in that a strong diminution in K could produce a lack of equilibration. Such a suggestion leaves unanswered the question of whether it is legitimate to assume that h is in equilibrium when is not. Nevertheless this mechanism has a certain connection to the following comment by Ross and Smettem (1999) regarding "traditionally assumed local equilibrium at the elementary volume scale": "The problem lies in choosing a convenient description scale. The usual response to this situation is to classify the system into two or more zones that exchange material with one another while maintaining local equilibrium internally." Ross and Smettem (1999) were referring to issues of soil structure, rather than vertical position, however.
We examine here the possibility that because of very low values of K, the time allotted for experiments could be inadequate to provide equilibrium drainage, and that the additional moisture content could lead to higher measured values of K than expected from theory. This scenario was found (Hunt and Gee, 2002a, 2002b; Hunt, 2004a) to be in general accord with experiments on Hanford site soils (Freeman, 1995; Khaleel and Relyea, 2001) in that more hysteresis was expected for K(h) than for K(). In particular it is hypothesized that the rather sudden crossover from the validity of Eq. [4] to the validity of Eq. [6] leads to a rather dramatic increase in equilibration times, and that if this crossover is not anticipated by the researcher, there will be a sudden onset of difficulties achieving equilibration.
The algorithm for calculating the modified or actual value, a, of the water content under these assumptions is as follows: We start at the moisture content = 1 = a. For t a 1, the actual drainage that is possible in an experiment will be a fraction of the drainage appropriate for the new value of the hydraulic head. That fraction will be the ratio of the hydraulic conductivity, Kp, predicted for percolation scaling, Eq. [6] to the hydraulic conductivity, Kf, predicted for fractal scaling, Eq. [4]. The amount of moisture predicted by this ratio is removed from the system, thus giving the new value for the water content. The conductivity then changes to the value, Kp, predicted using a, the presumed conductivity to Kf(a), and the procedure is repeated. The same equations used for predicting K() ( is the equilibrium water content) are used for predicting K(a), although the distribution of water in the pore space is not the same for a as for an equilibrium . The relation between the equilibrium water content, , and the actual water content, a, can be generated numerically in this manner, but in the limit of sufficiently small step sizes (in h), we can also write
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and integrate to obtain
 | [9] |
The right-hand integral is given in terms of the Gauss hypergeometric function, 2F1. If we define
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we can then write an implicit relationship for a
 | [11] |
that maps t a 1 to 0 1. The original, or presumed water retention function, h(), is then mapped to h[G(a)]. Note that since K vanishes at t, the moisture content t is never reached with this procedure. Figure 2
shows the results of the effects of incomplete equilibration on the water retention curve from this procedure for the same system parameters as chosen for Fig. 1. Although the procedure is simple, the results are robust. The procedure was also performed algorithmically for finite step sizes, h. For a relatively wide range of values of h investigated the resulting nonequilibrium portion of the water retention curve (not shown) is identical to that predicted by Eq. [10]. Note that the general result of not waiting long enough for water to drain requires a greater increase in hydraulic head to achieve a given water loss.
Figure 3
shows the impacts of incomplete equilibration on experimental results for K(h), again for the same system parameters as chosen for Fig. 1 and Fig. 2. At first the more rapid drop of K with diminishing water content causes the curve of K(h) to drop more steeply than predicted from Eq. [4], but in accord with Eq. [6], but very soon the tendency for the water content remaining in the medium to be higher than the equilibrium value causes K(h) to flatten relative to Eq. [6], and even in comparison with Eq. [4]. The tendency for the K(h) curve to flatten for large h is well known for coarse soils.
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Fig. 3. LogK vs. hydraulic head, h. Open triangles represent equilibrium K values. x values assume validity of Eq. [4] for K for entire range. The solid curve represents typical nonequilibrium values of K(h).
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An important limitation of a study that does not link the derived constitutive relations and their limitations from lack of equilibration with one-dimensional column modeling is that it is impossible to evaluate the effects of the lack of equilibration of the interfacial tension. The implicit assumption here that at least some of the column is in equilibrium with the new tension, even though the water content of the system is not, could contribute to errors. Thus our approach could be called a zero-dimensional model. Instead of addressing such questions theoretically, we make a direct comparison with experimental data.
We analyze a suite of water retention functions from the USDOE Hanford site, which was already considered in Hunt and Gee (2002b). The data, collected by Freeman (1995), were also reported in Khaleel and Relyea (2001). Note that the maximum experimental time scale of these experiments was given as approximately 6 wk, or about 1000 h. From the beginning discussion, however, this should probably be regarded as a minimum time scale. A description of the collection of soils and the methods used was transcribed from Freeman (1995) to the previous publications in this journal (Hunt and Gee, 2002b, 2003), and is not repeated here. Note, however, that one soil shown here is a silt loam, while the other soils are sands.
Hunt and Gee (2002b) showed that, statistically at least, the values for the onset of the deviation from fractal scaling of the water retention curves started at a moisture content linearly related to t. Later work (Hunt, 2004a) demonstrated that the onset of the deviation was, on the average, at = 1. Here we make a direct comparison of the predicted and observed water retention curves, following the same algorithm as outlined above. Instead of relying on calculated values of t, which require experimental values of the surface area/volume ratio, we take the lowest moisture content reached to be t. Then we calculate 1 from Eq. [7].
We apply the algorithm starting at the experimentally attained moisture content nearest 1. For programming ease, we apply a similar algorithm for > 1 as well, but because of difficulty with tensions less than the air entry value, it is necessary in this range to reinitialize the moisture content to the observed value following every step. This means that the predicted values according to the algorithm should not be compared with experiments for > 1, so the figures also depict the analytical fractal scaling result (Eq. [2]). While applying Eq. [2] requires use of hA as an adjustable parameter, D is calculated from Eq. [1] and the pore size distribution, as described in Hunt and Gee (2002b). Then, starting at = 1 any errors in the algorithm are cumulative because no correction from experimental data is used, thus allowing direct comparison of the algorithm with experiment in this region. It should be mentioned that the agreement with experimental data is optimized when Kf is evaluated two tension measurements before Kp. This is equivalent to optimizing with respect to an adjustable parameter. However, this parameter is the same for all the curves used.
Seven of the 13 curves for which the results tended to be most satisfactory were chosen for representation here (Fig. 4–10)
. In the other cases the results are as in Fig. 8 (ITS 2-2229), where the algorithm produced an underestimation of the moisture content for a range of moisture values. Considering that our treatment not only assumes the validity of the simplified physics, but also ultimately rests on the reliability and consistency of a hypothesized behavior pattern of experimental researchers, we find the agreement with experimental data to be exceptional.
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CONCLUSIONS
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We believe that this work demonstrates the usefulness of the hypothesis that continuum percolation theory is the appropriate framework for discussing flow properties of unsaturated porous media. In particular, the vanishing of the hydraulic conductivity, K, from capillary flow at a critical moisture content for percolation appears to have importance far beyond the expected measurements of K as a function of saturation. The results predicted here, which are at least in qualitative agreement with experimental data, include a "steepening" of the h() curve at small values of , and a flattening of K(h) at large h. The former result leads to a discrepancy between water retention curves predicted from fractal models and experimental values, whereas the latter has been shown (Hunt and Gee, 2002b) to lead to a misinferred lack of variability of Gardner's 
(Khaleel and Relyea, 2001) for coarse Hanford site soils. We do not claim that all deviations from fractal (volume) scaling are due to the mechanism that we have identified, but if the mechanism we have described here is not eliminated from other investigations, the effects of lack of equilibration could easily be mistaken for contributions to the water retention curve from surface films, for example. Thus we recommend that experiments done using ceramic plates be carefully analyzed to reduce ambiguities from nonequilibrium values of the moisture content. We also do not consider this study to be a final word on the theory of nonequilibrium, but, rather, a demonstration that even a lowest order analysis of some effects, which are typically ignored (e.g., lack of continuity of water-filled pore space) may actually be more important than a full understanding of one-dimensional column modeling that neglects the microscopic properties. More specifically, it appears that the comparison with experimental data supports our contention that an effectively zero-dimensional model with physically based constitutive relations may outperform column modeling with poorly constrained constitutive relations.
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ACKNOWLEDGMENTS
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The authors are very grateful to John Selker for providing results of estimates of equilibration times through standard simulations.
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REFERENCES
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ABSTRACT
INTRODUCTION
