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Water-Retention of Fractal Soil Models Using Continuum Percolation Theory

Tests of Hanford Site Soils

^a CIRES, Box 216, University of Colorado, Boulder, CO 80309-0216
^b Hydrology Group, P.O. Box 999, Pacific Northwest National Laboratory, 3200 Q. St., Richland, WA 99352-0999
* Corresponding author (ahunt{at}nsf.gov)

Received 6 February 2002.

	ABSTRACT

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For 43 Hanford site soils, we use fractal analysis and assume proportionality of pore radii to particle radii to generate water-retention curves, h(), from particle-size distributions. The air-entry head is used as an adjustable parameter to optimize the fit to experimental data for h(). At a low moisture content, _d, the predicted and observed water-retention curves deviate. It is shown here that the moisture content at which this deviation occurs is in most cases probably the same value, at which previous experiments found a vanishing of solute diffusion. Where this correlation is indicated, we interpret _d as a critical moisture content for percolation of capillary flow paths, and the relevance of other mechanisms of water transport, such as film flow, to equilibration at lower moisture contents. In other individual cases, however, the deviation is correlated with very low values of the hydraulic conductivity associated with capillary flow. In either case, we infer that the deviation from fractal predictions is due to the lack of equilibration of the medium. Our work thus exploits theoretical and analytical gains from percolation theory and fractal analysis to define the equilibrium limits on water retention curves.

Abbreviations: DOE, U.S. Department of Energy • pdf, probability density function • PSD, particle-size distribution

	INTRODUCTION

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THE HANFORD SITE in southeastern Washington has 177 buried waste storage tanks with 208 million liters of highly radioactive legacy waste from the Cold War years. Some 67 of these tanks are known or suspected leakers, from which an estimated 3.8 million liters of toxic fluids have seeped into the vadose zone. In addition, other less toxic waste fluids, in excess of three billion liters, have been discharged to ground at Hanford. The U.S. Department of Energy (DOE) has a program to monitor and ultimately control the spread of this contamination. To accomplish this objective, DOE has relied on vadose zone modeling of flow and transport through Hanford soils and sediments. This modeling assessment requires that the hydraulic properties (e.g., water retention and unsaturated hydraulic conductivity) of the sediments be adequately characterized.

A fundamental problem in unsaturated flow in porous media, and for characterization of the vadose zone at the Hanford site, is the lack of certainty in characterizing the pore space. One major question regards the potential relevance of fractal soil models (Turcotte, 1986; Baveye et al., 1998; Tyler and Wheatcraft, 1990; Rieu and Sposito, 1991; Wu et al., 1993; Sahimi, 1995; Bittelli et al., 1999; Posadas et al., 2001). The second question regarding the connectivity of the pore space can be addressed using percolation theory (Kirkpatrick, 1973; Stauffer, 1985; Stauffer and Aharony, 1994). We recently showed (Hunt and Gee, 2002) that using the framework of percolation theory for flow in a fractal medium increases understanding of the steady-state hydraulic conductivity in the DOE Hanford Site soils. A related problem, which we would like to address here, is the shape of the water-retention curve at moisture contents low relative to saturation, but still high enough so that the hydraulic conductivity is defined by capillary flow. In the process, we are able to evaluate the applicability of fractal models to the porous media typical of the Hanford site.

The general procedure is to use concepts from fractal theory and soil particle-size data to predict water retention curves; to record the (low) moisture content, _d, at which observed and predicted water retention characteristics diverge; and to compare _d with results of recent experiments for a threshold moisture content for solute diffusion, _t. Because _d and _t are correlated (R² 0.85), we conclude that an important physical change at this moisture content influences both solute diffusion and water retention characteristics. This physical change we infer to be the breakup of connected capillary flow paths into isolated regions, and we interpret the change within the framework of percolation theory.

	THEORY

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Pore Distribution, Fractal Dimension, and SA_vol Ratio
The fractal analysis used is described in detail in Hunt (2001) and Hunt and Gee (2002). Theory and associated analysis were adapted from the theory of Rieu and Sposito (1991), with a simplifying assumption derived from the Arya-Paris model (Arya and Paris, 1981). A probability density function (pdf), W(r), for pore sizes, r, is assumed, which has a continuous, power-law form (Hunt and Gee, 2002)

[1]

with D the fractal dimension of the pore space, and r₀ and r_m the radii marking the lower and upper limits of validity of the fractal description. As is often discussed (e.g., Bittelli et al., 1999), such limits are required for normalization to be possible. To be consistent with Eq. [1], the volume of a pore must be r³, so that an integral of r³W(r) over the range r₀ to r_m yields for the porosity (in agreement with Rieu and Sposito, 1991):

[2]

To obtain Eq. [2] for arbitrary geometry, the normalization factor must be modified according to the choice of pore geometry. We ignored geometric shape factors, consistent with assuming (from self-similarity) that their values are the same for pore radii, r₀ r r_m. This omission introduces no uncertainty into later results for h() or for the relative hydraulic conductivity, K(S)/K_S, as a function of saturation, S, or suction head, h, (in terms of h_A, the air entry pressure) because our technique exploits the self-similarity to develop what are essentially scaling relationships. However, accurate calculation of h_A itself (or K_S), is not possible without knowledge of the individual pore geometries.

As in Arya and Paris (1981) and Gevirtzman and Roberts (1991), the largest pore radius is proportional to the largest particle radius, r_m. Consistent with that condition, we assume that the smallest pore radius is proportional to the smallest particle radius, r₀, making the ratio r₀/r_m equal to the ratio of the smallest to the largest particle sizes over which the fractal description is valid. Then D can be found for the pore space from (Hunt and Gee, 2002)

[3]

where the radii involved refer to the particle-size distribution (PSD). The hydraulic conductivity as a function of saturation, S, was shown (Hunt, 2001) to be well approximated by,

[4]

where K_S is the hydraulic conductivity at full saturation, and _c is the critical volume fraction for percolation. This calculation employed critical-path analysis (Ambegaokar et al., 1971; Pollak, 1972) in a continuum percolation problem (Stauffer, 1985; Stauffer and Aharony, 1994). Equation [4] was shown to describe K(S) in Hanford soils (Hunt and Gee, 2002).

Since r₀/r_m is determined from the PSD and from the bulk density, D can be derived from physical measurements. Then K(S) is predicted in terms of K_S if _c can be obtained. We show that _c as obtained from the water-retention curve is in reasonable agreement with an experimental relationship for the threshold moisture content for solute diffusion in terms of the surface area/volume ratio, SA_vol (Moldrup et al., 2001). This means that it becomes possible to predict K(S) in terms of its saturated value and quantities that can be obtained without measurements of hydraulic properties. Then, using only the air-entry pressure from the water-retention curve (assuming that this curve agrees with theory), results for K(h) can be predicted as well.

In the absence of experimental data for SA_vol, it must be calculated. In our procedure, we calculate the total surface area of all the particles and divide by the volume of all the particles. For this purpose, we need first the fractal dimension of the solid volume, D_s. D_s is, by analogy with Eq. [2],

[5]

Also, using Eq. [1], (but with the solid fractal dimension, D_s) for the pdf for finding a particle with a radius within dr of r, SA_vol is found to be

[6]

Because Moldrup et al. (2001) report SA_vol in terms of the soil volume, Eq. [6] must be multiplied by the factor 1 - in order to compare with their results. Although representing SA_vol in units (m² g^-1) compatible with Moldrup et al. (2001) requires multiplication also by the particle density (assumed independent of particle size), we are concerned only with its dependence on D_s, r₀, r_m, and . Clearly, SA_vol is inversely proportional to the maximum radius, r_m. Since SA_vol increases with diminishing radii, the role of the remaining factors is to adjust its value appropriately for changing r₀. Thus, we take SA_vol proportional to the inverse of the geometric mean particle radius, R_m in an alternative test. Note that the Moldrup et al. (2001) determination of SA_vol by N₂ Brunauer–Emmett–Teller (see, e.g., Pennell et al., 1995) measurements will also include internal surface area for soils with high clay content, but the large majority of Hanford site soils contain negligible clay, and Eq. [6] should be a reasonable equivalent under the circumstances.

Water-Retention Characteristics
To derive the water-retention curve in terms of the air-entry pressure, h_A, it is necessary only to write the hydraulic head necessary to evacuate a pore of radius, r, in the form, h = -B/r. Then it is possible (Hunt and Gee, 2002) to write for the relative saturation,

[7]

and for the fully saturated medium,

[8]

The water-retention curve is then (Hunt and Gee, 2002)

[9]

which may be inverted to find h in terms of S

[10]

While the derivation of Eq. [9] and [10] was in terms of a negative matric potential, since h and h_A always appear together in the context of this paper, it is convenient to drop the negative and just report their absolute value as a suction head (reported here in units of centimeters). In its approximate form, Eq. [9] is equivalent to the Brooks and Corey (1964) parameterization, as already known (e.g., Tyler and Wheatcraft, 1990). Soils with more complicated structure, such as PSDs that correspond to different fractals in different size ranges, were treated in Hunt and Gee (2002). We note that other authors (Bittelli et al., 1999; Poulsen et al., 2002) have also found it necessary to treat different soil components with differing fractal dimensions (or related distinctions). Our results appear in general accord with those of Posadas et al. (2001), namely, that it is often sufficient to use fractal analysis, in contrast to multi-fractal analysis, for soils with <10% clay content.

Critical Volume Fraction for Percolation
The critical-volume fraction, _c, for percolation to occur represents the minimum moisture content that a particular porous medium can contain (under equilibrium conditions) that will allow continuous, infinitely long paths of capillary flow to develop. This same volume fraction was identified with the inflection point in porosimetry experiments corresponding to the pressure or pore-entry size of the first considerable fluid uptake (Katz and Thompson, 1986). It is advantageous to have a means to determine _c by more general empirical means. To this purpose we connect with diffusion measurements reported by Moldrup et al. (2001). In experiments discussed there, it was shown that the diffusion constant of solute in a given porous medium, divided by its value in water, vanished linearly with at a threshold moisture value, _t. _t was correlated with SA_vol thus,

[11]

An R² = 0.99 was obtained by Moldrup et al. (2001) for this correlation. Such a value of R² implies that the individual measurements were typically within 0.005 or less of the value expected by Eq. [11] for a range of _t from near 0 to 0.19, or higher. As guidelines, Moldrup et al. (2001) also gave values of 0.09 for sandy soils, 0.013 to 0.014 for loams, and 0.18 for clayey soils. Presuming that solute diffusion in soils virtually vanishes when capillary flow ceases, we identify _t with _c. In Hunt and Gee (2002), this identification was shown generally consistent with values of _c employed for the McGee Ranch and North Caisson soils (from Rockhold et al., 1988), while Eq. [11] generated exactly _c for the North Caisson soil. Also, if Eq. [6] is substituted into Eq. [11] for _t, it is found that _t is a function only of D and r_m/r₀ (since is expressible in those variables), the same functional dependence for the critical volume for percolation as found in two-dimensional, discrete "prefractals" by Sukop et al. (2002).

	MATERIALS, DATA SOURCES, AND METHODS

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Experimental Data
In our analysis, we used tabulated data for 43 soils from various locations on the Hanford site. These soils were part of a larger data set from the Westinghouse Geotechnical Engineering Lab and Battelle, Pacific Northwest Laboratory (Freeman, 1995; Khaleel and Freeman, 1995). The soil samples were collected in conjunction with drilling operations. In some cases in the original data sets, the samples were extracted from the borehole with the drive barrel; otherwise, the samples were obtained using a splitspoon (which allows the casing to be split and removed and thus avoids the necessity to repack the soil).

Prefixes used to identify Hanford sediment samples include: FLTF for Field Lysimeter Test Facility, VOC for Volatile Organic Carbon, ITS for Injection Test Site, USE for U.S. Ecology, and ERDF for Environmental Restoration Disposal Facility. Acronyms were classified in this manner because the soils exhibit systematic variations in texture, with FLTF soils typically loams, the VOC soils highly variable, the ITS soils sands to gravelly sands, the USE soils sands, and ERDF a coarse to medium sand (the two soils included are loamy sands). The ITS samples reported here were taken while boreholes were drilled at the site in 1979. The hydraulic property data were used in preliminary modeling of flow at the site (Sisson and Lu, 1984; Gee and Ward 2001). Two recent sediment samples collected at the 200 East area, B-BX tank farms and identified as B8814-135, and B8814-130B, were also used. The 218 W-5-005 and related soils were inadequately described for quantitative analysis. A list of sediment samples used and fractal analysis information is found in Table 1.

The VOC samples were derived from a splitspoon sampler, while the FLTF site soil samples were repacked, and the ITS soils samples were derived from air-rotary drill cuttings and collected in 18.9-L (5-gallon) plastic bags. Water retention characteristics were mostly measured in conjunction with the unsaturated hydraulic conductivity in a steady-state head control apparatus (Klute and Dirksen, 1986) with tensiometers built into side ports (Khaleel and Relyea, 2001). The moisture retention characteristics were believed to have achieved equilibrium conditions, but we dispute this here, particularly for the coarser soils under dry conditions. Porous media with very low K(S) may require more time to establish equilibrium than was allowed for during laboratory experiments.

Data Analysis
Extraction of Fractal Dimensions
For 43 soils, we used graphs of log cumulative mass vs. log particle size to find r_m and r₀, and thus D. Such graphs cannot be straight lines over an infinite range of particle size, and in fact must cut off at the radius for which the cumulative mass is 100%. For practical reasons, a minimum size must exist as well. The cut-off at small r cannot have the same appearance as that at large r because 0% is not visible on any log-log representation. But in fact, even at the smallest values of r investigated, some mass typically remains unaccounted for, so theoretical constraints appear to be less important in this case than practical ones. Thus, the present cumulative PSD data are mostly flat at large and small values of r, with a range in between that approximates a constant, non-zero slope, and small ranges of r at the boundaries where the constant non-zero slope region curves in to horizontal lines. In fact, the PSD data accessed is valid to only 1 to 5%, so the data tend to plot up as a fractal region cut-off by one horizontal line at near 100%, and by another horizontal line somewhere between 1 and 10%, such that outside of this region, no mass was observed to accumulate.

The curvature near the boundaries is best considered as due to the cut-offs, since using a statistical package to plot a straight line through the data proves most accurate if those marginal points are omitted. Thus, for a simply fractal soil, we replaced the observed PSD by three straight lines, one with a non-zero slope and two with zero slopes. These horizontal lines were chosen to intersect the abscissa at 100% and at the lowest cumulative mass reached through the hydrometry data. The values r₀ and r_m were then read off the ordinate at the two intersections with the horizontal lines.

In some soils (apparently) missing mass created difficulties. For example, in the FLTF (and McGee Ranch samples [Hunt and Gee, 2002]), hydrometry accounts consistently for only 92% ± 2% of the mass. If the PSD for a particular FLTF soil showed no evidence of a cut-off in mass accumulation at smaller r values, and if the fractional mass obtained at the smallest radius, for which data exist, was larger (say equal to 11%) than in the majority of the FLTF soils, then the PSD graph was continued down to 8% to estimate the lower cut-off, but in such situations extrapolating the PSD to orders of magnitude smaller than the smallest particles detected is inadvisable on account of the large differences in PSD that would be generated for closely related soils.

If the PSD appears more complex (of which we have 7 cases from 43), then it may be necessary to divide the PSD into two or more different regions with, in principle, different fractal dimensions and geometries. Although such a model may violate the simple assumptions regarding the relevance of effects of pore geometry on K, geometrical factors are insignificant compared with accurate determination of the powers 1/(3 - D) in the K(S) relationship, which can range from 6 to 150 (for 2.5 D 2.98).

Then, Eq. [10] was fitted to the experimental data using the value of D determined above (in two cases, where our determination of D was incorrect, we had to use D as a fit parameter too), and a single parameter, h_A. In nearly all cases, the theoretical curve was parallel to the experimental curve between a moisture content of approximately 0.15 and about 10 to 20% less than the porosity (A). Other investigations (Filgueira et al., 1999; Bird et al., 2000) also found that the water retention characteristics could be predicted from PSDs. The value of h_A was then chosen to make the two graphs coincident over as much of this range of moisture contents as possible. As water is drained from the medium, the continuous path for capillary flow will break down at a moisture content equal to the critical-volume fraction, _c, for percolation. We expect the water-retention curve to deviate from theoretical predictions near this value because, to achieve equilibration, film flow (Blunt and Scher, 1995; Tokunaga and Wan, 1997) will be required.

	RESULTS AND DISCUSSION

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Results for the PSDs of seven texturally different soils are shown in Fig. 1 and of the corresponding water-retention curves alongside the theoretical predictions in Fig. 2. The examples were chosen to demonstrate the following:

1. FLTF soils were finest, and, because _t was therefore largest, they have the shortest range of moisture contents for which the fractal model described the water retention curve.
   
2. The VOC soils are the most variable and the most complex as well, with many clearly incompatible with the same fractal dimension throughout the range of the PSD.
   
3. The ITS soils were rather uniformly coarse, and agreement between theory and experiment was best expressed over the longest range of .
   
4. When different fractal dimensions were appropriate for different size ranges, h() may become steeper (VOC 3-0654-2) or less steep (VOC 3-0649) with increasing tension, h. The double fractal representation was necessary to demonstrate that there is no _d obtainable for VOC 3-0649 (one of several).
   

View larger version (17K): [in this window] [in a new window]   Fig. 1. Experimental data for the particle size distributions of seven representative soils.

View larger version (13K): [in this window] [in a new window]   Fig. 2. Experimental data for the water-retention curves for the same seven soils as in Fig. 1 and comparison with theoretical calculations from assumed fractal geometry. Arrows denote the moisture contents, d, at which the experimental curves deviate from the predicted curves.

We compare experimentally determined values of _d with the calculated value of [SA_vol]^0.52 from Eq. [6] in Fig. 3. Because Eq. [6] is only a proportionality, it is thus implied that the comparison involves an unknown constant related to geometry. Since there is no requirement that the studied soils all have identical geometries, the lack of control of geometry will introduce some scatter. Linear regression is consistent with a straight line with intercept, 0.0586 (R² = 0.833). If we have calculated SA_vol correctly, and the data are accurate, according to Eq. [4], such a large intercept should not exist. The relatively low value of R² compared with results of Moldrup et al. (2001) (R² = 0.99) implies that we have determined either SA_vol or _c less precisely. In fact we have determined both _c = _d and SA_vol less precisely. Nevertheless, the existence of a finite intercept suggests that there is a systematic deviation from prediction as well.

View larger version (25K): [in this window] [in a new window]   Fig. 3. Plot of observed d vs. calculated SA0.52vol (with unknown proportionality constant A). The FLTF, VOC, and ITS soils are each denoted by an oval. The straight line fit through the origin is an approximate minimum value of d for a given SA0.52vol, and is, on account of its lack of a y-intercept, compatible with the Moldrup relation, Eq. [11], for the threshold moisture content for solute diffusion.

View larger version (23K): [in this window] [in a new window]   Fig. 4. Plot of observed d vs. calculated R-0.052m, where Rm is the geometric mean of r0 and rm. Note that similar parameters are obtained by the linear regression as in Fig. 3, as well as the value of R2.

The values for _d at which the fractal description of the water-retention curve deviates from experiment are mostly compatible with the Moldrup et al. (2001) relation. We suggest that this is additional evidence that this relation can be used to calculate the important parameter, _c. We further suggest that the deviations from Eq. [11] are indicative of equilibration problems in water-retention experiments when the hydraulic conductivity was 5 x 10^-8 cm s^-1. Many of our soils are common to those investigated by Khaleel and Relyea (2001), and our data come generally from the same sources. For the geometry of Fig. 1 of that paper, the time required to reduce of a soil by 0.01 with K = 5 x 10^-8cm s^-1 would be 5 wk, nearly equal to their longest experiment, 6 wk. Note that nearly all the K values in Fig. 3 of Khaleel and Relyea (2001) indeed exceed this lower limit of K.

Now invert Eq. [4] for K(S) to solve for ,

[12]

We check whether K(_d) for the FLTF soils can be so low that equilibrium would not be attained. Try K(_d) 5 x 10^-8 cm s^-1. K_S 10^-4 cm s^-1 so that K(_d) 5 x 10^-4 K_S, and log[5 x 10^-4] = -3.3. Then

[13]

Solving this equation (using the mean value of D = 2.78 for the FLTF soils and assuming that the mean _d = _c = 0.204) requires that

[14]

which is about 10% larger than a typical value of the porosity, 0.496, in these soils (using the correct would require _d to be less than _c by 0.049). So, in actuality, K(_d) in these soils should still be a little above 5 x 10^-8 cm s^-1, and _d is probably determined by percolation theory.

We try a comparable test of the ITS soils, which have a wide range of values of K_S. The ITS soils have mean fractal dimension D = 2.91, mean porosity, = 0.302, and geometric mean K_S = 0.0032, but many values as low as those of the FLTF soils. Using K_S = 0.0001 leads to

[15]

Here we have taken a typical _c as 0.05 in the ITS soils. The result, 0.05 + 0.05 = 0.1, is 125% of the expected value, 0.08, from the regression. Thus, using a value of _c, which is smaller than the measured _d by about 0.03, leads to an expected _d, which is larger than _c by about 0.05, and _d is controlled not by percolation theory, but by small values of K associated with capillary flow. Note that the ITS soils (1-1417, 1-1418, 2-1417, and 2-2230) with similar K_S values to the FLTF soils have _d values 0.088, 0.08, 0.037, and 0.11, three of which are among the four largest values of _d in the 15 ITS soils. Note, however, that taking the geometric mean K_S for the ITS soils (use -4.8 in place of -3.3 in the exponent of Eq. [15] above) leads to _d = _c - 0.028.

We should stress that if any value of _d required by a limit on equilibration due to small K (through Eq. [15]) is less than _c, the conditions for which Eq. [15] has been derived have been violated. In this case, _c must be the limiting value for _d. Thus, development of a scatter plot to find a relationship for _d due simply to a low K(S) should at least exclude the points for the FLTF soils, as their _d values were not determined by this limit of K. We show the results of this analysis in Fig. 5.

View larger version (21K): [in this window] [in a new window]   Fig. 5. Plot of observed d vs. calculated SA0.52vol (excluding the FLTF soils). The same straight line, which bounded all the measurements, is reproduced on this graph, and that and the linear regression are solved simultaneously for the moisture content at which the two lines cross, = 0.12.

For additional confirmation, we simply plot in Fig. 6 K(h_min) (h_min is the largest suction head, i.e., it corresponds to the most negative value of the matric potential) from theory for most of the investigated soils—note that K(h_min) can be found from substituting Eq. [9] into Eq. [4] using h_min from Table 1). None of the FLTF soils have K(h_min) less than 5 x 10^-8 cm s^-1, but approximately one-third of the ITS and one-half the VOC soils do. These low values of K(h_min) tend to have different causes in different soil groups.

View larger version (15K): [in this window] [in a new window]   Fig. 6. Predicted values of K(h), using experimentally determined values of threshold moisture content, air-entry pressure, and the first experimental head value at, or below, the deviation of h() from the fractal prediction. While the FLTF loams all have predicted K values above the minimum value for equilibration, roughly one-half of the sands or gravelly soils in the VOC and about one-third of those in the ITS regions are expected to have K values too low to equilibrate. Note that the "others" category denotes ERDF and B8814 loamy sands.

1. Larger maximum pore sizes tend to make equilibration easier at low moisture contents.
   
2. Smaller porosity tends to make equilibration easier.
   
3. Wider distributions of pore sizes tend to make equilibration more difficult.
   
4. Smaller critical-volume fractions tend to make equilibration more difficult.
   

With increasing coarseness, one may expect all of the four trends listed above, and it would be difficult, a priori, to judge what the net effect would be. Because some of these factors tend to produce opposing effects, there might not be any trend in many cases. In the present case, the largest maximum pore size scarcely increased with increasing coarseness of the soils, allowing the last two trends to more than compensate for the diminishing porosity. Insofar as this tendency for the largest pore sizes to be independent of texture in very coarse soils is general (note again the suggestion of Khaleel and Relyea [2001] regarding the need to treat gravelly soils differently), it might be true that the problem of equilibration due to low-capillary flow values of K is more frequently observed in coarse soils. In our case, the general trend of K with texture is rather weak, but as Fig. 6 shows, the variability of the coarser soils is higher (contradicting the analysis of Khaleel and Relyea [2001]), and this leads to more frequent violations of equilibrium criteria in the coarser soils, as well as (discussed below) a tendency to flatten curves of log[K(h)] vs. h, if equilibrium is not attained.

Although such general trends are difficult to identify, it is easy to see that the reason for low K(h_min) values in the ERDF and B-BX tank farm (B8814) and many of the ITS soils is because of the small values of K_S (typically 10^-5 cm s^-1 or less). On the other hand, the reason for the low values of K(h_min) in many VOC soils is their tendency toward complexity, and the reader is referred to Hunt and Gee (2002) for a discussion of how such structural complexity can produce a surprisingly rapid drop of K with diminishing S or larger (increasingly negative) values of h. This ability to identify and interpret potential problems with equilibration consistently with the deviations in _d from values expected from the Moldrup et al. (2001) is further evidence that the deviations from fractal predictions at low moisture contents are not due to deviations in structure from fractal descriptions, but lie outside the domain of structure and in the realm of dynamics.

The above discussion has important implications for measurements of the hydraulic properties of Hanford site soils. Suppose that the tension required to remove water from a soil is h_i, but that actually a significantly larger value, h_i + h, has been applied because applying h_i did not seem to reduce the water content over the previous value (because of the long equilibration time at a very low K). Then the water content that should be attained at h_i is not attained until h_i + h is applied. This tends to associate too large a value of h_i with a given moisture content, and measured conductivity values are thus assigned too large a value of h, stretching out the K(h) curve horizontally. The problem with equilibration appears most severe in the coarsest soils, so that these data for large h are most frequently stretched out when measurements of K(h) are not performed at equilibrium. This tendency is at least consistent with Fig. 3 of Khaleel and Relyea (2001), where the gravelly soils tend to flatten out noticeably, and with their conclusions about the lack of variability of the slope of log[K(h)] in the large (h) regime for gravelly soils. The lack of variability says more about the uniformity of measurement, however, than about the uniformity of the soils.

If it is a general problem that equilibration times for coarse-textured, poorly sorted soils common in arid regions can be extremely long, then field conditions are undoubtedly common, which would require using nonequilibrium values of K. However, this implies the necessity to determine a general time-dependence of K, and not just a single value related to a particular nonequilibrium state. Then future modeling studies would be able to select the appropriate K for the particular history of the soil.

We note that our assertion that the water-retention curves of the coarser soils at large h are not in equilibrium is supported by the frequent observation that these curves are multi-valued; clearly, an equilibrium relationship of h() should be a function and can thus only have a single value.

Finally, we believe it useful to show our predicted values of K(h) for the present suite of soils so as to make a concrete prediction, even though we cannot yet compare directly with experiment. The choice of representation is motivated by that of Khaleel and Relyea (2001)(their Fig. 7) to represent K as a function of h, as well as by our qualitative discussion of effects of nonequilibration on K(h). As seen in Hunt and Gee (2002), K(h) is quite insensitive to the value of _c. Where differences between theory and experiment exist, it will be an important task to identify the causes of these differences.

View larger version (24K): [in this window] [in a new window]   Fig. 7. Simultaneous plots of predicted K(h) for Hanford site soils. The solid circles are the gravelly sands or sandy gravels, the open squares are sands, and the dashes are loams.

	CONCLUSIONS

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Observed water-retention data deviated from the fractal model predictions for < _d, with _d defined graphically. In the finer soils, the deviation appeared to be due to the break-up of the capillary flow network below the critical moisture content for percolation. In some of the coarser soils, the lack of equilibration may have been due to small values of K. We found that the Moldrup et al. (2001) relationship for the threshold moisture content for diffusion, _t, correlated reasonably well with _d. Thus, the Moldrup et al. (2001) relationship can be used to calculate the critical volume fraction for percolation. Use of this critical volume fraction makes it possible to predict the magnitude of K(S)/K_S at any moisture content and soil texture using only the saturated hydraulic conductivity and physically obtained quantities.

	ACKNOWLEDGMENTS

 
The authors are grateful to Eugene Freeman for directing us to key Hanford soil data, including his Master's Thesis, and for helping us complete Table 1. This study was funded by the Vadose-Zone Transport Field Study, which is part of the Hanford Science and Technology Program of the Department of Energy's Richland Operations Office under Contract DE-AC06-76RL01830.

	REFERENCES

TOP ABSTRACT INTRODUCTION THEORY MATERIALS, DATA SOURCES, AND... RESULTS AND DISCUSSION CONCLUSIONS REFERENCES

 

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