HOME HELP FEEDBACK SUBSCRIPTIONS ARCHIVE SEARCH TABLE OF CONTENTS		QUICK SEARCH:   [advanced] Author: Keyword(s): Year:  Vol:  Page:

This Article Abstract Figures Only Full Text (PDF) Alert me when this article is cited Alert me if a correction is posted Services Similar articles in this journal Alert me to new issues of the journal Download to citation manager Citing Articles Citing Articles via HighWire Citing Articles via Google Scholar Google Scholar Articles by Hunt, A. G. Articles by Gee, G. W. Search for Related Content PubMed Articles by Hunt, A. G. Articles by Gee, G. W. GeoRef GeoRef Citation Agricola Articles by Hunt, A. G. Articles by Gee, G. W. Related Collections Hysteresis Water Retention/Capillary Pressure Fractal Approaches Soil Models

ORIGINAL RESEARCH PAPERS

Wet-End Deviations from Scaling of the Water Retention Characteristics of Fractal Porous Media

CIRES, Univ. of Colorado, Boulder, CO 80309
^* Corresponding author (allenghunt{at}msn.com).

Received 27 January 2003.

	ABSTRACT

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

 
A previous work investigated the dry-end deviation from fractal scaling of water retention characteristics of a suite of 43 U.S. Department of Energy (USDOE) Hanford site soils in relationship with the vanishing of solute diffusion at a moisture content _t. It was found that the deviation from fractal scaling of the water retention set on at a moisture content typically about 0.06 higher than the moisture content at which solute diffusion vanishes. Assuming that the vanishing of solute diffusion resulted from a lack of continuity of the water phase, we interpreted the deviation from fractal scaling in terms of a lack of water-phase continuity rather than as a breakdown of the fractal model. Now the wet-end deviations from fractal scaling of the same suite of soils are investigated. It is shown that the wet-end moisture contents at which the deviation occurs are correlated with the critical volume fraction for percolation and with the dry-end deviations. However, wet-end deviations occur at moisture contents closer to full saturation than do dry-end deviations from zero saturation. This contrast between the wet and dry ends of the water retention curve, h(), is suggested to be a result of the role of equilibration in the experimental determination of h(), and thus to be traceable to the much lower values of the hydraulic conductivity, K, at the dry end.

Abbreviations: ERDF, environmental restoration disposal facility • FLTF, field lysimeter test facility • ITS, injection test site • USDOE, United States Department of Energy • USE, United States ecology • VOC, volatile organic carbon

	INTRODUCTION

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

 
RECENTLY we applied continuum percolation theory to address the deviation from fractal scaling (Rieu and Sposito, 1991) of water retention curves at the dry end (Hunt and Gee, 2002a). The analysis revealed that the water content value _d, at which this deviation set on, was closely related to the water content _t, at which solute diffusion vanishes (Moldrup et al., 2001). The subscript t here stands for threshold. The _t was shown to be proportional to a power of the surface area to volume ratio, A/V. That solute diffusion, which requires water phase continuity, vanishes at a finite value of , implied the relevance of percolation theory to both solute diffusion and water retention. The latter relevance was assumed based on the importance of a continuous connected network of capillary flow paths to equilibration; in the absence of such a connected network, other much slower means of water transport such as film flow or vapor phase transport would be required for equilibration.

Various regressions employed (Hunt and Gee, 2002a) tended to the result

[1]

In Hunt and Gee (2002a), the existence of the intercept with value 0.06 was not explained. A subsequent work (Hunt, 2004) related the intercept to a change in the form of the hydraulic conductivity [K(S)] as a function of saturation (S) that is appropriate already above _t. This change in form was derived using the concept of a crossover from fractal scaling to percolation scaling of K(S) (to be summarized below). Then the restriction on the equilibrium measurements of water retention characteristics was assumed related to the sudden diminution in K at typical values of (for the same Hanford site soils) about 0.06 greater than _t.

Just as percolation theory may be assumed relevant for a minimum moisture content at which the water phase remains continuous, it should be relevant to a minimum air content as well; that is, that air cannot flow in unrestrictedly until a continuously connected network of pores, which could potentially be filled with air (according to the tension), actually exists. Accordingly, the point of the present note is to address the deviations from fractal scaling at the wet end.

We show that, in the USDOE Hanford Site soils investigated, the wet-end deviation from fractal scaling starts at an air fraction, which is related to the same function of A/V as is the water content, at which the deviation from fractal scaling starts at the dry end. The calculation of K(S) (Hunt, 2001) assumed that this critical volume fraction for percolation was independent of saturation. Experimental results for solute diffusion (Moldrup et al., 2001) also imply this constancy. Thus, one might first assume that the deviations from fractal scaling should be roughly symmetrical; that is, that they should set on at _t and at - _t. But the deviation at the dry end starts (typically) at about _t + 0.06, so that the wet-end deviation, which cannot be related to small K values, should thus occur at = - _t, much closer to saturation than _t + 0.06 is to zero water content.

	THEORY

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

 
Hydraulic Conductivity Crossover
In continuum percolation theory, the scaling of K(S) is expressed in the equation (Hunt and Gee, 2002b),

[2]

with K_S the value of K at full saturation, the porosity, _c the critical volume fraction for percolation, and D the fractal dimensionality of the pore space. The value of _c is not a fit parameter by virtue of the interpretation of continuum percolation theory and the associated identification, _c = _t. Moldrup et al. (2001) found that

[3]

where A/V was determined by N₂ BET measurements. Typical values of _t for sands fall in the range 0.03 to 0.09; for loams, 0.10 to 0.15; and for more clayey soils, 0.16 to 0.22.

Physically, Eq. [2] represents the saturation dependence of the minimum hydraulic conductance that cannot be avoided by water flowing on the most conductive flow paths. This dependence is essentially equal to the third power of the pore radius since with Poiseuille's law, the flow through a tube is proportional to the fourth power of the tube radius and inversely proportional to its length. The assumption of fractal media is consistent with self-similarity, meaning that pore lengths and pore radii should be assumed to be proportional to each other, leading to the power three. With reduction in saturation toward _c, however, the most important limitation to the hydraulic conductivity of the system is no longer the rate-limiting conductance of the most conductive paths. Near the critical volume fraction for percolation in an infinitely large system, the separation of paths, which carry water, must diverge. In a finite sample, where the percolation limit cannot be reached, the corresponding limit is to a single flow path that can just span the sample. The appropriate divergence is given by the correlation length of percolation theory, which may be expressed as (Stauffer, 1979)

[4]

with = 0.88, a critical exponent from percolation theory, and ₀ a fundamental-length scale on the order of a typical pore separation. The topological restriction on K requires that K tend to 0 in the limit _c, and this constraint must begin to dominate the minimum pore size restriction as this limit is approached. Thus, there exists a minimum moisture content at which the conductivity is dominated by a rate-limiting conductance (a fractal scaling), and below this moisture content K is determined by a diverging path separation (percolation scaling). The result for K(), consistent with percolation scaling, is given by

[5]

where K₀ is a hydraulic conductivity scale constant, ₁ is the value of at which the crossover from Eq. [2] to [5] occurs, and 1 < t 2 is a critical exponent from percolation theory (Golden, 1990). The value t = 1.76 arises from representing K as inversely proportional to the square of the path separation, or the number of water-carrying paths per cross-sectional area. But the uncertainty in ₁ arising from the uncertainty in the value of t is not large, and t = 1.76 will be used in the analysis. The onset of the percolation scaling regime where Eq. [5] becomes valid is found by applying the conditions that at that value of , both K and dK/d must be continuous, equations that together define K₀ and ₁. It can then be shown that (Hunt, 2004)

[6]

For a specific case, this crossover is shown pictorially in Fig. 1: D = 2.85, t = 2, _t = 0.08, and = 0.44 (not very different from the McGee Ranch silt loam).

View larger version (15K): [in this window] [in a new window]   Fig. 1. Results for log(K) (vertical) vs. S (horizontal) of a representative soil due to capillary flow using fractal scaling for large , and percolation scaling for near the percolation threshold. Here, D was chosen to be 2.85, t as 0.08, and as 0.44. From Eq. [6], t turns out to be t + 0.062. The inverted triangles represent the composite calculation for K. The open circles give the percolation scaling treatment extension into the fractal scaling regime, and the fractal scaling extension into the percolation scaling regime. The vertical line gives t, and the vertical arrow points to the crossover in validity from Eq. [2] to Eq. [4] for K at t as determined from Eq. [6].

The process of drying from saturation requires air entry into the porous medium, just as the process of saturation from the dry end requires water entry. We presented arguments in Hunt and Gee (2002a) regarding the deviation from fractal scaling at the dry end on a drying curve. To add water to a truly dry medium requires first that the matric potential be raised to a value such that those pores that water is allowed to occupy are actually integrated into an interconnected network of unbounded size (that reaches the edges of the sample). For the drying curve (in the absence of structure, or aggregations), air cannot enter throughout the medium until the tension is high enough, which we will refer to as the bubbling pressure, to allow air into sufficient number of pores that these pores are integrated into an interconnected network of unbounded size. These conditions are, respectively, the percolation of water and the percolation of air.

There are many implications of these concepts. First, we expect the deviations from fractal scaling at the wet and dry ends to be related. Second, if solute diffusion and gas diffusion are both measured on a drying curve, we expect that the gas diffusion will tend to vanish at zero air content (since no air can enter until the air phase is continuous), while solute diffusion vanishes at a finite moisture content (because the moisture that is left over is not connected below the percolation threshold) (Hunt and Ewing, 2003). This is precisely the case from observation (Moldrup et al., 2001). If these measurements were repeated during imbibition, we would expect the opposite. Then the gas diffusion should vanish when so little gas remained that the gas phase did not maintain continuity (air entrapment), while the solute diffusion would vanish at zero water content since essentially no water enters until the water phase can be continuous. Thus, the discussions here are relevant not only to the value of the critical volume fraction for percolation, and to the question of whether this value remains a constant across the entire range of saturations, but the entire interpretation of two-phase flow and equilibrium, with specific examples from various diffusion properties.

Summary of Percolation Concepts and Related Assumptions
1. The bubbling pressure of an initially saturated porous medium defines the establishment of an infinite interconnected network of pore space, allowed by the Young-Laplace equation to contain air.

2. Excluding the pore volume of the largest pores, which may coincidentally be open to the sample edges (a sample size-dependent fraction) in the absence of soil structure, or related spatial correlations in the positions of the largest pores, the bubbling pressure should equal the pressure at which air actually enters. However, the characteristic pressure in the water retention curve, usually called the air entry pressure, corresponds to the pressure at which the largest pores drain, if air can get to them.

3. The water entry pressure of an initially dry porous medium defines the establishment of an infinite interconnected network of pore space, allowed by the Young-Laplace equation to contain water. Here, effects of the smallest pores open to the edges must also be excluded as a sample size-dependent contribution.

4. Absent explicit reason to assume the contrary, the volume fractions defining the water entry pressure and the bubbling pressure are suggested to be the same, and this value is hypothesized to be the same volume that governs the vanishing of solute diffusion.

5. The most reliable relation for this volume, called by us the critical volume fraction for percolation, is given (Moldrup et al., 2001) in terms of the surface area to volume ratio of the medium.

6. In pressure saturation curves measured with, for example, ceramic plates, limitations on the relationship between the cumulative pore-size distribution and the water content at the wet and dry ends are placed by the effects of phase continuity given in the above concepts.

7. The effects of air and water phase continuity on deviations from expected (fractal) scaling do not manifest themselves equally on a drainage curve for two reasons: (i) because the hydraulic conductivity is a diminishing function of moisture content, and (ii) because of the asymmetry of the processes of air entry and water exit.

The complexity of natural porous media increases with high clay content. We anticipate that at least Assumption 4 may break down at high clay contents due to the importance of internal surface areas and bound water at both ends of the saturation spectrum. In the following investigation, Assumption 4 is not critically tested because Hanford site soils do not typically contain much clay. Further, real experiments on finite-sized systems will not generally satisfy Assumption 2 above.

	MATERIALS, DATA SOURCES, AND METHODS

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

 
Experimental Data
We used the same tabulated data sources for 43 Hanford soils as in a previous paper (Hunt and Gee, 2002a). We selected these soils because of programmatic interest related to improving our predictions of contaminant movement in the vadose zone at the USDOE's Hanford Site. Hydraulic property data at the Hanford Site is limited and methods to predict these properties from grain size and other available data sets are being sought. The samples were obtained from the Westinghouse Geotechnical Engineering Lab and the Pacific Northwest National Laboratory (Freeman, 1995; Khaleel and Freeman, 1995), and originally collected in conjunction with drilling operations. Samples were sometimes extracted from the borehole with a drive barrel, sometimes using a splitspoon sampler. Injection test site (ITS) soil samples were derived from air-rotary drill cuttings and collected in 18.9-L (5-gallon) plastic bags, while the volatile organic carbon (VOC) soils were obtained with a splitspoon sampler. The field lysimeter test facility (FLTF) soils were repacked. The hydraulic property data were used in modeling flow and transport at the site (Sisson and Lu, 1984; Gee and Ward, 2001). Water retention characteristics were typically measured in the laboratory in conjunction with the unsaturated hydraulic conductivity in a steady state head control apparatus (Klute and Dirksen, 1986) with tensiometers built into the side ports (Khaleel and Relyea, 2001). The remaining soils mentioned here are United States ecology (USE) and environmental restoration disposal facility (ERDF) soils. The VOC soils are usually sands with small amounts of silt and clay, and not infrequently, gravel mixed in; the FLTF soils are silt loams or loams; while the ITS soils are sands (two with gravel components). The USE soils are also mostly sands, while the ERDF soils included are loamy sands. For further details, the reader is referred to Hunt and Gee (2002a).

Fractal Dimension and the Deviation from Predicted Water Retention
The fractal dimension of the pore space, D, was obtained in Hunt and Gee (2002a) from the particle size distribution and the porosity. These values are given in Table 1 (from Hunt and Gee, 2002a). In Hunt and Gee (2002a), we applied the predicted water retention curves from Eq. [7],

[7]

[8]

This increase in h over h_A is then understood to correspond to the observed experimental increase in the primary drainage curve seen in nearly all the ITS soils except 2-1417, and exemplified by Fig. 2d). For comparison of the dry- and wet-end deviations from fractal scaling we will be able to use the deduced result for _t from the dry-end deviation.

View larger version (24K): [in this window] [in a new window]   Fig. 2. Determinations of wet-end moisture contents () at which deviation from fractal scaling of water retention occurs for four soils. The vertical axis is the tension (h). The open circles are theory, the solid circles experiment. The fractal dimensionality for the pore space was determined in Hunt and Gee (2002a) from the particle-size distribution and the porosity, and the air-entry head (hA) was used as an adjustable parameter. The wet-end deviations from fractal scaling are indicated with arrows. Two soils from Hunt and Gee (2002a) are used for which the wet-end deviation could clearly be seen, FLTF D11-06 (Fig. 2a) and VOC 3-0652 (Fig. 2b). For a number of the ITS soils, such as 2-2227 shown here, w was better determined from the pressure applying Eq. [8] as experimental values of h tend to rise above the theoretical curve at large .

	RESULTS AND DISCUSSION

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

Determination of Wet-End Deviation from Fractal Scaling, _w

Comparison of _w with _t
The value of - _w was then compared with _t derived from the experimentally measured _d for the various soil types in Fig. 3. We used Eq. [6] to find _t from ₁, the latter of which was assumed to be _d. Note that those four cases where Eq. [6] predicts _t < 0 were excluded from further analysis, as such an unphysical value implies an error either in the experiment or the analysis. A further soil (VOC 3-0650), for which _d was 0.37, was excluded because of the clear irrelevance of this value of _d. The same general procedure was followed for each of the three main soil groups (ITS, FLTF, VOC) and for the other soils separately. The complications that derived from the ambiguities with the ITS soils had to be dealt with separately, and are described in the following paragraph. The values of R² for the individual soil types were usually quite small, that is, approximately 0.25 or 0.3, but in the case of the ITS soils, the value was essentially zero. For individual soil groups, the correlations also failed standard statistical tests (e.g., normality). Some statistical quantities for the comparisons are given in Table 2. In Fig. 4, however, the same relationship for all the soils was plotted. In this case, R² 0.6 and the correlation passed the same standard statistical tests. What this appears to mean is that the correlation sought for is degraded by difficulties with either experimental procedure or with the analysis, so that if too small a sample (with too little variability of the critical volume fraction for percolation) is analyzed, the results tend to be ambiguous.

View larger version (21K): [in this window] [in a new window]   Fig. 3. Correlation of predicted and observed wet-end deviations from fractal scaling. (Upper right) VOC, (lower right) FLTF, (upper left) other, and (lower left) ITS soils. The predicted value is equal to t, which is deduced from d using Eq. [6].

View larger version (17K): [in this window] [in a new window]   Fig. 4. Correlation of predicted and observed wet-end deviations from fractal scaling for all soils. The predicted value is equal to t, which is deduced from d using Eq. [6].

The consistency of the values of the slopes and intercepts for the different soil groups in the experimental relationship between _t and - _w suggests that these values have some meaning. The intercept is pretty nearly zero, but the slope is less than 1. Provided that Eq. [6] is an accurate representation of the relationship between _t and _d, the tendency to generate a slope of between 0.7 and 0.8 is an indication that the critical volume fraction for percolation may be less for air than for water, which would suggest that the critical volume fraction for percolation is not quite a constant across the range of saturations. This slight inconsistency could be due to the presence of water on internal clay mineral surfaces, a contribution, which would not be required for air percolation on the wet end of the pressure-saturation curve. If this is the explanation for the slight inconsistency in the Hanford soils (with minimal clay content), however, then the equivalence of - _w and _t will break down more seriously in soils with high clay content.

For another perspective, Fig. 5 shows comparisons of the predictions of the Moldrup et al. relation for _t (Eq. 3) with both the observed _t = _t, derived from _d and - _w for all the studied soils. To make the comparison neater, the calculation of A/V (from Hunt and Gee, 2002b) was multiplied by the constant factor 0.16. Note that the correlation for the dry end has R² = 0.7, while the correlation at the wet end is somewhat lower, with R² = 0.6. For the dry end, the slope is 1.05, while for the wet end, it is 0.95. The intercepts are also both positive, 0.03 and 0.02, but they are small enough to be neglected. In this case, the slope for the wet-end deviation is only about 10% smaller rather than above 20%, but the implication is again that the critical volume fraction for percolation may be a little smaller for air than for water.

View larger version (19K): [in this window] [in a new window]   Fig. 5. Comparison of the values of - w and t (deduced from d) with c as predicted from Eq. [3] for the moisture content at which solute diffusion vanishes, and inferred to be the critical volume fraction for percolation. In this comparison the calculated value of A/V from the particle size distribution was used as described in Hunt and Gee (2002b), but the unknown proportionality constant was uniformly taken to be 1/6 to facilitate comparison.

That some water retention curves approach fractal scaling from below (with a negative curvature) could, in principle, be the result of structure in the soil, spatially correlating the positions of the largest pores with each other. If sufficiently well correlated, air could enter nearly all these pores at virtually complete saturation. Soil structure has been considered to have important effects on pressure-saturation curves near the wet end of the curve (Nimmo, 1997). In the context of percolation theory, the spatial correlations would imply that for water contents near saturation, the critical volume fraction for percolation of either phase is nearly zero, but that this value rapidly increases to the value given by Eq. [3] as the water is drained from the largest, highly connected, pores. Such a phenomenon would be expected to produce a very rapid diminution of K with diminishing water content near saturation as well, which is often observed. However, the soils investigated have very low carbon content, typically thought important for aggregation, and the particular FLTF soils, for which this tendency is most obvious, were repacked as well. Clearly, the normal understanding of structure should not apply to these soils.

Though the interpretation thus seems less simple at the wet end than at the dry end, we think there is sufficient evidence to suggest that the process of air entry has a relationship with percolation theory, and that the critical volume fraction for percolation as inferred from the Moldrup et al. (2001) relationship can be used, at least in the coarse soils of the Hanford site, to some advantage to understand the wet-end deviation from fractal scaling, as well as the dry end.

	CONCLUSIONS

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

 
We have shown that a predictable relationship between wet- and dry-end deviations from fractal scaling in the water retention characteristics of the USDOE Hanford site soils exists. Since the dry-end deviation was previously interpreted in terms of continuum percolation theory, we infer a role of percolation theory in the wet-end deviation as well. Because the actual values of the critical volume fraction for air entry and the value for air entry inferred from the critical water volume for water phase continuity are closely related, our evidence suggests that the critical volume fraction for percolation is at most weakly dependent on saturation, in accord with the comparison (Hunt and Gee, 2002b; Hunt, 2004) of the predicted hydraulic conductivity with experimentation (Rockhold et al., 1988). This conclusion might be altered, however, if our study were expanded to include a larger fraction of finer soils. Although the critical volume fraction is approximately independent of saturation, at least for coarse soils, this does not imply that the water retention curves should exhibit analogous (antisymmetric) deviations from fractal scaling at low and high water contents. The tendency for the deviation not to be antisymmetric has two causes: (i) At the dry end, K(S) drops to very low values just above the percolation threshold, meaning that the deviation starts typically at moisture contents (for the Hanford soils) of 0.06 above the critical volume fraction for percolation. But at the wet end, several orders of magnitude higher K values allow equilibration to occur much more rapidly, and the deviation from fractal scaling seems not, in general, to persist to values of < - _c. (ii) A pressure-saturation curve has a direction; in the present case, these curves were obtained by drainage and the resulting deviation, at least for the ITS soils, is to an excess water content at both ends. The result for the coarse soils is to add a positive curvature to h() at both the wet and dry ends of the curve. The positive curvature at the wet end leads to a small but significant deviation from fractal scaling. The magnitude of the curvature is related to an overshoot in tension [higher tensions than the scaling tension (h_A) are required for air entry] of the observed vs. the predicted water retention curve. This overshoot is also generally in accord with the predictions of percolation theory. The existence of the opposite (negative) curvature in the fine soils at the wet end is at least consistent with an inference that these soils are structured, even though little evidence for this presumption in the case of the Hanford soils actually exists. However, we found no alternative explanations for the relevance of percolation theory to the wet-end deviations in scaling for these soils that was also consistent with the curvature of the deviations.

	REFERENCES

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

 

• Freeman, E.J. 1995. Fractal geometries applied to particle size distributions and related moisture retention measurements at Hanford, Washington. M.A. thesis. Univ. of Idaho, Moscow.
• Gee, G.W., and A.L. Ward. 2001. Vadose zone transport field study: Status report. PNNL-13982. Pacific Northwest National Laboratory, Richland, WA.
• Golden, K. 1990. Convexity and exponent inequalities for conduction near percolation. Phys. Rev. Lett. 65:2923–2926.[ISI][Medline]
• Hunt, A.G. 2001. Applications of percolation theory to porous media with distributed local conductances. Adv. Water Resour. 24(3,4):279–307.
• Hunt, A.G. 2004. Percolative transport in fractal porous media. Chaos, Solitons Fractals (Jan., in press).
• Hunt, A.G., and R.P. Ewing. 2003. On the vanishing of solute diffusion in porous media at a threshold moisture content. Soil Sci. Soc. Am. J. 67:1701–1702.[Abstract/Free Full Text]
• Hunt, A.G., and G.W. Gee. 2002a. Water retention of fractal soil models using continuum percolation theory: Tests of Hanford site soils. Vadose Zone J. 1:252–260.[Abstract/Free Full Text]
• Hunt, A.G., and G.W. Gee. 2002b. Application of critical path analysis to fractal porous media: Comparison with examples from the Hanford site. Adv. Water Resour. 25:129–146.
• Khaleel, R., and E.J. Freeman. 1995. Variability and scaling of hydraulic properties for 200 area soils, Hanford site. Rep. WHC-EP-0883. Westinghouse Hanford Co., Richland, WA.
• Khaleel, R., and J.F. Relyea. 2001. Variability of Gardner's alpha for coarse-textured sediments. Water Resour. Res. 37:1567–1575.
• Klute, A., and C. Dirksen. 1986. Hydraulic conductivity and diffusivity: Laboratory methods. p. 687–734. In A. Klute (ed.) Methods of soil analysis. Part 1—Physical and mineralogical methods. SSSA Book Ser. 5. SSSA and ASA, Madison, WI.
• 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]
• Nimmo, J.R. 1997. Modeling structural influences on soil water retention. Soil Sci. Soc. Am. J. 61:712–719.[Abstract/Free Full Text]
• Rieu, M., and G. Sposito. 1991. Fractal fragmentation, soil porosity, and soil water properties: I. Theory. Soil Sci. Soc. Am. J. 55:1231–1238.
• Rockhold, M.L., M.J. Fayer, and G.W. Gee. 1988. Characterization of unsaturated hydraulic conductivity at the Hanford site. PNL-6488. Pacific Northwest Laboratory, Richland, WA.
• Sisson, J.B., and A.H. Lu. 1984. Field calibration of computer models for application to buried liquid discharges: A status report. RHO-ST-46 P. Rockwell Hanford Operations, Richland, WA.
• Stauffer, D. 1979. Scaling theory of percolation clusters. Phys. Rep. 54:1–74.

This article has been cited by other articles:

				A. G. Hunt Continuum Percolation Theory for Saturation Dependence of Air Permeability Vadose Zone J., February 1, 2005; 4(1): 134 - 138. [Abstract] [Full Text] [PDF]

				A. G. Hunt and T. E. Skinner Hydraulic Conductivity Limited Equilibration: Effect on Water Retention Characteristics Vadose Zone J., February 1, 2005; 4(1): 145 - 150. [Abstract] [Full Text] [PDF]

This Article Abstract Figures Only Full Text (PDF) Alert me when this article is cited Alert me if a correction is posted Services Similar articles in this journal Alert me to new issues of the journal Download to citation manager Citing Articles Citing Articles via HighWire Citing Articles via Google Scholar Google Scholar Articles by Hunt, A. G. Articles by Gee, G. W. Search for Related Content PubMed Articles by Hunt, A. G. Articles by Gee, G. W. GeoRef GeoRef Citation Agricola Articles by Hunt, A. G. Articles by Gee, G. W. Related Collections Hysteresis Water Retention/Capillary Pressure Fractal Approaches Soil Models