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Pacific Northwest National Laboratory, Richland WA
* Corresponding author (fred.zhang{at}pnl.gov).
Received 4 December 2002.
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ABSTRACT |
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 TOP ABSTRACT INTRODUCTIONTHEORYMATERIALS AND METHODSRESULTS AND DISCUSSIONSSUMMARY AND CONCLUSIONSREFERENCES |
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Abbreviations: TCT, tensorial connectivity–tortuosity • 2-D, two-dimensional
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INTRODUCTION |
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TOPABSTRACTINTRODUCTIONTHEORYMATERIALS AND METHODSRESULTS AND DISCUSSIONSSUMMARY AND CONCLUSIONSREFERENCES |
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The effects of saturation on soil anisotropy in unsaturated hydraulic conductivity have not been well described mathematically. In unsaturated media, it is often assumed that the anisotropy at moisture contents less than saturation is the same as at complete saturation (e.g., Neuman, 1973). This assumption was questioned by Zaslavsky and Sinai (1981), who analyzed the effect of sloping layered soil on the unsaturated flow pattern. They suggested that while each layer is uniform and isotropic, the whole soil profile behaves anisotropically. Using a two-layer soil of finite thickness, they calculated the lateral flux component and defined a coefficient of anisotropy. The authors concluded that the anisotropy coefficient can range from zero at extremely low rate of rainfall up to the order of 106 for high rainfall rate. Laboratory and field experiments have subsequently confirmed that anisotropy is saturation dependent (Stephens and Heermann, 1988; McCord et al. 1991). Recently, Ursino et al. (2000) conducted laboratory experiments in a sand tank model packed to be heterogeneous and anisotropic with a sandy soil. The ratio of correlation length of the structure was 1:10. Steady-state water contents were observed at three different water fluxes. They found that increasing the flux increased the mean water saturation and reduced the anisotropy factor of the hydraulic conductivity tensor.
Mualem (1984) proposed a conceptual model to quantify saturation-dependent soil anisotropy. The soil was assumed to consist of numerous thin parallel layers having different hydraulic properties. Variation of the saturated hydraulic conductivity (Ks) among the layers was described by a probability density distribution. The model indicates that the degree of anisotropy of unsaturated soil may vary considerably from its value at saturation. When the pressure head is reduced gradually from zero, the anisotropy factor first decreases to a minimum and then increases rapidly as the soil dries. In some cases, the anisotropy factor can reach values several orders of magnitude higher than at saturation. A wide range of permeability distribution of the layers may enhance anisotropy of unsaturated soil.
Theoretical analysis based on stochastic methods (Yeh et al., 1985) suggests that in a steady flow field the anisotropy of a stratified heterogeneous soil should increase as the mean pressure head (and moisture content) of the soil decreases. Khaleel et al. (2002) coupled the stochastic method with Monte Carlo simulations to investigate effective flow and transport properties for unsaturated coarse-textured media under relatively dry conditions. They also found that the macroscopic anisotropy increased with decreasing saturation. Using the stochastic approach, Polmann et al. (1991) presented a generalized model to account for tension-dependent anisotropy. They assumed that the soil unsaturated hydraulic conductivity function obeyed the Gardner (1958) relationship and the horizontal correlation scales were much larger than the vertical correlation scales. However, application of the Polmann et al. (1991) model requires the knowledge of the variance of ln(Ks), with Ks being the saturated hydraulic conductivity, the correlation between hydraulic parameters, and the vertical correlation length. Typically, such information is not readily available.
Recognizing that none of these studies considered anisotropy at the pore scale, Ursino et al. (2001) derived analytical expressions for the macroscopic anisotropic conductivity tensor for different pore-scale geometries. The expressions show that the geometry of the microstructure can lead to anisotropic behavior at the macroscopic scale. Anisotropy in the hydraulic conductivity switched from values <1 to values >1 or vice versa, depending on saturation. However, the existence of a switching point of anisotropy may not be always true.
There are different types of models available to describe the relationship between unsaturated soil hydraulic conductivity and pressure head or water content (e.g., Gardner, 1958; Burdine, 1953; Mualem, 1976). As was pointed out by Bear et al. (1987), essentially all models of unsaturated flow represent the effective hydraulic conductivity (K) of an anisotropic porous medium as a product of the saturated hydraulic conductivity tensor, and a relative hydraulic conductivity (Kr):
 | [1] |
The objective of this study was to extend existing hydraulic functions to allow description of saturation-dependent anisotropy by introducing a TCT concept. The TCT concept was tested using numerical experiments of two-dimensional (2-D) flow through synthetic soils with different degrees of heterogeneity and anisotropy.
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THEORY |
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TOPABSTRACTINTRODUCTIONTHEORYMATERIALS AND METHODSRESULTS AND DISCUSSIONSSUMMARY AND CONCLUSIONSREFERENCES |
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 | [2] |
 | [3] |
 | [4] |
s and r are the saturated and residual water content, respectively; is water content; he is the pressure head at air entry; 
and n are fitting parameters that characterize the width of pore size distribution; 
is a fitting parameter that is inversely proportional to the pressure head at air entry; and m is a constant that is commonly approximated by m = 1 - 1/n (van Genuchten, 1980). Equation [2] or [3] may also be used to describe the retention curve of a heterogeneous soil with all the parameters being replaced by their corresponding effective values.
Unsaturated Hydraulic Conductivity of Isotropic Soils
With the assumption of pore continuity and connectivity, Burdine (1953) and Mualem (1976) proposed relationships for unsaturated hydraulic conductivity in terms of water content or pressure head. A general expression of the two can be written as (Hoffmann-Riem et al., 1999)
 | [5a] |
 | [5b] |
are constants. Equations [5a] and [5b] reduce to the Burdine (1953) relationship when ß = 2 and = 1 and to the Mualem (1976) relationship when ß = 1 and = 2. Mualem (1976) noted that L might be positive or negative. He found that L = 0.5 was an optimal value for a data set of 45 disturbed and undisturbed samples. Schuh and Cline (1990) reported that L varied between -8.73 and +14.80 for a data set of 75 samples. Yates et al. (1992) found that L varied between -3.31 and values greater than +100 and the geometric mean of L was 0.63 with a 95% confidence interval between -0.88 and +2.44. Schaap and Leij (2000) estimated the hydraulic parameters of 235 samples from the UNSODA database by fitting them to the retention and conductivity data. They found that the values of L for all textural groups ranges from -6.97 to -1.22, with the lowest values for the loams and clays, although not all individual samples had a negative L. Nevertheless, the most common approach in modeling K(h) is to use a constant value of 0.5.
A Tensorial Connectivity–Tortuosity Concept
The saturated hydraulic conductivity of an anisotropic soil is dependent on the direction of measurement primarily because of directional differences in pore connectivities and tortuosities. As the soil is desaturated, the flow path becomes less connected and more tortuous than that when the soil is saturated. Since different soils have different pore size distributions, the degree of change in flow path tortuosity with desaturation can be expected to differ between soils. Due to different apparent particle size distributions in different directions, there is no reason to assume that the pore connectivity and tortuosity are the same in all directions. Many studies have shown that anisotropy under drier conditions is larger than that under wetter soil condition (e.g., Zaslavsky and Sinai, 1981; Stephens and Heermann, 1988; McCord et al., 1991). These studies show that when an anisotropic soil is desaturated the decrease of pore connectivity and tortuosity in the horizontal direction is different from those in the vertical direction. The effects of this variation of pore connection and tortuosity on K is characterized by the term SLe in Eq. [5a]. Hence, the value of parameter L could be dependent on flow direction in an anisotropic soil, suggesting that it would be better described by a tensor. Therefore, an unsaturated hydraulic conductivity tensor may be written as
 | [6] |
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MATERIALS AND METHODS |
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TOPABSTRACTINTRODUCTIONTHEORYMATERIALS AND METHODSRESULTS AND DISCUSSIONSSUMMARY AND CONCLUSIONSREFERENCES |
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Synthetic Heterogeneous Soils
The hypothetical soils were generated in such a way that each soil is composed of 125000 soil elements, each one of which is homogeneous and isotropic but with properties different from an adjacent element. The soil elements were assembled into a composite by defining different correlation lengths in the three principal directions. Therefore, a composite soil was macroscopically anisotropic and heterogeneous.
The hydraulic properties of a soil element at spatial location s are defined by a nonhysteretic water retention characteristic, h(; s) and an unsaturated hydraulic conductivity function K(h; s). To facilitate description of soil heterogeneity, the hypothetical soil is assumed to be Miller-similar (Miller and Miller, 1956) at the macroscopic scale. Thus, the entire domain is characterized by a reference state {
*h, *K h*(), K*()}, with being the characteristic length, and a unique scaling relationship between the hydraulic functions at different spatial locations (Roth, 1995). In Miller-similar media, the relationship between h() at s and the reference state is given by (Sposito and Jury, 1990)
 | [7] |
) and the reference state is given by
 | [8] |
To generate fields of h(, s), h(s) was treated as a stationary random function with unique probability density and autocovariance functions. The logarithm of the scaling factor, f = log (h/*h), is assumed to be normally distributed with zero mean and variance 
2f (Roth,1995). An exponential covariance model with correlation lengths along the three principal directions was assumed for the autocovariance function. The autocovariance function was generated using sequential Gaussian simulation (SGSIM) program from the GSLIB library (Deutsch and Journel, 1992) with different correlation lengths () in the three principal directions. The full scaling invariance of the Richards equation for the whole soil domain requires a power law dependence of hydraulic conductivity on matric potential (Snyder, 1996):
 | [9] |
The values of K were calculated using Eq. [9] with the assumption 
= 0.8 (M.L. Rockhold, personal communication, 2002).
In this study, soil heterogeneity was described by the variance of f, 2f. The geometric mean of Ks, Ksg, that of , g, and the variances of Y = ln(Ks), 2Y, and of A = ln(), 2A, were then calculated using the generated Ks and (Table 1). Soil anisotropy was described by the ratio (R) of the correlation length of f at the direction parallel (p) to and that normal (n) to soil strata. Four levels of soil heterogeneity (i.e., 2f = 0.1, 0.25, 0.5, and 1.0) and four levels of soil anisotropy (i.e., R = 1:1, 10:1, 50:1, and 100:1) were generated on a 1.0-m3 domain uniformly discretized with the grid spacing of 0.02 m. This produced a total of 16 synthetic soils. The van Genuchten (1980) model was chosen to describe the hydraulic properties of the soils. The statistics of soil parameters for the 16 synthetic sandy soils are summarized in Table 1. The remaining parameters were set being constants and equal to the typical values of a sandy soil; that is, n = 4.0, s = 0.40 m3 m-3, r = 0.0 m3 m-3, and L = 0.5.
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Each numerical experiment started at saturation after which the soil was dewatered gradually in 32 steps of pressure heads from zero to -2.0 m. To make sure the flow was at steady state at each pressure head, the durations for each step were between 0.5 and 10 yr. The simulation time was longer when the soil became drier since the flow became slower. We could tell if the system was in steady state by comparing the water fluxes at the top and bottom boundaries. Unequal fluxes meant the system had not reached steady sate, and the simulation time was extended.
After determining the unsaturated hydraulic conductivity in the direction normal to the soil strata, Kn(h), the 2-D soil slice was rotated 90°. The numerical simulation was repeated to determine the conductivity in the direction parallel to the soil strata, Kp(h).
After all the numerical experiments were completed, the mean values of h and for the whole 1 by 1 m soil domain were calculated for each step. In all, there were 64 pairs of h(), 32 pairs of Kp(h), and 32 pairs of Kn(h) data for each soil. We chose to use the van Genuchten (1980) model to describe soil water retention curve. The 64 pairs of h() data were used to optimize the effective value of , e, and effective value of n, ne, while parameters s and r were fixed. Using the already optimized e and ne as constants, the parameters in the direction parallel to the soil strata, Ksp and Lp, were then optimized with the Kp(h) data. Similarly, the parameters in the direction normal to soil strata, Ksn and Ln, were optimized to the Kn(h) data.
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RESULTS AND DISCUSSIONS |
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TOPABSTRACTINTRODUCTIONTHEORYMATERIALS AND METHODSRESULTS AND DISCUSSIONSSUMMARY AND CONCLUSIONSREFERENCES |
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e is positively correlated with 2Y of the soil, while the value of ne is negatively correlated with 2Y. The dependence of e and ne on soil heterogeneity is due to the nonlinear relationship between and h. Figures 2 and 3 shows that both e and ne are functions of soil heterogeneity but independent of soil anisotropy.
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2Y = 1.717 and R = 50 (Fig. 4c), when the soil was saturated, the saturated hydraulic conductivities parallel and normal to stratification were 1.56 x 10-4 and 5.08 x 10-5 m s-1, respectively, giving an anisotropy ratio Ksp/Ksn = 3.1. At a pressure head of -2.0 m, the hydraulic conductivities parallel and normal to stratification were 7.36 x 10-9 and 1.88 x 10-10 m s-1, respectively, resulting in Kp/Kn = 39.2. This means that unsaturated hydraulic conductivity is more anisotropic when the soils become drier. These results are consistent with the findings of previous studies (e.g., Stephens and Heermann, 1988; McCord et al. 1991).
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Connectivity–Tortuosity Coefficient
Values of L for the soils with different levels of heterogeneity and anisotropy are depicted in Fig. 6. The value of L decreased linearly as the value of ln(R*) increased, where R* is the ratio of the correlation length of f in the direction parallel to flow and that normal to flow. Note that R* is defined relative to flow direction, while R was defined relative to the direction of soil strata. Hence, R* = R if flow is parallel to soil strata and R* = 1/R if flow is normal to soil strata. The values of R* range from 0.01 to 100 for the synthetic soils. The results show that L is inversely proportional to ln(R*). Larger values of L are indicative of less connected pores and/or more tortuous flow paths. Thus, L was smaller when flow was parallel to soil strata [ln(R*) > 0] than when flow was normal to the soil strata [ln(R*) < 0]. The effect of stratification on L is stronger when the soil is more anisotropic. This means that the difference between Lp and Ln becomes larger when a soil is more stratified.
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As many researchers (e.g., Schuh and Cline, 1990; Yates et al., 1992; Kosugi, 1999; de Vos et al., 1999; Schaap and Leij, 2000) have found, the L parameter can be negative (Fig. 6). A negative L may be explained by the expected difference between the configuration of real soil pores and the ideal cylindrical pores whose lengths are proportional to their radii as assumed by Mualem (1976). This difference may introduce some error when Eq. [5b], A(Se,ß,), was derived. The second term on the right-hand side of Eq. [5a], SLe, accounts for the effect of flow path tortuosity and pore connectivity on K. Theoretically, L should never be less than zero, which means no effects of flow path tortuosity and pore connectivity. However, the error in A(Se,ß,) due to the difference between real soil pores and the ideal cylindrical pores complicates the determination of L. We found that the Mualem (1976) model exhibits larger change in A(Se,ß,) than the Burdine (1953) model as Se decreases because of the larger value in the former model ( = 2) than that in the latter model ( = 1). However, the effect of a larger on hydraulic conductivity can be compensated by smaller values of L and ß (Kosugi, 1999). As a result, for the same soil, the L value in the Mualem (1976) model (typically L = 0.5) is smaller than that in the Burdine (1953) model (typically L = 2). It is possible that the Mualem (1976) model overestimates the changes in A(Se,ß,) as Se decreases; hence, a smaller L value, even negative, is needed to best describe the hydraulic function. Consequently, an optimized L is actually a lumped parameter that accounts not only for flow path tortuosity and pore connectivity, but for pore configuration as well.
It should be pointed out that the above results were obtained using synthetic sandy soils generated to be Miller-similar and under 2-D flow conditions with constant-head boundary conditions. Further tests of the TCT model are warranted using natural soils under different types of flow conditions.
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SUMMARY AND CONCLUSIONS |
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TOPABSTRACTINTRODUCTIONTHEORYMATERIALS AND METHODSRESULTS AND DISCUSSIONSSUMMARY AND CONCLUSIONSREFERENCES |
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The L tensor describes all the effects of soil heterogeneity and spatial correlations on the effective hydraulic conductivity, which need to be specified a priori in the stochastic method (Polmann et al., 1991). The widely used hydraulic functions can be easily extended by allowing L to take different values for the three principal orthogonal coordinate directions of flow. Hence, numerical solutions of the Richards equation can be easily modified to include the TCT concept.
An advantage of using the TCT concept is that the L tensor can be determined by measuring the hydraulic functions at different directions using direct methods or be optimized using inverse methods.
The value of L is a function of both soil heterogeneity and anisotropy and can be negative with current statistical pore size conductivity models. A negative L is attributed to differences between the configuration of real soil pores and the ideal cylindrical pores assumed in its derivation. An optimized L can be thought to be a lumped parameter that accounts, not only for flow path tortuosity and pore connectivity, but for pore configuration as well.
Further research is warranted to evaluate the applicability of the TCT concept to natural soils. This will require measurement of the directional hydraulic conductivity and an investigation of the relationship between the connectivity–tortuosity coefficient and soil anisotropy.
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ACKNOWLEDGMENTS |
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REFERENCES |
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TOPABSTRACTINTRODUCTIONTHEORYMATERIALS AND METHODSRESULTS AND DISCUSSIONSSUMMARY AND CONCLUSIONSREFERENCES |
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. Water Resour. Res. 21:457–464.This article has been cited by other articles:
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