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a Department of Mathematics and Computer Science, Eindhoven University of Technology, 5600 MB Eindhoven, The Netherlands
b Pacific Northwest National Laboratory, Richland, WA 99352
* Corresponding author (fred.zhang{at}pnl.gov)
Received 26 January 2004.
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ABSTRACT |
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 TOP ABSTRACT INTRODUCTIONEXPRESSION FOR THE HYDRAULIC...THE HYDRAULIC CONDUCTIVITY...THE HYDRAULIC RESISTIVITY TENSOR...DISCUSSIONCONCLUSIONSREFERENCES |
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Abbreviations: TCT, tensorial connectivity–tortuosity
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INTRODUCTION |
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TOPABSTRACTINTRODUCTIONEXPRESSION FOR THE HYDRAULIC...THE HYDRAULIC CONDUCTIVITY...THE HYDRAULIC RESISTIVITY TENSOR...DISCUSSIONCONCLUSIONSREFERENCES |
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The simplest models assume that the degree of anisotropy is saturation independent (i.e., the degree of anisotropy is independent of the water content). The saturation-independent model has been widely used in numerical modeling, although Neuman (1973) already doubted its correctness and felt that it could only be justified given the limited knowledge at that time. Philip (1987) attempted to find arguments in favor of saturation independency as a reasonable approximation, although he admitted that it "is unlikely to be exact." The advantage of saturation independency is that, just like for saturated soils, the flow equation for the unsaturated anisotropic soil can be transformed to that for an isotropic unsaturated soil. Philip (1987) used such a transformation to work out the consequences of anisotropy for three-dimensional adsorption from cavities of various shapes.
It appears that all derivations of Darcy's law for unsaturated anisotropic soils reported in the literature result in saturation-dependent anisotropy, no matter what is the underlying source of the anisotropy. Basically there are two sources of anisotropy: the microscale pore structure and the mesoscale layering or heterogeneity with directionally dependent correlation lengths.
Bear et al. (1987) considered a microscale three-dimensional orthogonal network of capillary tubes and introduced anisotropy by associating with each of the three directions a different lognormal probability density distribution of tube radii. On the basis of computer experiments, they calculated the degree of anisotropy as a function of saturation. Ursino et al. (2000) considered pore structures with anisotropic average number of pores and/or anisotropic pore size distribution. They worked out the consequences for the degree of anisotropy of the hydraulic conductivity by assuming that the pore size distributions associated with the principal directions are similar in the sense of Miller and Miller (1956). A notable feature of the results from the models based on anisotropic microscale pore structure proposed by Bear et al. (1987) and Ursino et al. (2000) is the possible existence, for a given pair of principal directions, of a saturation at which the corresponding principal hydraulic conductivities are equal and around which the major and minor axes become reversed. In a tracer experiment, McCord and Stephens (1987)(1988) did not find experimental evidence for this feature, which is lacking in models based on mesoscale layering or heterogeneity with directionally dependent correlation lengths. Ursino et al. (2001) performed tracer experiments in a laboratory sand tank and, using image analysis, found that low saturation led to very large heterogeneity and strong preferential flow and that saturation-dependent macroscopic anisotropy is an essential element of transport in unsaturated media.
Zaslavsky and Sinai (1981) showed that flow in a two-layered sloping soil, with each of the layers being homogeneous and isotropic, exhibits anisotropy at the scale of the entire profile. Inspired by this, Mualem (1984) proposed a conceptual model to quantify saturation-dependent soil anisotropy. In his model the soil is assumed to consist of numerous thin parallel layers having different hydraulic properties. Variation of the hydraulic conductivity at saturation among the layers is described by a probability density distribution. The model predicts that, with water content decreasing from saturation, the degree of anisotropy decreases, but in many cases reaches a minimum at some intermediate value of water content. Stephens and Heermann (1988) and McCord et al. (1991a) show that Mualem's model underestimates the degree of anisotropy.
Perhaps the most fruitful approach has been the stochastic analysis of large-scale flow in spatially variable unsaturated soils, in which the anisotropy arises from the directional dependence of the correlation lengths of the soil properties. Yeh et al. (1985a)( 1985b, 1985c) initiated this approach with an analysis of steady flows. Mantoglou and Gelhar (1987a)(1987b, 1987c) extended the model to transient flow, while Mantoglou (1992) generalized the model further to make it applicable for limited flow domains and nonstationary soil properties. Yeh et al. (1985a)(1985b) and Mantoglou and Gelhar (1987a) show that the macroscale effective parameters depend on the means, variances, and correlation lengths of the mesoscale water retention and hydraulic conductivity characteristics and on the mean flow condition. At the mesoscale, they used the Gardner (1958) exponential dependence of the hydraulic conductivity on the pressure head. Yeh et al. (1985b) show that in a steady flow field the macroscale anisotropy of a stratified heterogeneous soil increases as the mean pressure head and water content of the soil decrease (i.e., the anisotropy is saturation dependent). They also found a dependence on the spatial gradient of the pressure head, and Mantoglou and Gelhar found a further weak dependence on the temporal gradient of the pressure head, implying a form of macroscale hysteresis. Polmann et al. (1991) compared means and variances of the pressure head predicted by the stochastic theory of Mantoglou and Gelhar with results obtained with a detailed three-dimensional numerical model. They found that for a simulated infiltration event the stochastic theory gives the overall trends of the heterogeneous distribution of the pressure head. Polmann et al. (1991) emphasize that the stochastic model requires the knowledge of the variance of ln Ks, with Ks being the hydraulic conductivity at saturation, the correlation between hydraulic parameters, and the vertical correlation length. Typically, such information is not readily available.
McCord et al. (1991a)(1991b) also provide support for the stochastic model, by demonstrating that it gives the right order of magnitude for the degree of anisotropy found in field experiments. Green and Freyberg (1995) computed macroscale saturation-dependent anisotropy for steady gravity drainage in perfectly stratified, periodic, and inclined Gardner soils. They found that the stochastic results of Yeh et al. (1985b) are satisfactory in the wet range only and that this limitation is related to a Taylor series approximation in terms of a group of statistical and geometric parameters. They also showed that two alternative forms of the Taylor series provide upper and lower bounds for the state-dependent anisotropy of relatively dry soils.
The main limitation of the stochastic model is the required data. Therefore, Zhang et al. (2003) proposed a simple tensorial connectivity–tortuosity (TCT) concept to describe the hydraulic conductivity of anisotropic unsaturated soil. The TCT concept assumes that, in the hydraulic conductivity of unsaturated anisotropic soil, the anisotropy is not merely expressed by a proportionality to the hydraulic conductivity at saturation, but also by three connectivity–tortuosity coefficients, Li = (L1,L2,L3), corresponding to the three principal directions. In other words, the TCT concept implies saturation-dependent anisotropy. This TCT concept was tested using synthetic soils with four levels of heterogeneity and four levels of anisotropy. The results show that, while the soil water retention curves are dependent on soil heterogeneity but independent of direction, the connectivity–tortuosity coefficients are functions of both soil heterogeneity and direction. The TCT model can describe the hydraulic functions of anisotropic soils and can be easily introduced into commonly used relative permeability functions for use in numerical solutions of the flow equation. Zhang et al. (2003) regarded the connectivity–tortuosity coefficient as a tensor, which suggest that Li = (L1,L2,L3) be the components of a symmetric connectivity–tortuosity tensor in the three principal directions. However, there is no mathematical basis for such a tensorial character, despite the fact that the triple (L1,L2,L3) reflects directional dependence. In this paper we show that, nevertheless, a relative connectivity–tortuosity tensor can be defined, namely the tensor T(Se,Li), with principal components Ti = SLie = 
. We present a mathematical formalization of this connectivity–tortuosity tensor, assuming that its principal axes coincide with those of the hydraulic conductivity tensor at saturation. The hydraulic conductivity tensor of such unsaturated anisotropic soils is shown to be the product of a saturation-dependent scalar variable, the connectivity–tortuosity tensor T(Se,Li), and a hydraulic conductivity tensor at saturation. Our aim is to interpret this hydraulic conductivity model with saturation-dependent anisotropy that is easily implemented in existing numerical models.
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EXPRESSION FOR THE HYDRAULIC CONDUCTIVITY TENSOR FOR UNSATURATED SOILS |
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TOPABSTRACTINTRODUCTIONEXPRESSION FOR THE HYDRAULIC...THE HYDRAULIC CONDUCTIVITY...THE HYDRAULIC RESISTIVITY TENSOR...DISCUSSIONCONCLUSIONSREFERENCES |
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, the pressure head h, and the hydraulic conductivity K define the hydraulic properties of a soil. Different classes of soils have been identified using different functions approximating the physical properties. Two groups of parametric expressions describing the hydraulic properties for isotropic soils are (Raats et al., 2002):
With regard to the second group, following Hoffman-Riem et al. (1999; see also Raats, 1993), Zhang et al. (2003) observe that the hydraulic conductivity characteristic is commonly defined by an expression of the form:
 | [1] |
 | [2] |
s is the volumetric water content at saturation and r is the residual volumetric water content, and A(Se,ß,
) is defined by:
 | [3] |
are empirical constants. There is a minimal domain size for which the concept of effective properties is appropriate. Therefore, the effective saturation defined by Eq. [2] may not be applicable to a local scale. For the soils of Zhang et al. (2003), the control volume for which Eq. [2] was defined was taken to be the whole soil domain (1 x 1 m). The same definition as Eq. [2] was also used by other researchers for heterogeneous media (for example, Russo, 1991).
The water retention curve h(Se) is a relationship between the scalar variables pressure head h and effective saturation Se, and Zhang et al. (2003) assumed that for anisotropic soils it is still described by a scalar relationship, such as the Brooks and Corey (1966) or the van Genuchten (1980) relationship. This assumption automatically implies particular expressions for the scalar variable A(Se,ß,) defined by Eq. [3]. Zhang et al. (2003) assumed that the volumetric flux vector f of the water in an unsaturated soil is given by:
 | [4] |
H being vectors, it follows from the so-called quotient law (McConnell, 1957) that K(Se) is a second-order tensor.
For the hydraulic conductivity characteristic K(Se) of an anisotropic unsaturated soil, Zhang et al. (2003) assumed that there exist at each location three principal directions i = 1,2,3 for each of which apply expressions analogous to Eq. [1]:
 | [5] |
 | [6] |
Equation [6] can also be written as:
 | [7] |
Equation [7] suggests that we can regard Ti
= SLie = as the principal components of the relative connectivity–tortuosity tensor T(Se,Li) corresponding to the three principal directions i = 1,2,3. Note that it is tacitly assumed that the principal axes of the hydraulic conductivity tensor Ks at saturation and the relative connectivity–tortuosity tensor T(Se,Li) coincide. With this interpretation, the hydraulic conductivity tensor K(Se) at the effective saturation Se is given as the product of three factors:
); ; and
 | [8] |
Note that at saturation the relative connectivity–tortuosity tensor T(Se,Li) reduces to the unit second-order tensor I, that is, T(Se = 1,Li) = I.
With the tensorial nature of the saturation state–dependent hydraulic conductivity tensor established, various other concepts are easily defined, generalizing concepts long known in the context of saturated soils. In the following we mainly generalize the presentation by Raats (1965)(Subsection 4.1.6) for saturated anisotropic soils to unsaturated anisotropic soils.
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THE HYDRAULIC CONDUCTIVITY VECTOR AND HYDRAULIC CONDUCTIVITY SCALAR |
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TOPABSTRACTINTRODUCTIONEXPRESSION FOR THE HYDRAULIC...THE HYDRAULIC CONDUCTIVITY...THE HYDRAULIC RESISTIVITY TENSOR...DISCUSSIONCONCLUSIONSREFERENCES |
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 | [9] |
The physical significance of kn(Se) becomes evident on dividing Eq. [4] by the magnitude |H| of the driving force H:
 | [10] |
Comparison of Eq. [9] and [10] shows that:
 | [11] |
H| = 1 in the n direction. Only for the principal directions, the direction of f, and hence kn(Se), will coincide with the direction of n.
The hydraulic conductivity scalar kn(Se) associated with the n direction is defined by:
 | [12] |
 | [13] |
The conductivity scalar kn(Se) associated with the n direction is the component of the conductivity vector kn(Se) in the n direction. Or in view of Eq. [11], the conductivity scalar kn(Se) associated with the n direction is the component of the flux in the n direction resulting from unit driving force in the n direction. In the context of the saturated case, Bear (1972) refers to kn(Se) as the directional hydraulic conductivity in the direction of the hydraulic gradient.
According to Eq. [13], a radial plot of 1/
gives a family of ellipsoids with Se as a parameter. These ellipsoids have semi-axes 1/
= 
. For stratified soils, such as the synthetic anisotropic soils of Zhang et al. (2003), two of the principal components of the hydraulic conductivity tensor are equal to each other. It is then sufficient to consider the family of ellipses with semi-axes 1/ = 
, where Kpar(Se) and Knor(Se) are the principal components of the hydraulic conductivity tensor corresponding to the directions parallel and normal to the strata. Figure 1
shows four families of such ellipses for the four sets of parameters considered by Zhang et al. (2003) in their Fig. 4. Note that the water retention characteristic was described using the van Genuchten (1980) model:
 | [14] |
is a fitting parameter that is inversely proportional to the pressure head at air entry. Parameters , n, Ksp, and Ksn are similar among the four sets, while Lp decreases and Ln increases from Fig. 1a through 1d. The individual ellipses are labeled by Se. Larger ellipses are for smaller saturation. The distance of a point on the ellipses to the center represents the magnitude of 1/ for the n direction coinciding with the direction of the hydraulic gradient. The minor axes of the ellipses in Fig. 1 correspond to the principal direction with larger hydraulic conductivity.
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THE HYDRAULIC RESISTIVITY TENSOR AND THE INVERSE RELATIVE CONNECTIVITY–TORTUOSITY TENSOR |
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TOPABSTRACTINTRODUCTIONEXPRESSION FOR THE HYDRAULIC...THE HYDRAULIC CONDUCTIVITY...THE HYDRAULIC RESISTIVITY TENSOR...DISCUSSIONCONCLUSIONSREFERENCES |
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 | [15] |
 | [16] |
From Eq. [8], [15], and [16] it follows that the hydraulic resistivity tensor K–1s
can be decomposed as:
 | [17] |
 | [18] |
The physical significance of the hydraulic resistivity tensor K–1(Se) becomes clear if one multiplies Eq. [4] by K–1(Se) and uses Eq. [15] to obtain:
 | [19] |
(h – z) = –H and the drag force –K–1(Se)f. For details regarding the interpretation of Darcy's law as a force or momentum balance, the reader is referred to Raats and Klute (1968).
In analogy with the hydraulic conductivity vector kn(Se) and the hydraulic conductivity scalar kn(Se), one can define the hydraulic resistivity vector rn(Se) and the hydraulic resistivity scalar rn(Se) by:
 | [20] |
 | [21] |
The hydraulic resistivity vector rn(Se) associated with the n direction is the driving force required to produce unit flux in the n direction. The hydraulic resistivity scalar rn(Se) associated with the n direction is the component of the resistivity vector rn(Se) in the n direction. In other words, the resistivity scalar rn(Se) associated with the n direction is the component of the driving force in the n direction needed to produce unit flux in the n direction.
Because rn(Se) is a scalar, the reciprocal hydraulic resistivity scalar k*n = r–1n exists, and is, according to Eq. [21], given by:
 | [22] |
 | [23] |
Equation [23] can also be written as:
 | [24] |
The reciprocal resistivity scalar k*n = r–1n associated with the n direction is the magnitude of the flux in the n direction produced by a driving force whose component in the n direction is of unit magnitude. In the context of the saturated case, Bear (1972) refers to kn*(Se) as the directional hydraulic conductivity in the direction of the flow. Note the difference between Eq. [13] and [23]. There is a distinct difference between the two expressions, reflecting the differences in hydraulic conductivity that arise when the direction of the hydraulic gradient and the flow are not the same.
According to Eq. [24], a radial plot of 
gives a family of ellipsoids with Se as a parameter. These ellipsoids have semi-axes = 
. For stratified soils, such as the synthetic anisotropic soils of Zhang et al. (2003), two of the principal components of the hydraulic conductivity tensor are equal to each other. It is then sufficient to consider the family of ellipses with semi-axes = 
, where Kp(Se) and Kn(Se) are the principal components of the hydraulic conductivity tensor corresponding to the directions parallel and normal to the strata. Figure 2
shows four families of such ellipses for the four sets of parameters considered by Zhang et al. (2003) in their Fig. 4. Again, the individual ellipses are labeled by Se. Smaller ellipses are for smaller saturations. The distance of a point on the ellipses to the center represents the magnitude of for the n direction coinciding with the flow direction. Contrary to Fig. 1, the minor axes of the ellipses in Fig. 2 correspond to the principal direction with smaller hydraulic conductivity.
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DISCUSSION |
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TOPABSTRACTINTRODUCTIONEXPRESSION FOR THE HYDRAULIC...THE HYDRAULIC CONDUCTIVITY...THE HYDRAULIC RESISTIVITY TENSOR...DISCUSSIONCONCLUSIONSREFERENCES |
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can be regarded as the directional hydraulic conductivities corresponding to the n direction. In general the two directional hydraulic conductivities are not equal to each other, that is:
 | [25] |
For the saturated case, Scheidegger (1954) originally assumed the equality of the two directional hydraulic conductivities ksn and k*sn. Later, Maasland noticed and Scheidegger admitted the error (Scheidegger, 1956). In his classic treatment of soil anisotropy and land drainage, Maasland (1957) only discusses the directional hydraulic conductivities k*sn. Dullien (1979) and Bear (1972) discussed both ksn and k*sn, as did Carslaw and Jaeger (1959)(Section 1.20) for the analogous process of thermal conduction.
In principle, the two directional hydraulic conductivities kn(Se) and k*n can be measured as follows (see Carslaw and Jaeger, 1959):
associated with the n direction can be determined by cutting in the n direction a narrow tube of material and measuring its hydraulic conductivity as a function of Se.
Jump conditions at interfaces between different unsaturated isotropic soils have been discussed by Raats (1972)(1973). The refraction of streamlines and equipotentials at interfaces between saturated anisotropic soils is analyzed in detail for the two-dimensional case in Raats (1972) and for the three-dimensional case in Raats (1973). The analysis for the three-dimensional case for saturated soils shows that whereas the planes of incidence and refraction of the hydraulic gradient H always coincide, the planes of incidence and refraction of the flux coincide only in some very special cases (Raats, 1973). It appears that the swirly streamlines found computationally by Hemker (2001), Bakker and Hemker (2002), Hemker and Bakker (2002), and Hemker et al. (2004) for flow in layered, anisotropic aquifers are directly related to noncoincidence of planes of incidence and refraction of the flux. Although in principle generalization to unsaturated anisotropic soils is straightforward, computational and observational implementation for three-dimensional cases with different principal directions at the two sides of the interface still presents quite a challenge.
The model is a straightforward extension to anisotropic soils of the widely used van Genuchten–Mualem characterization for isotropic soils. The big advantage of this approach is that, once the principal directions, and the corresponding components of the hydraulic conductivity tensor at saturation and the connectivity–tortuosity tensor are determined, it is easily implemented in existing numerical codes, such as Hydrus2-D (Simunek et al., 1999), FUSSIM2 (Heinen and de Willigen, 2001), and STOMP (White and Oostrom, 2004). Of course, establishing these parameters for anisotropic soils remains a big challenge. At least four approaches are possible to estimate flow path tortuosity and connectivity:
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CONCLUSIONS |
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TOPABSTRACTINTRODUCTIONEXPRESSION FOR THE HYDRAULIC...THE HYDRAULIC CONDUCTIVITY...THE HYDRAULIC RESISTIVITY TENSOR...DISCUSSIONCONCLUSIONSREFERENCES |
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ACKNOWLEDGMENTS |
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REFERENCES |
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TOPABSTRACTINTRODUCTIONEXPRESSION FOR THE HYDRAULIC...THE HYDRAULIC CONDUCTIVITY...THE HYDRAULIC RESISTIVITY TENSOR...DISCUSSIONCONCLUSIONSREFERENCES |
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. Water Resour. Res. 21:457–464.This article has been cited by other articles:
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, with kn being the hydraulic conductivity scalar at direction n, as a family of ellipses at different saturations for the four soils of 
, with k*n being the reciprocal hydraulic resistivity scalar at direction n, as a family of ellipses at different saturations for the four soils of