Upscaling for unsaturated flow for non-Gaussian heterogeneous porous media

Authors


Abstract

[1] Large-scale models of transient flow processes in the unsaturated zone require, in general, upscaling of the flow problem in order to capture the impact of heterogeneities on a small scale, which cannot be resolved by the model. Effective parameters for the upscaled models are often derived from second-order stochastic properties of the parameter fields. Such properties are good quantifications for parameter fields, which are multi-Gaussian. However, the structure of soil does rarely resemble these kinds of fields. The non-multi-Gaussian field properties can lead to strong discrepancies between predictions of upscaled models and the averaged real flow process. In particular, the connected paths of parameter ranges of the medium are important features, which are usually not taken into account in stochastic approaches. They are determined here by the Euler number of one-cut indicator fields. Methods to predict effective parameters are needed that incorporate this type of information. We discuss different simple and fast approaches for estimating the effective parameter for upscaled models of slow transient flow processes in the unsaturated zone, where connected paths of the material may be taken into account. Upscaled models are derived with the assumption of capillary equilibrium. The effective parameters are calculated using effective media approaches. We also discuss the limits of the applicability of these methods.

1. Introduction

[2] Modeling of flow in the unsaturated zone on large scales can become very demanding, as soil is heterogeneous. Upscaled models are often required to describe the averaged flow behavior. The purpose of deriving upscaled models is to predict the spatial average of the tension and the water saturation in the soil. The parameters for such models have mostly to be determined from incomplete knowledge about the detailed soil structure. Mostly only local information is available and the heterogeneity of the soil is characterized from local measurements.

[3] Upscaling is often performed with a stochastic approach. The effective model is obtained by deriving an approximate equation for the stochastically averaged state variables. The effective hydraulic conductivity is then, for example, obtained by calculating the averaged total head gradient and the averaged flux to second order in the fluctuating parameters. The effective conductivity tensor is obtained by the dyadic product of the inverse of the gradient vector and the flux vector. In this framework, heterogeneity of the soil is mostly characterized by second-order stochastic properties of the soil parameters. In principle that is the mean, the variance and an autocovariance model. As the autocovariance of the parameters is mostly assumed to have a finite integral scale, it is quantified by the integral scale for one direction and anisotropy factors for the integral scale for the remaining directions. The parameters of the upscaled flow equation are then functions of these stochastic parameters.

[4] Upscaling of unsaturated flow with stochastic methods has been studied since decades. It has been found already for the steady state problem that the nonlinearity of the flow equation can cause complex behavior of the effective parameters of the upscaled problem. Yeh et al. [1985] discussed the increase of anisotropy of the effective hydraulic conductivity in a layered system under unsaturated conditions. Because of the change of the unsaturated hydraulic conductivity with water saturation, the layering structure may have a stronger impact on the unsaturated hydraulic conductivity than it has on the saturated one. Russo [1992] has analyzed the effective unsaturated hydraulic conductivity for the more general case, where the upscaled problem is considered for a finite averaging size. The transient problem has been analyzed by Mantoglou and Gelhar [1987], who found hysteretic behavior of the effective unsaturated flow parameters, depending on the heterogeneity of the soil. The transient problem has also been analyzed by Zhang [1999]. In principle the effective hydraulic conductivity and water content function depend in a complex way on the stochastic parameters of saturated hydraulic conductivity and local parameters of the models for the water content function and the unsaturated hydraulic conductivity (such as the entry pressure in a Brooks-Corey model), as well as on the mean of the tension (which is the absolute value of the capillary pressure head) and its gradient in the soil. The parameters change strongly with the tension in the soil.

[5] An upscaled model can also be derived using homogenization theory. The upscaled model is derived for the spatially averaged state variables. For certain unsaturated flow problems this has been performed by Lewandowska and Laurent [2001], Lewandowska et al. [2004], or Neuweiler and Cirpka [2005]. The heterogeneous parameter field is here not necessarily characterized by its stochastic properties. The main difference to the upscaling approaches mentioned above is that in homogenization theory a scale expansion of the problem is performed. An upscaled equation is obtained by performing an expansion in the ratio between the typical small length scale (such as the unit cell of a periodic medium or the length scale of a macroscopic representative elementary volume) and the typical macroscopic length scale in the system and considering the limit that this ratio goes to zero. This method has the advantage, that it yields an upscaled equation directly. However, due to the expansion in the length scales, the upscaled problem is valid only for certain flow scenarios, defined by given ranges of the dimensionless numbers of the problem. In this way, the upscaled model is less general than the upscaled models derived by ensemble averaging the heterogeneous problem without performing a scale expansion. The method itself is explained, for example, by Hornung [1997].

[6] Effective parameters for the upscaled model need to be calculated from a given parameter field for a unit cell. In reality, the parameter distribution is unknown. Effective parameters can therefore only be approximated based on a given description of the soil heterogeneity. It is, for example, possible to calculate effective parameters for a homogenized model based on a stochastic description of the field, taking its second-order properties into account (as, e.g., by Lunati et al. [2002] or Neuweiler and Cirpka [2005]).

[7] The second-order statistic properties do not quantify a field completely. Only multi-Gaussian fields are completely characterized by their second-order properties. However, natural soils seldom resemble multi-Gaussian fields. Thus to characterize a general field completely, all higher moments would have to be taken into account, which is in practice not feasible. The neglect of higher-order terms will therefore lead to deviations from results from real soils. These deviations might not be very strong for linear problems such as single-phase flow, at least under the condition of small variance of the field distribution. However, even for single-phase flow or solute transport, the influence of the non-multi-Gaussian character of fields has been discussed recently [Zinn and Harvey, 2003; Western et al., 2001; Gomez-Hernandez and Wen, 1997; Knudby and Carrera, 2005]. For nonlinear flow problems, such as unsaturated flow, deviations from the second-order results are expected if the fields have non multi-Gaussian features. In addition to the results from stochastic theory, it would therefore be useful to have estimates for effective parameters based on a different characterization of heterogeneity.

[8] It is an open question how to characterize the most important features of a heterogeneous field. The topic is often discussed for pore-scale models, where binary fields (describing pore space and grain material) have to be characterized efficiently [e.g., Torquato, 2002; Hilfer, 2002]. These fields are also characterized by their stochastic properties, but also by geometric or topologic properties. Often it has been found, that it is important to get measures to quantify the connectivity of the pore space. On a larger scale, similar measures to quantify connected paths of different parameter ranges can be applied to the indicator field of the parameter fields for different values of indicator thresholds. A very good measure would in principle be the two-point cluster function, as defined by Torquato et al. [1988], applied to two-cut indicator fields. However, locally measured data on the Darcy scale will hardly be large enough to calculate these quantities. Another measure is the Euler characteristic [e.g., Vogel 2002], which is applied to a one-cut indicator field of the parameter field. It gives a good measure about the connectivity of a structure and can be calculated locally [see also Samouelian et al., 2007]. Connectivity is here not meant as the connectivity of a pore network, but as connectivity of certain parameter ranges in a continuously changing field. Such measures are useful to characterize fields, which have connected extreme values and are thus not multi-Gaussian.

[9] However, once such connected paths are determined, it is not clear, how such kind of information can be included into the calculation of effective model parameters. This is especially true as mostly very little data about the parameter fields are available. In this paper we want to test simple methods to estimate effective parameters for homogenized unsaturated flow models. The aim is to test these methods for fields which have connected extreme values. We proceed from the upscaled models derived by Lewandowska and Laurent [2001], Neuweiler and Eichel [2006], and Neuweiler and Cirpka [2005] with homogenization theory. We estimate for these models the effective unsaturated hydraulic conductivity using different effective medium theory concepts as discussed partly, for example, by Renard and de Marsily [1997], Dagan [1979], and Knudby et al. [2006] for single-phase flow. The important point is that some allow to take the existence of connected paths of the parameter field into account. We use the Euler number to quantify connected paths of different parameter fields [see Mecke and Wagner, 1991]. We discuss advantages and disadvantages of the different effective medium theory methods and compare their results for several test cases to the effective parameter functions obtained for these fields numerically. We also compare the effective parameter functions obtained for homogenized equations with the assumption of local capillary equilibrium to those obtained for the case that capillary forces do not dominate on the small scale.

2. Dimensionless Richards Equation

[10] Transient flow in the unsaturated zone is described by the Richards equation.

equation image

in which u [L/T] is the specific discharge of soil water, q [1/T] stands for volumetric sources and sinks, and Θ is the water content, which depends on the tension Ψ [L]. The water content Θ is the product of the saturation S (dimensionless) and the porosity nf (dimensionless), Θ = Snf. The water tension Ψ has positive values. Vectors and tensors are denoted by boldface symbols. The specific discharge follows Darcy's law:

equation image

in which Ks [L/T] is the saturated hydraulic conductivity, kr (dimensionless) is the relative hydraulic conductivity and ez (dimensionless) denotes the vertical unit vector pointing downward. As the water content function, the relative hydraulic conductivity depends also on the tension Ψ and on the soil properties. The product Kskr (Ψ) is in the following called the unsaturated hydraulic conductivity Ku (Ψ). The saturated hydraulic conductivity Ks is in the following assumed to be locally isotropic, Ks = Ks1, where 1 is the unity tensor.

[11] Different flow regimes are best analyzed in a dimensionless form of the Richards equation. A typical timescale T and a typical length scale L are related to the flow problem. The length scale L is set (arbitrarily) to a macroscopic length scale (such as a scale of the order of the distance between land surface and the water table of an aquifer). For a two-dimensional system there is in principle a typical length scale for each direction, Lx and Lz. There is also a typical value for the variable Ψ, which is a typical water tension H, defined for example from the moisture content–tension relation of the soil. Hilfer and Øren [1996] suggest, for example, to use the value of the tension where the saturation is S = 0.5. In a heterogeneous medium a typical value K0 for the saturated hydraulic conductivity has to be chosen. In a two-dimensional medium it would be reasonable to use the geometric mean of the field, as this is the best guess for the effective hydraulic conductivity if only the univariate parameter distribution is available. The saturated hydraulic conductivity is then written as

equation image

where κs is dimensionless. The tension, the space and time variables are made dimensionless with the typical values as

equation image

[12] Quantities with the star index are dimensionless. By choosing a typical large timescale we choose a form of the Richards equation that describes slow flow processes. This choice is important, as processes such as fast infiltration after a strong rainfall event are not captured by this. The analysis is restricted to slowly changing conditions. We assume here that the typical timescale is related to gravity forces on the large scale. That is, the typical timescale is the time that a tracer particle would need to travel over a length L due to gravity driven flow if the medium was homogeneous and water saturated, and if dispersion is neglected,

equation image

[13] Making the Richards equation dimensionless yields then

equation image

in which ∇equation image denotes the vector of derivatives with respect to the dimensionless coordinates xequation image, and qequation image = qT is the dimensionless source term. H/L is a dimensionless number in the equation.

[14] The soil is parameterized by the saturated hydraulic conductivity Ks and by a model for the water content function Ψ(Θ) and for a relative hydraulic conductivity kr (Θ). The models have several parameters, one of them having the dimensions of the water tension. As an example, for a Gardner-Russo model [Gardner, 1958; Russo, 1988] as well as for a Mualem–van Genuchten model [van Genuchten, 1980; Mualem, 1976] this would be the inverse of the parameter α, while for the Brooks-Corey model [Brooks and Corey, 1966] this would be the air entry pressure divided by density ρ and gravity constant g. The model depends (apart from the other, dimensionless, model parameters) on the product of this parameter and the tension. As the tension is here considered to be dimensionless, the equivalent parameter also has to be made dimensionless accordingly. For example, for a van Genuchten model with parameters α, n and m, would be written as

equation image

[15] In a heterogeneous medium all parameters can be space dependent.

[16] Star indices are omitted in the following in order to avoid complex notation. All parameters and variables are now assumed to be dimensionless.

3. Upscaled Model

[17] In this paper we want to estimate effective parameters for upscaled models for the Richards equation (6). The uncertainty related to their prediction, which is often a topic in stochastic theory, is not the topic of this paper. We proceed from upscaled models for the Richards equation, derived with homogenization theory. Such models have been derived by Lewandowska and Laurent [2001] and Neuweiler and Cirpka [2005]. The technical details are given in these papers. As these models are the basis for the effective parameters discussed here, a brief overview will be given and the results for two different flow regimes will be discussed below. The models describe the spatially averaged tension 〈Ψ〉 and the spatially averaged water content 〈Θ〉 in the soil. Angular brackets denote spatial averages.

[18] Homogenization theory is based on a scale analysis. There are assumed to be two clearly distinct length scales in the medium. One is a macroscopic length scale, which is of the order of the length scale of the medium itself. In the notation used here that would be L (where L is identical to the scale L used in equation (4)). There is another small length scale, which is the length scale of a unit cell or a macroscopic representative elementary volume [see Lewandowska and Laurent, 2001]. It is denoted as ℓ. A simple example of such a field is a periodic field, where spheres of one material are included periodically into a background material. The distance between the centers of the spheres is the small length scale ℓ. In a correlated random field the small length scale ℓ has to cover several correlation lengths. The ratio between the two length scales ℓ and L is denoted as ɛ = ℓ/L. ɛ is supposed to be very small and provides thus an expansion parameter for the system. All variables and functions of variables in the Richards equation (6) are expanded in terms of ɛ as

equation image

[19] The term Ψ(i) is the expansion coefficient of the ith term of the expansion of Ψ in ɛ. The space variable is split into two independent variables. One is made dimensionless with the small length scale ℓ and is denoted as equation image. The second one is made dimensionless with the large length scale L and is denoted as equation image. The dimensionless gradient is under this transformation split as

equation image

[20] If the expressions (8) and (9) are inserted into the dimensionless Richards equation (6), the equation contains terms of different orders of ɛ. If we assume that the two length scales L and ℓ are clearly distinct, we may assume that the equation is fulfilled for each order independently. In the end the limit ɛ → 0 is performed. In this limit the heterogeneities are not “seen” anymore, so that the equation is in this limit considered as the upscaled homogeneous model. The full expansion of the Richards equation in terms of ɛ is given in Appendix A.

[21] The upscaled flow equation is the flow equation of the lowest considered order in ɛ (here ɛ0), which has been spatially averaged over the unit cell. In this way the small-scale fluctuations due to the heterogeneities of the parameter field have been averaged out.

[22] Upscaled models derived with homogenization theory are subject to several assumptions and restrictions.

[23] 1. The space variables made dimensionless with the scales L and ℓ are treated as separate variables. These two scales have to be clearly separated and there have to be no length scales in between these two. The fields have thus to be statistically homogeneous.

[24] 2. In principle homogenization theory is performed for problems in periodic media. The parameter field has then a defined unit cell. All orders of the variable Ψ are assumed to be periodic over the unit cell. In a correlated random field they are assumed to be locally stationary.

[25] 3. All dimensionless numbers and parameter ratios have to be fixed in relation to the expansion parameter ɛ. Different orders of the dimensionless numbers lead to different upscaled models [Lewandowska et al., 2004]. The upscaled models are therefore only valid for specified flow regimes. The typical timescale is usually chosen as a large timescale. The upscaled models are therefore valid only for slow processes. Small-scale dynamics [e.g., Hassanizadeh et al. 2002] is thus not captured by the upscaled equations.

[26] To derive the upscaled model, the dimensionless numbers have to be scaled with the expansion parameter ɛ. We consider the case that capillary forces and gravity contribute equally on the large scale. If the dimensionless gradient ∇equation image Ψ(0) is of the order ɛ0, this is fulfilled if the typical tension is of the same order of magnitude as the large length scale, equation image ∝ ɛ0. This is met if the large length scale L is small (such as a lab scale) or if the porous material is very fine.

[27] We compare our results to the case that the typical tension is much smaller than the large length scale equation image ∝ ɛ1, which means the typical tension is equivalent to the small length scale. This case is more relevant for field applications.

3.1. Capillary Equilibrium

[28] For the case that the typical tension is of the same order as the large length scale (equation image ∝ ɛ0), capillary forces are very strong. As the capillary term appears in the Richards equation (6) with a second-order derivative, its highest-order contribution is due to the transformation of the gradient (9) of order ɛ−2. The highest-order contribution of the gravity term is only of order ɛ−1. That means capillary forces dominate on the small scale. The highest-order equation to be solved is a pressure equation (given by the terms of order ɛ−2 in equation (6)) for the tension on the small scale

equation image

[29] Because of the requirement of periodicity of the tension Ψ(0), this can only be fulfilled if the tension is independent on the small scale equation image, Ψ(0) = Ψ(0)(equation image, t). This is equivalent to the capillary equilibrium condition [e.g., Pickup and Stephen 2000]. It can be interpreted in the way that due to the dominance of capillary forces on the small-scale, fluctuations of the tension on the small scale are balanced out quasi instantaneously. This results on the small scale in a steady state situation for capillary driven flow, which is a constant tension.

[30] The capillary equilibrium has as a consequence, that the lowest order water content Θ(0) as well as the lowest-order relative hydraulic conductivity kr(0) is locally fixed and determined by the local value of the tension Ψ(0) only.

[31] The upscaled equation is given by the terms of order ɛ0, which have been averaged over the unit cell. It can be shown, that this equation is of the form

equation image

[32] The effective water content Θeff and the effective unsaturated hydraulic conductivity tensor κeff, u have to be calculated from the terms in the Richards equation of order ɛ−1. The details are discussed elsewhere [Lewandowska and Laurent, 2001]. Because of the capillary equilibrium, the procedure to calculate the effective curves is quite intuitive. A value of the effective water content function at a given tension Ψ(0) is the spatial average of the local water content function values at this tension over the unit cell, Θeff = 〈Θ(Ψ(0), equation image)〉. To obtain the effective curve, the average of the local water content function has to be calculated for a number of tension values Ψ(0). The effective unsaturated hydraulic conductivity is obtained from the local flow problem for the unsaturated hydraulic conductivity κs (equation image)kr(0), equation image) in the unit cell. A unit pressure gradient in horizontal and vertical direction and periodic boundary conditions are applied. From the average of the flux over the unit cell in horizontal and vertical direction the effective total hydraulic conductivity tensor for a given value of Ψ(0) can be obtained. The effective unsaturated hydraulic conductivity function is obtained by calculating the effective conductivities for a number of tension values Ψ(0).

3.2. Gravity-Dominated Flow

[33] For the case of small typical tensions, capillary forces and gravity forces are in equilibrium on the small scale. As the term, which contains in equation (6) the factor equation image, contributes an order ɛ to the capillary term, the highest-order terms in the Richards equation (6) are of order ɛ−1 (see Appendix A). The corresponding subequation reads

equation image

[34] Again we have to have equation image periodicity for the tension Ψ(0). This equation has no simple solution and has to be solved numerically. The tension Ψ(0) is this time not constant on the small scale equation image, (Ψ(0) = Ψ(0)(equation image, equation image,t)).

[35] The upscaled equation is here again the locally averaged equation of order ɛ0. It can be shown that this equation has to the leading order the shape

equation image

[36] The flow is to the leading order of ɛ gravity dominated on the large scale, although the effective parameters are via equation (12) influenced by capillary forces.

[37] To calculate the effective water content and the unsaturated hydraulic conductivity function for the gravity-dominated problem, the local problem (12) has to be solved given a value of Ψ(0) on one location at the boundary of the unit cell, otherwise assuming periodic boundary conditions for Ψ(0). The resulting solution Ψ(0)(equation image) and the corresponding local water content Θ (Ψ(0)) are averaged over the unit cell, yielding one point on the effective water content curve. The flux u(0) has to be calculated from (12) and averaged over the unit cell. The averaged flux and the averaged value of Ψ(0) give one point on the effective unsaturated hydraulic conductivity function. By using different values of Ψ(0) for the fixed value on the boundary, the effective water content curve and the effective unsaturated hydraulic conductivity curve are calculated.

3.3. Effective Parameters and Heterogeneous Fields

[38] The effective water content function and the effective unsaturated hydraulic conductivity function for the upscaled Richards equation (11) and (13) depend on the local parameter fields. In reality it is problematic to determine these effective parameter functions, as the small-scale parameter distribution is not known. We usually only have local samples, which have to be considered as representative for the whole medium. From the local samples we have to derive a characterization of the heterogeneity of the soil.

[39] The lack of knowledge of the detailed parameter distribution makes a stochastic approach suited for the soil description. Using a stochastic approach, we would assume a certain multimodal probability distribution for the model parameters. The effective parameters are then calculated by replacing the spatial averages over the unit cell by ensemble averages. Usually, second-order approximations are made for these calculations. A field is completely determined by its first and second moment only if it is multi-Gaussian.

[40] However, soil parameter are mostly not multi-Gaussian. As higher statistical moments are difficult to treat, other measures for heterogeneity might be better suited to characterize non-multi-Gaussian fields. A very important characteristic is the existence of connected paths of given parameter ranges. Properties of the material types, which are connected throughout the soil, will influence the flow processes much more than materials which form isolated structures.

[41] For multi-Gaussian fields with a Gauss-shaped two-point autocovariance the extreme values of the fields form isolated structures, while the intermediate values are connected. However, this is not a common structure of the subsurface. In contrasts, due to sedimentation or relocation processes, inclusions of fine or coarse textured materials are a typical feature. Such subsurface structures will not be well represented by multi-Gaussian fields. Such soils are better characterized by indicating the materials which are connected throughout the medium.

[42] To compare effective parameter functions for such fields, we will here analyze three different extreme types of fields. We will compare non-multi-Gaussian fields with connected extreme values to multi-Gaussian fields with connected intermediate values. The fields are two-dimensional. Except for the case of a layered medium, there can then be only one material, which is connected throughout the medium. Therefore, once this material is specified, it is clear that all other materials are isolated. This is different in the three-dimensional case, where more than one material can be connected throughout the medium. Also, in a three-dimensional medium, different materials can be connected in different directions. The analysis for the two-dimensional case can therefore only be transferred to three-dimensional media, where there is only one background material, while the other materials form isolated structures. In this way the two-dimensional analysis cannot be transferred to all possible three-dimensional scenarios, however, the analysis holds for a whole group of three-dimensional media. Possible extensions of the two-dimensional analysis to the general three-dimensional case are discussed in the conclusions.

[43] The topology of the heterogeneous fields is here quantified using the Euler number χ = NC which is defined for a two-dimensional binary structure (black/white). It is the number of isolated objects N minus the number of redundant connections or loops C. Hence χ has positive values for few connected structures and becomes negative for objects which are more intensely connected.

[44] However, the heterogeneous fields considered here are not binary. They are composed by a number of different materials represented by different grey values. To quantify the connectivity of these fields, we consider binary images obtained after segmentation of the grey image using various thresholds gt which correspond to the different materials. In this way we get a continuous function χ(gt) which quantifies the connectivity of materials above a given threshold. This function is referred to as the connectivity function as proposed by Vogel [2002].

[45] It was shown by Mecke and Wagner [1991] that for isotropic structures the point where the χ(gt) = 0 corresponds approximately to the percolation threshold of the structure. In percolation theory [Stauffer, 1985], the percolation threshold is defined as the critical point where the fraction of objects (typically sites or bonds on a lattice) is just large enough to form a sample spanning cluster. In our case, this is the critical grey threshold equation imaget for which a continuous path through the entire field exists. The threshold equation imaget corresponds to the lowest conductivity that has to be passed on the most efficient path through the heterogeneous field, and hence this quantity is expected to be relevant for the effective conductivity of the field. This approach corresponds to the concept of critical path analysis in percolation theory [Hunt, 2005, 2001]. The critical grey threshold could also be analyzed using the cluster density function as introduced by Torquato et al. [1988].

[46] An advantage of using χ(gt) is the fact that the Euler number can be estimated from local measurements. This is in contrast to the direct measurement of the percolation threshold where the entire structure has to be known. Although in this study the entire fields are actually known, this might be important for applications in natural systems where the analysis are typically restricted to a number of subsamples of the heterogeneous structure.

[47] For a statistically anisotropic field the information about the connectivity of different parameter ranges is not sufficient, as the Euler number is not related to the anisotropy of the field. However, stochastic properties of the field can be taken into account additionally to the connectivity. In stochastic theory the anisotropy ratio of a two-dimensional field is defined as the ratio between the integral scales in the two main directions of the field.

4. Estimates of Effective Parameters Using Effective Medium Theory

[48] We will discuss in the following simple approaches to calculate the effective parameter functions for the upscaled models described above. We will proceed from the upscaled model derived with the assumption of capillary equilibrium (see section 3.1.), as is allows for a relatively simple procedure to calculate the effective parameters. Applying the capillary equilibrium means, we go through a number of values for the tension Ψ(0) and calculate the corresponding saturation field S(0)) and the corresponding dimensionless total unsaturated hydraulic conductivity field κu(0)) = κskr(0)). The only information needed to obtain the effective water content curve is the univariate parameter distribution of the medium and the local water content functions. The effective water content curve is thus not dependent on the structure of the local unsaturated parameter field. The effective hydraulic conductivity tensor has to be calculated from the local total hydraulic conductivity field. As the detailed parameter distribution is not known, we would like to estimate the effective hydraulic conductivity based on a simple characterization of the structure of the parameter field, which can be obtained from local measurements.

[49] Using stochastic theory, the effective hydraulic conductivity for a statistically isotropic field would be the geometric mean of the local field with some correction term, which depends on the dimension of the problem and the variance of log hydraulic conductivity. Alternatively, effective medium theory can be used to determine the effective hydraulic conductivity [Torquato, 2002]. Effective medium theory approaches consider the porous medium a composite. The composite is compared to a medium with effective hydraulic conductivity by analyzing the perturbation that the composites of the material impose on an external pressure field.

[50] Some of these approaches are described in the following. They are mean field approaches and do not take two-point statistics into account. The composites are approximated as ellipsoids. By assigning the anisotropy ratio of the parameter field to the shape of the ellipsoids, statistical anisotropy of the fields can be reflected. We discuss first the self-consistent approach, which is the approach which is mostly used. We then discuss the Maxwell approach and the differential effective medium theory approach. These two approaches allow to use information about connected material properties, as they consider media with a defined background material.

4.1. Self-Consistent Approach

[51] Very often the self-consistent approach [Landauer, 1978; Dagan, 1979; Renard and de Marsily, 1997] is used to calculate the effective hydraulic conductivity. The heterogeneous medium is assumed to be composed of patches of different materials with conductivities κi. The different materials are treated symmetrically, which means there is no distinction between background and inclusion. To calculate the effective hydraulic conductivity of the patch composite, each patch is one by one embedded into a homogeneous material with the (unknown) effective conductivity κeff and the perturbation caused in an external pressure field is calculated (see Figure 1). The postulation that the sum of the perturbations caused by each single patch has to vanish, yields the effective hydraulic conductivity as an implicit scheme. This is a low-order approximation, as it is assumed that the patches do not interact. The total perturbation of the patches is assumed to be the sum of the single perturbations.

Figure 1.

Sketch for the self-consistent approach.

[52] In particular, the perturbation due to the single patches is assumed to be representable as the perturbation due to ellipsoids which are placed in the coordinate center.

[53] The heterogeneous medium is assumed to be composed of N different materials, each having a volume percentage of Φi and a conductivity κi. The local conductivity is here assumed to be a scalar. The coordinate system is assumed to be such that the effective hydraulic conductivity is a diagonal tensor. The j component of the effective hydraulic conductivity tensor is then determined as

equation image

[54] A is the ellipsoidal shape matrix. For an isotropic field (in 2d and in 3d) it is

equation image

[55] The only information about the structure of the material which is used with this approach is the volume percentage of the different materials.

4.2. Maxwell Approach

[56] Using the Maxwell approach (as, e.g., by Zimmerman et al. [1992]), the heterogeneous medium is also considered as a composite of materials with conductivity κi. However, one material (here assumed to be the material with conductivity κ1) is assumed to be the background material. To calculate the effective hydraulic conductivity two configurations are compared. On the one hand an ellipsoid of the heterogeneous material (background plus inclusions) is embedded into a homogeneous medium made of the background material. The only perturbations of the external field are due to the inclusions. It is then assumed, that the perturbation caused by the inclusions can be in the far field approximated as the sum of the perturbations caused by spheres or ellipsoids of the inclusion material which are placed in the center of the coordinate system.

[57] The resulting external pressure field is compared to the field where an ellipsoid with effective hydraulic conductivity κeff is included into the background material (see Figure 2). By setting the two far fields equal to each other an explicit rule for the effective hydraulic conductivity is obtained. The j component of the effective hydraulic conductivity is

equation image
Figure 2.

Sketch for the Maxwell approach.

[58] The index i runs from 1 to N, including the index for the background material. A is the ellipsoidal shape matrix. As in the self-consistent approach, the only information about the heterogeneity of the field is the volume percentage of the materials and the anisotropy ratio. However, as there is a distinct background material, the information about the connected material phase can be taken into account by assigning it to the background. The background material is treated in the Maxwell approach in a way that it is always connected and the ratio between material interface and area of the background is small, as the inclusions are ellipsoids. In the case that one material becomes unconductive, the percolation properties of the medium are thus not reproduced. The Maxwell approach thus overestimates the effective hydraulic conductivity if the background material is highly conductive and underestimate it if the background material is poorly conductive.

4.3. Differential Effective Medium Approach

[59] The differential effective medium approach is based on the same principle of the Maxwell approach. It is also formulated for a background material with inclusions, so that the connected material can be assigned to the background. The basic idea is to increase the volume percentage of inclusion material incrementally. From the incremental increase of the volume percentage of inclusion material the incremental change of the effective conductivity is derived. This yields in the limit that the increments become infinitesimally small a differential equation, which can be solved for the final configuration of inclusion material. That means, the background material properties are changed while the volume percentage of inclusion material is increased.

[60] Torquato [2002] demonstrates the method for the case that there is only one inclusion material with conductivity κ2 with volume percentage Φ2 embedded into the background material with conductivity κ1. The effective conductivity at the starting point of one step of incremental increase of volume percentage of inclusion material from Φ to Φ + ΔΦ is assumed to be known as κeff (Φ). As there is Φ per cent of inclusion material in the composite, a volume percentage of ΔΦ/(1 − Φ) of the material has to be replaced by inclusion material, in order to have after the replacement a total of Φ + ΔΦ of inclusion material. The Maxwell principle (16) can be applied to calculate the effective conductivity. Equation (16) is expanded to first order in ΔΦ. The background material with known effective conductivity κeff, j (Φ) is now treated as the material with κ1 in equation (16). The volume percentages of the inclusion materials are now (Φ +Δ Φ). This leads to

equation image

[61] A differential equation can be derived from this, which has to be integrated from the initial state, where we have κeff, j (Φ = 0) = κ1 to the final state, where we have κeff (Φ = Φ2). This yields the relation for the effective conductivity:

equation image

[62] In case that there is more than one inclusion material the method is more complex. In principle the incremental change of the conductivity depends on the path, on which the final configuration of inclusion materials has been reached [cf. Norris et al. 1985]. It would, e.g., be possible to increase the volume percentage of each material one after the next, while the other one is held constant. It would also be possible to increase the volume percentage of both materials with a constant rate.

[63] If the volume percentage of all inclusion materials is increased simultaneously with the same rate, each phase of the inclusion material has a volume percentage out of the total inclusion material of ρi = Φi/Φ, which is kept constant while the total volume percentage of inclusion material Φ is increased. This leads in the same way as explained above for a two-material composite to the differential equation

equation image

[64] A solution procedure for this problem is shown in Norris et al. [1985]. The equation (19) is integrated from Φ = 0 (with κeff (Φ = 0) = κ1) to Φ. This yields an equation, which then has to be solved for κeff.

equation image

where F is defined as

equation image

equation imagei are the roots of a function G(κ) and Ri are the residues of 1/G(κ). G is given as

equation image

[65] For the test cases in section 5 we compared this procedure to the procedure, where the volume percentage of the inclusion materials is increased one after the next. As we did not find significant differences for the results compared to the differences between the results of the different effective medium theory approaches, this is here not discussed further.

[66] In the differential effective medium theory approach the background material is in principle overemphasized as in the Maxwell approach. However, as the properties of the background material are changed incrementally, the overemphasis of the background is less pronounced. Compared to the Maxwell approach, the disadvantage of the differential effective medium approach is its complexity for composites of more than two materials.

5. Test Case

[67] In this section the effective water content functions and the effective unsaturated hydraulic conductivity functions are estimated with the different approaches discussed in the last sections for test fields. The results are compared to the numerically calculated parameters for these test fields.

5.1. Test Fields

[68] Effective parameter functions are compared for three different test fields. The fields are statistically anisotropic with an anisotropy ratio of λ1/λ2 = 3, where λi is the integral scale of the field in i direction. As required by the homogenization calculations, all fields are composed of periodic unit cells. At this stage, only the log saturated conductivity fk = ln(Ks) is assigned to the fields, the other parameters will be discussed later. The first field, fk, 1, is considered the reference case. It is a multi-Gaussian field, where the material with intermediate hydraulic conductivity value is connected. In the second and the third field, fk, 2 and fk, 3, the material with high and low saturated hydraulic conductivity are connected through the materials, respectively.

[69] The fields were generated starting from a standard periodic multi-Gaussian continuous field fnor with a Gaussian autocovariance function, generated as by Cirpka and Kitanidis [2002]. This field was transformed into two continuous field with connected high and low values, applying the method introduced by Vogel [2002] or Zinn and Harvey [2003],

equation image
equation image

[70] These fields have the same mean and variance as the original field fnor, however, the autocovariance function is changed under this transformation. To compare fields with identical second-order stochastic properties, a multi-Gaussian field (fk, 1) was generated from the spectrum of the transformed fields, which has the same univariate distribution and autocorrelation function as the fields fk, 2 and fk, 3.

[71] All fk fields have a mean of zero. The saturated hydraulic conductivity κs is obtained by taking the exponent of fk, so that all κs fields have the same geometric mean of one. Field fk, 1 is a multi-Gaussian field, fk, 2 and fk, 3 are not multi-Gaussian. The fields are shown in Figure 3. The fields with connected extreme values look somewhat artificial. Although the connected extreme values are a feature also found in real soils, they are not meant as prototypes of real soil structures. The crucial point about the fields is, that all three fields have an identical univariate distribution and autocovariance. That is, they would not be distinguishable with their second-order stochastic properties, and would therefore all have the same effective parameter functions derived with second-order stochastic theory. Yet they are clearly distinct in terms of their connectivity. They are therefore very useful in order to illustrate the influence of non-multi-Gaussian features and how they can be captured in the effective parameter functions.

Figure 3.

Unit cells of the test fields showing fk = ln(κs). (left) Multi-Gaussian field with the same mean, variance, and autocovariance as Figures 3 (middle) and 3 (right). (middle) High values connected. (right) Low values connected. All three fields have a mean of 〈fk〉 = 0, a variance of σfk2 = 2.0, and integral scales λx = 0.075, λy = 0.025. Note that the length of the field is set to unity.

[72] The different topology of the three fields is reflected by the connectivity function (Figure 4). For field 2, where high values of saturated hydraulic conductivity (dark grey values) are connected, the Euler number drops below zero first, followed by the multi-Gaussian field 1 and finally by field 3 where low conductivities are connected. This reflects the different connectedness of the structures. Hence the connectivity function based on the Euler number appears to be a suitable approach to quantify the topology of heterogeneous fields. For our analysis the entire structure is known, so that we could also calculate the lowest binarization threshold which lead to a continuous vertical path through the field along higher conductivities. This threshold corresponds to the percolation threshold and, as discussed above, the zero-point of the connectivity function should take similar values. Actually for 5 realizations for each field the mean percolation thresholds were found to be 133, 120 and 152 for field 1, field 2 and field 3 respectively. The corresponding zero-points of the connectivity functions were 133, 108, 142. Obviously, for the narrow and tortuous connected paths of field 2 and 3, the percolation threshold is underestimated while for field 1 the two quantities were found to be exactly the same.

Figure 4.

Connectivity function of field 1 (dotted line), field 2 (dashed line), and field 3 (solid line). Negative Euler numbers indicate high connectivity.

[73] In order to apply effective medium theory, the fields were segmented into five parameter classes by applying four threshold values and setting the values in between these thresholds constant (using the mean of the f values in between the thresholds). They were segmented in a way that each material has the same volume percentage. For the calculation of the effective conductivities with effective media, the fields were treated as five-material composites.

5.2. Local Parameter Distribution

[74] The mean of the log saturated conductivity is 〈fk〉 = 0, and its variance is σf2 = 2.0. Note that all parameters are here made dimensionless, therefore the hydraulic conductivity fields have to be considered as divided by their geometric mean.

[75] The local parameter functions (the relative hydraulic conductivity and water content function) were described by a van Genuchten model [van Genuchten, 1980], assuming the relation m = (n − 1)/n

equation image

where the porosity nf is assumed to be constant. The soil has thus the parameters fk, α and n.

[76] The parameter α is related to the entry pressure of the medium. It is here assumed to be related to the saturated hydraulic conductivity, the same way it would be for a Miller similar medium [Miller and Miller, 1956]. The relation is given by

equation image

where C is a constant. When making the Richards equation dimensionless, it is practical to choose the typical tension H such that it is the equivalent entry pressure which corresponds to the typical saturated hydraulic conductivity value K0. We choose the typical tension as

equation image

[77] This leads to the general form of the dimensionless parameter αequation image

equation image

[78] The parameter n is also assumed to be space dependent. Usually, finer materials with lower saturated hydraulic conductivity have a lower value of n (which corresponds to a large slope of the water content curve at intermediate saturation), while coarser materials have a higher value of n. It was assigned to the saturated log conductivity as n = 2.1 + (erf(fk/2) + 1)/2*1.6. In this way the n fields have a minimum of n = 2.1, a maximum of n = 3.7, and a mean of 〈n〉 = 2.9.

[79] As α and n are related to fk, the only heterogeneous parameter in the field is here fk. This is chosen for the test fields here, but it is not necessary to have these relations.

[80] The tension–unsaturated hydraulic conductivity functions for five values of fk (which correspond to the values chosen for the composites for the estimation of the effective parameters with effective medium theory) are shown in Figure 5. The unsaturated conductivity functions cross at different tensions, so that the medium, which has the highest hydraulic conductivity at saturated conditions, has at high tensions the lowest hydraulic conductivity and vice versa. However, because of the space dependency of the n parameter, the ratios of two of the conductivities of two successive materials at a given tension are not the same. At high tensions, the unsaturated hydraulic conductivity of the finest material (lowest fk) differs much less from that one of the intermediate material than the unsaturated hydraulic conductivity of the coarsest material (highest fk). In Figure 5 the variance of the logarithm of the unsaturated hydraulic conductivity is also shown as a function of the tension. Starting at saturated conditions, the variance decreases first with increasing tension. After reaching a minimum it increases again to very high values at dry conditions. This behavior of the variance of unsaturated hydraulic conductivity is also discussed, for example, by Roth [1995].

Figure 5.

(top) Tension-unsaturated hydraulic conductivity functions of the five composite materials of the fields. (bottom) Variance of the total hydraulic conductivity of all three fields.

5.3. Effective Water Content Functions

[81] The effective water content functions for all three fields shown in Figure 3 were calculated according to the homogenization results for the capillary equilibrium (11) and for the gravity-dominated flow regime (13). The effective water content function for the capillary equilibrium case can be easily obtained for a given value of the tension. For the gravity-dominated case the tension field has to be calculated numerically before it is averaged over a unit cell.

[82] The effective saturation–tension curves are obtained as described in sections 3.1 and 3.2. For the capillary equilibrium case a set of tension values is chosen, the saturation field is calculated for each value and averaged. For the gravity-dominated case the local tension field is solved according to (12). To solve the equation a Newton-Raphson scheme [Press et al., 1988] was used based on a finite volume scheme for the discretization of the flow equation. The Jacobian was calculated analytically. Newton-Raphson iterations were performed until a relative accuracy of 10−5 was obtained. The local saturation field is calculated from the tension field. The tension field and the saturation fields are averaged. This is repeated for a set of boundary values for the tension. As the scheme becomes difficult at saturation values close to one (where gradients diverge), the curves were only calculated for small tensions, up to 〈Ψ(0)〉 = 0.6.

[83] The effective curves for both flow regimes are shown for the three fields in Figure 6. The difference between the curves for capillary equilibrium and gravity-dominated flow are negligible. Although the tension in the gravity-dominated case is not constant on the small scale the fluctuations are not big compared to the fluctuations due to the heterogeneity of the field. The fluctuations caused in the corresponding saturation fields are therefore dominated by the heterogeneity. The averaged values are thus almost identical to the values of the capillary equilibrium case. This has also been found for layered media [Neuweiler and Eichel, 2006].

Figure 6.

Effective water content function for the three test fields. (top) Capillary equilibrium. (bottom) Gravity-dominated flow.

5.4. Effective Unsaturated Hydraulic Conductivity Functions

[84] The three effective medium theory approaches discussed in section 4 are applied to calculate the effective unsaturated hydraulic conductivity for the capillary equilibrium. The results obtained with the three approaches (self-consistent approach, Maxwell approach, differential effective medium theory) are compared to the “real” (numerically calculated) curves for the three test fields in Figure 7.

Figure 7.

The x component of the effective unsaturated hydraulic conductivity curves using the three different effective medium theory approaches and the capillary equilibrium results obtained with homogenization theory. (top) Field 1. (middle) Field 2. (bottom) Field 3. DEMT stands for differential effective medium theory.

[85] The curves were calculated from the parameter distributions of the fields. Different tension values Ψ(0) were chosen and the resulting dimensionless unsaturated hydraulic conductivity values, κu = κskr(0)) were calculated for each field. The effective hydraulic conductivity according to the self-consistent approach (14) can be calculated using this information only. For the Maxwell approach and the differential effective medium approach we need additionally the information, which material is the background material. It is a reasonable approach to treat the connected material as the background material. According to Figure 4 for field 1, this is the intermediate phase, for field 2 it is the phase with the highest saturated hydraulic conductivity, while for field 3 it is the phase with the lowest saturated hydraulic conductivity (see section 5.1). Including this information the effective hydraulic conductivity is then calculated for both methods using (16) and (19). The inclusions are assumed to be ellipsoids with the same anisotropy ratio as that one of the heterogeneous fields, λx/λz = 3. The anisotropy shape matrix A of the ellipsoids reads thus

equation image

[86] The effective curves for the x component of the effective hydraulic conductivity for the three fields are shown in a semilog plot in Figure 7. The z component has always smaller values, but as the curves show no qualitative differences from the curves of the x component, they are not shown here. The three effective medium theory approaches are shown together with the numerically calculated results of homogenization theory (see section 5.3.2). The tension values are chosen between Ψ(0) = 0 and Ψ(0) = 5. The numerically calculated curves are averages over 100 realizations. Because of the large variances at high tensions (see Figure 5) the sample-to-sample fluctuations are large, especially for field 2.

[87] The curves calculated with the self-consistent approach are similar for all three fields. This is to be expected, as the parameter distributions are more or less the same for all three materials. This hydraulic conductivity curve fits the homogenization result only well for field 1 and 3. However, for these fields the other two approaches (Maxwell and differential effective medium) are also not far away from the homogenization result.

[88] For field 2, where the high saturated hydraulic conductivity values are connected, the unsaturated hydraulic conductivity curve calculated with homogenization theory lies for low tensions above the curve for field 1. In this case, the self-consistent approach underestimates the hydraulic conductivity. At full saturation the (dimensionless) hydraulic conductivity is κs = 1.6, while the self-consistent approach would predict it to be κs = 1.3. On the other hand, the Maxwell approach overestimates the hydraulic conductivity at low tensions. For full saturation it would predict a hydraulic conductivity of κs = 2.0. The differential effective medium approach also overestimates the effective hydraulic conductivity, but not to such a high degree as the Maxwell approach. At full saturation it would predict a hydraulic conductivity of κs = 1.7. This is the best estimate for the saturated hydraulic conductivity. The difference between the effective conductivities for field 2 becomes more pronounced for high tensions. In this case the variance of the unsaturated conductivity is very high and the material which is connected throughout the medium has a hydraulic conductivity which is several orders of magnitude smaller than the mean conductivity. The effective hydraulic conductivity is highly overestimated by the self-consistent approach and highly underestimated with the Maxwell approach. The differential effective medium approach yields good result.

[89] For field 3, the saturated (dimensionless) hydraulic conductivity values are κs = 1.2 with homogenization theory, κs = 1.3 with the self-consistent approach, κs = 0.7 with the Maxwell approach and κs = 1.0 with the differential effective medium theory approach. At high tensions the situation is vice versa than in field 2. At high tensions the results obtained with the three approaches differ not as much as they do for field 2. The reason is, that although the variance of the field is extremely high, the difference between the conductivity of the connected material (fine material with the lowest value of fk) and the mean conductivity is not that large (see Figure 5). This nonsymmetric behavior of fields 2 and 3 is a consequence of the nonsymmetric distribution of the n value in the van Genuchten model.

5.5. Anisotropy Ratio

[90] The anisotropy ratio of the effective unsaturated hydraulic conductivity is an important feature. It has already been discussed by Yeh et al. [1985] that under unsaturated conditions the anisotropy ratio of the hydraulic conductivity does not only depend on the anisotropy ratio of the parameter field, but also on the external conditions, such as the mean tension. The anisotropy ratio κeff, u, xeff, u, z of the effective conductivities is shown here for different tensions in Figure 8. It is an ensemble average, and it should be noted that the fluctuations between the realizations are extremely high at high tensions, especially for field 2. The reason is the high variance of the unsaturated conductivity at high tensions. It is therefore only shown to tensions of Ψ(0) = 3. The self-consistent approach clearly overestimates the anisotropy ratio for all test fields at high tensions. The Maxwell approach underestimates it, while the differential effective medium approach yields reasonably good results.

Figure 8.

Anisotropy ratio of the effective hydraulic conductivity κeff, u, xeff, u, z. (top) Field 1. (middle) Field 2. (bottom) Field 3. DEMT stands for differential effective medium theory.

[91] The analysis of Yeh et al. [1985] of the anisotropy ratio of the effective conductivity under unsaturated conditions cannot be compared directly to the effective hydraulic conductivity calculated here, as it is derived for a Russo-Gardener model and a different relation between the α parameter in this model and to fk is than used here. However, Yeh et al. [1985] found in principle a similar behavior as found here: At low water tension the anisotropy ratios follow the variance of the total hydraulic conductivity κu and thus decrease at certain regions with the tension. This effect is not expected if the parameter α and the log saturated conductivity fk are uncorrelated [cf. Yeh et al., 1985]. The anisotropy ratio of the three test fields differ slightly at high tensions, where the variance of the unsaturated hydraulic conductivity is very high. The reason is the nonsymmetric behavior of the tension–conductivity curves of the different materials due to the nonconstant n parameters.

5.6. Capillary Equilibrium and Gravity-Dominated Flow

[92] The effective hydraulic conductivity functions for the capillary equilibrium and for the gravity-dominated flow regime were calculated according to the homogenization result. The local hydraulic conductivity field, κu = κskr(0)) was calculated proceeding from a constant tension or from the solution of equation (12). Then the effective hydraulic conductivity for the local hydraulic conductivity fields κu were calculated. For the capillary equilibrium case this was done by applying unit pressure gradients in both directions and solving the local pressure equation with periodic boundaries. The solution was done numerically using a finite volume scheme. The resulting averaged horizontal and vertical fluxes yield the effective hydraulic conductivity tensor. As the main axes of the covariance of the field are aligned with the coordinate system, the obtained tensor was always diagonal. We will therefore show here only the z entry. For the gravity-dominated case the vertical flux (and thus the hydraulic conductivity) is a by-product of the numerical calculation of the local tension field. The effective curves calculated according to the homogenization theory results are shown in Figure 9. The differences between the results for the capillary equilibrium case and the gravity-dominated case are marginal. The reasons are the same as discussed for the effective water content curves. We can therefore assume that the capillary equilibrium curves provide a reasonable estimate for the effective parameter functions also for the gravity-dominated flow regime. This has also been found for layered media [Neuweiler and Eichel, 2006].

Figure 9.

Effective hydraulic conductivity function for the four test fields. (top) Capillary equilibrium. (bottom) Gravity-dominated flow.

6. Discussion and Conclusions

[93] The purpose of this paper is to apply simple approaches to estimate effective unsaturated parameters, which reflect the existence of connected paths of different parameter ranges in the soil. The determination of the connected paths based on the Euler number, in combination with the anisotropy ratio of the field, might offer one possibility to quantitatively include topological information. Including the information of the connected paths is here considered as a method suitable for non-multi-Gaussian fields, where the extreme parameters are connected.

[94] The effective parameters were calculated for upscaled models derived with a scale expansion. The models are restricted to slow flow processes, where local dynamic effects can be neglected and nonequilibrium terms are not expected to appear in the upscaled equation. Two flow regimes were compared, the capillary equilibrium case, where capillary forces dominate on the small scale and the gravity-dominated domain, where capillary forces and gravity forces are locally in equilibrium. The first case leads to a simple method to calculate effective parameters, while the second one is more complex. However, comparing the results for several test cases showed that the effective parameter curves are not that different. The capillary equilibrium assumption is often a good approximation also for the gravity-dominated flow regime.

[95] Given the parameter field, the effective parameter functions can be calculated according to local problems applying periodic boundary conditions. However, as in reality the detailed parameter distribution is not known, we tested different effective medium theory methods to estimate the effective unsaturated conductivity for a model based on the assumption of capillary equilibrium. The methods were applied to three anisotropic test fields, where the constitutive relations of the soil parameters were modeled using a van Genuchten approach. The model parameters α and n were assumed to be heterogeneously distributed and related to the heterogeneous saturated hydraulic conductivity. The effective medium theory estimates are compared to the effective parameters obtained from homogenization theory, where the effective parameters are calculated for a given field explicitly.

[96] When applied to test fields, it was found that the self-consistent approach gives good results if the fields are multi-Gaussian. However, if the medium has connected structures of the extreme parameter ranges and if the variance of the local parameter field is large, the estimate of the effective conductivity with the self-consistent approach can fail. For the test fields this was the case for the medium where the coarse material is connected.

[97] Effective medium theory methods, where one material can be assigned as the background material (Maxwell approach and differential effective medium theory), were also tested. Knowledge about the background material can, for example, be obtained from local measurements via the Euler number, as introduced by Vogel [2002]. At tensions close to zero the Maxwell approach overestimates (or underestimates) the unsaturated hydraulic conductivity of material where connected structures of high (or low) permeable material are present, and vice versa at very high tensions. The differential effective medium approach shows the same trend, however, as the effective properties are adjusted in incremental steps with incremental increase of inclusion material, the overestimation and underestimation of the background material is less pronounced. The differential effective medium approach yielded good results for the different test fields. Its drawback is that for porous media with many material classes, the method becomes very complex to handle.

[98] The anisotropy ratio of the effective hydraulic conductivity could also best be reproduced using the differential effective medium theory approach. At low tensions (wet conditions) the anisotropy ratio of fields where high, low and intermediate saturated hydraulic conductivity values are connected are similar and are related to the variance of the total hydraulic conductivity of the field. At high tensions (dry conditions) the anisotropy ratio of the three test fields showed clear differences. It should, however, be noted that the variances of the parameter fields are here very high, so that realization-to-realization fluctuations are high.

[99] The type of non-multi-Gaussian fields discussed here was material with one connected parameter range, while all other parameter ranges form isolated structures. As discussed in section 3.3, this is in a two-dimensional nonlayered field always the case. As soon as two different materials are connected, the medium has to have a layered structure, where the layers may be distorted. This is different in a three-dimensional medium, where several parameter ranges can be connected and also the connected material ranges can be different in different directions. In this case the Maxwell approach and the effective medium theory approach are not suitable, as these methods proceed from one background material. The connected parameter ranges have then to be estimated from two-cut instead from one-cut indicator fields, for example by using critical path analysis [Hunt, 2001, 2005], or by using the Euler number for three-dimensional fields.

[100] It might be a reasonable approach to assign the arithmetic mean of the connected materials as background parameter, where the background hydraulic conductivity can be considered a tensor. The arithmetic mean is here a reasonable estimate, as all materials considered are connected. The Maxwell or the differential effective medium theory approach for anisotropic background properties, as outlined, for example, by Torquato [2002], could then be applied to estimate the effective conductivity of the field.

[101] The methods to calculate effective hydraulic conductivity curves presented here are simple and partly allow to take the aspect of connectivity into account. However, the results would certainly improve if more information about the geometric properties of the soil, such as the tortuosity of the connected phase, could be taken into account. Also, the two-point correlation structure of the soil, which is accounted for in the stochastic approach, is here neglected.

Appendix A:: Multiscale Expansion of the Richards Equation

[102] The expansion (8) and the transformation of the gradient (9) are inserted into the Richards equation (6). All other functions of the saturation are also expanded around the zeroth order of the saturation in terms of a Taylor expansion (Θ = Θ(0) +ɛ Θ(1) + ɛ2 Θ(2) + … and kr = kr(0) + ɛ kr(1) + ɛ2kr(2) + …), with

equation image

and analogous terms for kr. This yields

equation image

Acknowledgments

[103] This study was funded by the Deutsche Forschungsgemeinschaft within the Emmy Noether program under grant Ne 824/2-2.

Ancillary