## 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.