Upscaling for unsaturated flow including connectivity of the soil structure

Modelling of flow processes in the unsaturated zone on a large scale requires often upscaling of the flow problem. The soil parameter distribution is mostly not known, so that the upscaled problem is based on a characterization of the soil heterogeneity, which is obtained from local measurements. Effective flow parameters are therefore often derived in a stochastic framework, where the heterogeneity of a field is quantified by its second order stochastic properties (the mean, variance and a two-point covariance model with an integral scale and anisotropy parameters). A field has to be Gaussian to be quantified completely by these parameters. However, soil structure resembles rarely Gaussian fields. It has connected and isolated structures of different material types. The connectedness of material has a large influence on the upscaled models, yet these properties can in general not be derived from second order stochastic field parameters. Therefore upscaling methods are needed which take connectivity into account. In this contribution we first discuss how soil connectivity can be quantified from local data in a reasonable way. We then discuss how the information about connectivity can be taken into account into the upscaled models. This is done here by applying different effective medium approaches to derive unsaturated effective permeability. The methods are all derived assuming capillary equilibrium on the small scale. Effective parameters are obtained using the self-consistent approach, the Maxwell approach, the differential effective medium approach and a generalized percolation approach. We compare the methods and discuss their advantages and limits. The results obtained for different test fields are compared to the exact solution for the effective parameters, derived by homogenization theory.