• Richards equation;
  • unsaturated flow;
  • upscaling

[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.