Estimation of effective parameters for a two-phase flow problem in non-Gaussian heterogeneous porous media

In this paper we discuss estimates of effective parameters for an upscaled model for buoyant counter flow of DNAPL and water in a closed box filled with heterogeneous porous material. The upscaling procedure is based on the assumption that the flow is dominated by capillary forces on the small scale and that the fluids are segregated. The upscaled model has the same form as the usual two-phase flow model with an effective capillary pressure function and an effective mobility function Λ. Effective parameters are then estimated in two different ways. Stochastic theory can be applied to calculate the effective parameters to first order in the parameter fluctuations. This approach does not take into account that different parameter ranges of the heterogeneous field may be connected or isolated, yielding very different macroscopic residual saturations. Therefore, the second estimate of effective parameters takes connectivity of parameter ranges into account. In this case, the univariate parameter distribution of the heterogeneous field and the values that mark connected materials are the only information about heterogeneity that is used. Effective parameters are then estimated using mean field theory (the Maxwell approach). The upscaled model and the estimation of effective parameters are applied to a numerical test case. Buoyant counter flow in heterogeneous parameter fields with different structures is simulated numerically and compared to the solutions of the quasi-1d upscaled model with differently estimated parameters. It is demonstrated that connectivity of the different parameter ranges is an important information that determines typical time scales for the flow process and the macroscopic residual saturation. Even simple estimates of effective parameters based on little information may capture the typical time scales, provided that information about connected parameter ranges is taken into account.