Macro-scale effective constitutive relationships for two-phase flow processes in heterogeneous porous media with emphasis on the relative permeability–saturation relationship

Macro-scale simulations often play an important role in the assessment and remediation of contamination by dense non-aqueous phase liquids (DNAPLs) in the subsurface. Effective parameters for the macro scale are required for these simulations in order to avoid a detailed discretisation of the geological structures. Starting from the observed influence of heterogeneities on multiphase flow processes at the macro scale, we present an upscaling procedure from the local to the macro scale for the derivation of constitutive relationships for multiphase flow processes. The approach is based on the assumption of an equilibrium of (capillary) forces, which allows the application of a percolation model. This results in saturation distributions for different capillary pressures. Averaging these distributions gives rise to a macroscopic capillary pressure–saturation relationship. For the saturation distribution, relative permeabilities and effective conductivities are computed depending on the structure and the flow direction. These are averaged with the help of the renormalisation method. The evolving relative permeability–saturation relationship for the macro scale shows a saturation-dependent anisotropy and pronounced residual saturations of the nonwetting phase (which were not assumed for the local scale). The anisotropy reflects the underlying structure of the considered system that needs not to be known in detail.