Generalized convection-diffusion model for subgrid transport in porous media

Models for transport in heterogeneous subsurface formations usually require some type of treatment for subgrid effects. In this work, a generalized nonlinear convection-diffusion model for subgrid transport in two-dimensional systems is developed and applied. The model, although somewhat heuristic, is motivated by previous findings within both the stochastic and the deterministic frameworks. The numerical calculation of the diffusive and convective subgrid terms is described. The model is applied to several example cases involving heterogeneous permeability fields and different global boundary conditions. Both linear and nonlinear fine scale flux functions are considered. Coarse scale results for oil cut (fraction of oil in the produced fluid) and the global saturation field generated using the new subgrid model are shown to be in consistently better agreement with reference fine scale solutions than are coarse scale results using standard subgrid treatments. Extensions of the method required to treat more realistic subsurface systems are discussed.