Numerical modeling of subgrid heterogeneity in two phase flow simulations

A coarse-scale model governing two-phase viscous dominated flow in heterogeneous porous media is presented. The coarse-scale water transport equation (or saturation equation) is derived from a volume average of the underlying fine-scale saturation equation. A key component of the coarse-scale description is the subgrid model, which appears in the formulation as a nonlinear, nonlocal term in the coarse-scale saturation equation. This subgrid model captures the effects of the unresolved (fluctuating) components of saturation and velocity. An approximation is introduced which allows the nonlocal effect to be estimated using local fine-scale information (computed in a preprocessing step) coupled with global coarse-scale quantities, thus avoiding the need for any global fine-scale calculations. Results using the new model are presented for a variety of heterogeneous permeability fields in two dimensions over a range of fluid viscosity ratios. In all cases considered, computations using the new model are significantly more accurate, relative to the reference fine-scale results, than simulations in which subgrid effects are neglected. Further extensions of the formulation, which would enable the modeling of realistic systems, are discussed.