Numerical Subgrid Upscaling for Waterflood Simulations

Abstract

Abstract We present a subgrid-scale numerical technique for upscaling waterflood simulations. We scale up the usual parameters porosity and relative and absolute permeabilities, and also the location of wells and capillary pressure curves. Some of these are critical nonlinear terms that need to be resolved on the fine scale, or serious errors will result. Upscaling is achieved by explicitly decomposing the differential system into a coarse-grid-scale operator coupled to a subgrid-scale operator. The subgrid-scale operator is approximated as an operator localized in space to a coarse-grid element. An influence function (numerical Greens function) technique allows us to solve these subgrid-scale problems independently of the coarse-grid approximation. The coarse-grid problem is modified to take into account the subgrid-scale solution and solved as a large linear system of equations. Finally, the coarse scale solution is corrected on the subgrid-scale, providing a fine-grid scale representation of the solution. In this approach, no explicit macroscopic coefficients nor pseudo-functions result. The method is easily seen to be optimally convergent in the case of a single linear parabolic equation. The method is several times faster than solving the fine-scale problem directly, generally more robust, and yet achieves good results as it requires no ad hoc assumptions at the coarse scale and retains all the physics of the original multiphase flow equations.