Numerical Subgrid Upscaling of Two-Phase Flow in Porous Media
We present an approach and numerical results for scaling up fine grid information to coarse scales in an approximation to a nonlinear parabolic system governing two-phase flow in porous media. The technique allows upscaling of the usual parameters porosity and relative and absolute permeabilities, and also the location of wells and capillary pressure. 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, which we localize by imposing a closure assumption. We approximate the coarse-grid-scale operator with a mixed finite element method that has a second order accurate velocity coupled implicitly to the subgrid scale. The subgrid-scale operator is approximated locally by a first order accurate mixed method. A numerical Greens influence function technique allows us to solve these subgrid problems independently of the coarse-grid approximation. 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.
Keywordsupscaling subgrid numerical Greens functions porous media
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