Spatially-varying Compact Multi-point Flux Approximations for 3-D Adapted Grids with Guaranteed Monotonicity

We propose a new single-phase local transmissibility upscaling method for adapted grids in 3-D domains that uses spatially varying and compact multi-point flux approximations (MPFA), based on the VCMP method previously introduced for 2-D Cartesian grids. For each cell face in the coarse upscaled grid, we create a local fine grid region surrounding the face on which we solve three generic local flow problems. The multi-point stencils used to calculate the fluxes across coarse grid cell faces involve up to ten neighboring pressure values. They are required to honor the three generic flow problems as closely as possible while maximizing compactness and ensuring that the flux approximation is as close as possible to being two-point. The resulting MPFA stencil is spatially varying and reduces to two-point expressions in cases without full-tensor anisotropy. Numerical tests show that the method significantly improves upscaling accuracy as compared to commonly used local methods and also compares favorably with a local-global upscaling method. We also present a corrector method that adapts the stencils locally to guarantee that the resulting pressure matrix is an M-matrix. This corrector method is needed primarily for cases where strong permeability contrasts are mis-aligned with the grid locally. The corrector method has negligible impact on the flow accuracy. Finally, we show how the computed MPFA can be used to guide adaptivity of the grid, thus allowing rapid, automatic generation of grids that can account for difficult geologic features such as faults and fractures and efficiently resolve fine-scale features such as narrow channels.