Grid Adaption for Upscaling and Multiscale Method

A Dirichlet-Neumann representation method was recently proposed for upscaling. The method expresses coarse fluxes as linear functions of multiple discrete pressure values along the boundary and at the center of each coarse block. The number of pressure values can be adjusted to improve the accuracy of simulation results, and in particular to resolve important fine-scale details. Improvement over existing approaches is substantial especially for reservoirs that contain high permeability streaks or channels. Multiscale methods obtain fine-scale fluxes or pressures at the cost of solving a coarsened problem, but can also be utilized for flexible upscaling. We compare the DNR and a multiscale mixed finite-element method. Both can be expressed in mixed form, with local stiffness matrices obtained as inner products of basis functions with fine-scale subresolution determined from local flow problems. Piecewise linear Dirichlet boundary conditions are used for DNR and piecewise constant Neumann conditions for MsMFE. Adding discrete pressure points in the DNR method corresponds to subdividing coarse faces and hence increasing the number of basis functions in the MsMFE method. The methods show similar accuracy for 2D Cartesian cases, but the MsMFE method is more straightforward to formulate in 3D and implement for general grids.