Coarse graining on arbitrary grids

We focus on upscaling flow in heterogeneous porous media by means of the coarse graining method. In previous studies a numerical upscaling scheme was introduced to calculate an effective permeability tensor. It appeared that this scheme provides reasonable results, but this method is limited to regular square grids. We extend the numerical coarse graining scheme to arbitrary grids. In order to get a higher quality upscaling method, appropriate boundary conditions for the cell problem have to be used. Therefore, we introduce new boundary conditions suitable for arbitrary elements. Considering the local permeability K(x) being a stationary random field of lognormal distribution, we compare the extended upscaling scheme with simple upscaling methods (arithmetic and geometric upscaling) on different grids and scales, using various correlation lengths by calculating fluxes Q_i and solutions u_i of a test problem. In this comparison we also use various boundary conditions for the cell problem to evaluate their influence on the quality of the coarse graining method. For anisotropic fields, coarse graining with the newly introduced boundary conditions proves to be superior to simple upscaling methods.