Effective Transmissibilities of a Gridblock by Upscaling — Comparison of Direct Methods with Renormalization
Donald W. Peaceman · SPE Journal · 1997
Summary Previous methods for upscaling fine-scale permeability data for use in reservoir simulation produce effective permeabilities for each simulator gridblock. Since transmissibilities between neighboring gridblocks are required by the simulator, it is better to determine six "half-block transmissibilities" for each gridblock. These can be calculated directly by solving the finite-difference equations for pressure in each of six half-blocks, wherein uniform pressures are applied at two opposite faces and no-flow conditions are applied at the other four faces. Alternatively, in order to reduce computation time, renormalization has been proposed for calculating effective permeabilities of the gridblocks. These previous proposals ignore anisotropy and require that the elemental blocks and each coarse block be rectangular. In this paper, renormalization is modified to avoid these disadvantages. Renormalization is most easily implemented if there are N × N × N elemental blocks within each coarse block, where N is a power of 2. A more flexible variation of renormalization is also presented which reduces this restriction somewhat. However, the direct method is much more flexible with regard to the number of elemental blocks in each direction. While renormalization might be expected to be faster than the direct method, comparison of running times shows that a highly efficient iterative solver for the direct method is somewhat faster than renormalization.