Preferential Flow-Paths Detection for Heterogeneous Reservoirs Using a New Renormalization Technique

Abstract

We have devised a renormalization scheme which allows very fast determination of preferential flow-paths and of up-scaled permeabilities of 2D heterogeneous porous media. In the case of 2D log-normal and isotropically distributed permeability-fields, the resulting equivalent permeabilities are very close to the geometric mean, which is in good agreement with a rigorous result of Matheron. It is also found to work well for geostatistically anisotropic media when comparing the resulting equivalent permeabilities with a direct solution of the finite-difference equations. The method works exactly as King's does, although the renormalization scheme was modified to obtain tensorial equivalent permeabilities using periodic boundary conditions for the pressure gradient. To obtain an estimation of the local fluxes, the basic idea is that if at each renormalization iteration all the intermediate renormalized permeabilities are stored in memory, we are able to compute -- ad reversum -- an approximation of the small-scale flux map under a given macroscopic pressure gradient. The method is very rapid as it involves a number of calculations that vary linearly with the number of elementary grid blocks. In this sense, the renormalization algorithm can be viewed as a rapid approximate pressure solver. The ‘exact’ reference flow-rate map (for the finite-difference algorithm) was computed using a classical linear system inversion. It can be shown that the preferential flow paths are well detected by the approximate method, although errors may occur in the local flow direction.