Effective anisotropy tensor for the numerical solution of flow problems in heterogeneous porous media

The numerical solution of porous media flow equations often requires the computationof discrete interfacial average fluxes. Standard methods, such as Integrated Finite Differences(IFD) or Finite Volumes (FV), rely on the definition of average gradients of thesolution, and calculate the interface fluxes by means of appropriate averages of the conductivitytensor K. The question of how to choose a correct average for the conductivityin the anisotropic case is however still open. We derive a procedure based on conservationof flux and energy, which is particularly suited for non-asymptotical regimes, at afixed mesh (size). The resulting effective tensor provides standard arithmetic-harmonicmeans of tensor coefficients with respect to tangential and normal components of thegradient in simple cases. Moreover, this tensor is shown to coincide with a matrix arisingfrom homogenization theory, even though it has been obtained for different purposes,and following a different approach. The effectiveness of the proposed method is verifiednumerically.http://proceedings.cmwr-xvi.org