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 computation of discrete interfacial average fluxes. Standard methods, such as Integrated Finite Differences(IFD) or Finite Volumes (FV), rely on the definition of average gradients of the solution, and calculate the interface fluxes by means of appropriate averages of the conductivity tensor K. The question of how to choose a correct average for the conductivity in the anisotropic case is however still open. We derive a procedure based on conservation of flux and energy, which is particularly suited for non-asymptotical regimes, at a fixed mesh (size). The resulting effective tensor provides standard arithmetic-harmonic means of tensor coefficients with respect to tangential and normal components of the gradient in simple cases. Moreover, this tensor is shown to coincide with a matrix arising from homogenization theory, even though it has been obtained for different purposes,and following a different approach. The effectiveness of the proposed method is verified numerically.