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

Abstract

The numerical solution of strongly heterogeneous and anisotropic porous media flow equations often requires the definition of discrete interfacial average fluxes. Standard methods, such as Integrated Finite Differences (IFD) or Finite Volumes (FV), start from the definition of an average solution gradient and calculate the interface fluxes by means of appropriate averages of the conductivity tensor K. For a scalar K, the harmonic average is correct in the sense that it returns the same flux if multiplied by the average gradient in the direction normal to the control volume boundary. On the other hand, in terms of asymptotic convergence, any average procedure yields the correct rate. However, significant differences may arise when working on a fixed mesh size. The question of how to define a correct averaging procedure for the anisotropic case is however still open, in particular when the principal directions of anisotropy change in space. We derive a procedure based on variational principles of conservation of flux and energy. The resulting mean matrix calls into play the tangential component of the gradient and provides standard arithmetic or harmonic means for tangential and normal gradient components in simple cases. Moreover, the resulting tensor is shown to coincide with a matrix arising from homogeneization theory, even though it has been obtained using a different approach. The effectiveness of the proposed average is tested on a number of numerical examples on two-dimensional domains for both IFD and FV.