Finite Volume Solutions of Strongly Anisotropic Porous Media Flow

Anisotropic (i.e. direction dependent) porous media flow equations are characterized by conductivities that may be space dependend full rank tensors. This class of problems is also known as parameter dependent problems, the parameter being the anisotropy ratio, i.e. the ratio between the smallest and largest eigenvalues of the conductivity tensor. Efficient numerical discretization of strongly anisotropic problems is generally obtained by means of ad hoc, mesh dependent scheme modifications developed to overcome the problem known as parametric locking. Locking is experimentally observed when the discretization error does not decrease at the expected rate for limiting values of the parameter. This loss of convergence disappears for sufficiently fine discretizations, but may involve costly or even unfeasibly large calculations. The numerical solution of this type of problems requires careful consideration of the errors that may be introduced by the discretization scheme. We propose a modification of Finite Volume approach based on the definition of a diamond cell that makes optimal use of the reconstruction algorithm to yield an accurate discretization of tangential gradients, a key ingredient for achieving robustness of the numerical scheme for large values of the anisotropy ratio. Numerical results are presented to show the performance of the proposed scheme. Without introducing additional nonconsistent terms in the numerical scheme, as is typically done in these cases (the so called variational crimes), the proposed scheme robust with respect to the anisotropy ratio, as per the definition of Babuska et al, 1992. We also show that the region of convergence of the method is defined by a quadratic relationships between the anisotropy ratio and the mesh size parameter.