Analysis of control volume methods

The control volume methods, and in particular the two-point flux approximation (TPFA) method, are widely used in reservoir simulations to calculate the Darcy flux. A generalization of the TPFA method called the multi-point flux approximation (MPFA) method was proposed in the middle of the nineties. The aim of the MPFA methods is to obtain reliable results even when dealing with non-orthogonal grids, i.e. when the permeability tensor is not aligned with the grid. While numerical tests indicate the convergence of the MPFA methods, tests also show that there are limitations regarding grid deformity and permeability anisotropy. There are also different MPFA methods, with no method being optimal for all grids.

The aim of this project has been to provide the theoretical analysis for the MPFA methods to supplement the numerical results. The convergence proofs for the MPFA methods O and L (named after the form of their stencils in 2D) were obtained by drawing on the similarities between the MPFA methods and the mimetic finite difference methods. The proofs give explicit local conditions regarding the grid deformation and the permeability that will ascertain convergence. This gives us a tool to evaluate which MPFA method to use on which grid. The results have also led to a better understanding of the relationship between the MPFA methods and the mimetic finite difference method.

As an extension to multi-phase flow we have also looked at the difficulties regarding discretization which arise when wanting to consider a tensor relative permeability field (and not only a scalar). Finally, as the MPFA methods resolve for fluxes and not continuous velocity fields, we have looked at an interpolation valid on general polytopes and which reproduces uniform flow. The interpolation has been used for streamline tracing with success.