Effect of Grid Deviation on Flow Solutions

The two-point flux finite-volume method (2P-FVM) is the most widely used method for solving the flow equation in reservoir simulations. For 2P-FVM to be consistent, the simulation grid needs to be orthogonal (or k-orthogonal if the permeability field is anisotropic). It is well known that corner-point grids call introduce large errors in the now Solutions because of the lack of orthogonality in general. Multipoint flux formulations that do not rely on grid orthogonality have been proposed, but these methods add significant computational cost to solving the flow equation. Recently, 2.5D unstructured grids that combine 2D Voronoi areal grids with vertical projections along deviated coordinate lines have become an attractive alternative to corner-point gridding. The Voronoi grid helps maintain orthogonality areally and can mitigate grid orientation effects. However, experience with these grids is limited. In this paper, we present an analytical and numerical study of these 2.5D unstructured grids. We focus on the effect of grid deviation oil flow solutions in homogeneous, but anisotropic, permeability fields. In particular, we consider the grid deviation that results from gridding to sloping faults. We show that 2P-FVM does not converge to the correct solution as the grid refines. We further quantify the errors for some simple flow scenarios using a technique that combines numerical analysis and asymptotic expansions. Analytical error estimates are obtained. We find that the errors are highly flow dependent and that they can be global with no strong correlation with local nonorthogonality measures. Numerical tests are presented to confirm the analytical findings and to show the applicability of our conclusions to more-general flow scenarios.