Abstract
The two-point flux, finite volume method (FVM-2P) is the most widely used method for solving the flow equation in reservoir simulations. For FVM-2P to be consistent, the simulation grid needs to be orthogonal (or K-orthogonal if the permeability field is anisotropic). It is well known that cornerpoint grids can introduce large errors in the flow solutions due to the lack of orthogonality in general. Multi-point 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 to 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 on 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 FVM-2P does not converge to the correct solution as 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 they can be global with no strong correlation with local non-orthogonality measures. Numerical tests are presented to confirm the analytical findings and to show the applicability of our conclusions to more general flow scenarios.
