Implementation and Application of a Hybrid Multipoint Flux Approximation for Reservoir Simulation on Corner-Point Grids

Abstract

Abstract Accurate and robust discretization of the fluid flow equations is required to account for the extreme heterogeneity of oil and gas reservoirs, and the combined effect of anisotropy and grid distortion, necessary to adapt the grid to the geology. There is a growing need for handling distorted unstructured grids and full permeability tensors that appear after upscaling of the fine-scale permeability field. The classical cell-centered finite difference method results in a 7-point stencil and is insufficient to account for these effects. A number of closely related methods, coined as multipoint flux approximations (MPFA), have been proposed recently and are currently under active development. The basic principle of MPFA is that the flux across an interface between two gridblocks depends on the state variables (pressure and saturations) of more than two gridblocks. MPFA leads naturally to an enhanced finite volume method with a 27-point stencil. In this paper, we implement a variant of the MPFA method for corner-point geometry hexahedral grids. Motivated by the very high aspect ratio of gridblocks in typical reservoir models (ratios as high as 100:1 are not uncommon), we propose a hybrid method that employs a multipoint flux approximation in the areal direction and a two-point flux approximation in the vertical direction. This restricted MPFA method leads to an 11-point stencil, therefore reducing the computational effort significantly. We discuss the implementation of the method in detail. We evaluate its performance on a number of test cases and show that, for typical applications, this simplification does not greatly compromise the accuracy of the solution. Introduction Some of the challenges of reservoir simulation are related to the accurate and robust discretization of the governing equations, in order to account for the extreme heterogeneity of the medium, and the combined effect of anisotropy and grid distortion, necessary to adapt the grid to the geology. In this regard, there is a growing need for handling distorted corner-point grids1 —eventually fully unstructured grids— and full permeability tensors that appear after upscaling the fine-scale permeability field.[2] The classical 5-point stencil discretization with finite differences (7-point stencil in 3D) is insufficient to account for these effects. In the context of non-Cartesian grids, finite difference methods can be generalized to finite volume methods. These schemes establish mass conservation over a control volume (gridblock) by approximating the flux across the control boundary. Traditionally, a two-point flux approximation is used to discretize the flux across each face of the gridblock. Such a scheme leads, however, to a nonconvergent method when the permeability tensor is not aligned with the grid. Over the past decade, enhanced control-volume schemes have been proposed to properly account for full-tensor permeability fields and nonorthogonal grids. These methods are based on an extended flux molecule that uses more than two gridblocks to define the interface flux, and therefore receive the name of multipoint flux approximation (MPFA) methods. The first derivation of these methods was presented independently by Aavatsmark et al.[3] and Edwards and Rogers.[4] These methods have been extended to two-dimensional unstructured grids[5–7] and to three-dimensional hexahedral grids[8–11] that account for complex geological features such as faults, pinch-outs and deviated wells.[12, 13] These works demonstrate that there are many practical situations in which neglecting the effect of permeability anisotropy and grid nonorthogonality results in large errors in flow predictions.