MPFA for Faults and Local Refinements in 3D Quadrilateral Grids With Application to Field Simulations

Abstract

Abstract This paper extends the multipoint flux-approximation (MPFA) control-volume method to 3D quadrilateral grids for which the adjacent cells do not necessarily share corners. Examples are grids with faults and locally refined grids. This paper gives a derivation of the method for such grids. The difference between two-point flux-approximation (TPFA) results and MPFA results for faults is demonstrated for synthetic problems. Further, the results are compared with results from uniform fine-grid simulations. The effect of repeated fault patterns as well as anisotropy is investigated. Large errors may be found for the TPFA method for flow through a series of faults in an anisotropic medium. Finally, a comparison is done for a reservoir field application. Introduction Over the last years control-volume formulations for multiphase flow in the case of full permeability tensors and general corner-to-corner grids have received increased attention, Discretizations in which the flux is determined by a multipoint flux approximation (MPFA) have been proposed for both structured1,2,6,11,12,13,18 and unstructured grids3,4,5,14,16,17. This paper extends the MPFA method to 3D quadrilateral corner-point grids15 for which the adjacent cells do not necessarily share corners. Examples are grids with faults and grids with local refinements. A 2D-extension7 has previously been given. Discretization methods for nonmatching grids have also been developed using mixed finite elements and mortar methods8,19. In these papers convergence for arbitrary but fixed multiblock decompositions is proved. Flow through faults is not only influenced by the permeability in the layers, but also by the properties of the fault itself. Flux multipliers for the fault interfaces may easily be implemented in the suggested MPFA method. How these multipliers should be determined, is not the subject of this paper. For single-phase flow in a homogeneous medium, the proposed MPFA method has on a uniform grid a truncation error of O(h2). However, convergence analysis is principally of interest for nonuniform grids in heterogeneous media. The paper contains no attempt to give such a convergence analysis. In the first three sections of the paper, the proposed MPFA method is discussed and derived. The following section gives a review of the two-point flux approximation (TPFA), commonly used at faults15. In the next section the properties of the proposed MPFA method is investigated through synthetic examples. MPFA is compared to TPFA. In the final section MPFA and TPFA are compared for a 3D reservoir field case.