Implementation of a Flux-Continuous Finite-Difference Method for Stratigraphic, Hexahedron Grids

Abstract

Summary In this paper we present a 3D flux-continuous finite-difference formulation designed for flow simulation of models with nonorthogonal hexahedron grids with general tensor permeability. Our development follows that of Aavatsmark et al.,1 but we do not operate in transformed space. The new 27-point discretization formula has been implemented in a finite-difference reservoir simulator. This stencil has many desirable properties, including collapsing into a consistent form in two dimensions. We demonstrate that there are many practical situations in which neglecting the influence of nonorthogonality and general tensors results in firstorder errors in flow predictions. A rigorous implementation for this 27-point difference operator as a control-volume finite-difference method determines the upwinding of convection terms associated with multiphase computations. Results and issues associated with implementation of this operator in a conventional finite-difference reservoir simulator are discussed. As an alternative to directly solving the linear matrix associated with the 27-point stencil of the flux-continuous operator, we examine iterative methods that split the matrix into a 7-point stencil part and a remainder. The 7-point stencil part is solved by a direct or iterative method, with the remainder part updated from the previous timestep or iteration. This split operator may permit retention of the linear solver for the standard 7-point formulation while retaining nonorthogonal grid and tensor information.