Control-Volume Distributed Sub-Cell Flux Schemes for Unstructured and Flow Based Grids

Abstract

Abstract New locally conservative flux continuous formulations are presented for general grids comprised of quadrilateral and or triangular grid cells. The schemes guarantee that a symmetric positive definite discretization matrix is obtained for both an arbitrary polygonal control-volume distributed formalism and a new sub-cell distributed formalism. Introduction The derivation of algebraic flux continuity conditions for full tensor discretization operators has lead to efficient and robust locally conservative flux continuous finite volume methods for determining the discrete velocity field in subsurface reservoirs e.g [1–9]. This paper continues with recent developments [6, 8]. Further new locally conservative flux continuous formulations are presented for unstructured grids comprised of quadrilateral and or triangular grid cells. The new schemes guarantee that a symmetric positive definite discretization matrix is obtained for both an arbitrary polygonal control-volume distributed formalism and a new sub-cell distributed formalism. The relationships between the new formulation and earlier flux continuous schemes are given. M-matrix conditions of the schemes are outlined and the relationship with more traditional CVFE based methods is also presented [10]. A flow based grid generation process is also briefly described which involves concentrating grid nodes in high flow paths that are detected via a single phase flow response [6, 8, 11]. The new methods allow the flow based grids to be interpreted at both the cell and sub-cell level. Method performance is demonstrated on unstructured flow based grids for two phase flow problems.