Quasimonotonic Continuous Darcy-Flux Approximation for General 3D Grids of Any Element Type

Abstract

Abstract A family of flux-continuous, locally conservative, finite-volume schemes has been developed for solving the general geometry-permeability tensor pressure equation on structured and unstructured grids and are control-volume distributed [1,2]. The schemes are applicable to the general tensor pressure equation with discontinuous coefficients and remove the O(1) errors introduced by standard reservoir simulation (two-point flux) schemes when applied to full tensor flow approximation. The family of flux-continuous schemes is quantified by a quadrature parameterization and has been tested for ranges of quadrature points. Specific points have been observed to yield improved convergence for the family of flux-continuous schemes for structured and unstructured grids in two dimensions [3]. This paper presents a complete extension of the family of control-volume distributed (CVD) multi-point flux approximation (MPFA) flux-continuous schemes for general three dimensional grids comprised of any element type, hexahedra, tetrahedra, prisms and pyramid elements. Discretization principles are presented for each element. The pyramid element is shown to be a special case with unique construction of the continuity conditions. The Darcy flux approximations are applied to a range of test cases that verify consistency of the schemes. Convergence tests of the three-dimensional families of schemes are presented, with emphasis on use of quadrature parameterization. Monotonicity issues are also discussed and tests performed confirm optimal monotonicity of the schemes as determined by an M-matrix analysis [1]. Introduction The derivation of algebraic flux continuity conditions for full tensor pressure equation discretization operators has lead to efficient and robust locally conservative flux-continuous control-volume distributed CVD (MPFA) finite-volume schemes in two and three dimensions for determining the discrete velocity and pressure field in subsurface reservoirs [1–13]. The continuous Darcy flux schemes presented here are three-dimensional full-tensor generalizations of the two-dimensional flux-continuous schemes presented in [1, 2] to hexahedra, tetrahedra, prisms and pyramid elements. Previous work in the field of locally conservative flux-continuous schemes in three-dimension includes references [5,7,9], [14–19], where construction of flux-continuous schemes has been for either hexahedra and/or tetrahedral elements in 3-D and are specialised for quadrature q = 1 (e.g. [3]). Here families of locally conservative control-volume distributed CVD(MPFA) flux-continuous schemes are presented, which are quantified by a quadrature parameterization that defines a q-family of schemes for general three dimensional grids. The q-families are presented for different element types, thereby giving maximum flexibility with respect to application of the flux-continuous scheme for different three-dimensional geometries. While the formulation presented generalises quite naturally for hexahedra, tetrahedra and prism elements, the pyramid element requires a unique treatment. Details of the variations in element q-families of flux-continuous schemes are also presented. Monotonicity issues are also discussed and tests performed in 2-D and 3-D. An M-matrix analysis [1] is used to determine the route towards optimal schemes with respect to monotonicity. Flow Equations While the methods presented here are for both single phase and multiphase flow problems, specific discretization issues are considered which arise with the finite-volume approximation of the pressure equation on general three-dimensional grids. Further details of the discrete coupled formulation can be found in [8–9]. We shall begin with the integral form of pressure equation written as: Equation (1)