Symmetric Flux Continuous Positive Definite Approximation of the Elliptic Full Tensor Pressure Equation in Local Conservation Form

Abstract

Abstract A classical finite volume scheme well known in computational aerodynamics for solving the Transonic full potential equation is imported into reservoir simulation and applied to the full tensor pressure equation. Cell vertex discretisation is shown to be a natural framework for approximation. With permeability placed at the cell centres and potential at the vertices (cell corner points), of the grid the scheme is flux continuous and locally conservative. Analysis is presented which proves that the resulting discrete matrix is symmetric positive definite provided the full permeability tensor is symmetric elliptic. The discrete matrix is also diagonally dominant subject to a sufficient ellipticity condition. For a diagonal anisotropic tensor the discrete matrix is always symmetric positive definite and the scheme is up to 4th order accurate. A cell centred version of the scheme is indicated. Introduction Reservoir simulation pressure equation coefficients can jump by orders of magnitude in size from one cell to a neighbour, due to rapid variation in permeability. Flux and pressure continuity are physical requirements across interfaces in the domain. Conventional reservoir simulation employs a cell centred five-point scheme (7 in 3-D) for a approximating the discrete pressure equation. Flux and pressure continuity are readily incorporated into the standard discretisation by approximating interface coefficients with a harmonic mean of neighbouring grid block permeabilities. Unfortunately for an arbitrary heterogeneous domain the assumption of a diagonal tensor is not always valid at the grid block scale. Need for a full tensor pressure equation arises in two important cases;Upscaling (renormalizing) rock properties from fine scale diagonal tensor simulations to the grid block scale, since permeability anisotropy causes fluid to flow in a different direction to that of the imposed flow gradient. For example with a flow gradient imposed in the x direction and use of periodic Dirichlet boundary conditions for pressure in y, the solution of the fine scale problem results in a non-zero normal flux at the corresponding constant y boundaries unlike the usual zero flux boundary conditions presently employed. Thus incorporation of off diagonal (cross) terms in the effective permeability tensor is required to represent fine scale cross flow in the coarse grid cells. The approximate renormalization process can lead to non-symmetric tensors, although the tensor is symmetric in theory.Cross bedding, where the permeability is known in a local coordinate system and simulation is performed in some alternative rotated coordinate system, chosen according to physical constraints, such as a shale barrier or well orientation. The local tensor is symmetric under rotation. P. 553