Higher Order Flux Update Function Method for Reduction of Numerical Dispersion and Grid Orientation Effect in Reservoir Simulation

Abstract

This method reduces the numerical dispersion and the grid orientation effect simultaneously. Introduction Reservoir simulators have become one of the valuable tools which allow petroleum engineers to answer many important questions arising in the studies of petroleum recovery. They are the products of evolution over several decades and the technology is quite matured. There still exist some shortcomings, however. The most significant among them include:In an unfavorable mobility ratio piston-type displacement, the oil recovery is dependent upon the grid system employed,The error caused by the numerical dispersion inherent to low order numerical schemes can be significant in some cases. Much work has been done in the past to alleviate these problems. Since the literature is quite extensive, we will not attempt to review all of it here. The grid orientation effect was first reported by Todd et. al. and confirmed later by Coats et. al. Yanosik and McCracken proposed a nine-point scheme which has been proved to be effective in alleviating the grid orientation effect for some problems. The two point upstream weighting method, proposed by Todd et. al., reduces numerical dispersion. Although it also reduces the grid orientation effect for immiscible displacement, it still show serious grid orientation effect for unfavorable mobility-ratio piston-type displacement. Even though numerous other methods have been proposed in the past to address the grid orientation and the numerical dispersion problems, it appears to the best knowledge of the authors that the nine-point scheme of Yanosik and McCracken and its variations, and the two point upstream weighting method of Todd et. al. are probably the only methods which have been widely accepted by the petroleum industry. Unfortunately, however, none of these methods can simultaneously handle both grid orientation effect and numerical dispersion problems. The nine-point scheme is still of first order accuracy in mobility and hence has the numerical dispersion problem. The two point upstream weighting method, as was pointed by Yanosik and McCracken, can not reduce the grid orientation effect for piston-like displacement. We present in this paper a simple and new concept which can be effectively utilized to attack the grid orientation effect and numerical dispersion problems simultaneously. We also describe a feasible scheme to implement this concept. We will concentrate, for simplicity, on a two dimensional, two phase problem with uniform grid size, zero capillary pressure and each phase containing only one component. Extension to more complicated cases is straightforward. Basic Conditions Using Darcy's law and assuming the flow direction is from (i, j) to (i+1, j) (see Fig. 1), we obtain the flux of phase P across the block boundary BC as (1) The fluxes across three other block boundaries and for other phases are computed in a similar way. With the substitution of the finite difference approximation of equation (1) into the material balance equation, we have finite difference equations for pressure, saturation and, for compositional models, concentration. The resulting pressure difference equation can be represented in the following form: where n = 1,2, …, s are grid points adjacent to point (i, j). Since the pressure equation occurring in many reservoir simulations is either elliptic or near elliptic, the coefficients must satisfy the following conditions: Condition (1) - non-negativeness of the coefficients Condition (2) - diagonal dominance of the coefficient matrix with strict inequality for at least one point. Satisfaction of the above conditions is fundamental since they ensure that the difference equation satisfies a maximum principle. The resulting saturation equation on the other hand has the nature of a first order hyperbolic equation. It was pointed out by many investigators that the mobility should be monotonic for a numerical scheme to successfully produce a non-oscillating saturation front. Hence an additional condition we will require a numerical scheme to satisfy is: Condition (3) - the mobility should be monotonic between the nodal points.

A New Grid Block System for Reducing Grid Orientation Effect

“…The third group is concerned mainly with the numerical methods (Wocott et al, 1996;Shin and Merchant, 1993;Chen and Durlofsky, 1991) using a higher-order finite difference method, or generally known as the high-order techniques (HOT). It is generally believed that the GOE is partly caused by numerical dispersion.…”