A Finite-Element Method for Reservoir Simulation

Abstract

Abstract Several previous studies have applied finite-element methods to reservoir simulation problems. Accurate solutions have been demonstrated with these methods; however, competitiveness with finite difference has not been established for most nonlinear reservoir simulation problems. In this study a more efficient finite-element procedures is presented and tested. The method is Galerkin-based, and improved efficiency is obtained by combining Lagrange trial functions with Lobatto quadrature in a particular way. The simulation of tracer performance, ion exchange preflush performance, and adverse mobility ratio miscible displacements is considered. For the problems considered, the method is shown to yield accurate solutions with less computing expense than finite differences or previously proposed finite-element techniques. For the special case of linear trial functions, the method reduces to a five-point central difference approximation. In contrast to previously reported results, this approximation is found to simulate adverse mobility ratio displacements without grid orientation sensitivity, provided a sufficiently fine grid is used. Introduction In the past few years several studies have investigated the use of finite-element methods in reservoir simulation. These include single-phase two-component simulations in one1–3 and two4,5 spatial dimensions and two-phase immiscible calculations in both one6–8 and two9,10 dimensions. These studies have demonstrated that the method is capable of giving accurate solutions, particularly for small slug problems and adverse mobility ratio displacements. All these studies used what we term conventional Galerkin finite-element techniques,1 and, unfortunately, these methods have not proved to be cost competitive with finite differences for most nonlinear reservoir simulation problems. A reduction in computing requirements is, therefore, necessary to make finite-element methods truly useful for reservoir simulation. Relative to finite differences, the increased computing requirements of conventional Galerkin-based methods are due to the following.The approximation of time-derivative terms involves the same number of surrounding grid points as the approximation of flow terms; thus, implicit-pressure/explicit-saturation (IMPES) techniques are not possible (see Ref. 12, Chap. 7).The matrices which result from the approximation of flow terms are not nearly so sparse as in finite differences; thus, the solution of matrix problems requires more computation.The computational work required to generate matrix coefficients is considerably greater than with finite differences due to the number of numerical integrations which must be performed.