Simplified Finite-Element Models for Reservoir Flow Problems
Authors: Dalen Vilgeir 
Published: December 2007

Abstract
This paper summarizes some research that was conducted to construct finite-element models for reservoir flow problems. The models are based on Galerkin's method, but the method is applied in an unorthodox manner to simplify calculation of coefficients and to improve stability. Specifically, techniques of compatibility relaxation, capacity lumping, and upstream mobility weighting are used, and schemes are obtained that seem to combine the simplicity and high stability of conventional finite-difference models with the generality and modeling flexibility of finite-element methods.The development of a model for single-phase gas flow and a two-phase oil/water model is described. Numerical examples are included to illustrate the usefulness of finite elements. In particular, the triangular element with linear interpolation is shown to be an attractive alternative to the standard five-point, finite-difference approximation for two-dimensional analysis.

Introduction
During the past decades, finite-element methods have been developed to a high level of sophistication and have gained wide popularity within several branches of engineering science. In some fields, such methods have replaced to some extent the older finite-difference methods in engineering practice because they have been regarded as a more convenient tool for numerical analysis. An increasing interest in finite-element methods, or variational methods in general, also may be noticed in the field of numerical reservoir simulation, but so far no definitive breakthrough has occurred in this field.One reason for this probably is the complexity of reservoir flow problems. Reservoir flow equations in most cases are nonlinear, and for multiphase flow, they are usually found on the borderline between parabolic and hyperbolic equations. For such parabolic and hyperbolic equations. For such problems, the dissimilarities between problems, the dissimilarities between finite-difference and finite-element methods are much more pronounced than for linear problems of the elliptic type. This means that the finite-element method may not be looked upon as easily as an extension or generalization of finite-difference methods. Second, one can question whether all the advantages that are gained in other instances by using finite elements may be realized at all.Applications of variational methods to single-phase flow problems, or diffusion-type problems in general, have been studied extensively. The merits of finite elements for such problems are apparently well established, at least as far as linear problems are concerned.The literature on variational methods in multiphase flow is comparatively sparse, and so far the results are inconclusive regarding the relative advantages of variational methods and finite- difference methods in this field. In summary, variational methods offer the potential advantages of (1) easy implementation of higher-order approximations, (2) a more proper treatment of variable coefficients, and (3) greater modeling flexibility.Previously, attention was focused on Aspects 1 and 2. Several authors used cubic Hermitian basis functions.
SPEJ
P. 333