Galerkin Methods for the Numerical Solution of Boundary Value Problems
Authors: Cavendish J.C. , Price H.S. , Varga R.S. 
Published: December 2007

Abstract
This paper presents a new technique for solving some of the partial differential equations that are commonly used in simulating reservoir performance. The results of applying this technique to a simple problem show that one obtains accurate pressure problem show that one obtains accurate pressure values near wells, as well as accurate pressure gradients, which can be explicitly calculated. The method is completely rigorous in that convergence of the discrete numerical solution to the continuous solution for both pressure and pressure gradient is established. High-order, piecewise-polynomial approximations are used near the wells where pressure gradients are steep, while low-order, pressure gradients are steep, while low-order, piecewise-polynomial approximations are used piecewise-polynomial approximations are used elsewhere to reduce greatly the calculation time. This combination is shown to give a uniformly good approximation to the solution. These approximations, obtained by using a Galerkin process with suitable Hermite subs paces, are shown to be theoretically and numerically superior to the usual approximations obtained from standard finite-difference techniques. Not only are much greater accuracies obtained, but computer times are also greatly reduced. The application of this technique to multiphase flow problems (e.g., single well coning problems) would have considerable practical interest, but such extensions of this technique with full mathematical rigor have not been made as yet. However, the numerical methods presented here are general, and in principle extend to multidimensional, multiphase flow. Moreover, the preliminary results given in this paper are sufficiently encouraging that we feel the effort in attempting these extensions is justified.

Introduction
The problem of obtaining accurate pressure distributions and pressure gradients around wells is of considerable importance in the numerical simulation of reservoir performance. The most common approach to solving this problem is to use finite difference techniques (see McCarty and Barfield or Peaceman and Rachford). This approach, however, has many disadvantages, the major one being that many grid points are generally necessary for accurately describing the pressure distribution and the pressure gradient around wells. This need for a fine grid results in large computer times and often in prohibitively high costs. Besides investigating the method of finite differences, some authors, such as Welge and Weber and Roper, Merchant and Duvall, have considered a combination of analytical and numerical techniques with some success. These approaches, however, are all nonrigorous and quite -often cannot even be applied. In this paper, we present a numerical formulation of high-order present a numerical formulation of high-order accuracy, based on the Galerkin method, for solving this problem. We treat here only the partial differential equation that describes steady-state, single-phase flow. However, the methods presented are general and in principle extend to multidimensional, multiphase principle extend to multidimensional, multiphase flow. Specifically we treat special cases of the problem in two dimensions described by: problem in two dimensions described by:
(1)
(2)
where G is a rectangle (with sides parallel to the coordinate axis) with boundary G, / n denotes the outward normal, and and beta are non-negative constants such that + beta greater than 0.
SPEJ
p. 204