Alternating Direction Collocation Methods For Simulating Reservoir Performance
Authors: Bangia V.L. , Bennett C. , Reynolds A. , Raghavan R. , Thomas G. 
Published: December 2007

Abstract
This paper presents two new alternating-direction collocation methods for solving some of the partial differential equations encountered in simulating reservoir performance. Such methods differ from Galerkin methods insofar as quadratures are not required while apparently still retaining the high-order accuracy associated with Galerkin's procedure. Applications are made to single-phase flow problems where pressure approximations are obtained as linear combinations of bicubic Hermite polynomials. Results are presented for several reservoir problems including one having a fractured well with a uniform flux. Comparisons are also made to analytical solutions (when available) and to those obtained from alternating direction finite difference solutions. We conclude that collocation yields accurate values of pressure, even in the vicinity of wells where finite difference methods fail to do so. Furthermore, more accurate results can be obtained via collocation methods with a coarse grid than can be obtained with a refined grid using finite difference techniques. Consequently, the computation time required for collocation is considerably less, to achieve comparable accuracy.

Introduction
The widespread use of reservoir simulators for predicting field performance has generated considerable predicting field performance has generated considerable interest in efficient and accurate numerical methods for solving the partial differential equations describing fluid flow in porous media. Finite difference methods are the most common numerical techniques used, however, this approach has many disadvantages. In particular, excessive grid refinement is required to particular, excessive grid refinement is required to generate accurate solutions near wells or when the dependent variables change rapidly over short spatial distances. Because of this, interest has been directed toward techniques that inherently offer higher order accuracy. Cavendish, Price and Varga applied Galerkin's method to a steady-state single phase two-dimensional problem and found it superior to standard finite difference methods in terms of accuracy and computer time required. Settari, Price, and Dupont indicate Galerkin's method is competitive with, and in some cases superior to, finite differences for solving a system of coupled equations for miscible displacement processes. For a two-dimensional, two-phase immiscible flow problem, Spivak, Price and Settari found that a Galerkin model does not exhibit grid orientation effects, and qualitatively, gives more accurate saturation fronts than a finite difference model. However, they did not compare Galerkin and finite difference models in terms of the work required to achieve a specified accuracy. Despite the feasibility of Galerkin's method, it has some disadvantages. The most serious being the necessity to perform time consuming quadratures for nonlinear problems. Because of this, for multidimensional, multiphase flow problems, it appears that Galerkin methods are generally problems, it appears that Galerkin methods are generally not competitive with finite difference techniques in terms of the computer time required to achieve a desired accuracy (See McMichael and Thomas, Farrar, and Kebaili and Thomas.) Culham and Varga recommend interpolating the nonlinear coefficients. This reduces the total number of quadratures required, however, for multiphase flow problems, the finite difference approach is still computationally faster.
To minimize the number of quadratures, Douglas and Dupont formulated a Laplace-modified alternating direction Galerkin procedure. This requires the choice of a parameter, lambda, but it is not clear how it should be selected to guarantee an accurate solution.
The collocation methods presented here are alternatives to Galerkin's procedure. They have the advantage that quadratures are not required. Following Douglas and Dupont, our approach is to employ a splitting procedure to achieve alternating direction methods. The idea is to reap the advantage of treating multidimensional problems as a collection of I-D problems. But more than this, we found that a problems. But more than this, we found that a non-alternating Crank-Nicolson collocation scheme gave physically unrealistic results and yielded poor material physically unrealistic results and yielded poor material balances.