Solution of the Equations for Multidimensional, Two-Phase, Immiscible Flow by Variational Methods
Authors: Spivak A. , Price H.S. , Settari A. 
Published: December 2007

Abstract
This paper describes the solution of the equations for two-dimensional, two-phase, immiscible flow by variational methods. The formulation of the equations and the Galerkin procedure for solving the equations are given. procedure for solving the equations are given. The results of numerical experiments for one-dimensional, two-dimensional areal, and two-dimensional cross-sectional examples are presented. In each case, the results are compared with finite-difference solutions for the same problem. The ability to track sharp fronts is demonstrated by the variational approach. The time approximation used is shown to be stable for difficult problems such as converging flow and gas percolation. Also, the variational solution is shown to be percolation. Also, the variational solution is shown to be insensitive to grid orientation.

Introduction
In practical applications in the petroleum industry, the nonlinear, partial differential equations for fluid flow through a porous medium are currently solved almost exclusively by finite-difference methods. Variational or Galerkin (the terms are used interchangeably here) methods for solving these equations offer the potential advantage of higher-order accuracy at lower computational cost.This paper describes research on the solution of the equations for two-phase immiscible fluid flow using variational methods. The literature on the application of these methods to immiscible fluid flow is sparse. Douglas et al. describe solution of the one-dimensional immiscible displacement problem using cubic-spline basis functions and solving simultaneously for pressure and saturation as the dependent variables. They concluded pressure and saturation as the dependent variables. They concluded that the method was practical and that better answers are obtained with the same computational effort than by finite-difference methods. They also concluded that their choice of basis functions was probably not optimal. Verner et al. discuss the solution to the one-dimensional problem using "parabolic basis elements" (C degrees quadratic-basis problem using "parabolic basis elements" (C degrees quadratic-basis elements). Using the same data as was used by Douglas et al., they concluded that the parabolic, finite-element, spatial approximation gives results similar to the cubic splines for the same number of degrees of freedom. McMichael and Thomas solved the equations for three-phase, multidimensional immiscible flow. They solved simultaneously for the three-phase potentials as dependent variables. Although they stated that a general three-dimensional program with variable-basis function capability was developed, program with variable-basis function capability was developed, the examples they presented were two-dimensional areal. Also, piecewise linear basis (Chapeau) functions were used in their piecewise linear basis (Chapeau) functions were used in their example problems. The numerical experiments presented by McMichael and Thomas were limited to two relatively simple problems. They concluded that the Galerkin method requires significantly more work per time step than a finite-difference model, but that larger time steps could be taken. Vermuelen discussed the solution of the two-phase immiscible flow equations by simultaneously solving for the wetting- and nonwetting-phase pressures using a semi-implicit, first-order time approximation. Vermuelen's example problems used piecewise linear-basis functions. Based on one of these examples, piecewise linear-basis functions. Based on one of these examples, he concluded that the Galerkin technique appears to be less accurate than the finite-difference method for problems of water tongue displacement. In addition to the above work on two-phase immiscible flow through porous media, several authors have discussed the application of variational methods to miscible displacement problems and single-phase flow problems.
SPEJ
P. 27