Multigrid Methods for the Solution of Porous Media Multiphase Flow Equations

Abstract

Two solution methods for the solution of the equations

$$\begin{gathered} \varphi \frac{{\partial S_\omega }} {{\partial t}} = \nabla \cdot [\frac{{k_{r\omega } K}} {{\mu _\omega }}(\nabla P_\omega + \rho _\omega )] - Q_\omega \hfill \\ - \varphi \frac{{\partial S_w }} {{\partial t}}.[\frac{{k_{ro} K}} {{\mu _0 }}(\Delta P_0 + \rho _0 g)] - Q_0 \hfill \\ \end{gathered}$$

for incompressible, two phase flow in a porous medium were employed together with multigrid solvers. The IMPES method is implicit in pressure and explicit in saturation. The simultaneous solution (SS) method is a linearized fully implicit method. Both methods relied on special interpolation operators for multigrid transfers based on the discretized differential equation. This provided a method of overcoming the difficulties associated with the discontinuous coefficients. The application to the SS method was completely new and showed the usual increase of efficiency associated with multigrid methods.

Part of the Brazilian-German Cooperation in the Field of Informatics in the Area of Computational Mathematics; P. J. Paes-Leme, Brazilian Project Leader of ORESIM in Rio de Janeiro, Brazil.