American Institute of Mining, metallurgical, and Petroleum Engineers Inc.

Abstract

A finite element approximation to the differential equations for flow of two compressible fluids (oil and gas) through a porous medium is presented. Following a Galerkin approach, the algebraic equations are derived using matrix notation. Time integration is done by a simple finite difference scheme, and the resulting non-linear equations are solved by a chord slope technique. Some simple means for stabilizing the solution are pointed out and incorporated in the method.

Numerical results are presented for a displacement and a gas percolation problem, and the finite element results are shown to be slightly superior to those of a conventional model.

Introduction

The finite element method (FEM) has come to be recognized as an effective analysis tool for a wide range of problems. In particular, it has become the dominating numerical particular, it has become the dominating numerical method within static and dynamic structural analysis, for which it was originally developed some twenty years ago. The FEM gained immediate popularity because of its systematic formulation, ability to handle irregularly shaped boundaries and general appeal to engineers. It was soon recognized as an extension of the variational methods. The basic idea of variational calculus is that the solution is found as the function which results in an extremum value of some integral, the functional, often related to the energy of the system. The problem is to identify a functional which problem is to identify a functional which will yield the same result as direct solution of the differential equation. If a functional exists, an approximate solution can be found by finding the extremum value when the solution is restricted to linear combinations of a finite number of basis functions.

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