Application of the Finite Element Method to Transient Flow in Porous Media
Authors: Javandel Iraj , Witherspoon P.A. 
Published: December 2007

Abstract
The finite element method was originally developed in the aircraft industry to handle problems of stress distribution in complex airframe configurations. This paper describes how the method can be extended to problems of transient flow in porous media. In this approach, the continuum is replaced by a system of finite elements. By employing the variational principle, one can obtain time dependent solutions for the potential at every point in the system by minimizing a potential energy functional.
The theory of the method is reviewed. To demonstrate its validity, nonsteady-state results obtained by the finite element method are compared with those of typical boundary value problems for which rigorous analytical solutions are available. To demonstrate the power of this approach, solutions for the more complex problem of transient flow in layered systems with crossflow are also presented. The generality of this approach with respect to arbitrary boundary conditions and changes in rock properties provides a new method of handling properties provides a new method of handling problems of fluid flow in complex systems. problems of fluid flow in complex systems

Introduction
Problems of transient flow in porous media often can be handled by the methods of analytical mathematics as long as the geometry or properties of the flow system do not become too complex. When the analytical approach becomes intractable, it is customary to resort to numerical methods, and a great variety of problems have been handled in this manner. One such method relies on the finite difference approach Wherein the system is divided into a network of elements, and a finite difference equation for the flow into and out of each element is developed. The solution of the resulting set of equations usually requires a high speed computer. When heterogeneous systems of arbitrary geometry must be considered, however, this approach is sometimes difficult to apply and may require large amounts of computer time.
The finite element method is a new approach that avoids these difficulties. It was developed originally in the aircraft industry to provide a refined solution for stress distributions in extremely complex airframe configurations. Clough has recently reviewed the application of the finite element method in the field of structural mechanics The technique has been applied successfully in the stress analysis of many complex structures. Recognition that this procedure can be interpreted in terms of variational procedures involving minimizing a potential energy functional leads naturally to its extension to other boundary value problems. problems. In the field of heat flow, there recently have been introduced several approximate methods of solution that are based on variational principles. By employing the variational principle in conjunction with the finite element idealization, a powerful solution technique is now available for determining the potential distribution within complex bodies of arbitrary geometry. In the finite element approximation of solids, the continuum is replaced by a system of elements. An approximate solution for the potential field within each element is assumed, and flux equilibrium equations are developed at a discrete number of points within the network of finite elements. For the case of steady-state heat flow, the technique is completely described by Zienkiewicz and Cheung.
Since the flow of fluids in porous media is analogous to the flow of heat, Zienkiewicz et al. have employed the finite element method in obtaining steady-state solutions to heterogeneous and anisotropic seepage problems. Taylor and Brown have used this method to investigate steady-state flow problems involving a free surface. The work of Gurtin has been instrumental in laying the groundwork for the application of finite element methods to linear initial-value problems.
SPEJ
P. 241