Use of the Finite-Element Method in the Solution of Diffusion-Convection Equations

A variational principle can be applied to the transient heat conduction equation with heat-flux boundary conditions. The finite-element method is employed to reduce the continuous spatial solution into a finite number of time-dependent unknowns. From previous work, it was demonstrated that the method can readily be applied to solve problems involving either linear or nonlinear boundary conditions, or both. In this paper, with a slight modification of the solution technique, the finite-element method is shown to be applicable to diffusion-convection equations. Consideration is given to a one-dimensional transport problem with dispersion in porous media. Results using the finite-element method are compared with several standard finite-difference numerical solutions. The finite-element method is shown to yield satisfactory solutions. Introduction The problem of finding suitable numerical approximations for equations describing the transport of heat (or mass) by conduction (or diffusion) and convection simultaneously has been of interest for some time. Equations of this type, which will be called diffusion-convection equations, arise in describing many diverse physical processes. Of particular interest to petroleum engineers is the classical equation describing the process by which one miscible fluid displaces another in a one-dimensional porous medium. Many authors have presented numerical solutions to this rather simple presented numerical solutions to this rather simple diffusion-convection problem using standard finite-difference methods, method of characteristics, and variational methods. In this paper another numerical method is employed. A finite-element method in conjunction with a variational principle for transient heat conduction analysis is briefly reviewed. It is appropriate here to mention the recent successful application of the finite-element method to solve transient heat conduction problems involving either linear, nonlinear, or both boundary conditions. The finite-element method was also applied to transient flow in porous media in a recent paper by Javandel and Witherspoon. Prime references for the method are the papers by Gurtin and Wilson and Nickell. With a slight modification of the solution procedure for treating the convective term as a source term in the transient heat conduction equation, the method can readily be used to obtain numerical solutions of the diffusion-convection equation. Consideration is given to a one-dimensional mass transport problem with dispersion in a porous medium. Results using the finite-element method yield satisfactory solutions comparable with those reported in the literature. A VARIATIONAL PRINCIPLE FOR TRANSIENT HEAT CONDUCTION AND THE FINITE-ELEMENT METHOD A variational principle can be generated for the transient conduction or diffusion equation. Wilson and Nickell, following Gurtin's discussion of variational principles for linear initial value problems, confirmed that the function of T(x, t) that problems, confirmed that the function of T(x, t) that leads to an extremum of the functional...........(1) is, at the same time, the solution to the transient heat conduction equation SPEJ P. 139