Abstract

The equations of motion and the energy equation are developed for the prediction of temperature and velocity fields in the problem of natural convection in porous media. The equations are made suitable for numerical computation by introducing a vector potential which is the counterpart of stream function in two dimensions. This vector potential has previously been used by one of the authors to solve previously been used by one of the authors to solve the Navier-Stokes equation in three dimensions. The partial differential equations are solved by finite difference methods employing Alternating Direction Implicit Procedure [ADIP] and Successive Over Relaxation [SOR] methods. The model may be used when the heated side is vertical, horizontal or inclined. The detailed structure of three-dimensional cells predicted by this model is compared with previous two-dimensional studies conducted by the authors.

Analytical solutions are possible and available for the linearized partial differential equations for this problem. Such solutions only predict the onset of convection and fail to reveal what happens after convection sets in.

This is the first attempt in solving these equations for the case where nonlinear terms are not neglected and the predictions are carried out as a function of time and all three space directions.

The results are presented in the form of contour maps and show the interesting structure of convection cells in porous media.

Introduction

Natural convection is a phenomena conceivable in a petroleum reservoir subjected to fire or steam flooding. It is the result of adverse temperature gradients generating buoyancy forces that in turn cause fluid motion. A mathematical analog of the physical phenomena may be obtained by writing mass, force and energy balances on a differential volume of porous media. If Darcy's law is employed to porous media. If Darcy's law is employed to relate fluid flow to potential gradients and the Boussinesq approximations [fluid properties independent of temperature except where they contribute to buoyancy] are assumed to be applicable, the three balances reduce to two nonlinear partial differential equations. The two-dimensional form of these equations has been solved by a number of investigators. The reader is referred to Karra's thesis for a comprehensive literature review on the subject. In order to apply the mathematical model with some degree of confidence, an experimental verification is desirable if not necessary. As any experimental verification would presumably involve a three-dimensional apparatus, a study as to the effect of the third dimension on the solution was clearly indicated.

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