Abstract

The problem of transient natural convection inporous media is formulated and solved. Solutions for various boundary conditions are presented in the form of contour maps for stream function and temperature. It is possible that convective motions play an important role in many thermal recovery processes in petroleum reservoirs. This is an attempt into better understanding of this important but almost neglected phenomena. A new method is introduced for the numerical solution of such problems. This method is compared with the usual finite difference approach.

Introduction

Vertical temperature gradients exist in petroleum reservoirs in their natural states. In many thermal operations complex temperature fields may be artificially created. The motion of' fluids caused solely by temperature gradients is known as natural or free convection.

When a layer of fluid at rest is heated from below, the fluid has a tendency to move because the colder, more dense fluid is above the lighter, less dense fluid. However, convective motion can only start if the thermally induced buoyancy can overcome the effects of thermal diffusion and flow resistance. The conditions for the onset of convection are expressed in terms of a dimensionless temperature difference across the layer, known as the Rayleigh number, Re. If the fluid density varies according to the first order equation of' state Equations (Available in full paper)

where H is the separation between the two horizontal surfaces. The value of Rayleigh number so defined must exceed a certain critical value, Rc for the start, of natural convection. For a thin layer of fluid enclosed between horizontal rigid surfaces this value is well established at 1708. Count Rumford discovered this phenomena almost by accident in 1797 while he was observing the contents of the bore of a large thermometer:

" I saw the whole mass of the liquid in the tube in a most rapid motion running swiftly in two opposite directions, up and down at the same time. The bulb of the thermometer, which is of copper, had been made two years before I found leisure to being my experiments, and having been left unfilled without being closed with a stopple, some find particles of dust had found their way into it and these particles which were intimately mixed with the spirits of wine, on their being illuminated by the sun's beam, became perfectly visible...and by their motions discovered the violent motions by which the spirit of wine in the tube of the thermometer was agitated... On examining the motion of the spirits of wine with a lens, I found that the ascending current occupied the axis of the tube and that it descended by the sides of the tube. On inclining the tube a little, the rising current moved out of the axis and occupied the side of the tube which was uppermost, while the descending currents occupied the whole of the lower side of it."

The above quotation is reproduced from a paper by S. C. Brown (1957).

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