Convection in a Porous Medium Heated from Below: The Effect of Temperature Dependent Viscosity and Thermal Expansion Coefficient

Abstract

The natural convection of water through permeable formations is an important phenomenon governing the behavior of geothermal aquifers. In such situations the temperature of the fluid can vary by amounts of the order of 250 K, over which range the viscosity of the water may decrease by as much as a factor of 10, and the coefficient of thermal expansion may increase by an even greater factor. Although previous studies have examined the effect of this behavior on the onset conditions at which natural convective flow in the medium first occurs, the actual flow patterns that will result from the convection of a fluid with such widely varying properties has not been reported. To this end, numerical methods of solution are applied to this problem and a visualization of the flows obtained. By solving in terms of both pressure and stream function instead of stream function alone, it was possible to maintain semiconservative properties of the numerical scheme. The convective motion is unstable (in two dimensions) at even moderate values of the Rayleigh number and exhibits a fluctuating convective state analogous to the case of a fluid with constant viscosity and coefficient of thermal expansion. In some cases the acceleration of the flow in certain areas due to the decrease in viscosity causes localized thermal instabilities.