Convection Currents in a Porous Medium

Abstract

The problem is considered of the convection of a fluid through a permeable medium as the result of a vertical temperature-gradient, the medium being in the shape of a flat layer bounded above and below by perfectly conducting media. It appears that the minimum temperature-gradient for which convection can occur is approximately 4π2h2μ/kgρ0α D2, where h2 is the thermal diffusivity, g is the acceleration of gravity, μ is the viscosity, k is the permeability, α is the coefficient of cubical expansion, ρ0 is the density at zero temperature, and D is the thickness of the layer; this exceeds the limiting gradient found by Rayleigh for a simple fluid by a factor of 16D2/27π2kρ0. A numerical computation of this gradient, based upon the data now available, indicates that convection currents should not occur in such a geological formation as the Woodbine sand of East Texas (west of the Mexia Fault zone); in view of the fact, however, that the distribution of NaCl in this formation seems to require the existence of convection currents, and in view of the approximations involved in applying the present theory, it seems safe tentatively, to conclude that convection currents do exist in this formation and that the expression given above predicts excessive minimum gradients when applied to such a formation.