On the onset of convection in a water-saturated porous box: Effect of conducting walls

Abstract

The critical Rayleigh number for the onset of convection in a water-saturated porous box, heated from below, has been determined by the Galerkin method and by an approximate analytical technique. The present problem differs from previous ones which have been published in that we consider the effect of conducting vertical boundaries on the critical number and on the cell pattern at the critical number. Moreover, we emphasize fault or fracture zone-like box geometries; that is, geometries in which one box dimension is much shorter than the other and much shorter than the height. The results indicate that for fault/fracture zone geometries, in which the long vertical walls are conducting and the short end walls are insulated, the critical Rayleigh number is roughly four orders of magnitude greater than the critical number for a horizontal porous slab. The flow at the onset of convection takes the form of a roll with its axis along the long horizontal dimension of the box. There is, however, little difference between the critical numbers for two and three dimensional cell patterns. These results indicate that convection may occur in naturally occurring faults and fracture zones in the earth's crust only if the permeability is of the order of Darcies. In natural systems, the Rayleigh number would probably not exceed the critical number greatly, and the flow may tend to be fully three dimensional.