The effects of combined horizontal and vertical heterogeneity on the onset of convection in a porous medium

The effects of hydrodynamic and thermal heterogeneity, for the case of variation in both the horizontal and vertical directions, on the onset of convection in a horizontal layer of a saturated porous medium uniformly heated from below, are studied analytically for the case of weak heterogeneity. It is found that the effect of such heterogeneity on the critical value of the Rayleigh number Ra based on mean properties is of second order if the properties vary in a piecewise constant or linear fashion. The effects of horizontal heterogeneity and vertical heterogeneity are then comparable once the aspect ratio is taken into account, and to a first approximation are independent. For the case of conducting impermeable top and bottom boundaries and a square box, the effects of permeability heterogeneity and conductivity heterogeneity each cause a reduction in the critical value of Ra, while for the case of a tall box there can be either a reduction or an increase.