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

The effects of both horizontal and vertical hydrodynamic and thermal heterogeneity together with anisotropy of both permeability and thermal conductivity, on the onset of convection in a horizontal layer of a saturated porous medium, uniformly heated from below, are studied analytically using linear stability theory 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 a square enclosure, horizontal heterogeneity is invariably destabilizing, but vertical heterogeneity can be either stabilizing or destabilizing. For an enclosure whose aspect ratio is optimized to give the minimum value of the critical Rayleigh number, both horizontal and vertical heterogeneity are destabilizing, by an amount dependent on the ratio of the conductivity and permeability anisotropy measures.