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

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 now studied analytically for the case of moderate heterogeneity (rather than the weak heterogeneity previously studied), for the case of a square box where the properties vary in a piecewise constant or linear fashion, with conducting impermeable top and bottom boundaries and insulating impermeable sidewalls. In order to allow for the moderate heterogeneity the order of the Galerkin expansion employed has been increased, and the expansion of a determinant of high order has been avoided by the use of a least squares methodology to find the critical value of the Rayleigh number Ra. It is found that the effects of permeability heterogeneity and conductivity heterogeneity each cause a reduction in the critical value of Ra in all cases, and the effects of horizontal and vertical heterogeneity are still approximately additive.