The Onset of Convection in an Anisotropic Heterogeneous Porous Medium: A New Hydrodynamic Boundary Condition

The Horton–Rogers–Lapwood problem with strong heterogeneity and anisotropy is examined for a simple case, namely where the heterogeneity is provided by two layers, each of which is homogeneous and isotropic in a horizontal plane. We derived a new hydrodynamic boundary condition at the interface between two different porous media and then formulated and numerically solved the eigenvalue problem to determine the critical values of the wavenumber and Rayleigh number. We found that our approach works and gives the correct result for the homogeneous situation independent of the position of the interface. We also showed that for weak heterogeneity, by using modified anisotropy parameters weighted with the two layer depths, results obtained in Kvernvold and Tyvand (J Fluid Mech 90:609–624, 1979) for a single-layer problem can be used to approximate the critical Rayleigh number for the double-layer problem. We found that the agreement with the two-layer solution is best if a harmonic mean is used to define the mean permeability ratio and an arithmetic mean is utilized to define the mean conductivity ratio.