Solute Driven Convection with Dispersion

Abstract:

Convection in porous media controls mass and energy transport in many natural and engineered systems. The convective flux driven by solute transport in a porous medium with hydrodynamic dispersion is determined by the balance between buoyant driving forces and dissipation by either molecular diffusion or mechanical dispersion. Numerical results show that the dimensionless flux, given by the Sherwood number, only scales linearly with the Rayleigh number, if both dissipative mechanisms are included, Ra_h = 1/(1/Ra_m+1/Ra_d). Here Ra_m is the standard diffusive Rayleigh number and Ra_d=H/α_t is a dispersive Rayleigh number, where H is the domain height and α_t is the transverse dispersivity. Due to the nonlinear feedback induced by the velocity dependence of mechanical dispersion, Ra_d does not contain physical parameters from Darcy's law and depends only the thickness of the formation H and its transverse macrodispersivity α_t. The ratio of these Rayleigh numbers Δ=Ra_m/Ra_d determines the contribution of mechanical dispersion to dissipation and identifies diffusion limited (Δ«1) and dispersion limited (Δ»1) convection regimes. In both natural and laboratory systems, convection is commonly dispersion limited so that the solute flux is determined by Ra_d and not by Ra_m as commonly assumed in previous studies. Our results, therefore, change the fundamental governing paramter for solute driven convection in porous media and the determination of macrodispersivities is a key component for the reasonable estimate of the solute driven convective flux.