A study of hydrothermal convection in saturated porous media

Abstract

Abstract Because of its relevance to many geological and technical problems, hydrothermal convection is investigated here mainly with the aid of numerical models by a systematic analysis of the properties of this type of convection for a range of super-critical Rayleigh numbers. Calculations were performed for two-dimensional models with constant properties in a region of aspect ratio 2. The principal results in the case of temperature fixed at the impermeable top and bottom are the following for the Nusselt number Nu, the cell aspect ratio a, and the boundary layer thickness δ: Nu ≈ 1.7 R0.5, a ≈ 1.3 R−0.4, δc ≈ 0.4 R−0.4 for 2.5 R = R f /R f ∗ and Rf and R f ∗ are the ambient and critical filtration Rayleigh numbers, respectively, relevant to the hydrothermal problem. Beside the usual critical Rayleigh number R f ∗ two additional, new ones have been found: R f ∗∗ which characterizes the appearance of boundary layers, and R f ∗∗∗ which marks the transition to the non-steady state. A number of definitions of the boudary layer are analyzed, and a new definition is proposed based on the condition that the conductive heat flux is equal or greater than the convective one. The initial oscillations after the onset of convection in our model are analyzed and interpreted for the flow in porous media in comparison to the viscous case. Finally a diagram is constructed for the easy evaluation of the numerical accuracy as a function of grid size.