Onset and cessation of time-dependent, dissolution-driven convection in porous media

Motivated by convection in the context of geological carbon dioxide sequestration, we present the conditions for free, dissolution-driven convection in a horizontal, ideal porous layer from a time-dependent, pure-diffusion base state. We assume that solute as a separate phase is instantaneously placed in the pores above a given horizontal level at time zero, and gradually diffuses into the underlying liquid. As the concentration of dissolved solute in the liquid increases, its density increases and the system may eventually become gravitationally unstable and convection may begin. We define the amplitude of a perturbation as the mean square of the difference of the concentration profile and the pure-diffusion profile. To identify instability, we calculate the maximum possible instantaneous growth rate of the amplitude over all possible infinitesimal and finite perturbations. Instability exists where this growth rate is positive. We consider two scenarios. In the first scenario, the underlying liquid cannot penetrate into the upper region occupied by the separate, solute-rich phase. In this case, no instability is possible for thin porous layers corresponding to Rayleigh–Darcy numbers less than 32.50. For thicker layers, convection can occur at finite, nonzero times and wavenumbers. The earliest possible onset becomes independent of layer thickness for Rayleigh–Darcy numbers above about 75, and occurs at a time of approximately 47.9D(μϕ/[KΔρ(1−ci∗/cm∗)g])2 after solute placement, where D is the effective diffusivity, μ the liquid viscosity, ϕ the porosity and K the permeability of the medium, g the acceleration due to gravity, Δρ=ρm−ρ0 with ρm the density of saturated liquid and ρ0 the density of pure liquid, cm∗ the maximum solute concentration, and ci∗ the initial average concentration. In the second scenario, the liquid phase in the upper layer remains connected; the liquid beneath can penetrate the horizontal boundary between the two layers and immediately becomes saturated. In this case, the onset occurs sooner and increasingly so for thicker layers. We also present the mode profiles and discuss the implications for sequestration.