A step function density profile model for the convective stability of CO2 geological sequestration
The convective stability associated with carbon sequestration is usually investigated by adopting an unsteady diffusive basic profile. The method of normal modes is not applicable due to the time dependence of the nonlinear base profile. Therefore, the instability is quantified either in terms of critical times at which the boundary layer instability sets in or in terms of long time evolution of initial disturbances. This paper adopts an unstably stratified basic profile having a step function density with top heavy carbon saturated layer (boundary layer) overlying a lighter carbon free layer (ambient brine). The resulting configuration resembles that of the Rayleigh-Taylor problem with buoyancy diffusion at the interface separating the two layers. The discontinuous reference state satisfies the governing system of equations and boundary conditions and pertains to an unstably stratified motionless state. Our model accounts for anisotropy in both diffusion and permeability and chemical reaction between the carbon dioxide rich brine and host mineralogy. We proceed by supposing that the carbon dioxide that has accumulated below the top cap rock forms a layer of carbon saturated brine of some thickness that overlies a carbon-free brine layer. The resulting stratification remains stable until the thickness of the carbon saturated layer is sufficient to induce the fluid to overturn. With this formulation for the reference state, the stability calculations will be in terms of critical boundary layer thickness instead of critical times, although the two formulations are homologous. This approach is tractable by the classical normal mode analysis. Even though it yields only conservative threshold instability conditions, it offers the advantage for an analytically tractable linear and weakly nonlinear studies.