Modeling the convective stability of $CO_2$ sequestration by a discontinuous and unstably stratified density profile

The convective stability associated with carbon sequestration is modeled by adopting an unstably stratified basic profile having a step function density with top heavy carbon saturated layer overlying a lighter carbon free layer. The model takes into account the anisotropy in both permeability and carbon dioxide diffusion, and chemical reactions between the $CO_2$ rich brine and host mineralogy. We carry out a linear stability analysis to derive the instability threshold parameters for a variety of $CO_2$ boundary conditions. We solve for the minimum thickness of the carbon-rich layer at which convection sets in and quantify how its value is influenced by diffusion, anisotropy, permeability, reaction and type of boundary conditions. The discontinuity leads to convective concentration contours that have the shape of an asymmetric lens which we quantify by deriving and making use of the $CO_2$ flux expressions at the interface. The linear problem is extended to the nonlinear regime, the analysis of which leads to the determination of a uniformly valid super critical steady solution.