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Description
We examine the linear and weakly nonlinear stability analyses of the dissolution-driven convection induced by the sequestration of carbon dioxide in a geological formation. The mathematical model consists of Darcy's equation, the conservation of mass and the conservation of solute equations. The model accounts for anisotropy in both carbon diffusion and permeability which is modeled by a decaying exponential function of depth. The presence of a first order reaction between the carbon-rich brine and host mineralogy is also included. We prescribe either Neumann or Dirichlet boundary condition for the concentration of carbon dioxide at the rigid upper and lower walls that bound a layer of infinite horizontal extent. We consider a Rayleigh-Taylor-like base state consisting of a carbon-rich heavy layer overlying a carbon-free lighter layer and seek the critical thickness at which this configuration becomes unstable. With this approach, standard mathematical methods that were successfully used in the study of Rayleigh-Benard convection can be applied to this problem. We quantify the influence of carbon diffusion anisotropy, permeability dependence on depth and the presence of the chemical reaction on the threshold instability conditions and associated flow patterns using the classical normal modes approach. The critical Rayleigh number and corresponding wavenumber are found to be independent of the depth of the formation. The weakly nonlinear analysis is performed using long wavelength asymptotics, the validity of which is limited to small Damk\"{o}hler numbers. We derive analytical expressions for the solute flux at the interface, the location of which corresponds to the minimum depth of the boundary layer at which instability sets in. We show that the interface acts as a sink leading to the formation of a self-organized exchange between descending carbon-rich brine and ascending carbon free brine. Plots of the high order perturbation terms for the concentration successfully reproduce the fingering pattern that is typically observed in experiments and full numerical simulations. Using the derived interface flux conditions, we put forth differential equations for the time evolution of the upward migration of the interface as the dissolution process progresses. We solve for the terminal time when the interface reaches the top boundary thereby quantifying the time it takes for an initial amount of injected super-critical Carbon dioxide to be completely dissolved. We also consider the case where the interface migration is accompanied by interface deformations that conform to the convection pattern.
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