This thesis examines the problem of convection that occurs during the geological sequestration of carbon dioxide. The mathematical model accounts for both diffusion and convection in a geological formation that has anistropic permeability and anisotropic carbon diffusion. The permeability is modeled as a decaying exponential to describe its slow decrease with depth. We also account for a first order reaction between CO2 and porous medium. We investigate the onset of convection and its weakly nonlinear evolution. We consider a base state that mimics a Rayleigh-Taylor configuration with a carbon-rich heavy layer overlying a carbon-free lighter layer and determine the thickness at which the configuration becomes unstable. The analysis is performed using the classical normal modes and the weakly nonlinear analysis is performed using long-wavelength asymptotics. We derive the threshold instability conditions and associated flow patterns. Our analysis leads to the derivation of the convective flux conditions at the interface and the resulting fingering patterns. Finally, we put forth the conditions expressed in terms of formation and fluid parameters of parameters for the onset of convective shutdown.