Dynamics of convective carbon dioxide dissolution in a closed porous media system

Motivated by geological carbon dioxide (CO$_2$) storage, many recent studies have investigated the fluid dynamics of solutal convection in porous media. Here we study the convective dissolution of CO$_2$ in a closed system, where the pressure in the gas declines as convection proceeds. This introduces a negative feedback that reduces the convective dissolution rate even before the brine becomes saturated. We analyse the case of an ideal gas with a solubility given by Henry's law, in the limits of very low and very high Rayleigh numbers. The equilibrium state in this system is determined by the dimensionless dissolution capacity, $\Pi$, which gives the fraction of the gas that can be dissolved into the underlying brine. Analytic approximations of the pure diffusion problem with $\Pi>0$, show that the diffusive base state is no longer self-similar and that diffusive mass transfer declines rapidly with time. Direct numerical simulations at high Rayleigh numbers show that no constant flux regime exists for $\Pi > 0$; nevertheless, the quantity $F/C_s^2$ remains constant, where $F$ is the dissolution flux and $C_s$ is the dissolved concentration at the top of the domain. Simple mathematical models are developed to predict the evolution of $C_s$ and $F$ for high-Rayleigh-number convection in a closed system. The negative feedback that limits convection in closed systems may explain the persistence of natural CO$_2$ accumulations over millennial timescales.