CO$_2$ migration in saline aquifers: a gravity current model with capillary and solubility trapping

Injection of carbon dioxide into geological formations is widely regarded as a promising tool for reducing global atmospheric CO$_2$ emissions. While an accurate understanding of the post-injection spreading and migration of the plume of mobile CO$_2$ is essential, many aspects of the fundamental physics of CO$_2$ migration are poorly understood. Here, we develop a sharp-interface mathematical model for the post-injection migration of a CO$_2$ plume driven by groundwater flow in a sloping aquifer, subject to both residual trapping and CO$_2$ dissolution. We show that the interplay between dissolution and migration leads to three regimes, depending on how quickly the water beneath the plume saturates with dissolved CO$_2$. We develop some semi-analytical solutions to the migration equation when the water beneath the plume saturates very slowly or very quickly relative to plume motion, and we solve the migration equation numerically in general. We use these solutions to study the relative importance of capillary and solubility trapping, and the impacts of these physical mechanisms on the storage capacity of an aquifer.