We investigate the long time steady-state dissolution of CO$\mathrm{<mrow\; data-mjx-texclass="ORD">2</mrow>}$ in a deep saline aqquifer in the presence of a background hydrological flow. In steady-state, the distribution of CO$2$ in the groundwater upstream of the aquifer involves a balance between three competing effects: (i) the buoyancy-driven flow of CO$2$ saturated water; (ii) the diffusion of CO$2$ from saturated to under-saturated water; and (iii) the advection associated with the oncoming background flow. This leads to three limiting regimes. In the limit of very slow diffusion, a nearly static intrusion of dense fluid may extend a finite distance upstream, balanced by the pressure gradient associated with the oncoming background flow. In the limit of fast diffusion relative to the flow, a gradient zone may become established in which the along aquifer diffusive flux balances the advection associated with the background flow. However, if the buoyancy-driven flow speed exceeds the background hydrological flow speed, then a third, intermediate regime may become established. In this regime, a convective recirculation develops upstream of the anticline involving the vertical diffusion of CO$2$ from an upstream propagating flow of dense CO$2$ saturated water into the downstream propagating flow of CO$2$ unsaturated water. For each limiting case, we find analytical solutions for the distribution of CO$2$ upstream of the anticline, and test our analysis with full numerical simulations. A key result is that, although there may be very different controls on the distribution and extent of CO$2$ bearing water upstream of the anticline, in each case the dissolution rate is given by the product of the background volume flux and the difference in concentration between the CO$2$ saturated water and the original aquifer water upstream.