Self-similarity in porous convection

In geological carbon storage, the carbon dioxide injected into a saline formation is less dense than the resident brine and floats above it. However it is also slightly soluble in brine and progressively dissolves. Brine with dissolved CO$_2$ is slightly denser than ``pure" brine and there is the potential for convective overturning. The form of this convection changes as more and more CO$_2$ dissolves, but eventually a surprising regime is reached in which the rate at which CO$_2$ dissolves is constant. In this regime, we find that there is no characteristic length-scale and the horizontally averaged concentration field is self-similar. I will describe features of this regime and develop a system of almost-complete effective equations that describe its evolution.