  CO$_2$ sequestration in subsurface reservoirs is important for limiting
atmospheric CO$_2$ concentrations. However, a complete physical picture able to
predict the structure developing within the porous medium is lacking. We
investigate theoretically reactive transport in the long-time evolution of
carbon in the brine-rock environment. As CO$_2$ is injected into a brine-rock
environment, a carbonate-rich region is created amid brine. Within the
carbonate-rich region minerals dissolve and migrate from regions of high
concentration to low concentration, along with other dissolved carbonate
species. This causes mineral precipitation at the interface between the two
regions. We argue that precipitation in a small layer reduces diffusivity, and
eventually causes mechanical trapping of the CO$_2$. Consequently, only a small
fraction of the CO$_2$ is converted to solid mineral; the remainder either
dissolves in water or is trapped in its original form. We also study the case
of a pure CO$_2$ bubble surrounded by brine and suggest a mechanism that may
lead to a carbonate-encrusted bubble due to structural diffusion.
