Onset and cessation of porous convection in the context of geological carbon sequestration

In geological carbon sequestration, CO$_2$ injected into a saline aquifer is less dense than the resident brine and floats above it. It is slightly soluble in brine and progressively dissolves, making the brine slightly denser than ``pure" brine. Motivated by this, we consider conditions for free convection in a porous medium from a one-dimensional, time-dependent, pure-diffusion base state. This problem has been addressed in several previous studies using a variety of approximations. We present a simple but rigorous calculation, showing where in time and wavenumber space a perturbation exists (of infinitesimal or finite amplitude) whose mean square amplitude grows. The critical Rayleigh-Darcy number, $Ra$, below which instability cannot occur is $Ra=32.50$. Above $Ra\approx 100$, the earliest possible onset time becomes independent of porous-layer thickness. We discuss implications for realistic reservoir conditions.