Transient buoyant convection from a discrete source in porous media

The study of porous media filling box flows informs (i) the dissolution of non-aqueous phase liquids or sequestered CO$_2$ into potable groundwater, (ii) leakage of contaminants from waste piles, and (iii) enhanced oil recovery technologies. Here we examine the flow of a negatively buoyant, laminar plume in a box filled with a porous medium, which is connected to an infinite external ambient via upper and lower fissures. As $t \to \infty$, the box contains two uniform layers of different densities. However, the approach towards steady state is characterized by a lower (contaminated) layer that is continuously stratified and is governed by the ratios of the virtual origin correction and lower fissure depth to the box height, and the ratio, $\mu$, of the draining timescale to the filling timescale. Whereas the presence of a continuous stratification in the contaminated layer for finite time poses analytical challenges, we show that it is possible to derive bounds on the range of possible solutions. A separate component of our study considers time-variable forcing where the plume source strength is either abruptly altered or turned on and off with fixed half-period. Throughout, comparisons are drawn against filling boxes driven by turbulent free plumes.