Supercritical flow during extraction through a point sink in a porous medium

Fluid withdrawal through porous media plays a critical role in numerous applications, including oil recovery from underground reservoirs, extraction of fresh water from aquifers, and mineral leaching in mining operations. In many instances, these fluids exhibit density stratification due to differences in intrinsic properties (e.g., oil-water), salt content (freshwater-saltwater), or temperature. Supercritical flows into a point sink, where both fluid layers are entrained into the sink, are examined. These present a complex challenge due to the unknown interface shape and the singularity at the sink. By employing a spectral method, we can remove the singularity and obtain numerical solutions, including two exact solutions: one where the sink is positioned at the initial interface height, and another in the limit of large flow rates. For all other flow rates, numerical solutions are obtained. Our results show that as the sink moves away from the initial height of the interface, the critical flow rate required to induce coning and supercritical flow increases significantly, aligning with the behaviour observed in single-layer flow dynamics.