The radial slump of a gravity current in a confined porous layer

The flow of a gravity current widely occurs in many geophysical, environmental and industrial processes. We focus on the overlooked process of radial slump of a gravity current in confined porous layers, which is related to the post-injection flow dynamics of many subsurface displacement processes, initiated by fluid injection through vertical wells. Specifically, we describe the time evolution of the interface shape and location of the propagating front, governed by a second-order nonlinear partial differential equation (PDE). By solving a nonlinear eigenvalue problem, we identify a self-similar solution to describe the dynamics of confined flows at early times, when the interface attaches to both the top and bottom boundaries of the porous layer. As time progresses, the flow gradually becomes unconfined, as gravity-driven slump causes the interface to detach from one boundary; the interface evolution at late times is then described by a self-similar solution of a nonlinear diffusion equation. Time transition from the early-time confined towards late-time unconfined self-similar solutions is also verified by comparing the self-similar solutions and numerical solutions of the governing PDE for both the interface shape and the frontal location. Interestingly, two special vertical locations have also been identified where the best agreement appears between the PDE numerical solutions and the early-time self-similar solutions for the interface shape, which is a novel feature of the radial flow, and the degree of agreement is not impacted by the time evolution.