We study the coupled drainage mechanisms of a propagating viscous gravity current that leaks fluid through a permeable substrate and a fixed edge. Using both theoretical analyses and numerical simulations, we investigate the time evolution of the profile shape and the amount of fluid loss through each of the drainage mechanisms. For the case of a finite-volume release, asymptotic solutions are provided to describe the dynamics of the profile shapes. Specifically, for the case of buoyancy-driven drainage with finite-volume release, an early-time self-similar solution is obtained to describe the profile evolution and a late-time self-similar solution is approached in the limit of pure edge drainage. For the case of constant fluid injection, numerical and analytical solutions are given to describe the time evolution and the steady-state profile shapes, as well as the partition of the fluid loss through each mechanism. We also briefly discuss the practical implications of the theoretical predictions to the ${\mathrm{CO}}_2$ sequestration and leakage problems.

• Received 6 September 2016

DOI:https://doi.org/10.1103/PhysRevFluids.2.074101