Drainage of Gravity Currents from an Edge and a Permeable Substrate in a Porous Medium or Fracture with Variable Properties

Gravity-driven flow in porous media has been extensively investigated in recent years in connection with numerous environmental and industrial applications, including seawater intrusion, oil recovery, penetration of drilling fluids into reservoirs, contaminant migration such as NAPL spreading in shallow aquifers, and carbon dioxide sequestration in deep subsurface formations. Analytical and numerical solutions are available for various geometries and boundary conditions, including different drainage mechanisms. Their use can be extended to gravity-driven flow in narrow vertical fractures or cracks via the well-known Hele-Shaw (HS) analogy between parallel plate and porous media flow, with the aperture b squared being the analog of permeability k according to k = b²/12. The propagation of gravity currents in porous and fractured media is mainly governed by the interplay between viscous and buoyancy forces, typically with negligible inertial effects, and is affected by spatial heterogeneity of medium properties; permeability, porosity, and aperture gradients have been shown to affect the propagation distance and shape of gravity currents, with practical implications for remediation and storage. This paper is interested in the dynamics of gravity currents propagating in a porous medium with spatial properties varying parallel to the flow direction under the coupled drainage mechanisms of a fixed edge and a permeable substrate. Simultaneous permeability and porosity gradients parallel to the flow are considered: this is equivalent to a fracture with a horizontally variable aperture, as the Hele-Shaw analogy necessarily accounts for both permeability and porosity gradients. We consider a Newtonian fluid with a density of ρ+Δρ intruding into a porous medium and advancing in a fluid of density ρ under the sharp interface approximation, where a no-flow boundary condition is considered at the origin x = 0. We assume that the fluid drains away simultaneously from the permeable base and an edge and neglect vertical velocities in a long and thin current; this implies vertical equilibrium and, in turn, a hydrostatic pressure distribution within the advancing current. The final assumption is of vanishing height of the current at the draining edge after a relatively short adjustment time, favoured by increased permeability/porosity or aperture along the flow direction. Under these assumptions, a semi-analytical solution is derived for the height of the current h(x, t) in a self-similar form, valid as a late-time approximation modelling the drainage phenomenon after the influence of the initial condition has vanished. This allows transforming the nonlinear PDE governing the flow into a nonlinear ODE amenable to a numerical solution. Besides, the current profile yields the amount of fluid loss through each of the drainage mechanisms. Results are discussed as a function of model parameters, and an analysis of the conditions required to avoid an unphysical or asymptotically invalid result is presented.