Drainage of viscous gravity currents from the edge of a porous or fractured domain with variable properties

Gravity-driven flow in porous and fractured 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 subsurface formations. The propagation of such currents is typically governed by the interplay between viscous and buoyancy forces, with negligible inertial effects. For long and thin currents, the spreading can be described by similarity solutions for a variety of geometries, with topographic controls often playing a crucial role. These solutions can be extended to gravity-driven flow in vertical narrow fractures or cracks via the well-known Hele-Shaw (HS) analogy between parallel plate and porous media flow, with the aperture b (distance between fracture walls) squared being the analogue of permeability k according to k = b²/12.  

Buoyancy-driven spreading in porous and fractured media is also influenced by spatial heterogeneity of medium properties; permeability, porosity, and aperture gradients affect the propagation distance and shape of gravity currents, with practical implications for remediation and storage. In this paper we are interested in the coupled effect of heterogeneity and a fixed edge draining the current at one end of a finite domain. Simultaneous permeability and porosity gradients parallel to the flow are considered: this is equivalent to a wedge-shaped fracture, as the Hele-Shaw analogy necessarily accounts for both permeability and porosity gradients.

A current of density ρ+Δρ advances horizontally in a fluid of density ρ under the sharp interface approximation, and is drained by an edge at a distance x = L from the origin; a no-flow boundary condition is considered at x = 0. We neglect vertical velocities for an elongated current; this implies vertical equilibrium, and in turn an 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 the increase in 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. Knowledge of the current profile then yields the residual mass in the fracture and the drainage flowrate at the edge. A full sensitivity analysis to model parameters is performed, and the conditions required to avoid an unphysical or asymptotically invalid result are discussed. An extension to non-Newtonian rheology is then presented.