The Rayleigh–Taylor instability in a porous medium

Abstract

The classical Rayleigh–Taylor instability occurs when a heavy fluid overlies a lighter one, and the two fluids are separated by a horizontal interface. The configuration is unstable, and a small perturbation to the interface grows with time. Here, we consider such an arrangement for planar flow, but in a porous medium governed by Darcy’s law. First, the fully saturated situation is considered, where the two horizontal fluids are separated by a sharp interface. A classical linearized theory is reviewed, and the nonlinear model is solved numerically. It is shown that the solution is ultimately limited in time by the formation of a curvature singularity at the interface. A partially saturated Boussinesq theory is then presented, and its linearized approximation predicts a stable interface that merely diffuses. Nonlinear Boussinesq theory, however, allows the growth of drips and bubbles at the interface. These structures develop with no apparent overturning at their heads, unlike the corresponding flow for two free fluids.

An extended Boussinesq theory for interfacial fluid mechanics

“…which is consistent with the models of Trevelyan et al [20] and Forbes et al [21] for Rayleigh-Taylor flows in porous media. Finally, the Navier-Stokes equation of viscous fluid flow is approximated by the simpler equation…”

Fluid injection through a line source into a wet porous medium

“…As time progresses, the plume develops under the effects of gravity and fluid injection at the source by widening and moving downwards as expected. The close agreement with the large-time asymptotic solution (21) at the last time t = 25 shown strongly suggests that gravity and the fluid injection term (11) become the dominant features of the flow.…”

“…As was assumed by Trevelyan et al [12], De Paoli et al [13] and Forbes et al [11] we assume that the larger density ρ 1 of the fluid being injected into the porous medium is due to the addition of some solute with concentration C(r, θ, t) such that C → C 0 as r → 0. This makes it relevant to in situ mining, where a leaching lixiviant is pumped in, to extract a mineral of interest, an example of which is described by Forbes [4].…”

“…One such numerical method would be to use a finite-element method, a solid introduction to which can be found in the text by Wang and Anderson [9]. As an example of a different numerical approach, Hocking and Zhang [10] have applied boundary-integral techniques to find the shape of an up-coning More recently Forbes has used a spectral approach [11], which has advantages of allowing singularities and also of providing a straightforward relationship between solute concentration, stream function and velocity.…”

Axisymmetric plumes due to fluid injection through a small source in a wet porous medium

“…We instead assume that the fluid density ρ(r , φ, t) varies continuously across a narrow boundary of finite width from ρ 1 (the density of fluid injected at the source) to ρ 2 (the density of the fluid that saturates the ground far away from the source). Similarly to a number of other papers [14,[19][20][21], we assume that the higher density of the introduced fluid is due to the addition of some solute with concentration C(r , φ, t) such that the density varies linearly with the concentration and C → C 0 as ρ → ρ 1 ; that is…”