Fluid injection into a confined porous layer

Abstract

We present a theoretical and experimental study of viscous flows injected into a porous medium that is confined vertically by horizontal impermeable boundaries and filled with an ambient fluid of different density and viscosity. General three-dimensional equations describing such flows are developed, showing that the dynamics can be affected by two separate contributions: spreading due to gradients in hydrostatic pressure, and that due to the pressure drop introduced by the injection. In the illustrative case of a two-dimensional injection of fluid at a constant volumetric rate, the injected fluid initially forms a viscous gravity current insensitive both to the depth of the medium and to the viscosity of the ambient fluid. Beyond a characteristic time scale, the dynamics transition to being dominated by the injection pressure, and the injected fluid eventually intersects the second boundary to form a second moving contact line. Three different late-time asymptotic regimes can emerge, depending on whether the viscosity of the injected fluid is less than, equal to or greater than that of the ambient fluid. With a less viscous injection, the flow undergoes a slow decay towards a similarity solution in which the two contact lines extend linearly in time with differing prefactors. Perturbations from this long-term state are shown to decay algebraically with time. Equal viscosities result in both contact lines approaching the same leading-order asymptotic position but with a first-order correction to the distance between them that expands as t 1/2 due to gravitational spreading. For a more viscous injection, the distance between the contact lines approaches a constant value, with perturbations decaying exponentially. Data from a new series of laboratory experiments confirm these theoretical predictions.

Stratified gravity currents in porous media

“…In writing (2.2), we also assume that there are no capillary forces acting between the fluids, which could otherwise control the density field through a capillary fringe (Golding et al 2011). Under our assumption of a deep porous medium (h ≪ H), the results of this paper are equally applicable to situations with a confining or free upper boundary (Hesse et al 2007;Pegler et al 2014). It is also applicable to cases where a buoyant current flows along a free surface [as in our experiments presented in §5].…”

Shallow, gravity-driven flow in a poro-elastic layer

“…Figure 5 shows that, for M < 1 (ambient fluid is more viscous than injected fluid), both the height of the injected current and the height of the uplift differ significantly from when M = 1. The injected current extends further but the height is lower, as has been observed in non-deformable porous layers (Nordbotten & Celia 2006;Pegler, Huppert & Neufeld 2014), while the magnitude of the uplift increases. The uplift also becomes more localized to the position of the injected fluid rather than spreading ahead of the current.…”

Fluid invasion of an unsaturated leaky porous layer

“…Therefore, the dimensionless solutions are all equivalent before any contact with the outlet occurs, that is, while x L (t) < X. During this time, the flow evolves as a gravity current injected at a constant rate into a confined porous medium, corresponding to the limiting case of zero ambient fluid viscosity in the analysis of Pegler et al (2014a). Initially, while the current lies below the upper boundary, the evolution is the same as a gravity current injected into a porous medium without an upper boundary, which is described by a similarity solution of horizontal and vertical extents x L = 1.482 t 2/3 and h(0, t) = 1.296 t 1/3 , respectively (Huppert & Woods 1995).…”

“…It is notable that the difficulty described above does not arise in studies where ambient fluid viscosity is included (Pegler et al 2014a). In that case, conditions of vanishing thickness and flux of the ambient fluid at the contact line can be utilized to obtain an evolution equation describing its rate of propagation (given by (2.8b) in Pegler et al 2014a) using an analogous argument to that given in appendix A.…”