Fluid drainage from the edge of a porous reservoir

Abstract

We report theoretical and experimental studies to describe buoyancy-driven fluid drainage from a porous medium for configurations where the fluid drains from an edge. We first study homogeneous porous systems. To investigate the influence of heterogeneities, we consider the case where the permeability varies transverse to the flow direction, exemplified by a V-shaped Hele-Shaw cell. Finally, we analyse a model where both the permeability and the porosity vary transverse to the flow direction. In each case, a self-similar solution for the shape of these gravity currents is found and a power-law behaviour in time is derived for the mass remaining in the system. Laboratory experiments are conducted in homogeneous and V-shaped Hele-Shaw cells, and the measured profile shapes and the mass remaining in the cells agree well with our model predictions. Our study provides new insights into drainage processes such as may occur in a variety of natural and industrial activities, including the geological storage of carbon dioxide.

Influence of heterogeneity on second-kind self-similar solutions for viscous gravity currents

“…In soil mechanics, the propagation of a gravity current into an unsaturated soil is known as infiltration (Philip 1970;Bear 1972), and self-similar solutions of the first kind have also been adapted as approximations for infiltration into homogeneous layers with a pre-existing moisture distribution (Witelski 1998). A wealth of first-kind self-similar solutions also describe the early and late time propagation regimes through a homogeneous porous medium lying above an inclined impermeable boundary (see, e.g., Vella & Huppert 2006), over fractured horizontal substrates (see, e.g., Pritchard 2007) and for drainage from the edge of a finite porous reservoir (Zheng et al 2013). In confined porous media, on the other hand, a rarefaction wave self-similar solution was derived, and verified by numerical simulations, to describe the evolution of the immiscible interface between a fluid being injected at a constant rate and the more viscous fluid being displaced (Nordbotten & Celia 2006); seepage through the confining layer has also been considered (Woods & Farcas 2009).…”

A dipole solution for power-law gravity currents in porous formations

“…Newtonian flow in a uniform Hele-Shaw cell of thickness b models Darcy flow in a homogeneous medium of permeability k(b) = b 2 /12. Newtonian flow in a V-shaped Hele-Shaw cell reproduces Darcy flow in a heterogeneous medium with both permeability and porosity power-law variations along the vertical direction (Zheng et al 2013); the ratio between the exponents of permeability and porosity is necessarily equal to 3, a value appropriate to represent a fissured medium. The approximation was extended to non-Newtonian power-law fluids (e.g.…”

“…A vertical porosity gradient is added to the conceptual model, taking into account that in real media, variations in permeability and porosity are often interdependent (Phillips 1991;Dullien 1992). The presence of permeability and porosity gradients parallel or perpendicular to the flow direction has been shown to affect significantly the shape of gravity currents of Newtonian (Huppert & Woods 1995;Zheng et al 2013) or non-Newtonian (Di Federico et al 2014) fluids within porous domains.…”

Converging gravity currents over a permeable substrate

“…Also, various coating flows occur above porous substrates. Because the boundaries are porous, a variety of studies in recent years have analysed spreading gravity currents with leakage (Acton, Huppert & Worster 2001;Pritchard, Woods & Hogg 2001;Pritchard & Hogg 2002;Pritchard 2007;Neufeld, Vella & Huppert 2009;Spannuth et al 2009;Neufeld et al 2011;Vella et al 2011;Zemoch, Neufeld & Vella 2011;Zheng et al 2013). In this case, it is possible in some cases involving outward spreading to make a change of variables to arrive back at a PDE that again admits a similarity solution of the first kind (Pritchard et al 2001).…”