Dispersive entrainment into gravity currents in porous media

Abstract

The effects of dispersion acting on gravity currents propagating through porous media are considered theoretically and experimentally. We exploit the large aspect ratio of these currents to formulate a depth-averaged model of the evolution of the mass and buoyancy. Dispersion, acting predominantly at the interface between the current and the ambient, is velocity dependent and acts to entrain fluid into the gravity current, in direct analogy to turbulent mixing. Here, we show that when the gravity current is fed by a constant buoyancy and mass flux the buoyancy of the current is self-similar and recovers the classical solution for gravity currents in porous media. In contrast, the profile and the depth-averaged concentration of the current evolve in a non-self-similar manner. The total volume of the current increases with time as $t^{1/3}$ due to this dispersive entrainment. We test our theoretical predictions using a suite of laboratory experiments in which the evolution of the concentration within the current was mapped using a dye-attenuation technique. These experimental results show good agreement with the early-time limits of our theoretical model, and in particular accurately predict the evolution of the depth-averaged concentration profile. These results suggest that mixing within porous media may be modelled using an effective dispersive entrainment, the magnitude of which may be set by the underlying structure of the porous medium.