Filling box flows in porous media

Abstract

We report upon a theoretical and experimental investigation of a porous medium 'filling box' flow by specifically examining the details of the laminar descending plume and its outflow in a control volume having an impermeable bottom boundary and sidewalls. The plume outflow initially comprises a pair of oppositely directed gravity currents. The gravity currents propagate horizontally until they reach the lateral sidewalls at y = ±L. The flow then becomes of filling box type, with a vertically ascending 'first front' separating discharged plume fluid below from ambient fluid above. The flow details are described analytically by first deriving a new similarity solution for Darcy plumes with Pe > O(1), where Pe is the Péclet number. From the similarity solution so obtained, we then derive expressions for the plume volume flux and mean reduced gravity as functions of the vertical distance from the source. Regarding the plume outflow, a similarity solution adopted from Huppert & Woods (J. Fluid Mech., vol. 292, 1995, pp. 55-69) describes the height and front speed of the gravity currents, whereas a semi-implicit finite difference scheme is used to predict the first front elevation versus time and horizontal distance. As with high-Reynolds-number filling box flows, that studied here is an example of a coupled problem: the gravity current source conditions are prescribed by the plume volume flux and mean reduced gravity. Conversely, discharged plume fluid may be re-entrained into the plume, be it soon or long after reaching the bottom impermeable boundary. To corroborate our model predictions, analogue laboratory experiments are performed with fresh water and salt water as the working fluids. Our experiments consider as independent variables the porous medium bead diameter and the plume source volume flux and reduced gravity. Predictions for the gravity current front position and height compare favourably against analogue measured data. Good agreement is likewise noted when considering either the mean elevation or the profile of the first front. Results from this study may be adopted in modelling geological plumes. For example, our equations can be used to predict the time required for discharged plume fluid to return to the point of injection in the case of aquifers closed on the sides and below by impermeable boundaries.

Isolated buoyant convection in a two-layered porous medium with an inclined permeability jump

“…As in previous works (Huppert & Woods 1995;Sahu & Flynn 2015;Pegler et al 2016), the dynamics of the gravity currents were characterized by measuring their speed of advance. Here, measurements were made in both the up-and downdip directions and with reference to both the bulk and dispersed interfaces, see figure 14.…”

“…Recall that the density of the fluid supplied to the gravity currents slowly increases with time as a result of the gradual increase of . Wishing to account for this fact in the governing equations (2.5) and (2.10) we adopt (2.25) of Sahu & Flynn (2015), and write Here, the source correction term is evaluated considering the volumetric flow rate at the plume source. Also, is the transverse dispersivity, whose connection to the porous medium grain size is discussed below in § 5.1.…”

“…The plume volume flux per unit width, , supplied at , increases with distance from the source. Analogous to (2.11), can be expressed in terms of using see (2.24) of Sahu & Flynn (2015). The in (2.15) is divided into unequal up- and downdip components for .…”

“…Recall that the density of the fluid supplied to the gravity currents slowly increases with time as a result of the gradual increase of h. Wishing to account for this fact in the governing equations (2.5) and (2.10) we adopt (2.25) of Sahu & Flynn (2015), and write…”

Dispersive entrainment into gravity currents in porous media

“…The prevalence of dispersion in miscible gravity currents is apparent in previous work, for example in flows through homogeneous porous media by Sahu & Flynn (2015) and Pegler et al (2017) -see their figures 6 and 12, respectively, and the associated discussions. In multilayered porous media this mixing may be greatly enhanced as observed in the laboratory experiments of Huppert, Neufeld & Strandkvist (2013, figure 7) and Sahu & Flynn (2015, figure 4).…”