Analysis and computation of gravity-induced migration in porous media

Abstract

Motivated by the problem of gravity segregation in an inclined porous layer, we present a theoretical analysis of interface evolution between two immiscible fluids of unequal density and mobility, both in two and three dimensions. Applying perturbation theory to the appropriately scaled problem, we derive the governing equations for the pressure and interface height to leading order, obtained in the limit of a thin gravity tongue and a slightly dipping bed. According to the zeroth-order approximation, the pressure profile perpendicular to the bed is in equilibrium, a widely accepted assumption for this class of problems. We show that for the inclined bed two-dimensional problem, in the reference frame moving with the mean gravity-induced advection velocity, the interface motion is dictated by a degenerate parabolic equation, different from those previously published. In this case, the late-time behaviour of the gravity tongue can be derived analytically through a formal expansion of both the solution and its two moving boundaries. In three dimensions, using a moving coordinate along the dip direction, we obtain an elliptic–parabolic system of partial differential equations where the fluid pressure and interface height are the two dependent variables. Although analytical results are not available for this case, the evolution of the gravity tongue can be investigated by numerical computations in only two spatial dimensions. The solution features are identified for different combinations of dimensionless parameters, showing their respective influence on the shape and motion of the interface.

Three-dimensional buoyancy-driven flow along a fractured boundary

“…In such gravity-driven flow, provided the supply flux is sufficiently small, the current will become relatively long and thin so that the velocity is, to leading order, parallel to the boundary (Barenblatt 1996;de Loubens & Ramakrishnan 2011) and, in the limit in which we neglect capillary effects, to leading order, the pressure gradient in the current in the direction normal to the boundary becomes hydrostatic (cf. Barenblatt 1996;Huppert & Woods 1995;de Loubens & Ramakrishnan 2011).…”

“…Barenblatt 1996;Huppert & Woods 1995;de Loubens & Ramakrishnan 2011). Once the injected fluid has established a steady flow, the original fluid in the aquifer is stationary and therefore has a hydrostatic gradient in the direction both normal to and along the boundary, and this determines the dynamic pressure gradient in the buoyant current.…”

“…Numerous numerical models describing the motion of CO 2 gravity currents have been developed and have led to a series of fascinating results † Email address for correspondence: andy@bpi.cam.ac.uk 280 A. Farcas and A. W. Woods concerning the subsurface dispersal of a buoyant fluid of low viscosity displacing a second fluid (Pruess et al 2003;Obi & Blunt 2006;Juanes et al 2006;de Loubens & Ramakrishnan 2011). These numerical solutions include a wide range of physical processes.…”

“…Pritchard, Hogg & Woods (2001) and examined the impact of drainage through the upper boundary on the upslope propagation distance of such two-dimensional currents and examined how, after injection has ceased, the flow partitions into a component which is capillary trapped in the injected layer and a component which drains into the overlying permeable strata. Many of these papers have adopted an approximate description in which the cross-flow pressure gradients are assumed to be hydrostatic, based on the assumption that the current is long and thin; recently, de Loubens & Ramakrishnan (2011) have presented a detailed asymptotic analysis of the underlying governing equations, identifying conditions under which different approximations are appropriate.…”

Asymptotic solution of a nonlinear advection-diffusion equation

“…in water infiltration theory [3,11]. More recently, it has been shown to govern the interface shape evolution of a gravity tongue propagating up an inclined porous layer [9] with respect to a moving frame. Here u is the thickness of the gravity tongue, and α > 0 is a parameter proportional to the up-dip slope and the mobility ratio of the lighter to 390 R. DE LOUBENS AND T. S. RAMAKRISHNAN the heavier fluid.…”