Two-phase gravity currents resulting from the release of a fixed volume of fluid in a porous medium

We consider the instantaneous release of a finite volume of fluid in a porous medium saturated with a second, immiscible fluid of different density. The resulting two-phase gravity current exhibits a rich array of behaviours due to both the residual trapping of fluid as the current recedes and the differing effects of surface tension between advancing and receding regions of the current. We develop a framework for the evolution of such a current with particular focus on the large-scale implications of the form of the constitutive relation between residual trapping and initial saturation. Pore-scale hysteresis within the current is represented by families of scanning curves relating capillary pressure and relative permeability to saturation. In the resulting vertically integrated model, all capillary effects are incorporated within specially defined saturation and flux functions specific to each region. In the long-time limit, when the height of the current and the saturations within it are low, the saturation and flux functions can be approximated by mathematically convenient power laws. If the trapping model is approximately linear at low saturations, the equations admit a similarity solution for the propagation rate and height profile of the late-time gravity current. We also solve the governing partial differential equation numerically for the nonlinear Land’s trapping model, which is commonly used in studies of two-phase flows. Our investigation suggests that for trapping relations for which the proportion of trapped to initial fluid saturation increases and tends to unity as the initial saturation decreases, both of which are properties of Land’s model, a gravity current slows and eventually stops. This trapping behaviour has important applications, for example to the ultimate distance contaminants or stored carbon dioxide may travel in the subsurface.