Vaporizing gravity currents in a superheated porous medium

Abstract

We analyse the motion of a current of water migrating under gravity into a hot vapour-saturated porous rock accounting for the vaporization which occurs as the water invades the hot rock. We present a series of similarity solutions to describe the rate of advance of both planar and axisymmetric currents when the total mass of water injected after time t is proportional to t^γ. Three distinct cases arise. When γ>1/2 (planar) or γ>1 (axisymmetric), the depth of the current increases at all points from the source, and therefore vaporization occurs at all points on its surface. This case is described by a simple extension of the well-known similarity solutions for non-vaporizing currents. When 0<γ<1/2 (planar) or 0<γ<1 (axisymmetric), there is a region near the source where the depth of the current decreases. The depth only increases at the more distant points. Vaporization therefore only occurs in the leading part of the current where it is advancing into the superheated rock. In this case, we develop modified similarity solutions which account for the vaporization in the distal part of the current. The third case involves the finite release of fluid. Owing to the vaporization, the mass of the current decreases with time. Since there is no injection, the rate of advance of the current can no longer be found by comparing the exponents of time in the local and global equations for mass conservation. Instead, the motion is described by a class of similarity solutions of the second kind, analogous to those described by Barenblatt (1997), in which the total mass of the current is proportional t^γ, where γ is a function of the mass fraction which vaporizes, [Fscr ], such that γ→0 as [Fscr ]→0 and γ→−1 as [Fscr ]→1. The model is extended to include the effects of capillary retention of fluid in the pore spaces and we discuss the relevance of our results to the process of liquid reinjection in the geothermal power industry.