Infiltration of a liquid finger down a fracture into superheated rock

Abstract

A finger or ribbon of liquid water flowing down a fracture in an unsaturated rock matrix begins to boil when it passes the boiling‐point isotherm. The length of the infiltrating finger increases with time; boiling of the water in the fracture is maintained by heat conduction from the surrounding superheated rock. An intrinsic length scale in this problem is l3 = (ρlQ0h/kmβ)½, where Q0 is the volume flow rate of liquid across the boiling‐point isotherm, h is the enthalpy change on boiling, km is the thermal conductivity of the matrix rock, and β is the ambient temperature gradient. Initially, while (kt)½ <b, the length l(t) of the water finger increases rapidly, proportional to l3 (kt)¼/b½, where k is the thermal diffusivity of the matrix and b is the ribbon width. A corresponding result is given for a cylindrical finger. If water continues to infiltrate beyond the time b2/k, the liquid penetration ultimately stabilizes at a distance l of order l3 below the boiling‐point isotherm, even though the temperature distribution continues to evolve. Numerical calculations show that l depends only weakly on the ribbon width or on the radius of a cylindrical finger.