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=(&rgr;lQ0h/km&bgr;)1/2, 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 &bgr; is the ambient temperature gradient. Initially, while (&kgr;t)1/2<b, the length l(t) of the water finger increases rapidly, proportional to l3(&kgr;t)1/4/b1/2, where &kgr; 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/&kgr;, 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. ¿ American Geophysical Union 1996