Hydraulic crack propagation in a porous medium

We develop a model for the propagation of a fluid-filled crack in a porous medium. The problem is motivated by the mechanism whereby drainage networks may form in partially molten rock below the Earth‘s lithosphere. Other applications include the propagation of hydraulic fractures in jointed rocks and in oil drilling operations, and the formation of dessication cracks in soils. Motivated by the lithospheric problem, we study a situation in which gravity acts in the direction of crack propagation. The model couples the elastohydrodynamic equations of crack propagation with a pore pressure field in the porous rock, which drives the fluid flow which supplies the crack. The effect of the pore flow is to include a diffusional term in the evolution equation for the crack width, thus allowing a crack initiated at the base of the lithosphere to propagate down into the asthenosphere. Asymptotic and numerical solutions are presented for this crack evolution. However, the predicted drainage of melt into this crack is tiny compared with the upward percolative melt migration, and the predicted width of cracks (millimetres) is much too small to allow propagation of melt into the lithosphere without freezing. As a mechanism to explain magma fracturing in the lithosphere, the process described here therefore requires further refinement.