Buoyancy-driven crack propagation from an over-pressured source

The propagation of a liquid-filled crack from an over-pressured source into a semi-infinite uniform elastic solid is studied. The fluid is lighter than the solid and propagates due to its buoyancy and to the source over-pressure. The role of this over-pressure at early and late times is considered and it is found that the combination of buoyancy and over-pressure leads to significantly different behaviour from buoyancy or over-pressure alone. Lubrication theory is used to describe the flow, where the pressure in the fluid is determined by the elastic deformation of the solid due to the presence of the crack. Numerical results for the evolution of the crack shape and speed are obtained. The crack grows exponentially at early times, but at later times, when buoyancy becomes important, the crack growth accelerates towards a finite-time blow-up. These results are explained by asymptotic similarity solutions for early and late times. The predictions of these solutions are in close agreement with the full numerical results. A different case of crack geometry is also considered in order to highlight connections with previous work. The geological application to magma-filled cracks in the Earth's crust, or dykes, is discussed.