A numerical model of dyke propagation in layered elastic media

We develop a mathematical model describing dyke propagation in proximity of an elastic discontinuity of the embedding medium. The dyke is modelled as a fluid-filled crack in plane strain configuration employing the boundary element method. The pressure gradient along the crack is assumed proportional to the difference between the densities of the host rock and the fluid. Mass conservation is imposed during propagation and fluid compressibility is taken into account. The path followed by the crack is found by maximizing the total energy release, given by the sum of the elastic and gravitational contributions. The mathematical simulations provide a sort of 'refraction phenomenon', that is a sudden change in the direction of propagation when the crack crosses the boundary separating different rigidities: if the dyke enters a softer medium, its path deviates towards the vertical, if the dyke enters a harder medium its path deviates away from the vertical and may even become arrested as a horizontal sill along the interface, if the rigidity contrast is large. Gravitational energy plays a major role during propagation; in particular, in proximity of layer boundaries, this role is enhanced by the shift of the centre of mass due to changes of dyke shape. Mathematical results were validated by laboratory experiments performed injecting tilted air-filled cracks through gelatin layers with different rigidities.