Mathematical and analogue models of fluid filled fracture propagation in layered elastic media

In order to overcome the practical impossibility of direct observation of dike propagation within the crust, we develop mathematical and analogue models to describe the physical processes involved in their dynamics. We focus our attention on what happens when a dike approaches the boundary between two media with different rigidities. In our 2D mathematical models, a dike opens and propagates in an infinite elastic medium, made up of 2 welded half-spaces with different elastic parameters. Dikes are modelled as boundary element fluid-filled cracks. The pressure gradient along the crack is proportional to the difference between the densities of the host rock and the fluid. We take into account the compressibility of the fluid and a variable density in order to conserve the mass of the intrusion during its motion. The mathematical model allows to set a tectonic stress field and an arbitrarily tilted boundary separating different media. The growth, arrest and direction of propagation of the crack is governed by an energetic criterion: the motion of the dike is driven by the minimization of the elastic deformation energy plus the gravitational energy. Propagation is allowed when the energy release during the motion exceeds a fracture threshold. The output of this code gives us the path followed by the crack during the propagation, its shape and the stresses induced in the elastic medium. Interestingly, the mathematical simulations provide a sort of refraction phenomenon, that is a sudden change in direction of propagation when the crack crosses the boundary separating different rigidities. In order to validate our mathematical results, we perform laboratory experiments of air filled cracks propagating in gelatine. Gelatine represents well an elastic medium: it is brittle at refrigerator temperature and varying the concentration of dry gel powder dissolved in water we can control its rigidity. By injecting air from the bottom of a trasparent cylinder containing the gelatine, we obtain an air filled crack, tilted with respect to the vertical and propagating upwards toward the rigidity transition surface. The experiments confirm the main characteristics of the mathematical simulations.