A simplified model of fracking based on the topological derivative concept

This paper deals with a simplified hydraulic fracture model based on the concept of topological derivatives. It means that we consider a two dimensional idealization in which the rock is assumed to be impermeable, while the fracturing process is activated by a given pressure acting within the existing geological faults. The basic idea consists in adapting the Francfort-Marigo damage model to the context of hydraulic fracture. The Francfort-Marigo damage model is a variational approach to describe the behavior of brittle materials under the quasi-static loading assumption, focusing on the evolution of damaged regions under an irreversibility constraint. In our model problem, the loading comes out from a pressurized damaged region embedded into the rock, which is used to trigger the hydraulic fracturing process. In particular, a shape functional given by the sum of the total potential energy of the system with a Griffith-type dissipation energy term is minimized with respect to a set of ball-shaped pressurized inclusions by using the topological derivative concept. Thus, the topological asymptotic expansion of the shape functional with respect to the nucleation of a circular inclusion endowed with non-homogeneous transmission condition on its boundary is obtained. The associated topological derivative, which corroborates with the famous Eshelby theorem, is used to devise a simple topology optimization algorithm specifically designed to simulate the whole nucleation and propagation process of hydraulic fracturing. To assess our model, some numerical examples are presented, showing typical features of hydraulic fracture phenomenon, including the characterization of the fault-activation pressure and specific crack path growth, allowing for kinking and bifurcations.