Department of Mathematics and Center for Computation & Technology
Louisiana State University
Friday, October 12, 2012
The Variational Approach to Fracture: Application to Reservoir Stimulation
In many of its classical engineering applications, the focus of fracture mechanics is on preventing or mitigating crack nucleation and propagation. In contrast, the goal of reservoirs stimulation is to create large amounts of interconnected underground cracks leading to improved fluid flow, resource extraction, or heat transfer. In this context, many of the usual simplifying hypotheses of the classical methods based on Griffith’s theory, such as predetermined crack path or simple cracks geometry, are unreasonable.
The variational approach to brittle fracture, originally proposed by G. Francfort and J.-J. Marigo, addresses these issues by recasting Griffith’s theory into a global minimization principle while preserving its essence: restitution between surface and bulk energies. It can fully identify crack paths, predict the nucleation of new cracks, and account for the interactions between multiple cracks in two and three space dimensions. Of course, this has a price: its analysis relies on sophisticated mathematical techniques developed for the larger class of free discontinuity problems, and its numerical implementation requires carefully tailored techniques.
In my talk, I will first present the model, and its connections with the classical Griffith theory. Then, I will focus on its numerical implementation, and in particular on an approach based on approximation by elliptic functionals. I will illustrate the properties of the model and its approximation using simple numerical experiments. Finally, I will focus on validation and verification experiments, in the context of reservoir stimulation.