A Variational Approach to the Modeling and Numerical Simulation of Hydraulic Fracturing

Abstract

Most classical approaches to fracture mechanics focus on the growth of a single crack along a known path. In contrast, the predictive modeling and simulation of hydraulic fracturing requires the ability to handle interaction between multiple cracks --including pre-existing ones-- growing along unknown paths, and changes in crack topology and connectivity. We present an extension of Francfort and Marigo\\\'s variational approach to fracture to hydraulic stimulation. We recast Griffith\\\'s criteria into a global minimization principle, while preserving its essence, the concept of energy restitution between surface and bulk terms. More precisely, to any admissible crack geometry and kinematically admissible displacement field, we associate a total energy consisting of the sum of an elastic and a surface term. We focus on the quasi-static setting in which the reservoir state is then given as the solution of a sequence of unilateral minimizations of this total energy with respect to any admissible crack path and displacement field. The strength of this approach is to provide a rigorous and unified framework accounting for new cracks nucleation, existing cracks activation, and full crack path determination (including complex behavior such as crack branching, kinking, and interaction between multiple cracks) without any a priori knowledge or hypothesis. Of course, the lack of a priori hypothesis on cracks geometry is at the cost of numerical complexity. We present a regularized phase field approach where fractures are represented by a smooth function. This approach makes handling large and complex fracture networks very simple yet discrete fracture properties such as crack aperture can be recovered from the phase field. We compare variational fracture simulation results against several analytical solutions and demonstrate the approach\\\'s ability to predict complex fracture systems with example of multiple interacting fractures.