A Variational Approach to the Numerical Simulation of Hydraulic Fracturing

One of the most critical capabilities of realistic hydraulic fracture simulation is the prediction of complex (turning, bifurcating, or merging) fracture paths. In most classical models, complex fracture simulation is difficult due to the need for a priori knowledge of propagation path and initiation points and the complexity associated with stress singularities at fracture tips. In this study, we follow Francfort and Marigo's variational approach to fracture, which we extend to account for 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 given as the sum of the elastic and surface energies. In a quasistatic setting, 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 also demonstrate the approach's ability to predict complex fracture systems with example of multiple interacting fractures.