A VARIATIONAL APPROACH TO THE MODELING AND NUMERICAL SIMULATION OF HYDRAULIC FRACTURING UNDER IN-SITU STRESSES

Most hydraulic fracturing simulation approaches apply propagation criteria to an individual fracture along prescribed path(s). In practice, however, the modeling of hydraulic fracturing in reservoir stimulation requires handling interactions between multiple natural or induced fractures growing along unknown paths, and changes in fracture pattern and network. In this paper, we extend the work of Bourdin et al 2012 and focus on the effectof in-situ stresses on crack path in 2D and 3D, and revisit the verification work against analytic solution with more realistic units. The approach is based on Francfort and Marigo's variational approach to fracture (Francfort and Marigo, 1998). The main idea is to recast Griffith's criterion for a single fracture growth into a global energy minimization problem. The energy we consider consists of the sum of surface and bulk terms accounting for the energy dissipated by a growing crack and the mechanical energy, including the work of residual (in-situ) stresses and pressure force against the fracture walls. To be more specific, we search the minimum of the total energy under any admissible fracture sets and kinematically admissible displacement field. Our focus is on quasi-static crack propagation propagation encountered during hydraulic fracturing process, which we model as a rate independent process. This approach does not need any a priori knowledge of the crack path, or any additional hypotheses concerning fracture nucleation or activation. We claim that it provides a mathematically rigorous and mechanistically sound unified framework accounting, derived from first principles, and accounting for new fractures nucleation, existing fractures activation, and full fracture path determination such as branching, kinking, and interaction between multiple cracks. It is no surprise that having no a priori hypothesis or knowledge on fracture geometry comes at the cost of numerical complexity. To overcome the complexities associating with handling of large and complex fracture patterns, we propose an approach based on a regularized model where fractures are represented by a smooth function. In this paper, we first show series of comparison cases of the variational fracture simulation against analytical solutions. We demonstrate our approach's ability to predict complex behaviors such as turning fracture under in-situ earth stresses and the interactions of multiple fractures.

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