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. Introduction Conventionally, in most numerical modeling strategy of hydraulic fracturing, fracture propagation is assumed to be planar and perpendicular to the minimum reservoir stress (Adachi et al. 2007), which simplifies fracture propagation criteria to mode-I and aligns the propagation plane to the simulation grid. Restricting propagation mode search into one direction and prescribing fracture growth plane can greatly reduce computation overhead and make practical numerical modeling tractable. However, recent observations suggest creation of nonplanar complex fracture system during reservoir stimulation (Mayerhofer et al. 2010) or waste injection (Moschovidis et al. 2000). To address predictive capabilities of complex fracture propagation, several different approaches, namely, mixed-mode fracture growth criterion with a single fracture (Rungamornrat, Wheeler, and Mear 2005), multiple discrete fractures that grow with empirical correlations (Gu et al. 2012), and implicit fracture treatment with the idea of stimulated reservoir volume where averaged properties are estimated over an effective volume (Hossain, Rahman, and Rahman 2000) have been proposed. In this study, we propose to apply the variational approach to fracture (Francfort and Marigo 1998; Bourdin, Francfort, and Marigo 2008) to hydraulic fracturing. One of the strengths of this approach is to account for arbitrary numbers of pre-existing or propagating cracks in terms of energy minimization, without any a priori assumption on their geometry, and without restricting their growth to specific grid directions. The goal of this paper is to present early results obtained with this method. At this stage, we are not trying to account for all physical, chemical, thermal, and mechanical phenomena involved in the hydraulic fracture process. Instead, we propose a mechanistically sound yet mathematically rigorous model in an ideal albeit not unrealistic situation, for which we can perform rigorous analysis and quantitative comparison with analytical solutions. In particular, we neglect all thermal and chemical effects, we assume that the injection rate is slow enough that all inertial effects can be neglected, and place ourself in the quasi-static setting. Furthermore, we consider a reservoir made of an idealized impermeable perfectly brittle linear material with no porosity and assume that the injected fluid is incompressible. These assumptions imply that no leak-off can take place and that the fluid pressure is constant throughout the fracture system (infinite fracture conductivity), depending only on total injected fluid volume and total cracks opening respectively. Also, throughout the analyses presented, we only deal with dimensionless parameters, which are normalized by the Young’s moduli for mechanical parameters and by the total domain volume for volumetric parameters. The variational approach to hydraulic fracturing In classical approaches to quasi-static brittle fracture, the elastic energy restitution rate, G, induced by the infinitesimal growth of a single crack along an a priori known path (derived from the stress intensity factors) is compared to a critical energy rate Gc and propagation occurs when G = Gc, the celebrated Griffith criterion. The premise of the variational approach to fracture is to recast Griffith’s criterion in a variational setting, i.e. as the minimization over any crack set (any set of curves in 2D or of surface in 3D, in the reference configuration) and any kinematically admissible displacement field u, of a total energy consisting of the sum of the stored potential elastic energy and a surface energy proportional to the length of the cracks in 2D or their area in 3D. More specifically, consider a domain Ω in 2 or 3 space dimension, occupied by a perfectly brittle linear material with Hooke’s law A and critical energy release rate (also often referred to as fracture toughness) Gc. Let f(t, x) denote a time-dependent body force applied to Ω, τ(t, x), the surface force applied to a part ∂NΩ of its boundary, and g(t, x) a prescribed boundary displacement on the remaining part ∂DΩ. To any arbitrary crack set Γ and any kinematically admissible displacement set u, we associate the the total energy