On the Computation of Crack Surface Normals in the Phase‐Field Model of Hydraulic Fracture

ABSTRACT Hydraulic fracturing techniques have been used to extract oil from the Earth's crust and to enhance the efficiency of geothermal systems among other applications. Numerical simulation of hydraulic fracturing is essential for predicting the environmental impact of this method, as well as its efficiency in such applications. In the present work, our aim is to investigate fluid‐driven fracturing in porous materials by employing a phase‐field model of fracture within Biot's poroelasticity theory. In this context, a major challenge in the simulation of hydraulic fracturing is the accurate computation of the normals to the regularized crack surface. This challenge arises due to vanishing gradients of the phase‐field variable in fractured zones. To overcome this issue, several methods have been proposed to compute the normal of the crack surface. For instance, the –projection of the phase‐field gradients can be exploited to determine the crack normals in entirely fractured finite elements. Another approach identifies the crack normals as the principal directions of the strain tensor corresponding to its largest principal value. In the present contribution, we explore these approaches and show that, in certain scenarios, they yield spurious normals at the crack tip or at the fluid‐mass injection boundary. Our investigations are carried out within finite‐element implementations based on first‐ and second‐order Lagrangian interpolations ( and ), as well as second‐order divergence‐conforming Brezzi–Douglas–Fortin–Marini () interpolations. In addition to these investigations, we provide an alternative approach to improve the accurate computation of the normals at the crack surface.