Propagation of a planar hydraulic fracture perpendicular to the isotropy plane of a transversely isotropic material

We investigate the case of a hydraulic fracture (HF) propagating perpendicular to the isotropy plane of a transversely isotropic material: a relevant configuration for the growth of HFs in sedimentary rocks where fractures are propagating vertically across layers. We extend the implicit level set algorithm originally developed for the propagation of planar 3D HF in isotropic media to transverse isotropy (TI) of elasticity and toughness. Contrary to the isotropic case, the near-tip plane strain elastic relation depends on the angle α between the local propagation direction and the isotropy plane. We present an analytical solution for an elliptical HF in the toughness dominated regime and use it to benchmark our numerical solver. For a TI elastic material, we investigate HF growth for two different assumptions: isotropic material toughness or isotropic critical fracture energy. In both cases, we compare the fracture aspect ratio obtained by our numerical results with simplified estimations based on the minimization of the variation of local stress intensity factor (or energy release rate) under the assumption of an elliptical fracture. Our numerical results show that for both assumptions, the fracture aspect ratio inversely scales with the ratio of plane-strain elastic modulus in the two orthogonal directions of the material frame with a different exponent. However, the fracture is never strictly elliptical, except for a very peculiar form of toughness anisotropy.