Planar hydraulic fracture growth perpendicular to the isotropy plane in a transversely isotropic material

Abstract

The configuration of a hydraulic fracture (HF) propagating perpendicular to the isotropy plane of a transversely isotropic (TI) material is encountered in most sedimentary basins. We account for both elastic and fracture toughness anisotropy, and investigate fracture growth driven by the injection of a Newtonian fluid at a constant rate from a point source. In addition to the usual dimensionless parameters governing HF growth in isotropy, four dimensionless elastic parameters enter the problem for a TI material: the ratio β of elastic plane-strain modulus in the two orthogonal directions of the material frame, two Thomsen parameters ϵ, δ and the stiffness ratio C13/C11. Moreover, the ratio κ of fracture toughness in the two orthogonal directions as well as the details of the toughness anisotropy also plays a role on the development of the fracture geometry. We quantify HF growth numerically without any a-priori assumptions on the fracture shape. In doing so, we derive the exact expression for the near-tip elastic modulus as a function of propagation direction and extend to TI an implicit level set algorithm coupling a finite discretization with the near-tip solution for a steadily moving HF. A solution for a toughness dominated elliptical HF in a TI material is derived and used to verify our numerical solver. Importantly, the fracture shape is strictly elliptical only for a very peculiar form of toughness anisotropy. The evolution of the HF from the viscosity dominated regime (early time) to the toughness dominated regime (late time) results in an increase of the fracture elongation. The elongation of the fracture in the viscosity dominated regime scales as 0.76β−1/3 and increases as the propagation transition to the toughness dominated regime. We confirm the expressions for the transition time-scales in the two orthogonal directions of the material frame obtained from scaling considerations. The exact form of the toughness anisotropy plays a crucial role on the final fracture elongation in the toughness regime, which scales as β−2 for the case of an isotropic toughness, β−1 for an isotropic fracture energy and as (κ/β)2 for the peculiar case of an ‘elliptical’ fracture anisotropy. Our results also indicate that i) simplified approximations for the near-tip modulus previously derived are only valid for weak anisotropy (β < 1.2) and that ii) the other elastic parameters have a second order effect on HF growth (at most 10 percent).