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

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 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 for the case of an isotropic toughness, 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 

Propagation of fluid driven fractures in transversely isotropic material

Fatima-Ezzahra Moukhtari

The propagation of fluid driven fractures is used in a number of industrial applications (well stimulation of unconventional reservoirs, development of deep geothermal systems) but also occurs naturally (magmatic dyke intrusion). While the mechanics of hydraulic fractures (HF) in isotropic media is well established, the impact of the anisotropy of natural rocks on HF propagation is still far from being understood. Sedimentary rocks like shales and mudstones are ubiquitous in upper earth crust which are made of fine layers which result in transverse isotropy. In the framework of continuous mechanics, these rocks are commonly modelled as a transverse isotropic media (TI). In addition, a large number of fluids used in HF are non-Newtonian. They typically exhibit a shear-thinning behavior which can be reproduced by different rheological models with varying levels of accuracy (Carreau, power law, Ellis). In this thesis, we focus on hydraulic fractures in impermeable TI media. We assume that the fracture propagates normal to the isotropy plane without any further assumption on its shape. This configuration is relevant for normal and strike-slip stress regimes where the minimum in-situ stress is horizontal. We combine a boundary element and a finite volume method with an implicit level set scheme to model the growth of three dimensional planar HF. Both anisotropy of elasticity and fracture energy/toughness are accounted for. This algorithm couples a finite discretization of the fracture with the solution for a steadily moving hydraulic fracture in the tip region. We show that the near tip elastic operator has a similar expression than in the isotropy pending the use of a near-tip elastic modulus which now depends on the local propagation direction with respect to the isotropy plane. Using this numerical model we quantify the fracture elongation as a function of both the elastic and fracture toughness anisotropies. The elongation is maximal in the toughness dominated regime. The transition of the viscosity to the toughness regime occurs faster along the arrester direction, thus promoting fracture elongation. In parallel, we report laboratory experiments of HF growth in cubic samples of slate Del Carmen under true-triaxial confinement. We were able to propagate a planar HF perpendicular to the bedding planes only when the initial stress normal to bedding was 20 times larger than the other two stresses. For both regimes, the fracture surfaces are very rough with a self-affine behavior in the direction of the bedding and a stationary state in the direction normal to bedding. We also investigate the effect of a non-Newtonian rheology on HF growth. We solve the problem of steadily moving semi-infinite HF driven by a Carreau fluid. We use a Gauss-Chebyshev method for elasticity combined with finite differences for lubrication flow and solve the resulting non-linear system with the Newton Raphson method. The solution exhibits four asymptotic regions: a linear elastic fracture mechanics (lefm) asymptote near the tip, high-shear rate Newtonian and power law asymptotes in an intermediate region and a low-shear Newtonian asymptote in the far field. For the same dimensionless toughness, the fluid lag is smaller than for a Newtonian fluid of low shear rate viscosity. We show that simpler rheological models (Ellis and power law) cannot capture the complete solution, which accounts for the full rheological behavior.

EPFL2020

The tip region of a hydraulic fracture driven by a power law fluid

Brice Tanguy Alphonse Lecampion, Fatima-Ezzahra Moukhtari

We investigate the tip region of a hydraulic fracture driven by a power law fluid propagating at a constant velocity V in an impermeable linear elastic medium. We account for the presence of a fluid lag of a priori unknown length at the tip of the fracture. The fluid pressurized fracture propagates perpendicular to the far-field compressive minimum stress in pure opening mode (see Fig.1a). The solution of this semi-infinite hydraulic fracture problem combines elastic deformation, lubrication flow in the filled region of the fracture and the quasi-static fracture propagation condition. It exhibits a multiscale structure related to the strong fluid solid coupling at play near the fracture tip. For a Newtonian fluid, the solution derived by Garagash & Detournay (2000) transition from the classical linear elastic fracture mechanics asymptote near the tip to the viscous asymptote (Desroches et al., 1994) away from the tip (see Fig.1b). The fluid lag was shown to vanish for large values of a dimensionless fracture toughness encapsulating all the problem parameters.

2016

Tip asymptotic of a hydraulic fracture driven by a power law fluid accounting for the presence of a fluid lag

Brice Tanguy Alphonse Lecampion, Fatima-Ezzahra Moukhtari

We investigate the tip region of a hydraulic fracture driven by a power law fluid propagating at a constant velocity in an impermeable linear elastic isotropic medium. We account for the presence of a fluid lag of a priori unknown length at the tip of the fracture. The fluid pressurized fracture propagates perpendicular to the far-field compressive minimum stress in pure opening mode. The solution of this semi-infinite hydraulic fracture problem combines elastic deformation, lubrication flow in the filled region of the fracture and the quasi-static equilibrium fracture propagation condition. It exhibits a multiscale structure related to the strong fluid solid coupling at play near the fracture tip. For a Newtonian fluid, the solution derived by Garagash & Detournay (2000) transition from the classical linear elastic fracture mechanics asymptote near the fracture tip to the viscous asymptote (Desroches et al., 1994) away from the tip. The fluid lag was shown to vanish for large values of a dimensionless fracture toughness encapsulating all the problem parameters (viscosity, fracture velocity, plane- strain elastic modulus).

2017

Propagation of fluid driven fractures in transversely isotropic material

Fatima-Ezzahra Moukhtari

EPFL2020