A plane‐strain hydraulic fracture driven by a shear‐thinning Carreau fluid

Abstract

We study the propagation of a plane-strain hydraulic fracture driven by a shear thinning fluid following a Carreau rheology. We restrict to the impermeable medium case and quantify in details the impact on fracture growth of the shearthinning properties of the fluid between the low and high shear-rates Newtonian limits. We derive several dimensionless numbers governing the evolution of the solution. The propagation notably depends on the ratio between the two limiting viscosities, the fluid shear-thinning index, a dimensionless fracture toughness and a characteristic time-scale capturing the instant at which the fluid inside the fracture reaches the low-shear rate Newtonian plateau. We solve the problem numerically using Gauss-Chebyshev methods for the spatial discretization of the coupled hydro-mechanical problem and a fully implicit time integration scheme. The solution evolves from an early time self-similar solution equals to the Newtonian one for the large-shear rate viscosity to a late time self-similar solution equals to the low-shear rate Newtonian solution. The transition period (corresponding to the shear thinning part of the rheology) exhibits features similar to the power law rheology, albeit quantitatively different. Comparisons of hydraulic fracture growth predictions obtained with a power-law model confirm its inadequacy for realistic fluids used in practice compared to the more physical Carreau rheology: the Newtonian plateau at high and low shear rates cannot be neglected.

Effects of velocity-dependent apparent toughness on the pre- and post-shut-in growth of a hydraulic fracture

“…Appendix A. Discretization using the Gauss-Chebyshev quadrature Gauss-Chebyshev quadrature combined with the Barycentric interpolation techniques provide an efficient way to solve elastic boundary integral solutions arising in fracture problems (Viesca and Garagash, 2017;Liu and Brantut, 2022). It has been recently applied to semi-infinite (Garagash, 2019) and finite hydraulic fracture propagation problems (Liu et al, 2019;Kanin et al, 2021;Möri and Lecampion, 2021;Pereira and Lecampion, 2021), illustrating a spectral accuracy and efficiency in large time span semi-analytical investigation. In this paper, following Liu et al (2019) we use the first type Gauss-Chebyshev quadrature T k to discretize the fracture.…”

Numerical simulation of hydraulic fracturing: a hybrid FEM based algorithm

“…Moreover, its extension to the case of a penny shape fracture was done in [29,30]. This iterative scheme of computations comprises two basic modules: i) the module to compute the fluid velocity from the continuity equation ( 1), ii) the module for computing the crack opening from the boundary integral equation of elasticity (12). An additional subroutine, based on equation ( 9), is employed for the fracture front tracing.…”

“…Despite their geometrical simplicity, these models reflect properly the basic physical mechanisms that govern the HF process. Thus, they enabled e. g. to define the regimes of crack propagation [7,8,9], estimate the influence of the non-Newtonian rheology of fracturing fluid on the HF process [10,11,12], analyze the phenomenon of subcritical fracture growth [13] or investigate the effect of hydraulically induced tangential tractions exerted on the crack faces [14,15,16,17]. Moreover, solutions obtained for the simplifed models can be used for benchmarking purposes in more complex cases [18,19].…”

Numerical simulation of hydraulic fracturing: A hybrid FEM‐based algorithm

“…Despite their geometrical simplicity, these models reflect properly the basic physical mechanisms that govern the HF process. Thus, they enabled, for example, to define the regimes of crack propagation, [7][8][9] estimate the influence of the non-Newtonian rheology of fracturing fluid on the HF process, [10][11][12] analyze the phenomenon of subcritical fracture growth 13 or investigate the effect of hydraulically induced tangential tractions exerted on the crack faces. [14][15][16][17] Moreover, solutions obtained for the simplified models can be used for benchmarking purposes in more complex cases.…”