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

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 shear‐thinning 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.