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Publication# 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 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.

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Viscosity

The viscosity of a fluid is a measure of its resistance to deformation at a given rate. For liquids, it corresponds to the informal concept of "thickness": for example, syrup has a higher viscosity than water. Viscosity is defined scientifically as a force multiplied by a time divided by an area. Thus its SI units are newton-seconds per square metre, or pascal-seconds. Viscosity quantifies the internal frictional force between adjacent layers of fluid that are in relative motion.

Power-law fluid**NOTOC** In continuum mechanics, a power-law fluid, or the Ostwald–de Waele relationship, is a type of generalized Newtonian fluid (time-independent non-Newtonian fluid) for which the shear stress, τ, is given by where: K is the flow consistency index (SI units Pa sn), ∂u/∂y is the shear rate or the velocity gradient perpendicular to the plane of shear (SI unit s−1), and n is the flow behavior index (dimensionless). The quantity represents an apparent or effective viscosity as a function of the shear rate (SI unit Pa s).

Non-Newtonian fluid

A non-Newtonian fluid is a fluid that does not follow Newton's law of viscosity, that is, it has variable viscosity dependent on stress. In non-Newtonian fluids, viscosity can change when under force to either more liquid or more solid. Ketchup, for example, becomes runnier when shaken and is thus a non-Newtonian fluid. Many salt solutions and molten polymers are , as are many commonly found substances such as custard, toothpaste, starch suspensions, corn starch, paint, blood, melted butter, and shampoo.

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