Group invariant solution for a pre-existing fracture driven by a power-law fluid in impermeable rock

• The two-dimensional impermeable fracture driven by a power-law fluid is modeled..
• Group invariant solutions for a pre-existing hydraulic fracture are derived.
• Fractures driven by shear thinning, Newtonian, shear thickening fluids are compared.
• In a hydraulic fracture the width-averaged fluid velocity is physically significant.

The effect of power-law rheology on hydraulic fracturing is investigated. The evolution of a two-dimensional fracture with non-zero initial length and driven by a power-law fluid is analyzed. Only fluid injection into the fracture is considered. The surrounding rock mass is impermeable. With the aid of lubrication theory and the PKN approximation a partial differential equation for the fracture half-width is derived. Using a linear combination of the Lie-point symmetry generators of the partial differential equation, the group invariant solution is obtained and the problem is reduced to a boundary value problem for an ordinary differential equation. Exact analytical solutions are derived for hydraulic fractures with constant volume and with constant propagation speed. The asymptotic solution near the fracture tip is found. The numerical solution for general working conditions is obtained by transforming the boundary value problem to a pair of initial value problems. Throughout the paper, hydraulic fracturing with shear thinning, Newtonian and shear thickening fluids are compared.