Hydraulic fracture crack propagation driven by a non-Newtonian fluid

The problem of hydraulic fracture crack propagation in a porous medium is considered. The fracture is driven by an incompressible viscous fluid with a power-law rheology of the pseudoplastic type. The fluid seepage is described by an equation generalizing the Darcy law in the hydraulic approximation. It is shown that the system of governing equations has a power-law self-similar solution, whereas, in the limiting cases of low and high fluid saturation in the porous medium, there are some families of power-law or exponential self-similar solutions. The complete self-similar solution is constructed. The effect of the nonlinear rheology of the fracturing fluid on the behavior of the solution is studied. The problem is solved analytically for an arbitrary boundary condition at the crack inlet when the viscous stresses in the non-Newtonian fluid are close to a constant.