Hydraulic fracture crack propagation driven by a non-Newtonian fluid

The fluid driven fracture problem in a porous medium is considered. The fracture is driven by injecting an incompressible viscous fluid with a power-law rheology. The fluid filtration is described under the hydraulic approximation by an equation generalizing the Darcy's law. It is shown that the system of governing equations has a power-law self-similar solution, whereas, in the case of small or large fluid leak-off into the porous medium, there are the classes of power-law or exponential self-similar solutions. The complete self-similar solution is constructed and the effect of the nonlinear rheology of the fracturing fluid on the solution behavior is studied. The analytical solution to the problem is found for an arbitrary boundary condition at the crack inlet when the viscous stresses in the non-Newtonian fluid are close to a constant.