Self-Similar Solutions of the Problem of Formation of a Hydraulic Fracture in a Porous Medium

The problem of hydraulic fracture formation in a porous medium is investigated in the approximation of small fracture opening and inertialess incompressible Newtonian fluid fracture flow when the seepage through the fracture walls into the surrounding reservoir is asymptotically small or large. It is shown that the system of equations describing the propagation of the fracture has self-similar solutions of power-law or exponential form only. A family of self-similar solutions is constructed in order to determine the evolution of the fracture width and length, the fluid velocity in the fracture, and the length of fluid penetration into the porous medium when either the fluid flow rate or the pressure as a power-law or exponential function of time is specified at the fracture entrance. In the case of finite fluid penetration into the soil the system of equations has only a power-law self-similar solution, for example, when the fluid flow rate is specified at the fracture entrance as a quadratic function of time. The solutions of the self-similar equations are found numerically for one of the seepage regimes.