A mathematical model of hydraulic fracture growth and stabilization

This paper considers stationary and non-stationary models of hydraulic fracture growth and stabilization. For the first time, the length of an elliptical section fracture is determined in terms of the balance of fluid inflow into and outflow from the fracture. The stationary model assumes that the fracture formation time is much less than the characteristic time of water injection into the reservoir, and the rate of further fracture growth is much less than the rate of the outflow and is negligible. The non-stationary model takes into account the stage of the fracture growth. Both mathematical models are developed using the laws of conservation of mass and momentum. Darcy's law is applied to describe the leaks into the reservoir. The boundary conditions consider the constancy of the injected water flow rate and the balance between the fluid inflow into and outflow from the fracture. It is established that over time, the half-length of the fracture calculated by the non-stationary model gradually reaches a stationary value corresponding to that determined by the stationary model.