Analytical solution to the problem of injection or reduction of the formation pressure in the reservoir with a fracture

The problem of injection of Newtonian fluid at a constant flow rate through an injection well into an initially undisturbed infinite reservoir with an erosive vertical main fracture of constant width is considered. Using the Laplace transform method, analytical solutions are obtained for the pressure fields in the fracture and reservoir, the flow velocity in the fracture, as well as the equations for fluid trajectories in the reservoir and in the main fracture are derived. The solutions obtained are also applicable to the problem of fluid withdrawal into a production well intersected by a vertical main fracture. Nonstationary two-dimensional pressure fields in the reservoir, as well as the pressure and velocity fields in the fracture, are constructed.