In order to better understand the interaction between pore-fluid overpressure and failure patterns in rocks we consider a porous elasto-plastic medium in which a laterally localized overpressure line source is imposed at depth below the free surface. We solve numerically the fluid filtration equation coupled to the gravitational force balance and poro-elasto-plastic rheology equations. Systematic numerical simulations, varying initial stress, intrinsic material properties and geometry, show the existence of five distinct failure patterns caused by either shear banding or tensile fracturing. The value of the critical pore-fluid overpressure pc at the onset of failure is derived from an analytical solution that is in excellent agreement with numerical simulations. Finally, we construct a phase-diagram that predicts the domains of the different failure patterns and pc at the onset of failure.