The evaluation of geomechanical effects and fluid flow related to pressure transient phenomena in fractured Biot’s stress-sensitive oil reservoirs is essential to minimize the mechanical formation damage and extend the well-reservoir life cycle for economical production. Therefore, the management of the damage caused by effective permeability loss in this type of reservoir becomes essential to productivity maintenance. This paper proposes a new unsteady-state poroelastic solution for the nonlinear hydraulic diffusivity equation (NHDE) in Biot’s effective stress-sensitive reservoirs fully penetrated by fractured oil wells. The hydraulic fracture in the proposed mathematical modeling is finite with tip effects and crosses the whole reservoir net-pay. A new permeability stress-sensitive pseudopressure is developed, and the solution of the NHDE is derived in terms of this function. The NHDE is expanded in a first-order asymptotic series, and a poroelastic integro-differential solution coupled to a Green’s function is used to represent the source/sink term. A set of pore pressure and permeability data is used from geomechanical literature and transformed into effective stress through Biot’s equation. The effects of the Biot’s coefficient, overburden stress, oil flow rate, fracture’s tip, and proppant porosity arrangements are simulated. The results show that these parameters are essential to minimize formation damage. Model calibration is performed using a numerical oil flow simulator named IMEX®, widely used in the oil industry. The accuracy, ease of implementation, and low computational costs constitute the main advantages of the model addressed in this paper. Hence, it may be a valuable and attractive mathematical tool to identify flow regimes, providing permeability loss control and supporting well-reservoir management.