In this study, we analyze the flow filtration process of slightly compressible fluids in fractured porous media. We model the coupled fractured porous media system, where the linear Darcy flow is considered in porous media and the nonlinear Forchheimer equation is used inside the fracture. Flow in the fracture is modeled as a reduced low dimensional boundary value problem which is coupled with an equation for flow in the reservoir. We prove that the solution of the reduced model can serve very accurately to approximate the solution of the actual high-dimensional flow in reservoir fracture system because the thickness of the fracture is small. In the analysis, we consider two types of Forchheimer flows in the fracture: isotropic and anisotropic, which are different in their nature. Using the method of reduction, we develop a formulation for an optimal design of the fracture, which maximizes the capacity of the fracture in the reservoir with fixed geometry. Our method, which is based on a set point control algorithm, explores the coupled impact of the fracture geometry and β Forchheimer coefficient.