  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 BVP which is
coupled with an equation 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
Forchhemer flows in the fracture: isotropic and anisotropic, which are
different in their nature.
  Using method of reduction, we developed 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
beta-Forchheimer coefficient.
