  In this work, we analyze the flow filtration process of slightly compressible
fluids in porous media containing man made fractures with complex geometries.
We model the coupled fracture-porous media system where the linear Darcy flow
is considered in porous media and the nonlinear Forchheimer equation is used
inside the fracture. We develop a model to examine the flow inside fractures
with complex geometries and variable thickness, on a Riemannian manifold. The
fracture is represented as the normal variation of a surface immersed in
$\mathbb{R}^3$. Using operators of Laplace Beltrami type and geometric
identities, we model an equation that describes the flow in the fracture. A
reduced model is obtained as a low dimensional BVP. We then couple the model
with the porous media. Theoretical and numerical analysis have been performed
to compare the solutions between the original geometric model and the reduced
model in reservoirs containing fractures with complex geometries. We prove that
the two solutions are close, and therefore, the reduced model can be
effectively used in large scale simulators for long and thin fractures with
complicated geometry.
