We consider a fractured porous medium that is studied at a scale such that the fractures can be modeled individually. Models for flow in which the fractures are interfaces between subdomains are presented. These models take into account interactions between the fractures and the surrounding porous medium. Existence and uniqueness of the solution to the model problem is proved. Error estimates show convergence as O(max(h,d)) in the L2 norm, where h is the mesh size and d the fracture width. Numerical experiments confirm the theorical results.