We are interested in modeling and simulating surface runoff in urban areas during extreme rainfall events. We propose an extended Saint-Venant system which incorporates an additional rain term on the surface of the water with wind effects and infiltration of the water into the ground below. This model is based on the depth-averaged thin layer approximation of shallow water flows over non flat ”beds” and furthermore takes into account a possibly variable friction law depending on the soil material. We then propose an exact solution of the Riemann problem in a topography-rain linked coordinate system for all the possible Riemann problems. Next, we propose several numerical experiments that we compare to the kinetic scheme with reflections.