We propose a one-dimensional Saint-Venant (also known as open channel or shallow water) equation model for overland flows including a water input--output source term. We derive the model from the two-dimensional Navier--Stokes equations under the shallow water assumption, with boundary conditions including recharge via ground infiltration and runoff. We show that the energy-consistency of the resulting Saint-Venant model is strictly dependent upon the assumed level of rain- or recharge-induced friction. The proposed model extends most extant models by adding more scope depending on friction terms that depend on the rate of water entering or exiting the flow via recharge and infiltration. The obtained entropy relation for our model validate it both mathematically and physically. We compare both models computationally based on a kinetic finite volume scheme; in particular, we provide numerical evidence that the two models may show drastically different results, where the model conditioned on the flow velocity provides what may not be the physically relevant solution because of the lack of the appropriate friction terms depending on a master parameter ($\alpha$). We also look at a comparison with real-life data.