  We propose a one-dimensional Saint-Venant (open channel) model overland flows
including a water input--output source term modelling recharge via rainfall and
infiltration (or exfiltration). We derive the model via asymptotic reduction
from the two-dimensional Navier--Stokes equations under the shallow water
assumption, with boundary conditions including recharge via ground infiltration
and runoff. This new model recovers existing models as sepcial cases, and adds
more scope by adding a water-mixing friction terms that depends on the rate of
water recharge. We prospose a novel entropy function and its flux, that are
useful in validating the model's conservation or dissipation properties. Based
on this entropy function we propose a finite volume scheme extending a class of
kinetic schemes and provide numerical comparisons with respect to the newly
introduced mixing friction coefficient. We also provide a comparison with
experimental data.
