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We propose a one-dimensional Saint-Venant (open channel) model overland flowsincluding a water input--output source term modelling recharge via rainfall andinfiltration (or exfiltration). We derive the model via asymptotic reductionfrom the two-dimensional Navier--Stokes equations under the shallow waterassumption, with boundary conditions including recharge via ground infiltrationand runoff. This new model recovers existing models as sepcial cases, and addsmore scope by adding a water-mixing friction terms that depends on the rate ofwater recharge. We prospose a novel entropy function and its flux, that areuseful in validating the model's conservation or dissipation properties. Basedon this entropy function we propose a finite volume scheme extending a class ofkinetic schemes and provide numerical comparisons with respect to the newlyintroduced mixing friction coefficient. We also provide a comparison withexperimental data.

A Saint-Venant shallow water model for overland flows with precipitation and recharge