The overland flow equation for constant rainfall excess: an evaluation of runoff volume and time to equilibrium

In this paper we study the runoff volume V(t) per unit area generated by the overland flow in the time interval [0,t], under a stationary rainfall r and initially dry conditions. For all positive values of the rating exponent m we express V(t) in closed form in terms of the solution q(t) of the overland flow equation. We define a simpler, approximated value of V(t) and show that, for m≥1, the error is smaller than a quantity of the form c e^{-pt}. Finally, for 1≤m≤3, we find an explicit upper bound for the time to equilibrium t_e, by showing that q(t) differs from the equilibrium outflow r by a quantity of the form c e^{-pt}.