Analytical Model of Surface Flow on Hillslopes Based on the Zero Inertia Equations

Coming from the zero inertia (ZI) equations, an analytical model to describe sheet flow phenomena with a special focus on rainfall runoff processes is developed. A slight modification of the ZI equations, which draws upon the concept of a momentum-representative cross-section of the moving water body, leads—after comprehensive mathematical calculus—to an analytical solution describing essentially one-dimensional, shallow overland flow. In a test series, the analytical ZI model is applied together with three numerical models, one based on the Saint-Venant equations, one on the kinematic wave equations, and another one on diffusion wave equations. The test application refers to a typical rainfall runoff situation, i.e., rather shallow overland flow on a hillslope as a consequence of excess rainfall. Contrary to the analytical model, the comparative analysis clearly shows the difficulties of the numerical solutions in terms of exactness and robustness when approaching typical shallow water depths. This problem of numerical models is tackled by applying small time and space discretization, which, however, comes along with higher CPU execution times. Besides the good computational efficiency and freedom of any numerical inconvenience, the new analytical model outperforms the numerical models for typical overland flow simulations. This particularly refers to a highly satisfactory fulfillment of the mass balance and a nearly perfect match of peak flow rates.