Kinematic wave solutions for dam-break floods in non-uniform valleys

In non-uniform valleys, dam-break flood waves can be significantly affected by downstream variations in river width, slope, and roughness. To model these effects, we derive new analytical solutions to the kinematic wave equation, applicable to rating curves in the power law form and hydrographs of generic shape as long as they produce a single shock at the wave front. New results are first obtained for uniform channels, using the Gauss-Green theorem applied to characteristic-bounded regions of the (x, t) plane. The results are then extended to non-uniform valleys, using a change of variable that homogenizes river properties by rescaling the distance coordinate. The solutions are illustrated and validated for idealized cases, then applied to three historical dam failures: the 2008 breaching failure of the Tangjiashan landslide dam, the 1976 piping failure of Teton Dam, and the 1959 sudden failure of Malpasset Dam. In spite of the much reduced computational cost and data requirements, the results agree well with the field data and with more elaborate simulations. They also clarify how both river and hydrograph properties affect flood propagation and attenuation.