The objective of this three-part work is the formulation and rigorous
analysis of a number of reduced mathematical models that are nevertheless
capable of describing the hydrology at the scale of a river basin (i.e.
catchment). Coupled effects of surface and subsurface flows are considered.
In this third part, we focus on the development of analytical solutions and
scaling laws for a benchmark catchment model that models the river flow
(runoff) generated during a single rainfall. We demonstrate that for catchments
characterised by a shallow impenetrable bedrock, the shallow-water
approximation allows a reduction of the governing formulation to a coupled
system of one-dimensional time-dependent equations for the surface and
subsurface flows. Asymptotic analysis is used to derive semi-analytical
solutions of the model. We provide simple asymptotic scaling laws describing
the peak flow formation. These scaling laws can be used as an analytical
benchmark for assessing the validity of other physical, conceptual, or
statistical models of catchments