Olivier Brivois1, 2, Stéphane Bonelli1, 2, Roland Borghi1
1 LMA (UPR 7051 CNRS) 31, Chemin Joseph Aiguier - 13402 Marseille
2 CEMAGREF Le Tholonet BP 31 - 13612 Aix-en-Provence
Modelling of erosion and entrainment by flow of water over an earth dam

The breaking of dams built with rocks and natural materials, when there is an unexpected flooding, is a relatively classical event and the prediction of the delay to the complete breaking is important. The erosion of the slopes by the water charged with solid matter is an important process in these conditions, and a "macroscopic modelling" of the system composed of the two-phase flow, the wet soil and the rough surface is an interesting problem, at the interface of fluid and solid mechanics. Some models devoted to debris flows with St-Venant type equations have been proposed, but they include very empirical formulae for the friction with the ground, and it is very difficult to generalise this approach including erosion and injection of ground matter in the flow because it is to closely linked with this friction. On the other hand, experimental works have been devoted to erosion problems in the bed of rivers, by means of laboratory devices, and some empirical formulae have been found. But the experimental conditions are usually related to too small slopes, and the results are given under the form of global formula that cannot be directly applied to a given dam.

The purpose of the study here is to propose a set of equations, where the fluid and solid media are considered as continuous media in some statistically averaged sense. More or less elaborated models can be used for both media in this approach, but here we consider simply a non-deformable solid medium, with some water inside, and a two-phase mean turbulent fluid flow in which turbulent stresses and turbulent diffusion are playing. The flow along the ground surface is represented by the well known boundary layer equations, with the possibility of mass injection from the ground. The complete solution of this problem needs the prescription of a "local erosion criterion" that is able to give, after complete coupling with the flow field, the local eroded mass flow rate. Different criterions are possible depending on the nature of the processes, but here we consider a purely mechanical process where the ground adapts itself in such a way that the shear stress that it experiences is a given critical characteristic shear stress.

This very simple model is not difficult to be solved, with a very simple numerical code, in a quasi-steady state situation with respect to the surface of the ground. The findings are that the local eroded mass flow rate is very dependant of the upstream velocity profile in the water layer, and, for a given flow and ground material, it is possible to follow the downstream decrease of this local erosion rate until it stops. It is possible also to extract from these detailed calculations the global eroded mass flow rate, as function of some nondimensional parameters similar to the Shield number, Froude number...These global curves, in restricted cases, resembles the empirical formulae extracted from experiments.

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