L'approche débit-durée-fréquence : historique et avancées The flow-duration-frequency approach : former and new developments

Résumé

La prévention du risque d'inondation nécessite une connaissance détaillée du régime hydrologique en crue du bassin étudié. Dans ce but, l'approche débit-durée-fréquence (QdF), développée depuis déjà plusieurs années, a permis de définir un modèle statistique décrivant les crues observées en fonction de leur débit, de leur durée et de leur fréquence. Un récent travail a revisité cette approche. Grâce à son nombre réduit de paramètres, le modèle proposé, appelé modèle local convergent, peut être facilement ajusté pour chaque bassin. Dans l'ancienne approche, que nous appelons approche " bassin de référence ", l'ajustement local avait été effectué sur seulement trois bassins, dits de référence, et réputés être chacun représentatif d'une typologie d'écoulement différente. Ces trois paramétrisations types ont ensuite donné lieu à trois modèles adimensionnels, capables de caractériser la majorité des régimes observés. Le modèle adimensionnel correspondant au régime du bassin étudié devait être dénormé par deux caractéristiques locales du bassin : le débit instantané maximal de crue décennale et une durée caractéristique de crue. Une comparaison du nouveau modèle, appelé modèle local convergent, et de l'approche type " bassin de référence " a été effectuée sur une cinquantaine de bassins jaugés. Elle met en évidence la robustesse du modèle convergent et permet de discuter du choix du modèle relatif à l'approche " bassin de référence ". Le modèle local convergent autorise d'envisager le développement d'un modèle QdF régional, s'inspirant de différentes méthodes de régionalisation. Ceci permettra alors une application à des bassins peu ou non observés.

Mots-clés:

• Analyse fréquentielle,
• crue,
• approche débit-durée-fréquence,
• modèle régional,
• gradex

Abstract

Flood risk mitigation requires a good knowledge of hydrological flood regime, which can be described by a flow-duration-frequency (QdF) approach. New developments of this approach are presented and compared to the former method.

Usually, flood frequency analysis deals only with the maximum flood peak distribution or the maximum daily discharge distribution. The QdF approach analyses maximum average flows over different durations d (d =1, 3, …, N days). Similar to intensity-duration-frequency curves, each of the QdF curves represents the flood frequency distribution, for the duration d. QdF modelling aims to express QdF curves by a Q(d,T) function (d : the duration; T : the return period).

Before this present work, QdF modelling was associated with the "reference basin" approach. In this approach, QdF curves (plotted as a function of d, for fixed T) of many studied basins are converted into a dimensionless form. The two characteristics used are the 10-year peak flood, Q(d=0, T=10 years), and a characteristic flood duration (D) of the studied catchment, calculated from different flood hydrographs. Then, three different families are determined, grouping basins with similar dimensionless QdF curves. For each of these families, one reference basin is chosen. Their dimensionless curves are parameterised, in order to obtain a continuous formulation, as a function on T and d. By denormalising one of these dimensionless QdF models with the local parameters Q(0,10) and D, it is possible to obtain the continuous Q(d,T) formulation for the studied basin. The choice of the correct dimensionless model is made via a choice criterion. It involves Q(0,10), D and shape parameters of local maximal rainfall distributions (a Gumbel law is assumed), for different durations, d. These distributions are obtained according to the intensity-duration-frequency approach. If the studied basin is ungauged, local parameters Q(0,10) and D are estimated by regional formulas, involving significant variables such as catchment area and rainfall.

Recent work has improved this "reference basin" approach. A new QdF model, called convergent local, has been developed. For fixed T, the model assumes that the Q(d,T) is described by a hyperbolic form, as a function of d. This choice of the hyperbolic form is based on the observation of many catchments (about one hundred). It has also been observed that QdF curves, plotted for fixed d as a function of T, converge toward the same point, when T decreases. Using these observations as assumptions, the model is then able to calculate Q(d,T) for any return period T and any duration d.

If a two-parameter statistical law (such as the exponential law) is adopted, the model contains only 4 parameters. The first parameter is the limit of Q(d,T), when d tends to infinity. It is estimated by calculating the average value over the entire observed period of the Q(t) discharge time series. The second one gives the hyperbolas curvatures and is ∆. The ∆ parameter has a time dimension and is consequently a characteristic duration of the studied basin. The final two parameters are the location and shape parameters, x₀ (0) and a_q (0), of the exponential maximal flood distribution for d=0. x₀ (0), a_q (0) and ∆ parameters are directly adjusted on observed QdF curves of the studied basin.

The comparison between the convergent local model and the "reference basin" approach has been carried out on about 50 basins, drawn from different regions of France. For each basin, the two approaches have been tested. First, the two characteristic durations D and∆, defined respectively by the "reference basins" approach and the convergent local model, are compared. As mentioned earlier, ∆ characteristic duration is an adjusted parameter and its calculation does not depend on D. In spite of their different definitions, a strong correlation between these two parameters is observed. This shows a good coherence between the two tested approaches. Second, in order to compare results, a relative mean error between calculated and observed values is determined for each basin and each model. Only the observed domain (T ≤ 20 years) has been considered, because the extrapolations cannot been validated with observed data.

Concerning the "reference basin" approach, the three reference basin models are studied, and the choice criterion is applied. Results show that this choice criterion is not relevant. Concerning the convergent local model, the observed mean relative error is lower than in the "reference basin" approach. These good results are confirmed by a very small error dispersion. Consequently, the convergent local model is robust.

As a conclusion, this paper presents new developments of the QdF approach: the convergent local continuous model. This model, locally adjusted, yields very satisfactory results. The next step is to apply it on ungauged basins, as is possible in the "reference basins" approach. This could be done by adapting regional methods, such as the index flood method.

Keywords:

• Lead removal,
• sodium dodecylsulfate,
• Micelles,
• membrane process,
• water treatment,
• Micellar-Enhanced Ultrafiltration,
• acidic media