Un modèle débit-durée-fréquence pour caractériser le régime d'étiage d'un bassin versant A discharge-duration-frequency model adapted for low flows

Résumé

La méthodologie débit-durée-fréquence (QdF), appliquée ces dernières années aux étiages, a permis de définir quatre modèles types recouvrant l'ensemble des rivières étudiées. L'identification de la typologie du site étudié et l'estimation de deux descripteurs hydrologiques locaux suffisent au modèle, dit de référence, pour en déduire les courbes QdF (1j ≤ d ≤ 30j) en débit moyen minimum sur l'année (VCNd) ou débit seuil minimum annuel (QCNd) non dépassé sur ces mêmes durées. S'il est relativement aisé de définir les descripteurs hydrologiques, il est plus difficile d'identifier, sans observation de débit, le modèle à prendre en compte. En reconsidérant avec plus de rationalité la démarche d'identification des typologies, et en particulier les distributions multidurées relatives à chaque bassin, il est apparu possible d'évoluer vers un modèle unique pouvant être calé en chaque site observé. Ce nouveau concept de modélisation repose sur la propriété d'affinité des distributions, relatives aux échantillons de valeurs de durées d. Par souci de continuité avec l'approche QdF à référence typologique, la loi statistique log-normale à deux paramètres a été choisie. Le modèle, dont la conceptualisation est indépendante de la loi statistique choisie, aura dans le cas de la loi log-normale trois paramètres à ajuster sur les VCNd échantillonnés. Trente six sous bassins du bassin hydrographique de la Moselle ont été étudiés. Le modèle développé pour les débit moyens VCNd peut être appliqué aux débits seuils QCNd en conservant le même jeu de paramètres, grâce à une relation observée entre débits moyens et débits seuils. Cette nouvelle modélisation rationalise l'approche antérieure basée sur la typologie d'écoulement de basses eaux des bassins versants.

Mots-clés:

• Débits d'étiage,
• hydrologie statistique,
• modélisation de synthèse (débit-durée-fréquence)

Abstract

The flow-duration-frequency (QdF) concept, as applied in recent years to low flows, has made it possible to establish four reference models (GALEA et al., 1999a), corresponding to four typologies. The hydrological variables concerned are the minimum mean discharge of the year defined for various continuous durations d (1day ≤ d ≤ 30day), called VCNd, and the annual minimum threshold discharge not exceeded over these same durations, called QCNd, according to OBERLIN (1992). These QdF models allow a description of the temporal variability of low flows observed for a river basin, from a statistical point of view. The typology of the basin and two local hydrological descriptors have to be known. For ungauged basins, these two descriptors (GALEA et al., 1999b) are well estimated by various methods, such as multivariate analysis relating to the physiographic characteristics of the basin. Nevertheless, the choice of the reference model still remains contentious.

By reconsidering in a more rational manner the step of identification of typologies, and in particular the discharge distributions (for durations d) relating to each basin, it appeared interesting to establish a local model. This new model has a simpler formulation, thanks to a scale invariance assumption. This research (CHAPUT, 1999) was undertaken on 36 sub-basins of the Mosel basin. In order to ensure continuity with the earlier QdF models described above, the two-parameter log-normal law was chosen and adjusted on the distribution of mean discharges. The scale invariance assumption is deduced from the observed parallelism of distributions related to different durations, when discharges are represented in a logarithm scale. This observation means that all of the distributions can be translated to a common point, in order to obtain one "consolidated" distribution, independent on the considered duration. This parallelism has been observed on many basins, and seems to be a realistic assumption. Furthermore, these observations have been made on samples, and do not depend on the choice of statistical law. The methodology described in this paper makes it possible to adjust the local QdF model on sampled discharges. Only three parameters have to be determined: s^c, the "consolidated" standard deviation, ∆e the low flow characteristic duration and VCN(2,1), which represents the quantile of the one-day distribution, with the two-year return period (F=0.5).

This model is also useful for the determination of threshold discharges (QCNd). An observed property gives a relation between the VCN and QCN quantiles, for a fixed return period, considering different durations d: VCN quantiles can been deduced from QCN quantiles by integrating them, according to d. Consequently, the analytical formulation of the VCN model can be derived according to d, in order to obtain a QCN model. This model has the same three parameters s^c, ∆e and VCN(2,1) described above. The comparison between QCN quantiles adjusted on samples and QCN quantiles deduced from the VCN model by derivation shows good results.

As a conclusion, this new modelling approach unifies the typological approach for both mean discharges and threshold discharges. It is based on a local adjustment and avoids having to choose between one of the four former reference models. This local model opens up perspectives for a regional model, as it has been done for floods, for example by the Group of Research in Statistical Hydrology (1996). This will make it possible to estimate the low flow regime on an ungauged basin.

Keywords:

• Low flow,
• statistical hydrology,
• synthetic models (discharge-duration-frequency)