Prédétermination des débits maximaux de crue par simulation Monte-Carlo de la pluie nette Predetermination of maximum discharge runoff by Monte Carlo simulations of the effective rainfall

Résumé

Une méthodologie a été proposée pour la prédétermination des débits à partir de la pluviométrie. Il s’agit d’associer, en entrée, les simulations Monte Carlo de hyétogrammes de pluie nette avec un modèle d’hydrogramme unitaire à base de géomorphologie pour obtenir des hydrogrammes simulés en sortie. À la fin des simulations, l’analyse statistique des sorties permet d’apprécier l’étendue de la variabilité de la réponse du bassin aux événements pluvieux et de caractériser les débits et les temps de pointe. Un total de 44 simulations ont été menées pour chacun des 15 événements observés pour un petit bassin versant situé en Tunisie centrale. L’analyse des hydrogrammes générés a montré une assez faible dispersion des débits de pointe, d’une simulation à l’autre, pour un événement donné, et a mis en évidence la dissymétrie des distributions des débits et temps de pointe. L’exploitation des résultats de l’ensemble des simulations permet de dégager des relations empiriques caractérisant le comportement du bassin selon les débits de pointe, les temps de pointe, les temps de base et les volumes écoulés.

Mots-clés:

• Hydrogramme unitaire,
• géomorphologie,
• pluie nette,
• simulations de Monte Carlo,
• indice d’infiltration

Summary

The predetermination of peak discharges and flood volumes of ungauged basins is an important aspect of the management of surface waters, protection against floods, water supply, etc. In this study, a method is proposed for the predetermination of discharges from rainfall data. The method associates effective rainfall obtained from Monte Carlo Simulations (MCS) with a unit hydrograph based on geomorphology. The unit hydrograph (UH) based on geomorphology is selected knowing that the parameters can be obtained from topographic charts, soil charts and ground occupation charts, as well as from soil data. The UH used was produced from the Nash cascade model in which the scale and shape parameters were taken from the literature. These parameters depend on the hydrographical network, the Horton ratios and the average peak flow velocity, which is assumed to be constant throughout the network and with respect to time. The average peak flow velocity can be expressed as a function of 1) geomorphologic parameters such as the total surface area of the basin, the slope of the highest order stream, the Manning-Strickler coefficient, the width of the channel, the kinematic wave parameter of the highest order stream and the length of the main channel, and 2) the effective rainfall intensity and duration.

With respect to effective rainfall intensities, the idea is to consider the effective rainfall as a vector of the parameters of the hydrological model, and then to use the MCS method to generate the corresponding components. The proposed simulation framework includes: 1) the specification of the data for which the geomorphologic parameters and the time increments are fixed for all simulations, whereas the duration of the total rainfall and the effective rainfall volume vary from one event to another, and constitute constraints determining whether or not simulations should be rejected, 2) the random drawing of effective rainfall intensities and durations, 3) the computation of resulting hydrographs and 4) the analysis of the simulated hydrographs, where the hydrographs are first simulated for each event and then simulated in their entirety to highlight indicators to characterize outputs.

In order to statistically interpret the simulated hydrographs, the generated peak discharges were classified for each event, and their 25^th, 50^th and 75^th percentiles were analyzed. The same treatment was applied to the simulated times to attain peak values. The use of the 25^th and 75^th percentiles makes it possible to evaluate the extent of the 50% interval of the simulated discharges, whereas the median and the mode make it possible to position values representative of the distribution of the generated discharges. The hydrographs are assumed have the same “recurrence” as their peak discharge. Hydrograph generation by the MCS method is a two step process: 1) the generation of effective rainfall intensities based on the assumption that the total volume is observed, and 2) the convolution of the unit hydrograph resulting from each interval of effective rainfall.

The study site, Saddine1, is a small catchment with a surface area of 384 hectares. It is located adjacent to Makthar in Tunisia (northern latitude 35°48’06’’ and longitude 9°04’ 09’’) in a mountainous zone. The catchment is controlled by a small headwater dam and was monitored from 1992 to 1999. Observed over periods of five minutes, the maximum rainfall intensity was 324 mm/h and the minimal intensity was 10 mm/h. The maximum total rainfall recorded for an event was of 106 mm. The longest duration for an event was of approximately 5 hours (299 min) and shortest was 12 minutes. A great disparity in the volumes was also noted: the maximum volume observed was 67,200 m³ whereas the minimum was 1,275 m³. The peak discharges of the recorded hydrographs were very variable with a minimum/maximum ratio of about 1/1370. Indeed the maximum discharge observed was 85.6 m³/s, and the minimum discharge only 0.062 m³/s. The time to attain peak flows for the rainfall events varied from 10 to 120 minutes. The effective rainfall intensities were calculated using the infiltration index method, ϕ, which remains a method still largely used in spite of its rudimentary character. The effective rainfalls estimated for each event varied from 0.3 mm with 17.5 mm.

Before using the MCS, the model was calibrated. The results of the calibration analysis showed that the calculated hydrographs were reasonable comparable to the observed hydrographs. In addition to the shape, the peak discharge and the peak time reconstitutions were satisfactory. A total of 44 simulations were carried out for each of the 15 events observed, of which 13 allowed for the identification of the distributions of effective rainfall intensities and durations. The remaining two events were used for the validation of the approach. The analysis of the generated hydrographs showed a rather weak dispersion of the peak output from one simulation to another, for a given event. Moreover, the discharges and times to attain peak discharge resulting from the generated hydrographs followed a dissymmetrical distribution. The observed values of the peak discharges and times to attain peak discharge represent realisations of output simulations with different probabilities of occurrence. In order to capitalize on the model, relationships between simulated peak discharges, times to peak discharge, base times and volumes were constructed.

Key words:

• Unit Hydrograph,
• geomorphology,
• effective rainfall,
• Monte Carlo simulations,
• infiltration index