Analyses et simulations multifractales pour une meilleure gestion des eaux pluviales en milieu urbain et péri-urbain

摘要

The Universal Multifractals (UM) are a powerful tool which has been extensively used to analyze and simulate geophysical fields, such as rainfall, that are extremely variable over wide range of scales. It is based on the concept of cascade phenomenology that relies on the physical notion of scale invariance to explore the fundamental phenomenon of intermittency. In this framework the whole variability of a field is characterized with the help of only three parameters that are furthermore physically meaningful. In this PhD thesis we use this theoretical framework to quantify the impacts of small scale rainfall variability in urban hydrology. The first step consists in analysing radar rainfall space-time variability with the help of a simple anisotropic multifractal model. A variety of rainfall events are analyzed. It appears that a scaling behaviour was observed on two distinct ranges of scales separated by a break at roughly 16 km that is discussed. These data sets are in overall agreement with a simple space-time scaling model relying on single anisotropy exponent between space and time. The results hint at a possible universality of the UM parameters for rainfall. This thesis also explores another facet of intermittency, which is particularly important for long time series of precipitation, that of numerous zero rainfall measurements (a pixel or a time step with no recorded rainfall), i. e. long “dry” periods. We revisit the long lasting discussion on the source of this intermittency, e.g. whether it requires a specific modelling. First the effects of a threshold on a universal multifractal field are investigated and second a toy model that introduces some zeros within the cascade process conditioned by the field value is developed. This enables to explain most of the observed behaviour, e.g. the difference between event statistics and overall statistics. The impact of rainfall variability is investigated through the analysis of the sensitivity to the rainfall input of urban hydrologic-hydraulic models. Two predominantly urban catchments (a 3 400 ha one in Seine-Saint-Denis near Paris, and a 900 ha one in London) modelled with the help of operational semi-distributed models are used as case studies. The fully distributed model Multi-Hydro (under development at LEESU) is also tested on a 147 ha portion of the Paris case study. First the impact of unmeasured small scale rainfall variability (i.e. occurring at scales smaller than 1 km in space and 5 min in time which are available with C-band radar data) is evaluated. This is achieved by generating an ensemble of realistic downscaled rainfall fields by continuing the stochastic cascade process below the observation scale and then simulating the corresponding ensemble of hydrographs. It appears that the small scale rainfall variability generates significant hydrological variability that should not be neglected. Furthermore the Multi-Hydro model generates a larger variability not only during the peak flow, but during the whole event, i.e. for moderate rain rates. These findings highlight the need to implement X-band radars (whose resolution is hectometric) in urban areas. In a second part multifractal tools are used on both rainfall and simulated discharges that also exhibit a scaling behaviour. It appears that the rainfall drainage system basically transmits the rainfall variability without damping it, at least in terms of multifractal statistics