Stochastic Modeling of Anomalous Water Transport

This thesis investigates the application of stochastic models to anomalous water transport phenomena in heterogeneous and complex environments, highlighting their effectiveness to realistically represent water movement through heterogeneous media. Indeed, classical diffusion models do not always capture complex dynamics that arise in such environments, where mechanisms such as trapping, memory effects, and spatial variability give rise to anomalous diffusion. Stochastic approaches, including continuous time random walks and Lévy walks, provide suitable models to describe such anomalous transport. The thesis is structured into four chapters. The first two chapters provide the theoretical framework, covering advection-dispersion phenomena, subsurface flow equations, transit time distribution in catchments, and the use of natural tracers to estimate the young water fraction. An overview of stochastic models for anomalous diffusion is also provided, including continuous time random walks, Lévy walks, and semi-Markov models. The third chapter presents a study on subsurface water flow using a random velocity process with rests, where water particles alternate between moving times and waiting times. Simulations of particle movement provide the empirical transit time distribution and allow the estimation of the threshold ages of the young water fraction using stable water isotope concentrations for 22 Swiss catchments. The model has only one parameter of the Mittag-Leffler distribution for waiting times to be estimated. The fourth chapter focuses on soil moisture modeling through a spacedependent continuous time random walk with Mittag-Leffler waiting times. Applied to daily soil moisture measurements, the model captures vertical water movement across soil layers during dry spells, and the three parameters are estimated based on the observed data. The relationships between model parameters, initial soil moisture and potential evapotranspiration enable accurate prediction of soil moisture and validation of the model