Stochastic model for subsurface water flow in Swiss catchments

Abstract

Understanding water movement in catchments subsurface is crucial for numerous applications such as pollutant contamination, nutrient loss, water resource management and ecosystem functioning. Among the variables of particular interest, the transit times of water particles and their statistical distribution are a desirable output. Nevertheless, past approaches assume explicitly the form of the transit time distribution (TTD) to provide information on water age in catchments. In this study we adopt a different approach by making assumptions on the movement of water particles in the subsurface instead of assumptions on the transit time distribution. Hence we propose a model based on a random velocity process with rests, where a water particle alternatively moves with a constant velocity or it is trapped (with zero velocity) until it reaches the outlet of the catchment. We assume that the moving times are i.i.d. (independent and identically distributed) random variables with exponential distribution, while waiting times, i.e., times in which the water particle is trapped in subsurface cavities, are assumed to be i.i.d. random variables with Mittag-Leffler distribution of order $\alpha$, which is heavy tailed. At the catchment outlet, which is assumed here to be at a distance from the inlet equal to the catchment median flow path length $L$, the first passage time (or transit time) of the water particles is measured.

We applied the model to 22 Swiss catchments simulating, for each catchment, the movement of millions of water particles thus obtaining the corresponding empirical TTD. We search for the threshold age (${\tau }^{\star }$) that closely approximates the portion of the empirical TTD younger than ${\tau }^{\star }$, that is the young water fraction ($F_{yw}$). We use the complex modulus of the empirical characteristic function of the TTD: this quantity represents, in our model, the amplitude ratio of seasonal isotope cycles in stream water and precipitation. Our results reveal that ${\tau }^{\star }$ is comprised between 46 and 76 days, exactly in the range 2-3 months previously identified. Additionally, given the amplitude ratio of isotopic concentrations, we estimate the only parameter of the model, that is the $\alpha$ parameter of the Mittag-Leffler distribution, for each Swiss catchment using suitable catchments properties. In conclusion, our study reveals that random velocity processes with rests are stochastic transport processes useful for modeling water movement in heterogeneous catchments, with a limited number of assumptions.