On the relations between the hydrological dynamical systems of water budget, travel time, response time and tracer concentrations

Abstract

Using previous results on extended Petri Nets (EPN), we present the relations between various hydrological dynamical systems (HDSys) derived from the water budget. Once the water budget has been implemented, there is a consistent way of getting the equations for backward travel time distributions, for forward response time distributions and for the concentration of a solute or tracer. We show that the water budget has a correspondence of one to many with the backward travel time distributions. In fact, to any one of the water budget equations there correspond as many equations as there are input precipitation events. The backward travel time distributions are related to the response time distributions by Niemi's relationship and, if there are n outputs, by the definition of n − 1 partition functions. These determine what fraction of the water volume injected into the control volume at a specific time t in goes asymptotically to a specific output. Given the backward travel time distributions, the output concentration of a solute or tracer also depends on the input concentration. The paper clarifies the complicated relations described above by taking [Hydrology and Earth System Sciences, 20, 299-328] as an example from literature. Once the appropriate information is presented, it is shown how these HDSys can be solved simultaneously without duplicating calculations. Then, it is also shown that, under the hypothesis of uniform mixing of water ages within each reservoir, these systems can be solved exactly.

Transit Time Estimation in Catchments: Recent Developments and Future Directions

“…There have been reviews of how transit times form a link between hydrology and water quality (Hrachowitz et al., 2016), on the use of tracer data to improve rainfall‐runoff models (Birkel & Soulsby, 2015), and on multi‐tracer inference and its implications for transit time estimation (Abbott et al., 2016). Many papers reviewed the mathematics and theory behind water age concepts (e.g., Benettin, Rinaldo, & Botter, 2015; Calabrese & Porporato, 2017; Rigon & Bancheri, 2021; Rigon et al., 2016), and computed time‐variant transit time distributions (TTDs) from distributed hydrological models (Engdahl et al., 2016). Finally, the most recent review has been on the demographics of water age in the different compartments of the critical zone (Sprenger et al., 2019).…”

Using water age to explore hydrological processes in contrasting environments

“…The studies by Rodriguez et al (2020), Zarlenga and Fiori (2020) and Rigon and Bancheri (2021) provide nice theoretical advances in the characterization of water age distributions in rivers, and analyse the linkages between water ages, catchment structure and climatic forcing. In particular, Rodriguez et al (2020) explore the origin and the implications of multimodal age distributions of catchment-scale discharge.…”

“…The study shows that the unsteady nature of rainfall and climate is the primary driver of water ages in hillslopes, while the relationship between the age structure of the storage and that of the outflow depends on key topographic properties such as the bedrock slope and the hillslope shape. The relationship between age distributions of different components of a complex hydrological system is also the subject of the work contributed by Rigon and Bancheri (2021), in which the relationships among the internal structure of a complex hydrological systems and the ensuing age/response time distributions are studied starting from simple water budget equations which encapsulate the underlying catchment structure. The paper offers a framework to couple flow and transport models in a coherent manner, combining analytical results and a set of practical examples.…”

Consistent Modeling of Transport Processes and Travel Times—Coupling Soil Hydrologic Processes With StorAge Selection Functions

“…However, they also simplify the transient processes of solute transport in soils by merging spatially explicit processes into single statistical descriptors. In recent years, travel‐time‐based transport models using StorAge selection (SAS) functions (Rinaldo et al., 2015) have particularly emerged in solute transport studies (e.g., Asadollahi et al., 2020; Asadollahi et al., 2022; Kumar et al., 2020; Nguyen et al., 2022; Rigon & Bancheri, 2021). SAS models rely on time series of storage states (e.g., soil water content) and fluxes (e.g., streamflow) and represent underlying hydrologic processes in a lumped way.…”