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

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 tin 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.