Dynamical modelling of concentration–age–discharge in watersheds

Abstract

Abstract:There is now a wide literature on the use of tracer age and transit time distributions to diagnose transport in environmental systems. Theories have been proposed using idealized tracer age modelling for ocean ventilation, atmospheric circulation, soil, stream and groundwater flow. Most approaches assume a steady flow regime and stationarity in the concentration (tracer) distribution function for age, although recent work shows that this is not a necessary assumption. In this paper, dynamic model for flow, concentration, and age in volume-averaged and a spatially distributed watershed system are derived in terms of the moments of the underlying distribution function for tracer age, time, and position. Several theoretical and practical issues are presented: (1) The low-order moments of the age distribution function are sufficient to construct a dynamical system for the mean age and concentration under steady or transient flow conditions. (2) Solutions to the coupled system of equations for flow, concentration and age show that 'age' of solutes stored within the watershed or leaving the watershed is a dynamic process which depends on flow variations as well as the solute or tracer dynamics. (3) Intermittency of wetting and drying cycles leads to an apparent increase in the tracer age in proportional to the duration of the 'dry' phase. (4) The question of how mobile/immobile flow may affect the age of solutes is examined by including a low permeable, passive store that relaxes the well-mixed assumption. (5). A spatially distributed advective and dispersive transport solution for age evolution over a simple 1-D hillslope is developed to demonstrate the age theory for a distributed source of water and tracer, and the solution is shown to have very similar input-output behaviour when compared to the volume-average model for comparable parameters.

Spatially distributed characterization of soil-moisture dynamics using travel-time distributions

“…Such behavior is often assumed to be valid for TTDs in general such that they are consequently modeled using exponential or gamma distributions (Małloszewski and Zuber, 1982). Recent works, however, have questioned this generalization by emphasizing the time-dependent nature of TTDs (Duffy, 2010;Botter et al, 2011). The examples given in Fig.…”

“…As a result, such studies might fail to find long-term trends, establish connections between traveltime behavior and specific catchment properties or investigate the impact of certain hydraulic regimes that only occur rarely (e.g., extreme drought or storm events). The second category are theoretical studies that either use a very simplified computational model to focus on specific questions (Rinaldo et al, 2006;Duffy, 2010;Botter et al, 2010;van der Velde et al, 2012;Benettin et al, 2015a;Porporato and Calabrese, 2015) or employ more realistic hydrological models that provide a large data set typically not available in realworld sites (Sayama and McDonnell, 2009;Fenicia et al, 2010;McMillan et al, 2012). Such theoretical studies allow a more thorough and detailed analysis of the involved processes and their interdependence may suffer from an oversimplified model setup for influx and outflux generation.…”

Residence time distributions for hydrologic systems: Mechanistic foundations and steady-state analytical solutions

“…This paper focuses on steady-state solutions of RTD. While some of the limitations related to the steady-state assumption are discussed in section 5, a companion paper fully addresses the topic of transient RTDs which have received more attention in the recent years (Duffy, 2010;McDonnell and Beven, 2014;Rinaldo et al, 2015). The article is organized into three main sections, each of which is designed to be useful on its own.…”

“…The majority of techniques that can be applied to solute transport of a conserved mass can be applied to residence time, so there are many more possible approaches for (Duffy, 2010). It is important to note that these variations can be significant, therefore altering the physical interpretations of RTDs if the solution is based on the steady-state assumption (Schwartz et al, 2010).…”

The master transit time distribution of variable flow systems

“…Several numerical models have been proposed to capture the time-varying nature of the transit time of water [Dunn et al, 2007;McGuire et al, 2007;Sayama and McDonnell, 2009;Duffy, 2010]. Such models have two components: (1) a component that estimates the incoming effective rainfall and (2) a component that keeps track of the fate of these waters by determining the surface and subsurface flow paths and associated residence times in different storages.…”