Tracking residence times in hydrological systems: forward and backward formulations

In recent years, several papers contributed to the development and clarification of key theoretical issues underlying the formulation of transport by travel time distributions in catchments. Such a formulation provides a robust description of the temporal evolution of ages contained in the catchment storage and sampled by the output fluxes. In particular, special attention has been devoted to general time-variant dynamics that are likely to occur in complex systems like catchments. In this context, important theoretical and practical implications arise from a proper distinction between backward and forward age distributions, which are based on the definition of diverse reference variables. The 'age' of a water particle represents the time elapsed since a previous injection, and as such, it is intrinsically a backward time concept. A forward approach, instead, requires the introduction of the particle's 'life expectancy', which quantifies the time a water particle will spend within the system before being sampled by one of the outflows (e.g. stream discharge or evapotranspiration). The sum of age and life expectancy is the particle's travel time. Despite forward and backward approaches being different, and that they only coincide in the special case of stationary systems, a proper distinction of these formulations has been sometimes overlooked in the literature. In this contribution, we review recent backward formulations using a unified notation and discuss a novel forward formulation. This paper illustrates how age and life-expectancy distributions naturally evolve in response to unsteady hydrologic fluxes and presents numerical applications relevant to catchment-scale solute circulation. In both forward and backward formulations, the mixing of ages, which is modelled through age-selection functions, plays a central role in describing the fate of solutes introduced in the system and measured at the system outlets. Copyright (C) 2015 John Wiley & Sons, Ltd.

[1]  E. B. Nauman,et al.  Residence time distribution theory for unsteady stirred tank reactors , 1969 .

[2]  P. Perrochet,et al.  Groundwater age, life expectancy and transit time distributions in advective–dispersive systems: 1. Generalized reservoir theory , 2006, 1108.5483.

[3]  J. McDonnell,et al.  The future of applied tracers in hydrogeology , 2005 .

[4]  Heather Wickham,et al.  High-frequency water quality time series in precipitation and streamflow: from fragmentary signals to scientific challenge. , 2012, The Science of the total environment.

[5]  Doerthe Tetzlaff,et al.  What can flux tracking teach us about water age distribution patterns and their temporal dynamics , 2012 .

[6]  G. Botter.  Catchment mixing processes and travel time distributions , 2012 .

[7]  P. Maloszewski,et al.  A comparative modeling study of a dual tracer experiment in a large lysimeter under atmospheric conditions. , 2009 .

[8]  R. Uijlenhoet,et al.  Quantifying catchment‐scale mixing and its effect on time‐varying travel time distributions , 2012 .

[9]  Ciaran J. Harman,et al.  Time‐variable transit time distributions and transport: Theory and application to storage‐dependent transport of chloride in a watershed , 2015 .

[10]  Minseok Kim,et al.  An efficient tracer test for time‐variable transit time distributions in periodic hydrodynamic systems , 2014 .

[11]  Vladimir Cvetkovic,et al.  Water and solute transport along hydrological pathways , 2012 .

[12]  P. V. Danckwerts.  Continuous flow systems. Distribution of residence times , 1995 .

[13]  Enrico Bertuzzo,et al.  Transport in the hydrologic response: Travel time distributions, soil moisture dynamics, and the old water paradox , 2010 .

[14]  Kevin Bishop,et al.  Transit Times for Water in a Small Till Catchment from a Step Shift in the Oxygen 18 Content of the Water Input , 1996 .

[15]  Chantal Gascuel-Odoux,et al.  Solute transport dynamics in small, shallow groundwater-dominated agricultural catchments: insights from a high-frequency, multisolute 10 yr-long monitoring study , 2013 .

[16]  A. J. Niemi,et al.  Residence time distributions of variable flow processes , 1977 .

[17]  Malcolm R. Mackley,et al.  Experimental residence time distribution measurements for unsteady flow in baffled tubes , 1989 .

[18]  A. Rinaldo,et al.  Transport of fluorobenzoate tracers in a vegetated hydrologic control volume: 2. Theoretical inferences and modeling , 2015 .

[19]  Karsten Høgh Jensen,et al.  Multiple Tracing Experiments in Unsaturated Fractured Clayey Till , 2001 .

[20]  Michael S. K. Chen.  The theory of micromixing for unsteady state flow reactors , 1971 .

[21]  A. Rinaldo,et al.  Transport of fluorobenzoate tracers in a vegetated hydrologic control volume: 1. Experimental results , 2015 .

[22]  Jeffrey J. McDonnell,et al.  A look inside ‘black box’ hydrograph separation models: a study at the Hydrohill catchment , 2001 .

[23]  Transit time evaluation using a chloride concentration input step shift after forest cutting in a Japanese headwater catchment , 2009 .

[24]  J. Kirchner.  Catchments as simple dynamical systems: Catchment characterization, rainfall‐runoff modeling, and doing hydrology backward , 2009 .

[25]  Marcel G. Schaap,et al.  The Landscape Evolution Observatory: a large-scale controllable infrastructure to study coupled Earth-surface processes , 2015 .

[26]  Enrico Bertuzzo,et al.  Catchment residence and travel time distributions: The master equation , 2011 .

[27]  Sjoerd E. A. T. M. van der Zee,et al.  Chloride circulation in a lowland catchment and the formulation of transport by travel time distributions , 2013 .

[28]  A. Rodhe,et al.  Consequences of mixing assumptions for time‐variable travel time distributions , 2015 .

[29]  A. Rinaldo,et al.  Kinematics of age mixing in advection‐dispersion models , 2013 .