Chloride circulation in a lowland catchment and the formulation of transport by travel time distributions



[1] Travel times are fundamental catchment descriptors that blend key information about storage, geochemistry, flow pathways and sources of water into a coherent mathematical framework. Here we analyze travel time distributions (TTDs) (and related attributes) estimated on the basis of the extensive hydrochemical information available for the Hupsel Brook lowland catchment in the Netherlands. The relevance of the work is perceived to lie in the general importance of characterizing nonstationary TTDs to capture catchment transport properties, here chloride flux concentrations at the basin outlet. The relative roles of evapotranspiration, water storage dynamics, hydrologic pathways and mass sources/sinks are discussed. Different hydrochemical models are tested and ranked, providing compelling examples of the improved process understanding achieved through coupled calibration of flow and transport processes. The ability of the model to reproduce measured flux concentrations is shown to lie mostly in the description of nonstationarities of TTDs at multiple time scales, including short-term fluctuations induced by soil moisture dynamics in the root zone and long-term seasonal dynamics. Our results prove reliable and suggest, for instance, that drastically reducing fertilization loads for one or more years would not result in significant permanent decreases in average solute concentrations in the Hupsel runoff because of the long memory shown by the system. Through comparison of field and theoretical evidence, our results highlight, unambiguously, the basic transport mechanisms operating in the catchment at hand, with a view to general applications.

1. Introduction

[2] Travel time distributions (TTDs) are key descriptors of catchment-scale transport processes as they provide fundamental information on water storage dynamics, flow pathway heterogeneity, sources of water in space and time within a catchment and on the chemistry of water flows through the outlet. The early formulation of the so-called old-water paradox, first recognized that a notable part of the streamflow released in response to a given rainfall event is supplied by water volumes already in storage within the catchment prior to the event. Since that time, the issue of the age of runoff water has attracted many theoretical and observational studies [Rinaldo and Marani, 1987; Rinaldo et al., 1989; Beven, 2012, 2010; Weiler et al., 2003; Kirchner, 2003] and the spate of papers dedicated to the subject was justified by the true paradigm shift implied by the role of nonevent water.

[3] Multifaceted implications are brought by the emerging need for tracking the time spent by water particles traveling through a catchment, including the identification of flow pathways, catchment storage capacity, water quality and biogeochemistry [McGlynn et al., 2003; Liu et al., 2004; McGuire and McDonnell, 2006; McGuire et al., 2007; McDonnell et al., 2010]. Moreover, the age of water has also been recently introduced as an issue to constrain model identification [Fenicia et al., 2008; Birkel et al., 2010]. While early studies focused on data-driven, parametric identifications of stationary TTDs [Kirchner et al., 2001; Broxton et al., 2009; Tetzlaff et al., 2011], recent developments explicitly incorporated the time variability of climatic conditions [Hrachowitz et al., 2010; Brooks et al., 2010], topographic controls [McGuire et al., 2005] and the spatial heterogeneity of soil properties [Fiori and Russo, 2008; Russo and Fiori, 2009]. More recent developments [Botter et al., 2010, 2011; Rinaldo et al., 2011; van der Velde et al., 2010a, 2012; Harman et al., 2011; Hrachowitz et al., 2013] have focused on the role played by the intermittency of stochastic rainfall and soil moisture dynamics as sources of nonstationarity for TTDs. The last works helped to improve common practices used in travel time distribution analyses, with particular regard to conceptual and practical differences between the travel time distribution conditional on a given injection time and that conditional on a given sampling time at the outlet, jointly with the differences of both with the residence time distributions (RTDs) of water particles in storage within the catchment at any time. The related theoretical formulation of time-variant catchment transport processes introduces the need for a suitable analytical mixing scheme, whose limiting cases have already been defined [Botter, 2012]. However, the practical applications of such complex theoretical apparatus [see, Bertuzzo et al., 2013] are far from clear, and need to be better explored.

[4] The age of streamflow strongly impacts the chemical composition of river flows as it reflects the memory of hydrologic systems to rainfall or soil moisture compositions and is thus crucial for our understanding and modeling of the chemical composition of runoff waters. The issue has long been disregarded by hydrologists dealing with the quantification of input/output water fluxes, and has now rightfully become a cornerstone of hydrochemical studies. Typical applications include both atmospheric compounds entering the catchment through precipitation [Shaw et al., 2008; Godsey et al., 2010] or anthropogenic compounds injected onto the catchment, for example, due to farming activities [e.g., Basu et al., 2010; Rouxel et al., 2011; Kennedy et al., 2012]. Chlorides are an example of solutes which can be considered either geogenic or anthropogenic, because the mass input to the terrestrial part of the hydrologic cycle, whose proper quantification is sometimes problematic [Neal et al., 1988; Tetzlaff et al., 2007; Dunn and Bacon, 2008], originates either from atmospheric deposition and/or from the application of inorganic fertilizers. In extensively managed croplands, however, chloride loads associated to massive fertilizations typically cloud atmospheric deposition, leading to streamflow chloride concentrations exceeding math formula (mg/l) [van der Velde et al., 2010a; Rouxel et al., 2011; Molenat et al., 2013]. Depending on the biogeochemical properties of the catchment soils, high-chloride input rates may determine a transfer of acidity to surface waters [Farrell, 1995], with negative impacts on the ecosystem functions provided by river biomes [e.g., Ikuta and Kitamura, 1995].

[5] In this paper, we use analytical, time-variant TTDs to explicitly incorporate the variability of the input and output fluxes in modeling the leaching of chloride from a small and well-instrumented agricultural catchment in the Netherlands. The aim of the study is twofold: (i) demonstrate the suitability of the general formulation of transport by TTDs to describe a real-world case study where empirical hydrologic and chemical information is simultaneously available; (ii) identify the uncertainty in the model structure/parameters, and the additional information gathered when chemical data are added to the hydrologic data during calibration. The key role played by the time variance of transport processes (emerging at different temporal scales) will also be highlighted.

2. Methods

[6] Our formulation is based on considering the age of water particles traveling within a given control volume V (representing a catchment, a pipe or any type of variable flow system), that is forced by input (say, IN(t)) and output (OUT(t)) fluxes. The evolution of the water stored within V at each time t follows by continuity, that is, math formula. Particles entering the control volume at time ti and still within V at time t are characterized by a residence time math formula, (for math formula). At any given time, the storage is constituted by the trace of inputs entered at different times in the past, and it is thus characterized by the simultaneous presence of different ages. The probability distribution of the ages in storage at time t is termed, in general, the RTD math formula. The notation emphasizes that pRT is a function of the residence time tR and that it evolves in time (e.g., due to the progressive aging of the stored particles). Each pRT can be highly erratic due to the vagaries of the inputs and the obvious lack of residence times corresponding to periods devoid of input. To study the evolution of pRT in time it is useful to introduce the reference quantity math formula, which expresses at any time t the amount of storage with age tR. In a more formal vein, math formula is the stored volume that at any time t is made up by particles whose residence time is within the interval math formula.

[7] By definition math formula, so the residence time of the stored particles increases with the same speed as time (e.g., after a time interval of 10 days all the resident particles are 10 days older). For a system with no inputs and outputs, particles cannot leave V and can only get older. In this case, the aging of the particles is expressed by the relation:

display math(1)

that indeed expresses the fact that after a time interval τ all particles in storage get uniformly older by the same amount τ independent of its value. As the variation of math formula with respect to τ is null, the governing equation for a pure aging process becomes:

display math(2)

[8] Equation (2) takes on the form of a purely convective process equation, with unit speed in the travel time domain.

[9] In the more general case where inputs and outputs affect the storage, flows inject new water particles and remove others from the control volume. By removing water particles, outflows also remove their corresponding age contribution to storage. Obviously, ages that are not in storage cannot be drawn from the outflow. For the inputs, the residence time is linked to the way age is defined. In this case math formula, so at the time of injection math formula implying math formula and all particles entering the system have null residence time. As per the outputs, the function that describes the distribution of the ages in the outflows, conditional to sampling time, is the travel time distribution conditional to the exit time. Following previous notations [Niemi, 1977; Botter et al., 2011], such distribution is denoted by math formula. Note that sampled ages can only be among those available in storage at the sampling time and the particle age's selection can differ at any sampling time, thus making the distributions a function of both tR and t. If the quantity math formula expresses, for each residence time, the amount of particles that leave the system via OUT(t), equation (2) becomes

display math(3)

where the right-hand side is always negative, clearly indicating that the fate of any age group in storage is that of being depleted. This is a reformulation of the Mater Equation for the residence time probability density function (pdf) in time-variable flow systems developed by Botter et al. [2011]. A similar description of analogous concepts was also provided by later works [van der Velde et al., 2012; Hrachowitz et al., 2013]. A graphical representation of the physical meaning of equations (2) and (3) is provided in Figure 1. Note that inputs do not explicitly appear in equation (3), (provided that they only define the fraction of water particles with zero age) but only appear in the corresponding boundary condition:

display math(4)
Figure 1.

Graphical representation of the basic balances implied by the general master equation for residence and TTDs [after, Botter et al., 2011]. (a) Pure aging process (equation (2)). (b) Effect of input and output fluxes (equation (3)).

[10] In a catchment control volume of the type considered here, the only input is rainfall J, while output flow is made up of two different contributions: discharge Q and evapotranspiration (ET), that are characterized by distinct conditional TTDs, say math formula and math formula. Making use of the definition of math formula, equation (3) can thus be written as:

display math(5)

with the initial condition:

display math(6)

[11] Equation (5) can be interpreted as a balance of water volumes labeled by a given age. The volumes change because of two reasons. On one hand, particles get older with time, so keeping one age fixed, the water with that age will be drawn from different inputs at each time. Note that checking the relative volume of particles marked by a fixed age requires tracking different particles as time elapses. The aging effect can be recognized in the second term on the left-hand side of equation (5). On the other hand, new amounts enter the system (through rainfall) and others exit V (through discharge and ET) subtracting water particles from the storage together with their age contribution.

[12] In order to express equation (5) in terms of pRT, the derivatives at left-hand side must be expanded. Recalling that the water balance equation yields math formula, one gets the formulation of Botter et al. [2011]:

display math(7)

[13] Equation (7) is a first order partial differential equation in pRT. It is formally similar to equation (5) with the addition of a term that accounts for the normalization of the pdf owing to storage variations (the first term at right-hand side). Equation (7) is linear and nonhomogeneous, and can be solved analytically.

[14] As discharge and ET can only sample from existing resident ages, the corresponding travel time pdf's can be expediently expressed as a function of pRT, namely:

display math(8)

where math formula are called age functions [Botter et al., 2011] expressing the affinity of the outflows (owing to mixing or to selective factors like those possibly induced by plant physiology) for the ages in storage. Functions math formula are dimensionless and relative to pRT.

[15] After substitution we finally get the homogeneous equation:

display math(9)


display math(10)

[16] Equation (9) is based only on the assumption that the control volume, fluxes and age functions can be singled out, and it can be solved analytically in terms of the time-dependent fluxes/storages and of the age functions, as discussed in Botter [2012].

[17] The solution of the Master Equation can be directly used to calculate the outflowing concentration of a solute that is transported along the catchment. Let us consider the case of a solute input that enters the system through rainfall and moves across the control volume transported by the water fluxes before eventually leaving V through discharge or ET . Due to the presence of solutes, rainfall is characterized by an initial solute concentration Cin that is zero in case no mass is injected. During the transport process, solutes may undergo physical, chemical or biological reactions that may cause the initial concentration Cin to change over time or in time. Thus, the concentration of the outflowing water particles is, in general, a function of both t and tR and is here denoted by math formula. Discharge concentrations can then be computed by means of the following convolution operations:

display math(11)

where the convolution operators are the TTDs conditional on the exit time math formula (that can also be expressed as math formula as shown in equation (8)).

[18] Among the possible mixing schemes, a noteworthy case is the one where ages are randomly sampled by the output fluxes in the same proportion as they are stored in the control volume (random sampling). Under this hypothesis, the age distribution of the outflows is fully representative of the age distribution of the whole storage, and the solution to the master equation gets particularly simple and instructive [Botter, 2012]. This framework has been previously called complete mixing (or well-mixed case), which is however a misleading term in that it seems to imply spatially uniform concentrations in the control volume, as a byproduct of the instantaneous mixing of each input with the existing storage. Instead, even though a well-mixed system behaves by definition as a random sampler, yet there exist systems far from being completely mixed where the output fluxes may still sample stored ages proportionally to their relative abundance. Hence, we shall term this mixing scheme random sampling (RS), to emphasize that spatial gradients of resident ages (or solute concentrations) are not necessarily neglected. Physically, the situation is representative of control volumes where mixing of ages and macrodispersion are significant. The assumption of random sampling was proved to be quite robust in single hillslopes characterized by heterogeneous soils [Rinaldo et al., 2011]. Moreover, the nonpoint source nature of the inputs and the integrative nature of the network geometry [Rinaldo et al., 1991] seem to favor the robustness of this assumption also in larger catchments with complex network structures.

[19] Mathematically, the RS assumption implies that the distributions math formula and math formula coincide with pRT (i.e., math formula), leading to a major simplification in equation (7) that gives:

display math(12)

and whose general solution is readily available [Rinaldo et al., 2011]. Though relatively simple, the RS hypothesis has a broad range of applications, especially when the catchment is schematized as a combination of different water storages in series or in parallel (as customary on conceptual modeling of the interactions of, say, the root zone and groundwater). The RS scheme could then be applied to each compartment, providing an overall non-RS storage. The overall travel time pdf's would be obtained from those pertaining to each substorage by means of weighted averages (parallel reservoirs) or convolutions (in-line reservoirs). This makes the scheme flexible to different types of applications. In relatively small lowland catchments of the type considered in this paper, reasonable results can be obtained through the use of two storages, as shown in section 4.

[20] An advantage of applying RS schemes pertains the computation of the flux concentration of solutes transported through the hydrologic cycle. The equivalence between travel and RTDs implied by the RS hypothesis allows the use of pRT as the convolution operator in equation (11). Thus the flux concentration in the runoff is given by the composition of the different water particles' concentrations that are stored within the catchment at any time, yielding:

display math(13)

where math formula is the average storage concentration. This brings notable simplifications in the calculations because the outflowing concentration can now be computed as math formula, where MS is the solute mass contained within the storage and can be obtained, as well as S(t), from a mass balance. Equation (13) shows that behind such an apparently easy scheme there is a rather complex process, driven by time-variant TTDs. Therefore, the use of the mean concentration does not imply that all the water particles have the same concentration, but rather that the global concentration can be used to characterize the outflows. This makes transport modeling easier from a computational point of view, because all mass leaving a RS storage can be computed as the product between the output water flux Q(t) and the average storage concentration math formula. Were this approach extended to a multistorage system, each compartment would be characterized by a different average concentration that also characterizes the corresponding outflows.

3. Case Study: The Hupsel Brook Catchment

[21] The Hupsel Brook catchment is a small lowland watershed of about 6.5 km2 located in the eastern part of the Netherlands (Figure 2). Due to its geological and hydrogeological characteristics it has been used as a research catchment since the 1960s [Wosten et al., 1985; van Ommen et al., 1989; van der Velde et al., 2009, 2010a, 2010b; Brauer et al., 2011]. Data series regarding rainfall, discharge, solar radiation, temperature, chemical concentration, water levels, etc. are available at different time scales. In particular, more than 25 years of hourly discharge data at the outlet are available.

Figure 2.

Sketch of the Hupsel Brook catchment.

[22] Typical of much of the Netherlands, the climate of the study area is semihumid (annual rainfall about 700–1000 mm) with rare snowfall events in winter. Seasonality is quite marked. Summers are relatively dry and ET is predominant, while winter is wet and characterized by a soil water content that is close to saturation. Intense agricultural use dominates the hydrologic and biochemical characteristics of the catchment. The crop fields are densely drained by ditches and almost 50% of the land is artificially drained via a tile drain network. Overland flow is mainly due to saturation excess, which is frequently observed in winter and it is estimated to contribute, depending on the period, between 25% and 40% of total catchment discharge [van der Velde et al., 2010c, 2011]. The response of the catchment is quickened also by the high efficiency of the tile drainage network. Due to the small size of the catchment, discharge is relatively low (average flow is about 0.070 math formula) with highest peaks rarely reaching 2 math formula. Nutrients like nitrates, phosphates and chlorides are introduced into the soil in the form of manure and fertilizers during the fertilization period (March to October). Regional estimates of the inputs clearly show a decreasing trend in agricultural loads: chloride decreased from about 250 kg/ha in 1983 to less than 130 kg/ha in 2008 (−48%) [van der Velde et al., 2010a]. Chlorides stemming from atmospheric sources are expected to be around a few mg/l, providing a negligible contribution if compared to uncertainty in anthropogenic loads.

[23] In this study, one year of available chloride measurements in the discharge, from May 2007 to May 2008, are considered. Therein, samples have been taken with irregular frequencies of about one week. The mean over the measured period is about 30 mg/l, which is higher than the estimated input concentration during the preceding 5 years (about 18 mg/l). Since chlorides are known to be nonreactive, the apparent imbalance between input and output is to be first addressed. This is a classical problem when dealing with chlorides [Neal et al., 1988] and can be caused by a set of different reasons such as dry and occult deposition [McMillan et al., 2012; Page et al., 2007], presence of forested areas [Guan et al., 2010; Oda et al., 2009] or evapoconcentration during dry seasons. In the Hupsel Brook Catchment forests are almost absent and as wet deposition is very low (1–2 mg/l), also dry deposition is expected to be of the same order of magnitude, thus not sufficient to explain the 12 mg/l gap between output and input. As chlorides have potential toxicity on plants metabolism [Taiz and Zeiger, 2010], particularly at such high concentrations, we expected plants to transpire chlorides at a lower rate than water, resulting in a global increase of stored water concentration. Given annual water balance, according to which approximately 50% of annual rainfall goes into ET, if plants uptake concentration were on average not higher than 10 mg/l, discharge concentration would increase up to approximately 30 mg/l, which would be amply sufficient to close the mass balance. Following similar conjecture, a successful approach was adopted by van der Velde et al. [2010a] who assumed plants to uptake chlorides at a fixed concentration, whose value was derived through calibration. Resulting value was significantly lower than discharge concentration and allowed balance closure. On a similar approach, a time-dependent uptake rate was used here to simulate time-variant evapoconcentration dynamics, as explained in section 4. In addition to its theoretical basis, such an approach can be an effective tool to take into account the uncertainties in input estimation.

[24] Observed concentration, besides flat long-term mean, show short-term fluctuations that, when compared with discharge, suggest a negative correlation between Q and C (Figure 3). This is a typical behavior of chlorides in case water injections have lower concentration than base flow [see, Neal et al., 2012], indicating that most of the discharge right after intense rainfall events is comprised event water contributions. This further confirms the general affinity of lowland catchment responses for relatively new water suggested by van der Velde et al. [2012]. Finally, we note that concentration remains high even in winter period when fertilizations are suspended and discharges (that result in solute depletion) are higher.

Figure 3.

Measured discharge and chloride concentration from May 2007 to May 2008. Vertical bars highlight the matching between discharge peaks and flux concentration troughs.

4. Model

4.1. Hydrologic Model

[25] A key feature of the test catchment is the strongly nonlinear dependence of the outflows on soil moisture content [Brauer et al., 2011]. A soil moisture balance for the root zone [Rodriguez-Iturbe et al., 1999; Laio et al., 2001] was set up as:

display math(14)

where n is porosity, Zr the root zone depth [L], s the soil moisture content, I is infiltration into the root zone [L/T], L is the leakage to lower horizons [L/T] and ET is evapotranspiration flux [L/T]. The notation for L and ET stresses the dependence on s. The stored volume in the root zone ( math formula) is named Srz hereafter. As described in section 3, fast hydrological response is due to artificial tile drainage network and to overland flow (mainly driven by soil water saturation excess). In both these two processes, water flux can be associated with soil moisture content, so all runoff contributions flowing out of the root zone were grouped together in a single nonlinear leakage term, that was modeled through the Clapp-Hornberger equation [Clapp and Hornberger, 1978]. The redundancy of a separate modeling of the overland flow was confirmed by preliminary numerical experiments. Infiltration flux I was then set equal to rainfall intensity J.

[26] Starting from equation (14), four different models with increasing degree of nonlinearity were tested (Figure 4). Model (a) is made of a single root zone storage that directly leaks into discharge. The other three models include deep storage that accounts for groundwater flow. This is obtained by partitioning the leakage from the root zone into two components: one producing the subsurface discharge Qrz and one feeding the groundwater storage. Slow flows Qgw originating from groundwater are modeled according to the linear reservoir scheme: math formula, while the overall discharge is the sum of subsurface and deep components math formula. The key difference between the three multistorage models lies in leakage partitioning. Model (b) uses a constant partition, while model (c) uses a time-variant partition that is a function of the soil moisture content. Finally, model (d) has two separate leakages from the root zone, to separate the processes giving birth to fast response flows (preferential pathways, drainage network, etc) from those that produce deep percolation and groundwater flow. The main equations implemented within each model are summarized in Figure 5.

Figure 4.

Tested models representation. Mass terms are colored in red. Mass fluxes are always associated with a corresponding water flux (e.g., math formula is mass flux corresponding to Qrz).

Figure 5.

Summary of model equations for the four models tested in this paper.

4.2. Transport Model

[27] A transport model for chloride was coupled to the hydrologic model on the following premises. Solute main input to the catchment is the mass introduced through fertilization. Mass loads were calculated by dividing annual estimates into the total number of hours characterizing fertilization season. Loads were introduced during every hour of the fertilization season but were assumed to remain immobile in the topsoil until a new rainfall pulse mobilized them through infiltration. Therefore, the concentration of a new rainfall pulse was obtained as the total mass loaded since previous precipitation, divided by rainfall depth. An additional constant chloride concentration in rainfall (wet deposition), though very low compared to uncertainty in fertilization loads, was accounted for and set to 1.5 mg/l [van der Velde et al., 2010a].

[28] The main assumption of the transport model is that both root zone and groundwater storage are random sampled. This allows the numerical computation of the outflowing concentrations to be simplified by the introduction of mass balances that implicitly incorporate the time-variant structure of the convolution operations, as shown in equation (13). Such a convenient numerical scheme is used throughout the calibration process (see section 5). TTDs are then calculated on the calibrated model, to infer their time-variant properties.

[29] Under the RS hypothesis, the two modeled storages can be characterized by average concentrations math formula (root zone) and math formula (groundwater) respectively. Chlorides are assumed to be conservative (no reaction takes place along with the transport), but they are assumed to be only partially uptaken by plants because of their potential toxicity on plants metabolism [Taiz and Zeiger, 2010]. Plants are assumed to sample particles from the root zone with concentration CET that is proportional to the average concentration of soil moisture: math formula, with math formula. During ET, solutes leave the system at a lower rate than water, resulting in an increase of the average storage concentration. This yields a higher flux concentration in the runoff offering a possible explanation to the apparent chloride imbalance, as discussed in section 3.

[30] Mass balance in the root zone yields:

display math(15)

where Mrz is the chloride mass contained in the root zone and the terms math formula, math formula, and math formula are mass flows attached to rainfall, leakage and ET, respectively. All mass fluxes in equation (15) can be expressed as water fluxes times the corresponding flux concentrations. Recalling the RS hypothesis for the leakage flow ( math formula) and the selective ET assumption ( math formula) one gets:

display math(16)

where C0 is the initial concentration of rainfall pulses. By expanding the derivative of the product at left-hand side of equation (16) and rearranging the equation, one gets the differential equation that governs the average concentration math formula in the root zone storage as:

display math(17)

[31] Equation (17) shows that average storage concentration is increased/decreased when initial concentration of rainfall pulses is higher/lower than the resident concentration math formula (first term at right-hand side). Moreover, selective ET (second term at right-hand side) induces an increase in math formula (recall math formula which is stronger when ET is more intense.

[32] Models (b), (c), and (d) also employ a groundwater storage that is assumed to be randomly sampled with no ET occurring within the groundwater compartment. Provided that output fluxes do not change the storage concentration, the evolution of average concentration in the groundwater storage is due only to the leakage input:

display math(18)

and the final flux concentration is the weighted average of subsurface and groundwater flow concentrations:

display math(19)

[33] The related water TTDs are derived in detail in Appendix A.

5. Parameters, Calibration, and Ranking Methods

[34] Each of the four models tested in this paper requires the determination of the related parameters. Some of these parameters, common to all the models, were chosen according to previous works and are summarized in Table 1. The remaining free parameters (hydrologic ones plus the selective ET factor α) are in number 5 in model (a) and 6 in models (b, c, d) and have been derived through calibration.

Table 1. Constant Parameters of the Four Tested Models
ParameterSymbolModel (a)Model (b, c, d)
Soil porosityn0.350.35
Root zone depth (mm)Zr750750
Soil moisture at wilting pointsw0.110.11
Soil moisture for runoff triggering
display math
Saturated soil conductivity (mm/h)Ksat100100
Initial groundwater saturated depth (mm)
display math
Initial groundwater concentration (mg/l)
display math

[35] To fully explore the parameters space, a Montecarlo approach is employed [Beven and Freer, 2001]. For each model, 107 simulations were run with combinations of random parameters sampled from a uniform prior distribution. The length of the simulation period is 350 days (from May 2007 to May 2008), which, at an hourly time scale, results in 8400 simulated timesteps for each run. Initial groundwater average concentration math formula was derived from preliminary simulations starting approximately 22 years before the start of the measurements. The resulting value (40 mg/l) was observed to be quite stable under different reasonable model settings and initial conditions. Other details about the setup of the calibration procedure are summarized in Table 2. In order to rank model performances, the Nash-Sutcliffe model efficiency was first evaluated for discharge ( math formula) over 8400 discharge measurements. The Residual Sum of Squares was used instead for chloride concentration in discharge ( math formula) over the 50 available chloride concentration measurements.

Table 2. Setup of Montecarlo Calibrationa
ParameterSymbolParent ModelLower BoundUpper Bound
  1. a

    CH, Clapp-Hornberger; LS, Leakage Separation; GW, Groundwater.

Max ET in warm periods (mm/h)ETmaxwa-d0.050.15
Max ET in cold periods (mm/h)ETmaxca-d0.010.06
Plants selectivity coefficientαa-d00.6
CH parameter (mm/h)ba-c818
CH parameter (mm/h)brzd515
CH parameter (mm/h)bgwd1020
LS coefficientls1b0.10.9
LS coefficientls2c0.51.5
GW coefficient ( math formula)kb-d math formula math formula

[36] Models and parameters were accepted if math formula, and the accepted models were then ranked according to their root-sum-square (RSSC) score (the lowest RSSC corresponds to the best performance). The behavioral subset of simulations for the sensitivity analysis was defined by the first 100 performances of the ranking.

6. Results

[37] After calibration, each of the four models was associated with a ranking of the 100 best RSSC scores. This makes it easy to compare the models in order to select the most suitable one. A representation of the scores versus the ranking position is shown in Figure 6.

Figure 6.

Ranking of model performance according to first 100 residual sum of squares scores.

[38] Model (a) performed the worst (i.e., possessed the highest RSS scores), especially for the first positions of the ranking, showing that the single storage proves too rough a scheme to correctly reproduce the main mechanisms responsible for the measured concentration data. Moreover, the score function shown in Figure 6 is quite flat, meaning that the differences in score among different parameter combinations is low, thus implying that optimal parameter identification is uncertain. Remarkably, were the focus limited to discharge alone, model (a) would be as performant as the double storage models. This issue is further discussed in section 7.

[39] Models (b) and (c) have similar performances and clearly outperform the single RS (a) and the double-leakage (d) models. The pronounced slope of the score function in the region of the first positions of the ranking indicates that optimal parameter identification is reliable.

[40] Model (d) allows for a clear identification of its parameters, but the performances are not satisfactory. After the first 20 positions of the ranking it is even worse than the single RS model. Even though the double leakage was designed to be flexible enough to simultaneously capture fast and slow subsurface flow processes, the number of free parameters proved inadequate to meet this purpose.

[41] As model (c) has, on average, the best scores, it is selected as the choice model to describe the hydrochemistry of the Hupsel Brook catchment.

[42] The posterior parameter distribution of the selected model is represented in Figure 7, suggesting the robustness of the identification procedure in this case. The corresponding discharge and chloride concentration series from the best 100 model parameterizations are then compared to observations in Figures 8 and 9. Modeled discharge series shows a general agreement with the measurements. For low flows ( math formula) mm/h, approximately equivalent to 20 L/s) the model tends to underpredict discharge, with no significant effect on its chemical composition and TTDs, because of the negligible impact of low flows on the storage and RTDs. Modeled chloride concentration (Figure 9) is able to reproduce the two main observed features, that are the general fluctuations pattern and the mean value of about 30 mg/l. The mean value is a direct consequence of the employed evapoconcentration scheme, which proved effective in simulating the observed difference between input and output concentrations. The model is less accurate during periods where separation between fast and slow flow contributions to discharge is more uncertain. This is visible in the period June to July 2007, as also underlined by the larger deviations of model simulations among each others.

Figure 7.

Sensitivity plots for 100 best performances of the selected model. Parameter description and range is described in Table 2.

Figure 8.

Discharge series modeled according to the selected model (model C). All simulations have Nash-Sutcliffe (NS) score math formula 0.75. (a) Best performance in natural axis. (b) 100 best performances in semi log scale.

Figure 9.

Modeled chloride concentrations for the 100 best performances of the selected model.

[43] The reliability of the model allowed a meaningful analysis of the TTDs that enabled to match the observed concentrations. TTDs were first calculated for the two storages separately, and then combined together to get the overall distributions (equation (A(4))). As each distribution retains the memory of the hydrologic history of the system for years, the calibrated model was started 20 years before the beginning of the available measurements, to ensure the tails of the distributions were properly calculated. A startup period of 16 years was observed before discharge was made for more than 99% by water injected into the system after the start of the simulation. This assured that, over the period 2004–2008, TTDs were computed over more than 99% of their age domain. For this period, the mean value of each distribution was used for further analysis.

[44] A look into the temporal evolution of mean travel times in the two storages allows an assessment of the difference in the corresponding characteristic time scales (Figure 10). The root zone is characterized by a low ratio between storage and flows, meaning that it is relatively dynamic. Therefore, the storage has short memory of past rainfall events and mean travel times are easily affected by single events, producing high-frequency fluctuations in the transport features, including the mean (Figure 10). Groundwater, on the contrary, has a much larger ratio between storage and flows, meaning that it is only affected by long term variability of climate conditions and it is likely to be characterized by seasonal and interannual fluctuations.

Figure 10.

Mean travel time computed over a 4 year period.

[45] The TTDs of the root-zone and groundwater storages were then combined together to yield the overall TTDs . The mean of the overall TTD shows a rather irregular behavior, reflecting the continuous interplay between fast and slow flow contributions to discharge. In contrast with stationary distributions, that are characterized by a single mean value, we note here a whole pdf of mean travel times, whose coefficient of variation is 0.77 (Figure 11). The two peaks of the pdf of the mean travel times recall the fact that streamflows are a suitable combination of young ( math formula) water from the root zone and old ( math formula) water from groundwater. The average travel time, though not representative of the actual travel time of the transported water particles (its frequency is practically zero), is of the order of 1.45 years. This turns out to be slightly lower but very similar to the value of 1.8 years estimated by van der Velde et al. [2010a]. This difference is due to the different model structures and to the nonstationarity of the distributions over the years, (TTDs were calculated here during a shorter time window with respect to van der Velde et al. [2010a]).

Figure 11.

Empirical probability distribution function of the mean travel time in the Hupsel Brook catchment.

[46] The highly nonstationary behavior renders rather meaningless, for transport computations, the characterization of catchment travel times through stationary distributions, as frequently pursued in the literature. Nevertheless, we note that stationary distributions are entirely meaningful in the context of peak discharges and where no basin-scale transport is attached to the hydrological component (as the age paradox is immaterial as long as only the quantity of the runoff water is pursued).

7. Discussion

[47] The theoretical tools employed in this paper combine several recent advancements [Botter et al., 2010, 2011; Rinaldo et al., 2011; van der Velde et al., 2010a, 2012; Hrachowitz et al., 2013] in our understanding of the nonstationary character of catchment TTDs and the role of age distributions in the chemical composition of runoff into a single useful application. Our aims are rather of methodological nature, as the application highlights key features of any model of catchment-scale transport processes.

[48] Indeed, the analysis of TTDs allows for proper understanding of large-scale, multistate control volumes hosting transport processes. Here we show that root zone (i.e., near-surface) residence times are affected by short-term events, leading to the relatively fast release of solutes associated with regimes entailing transport times ranging from weeks to months. Moreover, solutes in the root zone (or whatever is conveniently modeling the behavior of a near-vertical water and solute circulation) are likely to be quickly diluted in case of intense rainfall events and almost completely removed in periods with no fertilization. On the contrary, solutes that reach groundwaters are released at lower rates and tend to accumulate and persist for much longer times. The combination of these separate behaviors, together with the time-variant partitioning between discharge contributions, fully captures the behavior of solutes transported along the catchment. Our framing of this particular model into the general scheme of the Master Equation for residence and TTDs clarifies a mathematical position that may seem arcane unless dressed by a proper physical meaning.

[49] After intense rainfall events, large amounts of new water enter the root zone. In this case new water is almost solute free (even in the presence of fertilization because solutes get largely diluted), so it causes the average concentration in the root zone outflows math formula to suddenly decrease. According to the model(s), because the main discharge component right after rainfall events is subsurface flow, solute concentrations in the global discharge also quickly decreases. This corresponds rather well with the observed concentration decreases (recall Figure 3). Note that this dilution mechanism could be simulated by a single storage model. However, in such a case, concentration would remain low until solutes are replenished by new fertilizations. Even so, this would take a long time to occur and would not happen at all in periods with no fertilization.

[50] What makes it possible for the model to allow regrowth of the flux concentration is the presence of deep storage. Solutes in the groundwater tend to persist and accumulate, so at any time chlorides mainly belong to older fertilizations from many years before. Also, during wet winter periods the average groundwater concentration remains high (around 40 mg/l) because past fertilizations are speculated to be more intense [van der Velde et al., 2010a]. As soon as subsurface contribution to discharge is depleted, the main discharge source is groundwater flow, which increases solute concentration again.

[51] The use of TTDs fosters a deeper understanding that can be of practical use. It often happens in rural catchments [Aquilina et al., 2012; Ruiz et al., 2002] that stopping fertilization loads for one or more years does not result in a significant permanent decrease of average stream concentrations. According to our results, the same would occur in the Hupsel Brook catchment. Ceasing fertilizations would affect the root zone storage, mainly because of its short hydrologic memory as underlined by its relatively short mean travel times. For the same reason, groundwater storage would hardly be affected. As the average concentration is the byproduct of long-term dynamics, it is mainly governed by the groundwater storage which hardly responds to just one or two nonfertilized seasons.

[52] The single-storage model was able to reproduce measured discharge with the same accuracy as double-storage models. This means that it is appropriate, as commonly held, in describing the hydrological response but not in describing transport, making clear the different nature of the two problems. Hence, the ability of the model to reproduce measured concentration data lies in the double-storage schematization. It seems reasonable for relatively small and densely drained catchments to schematize both the root zone and groundwater as separate but randomly sampled storages. Then the nonlinear and nonstationary partitioning among the storages makes the overall system strongly nonrandomly sampled. In hydrologic practice, however, more storages could be employed (e.g., a “saturated area” storage which can capture the very fast response of the catchment). This would potentially yield better model performance, but it is unclear whether improved performance would be justified by the added complexity, say via Akaike testing. It is remarked, however, that this goes beyond the scope of this study, centered, as it is, on the inferences of nonstationary travel time tools and comparative field data analysis on the basic transport mechanisms operating in a typical catchment setting.

8. Conclusions

[53] The analytical theory of transport by TTDs, recalled and adapted in section 2, is a powerful tool to understand and model transport at catchment scales. The Master Equation for the RTD, keeping track of the temporal evolution of the ages in storage, has the central role of generator of all the other involved distributions. We show here that what is needed for application is a reasoned choice of input-output flows, possible chemical reactions and a mixing scheme. We argue that the RS assumption employed in the paper is one of the many possible choices having some key operational advantages. Results can be tested with different hydrochemical models and mixing schemes, at no loss of generality.

[54] Besides theoretical aspects, the theory has found in the Hupsel Brook catchment a broad space for application. Modeling chloride concentration in discharge led to understanding and reproducing long-term and short-term transport dynamics. The schematization of the catchment by separate randomly sampled compartments is computationally easy and proved reliable in reproducing measured signals.

[55] As one of the first applications to real catchments, the model was tested on simple nonreactive solutes. Future applications can be extended to solutes that undergo different transport processes, by including proper chemical, physical or biological reactions. Another challenge for the future is the possibility to extend current approaches to relate a single catchment mixing function to different sources of mixing (e.g., dispersion, plants selectivity, and catchment connectivity).

[56] We conclude that the treatment of the problem and the methodological procedure comparing chemical and hydrological data with a novel conceptual framework represent an interesting advancement of our understanding of catchment-scale transport processes.

Appendix A: Equations for TTDs

[57] In section 2 it was shown that, under the hypothesis of randomly sampled storage, TTDs math formula are equal to RTDs pRT. Thus, the Master Equation (equation (9)) simplifies into equation (12).

[58] In case the catchment is made up of two storages (one for the root zone and one for the groundwater, Figures 4b–4d), each storage is characterized by its associated TTDs. Note that for particles that reach the groundwater storage, the travel time must be computed starting with entrance into the catchment (i.e., entrance into the root zone), so the TTDs are obtained as the convolution between the pdf's in the root zone and the pdf's in the purely groundwater storage. Finally, the overall math formula (that accounts for the contribution of the two storages to discharge Q) is computed by means of weighted averages between the distributions of the two storages.

[59] In order to simplify the notation, the following simbology is used:

display math
display math
display math

(e.g., the function math formula referred to the root zone is named math formula and so on). So the TTD equations for the root zone in the simplified notation are:

display math(A1)

where J is the input rainfall and Srz is the root zone storage. The equation for the purely groundwater storage has the same form, but in this case the input is given by the leakage to the groundwater Lgw and the storage is the deep Sgw:

display math(A2)

[60] To obtain the overall groundwater storage TTDs, the convolution between math formula and math formula is performed:

display math(A3)

[61] Finally, the global TTDs are:

display math(A4)


[62] The authors wish to thank Aldo Fiori for his valuable suggestion about using the terminology random sampling against complete mixing. A.R. wishes to thank support from SNF-FNS Projects 200021-124930/1 and 200021-135241.