J Marçais
email: jean.marcais@polytechnique.edu

J.-R De Dreuzy

T R Ginn

P Rousseau-Gueutin

S Leray

Inferring transit time distributions from atmospheric tracer data: Assessment of the predictive capacities of Lumped Parameter Models on a 3D crystalline aquifer model

Keywords: Transit time distributions, Lumped parameter models, Renewal times, Atmospheric anthropogenic tracers, Crystalline aquifer

While central in groundwater resources and contaminant fate, Transit Time Distributions (TTDs) are never directly accessible from field measurements but always deduced from a combination of tracer data and more or less involved models. We evaluate the predictive capabilities of approximate distributions (Lumped Parameter Models abbreviated as LPMs) instead of fully developed aquifer models. We develop a generic assessment methodology based on synthetic aquifer models to establish references for observable quantities as tracer concentrations and prediction targets as groundwater renewal times. Candidate LPMs are calibrated on the observable tracer concentrations and used to infer renewal time predictions, which are compared with

the reference ones. This methodology is applied to the produced crystalline aquifer of Ploemeur (Brittany, France) where flows leak through a micaschists aquitard to reach a sloping aquifer where they radially converge to the producing well, issuing broad rather than multi-modal TTDs. One, two and three parameters LPMs were calibrated to a corresponding number of simulated reference anthropogenic tracer concentrations (CFC-11, 85 Kr and SF 6 ). Extensive statistical analysis over the aquifer shows that a good fit of the anthropogenic tracer concentrations is neither a necessary nor a sufficient condition to reach acceptable predictive capability. Prediction accuracy is however strongly conditioned by the use of a priori relevant LPMs. Only adequate LPM shapes yield unbiased estimations. In the case of Ploemeur, relevant LPMs should have two parameters to capture the mean and the standard deviation of the residence times and cover the first few decades [0;50 years]. Inverse Gaussian and shifted exponential performed equally well for the wide variety of the reference TTDs from strongly peaked in recharge zones where flows are diverging to broadly distributed in more converging zones. When using two sufficiently different atmospheric tracers like CFC-11 and 85 Kr, groundwater renewal time predictions are accurate at 1 to 5 years for estimating mean transit times of some decades (10-50 years). 1-parameter LPMs calibrated on a single atmospheric tracer lead to substantially larger errors of the order of 10 years, while 3-parameter LPMs calibrated with a third atmospheric tracers (SF 6 ) do not improve the prediction capabilities.

Based on a specific site, this study highlights the high predictive capacities

Introduction

Transit times in groundwater flow systems result from local-scale advective and dispersive processes [START_REF] Bellin | Probability density function of nonreactive solute concentration in heterogeneous porous formations[END_REF][START_REF] Dagan | A solute flux approach to transport in heterogeneous formations .1. the general framework[END_REF][START_REF] Dentz | Concentration statistics for transport in heterogeneous media due to stochastic fluctuations of the center of mass velocity[END_REF][START_REF] Maloszewski | Lumped parameters models for interpretation of environmental tracer data[END_REF][START_REF] Simmons | A stochastic-convective transport representation of dispersion in one-dimensional porous-media systems[END_REF], as well as from global-scale watershed and geological structures [START_REF] Eberts | Comparison of particle-tracking and lumped-parameter age-distribution models for evaluating vulnerability of production wells to contamination[END_REF][START_REF] Goderniaux | Partitioning a regional groundwater flow system into shallow local and deep regional flow compartments[END_REF]McGuire & McDonnell, 2006;[START_REF] Rinaldo | Catchment travel time distributions and water flow in soils[END_REF]. Hence Transit Time Distributions (TTDs) are most generally broad, mixing times of different flow lines [START_REF] Engdahl | Approximating groundwater age distributions using simple streamtube models and multiple tracers[END_REF][START_REF] Ginn | On the distribution of multicomponent mixtures over generalized exposure time in subsurface flow and reactive transport: Foundations, and formulations for groundwater age, chemical heterogeneity, and biodegradation[END_REF] potentially acquired from various recharge areas [START_REF] Weissmann | Dispersion of[END_REF], within a succession of aquifers and aquitards [START_REF] Castro | Calculation of ground water ages -A comparative analysis[END_REF][START_REF] Phillips | Groundwater dating and residence-time measurements[END_REF]Zinn & Konikow, 2007), or through exchanges between high-flow zones like fractures and large diffusive zones like the matrix [START_REF] Cook | Groundwater ages in fractured rock aquifers[END_REF][START_REF] Haggerty | Multiple-rate mass-transfer for modeling diffusion and surface-reactions in media with pore-scale heterogeneity[END_REF][START_REF] Varni | Simulation of groundwater age distributions[END_REF]. With only scarce observations, it is out of reach to identify accurately which processes intervene where and how multiscale complexity of the flow structures generate the TTD. Nevertheless some characteristics of the TTD, also called "groundwater age distribution" [START_REF] Ginn | On the distribution of multicomponent mixtures over generalized exposure time in subsurface flow and reactive transport: Foundations, and formulations for groundwater age, chemical heterogeneity, and biodegradation[END_REF] or "groundwater residence time distribution" (McCallum et al., 2014a) may be inferred from measurements of isotopic tracer concentrations in the groundwater (McCallum et al., 2014b;[START_REF] Massoudieh | Bayesian evaluation of water systems[END_REF][START_REF] Massoudieh | The theoretical relation between unstable solutes and groundwater age[END_REF]Newman et al., 2010).

We refer to these data as "groundwater age data." Moreover classical renewal and flushing time applications can directly be derived from the TTD without going back to the transport processes [START_REF] Bohlke | Groundwater recharge and agricultural contamination[END_REF][START_REF] Katz | Timescales for nitrate contam-ination of spring waters, northern Florida, USA[END_REF][START_REF] Lerner | A simple analytical approach for predicting nitrate concentrations in pumped ground-water[END_REF][START_REF] Pinay | Upscaling nitrogen removal capacity from riparian 1 zone to the landscape scale: A new framework[END_REF][START_REF] Sebilo | Long-term fate of nitrate fertilizer in agricultural soils[END_REF].

In this framework the model for the TTD takes the critical role of translating groundwater age data to renewal or flushing predictions, that with any principle of parsimony should be simple enough to be fully determined by the scarce information available. It should also be representative in a broad sense of the underlying transport processes. This has promoted the development of simple TTD models known as Lumped Parameter Models (LPMs) that represent the multi-scale multi-factor dispersive processes all together in a parsimonious "lumped" way [START_REF] Cook | Environmental tracers in subsurface hydrology[END_REF][START_REF] Maloszewski | Lumped parameters models for interpretation of environmental tracer data[END_REF]. Numerous LPMs have been proposed following different primary objectives. Some have been developed to efficiently fit observed groundwater ages like the mono-or multi-modal Dirac distributions corresponding to one or several piston flow models [START_REF] Begemann | Continental water balance, ground water inventory and storage times, surface ocean mixing rates and world-wide water circulation patterns from cosmic-ray and bomb tritium[END_REF], or flexible generalized Gamma distributions [START_REF] Cvetkovic | A general memory function for modeling mass transfer in groundwater transport[END_REF]. Others are analytical solutions of simplified problems. This is the case of the Inverse Gaussian distribution solution to the 1D advection-dispersion equation [START_REF] Ginn | Notes on Groundwater Age in Forward and Inverse Modeling[END_REF], of the exponential model taken from chemistry of continuous stirred tank reactors [START_REF] Eriksson | The possible use of tritium for estimating groundwater storage[END_REF][START_REF] Haitjema | On the residence time distribution in idealized groundwatersheds[END_REF][START_REF] Kaufman | The natural distribution of tritium[END_REF], and of some more complex semi-analytical distributions for gently sloping aquifers [START_REF] Etcheverry | Direct simulation of groundwater transit-time distributions using the reservoir theory[END_REF] or dipole flow configurations [START_REF] Luo | Fluid residence times within a recirculation zone created by an extraction-injection well pair[END_REF]. Increasing their complexity to three or more parameters has been proposed by composing some of the previous distributions (Ozyurt & Bayari, 2003, 2005).

Globally, LPMs have been proposed following Bayes spirit either by favoring the goodness of fit of the groundwater ages (likelihood in Bayes theory) like for the multi-modal Dirac distributions, or by favoring some consistency with the a priori flow pattern (prior in Bayes theory) such as for the Inverse Gaussian or exponential models. LPMs are still extending to "shape-free" distributions precisely designed to find an optimal balance of the likelihood and prior terms within a Bayesian framework [START_REF] Massoudieh | Bayesian evaluation of water systems[END_REF][START_REF] Massoudieh | Assessment of the value of groundwater age time-series for characterizing complex steady-state flow systems using a bayesian approach[END_REF].

Lumped Parameter Models are highly attractive as they offer predictive capacities at limited observation and modeling costs. However, they raise a number of questions that should be carefully assessed: is the LPM approach appropriate for 3D complex aquifers? How should the LPM distribution be chosen? Is there any a priori more relevant LPM among the existing analytical solutions? When is the information content of the groundwater age data enough for establishing the targeted predictions like the renewal or flushing times?

Assessment studies have been undertaken on sampling zones located preferentially in the downstream convergence zones [START_REF] Eberts | Comparison of particle-tracking and lumped-parameter age-distribution models for evaluating vulnerability of production wells to contamination[END_REF][START_REF] Leray | Contribution of age data to the characterization of complex aquifers[END_REF], using extensively geographical information with simple transport solutions [START_REF] Basu | Evaluation of analytical and numerical approaches for the estimation of groundwater travel time distribution[END_REF], or with two parameters widely varying LPMs under a broad panel of observation and tracer scenarios [START_REF] Green | Accuracy of travel time distribution (ttd) models as affected by ttd complexity, observation errors, and model and tracer selection[END_REF]. In this article, we analyze the predictive capacities of LPMs against evolving quantity of tracer information. We adapt the LPM complexity through the number of its parameters to the informational content of the tracer data, in order to determine the relation between the informational content of the tracer data and the quality of the predictions. While our analysis is centered on LPMs, LPMs just appear as the necessary step to extract from the tracer data the information needed to establish predictions.

We first propose a methodology for a spatially extensive assessment of the LPMs (section 2). It is illustrated on a 3D crystalline aquifer model under steady-state convergent flow (section 3). Seven LPMs are compared at 73 locations of the crystalline aquifers (section 4). Results are further discussed in section 5 to determine the quantity of information required to establish accurate renewal time predictions.

Methodology: assessing LPMs on in-silico aquifers

We detail the methods for determining the predictive capacities of Lumped Parameter Models (LPMs) based on groundwater age information derived from tracer concentrations on in-silico aquifers. In-silico aquifers, also generally called synthetic aquifer models, are essential since the assessment procedure requires the knowledge of the true Transit Time Distributions (TTDs).

As TTDs cannot be obtained in the field from any measurement at the watershed scale, we access it through aquifer modeling. Both 2D and 3D theoretical models have brought up key issues [START_REF] Massoudieh | Bayesian evaluation of water systems[END_REF][START_REF] Park | Transport modeling applied to the interpretation of groundwater Cl-36 age[END_REF][START_REF] Varni | Simulation of groundwater age distributions[END_REF][START_REF] Weissmann | Dispersion of[END_REF]Zhang, 2004;Zinn & Konikow, 2007), as has direct modeling of age distributions in real 2D aquifer case (Woolfenden & Ginn, 2009). The most relevant aquifer models are however those closer to the 3D real aquifer cases calibrated using both hydraulic and atmospheric tracer information [START_REF] Eberts | Comparison of particle-tracking and lumped-parameter age-distribution models for evaluating vulnerability of production wells to contamination[END_REF][START_REF] Green | Accuracy of travel time distribution (ttd) models as affected by ttd complexity, observation errors, and model and tracer selection[END_REF][START_REF] Leray | Numerical modeling of the productivity of vertical to shallowly dipping fractured zones in crystalline rocks[END_REF].

In this section, we assume that transport simulations have previously been performed and that the necessary results are available. These results can take the form of particle paths or concentration fields depending on the numerical method chosen for solving the advective-dispersive equation [START_REF] Bear | Dynamics of Fluids in Porous Media[END_REF].

When using random walk methods [START_REF] Delay | Simulating solute transport in porous or fractured formations using random walk particle tracking[END_REF][START_REF] Kinzelbach | The random walk method in pollutant transport simulation[END_REF][START_REF] Tompson | Numerical simulation of solute transport in three-dimensional, randomly heterogeneous porous media[END_REF], results are particle paths derived from forward or backward simulations (Neupauer & Wilson, 2002). For a straightforward derivation of concentrations from particle densities under forward tracking, particles must be injected at densities proportional to the inflow boundary fluxes. In recharge and pumping cases, the density of injected particles must be weighted by flows entering the aquifer either by the recharge zones in forward studies or by the sampling areas in backward studies [START_REF] Leray | Contribution of age data to the characterization of complex aquifers[END_REF]. The method is general in the sense that it can be applied to any aquifer model with adequate transport simulation and injection conditions.

In this article, it will be applied to the crystalline aquifer model of Ploemeur described in section 3. We discuss application to other sites in section 5. Methods are organized in 5 steps. In the two first steps, the transit time distributions (TTDs) are used to establish the reference atmospheric tracer concentrations often displayed as groundwater ages and the reference predictions. The third step consists in defining the candidate LPMs. Several LPMs are concurrently considered. In this article, we consider seven LPMs with evolving number of parameters. This is a key point of our methodology for studying the evolution of the prediction capacities according to the quantity of available information in the atmospheric tracer concentrations. The number of informative parameters will be taken as a measure of the available information. We then calibrate the LPMs on the reference atmospheric concentrations and compare their predictions with the reference ones. This procedure is consistent with a field observation methodology where only atmospheric concentrations can be measured. There is no fitting of the TTD obtained from the numerical simulations by the candidate LPMs. This assessment methodology is not only practically appropriate but also conceptually relevant. In fact, LPMs are not designed to reproduce all the details of the TTD and underlying transport processes but capture them globally to be both consistent with the available data and effective in terms of predictions. They should be somehow statistically similar but not identical.

In the same spirit, LPMs are not thought to be excellent on every sampling zone of the aquifer, but should be good on average. It thus requires an extensive spatial assessment. We underline that, even if we eventually compare the performances of the LPMs at different sampling zones, the procedure to estimate the LPMs is always made point by point and never accounts simultaneously for observations coming from different sampling zones.

2.1. In-silico modeling approach 2.1.1. Reference TTDs and atmospheric tracer concentrations

The approach used in this article requires the use of a synthetic aquifer taken as a reference and providing flow weighted consistent transit time distributions (TTDs), apparent ages and predictions on renewal times. We derive the TTDs f ref everywhere in the watershed, so f ref is a distribution function of the transit time u conditioned by the position x. The TTDs obtained are the references to which will be compared the approximations obtained with LPMs.

From the TTDs, we derive the reference tracer concentrations C ref Tr . For a given time and position (t, x) these concentrations can be readily expressed from the TTD f ref determined previously, the atmospheric concentration chronicle of the tracer C Tr and its possible decay constant λ:

C Tr (t, x) = t 0 C Tr (t -u) e -λu f ref (u, x) du (1)
where t-u is the date when the tracer enters the aquifer. C ref Tr corresponds to the concentration C Tr (t, x) at a given sampling date t and at the position x.

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These concentrations computed for different environmental tracers such as CFC-11, SF 6 and 85 Kr (see Figure 1) are taken as observables. They closely match sampled concentrations in field studies.

The concentrations can readily be converted to apparent ages ÂTr : C Tr (t, x) = C Tr ÂTr (t, x) e -λ ÂTr (t,x) (2) Tracers are considered ideal , i.e. they do not have any interaction with the aquifer matrix [START_REF] Maloszewski | Lumped parameters models for interpretation of environmental tracer data[END_REF]. We also assume that tracers 150 are at their atmospheric concentration at the aquifer recharge C Tr (equations 1 and 2).

Reference predictions

From the reference TTDs f ref we also derive renewal times in the aquifer.

For a given sampling zone, the renewal period t y of a fraction y of the water to be renewed in aquifer is defined as

y = F ref (t y ) = ty 0 f ref (u) du (3)
where F ref is the cumulative density function. The quartiles

Q ref 1 (resp. Q ref 2 and Q ref 3
) of the reference TTD represent the renewal period of time for a given sampling zone in the aquifer so that 25, 50 or 75% of the water is renewed. They are determined by:

F -1 ref (25%) (resp. F -1 ref (50%) and F -1 ref (75%)).
In the following, they will be expressed in years. The renewal times at 25, 50 and 75 % will be determined everywhere in the aquifer and reference maps will be drawn for the in-silico model.

Inferring LPMs from environmental tracer concentrations

In the next step, we define candidate LPMs and calibrate them on the reference tracer concentrations determined in the in-silico modeling phase.

We underline that, at this stage, the tracer concentrations are the only data made available. LPMs are estimated without any knowledge of the TTDs.

Defining candidate LPMs

As we aim at determining the influence of the data amount on the prediction accuracy, we choose LPMs with evolving number of parameters to be calibrated on an increasing quantity of tracer concentrations. Basically, the number of parameters will be made equal to the number of tracers. As opposed to the approach of Corcho [START_REF] Alvarado | Constraining the age distribution of highly mixed groundwater using 39 Ar: A multiple environmental tracer (H-3/He-3, Kr-85, Ar-39[END_REF], the number of degrees of freedom is always equal to 0 as the number of free parameters is made equal to the number of observations. We choose seven LPMs that either correspond to prior information on the flow pattern or that are often used in hydrogeological studies. We sort them according to the number of parameters required to define them (see table 1) and detail in the following their physical basis. They are all defined only for t 0. [START_REF] Maloszewski | Lumped parameters models for interpretation of environmental tracer data[END_REF]. P e is related to macrodispersive watershed processes with potentially much smaller values than classical microscale hydrodynamic dispersion. The lower P e is, the broader and the more asymmetrical the TTD will be. The last 2-parameter model considered is the uniform distribution corresponding to the linear piston flow model, which can represent the TTD in a partly confined aquifer [START_REF] Cook | Determining timescales for groundwater flow and solute transport[END_REF]. It is characterized by the two extreme times t 0 and t 1 .

As 3-parameter models, we have taken the weighted sum of two Diracs typically used in karstic systems [START_REF] Long | Translating CFC-based piston ages into probability density functions of ground-water age in karst[END_REF] or when considering two independent poles or reservoirs characterized again by the two times t 0 and t 1 and by the relative weight of the poles ω. Alternatively, we also consider the Shifted Inverse Gaussian distribution corresponding to an Inverse Gaussian model with a shifted origin t 0 . Figure 2 shows the diversity of shapes of the 7 LPMs. All these LPMs have the same mean. LPMs with two and more parameters have the same variance. LPMs with three parameters have additionally the same skewness. Given to illustrate the diversity of shapes, these LPMs have not been calibrated with any tracer data.

Calibrating LPMs on reference tracer concentrations

The 7 LPMs are calibrated to fit the reference concentrations C ref Tr derived from the in-silico model. As previously underlined, the number of tracer concentrations in a sampling zone is made equal to the number of parameters of the model and is considered as a measure of the quantity of available information. For this analysis, we have taken the three most different atmospheric tracers available on the last 50 years period introducing them in this order (CFC-11, 85 Kr, SF 6 ) to get the best possible amount of potential information. In fact the more different the atmospheric concentration chronicles, the more complementary information can be extracted. For each N-parameter LPM (where N equals either 1, 2 or 3) we seek for the parameters (π 1 , • • • , π N ) that solve the system of N equations:

For n = 1 • • • N, C ref Trn = t 0 C Trn (t -u) f (π 1 ,••• ,π N ) (u) du (4)
where N is the number of parameters and Tr n is the nth tracer

(if n = 1, Tr 1 is CFC-11, if n = 2, Tr 2 is 85 Kr and if n = 3, Tr 3 is SF 6 ).
We use the Levenberg-Marquadt method of Matlab to find the set of

parameters (π 1 , • • • , π n ) that minimize the objective function χ: χ = N n=1 ν n (π 1 , • • • , π N ) C ref Trn 2 (5)
where:

ν n = C ref Trn (t) - t 0 C Trn (t -u) f (π 1 ,••• ,π N ) (u) du. (6)
We also choose the starting parameters for the Levenberg-Marquardt al- To evaluate the goodness of fit, we use the maximum of the relative residuals over the N reference tracer concentrations ρ:

ρ = max n∈J1;NK |ν n | C ref Trn . (7)
ρ, expressed in %, gives the relative error made on the tracer concentrations C ref Trn . If it is lower than 10%, we consider the fit acceptable as 235 experimental uncertainties on field measurements are of the same order of magnitude [START_REF] Cook | Recent advances in dating young groundwater: Chlorofluorocarbons, H-3/He-3 and Kr-85[END_REF]. We choose to express the residual with the maximum of the residuals obtained to calibrate one tracer so that the values taken by ρ do not depend on the number of parameters of the LPM.

The maximum defining ρ highlights more the possible deviations of the model than the quadratic norm of the errors.

Predictive performances of LPMs

We derive the quartiles of the calibrated LPMs. As for the reference TTD, the quartiles of the LPMs Q

(π 1 ,••• ,π N ) 1 , Q (π 1 ,••• ,π N ) 2 and Q (π 1 ,••• ,π N ) 3
correspond to the prediction of the renewal time at which 25% (resp. 50% and 75%) of the water in the aquifer is renewed. To evaluate the predictive performance of the LPMs we compare the reference renewal times given by TTDs to the prediction given by LPMs. We introduce the comparison criterion θ (given in years) :

θ = 1 3 3 i=1 |Q ref i -Q (π 1 ,••• ,π N ) i | (8)
θ assesses the quality of the predictions given by a LPM on the quartiles.

Small θ values indicate a good agreement. The analysis of the performance criterion θ will be made over all the aquifer with the different LPMs model.

This methodology is generic and can be applied to any aquifer model on which transport simulations can be performed. The most important limitation comes from the characteristic time ranges of the available atmospheric tracers. In the following, the analysis is relevant for recent circulations (less than 60 years in practice). Extension to longer times with different sets of tracers is discussed in section 5.

The 3D numerical model of Ploemeur pumping site

The site of Ploemeur has several advantages for this study. It is a 3D crystalline aquifer with circulations of the order of some tens of years. Although developing in complex fractured geological structures, the flow pattern is fully 3D but remains mostly convergent to the actual pumping site. Moreover, it has been studied for almost 20 years and calibrated flow models have been developed [START_REF] Carn | Mise en valeur des ressources en eau souterraine du socle breton[END_REF][START_REF] Le Borgne | Assessment of preferential flow path connectivity, and hydraulic properties at single-borehole and cross-borehole scales in a fractured aquifer[END_REF][START_REF] Touchard | Caractérisation hydrogéologique d'un aquifère en socle fracturé: Site de Ploemeur (Morbihan)[END_REF].

Description of the 3D crystalline aquifer model

This study is based on one of the models of the Ploemeur aquifer developed by [START_REF] Leray | Contribution of age data to the characterization of complex aquifers[END_REF]. The Ploemeur aquifer is located in the south of Brittany (France) within a crystalline basement. It is produced from a single well since 1991 at 110 m 3 /h for the drinking water supply of the nearby city.

The site is composed of four major structures (Figure 3). Two granitic plutons outcrop North and South of the site acting as no-flow boundaries.

The recharge of 200 mm/year in average leaks into semi impervious overlying micaschists. A gently dipping fractured zone further drains recharge to the pumping well. This transmissive structure at the contact between the southern granite and the micaschists, named the contact zone, constitutes the aquifer. Another transmissive structure, a subvertical fault oriented North 20˚, while clearly marked on the geological map, only slightly impacts the flow pattern [START_REF] Leray | Contribution of age data to the characterization of complex aquifers[END_REF]. The pumping well is located at the intersection between the subhorizontal contact zone and the subvertical North 20˚fault. Table 2 sums up the properties of the hydrogeological model used.

The flow and transport model used in this study refers to the "deep" model of [START_REF] Leray | Contribution of age data to the characterization of complex aquifers[END_REF]. It has been calibrated with data at the pumping well of the mean piezometric level (-5.5 m above sea level) and of the CFC-12 atmospheric concentration measured in 2009 (30 years) [START_REF] Leray | Contribution of age data to the characterization of complex aquifers[END_REF].

Transport simulations are performed by forward-tracking particles along the steady-state velocity field (advective transport only), mixing occurring by convergence of the flow lines within the sampling zones. Numerical methods were performed using the h2olab platform [START_REF] Erhel | A parallel scientific software for heterogeneous hydrogeoloy[END_REF] and are thoroughly described in [START_REF] Leray | Contribution of age data to the characterization of complex aquifers[END_REF]. TTDs display a wide variety of shapes (Figure 5) closer to the exponential distribution at the well (dW=0) and more Inverse Gaussian like at intermediary distance (dW=600 m) or shifted exponential further away (dW=1700 m). Whatever the sampling zone, we have checked that TTDs converged to a steady shape with the number of particles used. In practice, at least 400 particles should intersect the sampling zone to reach convergence. This tar-

In

305

get is attained with an overall injection of 10 6 flow-weighted particles on the free surface. Most of the time, the number of particles crossing the sampling zones is well above the 400 particle threshold, and of the order of 10 4 to 10 5 .

Derivation of the apparent ages per the in-silico model

Reference tracer concentrations are computed for CFC-11, SF 6 and 85 Kr in all the sampling zones at year 2010. Maps of the different apparent ages are built by interpolating these 73 reference tracer concentration values (Figure 6). Following the differences in the slopes of C Tr , we recover that ÂKr < ÂSF 6 < ÂCFC as already indicated by [START_REF] Waugh | Relationships among tracer ages[END_REF]. For the 85 Kr however, we had to consider the slope of the input tracer concentration corrected from its radioactive decay: C Tr e -λ• . In this case, the slope of the corrected atmospheric tracer concentrations

d dt C CFC < d dt C SF 6 < d dt C Kr e -λ• (9)
match the inverse of the order of the different ages. Finally, maps of the different renewal times (so that 25, 50 and 75% of the water is renewed) across the whole watershed are derived from the in-silico modeling (Figure 7). Q 1 , Q 2 and Q 3 all display the same overall South-North gradient of times corresponding to the deepening of the aquifer. Secondary tendencies come from the topography of the site. These are the riddles in Q 1 around 20 years where the relatively smaller renewal times around 20 years (cyan color) can be correlated with the topographic heights, which have also been shown to be correlated with higher gradients and faster circulations [START_REF] Leray | Contribution of age data to the characterization of complex aquifers[END_REF]. It is also the systematic deviation towards the west of the smaller renewal times following the delineation of the watershed identified by the purple line on Figure 4. The eastern limit of the watershed is well marked on Q 1 by the sharp east-west time gradient 500m and 1000m north of the well. It is also marked on Q 3 by the renewal time increase to over 60 years beyond the watershed limit.

Results

We present the results according to the number of atmospheric tracers used for specifying the TTD. As said in the methodological section 2, our primary objective is to determine how prediction capacities taken as aquifer renewal times depend on the quantity of available information in the atmospheric tracer data. The analysis is performed independently on the 73 TTDs obtained in the contact zone of the Ploemeur aquifer described in section 3 with the 7 LPMs models of section 2.2. Depending on the LPM shape, on the number of parameters, we determine the goodness of fit ρ in percentage (equation 7) and the prediction accuracy θ in years (equation 8). We present systematically the mean and standard deviations of ρ and θ over the 73 sampling zones denoted as ρ, θ, σ(ρ), σ(θ), as well as the percentage of the zones for which the discrepancy of the observed versus modeled groundwater ages exceeds 10% (P (ρ > 10%)) (table 3).

1-parameter LPMs calibrated on CFC-11

Statistics on ρ and θ are summarized in the upper part of table 3. The residual for the Dirac distribution ρ is always equal to 0. This is expected because by centering the Dirac on the apparent age given by the CFC-11, the residual is equal to 0 i. e. the Dirac distribution perfectly fits the CFC-11 tracer observable. At the opposite, for the exponential distribution, the mean and standard deviation of the residual (ρ and σ(ρ)) are very high. They are above the 10% acceptable limit for 45% of the sampling zones. The spatial distribution of ρ (Figure 8) shows that ρ is high away from the discharge zone Results for the Q 2 and Q 3 are very similar and are thus not displayed.

While highly attractive because characterized by only one parameter, the Dirac and exponential models are relevant only in very limited zones of the aquifer. For the Dirac distribution, these zones are very far away from the pumping zone. For the exponential model, the relevant zones are only those very close to the pumping well. In between these two cases, other LPMs should be applied with more than one parameter.

2-parameter LPMs calibrated on CFC-11 and 85 Kr

Fitting the 2-parameter models to CFC-11 and 85 Kr concentrations gives both much better fits and predictions (middle part of Table 3). For both models, the fit of the concentrations is excellent with only 3 to 4 zones out of the 73 sampling zones for which ρ is larger than 10%. It means that both models are good for fitting concentrations almost everywhere in the aquifer.

Statistics on the prediction accuracy θ gives surprisingly very similar results for the Inverse Gaussian, Shifted Exponential and Uniform models despite the broadly differing shapes of the distributions (Figure 2). Average prediction accuracy θ is of 3.5 years for groundwater ages of the order of decades (Table 3). Standard deviations of the accuracy σ(θ) are small and less than 2 years. Models do not perform well in only some restricted areas in the north of the aquifer far away from the pumping well where velocities are substantially smaller and travel times larger (Figure 10). Outside of these areas, predictions are two to three times more accurate than for the 1-parameter Exponential and Dirac models and lead to only small errors in absolute values. A few years of differences can be deemed acceptable compared to the broad time ranges of the TTD, which can be evaluated at 37 years by the span of the uniform distribution [t 0 ; t 1 ]. So the accuracy in terms of predictions is of the order of a few years on some decades.

The agreement with the reference tracer concentrations measured by ρ displays relatively more differences between the Inverse Gaussian, Shifted

Exponential and Uniform models. The inverse Gaussian gives on average much better agreements than the two other distributions. The uniform model leads to larger deviations than the 2 other models. 12% of the zones have more than 10% deviations to the reference concentrations (P (ρ > 10%) in Table 3). Maximum deviation is about 38%, more than twice larger than for the 2 other models. This is consistent with the characteristic shape of the TTDs that looks more inverse Gaussian than uniform, and in a less extent than the shifted exponential (Figure 5). While the inverse Gaussian model is more representative of the TTD, it does not yield higher predictive capability, as if predictions depend more on some basic characteristics of the distribution like its two first moments than on its precise shape. It is confirmed by the spatial distribution of the 25% renewal times (Figure 10). All three LPMs lead to very similar patterns. There are only somewhat steeper gradients for the exponential shape likely coming from the initial irregularity of its shape.

3-parameter

LPMs calibrated on CFC-11, 85 Kr and SF 6

The 3-parameter Shifted Inverse Gaussian leads to slightly less accurate calibration than the 2-parameter Inverse Gaussian (Table 3). It is relatively more difficult to find a 3-parameter distribution calibrated on the CFC-11, 85 Kr and SF 6 than a 2-parameter distribution calibrated on just the CFC-11 and 85 Kr. This confirms that there is an additional information contained in the SF 6 concentration compared to the sole CFC-11 and 85 Kr concentrations despite the apparent similarities of the SF 6 and 85 Kr atmospheric concentration chronicles. The additional information obtained using SF 6 improves the mean accuracy of the renewal time predictions θ by only 0.8 years while keeping its standard deviation σ(θ) unchanged (Table 3). The gain in terms of predictions is marginal. It does not necessarily mean that the additional information is itself marginal, as the accuracy with the 2-parameter models is already very good and any improvement can only be marginal.

The other 3-parameter Dirac combination can closely fit the atmospheric tracer concentrations. The fitting criterion ρ around 0,01 % is lower than the temporal resolution of the atmospheric tracer concentration and can thus be considered negligible. Predictions are however not as good as with the 2parameter inverse Gaussian or 3-parameter shifted inverse Gaussian. Spatial renewal time predictions especially show irregular variations of the predictions likely related to the inability of the Dirac distributions to match the smoother variations of the TTD (Figure 11). This shows that the Lumped Parameter Model should have a shape related to the real shape of the TTD.

As shown with the comparison of the 2-parameter models, the LPM does not have to follow all the variations of the TTD. It should not be however completely off as shown here. The shape of the distribution taken as the prior in the Bayesian framework remains a key factor of the prediction accuracy and should be carefully chosen depending on the hypothesized flow pattern likely to happen.

Discussion

The previous results show that the quality of the calibration does not grant the accuracy of the predictions. The lowest ρ values are obtained for the Dirac distribution (residual always equal to 0) and for the sum of two Dirac distributions. However, they give the worst predictions of renewal times. It is not a matter of the number of parameters as the sum of two Dirac has three parameters. It is rather the lack of relevance of the Dirac distribution away from the recharge zones, where it is in fact expected to be relevant. In the discharge zone, it still perfectly fits the observed CFC-11 concentration but it is no longer relevant. More generally, the concentration residual ρ obtained with the seven candidate LPMs cannot be linked to the accuracy of the predictions. While ρ is interesting because it is the sole quantitative criterion that can be obtained in the field, it is neither a necessary nor a sufficient condition for the accuracy of the predictions.

We use the quartiles of the renewal time distribution to assess the quality of the predictions rather than the full distribution. The quartiles correspond to the renewal time of (25, 50 or 75 %) of the water within the sampling zones.

This choice is motivated by the two following reasons. First, quartiles give numerical indicators of practical interest while comparing shapes of LPMs as in [START_REF] Eberts | Comparison of particle-tracking and lumped-parameter age-distribution models for evaluating vulnerability of production wells to contamination[END_REF] is more qualitative and remote. Quartiles can be straightforwardly used to build statistics on prediction errors as in table 3.

Quantitative indicators could still be derived from the comparison of the LPMs and TTDs shapes like the Root Mean Square Error (RMSE). However they can be artificially large because of the irregularities of the TTDs (see figure 2) that cannot be accurately reproduced by any reduced model [START_REF] Green | Accuracy of travel time distribution (ttd) models as affected by ttd complexity, observation errors, and model and tracer selection[END_REF]. Besides, these irregularities are of little influence to integrated predictions like renewal times. The second motivation for using quartiles is indeed their integrative nature. Management decisions depend on global indicators like renewal times, flushing and dilution capacities rather than on detailed but approximate distribution shapes. Quartiles correspond to the time after which 25, 50 or 75% of the water reaching a piezometer or well have been flushed. On top of the median (second quartile Q 2 ) Green et al.

(2014), we also use the first and third quartiles to enhance sensitivity of the predictions to the variance and skewness of the TTDs.

It is not the concentration fit nor the number of parameters of the model but rather the likelihood of the distribution shape that conditions the accuracy of the predictions (table 3) as also concluded by [START_REF] Green | Accuracy of travel time distribution (ttd) models as affected by ttd complexity, observation errors, and model and tracer selection[END_REF].

Thus "likely" distributions imply considering analytic distributions with at least two parameters. Indeed, LPMs with only one parameter (Dirac, exponential) give accurate predictions only in very narrow sections of the aquifer.

2-parameter LPMs on the contrary give already very accurate predictions that cannot be made much better using an additional third parameter. The 2-parameter LPMs capture in various ways the mean and standard deviation of the distribution, predictions becoming almost insensitive to higher order moments. Because they are flexible and can evolve continuously from Dirac to broad distributions, the 2-parameter LPMs are appropriate almost everywhere in the aquifer as shown by the similarity of Figures 7c and12b. There are only some spurious variations of Q 3 that are corrected by the 3-parameter shifted inverse Gaussian model. In fact the residence time distributions display some initial shift outside of the pumping area linked to the transit time of the water through the micaschist aquitard zone to get to the aquifer sampling zones (Figure 5). But even this striking distribution feature is not essential for establishing good predictions.

The distribution of Q 1 prediction errors represented on Figure 13 Appropriate distributions should be continuous and relatively broad covering a wide range of transit times. Beyond these general characteristics, the details of the distributions have little importance. The Inverse Gaussian and the Shifted Exponential distributions give the same prediction quality. We can show that it is a general result. We consider an Inverse Gaussian distribution with parameters T and P e respectively equal to 20 and 3/2. From equation 1 we compute the tracer concentrations and ages obtained with such a TTD and find ÂIG CFC = 1986 and ÂIG Kr = 1997. With the method developed in section 2.2.2, we determine the parameters T and t 0 of the shifted exponential distribution that fits best these concentrations (equation 5). The residual ρ is equal to 3%. The two distributions (Figure 14) are very close and their cumulative density function lead to extremely close predictions.

Similar conclusions have been drawn by [START_REF] Waugh | Relationships among tracer ages[END_REF] with an inverse Gaussian and a combination of two inverse Gaussian distributions.

As the number of parameters has been taken equal to the number of available concentrations, the practical consequence of the previous results is to seek for two tracers in the same age range to reach satisfying predictions. Indeed, one tracer concentration is not enough to determine two parameters. Two tracer concentrations are necessary to give the complementary information needed to inform the mean and standard deviation of the TTD in this given age range, consistent with the theoretical result of [START_REF] Massoudieh | The theoretical relation between unstable solutes and groundwater age[END_REF]. Redundant tracers remain of key interest for checking potential contamination, excess air and geogenic production, and for reducing the uncertainties inherently linked to measurement errors [START_REF] Green | Accuracy of travel time distribution (ttd) models as affected by ttd complexity, observation errors, and model and tracer selection[END_REF][START_REF] Massoudieh | Bayesian evaluation of water systems[END_REF]. To ensure complementarity, the atmospheric tracer chronicles should cover comparable date range and be as different as possible as shown by the convolution product (see equation 1) of C in T exp -λ• by the TTD f . [START_REF] Leray | Numerical modeling of the productivity of vertical to shallowly dipping fractured zones in crystalline rocks[END_REF] show that knowing apparent ages of two CFCs [START_REF] Broecker | Comparison of Ar-39 and C-14 ages for waters in the deep ocean[END_REF] first calibrated LPMs with both of these tracers. [START_REF] Waugh | Relationships among tracer ages[END_REF] studied the dependence of the [ ÂAr , ÂC ] to the mean and standard deviation of an Inverse Gaussian TTD. For much older water, [ 81 Kr, 36 Cl] (for transit times bigger than 10 000 years) should be tested. Illustrated here on the site of Ploemeur, the methodology developed in this article can be applied to a broad variety of aquifer models. We may expect different outcomes when mixing different characteristic transit times on a broader range (e.g. [0;1000] years) either from different aquifers or from an aquitard and an aquifer. In such cases the combination of Dirac distributions might be more appropriate. Two tracers on different age ranges (for example CFC-11 and 39 Ar) might be tested to characterize different parts of the TTD. This is a valuable method to identify potential mixing of younger and older waters in the aquifer but cannot be apply directly to determine the shape of the TTD. Indeed, the younger tracer characterizes the local flow (with transit times under 100 years) whereas the older tracer characterizes more regional flow with longer transit times. Other multitracer approaches could also be tested by considering the ratio of the same date ranging tracer concentrations.

Indeed, such ratios have the advantage to be independent of the mixing affecting the groundwater. With two ratios, in the same way as what have been done previously, one can characterize the piecewise TTD on the support of the date ranging of these tracers. Building two ratios however requires to provide at least three independent tracer concentrations and not two.

Moreover, the choice of predictive indicators gains in importance as soon as the TTD gets broader (i. e. with an increasing coefficient of variation, ratio of standard deviation upon average). In this case, more precise metrics (deciles or even centiles) instead of quartiles might be considered to get indicators with useful predictive power.

Conclusion

We develop a methodology to assess Lumped Parameter Model prediction capacities in realistic 3D aquifer cases. Lumped Parameter Models (LPMs) are simple analytical distributions designed to approximate natural transit time distributions (TTDs). LPMs are appealing for establishing predictions directly from groundwater age data. Predictions are generally framed in some kind of aquifer renewal times and do not require involved hydrogeological models. Our methodology assesses the prediction capacities of LPMs on realistic 3D aquifer synthetic models. Aquifer modeling cannot be avoided as reference TTDs as well as reference renewal time predictions cannot be directly measured on the field. Beyond LPMs assessment, we aim at finding the right balance between the information available in groundwater age data and the knowledge required to establish accurate predictions. We simply rate the amount of available information as the number of parameters that can be calibrated on a set of groundwater age data. When considering gas molecules like CFCs, SF 6 and 85 Kr with sufficiently different atmospheric concentration chronicles, the amount of available information is the number of parameter of the LPM chosen equal to the number of tracers.

We apply this methodology on the kilometric-scale crystalline aquifer of Ploemeur (France). The aquifer is located in a sloping contact zone between granites and micaschists overlain by a semi-pervious aquitard. Flow pattern is globally convergent to a well steadily produced for the nearby city. A previous model calibrated on head and age data shows that local topographical as well as geological features yield to complex 3D flow structures. It results in a variety of Transit Time Distributions ranging from Dirac-like close to recharge zones to exponential-like at the well. Outside of these zones, distributions cannot be evidently modeled as one of the classically used LPM like the composition of Diracs, the shifted exponential or the inverse Gaussian distributions. Extensive analysis on 73 sampling zones located everywhere in the aquifer however shows that LPMs can be highly effective for establishing accurate predictions. The 2-parameter inverse Gaussian or shifted exponential LPMs calibrated on CFC-11 and 85 Kr concentrations both yield almost everywhere accurate renewal times at less than 10% precision. By comparison, the 1-parameter exponential and Dirac LPMs calibrated on CFC-11 could not give any relevant predictions in most of the aquifer. The 3-parameter models calibrated with the additional knowledge of SF6 do not lead to any better prediction.

For the type of studied crystalline aquifer system, we conclude that two tracers with sufficiently different atmospheric concentration chronicles give highly accurate renewal time predictions, provided that hydrogeologically relevant LPMs be used. Relevant LPM shapes are also necessary to yield to unbiased estimations. In this case, the relevant LPMs should be broad and relatively smooth like the shifted exponential or inverse Gaussian distributions. More details on the distribution provided by additional tracers are not necessary for the type of renewal time predictions targeted. On the contrary, distributions made up of some Dirac combinations do not give any close predictions despite good agreements with the reference tracer concentrations. More extensive analysis should be undertaken in a wider variety of realistic aquifer cases to assess the representativeness and relevance of these conclusions. Uniform (Linear Piston Flow )

f (T, ) (t) = 1
for T -2 ≤ t < T + 2 , 0 otherwise.

3

Weighted sum of Diracs f (t 0 ,t 1 ,ω) (t) = ω δ(tt 0 ) + (1ω) δ(tt 1 ) Shifted Inverse Gaussian Table 1: LPMs used in this study with their number of parameters N, their expression and a sketch of the associated circulation. The name referring to this LPM according to [START_REF] Maloszewski | Lumped parameters models for interpretation of environmental tracer data[END_REF] is recalled in brackets.

  We have chosen the Dirac and exponential distributions as 1-parameter models. Physically, the Dirac distribution corresponds to the piston flow model (PFM), where tracers are advected along a single flowpath without any mixing within the flow line nor with the other flow lines. At the other extreme, the exponential model (EM) characterizes a well-mixed flow and is relevant to TTDs at convergent and outflow zones (pumping site, discharge). Both distributions are defined by a single parameter T equal in both cases to their mean. A straightforward extension to 2-parameter models is the Shifted Exponential distribution corresponding to the exponential piston flow model (EPM) and representing a combination of exponential and piston flow models. Such a distribution can represent a system where recharge occurs in an unconfined zone (EM) and tracers are then advected in a confined part of the aquifer (PFM). It is defined by the same exponential constant T and by the time lag t 0 . The other classical 2-parameter model is the Inverse Gaussian distribution. It is a solution to the classical advection dispersion equation (ADE) and is characterized by the distribution mean T and by the Peclet number P e

  1-parameter LPMs are calibrated on the reference CFC-11 tracer concentration (C ref CFC ). 2-parameter LPMs are calibrated on the reference CFC-11 and 85 Kr tracer concentrations (C ref CFC and C ref Kr ). 3-parameter LPMs are calibrated on the reference CFC-11, 85 Kr and SF 6 tracer concentrations (C ref CFC , C ref Kr and C ref SF 6 ).

  230gorithm not far from the global minimum of the objective function χ by pre-calibrating the first moments of the LPM. This ensures faster and more systematic convergence to the global minimum of χ.

  -silico reference results for the Ploemeur aquifer 3.2.1. Derivation reference TTDs per the in-silico model Following the methodology of section 2.1, reference TTDs are obtained using the previously described hydrogeological model. Sampling zones have been designed in such a way as to stay close to field measurements conditions (Figure4). They are parallelepiped that cross the contact zone over all its thickness to sample the aquifer. 73 sampling zones of characteristic width of 250 meters map the full extent of the aquifer contributing to the pumping zone. TTDs are built on all the flow lines crossing the sampling zones and are located at the center of the box. The issued TTDs fundamentally represent the residence time of the solutes flowing through the sampling zones around their center analogously with field sampling where abstracted water comes from a limited zone around the piezometer or well. Their size and number have been set in order to meet computational capacities with enough flow lines crossing the sampling zones and not too many sampling zones for fitting the seven LPMs.

  of actual renewal times per the in-silico model

  but always under 10% close to the convergence zone of the aquifer where a broad range of transit times are collected. The limits of the exponential distribution come from the large weight given to the more recent atmospheric tracer concentrations, which represent the largest values (Figure1). Considering that the CFC-11 atmospheric concentration is essentially increasing over the last 75 years, the minimum concentration that can be obtained with any exponential model is the one obtained when the exponential model tends to a uniform distribution over the last 75 years. This minimum concentration is thus equal to the mean atmospheric concentration of CFC-11 over the same period corresponding to an age given by equation 2 of about 40 years.Away from the discharge zone, apparent ages in the watershed larger than 40 years cannot be fitted by the exponential distribution. The same type of inadequacy occurs for all other atmospheric tracers like 85 Kr or SF 6 .Predictions of the renewal time of 25% of the groundwater to the sampling zones (Q 1 ) show that neither the exponential nor the Dirac distribution model (Figure9) can give everywhere accurate predictions of the reference renewal times displayed by Figure7a. For the Dirac distribution, the quartiles Q 1 , Q 2 and Q 3 are taken as the characteristic time of the Dirac distribution.The Dirac is predictive only very close to the diverging recharge areas while the exponential model is accurate only in the discharge converging areas.Conversely the Dirac LPM overestimates the renewal times while the exponential LPM underestimates them in most of the aquifer. None of the model can give even close predictions away from the pumping and recharge zones.

  for the tested distributions confirm these conclusions. 1-parameter LPM errors are much larger than those of the 2 and 3-parameter LPMs. Errors are considerably reduced from 1-parameter LPMs to 2-parameter LPMs. 3-parameter LPMs do not significantly improve the predictions. When the shape of the LPM deviates more from the reference TTD, errors become somewhat larger like for the Dirac composition and uniform distributions. The error distribution of Figure13additionally reveals that 1-parameter LPMs prediction errors are not symmetric (centered on 0). The Dirac LPM tends to underestimate renewal times while the exponential LPM tends to overestimate them. This comes from the specific shape of these LPMs. The exponential distribution overestimates the large transfer times to fit the right mean. On the opposite, the Dirac distribution underestimates the variability, underestimates Q 1 predictions and overestimates Q 3 predictions. In comparison to that, prediction errors of 2-parameter LPMs and of the 3-parameter shifted inverse Gaussian appear to have error in predictions centered on 0, which means that in average, predictions are not biased by a tendency to over-or under-estimate renewal times.

  did not bring much to calibrate an hydrological model since their atmospheric chronicles were too similar. Tracer couples with more different atmospheric chronicles should be preferred. For recent waters ([0;60] years) preferential couples are: [CFC, 85 Kr], [CFC,SF 6 ], [CFC, 3 H/ 3 He], [SF 6 , 3 H/ 3 He] or [ 85 Kr, 3 H/ 3 He]. For an aquifer with longer characteristic transit times, other couples of tracers should be considered despite the technological challenges. [ 39 Ar, 14 C] could be used in a range of time of [500 ; 2000] years. In oceanography,

Figure 2 :

 2 Figure 2: Comparison of the seven candidate LPM models as functions of the time t normalized by the mean time T (t/T). All distributions have the same mean and all distributions with more than one parameter have the same variance, but display quite different shapes.

Figure 3 :

 3 Figure 3: Conceptual hydrogeological scheme of the Ploemeur aquifer with the different geological units. Granites in the North and in the South are shown in brown. The micaschists (green) overly the contact zone (red) and all units are crosscut by the N20 fault (block dashed line). The pumping site is identified by the red triangle. Blue arrows sketch the characteristic flow direction.

Figure 4 :

 4 Figure 4: Map of numeric sampling strategy used to get observables and TTDs values overall the aquifer. Boxes are 250 x 250 m. Their heights are determined to span the contact zone.

Figure 5 :

 5 Figure 5: Transit time distributions obtained with the in-silico model of the Ploemeur site. dW is the distance between the sampling zone and the pumping well. NP is the number of particles intersecting the sampling zone used to build the transit times distributions.

Figure 6 :

 6 Figure 6: Map of the "apparent" recharge dates obtained with the in-silico model in the contact zone of the Ploemeur site. Contour of the map is determined by taking the supplying limit of the pumping well. From left to right: CFC-11, SF 6 and 85 Kr apparent ages. The pumping well is identified by the black square.

Figure 7 :

 7 Figure 7: Map of the different renewal times (from left to right, times at which 25, 50 and 75% of the water is renewed) obtained with the in-silico model. The pumping well is identified by the black square.

Figure 8 :

 8 Figure 8: Map of the residual ρ expressed in % obtained by fitting CFC-11 concentrations C ref CFC with concentrations obtained from an exponential TTD. The black line identifies the limit for which the residual ρ is equal to 10%. The Exponential model gives acceptable residual only on the lower southern part of the aquifer close to the pumping site identifiable by the black square.

Figure 9 :

 9 Figure 9: Characteristic renewal time of 25% of the water to the sampling zone (Q 1 ) for the 1-parameter (a)Dirac LPM and (b) exponential LPM. The insert in the topright corner of the figure shows a sketch of the corresponding LPM.

Figure 10 :

 10 Figure 10: Characteristic renewal time of 25% of the water to the sampling zone (Q 1 ) for the 2-parameter LPMs. From left to right, times derived by the shifted exponential model, by the inverse gaussian model and by the uniform model. The insert of each of the figure show on sketch of the LPM.

Figure 11 :

 11 Figure 11: Characteristic renewal time of 25% of the water to the sampling zone (Q 1 ) for the 3-parameter LPMs. From left to right, times derived by the sums of Dirac model and by the shifted inverse gaussian model. The insert of each of the figure show on sketch of the LPM.

Figure 12 :

 12 Figure 12: Characteristic renewal time of 75% of the water to the sampling zone (Q 3 ). From left to right, times derived by a 1-parameter LPM (exponential model), by a 2parameter LPM (inverse gaussian) and by a 3-parameter LPM (shifted inverse gaussian). The insert of each of the figure show on sketch of the LPM.

Figure 13 :

 13 Figure13: Histogram of the differences in predictions between reference Q 1 (25% renewal time) and Q 1 predictions given by the LPMs.

Figure 14 :

 14 Figure 14: Distribution and cumulative density function associated of an Inverse Gaussian distribution with (µ, λ)=(20,30) years and of the Shifted Exponential distribution that fits the observables C IG CFC and C IG Kr obtained with the Inverse Gaussian distribution.

groundwater age distribution using radioactive tracers and anthropogenic chemicals. Water Resources Research, 48 . doi:{10.1029/2012WR011815}.

McCallum, J., Cook, P., Simmons, C., & Werner, A. (2014a). Bias of apparent tracer ages in heterogeneous environments. Ground Water , 52 , 239-250.

McCallum, J., Engdahl, N., Ginn, T., & Cook, P. (2014b)

Parameters Values References

Potential recharge rate R [mm/year] 200 [START_REF] Carn | Mise en valeur des ressources en eau souterraine du socle breton[END_REF][START_REF] Touchard | Caractérisation hydrogéologique d'un aquifère en socle fracturé: Site de Ploemeur (Morbihan)[END_REF]

Highlights

-Analysis of Transit Time Distributions (TTDs) on a crystalline aquifer model.

-Prediction of renewal times of the aquifer from environmental tracers.

-High predictive capacities of lumped models with two parameters requiring two tracers.

-Shape of the TTDs is not critical beyond the first 2 moments of the distribution.!


B Nettier

L Dobremez

G Brunschwig Irstea

R Baumont

"Alpine Pasture-Farms System": a concept to analyse the management of alpine pastures

Keywords:

Adapting to climate change on alpine pastures requires finding flexibility at different levels. The articulation between the alpine pasture and the farms is where potential solutions might be activated. Yet research focusing on alpine pasture management usually considers the farm as a black box. The use of an alpine pasture -which is sometimes collective -nevertheless lends a number of specific characteristics to livestock farming systems. The distance between the alpine pasture and the farmsteads, the mixing up of herds/flocks, the necessary collective organisation, as well as the challenges stemming from the multiple use of these collective spaces, must be taken into account in an analysis of these systems. We propose the concept of the Alpine Pasture-Farms System to analyse the consequences of these specific characteristics in analyses at farm level and to gain a better understanding of interactions between the alpine pasture and farms.

I -Introduction

For many mountain and Mediterranean livestock farming systems, alpine pastures represent the main forage resource during summer. The pastoral resources sought out by livestock farming systems on the alpine pastures are in fact especially vulnerable to climate change and to the associated increase of extreme weather events, especially droughts. Management requires to be adapted, in order to prevent long-term deterioration that would jeopardise both the resource and biodiversity on alpine pastures.

With the challenges of biodiversity, alpine pastures are the focus of a number of concerns. However, climate change does not impact alpine pastures alone. The increasing occurrence of droughts has taken its toll on all spaces used by farms whose flocks and herds have been moved around the Alps since the start of the 2000s [START_REF] Nettier | Attitudes of livestock farmers and sensitivity of livestock farming systems to drought conditions in the French Alps[END_REF]. Livestock farmers have had to react and change their farm practices. Such changes may have repercussions on the alpine pastures. Conversely, the adaptations implemented to preserve alpine pastures may have an impact on the farms [START_REF] Nettier | Attitudes of livestock farmers and sensitivity of livestock farming systems to drought conditions in the French Alps[END_REF]. The current context questions the usual production models and calls for much more flexibility in livestock farming systems [START_REF] Dedieu | L'élevage En Mouvement. Flexibilité et Adaptation Des Exploitations D'herbivores[END_REF].

In order to seek out opportunities for flexibility faced with climate change, a systemic approach seems necessary. The relevant system for analysis is the system made up of the alpine pasture and the farms using it. This is because climate change impacts all of the components making up this system and changes to part of the system are likely to have an impact on the whole. The second reason is that opportunities for improving the flexibility of the system can be found at the articulation between the alpine pasture and the farms [START_REF] Nettier | Managing the Summer Alpine Pastures in a Context of Recurrent Droughts[END_REF]. We therefore propose to focus on the organisation of the entire "Alpine Pasture-Farms System" (APFS).

II -Differences between the alpine pasture-farms system and the livestock farming system

Research on the management of alpine pasture address the alpine pasture as a system [START_REF] Deffontaines | Pour un diagnostic d'alpages : Une traduction de l'expertise des bergers en langage cartographique. L'exemple de l'alpage du Saut du Laire[END_REF] and the farm(s) remain(s) a black box sending up a certain number of animals to the pasture for a certain period time. Conversely, research on livestock farming systems [START_REF] Darnhofer | Farming Systems Research into the 21st Century: The New Dynamic[END_REF], which studies the trio "Man-Herd-Resource" at the scale of the farm [START_REF] Landais | Introduction à l'étude des systèmes d'élevage extensifs[END_REF], does not consider the systems using alpine pastures as specific systems. Yet they are systems with specific characteristics, stemming at once from the very specific soil and climate context of alpine pastures, their land status as common areas, and the often collective organisation of summer grazing. Given these specific traits, we need to take a more nuanced approach to the "Man-Herd-Resource" model.

"Men": a specific work organisation

In the case of an APFS, the spatial and temporal disconnection between the alpine pasture and the farmstead leads to specific forms of work organisation. Furthermore, one or several herders may be employed. The herder(s) may only work on the alpine pasture, and is/are entrusted with the management of grazing and the state of health of the animals. Such specialisation for a worker in terms of time, space and tasks performed is rare in livestock farming systems.

The APFS is often made up of several farms and therefore several work collectives who from time to time are brought together up on the alpine pastures to perform various tasks (maintenance duties, animal drafting, etc.). Thus, the alpine pasture relieves the farm(s) of animal-related work, which can be essential in summer. However, the alpine pasture also sets a work scheduling, with an impact on the whole work organisation of the different farms.

Herds (or flocks): combining different types of livestock management

Using an alpine pasture has an impact on livestock management: issues such as mating time and batching are thought through in relation to the alpine pasture (whether the animals can be fattened, whether they can give birth, the quality of the feeding resources, work organisation and issues such as predation). In the case of collective APFS, the animals of several farms are mixed up together on the pasture. This has consequences in terms of health management: specific attention must be given to the health of the animals before they are taken up to the pasture to prevent them contaminating other herds or flocks, different specifications must be complied with (organic farming, quality labels, etc.) during health treatment. The diversity of resources is an essential component in the management of agropastoral systems. In the case of an APFS, this diversity is highly pronounced owing to the considerable altitudinal layering and geographic distance that may exist between the different areas. The combination of resources from several farms increases this diversity, especially given that the farms are located in different geographic contexts, use different types of vegetation, are subject to different climates and weather events, and have different productions. Diversity also stems from resource management modes: on the alpine pasture, unlike in most other areas of an APFS, grazing is often the only management "tool".

Resources: capitalising on their diversity

Management: combining common and individual areas

In most situations, the APFS combines the use of resources used individually on the farms, and common pastoral resources available on the alpine pasture. The latter resources are common owing to their pastoral use, which is sometimes collective, but also because of their land status (most often public areas). In consequence the plants grazed should not only be seen as pastoral resources for livestock farming, but also as biodiversity, landscape and cultural heritage and therefore belong to society as a whole. APFS management must therefore include these other issues, providing sustainable management of both individual and collective resources. In particular, the APFS management must pay attention to the preservation of the commons. The pastoral use of alpine pastures is thus subject to specific rules outlined in grazing agreements and agri-environmental contracts, which differ from the usage rules for individual areas. When an alpine pasture is collective, it is generally coordinated by a collective body representing the different farms. This coordination is essential in order to ensure the right kind of articulation between work, management of herds/flocks and resources, and between the different farms and the alpine pasture.

III For a range of flexibility in Alpine Pasture-Farms systems

To preserve the vegetation on alpine pastures in the face of climate change, without undermining the agropastoral systems, one major avenue that might be explored is how to make APFS operation more flexible. Potential flexibility may be applied to the alpine pasture, but also to the farmsteads and to the articulation between the alpine pasture and the farm(s) [START_REF] Nettier | Managing the Summer Alpine Pastures in a Context of Recurrent Droughts[END_REF]. These three sources of flexibility provide an analysis framework that was tested on 9 APFS in the Ecrins National Park in France, as part of the Alpages Sentinelles programme (a network of actors from the Alpine massif working to anticipate the impact of climate change and assess adaptation modes set up to face unexpected climate events).The observation of adjustment practices to counter different events over the past 5 years on these 9 APFS makes it possible to analyse their flexibility, and to position them on a graph (see figure 1) according to the level of flexibility implemented on the alpine pasture (vertical axis) and on the farms (horizontal axis), with or without adjustments at the articulation between the alpine pasture and the farm in order to preserve the alpine vegetation (see table 1 for two examples).

This study of APFS may be of interest to all agropastoral systems using a diversity of resources, articulating individual and collective management and using both common and individual areas.

Flexibility on farms

Long winters, important slopes: No room for manoeuvre: mechanised areas reserved for making hay, sloppy rangelands for spring grazing.

Important adjustments (in response to Mediterranean climatic variability): flocks management, pastoral areas used, grazing schedule.

Flexibility on alpine pasture

Size of the different range allotment reduced, buffer areas limited and best range allotment reserved for lambs: grazing schedule very rigid.

Despite abundant pastoral resource, many constraints limit and rigidify management : lower range allotment short, with late vegetation, higher one big but short grazing period, middle one exposed to predation (no hut).

Adjustments observed at interaction between alpine pastures and farms

Difficult to decrease the animal number or the time spent on alpine pasture. Few evolutions of the management on farms and on alpine pasture.

No flexibility on alpine pasture, or on farms, autonomous management of the different entities.

Important evolutions observed: (1) going up to alpine pasture delayed since 2005, (2) reorganization on farms, with new pastoral areas grazed (ski slopes, firebreaks…).

Lack of flexibility on alpine pasture. Structural adjustment made to preserve it on long-term. Annual adjustments made only on the farms.

  Nettier, B. ; Dobremez, L. ; Brunschwig, G. "Alpine Pasture-Farms System": a concept to analyse the management of alpine pastures. In : Forage resources and ecosystem services provided by Mountain and Mediterranean grasslands and rangelands. Baumont R., Carrère P., Jouven M., Lombardi G.,López-Francos A., Martin B., Peeters A., Porqueddu C. (eds.). Joint Meeting of the "Mountain Pastures, Mediterranean Forage Resources (FAO/ESCORENA-CIHEAM) and Mountain Cheese" Network, 24-26/06/2014 Clermont-Ferrand

Fig. 1 :

 1 Fig. 1: APFS flexibility analysis framework and relative position of 9 APFS "Sentinelles" in the Ecrins National Park. The boxes list ideas for improving flexibility.

Table 1 . Description and flexibility of 2 alpine pastures-farms systems

 1

		APFS 9	APFS 6
	Description	5 sheep farms in a mountain village,	3 pastoral sheep farms in South-East of France,
	of the APFS	that use a local alpine pasture. Spring	that transhume to a distant alpine pasture.
		and autumn lambing. Feeding-finishing	Spring lambing only. Lambs go up on alpine
		of spring born lambs on alpine pasture.	pasture. Farmers take turns on alpine pasture.
		Salaried shepherd on alpine pasture