Martin Le Mesnil
email: martin.lemesnil@gmail.com

Jean-Baptiste Charlier

Roger Moussa

Yvan Caballero

Investigating flood processes in karst catchments by combining Concentration-Discharge relationship analysis and lateral flow simulation

Keywords: Concentration-discharge, karst, storm, hysteresis, lateral flow

Introduction

Runoff at the catchment outlet has different origins depending on the water flow paths in the surface, subsurface, and underground compartments. The joint study of hydrological and hydrochemical signals through concentration-discharge (C-Q) relationships can help improving our understanding of these origins and associated hydrological processes. Depending on the context and nature of investigations, C-Q relationship analysis can be applied to different variables, the most frequently monitored parameters being major ions or continuous physicochemical variables such as electrical conductivity, turbidity, alkalinity, or pH (e.g. [START_REF] Rose | Comparative solute-discharge hysteresis analysis for an urbanized and a 'control basin' in the Georgia (USA) Piedmont[END_REF].

A synthesis by [START_REF] Knapp | Concentration–discharge relationships vary among hydrological events, reflecting differences in event characteristics[END_REF], based on results obtained in a Swiss mountainous catchment, 2 summarizes the main environmental drivers of C-Q relationships at the storm-event and longer time scales. The main approaches encountered in the literature for the study of these relationships are end-member mixing analysis [START_REF] Burns | Quantifying contributions to storm runoff through end-member mixing analysis and hydrologic measurements at the Panola Mountain Research Watershed (Georgia, USA)[END_REF][START_REF] Doctor | Quantification of karst aquifer discharge components during storm events through end-member mixing analysis using natural chemistry and stable isotopes as tracers[END_REF], statistical approaches [START_REF] Anderson | Concentration-discharge relationships in runoff from a steep, unchanneled catchment[END_REF] and hysteresis studies [START_REF] Evans | Component flow processes at four streams in the Catskill Mountains, New York, analysed using episodic concentration/discharge relationships[END_REF][START_REF] Evans | Component flow processes at four streams in the Catskill Mountains, New York, analysed using episodic concentration/discharge relationships[END_REF][START_REF] Rose | Comparative solute-discharge hysteresis analysis for an urbanized and a 'control basin' in the Georgia (USA) Piedmont[END_REF][START_REF] Rose | Concentration-discharge relationships describe solute and sediment mobilization, reaction, and transport at event and longer timescales[END_REF]. They led to advances in the understanding of aquifer recharge and vulnerability [START_REF] House | Hysteresis of the solute concentration/discharge relationship in rivers during storms[END_REF][START_REF] Huebsch | Mobilisation or dilution? Nitrate response of karst springs to high rainfall events[END_REF], streamflow contributions [START_REF] Ribolzi | Contribution of groundwater and overland ¯ows to storm ¯ow generation in a cultivated Mediterranean catchment. Quanti®cation by natural chemical tracing[END_REF], and storm-event spring discharge [START_REF] Toran | Using Stormwater Hysteresis to Characterize Karst Spring Discharge[END_REF].

Most C-Q analyses are interpreted in terms of a two-component system involving pre-event water (PEW) and event water (EW) [START_REF] Evans | Component flow processes at four streams in the Catskill Mountains, New York, analysed using episodic concentration/discharge relationships[END_REF][START_REF] Rose | Comparative solute-discharge hysteresis analysis for an urbanized and a 'control basin' in the Georgia (USA) Piedmont[END_REF], or include a third component defined as soil or rapid infiltration water [START_REF] Evans | Component flow processes at four streams in the Catskill Mountains, New York, analysed using episodic concentration/discharge relationships[END_REF][START_REF] Ribolzi | Contribution of groundwater and overland ¯ows to storm ¯ow generation in a cultivated Mediterranean catchment. Quanti®cation by natural chemical tracing[END_REF][START_REF] Burns | Quantifying contributions to storm runoff through end-member mixing analysis and hydrologic measurements at the Panola Mountain Research Watershed (Georgia, USA)[END_REF][START_REF] Toran | Using Stormwater Hysteresis to Characterize Karst Spring Discharge[END_REF]. While most C-Q studies use the concentration values of different streamflow source to perform the hydrograph separation, some use an a priori separation to infer source concentration values [START_REF] Neira | A combined mixing model for high-frequency concentration-discharge relationships[END_REF].

At the catchment scale, hysteresis analyses allow studying C-Q relationships, using C and Q data at a given monitoring station. Hysteresis loops are mainly characterized using classifications of loop shapes, such as the widely used 6-type proposed by [START_REF] Evans | Component flow processes at four streams in the Catskill Mountains, New York, analysed using episodic concentration/discharge relationships[END_REF], or more recent 9-type ones [START_REF] Butturini | Diversity and temporal sequences of forms of DOC and NO3-discharge responses in an intermittent stream: Predictable or random succession?[END_REF]Heathwaite and Bieroza, 2020) using hysteresis indices (see a synthesis by [START_REF] Lloyd | Technical Note: Testing an improved index for analysing storm discharge-concentration hysteresis[END_REF]. These approaches make it possible to investigate streamflow contributions during storm events, knowing concentration ranking of supposed end-members [START_REF] Rose | Comparative solute-discharge hysteresis analysis for an urbanized and a 'control basin' in the Georgia (USA) Piedmont[END_REF][START_REF] Vaughan | High-frequency dissolved organic carbon and nitrate measurements reveal differences in storm hysteresis and loading in relation to land cover and seasonality[END_REF]. When monitoring several gauging stations for Q and C along the stream, it is possible to assess C-Q relationship spatial variability at nested catchment scale. It is also possible to characterize it based on a reach scale approach (between two gauging stations). In that case, water origin and flood hydrological processes can be investigated by analyzing lateral exchanges, i.e. stream water gains or losses from or to groundwater. Such analyses were performed by [START_REF] Covino | Stream-groundwater exchange and hydrologic turnover at the network scale[END_REF] and [START_REF] Mallard | Lateral inflows, stream-groundwater exchange, and network geometry influence stream water composition[END_REF], using artificial tracing experiments to investigate stream composition changes. Analytical approaches also exist, such as the advection-diffusion equation (ADE), widely used to model conservative solute transport [START_REF] Runkel | Solution of the Advection-Dispersion Equation: Continuous Load of Finite Duration[END_REF][START_REF] Baeumer | Subordinated advection-dispersion equation for contaminant transport[END_REF][START_REF] Hauns | Dispersion, retardation and scale effect in tracer breakthrough curves in karst conduits[END_REF][START_REF] Luhmann | Comparing conservative and nonconservative tracers in karst and using them to estimate flow path geometry[END_REF]. This equation is mathematically similar to the diffusive wave equation (DWE) and both can be resolved using the Hayami analytical solution (1951), making it possible to simulate lateral flows Q and C [START_REF] Cholet | Assessing lateral flows and solute transport during floods in a conduit-flow-dominated karst system using the inverse problem for the advection-diffusion equation[END_REF].

Karst catchments are located in carbonate areas (including limestone and dolomite) and cover 20% of Europe land surface and 35 % in France [START_REF] Goldscheider | Global distribution of carbonate rocks and karst water resources[END_REF]. They are known to be complex permeable hydrosystems, involving high rainfall infiltration rates through open conduits and promoting fast groundwater flow and significant surface water-groundwater interaction [START_REF] Bakalowicz | Karst groundwater: a challenge for new resources[END_REF]. In such areas, electrical conductivity (EC) provides information on groundwater residence time and water origin, as it is controlled by bicarbonate, calcium, and magnesium concentrations resulting from carbonate rock dissolution [START_REF] Hess | Storm response of the karstic carbonate aquifer of southcentral Kentucky[END_REF][START_REF] Lambán | Isotopic and hydrogeochemical characterization of high-altitude karst aquifers in complex geological settings. The Ordesa and Monte Perdido National Park (Northern Spain) case study[END_REF][START_REF] Liu | Seasonal, diurnal and storm-scale hydrochemical variations of typical epikarst springs in subtropical karst areas of SW China: Soil CO2 and dilution effects[END_REF]. Because EC can be easily monitored at high-frequency time steps, it is a useful variable for the characterization of C-Q relationships, particularly in karst catchments. Indeed, the contrast between water end-members is notable: lowmineralization EW from surface runoff or fast groundwater flow, and high-mineralization PEW from groundwater present in the aquifer prior to the storm event. The range of EC values is variable, depending on groundwater residence time or mixing with fast rainwater infiltration, and allow identifying PEW and EW occurrence. In the particular case of karst catchments, an additional challenge is to distinguish EW from direct surface runoff on low permeability areas and EW from fast infiltration in open conduits feeding rivers through springs [START_REF] Hartmann | Risk of groundwater contamination widely underestimated because of fast flow into aquifers[END_REF]. Moreover, unexpected EC variations can occur during high-flow periods, as groundwater boundaries extend and incorporate areas with different EC values [START_REF] Ravbar | Anomalous behaviour of specific electrical conductivity at a karst spring induced by variable catchment boundaries: the case of the Podstenjšek spring, Slovenia[END_REF].

Various methodologies have been used to explore EC-Q relationships during storm events in karst catchments. Early studies mainly describe EC variations during storm events, an EC drop being interpreted as an increasing EW contribution in streamflow, and complex fluctuations corresponding to several simultaneous karst conduit contributions [START_REF] Hess | Storm response of the karstic carbonate aquifer of southcentral Kentucky[END_REF]. Statistically-based methods exist, such as EC frequency distribution analysis (Massei et al., 2007) which makes it possible to investigate the relative contributions of surface water and groundwater to streamflow and how they are influenced by hydrological conditions. Characterization of EC-Q hysteresis loops is also used, which makes it possible to study delays between discharge and solute concentration variations, bringing to light the changing hydrological processes at the scale of a storm event, or between different events [START_REF] Fournier | Using turbidity dynamics and geochemical variability as a tool for understanding the behavior and vulnerability of a karst aquifer[END_REF][START_REF] Toran | Using Stormwater Hysteresis to Characterize Karst Spring Discharge[END_REF]. Hysteresis loops of hydrographs at different locations within a karst catchment were also studied to identify hydrological flows at nested catchment scale [START_REF] Zhang | Using hysteretic behaviour and hydrograph classification to identify hydrological function across the "hillslope-depressionstream" continuum in a karst catchment[END_REF]. Analysis of EC and Q patterns at the storm-event time scale have also highlighted the importance of seasonal variability in the physico-chemical response of a karstic spring [START_REF] Fournier | Using turbidity dynamics and geochemical variability as a tool for understanding the behavior and vulnerability of a karst aquifer[END_REF]. Recently, EC-Q relationships at the reach scale were studied by [START_REF] Cholet | Assessing lateral flows and solute transport during floods in a conduit-flow-dominated karst system using the inverse problem for the advection-diffusion equation[END_REF] using lateral exchange modeling to investigate conduit-matrix relationship variability during storm events according to seasons. Each methodology helps better understanding hydrological processes, and we identify a strong interest in combining different approaches and to investigate both nested and reach scales to improve the conceptual model of karst catchments.

The aim of this article is to assess the potential of a combined approach to investigate the spatio-temporal variability of flood processes within karst catchments. To do that, we combine two spatial scales of investigation: at the nested catchment scale, a classification of C-Q hysteresis loops adapted to EC measurement in karst context; at the reach scale, an inverse modeling of lateral flow Q and C. Using this combined approach, we will define what can be learned from C-Q hysteresis loop analysis in karst catchment storm events, and what additional information can bring the modeling of lateral flow Q and C. To answer these questions, we apply the two-step approach on hourly records of Q and EC during storm events at 4 gauging stations in two karst catchments in France, characterized by contrasting climate and karst area extension.

Results are interpreted in terms of water origin using PEW and EW contributions during storm events. Finally, conceptual models are proposed, that summarize the main information obtained with the combined approach, regarding spatio-temporal variability of flood processes.

Methodology

General methodology

To propose a conceptual model of flood water origin spatio-temporal variability, a twopart methodology was applied. The parts correspond to two spatial scales: i) the nested catchment scale considering the topographic catchment upstream a gauging station, and ii) the reach scale considering the catchment delimited by two gauging stations. Both parts rely on hourly Q and C data collection during storm events, the difference being that reach scale analysis requires data from both inlet and outlet stations. To analyze temporal variability, the methodology was applied on storm-events grouped by seasons. Figure 1 presents this general methodology, from field measurement network and monitored variables to data processing.

At the nested catchment scale, C-Q loops are analyzed for each storm event according to the methodology described in section 2.2. This analysis is performed at each monitoring station of the hydrograph network and characterizes flood processes occurring in the whole upstream topographic catchment. It is thus a lumped approach at the catchment scale, and successive analyses towards the final outlet are interpreted in terms of flood process variability of nested catchments. Even so, a more spatially discrete approach at the reach scale is necessary to refine and spatialize this variability description.

At the reach scale, the hourly lateral flow discharge (QL) and concentration (CL) variations are simulated using the DWE inverse model, detailed in section 2.3, assuming that exchanges are uniformly distributed along river reaches. Although this may not be the case in karst systems, this approach allows a more spatialized investigation of the location of lateral flows, which commonly lacks in C-Q studies. This reach-scale approach, consists in comparing obtained mean CL values to PEW and EW end-member C values. More details regarding definitions of these end-members are given in section 3.3.

[Figure 1] A season-based storm-event typology is proposed, which allows interpreting hydrological response variability. Four seasons are defined, based on typical hydrological dynamics of the study sites: low water table and first storm events in September and October, high precipitation volume winter events from November to March, lower intensity events from April to May, and occasional summer storm events from June to August. Finally, a conceptual model that associates flood processes to different seasons and karst configurations is proposed, in addition to a schematic representation of the spatial variability of water origin along river reaches.

Nested catchment scale approach: C-Q loop analysis

This section presents a C-Q loop classification that allows characterizing hysteresis loops obtained from the monitoring of various physico-chemical parameters (concentration, turbidity, EC, …). [START_REF] Evans | Component flow processes at four streams in the Catskill Mountains, New York, analysed using episodic concentration/discharge relationships[END_REF] proposed a 6-type classification of C-Q loops based on the loop's general trend and its rotational direction. This classification was used in later C-Q studies [START_REF] Rose | Comparative solute-discharge hysteresis analysis for an urbanized and a 'control basin' in the Georgia (USA) Piedmont[END_REF][START_REF] Vaughan | High-frequency dissolved organic carbon and nitrate measurements reveal differences in storm hysteresis and loading in relation to land cover and seasonality[END_REF][START_REF] Zhang | Using hysteretic behaviour and hydrograph classification to identify hydrological function across the "hillslope-depressionstream" continuum in a karst catchment[END_REF]. Building on Evans and Davies (1998), we use a modified loop typology, enriched by the integration of a hysteresis index quantification and the inclusion of non-hysteresis pathways. Similar 9-type classifications have been proposed and shown to be relevant for C-Q relationship studies [START_REF] Butturini | Diversity and temporal sequences of forms of DOC and NO3-discharge responses in an intermittent stream: Predictable or random succession?[END_REF]Heathwaite and Bieroza, 2020). We calculated two indicators for loop classification: slope (s), obtained by applying a linear regression to the C-Q values, and hysteresis index (HI).

The hysteresis index proposed by [START_REF] Lloyd | Technical Note: Testing an improved index for analysing storm discharge-concentration hysteresis[END_REF] is used in this work. It provides information on both the loop direction and its amplitude. Moreover, it considers the whole range of discharge values and is calculated based on normalized Q and EC values, which facilitates consistent inter-event and inter-site comparison. For a given storm event, Q and EC values are normalized between 0 and 1, and at each 5% interval (i) of normalized Q, a local HIi is calculated following eq. 1:

= - (1)
Where ECnRi and ECnFi are normalized EC values at discharge interval i, taken on the rising limb and falling limb of the loop, respectively. Figure 2 [Figure 2]

Once threshold values are defined (values of slope and HI that delineate the classification types), measure slope and HI make it possible to relate any storm event to one of the nine categories of the classification presented in Figure 3, using its C-Q loop. The loop typology is built on two-character names, the first corresponding to HI sign (and consequently to rotational direction: C for clockwise, N for neutral and A for anticlockwise), and the second corresponding to the slope sign (-for negative s, 0 for nil s, and + for positive s). This classification based on [START_REF] Butturini | Diversity and temporal sequences of forms of DOC and NO3-discharge responses in an intermittent stream: Predictable or random succession?[END_REF] was adapted to the case of EC-Q in karst catchments.

In the case of karst areas, event water (EW) having a lower residence time than pre-event water (PEW), its EC value is lower. Indeed, EC is a reliable proxy of bicarbonates concentration, which is linked to limestone dissolution and thus residence time [START_REF] Peyraube | SIc-Abacus: An in-situ tool for estimating SIc and Pco2 in the context of carbonate karst[END_REF].

Therefore, we will use our EC-Q loop typology to determine patterns of successive contributions to streamflow (PEW and EW), assuming the EC ranking is known.

Each loop type is linked to a specific pattern of successive contributions to streamflow, based on the main findings in the literature of C-Q studies. [START_REF] Evans | Component flow processes at four streams in the Catskill Mountains, New York, analysed using episodic concentration/discharge relationships[END_REF] showed that, in a two-component system, when the solute concentration of PEW is higher than that of EW, anticlockwise C-Q loops indicate a dominant contribution of EW during the rising limb, and a dominant contribution of PEW during the falling limb (scenario 1). On the other hand, clockwise C-Q loops indicate a dominant PEW contribution during the rising limb and a dominant EW contribution during the falling limb (scenario 2). This C-Q loop taxonomic consistency was verified experimentally by [START_REF] Chanat | Consistency of patterns in concentration-discharge plots[END_REF], and was shown to be reliable when concentration values of end-members are distinct enough, which is the case here as surface runoff water and groundwater are compared (see values at end of section 3.3). Regarding the specific case of EC-Q hysteresis, [START_REF] Toran | Using Stormwater Hysteresis to Characterize Karst Spring Discharge[END_REF] observed anticlockwise loops during storm events at a karst spring discharge in Pennsylvania and verified that they were linked to Scenario 1 based on Ca 2+ and Mg 2+ concentration monitoring. [START_REF] Rose | Concentration-discharge relationships describe solute and sediment mobilization, reaction, and transport at event and longer timescales[END_REF] linked Scenario 2 to clockwise C-Q loops thanks to geogenic solute monitoring in a small USA catchment. In the case of monitoring of solutes in which EW is enriched, such as NO3 -from the organic horizon [START_REF] Evans | Component flow processes at four streams in the Catskill Mountains, New York, analysed using episodic concentration/discharge relationships[END_REF][START_REF] Huebsch | Mobilisation or dilution? Nitrate response of karst springs to high rainfall events[END_REF], the rotational directions are reversed.

Based on previously described findings from the literature and in our case of EC monitoring in a karst catchment where the PEW solute concentration is higher, our hypothesis is that clockwise EC-Q loops correspond to PEW preceding EW and anticlockwise EC-Q loops correspond to EW preceding PEW. This is particularly clear with loops that have slopes equal to zero (C0 and A0 loops), rising and falling limbs showing EC values exclusively above or below the initial value prior to a storm event. In the case of non-zero loop slopes, one limb may have mixed water origins. A mixed water signature, in the case of karst systems where fast infiltration can occur, may result from low residence time PEW recently stored in the karst (i.e., from several days or weeks) or from EW infiltrating through preferential pathways and mixing with highly mineralized PEW (i.e. several months or years old). Regarding non-hysteretic EC-Q paths, both limbs have a similar water origin, this latter being inferred from the slope (a negative slope indicating EW contribution whereas a positive slope indicating PEW contribution).

This methodology is applicable only when the initial EC value falls between the two endmember EC values. This observation was systematically verified in the framework of this study (the initial EC value always being higher than the EW end-member EC value in karst areas).

More information on end-member EC values is provided in section 3.3. On Figure 3 that presents the nine loop types of our classification, a framework of interpretation of associated streamflow contribution patterns is also proposed based on residence time, which is inferred from EC values.

Mixed water can be linked either to a mix of low and high residence time PEW or to a mix of EW and high residence time PEW.

[Figure 3]

Reach-scale approach: lateral flow simulation

To investigate the origin of lateral inflows along river reaches (i.e. between two gauging stations), we used the modeling approach proposed by [START_REF] Cholet | Assessing lateral flows and solute transport during floods in a conduit-flow-dominated karst system using the inverse problem for the advection-diffusion equation[END_REF] based on the diffusive wave equation (DWE) and extended to the advection-diffusion equation (ADE). Both equations are resolved using the Hayami analytical solution (1951) as proposed by [START_REF] Moussa | Analytical Hayami solution for the diffusive wave flood routing problem with lateral inflow[END_REF]. We provide in this section the key points of the approach. More details on this analytical approach are provided in the Appendix.

Separate modeling of lateral discharge QL and mass flux ML make it possible to assess lateral solute concentration CL by division, as shown in equation 2.

=

(2) QL is simulated using the solution of the inverse problem of the DWE and ADE, assuming that lateral flow is uniformly distributed along the river reach [START_REF] Moussa | Analytical Hayami solution for the diffusive wave flood routing problem with lateral inflow[END_REF]. ML is obtained by applying the DWE to mass flux, this latter being calculated as the product of QL and CL (equation 2). Assuming that total dissolved solids (TDS) consists mainly of conductive ionic compounds, EC is proportional to the TDS concentration (noted C, in g.m -3 ). Concentration C is therefore calculated from EC, using a constant factor of 0.64, in accordance with values found in the literature [START_REF] Lloyd | Natural inorganic hydrochemistry in relation to ground water[END_REF]:

= • 0.64 (3)
Mass flux M is then calculated as the product of concentration C and discharge Q:

= • (4)
Finally, the simulated lateral conductivity ECL is calculated as the ratio of simulated lateral mass flux ML to simulated lateral discharge QL, divided by a factor of 0.64:

= • . (5)
As some river reaches can show negative QL values, indicating lateral streamflow losses, ECL is calculated only for positive QL values. Indeed, negative QL values would lead to negative ECL values, which is impossible. ECL variations thus make it possible to compare water origin on successive streams within a catchment.

Study areas and data sets

Study areas

The previously described methodology was applied to two catchments in France, which are partially to strongly karstified, and located in different geological and hydrometeorological settings (figure 4). They are similar in size (~ 1000 km²) and both include four Q and EC measurement stations.

The Loue River basin is located in the Jura Mountains. Its outcrops consist primarily of extensively and homogeneously karstified Jurassic limestone and locally marl, making it a unary karst basin. The Lison River is the major left bank Loue tributary and oth rivers start as a karst spring. The Loue spring being partly fed by Doubs river losses that occur in the southeastern part of the catchment [START_REF] Charlier | Appui au SCHAPI 2014 -Module 1 -Rôle et contribution des eaux souterraines d'origine karstique dans les crues de la Loue à Chenecey-Buillon[END_REF], it is not a typically unary karst basin. The term of unary karst basin is used here as the allogenic recharge is also from a similar karstified area. The Loue and Lison springs are among the largest karst springs in Europe [START_REF] Chen | The World Karst Aquifer Mapping project: concept, mapping procedure and map of Europe[END_REF], with mean discharge of around 10 and 7 m 3 .s -1 , respectively, and maximum peak flows of around 80 and 70 m 3 .s -1 . Precipitation follows an elevation gradient; annual values range from 1600 mm on the upstream catchment at elevations of 900 m (a.s.l) to 1450 mm at the outlet, at an elevation of around 400 m (a.s.l).

The Cèze catchment is located in the eastern part of the Cévennes Mountains. It can be characterized as a binary karst basin with upstream areas that consist of non-karst outcrops (i.e.

hard rocks) where mean yearly precipitation is 1500 mm and median and downstream areas underlain by karst plateaus that receive precipitation of approximately 1000 mm/year. The karstified portion is composed of early Cretaceous limestone (dark green, figure 54b) that is mainly situated between the Tharaux and Laroque stations, and extends southward. More information on exchanges between river streams and karst systems in this area can be found in the work of [START_REF] Chapuis | Caractérisation, évaluation, modélisation des échanges entre aquifères karstiques et rivières -Application à la Cèze[END_REF].

[Figure 4]

Data sets

Detailed information on the data time series is available in the supplementary material.

Temporal data used in this paper are:

• Hourly streamflow data, available from the French public streamflow database 'Banque Hydro' (http://www.hydro.eaufrance.fr).

• Hourly rainfall data, available from the Antilope database [START_REF] Champeaux | Quantitative precipitation estimations using rain gauges and radar networks: inventory and prospects at Meteo-France[END_REF], a reanalysis produced by the French public meteorological service Météo

France (http://www.meteofrance.fr/).

• Sub-hourly EC measurements for the Cèze river, recorded by the BRGM using Schlumberger CTD diver probes from July 2019 to July 2020. Similar data recorded using OTT MS5 probes within the QUARSTIC network [START_REF] Charlier | Projet QUARSTIC : QUAlité des eaux et Réseau de Surveillance des rIvières Comtoises[END_REF] for the Loue River, from January 2016 to January 2020.

As available data time series on the studied catchments are not of identical length and because of contrasted hydro-climatic contexts, 58 storm events were extracted for the Loue catchment, against 8 for the Cèze catchment. In the Loue catchment, storm events have a median precipitation depth of 58 mm (min: 10 mm and max: 136 mm) and a median peak flow value of 78 m 3 .s -1 (min: 2 m 3 .s -1 and max: 529 m 3 .s -1 ). In the Cèze catchment, storm events have a median precipitation depth of 65 mm (min: 30 mm and max: 180 mm) and a median peak flow value of 128 m 3 .s -1 (min: 20 m 3 .s -1 and max: 780 m 3 .s -1 ).

EC end-member values

For the Loue catchment, the PEW end-member EC is defined as the signal for water having the longest residence time, thus having the highest EC value (at 550 μS.cm -1 ) recorded during the acquisition period at the spring monitoring stations (Ouhans and Nans, see figure 4a).

The Loue catchment EW end-member EC for surface runoff is defined at 250 μS.cm -1 , based on values recorded on an intermittent stream on marly areas. For the Cèze catchment, the PEW endmember is defined at 700 μS.cm -1 , based on highest values recorded at Cèze karst springs monitoring stations [START_REF] Chapuis | Caractérisation, évaluation, modélisation des échanges entre aquifères karstiques et rivières -Application à la Cèze[END_REF]. The Cèze catchment EW end-member EC for surface runoff is defined at 75 μS.cm -1 , based on minimum values recorded during the acquisition period at the upstream monitoring station (Tharaux, see figure 4b) which is only fed by non-karst areas. These illustrative events were chosen as they are representative of the generally observed EC and Q variations at each site. Regarding the Loue catchment, figure 6a shows a storm event associated with an EC drop that takes place mostly after peak flow and leads to a C-loop type indicating an event-water (EW) contribution that occurs mainly at the end of peak flow and during recession. Figure 6b shows that other storm events on the Loue catchment are associated with an EC increase, indicating pre-event water (PEW) mobilization. On the pictured stormevent, EC variations are quite similar to Q variations, leading to a flatter EC-Q loop.

Results

EC-Q relationships at the catchment scale

Nevertheless, EC values are slightly higher on the falling limb, leading to an identifiable A+ loop type.

Regarding the Cèze catchment, figure 6c illustrates a storm event associated with an EC decrease, indicating dilution by event water and leading to a C-loop type, as shown in figure 6a.

Finally, figure 6d represents a storm event with strongly anticorrelated EC and Q variations during both rising and falling limbs, leading to a non-hysteretic N-loop type.

[Figure 6] [Figure 7]

Table 1 summarizes the main EC-Q loop types on the Loue and Cèze catchments for all seasons. In the Loue catchment, predominantly clockwise EC-Q hysteresis loops (C-and C0)

occur during winter and spring, while anticlockwise loops (A0 and A+) dominate in summer and fall. Conversely, the Cèze catchment stations show mostly non-hysteretic behavior in winter (Nloops) and show clockwise loops (C-and C0) in summer and fall. As these results provide information for nested catchments, their analysis on successive stations can highlight spatial variability in flood water origin towards the final outlet. Even though EC-Q loop types are quite homogeneous for the four stations for a given catchment, some differences are observed. Regarding Loue catchment, the Nans and Chenecey stations have particularly low slope values. Indeed, except for summer events, most of their EC-Q loops have slopes lower than 1 (figure 7a). This indicates a greater EW effect at the Nans-Chenecey and Vuillafans-Chenecey (downstream reaches, see figure 4a). This is also the case regarding Cèze catchment, where lower slopes are characterized for Montclus. Because each nested catchment incorporates the upstream previous one, these intra-site trends are not totally straightforward and can hardly be quantified. ECL values decrease at the end of the period, indicating the decreased contribution of PEW.

EC of reach-scale lateral flow

[Figure 8] Figure 9 shows the distribution of simulated lateral flow EC on the Loue and Cèze River reaches. ECL mean values are calculated around the peak of lateral flow (QL > 0.75QL,max), so as to obtain a value that is representative of the major amount of lateral water inflow. Values of EW and PEW EC end members are also plotted on the graphs. As the simulation of ECL values is made using the DWE, it requires a simple reach with one input and one output. For this reason, the Nans-Chenecey reach is not represented (see the river confluence on figure 4a).

Regarding the Loue catchment (figure 9a), the upstream reach (Ouhans-Vuillafans) shows ECL values mainly above 400 μS.cm -1 , except for some winter storm events. Lateral water inflows are thus of mixed origin. On the downstream reach (Vuillafans-Chenecey), ECL values are lower, mostly less than 400 μS.cm -1 . This reach-scale approach provides more precise and discretized information of the streamflow contribution variability within catchment, and is consistent with the lower slope values measured at the Chenecey station in figure 7, indicating a higher EW signature. This can be explained by a higher contribution of either surface runoff downstream the canyon area (open valleys in the downstream parts of the catchment) or EW from fast infiltration pathways, or both. It is also noticeable that ECL distribution are more spreaded on the downstream reach, probably due to more intense or frequent water mixing.

On the Cèze catchment (figure 9b), ECL values span from 100 to 300 μS.cm -1 on the two median reaches (Tharaux-Montclus and Montclus-Laroque), whereas most ECL values on the downstream reach (Laroque-Bagnols) are lower (around 100 μS.cm -1 ). This pattern is consistent with its karst outcrop location, located between Tharaux and Laroque (figure 4b) and the EC river measurement (figure 5b).

[Figure 9]

Discussion

Conceptual model of flood water origin and process variability

Figures 10a to 10d show the main hysteresis loop types, associated hydrographs, and the scheme of flood water origin and processes for the Loue (unary) and Cèze (binary) karst catchments, for two contrasted periods in the hydrological cycle. Figures 10e and10f represent, within each catchment, the reach-scale variability of streamflow contribution. The conceptual model provides key insights into the influence of seasons and karst configuration on flood processes.

In the Loue unary karst catchment, mainly C-type EC-Q loops are observed from November to May (figure 10a), corresponding to a dominant contribution sequence of 1) mix of EW and PEW and 2) EW. This sequence can be explained by a two-step process involving 1) during the rising limb, EW infiltrating into karst drains and pushing PEW into the stream, associated with EW from fast infiltration or surface runoff, and 2) during the falling limb, EW that infiltrated into karst drains reaching the stream, associated with surface runoff. From June to October (figure 10b), mainly A+ type loops are observed, corresponding to a dominant contribution sequence of 1) mixed water and 2) PEW. This can be explained by a lower degree of saturation in the karst system, reducing surface runoff contribution, and showing a purely piston-type flow leaching the PEW initially stored in the aquifer.

In the Cèze binary karst catchment N-type loops are the most common between November and May (figure 10c), corresponding to a dominant contribution of EW during both rising and falling limbs. This pattern is explained by the lower storage capacity of smaller karst units compared to the Loue catchment. The high karst system saturation level leads to a blocked infiltration and promotes increased EW surface runoff. From June to October (figure 10d), C-EC-Q loops are the most common type observed, corresponding to a dominant contribution sequence of 1) mixed water and 2) EW. At this period of the year, the karst system is less saturated, which reduces the surface runoff signature associated with blocked infiltration, and shows PEW reaching streams by piston-type flow, associated with the EW influence, leading to similar processes as described for figure 10a.

[Figure 10] The surface runoff contribution to streamflow is less important in the Loue catchment compared to the Cèze one, as seen in previous sections. This fact is explained by the catchment's morphology, which is for the unary catchment a typical configuration of plateaus and canyons that promotes higher rainfall infiltration and groundwater contribution to surface flows (Le [START_REF] Mesnil | Interbasin groundwater flow: Characterization, role of karst areas, impact on annual water balance and flood processes[END_REF]. This type of surface water-groundwater interaction is promoted in other catchments made up of plateaus and canyons, where stream losses and/or lateral gains from springs occur according to hydrological conditions [START_REF] Bailly-Comte | Conceptualization and classification of groundwater-surface water hydrodynamic interactions in karst watersheds: Case of the karst watershed of the Coulazou River (Southern France)[END_REF][START_REF] De Waele | Flash flood hydrology in karstic terrain: Flumineddu Canyon, central-east Sardinia[END_REF][START_REF] Charlier | Quantifying peakflow attenuation/amplification in a karst river using the diffusive wave model with lateral flow[END_REF]. Our results confirm this behavior, which results in increased EC values and positive slope values of the hysteresis EC-Q loops. Because the saturation level of the karst aquifer also plays an important role on the mobilisation of PEW and EW, its low or high storage capacity can affect flood process variability. To interpret surface runoff contributions in the Cèze catchment, we need to refer to the two main types of flooding that are documented in the literature for karst areas: 1) infiltration excess runoff due to the low infiltration capacity of the karst medium [START_REF] Maréchal | Karst flash flooding in a Mediterranean karst, the example of Fontaine de Nîmes[END_REF], and 2) backflooding following infiltration with a simultaneous rapid rise of the aquifer water level due to a limited saturation capacity of the conduit network (Lopez-Chicano et al. 2002;[START_REF] Bonacci | Karst flash floods: an example from the Dinaric karst (Croatia)[END_REF][START_REF] Bailly-Comte | Time series analyses for Karst/River interactions assessment: Case of the Coulazou river (southern France)[END_REF]. The first process is linked to rainfall intensities, that are high under the specific Mediterranean climate of the Cèze river. The second one is more dependent on the storage capacity of the karst aquifer, which is small for the Cèze (compared to high elevation plateaux for the Loue). Superimposed on seasonal influence, physiographical factors also control intra-site variability of flood processes and catchment's response to storm events. Indeed, the extension of karst areas and topographic relief can both affect the water origin of streamflow. In the Loue catchment, the reach-scale lateral flow simulation (section 4.2) highlights a water origin spatial variability characterized by an increasing EW contribution to storm events in the downstream direction. This increased EW contribution is explained by the typical plateau-and-canyon morphology of the upstream portion grading into open valleys downstream, resulting in higher surface runoff.

The Cèze catchment, as shown in figure 4b, is characterized by a binary karst, that is, a delineated karstified area downstream of hard-rock headwater zones. Le [START_REF] Mesnil | Interbasin groundwater flow: Characterization, role of karst areas, impact on annual water balance and flood processes[END_REF]2021) showed that this specific karst location plays a role in the annual water budget, as well as in the flood pattern. In this study, EC monitoring shows that it also controls flood water origin, as illustrated by EC values increasing on the karst zone (figure 5b). This spatial variability within the Cèze catchment was detected by our reach-scale approach of lateral flow simulation, ECL values being increasing in the median reaches where the karst crops out. This approach shows that water being mobilized during floods has a higher PEW signature in this zone. Figures 10e and10f provide a sketch of the main features of streamflow contribution variability within each study site.

Pros and cons of the C-Q relationship analysis approach

Our approach makes it possible to differentiate storm events in which PEW may or may not be mobilized, thanks to EC monitoring. This approach is a suitable for assessing the role of groundwater in flooding, which in many cases is supposed to be mobilized without being quantified (e.g. [START_REF] Ascott | Improved understanding of spatio-temporal controls on regional scale groundwater flooding using hydrograph analysis and impulse response functions[END_REF]. For example, applying this methodology in the context of groundwater-induced flooding in chalk regions [START_REF] Finch | The spatial distribution of groundwater flooding in a chalk catchment in southern England[END_REF][START_REF] Pinault | Groundwater-induced flooding in macropore-dominated hydrological system in the context of climate changes[END_REF] or lowland karst areas (Jerome [START_REF] Morrissey | Modelling groundwater flooding in a lowland karst catchment[END_REF]) may be of great value, as well as for rivers that show complex and variable exchanges with underlying aquifers [START_REF] Bailly-Comte | Conceptualization and classification of groundwater-surface water hydrodynamic interactions in karst watersheds: Case of the karst watershed of the Coulazou River (Southern France)[END_REF][START_REF] Charlier | Use of a flood-routing model to assess lateral flows in a karstic stream: implications to the hydrogeological functioning of the Grands Causses area (Tarn River, Southern France)[END_REF]2019). Indeed, a better understanding of the groundwater role in flooding is essential for efficient management of flood hazards in karst areas. Several authors develop tools in this purpose [START_REF] Gill | Modeling a network of turloughs in lowland karst: Modeling a Network of Turloughs in Lowland Karst[END_REF][START_REF] Mayaud | Understanding flooding in poljes: A modelling perspective[END_REF]. A recent study of [START_REF] Mesnil | Impact of karst areas on runoff generation, lateral flow and interbasin groundwater flow at the storm-event timescale[END_REF] based on 108 gauging stations shows in the one hand that karst promotes generally peak-flow attenuation due partly to higher water infiltration from rivers, and in the other hand that the hydrological response is highly site-specific. The development of approaches investigating water origin as proposed in this paper gives interesting perspective to explore such variability and to facilitate their modeling, for example using semi-distributed models. Moreover, the proposed approach was kept generic enough to be applicable to a variety of contexts. This section provides a brief summary of the main features of this two-step methodology, what can be expected from its application, and what it requires.

The nested-catchment scale approach is integrative, as it provides information on the topographic catchment from a particular monitoring station to its upstream limit. Even so, analysis of results at successive monitoring stations provides information on water origin and flood process variability along the river. Loop slope and hysteresis index [START_REF] Lloyd | Technical Note: Testing an improved index for analysing storm discharge-concentration hysteresis[END_REF] of C-Q paths can be used to classify storm events through the use of a loop typology adapted and enriched from the work of [START_REF] Evans | Component flow processes at four streams in the Catskill Mountains, New York, analysed using episodic concentration/discharge relationships[END_REF]. Other hysteresis classifications have been proposed previously for karst areas, but include less types and do not allow to represent all storm processes: e.g. the 3-type T-C curve classification by [START_REF] Fournier | Using turbidity dynamics and geochemical variability as a tool for understanding the behavior and vulnerability of a karst aquifer[END_REF]. Our classification can be used with a variety of parameters (EC, concentrations, turbidity, …) involving different processes. In the framework of this study, it allowed determining flood water origin seasonal patterns on the two studied catchments. This first approach requires only concentration and discharge datasets at a consistent resolution (hourly for storm-event analysis, possibly wider time steps for other purposes) for one outlet station. It can thus be easily implemented at little cost (especially for EC continuous monitoring). This nested-catchment scale approach involves some limitations. First, the concentration ranking of potential contributing end-members must be known to make a consistent interpretation of the involved processes. Then, no more than two end members can be differentiated when monitoring one concentration variable. Finally, in the specific case of karst systems, the approach cannot always properly differentiate between EW that originates from surface runoff and EW that originates from fast infiltration through underground karst conduits. Indeed, localized infiltration in sinkholes for instance can be quick enough to keep water chemistry unchanged due to residence times of few hours/days. In this case, other techniques such as signal processing or baseflow separation might be necessary to figure out comprehensively which flood processes are involved, as well as analysis of alternative physicochemical data such as organic carbon to track infiltration water with short residence time [START_REF] Pronk | Percolation and Particle Transport in the Unsaturated Zone of a Karst Aquifer[END_REF][START_REF] Charlier | Conceptual hydrogeological model of flow and transport of dissolved organic carbon in a small Jura karst system[END_REF]. Some anomalous EC variations can also be witnessed during high-flow periods, as groundwater catchments expand and incorporate areas of different typical EC values [START_REF] Ravbar | Anomalous behaviour of specific electrical conductivity at a karst spring induced by variable catchment boundaries: the case of the Podstenjšek spring, Slovenia[END_REF].

The reach-scale approach is based on the inverse modeling approach [START_REF] Moussa | Analytical Hayami solution for the diffusive wave flood routing problem with lateral inflow[END_REF] using the diffusive wave equation. It allows simulating Q and C variations in lateral exchange flow of a river reach [START_REF] Cholet | Assessing lateral flows and solute transport during floods in a conduit-flow-dominated karst system using the inverse problem for the advection-diffusion equation[END_REF]. Analyzed along with end-member EC values, this analysis highlighted the increasing contribution of EW towards the downstream end of the Loue catchment and the major PEW contribution in the karst zone in the median area of the Cèze catchment. This analytical approach is very convenient as it allows investigating spatial variability of lateral exchanges without conducting extensive field work such as tracing tests (e.g. [START_REF] Covino | Stream-groundwater exchange and hydrologic turnover at the network scale[END_REF].

This second part is more data-dependent, as it requires similar datasets than the first part, but recorded at both inlet and outlet stations. Moreover, the approximate EC value of each end member must be known to interpret the results in terms of water origin mixing. Though, this reach-scale approach provides more precise and discretized information on spatial variability of streamflow contributions within catchments.

Conclusions

We applied a C-Q relationship analysis approach, using hysteresis loop classification at the nested catchment scale combined with lateral flow and concentration simulation at the reach scale. In the framework of this study, this combined method made it possible to establish a seasonal conceptual model of water origin during storm events for two contrasted karst catchments and to infer its spatial variability at the scale of the monitoring network.

Analysis of EC-Q hysteresis loops highlighted a flood water origin pattern, with successive contributions of pre-event water (PEW) and event water (EW) according to karst type (unary vs. binary) and seasonality (low flow periods vs. high flow periods). Simulation of lateral exchange flows led to a more detailed analysis of the water origin, spatializing it at the reach scale. This analysis highlighted the decreasing contribution of PEW in favor of EW as the canyons and plateaus that the Loue flows through grade into open valleys, the significant PEW effect on the karst area of the Cèze catchment, and the role of the aquifer saturation state in flood response.

Our results show that, in karst context the nested catchment scale analysis of C-Q hysteresis loops provides information on seasonality of flood processes, and trends of intracatchment variability. We also show that the reach scale lateral flow simulation provides a quantifiable information on intra-catchment variability of streamflow contributions. The tested approach is innovative and particularly suitable for partly karstified catchments, as it is semidistributed and uses EC measurements which are a reliable proxy of water residence time. Our results are encouraging to apply these complementary methodologies to a variety of sites of differing geology, with additional investigational purposes, such as water resource management and modeling. Indeed, such approach providing discretized information on flood processes within catchments could help refining lumped model structure, or facilitate the use of semidistributed models. Some perspectives are identified, as coupling the monitoring of multiple solutes, making it possible to differentiate additional end members. [START_REF] Moussa | Criteria for the choice of flood-routing methods in natural channels[END_REF][START_REF] Cholet | Assessing lateral flows and solute transport during floods in a conduit-flow-dominated karst system using the inverse problem for the advection-diffusion equation[END_REF][START_REF] Charlier | Quantifying peakflow attenuation/amplification in a karst river using the diffusive wave model with lateral flow[END_REF]. Therefore, c is assumed equal to time delay between peak discharges of QI and QO, and D is fixed in accordance with stream characterization proposed by [START_REF] Todini | The ARNO rainfall-runoff model[END_REF]. The DWE solution was validated experimentally under controlled conditions [START_REF] Moussa | Evaluating lateral flow in an experimental channel using the diffusive wave inverse problem[END_REF], and has been implemented on natural karst catchments [START_REF] Charlier | Use of a flood-routing model to assess lateral flows in a karstic stream: implications to the hydrogeological functioning of the Grands Causses area (Tarn River, Southern France)[END_REF][START_REF] Charlier | Quantifying peakflow attenuation/amplification in a karst river using the diffusive wave model with lateral flow[END_REF][START_REF] Cholet | Assessing lateral flows and solute transport during floods in a conduit-flow-dominated karst system using the inverse problem for the advection-diffusion equation[END_REF].

The lateral mass fluxes ML simulation by the ADE is done using the same analytical solution than DWE, as under some hypotheses, the physical equations of both the DWE and the ADE can lead to similar mathematical expressions. ML is simulated using MI and MO.

  represents local HIi values and the slope value for an illustrative storm event. The mean of the 19 HIi values (5% intervals, recommended by Lloyd et al, 2016) is used as the final HI value. As a result, HI values range from -1 to 1, absolute values near 0 and 1 corresponding to low and high hysteresis degrees, respectively (similar versus contrasting rising and falling paths). Negative HI values indicate anticlockwise loops whereas positive HI values indicate clockwise loops. Complex 8-shaped loops show an HI value corresponding to their primary shape (i.e. the widest loop part with one rotational direction).

Figure 5

 5 Figure 5 compares annual and event-based mean EC values for the Loue and Cèze

Figure

  Figure 7a and 7b show the seasonal slope value distribution in the Loue and Cèze

Figure 8

 8 Figure 8 shows examples of observed Q and EC at input (QIN) and output (QOUT) stations,

Figure 1 :

 1 Figure 1: Framework of the general methodology at the scales of nested catchments and river

Figure 2 :

 2 Figure 2: Calculation of local HIi indexes on an illustrative storm event (Chenecey, Jura,

Figure 3 :

 3 Figure 3: EC-Q loop classification, adapted from Evans and Davies (1998), and corresponding

Figure 4 :

 4 Figure 4: Location of gauging stations and river network in the studied catchments.

Figure 5 :

 5 Figure 5: Boxplots of mean EC values during storm events and corresponding annual values

Figure 6 :

 6 Figure 6: Examples of EC and Q variations recorded during storm events, with associated EC-Q loops and classifications. Slopes are expressed in µS.cm -1 .m -3 .s. HI is dimensionless. a: Nans

Figure 7 :

 7 Figure 7: Distribution of slope (a, b) and HI (c, d) values for the Loue (a, c) and Cèze (b, d)

Figure 8 :

 8 Figure 8: Variations of input, output and lateral simulated discharge (a, c) and electrical

Figure 9 :

 9 Figure 9: Distribution of simulated lateral flow EC on Loue (a) and Cèze (b) reaches, mean

Figure 10 :

 10 Figure 10: Conceptual model of main flood processes and water origin seasonality in the Loue

Table 1 :

 1 Commonly occurring EC-Q loop types on the Loue and Cèze catchments

	Catchment	Station	Sep-Oct Nov-Mar Apr-May Jun-Aug
	Loue	Nans	A+	C-	C-	N0
		Ouhans	A+	C-	C+	N+
		Vuillafans A+	C0	C+	A+
		Chenecey A0/A+	C0	N0	A+
	Cèze	Tharaux	C-	N-	-	C+
		Montclus	~	N-	-	C0
		Laroque	C-	N-	-	C-
		Bagnols	C0	~	-	C-
	-= no storm event and ~ = no clear dominant behavior

available in the literature, showing that it is more sensitive to parameter c than D(Moussa and

Acknowledgments

The work was funded by the French Governmental Administration for Risk Prevention (DGPR), the Service Central d'Hydrométéorologie et d'Appui à la Prévision des Inondations (SCHAPI), and the French Geological Survey (BRGM).

Author contribution

RM and JBC were involved in conceptualization, funding acquisition, and supervision.

MLM gathered the data and designed the methodology with the help of JBC, RM, and YC.

MLM prepared the manuscript with the help of all co-authors.

Hourly streamflow data were gathered from the "Banque hydro" database using a script available at http://doi.org/10.5281/zenodo.3744183. EC datasets of the Loue catchment were monitored within the QUARSTIC network funded by the BRGM, the Doubs Department, and the Rhône Mediterranean Corsica Water Agency, and the NUTRI-Karst project founded by BRGM and Rhône Mediterranean Corsica water agency.

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Appendix: Hayami analytical resolution of DWE and ADE

Diffusive wave equation

An inverse modelling approach is adopted for simulating lateral flow between two gauging stations. This approach simulates the lateral flow QL, based on measurements from two gauging stations QI and QO.

The diffusive wave equation (DWE), accounting for lateral flow, is an approximation of the St-Venant equation that can be written as:

where x [L] is the length along the channel, t [T] is the time, and celerity c(Q) [LT -1 ] and

represents the lateral flow distribution. The lateral hydrograph QL(t) is given by: . ( = / ( + 0 ( -/ ( 1 * 3 ( (A3)

with K(t) the Hayami Kernel function defined as:

and

The inverse problem Under Hayami's conditions and assuming that lateral flow is uniformly distributed along the channel, [START_REF] Moussa | Analytical Hayami solution for the diffusive wave flood routing problem with lateral inflow[END_REF] proposed a solution of the inverse problem; this enables evaluation of the temporal distribution of lateral flow QL(t) over the channel reach by knowing I(t) and O(t).

Knowing c, D and l, the lateral flow can be calculated using the following procedure: The DWE equation has two free parameters, namely celerity c (m.s -1 ) and diffusivity D (m².s -1 ) of the flood wave. Sensitivity analysis of the DWE to the two parameters is largely