The aim of this study is to present a framework that provides new ways to characterize the spatio-temporal variability of lateral exchanges for water flow and solute transport in a karst conduit network during flood events, treating both the diffusive wave equation and the advection–diffusion equation with the same mathematical approach, assuming uniform lateral flow and solute transport. A solution to the inverse problem for the advection–diffusion equations is then applied to data from two successive gauging stations to simulate flows and solute exchange dynamics after recharge. The study site is the karst conduit network of the Fourbanne aquifer in the French Jura Mountains, which includes two reaches characterizing the network from sinkhole to cave stream to the spring. The model is applied, after separation of the base from the flood components, on discharge and total dissolved solids (TDSs) in order to assess lateral flows and solute concentrations and compare them to help identify water origin. The results showed various lateral contributions in space – between the two reaches located in the unsaturated zone (R1), and in the zone that is both unsaturated and saturated (R2) – as well as in time, according to hydrological conditions. Globally, the two reaches show a distinct response to flood routing, with important lateral inflows on R1 and large outflows on R2. By combining these results with solute exchanges and the analysis of flood routing parameters distribution, we showed that lateral inflows on R1 are the addition of diffuse infiltration (observed whatever the hydrological conditions) and localized infiltration in the secondary conduit network (tributaries) in the unsaturated zone, except in extreme dry periods. On R2, despite inflows on the base component, lateral outflows are observed during floods. This pattern was attributed to the concept of reversal flows of conduit–matrix exchanges, inducing a complex water mixing effect in the saturated zone. From our results we build the functional scheme of the karst system. It demonstrates the impact of the saturated zone on matrix–conduit exchanges in this shallow phreatic aquifer and highlights the important role of the unsaturated zone on storage and transfer functions of the system.
Hydraulic transfers and solute transport processes in karst aquifers are
known to be very complex due to the organization of underground void
structures leading to preferential drainage axes through a conduit network
embedded in a less permeable fissured matrix
Natural and artificial tracers are commonly used in catchment hydrology to
better understand the spatial variability of lateral exchanges in order to
study flows and the corresponding solute exchanges between the main channel
and adjacent hydrological units.
The Saint-Venant equations (SVE) may be used to assess hydrodynamic processes
as they describe unsteady flow in partially filled conduits
In most practical applications, the acceleration terms in the Saint-Venant
equations can be neglected, and consequently by combining the differential
continuity equation and the simplified momentum equation, the Saint-Venant
system is reduced to a single parabolic equation: the diffusive wave equation
(DWE;
To model conservative solute transport along a 1-D flow path, the
advection–diffusion equation (ADE) is widely used in hydrology
For practical application in karst systems, the knowledge of the temporal distribution of lateral flows and concentrations allows for better characterization of interactions along the conduit during a flood and the hydrogeological functioning of the aquifer. In fact, most karst systems are only accessible locally, leaving large portions of the system inaccessible for direct observation. Thus, the modelling of the lateral exchanges by solving the inverse problem is a tool to decipher the hydrological functioning of such inaccessible conduits located between two monitoring stations. The total volume of water and mass balance of lateral exchanges can be easily estimated by the difference between input and output average values. Thus, the benefit of simulating temporal variability is that it allows for the characterization of the evolution of lateral flows and their mineralization during the flood. This is important because these exchanges can be successively positive or negative during a single flood. The existence or not of a complex dynamic of lateral exchanges cannot be identified without a temporal analysis.
The aim of this paper is to propose a new framework based on a solution of
the inverse problem for the advection–diffusion equations
The diffusive wave equation is an approximation of the Saint-Venant equations
used to model 1-D unsteady flow in open channels
Between two stations I (inflow) and O (outflow), the model is applied on the
flood component (
Equation (1) is of parabolic type and its resolution requires appropriate
initial and boundary conditions imposed at the limit of the solution domain
(
Equation (4) is then used to compare
By considering the existence of lateral flow exchanges along a channel reach,
we obtain the following (Moussa, 1996):
Diffusive wave equation to model lateral flood flow exchanges
along a channel reach of length
In the case of the diffusive wave equation with lateral flows, an analytical
resolution is proposed by
Note that Eq. (6) gives the general form of the diffusive wave equation for
any spatio-temporal distribution of
Then the term
The inverse problem enables the identification of the temporal distribution
of the lateral inflows or outflows
The solution of Eqs. (10) and (11) requires first the identification of
The Hayami analytical solution of the diffusive wave model assumes a
uniformly distributed flow rate
The classical 1-D advection–diffusion equation for steady-state flow
conditions is analogous to the DW Eq. (1), replacing discharge by solute
concentration, and celerity and diffusivity parameters by advective velocity
and diffusion parameters, respectively. In unsteady-state flow conditions,
and when lateral fluxes occur, the application of ADE is not so
straightforward. We propose here to assess lateral solute transport (defined
by Eq. 15) during a flood, applying the DW model as a transfer function to account for lateral
exchanges. Thus, the analytical solution of
As previously described for water flows, the model involves first the
determination of the flood (
Then, following the method described above for water flows, lateral solute
exchange
By analogy with Eq. (5), the kernel function for the solute transport,
As we use similar mathematical resolution technique for both DWE and ADE, the
resolution of the ADE needs two parameters
Framework to investigate lateral exchange dynamics of water flows and solute fluxes along a channel reach.
We propose in this section a step-by-step structure to help readers using our
framework to investigate the exchange dynamics of water flows and
conservative solute transport along a channel reach between two gauging
stations. To simulate lateral exchange flows
Collection of discharge ( Calculation of the total solute fluxes Determination of the base and flood components from the hydrograph ( Calculation of the lateral base component for water flow ( Modelling of the lateral flood exchanges using two steps.
(a) First, Eq. (4) is used to parametrize Simulation of the total lateral exchanges Calculation using Eq. (21) of the solute concentrations of lateral base and flood flows
The Fourbanne karst system:
The proposed framework is generic enough to explore saturated and unsaturated
conditions, base flow and floods, water and suspended particulate matter or
any other tracer concerned by the advection–diffusion equation, considering
the analogy with the diffusive wave equation. Moreover, an analytical
solution is used for the diffusive wave taking into account uniformly
distributed lateral flows (or solutes). In our study, the application of this
framework is done separately on various selected flood events. Hence, the two
parameter sets (
Summary of the flood event selection sorted as a function of spring baseflow condition.
The study site is located in the Doubs river valley at the northern limit of
the Jura Mountains in eastern France, near the village of Fourbanne
(
The climate is temperate with both oceanic and mountainous influence.
Rainfall averages 1200 mm yr
Discharge (
From the available time series, 7 flood events with complete data sets for
all stations and various rainfall intensities were selected. A synthetic
characterization of the 7 events is given in Fig. 4, where they are sorted
from events 1 to 7 as a function of decreasing baseflow at the system outlet
(station s3). The event duration varied from 24 to 52 h with total
precipitation amounts between 3.4 and 46.6
The EC is directly related to the total dissolved solids, assuming that
TDSs represent mainly conductive ionic compounds. The TDS values were
therefore calculated directly from EC by using a constant factor of 0.64
(1 mg L
To illustrate the model behaviour described theoretically in Sect. 2 and to help to define a parametrization strategy, this section presents a sensitivity analysis on a benchmark flood event. This event was defined in order to have similar characteristics (same range of magnitude and parameter's values) to those presented in the model application.
This analysis was carried out on the celerity
Sensitivity analysis of the model parametrization.
Graphs
Figure 5a and a' represent the routed input hydrograph and the simulated lateral flow,
respectively, varying
Figure 5a–b illustrate the application of the Eq. (4), simulating the
propagation of the input signal in order to fit the output signal with the
two-parameter set (
Figure 5a' and b' illustrate the simulation of lateral flows
using the solution of the inverse problem when input and output signals are known.
It shows that
From the sensitivity analysis, a parametrization strategy was defined for the
application of the Eq. (4) for the four parameters:
Framework application on the event no. 1 in high-flow condition
along the two reaches R1
Figure 6 illustrates the application of the framework (depicted in Fig. 2) for reaches R1 and R2 of the study site for flood event no. 1 with 21 mm of rainfall during high-flow conditions.
The lateral base exchanges calculated for reach R1 demonstrate that the output signal observed at station s2 for discharge and solutes cannot be entirely explained by the contribution of the input signal, but that it was due to lateral inflows between s1 and s2. The solute transport model shows, in addition, that the lateral inflows were strongly mineralized, with higher TDS values than for stations s1 and s2. The solute fluxes of the base component at station S2 were thus essentially derived from lateral inflows along reach R1. This pattern was different for reach R2, where lateral base inflows were half those of and with similar TDS values.
Regarding the flood components, the simulated lateral exchange indicates important lateral inflow and high solute influx along R1. The concentration estimations indicate a similar evolution than for station s2, which is characterized by a TDS dilution during the flood. In contrast, dynamics were totally different along R2, where outflows were simulated. The concentration estimations of the lateral component, deduced from outflows and solute outfluxes, are very low compared to the measurements at stations s2 and s3, suggesting the presence of more complex processes than simple outflows, as will be discussed later on in Sect. 5.2.
Base and flood analyses of the selected event set. Orange and purple
symbols correspond to the lateral-exchange modelling along R1 and R2,
respectively. Base analysis is performed on mean values (
Following the example detailed for flood event no. 1 in the previous section, this section aims to summarize results of the model application on all selected flood events in order to get information on the general hydrological functioning of the field site.
Figure 7a and c present lateral exchanges for base water flow and base solute fluxes for all events of reaches R1 (orange labels) and R2 (purple labels). When comparing the modelled mean lateral base flow exchange as a function of the measured mean base input, two distinct linear relationships can be observed (Fig. 7a). For both reaches, lateral exchanges were positive, indicating inflows that increased linearly with average base input. For reach R1, lateral water inflow was 2.6 times higher than mean input flow from station s1, whereas for reach R2 the mean lateral water inflow represented only 0.4 times the mean inflow from station s2. For solute transport (Fig. 7c) a very similar relationship was found. However, for R1 the slope of the correlation was much steeper than for water flow (4.3 against 2.6), indicating that the lateral inflow water was more mineralized than the input flow from station s1 already present in the system. In contrast, for reach R2 a slightly lower slope was found for solutes than for water flow (0.3 against 0.4), meaning that lateral inflow was probably a little less mineralized than input flow from station s2.
Figure 7b and d illustrate lateral water flows and solute transport calculated for flood flow. To summarize the dynamics of the lateral exchanges during the flood, minimum and maximum values are presented rather than average values in order to characterize intensities of both lateral gains and losses which may occur during the same event. Along reach R1, two distinct groups are observed depending on peak flow from input station s1 (Fig. 7b): (i) events 1 to 5 with low input values are characterized by lateral inflows, with high maxima and minima close to zero; (ii) events 6 and 7, with high input values from strong rain events during an extremely dry period, show maxima close to zero and strongly negative minima indicating important lateral losses. Comparing the slopes of the linear relationships for water flows (Fig. 7b) and solute flux (Fig. 7d), it appears that the water inputs of the first group were strongly mineralized, whereas the water losses of the second group were characterized by low mineralization. For reach R2, all maxima of water flow and solute flux are close to zero, whereas the minima are negatively correlated with input values, indicating increasing lateral losses with increasing peak flow. Comparing the slopes between water flood flow (Fig. 7b) and solute flood flux (Fig. 7d), it appears that lateral losses were systematically less mineralized than input water from station s2.
Parametrization analysis of the selected event set. Orange and purple symbols correspond to R1 and R2, respectively. Each parameter of the model is compared with the maximum intensity of the flood flow input signal (logarithmic scale).
In this section we present the distribution of values for the model parameters – celerity and diffusivity – as a function of maximum input flood flow intensities, with the aim to retrieve information on flow and transport dynamics (Fig. 8).
The
All events from reaches R1 and R2 had very low
The analysis of the parameter distributions showed distinct trends for R1 and R2, which can be attributed to the presence of the saturated zone in the lower parts of reach R2 (see Fig. 3 for the hydrogeological scheme of the main conduit). In fact, flood routing in the unsaturated zone is related to flow in open conduits, whereas in the saturated zone it is controlled by pressure transfer leading to an almost instantaneous propagation.
Consequently, the difference of flood routing between unsaturated and
saturated conduits along the system should be observed in the
Representation of the limit estimation between unsaturated and
saturated conduits along R2 as a function of the mean spring base flow.
This limit is supposed to fluctuate as a function of the baseflow condition.
Figure 9a represents the
Our study intends to present a new framework to quantify the temporal
evolution of lateral flows and their concentrations during floods in a
well-developed karst conduit networks. It uses the diffusive wave (DW) model,
which is a physically based, parsimonious and easy-to-use approach. The
inverse problem was used to identify both lateral flows and solute flux
under unsteady-state conditions following the assumption of uniformly
distributed lateral exchanges. Our modelling approach is not used with data
sets, allowing a validation of the computed lateral fluxes. Indeed, in the
study case, there is no monitoring of the tributaries or losses along the
two reaches of the study site. However, if additional variables are measured (for example piezometer levels or hydrographs on tributaries), a validation can
be undertaken by comparing the measured variables to those simulated when
solving the inverse problem. It has been proposed by Charlier et al. (2015),
in which the hydrograph dynamics of lateral springs is compared with the
simulated lateral exchanges. Regarding parametrization, the analysis of the
distribution of water-flow parameters (celerity
The proposed framework requires decomposition of the base and the flood
components in order to be consistent with the duality of flow processes
dynamics. As proposed by many authors
Our methodology quantifies total lateral water flow and solute flux exchange
for a given reach, but does not allow identification of simultaneously occurring
local lateral flows and fluxes, as has been already highlighted in previous
studies
The modelling framework is proposed as a diagnostic tool to assess the dynamics of lateral exchanges between the main conduit and the neighbouring compartments of the karst aquifers during floods. The selected events are typified by various initial baseflow condition and flood flow intensities, thus allowing characterization of both low-flow and high-flow periods and providing a rather generic view of the hydrological behaviour of the system. The model application on several contrasted events on the Fourbanne karst network provides several points of discussion about how to evaluate exchanges in various hydrological conditions and create a general hydrological functional scheme of the site. This scheme, presented in Fig. 10, highlights the lateral contributions for the base and the flood components. In fact, we assume that – as discussed before – base and flood components are mainly related to specific processes as conduit–matrix exchanges distinguished by slow and fast flow, respectively.
Hydrogeological functioning scheme of the Fourbanne karst aquifer, showing the contributions of lateral exchanges in terms of volume and mineralization for three hydrological conditions. For each graph, the representation of the lateral exchanges along the reaches (vertical black line) distinguishes the base component (left size) and the flood component (right size).
For major precipitation during high-flow periods (events nos. 1 and 2; Fig. 10a), we
showed the importance of lateral exchanges for both base and flood components. For the baseflow
component both reaches were fed by higher mineralized lateral inflows. However, for the flood
component the model yielded different results for the two reaches. Reach R1, located entirely
in the unsaturated zone, had lateral inflows which were more mineralized than the sinking stream
at the reach input. We relate these mineralized inputs to the arrival of tributaries from adjacent
sinkholes (see their location in Fig. 3). On the contrary, reach R2 shows less mineralized base
inflows and mainly slightly mineralized flood outflows, indicating high losses, which we relate
to mixing processes in the saturated part of the conduit. For minor precipitation during low-flow periods (events nos. 3, 4 and 5;
Fig. 10b), reach R1 showed similarly mineralized inflows from the base and the flood components, but with
a lower inflow amount in agreement with the lowest rainfall intensity of these events. On the
contrary, the baseflow component of reach R2 was mainly characterized by inflows, whereas weak
outflows were found for the flood component. This shows that the system was mainly influenced
by the drainage of water from the rock matrix even during low-flow periods. The weak outflows
found for the flood component of reach R2 probably occurred within the saturated zone. For major precipitation during extremely dry periods (events nos. 6 and 7;
Fig. 10c), quite different behaviour was observed. Reach R1 presented very low base
inflows with similar mineralizations to those for the input flow, probably from
sinking stream tributaries joining the unsaturated conduit network. However,
important outflows were observed for the flood component, indicating high
losses towards the rock matrix in the unsaturated zone. On the contrary,
reach R2 presented very low baseflow inputs of highly mineralized water. The
flood component was characterized mainly by lateral outflows followed by
lowest inflows. It is interesting to note this reversal of the lateral
flood flows from out- to inflows, indicating an evolution of the exchanges
during the flood event, that may be related to conduit–matrix interactions
in the saturated zone.
Even if we observed increasing lateral inflows with increasing baseflow conditions, a slow inflow component always remained present all along the network (i.e. in the unsaturated as well as in the saturated zone), whatever the hydrological conditions (see Sect. 4.2.1.). These constant inflows probably originate mainly from diffuse infiltration in the unsaturated zone and from lateral drainage systems in the saturated zone. However, the mineralization of the infiltrating water was higher along R1 compared to R2, reflecting probably different recharge mechanisms. It seems that R1 collected additional inflows from strongly mineralized secondary tributaries mainly localized in the upstream part of the aquifer (R1), in accordance with the presence of sinkholes in the north-eastern part of the study area (Fig. 3).
In Sect. 4.2.2., from the analysis of the events distribution according to
rainfall intensity and initial baseflow, we could distinguish two distinct
groups of events depending on the general hydrological context: (1) events 1
to 5 during periods of high and low flow and (2) events 6 and 7 from an
extremely dry period. The evolution of the lateral flood exchange for group
(1) increased linearly with flood intensity (maximum of the peak flood,
Fig. 7), suggesting that these lateral inflows, probably derived from
secondary conduits, were proportional to the discharge measured at the input
station. For reach R1, the mineralization of these inflows increased with the
maximum input of inflow, whereas for reach R2 the mineralization of outflows
decreased. For group (2), lateral flood outflows were observed along both
reaches, notably for R1, meaning that the inflows from secondary lateral
conduits observed for group (1) stopped during extreme dry periods in the
unsaturated zone. This change of behaviour of the distinct groups of events
showed the non-linearity of the lateral exchanges, depending on hydrological
conditions. This threshold effect in the hydrogeological response may be
related to the presence of epiphreatic conduits
In our opinion, from the available data, two types of outflows along the
karst conduit network can be described in our conceptual model. The first
type corresponds to outflows observed along R1 during the extremely dry
period. These outflows occurred in the unsaturated zone of the conduit at
extremely low baseflow conditions where only very few base inflows were
observed. Thus, during this period, the flood input following major
precipitation recharged the low permeability volume (or matrix) from the
conduit network. The second type corresponds to outflows observed along R2
for all baseflow conditions. These outflows seemed to occur in the saturated
zone and are related to flow reversal occurring in the saturated conduit, as
mentioned by several authors
Besides these results, the flood routing parameters are also indicators for
hydraulic processes in the conduit. They are used in Sect. 4.3.2. to estimate
the fluctuation of the limit between the unsaturated and the saturated zone
within reach R2. We evaluated that the saturated zone occupied 25 to 40 %
of R2, depending on flow conditions, corresponding to the 1.5 to 2.2 km of
conduit next to the outlet station s3. The Fourbanne karst system with its
total length of 8.5 km (3.1 km for R1 and 5.4 km for R2) is thus a
shallow-phreatic aquifer. Consequently, we demonstrated that besides the
impact of the saturated zone on the matrix–conduit exchanges, the unsaturated
zone plays a major role in the flood genesis in karst aquifers. Our results
confirm that the unsaturated zone essentially has a transfer function – as
it is commonly conceptualized
The study aims to propose a framework to characterize the spatio-temporal variability of lateral exchanges for flows and fluxes in a karst conduit network, known to have large amount of concomitant in- and outflows during flood events. The main interest in our study is the treatment of both phenomena, the diffusive wave equation and the advection–diffusion equation, with the same mathematical approach assuming uniform lateral flow and solutes, solving the inverse problem of the advection–diffusion equations using an analytical solution. In fact, as the model was applied for two different variables, the flow and the solute transport, a crossed analysis has been performed in order to characterize a functioning scheme of the studied karst system. We showed various lateral exchanges between both unsaturated and saturated zones, we estimated the fluctuation limit of the saturated zone in the conduit, and we illustrated the non-linearity of the hydrogeological response related to the initial hydrological conditions.
One of the main points was the ability of our approach to propose a deconvolution of the output hydrograph as well as a mass chemograph allowing quantification of the lateral contributions in terms of flows and mineralization. It was useful to identify water origin of lateral flows and make hypotheses on the flood generation in karst aquifers. The modelling approach uses all data available on the reach in both input and output time series, leading to use of our framework as a diagnostic tool to help decompose time series and investigate lateral exchanges more precisely. The results showed that this diagnostic step provides new ways to investigate the hydrogeological functioning of karst aquifer and demonstrates, for instance, that further hydrogeological model development for the case study has to take into account storage in the unsaturated zone and matrix–drain relationships during floods.
The datasets used in this article can be obtained by contacting Cybèle Cholet (cybele.cholet@univ-fcomte.fr) and Marc Steinmann (marc.steinmann@univ-fcomte.fr).
The authors declare that they have no conflict of interest.
The authors wish to thank Bruno Régent for his active contribution in the
field. Many thanks to the Speleology Association of Doubs Central (ASDC) for
the precious help in the field and their support in accessing the Fontenotte
river cave stream in the En-Versennes karst network. We thank Jacques Prost
for welcoming our monitoring equipment and giving us access to the
Fourbanne spring. We also thank the Verne municipality for letting us monitor
the Verne swallow hole. The Jurassic Karst hydrogeological observatory is
part of the INSU/CNRS national observatory of karstic aquifers, SNO KARST
(