HESSHydrology and Earth System SciencesHESSHydrol. Earth Syst. Sci.1607-7938Copernicus PublicationsGöttingen, Germany10.5194/hess-21-3635-2017Assessing lateral flows and solute transport during floods in a conduit-flow-dominated karst system using the inverse problem for the advection–diffusion equationCholetCybèleCharlierJean-Baptistehttps://orcid.org/0000-0002-1268-8848MoussaRogerSteinmannMarcDenimalSophieChrono-Environnement, UMR 6249 UBFC/CNRS, University of Burgundy Franche-Comté, Besançon, 25000, FranceBRGM, 1039 rue de Pinville, 34000 Montpellier, FranceINRA, UMR LISAH, 2 Place Pierre Viala, 34060 Montpellier, FranceCholet Cybèle (cybele.cholet@univ-fcomte.fr)18July20172173635365331October201611November20161June20172June2017This work is licensed under the Creative Commons Attribution 3.0 Unported License. To view a copy of this licence, visit https://creativecommons.org/licenses/by/3.0/This article is available from https://hess.copernicus.org/articles/21/3635/2017/hess-21-3635-2017.htmlThe full text article is available as a PDF file from https://hess.copernicus.org/articles/21/3635/2017/hess-21-3635-2017.pdf
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.
Introduction
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 .
Flow processes are driven by the spatial variability of the geometrical
elements that constitute the conduit network, such as full pipes, open channels, or
pipe constrictions , leading to rapid transitions from
free-surface to pressurized flows, after recharge events. In such
heterogeneous media, transport parameters are also dependent on scale
effects, as mentioned by who showed that retardation
factor is dominant at a local scale but vanishes at the benefit of an
increase in dispersivity with increased distances. In addition to the
interaction between conduit and matrix compartments , solute concentration evolutions in the conduit network are
strongly influenced by the mixing with inflows from tributaries
or with flooded areas . These
papers pointed out the various and complex lateral exchanges along the
conduit network, which remains an open question calling for new tools to
investigate it.
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. highlighted strong
channel losses despite storage exchange fluxes and lateral inflows. By
analysing exchanges between consecutive reaches in a mountainous headwater
stream, emphasized the importance of the geomorphological
context as a driver for the spatial variability of gaining and losing
reaches. Moreover, they demonstrated that many of the reaches studied were
concurrently losing and gaining. simulated exchanges
using the OTIS model , which is a 1-D
finite-difference model solving solute transport in the streams accounting
for transient storage. By testing the influence of different
conceptualizations of hydrologic exchanges on the estimation of transient
storage parameters, they showed the complexity of model tracer evolution.
This is related to the difficulty of modelling spatial patterns of tracer
concentrations as well as magnitudes of lateral inflows and outflows. The
same model was recently used by to simulate lateral
exchanges in karst conduits in both unsaturated (“river stretches”) and
saturated zones (“flooded area”). This study furthermore showed that the
parametrization of solute transport in the system had to account for
interactions within the saturated zone. However, all these works were
designed for low-flow periods and are therefore not suitable to investigate
the temporal evolution of lateral flows during flood events in conduit-flow-dominated karst systems with large-magnitude flash flows.
The Saint-Venant equations (SVE) may be used to assess hydrodynamic processes
as they describe unsteady flow in partially filled conduits
and are generally used to simulate discharge in
conduit-flow-dominated karst aquifers. As the conduit flow in the unsaturated
zone is mainly controlled by free-surface conditions, Manning's equations are
favoured, although they under-estimate head losses due to turbulent flows
. Equations to be used for the saturated zone should be
adapted for pressurized conditions, such as the Preissmann slot model in
ModBraC , or the Darcy–Weisbach equation in pipe-flow
models like the Storm Water Management Model (SWMM)
. However, the
application of these models is problematic, because detailed but often
unavailable information on hydraulic parameters is required, particularly
concerning the location and the geometry of conduits. As a consequence, such
physically based models are often applied in a degraded mode
and are sometimes over-parametrized even in the case of
relatively well-known conduit networks. An alternative approach to model
hydraulic processes would be to test the ability of simplified SVE with
parsimonious parameters. For that, the diffusive wave equation can be
considered as a relevant simplification of the full SVE
. This approach was used successfully by
to assess lateral flows in karst rivers with numerous
lateral in- and outflows. It is therefore a promising alternative to the more
complex approaches cited above.
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; ). The two parameters of the
equation, celerity and diffusivity, are usually taken as functions of the
discharge. Methods based on the finite-difference discretization techniques
are generally used to solve this equation, but these methods may induce
problems of stability and accuracy .
However, if celerity and diffusivity can be assumed constant, and if lateral
flow is negligible, the diffusive wave equation has an analytical solution:
the model. extended this analytical
solution to the case of uniformly distributed lateral flow which can be
either positive (lateral inflow) or negative (lateral outflow). In the
following, the model allows the calculation of the outflow
using as input the inflow and the lateral flow uniformly distributed, and
using the two parameters celerity and diffusivity. Moreover,
proposed an analytical solution which enables the calculation of the temporal distribution of the lateral flow (under the hypothesis
of uniformly distributed flow) by an inverse problem approach using
both the inflow and the outflow as input and using the two parameters celerity and
diffusivity. Contrary to classical modelling approach where the measured
output hydrograph is used only to validate a model, the inverse problem
developed herein uses all information available in both input and output data. In
comparison to numerical methods, the advantage of the Hayami solution
extended by is an easy-to-use analytical solution. The
solution of the inverse problem proposed is part of the hydrological model
MHYDAS Distributed Hydrological Modelling of AgroSystems;.
To model conservative solute transport along a 1-D flow path, the
advection–diffusion equation (ADE) is widely used in hydrology
and karst hydrology
. In agreement with the mass conservation law,
solute transport can be expressed by ADE. However, ADE is more challenging to
implement in the case of unsteady-state flow conditions, especially when lateral
exchanges occur. The diffusive wave equation and the advection–diffusion
transport equation have very similar mathematical expressions, but of course
do not describe the same processes. The diffusive wave equation is derived
from Saint-Venant continuity and momentum equations and can be applied to a
wide range of phenomena in different fields as exposed by
for the kinematic wave, such as flood routing model but also for solute
transport . The advection–diffusion equation is derived
from the mass conservation principle applied for matter dissolved in the
water and taking into account two basic processes of transport: advection and
diffusion in which the Fick's law leading to the diffusive term was applied.
Under some hypotheses, the physical equations of both the diffusive wave
equation and the advection–diffusion equation can lead to the same similar
mathematical expressions which can justify the use of the same resolution
approaches. In the present study, following , the diffusive
wave equation and the advection–diffusion equation are treated using the same
mathematical approach: the analytical solution extended
by to the case of uniformly lateral flow (and solute
transport) using an inverse problem approach.
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
to identify the temporal distribution of lateral flows and their
concentrations. We consider the general case where only flow (and eventually
solute concentration) is measured at gauging stations without any additional
information on lateral flow. The aim is to simultaneously use both measured
input and output hydrographs (and eventually input–output solute
concentrations) in order to identify lateral inflow–outflow temporal
distribution and solute concentration. The framework is tested on discharge
and total dissolved solids data from two reaches of a karst conduit in the
French Jura Mountains, for several flood events under various hydrological
conditions. The study case corresponds to the case generally encountered in
practice where the spatial variability is unknown. The model simulations –
describing the temporal variability of the lateral exchanges – are then used
to better characterize the interactions existing along the conduit during
floods.
Modelling approachAssessing lateral flowsDiffusive wave model without lateral flows
The diffusive wave equation is an approximation of the Saint-Venant equations
used to model 1-D unsteady flow in open channels
:
∂Q∂t+CQ∂Q∂x-DQ∂2Q∂x2=0,
where x(L) is the length along the channel, t (T) is the
time, and the celerity CQ (LT-1) and the diffusivity DQ
(L2T-1) are functions of the discharge
Q (L3T-1).
Between two stations I (inflow) and O (outflow), the model is applied on the
flood component (QI,flood(t) and QO,flood(t)) of the
total discharge (QI,tot(t) and QO,tot(t)), which can
be deduced by using the inflection point on the hydrograph recession and then
removing the baseflow components QI,base(t) and
QO,base(t):
QI,flood(t)=QI,tot(t)-QI,base(t),QO,flood(t)=QO,tot(t)-QO,base(t).
Equation (1) is of parabolic type and its resolution requires appropriate
initial and boundary conditions imposed at the limit of the solution domain
(0≤x≤L and t≥0, L being the channel length).
assumed the following domain 0≤x<∞ and t≥0. The initial condition is set for t=0, Q(x,t)=0 for x∈[0,∞] with the following boundary conditions: for x=0, Q(x,t)=δ(t) and for x→∞, Q(x,t)=0 where δ(t) is
the Dirac delta function. The resolution of Eq. (1) can be obtained using
either a numerical method or a convolution. We choose the convolution
approach instead of a numerical method for two reasons. First, an analytical
solution for the advection–diffusion equations proposed by
is available and easy to use. Moreover, the use of a
convolution enables us to avoid the need for choosing adequate space and time
steps in numerical methods (subdividing the reach into space steps dx, and
the time into time steps dt) which may introduce numerical instabilities.
Based on the Hayami assumptions , considering CQ and
DQ as constant parameters over time along a channel network of length
L, the diffusive wave equation without lateral exchange can be written as
follows :
QI,routed(t)=∫0pQI,flood(t-T)KQ(T)dT=QI,flood(t)*KQ(t),
where p is the time memory
of the system, and the symbol * represents the convolution operator. As
there is no problem of calculation time, the term p must be large in
comparison to the travel time on a channel reach. In Eq. (4), the Hayami
kernel function KQ(t) is expressed as follows:
KQ(t)=L2(πDQ)1/2exp[CQL4DQ(2-LCQt-CQtL)](t)3/2.
Equation (4) is then used to compare QI,routed(t) to
QO,flood(t) and to perform the parametrization of CQ and
DQ as described afterwards (Sects. 2.3 and 3.3.2).
Diffusive wave model with lateral flows
By considering the existence of lateral flow exchanges along a channel reach,
we obtain the following (Moussa, 1996):
∂Q∂t+CQ∂Q∂x-q-DQ∂2Q∂x2-∂q∂x=0,
where q(x,t) (L2T-1) is the lateral flow rate per unit length
as a function of distance along the channel reach x. The expression
q(x,t) may be positive or negative depending on the occurrence of lateral
inflow or outflow, respectively (Fig. 1).
Diffusive wave equation to model lateral flood flow exchanges
along a channel reach of length L. The black curve QI,flood
depicts the evolution of the flow rate with time at the beginning of the
channel (input), and the blue curve QO,flood at the end (output).
The dashed green curve QA,flood corresponds to the lateral flow
exchanges which are positive for lateral inflows or negative for lateral
outflows.
In the case of the diffusive wave equation with lateral flows, an analytical
resolution is proposed by based on the Hayami
assumptions, accounting for uniformly distributed lateral flow between two
gauging stations I (inflow) and O (outflow).
QI,routed(t)=ΦQ(t)+(QI,flood(t)-ΦQ(t))*KQ(t)withΦQ(t)=CQL∫0t(QA,flood(λ)-QA,flood(0))dλ
Note that Eq. (6) gives the general form of the diffusive wave equation for
any spatio-temporal distribution of q(x,t) while Eqs. (7, 8) give the
resolution of Eq. (6) in the particular case of uniformly distribution of
q(x,t) along the reach, under the hypotheses used in the Hayami model
(CQ and DQ constant).
Then the term QA,flood(t) is expressed as follows:
QA,flood(t)=∫0Lq(x,t)dx.
If the hypothesis of uniform lateral distribution is assumed, then q(x,t)
does not depend on x and the right-hand side of Eq. (9) can be simplified.
The inverse problem enables the identification of the temporal distribution
of the lateral inflows or outflows q(x,t) over the channel reach. According
to , by knowing QI,flood(t) and
QO,flood(t), it is possible to calculate
QA,flood(t). Equation (7) then gives the following:
AQ(t)=QO,flood(t)-QI,flood(t)*KQ(t),withΦQ(t)-ΦQ(t)*KQ(t)=AQ(t).
The solution of Eqs. (10) and (11) requires first the identification of
KQ(t) using Eq. (5) and consequently a predetermination of the two
parameters CQ and DQ, to calculate afterwards lateral flow
QA(t) as follows:
QA,tot(t)=QA,base(t)+QA,flood(t),withQA,flood(t)=LCQdΦQdt,andQA,base(t)=QO,base(t)-QI,base(t).
The Hayami analytical solution of the diffusive wave model assumes a
uniformly distributed flow rate q(x,t) along the reach, which is the
simplest hypothesis when no additional information from the field is
available. Moreover, under this hypothesis, and under the hypotheses used in
the Hayami model, the unknown q(x,t) is reduced to q(t) in the
analytical solution of the inverse problem. Even if the spatial distribution
of q(x,t) is unknown, the simulated q(t) under the hypothesis of
uniformly spatial distribution will give to the modeller important
information on the temporal variability of lateral exchanges, because it
enables us to distinguish three cases: (i) negative q(t) during the whole
event; (ii) positive q(t) during the whole event; (iii) alternating
positive and negative q(t) during an event. It also enables us to calculate
the temporal distribution of lateral flow q(t) and the maximum and minimum
values of q(t).
Assessing lateral solute transport
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
is used to resolve the conservative solute transport,
respecting the mass conservation law and accounting for uniformly distributed
lateral fluxes.
M(t)=S(t)⋅Q(t),
where M(t) is the solute flux rate (MT-1), S(t) is the
solute concentration (ML-3) and Q(t) is the discharge
(L3T-1).
As previously described for water flows, the model involves first the
determination of the flood (MI,flood(t) and
MO,flood(t)) and base (MI,base(t) and
MO,base(t)) components of the total fluxes (MI,tot(t)
and MO,tot(t)) based on mass-chemograph separation, corresponding
to the evolution of solute transport as a function of time:
MI,flood(t)=MI,tot(t)-MI,base(t)MO,flood(t)=MO,tot(t)-MO,base(t)
Then, following the method described above for water flows, lateral solute
exchange MA(t) is calculated by adapting Eqs. (5), (7), (10) and
(11) as follows:
MO,flood(t)=ΦM(t)+(MI,flood(t)-ΦM(t))*KM(t)withΦM(t)=CML∫0t(MA,flood(λ)-MA,flood(0))dλ
By analogy with Eq. (5), the kernel function for the solute transport,
KM(t), is expressed as:
KM(t)=L2(πDM)1/2exp[CML4DM(2-LCMt-CMtL)](t)3/2
As we use similar mathematical resolution technique for both DWE and ADE, the
resolution of the ADE needs two parameters CM and DM, corresponding
to the celerity and the diffusivity of solute flux, respectively. CM and
DM of the ADE play a similar role that the two parameters CQ and
DQ of the DWE. Finally, this modelling framework combining the
calculation of both lateral water flows QA(t) and solute fluxes
MA(t), allows the assessment of the solute concentration of the lateral
flows SA(t) in the case of positive or negative QA(t) and
MA(t):
SA(t)=MA(t)QA(t)
Framework to investigate lateral exchange dynamics of water flows
and solute fluxes along a channel reach.
Framework
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 QA and solute fluxes
MA during floods, the required data are discharge and solute time series
from both stations covering a complete flood event. The simulated lateral
flow and solute fluxes are then used to better characterize the temporal
variability of the exchanges occurring during the flood into the karst
conduit, giving additional information to better characterize the
hydrogeological functioning of the karst aquifer. We assume a priori
linearity of both processes: the diffusive wave equation for flow transfer
and the advection–diffusion equation for solute transfer. As a consequence of
such assumption, a superposition is valid (separation of the base flow and
the base solute transport can be done) as well as the convolution approach
can be applied. Consequently, because of the assumed linearity, both
problems, unsteady flow and unsteady solute transport, are analysed using a
uniform approach because both problems are described using the same type of
equation. Therefore, in this framework, both the diffusive wave equation and
the advection–diffusion equation are treated using the same mathematical
approach: the analytical solution extended by
to the case of uniformly lateral flow (and solutes). When
the observations at the upstream and downstream ends are known then
determination of the lateral inflow–outflow constitutes some kind of inverse
problem. The problem is solved using the same analytical techniques applied
to both the diffusive wave equation and the advection–diffusion transport
equation describing both flow and transport. The modelling of the lateral
flood component includes four parameters corresponding to CQ and DQ
for water flow and CM and DM for solute transport. A relationship
may exist between the diffusive wave celerity CQ and the flow velocity
CM in the advection–diffusion equation (for example for rectangular
sections, CQ=5/3CM). Figure 2 gives a graphical representation
of this framework whose 7 stages are listed below.
Collection of discharge (QI,tot and QO,tot) and concentration
(SI,tot and SO,tot) data from input and output
stations.
Calculation of the total solute fluxes MI,tot and MO,tot using Eq. (15).
Determination of the base and flood components from the hydrograph (QI,tot,
QO,tot) separation defined by Eqs. (2) and (3) and
mass-chemograph (MI,tot and MO,tot) separation
defined by Eqs. (16) and (17). The base and flood components are separated
with the constant slope method by using the inflection
point on the hydrograph recession. The inflection point determined on the
hydrograph is used to separate base and flood components for both hydrograph
and mass chemograph.
Calculation of the lateral base component for water flow (QA,base(t)) and solute fluxes (MA,base(t)), using Eq. (14).
Modelling of the lateral flood exchanges using two steps.
(a) First, Eq. (4) is used to parametrize CQ and DQ and CM and
DM with a trial-and-error optimization, calculating
QI,routed and MI,routed to get the best fit with
QO,flood and MO,flood, respectively. The model is
parsimonious with only two parameters (celerity and diffusivity) to optimize.
Nevertheless, if the user needs to optimize more parameters, an automatic
optimization procedure is necessary. (b) Afterwards, the inverse
problem is applied to simulate lateral flood exchanges
QA,flood(t) and MA,flood(t), using Eqs. (10), (11)
and (13).
Simulation of the total lateral exchanges QA,tot(t) and MA,tot(t), using Eq. (12).
Calculation using Eq. (21) of the solute concentrations of lateral base and flood flows
SA,base(t) and SA,flood(t), respectively. The
determination (if possible) of the total solute concentration
SA,tot(t) follows then.
The Fourbanne karst system: (a) geographical localization,
(b) hydrogeological map, (c) scheme of the main karst conduit network.
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 (CQ and DQ and CM and DM) are calibrated
for each event. Then, the relationships between the parameters and the
variations of water flow and solute transport are analysed. It can be
expected for example that CQ and DQ should increase with more
pronounced flood peaks. Furthermore, the model simulations performed along
the reaches allow estimating the temporal variability of the lateral
exchanges. The simulations can hardly be compared to punctual field
measurements, however they are used as a diagnostic tool to better
characterize the exchange dynamics occurring along the conduit during flood.
The compilation of all results leads us to define a functional scheme of the
studied karst system. Note that if additional information on lateral fluxes
is available, as for example punctual inputs/outputs on the reach,
information on the amplitude of the spatial distribution of lateral flows, or
measurements of solute concentrations on different points, the framework
proposed herein is generic and can be easily used. In this case, the studied
zone has to be subdivided into different reaches with eventually punctual
inputs/outputs on some nodes; then the inverse problem can be applied on each
reach. Moreover, if additional variables are measured, as for example
piezometer levels or hydrographs on tributaries (or concentrations in the
water table or tributaries), a validation can be undertaken by comparing the
measured variable to the simulated lateral flow hydrograph. But the gain in
understanding the complexity of the studied karstic systems is worth the
relative loss of ”lateral” precision, that is, soon as the tributaries are
not the key concern, i.e. the main reaches can be properly identified.
Summary of the flood event selection sorted as a function of spring
baseflow condition.
Study siteField situation
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
(47∘19′54′′ N, 6∘18′15′′ E, Fig. 3a). The Fourbanne
site is one of the experimental sites of the “Jurassic Karst”
hydrogeological observatory
(http://zaaj.univ-fcomte.fr/spip.php?article13) and monitored
continuously since December 2013. The local geological structure is
characterized by tabular Jurassic limestones and shales, crosscut by N–S
trending normal faults . The recharge area of the site
covers about 30 km2. The site was selected for its dominantly
allogenic recharge and its well-developed, partially accessible conduit
network. The upstream recharge area corresponds to a surface watershed
underlain by impervious lower Jurassic marls. The conduit network is fed by
swallow holes located at normal faults, bringing into contact lower Jurassic
shales and karstified middle Jurassic limestones (Fig. 3b). The Verne swallow
hole constitutes the main infiltration point of the Fourbanne karst system
and corresponds to monitoring station s1. The allochtonous recharge joins the
well-developed cave stream of the “En-Versennes” cave, which is explored
over a distance of about 8 km. An artificial well near the village of
Fontenotte gives direct access to the cave stream, where monitoring station
s2 was installed, about 5 km downstream of station s1. The Fontenotte cave
stream can be followed for another 2 km downstream of station s2,
where it disappears into an inaccessible conduit network. Finally, it joins
the Fourbanne spring, which is fed by a saturated siphon of 25 m depth and
explored by cave-divers up to the last 500 m. Monitoring station s3
was installed at the Fourbanne spring.
The climate is temperate with both oceanic and mountainous influence.
Rainfall averages 1200 mm yr-1, occurring mainly in autumn and winter,
but with slightly higher intensities in summer .
Based on long-term records in the Jura Mountains, the number of rainy days
per year is 140 on average , corresponding for the most
part to low-intensity rainfall events: 50 % of rainy days had less than 3 mm of rainfall, whereas days with between 15 and 30 mm of rainfall
represented only 10 % of rainy days.
Field monitoring and data processing
Discharge (Q) and electrical conductivity (EC) were monitored
continuously at 15 min intervals from January to June 2015 at the three
stations (Fig. 3b): the Verne swallow hole (station s1), the Fontenotte cave
stream (station s2,) and the Fourbanne spring (station s3). Stations s1 and
s2 were equipped with OTT CTD probes for conductivity, temperature, and water
level monitoring, whereas a, OTT Hydrolab DS5X multi-parameter probe was used
at station s3 for conductivity and water temperature, and a OTT Orpheus mini
probe for water level. The three stations divided the main conduit into two
reaches: R1 from s1 to s2 (3.1 km) and R2 from s2 to s3
(5.4 km, Fig 3c). Hourly precipitation were recorded by a fully
automatic Campbell BWS200 weather station installed next to monitoring
station s2 in the Fontenotte village.
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 mm. A progressive decrease
of the hydrogeological response with decreasing baseflow was found for events
1 to 5, whereas events 6 and 7, from an extremely dry period in June 2015,
behaved differently with an important inflow from the station s1.
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-1= 1 µS cm-1⋅ 0.64), which is commonly used by OTT CTD probes and is consistent
with the literature according to the water mineralization
range of the data set.
Model application to the study site
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.
Sensitivity analysis
This analysis was carried out on the celerity CQ and the diffusivity
DQ of the modelling approach both applied on water flows. The celerity
CQ and the diffusivity DQ describe the propagation and the
flattening of the flood peak, respectively. The model applied on solute
fluxes using CM and DM play similar roles as CQ and DQ.
Sensitivity analysis of the model parametrization.
Graphs (a–b) illustrate the simulation of the routed input
QI,routed (dashed black lines) without lateral exchange, while
graphs (a') and (b') illustrate the inverse problem approach which simulate lateral flows
QA,flood (dashed green lines). Graphs (a–a') and
(b–b') correspond to sensitivity test of CQ and DQ ,
respectively.
Figure 5a and a' represent the routed input hydrograph and the simulated lateral flow,
respectively, varying CQ from 0.1 to 0.3 ms-1 with a fixed
DQ of 0.1 m2s-1 (i.e. low DQ value). Similarly,
Fig. 5b and b' represent the same graphs, but
with a fixed CQ of 0.15 ms-1 (i.e. low CQ value) and
varying DQ from 0.1 to 10 m2s-1.
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 (CQ and DQ) by considering no lateral exchange.
The graphs clearly demonstrate that the routed input signal is much more
sensitive to celerity than diffusivity. Varying CQ by a factor of 3 has
a stronger impact on the results than varying DQ by a factor of 100. As
an example, a 10 % variation of CQ (with constant DQ) modifies
the maximum of QI,flood*KQ of 9 %, while 10 % variation of
DQ (with constant CQ) changes to vary the maximum of
QI,flood*KQ by 0.6 % only. Moreover, this sensitivity test
illustrates the impact of both parameters on the propagation velocity and the
shape of the flood peaks: increasing CQ values yield more rapidly
propagating and higher flood peaks, whereas increasing DQ values lead to
flattened peaks. These observations are in agreement with the literature
and
confirm that the lower the CQ and the higher the DQ, the lower the peak
flow intensity and the transfer velocity.
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 CQ and DQ influence the lateral flow rates as well as
the exchange direction (inflows or outflows) which could be reversed during a
similar flood event. Low CQ and DQ values lead to lateral inflows
followed by outflows, whereas high CQ and DQ values lead to
outflows followed by inflows.
Parametrization strategy
From the sensitivity analysis, a parametrization strategy was defined for the
application of the Eq. (4) for the four parameters: CQ and DQ for
discharge and CM and DM for solute flux. The parameters were
optimized by the trial-and-error method based on simulations of the routed
input signals and by using the following criteria: (i) C was determined
first from the peak delay between two succeeding gauging stations
(celerity = reach length divided by delay), whereas (ii) D was adjusted in order
to get a best fit of the shape of the hydrograph or mass chemograph with the
observed output signal.
Framework application on the event no. 1 in high-flow condition
along the two reaches R1 (a) and R2 (b). Total flow and fluxes, base
component and flood component are represented in the first, second, and
third column, respectively.
ResultsSimulation of lateral exchanges
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 (□)
exclusively whereas flood analysis takes into account minimum
(▽) and maximum (△) values of the calculated
lateral exchanges (QA,flood and MA,flood). Base (a) and flood (b)
lateral flow (QA,base and QA,flood) are compared to the mean base
and flood flow input (QI,base and QI,flood) and base (c)
and flood (d) lateral solute fluxes (MA,base and MA,flood) are
compared to the mean base and flood fluxes input (MI,base and
MI,flood).
Variability of lateral exchanges
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.
Lateral exchanges for the base component
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.
Lateral exchanges for the flood component
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).
Transport dynamics along the conduit networkDistribution of model parameters
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 CQ water-flow celerity parameter increased linearly with input
peak flow for events 1 to 5 for reaches R1 and R2 (Fig. 8a, graph in semi-log
scale). Reach R1, entirely located in the unsaturated zone, showed lower
celerities compared to reach R2, which span both the unsaturated and
saturated zone. Events 6 and 7, corresponding to the extreme dry period, did
not fit this trend for both reaches, and were characterized by lower CQ
values. This behaviour may be related to the low degree of water saturation
of the system at the beginning of the flood. The CM solute flux celerity
parameter showed a similar relationship with the flood flow intensities for
events 1 to 5, but only for reach R1 and again with a different behaviour for
events 6 and 7 (Fig. 8c). On the contrary, we did not find any relationships
for reach R2, suggesting a more complex behaviour for this part of the
system, most probably related to the presence of the saturated zone.
All events from reaches R1 and R2 had very low DQ diffusivities. Only
events 1–4 on reach R2 showed increasing DQ values with increasing
input peak flow (Fig. 8b). We note that these four data points were all from a
wet period and from reach R2, which localized in the non-saturated and the
saturated zone. The DM solute flux diffusivity parameter showed a very
similar distribution as DQ, i.e. again with high values only for events
1–4 on reach R2 (Fig. 8d). Finally, note that a relationship may exist
between the diffusive wave celerity CQ and the flow velocity CM in the
advection–diffusion equation, such as CQ≅5/3CM in the case of a
rectangular section. Fig. 8 shows that the ratio CQ/CM ranges
between 1.1 and 2.3.
Assessment of the saturated level of the conduit network
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 CQ
parametrization – characterizing the propagation of the flood peak along the
two reaches R1 and R2. In fact, by considering a mean constant slope of the
main karst conduit, we hypothesize a constant CQ value along the
unsaturated zone in R1 and R2. The relation between CQ values of R1 and
R2 for each event should therefore deliver information on the localization of
the limit between the saturated and the unsaturated zone. Based on these
hypotheses, the Eq. (22) is used to estimate the percentage of conduit length
of reach R2 located within the unsaturated zone, defined as the “U”
value.
U=CQ(R1)CQ(R2)⋅100
Representation of the limit estimation between unsaturated and
saturated conduits along R2 as a function of the mean spring base flow. U
calculation corresponding to the percentage of conduit length of R2 located
within the unsaturated zone.
This limit is supposed to fluctuate as a function of the baseflow condition.
Figure 9a represents the U values of all events as a function of the mean
baseflow at the system outlet in station s3 (Fourbanne spring). A linear
relationship is observed for events 1 to 6, showing – as expected – an
increasing U with a decreasing mean baseflow, which is coherent with the
saturation level of the system measured by discharge. Based on the calculated
U value and with an estimated total length of 5.4 km for R2, the saturated
conduit length is approximatively 1.5 to 2.2 km. Event 7 shows that the
system behaved differently for flood events during extreme dry periods. This
point will be discussed in the next Section.
DiscussionModelling framework
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 CQ and diffusivity
DQ) of seven flood events allowed us to characterize the variability of flood
routing in more detail for the reach R1, located in the unsaturated zone, and
the reach R2, covering both unsaturated and saturated zones. Both reaches
showed linearly increasing CQ values with increasing flood intensities
(Sect. 4.3.1.). The difference of CQ parametrization between the two
reaches was then used in Sect. 4.3.2 to quantify the fluctuation of the
unsaturated–saturated boundary within the karst conduit. Solute transport
parameters (CM and DM) are less obvious to interpret. Reach R1
showed a linear relationship of CM as a function of solute flux intensity,
whereas a comparable trend was absent for reach R2, attesting the difficulty
of characterizing transport processes within the saturated conduit network from
the available monitoring design.
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 among others,
the base component typifies slow flow in the system, which is strongly
influenced by interaction with the rock matrix (or low permeable volumes)
with high storage capacities, whereas the flood component typifies quick flow
within the conduit network with low storage capacities, which is strongly
linked with flood intensity. Since we used the baseflow to estimate base
exchanges and the diffusive wave flood routing model to assess flood
exchanges, our framework gives a deconvolution of the two components of the
lateral exchanges helping to interpret the involved processes.
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 results for the
reach R2 furthermore denote the difficulty of decomposing exchanges that
occurred in part in the unsaturated and in part in the saturated zone.
Lateral flood outflows for R2 were mainly combined with low flood outfluxes
with total mineralizations lower than those observed for the input and output
stations. This may be the result of complex exchanges occurring between
conduit and matrix (or low permeable volume) water within the saturated zone,
with out- and inflows occurring concomitantly during the same flood and
having contrasted concentrations. The DW model thus provides much
information about lateral exchanges in karst aquifers. However, we point out
the limits of our framework's ability to identify a single, unambiguous model structure
representing transport processes when boundary conditions are poorly
described . Its limitation is that it considers a reach
of a conduit system as a “homogeneous” unit with uniformly distributed
lateral exchanges. It is a diagnostic tool which naturally cannot be used to
decipher too much complexity in the case of too-remote monitoring stations
with highly variable lateral contributions. An interesting point would be to
discretize the conduit network in shorter reaches. Unfortunately, this would
require a large set of monitoring stations that are not always accessible in
karst systems. In this way, proposed a methodology to
identify the transfer function based on the Hayami analytical solution using
the topological elevations of a distributed-channel network flood-routing
model. However, in karst conduit systems, besides the partial access to
the conduit, the karst conduit elevations may not always be relevant to
describing flood routing propagation, specifically in the saturated zone.
Functional scheme
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 , leading
to time-variant limits of their recharge area .
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 who demonstrated the influence of the
contrasted kinetics fluctuations of the hydraulic head between the conduit
and the surrounding matrix during a flood event. This conduit–matrix
relationship in the saturated zone probably induces a complex water mixing
effect inside of karst conduit during the flood event.
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 – leading to a rapid
hydrological response at the spring due to the high connectivity of the
conduit networks. But in our case, we also noted that a large part of water
storage (assessed by the baseflow component in the different stations)
originate from this zone. This result highlighted the important storage
function of the unsaturated zone, something that was previously pointed out in works using
hydrochemical approaches , but is often
minimized in current conceptual karst models.
Conclusion
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).
SymbolsUnitsDefinitions*-represents the convolution operatorCM(ms-1)parameter controlling the celerity of solute fluxCQ(ms-1)flood wave celerityDM(m2s-1)parameter controlling the diffusivity of solute fluxDQ(m2s-1)flood wave diffusivity(KQ, KM)-Hayami kernel function for water-flow and solute transport modelling, respectivelym(kgs-1m-1)lateral solute flux per unit lengthM(kgs-1)solute flux(MI,base, MI,flood, MI,tot)(kgs-1)upstream base, flood and total solute flux, respectively(MO,base, MO,flood, MO,tot)(kgs-1)downstream base, flood and total solute flux, respectively(MA,base, MA,flood, MA,tot)(kgs-1)lateral base, flood and total solute flux exchanges, respectivelyq(m2s-1)lateral flow per unit lengthQ(m3s-1)discharge(QI,base, QI,flood, QI,tot)(m3s-1)upstream base, flood and total flow, respectively(QO,base, QO,flood, QO,tot)(m3s-1)downstream base, flood and total flow, respectively(QA,base, QA,flood, QA,tot)(m3s-1)lateral base, flood and total flow exchanges, respectively(R1, R2)-reach 1 (s1 to s2) and reach 2 (s2 to s3), respectively(s1, s2, s3)-monitoring stations 1, 2 and 3, respectively(SI,base, SI,flood, SI,tot)(gL-1)upstream solute base, flood and total concentrations, respectively(SO,base, SO,flood, SO,tot)(gL-1)downstream solute base, flood and total concentrations, respectively(SA,base, SA,flood, SA,tot)(gL-1)lateral solute base, flood and total concentrations, respectivelyt(s)timeU-percentage of the unsaturated conduit length along R2x(m)downstream distance
The authors declare that they have no conflict of
interest.
Acknowledgements
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
(http://www.sokarst.org/). The authors are very grateful to the Editor
Mauro Giudici and the four reviewers for their constructive comments on the
paper. This work was carried out with the financial support of the
Burgundi-Franche-Comté Region and the BRGM.
Edited by: Mauro Giudici
Reviewed by: four anonymous referees
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