HESSHydrology and Earth System SciencesHESSHydrol. Earth Syst. Sci.1607-7938Copernicus GmbHGöttingen, Germany10.5194/hess-19-893-2015Reducing the ambiguity of karst aquifer models by pattern matching of flow and transport on catchment scaleOehlmannS.sandra.oehlmann@geo.uni-goettingen.deGeyerT.LichaT.SauterM.Geoscience Center, University of Göttingen, Göttingen, GermanyLandesamt für Geologie, Rohstoffe und Bergbau, Regierungspräsidium Freiburg, Freiburg, GermanyS. Oehlmann (sandra.oehlmann@geo.uni-goettingen.de)12February20151928939127July20144August201417December201417January2015This work is licensed under a Creative Commons Attribution 3.0 Unported License. To view a copy of this license, visit http://creativecommons.org/licenses/by/3.0/This article is available from https://www.hydrol-earth-syst-sci.net/19/893/2015/hess-19-893-2015.htmlThe full text article is available as a PDF file from https://www.hydrol-earth-syst-sci.net/19/893/2015/hess-19-893-2015.pdf
Assessing the hydraulic parameters of karst aquifers is a challenge due to
their high degree of heterogeneity. The unknown parameter field generally
leads to a high ambiguity for flow and transport calibration in numerical
models of karst aquifers. In this study, a distributed numerical model was
built for the simulation of groundwater flow and solute transport in a
highly heterogeneous karst aquifer in south-western Germany. Therefore, an
interface for the simulation of solute transport in one-dimensional pipes
was implemented into the software COMSOL Multiphysics® and
coupled to the three-dimensional solute transport interface for continuum
domains. For reducing model ambiguity, the simulation was matched for
steady-state conditions to the hydraulic head distribution in the model
area, the spring discharge of several springs and the transport velocities
of two tracer tests. Furthermore, other measured parameters such as the
hydraulic conductivity of the fissured matrix and the maximal karst conduit
volume were available for model calibration. Parameter studies were
performed for several karst conduit geometries to analyse the influence of
the respective geometric and hydraulic parameters and develop a calibration
approach in a large-scale heterogeneous karst system.
Results show that it is possible not only to derive a consistent flow and
transport model for a 150 km2 karst area but also to combine the use
of groundwater flow and transport parameters thereby greatly reducing model
ambiguity. The approach provides basic information about the conduit network
not accessible for direct geometric measurements. The conduit network volume
for the main karst spring in the study area could be narrowed down to
approximately 100 000 m3.
Introduction
Karst systems play an important role in water supply worldwide (Ford and
Williams, 2007). They are characterized as dual-flow systems where flow
occurs in the relatively lowly conductive fissured matrix and in highly
conductive karst conduits (Reimann et al., 2011). There are a number of
process-based modelling approaches available for simulating karst aquifer
behaviour. Overviews on the various types of distributed process and
lumped-parameter models are provided by several authors (Teutsch and Sauter,
1991; Jeannin and Sauter, 1998; Kovács and Sauter, 2007; Hartmann et al.,
2014). In most cases, lumped-parameter models are applied, since they are
less demanding on input data (Geyer et al., 2008; Perrin et al., 2008;
Hartmann et al., 2013; Schmidt et al., 2013). These models consider neither
the actual flow process nor the heterogeneous spatial distribution of
aquifer parameters, but are able to simulate the integral aquifer behaviour,
e.g. karst spring responses. The spatial distribution of model parameters
and state variables, e.g. the hydraulic head distribution, need to be
addressed with distributed numerical models should the necessary field data
be available (e.g. Oehlmann et al., 2013; Saller et al., 2013). A
distributed modelling approach suited for the simulation of strongly
heterogeneous and anisotropic aquifers with limited data availability is the
hybrid modelling approach. The approach simulates the fast flow component in
the highly conductive karst conduit system in discrete one-dimensional
elements and couples it to a two- or three-dimensional continuum
representing the fissured matrix of the aquifer (Oehlmann et al., 2013).
Hybrid models are rarely applied to real karst systems because they have a
high demand of input data (Reimann et al., 2011). They are, however, regularly
applied in long-term karst genetic simulation scenarios (e.g. Clemens et
al., 1996; Bauer et al., 2003; Hubinger and Birk, 2011). In these models not
only groundwater flow but also solute transport is coupled in the fissured
matrix and in the karst conduits. Aside from karst evolution such coupling
enables models to simulate tracer or contaminant transport in the karst
conduit system (e.g. Birk et al., 2005). In addition to serving for
predictive purposes, such models can be used for deriving information about
the groundwater catchment itself (Rehrl and Birk, 2010).
A major problem for characterizing the groundwater system with numerical
models is generally model ambiguity. The large number of calibration
parameters is usually in conflict with a relatively low number of field
observations, e.g. different hydraulic parameter fields and process
variables may give a similar fit to the observed data but sometimes very
different results for prognostic simulations (Li et al., 2009). Especially
the geometric and hydraulic properties of the karst conduit system are
usually unknown and difficult to characterize with field experiments for a
whole spring catchment (Worthington, 2009). With artificial tracer test data
the maximum conduit volume can be estimated but an unknown contribution of
fissured matrix water prevents further conclusions on conduit geometry (Birk
et al., 2005; Geyer et al., 2008). It is well known that the use of several
objective functions, i.e. several independent field observations, can
significantly reduce the number of plausible parameter combinations (Ophori,
1999). Especially in hydrology (e.g. Khu et al., 2008; Hunter et al., 2005)
and also for groundwater systems (e.g. Ophori, 1999; Hu, 2011; Hartmann et
al., 2013), this approach has been successfully applied with a wide range of
observation types, e.g. groundwater recharge, hydraulic heads, remote
sensing and solute transport. Particularly, the simulation of flow and
transport is known to reduce model ambiguity and yield information on karst
conduit geometry (e.g. Birk et al., 2005; Covington et al., 2012; Luhmann et
al., 2012; Hartmann et al., 2013). Usually, automatic calibration schemes
performing a multi-objective calibration for several parameters are used for
this purpose (Khu et al., 2008). However, for complex modelling studies
calculation times might be large due to the high number of model runs needed
(Khu et al., 2008) and a precise conceptual model is essential as basis for
the automatic calibration (Madsen, 2003). In general, numerical models of
karst aquifers are difficult to build because of their highly developed
heterogeneity (Rehrl and Birk, 2010). Thus, automatic calibration procedures
are better suited for conceptual and lumped-parameter models, where
calibration parameters include effective geometric properties and no spatial
representation of the hydraulic parameter field and conduit geometry is
necessary. Complex distributed numerical approaches generally require longer
simulation times due to the necessary spatial resolution. Long simulation
times limit the number of model runs that can reasonably be performed and
manual calibration based on hydrogeological knowledge is necessary
(e.g. Saller et al., 2013). Therefore, applied distributed numerical models in
karst systems usually focus on a smaller number of objective functions. They
generally cannot simulate the hydraulic head distribution in the area,
spring discharge and tracer breakthrough curves simultaneously on catchment
scale. Some studies combine groundwater flow with particle tracking for
tracer directions (e.g. Worthington, 2009; Saller et al., 2013) without
simulating tracer transport. On the other hand there are studies simulating
breakthrough curves without calibrating for measured hydraulic heads
(e.g. Birk et al., 2005). For developing process-based models which can be used as
prognostic tools, e.g. for the delineation of protection zones, the
simulation should be able to reproduce groundwater flow and
transport within a groundwater catchment. Especially in complex
hydrogeological systems, this approach would reduce model ambiguity, which
is a prerequisite in predicting groundwater resources and pollution risks.
This study shows how the combination of groundwater flow and transport
simulation can be used not only to develop a basis for further prognostic
simulations in a heterogeneous karst aquifer with a distributed modelling
approach on catchment scale, but also to reduce model ambiguity and draw
conclusions on the spatially distributed karst network geometries and the
actual karst conduit volume. The approach shows the kind and minimum number
of field observations needed for this aim. Furthermore, a systematic
calibration strategy is presented to reduce the number of necessary model
runs and the simulation time compared to standard multi-objective
calibrations. For this purpose a hybrid model was built and a pattern
matching procedure was applied for a well-studied karst aquifer system in
south-western Germany. The model was calibrated for three major observed
parameters: the hydraulic head distribution derived from measurements in
20 boreholes, the spring discharge of six springs and the tracer breakthrough
curves of two tracer tests.
Modelling approach
The simulation is based on the mathematical flow model discussed in detail
by Oehlmann et al. (2013). The authors set up a three-dimensional hybrid
model for groundwater flow with the software COMSOL
Multiphysics®. As described by Oehlmann et al. (2013) the
simulation was conducted simultaneously in the three-dimensional fissured
matrix, in an individual two-dimensional fault zone and in one-dimensional
karst conduit elements to account for the heterogeneity of the system.
Results showed that the karst conduits widen towards the springs and
therefore, a linear relationship between the conduit radius and the conduit
length s [L] was established. Values for s start with zero at the point
farthest away from the spring and increase towards the respective karst
spring. In agreement with these results and karst genesis simulations by
Liedl et al. (2003), the conduit radius is calculated as
rc=ms+b,
where rc [L] is the radius of a conduit branch and m and b are the
two parameters defining the conduit size. b [L] is the initial radius of the
conduit at the point farthest away from the spring and m [–] is the slope
with which the conduit radius increases along the length of the conduit s.
In the following the equations used for groundwater flow and transport are
described. The subscript “m” denotes the fissured matrix, “f” the fault zone and
“c” the conduits hereby allowing a clear distinction between the respective
parameters. Parameters without a subscript are the same for all karst
features in the model.
Groundwater flow
Groundwater flow was simulated for steady-state conditions. This approach
seems appropriate since this work focuses on the simulation of tracer
transport in the conduit system during tracer tests, which are ideally
conducted under quasi-steady-state flow conditions. Therefore, the
simulations refer to periods with a small change of spring discharge,
e.g. base flow recession, and are not designed to predict conditions during
intensive recharge/discharge events. The groundwater flow in the
three-dimensional fissured matrix was simulated with the continuity equation
and the Darcy equation (Eq. 2a und b).
Qm=∇ρum,um=-Km∇Hm,
where Qm is the mass source term [M L-3 T-1], ρ the
density of water [M L-3] and um the Darcy
velocity [L T-1]. In Eq. (2b) Km is the hydraulic conductivity
of the fissured matrix [L T-1] and Hm the hydraulic head [L].
Two-dimensional fracture flow in the fault zone was simulated with
the COMSOL® fracture flow interface. The interface only allows
for the application of the Darcy equation inside of fractures, so laminar
flow in the fault zone was assumed. In order to obtain a process-based
conceptualization of flow, the hydraulic fault conductivity Kf was
calculated by the cubic law (Eq. 3):
Kf=df2ρg12μ,
where df is the fault aperture [L], ρ the density of
water [M L-3], g the gravity acceleration [L T-2] and μ the
dynamic viscosity of water [M T-1 L-1].
For groundwater flow in the karst conduits, the Manning equation was used (Eq. 4).
uc=1nrc223dHcdx,
where uc is the specific discharge in
this case equalling the conduit flow velocity [L T-1], n the Manning
coefficient [T L-1/3], rc/2 the hydraulic radius [L] and
dHc/dx the hydraulic gradient [–]. The Manning coefficient is an
empirical value for the roughness of a pipe with no physical nor measurable
meaning. The hydraulic radius is calculated by dividing the cross section by
the wetted perimeter, which in this case corresponds to the total perimeter
of the pipe (Reimann et al., 2011).
The whole conduit network was simulated for turbulent flow conditions. Due
to the large conduit diameters (0.01–6 m, Sect. 5) this assumption is a
good enough approximation. Hereby, strong changes in flow velocities due to
the change from laminar to turbulent flow can be avoided. At the same time,
the model does not require an estimation of the critical Reynolds number,
which is difficult to assess accurately.
The three-dimensional flow in the fissured matrix and the one-dimensional
conduit flow were coupled through a linear exchange term that was defined
according to Barenblatt et al. (1960) as
qex=αLHc-Hm,
where qex is the water exchange between conduit and fissured
matrix [L2 T-1] per unit conduit length L [L], Hm the hydraulic
head in the fissured matrix [L], Hc the hydraulic head in the conduit [L]
and α the leakage coefficient [L2 T-1]. The leakage
coefficient was defined as
α=2πrcKm,
where 2 πrc is the conduit perimeter [L]. Other possible
influences, e.g. the lower hydraulic conductivity at the solid–liquid
interface of the pipe and the fact that water is not exchanged along the
whole perimeter but only through the fissures are not considered. The exact
value of these influences is unknown and the exchange parameter mainly
controls the reaction of the karst conduits and the fissured matrix to
hydraulic impulses. Since the flow simulation is performed for steady-state
conditions this simplification is not expected to exhibit significant
influence on the flow field.
Solute transport
Transient solute transport was simulated based on the steady-state
groundwater flow field. COMSOL Multiphysics® offers a general
transport equation with its solute transport interface. This interface was
applied for the three-dimensional fissured matrix. In this work saturated,
conservative transport was simulated, with an advection–dispersion equation (Eq. 7)
∂∂tθmcm+∇umcm=∇DDm+De∇cm+Sm,
where θm is the matrix porosity [–], cm the solute
concentration [M L-3], DDm the mechanical
dispersion [L2 T-1] and De the molecular
diffusion [L2 T-1]. Sm is the source term [L3 T-1].
The solute transport interface cannot be applied to one-dimensional elements
within a three-dimensional model. COMSOL® offers a so-called
coefficient form edge PDE interface to define one-dimensional mathematical
equations. There, a partial differential equation is provided (COMSOL AB,
2012) which can be adapted as needed and leads to Eq. (8) in its application
for solute transport in karst conduits:
θc∂cc∂t+∇-Dc∇cc+uccc=f,
where θc is the conduit porosity which is set equal to 1,
Dc [L2 T-1] the diffusive/dispersive term
Dc= (DDc+De), f the source
term and uc [L T-1] the flow
velocity inside the conduits, which corresponds to the advective transport
component. Flow divergence cannot be neglected, as is often the case in
other studies (e.g. Hauns et al., 2001; Birk et al., 2006; Coronado et al.,
2007). Different conduit sizes and in- and outflow along the conduits lead
to significant velocity divergence in the conduit system. This needs to be
considered for mass conservation during the simulation. The mechanical
conduit dispersion DDc was calculated with Eq. (9) (Hauns et al., 2001).
DDc=εuc,
where ε is the dispersivity in the karst conduits [L].
The source term f [M T-1 L-1] in Eq. (8) equals in this case the
mass flux of solute per unit length L [L] due to matrix–conduit exchange of
solute cex:
f=cex=-De2πrcLcm-cc-qexci.
The first term of the right-hand side of Eq. (10) defines the diffusive
exchange due to the concentration difference between conduit and fissured
matrix. The second term is a conditional term adding the advective exchange
of solute due to water exchange. The concentration of the advective exchange
ci is defined as
ci=ccifqex>0cmifqex≤0.
When qex is negative, the hydraulic head in the fissured matrix is
higher than in the conduit (Eq. 5) and water with the solute concentration
of the fissured matrix cm enters the conduit. When it is positive, water
with the solute concentration cc of the conduit leaves the conduit and
enters the fissured matrix. Since one-dimensional transport is simulated in
a three-dimensional environment, the left-hand side of Eq. (8) is multiplied
with the conduit cross section πrc2 [L2]. These
considerations lead to the following transport equation for the karst conduits:
πrc2∂cc∂t+πrc2∇-Dc∇cc+uccc=-De2πrcLcm-cc-qexci.
Field site and model design
The field site is the Gallusquelle spring area on the Swabian Alb in south-western Germany. The size of the model area is approximately 150 km2,
including the catchment area of the Gallusquelle spring and surrounding
smaller spring catchments (Oehlmann et al., 2013). The Gallusquelle spring
is the main point outlet with a long-term average annual discharge of
0.5 m3 s-1. The model area is constrained by three rivers and
no-flow boundaries derived from tracer test information and the dip of the
aquifer base (Oehlmann et al., 2013) (Fig. 1).
The aquifer consists of massive and bedded limestone of the stratigraphic
units Kimmeridgian 2 and 3 (ki 2/3) (Golwer, 1978; Gwinner, 1993). The marly
limestones of the underlying Kimmeridgian 1 (ki 1) mainly act as an
aquitard. In the west of the area where they get close to the surface, they
are partly karstified and contribute to the aquifer (Sauter, 1992;
Villinger, 1993). The Oxfordian 2 (ox 2) that lies beneath the ki 1 consists
of layered limestones. It is more soluble than the ki 1 but only slightly
karstified because of the protective effect of the overlying geological
units. In the catchment areas of the Fehla-Ursprung and the
Balinger springs close to the western border (Fig. 1a) the ox 2 partly contributes to
the aquifer. For simplicity, only two vertical layers were differentiated in
the model: the aquifer and the underlying aquitard.
(a) Plan view of the model area. Settlements, fault zones
and rivers in the area are plotted, as well as the 20 observation wells used
for hydraulic head calibration, the six springs used for spring discharge
calibration and the two tracer tests employed for flow velocity calibration.
Catchment areas for the Gallusquelle spring and the Ahlenberg and
Büttnauquellen springs were simulated according to Oehlmann et al. (2013).
(b) Three-dimensional view of the model. The upper boundary is hidden to allow a
view of the karst conduit system and the aquifer base. The abbreviation
BC stands for boundary condition. At the hidden upper boundary, a
constant recharge Neumann BC is applied.
The geometry of the conduit system was transferred from the
COMSOL® model calibrated for flow by Oehlmann et al. (2013).
It is based on the occurrence of dry valleys in the investigation area and
artificial tracer test information (Gwinner, 1993). The conduit geometry for
the Gallusquelle spring was also employed for distributed flow simulations
by Doummar et al. (2012) and Mohrlok and Sauter (1997)
(Fig. 1). In this work, all highly conductive
connections identified by tracer tests in the field were simulated as
discrete one-dimensional karst conduit elements. The only exception is a
connection in the west of the area that runs perpendicular to the dominant
fault direction and reaches the Fehla-Ursprung spring at the northern
boundary (Fig. 1). While the element was regarded
as a karst conduit by Oehlmann et al. (2013) it is more likely that the
water crosses the graben structure by a transversal cross-fault (Strayle,
1970). Therefore, the one-dimensional conduit element was replaced by a
two-dimensional fault element (Fig. 1b). This leads to a small adjustment in
the catchment areas compared to the results of Oehlmann et al. (2013) (Fig. 1a).
While the discharge data for the Fehla-Ursprung spring are not as
extensive as for the other simulated springs, it is approximated to
0.1 m3 s-1, the annual average ranging from 0.068 to
0.135 m3 s-1. The fault zone aperture was calibrated
accordingly (Sect. 5).
Calibrated and simulated parameters for the best-fit simulations.
Literature values are given if available. TT 1 and TT 2 refer to the two
tracer tests.
a Jeannin (2001); b Geyer et al. (2008); c measurement
interval 1 min, simulation interval 2.7 h; d measurement interval 6 h,
simulation interval 2.7 h; e Sauter (1992); f Birk et
al. (2005); g Merkel (1991); h average for the interval between tracer
test 1 and the spring.
Hydraulic head distributions for different combinations of
geometric conduit parameters for scenario 1. b is the lowest conduit radius
and m the radius increase along the conduit. For comparison, a trend line is
fitted to the measured hydraulic head values showing the distribution of
hydraulic gradients from the Gallusquelle spring to the western border of
its catchment area.
Due to a large number of studies conducted in the area during the last
decades (e.g. Villinger, 1977; Sauter, 1992; Geyer et al., 2008; Kordilla et
al., 2012; Mohrlok, 2014) many data for pattern matching are available even
though the karst conduit network itself is not accessible. Since the
groundwater flow simulation was performed for steady-state conditions,
direct recharge, which is believed to play an important role during event
discharge (Geyer et al., 2008), was neglected. It is not expected that
recharge dynamics exhibit significant influence on the flow field during
recession periods. From Sauter (1992) the long-term average annual recharge,
ranges of hydraulic parameters and the average annual hydraulic head
distribution derived from 20 observation wells (Fig. 1a) are
available. Villinger (1993) and Sauter (1992) provided data on the
geometry of the aquifer base. Available literature values for the model
parameters are given in Table 1.
The observed hydraulic gradients in the Gallusquelle area are not uniform
along the catchment. Figure 2 shows a S-shaped distribution with distance to
the Gallusquelle spring. The gradient at each point of the area depends on
the combination of the respective transmissivity and total flow. The amount
of water flowing through a cross sectional area increases towards the
springs due to flow convergence. In the Gallusquelle area, the
transmissivity rises in the vicinity of the springs leading to a low
hydraulic gradient. In the central part of the area discharge is relatively
high while the transmissivities are lower leading to the observed steepening
of the gradient starting in a distance of 4000 to 5000 m from the
Gallusquelle spring. Towards the boundary of the catchment area in the west
the water divide reduces discharge in the direction of the Gallusquelle
spring leading to a smoothing of hydraulic gradients.
Geyer et al. (2008) calculated the maximum conduit volume for the
Gallusquelle spring Vc [L3] with information from the tracer test
that will be referred to as tracer test 2 in the following. Since the
injection point of the tracer test is close to the catchment boundary, it is
assumed that it covers the whole length of the conduit system. The authors
calculated the maximum volume at 218 000 m3. Their approach assumes the
volume of the conduit corresponds to the total volume of water discharged
during the time between tracer input and tracer arrival neglecting the
contribution of the fissured matrix.
The six springs that were monitored and therefore simulated are shown in
Fig. 1. Except for the Balinger spring, their discharges were fitted
to long-term average annual discharge data. For the Balinger spring
discharge calibration was not possible due to lack of data. It was included
as a boundary condition because several tracer tests provided a valuable
basis for the conduit structure leading to the spring.
Conceptual overview of the simulated scenarios. The conduit
geometry and the varying parameters are shown.
Tracer directions were available for 32 tracer tests conducted at
20 different tracer injection locations (Oehlmann et al., 2013). In all, 16 of the
tracer tests were registered at the Gallusquelle spring. For this work two
of them were chosen for pattern matching of transport parameters. Both of
them were assumed to have a good and direct connection to the conduit
network. Tracer test 1 (Geyer et al., 2007) has a tracer injection point at
a distance of 3 km to the Gallusquelle spring. Tracer test 2
(MV746 in Merkel, 1991; Reiber et al., 2010) was conducted at 10 km distance
to the Gallusquelle spring (Fig. 1a). Due to the flow conditions (Fig. 1a)
it can be assumed that tracer test 2 covers the total length of the conduit
network feeding the Gallusquelle spring. The recovered tracer mass was
chosen as input for the tracer test simulation. The basic information about
the tracer tests is given in Table 2.
Since the tracer tests were not performed at average flow conditions, the
model parameters were calibrated first for the long-term average annual
recharge of 1 mm d-1 and the long-term average annual discharge of
0.5 m3 s-1. For the transport simulations, the recharge was
then adapted to produce the respective discharge observed during the tracer
experiment (Table 2).
Parameter analysis
An extensive parameter analysis was performed in order to identify
parameters determining the hydraulic parameter field in the model area, as
well as their relative contributions to the discharge and conduit flow
velocities. The fitting parameters include the parameters controlling the
respective transmissivities of the fissured matrix and the karst conduit
system, i.e. the geometry and roughness of the conduit system, the hydraulic
conductivity of the fissured matrix and the fracture aperture for the
Fehla-Ursprung spring. Furthermore, the apparent dispersivities for the two
artificial tracer tests were calibrated (Table 1). Since all model runs were
performed for steady-state conditions parameters controlling the temporal
distribution of recharge were not considered. The parameter analysis was
performed with COMSOL Multiphysics® parametric sweep tool,
which sweeps over a given parameter range. Parameter ranges were chosen
according to literature values (Table 1). For the conduit geometry
parameters, lowest conduit radius b and slope of radius increase m, no
literature values are available. Therefore, the ranges were chosen so that
conduit volumes ranged below the maximum volume given by Geyer et al. (2008).
In addition to the variation of the fitting parameters, five basic
scenarios were compared. They correspond to different conceptual
representations of the area and are summarized in Fig. 3 and Table 3.
Field data of the simulated tracer tests.
TracerTracertest 1test 2Input mass (kg)0.7510Recovery (%)7250Distance to spring (km)310Spring discharge (m3 s-1)0.3750.76Sampling interval1 min6 hPeak time (h)4779.5
Specifics of the different scenarios. The bold writing indicates
the parameter that is analysed in the respective scenario. The results are
indicated by comparative markers. “+” means good, “o” means
average and “–” means bad compared to the other scenarios. Details
to the scenarios and results evaluation can be found in Sect. 4.
ParameterScenario 1Scenario 2Scenario 3Scenario 4Scenario 5Kcconstantlinear increaselinear increaselinear increaselinear increaseLateral networkminimalminimalextendedminimalminimalKmconstantconstantconstantvariableconstantIntersection radius rc2rc0rc0rc0rc0rc02+rc12Main results Hydraulic head fit+++++Fit of breakthrough–++++Model applicability+o––o
Three objective functions were employed for pattern matching: spring
discharge, hydraulic head distribution and flow velocities of the two tracer
tests (Sect. 3). The average spring discharge of the Gallusquelle spring was
set by the difference between simulated and the measured discharge. A
difference of 10 L s-1 was considered as acceptable. Parameter sets,
which could not fulfil this criterion, were not considered for parameter
analysis. The other low-discharge and less-investigated springs (Sect. 3)
were used to inspect the flow field and water balance in the modelling area,
i.e. they were only considered after parameter fitting to check the
plausibility of the deduced parameter set.
The fit of the tracer tests was determined by comparing the arrival times of
the highest peak concentration of the simulation with the measured value
(peak offset). Since tracer experiments conducted in karst conduits usually
display very narrow breakthrough curves, this procedure appears to be
justified. The quality of the fit was judged as satisfactory if the
peak offset was lower than either the simulation interval or the measurement interval.
The fit of the hydraulic head distribution was determined by calculating the
root mean square error (RMSE) between the simulated and the observed values
at the respective locations of the observation wells. Since the fit at local
points with a large-scale modelling approach generally shows large
uncertainties due to low-scale heterogeneities, an overall fit of
< 10 m RMSE was accepted. Furthermore, a qualitative comparison with
the hydraulic gradients in the area was performed (e.g. Fig. 2) to ensure
that the general characteristics of the area were represented instead of
only the statistical value.
Influence of the hydraulic conductivity of the fissured matrix on
the objective functions. (a) Influence on the root mean square error of the
hydraulic head distribution in relation to the conduit geometry. The conduit
geometry is represented by the parameter b/m (Eq. 1), which is the ratio of
the smallest radius to the slope of radius increase along the conduit
length. (b) Influence on the conduit flow velocity for tracer test 1.
(c) Influence on the conduit flow velocity for tracer test 2.
Scenario 1 – standard scenario
In scenario 1 all features were implemented as described in Sects. 2 and 3.
The parameter analysis shows that for each conduit geometry, defined by
their smallest conduit radii b and their slopes of radius increase along the
conduit length m (Eq. 1), only one value of the Manning coefficient n allows a
simulated discharge for the Gallusquelle spring of 0.5 m3 s-1.
The n value correlates well with that for the total conduit volume due to the
fact that the spring discharge is predominantly determined by the
transmissivity of the karst conduit system. The transmissivity of the
conduit system at each point in space is the product of its hydraulic
conductivity, which is proportional to 1/n, and the cross sectional area of
the conduit A. Thus, to keep the spring discharge at 0.5 m3 s-1
a higher conduit volume requires a higher calibrated n value (Eq. 4).
With scenario 1 it is possible to achieve a hydraulic head fit resulting in
a RMSE of 6 m that can be judged as adequate on
catchment scale. Regarding the conduit geometry, a good hydraulic head fit
can be achieved with small b values independently of the chosen m value
(Fig. 2a). The higher the b value, the higher the m value to reproduce the hydraulic
gradients of the area (Fig. 2). This implies that the hydraulic head fit is
independent of the conduit volume during steady-state conditions but depends
on the b/m ratio. The influence of the b/m ratio on the hydraulic head fit
depends on the hydraulic conductivity of the fissured matrix Km. For low
Km values of ca. 1 × 10-6 m s-1 the hydraulic head fit
is completely independent of the conduit geometry and the RMSE is very high
(Fig. 4a). For high Km values of ca. 5 × 10-4 m s-2
(Fig. 4a) the dependence is also of minor importance and the RMSE is relatively
stable at ca. 11 m. Due to the high hydraulic conductivity of the fissured
matrix the hydraulic gradients do not steepen in the vicinity of the spring
even for high b/m ratios. For Km values between the above values the RMSE
significantly rises for b/m ratios above 1000 m. For the range of acceptable
errors, i.e. lower than 10 m, it is apparent in Fig. 4a that the best-fit
Km value is approximately 1 × 10-5 m s-1
independent of the conduit geometry. However, no distinct best-fit conduit
geometry can be derived. There are several parameter combinations providing
a good fit for the Gallusquelle spring discharge and the hydraulic head distribution.
The goodness of the fit of the simulation of the tracer breakthrough is
mainly determined by the conduit geometry. The influence of the hydraulic
conductivity of the fissured matrix Km on flow velocities inside the
karst conduits is comparatively low and decreases even further in the
vicinity of the springs (Fig. 4b and c) leading to minor influences on
tracer travel times. Instead, the quality of the fit mainly depends on the
conduit volume and accordingly on the Manning coefficient n (Fig. 5). It is
possible to simulate only one of the two tracer experiments with this
scenario (Fig. 5). Given the broad range of geometries for which an adequate
hydraulic head fit can be achieved (Figs. 2 and 4) it is possible to simulate
one of the two tracer peak velocities and the hydraulic head distribution
with the same set of parameters. While the simulation of the breakthrough of
tracer test 1 requires relatively high n values, of ca. 2.5 s m-1/3,
that of tracer test 2 can only be calibrated with lower values of
ca. 1.7 s m-1/3 (cf. Fig. 5a and b). For every parameter set, where the
travel time of the simulated tracer test 2 is not too long, that of tracer
test 1 is too short. For the simulation of tracer test 2, the velocities at
the beginning of the conduits must be relatively high. To avoid the flow
velocities from getting too high in downgradient direction, the conduit size would have to
increase drastically due to the constant additional influx of water from the
fissured matrix. In the given geometric range, the conduit system has a
dominant influence on spring discharge. Physically, this situation
corresponds to the conduit-influenced flow conditions (Kovács et al.,
2005). Thus, conduit transmissivity is a limiting factor for conduit–matrix
exchange and a positive feedback mechanism is triggered, if the conduit size
is increased. A higher conduit size leads to higher groundwater influx from
the fissured matrix and spring discharge is overestimated. Therefore,
parameter analysis shows that scenario 1 is too strongly simplified to
correctly reproduce the complex nature of the aquifer.
Difference between peak concentration times vs. the Manning
n value for scenario 1. High n values correspond to high conduit volumes and
high cross sectional areas at the spring (a) for tracer test 1 and
(b) for tracer test 2.
Scenario 2 – conduit roughness coefficient Kc
In scenario 2 the Manning coefficient n was changed from constant to
laterally variable. In the literature, n is generally kept constant
throughout the conduit network (e.g. Jeannin, 2001; Reimann et al., 2011)
for lack of information on conduit geometry. However, it is assumed that the
Gallusquelle spring is not fed by a single large pipe. Rather there is some
evidence in the spring area that a bundle of several small-interconnected
pipes feed the spring. Since the number of individual conduits per bundle
is unknown and the regional modelling approach limits the resolution of
local details, the small diameter conduits, which the bundle consists of,
cannot be simulated individually. Therefore, each single pipe in the model
represents a bundle of conduits in the field.
It can be assumed that the increase in conduit cross section is at least
partly provided by additional conduits added to the bundle rather than a
single individual widening conduit. Therefore, while the cross section of
the simulated conduit, i.e. the total effective cross section of the conduit
bundle, increases towards the springs, it is not specified how much of this
increase is due to the individual conduits widening and how much is due to
additional conduits, not distinguishable in the simulation. If the simulated
effective cross sectional area increase is mainly due to additional conduits
being included in the bundle, the surface / volume ratio increases with the
cross section, contrary to what would be observed, if a single conduit in
the model would represent a single conduit in the field. The variation in
surface area / volume ratio implicitly leads to a larger roughness in the
simulation, even further enhanced by exchange processes between the
individual conduits. This effect again leads to an increase in the Manning
coefficient n in the downgradient direction towards the spring for a
simulated single conduit. Since the number and size of the individual
conduits is unknown, it is impossible to calculate the change of n directly
from the geometry. Thus, a simple scenario was assumed where the roughness
coefficient Kc, which is the reciprocal of n, was linearly and negatively
coupled to the rising conduit radius (Eq. 13).
Kc=1n=-mhrc+mhrc,max+bh,
where rc [L] is the conduit radius and rc,max [L] the maximum
conduit radius simulated for the respective spring, which
COMSOL® calculates from Eq. (1). mh [L-2/3 T-1] and
bh [L1/3 T-1] are calibration parameters determining the slope and the
lowest value of the roughness coefficient respectively.
For every conduit geometry several combinations of mh and bh lead to
the same spring discharge. However, hydraulic head fit and tracer velocities
are different for each mh–bh combination even if spring discharge
is the same. With the new parameters a higher variation of velocity profiles
is possible. This allows for the calibration of the tracer velocities of
both tracer tests. The dependence of tracer test 2 on mh is much higher
than that of tracer test 1 since it is injected further upgradient towards
the beginning of the conduit (Fig. 6). Therefore, tracer test 2 is
influenced more strongly by the higher velocities far away from the spring
introduced by high mh values and always shows a significant positive
correlation with mh (Fig. 6).
Hydraulic head errors and differences between peak concentration
times for both tracer tests for scenario 1. The example is shown for a
conduit geometry with a starting value b= 0.01 m and a radius
increase of m= 2 × 10-4. Each mh (m-2/3 s-1) value corresponds to a
respective value of the highest conduit roughness bh (m1/3 s-1) and each
combination results in the same spring discharge.
Calibrated values for the simulated scenarios. For scenarios 2, 3
and 5 (a, b, d) hydraulic head fit and the peak-offset times of both
tracer tests (referred to as TT 1 and TT 2) are shown in relation to conduit
volume. The thick grey bar marks the target value of zero. For scenario 4 (c)
the root mean square error of the hydraulic heads is given for two
different conduit geometries in relation to the hydraulic conductivity of
the fissured matrix Km. For the version with laterally variable matrix
conductivity the axis shows as an example the hydraulic conductivity of the
north-western part. The parameters for the two geometries are given in Table 3.
Since the slope of Kc is negative with respect to the conduit length,
the variable Kc leads to a slowing down of water towards the springs. As
discussed in detail by Oehlmann et al. (2013) a rise of transmissivity
towards the springs is observed in the Gallusquelle area. Therefore,
adequate hydraulic head fits can only be obtained, if the decrease of
Kc towards the spring is not too large and compensates the effect of
the increase in conduit transmissivity due to the increasing conduit radius.
This effect reduces the number of possible and plausible parameter
combinations. From these considerations a best-fit model can be deduced
capable of reproducing all objective functions within the given error ranges
(Fig. 7a). According to the model simulations, karst groundwater discharge
and flow velocities significantly depend on the total conduit volume as is
to be expected. It can be deduced from the parameter analysis that the
conduit volume can be estimated at ca. 100 000 m3 for the different
parameters to match equally well (Fig. 7a).
Scenario 3 – extent of conduit network
In scenario 3, a laterally further extended conduit system was employed,
assuming the same maximum conduit volume as in scenarios 1 and 2 but with
different spatial distribution along the different total conduit lengths.
The original conduit length for the Gallusquelle spring in scenarios 1 and 2
is 39 410 m, for scenario 3 it is 63 490 m; therefore, the total length was assumed
to be larger by ca. 50 % (Fig. 8). The geometry of the original network
was mainly constructed along dry valleys where point-to-point connections
are observed based on qualitative evaluation from artificial tracer tests.
Of the dry valleys without tracer tests, only the larger ones were included,
where the assumption of a high karstification is backed up by the occurrence
of sinkholes (Mohrlok and Sauter, 1997). Therefore, it represents the
minimal extent of the conduit network. For scenario 3 the network was
extended along all dry valleys within the catchment, where no tracer tests
were conducted.
The results of the parameter variations are comparable to those of scenario 2
(cf. Fig. 7a and b). While the hydraulic head contour lines are smoother
than for the original conduit length the general hydraulic head fit is the
same (Fig. 7b). It seems possible to obtain a good fit for all model
parameters but the scenario is more difficult to handle numerically.
Calculation times are up to 10 times larger compared to the other scenarios
and goodness of convergence is generally lower. Since the calibrated
parameters are not significantly different from those deduced in scenario 2
it is concluded that the ambiguity introduced by the uncertainty in total
conduit length is small if hydraulic conduit parameters and total conduit
volumes are the aim of investigation.
Scenario 4 – matrix hydraulic conductivity Km
In scenario 4, the homogeneously chosen hydraulic conductivity of the
fissured matrix Km was changed into a laterally variable conductivity
based on different types of lithology and the spatial distribution of the
groundwater potential. Sauter (1992) found from field measurements that the
area can be divided into three parts with different hydraulic
conductivities. Oehlmann et al. (2013) discussed that the major influence is
the conduit geometry leading to higher hydraulic transmissivities close to
the springs in the east of the area. It is also possible that not only the
conduit diameters change towards the spring but the hydraulic conductivity
of the fissured matrix as well, since the aquifer cuts through three
stratigraphic units (Sect. 3). These geologic changes are likely to affect
the lateral distribution of hydraulic conductivities (Sauter, 1992). Figure 9
shows the division into three different areas. Km values were varied
in the range of the values measured by Sauter (1992).
Extended conduit system for scenario 3. The conduit configuration
(extent) that is used for the other scenarios is marked in red.
Model catchment with spatially distributed hydraulic
conductivities. The model area is divided into three parts after geologic
aspects. For each segment different values of the hydraulic conductivity
were examined during parameter analysis in scenario 4.
It was expected that a laterally variable Km value has a major influence
on the hydraulic head distribution. All variations of scenario 2 that
produce good results for both tracer tests and have a high total conduit
volume above 100 000 m3 yield poor results for hydraulic head errors
and spatial distributions of the hydraulic heads (Fig. 7a). For scenario 4,
two different conduit configurations (geometries) were chosen that achieve
good results with respect to conduit flow velocities. Geometry G1 has a
conduit volume of 112 000 m3. G2 has a higher b value which leads to the
maximum conduit volume of ca. 150 000 m3. All parameters for the two
simulations are given in Table 4.
It was found that while the maximum root mean square error of the hydraulic
head fit is similar for both geometries, the minimum RMSE for the hydraulic
head is determined by the conduit system. It is not possible to compensate
an unsuitable conduit geometry with suitable Km values (Fig. 7c), which
assists in the independent conduit network and fissured matrix calibration.
This observation increases the confidence in the representation of the
conduits and improves the possibility to deduce the conduit geometry from
field measurements. For an adequate conduit geometry, laterally variable
matrix conductivities do not yield any improvement. The approach introduces
additional parameters and uncertainties because the division of the area
into three parts is not necessarily obvious without detailed investigation.
From the distribution of the exploration and observation wells (Fig. 1a) it
is apparent that especially in the south and west the boundaries are not
well defined.
Scenario 5 – conduit intersections
In scenario 5, the effect of the conduit diameter change at intersections
was investigated. In the first four scenarios the possible increase in
cross sectional area at intersecting conduits was neglected. In nature,
however, the influx of water from another conduit is likely to influence
conduit evolution and therefore its diameter. In general, higher flow rates
lead to increased dissolution rates because dissolution products are quickly
removed from the reactive interface. If conditions are turbulent the
solution is limited by a diffusion dominated layer that gets thinner with
increasing flow velocities (Clemens, 1998). Clemens (1998) simulated karst
evolution in simple Y-shaped conduit networks and found higher diameters for
the downstream conduit even after short simulation times. Preferential
conduit widening at intersections could further be enhanced by the process
of mixing corrosion (Dreybrodt, 1981). However, Hückinghaus (1998) found
during his karst network evolution simulations that the water from other
karst conduits has a very high saturation with respect to Ca2+ compared
to water entering the system through direct recharge. Thus, if direct
recharge is present, the mixing with nearly saturated water from an
intersecting conduit could hamper the preferential evolution of the conduit
downstream slowing down the aforementioned processes. In scenario 5 the
influence of an increase in diameter at conduit intersections was
investigated. Since the amount of preferential widening at intersections is
unknown, the cross sections of two intersecting conduits were added and used
as starting cross section for the downstream conduit. The new conduit radius
was then calculated according to Eq. (14) at each intersection.
rc2=rc02+rc12,
where rc2 is the conduit radius downstream of the intersection and
rc0 and rc1 the conduit radii of the two respective conduits
before their intersection.
Parameters for the two different conduit configurations compared in
scenario 4. b is the minimum conduit radius, m the slope of radius increase
towards the springs, bh the highest conduit roughness, mh the slope
of roughness decrease away from the spring and V the conduit volume.
Results are very similar to those of scenario 2 (cf. Fig. 7a and d). Both
simulations result in nearly the same set of parameters (Table 1). The
estimated conduit volume is even a little smaller for scenario 5 since
larger cross sections in the last conduit segment near the spring are
reached for a lower total conduit volume. The drastic increase of conduit
cross sections at the network intersections leads to higher variability in
the cross sections along the conduit segments. The differences between the
peak offsets of both tracer tests are higher compared to those of scenario 2.
While the peak time of tracer test 2 can be calibrated for large conduit
volumes, i.e. conduit volumes above 120 000 m3 (Fig. 7d), the peak time
of tracer test 1 is too late for large conduit volumes. This is due to the
fact that the injection point for tracer test 1 is much closer to the spring
than that for tracer test 2. In scenario 5 the conduit volume is spatially
differently distributed from that of scenario 2 for the identical total
conduit volume. The drastic increase in conduit diameters downgradient of
conduit intersections leads to rather high conduit diameters in the vicinity
of the spring. Therefore, while tracer transport in tracer test 2 occurs in
relatively small conduits with high flow velocities and larger conduits with
lower velocities, the tracer in tracer test 1 is only transported through
the larger conduits whose flow velocities are restricted by the spring
discharge. In Fig. 7d the parameter values for the best fit would lie well
below the lower boundary of the diagram at negative values below -10 h.
However, since the fit for conduit volumes around 100 000 m3 is similar to that
of scenario 2, the two scenarios can in this case not be
distinguished based on field observations.
Comparison of the best-fit simulations with field data for
scenarios 2 and 5. (a) Breakthrough curve of tracer test 1, (b) breakthrough
curve of tracer test 2 and (c) spring discharge.
Conclusions of the parameter analysis
Table 3 provides a comparison, i.e. the characteristics for all scenarios.
The parameter analysis shows that there is only a limited choice of
parameters with which the spring discharges (water balance), the hydraulic
head distribution and the tracer velocities can be simulated. Scenario 1 is
the only scenario that cannot reproduce the peak travel times observed in
both tracer tests simultaneously (Sect. 4.1). It underestimates the
complexity of the geometry and internal surface characteristics (e.g. roughness)
of the conduit system.
Scenario 4 introduces two additional model parameters. The best fit for this
scenario is, however, still achieved with all three Km values being equal,
which basically results in the parameter set of scenario 2. This implies
that the major influence leading to the differences in hydraulic gradients
observed throughout the area is the conduit system and not the variability
of the fissured matrix hydraulic conductivity. It was also shown that for the
Madison aquifer (USA), by Saller et al. (2013), a better representation
of the hydraulic head distribution can be achieved by including a discrete
conduit system even for reduced variability in the hydraulic conductivity of
the fissured matrix. Their conclusion complies very well with the findings
for scenario 4.
Scenario 3 simulates the presence of a couple of additional smaller
dendritic branches. The deduced parameter values and the fit of the
objective functions are similar to those of scenarios 2 and 5. Because of
long calculation times without additional advantage for the presented study,
scenario 3 is not considered for further analysis.
Scenarios 2 and 5 are both judged as suitable. Their parameters and the
quality of the fit are similar. Therefore, it is not possible to decide
which one is the better representation of reality. Regarding the different
processes interacting during karst evolution (Sect. 4.5) it is most likely
that the actual geometry ranges somewhat in between these two scenarios.
Table 1 summarizes all parameters of both simulations and Fig. 10 shows the
simulated tracer breakthrough curves and spring discharges.
DiscussionPlausibility of the best-fit simulations
The main objective of the model simulation is not only to reproduce the
target values but also to provide insight into dominating flow and transport
processes, sensitive parameters and to check the plausibility of the model
set-up. Possible ambiguities in parameterizations can also be checked,
i.e. different combinations of parameters producing identical model output.
For these aims model parameters and aquifer properties simulated with
scenarios 2 and 5 are compared to those observed in the field. As seen
in Table 1 most of the calibrated parameters
range well within values provided in the literature. The calibrated Manning
coefficients are relatively high compared to other karst systems. Jeannin (2001)
lists effective conductivities for several different karst networks
that translate into n values of between 0.03 and 1.07 s m-1/3,
showing that the natural range of n values easily extends across 2 orders of magnitude and the minimum n values of the simulation lie within
the natural range. The maximum n values are significantly higher than those
given by Jeannin (2001). This is not surprising since the calibrated
n value reflects the total roughness of the conduit bundles and therefore
includes geometric conduit properties in addition to the wall roughness that
it was originally defined for. This effect is specific for the Gallusquelle
area but it might be important to consider for other moderately karstified
areas as well where identification of conduit geometries is especially difficult.
The total conduit volume of the Gallusquelle spring derived from scenarios 2
and 5 is only 50 % of that estimated with traditional methods (Geyer et
al., 2008). Since the conduit transmissivity increases towards the spring
water enters the conduits preferably in the vicinity of the spring in the
Gallusquelle area. Therefore, the matrix contribution is high. In addition,
the travel time at peak concentration of tracer test 2, which was used for
the volume estimation by Geyer et al. (2008), is longer than 3 days,
during which time matrix–conduit water exchange can readily take place.
Based on the results of a tracer test conducted in a distance of 3 km to the
Gallusquelle spring Birk et al. (2005) estimated the error incurred by
deducing the conduit volume without taking conduit–matrix exchange fluxes
into account with a very simple numerical model. The authors found a
difference in conduit volumes of approximately 50 %. This fits well with
the results of the present simulation. Birk et al. (2005) also the simulated
equivalent conduit cross sectional area between their tracer injection point
and the spring to be 13.9 m2. For scenario 2 the simulated average
cross sectional area is 11.9 m2 and for scenario 5 13.4 m2, which
compares very well with the results of Birk et al. (2005).
It was not possible to match the shape of both breakthrough curves with the
same dispersivity. The apparent dispersion in the tracer test 2 breakthrough
is much higher compared to that of tracer test 1, while the breakthrough of
tracer test 1 shows a more expressed tailing (Fig. 10a and b). This
corresponds to the effect observed by Hauns et al. (2001). The authors found
scaling effects in karst conduits: the larger the distance between input and
observation point, the more mixing occurred. The tailing is generally
induced by matrix diffusion or discrete geometric changes such as pools,
where the tracer can be held back and released more slowly. Theoretically,
every water drop employs medium and slow flow paths if the distance is large
enough, leading to a more or less symmetrical, but broader, distribution and
therefore a higher apparent dispersion (Hauns et al., 2001). To quantify
this effect, exact knowledge of the geometric conduit shape such as the
positions and shapes of pools would be necessary. Furthermore, an additional
unknown possibly influencing the observed retardation and dispersion effects
is the input mechanism. The simulation assumes that all introduced tracers
immediately and completely enter the conduit system, which neglects effects
of the unsaturated zone on tracer breakthrough curves. In addition, the
shape of the breakthrough curve of tracer test 2 is difficult to deduce
since the 6 h sampling interval can be considered as rather low
leading to a breakthrough peak which is described by only seven measurement
points. Therefore, the apparent dispersivity was calibrated for both
breakthrough curves separately. Calibrated dispersivity ranges well within
those quoted in literature (Table 1). The mass
recovery during the simulation was determined to range between 98.4 and
99.9 % in all simulations. The slight mass difference results from a
combination of diffusion of the tracer into the fissured matrix and
numerical inaccuracies.
The spring discharge of the minor springs in the area (Sect. 3) was slightly
underestimated in most cases (Fig. 10c). For most springs the models of
scenarios 2 and 5 provide similar results. The underestimation of discharge
is in the order of < 0.05 m3 s-1and is not expected to
significantly influence the general flow conditions. It probably results
from the unknown conduit geometry in the catchments of the different minor
springs. The only case in which the two scenarios give significantly
different results is the spring discharge of the spring group consisting of
the Ahlenberg and Büttnauquellen springs (Fig. 10c). Scenario 2
overestimates and scenario 5 underestimates the discharge. This is due to
the fact that the longest conduit of the Ahlenberg and Büttnauquellen
springs is longer than the longest one of the Gallusquelle spring but the
conduit network has less intersections (Fig. 1).
Therefore, the conduit volume of the Ahlenberg and Büttnauquellen
springs is 134 568 m3 in scenario 2 and only 75 085 m3 in
scenario 5 leading to the different discharge values. It is reasonable to assume that
a better fit for the spring group can be achieved, if more variations of
conduit intersections are tested. An adequate fit for the Fehla-Ursprung
spring of 0.1 m3 s-1 was achieved for both scenarios with a
fault aperture of 0.005 m.
Uncertainties and limitations
The most important uncertainties regarding the reliability of the simulation
include the assumptions that were made prior to modelling. First, flow
dynamics were neglected. This approach was chosen because tracer tests are
supposed to be conducted during quasi-steady-state flow conditions. However,
this is only the ideal case. During both tracer tests spring discharge
declined slightly. The influence of transient flow on transport velocities
inside the conduits was estimated by a very simple transient flow simulation
for the best-fit models in which recharge and storage coefficients were
calibrated to reproduce the observed decline in spring discharges. The
transient flow only slightly affected peak velocities but lead to a larger
spreading of the breakthrough curves and therefore lower calibrated
dispersion coefficients. This effect occurred because the decline in flow
velocities is not completely uniform inside the conduits and depending on
where the tracer is at which time it experiences different flow velocities
in the different parts of the conduits, which leads to a broader
distribution at the spring. The same breakthrough curves can be simulated
under steady-state flow conditions with slightly higher dispersivity
coefficients. So, the calibrated dispersivities do not only represent
geometrical heterogeneities but also temporal effects as is the case for all
standard evaluations of dispersion from tracer breakthrough curves.
Flow velocities inside the main conduit branch of the
Gallusquelle spring during the simulation of tracer test 2. The best-fit
simulations for scenarios 2 and 5 are compared to simulations where a direct
recharge of 10 % is introduced.
The influence of rapid recharge is not to considered in the simulation of
baseflow conditions. However, there might be an influence on flow velocities
during the actual recharge events, i.e. if rapid recharge is intensive and
strong enough to lead to a reversal of the flow gradients between conduit
and fissured matrix. Therefore, an alternative simulation was performed for
tracer test 2, which was conducted during high flow conditions (Table 2)
after a recharge event. The maximum percentage of direct recharge of 10 %
estimated by Sauter (1992) and Geyer et al. (2008) was used for this
simulation. Neither for scenario 2 nor for scenario 5 a gradient reversal
between conduit and matrix occurred and the influence on flow velocities was
negligible (Fig. 11).
Furthermore, flow in all karst conduits was simulated for turbulent
conditions. Turbulent conditions can be generally assumed in karst conduits
(Reimann et al., 2011) and also apply to all calibrated model conduit
cross sections. Since the conduit cross section presents the total
cross section of the conduit bundle, the cross sections of the individual
tubes are uncertain, though. The high n values suggest that the
surface / volume ratio is relatively high, which implies that the individual
conduit cross sections are rather small. Therefore, laminar flow in some
conduits is likely. While laminar flow conditions in the conduits influence
hydraulic gradients considerably, this fact is believed not to influence the
overall results and conclusions of this study, i.e. the relative
significance of the parameters deduced from parameter analysis and the
deduced conduit volume, especially since flow is simulated for steady-state conditions.
For all distributed numerical karst simulations, uncertainties regarding the
exact positions and interconnectivities of the conduit branches still
remain. Due to the extensive investigations already performed in previous
work (Sect. 3) these uncertainties are reduced in
the Gallusquelle area and the above scenarios include the most probable
ones. However, the flexibility of the modelling approach allows for the
integration of any future information that might enhance the numerical model further.
Calibration strategy
For a successful calibration of a distributed groundwater flow and transport
model for a karst area on catchment scale certain constraints have to be set
a priori. The geometry of the model area, i.e. locations/types of boundary
conditions and aquifer base, fixed during calibration, has to be known with
sufficient certainty. Furthermore, the objective functions for calibration
have to be defined, i.e. the hydraulic response of the system and transport
velocities. In a karst groundwater model, these consist of measurable
variables, i.e. spring discharges, hydraulic heads in the fissured matrix
and two tracer breakthrough curves. The hydraulic head measurements should
be distributed across the entire catchment and preferably close to the
conduit system, should geometric conduit parameters be calibrated for as
well. It is expected that the influence of the conduits on the hydraulic
head decreases and the influence of matrix hydraulic conductivities
increases with distance to the conduit system. In the design of the tracer
experiment, the following criteria should be observed: for a representative
calibration, the dye should be injected at as large a distance to each other
as possible with one of them including the length of the whole conduit
system. Each tracer test gives integrated information about its complete
flow path. If the injection points lie close together, no information about
the development of conduit geometries from water divide to spring can be
obtained. Further, the dye should be injected as directly as possible into
the conduit system, e.g. via a flushed sinkhole, to obtain information on
the conduit flow regime and to minimize matrix interference. To ease
interpretation a constant spring discharge during the tests is desirable.
In this study, the flow field was simulated not only for the catchment area
of the Gallusquelle spring, but also for a larger area including the catchment
areas of several smaller springs (Fig. 1). This is in general not essential
for deducing conduit volumes and setting up a flow and transport model.
Simulating several catchments, however, helps to increase the reliability of the
simulation. The positions of water divides are majorly determined
by the hydraulic conductivity of the fissured matrix Km, so that the
simulated catchment areas of the different springs can be used to estimate
how realistic the simulated flow field is and decrease the range of likely
Km values. In this study, high Km values above
ca. 3 × 10-5 m s-1 made the simulation of the spring discharge of the
Fehla-Ursprung spring (Fig. 1) impossible because the water divide in the
west could not be simulated and most of the water in the area discharged to
the east towards the river Lauchert resulting in a very narrow and long
catchment area for the Gallusquelle spring.
There are eight parameters available for model calibration in this study.
Two of these parameters define the conduit geometry: b is the lowest conduit
radius and m the slope with which the conduit radius increases. One
parameter, df, defines the aperture of the fault zone. The hydraulic
conductivity of the fissured matrix is represented by the parameter
Km and the roughness of the conduit system by two parameters: bh
represents the highest roughness and mh the slope of roughness decrease
in upgradient direction from the spring. The last two parameters
ε1 and ε2 are the respective conduit
dispersivities obtained from the two artificial tracer experiments (Table 1).
For efficiency reasons it is important to know which of these parameters can
be calibrated independently. The apparent transport dispersivities
ε1 and ε2 are pure transport parameters,
which influence only the shape of the breakthrough curves and not the flow
field. The hydraulic model parameters influence the shape of the tracer
breakthrough curves as well. Therefore, dispersivities ε1
and ε2 should be calibrated separately after calibrating
the hydraulic model parameters.
Only for hydraulically dominant fault zones knowledge of the fault zone
aperture df is required. For the model area this parameter was required
for one fault zone lying in the west of the area feeding the Fehla-Ursprung
spring (Fig. 1). Since the Fehla-Ursprung spring has its own catchment area
the fault zone has only minor influence on the flow regime in the
Gallusquelle catchment. Its hydraulic parameters were calibrated at the
beginning of the simulation procedure to reproduce the catchment and the
discharge of the Fehla-Ursprung spring adequately and kept constant
throughout all the simulations. In the final calibrated models it was
rechecked, but the calibrated value was still acceptable.
The hydraulic conductivity of the fissured matrix Km can be calibrated
independently in principle as well. The influence on spring discharge is
relatively small. The best-fit Km value depends on the conduit
parameters, i.e. geometry and roughness, since the hydraulic conductivities
of the conduit system and of the fissured matrix define the total
transmissivity of the catchment area together. Nonetheless, the best-fit
value lies in the same range for different conduit geometries (Figs. 4a and 7c).
The greater the difference between the simulated conduit
geometries, the more likely is a slight shift of the best-fit
Km value. Therefore, it is advisable to calibrate it anew for
significant model changes, e.g. different scenarios, but to keep it constant
during the rest of the calibrations. For the best-fit configuration,
potentially used as a prognostic tool, the Km value needs to be checked
and adapted if necessary. This observation is, however, only valid for
steady-state flow conditions. The dynamics of the hydraulic head and spring
discharge might be highly sensitive to the matrix hydraulic conductivity,
the conduit–matrix exchange coefficient and the lateral conduit extent. This
work focuses on the conduits as highly conductive pathways for e.g. contaminant
transport, but the calibration of matrix velocities, e.g. by use
of environmental tracers, would likely be sensitive to the Km values as
well. Therefore, the choice of the flow regime and the objective functions
determines the strength of the interdependencies between fissured matrix and
conduit system parameters and therefore whether Km can be calibrated independently.
The conduit parameters for geometry and roughness, here four parameters
(lowest conduit radius b, slope of radius increase m, highest roughness
bh and slope of roughness decrease mh), have to be varied
simultaneously. All of them have a major influence on spring discharge and
cannot be varied separately without introducing discharge errors. For each
conduit geometry, there are a number of possible bh–mh combinations
that result in the observed spring discharge. In general, the slowest
transport velocities are achieved with a mh value of zero. So, to deduce
the range of geometric parameters that reproduce the objective functions, it
is advisable to check the minimum conduit volume for which the tracer tests
are not too fast for a value of mh equal to zero. For the Gallusquelle
area, transmissivities significantly increase towards the springs, which is
characteristic for most karst catchments. Therefore, low bh values oppose
the general hydraulic head trend: they increase the conduit roughness at the
spring leading to slower flow and higher gradients. The higher the conduit
volume, the higher bh is required to reproduce the observed transport
velocities. Therefore, the best-fit model likely has the smallest conduit
volume for which both tracer tests can be reproduced. In Fig. 7 this
condition can be seen to clearly range in the order of 100 000 m3 for
the Gallusquelle area. While the four conduit parameters allow for a good
model fit, they are pure calibration parameters. They show that the karst
conduit system has a high complexity, which cannot be neglected for
distributed velocity and hydraulic head representation. A systematic
simulation of the heterogeneities, e.g. with a karst genesis approach, would
be a process-based improvement to the current method and give more physical
meaning to the parameters.
Conclusions
The study presents a large-scale catchment-based distributed hybrid karst
groundwater flow model capable of simulating groundwater flow and
solute transport. For flow recession conditions this model can be used as a
predictive tool for the Gallusquelle area with relative confidence. The
approach of simultaneous pattern matching of flow and transport parameters
provides new insight into the hydraulics of the Gallusquelle conduit system.
The model ambiguity was significantly reduced to the point where an
estimation of the actual karst conduit volume for the Gallusquelle spring
could be made. This would not have been possible simulating only one or two
of the three objective functions, i.e. the spring discharge, the hydraulic
head distribution and two tracer tests.
The model allows for the identification of the relevant parameters affecting
karst groundwater discharge and transport in karst conduits and the
examination of the respective overall importance in a well-investigated
karst groundwater basin for steady-state flow conditions. While a
differentiated representation of the roughness values in the karst conduits
is substantial for buffering the lack of knowledge of the exact conduit
geometry, e.g. local variations in cross section and the number of
interacting conduits, variable matrix hydraulic conductivities cannot
improve the simulation. It was shown that the effect of the unknown exact
lateral extent of the conduit system and the change in conduit cross section
at conduit intersections is of minor importance for the overall karst
groundwater discharge. This is important since these parameters are usually
unknown and difficult to measure in the field.
For calibration purposes, this study demonstrates that for a steady-state
flow field and the observed objective functions the hydraulic conductivities
of the fissured matrix can practically be calibrated independently of the
conduit parameters. Furthermore, a strategy for the simultaneous calibration
of conduit volumes and conduit roughness in a complex karst catchment was developed.
As discussed in Sect. 5 the major limitation of the simulation is the
neglect of flow dynamics, which limits the applicability to certain flow
conditions. Therefore, transient flow simulation is the focus of on-going
work. This will enhance the applicability of the model as a prognostic tool
to all essential field conditions and lead to further conclusions regarding
the important karst system parameters, their influences on karst hydraulics
and their interdependencies. It can be expected that some parameters, which
are of minor importance in a steady-state flow field, e.g. the lateral
conduit extent and the percentage of recharge entering the conduits
directly, will exhibit significant influence for transient flow conditions.
Acknowledgements
The presented study was funded by the German Federal Ministry of Education
and Research (promotional reference no. 02WRS1277A, AGRO, Risikomanagement
von Spurenstoffen und Krankheitserregern in ländlichen
Karsteinzugsgebieten).
This open-access publication is funded by the University of Göttingen.
Edited by: M. Giudici
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