Supplement to Hartmann et al., re-submitted to Water Resources Research July 2014
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Supplement to
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Modeling spatio-temporal impacts of hydro-climatic extremes on groundwater recharge a
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Mediterranean karst aquifer
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Authors:
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Andreas Hartmann1,3, Matías Mudarra2, Bartolomé Andreo2, Ana Marín2, Thorsten Wagener1,
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Jens Lange3
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Affiliations:
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1: Department of Civil Engineering, University of Bristol, UK
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2: Department of Geology and Centre of Hydrogeology of the University of Malaga
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(CEHIUMA), Malaga 29071, Spain
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3: Chair of Hydrology, Freiburg University, Germany
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Corresponding Author:
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Andreas Hartmann
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e-mail: aj.hartmann@bristol.ac.uk
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Supplement to Hartmann et al., re-submitted to Water Resources Research July 2014
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The variability of soil depths is expressed by a mean soil depth Vmean,S [mm] and a distribution
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coefficient aSE [-]. The soil storage capacity VS,i [mm] for every compartment i is found by:
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Detailed description of the VarKarst model
VS ,i  Vmax, S
 i 
 
N
aS E
(1)
With Vmax,S [mm] being the maximum soil storage capacity that is derived from Vmean,S:
x
Vmax, S  
i1 2
aS E

N
x
0 Vmax, S  N  dx  0
2
N
aS E
dx
; Vmean,S
 i1 2 
 Vmax, S  
N

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aS E
.
Vmax, S  Vmean,S  2
(2)
 aS E 


 aS E 1 
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Where i1/2 is the compartment at which the volumes on the left equal the volumes on the right.
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Applying same distribution coefficient aSE, the epikarst storage distribution is derived by the
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mean epikarst depth Vmean,E [mm] (derivation of Vmax,E likewise to Vmax,S in Eq (3)):
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VE ,i  Vmax, E
 i 
 
N
aS E
(3)
At time t, actual evapotranspiration from each soil compartment Eact,i is found by:
E act ,i t   E pot t  

min VSoil,i t   Pt   QSurface,i t , VS ,i
V S ,i

(4)
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Hereby, Epot [mm] is the potential evapotranspiration derived by the Thornthwaite equation
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[Thornthwaite, 1948], Qsurface,i [mm] is the surface inflow that arrives from compartment i-1
Supplement to Hartmann et al., re-submitted to Water Resources Research July 2014
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(see Eq. (10)), and VSoil,i [mm] the water stored in the soil at time step t. Recharge from the
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soil to the epikarst REpi,i [mm] is calculated by:

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
REpi,i t   max VSoil,i t   Pt   QSurface,i t   Eact ,i t   VS ,i , 0
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(5)
Outflow from the epikarst is controlled by the epikarst storage coefficients KE,i [d]:
QEpi,i t  
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K E ,i
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
min VEpi,i t   REpi,i t ,VE ,i
K E ,i
 N  i 1
 K max, E  

 N


 t
(6)
aS E
(7)
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Where VEpi,i [mm] is the water stored in the epikarst at timestep t. Applying separate
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distribution coefficients for all variable model parameters, Hartmann et al. [2013] found that
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for Vmean,S, Vmean,E and Kmean,E, a single distribution coefficient was sufficient. Hence, Kmax,E is
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derived by the mean epikarst storage coefficient Kmean,E using the same distribution coefficient
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aSE:
a
SE
 x
N  K mean, E   K max, E   dx
N
0

K max, E  K mean, E  a SE  1
N
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(8)
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When soil and epikarst storage capacities are exceeded, surface flow to the next model
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compartment QSurf,i+1 [mm] initiates:
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

QSurf ,i 1 t   max VEpi,i t   REpi,i t   VE ,i , 0
(9)
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Outflow from every epikarst compartment is separated into diffuse (Rdiff,i [mm]) and
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concentrated groundwater recharge (Rconc,i [mm]) by a variable separation factor fC,i [-]:
Supplement to Hartmann et al., re-submitted to Water Resources Research July 2014
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Rconc,i t   fC ,i  QEpi,i t 
(10)
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Rdiff ,i t   1  f C ,i   QEpi,i t 
(11)
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f C ,i
 i 
 
N
a fsep
(12)
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afsep [-] represents the distribution coefficient of the groundwater separation factor. The sum
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of Rconc,i and Rdiff,i over all i represents the total recharge at time step t. All diffuse recharge
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reaches the groundwater compartments (i = 1…N-1) below, while concentrated recharge is
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routed to the conduit system (compartment i = N). Hence, with variable groundwater storage
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coefficients KGW,i [d], groundwater contributions –QGW,i [mm] represent the matrix system:
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QGW ,i t  
VGW ,i t   Rdiff ,i t 
K GW ,i
; i  1...N  1
(13)
KGW,i is calculated by:
KGW , i
 i 
 KC   
N
 a GW
(14)
The contribution of the conduit system, originates from compartment N:
N
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QGW ,i t  
VGW , N t    Rconc,i t 
i 1
KC
;i  N
(15)
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where KC is the conduit storage coefficient. Using the recharge area A [km²] and rescaling the
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dimensions to [l s-1], discharge of the main spring Qmain [l s-1] includes both the matrix and the
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conduit system:
Supplement to Hartmann et al., re-submitted to Water Resources Research July 2014
Qmain t  
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Amax N
  QGW ,i t 
N i 1
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(16)
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For NO3, solute transport follows the assumption of complete mixing for every model
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compartment. For SO4 and Cl, geogene contributions had to be considered. Likewise to
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Hartmann et al. [2013] their equilibrium concentrations vary according to:
GeoCl ,i / SO 4,i
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 N  i 1
 Geomax, Cl / max, SO 4  

 N

aGeo
(17)
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where aGeo is a variability constant and Geomax,Cl and Geomax,SO4 are derived from GeoCl
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[mg l-1] and GeoSO4 [mg l-1] according to Eq. (9). Table 1 in the main manuscript provides a
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summary of all model parameters.
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The Shuffled Complex Evolution approach that was used to find the model parameters in our
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study (subsection 3.2 in the revised manuscript also provides posterior distribution of each of
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the calibrated model parameters. Plotting them cumulatively for each parameter their
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deviation from a uniform distribution (1:1 line) can be used to assess its sensitivity (the higher
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the deviation the larger the sensitivity). In our case this analysis shows that, using only
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discharge, the parameters responsible for solute transport are not sensitive (Figure 1). When
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hydrochemical information is added, they show a strong deviation from a uniform distribution
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indicating that they have become sensitive. In addition some parameters show a shift of their
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distribution when hydrochemical data is added, which indicates a shift of model internal
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process behavior favoring the required multi-objective fit of simulated and observed
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discharge and hydrochemical data as described in subsection 3.2 and discussed in subsection
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5.2 in the main manuscript..
Sensitivity analysis
Supplement to Hartmann et al., re-submitted to Water Resources Research July 2014
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Figure 1: Posterior distribution of model parameters derived by SCEM for calibration using discharge
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only (blue) and using discharge and hydrochemistry (green); a deviation from the 1:1 line (grey) indicates
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a sensitive parameter.
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Hartmann, A., J. A. Barberá, J. Lange, B. Andreo, and M. Weiler (2013), Progress in the
hydrologic simulation of time variant recharge areas of karst systems – exemplified at a karst
spring in Southern Spain, Advances in Water Resources, 54, 149-160.
Thornthwaite, C. W. (1948), An Approach toward a Rational Classification of Climate,
Geographical Review, 38(1), 55-94.
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References