Hydrological Sciences Journal
ISSN: 0262-6667 (Print) 2150-3435 (Online) Journal homepage: http://www.tandfonline.com/loi/thsj20
Transit time determination for a riverbank
filtration system using oxygen isotope data and
the lumped-parameter model
Krisztina Kármán, Piotr Maloszewski, József Deák, István Fórizs & Csaba
Szabó
To cite this article: Krisztina Kármán, Piotr Maloszewski, József Deák, István Fórizs & Csaba
Szabó (2014) Transit time determination for a riverbank filtration system using oxygen isotope
data and the lumped-parameter model, Hydrological Sciences Journal, 59:6, 1109-1116, DOI:
10.1080/02626667.2013.808345
To link to this article: http://dx.doi.org/10.1080/02626667.2013.808345
Accepted author version posted online: 29
May 2013.
Published online: 02 Jun 2014.
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Date: 27 November 2015, At: 02:28
Hydrological Sciences Journal – Journal des Sciences Hydrologiques, 59 (6) 2014
http://dx.doi.org/10.1080/02626667.2013.808345
1109
Transit time determination for a riverbank filtration system using
oxygen isotope data and the lumped-parameter model
Krisztina Kármán1, Piotr Maloszewski2, József Deák3, István Fórizs1 and Csaba Szabó4
1
Institute for Geological and Geochemical Research, Research Centre for Astronomy and Earth Sciences, Hungarian Academy of
Sciences, H-1112 Budapest, Hungary
[email protected]
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2
Institute of Groundwater Ecology, Helmholtz Zentrum München, German Research Center for Environmental Health (GmbH), D-85764
Neuherberg, Germany
3
GWIS Ltd, H-2120 Dunakeszi, Hungary
4
Department of Petrology and Geochemistry, Lithosphere Fluid Research Laboratory, Eötvös University, H-1115 Budapest, Hungary
Received 28 August 2012; accepted 7 May 2013; open for discussion until 1 December 2014
Editor D. Koutsoyiannis; Associate editor M.D. Fidelibus
Citation Kármán, K., Maloszewski, P., Deák, J., Fórizs, I. and Szabó, C., 2014. Transit time determination for a riverbank filtration
system using oxygen isotope data and the lumped-parameter model. Hydrological Sciences Journal, 59 (6), 1109–1116. http://dx.doi.
org/10.1080/02626667.2013.808345
Abstract The mathematical description of the lumped-parameter Dispersion Model is presented, and its use for
estimating transit time and proportion of river water in a riverbank filtration system on Szentendre Island (in the
Danube River, Hungary) is assessed by applying stable oxygen isotope data. The modelling approach could only be
used to analyse single oxygen-isotope peaks observed during a 6-month-long time series, because the mean input
(Danube) concentration was very close to the output (well) concentration. The water level in the Danube varied, but
the lumped-parameter approach is only applicable for steady-state conditions. Isotope data observed during quasi
steady-state hydraulic conditions selected at medium and higher water levels were modelled. At low water levels
the method could not be used because the 18O content of the Danube water and of collector wells was similar.
Key words riverbank filtration; lumped-parameter model; oxygen isotope; Szentendre Island
Détermination du temps de transit dans un système de filtration par les berges grâce à un modèle
global paramétré utilisant des données isotopiques de l’oxygène
Résumé Nous présentons la description mathématique d’un modèle de dispersion global paramétré. Nous
évaluons son utilisation pour l’estimation du temps de transit et de la contribution de l’eau de la rivière dans
le système de filtration par les berges de l’île de Szentendre (Hongrie), en utilisant les données des isotopes
stables de l’oxygène. L’approche de modélisation n’a pu être utilisée que pour analyser les pics isotopiques
simples de l’oxygène, observés sur une série temporelle de six mois, parce que la concentration moyenne en
entrée (Danube) était très proche de la concentration en sortie (puits). Le niveau d’eau dans le Danube variait,
alors que l’approche paramétrée globale n’est applicable qu’en régime permanent. Nous avons modélisé les
données isotopiques observées dans des conditions hydrauliques de régime quasi-permanent, sélectionnées pour
des niveaux d’eau moyens et élevés. Pour les faibles niveaux d’eau, la méthode n’a pas pu être utilisée parce que
les contenus en 18O de l’eau du Danube et du puits collecteur étaient similaires.
Mots clefs filtration par berge ; modèle paramétré global ; isotope de l’oxygène ; île de Szentendre
1 INTRODUCTION
Tracers, especially the ones originating from environmental isotopes, are becoming increasingly important
in hydrology. The most widely used tracers are those
that are part of the water molecule (3H, 2H, 18O).
© 2014 IAHS Press
They have found wide application, for example, for
estimating recharge, mixing processes, the origin of
water, transit times of water and for calibrating flow
and transport models (Małoszewski et al. 1983,
Stichler et al. 1986, Katz et al. 1997, Vitvar and
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1110
Krisztina Kármán et al.
Balderer 1997, Kumar et al. 2001, Maloszewski et al.
2002, McGuire et al. 2002, Stichler et al. 2008,
Leibundgut et al. 2009). The stable isotopes of
water (18O and 2H) play a particularly important
role. The isotopic composition of precipitation
depends on latitude, seasonal, continental, altitude
and amount effects (Gat et al. 2001). The isotopic
composition of groundwater depends on the isotopic
composition of the infiltrating precipitation and/or
mixing and transport processes. In the present study,
δ18O data observed in the river water, pumping and
observation wells were used to model the proportion
of infiltrated Danube water and mean transit times
between the river and collector wells. The catchment
area of the Danube is huge, resulting in variability of
the δ18O composition of its water. The δ18O isotope
is conservative, thus its monitoring could help us to
gather direct information about the leaking of the
water. The most common method used to calculate
the transit time of water from isotopic data is the
application of the lumped-parameter approach proposed by Małoszewski and Zuber (1982). A short
description of this method is given in the
“Methodology” section.
In the present work, we used the lumped steadystate Dispersion Model (Stichler et al. 1986), but
under special circumstances. The observation site
(multiple collector well) is located very close
(45 m) to the river bank, while the water level of
the Danube is variable. Thus, the modelling of the
isotope data was performed by selecting single oxygen-isotope peaks observed during “steady-state”
water levels, instead of using the whole continuous
time series as in case studies by Amin et al. (1996),
Maloszewski et al. (2002) and Fórizs et al. (2005).
Every input/output oxygen-isotope peak was considered as an independent signal characterized by its
own travel time, dispersion parameter and Danube
water proportion.
The significance of the study area, Szentendre
Island, is that the majority of the good quality drinking water of Budapest comes from the riverbank
filtration system installed here (Fig. 1). For safe
operation of the system, it is important to quantify
Fig. 1 Location of Szentendre Island and the studied production well. Coordinates in EOV (Hungarian National Grid).
Transit time determination for a riverbank filtration system
the mean transit time and the proportion of Danube
water in the production wells.
In the area under investigation the Danube water
and the most important production wells were
sampled and the observed data were modelled.
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2 STUDY AREA
Szentendre Island (56 km2) lies to the north of
Hungary’s capital, Budapest, between two arms of
the Danube River (Fig. 1). The island is 31 km long
and its width varies between 2.3 and 3.5 km. The
elevation is between 100 and 124 m. Fields, orchards,
vineyards, meadows (partly river meadows) and forests (partly floodplain grove forests) cover the surface. Twenty five protected plant species and 205
migratory bird species are known from the region.
The climate is humid continental (OMSZ 2002). The
average annual temperature is 10°C, and average
annual precipitation is 600–650 mm. Some sections
of the island are parts of the Natura2000 network
and/or the Duna-Ipoly National Park.
Four settlements are situated on the island:
Kisoroszi (population 962), Tahitótfalu (population
5525, but approx. half living off the island),
Pócsmegyer (population 2245) and Szigetmonostor
(population 1838), and two recreation areas: Surány
and Horány, where the resident population is increasing.
The geological basement (Fig. 2) consists of
Tertiary marine sediments (Tard Clay). This formation
is dominantly argillaceous silt; its thickness varies
between 90 and 130 m (Császár 1997). No significant
discharge through this formation is expected, so here
we assume it can be neglected. The basement is topped
by fluvial gravels and sands of Pleistocene age
(Góczán 1955). The immediate subsurface consists
of hardly permeable formations such as muddy
sands, soils and floodplain horizons with semi-
1111
consolidated shifting sand layers. In the watercourse
of the Danube, the gravel and sand are covered by only
a thin filtering layer, which clears the river water
entering the porous sediments of physical and chemical as well as biological contaminants. Previous studies (Deák et al. 1992) proved that the exploited water
recharges largely (up to 96–97%) through the river
bed, while precipitation adds only about 3–4%.
The first wells on Szentendre Island were established in 1897; the production started in 1904 with 32
wells. At present, more than 500 operating wells are in
use yielding approx. 320 × 103 m3 d-1 of drinking water.
Production wells were drilled along the Danube on both
sides of the island. The water enters the production
wells from the Danube as well as from the groundwater.
The European Union (EU) Water Framework
Directive requires the identification and characterization of water bodies. In this classification, the
Szentendre Island is part of the “Szentendre Island
and other islands on the Danube” water body; the
identifier is AIQ652.
One of the most efficient pumping well groups,
Tahi I, containing multiple collector wells, was chosen
for modelling the transit time and proportion of
Danube water. The present study shows the results
obtained for the Tahi I/16 well (47°47′38.36″N;
19°03′24.06″E), which was most intensely sampled.
The average production of this well during the sampling period was 3.04 × 103 m3 d-1. Production deviating from this by more than ±1σ occurred on a total of
8 days during the sampling period, but was never zero.
3 METHODOLOGY
The lumped-parameter approach (Małoszewski and
Zuber 1982) is most widely used for the estimation
of the mean transit time from environmental tracer
data. Generally, this approach requires only a
Fig. 2 Conceptual geological model of the Szentendre Island (modified after Góczán 1955).
1112
Krisztina Kármán et al.
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minimum of information concerning the leakage zone
of the groundwater, and data of nonreactive tracer
concentrations measured in the input (recharge zone)
and output (observation or pumping wells) as a function of time (Małoszewski and Zuber 1982, 1993).
The aquifer is considered to be in a steady state and
is characterized by mean values of flow (transport)
parameters. According to Małoszewski and Zuber
(1982, 1996), the main parameter of the lumpedparameter approach is the mean transit time of
water (T), which is the ratio of the mobile water
volume (Vm) in the aquifer to mean water volumetric
flow rate through the aquifer (Q):
T ¼ Vm = Q
(1)
The relationship between input, Cin(t), and output, Cout
(t), concentrations is given by the convolution integral,
which for non-decaying tracers is defined as follows:
Cout ðtÞ ¼
ð1
Cin ðt τÞ gðτÞ dτ
(2)
0
where g(τ) is the transit time distribution function
(T is the main parameter of that function), while τ is
the transit time of a single tracer particle between
input and output. The integral over the g(τ) and all
transit times (τ) has to yield 1.
Assuming that the volume of stagnant water in
the considered system is so small as to be negligible
(zero), the transit time distribution function and its
parameters describe directly those of water
(Maloszewski et al. 2004).
In the case of bank filtration, the transit time
distribution function is defined by the lumped
Dispersion Model as shown by Stichler et al.
(1986). The g(τ) function is then defined as follows
(Małoszewski and Zuber 1982):
"
#
1
ð1 τ=T Þ2
gðτÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi exp (3)
4PD τ=T
τ 4πPD τ=T
where PD is the dimensionless dispersion parameter,
which describes the variance of the distribution. In
the case of bank filtration it can, in large approximation, be defined as:
PD ¼ αL =x
(4)
where αL is the dispersivity (m), and x is the mean
flow-distance between the input and output locations (m).
Water samples for oxygen isotopic measurements
were taken from the Danube River (47°39′32”;
19°06′56″) and the production well (Tahi I/16) four to
five times per week beginning on 1 September 2010.
Between 1 January 2011 and 31 March 2011, the
Danube was sampled daily. Although the distance
between the Danube sampling point and the production
well is significant, the oxygen isotopic composition of
the two arms of the Danube alongside Szentendre
Island is equal (Kármán et al. 2010). Additionally,
some monitoring wells were sampled and measured.
Assuming that some local groundwater can appear in
the production well, the conceptual model of water flow
was developed (Fig. 3). It shows that seepage through
the geological media modifies the oxygen isotopic
composition of Danube water (CDanube) according to
the dispersive transit time distribution function (equation (3)). The resulting concentration CDanube after leaking
appears in the collector well with the portion (γ). Local
groundwater with the portion (1–γ) also appears in that
well and is characterized by constant concentration
Clocal groundwater. Hence, the convolution integral (equation (2)) applied for the modelling can be modified as:
Cwell ðtÞ ¼ γ
ðt
CDanube ðτÞ gðt τÞ dτ
0
(5)
þ ð1 γÞ Clocal groundwater
where Cwell, CDanube and Clocal groundwater are the
oxygen isotopic composition of the collector well,
Fig. 3 Conceptual model of water flow to the collector well (CDanube is the variable oxygen-isotope composition in time in the
Danube River; CDanube after leaking is the variable oxygen-isotopic composition in time after leaking through the bank observed in
the collector well; Clocal groundwater is the constant oxygen-isotope composition in time in local groundwater; and γ is the
proportion of Danube River water in the collector well, while 1–γ is the proportion of local groundwater in that well).
1113
Transit time determination for a riverbank filtration system
the Danube water and the local groundwater,
respectively.
The proportion of Danube water (γ) can be calculated from the mean isotope contents calculated for
well, C well and for Danube River, C Danube according
to Stichler et al. (1986):
γ ¼
C well Clocal groundwater
C Danube Clocal groundwater
18 16 O
16 O
O
δ O¼
18
sample
18 O
16 O
O
standard
1000
(7)
standard
The uncertainty was ±0.2‰.
(6)
5 RESULTS OF MEASUREMENTS
As sampling from the studied well was not done at
regular time intervals throughout the sampling season, the values for the missing days up to the end of
2010 were calculated with a weighted average in
order to provide values for modelling that were equidistant in time. Although sometimes there were larger
differences between two consecutive days, this
approximation did not cause significant errors
because the long-term negative peaks lasted only
for several days. The short-term isotopic events lasting only 1 or 2 days could not be observed in the
output. The long-term changes in the isotopic composition are significant enough to allow the estimation with a weighted average. From 1 January 2011,
daily sampling was done from the Danube. The
model requires only the input data set to be equidistant. Input sampling does not have to be daily; the
four to five samples per week were enough for modelling. During the study period, the average oxygen
isotopic composition of the Danube was δ18O =
–10.60‰, and that of the collector well δ18O =
–10.54‰ (Fig. 4). After sampling some local monitoring wells, we estimated the oxygen-isotopic composition of the local groundwater to be δ18O =
–10.40‰.
4 DETERMINATION OF THE OXYGEN
ISOTOPIC COMPOSITION
Oxygen-isotope measurements were performed at the
Institute for Geological and Geochemical Research,
Research Centre for Astronomy and Earth Sciences,
Hungarian Academy of Sciences using a Liquidwater Isotope Analyser-24d (Los Gatos Research
Inc., USA). The instrument uses off-axis integrated
cavity output spectroscopy to measure absolute abundances of 2HHO, HH18O and HHO via laser absorption (IAEA 2009). We used self-made laboratory
standards: BWS-1, BWS-2 and BWS-3 calibrated
by Vienna Standard Mean Ocean Water (V-SMOW)
with International Certification (GISP and SLAP).
Standard BWS-1 (δ18O = –0.60 ± 0.1‰) was made
from sea water and locally sampled lake water;
BWS-2 (δ18O = –10.50 ± 0.1‰) was made from
tap water; and BWS-3 (δ18O = –20.00 ± 0.1‰) was
made from melted snow from the Tatra Mountains
and Budapest.
Isotopic ratios are reported in δ value with respect
to the international V-SMOW standard and in ‰,
according to the formula of McKinney et al. (1950):
Water Level
Multiannual Mean of Precipitation Oxygen Isotopic Composition
Local Groundwater Oxygen Isotopic Composition
107
Well Oxygen Isotope Composition
–9.7
–10.2
109
Precipitation
Danube Oxygen Isotope Composition
Modelled Oxygen Isotope Composition in the Well
Measured Well
Local Groundwater
105
103
–10.7
101
Modelled Well
–11.2
Danube
99
–11.7
97
28/03/2011
20/03/2011
04/03/2011
12/03/2011
16/02/2011
95
24/02/2011
08/02/2011
23/01/2011
15/01/2011
30/12/2010
07/01/2011
14/12/2010
22/12/2010
06/12/2010
20/11/2010
28/11/2010
12/11/2010
27/10/2010
04/11/2010
19/10/2010
11/10/2010
03/10/2010
17/09/2010
25/09/2010
09/09/2010
31/01/2011
Water Level
–12.2
01/09/2010
Water Level (ma.s.l.)
–9.2
δ18O (‰)
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18 Date
Fig. 4 Water level in the Danube River and observed oxygen-isotope compositions of the Danube water, precipitation,
local groundwater and water in the collector well, with modelled values of isotope content in the collector well for the time
period 1 September 2010–31 March 2011 (in the case of high amounts of snowmelt water, which yields higher water levels,
the oxygen isotopic composition is more negative).
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1114
Krisztina Kármán et al.
During the study period, two large amplitude
peaks of δ18O value were observed in the Danube
water. The first one was observed between 10
December 2010 (δ18O = –10.75‰) and 5 January
2011 (δ18O = –10.65‰), with a minimum value of
δ18O = –11.38‰ on 17 December 2011; its amplitude was 0.63–0.73 ± 0.2‰. The second one was
observed between 15 January 2011 (δ18O =
–10.39‰) and 29 January 2011 (δ18O =
–10.41‰), with a minimum value of δ18O
= –11.36‰ on 18 January 2011 (Fig. 4); its amplitude
was 0.95–0.97 ± 0.2‰. These two negative peaks
appeared in the collector well water with a time shift
as well as amplitude damping resulting from dispersive–advective flow and mixing. The first peak in the
well was observed between 14 December 2010 and 6
January 2011 with a minimum value of δ18O =
–10.68‰, and the second appeared between 21
January and 10 February 2011, with a somewhat
more negative δ18O value of –10.99‰.
6 MODELLING RESULTS AND DISCUSSION
In previous studies (e.g. Amin and Campana1996,
Maloszewski et al. 2002, Fórizs et al. 2005), isotope
data were observed for several years with a sampling
frequency of 1 week or 1 month, and the modelled
mean transit times represented an average value for
several years. As mentioned previously, only a short
time span with frequent (daily or daily calculated)
sampling could be modelled in the present study.
Because of the short water-sampling period, the
annual or multi-annual average behaviour of the
well could not be described adequately. However,
the sampled period could be examined with great
precision.
After starting the isotope data modelling it
became clear that it would not be possible to use
the existing data set as a continuous time series for
the whole observation period. The pumping rate of
the well did not change during the measured period.
However, a significant problem appeared due to the
variable water level of the Danube, which changed
between 96 and 102 m a.s.l. Consequently, it was
necessary to split the model calculation into several
“steady-state” cases characterized by different constant Danube water levels and hence different flow
conditions. At high water level (~102 m a.s.l.), a high
Danube water proportion in the production well and
high dispersion parameter, as well as a shorter travel
time resulted in the best fit calculation of modelledto-observed δ18O values. At medium water levels
(98–99 m a.s.l.), the Danube water proportion and
the dispersion parameter were lower, while the travel
time was longer. Unfortunately, the oxygen isotope
compositions of the Danube water were only significantly different from the local groundwater when
large amounts of alpine snowmelt water arrived in
the area under investigation, or when high winter
precipitation was discharged to the river in the headwater areas. Thus, at low water levels (<97 m a.s.l.)
the model was not usable, because it was not possible
to separate the Danube water from the local background water.
For the first oxygen-isotope peak (10 December
2010–5 January 2011) the modelling yielded a 12–14
days long mean transit time and PD = 0.7 (Table 1).
The proportion of Danube water in the production
well was calculated to be 60%. For the second peak
(15–29 January 2011), modelling yielded the best fit
with PD = 1.2 and mean transit time of 6–8 days. The
proportion of Danube water for that period was at
least 95%. Although both PD values seem to be high,
it should be taken into account that they represent
mainly the variance of the transit time distribution, in
the case considered, suggesting a longer river infiltration zone.
During low water-level periods (<97 m a.s.l.) of
the Danube, the oxygen isotope composition of the
well water was close to the local groundwater,
because the latter is composed primarily of previously infiltrated Danube river water. Only a smaller
part of this originates from precipitation, which has
the mean value of δ18O = –9.3‰ (Deák et al. 1992).
Table 1 Summary of the resulting parameters (mean transit time of water between the Danube River and the collector well,
and the proportion of Danube water in that well) found by modelling oxygen peaks in the Danube water in cases of medium
and high Danube water levels (see Fig. 4).
Water level in the Danube River
Mean transit time
Proportion of Danube water in the well
Dispersion parameter
Sigma
<97 m a.s.l.
98–99 m a.s.l.
102 m a.s.l.
not usable
12–14 days
6–8 days
~60%
~95%
0.7
1.2
0.0143
0.0473
Transit time determination for a riverbank filtration system
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Large amounts of precipitation could change the isotopic composition of the local groundwater, resulting
in isotopically heavier background water. Because
the composition of the background water is close to
the Danube water, and differs greatly from that of the
precipitation, the amount of precipitation in the background water can be neglected. When the oxygenisotope composition of the local groundwater is close
to the infiltrated Danube water, there is no possibility
of distinguishing these two water sources. Thus, we
cannot calculate the travel time and the proportions
of water from these two sources in the well.
1115
Acknowledgements The authors are grateful to the
International Atomic Energy Agency (Project RER/8/
016) and to the EU (Project 226536/GENESIS/FP-7ENV-2008-1) for supporting our research and the
staff of the Institute of Groundwater Ecology,
Helmholz Zentrum München, especially to Dr
Christine Stumpp for her help. Thanks go to the
staff of the Institute for Geological and
Geochemical Research, Hungarian Academy of
Sciences and Lithosphere Fluid Research Lab,
Eötvös Loránd University. The authors especially
acknowledge both reviewers: Dr Martin Kralik and
Dr Jacques Mudry for their valuable comments,
which helped to improve the manuscript.
7 CONCLUSIONS
The lumped-parameter “steady-state” dispersion
model has a very wide application range, especially
for all types of riverbank filtration systems. The
observation of time-dependent stable isotopes contents can be very successful. However, a strong contrast between the river water and local background
water is needed. The present study shows that it is
possible to apply the lumped dispersion model in
variable hydrodynamic conditions by selecting the
time periods during which the water level in the
river stays constant. The mean isotopic content of
the river water and groundwater were very close at
low water levels (<97 m a.s.l.) and prevented determination of the transit time and the proportion of
Danube to local groundwater in the well. The proposed selection of some long-term events with negative isotopic peaks in the Danube, resulting from
snow melt, and modelling of their transfer to the
production well yielded satisfactory results.
When a negative peak was observed in the
Danube oxygen isotope composition, there were significant differences between the input (Danube) and
the output (collector well) concentrations. This happened twice during the study period when the water
level was medium or high. The first peak indicates a
mean travel time of 12–14 days and 60% Danube
water in the well at water levels 98–99 m a.s.l.
(Table 1). The second peak indicates a mean travel
time of 6–8 days, and at least 95% Danube water in
the well at a water level of 102 m a.s.l. These transit
times for water moving from the Danube to the well
are very realistic, considering the works of Stichler
et al. (1986) and Maloszewski and Zuber (1996).
It is important to note that, in order to refine the
results, other models with different input parameters
will need to be used too.
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