Tracers and Modelling in Hydrogeology
(Proceedings of the TraM'2000 Conference
held at Liège, Belgium, May 2000). IAHS Publ. no. 262, 2000.
271
Lumped-parameter models as a tool for
determining the hydrological parameters of some
groundwater systems based on isotope data
PIOTR MALOSZEWSKI
GSF Institute of Hydrology, PO Box 1129, D-85758 Neuherberg,
Germany
e-mail: maloszewski@gsf.de
Abstract The use of lumped-parameter models for interpreting environmental
tracer data is based on the assumption that the transit time distribution
function of the tracer particles through the groundwater system under
consideration is known or can be assumed. These functions describe the whole
spectrum of the transit times of single tracer particles transported through the
system between the entrance (recharge area) and the exit (spring, stream,
pumping well). The main parameter of each model is the mean transit time of
water through the system. This parameter is determined by calibrating the
model using the environmental tracer concentrations measured in the output.
Additional useful hydrological information can be derived from the model
parameters. The paper presents the transit time distribution functions of some
commonly used lumped-parameter models and shows examples of their
application for solving bank filtration problems (drinking water supply for
Passau, Germany), and determining the water dynamics and water balance in a
heterogeneous karstic aquifer (Schneealpe, Austria).
INTRODUCTION
Recent progress in numerical methods and multilevel samplers focused the attention of
modellers on two- or three-dimensional solutions to the transport (dispersion)
equation. To solve the problem of pollutant transport, numerical models are generally
used which are based on the finite-element or the finite-difference methods. However,
to apply them in practice, an extensive database is required. These data are very often
not available, and when an extensive database is available, data are usually missing at
the beginning of most investigations. Lumped-parameter models require only a minimum of information concerning the turnover zone of the groundwater and the record of
non-reactive tracer concentrations measured in the input and the output from the
system as a function of time. These models can be simply validated with the available
hydrogeological data. It should be noted that if the lumped-parameter model is already
calibrated for the system under consideration, then this model can also be used to
evaluate other problems e.g. the effects of surface contamination on the underground
water.
The environmental isotopes e.g. tritium and stable isotopes ( O or deuterium) are
suitable for tracing the origin of water in the hydrological cycle because they are
constituents of the water molecule. Therefore, the use of environmental isotope
techniques may, in addition to conventional hydrological methods, provide further
insight into, for instance, determining the origin of water, the storage properties of a
ls
272
Piotr
Maloszewski
catchment; the water dynamics in a groundwater system or the reaction between
surface and groundwater by bank filtration. The quantitative evaluation of these data is
often based on using lumped-parameter models, which were developed for chemical
engineering and adopted by isotope hydrology.
BASIC CONCEPT OF LUMPED-PARAMETER MODELS
The principles and use of lumped-parameter models, also called black box models, are
well described e.g. by Maloszewski & Zuber (1982, 1993, 1996). Each model is
characterized by the transit time distribution function of tracer particles transported
between the input (recharge area) and the output (discharge area). This function has to
be known (assumed) based on hydrological knowledge of the system under
consideration. In most cases the transit time distribution functions have one or two
unknown (fitting) parameters which can be determined by calibrating the model to the
experimental data observed in the outflows from the groundwater system. The input
concentration of the environmental tracer is directly measured or has to be calculated
from the precipitation amount, infiltration coefficients and tracer content in
precipitation (Maloszewski & Zuber, 1982; Grabczak et al, 1984; Stichler et al,
1986).
The main parameter of all models is the mean transit time of water through the
system (7), which is related to the water volume in the system (V) and the volumetric
flow rate through the system (Q) according to the following relationship:
V=QT
(1)
The mean transit time represents the average of the number of flow times at the
individual streamlines in the aquifer, weighted with the amount of water flowing.
INPUT:
OUTPUT:
C (t)
c (t)
in
oul
MODEL 1
direct runoff
INPUT:
OUTPUT:
C,„(t)
c„ (t)
ut
MODEL 2
direct runoff
OUTPUT:
INPUT:
C (t)
in
C„ (t)
ut
upper
reservoir
lower
reservoir
MODEL 3
F i g . 1 T h e schematic presentation of the m o s t frequently u s e d c o n c e p t u a l m o d e l s of
the g r o u n d w a t e r s y s t e m (after M a l o s z e w s k i et al, 1983).
Lumped-parameter
models for determining
hydrological
parameters
273
Generally, it is assumed that the groundwater system can be considered as a closed
system, sufficiently homogeneous, being in the steady state, having a defined input
(recharge or infiltration area) and a corresponding output in the form of pumping
wells, springs or streams draining the system. However, the conceptual model must not
be limited to the "one-box" representation of the system. The conceptual model can
also be further developed as was shown in e.g. Maloszewski et al. (1983) and within
this paper. The schematic presentation of the most frequently used conceptual models
of the groundwater system is shown in Fig. 1.
To determine the mean transit time of water, the temporal variation of the
measured tracer input concentration, Q (t), is used to calculate the tracer output
concentration, C (0> which, in turn, is compared with the concentrations measured in
the output from the system. The relationship between input and output concentrations
is described by the well known convolution integral:
n
QUt
00
C
(0 = JC,„ (t -
oul
(2)
T)g(T)eX (-A.T)dT
P
0
with X being the decay constant of radioactive tracers.
The most common kinds of transit time distribution functions, g(x), which
automatically define the lumped-parameter model (3), are:
Piston-flow model
g(T)
= 8(x-T)
(3.1)
Exponential model
g(x)
X
= ^exp|
'
(3.2)
T '
Dispersion model
(1-x/r)
^4%P x/T
D
x
2
(3.3)
AP xlT
D
where P is the dispersion parameter.
Each model has one or two unknown (fitting) parameters which can be found by
solving equation (2) with the assumed lumped-parameter model (3). To solve the
inverse problem, e.g. to find model parameters, the user friendly software "FLOWPC"
described in detail in Maloszewski & Zuber (1996) can be applied. The selection of the
model depends on the hydrogeological conditions in the groundwater system under
consideration. Generally, the dispersion and piston flow models are applicable for
confined or partially confined aquifers, while the exponential model can be used only
for unconfmed aquifers. A detailed description of the model applicability can be find
in Maloszewski & Zuber (1982, 1996) or Zuber (1986).
D
EXAMPLES OF MODEL APPLICATION
A number of case studies have been presented and well reviewed, e.g. Maloszewski
(1994), Maloszewski & Zuber (1982, 1993, 1996) and Zuber (1986). Only two
selected examples of lumped-parameter model application are presented below.
274
Piotr
Maloszewski
River bank filtration
In river bank filtration studies, advantage is taken of the seasonal variations in the
stable isotope composition of river water, and of the difference between its mean value
and the mean value of local groundwater, e.g. Hôtzl et al. (1989), Maloszewski et al.
(1990) and Stichler et al. (1986). Measurements of the 0-content in river water, local
groundwater and pumping wells were used to determine the fraction and the flow times
of the Danube bank filtrate. The study was performed on a small island called
Soldatenau (0.3 Ion ), where the drinking water supply has been constructed for the
city of Passau in Germany (Stichler et al, 1986). The infiltrating Danube water was
mixed with the local groundwater, which under natural flow conditions is drained by
the river. In Fig. 2 a schematic presentation of the flow pattern and its lumpedparameter model is shown. The portion of the river water (p) was calculated from the
l8
2
mean tracer content of all components (C) applying the following formula:
C
-C
DR
LG
where subscripts: PW, LG, and DR are for pumping well, local groundwater and
Danube River water, respectively.
Due to the fact that the local groundwater had no seasonal variation, Q,G(0 ~
constant, the following formula is adequate for the situation shown in Fig. 2:
CO
C
PW
(0 = p JC
DR
(t - x)g(x)dx + {l-p)CT
(5)
G
o
i 8
where the input function CDR(1) was taken as the weighted monthly means of the 0 content in Danube River water for a period of 36 months.
Using equation (4) the portion of river water in the pumping well was calculated to
be p = 0.80. Due to the hydrogeological conditions the dispersion model (3.3) was
applied and yielded the best fit (Fig. 3) with mean transit time of water T = 60 days
and PD = 0.12 (corresponding to oti = 25 m). This calibrated model was later used to
predict the response of the pumping well to a hypothetical pollutant in the river water.
PUMPING WELL
CpwW
A Q
DANUBE RIVER
LOCAL GROUNDWATER
CDR(D
îH
T,P
D
r,
p
d
he
(1-p)Q
pQ
F i g . 2 C o n c e p t u a l m o d e l of the D a n u b e - R i v e r b a n k filtration at P a s s a u (after Stichler
etal,
1986).
Water dynamic in a karstic aquifer
The Schneealpe karst massif of Triassic limestones and dolomites (altitude up to
1800 m a.s.l.) is situated 100 km southwest of Vienna in Kalkalpen, and is the main
drinking water resource for the city (Rank et al, 1992). The catchment area of about
Lumped-parameter
models for determining
parameters
275
1 -11
i
-11
hydrological
z
1111111111111111111111 T 111111111111111
0
6
12
18
24
30
_-|4
36
MONTHS ( J a n . 80 - D e c . 82)
Fig. 3 O b s e r v e d (triangles) and fitted with dispersion m o d e l (solid line) O output
concentrations in the p u m p i n g well P S I (T = 60 days, P = 0.12, p = 0.8) at the island
Soldatenau (after Stichler et ah, 1986).
l s
D
2
1
23 km is drained by two springs: the Wasseralmquelle (2001 s" ) and the
Siebenquellen (310 1 s" ). This karstic aquifer is approximated by two interconnected
parallel flow systems of a fissured-porous aquifer and karstic channels. The fissuredporous aquifer with a high storage capacity contains mobile water in the fissures and
stagnant water in the porous matrix. The water enters this system at the surface and
flows through it to the drained channels which are regarded as a separate flow system
finally draining to both springs. The channels are connected with sinkholes which
introduce additional water directly from the surface. The conceptual model of water
flow in this basin is shown in Fig. 4.
It was assumed that for baseflow there is no flow in the drainage channels. Then,
tritium sampled in the springs represented the flow through the fissured-porous
aquifer. During high flow rates, the seasonal variations in 0 were assumed to
represent the fast flow through the channels. Fitting of the dispersion model to the
tritium data (Fig. 5) and the piston flow model to the 0 data, yielded the mean transit
times and volumes of water in both flow systems, and additionally, the portion of
water flowing directly from the sinkholes through the drainage channels. The volume
of water stored in the karstic channels (V ) was found to be insignificantly small (less
1
1 8
1 8
c
Input: infiltrated water Q(t), C (t)
in
Qç(t)|
j Qp(t)
Fissured-porous aquifer
T , (Pp)
p
p
6
V =246 1 0 m
Q =420 Us
3
p
p
Drainage channels T
Q =90L/s
c
c
6
V =1.6 1 0 m
3
c
Output: karstic springs
Q(t)=Q (t)+Q (t)
c
p
F i g . 4 C o n c e p t u a l m o d e l of w a t e r flow in S c h n e e a l p e karst m a s s i f and
m o d e l l i n g results (after M a l o s z e w s k i et al, 2 0 0 0 ) .
some
Piotr
276
Maloszewski
1970 1975 1980 1985 1990 1995
CALENDAR YEARS
F i g . 5 M e a s u r e d tritium concentrations (triangles) and fitted w i t h dispersion m o d e l
(solid line) values for baseflow in the W a s s e r a l m q u e l l e spring (T= 2 6 years, P = 0.8)
draining the S c h n e e a l p e karst m a s s i f (after M a l o s z e w s k i et al, 2 0 0 0 ) .
D
than 1%) in comparison to the volume of water stored in the fissured-porous aquifer
(V ). On the other hand through the karstic channels the whole volumetric water flux
(Q) flows. The volumetric flow rate of water entering the channels through the
sinkholes (Q ) was relatively large (17.5% of total discharge). This water flux, having
a very short transit time to the springs (about one month), constitutes a potential threat
for the water quality in the springs in the case of possible contamination at the surface
of the catchment. The resulting hydrological parameters are shown in Fig. 4.
Similar tracer studies are presented in e.g. Maloszewski et al. (1992).
p
c
REFERENCES
Grabczak, J., Maloszewski, P., Rozanski, K. & Zuber, A. (1984) Estimation of the tritium input function with the aid of
stable isotopes. Catena 1 1 , 105-114.
Hôtzl, H., Reichert, B., Maloszewski, P., Moser, H. & Stichler, W. (1989) Contaminant transport in bank filtration—
determining hydraulic parameters by means of artificial and natural labelling. In: Contaminant Transport in
Groundwater (ed. by H. E. Kobus & W. Kinzelbach), 65-71. Balkema, Rotterdam.
Maloszewski, P. (1994) Mathematical modelling of tracer experiments in fissured aquifers. Freiburger Schriften zur
Hydrologie 2, 1-107. Universitât Freiburg, Germany.
Maloszewski, P. & Zuber, A. (1982) Determining the turnover time of groundwater systems with the aid of environmental
tracers. I. Models and their applicability. J. Hydrol. 57, 207-231.
Maloszewski, P. & Zuber, A. (1993) Principles and practice of calibration and validation of mathematical models for the
interpretation of environmental tracer data in aquifers. Adv. Wat. Resour. 16, 173-190.
Maloszewski, P. & Zuber, A. (1996) Lumped parameter models for the interpretation of environmental tracer data. In:
Manual on Mathematical Models in Isotope Hydrology, 9-58. IAEA-TECDOC-910, International Atomic Energy
Agency, Vienna.
Maloszewski, P., Harum, T. & Zojer, H. (1992) Modelling of environmental tracer data. In: Transport Phenomena in
Different Aquifers, 136-143. Steirische Beitrâge zur Hydrologie 43.
Maloszewski, P., Moser, H., Stichler, W., Bertlef, B. & Hedin, K. (1990) Modelling of groundwater pollution by river
bank filtration using oxygen-18 data. In: Groundwater Monitoring and Management (ed. by G. P. Jones) (Proc.
Dresden Symp., March 1987), 153-161, IAHS Publ. no. 173.
Maloszewski, P., Rauert, W., Stichler, W. & Herrmann, A. (1983) Application of flow models in alpine catchment using
tritium and deuterium data. J. Hydrol. 66, 319-330.
Maloszewski, P., Stichler, W. & Rank, D. (2000) Combined application of lumped-parameter models to environmental
tracer data for determining of transport and hydraulic parameters in karstic aquifer of Schneealpe (Austria).
J. Hydrol. (in preparation).
Rank, D., Vôlkl, G., Maloszewski, P. & Stichler, W. (1992) Flow dynamics in an Alpine karst massif studied by means of
environmental isotopes. In: Isotope Techniques in Water Resources Development 1991, DXI—DAZ. International
Atomic Energy Agency, Vienna.
Stichler, W., Maloszewski, P. & Moser, H. (1986) Modelling of river water infiltration using oxygen-18 data. J. Hydrol.
83, 355-365.
Zuber, A. (1986) Mathematical models for the interpretation of environmental radioisotopes in groundwater systems. In:
Handbook of Environmental Isotope Geochemistry (ed. by P. Fritz & J. Ch. Fontes), vol. 2, part B, 1-59. Elsevier,
Amsterdam.