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A case study for a numerical box model with lagging argument in order to
simulate transportation of pollution in Dnieper reservoirs and Loire River
Vladimir P. Sizonenko
Institute of General Energy, Antonovicha 172, 03150, Kiev, Ukraine
INTRODUCTION
The focus of this study was to describe radionuclide pollution transport in the surface water
using a box model with lagging argument (UNDBE). The model takes into account the time
of transport of polluted water and partial mixing of pollution in the box. It is hypothesized
that this model will increase the accuracy of the prediction of pollutant concentration at the
outlet of the box. In addition, the complexity of the mathematical model and programming
tools are comparable to the simplest of current models for predicting pollution concentrations
in rivers. There are no additional requirements to obtain the quantity compared to full-scale
measurements. Transportation 90Sr was learned for reservoirs of the Dnieper River (Ukraine).
The research of passing of ejections 3H in 350 kilometers channel Loire River (France) on
stretch a half-year was made in the framework of the IAEA program EMRAS.
MATERIALS AND METHODS
A set of models of varying complexities for three-dimensional boxes was developed to
simulate transport of radionuclide pollution. The simplest of these box models include one
which considers complete mixing and averages volume variables; however, the prediction is
less accurate than more advanced models. In addition, the simplest model is less sensitive to
the quantity of initial data than the existing more complex 1, 2, or 3-dimensional models. The
use of latter models imposes fundamental constraints on the possibility of obtaining accurate
predictions because these require large quantities of accurate initial and boundary data, as
well as considerable computer time.
Model UNDBE takes into account that the transposition flows past in two stages. In the first
stage, each portion of the water and pollution moves through the reservoir until the outflow. In
the second step only at the end of transportation each portion of water and pollution mixes up
in a certain part of compartment volume and interacts there with sediments and bottom
depositions as in ordinary box model. This approach takes into account time of transporting of
water masses and intermixing of contamination in some part of volume of the camera at the
moment of termination of transporting. In addition, all transformations of pollutants during
transposition are equivalent to transformations in volume. These assumptions gives a model
which is described by a system of the usual differential equations with time lag - time of
contamination transposition.
The difference between the conventional box model and UNDBE box model is the impulse
response to contamination at the inflow of the box. Figure 1 illustrate the impulse of the
inflow at the first camera and the outflow response of the joint boxes. The conventional box
model gives an immediate response on outflow of the box where it is stipulated that
intermixing is instantaneous at the inflow. Whereas, the UNDBE box model with lagging
argument accounts for the time of contamination transposition.
0.10 0.10
Inflow Inflow
0.08 1-st box 1-st box
0.08
2-nd box 2-nd box
C o n cen tra tio n
C o n cen tra tio n
0.06 0.06
0.04 0.04
0.02 0.02
0.00 0.00
27 29 31 33 35 37 39 41
27 29 31 33 35 37 39 41 43 45 47 49 51
Time Time
Figure 1. Response of the conventional box model and the box model with lagging argument.
Case study reservoir
The floodplain areas near the Chernobyl Nuclear Power Plant and surrounding catchments are
heavily contaminated with radionucleotides, especially 90Sr. The major fraction of the
radionucleotides wash-off comes from the watershed of the Pripyat River, the right-hand
tributary of the River Dnepr. The 90Sr run-off from these watersheds is transported to the
Black Sea through a system of six reservoirs located along the Dnepr River. The largest part
of the annual runoff goes through the reservoirs from March until June during flood season.
The upper reservoir is Kiev reservoir which has a capacity of 3.7 km3, the length is 70 km, the
maximum depth is 14.5 m, the average depth is 4 m, and there are four main tributaries
including Pripyat River. An ice jam took place in the Pripyat River in 1994 and water covered
heavily contaminated area and inflow concentration of 90Sr increased rapidly up to 5920
Bq/m3 during the period 10–14 February 1994. At that time water discharge in the Pripyat was
low – 520 m3/sec. The spring flood occurred from 27 March to 22 April and concentrations of
90
Sr had increased up to 2553 Bq/m3. Water discharge in the latter period reached 1700
m3/sec. Daily data were used for water discharges and values of concentrations
(Voitsekhovitch et al., 1997). One box was used for modeling only.
Figure 2 presents conventional box model WATOX (Zheleznyak et al., 1992) results in a
comparison with daily data for 90Sr concentrations in the outflow of the Kiev reservoir.
The results usage of the new model with lagging argument for the same situation is in
Figure 3.
The next examples are when high spring flood in the Pripyat River took place during
February through June 1991 and 1999.
Case study River Loire
This study consists in modeling the dispersion of 3H in the Loire River has been done in the
framework of EMRAS project (EMRAS, 2006). The testing area is approximately 350 km in
length for a period of 6 months from the July 1 through December 31, 1999. This study takes
into account water discharges from 4 main tributaries and 3H discharges from 5 nuclear power
plants (14 reactors) (by using real hydraulic conditions of the year 1999).
The hydraulic boundary conditions and 3H discharges for each nuclear power plant were
given with a time step of one hour. Channel of the River Loire in length 350 kilometers, was
broken on 33 sequential cameras.
The results of the modeling (temporal series of 3H concentration with a time step one hour)
were compared to measurements of tritium concentration made in Angers, a city along the
Loire river, located downstream of all 3H discharges. The outcomes obtained with the help of
the model UNDBE in Angers and compared to measurements, are represented on Figure 4.
1300
Kiev Reservoir Outflow
Concen tration Sr (Bq/m )
3
Mod el WATOX
900
90
500
100
28/02/94
20/03/94
09/04/94
29/04/94
08/06/94
28/06/94
19/05/94
Figure 2. Dynamics of 90Sr concentrations in the outflow of the Kiev reservoir and ordinary box
model WATOX predictions.
1300
C once ntratio nSr (Bq/ m )
Kiev R eservo ir O utflow
3
900 Mod el UNDBE
90
500
100
2 8/0 2 /9 4
20/0 3 /9 4
0 9/0 4 /9 4
2 9/0 4 /9 4
1 9/ 05 /9 4
0 8/0 6 /9 4
2 8/0 6 /9 4
Figure 3. Dynamics of 90Sr concentration in the outflow of the Kiev reservoir in 1994 and UNDBE
model predictions.
RESULTS AND CONCLUSIONS
The box model with lagging argument (UNDBE) taking into account time of water transport
and partial mixing in box, gives the opportunity to increase the accuracy of the box model
without small spatial discretization. The comparison of results shows that box model UNDBE
gives good coincidence with measurements.
Figure 4. Comparison between calculated and measured tritium concentration at Angers.
ACKNOWLEDGEMENT
The author kindly thanks to the researchers of Ukrainian Hydrometeorological Institute, the
Hydrological Forecasting Department of the HydroMet Center in Kiev, the DIREN Centre
and EDF (France) for radiological and hydrological data.
REFERENCES
Voitsekhovitch O., Kanivets V., Laptev G., 1997. Radioecology of Water Objects the Chernobyl zone, Vol.1,
Chernobylinterinform, Kiev, (1997), pp. 60-96.
Zheleznyak M., Demchenko R., Khursin S., Kuzmenko Y., Tkalich P., Vitjuk N., 1992. Mathematical Modeling
of Radionuclide Dispersion in the Pripyat-Dnieper Aquatic System after the Chernobyl Accident, The
Science of the Total Environment, Vol. 112, (1992), pp. 89-114.
EMRAS Working Group on Model Validation for Radionuclide Transport in the Aquatic Systems. 2006.
http://www-ns.iaea.org/projects/emras/.
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