threetox — a computer code to simulate three-dimensional dispersion ...

Radiation Protection Dosimetry Vol. 73, Nos 1–4, pp. 177–180 (1997) Nuclear Technology Publishing

THREETOX — A COMPUTER CODE TO SIMULATE THREE-DIMENSIONAL DISPERSION OF RADIONUCLIDES IN STRATIFIED WATER BODIES N. Margvelashvily, V. Maderich and M. Zheleznyak Institute of Mathematical Machine and System Problems Pr. Glushkova 42, Kiev 252207, Ukraine Abstract — To simulate the radionuclide transport in stratified water bodies the THREETOX three-dimensional time-dependent flow and transport code has been developed. The prognostic variables are the three components of the velocity field, temperature, salinity, surface elevation, suspended sediment concentration, the radionuclide concentration in the solute, suspended sediments and bottom deposition. The model was applied to the Kiev reservoir and Dnieper–Bug estuary to predict the radionuclide fields of Chernobyl origin. Results revealed that THREETOX appears to simulate well the radionuclide migration in the stratified and homogeneous water bodies with complicated geometry and time-dependent flow.

INTRODUCTION Mathematical modelling of water systems is an effective and, on frequent occasions, the only tool to support decision making concerning remedial measures after an accidental radioactivity release. The complexity of the models should therefore be in conformity with the scale of the problem and the feasibility of a decision support system. It is reasonable to provide three-dimensional modelling of radionuclide transport in water bodies under the conditions of a complex configuration of a coast line and bottom topography, and large vertical and horizontal gradients of hydrophysical and pollution fields. Such conditions could be present in the vicinity of the heavily contaminated bottom and area of release, or might be found in stratified water bodies such as cooling ponds, estuaries and reservoirs. The three-dimensional code THREETOX, which attempts to incorporate dominant processes in water bodies to simulate radionuclide dispersion, is presented. It is a 3-D extension of the box, 1-D and 2-D models developed in IPMMS after the Chernobyl accident (1). Hydrodynamic and radionuclide transport models are briefly described. The case studies are developed on the basis of measured radionuclide concentrations for the Kiev reservoir and the Dnieper–Bug estuary, which are at the beginning and the end of the radionuclides’ path from the Chernobyl accident area to the Black Sea. MODEL The THREETOX code was developed to simulate 3D hydrodynamic fields, suspended sediment and radionuclide transport in water bodies. The hydrodynamics are simulated on the basis of the three-dimensional, time-dependent, free surface, primitive equation, Princeton Ocean Model (2). The model equations are written in Cartesian coordinates (x,y,z). The prognostic variables of the hydrodynamics code are the three components of the velocity fields (u,v,w), temperature, salinity and

surface elevation ␩. The water body is assumed to be hydrostatic and incompressible. The concept of eddy viscosity/diffusivity and Prandtl’s hypothesis with the variable turbulence length scale are used to define the turbulence stresses. At the free surface all fluxes (momentum, heat, etc.) are prescribed. At the bottom and the land boundaries the condition of no diffusive fluxes of any property is used. The open lateral boundary conditions are modified radiation conditions. The radionuclide migration code includes the model equations that describe the processes governing the specific radionuclide’s exchange between water, suspended sediment, and bottom deposits. The advection–diffusion equation governing the transport of suspended sediment is ⭸S ⭸S ⭸S ⭸S +U +V + (W − W0) ⭸t ⭸x ⭸y ⭸z =

冉 冊

⭸S ⭸ v + A(⌬S) ⭸z ⭸z

(1)

where S is sediment concentration, W0 the settling velocity of the particles, v and A are vertical and horizontal diffusivities, respectively. At the free surface no vertical flux of the sediments is assumed, i.e. (W − W0)S = v

⭸S , ⭸z

z=␩

The vertical flux of suspended sediments at the bottom is assumed to be equal to the difference of the resuspension and sedimentation rates. It yields the following formulation of the boundary condition: v

⭸S + W0S = qs − qb, ⭸z

z = −h − a

where a is the level of boundary between suspended sediment and bottom sediment transport, qs and qb are sedimentation and resuspension rate, respectively. It is assumed that in the case of fine noncohesive sediments these rates may be estimated as

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qs = qb =

N. MARGVELASHVILY, V. MADERICH and M. ZHELEZNYAK

再 再

W0(Sⴱ − S0), S0 ⬎ Sⴱ 0, S0 ⬍ Sⴱ

(W − W0)SCS − v

0, S0 ⬎ Sⴱ ErW0(Sⴱ − S0), S0 ⬍ Sⴱ

W0SCS + v

Here Er is the erodibility coefficient, S0 is the actual sediment concentration at the bottom level z = −h − a, Sⴱ is the near-bottom equilibrium sediment concentration that corresponds to the sediment capacity of the steady and uniform flow with the same local parameters. The sub-model of radionuclide transport describes the radionuclide concentration in solute C, the concentration in the suspended sediments Cs and the concentration in the bottom deposition Cb. The exchanges between these forms have been described as adsorption–desorption and sedimentation–resuspension processes. The governing equations are similar to those used in the FLESQOT model (3). The transport equation for dissolved radionuclides is

冉 冊

⭸C ⭸C ⭸C ⭸ ⭸C ⭸C +U +V +W = v ⭸t ⭸x ⭸y ⭸z ⭸z ⭸z + A(⌬C) − ␭C − a1,2S(KdC − CS)

(2)

where Kd is the distribution coefficient in a solid particle–water system in equilibrium conditions, Kd = CS/C when t → ⬁ in hydraulic steady conditions; a1,2 is the rate of water-suspended sediment exchange; ␭ is the decay constant for the radionuclide under consideration. The boundary condition at the free surface is v

⭸C =WC, ⭸z

⭸C = ␳S(1 − ε)Zⴱa1,3(KdbC − Cb), ⭸z z = −h − a

where ε is the bottom porosity, Zⴱ the effective thickness of the contaminated upper bottom layer, a1,3 is the rate of water–bottom exchange, ␳S is sediment density, Kdb the distribution coefficient for the bottom sediments. The radionuclide transport by suspended sediments is described by the equation ⭸SCS ⭸SCS ⭸SCS ⭸SCS +U +V + (W − W0) ⭸t ⭸x ⭸y ⭸z =

冉

冊

⭸SCS ⭸ v + A(⌬SCS) − ␭SCS ⭸z ⭸z

+ a1,2S(KdC − CS)

⭸SCS = CSqS − Cbqb , ⭸z

z=␩; z = −h − a

The thickness of the upper layer of the contaminated bottom deposition is governed by the equation of the bottom deformation ␳S(1 − ε)

⭸Zⴱ = qS − qb ⭸t

(4)

Radionuclide concentration in the upper bottom layer is described by the equation ⭸(ZⴱCb) = a1,3Zⴱ(KdbC − Cb) ⭸t −

1 (CSqb − CbqS) ␳S(1 − ε)

(5)

Sigma co-ordinates are used to avoid difficulties in the numerical solution of the problem for realistic bottom topography. Splitting of the barotropic and baroclinic modes is imposed in the code. The governing equations together with the boundary conditions were solved by finite difference techniques. A horizontally and vertically staggered mesh was used for the computations. The model runs under UNIX on workstation HP-9000/735. CASE STUDIES Kiev reservoir

z=␩

The diffusion flux into the bottom is taken as v

⭸SCS =0, ⭸z

(3)

The boundary conditions are no flux of the CS through the water surface, and its flux equal to the difference of amount of eroded and deposited pollutants at the bottom boundary:

The Kiev water reservoir is situated 20 km downstream from the Chernobyl nuclear power plant. The reservoir is 110 km long and 16 km wide, with maximum depth 15 m. The north part of the basin is relatively shallow with a depth of about 2–3 m. After the Chernobyl accident the bottom of the reservoir was heavily contaminated and contained about 70 TBq of 137 Cs. Since the accident, only low and moderate spring floods have occurred. The question arises of whether there is a strong influence of extremely high floods on the level of radioactive contamination in the reservoir. First, the moderate spring flood of April–May 1987 was simulated to compare the results with the available measurement data. Inlet measured reservoir data consist of water and suspended sediment discharge and 137Cs concentrations over time periods of 10 days. The outlet data were the same with time periods of one month. The realistic bottom contamination distribution (4) was used as an initial condition. Distribution of 137Cs at the surface after a two month simulation is depicted in Figure 1. Deviation of the simulated data from the measurement data does not exceed 10% at the end of the first month and 31% at the end of the second month: this is within the range of uncertainties of the measured data.

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THREETOX COMPUTER CODE FOR WATER BODIES

The influence of an extremely high flood (with maximum water discharge of 15,000 m3.h−1) on radionuclides distribution was studied. Figure 2 shows the bottom radionuclide distribution before (a) and after (b) 10 day simultation. A reduction of the concentration in the north, shallow, part of the basin and radionuclide deposition in the south, deep, part is observed. An increase of the concentration of radionuclides adherent to suspended sediments by more than an order of magnitude (from 90 to 2100 Bq.m−3 at the surface) was found, while the concentration of radionuclides in solution was not much changed.

a slightly sloping shelf exists with a maximum depth of 28 m in the south-western side of the basin. The water of the estuary and adjacent shelf area are generally stratified and are characterised by very inhomogeneous lateral and vertical flow distributions (Figure 3) resulting in a complex distribution of radionuclides. The THREETOX code was used to model 90Sr distribution in August 1991. The radionuclide concentration in the mouth of the Dnieper and Bug rivers was prescribed according to the measurement data (5). Figure 4(a) shows the simulated (solid lines) and measured (dashed lines) distribution of 90Sr at the surface.

Dnieper–Bug estuary

(a)

Chernobyl’s radionuclides from the Dnieper River reservoir system are transported into the Black Sea through the Dnieper–Bug estuary. The east–west dimension of this basin is 147 km, and north-south it is 53 km. The typical depth of the Dnieper–Bug estuary is about 5 m. In the adjacent north-western area of the Black Sea

Odessa N 0.35 m.s–1

S

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(km)

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10 5

0

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420 480 540 600

420 360 480 540

(b)

300 420 480 540

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0.18 m.s–1

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Figure 1. The computed Cs distribution (Bq.m−3) at the surface of Kiev reservoir for 30 April 1987. 137

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Figure 3. Simulated velocities distribution at the surface (a) and the near-bottom layer (b). North-East wind, U = 8 m.s−1.

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(b)

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(km)

Figure 2. The distribution of 137Cs concentration (Bq.kg−1) in the bottom deposition at initial moment of time (a) and produced by a 10 day model run (b) under the condition of extremely high flood water.

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Figure 4. The 90Sr fields (Bq.m−3) for August 1991 at the surface (a) and the near bottom level (b). Solid lines are computed concentrations, dashed lines are observed values.

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N. MARGVELASHVILY, V. MADERICH and M. ZHELEZNYAK

The distribution of the radionuclide at the near bottom layer is plotted in the Figure 4(b). The computed radionuclides distribution is characterised by strong lateral and vertical variability and correlates well with the observations. CONCLUSIONS (1) The THREETOX code, constructed from basic dynamic and physical considerations, appears to simulate well the flows and radionuclide dispersion in the stratified water bodies with complicated geometry and time-dependent spatially inhomogeneous flow.

(2) Its main advantages lie in the use of the well verified Princeton Ocean Model to the flow prediction and in the accuracy by which the various processes governing the specific radionuclide’s exchange between water, suspended sediment, and bottom deposits are modelled. (3) It was shown that even extremely high flood water does not greatly change the 137Cs bottom contamination of the Kiev water reservoir. Radionuclides released from the Dnieper–Bug estuary are characterised by strong vertical variability of 90Sr (more than 50% difference between surface and near bottom layer) observed up to Odessa coastal areas.

REFERENCES 1. Zheleznyak, M., Demchenko, R., Khursin, S., Kuzmenko, Yu, Tkalich, P. and Vitiuk, N. Mathematical Modeling of Radionuclide Dispersion in the Pripyat-Dnieper Aquatic System after the Chernobyl Accident. Sci. Total Environ. 112, 89–114 (1992). 2. Blumberg, A. F. and Mellor, G. L. Diagnostic and Prognostic Numerical Circulation Studies of the South Atlantic Bight. J. Geophys. Res. 88, 4579–4592 (1983). 3. Onishi, Y., Dummuller, D. C. and Trent, D. S. Preliminary Testing of Turbulence and Radionuclide Transport Modeling in Deep Ocean Environment. PNL-6853 (Pacific Northwest Laboratory, Richland, Washington) (1989). 4. Voitsekhovich, O., Prister, B., Nasvit, O., Los, I. and Berkovski, V. Present Concept on Current Water Protection and Remediation Activities for the Areas Contaminated by the 1986 Chernobyl Accident. Health Phys. 71(1), 19–29 (1996). 5. Batrakov, G. F., Eremeev, V. N., Chudinovskikh, T. V. and Zemlyanoy, A. D. Radioactivity of the Black Sea (NAS Marine Hydrophysical Institute, Sebastopol) pp. 2–216 (in Russian) (1994).

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