**3. Mathematical calculations of the mass transport of technogenic radionuclides in the water flow of the River Yenisei in the impact zone of the Mining and Chemical Combine**

In the chapter, the results radiation-chemical situation in the middle reach of the Yenisei River located in the nearest zone of the influence of the Mining and Chemical Combine of Rosatom have been described. It has been shown that a wide range of radionuclides, heavy metals and organic substances of different genesis flow into the waters of the Yenisei River. It has been demonstrated that radionuclides and other pollutants are transported by the water flow in the form of molecular solution or colloids or with suspended matter. In this case, the suspended matter consists of pelitic finely dispersed mineral particles, plant and organic detritus and amounts of living biological objects.

Calculations have been made according to the described method in the area of the River Yenisei from the estuary of the river Plosky up to the island Atamanovsky. Assuming the water discharge to be *Q* = 4085 m3 /s the river depth *H* ≈ 7 m and the flow rate *v* = 1.25–1.8 m/s in the given section are estimated based on the hydraulic model. According to an earlier estimation, the stream with the technogenic admixtures propagates along the right bank, not far than one tenth of the river width, i.e. along the flood plain where the flow rate and depth are several times lower than those calculated based on the hydraulic model. According to the calculations: *H<sup>п</sup>* ≈ 2.5 m, *v<sup>п</sup>* ≈ 0.38–0.44 m/s.

Transport of radionuclide along the Yenisei River is based on a modified one-dimensional model proposed by Schnoor et al. [30]. For the whole length of the Yenisei, a homogeneous distribution of radionuclides over the cross-section is presupposed. It is assumed that both in the water column and in the active sediment layer the radionuclides are present in two forms: soluble and adsorbed forms. The most important processes influencing the behaviour of radionuclides include adsorption and desorption, sedimentation of suspended particles from the river water and resuspension from the active sediment layer, activity exchange between the pore water of the sediment and overlying water due to diffusion through the boundary and radioactive decay.

The calculations presented in this chapter are limited to the abiotic form of substance transport since the contribution of the biogenic component is considered to be insignificant [9].

Complex fresh water systems, such as large rivers, are assumed to be composed of a chain of interconnected 'elementary segments (ES)' that are comprised of: (a) the water column, (b) an upper sediment layer strongly interacting with water ('interface layer'), (c) an intermediate sediment layer below the 'interface layer' ('bottom sediment'), (d) a sink sediment layer below the 'bottom sediment', (e) the right and left sub-catchments of each ES.

Depending on the water discharge rate and geometry of the river bed the stream velocity varies which determines the transport of the sediment suspensions and sediment disturbancesedimentation. To estimate the accumulation of radionuclides in the bottom sediments, a mathematical model described by Belolipetsky and Genova was used [31].

The concentrations of radionuclides on solid particles were assumed to be proportional to the area of the particle surface. We used the field data the fraction distribution of radionuclides in the initial solution. Then, the particle transport and sedimentation along the river bed was estimated. In the channels and floodplain (in the areas with small stream velocities) there occurs sedimentation of the sediment suspensions. During the periods of the increased water discharge rate (spring floods, increased volume of the hydroelectric station), the sediment disturbance is also possible as well as transport of impurities downstream (secondary pollution).

To describe the sediment suspension transport in a turbulent flow of non-compressible liquid a simplified equation is used:

$$\text{ĐSi/Đt + us đSi/Đx = } qSi/h + q/\omega \cdot \text{Siq} \tag{1}$$

where *Si* is the concentration of the *i*th fraction [kq/m3 ]; *Siq* is the concentration of an impurity of the *i*th fraction, entering with the tributary on the way *q*; *qSi* is sediment disturbancesedimentation of the impurity of the *i*-th fraction; *t* is time; *x* is a coordinate directed along the current; *Q* is the discharge rate; *ω* is the cross-section area of the river bed; *u<sup>в</sup> = Q/ω* is the cross-section; average velocity *h* is the depth.

The bottom exchange is determined by the formula

$$q\text{Si} \quad = \text{ (Si } tr-\text{SiO}\text{)} \quad \text{- } w\text{gj, Si } tr = 0.01 \cdot ai \cdot \text{Str, } q\text{S} \quad = \text{ } \Sigma \text{ qS}\text{j} \tag{2}$$

The Behaviour of Natural and Artificial Radionuclides in a River System: The Yenisei River, Russia as a Case Study http://dx.doi.org/10.5772/65743 371

$$S\_{\nu} = \begin{cases} 0.2 \cdot u\_{\ast} / gh \, w\_{\beta'} \text{ if } w\_{\beta} < w\_{\ast} \\ \cdot \, w\_{\beta} = (\rho\_{\beta} - \rho\_{\ast}) / \rho\_{\ast} \cdot g / 18 \nu \cdot d^{2} \,\_{\text{op}} \\ 0. \, f \, w\_{\beta} \ge \, w\_{\ast} \end{cases} \tag{3}$$

The transport capability of the flow *Str* depends on the depth-average flow velocity, depth and hydraulic coarseness; *qS* is the mass exchange with the bottom.; *Si*<sup>0</sup> is the concentration of the *i*-th fraction near the bottom; *α<sup>i</sup>* is the percent content of the *i*-th fractions in the bottom sediments. When calculating *Si tr* using Eq. (3) it should be taken into account that *Si tr* cannot exceed the concentration of the *i*-th fraction in the bottom sediments (*Si* day), therefore, when *Si tr* > S*<sup>i</sup>* day it is assumed that *Si tr* = *Si* day. If the concentration of the *i*-th fraction in the bottom sediments is equal to zero, then *Si tr =* 0.

The main change in the bottom sediment composition is assumed to be due to sediment disturbance and sedimentation. When *qs >* 0, the bottom sediments enter the flow (washing out, sediment disturbance) and when *q*<sup>s</sup> < 0 the silting of the river bed is observed (sedimentation of the suspended particles).

Let *z\** be the thickness of the active layer of the bottom sediments. Assuming that the formation of the upper layer of the bottom sediments (with the thickness *z\** ) results in the sediment disturbance-sedimentation, the mass conservation equation for the *i*-th fraction in the bottom sediments *Si* day is written as follows:

$$
\partial \langle \mathbf{z}\_\* \cdot \mathbf{S}\_{\vert\_{\partial \partial y}} \rangle / \partial t \quad = \quad -q\_{\vert \dot{s}} \tag{4}
$$

Since *∑qSi = qS* , *∑Si* day = *ρ*, from Eq. (4) one obtains the equation to find *z\** :

are several times lower than those calculated based on the hydraulic model. According to the

Transport of radionuclide along the Yenisei River is based on a modified one-dimensional model proposed by Schnoor et al. [30]. For the whole length of the Yenisei, a homogeneous distribution of radionuclides over the cross-section is presupposed. It is assumed that both in the water column and in the active sediment layer the radionuclides are present in two forms: soluble and adsorbed forms. The most important processes influencing the behaviour of radionuclides include adsorption and desorption, sedimentation of suspended particles from the river water and resuspension from the active sediment layer, activity exchange between the pore water of the sediment and overlying water due to diffusion through the boundary

The calculations presented in this chapter are limited to the abiotic form of substance transport since the contribution of the biogenic component is considered to be insignificant [9].

Complex fresh water systems, such as large rivers, are assumed to be composed of a chain of interconnected 'elementary segments (ES)' that are comprised of: (a) the water column, (b) an upper sediment layer strongly interacting with water ('interface layer'), (c) an intermediate sediment layer below the 'interface layer' ('bottom sediment'), (d) a sink sediment layer below

Depending on the water discharge rate and geometry of the river bed the stream velocity varies which determines the transport of the sediment suspensions and sediment disturbancesedimentation. To estimate the accumulation of radionuclides in the bottom sediments, a

The concentrations of radionuclides on solid particles were assumed to be proportional to the area of the particle surface. We used the field data the fraction distribution of radionuclides in the initial solution. Then, the particle transport and sedimentation along the river bed was estimated. In the channels and floodplain (in the areas with small stream velocities) there occurs sedimentation of the sediment suspensions. During the periods of the increased water discharge rate (spring floods, increased volume of the hydroelectric station), the sediment disturbance is also possible as well as transport of impurities downstream (secondary pollution). To describe the sediment suspension transport in a turbulent flow of non-compressible liquid

∂ *Si*/ ∂ *t* + *uв*  ∂ *Si*/ ∂ *x* = *qSi*/*h* + *q*/*ω* ⋅ *Siq* (1)

rity of the *i*th fraction, entering with the tributary on the way *q*; *qSi* is sediment disturbancesedimentation of the impurity of the *i*-th fraction; *t* is time; *x* is a coordinate directed along the current; *Q* is the discharge rate; *ω* is the cross-section area of the river bed; *u<sup>в</sup> = Q/ω* is the

*qSi* = (*Si tr* – *Si*0 ) ⋅ *wgi*, *Si tr* = 0.01 ⋅ *αi* ⋅ *Str*, *qS* = ∑ *qSj* (2)

]; *Siq* is the concentration of an impu-

≈ 0.38–0.44 m/s.

the 'bottom sediment', (e) the right and left sub-catchments of each ES.

mathematical model described by Belolipetsky and Genova was used [31].

is the concentration of the *i*th fraction [kq/m3

cross-section; average velocity *h* is the depth.

The bottom exchange is determined by the formula

calculations: *H<sup>п</sup>*

370 Water Quality

and radioactive decay.

a simplified equation is used:

where *Si*

≈ 2.5 m, *v<sup>п</sup>*

$$
\partial z\_\ast \partial t \; \equiv \; \; \; \neg q\_\diamond \prime \rho \tag{5}
$$

The calculation algorithm for the suspended and bottom sediment dynamics consists of the following stages:

Stage 1. The water flow rates *uw* are determined as well as the depth *h* from the solution of the Saint-Venant equation.

Stage 2. Determination of the initial conditions. The granulometric composition of the bottom sediments in the section *X = Xj* is taken to be (*d*<sup>i</sup> , *a0 i*day,*j* ), where *di* is the diameter of the *i*th fraction particle (mm), *a0 i*day,*j* is the percentage of the *i*-th fraction in the bottom sediments, *i* = 1, 2, …, *n*.

Stage 3. Establishment of the boundary condition in the initial section (*X = X*<sup>0</sup> ). In the initial section, *Sn i*day,0 are determined using relations employed for the second stage, *Sn i*,0 are estimated using the field data.

Stage 4. Estimation of the mass exchange between the bottom water and water flows. From the condition *wgi ≤ w\** , *w\** = 0.4*u*\* one determines the fractions which are suspended. Let the suspended fractions be assigned the following index *i =* 1, 2,…,*i \** , *ai,j* is the percentage of the suspended fractions in the section. The percentage of all the suspended fractions is *rj = a*1,day,*<sup>j</sup> + a*2,day,*<sup>j</sup> +…+ ai*,day,*<sup>j</sup>* . Then, the percentage of the suspended *i*th fraction is

$$a\_{ij} = 100 \cdot r\_{\mid}^{-1} \cdot a\_{i, \text{du}, \text{f}} \text{ i = 1, 2, \dots, i} \tag{6}$$

If *rj* = 0 (the suspended fractions are absent), then, all *ai,j* = 0.

Stage 5. Estimation of the concentrations of the suspended and bottom sediments as well as the location of the water-bottom interface.

Stage 6. Calculation of the granulometric composition of the bottom sediment:

$$a\_{i, \partial M\_{\theta}} = S^{v+1} \cdot \rho^{-1} \cdot 100 \tag{7}$$

Stage 7. Estimation of the bottom sediment radioactive contamination in the calculation sections.

Each fraction is assumed to be uniformly contaminated by radionuclides:

$$\mathcal{R}^{\iota}{}\_{\iota\iota} = \lambda\_{\iota} \mathcal{S}^{\iota}{}\_{\iota\iota} \tag{8}$$

Knowing the contamination level in the initial section *Rn <sup>i</sup>*,0 *= λ<sup>i</sup> Sn <sup>i</sup>*,0, it is possible to estimate the level of the radionuclide contamination in the sections downstream the river

$$\mathcal{R}^{\
u}\_{\
u\phi} = \mathcal{S}^{\
u}\_{\
u\phi} \cdot (\mathcal{S}^{\
u}\_{\
u\phi})^{-1} \cdot \mathcal{R}^{\
u}\_{\
u\phi} \tag{9}$$

In the next time interval, the calculations are repeated (from stage 3 to stage 7).

The influence of the suspension-sedimentation processes on the admixture transport in the river flow close to the right bank of the River Yenisei in the studied area has been estimated.

The calculations made show that the concentrations of the lightest fraction in the calculation area almost do not change, while for the heavier fractions the decline of the suspended sediment concentrations is observed and the level of the radionuclide contamination also decreases (**Table 3**).

In the field data, the increase of the coarse fraction concentration is observed which is not connected with the suspension-sedimentation process. (*S\_nat, R\_nat* are the measured values, *S\_calc, R\_calc* are the calculated ones)


**Table 3.** Concentrations of particulate matter size fractions: real and calculated data.

Thus, the abiogenic mass-transport of the technogenic radionuclides, metals being among them, occurs mainly due to the coagulation of the suspended particles and contamination redistribution into bigger fragments.

Our calculations show that the concentration of the lightest fraction of the water on the current site remains virtually unchanged. However, we observed that concentrations of suspended sediment had decreased for heavier fractions and, consequently, decreased the level of contamination. In addition, our field data indicated an increase in the concentration of coarse fraction, which is associated not only with the resuspension-deposition, but also with the coagulation of suspended solids.

Thus, the data on artificial radionuclides entering the Yenisei River water obtained by longterm monitoring, which is likely to be connected with the activity of the industrial enterprises located on the river's banks of the studied area.
