ISSN 00405795, Theoretical Foundations of Chemical Engineering, 2013, Vol. 47, No. 6, pp. 702–708. © Pleiades Publishing, Ltd., 2013. Original Russian Text © O.B. Butusov, V.P. Meshalkin, D.V. Popov, D.A. Tyukaev, 2013, published in Teoreticheskie Osnovy Khimicheskoi Tekhnologii, 2013, Vol. 47, No. 6, pp. 639–645.
ComputerAided Simulation of Radioactive Pollution of Environment upon Destruction of Geologic Repositories of Radioactive Wastes with Allowance for Uncertainty O. B. Butusova, V. P. Meshalkinb, D. V. Popova, and D. A. Tyukaevc a
Moscow State University of Mechanical Engineering, ul. Bol’shaya Semenovskaya 38, Moscow, 107023 Russia b Mendeleev Russian University of Chemical Technology, Miusskaya pl. 9, Moscow, 125047 Russia c Rosenergoatom Concern, Ferganskaya ul. 25, Moscow, 109507 Russia email: butusov
[email protected] Received October 1, 2012
Abstract—Some mathematical and computer models of the propagation of radioactive pollutants in the envi ronment upon the destruction of geologic repositories of radioactive wastes have been developed with allow ance for the interval uncertainty of some physicochemical parameters of the propagation of radioactive pol lutants in the environment (the velocity of the rise of underground water from radioactive wastes and the dif fusion coefficients of radioactive pollutants in a geologic horizon). Computeraided simulation results show that the halflife of radioactive wastes, the rate of the leak of radioactive pollutants from a repository, and the velocity of the advection of radioactive pollutants towards the ground are the principal characteristics of the propagation of radioactive pollutants in the environment. These characteristics of the propagation of radio active pollutants in the environment must be taken into account in the design of safe geologic repositories of radioactive wastes. As a result of the computeraided simulation of the propagation of radioactive pollutants, it has been established that the minimization of the radioactive pollution dose requires that the hermeticity periods of geologic repositories exceed or be nearly equal to the halflives of radioactive pollutants. DOI: 10.1134/S0040579513060031
INTRODUCTION The problem of providing the safety of geologic repositories of radioactive wastes is currently of topical importance due to the enhancement of nuclear power capacities, which leads to the continuous growth and accumulation of the amount of radioactive wastes. Geologic repositories of radioactive wastes provide their safe disposal, as well as prevent the withdrawal of radioactive pollutants and the possibility that they will propagate into the environment. The importance of both the medium and longterm monitoring of geo logic repositories of radioactive wastes is noted in the review of the Organization for Economic Cooperation of Nuclear Energy Agency (OECD NEA) [1]. The destructions of geologic repositories of radio active wastes are caused by the formation of faults, cracks, or crushed zones in rock layers with the result ing change of the insulating, sorption, and thermo physical properties of different rock blocks, in which repositories are located. Hereinafter, we shall use the term “dehermetization” for the generalized notion “the destruction of a geologic repository of radioactive waste.” The destruction or dehermetization of radioactive waste repositories is produced in many respects by the
generation of a great deal of heat as a result of radioac tive reactions in repositories. There is no doubt that the generation of heat in radioactive wastes has an effect on the hermeticity period of a repository [1]. Depending on the heatgenerating capacity of radio active wastes, three types of radioactive waste reposito ries are considered [1], namely, highlevel waste (HLW) repositories for the deep geologic burial of radioactive wastes with a heat generation capacity of more than 2 kW/m3, low and intermediatelevel waste (LILW) repositories for the intermediate geo logic burial of radioactive wastes with a heat genera tion capacity of less than 2 kW/m3, and low and inter mediatelevel waste surfacelevel (LILWSL) reposi tories for the shallow or superficial burial of radioactive wastes with a heat generation capacity of lower than 2 kW/m3 and a short halflife. From the viewpoint of safety provision, the reposi tories of radioactive wastes in the geologic horizons of rock salt are most promising [2] due to the absence of underground water in these monolithic horizons. We should note that many geologic repositories of radio active wastes are in continuous contact with under ground water [3], which increases the risk of their per colation onto the ground together with underground water. In addition to rock salt, the major geologic hori
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zons for the burial of radioactive wastes are massive rocks and clay, the physicochemical properties of which must be taken into account in the analysis and mathematical simulation of the propagation of radio active wastes together with underground water onto the ground and into the biosphere [3]. The objective uncertainty of some physicochemical and geometric parameters of radioactive waste repositories has a con siderable effect on the results of the mathematical and computeraided simulation of the processes of the propagation of radioactive pollutants into the bio sphere, which is the principal reason for the low ade quacy of the models of the propagation of radioactive pollutants from radioactive waste repositories into the environment [1]. When radioactive wastes are buried into massive rocks, it is necessary to take into account the structural heterogeneity of the composition and the discreteness of the physicochemical characteristics of geologic rock materials, whose tectonic motion may lead to the destruction (or, in a general sense, deher metization) of radioactive waste repositories [4, 5]. MATHEMATICAL MODEL OF THE PROPAGATION OF RADIOACTIVE POLLUTANTS INTO THE ENVIRONMENT Let us consider the complex mathematical model proposed in [3] for the propagation of radioactive pol lutants from HLW repositories of deep geologic burial into the biosphere upon the dehermetization of a repository and incorporates the three integrated blocks for (1) the leak of radioactive wastes from a repository, (2) the propagation of radioactive wastes onto the ground together with underground water, and (3) the propagation of radioactive wastes over the bio sphere. The scheme of the regions of the propagation of radioactive pollutants into the environment upon the dehermetization of a repository is shown in Fig. 1. Let us consider the form of equations for each block of this complex mathematical model. (1) The equation of the mathematical model of the nuclear decay of radioactive wastes during the period of the geophysical integrity or hermeticity of a geo logic repository has the following form:
dM = −λ M , t ≤ T 1 dt1 M ( 0 ) = M 0.
(1)
The analytical solution of Eq. (1) is
M (t1 ) = M 0 exp ( −λ t1 ) , t1 ≤ T .
(2)
703
x
Biosphere x2 second horizon
x1
first horizon s 0
Geologic radioactive waste repository with high heat generation Fig. 1. Simplified scheme of regions of propagation of radioactive pollutant upon destruction of geologic radio active waste repository.
(2) The equation of the mathematical model of nuclear decay after the destruction or dehermetization of a repository has the following form: dM = −λ M − kM , t > 0 (3) dt M ( 0) = M 0 exp ( −λ T ) . The analytical solution of Eq. (3) is (4) M (t ) = M 0 exp ( −λ (t + T ) − kt ) , t > 0. According to [3], the parameters T and k should be considered as uncertain interval numbers. Hence, M(t) is also an interval function. Let us denote fuzzy interval numbers with the upper symbol tilde, interval boundaries with underline and overline symbols, and an interval with brackets. In this case, M (t ) = [M (t ), M (t )], where the underline and overline symbols denote the lower and upper interval boundaries, respectively, which are calculated by the following for mula [6]:
M (t) = [M 0 exp ( −λ (t + T ) − kt ) ,
(5) M 0 exp ( −λ (t + T ) − kt )]. (3) The equation of the mathematical model of the propagation of radioactive pollutants over the first horizon of a geologic repository of radioactive wastes. The propagation of radioactive pollutants from a repository to the ground occurs at the expense of the motion of underground water mainly due to the two processes, such as translational upward motion and diffusion. In the first approximation, let us accept
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constant values for the parameters of these processes within each of the two geologic horizons. In [3], it is proposed to use the following equation for the propa gation of radioactive pollutants: (1) ∂ C
(1) ∂ C
(1)
(1)
(1) ∂
C (1) = −λ R (1)C (1). (6) R +V −d V ∂t ∂x ∂x 2 Dividing Eq. (6) by R(1), we reduce this equation to the normalized form (1)
2
∂C (1) + V (1) ∂C (1) − D (1) ∂ 2C (1) = −λ C (1). (7) N N ∂t ∂x ∂x 2 To perform numerical calculations, we used the parameters proposed in [3] for the prediction of the propagation of a radioactive pollutant, i.e., the I129 iso tope, from a repository. Most of the parameters are uncertain interval numbers, for example, V (1) = ⎡⎣V (1),V (1) ⎤⎦ = [10–3, 10–1], R (1) = ⎡⎣R (1), R (1) ⎤⎦ = [1, 5]. The hydrodynamic models with allowance for uncertainty differ from the models of deterministic hydrodynam ics, for example, from the models considered in [7–9], by that the result of simulation can be expressed as intervals with crisp upper and lower boundaries. In this case, it is necessary to take into account that the size of an interval continuously increases [10]. Let us write the interval forms for the following physicochemical model parameters: the normalized velocity and the normalized diffusion coefficient [6]
(1) (1) VN = V (1) = R
(1) (1) (1) (1) (1) (1) (1) (1) = ⎡⎣min (V R ,V R ,V R ,V R ) ,
max (V
(1)
(1)
(1)
(1)
(1)
(1)
R ,V R ,V R ,V = [0.0002,0.1]
(1)
R
(1)
)⎤⎦
(8)
(1) (1) D N(1) = d V(1) = d (1)[0.0002,0.1] R = 10 × [0.0002,0.1] = [0.002,1].
Fuzzy interval parameters and fuzzy boundary conditions lead to the formulation of a fuzzy boundary problem for the first geologic horizon in the following form:
∂C (1) + V (1) ∂C (1) − D (1) ∂ 2C (1) = −λC (1), N N ∂t ∂x ∂x 2 C (1) ( x,0) = 0, C (1) ( 0, t ) = [M 0 exp ( −λ (t + T ) − kt ) , (9) M 0 exp ( −λ (t + T ) − kt )],
∂C (1) ( L1, t )
= 0.
∂x The secondkind boundary condition (steadystate regime) was accepted in boundary problem (9) at the upper boundary x = L1 of the first horizon.
To simplify boundary problem (9), let us introduce the substitution of variables (10) C (1) ( x, t ) = B (1)(x, t )exp(−λ t ). After the substitution of variables, boundary problem (9) takes the following form: (1) (1) 2 (1) ∂B = −V (1) ∂B + D (1) ∂ B , N N ∂t ∂x ∂x 2 B (1) ( x,0) = 0, (11) B (1) ( 0, t ) = ⎡⎣M 0 exp ( −λT − kt ) , M 0 exp ( −λT − kt )⎤⎦ , ∂B (1) ( L1, t ) = 0. ∂x Boundary problem (11) contains four fuzzy num bers (N = 4), namely VN(1), D N(1), M (t ), L1 , and must be solved for various combinations of the lower and upper boundaries of fuzzy numbers, i.e., for 2N different vari ants. Then, the boundaries of the interval solution B (1)(x, t ) are determined from the condition of mini mum and maximum. (4) The mathematical model of the propagation of a radioactive pollutant over the second geologic horizon. The mathematical model of the propagation of radioactive pollutants over the second geologic hori zon differs from the mathematical model of the prop agation of radioactive pollutants over the first horizon only by the boundary condition at the lower boundary, which is determined from the solution obtained for the first horizon. Hence, the mathematical model of the propagation of a radioactive pollutant over the second geologic horizon has the following form: ∂B (2) = −V (2) ∂B (2) + D (2) ∂ 2B (2) , N N ∂t ∂x ∂x 2 (12) B (2) ( x,0) = 0, B (2) ( 0, t ) = B (1)(L1, t), ∂B (2) ( L2, t ) = 0. ∂x The first and second boundary problems are solved on the intervals x ∈ [0, L1] and x ∈ [0, L2]., respectively. The total time beginning from the moment of the dehermetization of a radioactive waste repository is used to solve the boundary problems. (5) The mathematical model of the propagation of radioactive pollutants over the biosphere. The mentioned mathematical model allows us to calculate the radiation dose acquired by the popula tion for the time t as t
∫
D z = βσ C
(2)
( x 2, t ) dt.
(13)
0
In the International System of Units, the equivalent dose is measured in sieverts (Sv).
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COMPUTER MODEL OF PROPAGATION OF RADIOACTIVE POLLUTANT The computer model allows for the partial solution of the system of two boundary problems as follows:
∂B = −α ∂B + μ ∂ B , 1 1 2 ∂t ∂x ∂x B (1) ( x,0) = 0, B (1) ( 0, t ) = M 0 exp ( −λ T − kt ) , (1)
∂B
(1)
( L , t ) = 0,
(1)
2
(1)
(1)
∂x
α1 = V
(1)
R , μ1 = V d (1)
(1) (1)
R
(1)
∂B (2) = −α ∂B (2) + μ ∂ 2B (2) , 2 2 ∂t ∂x ∂x 2 (2) (2) (1) (1) B ( x,0) = 0, B ( 0, t ) = B L , t ,
(
∂B
(2)
( L , t ) = 0,
(14)
)
(2)
α 2 = V (2) R (2) , μ 2 = V (2)d (2) R (2) .
∂x Parameter values are selected at the middle of cor responding uncertainty intervals. The first and second boundary problems are solved on the intervals x ∈ [0, L(1)] and x ∈ [0, L(2)], respec tively, where L(1) and L(2) are crisp numbers at the boundaries of an uncertainty interval. To obtain a numerical solution of equation system (14), we shall use the Crank–Nicolson and Thomas algorithms [11]. The Crank–Nicolson algorithm leads to the following finitedifference equation for the pth geologic horizon: −DBk( p+)1, j +1 + (1 + 2D ) Bk( p+)1, j − DBk( p+)1, j −1 = Bk(,pj)
(
)
(
)
− Ku Bk(,pj) − Bk(,pj)−1 + D Bk(,pj)+1 + Bk(,pj)−1 − 2Bk(,pj) .
(15)
Equation (15) was solved by the sweep method using the Thomas algorithm and the MatLab software suit. RESULTS OF COMPUTER ANALYSIS OF PROPAGATION OF RADIOACTIVE POLLUTANT The results of the computer analysis of the motion of a pollution wave from a dehermetized repository to the ground for the I129 isotope, a radioactive pollutant with a radioactive decay constant λ = 4.41 × 10–8 yr–1 are illustrated in Fig. 2. The overall pollution shown in Fig. 2 is the sum of the pollutions in the first (0– 200 m) and second (200–300 m) geologic horizons. The velocity of the rise of a radioactive pollutant to the ground generally depends on the advection velocities in horizons and the rate of the leak of a radioactive pollutant from a repository. The advection velocity in the first and second horizons is V (1) = 0.05 m/yr and V (2) = 0.1 m/yr, respectively. At these velocities, the radioactive pollution wavefront passage time is nearly
705
4000 years for the first horizon and 1000 year for the second horizon (see Fig. 2a). After a repository is dehermetized, radioactive pol lution begins to move towards the ground together with underground water. The concentration distribu tion of a radioactive pollutant for different time peri ods is illustrated in Fig. 2. As follows from Fig. 2a, the concentration maximum is initially located near a repository and then begins to uniformly propagate over the entire depth of a geologic burial of radioactive wastes. The distribution of a radioactive pollutant for a stable steadystate regime is illustrated in Fig. 2b (the shapes of the plots in Fig. 2b are geometrically simi lar). The radioactive pollutant distribution maximum is located at the boundary between the geologic hori zons. This is due to the fact that the advection velocity of a radioactive pollutant in the second geologic hori zon is higher, which leads to the fast withdrawal of a radioactive pollutant from the horizon interface. A gradual decrease of maximum is due to both the tran sition to a more uniform distribution and the flow of a radioactive pollutant onto the ground. The time of the leak of a radioactive pollutant from a repository can be estimated by dividing the initial amount of radioactive wastes (M0 ≈ 100 mol) by the leak rate (k ≈ 0.01 yr–1) to obtain 10000 years. The obtained estimate corresponds to the characteristic points of the radioactive pollutant distribution plots shown in Fig. 2b. The time dependence of the equivalent dose per person per year after the dehermetization of a geologic repository containing a radioactive pollutant, i.e., the I129 isotope, is shown in Fig. 3. The maximum value is attained at t ≈ 2000–3000 years, which is in good agreement with the aboveobtained estimate of the time of the flow of a radioactive pollutant onto the ground t = 5000 years. It follows from Fig. 3 that the average I129 isotope dose gradually grows after dehermetization, which increases by many times and only begins to decrease after the predominant fraction of a radioactive pollut ant decays. The important parameters of any geologic reposi tory of radioactive wastes are its hermeticity period and the decay constant of a radioactive pollutant. Let us consider the results of the computer analysis of the dependence of the equivalent radiation dose for person per year on these parameters (see Fig. 4). Let us select the range of repository hermeticity periods from the interval of 10–105 years and the range of radioac tive decay constants from the interval of 10–105 yr–1. A more illustrative picture of the distribution of the equivalent dose of a radioactive pollutant can be obtained from the twodimensional sections of the threedimensional distribution shown in Fig. 4. The
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(а) 1 2 3 4
1.6
1.2
0.8
0.4
0 50
100
C × 104, mol/m3 3.5 3.0 2.5
150
200
250
300
(b) 1 2 3 4
2.0 1.5 1.0 0.5 0 50
100
150
200
250
300 x, m
Fig. 2. Propagation of a radioactive pollution wave from a repository to the ground for the I129 radioactive isotope and (a) periods t of (1) 1000, (2) 2000, (3) 3000, and (4) 4000 years and (b) periods t of (1) 10000, (2) 11000, (3) 12000, and (4) 13000 years.
result is illustrated in Fig. 5 as the dependence of the equivalent I129 isotope radiation dose on the hermetic ity period of a repository. It follows from Fig. 5 that the dose for the I129 isotope with a long halflife on the interval of nearly 10000 years depends slightly on the time and only begins to decrease for the repository hermeticity periods comparable with the halflife of the radioactive pollutant. Let us consider the similar plots obtained at differ ent radioactive decay constants (see Fig. 6). The halflife of a radioactive pollutant is calculated by the formula T = 1/λ. The plots have the shape of logistic functions. The plots of doses are slightly trans
formed at both short and long repository hermeticity periods. The middle region of the plots corresponds to the interval, on which the hermeticity period is nearly equal to the halflife of the radioactive pollutant. Hence, the hermeticity period of a repository must exceed or be nearly equal to the halflife of a radioac tive pollutant to minimize the dose. CONCLUSIONS In the present work, the computeraided simula tion of the propagation of the I129 isotope, a radioactive pollutant with a long halflife, into the environment upon the dehermetization of a radioactive waste
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707
4.5 4 3.5
10–10
3 2.5
10–14
2
07 0 1e 2 1 2.1
2 4.2 6.3 243 .1121e e0 00 36 07 4e 7 0 07 6.3 4. 2 8.4 486 364 243 e0 e e 07 00 00 7 7
10–6
8.44 86e 007
log λ [yr–1] 5
Dz, Sv/yr 10–4
07 0 6e 8 4 8.4
1.5 1
10–18 102
3
4
10
10
10 t, years
5
Fig. 3. Change in average equivalent I129 isotope dose per person per year after dehermetization of radioactive waste repository.
1
2
1.5
3
2.5
3.5
4 4.5 5 log T [years]
Fig. 4. Distribution of average equivalent radiation dose per person per year at different combinations of repository hermeticity periods and radioactive pollutant decay con stants.
Dz × 106, Sv/yr Dz × 107, Sv/yr 1.4
1.0562
1.2 1.0
1.0558
0.8 0.6 1.0554
3
2
1
4
5
0.4 0.2
1.0550 101
102
103
104
105 T, years
Fig. 5. Average equivalent I129 isotope dose versus reposi tory hermeticity period.
repository has been performed with allowance for the uncertainty of simulation parameters. The results of computeraided analysis show that the radiation dose strongly depends on the halflife of a radioactive pol lutant, the rate of the leak of a radioactive pollutant from a repository, and the velocity of the advection of a radioactive pollutant towards the ground due to the hydrodynamics of underground water. The radioactive pollutant leak rate and the advec tion velocity are important characteristics that depend
0 100
101
102
103
104 105 T, years
Fig. 6. Average equivalent dose versus repository hermetic ity period at a radioactive pollutant decay constant λ of (1) 10–1, (2) 10–2, (3) 10–3, (4) 10–4, and (5) 10–5.
on a number of uncertain parameters. For this reason, it is necessary to use more adequate mathematical models to estimate these characteristics of a radioac tive pollutant with allowance for the uncertainty of the interval in the design of radioactive waste repositories. When geological repositories of radioactive wastes with a long halflife are dehermetized, the bulk of a radioactive pollutant reaches the ground and contam inates the biosphere even in the case of ultimately low advection velocities. The results of the computer
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aided simulation of the propagation of a radioactive pollutant upon the dehermetization of geologic repos itories show that the hermeticity period of a repository must exceed or be nearly equal to the halflife of a radioactive pollutant to minimize the dose of a radio active pollutant in the biosphere. NOTATION B (p)—reduced concentration of a radioactive pollut ant in the pthgeologic horizon; C—summary concentration of a radioactive pollutant in the horizons, mol/m3;
β—coefficient of the conversion of the I129 concentra tion into the equivalent radiation dose acquired by the population; λ—decay constant of a radioactive pollutant, yr–1; σ—share of water consumed by the population; Ku = αΔ t —Courant–Friedrichs–Lewy parameter. Δx SUBSCRIPTS AND SUPERSCRIPTS j—spatial node of a calculation grid; k—time iteration.
C(p)—concentration of a radioactive pollutant in the pthgeologic horizon (p = 1, 2), mol/m3; D=
μΔ t —grid dimensionless diffusion coefficient; 2Δ x 2
( p) ( p) D N( p) = d V( p) —normalized diffusion coefficient in R the pthgeologic horizon;
Dz—radiation dose acquired by the population, Sv/yr;
REFERENCES 1. Advanced Nuclear Fuel Cycles and Radioactive Waste Management, NEA no. 5990, London: OECD NEA, 2006. 2. Dlouhy, Z., Disposal of Radioactive Wastes, Studies in Environmental Science, vol. 15, Amsterdam: Elsevier, 1982.
d (p)V (p)—diffusion coefficient in the pthgeologic horizon;
3. Uncertainty in Industrial Practice: A Guide to Quantita tive Uncertainty Management, de Rocquigny, E., Devic tor, N., and Tarantola, S., Eds., New York: Wiley, 2008.
k—rate of the leak of a radioactive pollutant from a repository, yr–1;
4. Pusch, R., Geological Storage of Highly Radioactive Waste, Berlin: Springer, 2008.
Lp—upper boundary of the pth horizon; M—amount of radioactive wastes in a repository, mol;
5. Stability and Buffering Capacity of the Geosphere for LongTerm Isolation of Radioactive Waste, NEA no. 5303, London: OECD NEA, 2004.
M0—initial amount of radioactive wastes in a reposi tory, mol;
6. Alefeld, G. and Herzberger, J., Introduction to Interval Computations, New York: Academic, 1983.
R(p)—time scale in the pthgeologic horizon;
7. Butusov, O.B. and Meshalkin, V.P., Texture and fractal methods for analyzing the characteristics of unsteady gas flows in pipelines, Theor. Found. Chem. Eng., 2006,vol. 40, p. 291.
T—repository hermeticity period, years; t = t1 – T—time after the dehermetization of a repos itory, years; t1—duration of the residence of radioactive wastes in a repository, years; Δt—calculation grid step, years; V(p)—rise velocity of underground water in the pth geologic horizon, m/yr; ( p)
V N( p) = V ( p) —normalized rise velocity of underground R water in the pthgeologic horizon, m/yr; x—distance from a repository, m; Δx—calculation grid step, m;
8. Butusov, O.B. and Meshalkin, V.P., Computer simula tion of transient gas flows in complex round pipes, Theor. Found. Chem. Eng., 2008, vol. 42, p. 85. 9. Butusov, O.B. and Meshalkin, V.P., Computation of the integral parameters of turbulent structures for the tran sient gas flows in pipes using wavelet transforms, Theor. Found. Chem. Eng., 2008, vol. 42, p. 160. 10. Datta, D., Nonprobabilistic uncertainty analysis of analytical and numerical solution of heat conduction, Int. J. Energy Inf. Commun., 2011, vol. 2, no. 2, p. 143. 11. Fletcher, C.A.J., Computational Techniques in Fluid Dynamics, vol. 2: Specific Techniques for Different Flow Categories, Berlin: Springer, 1988. Translated by E. Glushachenkova
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