- Email: [email protected]

Modeling the eﬀects of the engineered barriers of a radioactive waste repository by Monte Carlo simulation Marzio Marseguerraa,*, Enrico Zioa, Edoardo Patellia, Francesca Giacobboa, Piero Risolutib, Giancarlo Venturab, Giorgio Mingroneb a

Department of Nuclear Engineering, Polytechnic of Milan, Via Ponzio 34/3, 20133 Milan, Italy ENEA (Italian Agency for Energy and Environment), Task Force Sito, Via Anguillarese 301 - 00060 Rome, Italy

b

Received 27 April 2002; accepted 21 June 2002

Abstract In the current conception of some permanent repositories for radioactive wastes, these are trapped, after proper conditioning, in cement matrices within special drums. These drums, in turn, are placed in a concrete container called a ‘‘module’’, in which the space between the drums is back-ﬁlled with grout. Finally, several modules are stacked within the concrete walls of the repositories. Through this multiple barrier design, typical of the nuclear industry, the disposal facility is expected to ensure adequate protection of man and environment against the radiological impacts of the wastes by meeting various functional objectives which aim at limiting the release of radionuclides. Because one of the principal mechanisms of release of radionuclides is through water inﬁltration into the various constituents of the repository and subsequent percolation into the groundwater system, it is of utmost importance to study the phenomena of advection and dispersion of radionuclides in the artiﬁcial porous matrices hosting the waste (near ﬁeld) and, subsequently, in the natural rock matrix of the host geosphere (far ﬁeld). This paper addresses the issue of radionuclide transport through the artiﬁcial porous matrices constituting the engineered barriers of the repository’s near ﬁeld. The complexity of the phenomena involved, augmented by the heterogeneity and stochasticity of the media in which transport occurs, renders classical analytical-numerical approaches scarcely adequate for realistic representation of the system of interest. Hence, we propound the use of a Monte Carlo simulation method based on the Kolmogorov and Dmitriev theory of branching stochastic processes. # 2002 Elsevier Science Ltd. All rights reserved. * Corresponding author. Fax: +39-02-2399-6309. E-mail address: [email protected] (M. Marseguerra). 0306-4549/03/$ - see front matter # 2002 Elsevier Science Ltd. All rights reserved. PII: S0306-4549(02)00072-5

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1. Introduction Nuclear activities for power generation have been conducted in Italy for nearly thirty years, from the early 1960s to the end of the 1980s. These activities, together with other non-energy related applications of ionising radiation and radionuclides in the industrial and medical ﬁelds, have led to the production of a non-negligible amount of radioactive wastes. Considerably larger quantities of radioactive wastes will be produced during the decommissioning of the Italian, Ente Nazionale energia Elettrica ENEL, power stations and of the Italian agency for energy and environment, Ente Nazionale per le nuove tecnologie, l’Energia e l’Ambiente ENEA, fuel cycle plants. The waste thus far produced is currently stored at its site of origin, waiting to be transferred to a ﬁnal disposal facility when this shall come into operation. This disposal facility must be designed so as to guarantee isolation of the radionuclides from the biosphere for the entire period during which they remain of potential radiological signiﬁcance. This isolation is achieved by means of a multiple barrier system designed to limit the release and transport of radionuclides into the environment (PAGIS, 1998). The long time scales of potential radiological signiﬁcance associated with radioactive wastes require that the assessment of the isolation performance of the disposal site and facility be obtained by applying models which simulate the migration of radionuclides from the disposal facility to the biosphere through the various artiﬁcial and natural barriers. The results of the simulations are used as an input to the design of the disposal system to ensure that the radiological impact of the disposal system meets with the safety criteria issued by the national and international regulatory agencies (ICRP, 1977; G.T.26, 1997; Savage, 1995). The main vector for the transport of radionuclides through the barriers of the repository is water, which may percolate into the system and advectively transport dissolved radioactive material to and in groundwater. Diﬀerent approaches have been proposed for, and applied to, water-driven contaminant transport, such as the use of classical advection–dispersion theory (Freeze and Cherry, 1979; Bear, 1972) and its extension to the theory of stochastic transport in random ﬁelds (Dagan, 1989), random walk theory, the transport theory inherited from nuclear reactor physics (Williams, 1992) and Monte Carlo simulation techniques (Marseguerra et al., 1998; Marseguerra et al., 2001a,b). In this paper, we apply a stochastic approach based on the Kolmogorov–Dmitriev theory of branching stochastic processes (Kolmogorv and Dmitriev, 1947) to the problem of modelling radionuclide transport through diﬀerent engineered barriers. The corresponding model is evaluated by Monte Carlo simulation, where a large number of particles of solute are followed in their travel through the barriers, using appropriate probability distribution functions to characterise their transport. Diﬀerent kinds of particles are introduced to represent the contaminant in the various possible locations and physico-chemical states, and the processes occurring during transport are represented as transitions from one particle state to another, with given rates of occurrence. The main advantage of the proposed model is its ﬂexible structure. This allows one to consider

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multidimensional geometries and to describe a wide range of phenomena, by accounting for the individual interactions which each particle may undergo during its transport. In the next section, we describe the main characteristics of the underlying stochastic model, developed by some of the present authors (Marseguerra and Zio, 1997), to simulate contaminant transport through porous media. The estimation of model parameters is addressed by means of a comparison with the advection–dispersion formulation in ﬁnite-diﬀerence form. In the following section, the model is veriﬁed with respect to a literature case regarding the transport of a radioactive chain of three radionuclides (Lee and Lee, 1995). In Section 4, the model is applied to the evaluation of the performance of the transport through the engineered barriers of the proposed future ENEA waste repository (ENEA, 2000).

2. The stochastic model We consider the transport of contaminants through a saturated porous medium, which in principle can be one, two or three dimensional. The space domain in which the transport process occurs is subdivided in Nz zones, z=1, 2, . . ., Nz. Our objective is to determine the distributions of contaminant particles in time and space. We introduce the following two categories of particles: ‘solutons’, which are free particles of contaminant mobile within the water ﬂowing through the matrix pores; ‘trappons’, which are immobile particles of contaminant adsorbed on the solid matrix. We use Sp(z,t) and Tp(z,t) to denote a particle of contaminant of type p residing in zone z at time t, for solutons and trappons, respectively. In the illustration of the model, we refer, for simplicity, to a 1-D space domain and we describe the possible transitions that each p-contaminant particle may undergo, within an inﬁnitesimal interval of time dt, during the transport process. The soluton is free to move to other zones of the medium: for simplicity, we consider only transfers backward and forward to adjacent zones, occurring with rates (z!z1,t) and (z!z+1,t), respectively (Fig. 2.1). The soluton Sp (z,t) of type p contaminant may also be subject, with transition rate ads(p,z,t), to reversible adsorption on the solid matrix of the host medium, thus transforming into a trappon Tp(z,t). Moreover, the soluton Sp(z,t) may, due to chemical reactions, transform into a diﬀerent particle p00 with rate gp!p00 . Finally, if the contaminant particle under consideration is a radionuclide, it may transform into a diﬀerent p0 -contaminant according to its characteristic decay rate lp!p0 [Fig. 2.2 (a)].

Fig. 2.1. Schematics of the soluton particles transport.

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The trappon may be desorbed from the host matrix, with rate des(p,z,t), becoming a soluton, or, analogously to the soluton, it may transform into a diﬀerent particle due to chemical reactions or radioactive decay with rates p!p00 and lp! p0 , respectively [Fig. 2.2(b)]. Table 2.1 summarizes the various transitions that the particles may undergo during their transport, and the corresponding rates. The processes above described may be formalized within the Kolmogorov and Dmitriev theory of branching stochastic processes (Kolmogorv and Dmitriev, 1947). Considering the Green’s functions for all the particles involved, by working in the domain of the probability generating functions (pgfs) and then exploiting their properties we arrive at the following system of 2.p.Nz partial diﬀerential equations for the expected number of solutons and trappons particles where we denote by NSp ðz; tÞ, p=1,2, . . .,np, z=1,2, . . ., Nz, the expected number of soluton particle Sp of p kind in zone z at time t, by NTp ðz; tÞ the expected number of trappon particle Tp of p kind in zone z at time t:

Fig. 2.2. (a) Transformations of a soluton. (b) Transformations of a trappon.

Table 2.1 Particles transitions and corresponding rates Particle

Transitions

Rate

Sp

Forward transfer to adjacent zone (z!z+1) Backward transfer to adjacent zone (z!z1) Adsorption on the porous host matrix Chemical transformation Radioactive decay

(z!z+1,t) (z!z1,t) ads(p,z,t) p!p00 lp! p0

Tp

Desorption from the porous host matrix Chemical transformation Radioactive decay

des(p,z,t) p!p00 lp!p0

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@NSp ðz; tÞ ¼ ½ðz ! z þ 1; tÞ þ ðz ! z 1; tÞ NSp ðz; tÞ @t þ ðz þ 1 ! z; tÞNSp ðz þ 1; tÞ þ ðz 1 ! z; tÞNSp ðz 1; tÞ X adsðp; z; tÞNSp ðz; tÞ þ desðp; z; tÞNTp ðz; tÞ NSp ðz; tÞ lp ! p0 X X NSp ðz; tÞ p ! p00 þ lp0 ! p NSp0 ðz; tÞ p00

X þ p00 ! p NSp00 ðz; tÞ

p0

p0

ð1Þ

p00

X @NTp ðz; tÞ ¼ þadsðp; z; tÞNSp ðz; tÞ desðp; z; tÞNTp ðz; tÞ NTp ðz; tÞ lp ! p0 @t p0 X X NTp ðz; tÞ p ! p00 þ lp0 ! p NTp0 ðz; tÞ p00

X p00 ! p NTp00 ðz; tÞ þ

p0

ð2Þ

p00

These are balance equations describing the production and loss processes of particles in zone z. The ﬁrst two terms on the right hand side of Eq. (1) describe the disappearance of solutons Sp due to the transfer to an adjacent zone; the third and fourth terms describe the appearance of solutons Sp in zone z because of transfer from adjacent zones; the ﬁfth term represents the transformation of solutons Sp into trappons Tp due to the adsorption on the host matrix; the sixth term represents the production of solutons Sp due to desorption of trappons Tp from the host matrix; the seventh and eighth terms account for the disappearance of solutons Sp by transformations due to radioactive decay and chemical transformations; the last two terms describe the production of solutons Sp due to decay and chemical transformations of other species of solutons into the p-kind. Similar balance considerations apply in Eq. (2). Of course, these equations must be supplemented with the proper initial conditions. Substitution of Eq. (2) in Eq. (1) yields:

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@ NSp ðz; tÞ þ NTp ðz; tÞ ¼ ½ðz ! z þ 1; tÞ þ ðz ! z 1; tÞ NSp ðz; tÞ @t þ ðz þ 1 ! z; tÞNSp ðz þ 1; tÞ þ ðz 1 ! z; tÞNSp ðz 1; tÞ NSp ðz; tÞ þ NTp ðz; tÞ "

X X lp ! p 0 þ p ! p00 p0

þ

ð3Þ

#

p00

h i X X lp0 ! p NSp0 ðz; tÞ þ NTp0 ðz; tÞ þ p00 ! p p0

p00

h i NSp00 ðz; tÞ þ NTp00 ðz; tÞ

It can be seen that the transport phenomena are described explicitly in probabilistic terms, thus allowing for substantial ﬂexibility. Obviously, the eﬀectiveness of this representation is conditioned by the capability of estimating the values of the speciﬁed transition rates that govern the modelled processes. Finally, the importance of the formulation in terms of probability generating functions (here not reported for the sake of brevity) is that it allows deﬁnition, in a quite straightforward manner, of equations not only for the expected values of particle number, but also for the higher-order moments of the distributions (Kolmogorv and Dmitriev, 1947; Marseguerra and Zio, 1997). 2.1. Parameter determination In order to estimate the values of the various transition rates appearing in the stochastic model of Section 2, we make a comparison between Eqs. (1) and (2) and the corresponding classical advection–dispersion equations, with the aim of establishing a relationship between the transition rates pertaining to our stochastic model and the parameters of the classical model. To illustrate the procedure, we consider the case of a 1-D transport of a nonradioactive p-contaminant through a homogeneous porous medium, in a uniform ﬂow ﬁeld, with no interchange between the solute and the solid phases and no chemical transformations. The corresponding advection–dispersion equation is: X X @Cp ðz; tÞ @2 Cp ðz; tÞ @Cp ðz; tÞ 0 þ ¼D C v ð z; t Þ l lp0 ! p Cp0 ðz; tÞ p p ! p @t @z2 @z p0 p0

ð4Þ

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where Cp(z,t) represents the concentration of mobile p-contaminant particles in zone z at time t; v is the pore velocity and D is the hydrodynamic dispersion coeﬃcient. These two latter parameters are given by the following expressions: v¼

q ne

q ¼ K

ð5Þ @h @z

ð6Þ

D ¼ L v þ Ddiff

ð7Þ

where q is Darcy’s velocity, K the hydraulic conductivity, h the hydraulic head, ne the eﬀective porosity of the medium, L the longitudinal dispersivity coeﬃcient and Ddiﬀ the molecular diﬀusion coeﬃcient (Freeze and Cherry, 1979; Bear, 1972). The phenomena of interchange between the solute and the solid phases are generally represented using the simpliﬁcation of a linear equilibrium isotherm which establishes a proportionality relationship between the fraction of p-contaminant adsorbed on the matrix and the fraction of p-contaminant present in the liquid phase. Then the governing equation of the advection–dispersion model, with no chemical transformation, becomes: X X @Cp ðz; tÞ D @2 Cp ðz; tÞ v @Cp ðz; tÞ ¼ Cp ðz; tÞ lp ! p0 þ lp0 ! p Cp0 2 @t @z @z Rp Rp p0 p0

ð8Þ

ðz; tÞ where the constant Rp, called the ‘‘retardation factor’’, accounts for the delay in the transport of the p-contaminant due to adsorption/desorption processes and is deﬁned as: Rp ¼ 1 þ

b kd n

ð9Þ

where b is the porous media bulk density, kd is the partition coeﬃcient and n the total porosity. By analogy, in our stochastic model we introduce the same linear equilibrium hypothesis in the adsorption/desorption processes involving solutons Sp and trappons Tp: NTp ðz; tÞ ¼ p NSp ðz; tÞ

ð10Þ

where p is the proportionality coeﬃcient for p-contaminant. Substituting in Eq. (3) we obtain:

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@NSp ðz; tÞ ¼ p ðz ! z þ 1; tÞ þ p ðz ! z 1; tÞ NSp ðz; tÞ þ p @t ðz 1 ! z; tÞNSp ðz 1; tÞ þ p ðz þ 1 ! z; tÞNSp "

X X ðz þ 1; tÞ NSp ðz; tÞ lp ! p 0 þ p ! p00 p0

#

p00

X X lp0 ! p NSp0 ðz; tÞ þ p00 ! p NSp00 ðz; tÞ þ p0

ð11Þ

p00

ðz;tÞ is the eﬀective transition rate of transfer to an adjacent zone, ð1þp Þ speciﬁc for a p-contaminant.

where p ðz; tÞ ¼

In order to ﬁnd a relationship between the parameters D, v and Rp of the advection–dispersion model and the transition rates of the stochastic model we re-write Eq. (8) in a central ﬁnite diﬀerence approximation scheme: 0

1 0 1 D v D v B Rp B C Rp C @Cðz; tÞ CCðz þ 1; tÞ þ B Rp þ Rp CCðz 1; tÞ ¼ þB @ A @ 2 2 @t 2z 2zA ðzÞ ðzÞ 0

1 2D X X B Rp C B lp ! p0 C lp0 ! p Cp0 ðz; tÞ @ðzÞ2 þ ACðz; tÞ þ p0 p0

ð12Þ

Comparing term by term with Eq. (11) we obtain the following relationships among the parameters of the two models: D v Rp Rp p ð z ! z 1Þ ¼ ðzÞ2 2z D v Rp Rp p ð z ! z þ 1Þ ¼ þ ðzÞ2 2z

ð13Þ

ð14Þ

Since the transition rates must be positive by deﬁnition, from Eqs. (13) we obtain the following limitation on the spatial discretization step:

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2D max½z ¼ v

ð15Þ

The p and lp rates that govern the chemical and radioactive transformations of a p-contaminant into a diﬀerent contaminant can be estimated directly from the chemical and nuclear characteristics of the p-contaminant.

3. Validation on a literature case In order to verify the stochastic model of transport developed here and the corresponding Monte Carlo code, we have carried out a comparison with the results of a case from the literature in which the 1-D transport through a fractured porous medium of a chain of three radionuclides was analysed by simulating a Markov process, continuous in time (Lee and Lee, 1995). In this literature case, movement of Table 3.1 Transport of a chain of three radionuclides: parameters values in Lee and Lee, (1995) Parameter Initial concentration of nuclide (1), C(1) 0 Initial concentration of nuclide (2), C(2) 0 Initial concentration of nuclide (3), C(3) 0 Retardation coeﬃcient of nuclide (1), R(1) p Retardation coeﬃcient of nuclide (2), R(2) p Retardation coeﬃcient of nuclide (3), R(3) p Distance, L (m) (1) Decay constant of nuclide (1), l (1/y) Decay constant of nuclide (2), l(2) (1/y) Decay constant of nuclide (3), l(3) (1/y) Pore velocity, q/n (m/y) Dispersion coeﬃcient, D (m2/y)

Value in case A

Value in case B 1.0 0.0 0.0 100 100 100 100 1.60 103 4.62 102 1.06 104 10

2.5

25

Table 3.2 Transport of a chain of three radionuclides: rates of the present stochastic model Parameter Decay constant of nuclide (1), l(1) (1/y) Decay constant of nuclide (2), l(2) (1/y) Decay constant of nuclide (3), l(3) (1/y) Delta z (m) Forward rate (1/y) Backward rate (1/y) Final time (y) Number of time channels Number of trials CPU time (Pentium III, 800 Mhz)

Value in case A

Value in case B 1.60 103 4.62 102 1.06 104

0.25 0.6 0.2

1.0 0.3 0.2 2 10+3 500 1 10+6

0 h 9 min 33 s

0 h 9 min 12 s

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the radionuclides through the fractured medium was delayed by sorption processes on the fracture walls. Two cases, here referred as A and B, were considered in (Lee and Lee, 1995), diﬀering only in the values used for the dispersion coeﬃcient.

Fig. 3.1. (a) Case A proﬁles of the concentrations of the radionuclides (1), (2) and (3) at time t=100 year obtained in (Lee and Lee, 1995) and the corresponding analytical solutions. (b) Proﬁles of the concentrations of the three radionuclides obtained with our Monte Carlo code.

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Fig. 3.2. (a) Case B breakthrough curves at 20 m (clear dots) and at 50 m (black dots) obtained in (Lee and Lee, 1995) for radionuclide (1), and corresponding results of our Monte Carlo code (solid line). (b) Breakthrough curves at 20 m (clear triangles) and at 50 m (black triangles) obtained in (Lee and Lee, 1995) for radionuclide (3), and corresponding results of our Monte Carlo code (solid line).

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In Table 3.1, we report the parameter values assumed in (Lee and Lee, 1995). In Table 3.2, we provide the corresponding values of the stochastic model parameters obtained as explained in Section 2.1. By analogy with the initial condition of (Lee and Lee, 1995), in our simulation we assume that at time t=0 y, in zone z=0, there is one particle of the initial radionuclide (1) and zero particles of radionuclides (2) and (3). In Fig. 3.1(a) we report the proﬁles of the concentrations of the radionuclides (1), (2) and (3) at time t=100 y, and the corresponding analytical solutions for case A obtained in (Lee and Lee, 1995). In Fig. 3.1(b) we report the corresponding results obtained with our Monte Carlo code: the agreement is considered satisfactory. Fig. 3.2 compares, for case B and radionuclides (1) and (3), the breakthrough curves (time distributions of the concentrations at a given location) at distances of 20 and 50 m from the source, as obtained with the model described in the literature and with our Monte Carlo code. Again, the agreement is satisfactory. Radionuclide (2) is not reported because its decay leads to negligible concentrations.

4. Application of the model to the ENEA design for a LLW repository In the current conceptualization of the LLW repository under study by ENEA, the wastes are ﬁrstly conditioned and then incorporated within cement matrices in special drums made of steel (ENEA, 2000). The drums are, in turn, placed in a concrete container, called a ‘‘module’’ and here referred to as ‘‘ENEA module’’, in which the space between the drums is back-ﬁlled with grout. Through incorporation of this multiple barrier design, the disposal facility is expected to limit the radiological and other environmental impacts of the wastes within the pre-speciﬁed regulatory thresholds. Since one of the principal mechanisms of radionuclide release to the environment is water inﬁltration through the various constituents of the repository and subsequent percolation through the groundwater system, it is of utmost importance to study the phenomena of advection and diﬀusion of radionuclides in the artiﬁcial porous matrices hosting the waste (near ﬁeld) and subsequently in the natural rock matrix of the host geosphere (far ﬁeld). The scope of the present application is limited to the simulation of the transport of the radionuclides through the ENEA module. 4.1. Features of the Monte Carlo simulation Our code is based on the standard Monte Carlo transport procedure which requires the simulation of a large number of independent particle histories by sampling a particle transition time from the ‘free ﬂight kernel’ and then determining the particular transition occurring at that time, from the ‘collision kernel’ (Cashwell and Everett, 1959; Kalos and Whitlock, 1986; Lux and Koblinger, 1991). In the present work, we simulate the transport of Pu-238 and its progeny through the ENEA module, within a temporal period of 1000 years discretized into 500 time channels (each time channel corresponding to two years). The simulated Pu-238

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chain is limited to Pu- 238 and its immediate descendent U-234, because of the extremely low decay rate of the U- 234 (T 12=244 500 years). The source term is represented by one trappon particle of Pu-238 inside the waste drum at time t=0 y. 4.2. Schematization of the module We consider the current design of the ENEA module with reference to a standard waste drum of 400 l. The layout of the system is reported in Fig. 4.1. Three types of diﬀerent materials make up the module: concrete, grout and waste drum. To satisfy the condition (15) of non-negativity of the transition rates, we have assumed a discretization step of 2103 m for the waste drum and of 2104 m for grout and concrete, for a total of nz=3500 zones (Table 4.1). 4.3. Scenario For the determination of the Monte Carlo parameters necessary to simulate the migration of the radionuclides through the ENEA module, we have conservatively assumed that the module is fully saturated with water, with a constant hydraulic head of 60 cm on top of the module which establishes a 1-D ﬂow towards the bottom of the module. This scenario is extremely cautious and unlikely, and, in any case, it could occur only after the end of the institutional control period (300 years from the closure of the repository) and upon complete failure of the cover. We further assume that the radionuclides are initially uniformly distributed throughout the waste drum. Another cautious hypothesis that we have introduced is that of a direct contact between the radionuclides contained in the waste drum and the water ﬁltering through the module, thus ignoring the resistance oﬀered by the steel barrier of the drum.

Fig. 4.1. Layout of ENEA module. Dimensions are expressed in mm. The right sketch corresponds to the spatial discretization used in our Monte Carlo code.

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Table 4.1 Discretization of the ENEA module 1. Concrete (cover) Discretization step [m] Thickness zone [m] Zones number

1. Concrete (bottom)

4

210 0.17 850

210 0.15 750

4

2. Grout 4

210 0.135 675

3. Waste drum 2103 1.1 550

nznm From:

To:

Type of medium

1 851 1526 2076 2751

850 1525 2075 2750 3500

1 Concrete 2 Grout 3 Waste drum 2 Grout 1 Concrete

4.4. Material degradation behavior The hydraulic characteristics of the engineered barriers are expected to change during the long period of interest for our simulation (1000 years). The knowledge of material behaviour on such a long time span is scarce and has to rely mainly on natural analogs and accelerated tests (Miller et al., 2000). Subjective assumptions need to be made to model the behaviour of the materials. In our case, the parameters describing the physical properties of the engineered barriers are considered to change in time, from intact to totally degraded conditions, according to a stepwise degradation model (ISAM, 2000) (Table 4.2). In order to avoid abrupt changes, we sigmoidally smoothed the steps with the following expressions (Fig. 4.2): yð t Þ ¼ y1 þ

y2 y1 1þe

Þ ðt200 g 1

þ

y3 y2 1þe

Þ ðt500 g

; g1 ¼ t12 =10; g2 ¼ t23 =10

ð16Þ

2

where t is the time, y1, y2 and y3 are the parameter values in the three conditions of material: intact, partially degraded and totally degraded, t12=100 years is the duration of the transition of the material characteristics from intact to partially Table 4.2 Steps of the material degradations process (ISAM, 2000) Temporal period [y]

State of material

0—200 200—500 > 500

Intact Partially degraded Totally degraded

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487

Fig. 4.2. The sigmoidal degradation model applied in our simulation.

degraded, t23=300 years is the duration of the transition from partially to totally degraded conditions. 4.5. Parameters of the Monte Carlo model In Table 4.3 we report the complete set of physical and hydraulic parameters characterizing the transport of Pu-238 and U-234 through the module system under study. Substituting these parameters in Eqs. (13) and (14), we obtained the Monte Carlo transition rates needed for the simulation of the transport processes through the ENEA module (Table 4.4). At the early times, the forward, p(z!z+1,t), and backward, p(z!z*1,t), transition rates are very close in value because, with the material intact, the contaminant transport process is limited to a slow molecular diﬀusion with essentially no advection. As time goes by, the material retention properties degrade and the advective transport process begins to dominate, giving rise to signiﬁcant forward transport. 4.6. Leaching process We assume that the release of radionuclides from the waste drum, that is the transformation of trappons in solutons, is due only to the leaching process, ignoring releases due to dissolution and diﬀusion. Leaching release occurs when the inﬁltrating water removes radionuclides from the surface of the conditioned waste. As previously mentioned, the release of radionuclides into pore water is limited or retarded by several geochemical processes such as adsorption and ion-exchange. The resulting retardation is expressed by using the partition coeﬃcient kd.

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Table 4.3 Physical and hydraulic parameters of the model Temporal period [y]

1. Concrete

2. Grout

3. Waste drum

Density (rb) [kg/m ] 0–200

1600

1600

1500

Hydraulic conductivity (K) [m/y] 0–200 200–500 > 500

3.15103 3.15101 1.5710+3

3.15103 3.15101 1.5710+3

3.15101 1.2610+2 1.5710+3

Porosity (n) 0–200 200–500 > 500

0.18 0.25 0.35

0.18 0.25 0.35

0.30 0.35 0.35

Partition coeﬃcient kd [m3/kg]- Pu 0–200 200–500 > 500

2.00 2.00 1.00

2.00 2.00 1.00

2.00 2.00 2.00

Partition coeﬃcient kd [m3/kg]- U 0–200 200—500 > 500

5.00 5.00 1.00

5.00 5.00 1.00

2.00 2.00 2.00

3.15102 3.27102 1.60

3.16102 3.83102 1.1910+1

3

Hydrodynamic dispersion (D) [m2/y] 0—200 3.15102 200—500 3.27102 > 500 1.60 Geometric dimension [m] 0.150 / 0.170

0.135

1.1

Table 4.4 Monte Carlo transition rates p(z!z+1) Temporal period Pu- 238 0–200 200–500 > 500 z (m) U- 234 0–200 200–500 > 500 z (m)

Transition rate p [1/y] p(z!z1) p(z!z+1)

1. Concrete

p(z!z1)

2. Grout

p(z!z+1)

p(z!z1)

3. Waste drum

17.724 25.956 12079 2104

17.716 25.199 5440.6

17.724 25.956 12079 2104

17.716 25.199 5440.6

0.7900 1.2172 525.27 2103

0.7880 1.0153 171.21

44.308 64.886 12079 2104

44.290 62.994 5440.6

44.308 64.886 12079 2104

44.290 62.994 5440.6

7.8931 12.160 5247.1 2103

7.8732 10.143 1710.3

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489

Table 4.5 Leaching rates for Pu-238 and U-234 Radionuclide

Time [y]

Leaching rate ll [year1]

Pu-238

0–200 200–500 > 500 0–200 200–500 > 500

3.6204106 3.6708104 6.4374101 3.6172105 3.6669103 6.4307

U-234

Assuming that leaching of radionuclides, occurs by a steady-state inﬁltration through the waste, the following expression can be derived (ISAM, 2000): ll ¼

q hw ð#w þ w kd Þ

ð17Þ

where q is the Darcy’s velocity through the conditioned waste; w is the moisture content of waste; w is the bulk density of the waste; kd is the partitioning coeﬃcient end hw is the height of the waste drum. In Table 4.5 we report the values of the leaching rates for Pu-238 and U-234 assuming complete saturation of the waste bulk ( w=n).

5. Results In this section, we present and discuss the results obtained using our Monte Carlo code for the transport of Pu-238 and its immediate descendent U-234 through the module. In Fig. 5.1, we report the time-probability distribution of ﬁnding a particle of Pu238 present in the spatial region of the waste drum (z=1526 to 2075). Fig. 5.1(a) shows the probability of ﬁnding a particle of Pu-238 trapped anywhere in the waste drum, whereas Fig. 5.1(b) shows the time-probability distribution of ﬁnding a trapped particle of Pu-238 in the various zones of the waste drum. The assumption of spatial homogeneity implies that the probability distribution remains spatially uniform at all times. The probability of ﬁnding a particle of Pu-238 trapped inside the waste drum decreases in time due to radioactive decay and leaching. During the ﬁrst 200 time channels the probability decreases slowly, principally due to decay that transforms the Pu-238 particles into U-234 ones. Then, the decrease becomes more rapid due to the leaching process which becomes more signiﬁcant in driving the Pu-238 out of the waste drum region. In Fig. 5.2(a) and (b), we report the probability distributions of ﬁnding a soluton particle of Pu-238 inside the whole module. This shows that Pu-238 is conﬁned in the waste drum for the ﬁrst 200 years, thanks to the good conﬁning properties of the

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intact materials of the module. During this period, the movement of the Pu-238 inside the module is governed essentially by diﬀusion. Then, as a consequence of the degradation process, the ﬂow of water through the module grows, thus increasing the transport of the mobile radionuclide solutons by advection. This behaviour is clearly shown in Fig. 5.2(b), where we report the probability distribution of ﬁnding soluton particles in the various zones of the module at ﬁve diﬀerent temporal instants: Pu-238 is uniformly distributed in the area of the waste drum for the ﬁrst 100 time channels; then, with the increasing water ﬂow, the solutons are transported through the bottom of the waste drum into the grout and, later on (200 and 225 time channels), through the concrete and out of the module. In Fig. 5.3(a) we report the time-probability distribution for a particle of Pu-238 to escape from the bottom of the module: the exit times of Pu-238 are concentrated between the 170th and the 240th time channel, i.e. in the time interval (340,480) y, during which signiﬁcant degradation of the material washes out all Pu-238. In Fig. 5.3(b) we report the integrated time-probability distribution of ﬁnding a particle of Pu-238 outside the module. The signiﬁcant decrease at long times is due to the decay process.

Fig. 5.1. (a) Time probability distribution of ﬁnding a particle of Pu-238 trapped anywhere in the waste drum. (b) Time probability distribution of ﬁnding a trappon particle of Pu-238 in the various zones of the waste drum.

M. Marseguerra et al. / Annals of Nuclear Energy 30 (2003) 473–496

491

Fig. 5.2. (a) Time probability distribution of ﬁnding a soluton particle of Pu-238 inside the module. (b) Space probability distributions of ﬁnding a soluton particle of Pu-238 inside the module at ﬁve diﬀerent temporal instants.

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Fig. 5.3. (a) Time distribution of the probability of a particle of Pu-238 escaping from the bottom of the module. (b) Integrated time-probability distribution of ﬁnding a particle of Pu-238 outside the module.

In Fig. 5.4(a) we report the time-probability distribution of ﬁnding a particle of U234, generated from the Pu-238 decay, trapped anywhere in the waste drum (z=1526 to 2075). We observe a build-up period, during the ﬁrst 100 time channels, in which the probability of ﬁnding a trappon particle of U-234 in the waste drum increases due to the radioactive decay of the Pu-238, the probability of trappon particles being released by leaching being very low. Then, the probability decreases because of the increased leaching of both U-234 and Pu-238 particles, due to the changes in the physical characteristics of the module barriers. In Fig. 5.4(b), the homogeneity of the material means that the U-234 particles are trapped uniformly in the various zones. The soluton particles of U-234, Fig. 5.5(a) and (b), show a similar behaviour to that of Pu-238 [Fig. 5.2(a) and (b)], except that the probability values are much larger for U-234. This is due to the fact that the rate of decay of the U-234 can be considered negligible compared to the rate of decay of Pu-238, which transforms more readily to its progeny. As a ﬁnal result, in Fig. 5.6(a) we report the time distribution of the probability of one particle of U-234 escaping from the bottom of the module. All exits of U-234 occur between the 170th and the 200th time channels with a peak probability of the order of 8102 per time channel width. In Fig. 5.6(b) we report the integrated timeprobability distribution of ﬁnding a particle of U-234, generated from Pu-238, out-

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493

Fig. 5.4. (a). Time probability distribution of ﬁnding a particle of U-234 trapped anywhere in the waste drum. (b) Time probability distribution of ﬁnding a trappon particle of U-234 in the various zones of the waste drum.

side the module. The build-up does not reach unity because a small number of particles of Pu-238 escapes from the module before decaying into U-234. Notice that, contrary to before, in this case the probability reaches a plateau, due to the negligible decay rate of U-234.

6. Conclusions The worldwide problem of adequately conﬁning the radioactive wastes produced in industrial applications is of paramount importance for the future exploitation of the advantages of ionising radiation and radioisotopes, in both the energy and nonenergy related ﬁelds. The Italian Agency ENEA is carrying out a signiﬁcant amount of work to identify an appropriate site and design for a repository for low level waste (LLW). Both the characterization of the site and of the design rely heavily on mathematical models for the prediction of contaminant dispersion under various scenarios. In the present work, we have applied a stochastic model, based on the theory of branching stochastic processes developed by Kolmogorov and Dmitriev, to the characterization of the design of a module of the deposit. The explicit simulation of

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Fig. 5.5. (a) Time probability distribution of ﬁnding a soluton particle of U-234 inside the module. (b) Space probability distributions of ﬁnding a soluton particle of U-234 inside the module at four diﬀerent temporal instants.

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495

Fig. 5.6. (a) Time distribution of the probability of a particle of U-234 escaping from the bottom of the module. (b) Integrated time-probability distribution of ﬁnding a particle of U-234 outside the module.

all the processes aﬀecting the transport of contaminant through the various media constituting the module allows a realistic representation of the actual situation. For example, the modeling of the degradation behaviour of the conﬁning materials does not pose particular additional complications to the simulation. An uncertainty analysis could also be included in a quite straightforward manner, by a pre-sampling of the vector of uncertain parameters. The approach seems to be promising for the full characterization of both the engineered deposit and the natural geologic barriers provided by the site itself.

References Bear, J., 1972. Dynamics of Fluids in Porous Media. Elsevier, NY. Cashwell, E.D., Everett, C.J., 1959. A Practical Manual on the Monte Carlo Method for Random Walk Problems. Pergamon Press, NY. Dagan, G., 1989. Flow and Transport in Porous Formations. Springer-Verlag, NY. Centro di deposito deﬁnitivo dei riﬁuti a bassa attivita`, ENEA 2000, in Italian. Freeze, R.A., Cherry, J.A., 1979. Groundwater. Prentice-Hill, Englewood Cliﬀs, New Jersey. GUIDA TECNICA n. 26. Gestione dei riﬁuti radioatttivi, 1997, in Italian. ICRP PUBBLICATION 26. Recommendations of the International Commission on Radiological Protection, 1977. ISAM, MDWG, WD01, Working Material. Model Formulation, Implementation and Data for Safety Assessment of Near Surface Disposal Facilities. Reproduced by the IAEA. Vienna, Austria 2000.

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Kalos M.H., Whitlock P.A., Monte Carlo Methods. Volume I : basics, Wiley, 1986. Kolmogorov A.N., Dmitriev N.A., 1947. C. r. Acad. Sci. URSS, 56, 1. Lee, Y., Lee, K.J., 1995. Nuclide transport of decay chain in the fractured rock medium: A model using continuous time Markov process. Ann. Nucl. Energy 22 (2), 71–84. Lux, I., Koblinger, L., 1991. Monte Carlo particle transport methods: neutron and photon calculations, CRC Press. Marseguerra, M., Zio, E., 1997. Modelling the transport of contaminants in groundwater as a branching stochastic process. Ann. Nucl. Energy 24 (8), 625–644. Marseguerra, M., Zio, E., 1998. Contaminant transport in bidimensional porous media via biased Monte Carlo simulation. Ann. Nucl. Energy 25 (16), 1301–1316. Marseguerra, M., Patelli, E., Zio, E., 2001. Groundwater contaminant transport in presence of colloids I: a stochastic nonlinear model and parameter identiﬁcation. Ann. Nucl. Energy 28 (2001), 777–803. Marseguerra, M., Patelli, E., Zio, E., 2001. Groundwater contaminant transport in presence of colloids II. Sensitivity and uncertainty analysis on literature case studies. Ann. Nucl. Energy 28 (2001), 1779–1807. Miller, W., Alexander, R., Chapman. N., McKinley, I., Smellie, J., 2000 Geological disposal for radioactive waste and natural analogues. Waste Management Series, Vol. 2, Pergamon. PAGIS, 1988. Performance assessment of geological isolation systems for radioactive waste, Commission of the European Communities. Savage D., 1995. The scientiﬁc and regulatory basis for the geological disposal of radioactive waste. John Wiley & Sons. Williams, M.M.R., 1992. A new model for describing the transport of radionuclides through fractured rock. Ann. Nucl. Energy 19 (10), 791–894.

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