modeling the effects of the engineered barriers of a radioactive waste repository by monte carlo...

Modeling the effects of the engineered barriersof a radioactive waste repository by

Monte Carlo simulation

Marzio Marseguerraa,*, Enrico Zioa, Edoardo Patellia,Francesca Giacobboa, Piero Risolutib, Giancarlo Venturab,

Giorgio Mingroneb

aDepartment of Nuclear Engineering, Polytechnic of Milan, Via Ponzio 34/3, 20133 Milan, ItalybENEA(ItalianAgency forEnergyandEnvironment),TaskForceSito,ViaAnguillarese 301 - 00060Rome, Italy

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, inturn, are placed in a concrete container called a ‘‘module’’, in which the space between thedrums is back-filled with grout. Finally, several modules are stacked within the concrete wallsof 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 againstthe radiological impacts of the wastes by meeting various functional objectives which aim atlimiting 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 andsubsequent percolation into the groundwater system, it is of utmost importance to study thephenomena 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 (farfield). This paper addresses the issue of radionuclide transport through the artificial porousmatrices constituting the engineered barriers of the repository’s near field. The complexity of thephenomena involved, augmented by the heterogeneity and stochasticity of the media in which

transport occurs, renders classical analytical-numerical approaches scarcely adequate for realisticrepresentation of the system of interest. Hence, we propound the use of aMonte Carlo simulationmethod based on the Kolmogorov and Dmitriev theory of branching stochastic processes.

# 2002 Elsevier Science Ltd. All rights reserved.

Annals of Nuclear Energy 30 (2003) 473–496

www.elsevier.com/locate/anucene

0306-4549/03/$ - see front matter # 2002 Elsevier Science Ltd. All rights reserved.

PI I : S0306-4549(02 )00072 -5

* Corresponding author. Fax: +39-02-2399-6309.

E-mail address: [email protected] (M. Marseguerra).

1. Introduction

Nuclear activities for power generation have been conducted in Italy for nearlythirty years, from the early 1960s to the end of the 1980s. These activities, togetherwith other non-energy related applications of ionising radiation and radionuclides inthe industrial and medical fields, have led to the production of a non-negligibleamount of radioactive wastes. Considerably larger quantities of radioactive wasteswill be produced during the decommissioning of the Italian, Ente Nazionale energiaElettrica ENEL, power stations and of the Italian agency for energy and environ-ment, Ente Nazionale per le nuove tecnologie, l’Energia e l’Ambiente ENEA, fuelcycle 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 oper-ation.This disposal facility must be designed so as to guarantee isolation of the radio-

nuclides from the biosphere for the entire period during which they remain ofpotential radiological significance. This isolation is achieved by means of a multiplebarrier system designed to limit the release and transport of radionuclides into theenvironment (PAGIS, 1998). The long time scales of potential radiological sig-nificance associated with radioactive wastes require that the assessment of the iso-lation performance of the disposal site and facility be obtained by applying modelswhich simulate the migration of radionuclides from the disposal facility to the bio-sphere through the various artificial and natural barriers. The results of the simula-tions are used as an input to the design of the disposal system to ensure that theradiological impact of the disposal system meets with the safety criteria issued by thenational 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 transportdissolved radioactive material to and in groundwater. Different approaches havebeen proposed for, and applied to, water-driven contaminant transport, such as theuse 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 reactorphysics (Williams, 1992) and Monte Carlo simulation techniques (Marseguerra etal., 1998; Marseguerra et al., 2001a,b). In this paper, we apply a stochastic approachbased on the Kolmogorov–Dmitriev theory of branching stochastic processes (Kol-mogorv and Dmitriev, 1947) to the problem of modelling radionuclide transportthrough different engineered barriers. The corresponding model is evaluated byMonte Carlo simulation, where a large number of particles of solute are followed intheir travel through the barriers, using appropriate probability distribution func-tions to characterise their transport. Different kinds of particles are introduced torepresent the contaminant in the various possible locations and physico-chemicalstates, and the processes occurring during transport are represented as transitionsfrom one particle state to another, with given rates of occurrence. The main advan-tage of the proposed model is its flexible structure. This allows one to consider

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

multidimensional geometries and to describe a wide range of phenomena, byaccounting for the individual interactions which each particle may undergo duringits transport.In the next section, we describe the main characteristics of the underlying sto-

chastic model, developed by some of the present authors (Marseguerra and Zio,1997), to simulate contaminant transport through porous media. The estimation ofmodel parameters is addressed by means of a comparison with the advection–dis-persion formulation in finite-difference form. In the following section, the model isverified with respect to a literature case regarding the transport of a radioactivechain 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 wasterepository (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 whichthe transport process occurs is subdivided in Nz zones, z=1, 2, . . ., Nz. Our objec-tive 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 solidmatrix. We use Sp(z,t) and Tp(z,t) to denote a particle of contaminant of type presiding 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 mayundergo, 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 con-

sider only transfers backward and forward to adjacent zones, occurring with rates�(z!z�1,t) and �(z!z+1,t), respectively (Fig. 2.1). The soluton Sp (z,t) of type pcontaminant may also be subject, with transition rate ads(p,z,t), to reversibleadsorption on the solid matrix of the host medium, thus transforming into a trap-pon Tp(z,t). Moreover, the soluton Sp(z,t) may, due to chemical reactions, transforminto a different particle p00 with rate gp!p00. Finally, if the contaminant particle underconsideration is a radionuclide, it may transform into a different p0-contaminantaccording to its characteristic decay rate lp!p0 [Fig. 2.2 (a)].

Fig. 2.1. Schematics of the soluton particles transport.

M. Marseguerra et al. / Annals of Nuclear Energy 30 (2003) 473–496 475

The trappon may be desorbed from the host matrix, with rate des(p,z,t), becominga soluton, or, analogously to the soluton, it may transform into a different particledue to chemical reactions or radioactive decay with rates �p!p00 and lp! p0, respec-tively [Fig. 2.2(b)]. Table 2.1 summarizes the various transitions that the particlesmay 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 thedomain of the probability generating functions (pgfs) and then exploiting theirproperties we arrive at the following system of 2.p.Nz partial differential equationsfor the expected number of solutons and trappons particles where we denote byNSp z; tð Þ, p=1,2, . . .,np, z=1,2, . . ., Nz, the expected number of soluton particle Sp ofp kind in zone z at time t, by NTp z; tð Þ the expected number of trappon particle Tp ofp 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) �(z!z+1,t)

Backward transfer to adjacent zone (z!z�1) �(z!z�1,t)

Adsorption on the porous host matrix ads(p,z,t)

Chemical transformation �p!p00

Radioactive decay lp! p0

Tp Desorption from the porous host matrix des(p,z,t)

Chemical transformation �p!p00

Radioactive decay lp!p0


@NSp z; tð Þ

@t¼ � � z ! zþ 1; tð Þ þ � z ! z� 1; tð Þ½ NSp z; tð Þ

þ � zþ 1 ! z; tð ÞNSp zþ 1; tð Þ þ � z� 1 ! z; tð ÞNSp z� 1; tð Þ

� ads p; z; tð ÞNSp z; tð Þ þ des p; z; tð ÞNTp z; tð Þ �NSp z; tð ÞXp0

lp! p0

�NSp z; tð ÞXp00

�p! p00 þXp0

lp0 ! pNSp0 z; tð Þ

þXp00

�p00 ! pNSp00 z; tð Þ ð1Þ

@NTp z; tð Þ

@t¼ þads p; z; tð ÞNSp z; tð Þ � des p; z; tð ÞNTp z; tð Þ �NTp z; tð Þ

Xp0

lp! p0

�NTp z; tð ÞXp00

�p! p00 þXp0

lp0 ! pNTp0 z; tð Þ

þXp00

�p00 ! pNTp00 z; tð Þ ð2Þ

These are balance equations describing the production and loss processes of par-ticles in zone z. The first two terms on the right hand side of Eq. (1) describe thedisappearance of solutons Sp due to the transfer to an adjacent zone; the third andfourth terms describe the appearance of solutons Sp in zone z because of transferfrom adjacent zones; the fifth term represents the transformation of solutons Sp intotrappons Tp due to the adsorption on the host matrix; the sixth term represents theproduction 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 bytransformations due to radioactive decay and chemical transformations; the last twoterms describe the production of solutons Sp due to decay and chemical transfor-mations of other species of solutons into the p-kind. Similar balance considerationsapply in Eq. (2). Of course, these equations must be supplemented with the properinitial conditions.Substitution of Eq. (2) in Eq. (1) yields:


@

@tNSp z; tð Þ þNTp z; tð Þ� �

¼� � z ! zþ 1; tð Þ þ � z ! z� 1; tð Þ½ NSp z; tð Þ

þ � zþ 1 ! z; tð ÞNSp zþ 1; tð Þ

þ � z� 1 ! z; tð ÞNSp z� 1; tð Þ

� NSp z; tð Þ þNTp z; tð Þ� �

Xp0

lp! p0 þXp00

�p! p00

" #

þXp0

lp0 ! p NSp0 z; tð Þ þNTp0 z; tð Þ

h iþXp00

�p00 ! p

� NSp00 z; tð Þ þNTp00 z; tð Þ

h i

ð3Þ

It can be seen that the transport phenomena are described explicitly in probabil-istic terms, thus allowing for substantial flexibility. Obviously, the effectiveness ofthis representation is conditioned by the capability of estimating the values of thespecified 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 aquite straightforward manner, of equations not only for the expected values of par-ticle number, but also for the higher-order moments of the distributions (Kolmo-gorv and Dmitriev, 1947; Marseguerra and Zio, 1997).

2.1. Parameter determination

In order to estimate the values of the various transition rates appearing in thestochastic model of Section 2, we make a comparison between Eqs. (1) and (2) andthe corresponding classical advection–dispersion equations, with the aim of estab-lishing a relationship between the transition rates pertaining to our stochastic modeland the parameters of the classical model.To illustrate the procedure, we consider the case of a 1-D transport of a non-

radioactive p-contaminant through a homogeneous porous medium, in a uniformflow field, with no interchange between the solute and the solid phases and no che-mical transformations. The corresponding advection–dispersion equation is:

@Cp z; tð Þ

@t¼ D

@2Cp z; tð Þ

@z2� v

@Cp z; tð Þ

@z� Cp z; tð Þ

Xp0

lp! p0 þXp0

lp0 ! pCp0 z; tð Þ ð4Þ


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

v ¼q

neð5Þ

q ¼ �K@h

@zð6Þ

D ¼ �LvþDdiff ð7Þ

where q is Darcy’s velocity, K the hydraulic conductivity, h the hydraulic head, nethe effective porosity of the medium, �L the longitudinal dispersivity coefficient andDdiff the molecular diffusion coefficient (Freeze and Cherry, 1979; Bear, 1972).The phenomena of interchange between the solute and the solid phases are gen-

erally represented using the simplification of a linear equilibrium isotherm whichestablishes a proportionality relationship between the fraction of p-contaminantadsorbed on the matrix and the fraction of p-contaminant present in the liquidphase. Then the governing equation of the advection–dispersion model, with nochemical transformation, becomes:

@Cp z; tð Þ

@t¼D

Rp

@2Cp z; tð Þ

@z2�v

Rp

@Cp z; tð Þ

@z� Cp z; tð Þ

Xp0

lp! p0 þXp0

lp0 ! pCp0

z; tð Þ

ð8Þ

where the constant Rp, called the ‘‘retardation factor’’, accounts for the delay in thetransport of the p-contaminant due to adsorption/desorption processes and isdefined as:

Rp ¼ 1þ�bkdn

ð9Þ

where �b is the porous media bulk density, kd is the partition coefficient and n thetotal porosity.By analogy, in our stochastic model we introduce the same linear equilibrium

hypothesis in the adsorption/desorption processes involving solutons Sp and trap-pons Tp:

NTp z; tð Þ ¼ �pNSp z; tð Þ ð10Þ

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


@NSp z; tð Þ

@t¼� �p z ! zþ 1; tð Þ þ �p z ! z� 1; tð Þ

� �NSp z; tð Þ þ �p

z� 1 ! z; tð ÞNSp z� 1; tð Þ þ �p zþ 1 ! z; tð ÞNSp

zþ 1; tð Þ �NSp z; tð ÞXp0

lp! p0 þXp00

�p! p00

" #

þXp0

lp0 ! pNSp0 z; tð Þ þXp00

�p00 ! pNSp00 z; tð Þ ð11Þ

where �p z; tð Þ ¼� z;tð Þ

1þ�pð Þis the effective transition rate of transfer to an adjacent zone,

specific for a p-contaminant.

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

@C z; tð Þ

@t¼ þ

D

Rp

�zð Þ2�

v

Rp

2�z

0BB@

1CCAC zþ 1; tð Þ þ

D

Rp

�zð Þ2þ

v

Rp

2�z

0BB@

1CCAC z� 1; tð Þ

�

2D

Rp

�zð Þ2þXp0

lp! p0

0BB@

1CCAC z; tð Þ þ

Xp0

lp0 ! pCp0 z; tð Þ ð12Þ

Comparing term by term with Eq. (11) we obtain the following relationshipsamong the parameters of the two models:

�p z ! z� 1ð Þ ¼

D

Rp

�zð Þ2�

v

Rp2�z

ð13Þ

�p z ! zþ 1ð Þ ¼

D

Rp

�zð Þ2þ

v

Rp

2�zð14Þ

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


max �z½ ¼2D

v

� �ð15Þ

The �p and lp rates that govern the chemical and radioactive transformations of ap-contaminant into a different contaminant can be estimated directly from the che-mical 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 corre-sponding Monte Carlo code, we have carried out a comparison with the results of acase from the literature in which the 1-D transport through a fractured porousmedium of a chain of three radionuclides was analysed by simulating a Markovprocess, 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 Value in case A Value in case B

Initial concentration of nuclide (1), C0(1) 1.0



Retardation coefficient of nuclide (1), Rp(1) 100



Distance, L (m) 100

Decay constant of nuclide (1), l(1) (1/y) 1.60 � 10�3



Pore velocity, q/n (m/y) 10

Dispersion coefficient, D (m2/y) 2.5 25

Table 3.2

Transport of a chain of three radionuclides: rates of the present stochastic model

Parameter Value in case A Value in case B




Delta z (m) 0.25 1.0

Forward rate (1/y) 0.6 0.3

Backward rate (1/y) 0.2 0.2

Final time (y) 2 � 10+3

Number of time channels 500

Number of trials 1 � 10+6

CPU time (Pentium III, 800 Mhz) 0 h 9 min 33 s 0 h 9 min 12 s


the radionuclides through the fractured medium was delayed by sorption processeson the fracture walls. Two cases, here referred as A and B, were considered in (Leeand 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 con-

centrations of the three radionuclides obtained with our Monte Carlo code.


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).


In Table 3.1, we report the parameter values assumed in (Lee and Lee, 1995). InTable 3.2, we provide the corresponding values of the stochastic model parametersobtained 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 radio-nuclide (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 Aobtained in (Lee and Lee, 1995). In Fig. 3.1(b) we report the corresponding resultsobtained 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 of20 and 50 m from the source, as obtained with the model described in the literatureand 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 inspecial drums made of steel (ENEA, 2000). The drums are, in turn, placed in aconcrete container, called a ‘‘module’’ and here referred to as ‘‘ENEA module’’, inwhich the space between the drums is back-filled with grout. Through incorporationof this multiple barrier design, the disposal facility is expected to limit the radi-ological and other environmental impacts of the wastes within the pre-specifiedregulatory thresholds. Since one of the principal mechanisms of radionuclide releaseto the environment is water infiltration through the various constituents of therepository and subsequent percolation through the groundwater system, it is ofutmost importance to study the phenomena of advection and diffusion of radio-nuclides in the artificial porous matrices hosting the waste (near field) and subse-quently in the natural rock matrix of the host geosphere (far field). The scope of thepresent application is limited to the simulation of the transport of the radionuclidesthrough the ENEA module.

4.1. Features of the Monte Carlo simulation

Our code is based on the standard Monte Carlo transport procedure whichrequires the simulation of a large number of independent particle histories by sam-pling a particle transition time from the ‘free flight kernel’ and then determining theparticular transition occurring at that time, from the ‘collision kernel’ (Cashwell andEverett, 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 timechannels (each time channel corresponding to two years). The simulated Pu-238


chain is limited to Pu- 238 and its immediate descendent U-234, because of theextremely 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 standardwaste drum of 400 l. The layout of the system is reported in Fig. 4.1. Three types ofdifferent 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 2�10�3 m for the waste drum and of 2�10�4 m forgrout 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 themigration of the radionuclides through the ENEA module, we have conservativelyassumed that the module is fully saturated with water, with a constant hydraulichead of 60 cm on top of the module which establishes a 1-D flow towards the bot-tom of the module. This scenario is extremely cautious and unlikely, and, in anycase, it could occur only after the end of the institutional control period (300 yearsfrom 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 filteringthrough the module, thus ignoring the resistance offered by the steel barrier of thedrum.

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.


4.4. Material degradation behavior

The hydraulic characteristics of the engineered barriers are expected to changeduring 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). Sub-jective assumptions need to be made to model the behaviour of the materials. In ourcase, the parameters describing the physical properties of the engineered barriers areconsidered to change in time, from intact to totally degraded conditions, accordingto 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):

y tð Þ ¼ y1 þy2 � y1

1þ e�t�200ð Þ

g1

þy3 � y2

1þ e�t�500ð Þ

g2

; g1 ¼ �t12=10; g2 ¼ �t23=10 ð16Þ

where t is the time, y1, y2 and y3 are the parameter values in the three conditions ofmaterial: intact, partially degraded and totally degraded, �t12=100 years is theduration of the transition of the material characteristics from intact to partially

Table 4.1

Discretization of the ENEA module

1. Concrete (cover) 1. Concrete (bottom) 2. Grout 3. Waste drum

Discretization step [m] 2�10�4 2�10�4 2�10�4 2�10�3

Thickness zone [m] 0.15 0.17 0.135 1.1

Zones number 750 850 675 550

nznm

From: To: Type of medium

1 850 1 Concrete

851 1525 2 Grout

1526 2075 3 Waste drum

2076 2750 2 Grout

2751 3500 1 Concrete

Table 4.2

Steps of the material degradations process (ISAM, 2000)

Temporal period [y] State of material

0—200 Intact

200—500 Partially degraded

> 500 Totally degraded


degraded, �t23=300 years is the duration of the transition from partially to totallydegraded conditions.

4.5. Parameters of the Monte Carlo model

In Table 4.3 we report the complete set of physical and hydraulic parameterscharacterizing the transport of Pu-238 and U-234 through the module system understudy. Substituting these parameters in Eqs. (13) and (14), we obtained the MonteCarlo transition rates needed for the simulation of the transport processes throughthe 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 con-taminant transport process is limited to a slow molecular diffusion with essentiallyno advection. As time goes by, the material retention properties degrade and theadvective transport process begins to dominate, giving rise to significant forwardtransport.

4.6. Leaching process

We assume that the release of radionuclides from the waste drum, that is thetransformation of trappons in solutons, is due only to the leaching process, ignoringreleases due to dissolution and diffusion. Leaching release occurs when the infiltrat-ing water removes radionuclides from the surface of the conditioned waste. As pre-viously mentioned, the release of radionuclides into pore water is limited or retardedby several geochemical processes such as adsorption and ion-exchange. The result-ing retardation is expressed by using the partition coefficient kd.

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


Table 4.3

Physical and hydraulic parameters of the model

Temporal period [y] 1. Concrete 2. Grout 3. Waste drum

Density (rb) [kg/m3]0–200 1600 1600 1500

Hydraulic conductivity (K) [m/y]

0–200 3.15�10�3 3.15�10�3 3.15�10�1

200–500 3.15�10�1 3.15�10�1 1.26�10+2

> 500 1.57�10+3 1.57�10+3 1.57�10+3

Porosity (n)

0–200 0.18 0.18 0.30

200–500 0.25 0.25 0.35

> 500 0.35 0.35 0.35

Partition coefficient kd [m3/kg]- Pu

0–200 2.00 2.00 2.00

200–500 2.00 2.00 2.00

> 500 1.00 1.00 2.00

Partition coefficient kd [m3/kg]- U

0–200 5.00 5.00 2.00

200—500 5.00 5.00 2.00

> 500 1.00 1.00 2.00

Hydrodynamic dispersion (D) [m2/y]

0—200 3.15�10�2 3.15�10�2 3.16�10�2

200—500 3.27�10�2 3.27�10�2 3.83�10�2

> 500 1.60 1.60 1.19�10+1

Geometric dimension [m]

0.150 / 0.170 0.135 1.1

Table 4.4

Monte Carlo transition rates

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

Temporal period 1. Concrete 2. Grout 3. Waste drum

Pu- 238

0–200 17.724 17.716 17.724 17.716 0.7900 0.7880

200–500 25.956 25.199 25.956 25.199 1.2172 1.0153

> 500 12079 5440.6 12079 5440.6 525.27 171.21

�z (m) 2�10�4 2�10�4 2�10�3

U- 234

0–200 44.308 44.290 44.308 44.290 7.8931 7.8732

200–500 64.886 62.994 64.886 62.994 12.160 10.143

> 500 12079 5440.6 12079 5440.6 5247.1 1710.3

�z (m) 2�10�4 2�10�4 2�10�3


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

ll ¼q

hw #w þ �wkdð Þð17Þ

where q is the Darcy’s velocity through the conditioned waste; w is the moisturecontent of waste; �w is the bulk density of the waste; kd is the partitioning coefficientend 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 Carlocode for the transport of Pu-238 and its immediate descendent U-234 through themodule.In Fig. 5.1, we report the time-probability distribution of finding a particle of Pu-

238 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 wastedrum, whereas Fig. 5.1(b) shows the time-probability distribution of finding a trap-ped particle of Pu-238 in the various zones of the waste drum. The assumption ofspatial homogeneity implies that the probability distribution remains spatially uni-form 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 timechannels the probability decreases slowly, principally due to decay that transformsthe Pu-238 particles into U-234 ones. Then, the decrease becomes more rapid due tothe leaching process which becomes more significant in driving the Pu-238 out of thewaste 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 inthe waste drum for the first 200 years, thanks to the good confining properties of the

Table 4.5

Leaching rates for Pu-238 and U-234

Radionuclide Time [y] Leaching rate ll [year�1]

Pu-238 0–200 3.6204�10�6

200–500 3.6708�10�4

> 500 6.4374�10�1

U-234 0–200 3.6172�10�5

200–500 3.6669�10�3

> 500 6.4307


intact materials of the module. During this period, the movement of the Pu-238inside the module is governed essentially by diffusion. Then, as a consequence of thedegradation process, the flow of water through the module grows, thus increasingthe transport of the mobile radionuclide solutons by advection. This behaviour isclearly shown in Fig. 5.2(b), where we report the probability distribution of findingsoluton particles in the various zones of the module at five different temporalinstants: Pu-238 is uniformly distributed in the area of the waste drum for the first100 time channels; then, with the increasing water flow, the solutons are transportedthrough the bottom of the waste drum into the grout and, later on (200 and 225 timechannels), 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 concentratedbetween 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. InFig. 5.3(b) we report the integrated time-probability distribution of finding a particleof Pu-238 outside the module. The significant decrease at long times is due to thedecay 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.


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.


In Fig. 5.4(a) we report the time-probability distribution of finding a particle of U-234, 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 drumincreases due to the radioactive decay of the Pu-238, the probability of trapponparticles being released by leaching being very low. Then, the probability decreasesbecause of the increased leaching of both U-234 and Pu-238 particles, due to thechanges in the physical characteristics of the module barriers. In Fig. 5.4(b), thehomogeneity of the material means that the U-234 particles are trapped uniformly inthe 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 lar-ger for U-234. This is due to the fact that the rate of decay of the U-234 can beconsidered negligible compared to the rate of decay of Pu-238, which transformsmore 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-234occur between the 170th and the 200th time channels with a peak probability of theorder of 8�10�2 per time channel width. In Fig. 5.6(b) we report the integrated time-probability distribution of finding a particle of U-234, generated from Pu-238, out-

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.


side the module. The build-up does not reach unity because a small number of par-ticles 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 negli-gible decay rate of U-234.

6. Conclusions

The worldwide problem of adequately confining the radioactive wastes producedin industrial applications is of paramount importance for the future exploitation ofthe advantages of ionising radiation and radioisotopes, in both the energy and non-energy 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 thecharacterization of the site and of the design rely heavily on mathematical modelsfor 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 thecharacterization of the design of a module of the deposit. The explicit simulation of

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.


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.


all the processes affecting the transport of contaminant through the various mediaconstituting the module allows a realistic representation of the actual situation. Forexample, the modeling of the degradation behaviour of the confining materials doesnot pose particular additional complications to the simulation. An uncertainty ana-lysis could also be included in a quite straightforward manner, by a pre-sampling ofthe 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.

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