Monte Carlo simulation of radioactive contaminant transport in groundwater

Transcription

1 Politecnico di Milano Diartimento di Ingegneria Nucleare Dottorato di Ricerca in Scianza e Tecnologia delle Radiazioni Monte Carlo simulation of radioactive contaminant transort in groundwater Tesi di dottorato di: Edoardo Patelli Relatore: Prof. Enrico Zio Correlatore: Dott.ssa Francesca Giacobbo Tutor: Prof. Marzio Marseguerra Coordinatore del rogramma di dottorato: Prof. Marzio Marseguerra XVIII ciclo

2 Politecnico di Milano Diartimento di Ingegneria Nucleare Dottorato di Ricerca in Scianza e Tecnologia delle Radiazioni Monte Carlo simulation of radioactive contaminant transort in groundwater Ph.D. Dissertation of: Edoardo Patelli Advisor: Prof. Enrico Zio Coadvisor: Dott.ssa Francesca Giacobbo Tutor: Prof. Marzio Marseguerra Suervisor of the Ph.D. Program: Prof. Marzio Marseguerra XVIII edition

3 A Claudia, Mamma, e Paà...

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6 Acknowledgements-Ringraziamenti 1.Enrico Zio (advisor) 2.Francesca, Cristina, & Edo (Milano, 2005 ) 3.Marzio Marseguerra (tutor) 4.Francesca Giacobbo (coadvisor) 5.A Sveva, Edo, & Claudia (Tarifa, Sain 2003 ) 6.Claudia, & Edo (Burren, Ireland 2004 ) 7.Stockholm (Sweden, 8.Piero (Halden, Norway, 9.Andrew, & Edo (Jul 2004 ) 2004 ) Party, Stockholm 2004 ) 10.Andrea, Fabio, Simona, 11.Mondo, Michele, & 12.Nicola, Giulio (Milano, Alo, Barbara, Edo (Festa dei crotti, 2005 Elena, Massimo, Edo, 2004 Alessia, Ambra, & Jonn (Bergamo, 2004 ) 13.Piero, & Bea (Mantova, 14.Andrew (Jul Party, 15.Marco, & Pablo (Man ) Stockholm 2004 ) tova, 2004 ) 16.Claudia, & Edo (Mantova, 17.Pablo, Edo, & Alessan- 18.Edo, Massi, & Alessia 2004 ) dro ((Stockholm, 2004 ) (Colli di San Fermo, 2004 ) 19.Edo, Barbara, Claudia, 20.Claudia Caadocia, 21.Piero, & Edo (Al- & Elena Festa dei Turkey 2005 ) barella, 2004 ) crotti, 2004 ) 22.Enrico, Bea, Claudia, 23.Claudia, & Edo (Albarella, 24.Elena, Edo, Barbara, Edo, Piero, Giorgia, & 2005 ) Jonn & Ambra (Carona, Aurora (Albarella, 2004 ) 2005 ) 25.Andrea, & Edo (Stockholm, Sweden 2004 ) 26.Claudia (Pont du Gar, France 2005 ) 27.Max, & Crenguta (Albarella, 2005 ) 28.Alo, & Laura (Albarella, 29.Podo, Cado, & Edo 30.Edo, Piero, & Claudia 2005 ) (Siwa desert road, Egyt 2006 ) (Siwa's Desert, Egyt Claudia, Edo (Scotland, 2006 ) 32.Mondo, & Edo (Giant Causeway, Ireland 2004) 33.Cado, & Edo (Giza, Egyt 2006 ) 34.Thomas, Piero, Podo, 35.Edo, & Raka (Edinburgh, 36.Total solar eclise (Sal- Edo, Claudia, Katia, 2006 ) lum, Egyt 2006 ) Cado, & Vale (Sallum, Egyt 2006 ) 37.Raka, Barbara, Elena, Chiara, Edo & Claudia (Edinburgh, 2006 ) 38.Pendolari del treno MI-BG and thanks to all. i

7 Contents List of Paers and Reorts vii List of Figures ix List of Tables x List of acronyms xi Part I: Thesis 1 1. Introduction Problem statement Disosal of radioactive waste Origin and classication of radioactive wastes Multi-barrier concet Performance assessment Modelling radionuclide transort in groundwater Objective and aims of the resent work Dissertation outline Transort in orous media Introduction Deterministic aroach: the Advection-Disersion Equation Linear isotherm Freundlich isotherm Langmuir isotherm Limits of the Advection-Disersion aroach Is disersion a Fickian rocess? Transort in heterogeneous media Need of stochastic aroach Monte Carlo simulation of transort in orous media Introduction The stochastic model of contaminant transort Nonlinear stochastic model ii

8 Contents Parameters determination of the model Monte Carlo simulation Nonlinear Monte Carlo simulation Verication of the model Numerical alications Radionuclides transort from a HLW built in clay formation Transort in fractured media Introduction State of the Art Water ow modelling in fractured rock Solute transort modelling in fractured rock Summary Monte Carlo simulation transort in fractured media Introduction Stochastic Model Monte Carlo solution of the transort equation Alication to a case with adsortion and decay The Hybrid model Reactive transort Parameter determination from DFN simulations Verication of the hybrid model Alication: uscaling DFN simulations Numerical test of uscaling Comments Conclusions 53 A. Parameters determination of the stochastic model in orous media 55 A.1. Determination of motion rates, π, in saturated steady ow eld A.1.1. Parameters determination in uniform medium A.1.2. Parameters determination in layered medium A.1.3. Determination of motion rates π, under transient unsaturated ow A.2. Determination of the sortion rates ads, des A.2.1. Linear isotherm hyothesis A.2.2. Nonlinear isotherm hyothesis A.3. Determination of the chemical transformations γ and radioactive decay rates λ B. Discrete Fracture Network 65 B.1. Introduction B.2. Theoretical background: 2D Fracture Network B.2.1. Algorithm of the DFN B.2.2. Particle Source B.2.3. Mixing function iii

9 Contents B D-Discrete Fracture Network C. Literature Review for Fracture Parameters 74 C.1. Fracture Size C.2. Aerture C.3. Orientation C.4. Location C.5. Linear Frequency, Sacing, and Number Density C.6. Summary References 80 Part II: Paers and Reorts 87 Paer I 89 Paer II 98 Paer III 109 Reort I 120 Reort II 140 Reort III 157 iv

10 Contents List of aers Paer I Marseguerra, M., Padovani, E., Zio, E., Giacobbo, F., Patelli, E., A Nuclear Measurement Technique of Water Penetration in Concrete Barriers. Monte Carlo Methods and Alications 10 (3-4), Paer II Giacobbo, F., Patelli, E., Zio, E., Monte Carlo Simulation of Radioactive Contaminant Transort in Unsaturated Porous media. In: M&C 2005, Avignon-France Setember Paer III Giacobbo, F., Patelli, E., Zio, E., Monte Carlo Simulation of Radionuclides Groundwater Transort in resence of heterogeneities and nonlinearities. In: MIGRATION'05, Avignon-France Setember List of reorts Reort I Patelli, E., Giacobbo, F., Zio, E., Monte Carlo simulation of nonlinear reactive transort in unsaturated orous media. Reort I Patelli, Giacobbo, F., E., Zio, E Monte Carlo simulation of radionuclides transort in fractured media: the stochastic model. Reort II Patelli, E., Framton A., Monte Carlo simulation of radionuclides in fractured media: arameters determination from Discrete Fracture Network. v

11 List of Figures 1.1. Possible layout of the SKB reository Sources of radioactive waste in Switzerland and the waste management strategy considered after [NAGRA, 2005] Multile barriers revent the radionuclides from reaching the ground surface Postclosure erformance of the reository Radioactive contaminant transort rocesses studied in this thesis Schematics of the soluton articles rocesses Schematics of the traon articles rocesses Aroximation of the nonlinear rate π k with standard xed time ste rocedure (gure on the left) and with the burn-u technique (gure on the right) The scheme of the algorithm for automatically udated the time ste imlemented in the MASCOT code Examle of nonlinear MC simulations adoting the xed time ste and ATS rocedure. The numerical solution of the AD equation is reorted for reference Geological structure of the sedimentary sequences in the vicinity of the Benken borehole in the Zürcher Weinland, based on 3-D seismic data (left-hand gure), and schematic geological roles from SW to NE (to right-hand) and NW to SE (bottom right-hand) through the sedimentary rocks. The otential host rock consists of the Oalinus Clay formation and the Murchisonae beds in Oalinus Clay facies, after [NAGRA, 2002] Potential groundwater transort aths of the radionuclide from the reository to the bioshere Discretization of the otential groundwater transort ath used in MC simulation vi

12 List of Figures 3.9. MASCOT simulation results of 243 Am: normalized distribution in sace mobile in solution (left Figure) and traed in orous media (right Figure) at dierent time from the failure of the canister MASCOT simulation results of 239 Pu: normalized distribution in sace mobile in solution (left Figure) and traed in orous media (right Figure)at dierent time from the failure of the canister MASCOT simulation results of 235 U: normalized distribution in sace mobile in solution (left Figure) and traed in orous media (right Figure) at dierent time from the failure of the canister Water ow modelling in fractured rock, after [Geier et al., 1992] CDF, CCDF (left-hand gure) and PDF (right-hand gure) of advective travel time τ calculated by DFN (blue oints) and MASCOT simulation (red lines) with ux injection and static mixing hyothesis CDF, CCDF (left-hand gure) and PDF (right-hand gure) of advective travel time τ calculated by DFN (blue oints) and MASCOT simulation (red lines) with ux injection and comlete mixing hyothesis CDF, CCDF (left-hand gure) and PDF (right-hand gure) of advective travel time τ calculated by DFN (blue oints) and MASCOT simulation (red lines) with resident concentration and comlete mixing hyothesis CDF, CCDF (left-hand gure) and PDF (right-hand gure) of advective travel time τ calculated by DFN (blue oints) and MASCOT simulation (red lines) with resident concentration and static mixing hyothesis CDF, CCDF (left-hand gure) and PDF (right-hand gure) of retention arameter β calculated by DFN (blue oints) and MASCOT simulation (red lines) with ux injection and static mixing hyothesis CDF, CCDF (left-hand gure) and PDF (right-hand gure) of retention arameter β calculated by DFN (blue oints) and MASCOT simulation (red lines) with ux injection and comlete mixing hyothesis CDF, CCDF (left-hand gure) and PDF (right-hand gure) of retention arameter β calculated by DFN (blue oints) and MASCOT simulation (red lines) with resident concentration and comlete mixing hyothesis CDF, CCDF (left-hand gure) and PDF (right-hand gure) of retention arameter β calculated by DFN (blue oints) and MASCOT simulation (red lines) with resident concentration and static mixing hyothesis Verication of the MASCOT uscaling for the τ arameter at 50 meters from the inlet with ux injection and comlete mixing hyothesis. Left-hand gure shows the CDF and CCDF of advective arameter τ. Right-hand gure shows the PDF of the advective arameter τ vii

13 List of Figures Verication of the MASCOT uscaling for the β arameter at 50 meters from the inlet with ux injection and comlete mixing hyothesis. Lefthand gure shows the CDF and CCDF of β and the right-hand gure shows the PDF of β Verication of the MASCOT uscaling for contaminant disersion along the y-axis, y, at 50 meters from the inlet with ux injection and comlete mixing hyothesis A.1. Sace discretization at the layer interface l jk B.1. Scheme of a lain fracture B.2. Examle of a 2D Fracture Network realization over a domain of 10x10 meters B.3. Channel Network comuted from the Fracture Network realization of Figure (B.2). The line size are roortional to the ux through the fracture B.4. Particle tracking in the Channel Network of Figure (B.3) with ux injection boundary condition and erfect mixing at each nodes of the network B.5. Examle of two fracture intersection B.6. Possible streamline distributions. In left side Figure the case with q j > q i and in the Figure on the right the case with q j < q j C.1. Probability density function of lognormal distribution for length of fracture for the crystalline rock based on literature review (left hand gure) and adoted in DFN simulations reorted in this thesis (right hand gure) C.2. Probability density functions of lognormal distribution for aerture based on literature review (left hand gure) and adoted in DFN simulations reorted in this thesis (right hand gure) C.3. Probability density function for fracture aerture used in this study.. 78 viii

14 List of Tables 2.1. Retardation factor for dierent isotherm sortion Particles' transitions and corresonding rates Physical and hydraulic arameters of the Bentonite and clay layers of the otential reository of HLW in Zürchel Wienland [NAGRA, 2005] Transition rates of the stochastic model fot the otential reository of HLW in Zürchel Wienland Classication of transort models in fractured media B.1. Main DFN inut arameters used in Figures (B.3-B.2.1) C.1. Statistics of fracture geometry arameters used in DFN simulation ix

15 List of acronyms AD Advection-Disersion ATS Automatic Time Ste BTE Boltzmann transort equation CCDF Comlementary Cumulative Distribution Function CDF Cumulative Distribution Function CN Channel Network DFN Discrete Fracture Network EBS Engineered Barriers System FDM Finite Dierence Method FEM Finite Element Method HLW High Level Wastes ILW Intermediate-level waste KD Kolmogorov-Dimitriev LLW Low-level waste MASCOT Montecarlo Analysis of Subsurface COntaminant Transort MC Monte Carlo PDF Probability Distribution Function RW Random walk SC Stochastic Continuum URL Underground Research Laboratory x

16 Part I Thesis 1

17 An engineer is someone who does list rocessing in FORTRAN V. Michael Powers Chater 1 Introduction 1.1. Problem statement Throughout history, eole have disosed of most tyes of solid wastes by either burning them or burying them. All too often this has resulted in a hasty and convenient shallow grave for all kinds of environmentally unfriendly materials. As a consequence, the ercetion of waste burial is often dirty, old-fashioned landll sites, strewn with garbage, and of contaminated lakes and rivers. So, when eole hear of lans to bury radioactive wastes underground, they are understandably concerned for the safety of local inhabitants and for the environment. However, the reality of radioactive waste disosal is so far removed from the images of common waste tis as to bear no comarison. Indeed, in the last decades, the concet of underground waste disosal using urose-build, engineered facilities has been develoed to a degree far in advance of any other disosal ractice adoted in any other industry, reecting the high standards of safety which the nuclear industry is exected and legally required to achieve [Miller et al., 2000]. The location of a reository, its design and the deth burial deend very much on the tye of waste it is intended to contain, in terms of its level of radioactivity and hysical and chemical roerties. The waste materials, and the engineered barriers which initially contain them within the reository, are exected eventually to degrade and it is anticiated that some residual waste radionuclides might return to the surface in very low concentrations at some time in the distant future as art of the natural rocesses of groundwater movement and environmental change. One of the challenges facing the nuclear industry is to demonstrate condently that a reository will contain wastes for so long that any release that might take lace in the future will ose no signicant health or environmental risk. The long time scales of otential danger associated with radioactivity require that the assessment of the isolation erformance of the disosal site and facility be obtained by alying models which simulate the migration of radionuclides from the disosal facility to the bioshere through the various articial and natural barriers [ Cashwell and Everett, 1959; Tomson et al., 1987]. The erformance and the security of these 2

18 1.2 Disosal of radioactive waste Figure 1.1.: Possible layout of the SKB reository. barrier systems are evaluated by means of the redictive models that can simulate the transort of the radionuclides from the reository to the bioshere. In this contest, dierent mathematical model have been develoed which can deal with the comlex hysical structure and irregular geometric shaes of the reository system and they can describe the comlex hysical biological henomena that tyically occur in the dierent barriers system. The results of the simulations are used to design the disosal system in such a way that the nal radiological environmental imact of the disosal system meets with the safety criteria issued by the national and international regulatory agencies such as the IAEA (The International Atomic Energy Agency) and the OECD/NEA (Nuclear Energy Agency of the Organization for Economic Co-oeration and Develoment). The disosal of nuclear waste is a multifaceted subject involving olitics, legislation, international treaties, engineering technology, ublic accetance, but above all the decisive factor lies in the assurance of long-term safety Disosal of radioactive waste The safe long-term disosal of High Level Wastes (HLW) is a ressing issue for many countries that rely on nuclear ower [reort by the Uranium Institute Waste MAnagment & Decommission Working Grou, 1999; Savage, 1995] and many countries lan dee underground storage for their nuclear waste. For examle, Sweden, after a survey of feasibility studies, is now in the hase of site investigations. Two sites close to Äsö, Simevar and Laxemar [SKB, 2000], and one site in Forsmark [SKB, 2000] are currently under investigation to establish a foundation for deciding the site for dee storage. A ossible layout of the dee reository for HLW develoed by SKB is shown in Figure 1.1. Condence in redicting subsurface geohydraulic conditions form surface and borehole investigations has been gained form Underground Research Laboratory (URL) [NEA, 2001]. The URLs, which also serve as dress 3

19 1.2 Disosal of radioactive waste Figure 1.2.: Sources of radioactive waste in Switzerland and the waste management strategy considered after [NAGRA, 2005] rehearsals for future reository construction, facilitate in-situ exeriments and sulementary rock characterization, which is indisensable for understanding rocess in the reository environments. Major URL rogrammes have undertaken during last decades, including Stria [Olsson, 1992] and Äsö hard rock laboratory [Rhén et al., 1997] in Sweden, Whiteshell URL in Canada, Grimsel Test site (GTS) in Switzerland, Fanay-augies mine in France, and Kamaishi and Tono mines in Jaan Origin and classication of radioactive wastes The majority of the radioactive wastes created around the world are the unwanted by roducts of electricity generation using nuclear ower, and of military activities; however, there is also a large number of industrial, medical and scientic research activities which use radioactive materials and create radioactive wastes, albeit in relatively small amounts. In Figure (1.2) are reresented all the nuclear sources of radioactive waste in Switzerland and the waste management strategy considered. These last uses of radioactive materials mean that many countries than just those with nuclear ower have a waste roblem to address, although the magnitude of the roblem is much smaller in countries without a nuclear ower rogramme. There are several systems of nomenclature in use, but the following is generally acceted: Exemt waste Excluded from regulatory control because radiological hazards are negligible. Low-level waste (LLW) Contains enough radioactive material to require action for the rotection of eole, but not so much that it requires shielding in handling or storage. Intermediate-level waste (ILW) requires shielding. If it has more than 4000 Bq/g of long-lived (over 30 year half-life) α emitters it is categorized as "long-lived" and requires more sohisticated handling and disosal. 4

20 1.2 Disosal of radioactive waste Figure 1.3.: Multile barriers revent the radionuclides from reaching the ground surface. High-level waste (HLW) suciently radioactive to require both shielding and cooling, generates > 2 kw/m 3 of heat and has a high level of long-lived α-emitting isotoes Multi-barrier concet The reository is based on the concet of multile-barrier system (see Figure 1.3) so that the safety of the reository does not rely on a single barrier, whose failure alone could comromise these objectives. A multi-barrier aroach ensures this isolation, combining a series of lanned Engineered Barriers System (EBS) situated dee below the surface within a suitable stable geological location (Dee Geological Disosal). The EBS is one art of this assive multi-barrier system aroach. The Multi-Barrier Safety System has several dened zones: Near Field - The near eld encomasses the EBS and disturbed zones due to tunneling work carried out to create a reository. Far Field (Geoshere) - The far eld is the geological mass surrounding the near eld. Bioshere - The bioshere is the near surface, organic zone that contains life. It is exected that most of the activity initially resent decays while the wastes are totally contained within the rimary waste containers, articularly in the case of HLW, for which the high integrity steel canisters are exected to remain unbreached for at least years. Even after the canisters are breached, the stability of the HLW waste forms in the exected environment, the slowness of groundwater ow and a range of geochemical immobilization and retardation rocesses ensure that radionuclides continue to be largely conned within the EBS and the immediately surrounding rock, so further radioactive decay takes lace. 5

21 1.3 Performance assessment Although comlete connement cannot be rovided over all relevant times for all radionuclides, release rates of radionuclides from the waste forms are low, articularly from the stable HLW waste forms. Furthermore, a number of rocesses attenuate releases during transort towards the surface environment, and limit the concentrations of radionuclides in that environment. These include radioactive decay during slow transort through the barrier rovided by the host rock and the sreading of released radionuclides in time and sace by, for examle, diusion, hydrodynamic disersion and dilution Performance assessment A safety assessment is a comlex matter in itself. It can be summarized as a detailed system descrition of the ost-closure reository state, for which the consequences of otential future develoments are evaluated with regard to long-term safety [NEA, 2001; SKB, 1999]. A art of the safety assessment is the erformance assessment, it is more technical and limited to the rocesses relevant for the liability of the disosal system. Its urose is to evaluate if a otential site meets the requirements in functionality for hosting a reository [Andersson et al., 1998; Tsang et al., 1988]. In order for a erformance assessment to be able to calculate the exected dose and risk for a reository, it is necessary to reresent understanding of the system behaviour by a series of concetual models which can be converted into simle mathematical models for comutational uroses (Figure 1.3). At the most basic, a erformance assessment will need to include concetual and mathematic model for radionuclide migration with the groundwater through the articial matrices hosting the waste (near eld) and the natural rock matrix of the host geoshere (far eld) [Savage, 1995]. The modeling task is rendered articularly dicult by the inherent heterogeneity of the articial and natural media and by the comlexity of the involved transort henomena, so that the classical aroaches are forced to resort to aroximations which reduce the adherence of the model to reality Modelling radionuclide transort in groundwater The most common aroach adated to modelling contaminant (radionuclide) transort in groundwater is the classical Advection-Disersion (AD) aroach which is based on a mass balance over a control volume and on the assumtion of validity of Fick's law to describe the mechanisms of hydrodynamic disersion. The resulting artial dierential equations need to be couled to the equation describing the ow of groundwater in the medium to determine the groundwater velocity eld, i.e. the advective velocity. Solution to the ow equation also allows to determine the tensor disersion coecient whose longitudinal and transverse comonents are found to deend linearly, to a rst aroximation, on the local Darcy's velocity. The factor that relates the disersion coecient to Darcy's velocity is usually referred to as disersivity. Unfortunately, the disersivity so introduced as a local arameter does not satisfactorily account for the exerimental data since it actually turns out to be an integral arameter deendent on the whole system under investigation. In other 6

22 1.4 Modelling radionuclide transort in groundwater Figure 1.4.: Postclosure erformance of the reository. words, the identied values of the disersivity deend on the distance or time of tracer roagation, this deendence is called scale eect. This fact reveals the inconsistency of considering disersivity as a hysical roerty of the medium only [ Gelhar and Axness, 1983]. Many researchers have attemted to exlain the scale eect henomenon. One exlanation is that macroscoic heterogeneity and anisotroy can cause the solute to be transorted much faster then the corresonding results of model calculations based on the mean velocity. To account for these asects of heterogeneity and anisotroy, the statistical theory of mass transort has been develoed [Dagan, 1989; Gelhar and Axness, 1983], based on the re-interretation of the equations as stochastic dierential equations in which the advective velocity eld is a random eld. This leads to a situation in which Fick's law is no longer alicable and a varying disersivity arameter is obtained which can exlain, in theory, the exerimental observations. However this way of treating the contaminant transort roblem has not yet been roved useful in ractical assessment as it results in comlex integro-dierential equation of dicult solution. Another exlanation is that the discreancy derives form the fact that the exerimental observations are interreted through an oversimlied model [ Hutton and Lighfoot, 1984]. In fractured media, the heterogeneity and stochasticity of the media in which the disersion henomenon takes lace renders classical analytical-numerical aroaches scarcely adequate in ractice. Moreover consider the host matrix as a homogeneous continuum is an unaccetable modelling simlication [Sahimi, 1995]. Particle tracking in stochastically generated networks of discrete fractures rovides an alternative to the conventional AD descrition of transort henomena [Berkowitz, 2002; Bodin et al., 2003; Moreno and Neretnieks, 1993; Sahimi, 1995]. Numerous Discrete Fracture Network (DFN) models have been develoed and tested [Framton et al., 2005; Mo et al., 1998]. However, DFN are comutationally intensive and thus are not feasible for alications involving large rock volumes [Berkowitz, 2002; Painter and Cvetkovic, 2005; Sahimi, 1995]. Given the above observations, the resent 7

23 1.5 Objective and aims of the resent work research aims at develoing new stochastic models for the descrition of contaminant transort in groundwater system. The stochastic aroach adoted for the descrition of radionuclide transort through orous media is based on the Kolmogorov-Dimitriev (KD) theory of branching stochastic rocesses [Kolmogorov and Dmitriev, 1947]. The aroach is based on a discretization of the sace variable into comartments and on exlicit descrition of the individual transitions that the contaminant articles may undergo during the transort rocess, according to secied transition robabilities [Marseguerra and Zio, 1997]. A rst advantage of the aroach is that it has a exible structure which is well suited for dealing with multidimensional geometries and also for modelling various henomena. An obvious limitation lies in the fact that the larger is the variety of contaminant articles considered the higher the order of the system couled dierential equations to be solved. For the simulation of radionuclide transort in fractured media, a stochastic aroach, largely mutuated from the transort of neutron in non-multilying media and governed by the linear Boltzmann transort equation (BTE) [Williams, 1992, 1993], has been develoed. The roosed model interrets the oints of branching of the fractures as collision oints in corresondence of which a seudo-scattering reaction occurs and the average distance between successive breaching oints is treated as an eective mean free ath. The balance equation for the mass concentration results into as integro-dierential transort equation for its exected values formalized into the BTE. This aroach has many similarities with the time-deendent neutron transort equation. The comlexity of the transort henomena and the features of the roosed aroaches lead naturally to Monte Carlo (MC) simulation. In MC methods, a large number of histories of article motion are generated by random samling aroriate distribution functions. The article are then followed in their movements according to equation describing their motion and other hysical-chemical-biological rocesses such as adsortion and desortion from the host matrix [De Marsily, 1986]. In this view, a code named Montecarlo Analysis of Subsurface COntaminant Transort ( MASCOT) is being imlemented and veried in dierent cases. The MASCOT code, written in FORTRAN 90 standard, has a very exible structure. This feature allows one to simulate the transort through multidimensional comlex geometries and to describe a wide range of henomena, such as chemical reactions and radioactive decay, by accounting the individual interactions that each article may undergo during the transort Objective and aims of the resent work The overall objective of this thesis is to contribute to the advancement in the modelling of radionuclide groundwater transort. The study of radionuclide transort through the dierent barrier of the reository is very comlex and multidiscilinary. A ossible simlied concetualization is here sketched. At rst, the water inltrates through the EBS. Then, the radionuclides are released into seeage water contacting breached waste ackages in the reository. The 8

24 1.5 Objective and aims of the resent work Figure 1.5.: Radioactive contaminant transort rocesses studied in this thesis. radionuclides in solution migrate downward through the unsaturated zone to the water table (transort in unsaturated orous media). At that oint, the radionuclides enter the saturated zone and migrate down-gradient over large distances through saturated fracture media, until the utake into the accessible environment (bioshere). In this thesis the focus is on modelling and simulating the transort of radionuclides within the groundwater system through both orous and fractured media. In articular, the asects studied are (Figure 1.5): 1. Measurement of Water Inltration Through Engineered Barrier Systems ( Paer I); 2. Transort in Unsaturated Media and in Transient Flow Field Conditions ( Paer II and Reort I); 3. Transort in resence of Nonlinear Processes in Porous Media (Paer III and Reort I); 4. Transort in Fractured Media (Reort II and III) Dissertation outline The outline of the thesis is as follows. A brief review of the classical AD model of radioactive contaminant transort in orous media is resented in Chater 2. In Chater 3, the stochastic aroach to radioactive contaminant transort in orous media is derived from the KD theory of branching stochastic rocesses. The original stochastic model reviously develoed by [Marseguerra and Zio, 1997] has been generalized to allow the simulation in unsaturated and non-uniform ow eld, in steady/unsteady ow eld and in resence of linear and nonlinear sortion rocesses. 9

25 1.5 Objective and aims of the resent work Moreover, an alication regarding the modeling of radionuclide release from a otential reository in clay formation (orous media) is resented. In Chater 4 the state of the art of modelling radionuclide transort in fractured media is resented. A MC aroach to simulating the radionuclide transort in fractured rock based on an analogy with neutron transort, is resented in Chater 5. A otential diculty arises in the determination of the values of arameters required by the model. To overcome this diculty, the stochastic model has been transformed into a so-called hybrid model, in which the DFN simulation is adoted to comute the arameters required by the MC simulation of the stochastic transort. Finally, the conclusions drawn from this research exerience are shared in Chater 6. 10

26 Basic research is what I am doing when I don't know what I'm doing Werner von Braun Chater 2 Transort in orous media 2.1. Introduction Solute transort in groundwater is generally a result of comlex interactions between advection of the moving groundwater and various hysical, chemical and biological mechanisms that act to further immobilize and transform the solutes. The transort is also signicantly inuenced by the natural heterogeneity of aquifer roerties, such as the satially random distribution of the hydraulic conductivity, e.g., [ Dagan, 1989; Gelhar and Axness, 1983]. In tyical heterogeneous aquifers, hydraulic conductivity values can vary by orders of magnitude over a few meters, e.g., [Freeze, 1975]. Physically, advection is the main rocess resonsible for solute transort. In addition to this bulk motion with water ow, a sreading henomenon exists, called hydrodynamic disersion. This is due to the concomitance of molecular diusion and mechanical disersion. The former henomenon deends on the thermal-kinetic energy of the solute articles, it is imortant only at low velocity and causes the contaminant to move from higher concentration zones to lower concentration zones. Mechanical disersion, instead, is caused entirely by the motion of uid in the randomly oriented ores of the medium and consists of a change, in direction and magnitude, of water velocity. The travel of the solute is further comlicated by the fact that a fraction of it is adsorbed in the solid matrix of the orous medium and subsequently desorbed with a given characteristic time which may be very large. This henomenon gives rise to dynamic exchanges between the free and the traed contaminant fractions, which determine an overall decrease of the contaminant seed, i.e. a retardation in contaminant roagation. Dierent aroaches have been roosed for modelling the transort of contaminant in groundwater; the most widely alied is the classical AD. During the last two decades many stochastic theories have been develoed for redicting the fate and transort of solute in heterogeneous aquifers, and are reviewed in several recent textbooks [Dagan, 1989; Miller et al., 2000; Rubin, 2003]. The aim of these stochastic models is to overcome the limitation of the AD aroach. In ( 2.2) the classical aroach of the AD is resented. The limitation of this 11

27 2.2 Deterministic aroach: the Advection-Disersion Equation aroach are resented in ( 2.3) and the stochastic model roosed to overcome these limitation are resented in ( 2.4) Deterministic aroach: the Advection-Disersion Equation Most attemts at quantifying contaminant transort have relied on a solution of some form of a well-known governing equation referred to as the AD equation. The term advection is used here in reference to the term convection because convection often carries the connotation of transort in resonse to temerature- induced density gradients. The AD equation is derived by combining a mass-balance equation with an exression for the gradient of the mass ux (see [Bear, 1972]). The equation that govern the radionuclides movement through rock and other subterranean materials may be written as [Freeze and Cherry, 1979]: C t = D C UC λc 1 f (F, C) (2.1) n where C = C(r, t) is the radionuclide concentration at osition r and time t in the water, D is the disersion coecient tensor, U is the advective water velocity, λ the decay constant of the radionuclide, n is the orosity of the medium (ratio between the volume of liquid and the total volume) and f (F, C) reresent the exchange of the radionuclide between the rock and the water. The balance equation for the radionuclide adsorbed on the rock matrix can be written as: F t = λf + 1 f (F, C) (2.2) 1 n where F = F (r, t) is the radionuclide concentration at osition r and time t adsorbed on the rock matrix. Dierent exressions exist for modelling dierent adsortion henomena as the linear isotherm relationshi (Ÿ2.2.1), the Freundlich isotherm (Ÿ2.2.2) and the Langmuir isotherm (Ÿ2.2.3) Linear isotherm The oular linear isotherm relationshi assumes the contaminant concentration in the solid hase, F, directly roortional to that in the liquid hase, C. The linear isotherm is exressed as: F = K d C (2.3) where K d, called distribution coecient of linear isotherm, is exressed tyically in [mg/g] or in [l/kg]. The K d aroach may adequately describe contaminant migration and reversible sortion in groundwater system in which the domain has uniform mineralogical and chemical comositions and remains uniform through the time scale of interest. The K d aroach could not roerly describe contaminant migration in groundwater systems undergoing dynamic chemical evolution. Furthermore this aroach is valid for 12

28 2.2 Deterministic aroach: the Advection-Disersion Equation dissolved secies that are resent at concentration less than one-half of its solubility [Lyman et al., 1982]. Under the hyothesis of isotherm linear equilibrium between the contaminant resent in the liquid hase, C, and the contaminant adsorbed on the host medium, F, the Eq. (2.1) becomes: [ 1 + ρk ] [ d C = D C UC λ 1 + ρk ] d C (2.4) t n n The equation (2.4) may be re-written as: t R dc = D C UC λr d C (2.5) where R d = 1 + ρk d n is the so-called retardation factor (always greater then unity) which describes the delay in the roagation on the concentration front caused by the adsorbed henomena Freundlich isotherm The Freundlich isotherm relates the mass of the solute secies adsorbed on the host matrix, F, and the solute concentration, C according to the follow relationshi: log F = n f log C + log K f (2.6) or F = K f C n f (2.7) where n f is the Freundlich exonent and K f [L 3n /M] the Freundlich artitioning coecient. Under the hyothesis of the Freundlich isotherm the Eq. ( 2.1) becomes: [ where R d = 1 + ρk f n Langmuir isotherm t R dc = D C UC λr d C (2.8) ] n f C n f 1 is the retardation factor. The Langmuir model is adoted to describe the deendence of the surface coverage of an adsorbed contaminant on the concentration of solute at a xed temerature. The equation was develoed by Irving Langmuir in 1916 and is based on the following assumtions: each surface site can be singly occuied there are no lateral interactions between adsorbed secies the enthaly of adsortion is indeendent of surface coverage there is dynamic equilibrium between the adsortion and desortion rocesses 13

29 2.3 Limits of the Advection-Disersion aroach Table 2.1.: Retardation factor for dierent isotherm sortion. Sortion isotherm Linear Freundlich Langmuir h 1 + Retardation factor, R d 1 + ρk d n 1 + ρk f n n f C n f K l C K l C (1 + K l C) 2 i Q K l Mathematically the Langmuir isotherm can be written as: F = K l C Q 1 + K l C (2.9) where K l is the equilibrium constant for the adsortion reaction and Q is the number of sortion sites (i.e. the maximum amount of sorbed contaminant). Deriving the Eq. (2.9) resect to time and substituting in Eq. 2.1 one obtains: C(z, t) R d = D 2 C(z, t) t z 2 v [ C(z, t) z { where the retardation factor Rd = 1 + The retardation factors are summarized in Table 2.1. ] 1 1+K K l C l l C (1+K l C l ) 2 λr d C (2.10) Q K l } Limits of the Advection-Disersion aroach In the AD model, the characteristic henomena of ollutant transort are described by average, directly measurable, arameters such as the mean ow velocity, the retardation factor and the hydrodynamic disersion coecient. Unfortunately, this last coecient deends on the distance and time of trace roagation in such a way that, often, the exerimental values measured in laboratory are lower, even of some orders of magnitude, than those measured in situ [Geier et al., 1992]. This so called scale eect reveals the inconsistency of considering this arameter as a local characteristic reresentative of the system. In fact, if the disersion were a fundamental arameter of the system, its value should indeendent of the system size or time over which measurement are made. Moreover, the standard aroach in analyzing disersion, which is embodied in the AD, is to use an average linear velocity. A number of investigators maintain that this aroach is reasonable because it will never be ractical to dene the velocity eld in detail. If the AD equation is to be used to evaluate transort in groundwater, two questions must be addressed: how to overcome the roblem of the scale eect and is 14

30 2.3 Limits of the Advection-Disersion aroach it valid to assume that the disersion comonent of Eq. (2.1) can be reresented by a form of Fick's law? Is disersion a Fickian rocess? Much of the theory on which the analysis of disersion in orous material is based stems from the ioneering work of [Taylor, 1921, 1953], who suggested that disersion could be reresented as a Fickian rocess. This assumtion as alied to orous media has since been questioned by several researchers. [Fried, 1975] cites several examles of laboratory exeriments in which the exerimental results did not t theoretical curves derived from solutions of the advection-disersion equation, and [Dagan, 1987] noted that there is no a riori reason to believe that the diusion tye equation is valid at all for contaminant transort through orous media. Taylor and others, who continued the develoment of the theory of disersion clearly recognized that the Fickian assumtion is valid only after a certain length of time has elased in which the disersion rocess develos. A rocedure for redicting the length of this initial develoment eriod has not yet been erfected. Indeed, until recently it was not recognized that for the heterogeneous systems encountered in the eld this develoment rocess requires substantial transort from the source. However, the growth of the variance of the concentration distribution during the non-gaussian eriod did not follow a Fickian model. If disersion is a Fickian rocess, then the variance of the concentration distribution should increase linearly with time or distance, and the longitudinal disersion coecient and the disersivity will be a constant for constant velocity. Ref. [Dagan, 1982] concluded that longitudinal disersion can be reresented asymtotically by a Fickian equation, with disersivity much larger than ore-scale disersivity, but given the uncertainties involved in dening the hydrogeologic system a stochastic aroach is necessary and the traditional aroach of redicting solute concentrations by solving deterministic artial dierential equations is highly questionable in the case of heterogeneous formations Transort in heterogeneous media The imact of small-scale satial variability uon eld-scale transort is widely recognized as a key roblem in groundwater research and ractice [Gelhar and Axness, 1983]. In fact, the subsurface environment is very heterogeneous and its roerties vary dramatically over sace and consequently the inut arameters of the AD equation such as the ow velocity, the retardation factor and the hydrodynamic disersion coecient result functions of the sace. It is generally infeasible and cost-rohibitive to conduct detailed site characterization to obtain exact information to dene this satial variability. Hence, the distributed values of these arameters have to be inferred from observations or measurements made at discrete set of samling oints or observations that in general are sarsely distributed over the aquifer. The uncertainty associated with these undened inut arameters results in model rediction errors. Hence, it is necessary to incororate arameter uncertainty into models to increase condence in redictions. 15

31 2.4 Need of stochastic aroach 2.4. Need of stochastic aroach A site investigation must minimize its detrimental inuence on safety for future disosal at the site [Tsang et al., 1994]. Thus, due to the destructive nature of borehole samling, the data available for a site is tyically scarce in relation to the domain that needs to be characterized [Berkowitz, 2002], which causes a source of uncertainty. It is however imortant to distinguish between variability and uncertainty: a quantied roerty may exhibit satial or temoral variability, although roerly dened for given sace-time oints, while uncertainly reects incomlete characterization in sace and/or time [NEA, 1997]. This necessitates the use of a stochastic framework to estimate the range of ossible outcomes for a rocess that deends on an incomletely characterized and satially variable roerty. A stochastic descrition treats a roerty as a random variable and its observed values at various oints as realizations of a random rocess [Gelhar, 1993; Kitanidis, 1997; Mishra, 2002; Niemi, 1994]. A data set rovide an estimate of the roerty ensemble that denes the ossible outcomes of the random variable at any given oint in sace. Stochastic modelling then treats multile realizations of the random variable to estimate the range of ossible outcomes for a rocess that deends on the medium roerty. During the ast two decades, hydrogeologic studies have commonly used stochastic tools to incororate the eects of satial variability of hydrologic roerties and arametric uncertainty into the redictive caabilities of numerical groundwater ow and contaminant transort models. These studies have made it clear that inadequate and insucient data limit the ability of these models to redict system behavior without substantial uncertainty [Pohll et al., 1999; Pohlmann and Chaman, 2000]. Uncertainty is always inherent in the model rediction and is the result of the inability to fully characterize the subsurface environment and the rocesses controlling the system behavior. Stochastic aroaches that assign robability distributions to these arameters rovide a means to deal with the arameters and rediction uncertainties. These aroaches roosed in the early eighties have increasingly been acceted in subsurface studies. Stochastic-based modeling rovides a ossible range of solutions (e.g. contaminant concentrations and hydraulic heads) to groundwater-related roblems accounting for uncertainty associated with ow and transort arameters. Successful alication of stochastic methods in eld-scale modeling of ows and transort, and risk analysis has been reorted in literature [Dagan, 1982; Elfeki et al., 1995; Rubin, 2003]. For these reasons the stochastic aroach to simulate the radionuclide transort in orous media has been adoted in this thesis (see Chater 3). 16

32 If the facts don't t the theory, change the facts. Albert Einstein Chater 3 Monte Carlo simulation of transort in orous media 3.1. Introduction In this chater the stochastic model develoed to simulate contaminant transort in orous media is resented. The original develoment of the aroach is due to [Marseguerra and Zio, 1997], for the case of transort in homogeneous saturated orous media under steady ow conditions. In this research activity, the stochastic model has been generalized to simulate the contaminant transort in more realistic condition such as the transort in heterogeneous media, in saturated and unsaturated conditions, in resence of steady-state or transient ow eld and in resence of linear and nonlinear reactive rocesses. In ( 3.2) the stochastic aroach, based on the KD theory of branching stochastic rocesses, is resented. The MC aroach adoted to evaluate the stochastic model is resented in ( 3.3). ( 3.4) resents the verications of the stochastic model and the corresonding MC code in dierent conditions and nally, alications of the stochastic model are resented in ( 3.5) The stochastic model of contaminant transort The stochastic aroach adoted derives from the KD theory of branching stochastic rocesses [Kolmogorov and Dmitriev, 1947]. In its original formulation, the aroach, bears similarities with that of [Lee and Lee, 1995; M.Antonooulos-Domis et al., 1995]. The model is based on the discretization of the sace variable into comartments and the individual transitions that the contaminant articles may undergo during the transort rocess are modeled exlicitly by secied transition robabilities. The aroach diers from the classic random walk rocedure in that also the advective comonent, as well as the diusive one, is incororated into the stochastic descrition. According to this model, several kinds of articles corresonding to the dierent contaminant locations and states are introduced and, then, suitable systems of couled 17

33 3.2 The stochastic model of contaminant transort Table 3.1.: Particles' transitions and corresonding rates. Particle Transition Rate Forward transfer to adjacent zone (i i + 1) π (i i + 1, t) Backward transfer to adjacent zone (i i 1) π (i i 1, t) S Adsortion on the orous host matrix ads (i, t) Other chemical transformation γ (i, t) Radioactive decay λ (i, t) Desortion from the orous host matrix des (i, t) T Other chemical transformation γ (i, t) Radioactive decay λ (i, t) ordinary dierential equations for the number of articles of each kind are derived. The roblem is, thus, tackled in a very direct and straightforward manner and the analysis is signicantly simlied by working in terms of robability distribution functions. For illustration uroses, let us refer to a 1 D sace domain, subdivided in N i zones, i = 1, 2,..., N i. Two classes of articles are introduced: the solutons, which are articles of contaminant free to move with and within the water owing through the matrix ores; the traons, which are immobile articles of contaminant adsorbed on the solid matrix. Let S (i, t) (T (i, t)) denote a soluton (traon) of tye (e.g. chemical secies or isotoe), = 1,... N, residing in zone i at time t. Table 3.1 summarizes the various transitions that the two classes of articles may undergo during their transort and the corresonding rates. The soluton S (i, t) is free to move to other zones of the medium: for simlicity, we consider only transfers backward and forward to adjacent zones, whose occurrence in times is described robabilistically by the transition rates π (i i 1, t) and π (i i + 1, t), resectively (Figure 3.2). The soluton may also be subject, with transition rate ads (i, t), to reversible adsortion on the solid matrix of the host medium, thus transforming into a traon T (i, t). Moreover, it may transform into a dierent article, due to a chemical reactions, with rate γ. Finally, if the soluton is a radionuclide, it may transform into a dierent -contaminant according to its characteristic decay rate λ (Figure 3.2). The traon becomes a soluton by desortion from the host matrix, with rate des (i, t). Otherwise, it may transform into a dierent tye of traon due to chemical reactions or radioactive decays with rates γ and λ, resectively (Figure 3.2). The transition rocesses above described may be framed within the KD theory of branching stochastic rocesses [Kolmogorov and Dmitriev, 1947] from which the following system of 2 N N i artial dierential equations is obtained for the exected numbers of solutons N S (i, t) and traons N T (i, t) of -kind in zone i at time t 18

34 3.2 The stochastic model of contaminant transort π (i i 1, t) γ T (i, t) ads (i, t) π (i i + 1, t) S (i 1, t) S (i, t) S (i + 1, t) λ S (i, t) S (i, t) Figure 3.1.: Schematics of the soluton articles rocesses. T (i, t) T (i, t) γ T (i, t) des (i, t) S (i, t) λ Figure 3.2.: Schematics of the traon articles rocesses. 19

35 3.2 The stochastic model of contaminant transort [Marseguerra and Zio, 1997]: N S (i, t) t = [π (i i + 1, t) + π (i i 1, t)] N S (i, t) + π (i + 1 i, t)n S (i + 1, t) + π (i 1 i, t)n S (i 1, t) + des (i, t)n T (i, t) ads (i, t)n S (i, t) + λ N S (i, t) λ N S (i, t) (3.1) + γ N S (i, t) γ N S (i, t) N T (i, t) t = des (i, t)n T (i, t) + ads (i, t)n S (i, t) + λ N T (i, t) λ N T (i, t) (3.2) + γ N T (i, t) γ N T (i, t) These are balance equations describing the roduction and loss rocesses of articles in zone i. The rst two terms on the right hand side of Eq. (3.1) describe the disaearance of solutons of tye due to the transfer to an adjacent zone; the third and fourth terms describe the aearance of solutons S in zone i because of transfer from adjacent zones; the fth term reresents the transformation of solutons S into traons T due to the adsortion on the host matrix; the sixth term reresents the roduction of solutons S due to desortion of traons T from the host matrix; the seventh and eighth terms account for the disaearance of solutons S by transformations due to radioactive decay and chemical transformations; the last two terms describe the roduction of solutons S due to decay and chemical transformations of other secies of solutons into the -kind. Similar balance considerations aly to Eq. (3.2). Of course these equations must be sulemented with the roer initial and boundary conditions. In Eqs. ( ), the transort henomena are reresented exlicitly in robabilistic terms, this rovides the modeling exibility necessary for including time deendence and nonlinear henomena Nonlinear stochastic model The stochastic model of radioactive contaminant transort in orous media can be extended to simulate the transort in resence of nonlinear rocesses. The nonlinear rocesses are described in the stochastic model by means nonlinear transition rates. The transitions rates result functions of the articles number resent in that zone at that time. For examle, in case of nonlinear adsortion rocess governed by the Freundlich isotherm, the transition rate of adsortion, ads, deends of the number of soluton articles resent in that zone at that time, ads ( i, t, NS (i, t) ). Details of the 20

36 3.3 Monte Carlo simulation nonlinear stochastic model for radionuclide transort in orous media are reorted in Paer II and in Reort I Parameters determination of the model The eectiveness of this reresentation is conditioned by the caability of estimating the values of the transition rates that govern the modeled rocess. Ref. [Ferrara et al., 1999] has shown that the values of the transition rates of the stochastic model ( ) can be estimated by a comarison with the equations of the corresonding classical AD model written in nite dierence form. The arameters determination obtained by [Ferrara et al., 1999] was limited to the case of transort in saturated homogeneous orous media steady state ow eld. In Paers II, III and in Reort I the values of the arameter of the stochastic model has been determinated for the cases of transort in unsaturated and non-uniform ow eld, in steady/unsteady ow eld and in resence of linear and nonlinear sortion rocesses and summarized in Aendix A Monte Carlo simulation The comlexity of the transort henomena and the features of the roosed aroach lead naturally to Monte Carlo simulation [Cashwell and Everett, 1959; Sanier and Gelbard, 1969]. In this view, the imlementation of a code named MASCOT has been undertaken. In standard MC simulation aroach the individual fates of a large number of articles are followed, one at the time, from the birth to the dead of the articles due to the escae of the interesting zones or due to the transformations in a article kind that are not worthy of further analysis [Lux and Koblinger, 1991; Marseguerra and Zio, 2002]. At a given time t, the state of contaminant, i.e. the kind and the localization of the contaminant, is reresented by a oint (k; i; t) in the hase sace. Each trial of a standard MC simulation consists in generating a random walk which guides the contaminant from one state to another, at dierent times [ Marseguerra and Zio, 2002]. The aroach consists in samling rst the time t of a article transition from the corresonding conditional robability density T (t t, k ), called free ight kernel, of the article erforming one of its ossible transitions at time t given that the revious transition occurred at time t and that the article, as a consequence of that transition, entered in state k. Then, the transition to the new state k actually occurring is samled from the conditional robability C(k t, k ), called collision kernel, that the article enters the new state k given that a transition has occurred at t when the article was in state k. The rocedure then reeats from k at time t to the next transition. Aroriate counters are introduced which accumulate the contributions to contaminant concentrations. After erforming all the MC histories, the content of each counter, roerly normalized, gives an estimate of the mean contaminant concentration in that zone at that time. This rocedure corresonds to erforming an ensemble average of the realizations of the stochastic rocess governing the transort rocess. 21

37 3.3 Monte Carlo simulation non linear rate dτ=250 dτ=100 dτ= Non Linear rate dτ=250 with effective arameter π k 0.5 π k Time Time Figure 3.3.: Aroximation of the nonlinear rate π k with standard xed time ste rocedure (gure on the left) and with the burn-u technique (gure on the right) Nonlinear Monte Carlo simulation For treating nonlinear transort rocesses, the standard MC rocedure, linear by denition in that it builds the solution by ensemble-averaging the transort fates of the individual contaminant articles, has been roerly modied. In ( ) dierent algorithms adoted to simulate the nonlinear rocesses are resented Fixed time ste A classical method to simulate nonlinear rocesses is to follow the articles roagation in successive time ste of a given length dτ. All the articles resent in the system are rocessed one at a time at each time ste and the transition rates are udated according to the satial distribution of articles currently resent in the system; then, the simulation moves on to the next time ste. For a generic nonlinear rate, π k, the rate is udated according the following rule: { πk (i, t) = f ( N x (i, t i ) ) t i < t t i + dτ (3.3) where we have indicate with N x the number of article of kind x that governs the evolution of the nonlinear rate π k and f reresent the generic relationshi between the nonlinear rate and the number of x -article resent at time ste t i. A dierent algorithm, mutuated from the burn-u simulation of the fuel in the nuclear reactors, has been adoted and imlemented in the MASCOT code to imrove the accuracy of the MC nonlinear simulation. All the articles resent in the system are simulated for the time ste dτ adoting the eective arameter π eff k instead of the real values of the nonlinear rate π k. Dierent strategies can be adoted to comute these eective rates. For examle, the eective rate π eff k can be comuted as the 22

38 3.4 Verication of the model mean of the values of nonlinear rate values, π k, assumed at the beginning and at the end of the time ste π eff k (i, t i ) = [π k (i, t i 1 ) + π k (i, t i )] /2 Of course the value of the nonlinear rate at the end of the time ste dτ is known only after the MC simulation has been erformed whit a trial value of the eective arameter. In other word, the burn-u rocedure is an iterative methods and the iteration can be reeated until the eective rates reach a stable value. In Figure 3.3 shows the dierence in the aroximation of non linear rate adoting the standard rocedure and the burn-u technique Automatic time ste The main roblem of the classical nonlinear simulation and the burn-u techniques which adot a xed time ste dτ resides in the aroriate selection of the length of time ste. In fact, the time ste should be as small as ossible for accuracies of the simulations while should be as bigger as ossible to reduce the comutation time required by the nonlinear simulations. To overcome this roblem an algorithm to automatically select and adat the time ste dτ during the MC simulation has been develoed. In the Automatic Time Ste (ATS) algorithm two thresholds for each nonlinear rate, k, are introduced: the lower threshold l th (π k ) and the higher threshold h th (π k ). During the simulation, at the end of each time ste, the variation of each nonlinear rate ( π k ) is evaluated. If the variation of each nonlinear rate is below the corresonding lower thresholds, π k l th (π k ), the time ste of the simulation increases (dτ = K 1 dτ + K 2, where K 1 and K 2 are constant dened by the user). If l th (π k ) < π k < h th (π k ) the time ste is ket constant. Otherwise, if at least one variation of nonlinear rates exceed its higher threshold, π k > h th (π k ), the time ste is reduced at its half value (dτ = dτ/2). The scheme of the algorithm for automatically udated the time ste imlemented in the MASCOT code is shown in Figure 3.4. The ATS rocedure allows one to reduce consistently the time required by the nonlinear MC simulation. Figure 3.5 reorts an examle of nonlinear MC simulation of the transort of contaminant in orous media adoting the xed time ste rocedure and the ATS algorithm. As shown in Figure, the CPU time required by MC simulation is drastically reduced by the ATS algorithm; however the key roblem remain the correct selection of the thresholds h th and l th Verication of the model The stochastic model develoed, based on the theory of branching stochastic rocesses, and its imlementation in the MASCOT code, has been veried in dierent cases by comarison with: the standard FEMWASTE code [Yeh and Ward, 1981], based on the classical AD equation, on a case of non-reactive transort (Paer II) and on a case of linear reactive transort (Reort I) in unsaturated transient ow; the ow eld and the water content in the medium, as a function of location and time, are 23

39 3.5 Numerical alications Figure 3.4.: The scheme of the algorithm for automatically udated the time ste imlemented in the MASCOT code. obtained by the FEMWATER code [Yeh and Ward, 1980] based on Richards' equation [Freeze and Cherry, 1979]; the numeric solution of the classical AD equation obtained by the PDE tool of MatLab a case of nonlinear reactive transort under steady saturated ow in which the sortion rocess is governed by the Freundlich isotherm [Freeze and Cherry, 1979] (Paer III) Numerical alications Numerical alications of the stochastic model to simulate the contaminant transort in unsaturated transient ow eld through a column lled with orous media are resented in Paer II and in Reort I to illustrate the eects due to non-equilibrium linear interchange rocesses and the eects of nonlinear interchange rocessed governed by the Freundlich isotherm, resectively. An alication regarding the mod- 24

40 3.5 Numerical alications Automatic Time Ste verification ADE Pde of MatLab MASCOT Fixed time ste ( τ = 0.001) CPU Time = 16 h 20m MASCOT ATS (l th =0.05; h th =0.15; k 1 =1.2; k 2 =0) CPU time = 3m 31s MASCOT ATS (l th =0.05; h th =0.25; k 1 =1.2; k 2 =0) CPU time = 2m 39s C/C T Figure 3.5.: Examle of nonlinear MC simulations adoting the xed time ste and ATS rocedure. The numerical solution of the AD equation is reorted for reference. elling of radionuclide release from a otential reository in clay formation is resented in ( 3.5.1) Radionuclides transort from a HLW built in clay formation This case study resents the simulations of the some radionuclides transort from a dee geological reository sited in the Oalinus Clay of the Zürcher Wienland region in northern Switzerland that is designed for the disosal for the sent fuel, in the form of fuel assemblies containing UO 2 or mixed-oxide (MOX) fuel, the vitried HLW from the rerocessing of sent fuel and for the long-lived intermediate-level waste. The simulations of radionuclide migration through the dierent barriers of the reository are necessary for evaluate the long term safety of this kind of reository The geological environment of the reository The Oalinus Clay was deosited some 180 million years ago by sedimentation of ne clay, quartz and carbonate articles in a shallow marine environment. It is art of a thick sequence of Mesozoic and Tertiary sediments in the Molasse Basin (see Figures 3.6), which overly Palaeozoic sediments and crystalline basement rocks. The overlying Tertiary sediments thicken considerably into the Molasse Basin to the south. In the Zürcher Wienland the Mesozoic sediments containing the Oalinus Clay are of uniform thickness over several kilometres, almost at-lying (diing gently to the south east) and little aected by faulting. To the north-east, towards the Hegau- Bodensee Graben structure and in the Jura Mountains to the west, the sedimentary 25

41 3.5 Numerical alications Figure 3.6.: Geological structure of the sedimentary sequences in the vicinity of the Benken borehole in the Zürcher Weinland, based on 3-D seismic data (left-hand gure), and schematic geological roles from SW to NE (to right-hand) and NW to SE (bottom right-hand) through the sedimentary rocks. The otential host rock consists of the Oalinus Clay formation and the Murchisonae beds in Oalinus Clay facies, after [NAGRA, 2002]. rocks become faulted and folded [NAGRA, 2002]. The Zürcher Wienland consequently reresents a structurally simle region on the northern edge of a dee basin, bounded by deformed sedimentary rocks to the north-east and west. Nagra studies in the Zürcher Wienland and Oalinus Clay Regional geological investigations in the northern Switzerland over the ast 20 years [NAGRA, 1980, 2005], including more focused studies in the area over the last ten years, have rovided a clear icture of the geological and hydrogeological structure and roerties of the Zürcher Wienland. This information has come rincially from intensive work in a 1000 m dee borehole at Benken and 3-D seismic survey of the surrounding area. The seismic camaign conrmed the remarkable homogeneity and lateral extent of the Oalinus Clay. In addition, geological, hydrogeological, hydrochemical and isotroic data have been synthesized on a regional basis and a new, local hydrodynamic model has been develoed. The otential reository host rock identied consists of the Olainus Clay and the Murchisonae beds within the Olainus Clay facies, and is more that 100 m thick. The reository is foreseen as consisting of a series of waste emlacement tunnels constructed roughly in the mid-lane of the Oalinus Clay. The host rock, a well comacted, moderately over-consolidated claystone, is exected to have excetional isolation roerties resulting from its very low hydraulic conductivity. The host formation is overlain and underlain by further thicknesses (100 to 150 m) of clay rich formations (conning units) which, although not as imermeable and homogeneous as Oalinus Clay (they contain thin, robably discontinuous sandstones and carbonate rocks), also ossess good isolation roerties, with limited groundwater movement and good sortion otential. Above and below these clay-rich formations, the se- 26

42 3.5 Numerical alications Figure 3.7.: Potential groundwater transort aths of the radionuclide from the reository to the bioshere. quence contains large, regional aquifers in the carbonate rocks of the Malm and the Muschelkalk. To summarized, the geological comonent of the isolation concet is thus as follows: i) the absence of signicant advective groundwater ow in the host formation, which is thick enough to extend for more then 40 m above and below reository, [NAGRA, 2002], will ensure that the rate of movement of radionuclides out of the engineered barriers and through the undisturbed host rock will be extremely small; ii) The surrounding clay-rich sediments are rocks of the conning units and have the additional otential to retard the movement of any radionuclides that escae from the host rock. Although there are thin, more ermeable horizons in these clays based on isotroic and hydrochemical evidence, ows are rather small due to limited hydraulic interconnectedness, and otential athways to the bioshere are long ( km, if they exist). Furthermore, the surrounding formations (Uer Dogger/ Lower Malm above, and Lias/Keuer below) have good sortion roerties. Any radionuclides that migrate through the clay-rich formations of the Uer Dogger/ Lower Malm above, and Lias/Keuer, will enter the regional aquifers of the Malm (above) and the Muschelkalk (below). If radionuclides enter the regional aquifers, they will be significantly disersed and diluted. An additional stage of dilution will occur when the dee aquifers discharge to the more dynamic freshwater ow systems of near surface gravel aquifers, or to river waters Results of the Monte Carlo simulation The radionuclide migration through the dierent barriers of the reository has been simulated by means the MASCOT code based on the stochastic aroach resented in ( 3.2). Figure 3.7 shows the otential groundwater transort aths of the radionuclide based on sitting information and hydrodynamic modelling-discharge. For simlicity, the radionuclide transort along only the vertical direction (1D simulation) from the 27

43 3.5 Numerical alications Canister (source) z(1) z(2) z(14) z(15) z(814) z(815) z(2814) Acquifer Bentonite (0.86 m) Oalineos Clay (40 m) Uer Clay (100 m) Figure 3.8.: Discretization of the otential groundwater transort ath used in MC simulation. reository to the uer aquifer is analyzed. In the stochastic model the following discretization has been used: the Bentonite layer of 0.65 m is discretized in 13 zones with z = 0.05 m; the Oalinus Clay layer of 40 m is discretized in 800 zones with z = 0.05 m and the Claystone (uer conning) layer of 100 m is discretized in 2000 zones with z = 0.05 m (Figure 3.8). In this case study, the connement caacity of the canister has been neglected. Thus, the initial time in the MASCOT simulation corresond to the time necessary to the canister failure that can occur at least 10 4 years after the closure of the reository. The source term is reresented by the radionuclide 243 Am that decays (α decay) as follows: 243 Am 239 Pu 235 U. The half-live of the radionuclide 243 Am is T 1/2 = 7380 y. The daughter of 243 Am is 239 Pu with a half-live of T 1/2 = y. The extremely long half-life of 235 U (T 1/2 = y) limits our simulation to those three radionuclides. The initial concentrations of the radionuclides in the source zone ( i = 1) resent in solution are: C( 243 Am, i = 1) = C 0 = 1 and C( 239 Pu) = C( 235 U) = 0. For the determination of the arameters of the stochastic model necessary to simulate the migration of the radionuclides from the dee reository of HLW, we have conservatively assumed that the reository and the surrounding host rocks are fully saturated with water. Table 3.2 reorts the comlete set of hysical and hydraulic arameters characterizing the transort of the radionuclides considered through the EBS, the Oalinus Clay and the Claystone. Table 3.3 reorts the values of the transition rates used in the simulation of the radionuclides transort and calculated according to the relationshis reorted in Aendix A. The results of the case study simulation are reorted in Figures Figure 3.9 shows the normalized concentration role for 243 Am resent in solution and adsorbed on the host matrix. After years (from the canister failure) the normalized concentration for 243 Am resent in solution reaches the very low value of and the radionuclide results conned inside the Bentonite layer resent around the canister of the 243 Am. Figure 3.10 shows the normalized concentration role for 239 Pu resent in solution and traed in orous medium. All the 239 Pu roduced from the decay of 243 Am is ket in the Bentonite buer and in the rst zones of the Oalinus Clay also at very long time from the failure of the canisters. In Figure 3.11 is reorted the normalized distribution of 235 U resent in solution and traed in the host matrix at dierent times. The increasing of the 235 U in the 28

44 3.5 Numerical alications Table 3.2.: Physical and hydraulic arameters of the Bentonite and clay layers of the otential reository of HLW in Zürchel Wienland [NAGRA, 2005] Parameters Bentonite Oalinus Clay Claystones 243 Am D [m 2 /s] q [m/y] θ ρ[kg/m 3 ] K d [m 3 /kg] R d Pu D [m 2 /s] q [m/y] θ ρ[kg/m 3 ] K d [m 3 /kg] R d U D [m 2 /s] q [m/y] θ ρ[kg/m 3 ] K d [m 3 /kg] R d Table 3.3.: Transition rates of the stochastic model fot the otential reository of HLW in Zürchel Wienland Parameters Bentonite Oalinus Clay Claystones 243 Am π(i i + 1) [y 1 ] π(i i 1) [y 1 ] ads [y 1 ] des [y 1 ] λ 12 [y 1 ] Pu π(i i + 1) [y 1 ] π(i i 1) [y 1 ] ads [y 1 ] des [y 1 ] λ 23 [y 1 ] U π(i i + 1) [y 1 ] π(i i 1) [y 1 ] ads [y 1 ] des [y 1 ] λ 3 [y 1 ]

45 3.5 Numerical alications 10 2 AM 243 in solution 10 0 AM 243 traed in the orous medium T= y T= y 10 4 T= y T= y T= y 10 2 T= y T= y T= y C/C C/C Distance from the source [m] Distrance from the source [m] Figure 3.9.: MASCOT simulation results of 243 Am: normalized distribution in sace mobile in solution (left Figure) and traed in orous media (right Figure) at dierent time from the failure of the canister. bentonite with the time is due to the decay of 239 Pu. Thanks to the good retention roerties of the Bentonite and the Oalinus Clay also the 235 U results substantially immobile and conned in the Oalinus Clay Comments The radionuclide migration through the Bentonite buer (EBS) and the Oalinus Clay (geoshere) of a candidate site for the HLW reository has been tacked by means the stochastic model. The corresonding model is evaluated by the MASCOT code, where a large number of dierent kind articles are followed in their travel through the barriers, using aroriate robability distribution functions that characterize their transort. From the results of the simulation, the nuclear waste reository roosed by Nagra seem to be adequate to conning the radionuclides 243 Am, 239 Pu and 235 U in the roximity of the reository for very long time. Any release that might take lace after the ending time of the canisters will ose no signicant risk for the human health and for the environment due to the very low level of the contaminant concentrations. The aim of this alication is to show the otentiality of this aroach and its exibility that allows to simulate radionuclide transort through the dierent barriers that constitute the reository. 30

46 3.5 Numerical alications 10 4 Pu 239 in solution 10 2 Pu 239 traed in the orous medium T= y T= y T= y T= y 10 6 T= y T= y 10 0 T= y T= y T= y T= y T= y 10 2 T= y C/C C/C Distrance from the source [m] Distrance from the source [m] Figure 3.10.: MASCOT simulation results of 239 Pu: normalized distribution in sace mobile in solution (left Figure) and traed in orous media (right Figure)at dierent time from the failure of the canister U 235 in solution 10 0 U 235 traed in the orous medium T= y T= y 10 5 T= y T= y 10 1 T= y T= y T= y T= y 10 6 T= y T= y 10 2 T= y T= y C/C C/C Distrance from the source [m] Distrance from the source [m] Figure 3.11.: MASCOT simulation results of 235 U: normalized distribution in sace mobile in solution (left Figure) and traed in orous media (right Figure) at dierent time from the failure of the canister. 31

47 The dierence between genius and stuidity is that genius has its limits. Anonymous Chater 4 Transort in fractured media 4.1. Introduction Fractures are mechanical breaks in rocks; they originate from strains that arise from stress concentrations around aws, heterogeneities, and hysical discontinuities and occur at a variety of scales, from microscoic to continental. Such fractures are conduits for uid ow and are connected to other hydraulically conductive fractures to form systems or networks. Conductive fracture networks may include a large number of inter-connected hydraulically active features or may be limited to a very small roortion of the total fractures in the rock mass. The requirements vary from site to site and from alication to alication. Fractured rock is highly heterogeneous in comarison to orous media. Deending on the aerture of a fracture, its ermeability can exceed that of the ambient rock matrix by many order of magnitude. Therefore hydraulic ow in fractured media is mainly governed by the geometry and connectivity of its fracture network and by the individual fracture aertures. Thus, hydraulic data of fractured media tyically dislays strong variability in measurements at dierent locations deending on resence/absence of conducting and connected fractured State of the Art Mathematical models fall into one of three broad classes: (1) equivalent continuum models, (2) discrete network simulation models, and (3) hybrid techniques. The models dier in their reresentation of the heterogeneity of the fractured medium. They may be cast in either a deterministic or stochastic framework. The scale at which heterogeneity is resolved in a continuum model can be quite variable, from the scale of individual acker tests in single boreholes to eective ermeability averaged over large volumes of the rock mass. Discrete network models exlicitly include oulations of individual fracture features or equivalent fracture features in the model structure. They can reresent the heterogeneity on a smaller scale than is normally considered in a continuum model. Some of the more recent innovations in mathematical simulation 32

48 4.2 State of the Art Figure 4.1.: Water ow modelling in fractured rock, after [Geier et al., 1992]. are best classied as hybrid techniques, which combine elements of both discrete network simulation and continuum aroximations Water ow modelling in fractured rock The dierent fractures build u a network of fractures, more or less interconnected. The ow in this fractures system not only deend on the hydraulic conductivity of the individual fractures but it also deends on the orientation, size, fracture density, and degree of connectivity of the fractures. Dierent aroaches of calculation of the ow in fracture systems are discussed ( ). Figure 4.1 shows dierent models used to simulate the ow through fractured media Equivalent continuum simulation model In a conventional equivalent continuum model, the heterogeneity of the fractured rock is modeled by using a limited number of regions, each with uniform roerties. Individual fractures are not exlicitly treated in the model, excet when they exist 33

49 4.2 State of the Art on a scale large enough to be considered a searate hydrological unit. At the scale of interest, hydraulic roerties of the rock mass are reresented by coecients, such as ermeability and eective orosity, that exress the volume-averaged behavior of many fractures. If the coecients in the uid ow and solute transort equations are viewed as being known with certainty, or if the most likely values of the variables are used, the model is deterministic. Conventional forms of the groundwater ow equation, which were develoed originally for orous media, can then be adoted. The use of continuum aroximations in a deterministic framework has been the common ractice. If the coecients are viewed as a satially variable random eld, characterized by a robability distribution, the model is stochastic. The magnitude of uncertainty in the inut arameters deends on the natural variability of the medium, knowledge of which may be limited by the number and tyes of measurements available to ma the heterogeneity. Averaging volumes associated with a deterministic continuum aroximation must be large enough to encomass a statistically reresentative samle of the oen, connected fractures and their variable inuences on ow and transort behavior. Effectively, this means that uid ux and solute transort are not inuenced to any signicant degree by any individual fracture or its interconnections with other fractures that form the conducting network. The reresentation of a ow region using uniform ow roerties is best alied to cases where the scale of the roblem is large, the fractures are highly interconnected, and the interest is rimarily on volumetric ow, such as in groundwater withdrawal for water suly. If, however, fracture density is low or many fractures are sealed by mineral reciitates, fracture connectivity may be low and continuum assumtions less likely to be adequate. Equivalent continuum models for fractured media are of two general tyes, single and dual orosity. In a single-orosity model all orosity is assumed to reside in the fractures; orosity in the matrix blocks between the conducting fractures is neglected. In a dual-orosity (fracture lus matrix) model the matrix blocks are assigned a value of orosity greater than zero. Single-orosity models reresent hydrology in terms of a single continuum; dual-orosity models are based on two overlaing continua. For uid ow roblems involving steady-state conditions (i.e., no changes in uid storage), single-orosity models are usually adoted. The uid ux through the rock matrix is assumed to be negligible in comarison to that in the fracture network. For roblems involving transient ow (i.e., a change in uid storage), both singleand dual-orosity models have been used. For roblems involving solute transort in geological media with high matrix orosities or roblems with long time scales, diusion of mass between the fractures and the rock matrix (so-called matrix diusion) can be an imortant rocess. Matrix diusion is normally simulated by using a dualorosity model. The strength of the continuum aroach lies in its simlicity; it reduces the geometric comlexity of ow atterns in a fractured rock mass to a mathematical form that is straightforward to imlement. For most alications that are encountered in ractice, some tye of continuum aroach remains the referred alternative. 34

50 4.2 State of the Art Discrete network simulation models Discrete fracture network A DFN model is built u from a statistical descrition of fracture geometry and hydraulic roerties such as the location, size, orientation, satial structure, transmissivity and intensity. The ow aths are assumed to result from networks of interconnecting fractures. The most imortant advantage of the DFN model are the ossibility to incororate fracture geometry data in the model and exlicitly reresent roerties of fractures and fracture zones. On the other hand, fracture data over the total region that would be modelled is required. Therefore, it might be necessary to simlify the fracture attern, both because of the comlexity of the network and because fracture data are unknown. Details of DFN model are reorted in Aendix B. Channel network models The concet of a Channel Network (CN) model is that the ow within a fracture network is conned to discrete channels, which intersect at various intervals. Like the DFN model, CN models are based on the discrete nature of the fracture athways. Using a network comosed of eectively one-dimensional elements the comutational diculty can be reduced and, thus, this aroach simlies the modelling of comlex rocesses. The data requirements for a CN model are also somewhat less than for a DFN model, because the characteristics of fracture geometry are ignored. This is one of the major dierences between the DFN and the CN is that the system by the DFN aroach conducts water only if the system is ercolated, whereas the system by the CN aroach always conducts water. This is because the hydraulic conductance for the CN aroach is determined stochastically not by the fracture geometries but by assigning for each channel from the distribution Solute transort modelling in fractured rock The models for solute transort in a fractured rock are summarized based on literature survey for the cases where ow velocity role in the rock is known and not aected by the solute transort Solute transort modelling in discrete network models In the discrete network models such as the DFN and CN models, solute transort through a network of fractures or ies was modeled by reresenting the tracer as a nite number of articles. This method is called article-following method [ Cacas et al., 1990b; Gylling, 1997; Moreno et al., 1997; Robinson, 1984; Tsang et al., 1988], or article tracking method [Dverstor et al., 1992; Schwartz et al., 1983; Yamashita and Kimura, 1990]. A article-following method has been develoed to simulate the mass transort through the DFN [Robinson, 1984; Schwartz et al., 1983]. In this method, a large number of articles are added initially either to a single fracture at the ustream boundary of the system [Schwartz et al., 1983] or to the intersections along a side of the network [Robinson, 1984]. 35

51 4.2 State of the Art Particles move by advection through the fracture network based on water velocity for individual fracture segments. At the fracture intersection, dierent mixing hyotheses can be adoted (see Section B.2.3). For examle, under the hyothesis of comlete mixing where the incoming mass of contaminant is comletely mixed at intersection and the concentration leaving the node is the same for any segment, the robability of a article to take a fracture of kind k is w k = q k Q where q k reresents the ow through the fracture of kind k and Q is the outgoing total ow from the node. As a consequence of the ow re-mixing at the intersections, mass diserses as it moves by advection through the network. Thus, large-scale disersion is modeled in this way. However, the article-following method by [Schwartz et al., 1983] and [Robinson, 1984] does not take into account the matrix diusion, which is one of the most imortant mechanisms for retardation of the radionuclides [ Neretnieks, 1980]. [Yamashita and Kimura, 1990] introduced a article-following method incororating the matrix diusion for a single fracture model with an innite orous rock matrix. They described the eect of matrix diusion in terms of the residence times of articles because of the fact that a article may reside in the matrix for some time in addition to its residence time in the water in the fracture. The residence time of each article is determined by a robability density function of the article residence time, which is obtained by normalizing the mass ux. This method is adoted by [Moreno et al., 1997] to simulate the solute transort in the CN model. [Dershowitz and Miller, 1995] described an imlementation of a DFN model in which the matrix diusion was modeled by a robabilistic article-following technique. The solute concentration is obtained analytically from the one-dimensional diusion equation with a constant eective diusivity. Ref. [Cvetkovic and Dagan, 1994] adoted a stochastic Lagrangian framework to comute the robability that the contaminant will be retained by the rock. The robabilistic solution of the transort roblem is based on the statistics of two Lagrangian variables: τ, the travel time of an imaginary tracer moving with the owing water, and β, a suitably normalized surface area available for retention Solute transort modelling for continuum based models For continuum based models such as the Stochastic Continuum (SC) model, two basic aroaches were used to simulate the solute transort: (1) the Finite Dierence Method (FDM) or Finite Element Method (FEM), and (2) the Random walk (RW). The FDM or FEM is a numerical aroach to solve a dierential equation by exressing the dierential equation as a set of linear equations by domain discretization. Although this method is well develoed for solving the transort equation in a mathematical sense, there are disadvantages. First, comutational work to solve the N eq equations er time ste is intensive. Also, the solution may include the arts of domain, which never traversed by solutes [Tomson et al., 1987]. Second, the use of nite dierence or nite element causes numerical errors due to the numerical disersion and/or oscillation [Bear et al., 1993; Kinzelbach, 1988; Tomson et al., 1987; Unk, 1988]. The numerical disersion is caused by the aroximation of the rst-order derivative, which involves error of the order of magnitude of the second-order deriva- 36

52 4.2 State of the Art Table 4.1.: Classication of transort models in fractured media Reresentation of Fractured media References Equivalent Continuum Models Single orosity [Carrera et al., 1990] Double orosity [Pruess and Narasimhan, 1988] Stochastic continuum [Neuman and Dener, 1988; Williams, 1992] Discrete Network model Discrete Fracture Network [Andersson and Dverstor, 1987; Geier et al., 1992; Long et al., 1982; Schwartz et al., 1983] Channel Network [Gylling, 1997; Moreno and Neretnieks, 1993] Hybrid Models Boltzmann Transort equation Reort II and III Statistical continuum transort [Smith et al., 1990] Fractal Models Equivalent discontinuum [Long et al., 1992] tive [Bear and Verruijt, 1987]. The numerical disersion is severe when the advection is dominant (Peclet number, P e > 2 10) [Tomson et al., 1987]. The numerical error caused by the numerical disersion can be made small by decreasing the size of mesh [Bear and Verruijt, 1987; Kinzelbach, 1988; Tomson et al., 1987]. However, a ne mesh is not always ossible and sometimes the most imractical otion [ Tomson et al., 1987]. Various nite dierence methods to overcome the numerical disersion have been summarized by [Huyakorn and Nikuha, 1979]. The RW is an analogue rocess generally based on the transient AD equation and its advantages are as follows. First, the algorithm is simle and can easily be alied in large, 3 D as well as unsaturated roblem [Kinzelbach, 1988; Marseguerra and Zio, 1997; Tomson et al., 1987]. Also, this method can be easily combined with any ow model [Kinzelbach, 1988]. Second, there is no numerical disersion since the random walk method is not a direct numerical solution of the dierential equation [ Kinzelbach, 1988; Tomson et al., 1987; Unk, 1988]. Third, the comutational eort er time ste is roortional not to the number of nodes but to the number of articles. Hence, the storage requirement will be dramatically reduced from any nite element or nite dierence method [Tomson et al., 1987]. There are disadvantages of RW. A solute mass is reresented as a large collection of articles. In order to obtain consistent and reliable results, a large number of articles are required [Tomson et al., 1987]. The statistical uctuation of numerical results can be avoided by increasing the number of articles. However, increasing total number of articles does not show the same degree as an imrovement of results because this uctuation is roortional to the square root of the number of articles [Kinzelbach, 1988] Summary In Table 4.1 the single ow and transort models on the fractured media are summarized. 37

53 Every small honesty is better than a big lie Leonardo da Vinci Chater 5 Monte Carlo simulation transort in fractured media 5.1. Introduction In this Chater the stochastic model based on the analogues with neutron transort, originally roosed by Williams [Williams, 1992, 1993] and formalized into the linear Boltzmann transort equation is resented. In this aroach, the straight ath between intersection of fractures is considered as a seudo mean free ath and the re-distribution in transort direction which occurs at an intersection, as a seudoscattering event. This chater is organized as follow. In Section 5.2, the main characteristics of the underlying stochastic model to simulate contaminant transort through fractured media is described. In Session 5.3, the continuum model is converted into a hybrid aroach that combines the convenience and comutational exediency of a continuum stochastic model with the exibility of DFN simulation. Finally, Section 5.4 reorts the alication of the hybrid model for the uscaling the results of DFN simulations Stochastic Model Williams [Williams, 1992] suggested using the linear BTE to model transort in fractured rock. The linear version of BTE, a classical model from the kinetic theory of gases, is a natural aroach for describing the collective motion of a large number of noninteracting articles subject to random dislacements. The linear BTE is used to describe hoton transort in the atmoshere and in astrohysics alications. It is widely used by nuclear engineers for neutron transort in reactor design and radiation shielding calculations The transort of radionuclide through fractured media can be seen as a series of straight movement between fracture intersections with suddenly change of direction and velocity. Doing an analogy with the transort of neutron in a non-multilying 38

54 5.2 Stochastic Model medium, in articular the aths within the fractures are assumed to be straight and treated as seudo-mean free aths and the variation in direction of a contaminant article motion, which occurs at an intersection of fractures, is treated as a seudoscattering event. In the mathematical formulation of the radionuclide transorts in fractured media, let C(r, Ω, v, t)drdωdv be the mean density of radionuclides at osition r, in volume element dr, moving in the solid angle dω centered about the direction Ω, with velocity between v and v+dv, at time t. In the case of a single radioactive nuclide, the quantity C satises the following transort equation [Williams, 1992]: [ ] R d t + vω. + vσ(r, Ω) + λr d C(r, Ω, v, t) = (5.1) dω dv v Σ(r, Ω )g(r; v v; Ω Ω)C(r, Ω, v, t) + S(r, Ω, v, t) where R d is the retardation factor, Σ is the robability density of the branching oints so that 1/Σ is the seudo mean free ath, g(...) is the re-distribution in seed and direction of the articles at the intersection, λ is the decay constant of the radionuclide and S is an indeendent source of radionuclides. The transfer function g(...) describes the motion in direction and seed of the contaminant after the occurrence of a fracture intersection. The arameters distributions of Σ, g and S in equation (5.1) encode information about geometry of the fracture network and the hydrodynamics of water ow in the network. The deendence of the number of articles solely on osition and time, called the number density, is an insucient characterization for transort in fractured rock; more variables are often needed [Duderstadt and L, 1976]. The deendencies on article seed and direction of travel (i.e., velocity) were added for a more detailed characterization, called the angular article density. In other words the BTE is based on the angular density concet and diers from the AD equation in that the number of articles is also deendent on article seed and direction of travel Monte Carlo solution of the transort equation Modern methods for numerical solution of the linear BTE fall into two main grous: MC methods and discrete-ordinates methods, e.g. [Duderstadt and L, 1976]. The MC method tracks the motion of a large number of individual articles moving through the medium by simulating the rocesses causing changes in the article velocity and direction. The angular article density can be determined from the individual article histories. The discrete-ordinates method, by contrast, uses numerical methods to solve directly for the angular density by discretizing article seed, direction of travel, time, and satial variables. The result is a couled set of discrete mass conservation equations, which are solved through iteration. Here, we roose the solution of the transort equation by means MC simulation. The corresonding code called MASCOT (Montecarlo Analysis of Subsurface COntaminant Transort) has been vericated by means comarison with the analytic and numerical solutions (Reort II) of the BTE. The roosed aroach is very exible as it is alicable to one-, two- or three-dimensional roblems and it allows to account 39

55 5.2 Stochastic Model for dierent hysical rocesses and for the inhomogeneities in time, simly varying the arameter values with time. In the following we describe briey how to solve the transort equation by means MC simulation in simlied case of homogeneous transort (i.e. the arameter of the transort are constant in the time) and that only one kind of article is considered (i.e. all the articles reresent the same contaminant in the same hysico-chemical state) and no decay or exchange rocess are considered. The MC rocedure consists in simulating a large number of indeendent article histories in each of which a article is generated from a suitable source and then followed u through the fractured medium until the end of the time interval of concern. A article moves within a fracture dragged from the water ow and, if not adsorbed by the fracture wall, it arrives at the next intersection oint. At intersection oint the article erforms a collision and enters in a new fracture with features samled from relevant distributions. We consider the generic trajectory of the article: starting from a generic oint P n = (r n, v n 1, Ω n 1,...) in the hase sace where the article was entered after the n-collision, the oint after the n + 1 collision, P n+1, is samled from the df of the conditioned robability K(t, k k, t ) called Transort Kernel where k reresent the state of the article (i.e. its osition, velocity, ight directions, chemical and hysical state etc...). The Transort Kernel can be exressed by the roduct of two conditioned robabilities, the Free Flight Kernel, T, and the Collision Kernel, C: K(t, k k, t ) = T (t k, t ) C(k k, t) (5.2) The Free Flight Kernel, T (t k, t ), is the conditioned robability that the article of kind k undergoes a transition, e.g. meets a branching oint, between t and t + dt, given that the revious transition has occurred at time t. the Free Flight Kernel can be written as: T (t k, t ) = Λ k dt ex Λ k t (5.3) where Λ k is: Λ k = v k Σ k (5.4) The hysical meaning of (vσ) 1 is the average travel time of the radionuclide between fracture intersections (when the adsortion of the fracture walls is neglected). The Collision Kernel, C(k k, t), is the the conditional robability that the article, by eect of the transition that occurred at time t, when it was in state k, undergone the transition k k. C(k k, t) = Λ kk (5.5) Λ k where Λ k reresents the robability to exit from the state k and Λ kk reresents the robability of transitions from state k and state k. When no article transition are allowed, the Collision Kernel is reduced to a scattering function called Redistribution Kernel R. C(k k, t) = g(k k) = R(v v; Ω Ω; b b) (5.6) The Redistribution Kernel, is the conditional robability that the article, by eect of the transition of scattering (the article reach a redistribution oint) undergone at 40

56 5.3 The Hybrid model time t, while traveling along the fracture with orientation Ω, aerture b and velocity v, enters in a new fracture with direction Ω, aerture b and ow velocity v. Details of MC simulation of radionuclide transort in fractured media can be found in Reort II and III Alication to a case with adsortion and decay In real case, the radionuclides during the transort in fractured medium may interact with the rock and be adsorbed on the fracture surfaces. They may also diuse into the stagnant water in the ores of the rock and be sorbed in the inner ore surfaces and eventually, over long or very long times scale of interest in the erformance assessment of the radioactive reository, decay. In order to simulate by mean the MC the reactive contaminant through fractured media, we need to introduce the following two categories of articles: the solutons which are free articles of contaminant mobile within the water owing through the fractures and the traons which are immobile articles of contaminant that reresent the fraction of contaminant adsorbed on the rock or resent in the immobile water as done with the stochastic model develoed for the orous media (see Chater 3). The results of this alication to a case with adsortion and decay is reorted in Reort II The Hybrid model Models adoted to simulate the transort of contaminant through fractured rock are generally based on discrete or continuum aroach. Among the discrete aroach, the article tracking through stochastically generated DFN is largely used. However, DFN simulations are comutationally intensive and therefore usually limited to small scales. An alternative to the DFN is rovided by the continuum models as shown in Chater 4. Unfortunately the arameters required by the continuum models are dicult to be determinated. For examle, Williams [Williams, 1993] attemted to derive the arameters aearing in the BTE analytically from a urely geometric descrition of the fractures networks, which limited its alicability. Indeed the arameters aearing in the BTE are determinated not only by the network geometry but also by the uid dynamics through the interconnected network. To overcome this diculties, a hybrid aroach has been develoed in which a DFN simulation on a small scale domain is used to comuted the arameters for the simler and less comutationally demanding model such as the continuum stochastic model based on the BTE. Particle tracking is the core of the method. Solute transort is modeled by rst collecting statistics on article motion in a subdomain using a DFN simulation and then using these statistics in a continuum model to simulate transort at a larger scale. Particle tracking is also used in the continuum stochastic model. In essence, the articles are educated in the discrete model. Then, in the continuum model, 41

57 5.3 The Hybrid model they are able to mimic the eects of the interaction between network geometry and conductance, the orientation of the hydraulic gradient, and the resulting sreading atterns Reactive transort The hybrid model develoed can be alied to simulate the reactive transort through fractured media adoting the framework roosed by [Cvetkovic and Dagan, 1994]. In fact, a recent theory [Cheng and Cvetkovic, 2003; Cvetkovic and Dagan, 1994; Cvetkovic and Haggerty, 2002; Cvetkovic et al., 2002] has shown that the robabilistic measures of the transort roerties of the geoshere (i.e. the radionuclide discharge from crystalline rock volume) is comletely characterized by two Lagrangian quantities: τ and B, the advective residence time and a arameter which quanties the hydrodynamic control of retention, resectively [Cvetkovic and Haggerty, 2002; Cvetkovic et al., 2002] The mathematical descrition of contaminant transort in the geoshere Let q(m/t ) denote the radionuclide discharge from an elementary rock volume into the accessible environment over a given discharge surface. For instantaneous injection, it can be shown that q(t, τ) = m 0 γ(t, τ), where γ(1/t ) coules advection, retention, and decay and τ is the advective residence time along a trajectory [Cvetkovic and Dagan, 1994]. The function γ can be interreted as the Probability Distribution Function (PDF) of the radionuclide article residence time in the rock volume and can be obtained in dierent ways, e.g. by a robabilistic aroach as done in this work. Considering a network of N fractures, let i denote a given fracture and τ i the corresonding advective time, i = 1, 2,...N. If the retention is a linear rocess, and neglecting disersion, then the solution for γ is [Cvetkovic and Dagan, 1994]: H(t τ)b γ(t, τ) = 2 π(t τ) 3/2 ex where H( ) is the Heaviside ste function and: B = [ B 2 ] 4(t τ) λt (5.7) N β i κ i ; (5.8) i=0 β i = τ i ; b i (5.9) κ i (θ i D i R di ) (5.10) where θ i is the orosity of the rock matrix, D i is the eective diusivity into the rock matrix, R di is the retardation coecient and b i is the aerture of the fracture. Once the distributions τ and B are given any of several ossible robabilistic measures of the transort roerties of the geoshere can be retrieved [Cvetkovic and Haggerty, 2002]. 42

58 5.3 The Hybrid model Transort arameters τ and β The transort arameters τ and β can be directly comuted in the MC simulation. The advective travel time τ j is the time necessary to the j-article to reach the outut control anel and the travel time of the j-article can be comuted as follows: N j τ j = τ ji = i=1 N j i=1 l ji v ji (5.11) where the index i reresents the channel crossed by the article from the source to the outut. In the simle case where no decay and article transformation are considered, the advective travel time τ can be comuted directly during the MC simulation collecting the time samled from the Free Flight Kernel. The arameter β j, called also retention arameter, can be comuted in the following way: β j = N i=0 τ ji b ji = N i=0 l ji v ji b ji (5.12) From the values of the arameter τ j and β j for each article it is ossible to obtain the distribution of the arameters τ and β Parameter determination from DFN simulations The MC simulation of contaminant transort through fractured media requires dierent sets of model inuts as the conditioned robability of transort (transort kernel) and the source. These inuts, that corresond to discrete robabilities distribution functions, deends on the geometry and the uid dynamic in the interconnected fracture networks but also on the solute source and the solute mixing at each nodes of the network. Therefore, to relate the characteristics of the fracture network to the Kernels and source term it is necessary to take into account the source and mixing rocess of the solute in the network. The Mixing Function describes the redistribution of contaminant at the intersection oint and the Source Function governs the article injection into the system Source term In the MASCOT code, the source term is reresented by the articles injected in the system and is comuted as the discrete robabilities for the ow velocity, orientation and aerture of the rst fracture. Time and sace distributions of the source term are generally assigned by the user and are indeendent of the fracture network while the robability of the article to be released in a fracture of kind k, called source distribution S k, deends on the geometry of the fracture network and the boundary condition alied. In fact, the articles enter only in the fractures the intersected the source zone while the boundary conditions describe how the source is redistributed between these fractures. The source term distribution, S k, is the ratio between the number of fracture of kind k, N 0k, and the total number of fractures, N 0, that intersect the source zones 43

59 5.3 The Hybrid model weighted by the Source Function w k0 that describes dierent boundary conditions: S k = N 0kw k0 N 0 (5.13) There are two main kind of source mode: ux injection and resident injection. In the rst one, the contaminant concentration in each fracture that intersect the source area is assumed to be constant and therefore the robability of the article to be released in a fracture is roortional to its ow. The Source Function can be written as: w k0 = Q k0 Q 0 = Nk0 i=1 q k0 i k Nk0 i=1 q k0 i (5.14) where Q 0 is the ow through the source area, Q k0 is the ow through the source fracture of kind k and N k0 is the number of source fracture of kind k. With the resident concentration the article has the same robability to be released in each fracture indeendently to its ow (i.e. in each fracture enter the same amount of contaminant) and the Source Function is: where N 0 is the number of source fractures. w k0 = w 0 = 1 N k0 (5.15) Kernels The Free Flight Kernel is given by the equation (5.3) and is comletely determined by the total transition rate (eq. 5.4). The total transition rate can be obtained from the DFN simulation comuting the mean lengths (Σ 1 k ) of the dierent fracture kind k. The Free Flight Kernel encodes geometric and hydrodynamic roerties of the fracture network because deends on the fracture intersections and the ow velocity through the fracture. As shown in ( 5.2.1), the Collision Kernel under the hyothesis of no decay and no transition between article families, corresond to the redistribution Kernel. C(k k, t) = R kk = R(v v, θ θ, b b) (5.16) The values of the Redistribution Kernel R kk is calculated by counting all the fracture of kind k (i.e. the fracture with ow velocity v k and orientation θ k and aerture b ) connected to the fracture of kind k roely weighted by the mixing function. This number divided by the total number of fracture connected to the fracture k gives the conditioned robability R kk. The Redistribution Kernel can be written as: R(v v, θ θ, b b) = N f kk w kk N fk (5.17) where w reresents the mixing function of contaminant at the intersection oint. 44

60 5.4 Alication: uscaling DFN simulations Mixing function The most imortant rocess to be taken into account in order to simulate the transort of contaminant through fractured media is the way in which the incoming contaminant at intersection oint is distributed among the outgoing uxes. From the hysics oint of view there are two extreme aroaches for describing mass mixing at the intersection oint: the comlete mixing and the streamline routing (see Aendix B.2.3). The mixing function, w, that reroduce the comlete mixing model, i.e. the contaminant concentration is constant in the outgoing ux, can be written as: w k k = q kk k q kk (5.18) where q kk reresents the outgoing ow of the fracture of kind k connected to the fractures of kind k. In this comarison with DFN simulation, we have used also a dierent mixing function called static mixing where we have assumed the contaminant mixing is indeendent of the water ow in the channels, thus the contaminant article at the intersection oint have the same robability to take any outgoing segment. The static mixing function results: 1 w kk = k N = 1 (5.19) kk N k where N k reresents the number of outgoing segments connected to the fractures of kind k. The mixing function have a strong inuence on the Redistribution Kernel and consequently on the MC simulations as shown in Reort III Verication of the hybrid model The hybrid model has been veried by means of a comarison of the results obtained by DFN simulation in 2-D sace domain of [m]. The results obtained by means of MC simulation by the MASCOT code have been comared with the article tracking in DFN with dierent boundary conditions alied (i.e. how the contaminant are injected into the system) and with dierent hyothesis of solutes mix at fracture junctions where waters containing dierent concentrations of solutes interact. The agreement is very satisfactory as shown in Figures Details of the verication of the hybrid model are reorted in Reort III Alication: uscaling DFN simulations A tyical roblem in long-term reository is to quantify how much of the radionuclides which enter fractured rock is eventually discharge from the media. As shown in ( ) the discharge of contaminant can be characterized by the two Lagrangian variables τ and β [Cvetkovic et al., 2002]. However this arameters can be comuted by means of a article tracking through DFN only for a limited domain due to the comutational cost of the DFN simulations. 45

61 5.4 Alication: uscaling DFN simulations 10 0 Comarison DFN MASCOT (Flux injection & Static Mixing) 10 6 Comarison DFN MASCOT (Flux injection & Static Mixing) DFN DFN MASCOT MASCOT cdf & ccdf 10 2 df τ [s] τ [s] Figure 5.1.: CDF, CCDF (left-hand gure) and PDF (right-hand gure) of advective travel time τ calculated by DFN (blue oints) and MASCOT simulation (red lines) with ux injection and static mixing hyothesis Comarison DFN MASCOT (Flux Injection & Comlete Mixing) DFN MASCOT 10 6 Comarison DFN MASCOT (Flux Injection & Comlete Mixing) DFN MASCOT cdf & ccdf 10 2 df τ [s] τ [s] Figure 5.2.: CDF, CCDF (left-hand gure) and PDF (right-hand gure) of advective travel time τ calculated by DFN (blue oints) and MASCOT simulation (red lines) with ux injection and comlete mixing hyothesis. 46

62 MONTE CARLO SIMULATION OF RADIONUCLIDES IN FRACTURED MEDIA: PARAMETERS DETERMINATION FROM DISCRETE FRACTURE NETWORK EDOARDO PATELLI AND ANDREW FRAMPTON Abstract. Fractures are the rincial athways for the groundwater-driven disersion of radioactive contaminants which may escae from subsurface waste reositories. The heterogeneity and stochasticity of the media in which the disersion henomenon takes lace transort occurs, renders classical analytical-numerical aroaches scarcely adequate in ractice. In this aer a stochastic aroach, based on an analogy with neutron transort, is adoted, to treat the transort of radionuclides through fractured media. The resulting stochastic model is evaluated by means of the Monte Carlo (MC) simulation technique whose exibility allows considering exlicitly the dierent rocesses that govern the transort as well as dierent boundary conditions. The model arameters are derived directly from the geometric conguration of the fractures and from the dynamics of the driving uid system, as quantitatively described by means of the Discrete Fracture Network (DFN) simulation. A comarison with article tracking through stochastically generated networks of discrete fractures is resented. 1. Introduction The goal of a reository for nuclear waste is to isolate the radionuclides from the bioshere for the long time scales required by the radionuclides to decay to levels of negligible radiological signicance. This is achieved through a multi-barrier system made of the waste ackage, a series of engineered containments and the natural rock in which the reository is laced. Because one of the rincial mechanisms of release of radionuclides is through water inltration into the various constituents of the reository and subsequent ercolation into the groundwater system, it is of utmost imortance to study the henomena of advection and disersion of radionuclides in the articial matrices hosting the waste (near eld) and, subsequently, in the natural rock matrix of the host geoshere (far eld). When studying the transort in the far eld, consideration must be given to the fact that natural fractures act as major heterogeneities that control ow of contaminant in many dee geological formations, e.g. those consisting of crystalline rock generally considered stable environments suitable for disosal of highly radioactive wastes. In fractured media, uid ows mainly through fractures so that considering the host matrix as a homogeneous continuum is an unaccetable modelling simlication (Sahimi, 1995). Particle tracking in stochastically generated networks of discrete fractures rovides an alternative to the conventional advection-disersion descrition of transort henomena (Sahimi, 1995; Berkowitz, 2002; Bodin et al., 2003; Moreno and Neretnieks, 1993). Numerous discrete fracture network (DFN) models have been develoed and 158

63 MC simulation of radionuclides in fractured media: arameters determination from DFN tested (Framton et al., 2005; Mo et al., 1998). However, the article tracking simulation in discrete fracture networks becomes comutationally intensive even for small scales (Sahimi, 1995; Berkowitz, 2002). In this aer, we adot a stochastic aroach largely mutuated from the transort of neutrons in non-multilying media and governed by the linear Boltzmann transort equation (Williams, 1992, 1993). The corresonding model is evaluated by means of Monte Carlo (MC) simulation (Giacobbo et al., 2005), where a large number of articles of solute are followed in their travel through the fractures, using aroriate robability distribution functions to characterize their transort rocesses. The main advantage of the roosed model is its exible structure which allows one to consider multidimensional geometries and to describe a wide range of henomena, by accounting for the individual interactions which each article may undergo. A otential ractical diculty lies in the determination of the values of some model arameters. In (Williams, 1993) the relevant arameters were derived analytically from a simlied geometric descrition of the fracture networks and under simlifying hysical assumtions. In reality, the arameters governing the Boltzmann equation for fracture network transort deend also on the uid-dynamics through the interconnected network. The task of relating analytically the fractures geometric roerties (i.e. distributions of the fracture lengths, orientations, aertures etc.) with the velocity eld in the fracture network is imracticable in realistic settings. To overcome this diculty, we resort to a numeric solution of the uid-dynamics through the method of DFN simulation (Framton et al., 2005). Furthermore, the arameters in the transort Boltzmann equation deend also on the redistribution of the contaminant at the intersection oints (i.e. how the solute is redistributed on the outgoing ow channels) and on the source term (e.g. the radionuclide release from one or several hyothetical canisters into a fractured crystalline geoshere). Dierent hyotheses can be adoted to describe the redistribution of the contaminant at the intersection oints such, as the comlete mixing and the streamline routing (Berkowitz, 2002), and for the source term, such as the ux injection and resident concentration (Moreno and Rasmuson, 1986). The aer is organized as follows. In the next Section we introduce a robabilistic descrition of contaminant transort in the geoshere. In Section 3, the main characteristics of the DFN simulation are summarized. In Section 4, we describe the main roerties of the stochastic model for the simulation of contaminant transort through fractured media. Also, the estimation of the model arameters is erformed based on the statistical distributions of the fractures and on the resulting ow eld and reorted in Section 5. In Section 6, the model is veried by comarison with dierent article tracking engines under dierent boundary conditions. 2. The mathematical descrition of contaminant transort in the geoshere Let q(m/t ) denote the radionuclide discharge from an elementary rock volume into the accessible environment over a given discharge surface. The robabilistic descrition of the transort roerties of the geoshere which governs the discharge is comletely characterized by two Lagrangian quantities: the advective residence time, τ, and a arameter, B, which quanties the hydrodynamic retention (Cvetkovic et al., 2002; Cvetkovic and Haggerty, 2002). For instantaneous injection, it can be shown that q(t, τ) = m 0 γ(t, τ), where γ(1/t ) coules advection, retention, and decay and τ is the advective residence time along 159

64 MC simulation of radionuclides in fractured media: arameters determination from DFN Table 1. Main arameters used in DFN simulation of Figures (1-3). Parameter Value Fracture number 242 [-] Fracture osition uniform [-10, 10] [m] Fracture length lognormal (5, 0.25) [m] Fracture orientation uniform [0, 2π) [rad] Fracture aerture lognormal (1e-4,0.25) [m] Domain 2D: 20x20 [m] X Hydraulic gradient 5 [-] Y Hydraulic gradient 0 [-] number of realizations 5000 [-] Particle tracking Boundary condition ux injection Mixing mode comlete mixing a trajectory (Cvetkovic and Dagan, 1994). The function γ can be interreted as the robability density function (df) of the radionuclide article residence time in the rock volume and can be obtained in dierent ways, e.g. by a robabilistic aroach as done in this work. Considering a network of N fractures, let i denote a given fracture and τ i the corresonding advective time, i = 1, 2,...N. If the retention is linear, and neglecting disersion, then the solution for γ is (Cvetkovic and Dagan, 1994): [ ] H(t τ)b B 2 γ(t, τ) = 2 ex π(t τ) 3/2 4(t τ) λt (2.1) where H( ) is the Heaviside ste function and: N B = β i κ i ; (2.2) i=0 β i = τ i b i ; (2.3) κ i (θ i D i R di ) is a constant that deends on the retention arameters, θ i the orosity of the rock matrix, D i the eective diusivity into the rock matrix and R di the retardation coecient; b i is the aerture of the fracture. Once the distributions τ and B are given any of several ossible robabilistic measures of the transort roerties of the geoshere can be retrieved (Cvetkovic and Haggerty, 2002). 3. The Discrete Fracture Network aroach The DFN aroach is largely used to model groundwater ow and transort of contaminants through fractured rocks (Berkowitz, 2002; Framton et al., 2005). The resulting numerical model describes the ow and transort of individual fractures within a network. In the following, we briey describe the algorithm underlying the DFN aroach. Starting from the statistical distribution of the fracture roerties, such as length, osition, orientation and aerture (or transmissivity), a random realization of fractures is generated by Monte Carlo samling (Figure 1). This is called fracture network. The intersections between the samled fractures are identied as nodes of the network. Then, the channel network, containing those fractures connected to both 160

65 MC simulation of radionuclides in fractured media: arameters determination from DFN 15 Fracture Network 10 5 Y [m] X [m] Figure 1. Examle of a 2D Fracture Network realization over a domain of 10x10 meters. 15 Channal Network 10 5 Y [m] X [m] Figure 2. Channel Network comuted from the Fracture Network realization of Figure (1). The line size is roortional to the ux through the fracture. 161

66 MC simulation of radionuclides in fractured media: arameters determination from DFN 15 Particle track (flux injection and erfect mixing) 10 5 Y [m] Particle 1 Particle 2 Particle 3 Particle 4 Particle 5 Particle 6 Particle 7 Particle 8 Particle 9 Particle X [m] Figure 3. Particle tracking in the Channel Network of Figure (2) with ux injection boundary condition and comlete mixing at each nodes of the network. ustream and downstream boundaries, is identied. Hence, isolated fractures and fractures with dead ends are eliminated form the channel network (Figure 2). Given the network boundary conditions, the hydraulic heads at each connected nodes of the channel network are calculated solving a system of algebraic linear equations obtained from the mass conservation rincile alied at the fracture intersections (i.e., Kircho's law) couled with a laminar ow law that leads to the ux in a fracture being roortional to the dierence in hydraulic heads between its bounding intersections. Hence, the ow rate and uid velocity in each fracture channel between two nodes of the network are directly comuted from the hydraulic heads (Figure 2). Finally, a article tracking algorithm is alied to simulate the contaminant transort through the fracture network (Figure 3). Table 3 summarizes the main arameters used in the DFN simulation of Figures The stochastic model for transort in fracture network The transort of radionuclides through fractured media can be seen as a series of straight movements between fracture intersections with sudden changes of direction and seed. By analogy with the transort of neutrons in a non-multilying medium (Williams, 1992), the aths in the fractures can be treated as seudo-mean free aths and the variations in direction and seed at the intersections can be treated as seudoscattering events. Let C(r, Ω, v, t)drdωdv be the mean density of radionuclides at osition r, in volume element dr, moving in the solid angle dω centered about the direction Ω, with velocity between v and v + dv, at time t. In the case of a single radioactive 162

67 MC simulation of radionuclides in fractured media: arameters determination from DFN nuclide, the quantity C satises the following transort equation (Williams, 1992): [ ] R d t + vω. + vσ(r, Ω) + λr d C(r, Ω, v, t) = (4.1) dω dv v Σ(r, Ω )g(r; v v; Ω Ω)C(r, Ω, v, t) + S(r, Ω, v, t) where R d is the retardation factor, Σ is the robability density of the branching oints (fracture intersections) so that 1/Σ is the seudo-mean free ath, g(...) is the re-distribution in seed and direction of the articles at the intersection, λ is the decay constant of the radionuclide and S is an indeendent source of radionuclides. The distributions Σ, g and S in equation (4.1) encode information about the geometry of the fracture network and the hydrodynamics of the water ow. In articular, the transfer function g(...) describes robabilistically the direction and seed of motion of the contaminant after a fracture intersection Monte Carlo solution of the transort equation. The transort equation (4.1) can be solved analytically only under simlifying assumtions. A owerful alternative, commonly used in neutron transort roblems, is that of using Monte Carlo simulation. In this resect, a Monte Carlo aroach to simulate the Boltzmann transort of radionuclides along fractured media has been roosed by the some of the resent authors and the corresonding code MASCOT (Montecarlo Analysis of Subsurface COntaminant Transort) has been veried by comarison with analytic and numerical solutions (Giacobbo et al., 2005). The roosed aroach is very exible as it is alicable to one-, two- and three-dimensional roblems and it allows accounting for the dierent hysical rocesses occurring during the transort and for the inhomogeneities in time and sace. In the following, the aroach is briey summarized referring, for simlicity, to the simlied case of only one kind of contaminant article (i.e. all the simulated articles reresent the same contaminant and remain in the same hysico-chemical state), no radioactive decay and no exchange rocesses with the fractured host matrix. In few words, the Monte Carlo aroach consists in simulating a large number of indeendent travel histories in each of which a article is generated from a suitable source and then followed u through the fractured medium until the end of the time interval of concern. A article moves within a fracture dragged by the water ow. If not adsorbed by the fracture wall or transformed by radioactive decay, it arrives at the next intersection oint where it erforms a collision (change in direction and seed), entering a new fracture with geometric and ow features samled from the relevant distributions. Consider a generic trajectory of the reresentative oint P (k, t), in the hase sace, where k describes the article state with resect to its osition, direction, seed of ow and fracture aerture: starting from the oint P = (r, Ω, v, b, t ) (k, t ) of exit from the revious fracture intersection, the next collision oint, P (k, t), is samled from the conditional df K(k, t k, t ) called Transort Kernel, which is the roduct of the Free Flight Kernel, T, and the Collision Kernel, C: K(t, k k, t ) = T (t k, t ) C(k k, t) (4.2) The former is the conditional robability density function that the article undergoes a collision, i.e. meets an intersection branching oint, at time t given that the revious transition has occurred at time t and that the article entered state k as a result of the collision. In our illustrative model considered, the Free Flight Kernel is written 163

68 MC simulation of radionuclides in fractured media: arameters determination from DFN as: T (t k, t ) = Λ k ex Λ k (t t ) (4.3) where Λ k = v k Σ k. The recirocal of Λ k gives the mean travel time of the article between fracture intersections while in state k, when adsortion on the fracture walls and radioactive decay are neglected. The Collision Kernel, C(k k, t) is the the conditional robability that the article enters the new state k by eect of the transition entered at time t: C(k k, t) = Λ kk (4.4) Λ k where Λ kk reresents the rate of transitions from state k to state k. In the case considered of no hysico-chemical state change and no radioactive decay, the Collision Kernel reduces to a scattering function called Redistribution Kernel, R: C(k k, t) = g(t; k k) = R(t; Ω Ω; v v; b b) (4.5) which gives the conditional robability that the article by eect of scattering undergone at time t while traveling with velocity v along the fracture with orientation Ω and aerture b, enters in a new fracture in direction Ω, with aerture b and ow velocity v Transort arameters τ and β. The arameters τ and β governing the transort in the fractured media (Section 2) can be estimated directly within the Monte Carlo simulation. The advective travel time τ j is the time necessary for the j-th simulated article to travel through the N j fracture channels forming the trajectory and reach the exit boundary. It can be comuted as follows: N j τ j = τ ji = i=1 N j i=1 l ji v ji (4.6) where τ ji is the advective travel time in the i-th fracture of length l ji crossed by the j-th article at seed v ji. In the simle case considered of no radioactive decay and hysico-chemical transformations, the advective travel time τ can be comuted directly during the Monte Carlo simulation collecting the times samled from the Free Flight Kernel along the trajectory. The arameter β j characterizes the hydrodynamic retention of the fractures crossed by the j-article along its trajectory and can be comuted in the following way: β j = N j i=1 τ ji b ji = N j i=1 l ji v ji b ji (4.7) where b ji is the aerture of the i-th fracture crossed by the j-th article. 5. Parameter determination from Discrete Fracture Network The Transort Kernel governing the contaminant transort through fractured media deends hysically on the geometry of, and the uid-dynamics in the interconnected fracture networks but also on the solute source and solute mixing at each nodes of the network. In this Section, we show how the Transort Kernels and the Source Term, required by MC simulation, can be obtained from the results of standard DFN simulations with dierent boundary conditions and dierent hyotheses on the contaminant mixing at the fracture intersections. For ractical urose, the statistics of the fracture networks has been obtained discretizing the ow velocity, fracture orientation and 164

69 MC simulation of radionuclides in fractured media: arameters determination from DFN fracture aerture in N v, N θ, N b channels, resectively. Thus, the tye of fracture is univocally identied by the vector m (v g, θ g, b g ), with v g = 1,..., N v, θ g = 1,..., N θ, b g = 1,..., N b Particle Source. The Source Term is described by the robability distribution S(v g, θ g, b g ) S m for the ow velocity, fracture orientation and aerture, hereafter called source distribution. Time- and sace-distributions of the article source are generally assigned by the user and are indeendent of the fracture network. Then, the source articles enter only those fractures which intersect the source zone, with the boundary conditions determining how the injected articles are redistributed between these fractures. There are two main kinds of source terms: ux injection and resident injection (Kreft and Zuber, 1978; Demmy, 1999). In the rst one, the contaminant concentration in each fracture that intersects the source zone is assumed to be constant and therefore the robability of the article to be released in a fracture is roortional to its ow. The source distribution, S m, can be comuted as the ratio between the water ow through fractures of tye m, Q m0, and the total water ow through fractures that intersect the source zone, Q 0 : S m = Q m0 Q 0 = N m0 i=1 N m0 q m0i (5.1) q m0i where q m0i = is the ow through the i-th source fracture and N m0 is the number of source fractures of tye m. With the resident concentration the contaminant mass is injected uniformly in the source fractures. Thus, the article has the same robability to be released in each fracture indeendently to its ow (i.e. in each fracture enters the same amount of contaminant) and the Source distribution S m is: N m0 m i=1 S m = = N m0 (5.2) N m0 N 0 m 5.2. Kernels. The Free Flight Kernel is given by equation (4.3) and is comletely determined by the total transition rate Λ k = Λ m = v g Σ(v g, θ g, b g ) which can be obtained from the results of DFN simulations. This rate, which deends on the fracture tye m, can be comuted as the roduct between the ow velocity and the recirocal of the mean distance between fracture intersections. The Free Flight Kernel encodes geometric and hydrodynamic roerties of the fracture network. As shown in Section 4.2, the Collision Kernel under the hyothesis of no radioactive decay and no hysico-chemical transitions, corresonds to the redistribution Kernel: C(k k, t) = R kk = R mm = R(v g v g, θ g θ g, b g b g ) (5.3) The most imortant rocess to know to simulate the transort of contaminant through fractured media is how the incoming mass of contaminant is distributed among the outgoing uxes. From the hysical oint of view, there are two dual aroaches for describing mass mixing at the intersection oint, i.e. how the contaminant incoming at intersection oint is distributed among the outgoing uxes: the comlete mixing and the streamline routing (Berkowitz and Smith, 1994). 165

70 MC simulation of radionuclides in fractured media: arameters determination from DFN x 105 velocity distributions Comlete mixing Static mixing df velocity [m/s] x 10 5 Figure 4. Probability density functions of the ow velocity in the outgoing fractures for the comlete and static mixing functions. The Redistribution Kernel R mm for the comlete mixing model, is such that the contaminant concentration is constant in the outgoing uxes, is calculated as the ratio between the water ow through fracture of tye m connected to the fracture of tye m, Q mm, and the total water ow through fracture connected to the fracture of kind m: R mm = Q mm Q m = m N mm q mm i i=1 N mm i=1 q mm i (5.4) where N mm reresents the number of fractures of tye m intersecting fractures of tye m and q mm i is the water ow through the i-th fracture of kind m connected to fracture of kind m, resectively. In the comarison with DFN simulation, a dierent mixing hyotesis called static mixing has also been adoted, according to which the contaminant mixing is indeendent of the water ow in the channels and a contaminant article reaching an intersection oint has the same robability to take any of the outgoing channels. Under this mixing hyothesis the Redistribution Kernel deends only on the geometry of the fracture network: R mm = N mm m N mm (5.5) where N mm reresents the number of channels of tye m connected to the fractures of tye m. 166

71 MC simulation of radionuclides in fractured media: arameters determination from DFN Angle distibutions Comlete mixing Static mixing df θ [rad] Figure 5. Probability density functions of the outgoing fractures orientations for the comlete and static mixing functions. The dierent hyotheses that describe the redistribution of the solute contaminant at the intersection oint has a strong inuence on the Redistribution Kernel and consequently on the Monte Carlo simulations. For examle, comuting the marginal distribution of the Redistribution Kernel for the outgoing velocity (i.e. integrating over θ g and b g ) one obtains the robability that the contaminant takes a fracture with velocity v g : using static mixing, the contaminant has a greater robability to take the outgoing fractures with very small ow velocities whereas with comlete mixing the contaminant has a greater mean velocity and very small robability to take fractures with small ow velocities as shown in Figure 4. Figures (5-6) show the inuence of the mixing function on the marginal dfs of the fracture orientations θ g and aertures b g. 6. Numerical results A tyical question addressed in long-term geologic reository design is how much of the radionuclides which enter the fractured rock under accidental scenarios is eventually discharged. As shown in Section 2, the discharge of contaminant can be characterized by the two Lagrangian variables τ and β. In what follows, a case study is resented and the results obtained by MASCOT simulations are comared with those of article tracking through fracture network by DFN. As exlained in Section 5, the transort of contaminant through fractures deends on the boundary conditions alied (i.e. how the contaminant is injected into the system) but also on how contaminant solutes mix at the fracture intersections where waters containing dierent concentrations of solutes interact. The boundary conditions here considered are resident concentration and ux injection (Framton et al., 2005). The contaminant mixing at the intersection oints is modeled in two ways: 167

72 MC simulation of radionuclides in fractured media: arameters determination from DFN Aerture distributions Comlete mixing Static mixing df aerture [m] x 10 4 Figure 6. Probability density functions of the fractures aertures for the comlete and static mixing functions. comlete mixing and static mixing. These two models are actually oosite and extreme cases, both neglecting the ossibility of artial or no mixing. As exlained in Section 5.2, in the comlete mixing model the contaminant redistribution is governed by the velocity ow eld while in the static mixing model the redistribution of the contaminant deends only on the geometry of the fracture network Comlete mixing. In this Section, we comare the results obtained by means of Monte Carlo simulation by the MASCOT code with the article tracking in DFN under the hyothesis of comlete mixing of the contaminant in each nodes of the network Resident concentration. Figures 7 and 8 show the cumulative distribution functions (cdfs) and the comlementary distribution functions (ccdfs) of τ and β, resectively. The agreement is very satisfactory Flux injection. The cumulative distribution functions and the comlementary distribution functions of τ and β are comared in Figures 9 and 10 for the DFN and MASCOT, resectively. There is very good agreement between the results of the Monte Carlo simulation and those of the article tracking through the fracture network Static mixing. In this Section, we comare the results obtained by means of the Monte Carlo simulation by the MASCOT code with the article tracking in DFN under the hyothesis of static mixing of the contaminant in each nodes of the network. The consequence of this hyothesis of contaminant redistribution at the fracture intersection is a little increasing of the advective travel time τ and arameter β resect to the case of comlete mixing. 168

73 MC simulation of radionuclides in fractured media: arameters determination from DFN 10 0 Comarison DFN MASCOT (Resident Concentration & Comlete Mixing) DFN MASCOT 10 1 cdf & ccdf Figure 7. Cumulative distribution function (cdf) and the comlementary distribution function (ccdf) of advective travel time τ calculated by DFN (oints) and MASCOT simulation (lines) with resident injection and comlete mixing. τ [s] Resident concentration. The cumulative distribution functions and the comlementary distribution functions of τ and β are comared in Figure 11 and 12, resectively. The agreement is satisfactory. A small dierence are resent in the tails of the ccdfs for τ and β and are likely due to the discrete binning used in MC simulations Flux injection. The cumulative distribution function and the comlementary distribution function of τ and β comuted are comared in Figure 13 and Figure 14 for the article tracking by DFN and MASCOT, resectively. There is very good agreement between the results of the Monte Carlo simulation and those of the article tracking through the fracture network. 7. Conclusions In this study, a stochastic aroach largely mutuated from the transort of neutrons in non-multilying media and governed by the linear Boltzmann transort equation (Williams, 1992, 1993) is adoted to simulate the transort of radionuclide through fractured media. The corresonding model is evaluated by means of MC simulation (Giacobbo et al., 2005), where a large number of articles of solute are followed in their travel through the fractures, using aroriate robability distribution functions to characterize their transort rocesses. A otential ractical diculty lies in the determination of the values of some model arameters. In (Williams, 1993) the relevant arameters were derived analytically from a simlied geometric descrition of the fracture networks and under simlifying 169

74 MC simulation of radionuclides in fractured media: arameters determination from DFN 10 0 Comarison DFN MASCOT (Resident Concentration & Comlete Mixing) DFN MASCOT 10 1 cdf & ccdf β [s/m] Figure 8. Cumulative distribution function (cdf) and the comlementary distribution function (ccdf) of retention arameter β calculated by DFN (oints) and MASCOT simulation (lines) with resident injection and comlete mixing Comarison DFN MASCOT (Flux Injection & Comlete Mixing) DFN MASCOT 10 1 cdf & ccdf τ [s] Figure 9. Cumulative distribution function (cdf) and the comlementary distribution function (ccdf) of advective travel time τ calculated by DFN (oints) and MASCOT simulation (lines) with ux injection and comlete mixing. 170

75 MC simulation of radionuclides in fractured media: arameters determination from DFN 10 0 Comarison DFN MASCOT (Flux Injection & Comlete Mixing) DFN MASCOT 10 1 cdf & ccdf β [s/m] Figure 10. Cumulative distribution function (cdf) and the comlementary distribution function (ccdf) of retention arameter β calculated by DFN (oints) and MASCOT simulation (lines) with ux injection and comlete mixing Comarison DFN MASCOT (Resident Concentration & Static Mixing) DFN MASCOT 10 1 cdf & ccdf τ [s] Figure 11. Cumulative distribution function (cdf) and the comlementary distribution function (ccdf) of advective travel time τ calculated by DFN (oints) and MASCOT simulation (lines) with resident concentration and static mixing. 171

76 MC simulation of radionuclides in fractured media: arameters determination from DFN 10 0 Comarison DFN MASCOT (Resident Concentration & Static Mixing) DFN MASCOT 10 1 cdf & ccdf β [s/m] Figure 12. Cumulative distribution function (cdf) and the comlementary distribution function (ccdf) of retention arameter β calculated by DFN (oints) and MASCOT simulation (lines) with resident concentration and static mixing Comarison DFN MASCOT (Flux injection & Static Mixing) DFN MASCOT 10 1 cdf & ccdf τ [s] Figure 13. Cumulative distribution function (cdf) and the comlementary distribution function (ccdf) of advective travel time τ calculated by DFN (oints) and MASCOT simulation (lines) with ux injection and static mixing. 172

77 MC simulation of radionuclides in fractured media: arameters determination from DFN 10 0 Comarison DFN MASCOT (Flux injection & Static Mixing) DFN MASCOT 10 1 cdf & ccdf β [s/m] Figure 14. Cumulative distribution function (cdf) and the comlementary distribution function (ccdf) of retention arameter β calculated by DFN (oints) and MASCOT simulation (lines) with ux injection and static mixing. hysical assumtions. Here, the DFN simulation has been adoted to collecting statistics of the fracture geometry and the ow eld and comute the robability distributions required by the MC simulation under dierent mixing hyothesis and solute source conditions. The MC simulation has been veried by means a comarison with article tracking through stochastically generated networks of discrete fracture in the simlied case of no radioactive decay and no exchange rocesses with the fractured host matrix. The agreement between the results of DFN article tracking simulations and MC simulations are very satisfactory. The main advantage of the roosed model is its exible structure which allows one to consider multidimensional geometries and to describe a wide range of henomena, by accounting for the individual interactions which each article may undergo. Thus, comlex retention rocesses, such as kinetically controlled sortion, radioactive decay and decay chains, can be included in a straightforward manner. Moreover, the comutation requirements for the MC simulations are a tiny fraction of those required for DFN simulation (Benke and Painter, 2003). Thus, the main alication of the Monte Carlo simulation would be as an extraolation or u-scaling method where DFN simulations would be erformed on a modest satial scale to collect the statistics required by the MC simulations. Therefore, the MC simulation is an ecient tool to simulate of radionuclide transort through fracture network under dierent mixing hyothesis and solute source conditions and alied to simulate the transort at the satial scale of a reository geoshere, which would be imractical for DFN methods alone. 173

78 MC simulation of radionuclides in fractured media: arameters determination from DFN References Benke, R., Painter, S., Modelling conservatve tracer transort in fracture networks with a hybrid aroach based on the boltzmann transort equation. Water Resources Research 39 (11). Berkowitz, B., Characterizing ow and transort in fractured geological media: A review. Berkowitz, B., C. N., Smith, L., Mass transfer at fracture intersections: An evaluation of mixing models. Water Resources Research 30 (6), Bodin, J., Porel, G., Delay, F., Simulation of solute transort in discrete fracture networks using the time domain random walk method. Cvetkovic, V., Dagan, G., Transort of kinetically sorbing solute by steady random velocity in hererogeneus ouros formations. Journal of Fluid Mechanics 265, Cvetkovic, V., Haggerty, R., May Transort with multile-rate exchange in disordered media. Physical Review E 65 (5-1). Cvetkovic, V., Painter, S., Selroos, J., Comarative measures of radionuclide containment in the crystalline geoshere. Nuclear Science and Engineering 142, Demmy, George & Berglund, S.. G. W., July Injection mode imlications for solute transort in orous media: Analysis in a stochactic lagrangian framework. Water Resources Research 35 (7), Framton, A., Cvetkovic, V., Patelli, E., Discrete frecture network simulation...unublished. Giacobbo, F., Zio, E., Patelli, E., Monte carlo simulation of radionuclides transort through fractured media (unublished). Kreft, A., Zuber, A., On the hysical meaning of the disersion equation and its solutions for dierent initial and boudary conditions. Chemical Engineering Science 33, Mo, H., Bai, M., Lin, D., Roegiers, J.-C., Study of ow and transort in fracture network using ercolation theory. Moreno, L., Neretnieks, I., Fluid ow and solute transort in a network of channels. Journal of Contaminant Hydrology 14 (3-4), Moreno, L., Rasmuson, A., Contaminant trasort through a fractured orous rock: imact of the inlet boundary conditions on the concentration role in the rock matrix. Water Resources Research 22 (12), Sahimi, Flow and Transort in Porous Media and Fracturated Rock: from classical methds to modern aroaches. VCH Verlagsgesellschaft mbh. Williams, M., A new model for desribing the transort of radionuclide through fracturated rock. Ann. Nucl. Energy 19, 791. Williams, M., Radionuclide transort in fractured rock a new model: alication and discussion. Annals of Nuclear Energy 20 (4), Diartimento Ingegneria Nucleare - Politecnico di Milano, via Ponzio 34/3, Milan, Italy address: edoardo.atelli@olimi.it URL: htt://lasar.cesnef.olimi.it Det. of Water Resources Engineering - The Royal Institute of Technology, Brinellv 32, Stockholm, Sweden address: edoardo.atelli@olimi.it 174

79 5.4 Alication: uscaling DFN simulations 10 0 Comarison DFN MASCOT (Resident Concentration & Comlete Mixing) 10 6 Comarison DFN MASCOT (resident concentration and comlete mixing) DFN MASCOT DFN MASCOT cdf & ccdf 10 2 df β [s/m] τ [s] Figure 5.3.: CDF, CCDF (left-hand gure) and PDF (right-hand gure) of advective travel time τ calculated by DFN (blue oints) and MASCOT simulation (red lines) with resident concentration and comlete mixing hyothesis Comarison DFN MASCOT (Resident Concentration & Static Mixing) 10 6 Comarison DFN MASCOT (Resident Concentration & Static Mixing) DFN MASCOT DFN MASCOT cdf & ccdf 10 2 df τ [s] τ [s] Figure 5.4.: CDF, CCDF (left-hand gure) and PDF (right-hand gure) of advective travel time τ calculated by DFN (blue oints) and MASCOT simulation (red lines) with resident concentration and static mixing hyothesis. 47

80 5.4 Alication: uscaling DFN simulations 10 0 Comarison DFN MASCOT (Flux injection & Static Mixing) Comarison DFN MASCOT (Flux injection & Static Mixing) DFN MASCOT DFN MASCOT cdf & ccdf 10 2 df β [s/m] β [s/m] Figure 5.5.: CDF, CCDF (left-hand gure) and PDF (right-hand gure) of retention arameter β calculated by DFN (blue oints) and MASCOT simulation (red lines) with ux injection and static mixing hyothesis Comarison DFN MASCOT (Flux Injection & Comlete Mixing) Comarison DFN MASCOT β (flux injection & comlete mixing) DFN MASCOT DFN MASCOT cdf & ccdf 10 2 df β [s/m] β [s]/[m] Figure 5.6.: CDF, CCDF (left-hand gure) and PDF (right-hand gure) of retention arameter β calculated by DFN (blue oints) and MASCOT simulation (red lines) with ux injection and comlete mixing hyothesis. 48

81 5.4 Alication: uscaling DFN simulations 10 0 Comarison DFN MASCOT (Resident Concentration & Comlete Mixing) Comarison DFN MASCOT (resident concentration and comlete mixing) DFN MASCOT DFN MASCOT cdf & ccdf 10 2 df β [s/m] β [s]/[l] Figure 5.7.: CDF, CCDF (left-hand gure) and PDF (right-hand gure) of retention arameter β calculated by DFN (blue oints) and MASCOT simulation (red lines) with resident concentration and comlete mixing hyothesis Comarison DFN MASCOT (Resident Concentration & Static Mixing) DFN MASCOT Comarison DFN MASCOT (Resident Concentration & Static Mixing) DFN MASCOT cdf & ccdf 10 2 df β [s/m] β [s/m] Figure 5.8.: CDF, CCDF (left-hand gure) and PDF (right-hand gure) of retention arameter β calculated by DFN (blue oints) and MASCOT simulation (red lines) with resident concentration and static mixing hyothesis. 49

82 5.4 Alication: uscaling DFN simulations 10 0 Uascling DFN (Flux Injection & Comlete Mixing) 10 4 Uascling DFN (Flux Injection & Comlete Mixing) DFN 50x50m MASCOT 10x10m > 50x50m DFN 50x50m MASCOT 10x10m > 50x50m cdf & ccdf 10 2 df τ [s] τ [s] Figure 5.9.: Verication of the MASCOT uscaling for the τ arameter at 50 meters from the inlet with ux injection and comlete mixing hyothesis. Lefthand gure shows the CDF and CCDF of advective arameter τ. Righthand gure shows the PDF of the advective arameter τ. Many alications, articularly those involving geological disosal of HLW, involve satial scales of a few hundred meters to a few kilometers. Thus there is a need for methods to bridge the ga between the satial scale at which DFN simulation are tractable (generally limited to tens of meters) and the reository geoshere scale. One aroach for binding this ga in scales is to adot the hybrid model develoed in this research activity and resented in ( 5.3) Numerical test of uscaling In the aroach here considered, the DFN model has a central role in transferring detailed-scale data to the stochastic model and for the verication of the validity of the stochastic aroach for the uscaling. DFN simulations were used to test the uscaling methods. The aim is to simulate the transort of a radionuclide through a domain of L = meters by means of the hybrid model based on the BTE. The stochastic model has been calibrated on the base of the DFN results obtained in a domain of l = meters and then alied to simulate the radionuclide transort on the domain L. In order to verify the uscaling rocedure, the distributions of the arameters τ and β comuted by means of the MASCOT code for the scale L, calibrated on the small scale l, are comared with those obtained by DFN simulation erformed on the same scale and reorted in Figures Moreover in Figure 5.11 the comarison between the traversal disersion of contaminant obtained by means MC uscaling simulation and DFN simulation is reorted Comments The develoed aroach for uscaling the τ and β distributions to the eld scale has been veried successfully. This aroach allows mechanistic understanding of reten- 50

83 5.4 Alication: uscaling DFN simulations 10 0 Uscaling MASCOT 10 8 Uascling DFN (Flux Injection & Comlete Mixing) DFN 50x50m MASCOT 10x10m > 50x50m DFN 50x50m MASCOT 10x10m > 50x50m cdf & ccdf 10 2 df β [s/m] β [s/m] Figure 5.10.: Verication of the MASCOT uscaling for the β arameter at 50 meters from the inlet with ux injection and comlete mixing hyothesis. Lefthand gure shows the CDF and CCDF of β and the right-hand gure shows the PDF of β Uscaling MASCOT DFN 50x50m MASCOT 10x10m > 50x50m 10 1 cdf & ccdf dy [m] Figure 5.11.: Verication of the MASCOT uscaling for contaminant disersion along the y-axis, y, at 50 meters from the inlet with ux injection and comlete mixing hyothesis. 51

84 5.4 Alication: uscaling DFN simulations tion rocesses to be incororated in a direct way. In fact, once τ and β distributions are calculated, multile retention models can be evaluated. This latter feature is articularly convenient for nuclear waste reository studies, which often require that alternative models be evaluated. Another new asect of this work is that the individual fracture roerties from DFN are used in the MC simulation, without tting model distribution. In conclusion, direct uscaling of DFN simulations rovides an alternative to sitescale continuum transort models. The suggested rocedure is to rst erform smallscale DFN simulations utilizing site-secic information on the fracture network, and then to use the results collected from these DFN simulations in a MC calculation to obtain transort results at the eld scale. The aroach avoids volume averaging and other assumtions inherent in the continuum aroach. It reserves the highly non- Gaussian velocity statistics and the satial correlation in velocity that are observed in DFN simulations. 52

85 I can live with doubt and uncertainty and not knowing. I think it is much more interesting to live not knowing than to have answers that might be wrong Anonymous Richard Feynman Chater 6 Conclusions The worldwide roblem of adequately conning the radioactive wastes roduced in industrial alications is of aramount imortance for the future exloitation of the advantages of ionizing radiation and radioisotoes, in both the energy and non-energy related elds. Many environmental management and remediation lanning and control activities make use of redictive comuter models to describe the evolution of contaminant in groundwater systems. A tyical examle is that of the erformance assessment of a high level radioactive waste reository for which intensive reliminary studies for site characterization are erformed. Indeed, natural groundwater transort is a henomenon of articular interest because groundwater is a rincial vector through which hazardous contaminant can circulate to the bioshere, thus osing a signicant risk to the health of the human oulation and to the safety of environment. Several aroaches have been roosed for the modellization of radioactive contaminant transort through the articial and natural barriers of a disosal system of the radioactive wastes. In this work of thesis the focus was on modeling radionuclide transort through: i) the orous media by means of a stochastic model based on the theory of branching stochastic rocesses of Kolmogorov and Dmitriev and ii) fractured media by means of a stochastic model based on the Boltzmann transort equation. In articular the research work has regarded the following toics: 1. Measurement of water Inltration A non-destructive and non-invasive technique for measuring the deth of water enetration in a homogeneous slab has been roosed. The roosed device consists of a 252 Cf sontaneous ssion source and a set of 3 He detectors located on the same side of a concrete slab subject to water inltration. The eectiveness of this technique has been investigated by MC simulation by means of MCNP4C code (Paer I). 2. Transort in transient unsaturated ow eld conditions in orous media The stochastic model, based on the KD theory of branching stochastic rocesses, has been generalized to allow the simulation in more realistic conditions as in res- 53

86 ence of unsaturated and non-uniform ow eld, in steady/unsteady ow eld. The aroach has been veried with resect to a case of non-reactive transort under transient unsaturated ow eld conditions by a comarison with a standard code based on the classical AD equations (Paer II). Alications regarding linear reactive transort (Paer II) and radionuclide release from a otential reository build in clay formation ( 3.5.1) have been erformed. 3. Nonlinear transort in orous media The stochastic model of contaminant transort, based on the KD theory of branching stochastic rocesses, has been suitably extended to account for the nonlinear sortive henomena (Paer III). The verication of the corresonding MC simulation aroach has been erformed with resect to the quantication of dierent case studies (Reort I). 4. Transort in fractured media A MC aroach to the simulation of radionuclide transort in fractured rock, based an analogy with neutron transort and formalized into the Boltzmann transort equation, has been develoed. The MC aroach has been veried by means of a comarison with the analytic solution of the Williams model. A numerical alication of a more comlex and more realistic case has been erformed to show the exibility and the otential of the roosed aroach (Reort II). The model arameters are derived directly from the geometric conguration of the fractures and from the dynamics of the driving uid system, as quantitatively described by means of the DFN simulation. By so doing, the continuum stochastic model is transformed into a hybrid model. A comarison with article tracking through stochastically generated networks of discrete fractures has been erformed in order to verify the model (Reort III). 5. Uscaling DFN results The most romising alication of the roosed aroach called hybrid model is for the direct uscaling of DFN simulations. The latter is a very imortant issue in risk assessment of nuclear waste disosals, because, whereas exeriments are erformed at the meter and day scale, large-scale (kilometer) long-term (thousand of years) redictions are needed. The rocedure consists in erforming small-scale DFN simulations on the basis of site-secic information regarding the fracture network, and in using the results collected form these DFN simulations in a MC calculation to obtain transort results at the eld scale. The aroach avoids volume averaging and other assumtions inherent in the continuum aroach. It reserves the highly non-gaussian velocity statistics and the satial correlation in velocity and orientation that are observed in DFN simulations. Moreover, it also allows mechanistic models for retention rocesses to be incororated directly, including the eects of satial variability in retention rorieties. The uscaling rocedure has been veried ( 5.4.1) and the erformed simulation of the radioactive contaminant transort at the geoshere scale required relatively modest comutational eort. In summary, the stochastic aroaches resented in this thesis aear to be romising for modelling radionuclide transort through the dierent barriers of the radioactive reository under realistic conditions. 54

87 Aendix A Parameters determination of the stochastic model in orous media A.1. Determination of motion rates, π, in saturated steady ow eld A.1.1. Parameters determination in uniform medium In uniform medium and in saturated steady-state condition, Ref. [Ferrara et al., 1999] has shown that the values of the motion rates (π ) of the stochastic model (3.1) can be estimated by a comarison with the equations of the corresonding classical AD equation written in nite dierence form. The relationshis between the motion rates and the hydrodynamic disersion coecient of the contaminant of kind,d, and the water ore velocity, v, are: π (i i + 1) = D 2 + v 2 z π (i i 1) = D 2 v 2 z (A.1) (A.2) A.1.2. Parameters determination in layered medium We consider the heterogeneous medium made of homogeneous layers (l = 1, 2, 3,..., n l ), so that the study of the heterogeneous medium is resumed to the study of each homogeneous layer. For the arameters determination of the stochastic model in heterogeneous medium we resort to a well known characteristic of the linear system, which fulls the rincile of linear suerimosition: a linear combination of the inuts will roduce the same linear combination of the oututs. Alying this suerimosition rincile we can determinate the values of the transition rates as the sum of the values due to the single hysical henomena. For examle, the values of the transition rates of movements 55

88 A.1 Determination of motion rates, π, in saturated steady ow eld (forward and backward) can be exressed as the sum of the transition rates due to the diusion rocess (π D ) and the rates due to the advection rocess (π A ): π = π D + π A (A.3) In the follow we indicate with j the last zone of layer l j and with k the rst zone of the next layer l k. The net ow of contaminant of kind between the node j and the node k, indicated with φ (j k) = φ (l jk ) reresents the ux at the interface between layer l j and the layer l k. This ux can can be written as: φ (j k) = φ + (j) φ (k) (A.4) where we have indicated with φ + (j) and φ (k) the ux of -contaminant at node j along the ositive direction and at node k along the negative direction, resectively. In the ( A.1.2.1) is exlained how to determinate the value of the transition rates of the stochastic model at the interfaces of the layers. In ( A.1.2.1) we consider layers with dierent diusion coecient, in ( A.1.2.2) we consider layers with dierent orosity and in ( A.1.2.3) we analyze the layers with dierent sace discretization. A Layer with dierent diusion coecient Here we analyze the solute diusion in a layered medium characterized of dierent values of diusion coecient. Diusion rates The driving force for diusion of solutes is the thermal movement of small articles such as water molecules, ions, susended articles or colloids in solution (Brownian motion). In a ideal system, the random microscoic movement of the articles leads to a concentration dierence reduction with time and an increasing of the entroy of the system. Without external chemical otential alied on the boundary, in the asymtotic time limit the solute concentration is uniform and constant in the whole volume under consideration and consequently the net ow of contaminant (rate of solute ow er unit area) is zero in each zone of the medium. In the stochastic model the net ow of solute articles between the node j and the node k reresents the contaminant ux at the interfaces between layer l j and layer l k (Figure A.1): φ (j k) = N S (j) π D (j k) N S (k) π D (k j) = 0 (A.5) where N S reresents the mean number of soluton articles of kind and π D reresents the motion rate due to diusion rocess. Rearranging equation ( A.5) we obtain: N S (j) π D (j k) = N S (k) π D (k j) (A.6) At the asymtotic time limit the concentrations at node j and k reach the same value and therefore the mean number of solutons in zone j and in zone k should be the 56

89 A.1 Determination of motion rates, π, in saturated steady ow eld Figure A.1.: Sace discretization at the layer interface l jk. same, N S (j) = N S (k). From equation (A.6) we can write the rst relationshi of the motion rates at the interfaces between layers: π D (j k) = π D (k j) (A.7) On the macroscoic scale, diusion is driven by a gradient in the chemical otential alied on the boundary and the solute ux through the layered medium is: φ = D eq, C = D eq, (C (0) C (L)) L = D l, C l, (A.8) where D l, is the diusion coecient in layer l, C l, the solute gradient of contaminant ok kind in the layer l and D eq, is the equivalent diusion coecient that can be comuted using the following relationshi: L D eq, = n l L l D l, l=1 (A.9) At the steady state condition the ux through each layer of the medium should be the same: φ = φ (l) = φ (l j l k ) (A.10) where we have indicated with φ (l) is the ux through the l-layer and with φ (l jk ) the ow at the interface between layer l j and l k as shown in Figure A.1. In the stochastic model the ux can be comuted alying the Eqs. (A.5-A.7): φ (l jk ) = φ (j k) = π D (j k) (N S (j) N S (k) ) (A.11) The solute gradient at the interface can be comuted in the following way: N S (l jk ) = ( NS (j) N S (k) ) z = N S (j) N S (j 1) z + N S (k+1) N S (k) z 2 (A.12) 57