Analyses of Radionuclide Migration in Geologic Media Using Compartment Models

Transcription

1 UCB-NE-5017 Dissertation for the degree of Doctor of Philosophy Analyses of Radionuclide Migration in Geologic Media Using Compartment Models Daisuke Kawasaki Department of Nuclear Engineering Univerisity of California, Berkeley December 19, 2005

2 The author invite comments and would appreciate being notified of any errors in the report. Joonhong Ahn Department of Nuclear Engineering University of California Berkeley, CA USA The Regents of the University of California hold the copyright of the computer programs developed in the work described in this report. Redistribution of the programs in any form is prohibited without written consent. Disclaimer: Neither the Regents of the University of California nor any of their employees make any warranty, express or implied, or assumes any legal liability of responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial products process, or service by its trade name, trademark, manufacturer, or otherwise, does not necessarily constitute or imply its endorsement, recommendation, or favoring by the Regents of the University of California. The views and opinions of authors expressed herein do not necessarily state or reflect those of the Regents of the University of California and shall not be used for advertising or product endorsement purposes. 2

3 CONTENTS 1 Introduction Introduction State of the Art Nuclide Migration Models for a Water-Saturated Repository Nuclide Migration Model for an Unsaturated Repository Compartment Models Summary Objectives of the Study Scope and Summary Performance Assessment of a Repository in the Unsaturated Zone Introduction Physical Processes Mathematical Formulation Unsaturated Zone Saturated Zone Mass in the Environment Performance Measures Numerical Implementation Comparison with SAGE Model Input Data Numerical Results and Discussions Mean Residence Time Exposure Dose Rate Environmental Impact Effects of Vault Array Configuration Effects of Distance between NFI and GBI Discussions Conclusion Congruent Release of a Radionuclide from a Water-Saturated Repository Introduction Model Repository Structure Waste-Matrix Region Buffer Region Near-Field Rock Region i

4 3.2.5 Release into the Far Field Conversion into Dimensionless System Numerical Results and Observations Exit Concentration for N x < τ L Exit Concentration for N x > τ L Peak Exit Concentration and Its Upper Bound Release Rate from the Repository Discussions Upper-Bound Concentration and Upper-Bound Release Rate Expansion of Repository Footprint Effects of Leach Time Effects of Radioactive Decay Applicability to Other Nuclides Conclusions Solubility-Limited Release of a Radionuclide from the Water-Saturated Repository Introduction Physical Process Mathematical Formulation Waste-Matrix Region Buffer Region Near-Field Rock Region Radionuclide Masses Numerical Results and Discussion Input Data Transport in a Compartment Effects of Configuration Effects of Mass Reduction Mass-Based Performance Measures Future Extensions Conclusions Effects of Electro-Chemical Reduction Process on the Environmental Impact of Geologic Disposal Introduction Performance Measure for the Analysis Methods and Input Data Conditions on the Comparison Peak Radiotoxicity Release Rate Sodalite Waste Repository Results Limitations of the Analysis Conclusion ii

5 6 Markov Chain Model in a Two-Dimensional Array of Compartments Introduction Model Transition Probabilities for a Compartment Migration of a Single Particle through an Array of Compartments Migration of Multiple Particles Illustration 1: Migration at the Repository Scale Overview of Illustration Homogeneous Medium with All Compartments Connected Heterogeneous Medium with Random Connectivity Illustration 2: Comparison with a Continuum Model Continuum Model Application of the Markov Chain Model Ranges of the Parameters Illustration 3: Transport around a Waste Cylinder Groundwater Flow and Nuclide Migration around an Infinite Cylinder Release of Nuclide from the Waste Solid Results Illustration 4: Tilted Water Stream and Exit Concentration Application of the Model Observation of Exit Concentration Conclusion Distributions of Residence Times in the Compartment Model Introduction Framework of the Stochastic Model Migration of a Single Particle Migration of Multiple Particles Release from Multiple Waste Forms Illustration RTD for the Downstream Compartments RTD for the First Compartment Fractional Release Rate from a Single Waste Form Release Rate from the Entire Array Illustration Assignment of Residence Time Distributions Numerical Results Discussion Conclusion iii

6 8 Applicability and Limitations of the Models Streamline of Groundwater in the Repository Failure Time of Canisters and Matrix Degradation Rate Infiltration Rate of Water Determination of the Release Mode Application to Yucca Mountain Repository Conclusions 123 A Dispersion Effect in Compartment Models 125 A.1 Dispersion in an Array of Compartments A.2 Dispersion in a Continuous Medium A.3 Size of a Compartment and Dispersivity B True Peak Value of the Exit Concentration with Radioactive Decay Effect 128 C Determination of Release Mode 130 C.1 Concentration in the Buffer C.2 Peak Concentration at the Waste/Buffer Boundary C.3 Determination of Release Mode iv

7 LIST OF FIGURES 1.1 Nuclide migration models used (a) in [2, 3, 4, 6], (b) in [1], and (c) in Chapters 3 and 4 of this dissertation Configurations considered for an unsaturated repository (a) in SAGE and (b) in Chapter 2 of this dissertation Three schematic diagrams of vault array configuration and compartment arrangement Exposure dose rates of radionuclides for case C Radiotoxicity index of radionuclides released in the environment for case C Exposure dose rates of 239 Pu chain for cases A, B, and C Exposure dose rates of 129 I for cases A, B, and C Exposure dose rates of 14 C for cases A, B, and C Peak exposure dose rates for case C as functions of the distance between NFI and GBI Peak exposure dose rates of 129 I for cases A, B, and C Schematic diagram of repository structure and radionuclide transport considered in the compartment model Dimensionless concentration χ Nx (τ) of cesium in the groundwater at the repository exit obtained by numerical calculations with VR code Dimensionless concentration χ Nx (τ) of cesium in the groundwater at the repository exit obtained by numerical calculations with VR code Spatial distribution of the dimensionless concentration χ n (τ) of cesium in the NFR regions along a compartment row for N x = 64 and τ L = 7.3 (T L = 10 4 [yr]) Spatial distribution of the dimensionless concentration χ64 b (θ, τ) of cesium in the buffer region of the 64th compartment The peak exit concentration χ peak N x of cesium obtained from analytical formulae (3.24) and (3.25) for τ L = 7.3 (solid line) and τ L = 73 (dashed line) Contour plot of the peak exit concentration as a function of the dimensionless leach time τ L and N x The peak release rate φ peak N x,n y of cesium from the entire repository obtained from analytical formula (3.27) for τ L = 7.3 (solid line) and τ L = 73 (dashed line) Effect of radioactive decay on the exit concentration Profiles of 237 Np concentration in the buffer Time Tn dep when the radionuclide completely depletes from the waste Normalized concentration of 237 Np in the groundwater leaving the repository Masses M1 w(t), Mb 1 (t), Mr 1 (t), Mint (t), and M ext (t) of 237 Np in configuration B.. 62 v

8 4.5 Normalized masses of 237 Np in configuration A with 64 canisters, compared with those in configuration B Effect of the initial 237 Np mass reduction on the normalized 237 Np concentration C 1 (t)/c in the water leaving the repository for configuration B Effect of the initial 237 Np mass reduction on the normalized 237 Np concentration C 64 (t)/c in the water leaving the repository for configuration A Effect of the initial 237 Np mass reduction on M int (t)/64m and M ext (t)/64m for configuration B Effect of the initial 237 Np mass reduction on M int (t)/64m and M ext (t)/64m for configuration A Exposure dose rate obtained for the performance assessment of the reference repository Radiotoxicity index of radionuclides released from the repository Peak radiotoxicity release rate from the repository Peak radiotoxicity release rate from the repository with extended capacity (10 times) The transition probabilities for a particle migrating from the shaded compartment An N x N y array of compartments Probability a n (k) of the particle existence in each compartment in a array Fraction of the nuclide in the environment as a function of time Nuclide distribution at t = 60 t for one realization of randomly generated connectivity between compartments (P connect = 0.8) Points A and B, and the non-zero transition probabilities Dimensionless concentration χ at (x, y) = (L, 0) Range of the parameters Streamlines of steady flow of water around an infinite cylinder of radius r Distribution of a nuclide around a single waste cylinder Array of compartments considered in Illustration Dimensionless concentration at compartment A and B when water flow direction is parallel to x axis (θ = 0 ) Dimensionless concentration at compartment A when water flow direction is tilted by θ = Dimensionless concentration at compartment B when water flow direction is tilted by θ = Comparison of dimensionless concentrations at compartment B between θ = 0 and θ = Array of compartments and the migration of a single particle Relationship among the particle position index X (t), the time of transition T n, and the residence time U n Concept of superposition of nuclide streams from multiple waste forms vi

9 7.4 Fractional release rate f TN (t) of cesium initially contained in a single compartment based on Eq. (7.28) for N = 1, 2,..., Normalized release rate φn (t) of cesium from the entire array based on Eq. (7.31) Fractional release rate f TN (t) of cesium initially contained in a single compartment Normalized release rate φn (t) of cesium from the entire array Fractional release rate f TN (t) of cesium initially contained in a single compartment Normalized release rate φn (t) of cesium from the entire array Groundwater stream tilted vertically off the plane of canister array by angle θ A.1 Array of compartments connected by the groundwater flow vii

10 LIST OF TABLES 1.1 Categorization of radionuclide migration models by model assumptions and canister configurations Initial Inventory of Four Waste Vaults Radionuclides and Decay Chains Considered in the Numerical Demonstration Radioactive Decay Constants of Nuclides Sorption Distribution Coefficients Properties of Compartments Darcy Velocity of Water in the Unsaturated Zone and the Aquifer Flux-to-Dose Conversion Factor g (i) and Maximum Permissible Concentration C (i) MPC of Radionuclides Mean Lifetime of Radionuclides and Mean Residence Times in Compartments Assumed Parameters for Cesium and the Repository Assumed Parameters for Neptunium and the Repository Parameter Values for the Repository Parameter Values for the Repository Fission Products in the Waste from High-Burnup (48 GWd/t) PWR Transition Probabilities for Illustration viii

11 CHAPTER 1 INTRODUCTION 1.1 Introduction Radioactive wastes are being generated as a by-product of electricity generation in nuclear power plants. Some radionuclides in the high-level radioactive waste (HLW) are relatively shortlived and are radiologically toxic for a few years to a few hundred years, while some other radionuclides are long-lived and remain hazardous for more than 10,000 years. This waste must be safely isolated from the environment until it no longer poses a significant risk to human health and the environment. In many countries, geologic disposal is considered to be the method for the safe isolation of the HLW for a long period of time. A geologic repository confines HLW by the engineered barrier system (EBS) and the surrounding near-field rock (NFR). The EBS consists of the waste form, the canister, and the buffer that surrounds the canister. The confinement by the EBS eventually fails, and the radionuclides in the waste will be transported by groundwater through the geologic medium (geosphere) and released in the environment. In order to demonstrate that the repository can safely confine the radioactivity, the performance assessment is done. In a performance assessment, migration of radionuclides is modeled for the EBS, the near field, the geosphere (far field), and the biosphere to calculate the exposure dose estimates for individuals [1]. In previous performance assessments for HLW repository concepts, such as [2] for a geologic medium partially saturated with water and [1, 3, 4] for water-saturated media, the radionuclide transport in the groundwater in the EBS was first analyzed for a single-canister configuration (independent of other canisters) to determine the inlet boundary condition for the radionuclide transport in the far-field region. The repository was regarded as a collection of such independent single canisters, and the release rate of radionuclides from the entire repository was obtained by multiplying the total number of canisters by the release rate from a single canister. The size of the repository footprint and the position of each waste canister in the repository were not explicitly taken into account. A HLW repository, however, consists of thousands of waste canisters in a two-dimensional array. In a water-saturated repository where groundwater flows horizontally, groundwater would flow over multiple canisters, being contaminated before it flows out from the repository if there are failed canisters in the stream. The number and the array configuration of the waste canisters and the initial mass loading of toxic radionuclides in a canister could have significant effects on the release rate of radionuclides from the repository and radionuclide concentrations in the groundwater leaving the repository region. These quantities primarily determine the repository performance, which is measured by the radiological exposure dose rate to a human, who is assumed to live at a location 1

12 Table 1.1 Categorization of radionuclide migration models by model assumptions and canister configurations. Arrows represent water flow stream. Cylinders represent waste canisters. Configuration A: Parallel to water flow Configuration B: Perpendicular to water flow Independent-canister model (characterized by Eq. (1.1)) Connected-canister model (developed in Chapters 3 and 4) Release into the far field is spread out over time because of various migration distance depending on the Release into the far filed occurs in the same time frame for radionuclides from all canisters. canister position. Canister interaction is not observed. Assumed in [3, 2, 6, 4]. Canister interaction is observed Assumed in [1]. for solubility-limited release of radionuclides. downstream from the repository [5]. The point of interest of this dissertation is the analysis of radionuclide transport in geologic media with the repository footprint and the canister array configuration taken into account. 1.2 State of the Art Nuclide Migration Models for a Water-Saturated Repository Previous radionuclide migration models such as in [1, 2, 3, 4, 6] were developed for a single, noninteracting canister. Multiplicity of canisters was taken into account by multiplying the release rate from a single waste canister by the number of canisters in the repository. Consider that there are multiple identical canisters in the repository. We can assume two extreme configurations to take into account the canister multiplicity (see Table 1.1). In configuration A, N canisters are lined up in the direction parallel to, and included in, the water flow in the NFR. In configuration B, N canisters are lined up in the direction perpendicular to the water flow. In a water-saturated repository, groundwater is considered to flow horizontally. In such a repository, there will be multiple waste canisters in a water stream as configuration A. In configuration A, the nuclide migration distance in the NFR varies depending on the position of waste canister in the array. The time spent for the migration in the NFR therefore varies from canister to canister, and the radionuclide release into the far field would be spread out over time. It is also considered that, as the groundwater flows by multiple failed canisters, radionuclides accumulate in the groundwater. 2

13 Thus, the concentration of the radionuclide in the water stream increases as the water flows along the array. In configuration B, the migration distance to the far field is the same for all the canisters. Thus, the release into the far field occurs in the same time frame for radionuclides from all canisters. In contrast to configuration A, there is no accumulation of radionuclides in the groundwater since canisters are in separate streams. The radionuclide concentration in the water entering the far field is the same for any number of canisters. As the number of canisters in configuration B increases, the cross-sectional area of the contaminated water flow in the far-field region would also increase, resulting in an increase of the contaminated groundwater entering the biosphere. Configuration B can be considered as a collection of N single canister configurations, and is equivalent to the configuration considered in the previous models [1, 2, 3, 4, 6]. In [1], all waste canisters were assumed to be located 100 m from a major water-conducting fault (MWCF), ignoring the physical size of the repository. In their migration model, radionuclides released from the EBS immediately starts the migration through the 100 m pathway in the fractured rock. The length of the transport pathway through the host rock in the repository region, which could be more than 1000 m, 1 was neglected. Another aspect of the previous models is characterized by the boundary condition used at the outer buffer boundary (see Figure 1.1). An analysis for radionuclide transport in the EBS determines the mass release rate Q of a radionuclide from the EBS into the exterior region. In the buffer, molecular diffusion is considered to be dominant, and advection is ignored. At the outer boundary of the buffer, the radionuclide concentration is often assumed to be zero [2, 3, 4, 6], in order to decouple the transport problem inside the EBS from that in the exterior region [see Figure 1.1(a)]. For the far-field transport analysis, the inlet boundary concentration C is determined by C = Q/F, (1.1) by assuming a prescribed water flow rate F. We refer to the model characterized by the assumption of the zero concentration at the outer boundary of the buffer and by Eq. (1.1) as the independentcanister model. In [1], the previous zero-concentration boundary condition in the independent-canister model was replaced by the concentration continuity condition [see Figure 1.1(b)]. This was done by assuming that the concentration at the outer boundary of the buffer is equal to the concentration in the surrounding NFR, where the radionuclides are assumed to be mixed instantaneously. We refer to the model with the concentration continuity condition as the connected-canister model. The single canister configuration, which is equivalent to Configuration B, was still used in [1]. For actinide elements, the concentration in water at the waste-form surface is limited by the solubility. For a long-lived actinide such as 237 Np, the concentration profile in the buffer eventually reaches a steady state, ranging between the solubility limit at the inner boundary and the assumed zero concentration at the outer boundary (for independent-canister models). In configuration B, the 1 In one of the repository configurations suggested in [1], the waste canisters are placed on the rectangular plane whose dimensions are 1134 m 541 m. 3

14 release rate from the entire repository would be thus determined by the solubility, and would be proportional to the number of canisters. The combination of configuration A and the independent-canister model can also be considered, although this has never been applied in any previous models. The radionuclide concentration in the water leaving the repository increases proportionally with an increase of the number of canisters included in the same water stream because the identical release rate Q is added to the water stream by each waste canister. With this combination, the concentration of the radionuclide would increase indefinitely with the number of canisters. However, the concentration cannot exceed the solubility. This caveat results from the assumption that the concentration at the outer boundary of the buffer is equal to zero in the independent-canister model for decoupling the transport in the EBS from that in the NFR. In the combination of configuration A and the connected-canister model, if the concentration at the waste-form surface is limited by solubility, the concentration gradient in the buffer region decreases as the concentration in the NFR increases. Because the concentration becomes higher as the nuclides accumulate in the water stream, the release rate Q becomes smaller for downstream canisters. Thus, the radionuclide release from a canister is affected by other canisters ( canister interaction ). Figure of this dissertation. Nuclide migration models used (a) in [2, 3, 4, 6], (b) in [1], and (c) in Chapters 3 and 4

15 1.2.2 Nuclide Migration Model for an Unsaturated Repository Safety Assessment Groundwater Evaluation (SAGE) model [7, 8, 9] was previously developed for performance assessment for a generic unsaturated repository concept. The water, and hence radionuclides, flows vertically in the unsaturated media and eventually enter the aquifer underneath the repository. Groundwater flows horizontally in the aquifer. In SAGE model, the near field is divided into a series of compartments that represent the waste form, the EBS, and the unsaturated soil, which are connected by vertical mass flux of radionuclides [see Figure 1.2]. The aquifer in the far field is divided into a series of compartments connected by horizontal mass flux of radionuclides. The radionuclide flux at the bottom of the soil compartment was directly connected to the entrance of the far-field aquifer, regardless of the position of the waste vault. The pathway in the aquifer underneath the repository was not considered. This situation is similar to configuration B in Table 1.1 because the length of the migration pathway is the same for all the radionuclide in the repository Compartment Models Romero et al. [10] developed a compartment model for radionuclide migration in the Swedish repository concept. 2 The different regions such as the damage in the canister, the different parts of the backfill in the tunnel, the rock, and fractures are modeled as a number of compartments. The 2 The independent-canister model with single-canister configuration (see Table 1.1) was used in [10]. Figure 1.2 Configurations considered for an unsaturated repository (a) in SAGE and (b) in Chapter 2 of this dissertation. Boxes represent the compartments that form migration pathways in the system. Solid-line arrows indicate the directions of the radionuclide flow. The dashed-line arrow in (a) indicates that the unsaturated soil compartments are directly connected to the far-field. 5

16 discretization into compartments is similar to that of finite difference models. The main difference is that the compartment model uses much fewer cells or compartments. Radionuclide migration is represented by the mass in each compartment and diffusive and advective mass flow between compartments. The concept of compartment models is very useful when the transport is through materials with different properties and the geometry of the system is very complex. With a coarse discretization, the numerical calculation can be performed within a relatively short time compared to finite difference models with finer discretizations. The drawback to this compartment model approach is the compromised accuracy due to the effect of numerical diffusion introduced by the coarse discretization. In order to achieve a high accuracy, a fine discretization is necessary. Romero et al. [11] utilized analytical solutions to approximate the sensitive regions in the compartment model in order to avoid such fine discretizations Summary The effects of the canister array configuration on the repository performance were not observed in the previous performance assessments because release of radionuclides from each waste was treated independently and the length of radionuclide migration pathway was assumed to be the same for all radionuclides in the repository. The radionuclide migration model that explicitly takes into account the canister array configuration and the canister interaction has not been developed. In order to design a repository configuration, we need to study on the effects of the canister array configuration with such models. 1.3 Objectives of the Study Objectives of the present study are (1) to develop models for radionuclide transport in the repository and the surrounding geologic media, taking into account the repository footprint and canister array configuration, and (2) to observe the effects of the array configuration of waste canisters on the performance assessment of the repository. 1.4 Scope and Summary Three compartment models for radionuclide transport are developed in Chapters 2, 3, and 4 in order to observe the effects of canister array configurations. The main differences among the three models are the types of the repository being considered and the assumed modes of radionuclide release from waste forms. If we consider the pathways in the aquifer underneath the repository as shown in Figure 1.2(b), there would be various nuclide migration lengths in the aquifer because of the repository footprint. This situation corresponds to configuration A in Table 1.1. The radionuclide migration model characterized by Figure 1.2(b) is developed in Chapter 2. In Chapter 2, a compartment model is developed for transport of radionuclides released from a repository in the unsaturated zone. By considering the pathways in the aquifer underneath the 6

17 repository as shown in Figure 1.2(b), various lengths of nuclide migration in the aquifer due to the repository footprint is taken into account. The model thus incorporates the effects of the vault array configuration and the repository footprint. Instead of using a fine discretization, the numerical diffusion effect introduced by the coarse discretization in the compartment model is used to simulate the hydraulic dispersion effect. As a performance measure, the individual exposure dose rate is evaluated based on radionuclide release rates at the geosphere/biosphere interface (GBI). The effect of far field as a natural barrier and the effect of vault array configuration are investigated. In Chapters 3 and 4, a connected-canister model is developed for transport of radionuclides in water-saturated repository. Both configuration A and B discussed in Table 1.1 are considered. In Chapter 3, the radionuclide is assumed to be released congruently with the waste matrix degradation without the limitation of solubility. In Chapter 4, release of the radionuclide is assumed to be limited by its solubility. For numerical illustrations, 135 Cs (congruent release) and 237 Np (solubility-limited release), which are the two major contributors to the exposure dose, are used. It has been found that the exit concentration of 135 Cs would increase proportionally with the number of canisters aligned in the water flow direction if the number is smaller than a threshold value, and the exit concentration would remain the same regardless of the number of canisters if the number is greater than this threshold. The exit concentration of 237 Np would increase nonlinearly with the number of canisters aligned in the water flow direction. In Chapter 5, the results obtained in Chapters 3 and 4 are applied to evaluate how electrochemical reduction process [12] can affect the performance of the repository. With electro-chemical reduction process, the actinide elements in the spent fuel are recovered and recycled to produce fast reactor fuels. The mass of actinides loaded in a waste repository can be significantly reduced by this process. It is observed that the toxicity release rate from the repository due to 237 Np can be significantly reduced if the mass loading of 237 Np becomes less than 0.4 mol/canister by application of electro-chemical reduction process. However, if the mass loading of 237 Np is reduced below 0.1 mol/canister, the overall toxicity release rate would not be reduced because 135 Cs becomes the dominant contributor. It is observed in Chapters 2 through 5 that repository footprint and the canister array configuration have significant effects on the performance of a nuclear waste repository. Radionuclide migration analyses at the repository scale is necessary. As the scale of the domain becomes greater, it is preferred that the compartment size also becomes greater. However, the details of the migration process should not be lost in the repository-scale analysis. In Chapter 6, a new model is developed where particle migration is described by a discrete-time, discrete-state Markov chain. The goal of this model is to enable analyses of radionuclide migration at the repository scale based on the information obtained from a smaller-scale detailed analysis. In an illustration, a condition is posed on the size of a compartment, the size of a time step, and the diffusion coefficient in order to simulate diffusion effect. In Chapter 7, a particle migration model is developed using a continuous-time, discrete-state stochastic process. The effect of various lengths of migration pathways in a repository is revisited with an observation of distributions of the residence times in the repository. The size of a compartment is decoupled from the restriction due to dispersion effect. 7

18 CHAPTER 2 PERFORMANCE ASSESSMENT OF A REPOSITORY IN THE UNSATURATED ZONE 2.1 Introduction In this chapter, 1 the effects of the geometric configuration of a waste repository in the unsaturated zone are investigated with a compartment model. As depicted in Figure 1.2(a), the length of the migration pathway in the aquifer regions below the repository was not considered in the SAGE model [7, 8, 9]. As the result, the performance of the repository would be evaluated the same regardless of the configuration of the waste vault array. The compartment model (VR-KHNP) have been developed in this chapter by taking into account the nuclide migration through the unsaturated zones, the aquifer region below the repository and the aquifer in the far field. Observations on the repository performance are made with the radiological exposure dose rates and with the radiotoxicities in the environment. 2 The effects of vault array configuration and the effects of the migration length in the far field are observed. 2.2 Physical Processes The LILW repository in consideration consists of four waste vaults placed in an unsaturated soil. Dimensions of the considered LILW repository are shown in Figure 2.1. The waste vaults are first placed near the ground surface. Layers of materials cover the vaults. Three types of radioactive waste are contained in these four waste vaults. Radionuclide transport in and release from the repository are initiated by water contact. At the beginning of water entering the waste vault (i.e., t = 0), the waste contained in the vault is assumed to be instantaneously degraded into a porous medium partially saturated with water. Radionuclides originally contained in the solid phase in the vault dissolve into the water in the pores in the vault. It is assumed that the radionuclide concentrations in the pore water are instantaneously equilibrated with the concentrations in the solid phase after t = 0. The concentrations of dissolved 1 The material in this chapter is the results of the collaborative research project performed by Nuclear Environment Technology Institute, Korea Hydro & Nuclear Power Co. Ltd. (NETEC-KHNP) and the Department of Nuclear Engineering, University of California, Berkeley (UCB-NE) for a performance assessment of a low- and intermediate-level radioactive waste (LILW) repository. NETEC-KHNP is currently undertaking the performance assessment study for the LILW repository and is developing a conceptual design of the LILW repository. 2 In this chapter, the far field and the geosphere are used interchangeably. The environment and the biosphere are also used interchangeably. The near field is considered the geologic formation within the projected region of the repository footprint. 8

19 Figure 2.1 Three schematic diagrams of vault array configuration and compartment arrangement. The unsaturated zone consists of waste vault, concrete, and soil compartments. The vertical arrows through three unsaturated regions and the horizontal arrows through the aquifer compartments show the water flow directions. In case A, the long sides of the vaults are perpendicular to perpendicular to the groundwater flow in the aquifer. In cases B and C, the long sides are parallel to the groundwater flow. Difference between cases B and C is the positions of vaults 1 and 2. 9

20 nuclides in the pore water in the vaults are assumed to be lower than their solubilities. Limitation of radionuclide dissolution due to a low solubility is not considered in the present model. Water infiltrates through the waste vault at a rate gradually increasing with time because of gradual failure of the cover. In the present model, this evolution is described by a piecewise step function of time for the infiltration rate of water flowing down through the unsaturated zone in the vertical direction. Eventually, the infiltration rate reaches an ambient infiltration rate. It is assumed that the infiltration rate through the unsaturated zone is uniform in space. Radionuclides in the pore water in the waste vaults are transported vertically downward into the concrete region, into the soil region, and then into the aquifer via advection. Effect of molecular diffusion is neglected. Groundwater flows in a horizontal direction at a constant and uniform velocity in the aquifer. Radionuclides that have entered the aquifer are transported by advection. Radionuclide transport in the unsaturated zone and in the aquifer is modeled by considering a set of hypothetical compartments as shown in Figure 2.1. Properties of the medium such as the porosity, the water saturation, the density, and the sorption distribution coefficients, and radionuclide concentrations are assumed to be spatially uniform in each of these compartments. Three unsaturated regions, i.e., the waste vault, the concrete, and the soil, are represented by three compartments. The concrete and the soil compartments underneath the l th waste vault have the same cross-sectional area A l [m 2 ] perpendicular to water infiltration. The aquifer is divided into a series of compartments of identical dimensions. The dimension of the aquifer compartment in the flow direction is determined by considering the longitudinal dispersivity (see discussion in Section 2.3.2). To observe the effects of vault array configuration on the repository performance, three configurations are considered. In case A, the groundwater flow in the aquifer is perpendicular to the long sides of the vaults (see the top figure in Figure 2.1). In cases B and C, the groundwater flow in the aquifer is parallel to the long sides. In case B, waste vault 2 is in the downstream side relative to vault 1. Vault 2 contains greater inventory of radionuclides than vault 1 as shown in Table 2.1. Case C is the reverse of case B. 2.3 Mathematical Formulation Unsaturated Zone The mass balance equations in the three unsaturated compartments, including the waste vault, the concrete, and the soil, are written as follows in terms of the concentration C (i) lm [mol/m3 ] of radionuclide i in the water in the pores of the m th compartment in the unsaturated zone under the l th vault: with R (i) lm s lm ɛ lm V dc (i) lm lm dt = A l q k C (i) lm 1 (t) A l q k C (i) lm (t) + λ (i 1) R (i 1) lm C (0) lm (t) 0, s lm ɛ lm V lm C (i 1) lm (t) λ (i) R (i) lm s lm ɛ lm V lm C (i) lm (t), (2.1) 10 C (i) l0 (t) 0,

21 Table 2.1 Initial Inventory of Four Waste Vaults Initial Inventory [mol] a Nuclide Total Vault 1, M 0(i) 1 Vault 2, M 0(i) 2 Vault 3, M 0(i) 3 Vault 4, M 0(i) 3 4 H C Co Ni Ni Sr Nb Tc I Cs Pb Po Ra Ac Th Pa U U U Pu Pu a The initial inventories were originally given in terms of radioactivity [Bq] in [21]. for t k < t < t k+1, i = 1, 2,..., i 0, k = 0, 1, 2, l = 1,..., 4, m = 1, 2, 3, t 0 = 0. Superscript i denotes the i th member nuclide in a radioactive decay chain. Subscript k denotes the k th time interval in the piecewise step function for the infiltration rate. The infiltration rate of groundwater in the unsaturated zone is constant at q k [m/yr] during the time interval t k < t < t k+1. Subscript l indicates that the compartment of interest is connected with the l th waste-vault compartment. Subscript m denotes the three unsaturated compartments: the waste vault (m = 1), the concrete (m = 2), and the soil compartment (m = 3). The first and the second terms on the right side of Eq. (2.1) are the advective mass transfer rate of the radionuclide i into and out of the compartment, respectively. The third and the fourth terms on the right side are the gain and the loss of radionuclide i per unit time due to radioactive decay of its parent nuclide i 1 and nuclide i, respectively. The symbol λ (i) [yr 1 ] denotes the radioactive-decay constant of nuclide i. s lm and ɛ lm are the water saturation and the porosity, respectively, in the compartment designated by lm. With the cross-sectional area A l and the vertical dimension L lm [m] of the compartment, the volume, V lm [m 3 ], of the compartment 11

22 m is written as V lm = A l L lm. (2.2) By the assumed linear isotherm between the solid phase and the water phase, the retardation factor R (i) lm is defined as R (i) lm 1 + (1 ɛ lm )ρ lm K (i) dlm, (2.3) s lm ɛ lm (i) where K d lm [m3 /kg] and ρ lm [kg/m 3 ] are the sorption distribution coefficient of the i th nuclide and the density of the solid, respectively, in the compartment designated by lm. The total mass M (i) lm [mol] of the i th radionuclide in the compartment designated by lm is written as M (i) lm (t) R(i) lm s lm ɛ lm V lm C (i) lm (t). (2.4) With Eq. (2.4), the mass balance equation (2.1) can be rewritten as with for dm (i) lm dt = µ (i) klm 1 M(i) lm 1 (t) µ(i) klm M(i) lm (t) + λ(i 1) M (i 1) lm M (0) lm (t) 0, M(i) l0 (t) 0, (t) λ (i) M (i) (t), (2.5) lm t k < t < t k+1, i = 1, 2,..., i 0, k = 0, 1, 2, l = 1,..., 4, m = 1, 2, 3, t 0 = 0. All terms in Eq. (2.5) correspond to the terms in Eq. (2.1) in the same order. Rate coefficient µ (i) klm [yr 1 ] is defined as µ (i) klm A l q k R (i) lm s lm ɛ lm V lm = v klm R (i) lm L, (2.6) lm where v klm [m/yr] is the pore velocity of the water infiltrating in the compartment designated by lm during the k th time interval. The equality in Eq. (2.6) is obtained by substituting Eq. (2.2) and the following relationship: q k = s lm ɛ lm v klm. (2.7) The quantity 1/µ (i) klm [yr] is interpreted as the mean time for the i th nuclide to migrate through the length L lm by advection in the compartment designated by lm, or the mean residence time of the nuclide in the compartment. The greater the mean residence time is, the slower the nuclide migration is. We assume that at t = 0 the mass of radionuclide i in the waste vault l is equal to M 0(i) l [mol]. We also assume that no radionuclide initially exists in the concrete compartment or in the soil compartment. These are described by the following initial conditions for M (i) lm : M (i) l1 (0) = M0(i) l, M (i) l2 (0) = M(i) l3 (0) = 0, (2.8) for i = 1, 2,..., i 0, l = 1, 2,..., 4. The values of M 0(i) l assumed in this study are listed in Table

23 2.3.2 Saturated Zone Radionuclides are transferred from the soil compartment into the aquifer compartments located directly below the soil compartment. Groundwater flows in the horizontal direction in the aquifer. Radionuclides in the aquifer are transported from a compartment to another by advection. The aquifer is assumed to be saturated with water. The mass balance of the i th nuclide in the n th compartment in the aquifer is formulated as with for R sz (i) s sz ɛ sz V dc sz,n (i) sz dt = a ln A l q k C (i) l3 + A sz q sz C (i) sz,n 1 (t) A sz q sz C sz,n (i) (t) l + λ (i 1) R sz (i 1) s sz ɛ sz V sz C sz,n (i 1) (t) λ(i) R sz (i) s sz ɛ sz V sz C (i) (t), (2.9) C (i) (0) sz,0 (t) = 0, C sz,n (t) = 0, t k < t < t k+1, i = 1, 2,..., i 0, k = 0, 1, 2, n = 1, 2,..., N, t 0 = 0. Subscript SZ denotes the saturated zone, or the aquifer. Subscript n indicates the position of the compartment in the aquifer; the compartment denoted by n = 1 is located at the upstream end of the array, and n = N at the downstream end. The concentration of the i th radionuclide in the water phase in compartment n is denoted as C sz,n (i) [mol/m3 ]. The first term in the right side of Eq. (2.9) is the sum of the rates of mass transfer for the i th nuclide from the soil compartments into the n th compartment in the aquifer. In case a soil compartment has direct contact with multiple aquifer compartments, the water infiltration from the soil compartment into the aquifer is distributed over the multiple aquifer compartments. The fraction a ln of water infiltration into the n th aquifer compartment is proportional to the interfacial area between the n th aquifer compartment and the soil compartment. 3 The factor a ln A l q k represents the rate of water flowing into the n th compartment from the unsaturated soil compartment below the waste vault l. The second and the third terms in the right side of Eq. (2.9) represent the advective massflow rate of the i th nuclide into and out of the compartment n, where q sz [m/yr] is the Darcy velocity of water in the aquifer, and A sz [m 2 ] is the cross-sectional area of the aquifer compartments perpendicular to groundwater flow. The Darcy velocity q sz and the pore velocity v sz in the aquifer are connected by the following relationship: sz,n q sz = s sz ɛ sz v sz, (2.10) where s sz and ɛ sz are saturation and porosity in the aquifer, respectively. Because the aquifer is assumed to be fully saturated with water, s sz = 1. The Darcy velocity q sz in the aquifer compartments is assumed to be the same for all compartments and constant in time. 3 For example, in case A shown in Figure 2.1, the first compartment (n = 1) in the aquifer is connected with two rows of unsaturated-zone compartments, i.e., those including vault 2 and vault 3. Therefore, the summation in Eq. (2.9) includes l = 2 and l = 3. Footprints of both rows are completely included in aquifer compartment 1. Therefore, a 21 = 1, and a 31 = 1. 13