UCB-NE-5017
Dissertation for the degree of Doctor of Philosophy
Analyses of Radionuclide Migration inGeologic Media Using Compartment
Models
Daisuke KawasakiDepartment of Nuclear EngineeringUniverisity of California, Berkeley
December 19, 2005
The author invite comments and would appreciate being notified of any errors in the report.
Joonhong AhnDepartment of Nuclear EngineeringUniversity of CaliforniaBerkeley, CA 94720-1730USA
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CONTENTS
1 Introduction 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 State of the Art . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2.1 Nuclide Migration Models for a Water-Saturated Repository . . . . . . . . 21.2.2 Nuclide Migration Model for an Unsaturated Repository . . . . . . . . . . 51.2.3 Compartment Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.2.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.3 Objectives of the Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.4 Scope and Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2 Performance Assessment of a Repository in the Unsaturated Zone 82.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.2 Physical Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.3 Mathematical Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.3.1 Unsaturated Zone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.3.2 Saturated Zone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.3.3 Mass in the Environment . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.3.4 Performance Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.3.5 Numerical Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . 152.3.6 Comparison with SAGE Model . . . . . . . . . . . . . . . . . . . . . . . 16
2.4 Input Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.5 Numerical Results and Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.5.1 Mean Residence Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.5.2 Exposure Dose Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.5.3 Environmental Impact . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.5.4 Effects of Vault Array Configuration . . . . . . . . . . . . . . . . . . . . . 222.5.5 Effects of Distance between NFI and GBI . . . . . . . . . . . . . . . . . . 252.5.6 Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3 Congruent Release of a Radionuclide from a Water-Saturated Repository 303.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303.2 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.2.1 Repository Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313.2.2 Waste-Matrix Region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333.2.3 Buffer Region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333.2.4 Near-Field Rock Region . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
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3.2.5 Release into the Far Field . . . . . . . . . . . . . . . . . . . . . . . . . . 363.2.6 Conversion into Dimensionless System . . . . . . . . . . . . . . . . . . . 36
3.3 Numerical Results and Observations . . . . . . . . . . . . . . . . . . . . . . . . . 393.3.1 Exit Concentration for Nx < τL . . . . . . . . . . . . . . . . . . . . . . . 393.3.2 Exit Concentration for Nx > τL . . . . . . . . . . . . . . . . . . . . . . . 403.3.3 Peak Exit Concentration and Its Upper Bound . . . . . . . . . . . . . . . . 433.3.4 Release Rate from the Repository . . . . . . . . . . . . . . . . . . . . . . 45
3.4 Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 463.4.1 Upper-Bound Concentration and Upper-Bound Release Rate . . . . . . . . 463.4.2 Expansion of Repository Footprint . . . . . . . . . . . . . . . . . . . . . . 463.4.3 Effects of Leach Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473.4.4 Effects of Radioactive Decay . . . . . . . . . . . . . . . . . . . . . . . . . 473.4.5 Applicability to Other Nuclides . . . . . . . . . . . . . . . . . . . . . . . 49
3.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
4 Solubility-Limited Release of a Radionuclide from the Water-Saturated Repository 514.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 514.2 Physical Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 514.3 Mathematical Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
4.3.1 Waste-Matrix Region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 524.3.2 Buffer Region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 524.3.3 Near-Field Rock Region . . . . . . . . . . . . . . . . . . . . . . . . . . . 544.3.4 Radionuclide Masses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4.4 Numerical Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . 574.4.1 Input Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 574.4.2 Transport in a Compartment . . . . . . . . . . . . . . . . . . . . . . . . . 574.4.3 Effects of Configuration . . . . . . . . . . . . . . . . . . . . . . . . . . . 584.4.4 Effects of Mass Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . 634.4.5 Mass-Based Performance Measures . . . . . . . . . . . . . . . . . . . . . 664.4.6 Future Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
4.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
5 Effects of Electro-Chemical Reduction Process on the Environmental Impact of Geo-logic Disposal 715.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 715.2 Performance Measure for the Analysis . . . . . . . . . . . . . . . . . . . . . . . . 715.3 Methods and Input Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
5.3.1 Conditions on the Comparison . . . . . . . . . . . . . . . . . . . . . . . . 745.3.2 Peak Radiotoxicity Release Rate . . . . . . . . . . . . . . . . . . . . . . . 755.3.3 Sodalite Waste Repository . . . . . . . . . . . . . . . . . . . . . . . . . . 76
5.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 785.5 Limitations of the Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 805.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
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6 Markov Chain Model in a Two-Dimensional Array of Compartments 826.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 826.2 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
6.2.1 Transition Probabilities for a Compartment . . . . . . . . . . . . . . . . . 836.2.2 Migration of a Single Particle through an Array of Compartments . . . . . 836.2.3 Migration of Multiple Particles . . . . . . . . . . . . . . . . . . . . . . . . 86
6.3 Illustration 1: Migration at the Repository Scale . . . . . . . . . . . . . . . . . . . 876.3.1 Overview of Illustration . . . . . . . . . . . . . . . . . . . . . . . . . . . 876.3.2 Homogeneous Medium with All Compartments Connected . . . . . . . . . 886.3.3 Heterogeneous Medium with Random Connectivity . . . . . . . . . . . . . 89
6.4 Illustration 2: Comparison with a Continuum Model . . . . . . . . . . . . . . . . 916.4.1 Continuum Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 916.4.2 Application of the Markov Chain Model . . . . . . . . . . . . . . . . . . . 916.4.3 Ranges of the Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
6.5 Illustration 3: Transport around a Waste Cylinder . . . . . . . . . . . . . . . . . . 956.5.1 Groundwater Flow and Nuclide Migration around an Infinite Cylinder . . . 956.5.2 Release of Nuclide from the Waste Solid . . . . . . . . . . . . . . . . . . 976.5.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
6.6 Illustration 4: Tilted Water Stream and Exit Concentration . . . . . . . . . . . . . 996.6.1 Application of the Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 996.6.2 Observation of Exit Concentration . . . . . . . . . . . . . . . . . . . . . . 101
6.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
7 Distributions of Residence Times in the Compartment Model 1057.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1057.2 Framework of the Stochastic Model . . . . . . . . . . . . . . . . . . . . . . . . . 105
7.2.1 Migration of a Single Particle . . . . . . . . . . . . . . . . . . . . . . . . 1077.2.2 Migration of Multiple Particles . . . . . . . . . . . . . . . . . . . . . . . . 1097.2.3 Release from Multiple Waste Forms . . . . . . . . . . . . . . . . . . . . . 110
7.3 Illustration 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1117.3.1 RTD for the Downstream Compartments . . . . . . . . . . . . . . . . . . 1117.3.2 RTD for the First Compartment . . . . . . . . . . . . . . . . . . . . . . . 1127.3.3 Fractional Release Rate from a Single Waste Form . . . . . . . . . . . . . 1137.3.4 Release Rate from the Entire Array . . . . . . . . . . . . . . . . . . . . . 114
7.4 Illustration 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1157.4.1 Assignment of Residence Time Distributions . . . . . . . . . . . . . . . . 1157.4.2 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
7.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1177.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
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8 Applicability and Limitations of the Models 1198.1 Streamline of Groundwater in the Repository . . . . . . . . . . . . . . . . . . . . 1198.2 Failure Time of Canisters and Matrix Degradation Rate . . . . . . . . . . . . . . . 1208.3 Infiltration Rate of Water . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1218.4 Determination of the Release Mode . . . . . . . . . . . . . . . . . . . . . . . . . 1218.5 Application to Yucca Mountain Repository . . . . . . . . . . . . . . . . . . . . . 122
9 Conclusions 123
A Dispersion Effect in Compartment Models 125A.1 Dispersion in an Array of Compartments . . . . . . . . . . . . . . . . . . . . . . . 125A.2 Dispersion in a Continuous Medium . . . . . . . . . . . . . . . . . . . . . . . . . 126A.3 Size of a Compartment and Dispersivity . . . . . . . . . . . . . . . . . . . . . . . 127
B True Peak Value of the Exit Concentration with Radioactive Decay Effect 128
C Determination of Release Mode 130C.1 Concentration in the Buffer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130C.2 Peak Concentration at the Waste/Buffer Boundary . . . . . . . . . . . . . . . . . . 131C.3 Determination of Release Mode . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
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LIST OF FIGURES
1.1 Nuclide migration models used (a) in [2, 3, 4, 6], (b) in [1], and (c) in Chapters 3and 4 of this dissertation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2 Configurations considered for an unsaturated repository (a) in SAGE and (b) inChapter 2 of this dissertation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.1 Three schematic diagrams of vault array configuration and compartment arrange-ment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2 Exposure dose rates of radionuclides for case C . . . . . . . . . . . . . . . . . . . 212.3 Radiotoxicity index of radionuclides released in the environment for case C . . . . 232.4 Exposure dose rates of 239Pu chain for cases A, B, and C . . . . . . . . . . . . . . 242.5 Exposure dose rates of 129I for cases A, B, and C . . . . . . . . . . . . . . . . . . 242.6 Exposure dose rates of 14C for cases A, B, and C . . . . . . . . . . . . . . . . . . 252.7 Peak exposure dose rates for case C as functions of the distance between NFI and
GBI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262.8 Peak exposure dose rates of 129I for cases A, B, and C . . . . . . . . . . . . . . . 27
3.1 Schematic diagram of repository structure and radionuclide transport considered inthe compartment model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.2 Dimensionless concentration χNx (τ ) of cesium in the groundwater at the repositoryexit obtained by numerical calculations with VR code . . . . . . . . . . . . . . . 39
3.3 Dimensionless concentration χNx (τ ) of cesium in the groundwater at the repositoryexit obtained by numerical calculations with VR code . . . . . . . . . . . . . . . 41
3.4 Spatial distribution of the dimensionless concentration χn(τ ) of cesium in the NFRregions along a compartment row for Nx = 64 and τL = 7.3 (TL = 104 [yr]) . . . . 42
3.5 Spatial distribution of the dimensionless concentration χb64(θ, τ ) of cesium in the
buffer region of the 64th compartment . . . . . . . . . . . . . . . . . . . . . . . . 423.6 The peak exit concentration χ
peakNx
of cesium obtained from analytical formulae(3.24) and (3.25) for τL = 7.3 (solid line) and τL = 73 (dashed line) . . . . . . . . 43
3.7 Contour plot of the peak exit concentration as a function of the dimensionless leachtime τL and Nx . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.8 The peak release rate φpeakNx ,Ny
of cesium from the entire repository obtained fromanalytical formula (3.27) for τL = 7.3 (solid line) and τL = 73 (dashed line) . . . . 45
3.9 Effect of radioactive decay on the exit concentration . . . . . . . . . . . . . . . . 48
4.1 Profiles of 237Np concentration in the buffer . . . . . . . . . . . . . . . . . . . . . 584.2 Time T dep
n when the radionuclide completely depletes from the waste . . . . . . . 594.3 Normalized concentration of 237Np in the groundwater leaving the repository . . . 604.4 Masses Mw
1 (t), Mb1 (t), M r
1(t), M int(t), and Mext(t) of 237Np in configuration B . . 62
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4.5 Normalized masses of 237Np in configuration A with 64 canisters, compared withthose in configuration B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.6 Effect of the initial 237Np mass reduction on the normalized 237Np concentrationC1(t)/C∗ in the water leaving the repository for configuration B . . . . . . . . . . 64
4.7 Effect of the initial 237Np mass reduction on the normalized 237Np concentrationC64(t)/C∗ in the water leaving the repository for configuration A . . . . . . . . . 65
4.8 Effect of the initial 237Np mass reduction on M int(t)/64M◦ and Mext(t)/64M◦ forconfiguration B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
4.9 Effect of the initial 237Np mass reduction on M int(t)/64M◦ and Mext(t)/64M◦ forconfiguration A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
5.1 Exposure dose rate obtained for the performance assessment of the reference repos-itory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
5.2 Radiotoxicity index of radionuclides released from the repository . . . . . . . . . 725.3 Peak radiotoxicity release rate from the repository . . . . . . . . . . . . . . . . . 795.4 Peak radiotoxicity release rate from the repository with extended capacity (10 times) 80
6.1 The transition probabilities for a particle migrating from the shaded compartment . 846.2 An Nx × Ny array of compartments . . . . . . . . . . . . . . . . . . . . . . . . . 846.3 Probability a(k)
n of the particle existence in each compartment in a 30 × 30 array . 886.4 Fraction of the nuclide in the environment as a function of time . . . . . . . . . . 896.5 Nuclide distribution at t = 601t for one realization of randomly generated connec-
tivity between compartments (Pconnect = 0.8) . . . . . . . . . . . . . . . . . . . . 906.6 Points A and B, and the non-zero transition probabilities . . . . . . . . . . . . . . 926.7 Dimensionless concentration χ at (x, y) = (L , 0) . . . . . . . . . . . . . . . . . . 946.8 Range of the parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 956.9 Streamlines of steady flow of water around an infinite cylinder of radius r0 . . . . 966.10 Distribution of a nuclide around a single waste cylinder . . . . . . . . . . . . . . . 986.11 Array of compartments considered in Illustration 4 . . . . . . . . . . . . . . . . . 996.12 Dimensionless concentration at compartment A and B when water flow direction is
parallel to x axis (θ = 0◦) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1026.13 Dimensionless concentration at compartment A when water flow direction is tilted
by θ = 30◦ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1026.14 Dimensionless concentration at compartment B when water flow direction is tilted
by θ = 30◦ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1026.15 Comparison of dimensionless concentrations at compartment B between θ = 0◦
and θ = 30◦ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
7.1 Array of compartments and the migration of a single particle . . . . . . . . . . . . 1067.2 Relationship among the particle position index X (t), the time of transition Tn , and
the residence time Un . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1067.3 Concept of superposition of nuclide streams from multiple waste forms . . . . . . 111
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7.4 Fractional release rate fTN ′ (t) of cesium initially contained in a single compartmentbased on Eq. (7.28) for N ′
= 1, 2, . . . , 16 . . . . . . . . . . . . . . . . . . . . . . 1137.5 Normalized release rate φ◦
N (t) of cesium from the entire array based on Eq. (7.31) 1137.6 Fractional release rate fTN ′ (t) of cesium initially contained in a single compartment 1167.7 Normalized release rate φ◦
N (t) of cesium from the entire array . . . . . . . . . . . 1167.8 Fractional release rate fTN ′ (t) of cesium initially contained in a single compartment 1177.9 Normalized release rate φ◦
N (t) of cesium from the entire array . . . . . . . . . . . 117
8.1 Groundwater stream tilted vertically off the plane of canister array by angle θ . . . 120
A.1 Array of compartments connected by the groundwater flow . . . . . . . . . . . . . 125
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LIST OF TABLES
1.1 Categorization of radionuclide migration models by model assumptions and canisterconfigurations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
2.1 Initial Inventory of Four Waste Vaults . . . . . . . . . . . . . . . . . . . . . . . . 112.2 Radionuclides and Decay Chains Considered in the Numerical Demonstration . . 172.3 Radioactive Decay Constants of Nuclides . . . . . . . . . . . . . . . . . . . . . . 172.4 Sorption Distribution Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . 182.5 Properties of Compartments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.6 Darcy Velocity of Water in the Unsaturated Zone and the Aquifer . . . . . . . . . 192.7 Flux-to-Dose Conversion Factor g(i) and Maximum Permissible Concentration C (i)
MPCof Radionuclides . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.8 Mean Lifetime of Radionuclides and Mean Residence Times in Compartments . . 20
3.1 Assumed Parameters for Cesium and the Repository . . . . . . . . . . . . . . . . . 38
4.1 Assumed Parameters for Neptunium and the Repository . . . . . . . . . . . . . . . 56
5.1 Parameter Values for the Repository . . . . . . . . . . . . . . . . . . . . . . . . . 745.2 Parameter Values for the Repository . . . . . . . . . . . . . . . . . . . . . . . . . 745.3 Fission Products in the Waste from High-Burnup (48 GWd/t) PWR . . . . . . . . 77
6.1 Transition Probabilities for Illustration 1 . . . . . . . . . . . . . . . . . . . . . . . 88
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CHAPTER 1
INTRODUCTION
1.1 Introduction
Radioactive wastes are being generated as a by-product of electricity generation in nuclearpower plants. Some radionuclides in the high-level radioactive waste (HLW) are relatively short-lived and are radiologically toxic for a few years to a few hundred years, while some other radionu-clides are long-lived and remain hazardous for more than 10,000 years. This waste must be safelyisolated from the environment until it no longer poses a significant risk to human health and theenvironment.
In many countries, geologic disposal is considered to be the method for the safe isolation ofthe HLW for a long period of time. A geologic repository confines HLW by the engineered barriersystem (EBS) and the surrounding near-field rock (NFR). The EBS consists of the waste form, thecanister, 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 performanceassessment is done. In a performance assessment, migration of radionuclides is modeled for theEBS, the near field, the geosphere (far field), and the biosphere to calculate the exposure doseestimates for individuals [1].
In previous performance assessments for HLW repository concepts, such as [2] for a geologicmedium partially saturated with water and [1, 3, 4] for water-saturated media, the radionuclidetransport in the groundwater in the EBS was first analyzed for a single-canister configuration (inde-pendent of other canisters) to determine the inlet boundary condition for the radionuclide transportin the far-field region. The repository was regarded as a collection of such independent single can-isters, and the release rate of radionuclides from the entire repository was obtained by multiplyingthe total number of canisters by the release rate from a single canister. The size of the repositoryfootprint and the position of each waste canister in the repository were not explicitly taken intoaccount.
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 overmultiple canisters, being contaminated before it flows out from the repository if there are failedcanisters in the stream. The number and the array configuration of the waste canisters and the initialmass loading of toxic radionuclides in a canister could have significant effects on the release rateof radionuclides from the repository and radionuclide concentrations in the groundwater leavingthe repository region. These quantities primarily determine the repository performance, which ismeasured by the radiological exposure dose rate to a human, who is assumed to live at a location
1
Table 1.1 Categorization of radionuclide migration models by model assumptions and canisterconfigurations. Arrows represent water flow stream. Cylinders represent waste canisters.
Configuration A: Parallel to waterflow
Configuration B: Perpendicular towater flow
Release into the far field is spreadout over time because of various mi-gration distance depending on thecanister position.
Release into the far filed occurs inthe same time frame for radionu-clides from all canisters.
Independent-canistermodel (characterized byEq. (1.1))
Canister interaction is not observed. Assumed in [3, 2, 6, 4].
Connected-canistermodel (developed inChapters 3 and 4)
Canister interaction is observedfor solubility-limited release of ra-dionuclides.
Assumed in [1].
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
1.2.1 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 releaserate 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 extremeconfigurations to take into account the canister multiplicity (see Table 1.1). In configuration A, Ncanisters are lined up in the direction parallel to, and included in, the water flow in the NFR. Inconfiguration 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 repos-itory, there will be multiple waste canisters in a water stream as configuration A. In configurationA, the nuclide migration distance in the NFR varies depending on the position of waste canister inthe array. The time spent for the migration in the NFR therefore varies from canister to canister, andthe 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
Thus, the concentration of the radionuclide in the water stream increases as the water flows alongthe 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 sincecanisters are in separate streams. The radionuclide concentration in the water entering the far fieldis 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. ConfigurationB can be considered as a collection of N single canister configurations, and is equivalent to theconfiguration 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-conductingfault (MWCF), ignoring the physical size of the repository. In their migration model, radionuclidesreleased from the EBS immediately starts the migration through the 100 m pathway in the fracturedrock. The length of the transport pathway through the host rock in the repository region, whichcould be more than 1000 m,1 was neglected.
Another aspect of the previous models is characterized by the boundary condition used at theouter buffer boundary (see Figure 1.1).
An analysis for radionuclide transport in the EBS determines the mass release rate Q of a ra-dionuclide from the EBS into the exterior region. In the buffer, molecular diffusion is consideredto be dominant, and advection is ignored. At the outer boundary of the buffer, the radionuclideconcentration is often assumed to be zero [2, 3, 4, 6], in order to decouple the transport probleminside 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 assumptionof the zero concentration at the outer boundary of the buffer and by Eq. (1.1) as the “independent-canister model.”
In [1], the previous zero-concentration boundary condition in the independent-canister modelwas replaced by the concentration continuity condition [see Figure 1.1(b)]. This was done by as-suming that the concentration at the outer boundary of the buffer is equal to the concentration inthe surrounding NFR, where the radionuclides are assumed to be mixed instantaneously. We referto the model with the concentration continuity condition as the “connected-canister model.” Thesingle 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 thesolubility. For a long-lived actinide such as 237Np, the concentration profile in the buffer eventuallyreaches a steady state, ranging between the solubility limit at the inner boundary and the assumedzero concentration at the outer boundary (for independent-canister models). In configuration B, the
1In one of the repository configurations suggested in [1], the waste canisters are placed on the rectangular plane whosedimensions are 1134 m × 541 m.
3
release rate from the entire repository would be thus determined by the solubility, and would beproportional 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 thewater leaving the repository increases proportionally with an increase of the number of canistersincluded in the same water stream because the identical release rate Q is added to the water streamby each waste canister. With this combination, the concentration of the radionuclide would increaseindefinitely 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 isequal to zero in the independent-canister model for decoupling the transport in the EBS from thatin the NFR.
In the combination of configuration A and the connected-canister model, if the concentrationat the waste-form surface is limited by solubility, the concentration gradient in the buffer regiondecreases as the concentration in the NFR increases. Because the concentration becomes higher asthe nuclides accumulate in the water stream, the release rate Q becomes smaller for downstreamcanisters. Thus, the radionuclide release from a canister is affected by other canisters (“canisterinteraction”).
Figure 1.1 Nuclide migration models used (a) in [2, 3, 4, 6], (b) in [1], and (c) in Chapters 3 and4 of this dissertation.
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1.2.2 Nuclide Migration Model for an Unsaturated Repository
Safety Assessment Groundwater Evaluation (SAGE) model [7, 8, 9] was previously developedfor performance assessment for a generic unsaturated repository concept. The water, and henceradionuclides, flows vertically in the unsaturated media and eventually enter the aquifer underneaththe repository. Groundwater flows horizontally in the aquifer.
In SAGE model, the near field is divided into a series of compartments that represent the wasteform, 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 byhorizontal mass flux of radionuclides. The radionuclide flux at the bottom of the soil compartmentwas directly connected to the entrance of the far-field aquifer, regardless of the position of the wastevault. The pathway in the aquifer underneath the repository was not considered. This situation issimilar to configuration B in Table 1.1 because the length of the migration pathway is the same forall the radionuclide in the repository.
1.2.3 Compartment Models
Romero et al. [10] developed a compartment model for radionuclide migration in the Swedishrepository concept.2 The different regions such as the damage in the canister, the different parts ofthe backfill in the tunnel, the rock, and fractures are modeled as a number of compartments. The
2The 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 Chapter2 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
discretization into compartments is similar to that of finite difference models. The main differenceis that the compartment model uses much fewer cells or compartments. Radionuclide migrationis represented by the mass in each compartment and diffusive and advective mass flow betweencompartments.
The concept of compartment models is very useful when the transport is through materials withdifferent properties and the geometry of the system is very complex. With a coarse discretization, thenumerical calculation can be performed within a relatively short time compared to finite differencemodels with finer discretizations.
The drawback to this compartment model approach is the compromised accuracy due to theeffect of numerical diffusion introduced by the coarse discretization. In order to achieve a highaccuracy, a fine discretization is necessary. Romero et al. [11] utilized analytical solutions to ap-proximate the sensitive regions in the compartment model in order to avoid such fine discretizations.
1.2.4 Summary
The effects of the canister array configuration on the repository performance were not observedin the previous performance assessments because release of radionuclides from each waste wastreated independently and the length of radionuclide migration pathway was assumed to be thesame for all radionuclides in the repository. The radionuclide migration model that explicitly takesinto 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 arrayconfiguration with such models.
1.3 Objectives of the Study
Objectives of the present study are (1) to develop models for radionuclide transport in the repos-itory and the surrounding geologic media, taking into account the repository footprint and canisterarray configuration, and (2) to observe the effects of the array configuration of waste canisters onthe performance assessment of the repository.
1.4 Scope and Summary
Three compartment models for radionuclide transport are developed in Chapters 2, 3, and 4 inorder to observe the effects of canister array configurations. The main differences among the threemodels are the types of the repository being considered and the assumed modes of radionucliderelease 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 char-acterized by Figure 1.2(b) is developed in Chapter 2.
In Chapter 2, a compartment model is developed for transport of radionuclides released froma repository in the unsaturated zone. By considering the pathways in the aquifer underneath the
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repository as shown in Figure 1.2(b), various lengths of nuclide migration in the aquifer due tothe repository footprint is taken into account. The model thus incorporates the effects of the vaultarray configuration and the repository footprint. Instead of using a fine discretization, the numericaldiffusion effect introduced by the coarse discretization in the compartment model is used to simulatethe hydraulic dispersion effect. As a performance measure, the individual exposure dose rate isevaluated based on radionuclide release rates at the geosphere/biosphere interface (GBI). The effectof 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 radionuclidesin water-saturated repository. Both configuration A and B discussed in Table 1.1 are considered. InChapter 3, the radionuclide is assumed to be released congruently with the waste matrix degradationwithout the limitation of solubility. In Chapter 4, release of the radionuclide is assumed to be limitedby its solubility. For numerical illustrations, 135Cs (congruent release) and 237Np (solubility-limitedrelease), which are the two major contributors to the exposure dose, are used. It has been found thatthe exit concentration of 135Cs would increase proportionally with the number of canisters aligned inthe water flow direction if the number is smaller than a threshold value, and the exit concentrationwould remain the same regardless of the number of canisters if the number is greater than thisthreshold. The exit concentration of 237Np would increase nonlinearly with the number of canistersaligned in the water flow direction.
In Chapter 5, the results obtained in Chapters 3 and 4 are applied to evaluate how electro-chemical reduction process [12] can affect the performance of the repository. With electro-chemicalreduction process, the actinide elements in the spent fuel are recovered and recycled to produce fastreactor fuels. The mass of actinides loaded in a waste repository can be significantly reduced bythis process. It is observed that the toxicity release rate from the repository due to 237Np can besignificantly reduced if the mass loading of 237Np becomes less than 0.4 mol/canister by applicationof electro-chemical reduction process. However, if the mass loading of 237Np is reduced below0.1 mol/canister, the overall toxicity release rate would not be reduced because 135Cs becomes thedominant contributor.
It is observed in Chapters 2 through 5 that repository footprint and the canister array config-uration have significant effects on the performance of a nuclear waste repository. Radionuclidemigration 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 migrationprocess 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 migrationat the repository scale based on the information obtained from a smaller-scale detailed analysis. Inan illustration, a condition is posed on the size of a compartment, the size of a time step, and thediffusion coefficient in order to simulate diffusion effect.
In Chapter 7, a particle migration model is developed using a continuous-time, discrete-statestochastic process. The effect of various lengths of migration pathways in a repository is revisitedwith an observation of distributions of the residence times in the repository. The size of a compart-ment is decoupled from the restriction due to dispersion effect.
7
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 unsatu-rated zone are investigated with a compartment model.
As depicted in Figure 1.2(a), the length of the migration pathway in the aquifer regions belowthe repository was not considered in the SAGE model [7, 8, 9]. As the result, the performance ofthe 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 ac-count the nuclide migration through the unsaturated zones, the aquifer region below the repositoryand the aquifer in the far field. Observations on the repository performance are made with the radi-ological exposure dose rates and with the radiotoxicities in the environment. 2 The effects of vaultarray 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 unsaturatedsoil. Dimensions of the considered LILW repository are shown in Figure 2.1. The waste vaults arefirst placed near the ground surface. Layers of materials cover the vaults. Three types of radioactivewaste are contained in these four waste vaults.
Radionuclide transport in and release from the repository are initiated by water contact. Atthe beginning of water entering the waste vault (i.e., t = 0), the waste contained in the vault isassumed to be instantaneously degraded into a porous medium partially saturated with water. Ra-dionuclides originally contained in the solid phase in the vault dissolve into the water in the poresin the vault. It is assumed that the radionuclide concentrations in the pore water are instantaneouslyequilibrated with the concentrations in the solid phase after t = 0. The concentrations of dissolved
1The material in this chapter is the results of the collaborative research project performed by Nuclear EnvironmentTechnology Institute, Korea Hydro & Nuclear Power Co. Ltd. (NETEC-KHNP) and the Department of Nuclear Engi-neering, University of California, Berkeley (UCB-NE) for a performance assessment of a low- and intermediate-levelradioactive waste (LILW) repository. NETEC-KHNP is currently undertaking the performance assessment study for theLILW repository and is developing a conceptual design of the LILW repository.
2In 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 therepository footprint.
8
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 arrowsthrough three unsaturated regions and the horizontal arrows through the aquifer compartments showthe water flow directions. In case A, the long sides of the vaults are perpendicular to perpendicular tothe groundwater flow in the aquifer. In cases B and C, the long sides are parallel to the groundwaterflow. Difference between cases B and C is the positions of vaults 1 and 2.
9
nuclides in the pore water in the vaults are assumed to be lower than their solubilities. Limitation ofradionuclide 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 ofgradual failure of the cover. In the present model, this evolution is described by a piecewise stepfunction of time for the infiltration rate of water flowing down through the unsaturated zone in thevertical direction. Eventually, the infiltration rate reaches an ambient infiltration rate. It is assumedthat 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 theconcrete region, into the soil region, and then into the aquifer via advection. Effect of moleculardiffusion 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 consideringa set of hypothetical compartments as shown in Figure 2.1. Properties of the medium such as theporosity, the water saturation, the density, and the sorption distribution coefficients, and radionuclideconcentrations 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 bythree compartments. The concrete and the soil compartments underneath the l’th waste vault havethe same cross-sectional area Al [m2] perpendicular to water infiltration.
The aquifer is divided into a series of compartments of identical dimensions. The dimension ofthe aquifer compartment in the flow direction is determined by considering the longitudinal disper-sivity (see discussion in Section 2.3.2).
To observe the effects of vault array configuration on the repository performance, three config-urations are considered. In case A, the groundwater flow in the aquifer is perpendicular to the longsides of the vaults (see the top figure in Figure 2.1). In cases B and C, the groundwater flow in theaquifer is parallel to the long sides. In case B, waste vault 2 is in the downstream side relative tovault 1. Vault 2 contains greater inventory of radionuclides than vault 1 as shown in Table 2.1. CaseC is the reverse of case B.
2.3 Mathematical Formulation
2.3.1 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] ofradionuclide i in the water in the pores of the m’th compartment in the unsaturated zone under thel’th vault:
R(i)lm slmεlm Vlm
dC (i)lm
dt= AlqkC (i)
lm−1(t) − AlqkC (i)lm (t)
+ λ(i−1) R(i−1)lm slmεlm VlmC (i−1)
lm (t) − λ(i) R(i)lm slmεlm VlmC (i)
lm (t) , (2.1)
withC (0)
lm (t) ≡ 0 , C (i)l0 (t) ≡ 0 ,
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Table 2.1 Initial Inventory of Four Waste Vaults
Initial Inventory [mol] aNuclide
Total Vault 1, M0(i)1 Vault 2, M0(i)
2 Vault 3, M0(i)3 Vault 4, M0(i)
43H 2.50 × 10−2 2.78 × 10−4 8.62 × 10−3 8.06 × 10−3 8.06 × 10−3
14C 7.33 × 10+0 1.16 × 10−1 7.18 × 10−1 3.25 × 10+0 3.25 × 10+0
60Co 6.89 × 10−2 7.22 × 10−3 4.32 × 10−2 9.20 × 10−3 9.20 × 10−3
59Ni 2.08 × 10+1 1.91 × 10+0 6.90 × 10+0 5.98 × 10+0 5.98 × 10+0
63Ni 6.91 × 10−1 9.36 × 10−2 3.81 × 10−1 1.08 × 10−1 1.08 × 10−1
90Sr 3.05 × 10−3 2.68 × 10−4 2.55 × 10−3 1.17 × 10−4 1.17 × 10−4
94Nb 1.54 × 10−1 2.78 × 10−2 1.12 × 10−1 7.05 × 10−3 7.05 × 10−3
99Tc 6.55 × 10−1 8.73 × 10−2 4.27 × 10−1 7.04 × 10−2 7.04 × 10−2
129I 1.49 × 10+1 7.27 × 10−1 4.68 × 10+0 4.75 × 10+0 4.75 × 10+0
137Cs 1.38 × 10−1 1.52 × 10−2 1.19 × 10−1 2.11 × 10−3 2.11 × 10−3
210Pb 0 0 0 0 0210Po 0 0 0 0 0226Ra 0 0 0 0 0227Ac 0 0 0 0 0230Th 0 0 0 0 0231Pa 0 0 0 0 0234U 0 0 0 0 0235U 1.01 × 10+1 2.39 × 10−1 1.92 × 10+0 3.99 × 10+0 3.99 × 10+0
238U 1.61 × 10+4 4.46 × 10+3 1.15 × 10+4 5.81 × 10+1 5.81 × 10+1
238Pu 8.47 × 10−4 1.82 × 10−5 2.58 × 10−4 2.86 × 10−4 2.86 × 10−4
239Pu 1.07 × 10−1 9.51 × 10−3 7.42 × 10−2 1.18 × 10−2 1.18 × 10−2
a The initial inventories were originally given in terms of radioactivity [Bq] in [21].
for
tk < t < tk+1 , i = 1, 2, . . . , i0 , k = 0, 1, 2 , l = 1, . . . , 4 , m = 1, 2, 3 , t0 = 0 .
Superscript i denotes the i’th member nuclide in a radioactive decay chain. Subscript k denotesthe k’th time interval in the piecewise step function for the infiltration rate. The infiltration rateof groundwater in the unsaturated zone is constant at qk [m/yr] during the time interval tk < t <
tk+1. Subscript l indicates that the compartment of interest is connected with the l’th waste-vaultcompartment. 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 rateof 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 perunit time due to radioactive decay of its parent nuclide i −1 and nuclide i , respectively. The symbolλ(i) [yr−1] denotes the radioactive-decay constant of nuclide i . slm and εlm are the water saturationand the porosity, respectively, in the compartment designated by lm. With the cross-sectional area Aland the vertical dimension L lm [m] of the compartment, the volume, Vlm [m3], of the compartment
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m is written asVlm = Al L lm . (2.2)
By the assumed linear isotherm between the solid phase and the water phase, the retardation factorR(i)
lm is defined as
R(i)lm ≡ 1 +
(1 − εlm)ρlm Kd(i)lm
slmεlm, (2.3)
where Kd(i)lm [m3/kg] and ρlm [kg/m3] 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 writ-ten as
M (i)lm (t) ≡ R(i)
lm slmεlm VlmC (i)lm (t) . (2.4)
With Eq. (2.4), the mass balance equation (2.1) can be rewritten as
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 (t) − λ(i)M (i)lm (t) , (2.5)
withM (0)
lm (t) ≡ 0 , M (i)l0 (t) ≡ 0 ,
for
tk < t < tk+1 , i = 1, 2, . . . , i0 , k = 0, 1, 2 , l = 1, . . . , 4 , m = 1, 2, 3 , t0 = 0 .
All terms in Eq. (2.5) correspond to the terms in Eq. (2.1) in the same order. Rate coefficientµ(i)
klm [yr−1] is defined as
µ(i)klm ≡
Alqk
R(i)lm slmεlm Vlm
=vklm
R(i)lm L lm
, (2.6)
where vklm [m/yr] is the pore velocity of the water infiltrating in the compartment designated by lmduring the k’th time interval. The equality in Eq. (2.6) is obtained by substituting Eq. (2.2) and thefollowing relationship:
qk = slmεlmvklm . (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 ofthe nuclide in the compartment. The greater the mean residence time is, the slower the nuclidemigration is.
We assume that at t = 0 the mass of radionuclide i in the waste vault l is equal to M0(i)l [mol].
We also assume that no radionuclide initially exists in the concrete compartment or in the soilcompartment. 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, . . . , i0 , l = 1, 2, . . . , 4 .
The values of M0(i)l assumed in this study are listed in Table 2.1.
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2.3.2 Saturated Zone
Radionuclides are transferred from the soil compartment into the aquifer compartments locateddirectly 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. Theaquifer 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
R(i)sz sszεszVsz
dC (i)sz,n
dt=
∑l
aln AlqkC (i)l3 + AszqszC
(i)sz,n−1(t) − AszqszC
(i)sz,n(t)
+ λ(i−1) R(i−1)sz sszεszVszC
(i−1)sz,n (t) − λ(i) R(i)
sz sszεszVszC(i)sz,n(t) , (2.9)
withC (i)
sz,0(t) = 0 , C (0)sz,n(t) = 0 ,
fortk < t < tk+1 , i = 1, 2, . . . , i0 , k = 0, 1, 2 , n = 1, 2, . . . , N , t0 = 0 .
Subscript SZ denotes the saturated zone, or the aquifer. Subscript n indicates the position of thecompartment in the aquifer; the compartment denoted by n = 1 is located at the upstream end ofthe array, and n = N at the downstream end. The concentration of the i’th radionuclide in the waterphase in compartment n is denoted as C (i)
sz,n [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 soilcompartment has direct contact with multiple aquifer compartments, the water infiltration fromthe soil compartment into the aquifer is distributed over the multiple aquifer compartments. Thefraction aln of water infiltration into the n’th aquifer compartment is proportional to the interfacialarea between the n’th aquifer compartment and the soil compartment.3 The factor aln Alqk representsthe rate of water flowing into the n’th compartment from the unsaturated soil compartment belowthe waste vault l.
The second and the third terms in the right side of Eq. (2.9) represent the advective mass-flow rate of the i’th nuclide into and out of the compartment n, where qsz [m/yr] is the Darcyvelocity of water in the aquifer, and Asz [m2] is the cross-sectional area of the aquifer compartmentsperpendicular to groundwater flow. The Darcy velocity qsz and the pore velocity vsz in the aquiferare connected by the following relationship:
qsz = sszεszvsz , (2.10)
where ssz and εsz are saturation and porosity in the aquifer, respectively. Because the aquifer is as-sumed to be fully saturated with water, ssz = 1. The Darcy velocity qsz in the aquifer compartmentsis assumed to be the same for all compartments and constant in time.
3For example, in case A shown in Figure 2.1, the first compartment (n = 1) in the aquifer is connected with tworows 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, a21 = 1,and a31 = 1.
13
The fourth and the fifth terms in Eq. (2.9) represent the mass gain of the i’th nuclide by radioac-tive decay of the parent nuclide i − 1 in the decay chain, and the loss by radioactive decay of thei’th nuclide itself, respectively. The volumes of an aquifer compartment, Vsz [m3], is determined by
Vsz ≡ AszLsz , (2.11)
where Lsz [m] is the length of an aquifer compartment which is determined by the longitudinalhydrodynamic dispersivity α as4
Lsz = 2α . (2.12)
The derivation of Eq. (2.12) is explained in Appendix A.The retardation factor R(i)
sz is defined as
R(i)sz ≡ 1 +
(1 − εsz)ρsz Kd(i)sz
sszεsz, (2.13)
where ρsz [kg/m3] is the density of the solid phase in the aquifer, and Kd(i)sz [m3/kg] is the sorption
distribution coefficient of the i’th nuclide in the aquifer.The mass [mol] of the i’th radionuclide in the n’th aquifer compartment is written as
M (i)sz,n(t) ≡ R(i)
sz sszεszVszC(i)sz,n(t) . (2.14)
Substituting Eq. (2.14) into Eq. (2.9), the mass balance equation in the n’th aquifer compartment isrewritten in terms of M (i)
sz,n as
dM (i)sz,n
dt=
∑l=1
alnµ(i)kl3 M (i)
l3 (t) + µ(i)sz M (i)
sz,n−1(t) − µ(i)sz M (i)
sz,n(t) + λ(i−1)M (i−1)sz,n (t) − λ(i)M (i)
sz,n(t) ,
(2.15)with
M (0)sz,n(t) ≡ 0 , M (i)
sz,0(t) ≡ 0 ,
fortk < t < tk+1 , i = 1, 2, . . . , i0 , k = 0, 1, 2 , n = 1, 2, . . . , N , t0 = 0 .
All terms in Eq. (2.15) correspond to the terms in Eq. (2.9) in the same order. Rate coefficientµ(i)
sz [yr−1] is defined as
µ(i)sz ≡
Aszqsz
R(i)sz sszεszVsz
=vsz
R(i)sz Lsz
, (2.16)
where the second equality is obtained by substituting Eqs. (2.10) and (2.11). 1/µ(i)sz [yr] is inter-
preted as the mean residence time of the nuclide in an aquifer compartment.It is assumed that there is initially no radionuclide in the aquifer. The governing equations (2.15)
are therefore subject to the following initial conditions:
M (i)sz,n(0) = 0 , i = 1, 2, . . . , i0 , n = 1, 2, . . . , N . (2.17)
4In the present study, sizes of compartments in the unsaturated zone are not determined in the same manner. They areset to the physical sizes of the waste vaults, concrete, and soil.
14
2.3.3 Mass in the Environment
The radionuclides released from the last aquifer compartment (n = N ) through the geosphere/biosphereinterface (GBI) are considered to enter the environment. The mass of the radionuclide in the envi-ronment is controlled by the flux at the GBI and the radioactive decay:
dM (i)env
dt= µ(i)
sz M (i)sz,N (t) + λ(i−1)M (i−1)
env (t) − λ(i)M (i)env(t) . (2.18)
It is assumed that no radionuclide initially exists in the environment;
M (i)env(0) = 0 . (2.19)
2.3.4 Performance Measures
As a performance measure, we calculate the exposure dose rate of all members of each decaychain based on the release rate at the GBI. The exposure dose rate HN (t) [Sv/yr] for a decay chainis obtained as
HN (t) ≡
i0∑i=1
g(i)λ(i)µ(i)sz M (i)
sz,N (t) ×NA
3.15 × 107 s/yr, t > 0 (2.20)
where g(i) [(Sv/yr)/(Bq/yr)] is the flux-to-dose conversion factor determined by an analysis forradionuclide migration in the biosphere. The value of N represents the index number of the lastcompartment in the aquifer (N = 7 for case A, and N = 12 for cases B and C; see Figure 2.1). Thesymbol NA denotes Avogadro’s number (6.02 × 1023 atoms/mol).
As a supplemental performance measure, we consider the environmental impact of the LILWrepository, which is defined as radiotoxicity index of the radionuclides that exist in the environment[13, 14]. The radiotoxicity of a radionuclide is obtained by dividing the radioactivity by the maxi-mum permissible concentration C (i)
MPC [Ci/m3] for ingestion [15]. The environmental impact of theradionuclide is formulated as
environmental impact [m3] =λ(i)M (i)
env(t)
C (i)MPC
×NA
(3.15 × 107 s/yr) · (3.7 × 1010 Bq/Ci). (2.21)
The environmental impact has the unit of water volume, with which the radionuclide concentrationis diluted to its maximum permissible concentration.
2.3.5 Numerical Implementation
The governing equations for the radionuclide mass in the unsaturated-zone compartments, aquifercompartments, and the environment are described by Eqs. (2.5), (2.15), and (2.18) with initial con-ditions (2.8), (2.17), and (2.19). All these equations are linear and can be rewritten in the followingmatrix form:
dM(t)dt
= AkM(t) , tk < t < tk+1 , k = 0, 1, 2 , t0 = 0 , (2.22)
M(0) = M0 , (2.23)
15
where M(t) is a vector that consists of the masses of all nuclides in all compartments, M (i)lm (t)
and M (i)sz,n(t), and the masses in the environment, M (i)
env(t). The vector M0 consists of initial valuesgiven by Eqs. (2.8), (2.17), and (2.19), corresponding to M(t). The constant matrix Ak consists ofthe parameter values λ(i), µ(i)
klm , µ(i)sz , and aln for the time domain k. The solution to Eq. (2.22) is
explicitly obtained as
M(t) = M(tk)e−Ak t , tk < t ≤ tk+1 , k = 0, 1, 2 , t0 = 0 . (2.24)
The numerical code VR-KHNP5 has been developed based on formula (2.24) using GNU Oc-tave [17]. With formula (2.24), the mass distribution at time t can be directly obtained withoutcalculation of a history of mass distributions between time 0 and t . All the numerical results shownhereafter have been obtained with the VR-KHNP code.
2.3.6 Comparison with SAGE Model
The present VR-KHNP model differs from the SAGE model [9] in several aspects. SAGE takesinto account precipitation of a radionuclide when the radionuclide concentration in the water reachesits solubility limit, whereas VR-KHNP assumes that no radionuclides will precipitate. SAGE takesinto account transport from one compartment to another by molecular diffusion in addition to thatby advection, whereas VR-KHNP only considers transport by advection because we are primarilyinterested in cases where Peclet number is relatively large in all compartments. In SAGE all ra-dionuclides are transferred into a single aquifer compartment from the soil compartment, whereasin VR-KHNP nuclides could be transferred into multiple aquifer compartments according to thedimension of the waste vault.
A benchmark test has been performed by setting up input parameter sets for VR-KHNP and forSAGE, such that they become equivalent to each other. As the result of benchmark, the release ratesof radionuclides obtained from both codes have shown a good agreement [18, 19, 20].
2.4 Input Data
Radionuclides and the decay chains considered in the present study are listed in Table 2.2.Some short-lived radionuclides in the multi-member decay chains are omitted. For example, 238U istreated as if it directly decays into 234U, skipping intermediate nuclides 234Th and 234Pa. Parametervalues for each radionuclide are shown in Tables 2.3 and 2.4 [21].
Three types of radioactive wastes are placed in four waste vaults. The first type of the wasteis contained in waste vault 1, the second in vault 2, and the third in vaults 3 and 4. These threewaste types have different radionuclide compositions and different physical properties. See Table2.1 for the initial inventory of radionuclides in each waste vault, and Table 2.5 for the dimensionsand properties of the compartments. Note that waste vault 2 contains greater masses than wastevault 1 for all radionuclides. We consider three vault array configurations, cases A, B, and C, shownin Figure 2.1.
5The numerical codes developed for analyses in this dissertation are archived in [16].
16
Table 2.2 Radionuclides and Decay Chains Considered in the Numerical Demonstration
Radionuclides of Single Member Decay Chains3H, 14C, 60Co, 59Ni, 63Ni, 90Sr, 94Nb, 99Tc, 129I, 137Cs
Multi-Member Decay Chains a
(238U chain) 238U →234U →
230Th →226Ra →
210Pb →210Po →
(238Pu chain) 238Pu →234U →
230Th →226Ra →
210Pb →210Po →
(239Pu chain) 239Pu →235U →
231Pa →227Ac →
a Only the member nuclides with relatively long half-lives are shown and consideredin the calculation. Member nuclides in the decay chain that are not shown are notincluded in the calculation.
Table 2.3 Radioactive Decay Constants of Nuclides
Nuclide Decay Constant, λ(i) [yr−1]3H 5.59 × 10−2
14C 1.21 × 10−4
60Co 1.32 × 10−1
59Ni 9.19 × 10−6
63Ni 7.22 × 10−3
90Sr 2.38 × 10−2
94Nb 3.41 × 10−5
99Tc 3.25 × 10−6
129I 4.41 × 10−8
137Cs 2.31 × 10−2
Nuclide Decay Constant, λ(i) [yr−1]210Pb 3.11 × 10−2
210Po 1.83 × 10+0
226Ra 4.33 × 10−4
227Ac 3.18 × 10−2
230Th 9.00 × 10−6
231Pa 2.11 × 10−5
234U 2.83 × 10−6
235U 9.85 × 10−10
238U 1.55 × 10−10
238Pu 7.90 × 10−3
239Pu 2.88 × 10−5
Values for Darcy velocities of water in the unsaturated zone and the aquifer are shown in Table2.6. Three time intervals are considered for evolution of water infiltration rate in the unsaturatedzone.
The flux-to-dose conversion factors [22] and the maximum permissible concentrations [15] usedfor obtaining exposure dose rates and toxicity indices, respectively, are listed in Table 2.7.
2.5 Numerical Results and Discussions
2.5.1 Mean Residence Time
Based on the input parameter values shown in Tables 2.3 through 2.6, the mean lifetime and themean residence times in compartments for each radionuclide are calculated and listed in Table 2.8.For the unsaturated zone, only the mean residence times for t > 500 yr are shown in the table. Themean residence time is regarded as the time for the radionuclide to migrate in the compartment(s).The time the radionuclide is released into the environment at the GBI can be estimated by the sum ofthe residence times in the unsaturated zone and the aquifer along the trajectory of the radionuclide.
17
Table 2.4 Sorption Distribution Coefficients [21]
Distribution Coefficient [m3/kg]Element
Waste, Kd(i)l1 Concrete, Kd
(i)l2 Soil, Kd
(i)l3 Aquifer, Kd
(i)sz
H 0 0 0 0C 2.50 × 10+0 2.50 × 10+0 5.00 × 10−3 1.00 × 10−2
Co 2.00 × 10−2 2.00 × 10−2 1.50 × 10−2 1.00 × 10+0
Ni 2.00 × 10−2 2.00 × 10−2 4.00 × 10−1 1.00 × 10+0
Sr 2.50 × 10−3 2.50 × 10−3 1.50 × 10−2 2.00 × 10−2
Nb 5.00 × 10−1 5.00 × 10−1 0 a 1.00 × 10+0
Tc 5.00 × 10−1 6.00 × 10−1 1.00 × 10−4 1.00 × 10+2
I 6.00 × 10−4 6.00 × 10−4 1.00 × 10−3 5.00 × 10−3
Cs 2.50 × 10−4 2.50 × 10−4 3.00 × 10−1 1.00 × 10−1
Pb b 2.00 × 10+0 2.00 × 10+0 0 1.00 × 10+2
Po b 2.00 × 10+0 2.00 × 10+0 0 1.00 × 10+2
Ra b 2.00 × 10+0 2.00 × 10+0 0 1.00 × 10+2
Ac b 2.00 × 10+0 2.00 × 10+0 0 1.00 × 10+2
Th b 2.00 × 10+0 2.00 × 10+0 0 1.00 × 10+2
Pa b 2.00 × 10+0 2.00 × 10+0 0 1.00 × 10+2
U 2.00 × 10+0 2.00 × 10+0 0 a 1.00 × 10+2
Pu 4.00 × 10+0 4.00 × 10+1 0 a 5.00 × 10+0
a Distribution coefficients in the soil for Nb, U, and Pu were assumed to be zeroin [21]. As a consequence, it is considered that the mean residence times in theunsaturated zone shown in Table 2.8 have become shorter and the peak exposuredose rates are conservatively overestimated.b In the present study, distribution coefficients for these elements are assumed to bethe same as those for U.
Table 2.5 Properties of Compartments [21]
Dimensiona [m] Porosity Saturation Solid Material Density [kg/m3]
Waste Vault 1 L11 = 8.2 ε11 = 0.3 s11 = 0.5 ρ11 = 2857Waste Vault 2 L21 = 8.2 ε21 = 0.12 s21 = 0.5 ρ21 = 2840Waste Vault 3 L31 = 8.2 ε31 = 0.12 s31 = 0.5 ρ31 = 2840Waste Vault 4 L41 = 8.2 ε41 = 0.12 s41 = 0.5 ρ41 = 2840Concrete Ll2 = 0.5 εl2 = 0.12 sl2 = 0.5 ρl2 = 2840Soil Ll3 = 3.0 εl3 = 0.3 sl3 = 0.7 ρl3 = 2571Aquifer Lsz = 40.0 b εsz = 0.25 ssz = 1.0 ρsz = 3333a The dimension of a compartment is measured along the flow direction of ground-water.b The dimension of an aquifer compartment is determined by Eq. (2.12) with thelongitudinal dispersivity α = 20 m.
18
Table 2.6 Darcy Velocity of Water in the Unsaturated Zone and the Aquifer [21]
Darcy velocity in unsaturated zone (waste vault,concrete, and soil), qk [m/yr]
3.5 × 10−4, k = 0 ( 0 < t ≤ 100 yr)3.5 × 10−2, k = 1 (100 < t ≤ 500 yr)3.5 × 10−1, k = 2 (t > 500 yr)
Darcy velocity in aqfuier, qsz [m/yr] 1.0 × 101
Table 2.7 Flux-to-Dose Conversion Factor g(i) and Maximum Permissible Concentration C (i)MPC of
Radionuclides [15, 22]
Nuclide g(i) [(Sv/yr)/(Bq/yr)] C(i)MPC [Ci/m3]
3H 4.60 × 10−20 1 × 10−3
14C 3.79 × 10−19 3 × 10−5
60Co 1.71 × 10−18 3 × 10−6
59Ni 2.39 × 10−20 3 × 10−4
63Ni 5.64 × 10−20 1 × 10−4
90Sr 1.01 × 10−17 5 × 10−7
94Nb 7.38 × 10−19 1 × 10−5
99Tc 2.68 × 10−19 6 × 10−5
129I 5.23 × 10−17 2 × 10−7
137Cs 6.43 × 10−18 1 × 10−6
239Pu 1.96 × 10−16 2 × 10−8
235U 2.01 × 10−27 3 × 10−7
231Pa 4.95 × 10−27 6 × 10−9
227Ac 3.61 × 10−27 5 × 10−9
238Pu – 2 × 10−8
238U – 3 × 10−7
234U 9.35 × 10−25 3 × 10−7
230Th 3.60 × 10−25 1 × 10−7
226Ra 2.90 × 10−25 6 × 10−8
210Pb – 1 × 10−8
210Po – 4 × 10−8
19
Table 2.8 Mean Lifetime of Radionuclides and Mean Residence Times in Compartments
Nuclide Mean Lifetime [yr] Mean Residence Time [yr]Unsaturated Zone a Aquiferb
1/λ(i) ∑3m=1 1/µ
(i)klm 1/µ
(i)sz
l = 1 l = 2, 3, 43H 1.79 × 10+1 5.40 × 10+0 3.29 × 10+0 1.00 × 10+0
14C 8.26 × 10+3 1.26 × 10+5 1.55 × 10+5 1.01 × 10+2
60Co 7.58 × 10+0 1.25 × 10+3 1.48 × 10+3 1.00 × 10+4
59Ni 1.09 × 10+5 7.18 × 10+3 7.42 × 10+3 1.00 × 10+4
63Ni 1.38 × 10+2 7.18 × 10+3 7.42 × 10+3 1.00 × 10+4
90Sr 4.20 × 10+1 3.63 × 10+2 3.90 × 10+2 2.01 × 10+2
94Nb 2.93 × 10+4 2.52 × 10+4 3.11 × 10+4 1.00 × 10+4
99Tc 3.08 × 10+5 2.56 × 10+4 3.14 × 10+4 1.00 × 10+6
137Cs 4.33 × 10+1 4.65 × 10+3 4.65 × 10+3 1.00 × 10+3
238Pu 1.27 × 10+2 3.30 × 10+5 3.77 × 10+5 5.00 × 10+4
210Pb 3.22 × 10+1 1.01 × 10+5 1.24 × 10+5 1.00 × 10+6
210Po 5.46 × 10−1 1.01 × 10+5 1.24 × 10+5 1.00 × 10+6
226Ra 2.31 × 10+3 1.01 × 10+5 1.24 × 10+5 1.00 × 10+6
227Ac 3.14 × 10+1 1.01 × 10+5 1.24 × 10+5 1.00 × 10+6
230Th 1.11 × 10+5 1.01 × 10+5 1.24 × 10+5 1.00 × 10+6
231Pa 4.74 × 10+4 1.01 × 10+5 1.24 × 10+5 1.00 × 10+6
234U 3.53 × 10+5 1.01 × 10+5 1.24 × 10+5 1.00 × 10+6
235U 1.02 × 10+9 1.01 × 10+5 1.24 × 10+5 1.00 × 10+6
238U 6.45 × 10+9 1.01 × 10+5 1.24 × 10+5 1.00 × 10+6
239Pu 3.47 × 10+4 3.30 × 10+5 3.77 × 10+5 5.00 × 10+4
129I 2.27 × 10+7 5.11 × 10+1 5.60 × 10+1 5.10 × 10+1
a Sum of the residence times in the waste compartment, the concrete compartment,and the soil compartment during the last time domain (t > 500 [yr]) is shown asthe residence time in the unsaturated zone. Residence times for t ≤ 10 [yr] and10 < t ≤ 500 [yr] are greater than the shown values by a factor of 1000 and 10,respectively [see Eq. (2.6) and Table 2.6]. Results for vault 1 are shown in a separatecolumn because vault 1 has a porosity different from other three vaults as shown inTable 2.5.b Mean residence time in a single aquifer compartment is shown.
20
The time spent in the aquifer can be obtained by multiplying the mean residence time in a singleaquifer compartment by the number of aquifer compartments.
During this migration time in the unsaturated zone and the aquifer, radionuclides undergo ra-dioactive decay. Radionuclides with their mean lifetimes shorter than mean residence times byorders of magnitude, such as 60Co, 63Ni, 90Sr, 137Cs, and 238Pu, undergo significant radioactivedecay during the migration. These radionuclides are not of concern in performance assessmentsbecause they decay out before entering the environment.
Radionuclides with their mean lifetimes longer than their mean residence times by orders ofmagnitude such as 129I and 238U, on the other hand, survive the migration. Most of these radionu-clides initially placed in the repository are released into the environment.
One of the effects of the vault array configuration appears through the amount of radioactivedecay during the migration. Since a difference in the vault array configuration results in a differencein migration time in the aquifer, the amount of radionuclides that survive the migration would beaffected by the configuration.
2.5.2 Exposure Dose Rate
The exposure dose rates of radionuclides for case C are plotted in Figure 2.2. As shown later,the peak values of the exposure dose rates in case C are the lowest among the three cases.
The peak exposure dose rate of 129I at around t = 900 yr is the highest among all the ra-dionuclides. Before t = 500 yr, release of 129I into the environment is not significant because the
10-14
10-13
10-12
10-11
10-10
10-9
10-8
101 102 103 104 105 106 107
Exp
osur
e D
ose
Rat
e, H
12 [
Sv/y
r]
Time, t [yr]
3H 14C
59Ni
94Nb
129I 238U chain
239Pu chain
Figure 2.2 Exposure dose rates of radionuclides for case C. For 238U and 239Pu chains, the sum ofexposure dose rates of all the decay chain members is shown.
21
infiltration rate is small and the residence time in the unsaturated zone is longer than 500 yr.6
Because of zero sorption anywhere assumed for H, the residence times of 3H are significantlyshorter than those of other radionuclides. Thus, 3H is released into the environment earlier thanothers, and the peak of its exposure dose rate is observed at t = 120 yr. The steep increase in theprofile of 3H observed at t = 100 yr is due to the stepwise change in the infiltration rate qk assumedin the unsaturated zone (see Table 2.6).
For decay chains of multiple members, the sum of the exposure dose rates of all members areplotted in Figure 2.2. Two peaks are observed in the profile of 239Pu decay chain. The first peak isdue to 239Pu, and the second due to 235U and its decay daughters at a secular equilibrium.
As mentioned earlier, radionuclides (such as 60Co, 63Ni, 90Sr, 137Cs, and 238Pu) with their meanlifetimes shorter than their mean residence times under go significant radioactive decay during themigration, and their exposure dose rates are below 10−14 Sv/yr.
2.5.3 Environmental Impact
The radiotoxicity index of each radionuclide in the environment is shown in Figure 2.3 for caseC. Toxicity index of each radionuclide increases as the radionuclide is released into and accumulatesin the environment, and eventually decreases because of radioactive decay. It shows that environ-mental impact of the LILW repository is dominated by 129I at the early times (from around 300 yruntil 106 yr). After 106 yr, toxicities of 238U and its daughters become significant. By this time,radioactivity of all the daughter nuclides of 238U has reached a secular equilibrium and have thesame radioactivity as 238U. The greatest contributor to the total toxicity of the 238U chain is 210Pb,which has the smallest C (i)
MPC value among the member nuclides in the chain (see Table 2.7).
2.5.4 Effects of Vault Array Configuration
To observe the effects of vault array configurations, the exposure dose rates of 239Pu chain,129I, and 14C for all three cases of vault array configuration are plotted in Figures 2.4, 2.5, and 2.6,respectively.
Remarkable difference among cases A, B, and C is observed in the profiles of 239Pu chain in Fig-ure 2.4. The configuration difference results in differences in the migration distance in the aquifer.The radionuclide migration in the unsaturated zone is not affected by vault array configurations.Thus, the difference results from the vault array configuration relative to the groundwater flow di-rection in the aquifer.
In case A, radionuclides enter either the first aquifer compartment (n = 1) or the second (n = 2).In cases B and C, radionuclides entering the aquifer are distributed over seven aquifer compartments(n = 1, . . . , 7). This long series of near-field aquifer compartments in cases B and C leads to a broadrange of residence times elapsed in the aquifer. Therefore, in cases B and C, the radionuclide releaseat the GBI tends to be at a lower rate compared to that in case A.
The difference between cases B and C is due to the arrangement order of vaults 1 and 2. Vault2, which has the inventory greater than vault 1 for all radionuclides, is located further away from
6For example, the residence time of 129I in the unsaturated zones including the vault 1 (l = 1) for t ≤ 10 [yr] is51100 yr [see Eq. (2.6) and Table 2.6]. Thus, 129I will be retained in the unsaturated zone in the first 10 yr.
22
the GBI in case C. Therefore, the peak values in case C appear lower than in case B because of thelonger distance to the GBI.
The difference in the peak height among the three cases would become more prominent forradionuclides with a residence time in the aquifer greater than its residence time in the unsaturatedzone and its lifetime. For example, in Figure 2.4, a difference by one order of magnitude betweencases A and C is observed in the first peak of the exposure dose rate at around t = 105 yr, due to239Pu. The difference is not as significant in the second peak (due to 235U and its decay daughters)because there is negligible effect of radioactive decay due to the long lifetime of 235U. Note that thesorption distribution coefficient for Pu is assumed to be significantly smaller than those for 235U andits decay daughters in the aquifer (see Table 2.4).
The similar effect of vault array configuration is observed for 129I in Figure 2.5. The differencebetween the three cases is small because radioactive decay loss is negligible while it is transportedthrough the unsaturated zone and the aquifer.
Figure 2.6 shows that difference in the exposure dose rate of 14C is negligible among the threecases. As observed in Table 2.8, the residence time of 14C in an aquifer compartment (∼ 102 yr) issignificantly shorter than that in the unsaturated zone (of the order of 105 yr after t = 500 yr). Forsuch a nuclide, vault array configuration has negligible effects on the total migration time since it ispredominantly determined by the residence time in the unsaturated zone. (In contrast, for 129I and239Pu, the total residence time in the aquifer is not negligible.)
100101102103104105106107108109
101 102 103 104 105 106 107 108
Radi
otox
icity
Inde
x [m
3 ]
Time, t [yr]
3H
14C 59Ni
94Nb
129I
238U chain
238Pu chain
239Pu chain
Figure 2.3 Radiotoxicity index of radionuclides released in the environment for case C. The dis-tance between NFI (near-field/far-field interface) and GBI (geosphere/biosphere interface) is as-sumed to be 200 m. For 238U, 238Pu, and 239Pu chains, the sum of radiotoxicity of all the decaychain members is shown. See Eq. (2.21) for the definition of the radiotoxicity.
23
10-12
10-11
10-10
10-9
10-8
101 102 103 104 105 106 107
Exp
osur
e D
ose
Rat
e, H
N [
Sv/y
r]
Time, t [yr]
239Pu chain
239Pu
235U and its daughters
Case ACase BCase C
Figure 2.4 Exposure dose rates of 239Pu chain for cases A, B, and C. The first peak observedaround 105 yr is mainly due to 239Pu, and the second peak around 107 yr is due to 235U and itsdaughters.
10-12
10-11
10-10
10-9
10-8
101 102 103 104 105 106 107
Exp
osur
e D
ose
Rat
e, H
N [
Sv/y
r]
Time, t [yr]
129ICase ACase BCase C
Figure 2.5 Exposure dose rates of 129I for cases A, B, and C.
24
14C migrates through and is released slowly from the unsaturated zone. Once it enters theaquifer, it is swept away by the groundwater flow and quickly reaches the GBI regardless of thevault array configuration. Thus, the configuration of the waste vaults has negligible effects on theexposure dose rate of 14C.
In general, the vault array configuration has significant effects on the exposure dose rates if theradionuclide has a relatively long residence time in the aquifer compared with that in the unsaturatedzone and the lifetime of the radionuclide. For such radionuclides, the peak exposure dose rate incase C was observed to be smallest among the three considered configurations.
2.5.5 Effects of Distance between NFI and GBI
To observe effects of the far-field barrier, the distance between NFI and GBI (NFI-GBI distance)in case C is changed to values between 0 m and 1200 m. Figure 2.7 shows the peak value of theexposure dose rate for each radionuclide as a function of the NFI-GBI distance. Values at 200 mcorrespond to the peak values of the profiles for case C in Figure 2.2. Note that the time of thepeak appearance is different for each distance and for each radionuclide although it is not shown inFigure 2.7.
The peak exposure dose rates for all radionuclides are observed to decrease with the NFI-GBIdistance. This is due to the effect of radioactive decay and the effect of longitudinal dispersion.The effect of the far-field barrier is remarkable for 238Pu chain, 137Cs, 90Sr, 99Tc, and 63Ni. Thesenuclides decay to a negligible level with a far-field barrier as short as 200 m.
In contrast, the peaks of 3H, 14C, 129I, and 238U chain do not decrease as much by having alonger far-field barrier. Because their mean lifetimes are significantly longer than their residence
10-12
10-11
10-10
10-9
10-8
101 102 103 104 105 106 107
Exp
osur
e D
ose
Rat
e, H
N [
Sv/y
r]
Time, t [yr]
14CCase ACase BCase C
Figure 2.6 Exposure dose rates of 14C for cases A, B, and C.
25
times in the aquifer compartments, the amount of radioactive decay loss during migration is small.The slight decrease with the distance is mainly due to the longitudinal dispersion effect.
For 239Pu chain, two regimes are observed. For the NFI-GBI distance shorter than 120 m, thepeak exposure dose rate is due to 239Pu, whereas for the distance longer than 120 m, the peak isdue to 235U and its daughters. This indicates that, to confine 239Pu, which could give a dose ratecomparable to 129I, the NFI-GBI distance is recommended to be 200 m or greater for the parametervalues assumed in the present study.
The peak exposure dose rates of 129I for cases A, B, and C are plotted in Figure 2.8 with respectto the NFI-GBI distance. While the peak exposure dose rate in case C is lowest for all distances, thedifference among the three cases becomes smaller as the distance increases.
2.5.6 Discussions
Instantaneous dissolution of radionuclides in the waste vault is assumed in the present model. Itis considered that this assumption leads to a conservative overestimation of the peak exposure doserates. If, instead, it takes some time for a radionuclide to dissolve in the infiltrating water in thewaste vault compartment, the residence time in the unsaturated zone is increased. If this increasechanges the residence time by orders of magnitude, it would affect the profiles of exposure dose ratesignificantly. Results for radionuclides with relatively short residence times in the unsaturated zone(such as 3H and 129I) are considered to be thus affected significantly by this assumption.
As observed in Figure 2.2, the exposure dose rate of 3H has an sudden increase due to the step-
10-1710-1610-1510-1410-1310-1210-1110-1010-910-810-7
0 200 400 600 800 1000 1200
Peak
Exp
osur
e D
ose
Rat
e [S
v/yr
]
Distance between NFI and GBI [m]
3H 14C59Ni
63Ni
90Sr94Nb
99Tc
129I
137Cs
238U chain
238Pu chain
239Pu chain
Figure 2.7 Peak exposure dose rates for case C as functions of the distance between NFI and GBI.Values at 200 m correspond to the peaks observed in Figure 2.2. The time of the peaks are differentfor different radionuclides and for different NFI-GBI distances.
26
wise change in the water infiltration rate at t = 100 yr introduced by the present model. Similareffect is observed in the profile of 129I at t = 500 yr although it is less obvious in Figures 2.2 and 2.5.Since 129I determines the overall peak exposure dose rate of the repository, it is desirable to eval-uate its profile with more realistic assumptions for the water infiltration rate in future performanceassessment.
We have observed that the vault array configuration has significant effects on the exposure doserates resulting from two major contributors, 129I and 239Pu chain. As observed in Figure 2.8, thepeak exposure dose rate of 129I is the smallest for case C, where the long sides of the waste vaultsare parallel to groundwater flow direction and loading the more radioactive waste further away fromthe GBI. While the peak exposure dose rate of 129I can be reduced further by increasing the NFI-GBI distance, reduction of its release rate from the vault by reducing the initial inventory of 129I orby improving the EBS would be more effective to reduce the 129I peak dose.
As observed in Figures 2.4 and 2.7, the vault array configuration and the NFI-GBI distance alsoaffect the peak dose due to 239Pu chain significantly.
It has been observed in Figure 2.3 that 129I and 238U chain are the main contributors to the en-vironmental impact. Because of the long half-lives of these nuclides, almost all of 129I and 238Uinitially placed in the repository would be released into the environment before they undergo ra-dioactive decay. For such nuclides as 129I and 238U, whose lifetimes are significantly long, the vaultarray configuration and the NFI-GBI distance have almost no effect on the peak values of the en-vironmental impact. In order to reduce contributions of these radionuclides to the environmentalimpact, we therefore need to reduce the initial inventories of these radionuclides in the repository.
The acceptance guideline of the LILW repository in Korea is given as the maximum individualexposure dose rate of 10−5 Sv/yr. As observed in Figure 2.2, the overall peak exposure dose rate
0×100
1×10-9
2×10-9
3×10-9
0 200 400 600 800 1000 1200
Peak
Exp
osur
e D
ose
Rat
e [S
v/yr
]
Distance between NFI and GBI [m]
Case ACase BCase C
Figure 2.8 Peak exposure dose rates of 129I for cases A, B, and C.
27
in case C (1.2 × 10−9 Sv/yr), which is predominantly determined by 129I, is far smaller than theguideline.
The major objective of the present study is to develop a deterministic model, with which theeffects of repository configuration are investigated. Numerical evaluations have been performedonly for one specific set of parameter values; uncertainties of the parameter values are not takeninto account. If the exposure dose rate is used as the performance measure for the repository, itmust be shown with the uncertainty associated with it. In order to finally conclude if the differencein vault array configuration has statistically meaningful effects on the performance, a probabilisticassessment needs to be made with this model by knowing probability distribution functions for theparameters.
2.6 Conclusion
A compartment model has been developed for radionuclide transport by groundwater flowthrough an unsaturated repository and the surrounding media. The VR-KHNP code has been devel-oped, based on the model.
Migration of radionuclides with consideration of the vault array configuration can be charac-terized by the three key parameters: the mean lifetime, the mean residence time in the unsaturatedzone, and the mean residence time in an aquifer compartment.
The vault array configuration has significant effects on the exposure dose rates if the radionu-clide has a relatively long residence time in the aquifer compared with that in the unsaturated zoneand the lifetime of the radionuclide. For such radionuclides, the peak exposure dose rate in case Cis observed to be smallest among the three considered configurations in this study. In this case-Cconfiguration, the long sides of the waste vaults are parallel to the groundwater flow direction inthe aquifer. The exposure dose rate of such radionuclides can be further reduced by increasing theNFI-GBI distance.
Among such radionuclides, 60Co, 63Ni, 90Sr, 99Tc, 137Cs, and 238Pu chain have mean lifetimessignificantly shorter than mean residence times in the repository and in the far field. They decay outbefore reaching the biosphere, resulting in negligible contributions to the exposure dose rates andto the environmental impacts. Any combination of the vault array configuration and the NFI-GBIdistance would effectively confine radionuclides of this category within the far-field barrier.
For the major contributors to the exposure dose rate and the environmental impact, such as 129Iand 238U decay chain, the effects of vault array configuration and the NFR-GBI distance have beenfound to be still significant although the their lifetimes are significantly longer than the residencetimes. On the other hand, the environmental impacts of these radionuclides are not significantlyaffected by the vault array configuration and the NFI-GBI distance because of their long lifetimes.The exposure dose rates and the environmental impacts resulting from these radionuclides can bereduced by reducing their release rates from the waste vaults, which can be achieved by reducing theinitial loadings of these radionuclides in the waste vaults and by improving the engineered-barriersystem.
The residence time of 14C in the aquifer is significantly shorter than that in the unsaturatedzone. The total migration time is predominantly determined by the residence time in the unsaturated
28
zone for such a nuclide, and therefore the vault array configuration and the NFI-GBI distance havenegligible effects.
To finally conclude the effects of the vault array configuration, these performance measures needto be presented with uncertainty due to parameter variations. The model and the code developed inthis study can be utilized for such probabilistic performance assessments.
In the following two chapters, the effects of canister array configuration are investigated forwater-saturated repository, also considering the residence time in a compartment.
29
CHAPTER 3
CONGRUENT RELEASE OF A RADIONUCLIDE
FROM A WATER-SATURATED REPOSITORY
3.1 Introduction
In this chapter, effects of the canister array configurations are examined for mass release of along-lived radionuclide from a water-saturated repository.
In Chapter 2, it has been found that, if the mean lifetime of a radionuclide is significantly shorterthan its migration time in the aquifer, the radionuclide decays out during the migration and is notreleased into the biosphere. The major contributors to the exposure dose are the radionuclides withlifetimes longer than the migration times. Therefore, analysis is made for a long-lived radionuclidein this chapter.
The concentration and the release rate of the radionuclide from the downstream side of therepository region are numerically calculated. This concentration and the release rate can then beused as an input for a transport analysis in the far field and biosphere to complete the performanceassessment. The far-field analysis is not performed in this chapter because its qualitative character-istics are considered similar to those discussed in Section 2.5.5.
As a numerical illustration, we consider the case of release of 135Cs congruent with matrixdissolution in the Japanese repository concept.1 The case of congruent release is of special interestbecause:
• Major contributors to the exposure dose rate to a human are fission-product (FP) nuclides aswell as solubility-limited actinides such as 237Np. Most of FP nuclides, such as 135Cs and129I,2 are assumed to be released congruently with matrix dissolution. Thus, it is important toevaluate the concentrations and release rates of these long-lived FP nuclides at the exit of therepository for reliable repository performance assessment.
• Advanced fuel cycle systems such as studied in [23, 24, 25] are considered for their effective-ness in reducing masses of actinide isotopes (Pu, Np, Am, and Cm) in HLW and in decreasingthe heat emission and the radiotoxicity from the HLW. Major constituents of HLW from suchsystems would be fission products and actinides with reduced masses and concentrations. Asdiscussed later in Chapter 4, if the initial mass loading of 237Np is reduced by a factor of 1000
1In the Japanese repository, 40,000 canisters containing vitrified wastes will be placed in a water-saturated graniticrock. The nuclide 135Cs is considered to contribute to the peak exposure dose rate [1].
2FP nuclide 99Tc is also an important contributor to the repository performance. The release of 99Tc could be ei-ther solubility-limited or congruent, depending on the geochemical conditions in the repository. In a water-saturatedrepository, its release is considered to be solubility-limited [1].
30
in the vitrified waste of reprocessed light water reactor spent fuel, 237Np would be congru-ently released with dissolution of borosilicate glass. For a reliable evaluation and comparisonof various fuel cycle schemes from the viewpoint of repository performance, better modelingfor congruent release is needed.
Previously, it was assumed that the concentration of a congruently released radionuclide ingroundwater increases indefinitely and proportionally with the number of failed canisters in thewater stream. In this chapter, we observe that there is an upper-bound concentration regardless ofthe number of failed canisters.
3.2 Model
3.2.1 Repository Structure
In the present model, radionuclide transport in a hypothetical repository in a water-saturatedgeologic formation is modeled by considering an array of compartments each containing a single
������������������������������
������������������������������
n−1 n+1
− 1n n−th compartment n + 1
Nx−1 Nx
Ny
Waste matrix
Buffer
Buffer
NFR
NFR
Repository
2
Far field
flowGroundwater
1 n
A /2
F/2
S/2
S/2
F/2
A /2 d
L
L
Figure 3.1 Schematic diagram of repository structure and radionuclide transport considered inthe compartment model. Horizontal arrows in the near-field rock (NFR) regions represent advectivetransport of a radionuclide by groundwater flow. Vertical arrows in the buffer regions represent theradionuclide transport by diffusion. Dashed lines show the boundaries through which the radionu-clide can migrate. Solid lines show the boundaries which the radionuclide does not pass across.
31
waste canister. The waste canisters are placed in a two-dimensional array fashion inside the repos-itory (Figure 3.1). The repository consists of as many compartments as the number of the wastecanisters placed in it. The region interior to all compartments is referred to as the repository region.
Schematic diagram of repository structure considered is shown in Figure 3.1. The repository ispartitioned into an Nx × Ny array of compartments. Each compartment consists of a waste-matrixregion, a buffer region, and a near-field rock (NFR) region.
The waste matrix is transformed into a slab geometry with the same interfacial area S [m2] asthat of the original cylindrical waste canister. The slab waste matrix is assumed to have a widthd [m], which is the distance between two adjacent waste canisters. To make the surface area ofboth sides of the slab equal to the original surface area S, the hypothetical slab height H [m] isdetermined by
2Hd = S . (3.1)
The buffer is considered to be a homogeneous mixture of the corroded metal canister and thebackfill material such as bentonite. Each buffer region has the thickness of L [m].
Because of disturbance by excavation of disposal tunnels and emplacement of the waste, theNFR differs from the far-field rock. The NFR is assumed to be homogeneously porous. Groundwa-ter is assumed to flow through the NFR at a constant, uniform velocity v [m/yr] in the x direction.3
The groundwater flows into the NFR region through one of the interface between two adjacent com-partments at rate F [m3/yr], and flows outward through the other at the same rate. The groundwaterflowing out from a compartment flows into the NFR region of the next adjacent compartment inthe array. The compartments in the repository are thus connected by groundwater flow through theNFR regions. The volumetric flow rate F is given by
F = vεr A , (3.2)
where v [m/yr] is the pore velocity of the groundwater and εr is the effective porosity of the NFR.The cross-sectional area A [m2] between the NFR regions of two adjacent compartments is definedby
A ≡ Hd =12 S . (3.3)
The volumetric flow rate F is assumed to be constant with time and uniform over the NFR regionsof the entire repository. Groundwater in the buffer region is assumed to be stationary, and thus nowater flow in the buffer region is considered. The volume V [m3] of the NFR region is defined by
V ≡ Hd2 . (3.4)
The geometry of the repository is considered to be stationary for simplicity, and thus all the dimen-sions of the regions including S, L , and V are time-independent.
The buffer and the NFR are considered homogeneously porous. Any void space in the repositoryis assumed to be fully saturated with water. Temperature in the repository would settle down to theambient temperature [1] by the time the radionuclide release from the waste matrix starts. It isassumed that the ambient temperature in the repository is constant with time.
3In the case that the water flow is tilted with respect to the compartment array, the two dimensional water flow canbe decomposed into x- and y-components. The basic concept of the present model can then be applied to each flowcomponent. The effect of the tilted water flow is discussed in Section 6.6.
32
3.2.2 Waste-Matrix Region
In the present study, transport of a single radionuclide, highly soluble in groundwater, is con-sidered. Precursors of a radioactive-decay chain and effects of other isotopes of the element areneglected.4
It is assumed that groundwater begins to leach the waste matrix in all the compartments at timet = 0 [yr]. Each waste matrix degrades at a constant volumetric rate until the matrix is completelyleached at time t = TL. The matrix degradation is assumed to occur uniformly in all waste formsfor simplicity.
There initially exists M◦ [mol] of the radionuclide in the waste matrix region of each com-partment, and not in the other regions. As the waste matrix degrades, the radionuclide is releasedcongruently from the matrix. The radionuclide released from the matrix is assumed to dissolveimmediately in the water phase at waste-matrix/buffer interface. The nuclide is released uniformlyover the interfacial area S.
Mass of the radionuclide in the waste-matrix region, Mw(t) [mol], decreases due to radioactivedecay and release into the buffer region, and is given as
Mw(t) =
{M◦ (1 − t/TL) e−λt , 0 ≤ t < TL ,
0 , t ≥ TL ,(3.5)
where λ [yr−1] is the radioactive-decay constant of the radionuclide. The congruent release rateq(t) [mol/yr] from a single waste form is written as
q(t) =
{M◦e−λt/TL , 0 < t < TL ,
0 , t ≥ TL .(3.6)
3.2.3 Buffer Region
In the buffer region, transient molecular diffusion of the nuclide is treated in a one-dimensionalslab geometry. Advection in the buffer region is neglected because of assumed low permeability inthe buffer region. The concentration Cb
n(ξ, t) [mol/m3] of the radionuclide in the buffer is governedby the following diffusion equation:
Rb∂Cb
n
∂t= D
∂2Cbn
∂ξ 2− λRbCb
n , 0 < t , 0 < ξ < L , n = 1, 2, . . . , Nx , (3.7)
where D [m2/yr] is the molecular diffusion coefficient of the nuclide, which is assumed to be con-stant and identical for buffer regions of all compartments. Subscript n denotes the compartmentnumber relative to the upstream side of the repository. It is assumed that all Ny rows of compart-ments have identical properties, and hence the rows are not specified by notation. Variable ξ isthe distance from the waste-matrix/buffer boundary. The outer boundary ξ = L is located at thebuffer/NFR interface.
4This applies for most of long-lived FP nuclides. For actinides, a multiple member decay chain may need to beincorporated, depending on the half-lives and sorption retardation of precursors.
33
Sorption equilibrium between the solid phase and the pore-water phase is assumed. The retar-dation factor Rb is defined as
Rb = 1 +1 − εb
εbρb Kdb , (3.8)
where ρb [kg/m3] and εb are the density of the solid matrix material and the porosity in the bufferregion, respectively. Kdb [m3/kg] is the sorption distribution coefficient of the nuclide in the buffer.Parameters ρb, εb, Kdb, and hence Rb, are assumed to be constant with time and identical in thebuffer for all compartments.
It is assumed that there is no radionuclide initially in the buffer. The initial condition for Eq. (3.7)is written as
Cbn(ξ, 0) = 0 , 0 < ξ < L , n = 1, 2, . . . , Nx . (3.9)
At the waste-matrix/buffer interface (ξ = 0), the radionuclide is released into the buffer regionat rate q(t) until time TL. After time TL, there is no mass transfer through the interface. Thus, theboundary condition for Eq. (3.7) at the waste-matrix/buffer interface is written as
−Sεb D∂Cb
n
∂ξ
∣∣∣∣ξ=0
= q(t) , t > 0 , n = 1, 2, . . . , Nx . (3.10)
Considering the concentration continuity in the water phase at the buffer/NFR interface (ξ = L),the other boundary condition is written as
Cbn(L , t) = Cn(t) , t > 0 , n = 1, 2, . . . , Nx (3.11)
where Cn(t) [mol/m3] is the uniformized concentration of the nuclide in the pore-water in the NFRregion of the n’th compartment. The nuclide is released from the buffer into the NFR region ac-cording to the diffusive mass flux at the buffer/NFR interface.
3.2.4 Near-Field Rock Region
The nuclide in the NFR region is transported from a compartment to another by advection dueto the groundwater flow. The concentration of the nuclide in the NFR region of a compartment isuniformly represented by the average concentration over the pore-water volume in the whole region,Cn(t) [mol/m3]. The governing equation for the uniformized concentration Cn(t) in the NFR regionis formulated by considering the advective transport through the NFR region:
RrεrVdCn
dt= −λRrεrV Cn + FCn−1 − FCn + Q(Cn, t) ,
t > 0 , n = 1, 2, . . . , Nx , (3.12)
where εr is the effective porosity of the NFR region. Volume V of the NFR region of a compartmentand volumetric flow rate of groundwater, F , are calculated from Eqs. (3.4) and (3.2), respectively.The groundwater flowing into the first compartment is assumed to be uncontaminated, i.e.,
C0(t) = 0 , t > 0 . (3.13)
34
The retardation factor Rr is defined as
Rr = 1 +1 − εr
εrρr Kdr , (3.14)
where ρr [kg/m3] is the density of the solid matrix of the porous rock, and Kdr [m3/kg] is the sorp-tion distribution coefficient of the radionuclide for the rock matrix of the NFR. In this formulation,V , εr, ρr, Kdr, and thus Rr are assumed to be constant with time and identical for all compartments.
The first term on the right side of Eq. (3.12) represents the change of the mass of the radionu-clide in the NFR due to radioactive decay. The second and the third terms on the right side representthe mass of the radionuclide flowing in from compartment n − 1 to compartment n per unit time,and the mass flowing out of compartment n to compartment n +1 per unit time by advection. Effectof hydrodynamic dispersion of the nuclide in the flow direction is reproduced by the immediate uni-formization of the concentration within an NFR region and the sequence of multiple compartments.The term Q(Cn, t) [mol/yr] in Eq. (3.12) is the release rate of the radionuclide at the buffer/NFRinterface into the NFR region, which is written as
Q(Cn, t) = −Sεb D∂Cb
n
∂ξ
∣∣∣∣ξ=L
, t > 0 . (3.15)
The right side of Eq. (3.15) is the diffusive mass release rate at the buffer/NFR interface. TheCn-dependency of Q(Cn, t) arises from that of Cb
n , which is determined subject to the boundarycondition (3.11). It is assumed that there is no net mass exchange between two adjacent rowsof Nx compartments since a radionuclide is released equally in both rows and there will be noconcentration gradient between the rows. The governing equation (3.12) is subject to the followinginitial condition:
Cn(0) = 0 , n = 1, 2, . . . , Nx . (3.16)
The radionuclide released into the NFR region of the first compartment is swept downstream bythe uncontaminated water. The radionuclide concentration in the NFR of the second compartmentis hence somewhat higher than that of the first compartment since the incoming water is alreadycontaminated. In this fashion, the groundwater flowing in the array through the NFR is assumed tobe increasingly contaminated before it finally flows out of the repository.
The difference between the independent-canister model represented by Eq. (1.1) and the presentmodel for Nx = 1 represented by Eq. (3.12) can be considered as follows. In the independent-canister model, the mass release rate Q is obtained by assuming that the concentration in the NFRis zero. In the notation of Eq. (3.12), Q = Q(0, t). To assume the zero concentration at the interfacebetween the EBS and the NFR, it is assumed that the water flow rate F through the NFR is infinitelylarge. If the radionuclide released from the EBS into the NFR is assumed to be instantaneouslyswept by an infinitely large water flow F in the NFR, maintaining C1 at zero for all time, the termon the left side and the first term on the right side in Eq. (3.12) are dropped. The second term onthe right side of Eq. (3.12) is zero for n = 1 by definition. Then, provided that FC1 is finite andnonzero, Eq. (3.12) is reduced to 0 = −FC1 + Q(0, t), which is formally equivalent to Eq. (1.1).5
5This comparison shows that determination of the flow rate F in the previous models [2, 3, 4] is arbitrary and incon-sistent because of the following reason: C1 must be nonzero in order to have a nonzero far-field concentration; Q(0, t) isfinite and nonzero. However, C1 becomes zero because F is assumed to be infinitely large.
35
3.2.5 Release into the Far Field
From the downstream side of the repository region, i.e., from compartment Nx , the radionuclideis released into the far field. The concentration of the radionuclide in the groundwater being releasedinto the far field is CNx (t). Since the groundwater flows at a volumetric rate F through each of Ny
rows of compartments, the radionuclide release rate JNx ,Ny (t) [mol/yr] from the entire repository isgiven as
JNx ,Ny (t) = Ny FCNx (t) . (3.17)
Because the repository-exit concentration CNx (t) is determined by the mass transport in the otherupstream compartments by Eq. (3.12), JNx ,Ny (t) determined by Eq. (3.17) include the details of themass transport in the repository through CNx (t).
3.2.6 Conversion into Dimensionless System
We convert the governing equations into dimensionless forms. Time t and the position in thebuffer, ξ , are non-dimensionalized as
τ ≡ t/T1 and θ ≡ ξ/L , (3.18)
where T1 [yr] is defined as
T1 =RrεrV + RbεbSL
F. (3.19)
This time length T1 is physically interpreted as the average time that a nuclide, which has enteredcompartment n from compartment n − 1, spends in compartment n before it enters downstreamcompartment n + 1, for arbitrary n. The average time TNx for the nuclide to migrate across thelength of Nx compartments is written as
TNx = Nx T1 ,
and the corresponding dimensionless quantity is
τNx ≡ TNx /T1 = Nx . (3.20)
The waste-matrix leach time is non-dimensionalized with T1 as
τL ≡ TL/T1 . (3.21)
The mass, concentrations, and release rates are non-dimensionalized as follow: for n = 1, 2, . . . , Nx ,
9w(τ ) ≡ Mw(t)eλt/M◦ ,
χbn (θ, τ ) ≡ Cb
n(ξ, t)eλt/C◦ , χn(τ ) ≡ Cn(t)eλt/C◦ ,
ϕw(τ ) ≡ q(t)T1eλt/M◦ , ϕb(χn, τ ) ≡ Q(Cn, t)T1eλt/M◦ ,
φNx ,Ny (τ ) ≡ JNx ,Ny (t)T1eλt/M◦ ,
36
whereC◦
≡M◦
RrεrV + RbεbSL. (3.22)
Concentration C◦ can be interpreted as the nuclide concentration for the case that initial mass M◦
is uniformly dispersed over the water phase in the pores of the buffer and the NFR of a singlecompartment with sorption equilibria between the water phase and the solid phase of the buffer andthe NFR.
The dimensionless governing equations are summarized as follow:
9w(τ ) =
{1 − τ/τL , 0 ≤ τ < τL ,
0 , τ ≥ τL .(3.5′)
ϕw(τ ) =
{1/τL , 0 < τ < τL ,
0 , τ ≥ τL ,(3.6′)
For τ > 0, n = 1, 2, . . . , Nx ,
∂χbn
∂τ= α
∂2χbn
∂θ2, 0 < θ < 1 , (3.7′)
βdχn
dτ= χn−1(τ ) − χn(τ ) + ϕb(χn, τ ) , (3.12′)
ϕb(χn, τ ) = −(1 − β)α∂χb
n
∂θ
∣∣∣∣θ=1
, (3.15′)
andφNx ,Ny (τ ) = NyχNx (τ ) , τ ≥ 0 , (3.17′)
where dimensionless coefficients are defined as
α ≡T1 DRbL2
, β ≡RrεrV
RrεrV + RbεbSL. (3.23)
The constant parameter β is interpreted as the fraction of the radionuclide mass in the NFR regionwhen a certain mass of the radionuclide is dispersed at a uniform concentration over the buffer andthe NFR region. The constant α is the Fourier number for the diffusion in the buffer region for thetime duration of T1.
The boundary conditions and the initial conditions are similarly converted into the dimension-less forms. Note that the radioactive-decay terms are not included in the dimensionless equations.A solution set for the dimensionless system is valid for all isotopes of an element with differenthalf-lives.
37
Table 3.1 Assumed Parameters for Cesium and the Repository
Symbol Description Value
Nx Number of compartments in each row parallel to the ground-water flow
Ny Number of rows of Nx compartments each in the repositoryτL Normalized duration for the waste-matrix dissolutionβ See Eqs. (3.12′) and (3.23) 0.96α See Eqs. (3.7′) and (3.23) 1.3
Rb Retardation factor of Cs in the buffer region 50 a
Rr Retardation factor of Cs in the NFR region 131 a
TL Duration for the waste-matrix dissolutionD Diffusion coefficient for Cs in the buffer 0.046 m2/yr a,b
F Volumetric flow rate of groundwater through the interfacebetween two adjacent compartments in a row [see Eq. (3.2)]
0.45 m3/yr
εb Porosity in the buffer region 0.3 b
εr Effective porosity in the NFR region 0.5 b
ρb Density of the solid material in the buffer region 2100 kg/m3 b
ρr Density of the solid material in the NFR region 2600 kg/m3 b
V Volume of the NFR region in a compartment 9.05 m3 b
v Pore velocity of groundwater 1 m/yr b
S Surface area of a single waste matrix 1.81 m2 b
d Distance between waste canisters 10 m b
L Thickness of the buffer region 0.98 m b
T1 Average time for the nuclide to migrate across the length ofa single compartment [see Eq. (3.19)]
1370 yr
C◦ Uniform concentration in the water phase of the buffer andthe NFR when the initial mass M◦ of the nuclide is dispersedover the buffer and the NFR [see Eq. (3.22)]
5.15 × 10−3 mol/m3
a See Ref. [1].b See Ref. [26].
38
3.3 Numerical Results and Observations
The computer code VR [27] has been used for numerical calculations. The present model isapplicable to any long-lived radionuclide that is congruently released from waste forms. The nuclide135Cs has been chosen for numerical illustration, since it is considered a major contributor to thepeak exposure dose rate in the performance assessment of the Japanese repository and it is solublein groundwater [1]. Parameter values used in the calculation are listed in Table 3.1. As seen inEqs. (3.5′) through (3.17′), the first five parameters in Table 3.1 determine the dimensionless systemuniquely. Values for these parameters are calculated according to the rest of the parameter values inTable 3.1.
3.3.1 Exit Concentration for Nx < τL
The dimensionless concentration χNx (τ ) of cesium at the exit of the repository has been obtainedby the numerical calculation, and is depicted in Figure 3.2 for the dimensionless leach time τL = 73(TL = 105 [yr]) and various array configurations. The exit concentration χNx (τ ) is common to anyvalue of Ny since it does not depend on Ny . The top axis in Figure 3.2 shows the real physical timet corresponding to the dimensionless time τ for the particular parameter values shown in Table 3.1.
For Nx = 1, i.e., when there is only one compartment in the water-flow direction, the con-centration χ1(τ ) increases with time in the beginning, and gradually levels off until it reaches a
10-2
10-1
100
10-1 100 101 102 103
103 104 105 106
Dim
ensi
onle
ss e
xit c
once
ntra
tion,
χN
x(τ)
Dimensionless time, τ
Physical time, t [yr]
τ = τL
Nx = 1
2
4
8
16
32
64
χ1peak
χ2peak
χ4peak
χ8peak
χ16peak
χ32peak
χ64peak
Figure 3.2 Dimensionless concentration χNx (τ ) of cesium in the groundwater at the reposi-tory exit obtained by numerical calculations with VR code. Dimensionless leach time τL = 73(TL = 105 [yr]). Canister-array configuration Nx = 1, 2, 4, 8, 16, 32, and 64. Note that the exitconcentration χNx (τ ) does not depend on Ny . See Table 3.1 for other parameter values.
39
steady-state plateau. This steady-state concentration is the peak of χ1(τ ), and we will refer to thepeak level as χ
peak1 . During the steady state, mass balance in the NFR region [see Eq. (3.12′)] is
maintained between the mass release from the buffer into the NFR, ϕb(χ1, τ ), and the release intothe far field, χ1(τ ). At τ = τL, the congruent release from the waste matrix completes. The exitconcentration χ1(τ ) decreases thereafter, as the radionuclide remaining in the buffer and the NFRis diluted by the incoming uncontaminated water and is released into the far field.
For Nx = 2, a concentration profile similar to that for Nx = 1 is observed. The exit concen-tration χ2(τ ) increases in the beginning as cesium is released from the waste matrix in the secondcompartment. The concentration χ2(τ ), however, reaches a higher value than χ1(τ ) because cesiumaccumulates in the flowing water when cesium released from the first compartment flows into thesecond compartment. Thus, the exit concentration χ2(τ ) increases for a longer time than χ1(τ ), andreaches the steady-state level χ
peak2 twice as high as χ
peak1 . The steady state lasts until τ = τL and
the exit concentration χ2(τ ) decreases thereafter as observed in χ1(τ ).The same discussion applies to Nx = 4, 8, 16, 32, and 64. Because of cesium accumulation
in the water over Nx compartments, the exit concentration χNx (τ ) increases until the cesium thathas been released from the waste form in the first compartment reaches the repository exit aftermigration through Nx compartments. Considering that, from Eq. (3.20), τNx represents the dimen-sionless time it takes for cesium to migrate across the length of Nx compartments, χNx (τ ) thusincreases until τ ≈ Nx , and its steady state is observed within a time interval Nx < τ < τL. ForNx = 64, the cesium release from the waste matrices ceases soon after χ64(τ ) reaches its peak, andthus no time span of a plateau is observed. The peak exit concentration χ
peakNx
, corresponding to thesteady-state level of χNx (τ ), is Nx times as high as χ
peak1 due to the accumulation of cesium over the
compartments.For the steady state observed during Nx < τ < τL, the time-derivative terms in Eqs. (3.7′) and
(3.12′) are equal to zero, and hence the release rate from the buffer is obtained as ϕb= ϕw(τ ) =
1/τL. The analytical formula for the dimensionless steady-state concentration, or the peak exitconcentration χ
peakNx
, is obtained as
χpeakNx
= Nx/τL , Nx < τL . (3.24)
3.3.2 Exit Concentration for Nx > τL
The dimensionless exit concentration χNx (τ ) obtained by numerical calculation is plotted inFigure 3.3 for the leach time τL = 7.3 (TL = 104 [yr]).
For Nx = 1, 2, and 4, the cesium released from the first compartment reaches the repository exitbefore the congruent release from the waste matrices ceases, i.e., Nx < τL. The exit concentrationχNx (τ ) approaches a steady-state level, and rapidly decreases after τ = τL as observed for the caseof τL = 73 in Figure 3.2. The peak concentration χ
peakNx
proportionally increases with Nx for theseconfigurations, and is formulated as Eq. (3.24).
For Nx = 16, 32, and 64, the exit concentration increases until the congruent release from thewaste matrix completes at τ = τL, and is maintained at a plateau level for some time after τL. Theplateau levels for Nx = 16, 32, and 64 are observed identical to each other. The exit concentrationbegins to decrease at τ ≈ Nx .
40
The configuration of Nx = 8 shows an intermediate concentration profile between the afore-mentioned two groups.
The spatial distribution of the concentration χn(τ ) in the NFR regions along the compartmentrow is shown in Figure 3.4 for Nx = 64 and τL = 7.3. During 0 < τ < τL (shown by dashedcurves), the dimensionless concentration χn(τ ) increases uniformly over the compartments near therepository exit (downstream compartments) as cesium is released from the waste matrices at thesame rate in each compartment (τ = 0.2τL and 0.5τL). During this early time domain, there isno concentration difference between two adjacent compartments near the repository exit, i.e., thedifference χn−1(τ ) − χn(τ ) in Eq. (3.12′) is equal to zero. Therefore, increase in χn(τ ) with time isonly due to the mass release ϕb from the buffer region. When the congruent release from the wastematrices completes at τ = τL, the amount of cesium equivalent to its initial mass per canister existsin the buffer and the NFR of a single compartment.
In Figure 3.5, a uniform concentration is observed in the buffer and the NFR of the downstreamside of the repository after τ = τL. The right side of Eq. (3.12′) is zero in such a case, and χNx (τ ) isthus maintained at a constant level (τ = 2τL, 4τL, and 6τL). This constant concentration correspondsto the peak exit concentration χ
peak64 shown in Figure 3.3, which forms the plateau observed during
τL < τ < Nx . The peak concentration χpeakNx
for Nx > τL is formulated by assuming that the initialmass M◦ is uniformly dispersed over the buffer and the NFR. It is obtained as
χpeakNx
= 1 , Nx > τL . (3.25)
This dimensionless concentration χpeakNx
= 1 corresponds to C◦ [mol/m3], which is defined in
10-2
10-1
100
10-1 100 101 102 103
103 104 105 106
Dim
ensi
onle
ss e
xit c
once
ntra
tion,
χN
x(τ)
Dimensionless time, τ
Physical time, t [yr]
τ = τL
Nx = 1
2
4
8 16 32 64
χ1peak
χ2peak
χ4peak
χ8peak
χ16peak, χ32
peak, χ64peak
Figure 3.3 Dimensionless concentration χNx (τ ) of cesium in the groundwater at the repositoryexit obtained by numerical calculations with VR code. Dimensionless leach time τL = 7.3 (TL =
104 [yr]). Canister-array configuration Nx = 1, 2, 4, 8, 16, 32, and 64.
41
0.0
0.5
1.0
1.5
0 10 20 30 40 50 60Dim
ensi
onle
ss c
once
ntra
tion
in N
FR, χ
n(τ)
Compartment number, n
τ = τL
τ = 2τLτ = 4τL
τ = 6τLτ = 8τL
τ = 0.5τL
τ = 0.2τL
Figure 3.4 Spatial distribution of the dimensionless concentration χn(τ ) of cesium in the NFRregions along a compartment row for Nx = 64 and τL = 7.3 (TL = 104 [yr]). The dashed lines andthe solid lines represent concentration profiles for τ ≤ τL and for τ > τL, respectively.
0.0
0.5
1.0
1.5
2.0
2.5
3.0
0.0 0.2 0.4 0.6 0.8 1.0
Dim
ensi
onle
ss c
once
ntra
tion,
χb 64
(θ,τ
)
Dimensionless distance from matrix/buffer boundary, θ
τ = 2τL, 4τL, 6τL
τ = 8τL
τ = 0.2τL
τ = 0.5τL
τ = τL
χ64peak
Figure 3.5 Spatial distribution of the dimensionless concentration χb64(θ, τ ) of cesium in the buffer
region of the 64th compartment. Nx = 64 and τL = 7.3. The dashed lines and the solid linesrepresent concentration profiles for τ ≤ τL and for τ > τL, respectively.
42
Eq. (3.22) and has been used for non-dimensionalization, when the effect of radioactive decay isnegligible.
In the upstream side of the compartment array, cesium is swept away towards downstreamcompartments by the fresh water entering the repository. The concentration χn(τ ) decreases to zeroafter the cesium from the first compartment passes through the n’th compartment. The concentrationχ64(τ ) in the last compartment of the repository is kept constant until the cesium that has beenreleased from the first compartment reaches the last compartment at τ ≈ 64. The flow rate of cesiumfrom the upstream compartment n = 63 rapidly decreases thereafter, and the exit concentrationχ64(τ ) decreases accordingly.
3.3.3 Peak Exit Concentration and Its Upper Bound
Two curves of χpeakNx
for τL = 7.3 and 73 are shown in Figure 3.6. From Eqs. (3.24) and(3.25), there are thus two regimes of χ
peakNx
depending on the values of Nx and τL. The peak exitconcentration χ
peakNx
increases proportionally with Nx for small Nx values that satisfy Nx < τL
[Eq. (3.24)], whereas it becomes constant (i.e., independent of Nx ) if Nx > τL [Eq. (3.25)]. Itfollows that the peak exit concentration χ
peakNx
for any value of Nx never exceeds the value givenby Eq. (3.25). (This can also be confirmed with the numerical results shown in Figures 3.2 and3.3.) Therefore, χ
peakNx
given by Eq. (3.25) can be considered as a theoretical upper bound of theexit concentration for any number of connected compartments.6 Considering the concentration C◦
6In previous performance assessments, it was considered that the concentration of a congruently-released radionuclideincreases proportionally as the number of canisters, without such an upper bound.
10-2
10-1
100
101
100 101 102
Dim
ensi
onle
ss p
eak
exit
conc
entr
atio
n, χ
Nx
peak
Nx
Upper bound
Nx = τL(τL = 73)
Nx = τL(τL = 7.3)
Figure 3.6 The peak exit concentration χpeakNx
of cesium obtained from analytical formulae (3.24)and (3.25) for τL = 7.3 (solid line) and τL = 73 (dashed line).
43
in Eq. (3.22) used for non-dimensionalization, the upper-bound concentration depends on the initialinventory of the radionuclide, sorption properties, and the volumes of the NFR and buffer region ofa compartment.
If the repository size (Nx ) is greater than a threshold value (τL), the peak concentration at therepository exit is equal to the constant upper bound given by Eq. (3.25). This result implies that thereis no difference in the concentration of the radionuclide in out-going water once the repository size(Nx ) exceeds the threshold (τL). We may be able to expand the repository size without increasingits impact on the environment.
If τL is elongated from 7.3 to 73 by waste-form improvement, the Nx -dependent region (Nx <
τL) becomes larger accordingly, and χpeakNx
within this region becomes smaller. However, for theconfigurations Nx > τL, the peak exit concentration χ
peakNx
remains the same value as for τL = 7.3.The same discussion applies for any value of τL, and thus χ
peakNx
for Nx > τL, or the upper-boundconcentration, is independent of the leach time τL, as confirmed with Eq. (3.25).
The Nx - and τL-dependence of the peak exit concentration is depicted in Figure 3.7 as a contourplot of χ
peakNx
. An (Nx , τL)-dependent region (upper left) and an (Nx , τL)-independent region (lowerright) are observed in the figure. In the (Nx , τL)-dependent region, χpeak
Nxis determined by Eq. (3.24),
and it decreases as τL becomes greater and/or Nx becomes less. In the (Nx , τL)-independent region,χ
peakNx
is determined by Eq. (3.25), and is unity (the upper-bound concentration) over the region. Thestraight line Nx = τL separates the two regions.
Figure 3.7 Contour plot of the peak exit concentration χpeakNx
as a function of the dimensionlessleach time τL and Nx . The contour levels are shown in the logarithmic scale. The value of χ
peakNx
isequal to the constant upper bound given by Eq. (3.25) in the lower-right region (Nx > τL).
44
The upper-bound concentration can be used as the input boundary condition for far-field trans-port analyses of a congruently released radionuclide, which gives conservatively overestimated re-sults for the concentration in the far field. The upper bound as an estimate of the peak exit concen-tration is most realistic if τL < Nx since the peak concentration becomes equal to the upper boundin such cases. It becomes more conservative as τL increases and as Nx decreases.
3.3.4 Release Rate from the Repository
The dimensionless peak release rate from the entire repository, φpeakNx ,Ny
, can be derived fromEq. (3.17′) as
φpeakNx ,Ny
= NyχpeakNx
. (3.26)
Formulae of φpeakNx ,Ny
are obtained by substituting the analytical formulae (3.24) and (3.25) into theabove equation giving
φpeakNx ,Ny
=
{Nx Ny/τL , Nx < τL ,
Ny , Nx > τL .(3.27)
These are plotted in Figure 3.8 for the fixed number of canisters in the entire repository, i.e., Nx Ny =
100.In contrast to the peak exit concentration χ
peakNx
shown in Figure 3.6, the peak release rate φpeakNx ,Ny
is independent of the canister-array configuration for Nx < τL, and decreases with Nx for Nx >
τL. It follows that, with the total number Nx Ny fixed, φpeakNx ,Ny
takes the upper-bound value in the
10-1
100
101
102
100 101 102
100101102
Dim
ensi
onle
ss p
eak
rele
ase
rate
, φpe
akN
x,N
y
Nx
Ny
Nx = τL(τL = 73)
Nx = τL(τL = 7.3)
Figure 3.8 The peak release rate φpeakNx ,Ny
of cesium from the entire repository obtained from ana-lytical formula (3.27) for τL = 7.3 (solid line) and τL = 73 (dashed line). The fixed total number of100 waste canisters are considered (Nx Ny = 100).
45
configurations that satisfy Nx < τL. On the other hand, the peak release rate φpeakNx ,Ny
becomes thesmallest when Nx = 100 and Ny = 1, because it is proportional to Ny for Nx < τL by Eq. (3.27). Itis also shown in Eq. (3.27) that the upper bound is inversely proportional to τL.
3.4 Discussions
3.4.1 Upper-Bound Concentration and Upper-Bound Release Rate
The exit concentration χNx (τ ) and the release rate φNx ,Ny (τ ) are found not only affected by thetotal number of waste canisters in the repository but also by the canister-array configuration (Nx andNy).
It has also been observed that the dimensionless concentration χNx (τ ) has a fixed upper boundof unity which it never exceeds regardless of the number of canisters or of their configuration. If thecorresponding upper-bound dimensional concentration with dimensions given by C◦ [Eq. (3.22)]is used as the input boundary condition for a transport analysis in the far field, the concentrationin the far field can be conservatively overestimated. Since the upper bound is for all canister-arrayconfigurations, this conservative concentration in the far field can be estimated without specifyingthe array configuration or the number of canisters in the repository. From Eq. (3.22), the upper-bound concentration is determined by the initial mass loading of the radionuclide in a single canister,the volumes of the pore water in the buffer region and the NFR region of a compartment, and thesorption properties of the radionuclide in the media. Since it does not depend on the leach time TL
or on the groundwater-flow rate F , it is free from the uncertainties associated with these parametervalues.
In the previous study [1], the release rate from the entire repository has been obtained by mul-tiplying the total number of canisters by the release rate from a single canister. The release rateobtained in such a manner is independent of canister-array configuration. The present model, withthe canister-array configuration taken into account, also shows that the peak release rate is indepen-dent of the configuration if the number of canisters in the flow direction, Nx , is less than a thresholdvalue. The analytical formula for this upper-bound release rate in Eq. (3.27) shows that it is similarto the release rate in the previous study since it is expressed as the product of the total number of can-isters and the release rate from the single-canister configuration. If Nx is greater than the thresholdvalue τL, however, the peak release rate in the present model becomes dependent on the configura-tion, and is smaller than the upper-bound release rate. In this case, the configuration-independentrelease rate in the previous study gives a more conservatively overestimated value than the presentmodel. The present model is suitable in assisting in the design of a repository since the effects ofthe canister-array configuration is reflected by the peak release rate and the peak exit concentration.
3.4.2 Expansion of Repository Footprint
If the number of canisters in the groundwater-flow direction, Nx , is greater than the thresholdvalue τL, defined by Eq. (3.21), then the peak concentration of the radionuclide at the repositoryexit is equal to the upper-bound level regardless of the number of canisters or of the canister-arrayconfiguration. τL is determined by the matrix leach-time and properties in the buffer and the NFR
46
region. In such cases, the peak concentration at the repository exit would not increase any furthereven if the repository footprint is extended by increasing the total number of waste canisters loadedin the repository. Thus, the repository footprint can be extended in both directions parallel to andperpendicular to the groundwater flow without increasing the peak exit concentration. The peakrelease rate from the repository, however, increases in any case proportionally with the number ofwaste canisters Ny in the direction perpendicular to the groundwater flow as shown in Eq. (3.27).Footprint extension in the direction perpendicular to the groundwater flow (i.e., increasing Ny)would therefore lead to increase in the peak release rate even if there is no change in the peak exitconcentration. The peak release rate is not increased if the repository footprint is extended parallelto the groundwater flow instead (i.e., increasing Nx ) when Nx is greater than the threshold.
If the number of canisters in the groundwater-flow direction, Nx , is less than the threshold valueτL, it is found that the exit concentration is proportional to Nx and inversely proportional to thematrix leach-time τL. The peak release rate depends on the total number of waste canisters in therepository (Nx Ny) in such a case, and does not depend on the canister-array configuration.
In the comparison among different canister-array configurations, it is observed that the peak exitconcentration is minimized by arranging canisters in a line perpendicular to the groundwater-flowdirection, while the peak release rate from the entire repository is minimized by arranging canistersin a line parallel to the flow direction.
3.4.3 Effects of Leach Time
Effects of leach-time elongation on the peak exit concentration are not explicit in the dimension-less system, particularly when the repository size Nx is greater than τL , for which the dimensionlesspeak exit concentration is unity by Eq. (3.25). However, note that the threshold value τL itself in-creases proportionally with the leach time by Eq. (3.21). When Nx < τL, then by Eq. (3.24), thepeak exit concentration decreases inversely proportional to the leach time. This observation impliesthat, to have visible effects on the repository performance by improvement of the waste matrix (i.e.,increasing the leach time), the relationship between the repository size and the leach time needsto be taken into account. From the present study, it can be said that if hundreds of canisters are“connected” by advective transport in the near-field rock, then increasing the leach time would notdecrease the peak exit concentration (see the bottom right region of Figure 3.7).
3.4.4 Effects of Radioactive Decay
The dimensionless system given by Eqs. (3.5′) through (3.23) does not show the informationabout radioactive decay of the radionuclide. A single solution for the dimensionless equationsrepresents for all values of half-life. With radioactive decay, the peak exit concentration in thedimensionless system does not necessarily correspond to the peak value of the exit concentrationwith dimension.
In order to observe the effect of radioactive decay, the exit concentration C64(t) for 135Cs withits half-life 2.3 × 106 yr is depicted as Figure 3.9(b) together with that for the stable isotope 133Csand for hypothetical isotopes with half-life 2.3 × 104 yr and 2.3 × 103 yr. Note that CNx (t)/C◦
for the stable isotope (λ = 0) is identical to χNx (τ ) in the dimensionless system. Thus, the curve
47
C64(t)/C◦ for 133Cs in Figure 3.9 is identical to χ64(τ ) in Figure 3.3. Since the half-life of 135Cs(2.3×106 yr) is far greater than the time scale of its release, the effect of radioactive decay is hardlyobserved in C64(t) for 135Cs, and the curve C64(t) for 135Cs overlaps with that for the stable isotope.It follows that the discussions on χNx (τ ) and φNx ,Ny (τ ) in the dimensionless system can be directlyapplied to the real quantities CNx (t) and φNx ,Ny (t) for a long-lived radionuclide if the decay effectis negligible during the time domain of release.
It is observed that the concentration profile deviates from that for the stable isotope 133Cs asthe half-life becomes shorter. The plateau regime observed in the dimensionless system becomesa decreasing function of time for half-lives 2.3 × 104 yr and 2.3 × 103 yr because of the effect ofradioactive decay. For half-life 2.3 × 103 yr, the real peak of the exit concentration is observedbefore the leach time TL. In this case, the peaks for C64 and χ64 appear at different times. However,the peak exit concentration given in Eq. (3.25), which corresponds to C◦ [mol/m3], still gives theconservative, upperbound value regardless of the half-life. The upperbound value C◦ is realistic forlong-lived radionuclides, and is more conservative for short-lived radionuclides.
The analytical formulation of true peaks of the exit concentration with radioactive decay isshown in Appendix B.
10-2
10-1
100
10-1 100 101 102 103
103 104 105 106
Nor
mal
ized
exi
t con
cent
ratio
n, C
Nx(
t)/C
o
Normalized time, τ
Physical time, t [yr]
t = TL
(a),(b)
(c)
(d)
Figure 3.9 Effect of radioactive decay on the exit concentration. The normalized exit concentra-tion CNx (τ )/C◦ is plotted for (a) the stable isotope (133Cs), (b) isotope with half-life of 2.3 × 106 yr(135Cs), (c) isotope with the hypothetical half-life 2.3 × 104 yr, and (d) 2.3 × 103 yr. Matrix leach-time τL = 7.3 is used.
48
3.4.5 Applicability to Other Nuclides
While the calculation results in the present study have been illustrated for 135Cs and parametervalues based for a Japanese repository, the model can be applied to other water-saturated repositoriesif there is a diffusion buffer region around each waste canister and if groundwater flows through thecanister array. It can also be applied to other congruently released radionuclides. For example, ifthe retardation factors Rb and Rr for some radionuclide X are smaller than those for 135Cs (i.e.,the radionuclide is more mobile in the media), the characteristic time T1 [defined in Eq. (3.19)] fornuclide X is less than that for 135Cs. The dimensionless leach time τL [defined in Eq. (3.21)] fornuclide X is hence greater than that for 135Cs, even though the physical leach time TL is common toboth nuclides. This implies that, even if the peak exit concentration of 135Cs is at its upper boundand is insensitive to a slight change of canister-array configuration (this happens when Nx > τL),the peak exit concentration of nuclide X can be sensitive to the change of the array configuration ifτL > Nx for nuclide X.
3.5 Conclusions
Radionuclide transport in a water-saturated repository has been investigated for a radionuclidecongruently released from a waste matrix, with a model which takes into account spatial array con-figurations of multiple waste canisters. By numerical and analytical calculations, the concentrationof the radionuclide at the repository exit is observed for various canister-array configurations. Thepeak concentration and the peak release rate at the repository exit have been analytically formulated.
By numerical evaluations for 135Cs, we have found the following trends:
• If for example the number n of connected canisters is smaller than the dimensionless leachtime τL = 73, then the temporal profile of the concentration in the near-field rock region ineach compartment attains a steady-state plateau until the end of the leach time (Figure 3.2).The level of the plateau is proportional to the number of canisters up to the n’th compartment.
• If for example the total number Nx of connected canisters in the water stream of interest isgreater than the dimensionless leach time τL = 7.3, then the temporal profile of the concen-tration in the near-field rock region in the n’th compartment (τL < n ≤ Nx ) shows a plateauuntil nT1, which is the radionuclide transport time between the first and the n’th compart-ments. The level of the plateau is at C◦ given by Eq. (3.22) for any number of canisters if thisnumber is greater than τL (Figures 3.3 and 3.4).
• There are upper bounds for the peak exit concentration and the peak release rate from therepository. The upper bound for the peak exit concentration is C◦, regardless of the valuesof the leach time and the canister-array configuration (Figure 3.6). If the total number ofcanisters in the repository, Nx Ny , is kept the same, the upper bound for the peak release ratefrom the repository is inversely proportional to the leach time by Eq. (3.27).
• If the number of canisters connected in the flow direction increases, the effect of improvingthe waste matrix by increasing the leach time would be less effective in decreasing the peak
49
exit concentration because the peak concentration is maintained at the upper-bound value(Figure 3.7).
Thus, the present study shows that the peak exit concentration of a congruently released radionuclidewill not always be proportional to the number of canisters, but is affected by the canister-arrayconfiguration and the leach time. If the radionuclide transport time in the repository is longer thanthe leach time, the peak exit concentration becomes insensitive to the canister-array configurationand the leach time.
In the next chapter, the solubility-limited release of a radionuclide is considered, instead of thecongruent release considered in this chapter, for the same water-saturated repository.
50
CHAPTER 4
SOLUBILITY-LIMITED RELEASE OF A
RADIONUCLIDE FROM THE WATER-SATURATED
REPOSITORY
4.1 Introduction
In this chapter, the effects of canister array configurations in the water-saturated repository areexamined for the case where release of the radionuclide from the waste form is limited by its solubil-ity. In the mathematical model, the same compartmentalization scheme as Chapter 3 is used exceptthat the formulation for the release from the waste matrix is modified to account for solubility limit.
In Chapter 3, we observed that, because of radionuclide accumulation in the water stream, thepeak exit concentration of the congruently released radionuclide is proportional to the number ofcanisters connected in the flow direction if the migration time over the repository is shorter than thematrix leach time. And the peak exit concentration was observed to become constant at the upperbound if the migration time is longer than the matrix leach time.
The similar effects are expected with the solubility-limited release of a radionuclide. In ad-dition, the accumulation of the radionuclide affects the release from downstream canisters. Withthe solubility-limited release of a radionuclide, the radionuclide concentration at the waste/bufferinterface is at the solubility limit, and the concentration at the buffer/NFR boundary is equal to theconcentration in the NFR. Because of accumulation of radionuclide in the water stream in the NFR,the concentration gradient in the buffer becomes smaller as the number of waste canisters in the wa-ter stream increases, and thus the radionuclide release rate from the downstream canisters becomessmaller.
4.2 Physical Process
The model considered in this chapter is an extension of the model developed for the congruentlyreleased radionuclides discussed in Chapter 3. The same structure of the compartments shown inFigure 3.1 is used in the present model.
The concentrations of actinide species in the pore water at the waste-matrix dissolution locationare limited by their solubilities. Once the concentration reaches the solubility, the dissolution rate ofthe actinide is determined by the concentration gradient at the inner boundary of the buffer region.
As considered in Chapter 3 for congruently released species, the groundwater flowing in thearray through the NFR is assumed to be increasingly contaminated before it finally flows out of the
51
repository from the NFR of the Nx ’th compartment.The concentration gradient in the buffer is the largest in the first compartment because uncon-
taminated water flows into its NFR, resulting in the largest mass flux from the buffer. While adecrease in the radionuclide mass in the waste matrix occurs for all compartments, the completedepletion of radionuclide from the waste matrix occurs at the earliest time in the first compartmentin the array. Then, the second compartment is exposed to the highest mass flux among the remain-ing ones. In this fashion, the radionuclide mass depletes from the upstream compartment to thedownstream end of the array.
4.3 Mathematical Formulation
4.3.1 Waste-Matrix Region
In the waste-matrix region, the mass of a radionuclide changes because of radioactive decayand its release into the buffer region. A balance equation is written for the mass Mw
n (t) [mol] ofthe radionuclide in the waste-matrix region in the n’th compartment located relative to the upstreamside of the repository as
dMwn (t)
dt= −λMw
n (t) − qn(t) , 0 < t < T depn , (4.1)
subject to
Mwn (0) = M◦ , n = 1, 2, . . . , Nx ,
Mwn (t) = 0 , t ≥ T dep
n , n = 1, 2, . . . , Nx . (4.2)
Superscript ‘w’ represents the waste-matrix region and λ [yr−1] is the radioactive decay constantof the radionuclide. The term qn(t) [mol/yr] is the release rate of the radionuclide from the wastematrix into the buffer in compartment n; qn(t) is given by Eqs. (4.11) through (4.14), discussedbelow. T dep
n is the time when the radionuclide depletes completely from the waste-matrix region incompartment n; T dep
n is determined numerically by finding the time when Mwn (t) becomes zero, i.e.,
Mwn (T dep
n ) = 0. M◦ [mol] is the initial mass of the radionuclide in the waste matrix in a canister. Itis assumed that the same mass of the radionuclide is initially in all canisters.
4.3.2 Buffer Region
Nuclide transport in the buffer region is described by the diffusion equation (3.7), i.e.,
Rb∂Cb
n
∂t= D
∂2Cbn
∂ξ 2− λRbCb
n , t > 0 , 0 < ξ < L , n = 1, 2, . . . , Nx , (4.3)
subject to the initial condition
Cbn(ξ, 0) = 0 , 0 < ξ < L , n = 1, 2, . . . , Nx . (4.4)
52
There is no advection in the buffer. Sorption equilibrium between the solid phase and the pore-waterphase of the buffer is assumed. The retardation coefficient Rb is defined in (3.8).
At the buffer/NFR interface, the boundary condition is written in terms of the uniformized con-centration in the NFR as
Cbn(L , t) = Cn(t) , t > 0 , n = 1, 2, . . . , Nx . (4.5)
The boundary conditions for Cbn(ξ, t) at the interface ξ = 0 between the waste matrix and the
buffer are more complex than at ξ = L and are determined by the mechanisms of radionucliderelease from the matrix. The radionuclide concentration at ξ = 0 is assumed to be zero at t = 0.In the early-time period when the concentration of the radionuclide at ξ = 0 is smaller than thesolubility, it is assumed that the radionuclide in the waste matrix is released congruently with thematrix dissolution. For this, the boundary condition for Eq. (4.3) at ξ = 0 is
−Sεb D∂Cb
n
∂ξ
∣∣∣∣ξ=0
= mn(t) , n = 1, 2, . . . , Nx , 0 < t < T ∗
n , (4.6)
where T ∗
n is the time when the boundary concentration Cbn(0, t) reaches the solubility C∗ and the
precipitate of the radionuclide occurs. The left side of Eq. (4.6) represents the diffusive mass releaserate at ξ = 0 into the buffer region; mn(t) [mol/yr] is the congruent release rate of the radionuclidefrom the waste matrix in compartment n. Assuming that the waste matrix dissolves at a constantrate in the time period 0 < t < TL, mn(t) is given as [28]
mn(t) =Mw
n (t)TL − t
, 0 ≤ t < TL , n = 1, 2, . . . , Nx . (4.7)
Here, it is assumed that the constant TL is identical for all waste matrices in the array. Note thatthere is a mathematical coupling among Eqs. (4.1), (4.3), (4.6), and (4.7) to determine Cb
n(ξ, t).Depending on the magnitudes of T ∗
n and TL, there are two cases:Case 1: If Cb
n(0, t) increases with time and reaches the solubility at T ∗
n < TL, then Cbn(0, t) is
assumed to remain at the solubility C∗ until the radionuclide mass Mwn (t) in the waste-matrix region
depletes at t = T depn . Thus, the boundary condition at the waste-matrix/buffer interface is written as
Cbn(0, t) = C∗ , T ∗
n < t < T depn , n = 1, 2, . . . , Nx . (4.8)
The value of T depn could be greater than that of TL. In such a situation, the waste matrix has com-
pletely dissolved at TL, leaving a precipitate of the radionuclide in the waste-matrix region. Theprecipitate dissolves until T dep
n . The rate of precipitate dissolution is determined by the diffusivemass flux at ξ = 0, which is formulated by Eq. (4.12).
Case 2: If Cbn(0, t) does not exceed the solubility C∗ before t = TL, it is assumed that the
radionuclide is released congruently with the waste matrix. In this case, the radionuclide transportbecomes equivalent to that treated in Chapter 3. The radionuclide is depleted from the waste-matrixregion of the compartment n at the same time as when the waste matrix is depleted; i.e., Mw
n (t)becomes zero at t = TL, and T dep
n = TL. The boundary condition at ξ = 0 is written by
−Sεb D∂Cb
n
∂ξ
∣∣∣∣ξ=0
= mn(t) , 0 < t < T depn , n = 1, 2, . . . , Nx . (4.9)
53
Note that Eq. (4.9) is applied to 0 < t < T depn .
At t = T depn , the radionuclide release from the waste matrix region ceases. By Eq. (4.2), it is
assumed that there exists no radionuclide in that region for t > T depn . The boundary condition at the
matrix/buffer interface is written as
∂Cbn
∂ξ
∣∣∣∣ξ=0
= 0 , t > T depn , n = 1, 2, . . . , Nx . (4.10)
By this boundary condition, the effects of the waste-matrix region after T depn are ignored in the model
by neglecting possible back-diffusion of radionuclides from the buffer region to the waste-matrixregion.
Based on the aforementioned formulation for the boundary conditions at the interface betweenthe waste matrix and the buffer, the mass release rate qn(t) from the waste-matrix region into thebuffer region, which appears in Eq. (4.1), is written as follows:
If T ∗
n < TL, then
qn(t) = mn(t) , 0 < t < T ∗
n , n = 1, 2, . . . , Nx , (4.11)
qn(t) = −Sεb D∂Cb
n
∂ξ
∣∣∣∣ξ=0
, T ∗
n < t < T dep, n = 1, 2, . . . , Nx , (4.12)
andqn(t) = 0 , t > T dep
n , n = 1, 2, . . . , Nx . (4.13)
The formula (4.11) indicates that the mass release rate is the same as the congruent release rate.Equation (4.12) indicates that the release rate is determined by the diffusional flux at the interfacewhen the precipitate of the radionuclide exists there. Equation (4.13) indicates that no radionuclideis released from the waste-matrix region after the time T dep
n .If T ∗
n ≥ TL, then T depn = TL, and
qn(t) = mn(t) , 0 < t < TL , n = 1, 2, . . . , Nx , (4.14)
and Eq. (4.13) applies for t > TL. Equation (4.14) indicates that the release from the waste-matrixregion is the same as the congruent release rate until the radionuclide is depleted from that region.
4.3.3 Near-Field Rock Region
In the NFR region, the governing equation for Cn(t) same as Eq. (3.12) is used:
RrεrVdCn
dt= −λRrεrV Cn(t) + FCn−1(t) − FCn(t) + Q(Cn, t) , (4.15)
Cn(0) = 0 , C0 ≡ 0 , t > 0 , n = 1, 2, . . . , Nx .
The water flowing into the first compartment is uncontaminated (C0 ≡ 0). Sorption equilibriumbetween the solid phase and the pore-water phase is assumed.
54
The first term on the right side of Eq. (4.15) represents the change of the mass [mol] of the ra-dionuclide in the NFR due to radioactive decay. The second and the third terms on the right side rep-resent the mass of the radionuclide per unit time flowing in from compartment n−1 to compartmentn and the mass flowing out of compartment n to compartment n +1 by advection. Q(Cn ,t) [mol/yr]in the right side of Eq. (4.15) is the release rate of the radionuclide at the buffer/NFR interface intothe NFR, which is written as Eq. (3.15).
4.3.4 Radionuclide Masses
The mass of the radionuclide in various regions can be obtained based on the quantities definedabove. The mass Mb
n (t) [mol] of the radionuclide in the buffer in compartment n is written as
Mbn (t) ≡ RbεbS
∫ L
0Cb
n(ξ, t)dξ , t ≥ 0 , n = 1, 2, . . . , Nx . (4.16)
Similarly, the mass M rn(t) [mol] of the radionuclide in the NFR in compartment n is written as
M rn(t) ≡ RrεrV Cn(t) , t ≥ 0 , n = 1, 2, . . . , Nx . (4.17)
Then, the mass Mcn(t) [mol] of the radionuclide in compartment n is defined as
Mcn(t) ≡ Mw
n (t) + Mbn (t) + M r
n(t) , t ≥ 0 , n = 1, 2, . . . , Nx , (4.18)
where Mwn (t) is given by the solution of Eq. (4.1). The mass M int(t) of the radionuclide in the entire
array of Nx compartments is defined as
M int(t) ≡ Ny
Nx∑n=1
Mcn(t) = Ny
Nx∑n=1
[Mw
n (t) + Mbn (t) + M r
n(t)]
, t ≥ 0 . (4.19)
From the downstream side, i.e., from compartment Nx , the radionuclide is released into the farfield. The mass Mext(t) [mol] of radionuclide existing in the far field at time t is obtained by solvingthe mass balance equation
dMext
dt= −λMext(t) + Ny FCNx (t) , t > 0 , (4.20)
subject toMext(0) = 0 . (4.21)
Since the concentration CNx (t) in the water leaving the last compartment Nx and entering the farfield is determined by the mass of the other upstream compartments by Eq. (4.15), Mext(t) deter-mined by Eq. (4.20) includes the details of the mass transport in the repository through the termCNx (t).
55
Table 4.1 Assumed Parameters for Neptunium and the Repository
Symbol Description Value
Nx Number of compartments in the array parallel to the ground-water flow
λ Decay constant of 237Np 3.2 × 10−7 yr−1
Rb Retardation factor of Np in the buffer region 4900 a
Rr Retardation factor of Np in the NFR region 2600 a
C∗ Solubility of Np in the groundwater 2.0 × 10−5 mol/m3 a
M◦ Initial mass of 237Np in a single waste matrix 3.93 mol b
TL Duration for the waste-matrix dissolution 10000 yrD Diffusion coefficient for Np in the buffer 0.03 m2/yr c
F Volumetric flow rate of groundwater through the interfacebetween two adjacent compartments in a row
0.45 m3/yr d
εb Porosity in the buffer region 0.3 b
εr Effective porosity in the NFR region 0.5 b
ρb Density of the solid material in the buffer region 2100 kg/m3 b
ρr Density of the solid material in the NFR region 2600 kg/m3 b
V Volume of the NFR region in a compartment 9.05 m3 b
v Pore velocity of groundwater 1 m/yr b
S Surface area of a single waste matrix 1.81 m2 b
d Distance between waste canisters 10 m b
L Thickness of the buffer region 0.98 m b
Maximum permissible concentration (MPC) of 237Np 2 × 10−8 Ci/m3 e
Bare-sphere critical mass of 237Np 56 kg f
a See [1].b See [26].c See [39].d See Eq. (3.2).e See [15].f See [34].
56
4.4 Numerical Results and Discussion
4.4.1 Input Data
The radionuclide transport is investigated with 237Np. The two extreme configurations discussedin Chapter 1 are considered. In configuration A, canisters are lined up in the direction parallel tothe water flow in the NFR. In configuration B, canisters are lined up in the direction perpendicularto the water flow (see Table 1.1). The data for the radionuclide and for the repository are shownin Table 4.1. The values for the dimensions of the barriers in a compartment are the same as thoseselected in Chapter 3. The values of the neptunium solubility and the initial mass of neptunium inone waste canister are taken from [1]. The value M◦ of neptunium mass in one canister is the oneat 1000 yr after the emplacement of the canister in the repository, when the canister fails and therelease of radionuclide into the buffer starts. The values of the retardation factors, Rb and Rr, aredetermined from Eqs. (3.8) and (3.14) with the values of the sorption distribution coefficients Kdb
and Kdr given in [26] and the values for the densities and the porosities of the buffer and the NFRshown in Table 4.1. The time step for the time-marching calculations is set to 1 yr.
4.4.2 Transport in a Compartment
Figure 4.1 shows the concentration profiles for 237NP in the buffer in the first compartment. Theconcentration is normalized with the solubility. Calculations show that the concentration Cb
1(0, t)reaches the solubility as early as at the first time step with the parameter values given in Table4.1. Thus, the value for T ∗
1 is equal to 1 yr. It is confirmed by numerical results that also in thedownstream compartments, n > 1, the inner boundary concentration Cb
n(0, t) reaches the solubilityat the first time step. Thus, for the numerical results shown in Figures 4.1 through 4.5, the boundarycondition (4.8) is applied from the beginning of the neptunium release from the waste matrix. Forthe results for the no-, 1
10 -, and 1100 -mass-reduction cases shown in Figures 4.6 through 4.9, the
boundary condition (4.8) is applied from the first time step onward, whereas for those for the 11000 -
mass-reduction cases, the inner boundary concentration Cbn(0, t) does not reach the solubility, so
that the boundary condition (4.6) is applied until t = TL; TL = 10000 yr is given in Table 4.1.Although it cannot be seen from Figure 4.1, the front of the concentration profile reaches the
outer boundary, ξ = L , between 1000 and 10000 yr. This implies that the release of the radionuclidefrom the buffer to the NFR starts in this time span. (It is observed in Figure 4.4 that the mass M r
n(t)in the NFR starts to increase at around 2000 yr.) The concentration profile eventually reaches asteady state by 100000 yr.
Figure 4.2 shows the time T depn when the radionuclide depletes from the waste-matrix region in
the n’th compartment for configuration A. (In configuration A, groundwater flows parallel to thearray of canisters. See Table 1.1 for configurations A and B.) Until the time T dep
n , the boundary con-dition (4.8) is applied in the n’th compartment. After this time point, the boundary condition (4.10)is applied. For configuration B, the value for n = 1 applies. Figure 4.2 shows that 237Np remainsin the waste-matrix region for around 5000000 yr or longer for both configurations A and B. This isdue to the long half-life and small solubility of neptunium. The time T dep
n increases as n increasesbecause the water flowing through the NFR of these compartments gets more contaminated as n
57
increases, resulting in smaller Q(Cn, t) for a greater n.
4.4.3 Effects of Configuration
In this section, we discuss the concentration CNx (t) of 237Np in the groundwater leaving the lastcompartment and the masses, M int(t) in the array and Mext(t) in the far field for the two differentconfigurations A and B.
4.4.3.1 Radionuclide Concentration in NFR
In Figure 4.3, the normalized concentration CNx (t)/C∗ of 237Np in the groundwater at the repos-itory exit is plotted as a function of time for different values of Nx . This concentration CNx (t) isof particular importance because this determines the mass release rate from the repository to the farfield [see Eq. (4.20)].
For configuration A, the curves for Nx = 1, 2, 4, 8, 16, 32, and 64 compartments are shown inFigure 4.3. For configuration B, i.e., the perpendicular array, the transport from each compartmentis identical and corresponds to the calculation done for Nx = 1. This result applies for any numberof compartments in configuration B.
For any value of Nx , the concentration CNx (t) increases at early times until it reaches a steady-state plateau. The plateau concentration is maintained until the complete depletion of the radionu-clide in the waste-matrix region occurs in one compartment after another starting at the upstream
0.0
0.5
1.0
0.0 0.5 1.0
Nor
mal
ized
con
cent
ratio
n, C
1b (ξ,t)
/C*
Distance from the Waste Surface, ξ [m]
102yr
103yr
104yr
105yr,106yr
Figure 4.1 Profiles of 237Np concentration in the buffer, normalized with the solubility, for thefirst compartment. Time is measured form the beginning of the waste-matrix dissolution.
58
side of the array. For a greater Nx , the complete depletion in the waste matrix in the Nx ’th com-partment occurs at a later time as observed in Figure 4.2, and hence, the steady-state plateau lastslonger. It is observed that the plateau concentration increases as Nx increases, but not proportionallywith Nx . For example, the plateau concentration for Nx = 64 (0.75 in Figure 4.3) is less than 64times that for Nx = 1 (0.035 in Figure 4.3).
Suppose that the water flows in the direction parallel to the array axis but that the concentrationC1(t) based on configuration B is used as the inlet boundary condition for the far-field transportanalysis instead of C64(t) obtained for configuration A. Then, the far-field concentration would beunderestimated. On the other hand, the mass release rate from the 64 canisters for configuration Ais FC64(t), whereas that for configuration B is 64FC1(t). Note that FC64(t) < 64FC1(t). Thus,if the mass release rate 64FC1(t) is used for a repository with the parallel flow configuration as theinlet boundary condition for the far-field transport analysis, then, the far-field concentration wouldbe overestimated. The amount of overestimation is dependent on radionuclide properties.
4.4.3.2 Plateau Concentrations
In order to understand the transport mechanisms for the plateau time domain, we consider thefollowing situation. First, assuming that diffusional transport in the buffer reaches a steady state, we
0×100
2×106
4×106
6×106
8×106
1×107
0 10 20 30 40 50 60
Tim
e w
hen
237 N
p de
plet
esfr
om th
e w
aste
, Tnde
p [yr
]
Compartment number, n
T1dep
Figure 4.2 Time T depn when the radionuclide completely depletes from the waste in the n’th com-
partment for configuration A. For configuration B, the value represented by T dep1 (the leftmost point)
applies to all compartments. Time is measured from the beginning of the waste matrix dissolution.
59
set the time derivative in Eq. (4.3) to zero. The concentration at the waste-matrix/buffer interface isassumed to be the solubility C∗. Second, in the NFR, it is assumed that advective transport betweenadjacent compartments, radioactive decay loss, and the release from the buffer balance, so that thenet rate of change of the radionuclide mass in time in the NFR is set equal to zero in Eq. (4.15).We denote the radionuclide concentration in the NFR for this situation as C ss
n for compartment n.With C∗ and C ss
n as the inner and outer boundary concentrations, respectively, the diffusion equation(4.3) without the time derivative term can be solved analytically [28]. With the analytical solutionfor the steady-state profile for the concentration in the buffer region, the expression for the releaserate Q(C ss
n ) to the NFR is calculated from Eq. (3.15). Then, Eq. (4.15) (without the time derivativeterm) can be solved for C ss
n successively from the first compartment to the last compartment. Theexpression for the concentration in the NFR in the last compartment, normalized to the solubility isobtained as
C ssNx
/C∗= G f (Nx ; γ ) , Nx = 0, 1, 2, . . . , (4.22)
where
f (Nx ; γ ) ≡
1 − γ Nx
1 − γ, Nx = 0, 1, 2, . . . , if γ 6= 1 ,
Nx , Nx = 0, 1, 2, . . . , if γ = 1 ,(4.23)
0.0
0.5
1.0
0×100 5×106 1×107
Nor
mal
ized
con
cent
ratio
n, C
Nx(
t)/C
*
Time, t [yr]
Nx = 1
Nx = 2Nx = 4
Nx = 8
Nx = 16
Nx = 32
Nx = 64 0.75
0.035
Figure 4.3 Concentration of 237Np, normalized to the solubility C∗, in the groundwater leavingthe repository. Configuration B corresponds to the case with Nx = 1, whereas configuration Acorresponds to the cases with Nx = 2, 4, . . . , 64.
60
α ≡
√Rbλ
D, β ≡
Sεb Dα
sinh αL, γ ≡
FλRrεrV + F + β cosh αL
, (4.24)
G ≡β
λRrεrV + F + β cosh αL, (4.25)
All parameters in G, α, β, and γ are positive. Note that for Eq. (4.24) γ is less than or equal tounity.
The parameter values shown in Table 4.1 give the values G = 0.035 and γ = 0.96. In this case,the solution (4.22) is bounded for Nx = 1, 2, . . .. In the case of Nx = 1, the function f (1; γ ) in Eq.(4.22) becomes unity, and C ss
Nx/C∗ is obtained as 0.035. This value is in good agreement with the
numerical results shown in Figure 4.3. Similar agreement has been confirmed for Nx = 2, 4, 8, 16,and 64.
The function f (Nx ; γ ) defined in Eq. (4.23) represents the effect of radionuclide accumulationin the NFR due to multiple compartments. From the analytical expression (4.23) for f (Nx ; γ ), thesteady-state concentration C ss
Nxdoes not increase proportionally with Nx unless γ = 1 as discussed
below. It can be readily shown that for γ = 0.96, C ssNx
/C∗ < NxC ss1 /C∗ for Nx = 2, 3, . . .. This
is also confirmed by the numerical results shown in Figure 4.3. As Nx tends to infinity, C ssNx
/C∗
approaches a limit, independent of Nx , as follows:
limNx →∞
C ssNx
/C∗= G f∞(γ ) , (4.26)
wheref∞ ≡
11 − γ
, 0 < γ < 1 . (4.27)
Here, G f∞(γ ) is calculated as 0.88 for the parameter values shown in Table 4.1. Existence of thelimit independent of the number Nx of compartments shows that the radionuclide concentration inthe water stream in the NFR tends to a value of 0.88C∗. The limit value may be different if the timederivative terms in Eqs.(4.3) and (4.15) are not zero.
In order for γ to be unity,λRrεrV
F+
β
Fcosh αL = 0 ,
orλRrεrV
F+
Sεb DF L
· αL coth αL = 0
must be satisfied from the definitions of γ and β given in Eq. (4.24). Since αL coth αL ≥ 1, itmust be that λRrεrV/F = 0 and Sεb D/F L = 0. This occurs when F → ∞. Notice that fromthe discussion shown in Section 4.3.3, this situation corresponds to the independent-canister model.With γ = 1, as given in Eq. (4.23), the concentration in the water leaving the array increasesproportionally with Nx ; i.e., C ss
Nx/C∗
= NxC ss1 /C∗. This has also been discussed in Section 1.2.1
and Table 1.1.
4.4.3.3 Radionuclide Mass Distribution
In Figure 4.4, the masses Ny Mw1 , Ny Mb
1 , Ny M r1 of 237Np in the compartments, M int in the array,
and Mext in the far field, all normalized with the initial mass Ny M◦ of the radionuclide in the waste
61
matrices, are plotted as a function of time. Figure 4.4 shows the results for configuration B with anyvalue of Ny .
The normalized mass in the buffer increases immediately after t = 0 when 237Np release fromthe waste matrix starts. Having a sufficiently long half-life to survive its diffusion time in thebuffer, 237Np reaches the outer boundary of the buffer after 2000 yr, and starts being released intothe NFR. At the same time, mass release into the far field starts (Figure 4.4). Release into the farfiled continues until it completely depletes in the waste at 5.5 × 106 yr when a rapid drop in thenormalized masses in the waste matrix is observed. After this rapid drop, 237Np exists only in thefar field, and its mass decreases exponentially by radioactive decay. The maximum mass of 237Npin the far field at 5.5 × 106 yr is observed to be 20 % of the initial mass.
In Figure 4.5, a comparison of configuration B with configuration A is shown for the massesM int in the array and Mext in the far field for 64 canisters. The results for configuration B areidentical to those shown in Figure 4.4. For both configurations, the mass in the far field increaseswith time, while the mass in the array decreases rapidly after 106 yr. The mass in the array tends tozero at 5.5 × 106 yr for configuration B and 9 × 106 yr for configuration A. Most of the initial massof 237Np remains in the repository until 106 yr. For configuration B, about four-fifths of the initialmass of 237Np decays in the array, and one-fifth accumulates in the far field at 5.5 × 106 yr. Forconfiguration A, at the maximum, 8% of the initial mass in the array accumulates in the far field at9 × 106 yr. Because the release rate from the array is FC64 for configuration A, which is smallerthan 64FC1 for configuration B, more 237Np decays while it is in the array for configuration A.
10-6
10-5
10-4
10-3
10-2
10-1
100
101
102
102 103 104 105 106 107 108
Nor
mal
ized
mas
s
Time, t [yr]
M1w(t)/Mo
M1b(t)/Mo
M1r(t)/Mo
Mint(t)/NyMo
Mext(t)/NyMo
0.2
Figure 4.4 Masses Mw1 (t), Mb
1 (t), M r1(t), M int(t), and Mext(t) of 237Np in configuration B, nor-
malized to the initial mass M◦ in a waste matrix.
62
4.4.4 Effects of Mass Reduction
Previous analyses, such as in [3, 29, 30, 31, 32], of the impacts of partitioning-and-transmutation(PT) systems on the performance of geologic disposal conclude that PT systems are not justified be-cause of their high costs and modest improvements in radiological safety. This conclusion regardingradiological-safety improvements is primarily based on the observation that, because of low solu-bilities of actinides in groundwater, a decrease in the masses of actinides in the waste does notsignificantly decrease the radiological impact of actinides.
By decreasing the mass of the radionuclide in the waste form, the duration of the radionucliderelease into the surrounding NFR is reduced, but not the magnitude of the release rate. This is be-cause the release of the radionuclide from the waste form is limited by its solubility in groundwater.The decrease in the radionuclide mass does not affect the release rate unless the mass reduction isso significant that the release is not limited by the solubility.
Thus, if the independent-canister model is used and configuration B is assumed (see Table 1.1)as in previous studies [2, 3, 4, 6], no effects on the exposure dose rates would be observed byreducing the mass of radionuclide in individual canisters or by reducing the number of canisters.We consider that this is the primary reason for the insensitivity observed in the previous studies forPT effects on repository performance [3, 29, 31, 32].
In this section, we explore if this observation is still valid with the present model. The effects
10-6
10-5
10-4
10-3
10-2
10-1
100
101
102
102 103 104 105 106 107 108
Nor
mal
ized
mas
s
Time, t [yr]
Mint(t)/64Mo
Mext(t)/64Mo
Configuration AConfiguration B
Figure 4.5 Normalized masses of 237Np in configuration A (solid lines) with 64 canisters, com-pared with those in configuration B (dashed lines). Mext/64M◦ and M int/64M◦ are calculated fromEqs. (4.19) and (4.20), respectively.
63
of reduction in the initial mass loading of the waste canister are investigated by decreasing radionu-clide mass in each canister by the same amount. The number of canisters is kept unchanged at 64hereafter. In the no-mass-reduction case, the initial mass placed in the array is 3.93×64 = 252 mol.For the 1
10 -mass reduction case, the initial mass placed in the row in 64 canisters is 25.2 mol. The1
100 - and 11000 -reduction cases are defined similarly.
4.4.4.1 Radionuclide Concentration in NFR
In Figure 4.6, the change of the concentration C1(t) as a function of time is shown for configu-ration B. For the cases with no, 1
10 , and 1100 reduction, the magnitude of the peak concentration is of
the same order. The concentration time span decreases as the mass is reduced. The release rate fromthe waste matrix is the same for the three cases because it is limited by the solubility of neptunium.For the 1
1000 -reduction case, the concentration is substantially smaller than in the other three casesbecause the release from the waste matrix is congruent to the waste-matrix dissolution.
The concentration depicted in Figure 4.6 is the inlet boundary condition for the far-field model.Because the order of the magnitude of the peak inlet boundary concentration is not changed for theno-, 1
10 -, and 1100 -reduction cases, the peak far-field concentration would not differ by reducing the
radionuclide mass in the waste with configuration B.In Figure 4.7, the change in the concentration C64(t) as a function of time is shown for configu-
10-4
10-3
10-2
10-1
100
101
103 104 105 106 107 108
Nor
mal
ized
con
cent
ratio
n, C
1(t)
/C*
Time, t [yr]
Configuration B
11/101/100
1/1000
Figure 4.6 Effect of the initial 237Np mass reduction on the normalized 237Np concentrationC1(t)/C∗ in the water leaving the repository for configuration B. The fractions indicate the massreduction. Total of 64 compartments are considered.
64
ration A. Because of accumulation during the transport through 64 compartments, the concentrationC64(t) is close to the solubility for the mass reduction of 1 and 1
10 ; C64(t) decreases clearly in the1
100 - or 11000 -mass-reduction cases. Therefore, with C64(t) as the inlet boundary condition for the
far-field analysis, effects of mass reduction are expressed more clearly than with C1(t).
4.4.4.2 Radionuclide Mass
The temporal change of the mass M int(t) in 64 compartments and the mass Mext(t) in the farfield is shown in Figure 4.8 for configuration B and in Figure 4.9 for configuration A, for fourdifferent initial masses. Masses are normalized with 64M◦
= 252 mol.For the case of no mass reduction, the curves of M int and Mext are the same as those depicted
in Figure 4.5. With mass reduction by a factor of 110 , 1
100 , or 11000 , the mass in the array is reduced
by the corresponding factor at early times. With a smaller initial mass in the array, M int(t) tendsto zero earlier. Virtually all neptunium initially contained in the array is released into the far field.However, because the initial mass is reduced, the peak values of Mext(t) are also reduced by thesame factor.
For the no-, 110 -, and 1
100 -mass-reduction cases, the far-field-mass curves have the same shape atearly times. This is because for these three cases, the release from the waste matrix is limited bythe solubility. The far-field-mass curve for the 1
1000 -reduction case, however, lies below the other
10-4
10-3
10-2
10-1
100
101
103 104 105 106 107 108
Nor
mal
ized
con
cent
ratio
n, C
64(t
)/C
*
Time, t [yr]
Configuration A
11/10
1/100
1/1000
Figure 4.7 Effect of the initial 237Np mass reduction on the normalized 237Np concentrationC64(t)/C∗ in the water leaving the repository for configuration A. The fractions indicate the massreduction. Total of 64 compartments are considered.
65
three curves because the concentration C64 is about one order of magnitude smaller than those forthe other three cases, as observed in Figure 4.7.
In Figure 4.9, the same quantities shown in Figure 4.8 are plotted for configuration A. Massreleased in the upstream compartments accumulates as water flows in the array. Release from EBSin downstream compartments becomes smaller, so that the release from the repository is smallerthan the release from the repository in configuration B.
In Figures 4.8 and 4.9, for each case, the time when the mass M int in the array becomes equal tothe mass Mext in the far field is shown. Before this time the majority of the radionuclide mass existsin the array, whereas after this time the majority exists in the far field. It can be considered that thistime value expresses a residence time of the radionuclide in the array.
With configuration B (Figure 4.8), the effect of reduction of the initial mass loading on theresidence time is remarkable; 3.4 × 106 yr for the no-mass-reduction case is decreased to 90000 yrfor the 1
1000 -mass-reduction case. With configuration A (Figure 4.9), the residence time rangesbetween 6.1 × 106 yr and 1.1 × 106 yr. The radionuclide release from the array is “accelerated”by reducing the mass loading for configuration B, whereas the release from the array is smallerfor configuration A. These two figures indicate that the canister-array configuration has importanteffects on radionuclide confinement in a repository, i.e., the radionuclide residence time, and thatthe effects of reduction in the initial mass loading of the radionuclide in the waste appears differentfor different canister-array configurations. This illustrates the need that a PT system design shouldbe developed in conjunction with a repository design.
4.4.5 Mass-Based Performance Measures
The evaluated radionuclide concentration CNx (t) in water entering the far field is significantlydifferent, depending on which configuration, A or B, of multiple canisters is assumed. The peakvalue of the concentration CNx (t) is insensitive to mass reduction if evaluated with configuration Bwhereas it is affected if evaluated with configuration A. The radionuclide masses M int and Mext aresensitive to the initial mass reduction for both configurations. This implies that to observe the effectsof reduction of the initial mass loading, a performance measure based on the radionuclide mass isuseful. Here, two examples of measures for the repository performance are shown by utilizing themasses M int and Mext.
For environmental impact of the conceptual repository, the mass Mext could be used as a basis.For this, the radioactivity calculated from Mext is divided by the maximum permissible concentra-tion (MPC) for ingestion of that radionuclide [15]. The resultant quantity is known as the toxicityindex [33], which can be interpreted as the water volume with which the radionuclide must bediluted so that drinking the water will result in a radiation dose not exceeding the regulatory limit.
With the initial mass loading per canister, M◦= 3.93 mol, given in Table 4.1, the normal-
ized mass equal to one in Figures 4.8 and 4.9 corresponds to 64M◦ [mol] = 64 × 3.93 [mol] ×
0.237 [kg/mol] = 59.6 kg of 237Np in 64 canisters. The radioactivity of this mass is calculated to be41.5 Ci. Thus, with the value of 2×10−8 Ci/m3 [15] for MPC for 237Np, the normalized mass equalto one in Figures 4.8 and 4.9 corresponds to the toxicity index of 2.08 × 109 m3. Assuming that onewaste canister containing 237Np with the mass of M◦
= 3.93 mol is originated from 1 metric ton ofUO2 spent fuel [1], the toxicity index for the ore used to fuel the reactor to generate 64 canisters
66
of waste is calculated to be 3.5 × 108 m3.1 In Figures 4.8 and 4.9, the toxicity index relative tothe ore toxicity is indicated by the right axis. It is observed that in any case the toxicity index of237Np that exists in the far field would not exceed the toxicity of the ore and that by reduction ofthe initial mass loading the toxicity index is clearly reduced. It is also observed for the effects ofcanister-array configurations that the toxicity index in the far field is smaller with configuration Athan with configuration B and that the environmental impact measured by the toxicity index appearsearlier with configuration B than with A.
The toxicity index still does not measure hazards and risk of geologic disposal because it doesnot take into account the mechanisms by which the radionuclides could be transported to and takenby humans. However, the toxicity index obtained from Mext does take into account the mechanisms
1The ore toxicity is mainly due to 226Ra, which is in secular equilibrium. One gram of 238U has radioactivity of0.334 µCi; 226Ra in the uranium ore has the same radioactivity. The MPC for 226Ra is 6 × 10−8 Ci/m3 [15]. Thus, theore containing 1 g of 238U has the toxicity index of 5.5 m3. For 64 metric tons of uranium, the toxicity index is calculatedto be 3.5 × 108 m3.
10-6
10-5
10-4
10-3
10-2
10-1
100
101
102
103
103 104 105 106 107 10810-5
10-4
10-3
10-2
10-1
100
101
102
103
Nor
mal
ized
mas
s
Rel
ativ
e to
xici
ty in
dex
of 23
7 Np
Time, t [yr]
Configuration B
1
1/10
1/100
1/1000
9.0×104yr
1.1×105yr
6.1×105yr
3.4×106yr
Mint(t)/64Mo
Mext(t)/64Mo
Figure 4.8 Effect of the initial 237Np mass reduction on M int(t)/64M◦ and Mext(t)/64M◦ forconfiguration B. For each case, the time when M int(t)/64M◦ becomes equal to Mext(t)/64M◦ isshown. The left axis also represents the mass of 237Np relative to its bare-sphere critical mass. Ifthe value of M int(t)/64M◦ is read in this context, it could be used as a measure for proliferationresistance of the repository. The right axis represents the toxicity index of 237Np, relative to that ofthe ore that contains 64 metric tons of uranium. The value of Mext(t)/64M◦ read with the right axiscould be a measure for the environmental impact of the repository.
67
by which the radionuclides could be released from the repository. In this respect, it is remarkablydifferent from using a toxicity index obtained from the initial mass loading in the waste to expresseffects of PT system application in waste management [34].
With respect to proliferation, the mass M int of weapons-usable radionuclides remaining in thewaste canisters could be the basis for a measure. It is reported [35] that the bare sphere critical massfor 237Np is 56 kg, which is nearly equal to 64M◦
= 59.6 kg. Therefore, the left vertical axis inFigures 4.8 and 4.9 can be read approximately as the ratio of M int to the bare sphere critical mass.Figures 4.8 and 4.9 show that if, for example, the initial mass loading is reduced by a factor of1000, M int is also reduced by the same factor. Thus, 64000 canisters containing the 237Np mass ofM◦/1000 would be necessary to collect 59.6 kg of 237Np. Considering a typical repository capacity,such as 40000 vitrified HLW canisters in a proposed Japanese repository [1], the reduction of theinitial mass loading by a factor of 1000 could virtually eliminate the proliferation concern withrespect to 237Np.
4.4.6 Future Extensions
In the present study, the value for the solubility of neptunium reported in [1], which is relativelysmall, has been used. This value has been used to clearly show the effect of solubility limit on thetransport phenomena with multiple canisters. If the solubility is greater than that used in the presentstudy, then the radionuclide release rate into NFR becomes greater. The radionuclide depletes from
10-6
10-5
10-4
10-3
10-2
10-1
100
101
102
103
103 104 105 106 107 10810-5
10-4
10-3
10-2
10-1
100
101
102
103
Nor
mal
ized
mas
s
Rel
ativ
e to
xici
ty in
dex
of 23
7 Np
Time, t [yr]
Configuration A
1
1/10
1/100
1/1000
1.1×106yr
1.1×106yr
1.8×106yr
6.1×106yr
Mint(t)/64Mo
Mext(t)/64Mo
Figure 4.9 Effect of the initial 237Np mass reduction on M int(t)/64M◦ and Mext(t)/64M◦ forconfiguration A. See also the caption for Figure 4.8.
68
the waste-form region earlier. (The plateau observed in Figures 4.3 and 4.6 would last for a shortertime period.) With the value assumed in the present study, the congruent release has been observedonly for the case with 1
1000 reduction of the initial mass. If the 100-times-greater value is assumedfor the solubility, the release would become congruent with the waste matrix for the cases with 1
10 ,1
100 , and 11000 reduction. Colloid-facilitated transport should be studied in detail. It would create
the fraction of radionuclides moving faster than those in the solute phase and might invalidate theassumption of linear sorption isotherm.
The present study has addressed the issue of a new kind of model uncertainty in the performanceassessment. Attention should also be paid to parameter uncertainties. Realistically, the canisterfailure time should follow some probability distribution function, as well as other parameters, suchas the solubility, the water velocity, and the sorption distribution coefficients.
To verify and validate the present model, especially the central assumption of the connectedcanisters in the water stream, a laboratory-scale experiment or a natural-analog study is highlyrecommended.
4.5 Conclusions
The effects of canister array configurations in the water-saturated repository have been examinedfor the case where release of the radionuclide from the waste form is limited by its solubility.
The relation ship among the repository performance, canister-array configuration, and the ra-dionuclide mass in the waste has been investigated by observing (a) the radionuclide concentrationin the groundwater leaving the array of multiple canisters and (b) the masses of the radionuclide inthe regions interior and exterior to the array of multiple canisters. The following observations canbe made:
1. For the parallel-to-flow configuration A, the concentration in the water leaving the array in-creases as the number Nx of compartments in the array increases. It is remarkable that theconcentration in the steady state does not increase linearly with Nx , but less rapidly. For theperpendicular-to-flow configuration B, the concentration in the water leaving the array is thesame for any number of canisters.
2. For configuration A, the peak concentration in the water leaving the array decreases as theinitial mass loading in a canister is reduced, whereas for configuration B, the peak concentra-tion is not affected by the reduction in the initial mass loading as long as solubility limits areexceeded at the boundary of the waste form and the buffer. The length of time that the peakconcentration is maintained is affected greatly by waste loading.
3. In either configuration A or B, the radionuclide masses in the array and in the far field decreaseclearly by decreasing the radionuclide mass in the waste.
4. A greater mass of the radionuclide would exist in the far field with configuration B than withA because of a faster release into the far field.
From the viewpoint of obtaining more comprehensive results for repository performance, it isessential to select a proper canister configuration that represents actual repository design, especially
69
when the mass-reduction effects are of interest (see observations 2, 3, and 4 above). From theviewpoint of designing PT systems, because the mass-reduction effects appear in a different mannerfor a different canister-array configuration, PT system design should be considered in conjuctionwith repository design.
The formula (4.22) of the steady-state concentration for solubility-limit release corresponds tothe formula (3.24) of the steady-state concentration for congruent release case. These formulas areused in the next chapter to evaluate and compare the performance of two different options of HLWdisposal system.
70
CHAPTER 5
EFFECTS OF ELECTRO-CHEMICAL REDUCTION
PROCESS ON THE ENVIRONMENTAL IMPACT OF
GEOLOGIC DISPOSAL
5.1 Introduction
In this chapter, the models discussed in Chapters 3 and 4 are applied to evaluation of the envi-ronmental impact of a HLW repository.
The following two types of the radioactive waste are considered and compared: (1) The firstwaste is the vitrified waste from PUREX reprocessing of spent fuel assumed as the basis for theJapanese repository concept in [1]. We refer to this waste as “H12 waste” hereafter. (2) The secondwaste is the sodalite waste from the electro-chemical reduction process of the high-burnup PWRspent fuel. We refer to this waste as “sodalite waste” hereafter.
Electro-chemical reduction process1 is a technology to extract part of fission products and oxy-gen from the oxide spent fuel. During the process, fission products such as alkaline and alkalineearth metals dissolve in the molten salt, whereas other fission products and heavy metal elementsremain in the solid and are reduced to metal. The reduced heavy metal may be recovered to producefuels for fast reactors. The fission products that dissolved in the molten salt are considered to besolidified with sodalite and eventually be disposed of in a deep geologic repository.
Since the actinides are recovered through this reduction process, mass of actinides (including237Np) in the waste repository is reduced. In Chapter 4, we observed that the mass reduction of 237Npcan significantly reduce its release from the repository depending on the canister array configuration.Electro-chemical reduction process may thus improve the performance of waste disposal system.
In the present study, in order to observe the effects of electro-chemical reduction process ongeologic disposal of waste, the peak radiotoxicity release rate from the repository is comparedbetween the sodalite wastes and the H12 wastes.
5.2 Performance Measure for the Analysis
According to the result of the performance assessment for H12 reference repository [37] shownin Figure 5.1, the total exposure dose rate to the public individuals is mainly determined by the dose
1Reductive extraction process [36] and electro-chemical reduction process [12] are currently being developed by Cen-tral Research Institute for Electric Power Industries (CRIEPI). Reductive extraction process is a technology for recoveringactinides from the high-level liquid waste from PUREX process.
71
Surface EnvironmentGeosphereEngineered Barrier System
10-8
10-7
10-6
10-5
10-4
10-3
10-2
10-1
100
101
102
103
104
Do
se
[S
v/y
]
100 101 102 103 104 105 106 107 108
Time after disposal [y]
Th-229
Cs-135
Se-79 Pb-210
U-238U-234
Total
Np-237
Time after disposal [y]
Ca
lcu
late
d d
os
e[S
vy
-1]
LifestyleSurface EnvironmentGeosphereEngineered Barrier System
10-8
10-7
10-6
10-5
10-4
10-3
10-2
10-1
100
101
102
103
104
Do
se
[S
v/y
]
10-8
10-7
10-6
10-5
10-4
10-3
10-2
10-1
100
101
102
103
104
Do
se
[S
v/y
]
100 101 102 103 104 105 106 107 108
Time after disposal [y]
100 101 102 103 104 105 106 107 108
Time after disposal [y]
Th-229
Cs-135
Se-79 Pb-210
U-238U-234
TotalTotal
Np-237
Time after disposal [y]
Ca
lcu
late
d d
os
e[S
vy
-1]
Ca
lcu
late
d d
os
e[S
vy
-1]
Lifestyle
Figure 5.1 Exposure dose rate obtained for the performance assessment of the reference reposi-tory [37].
103
104
105
106
107
108
109
1010
1011
1012
1013
1014
1015
Tox
icity
inde
x, c
ubic
met
er o
f wat
er
101
102
103
104
105
106
107
Time after canister failure, year
Cs135
I 129
Np237
Tc99
Sum of toxicity indices of radionuclides both in the repository and in the far field
Sum of toxicity indices of radionuclides existing in the far field
Figure 5.2 Radiotoxicity index of radionuclides released from the repository [39].
72
rates of 135Cs and 237Np and its daughters. The highest peak in the dose rate observed at around106 yr is formed by 135Cs, and the second highest peak at around 107 yr is formed by 229Th. Since237Np, which is a precursor of 229Th, has its peak at the same time as 229Th, it is considered that237Np migrates to a point where it is ingested by human, and that 229Th in a secular equilibriumfollows 237Np. Thus, the dose rate of 229Th is determined by migration of 237Np. To observe theperformance of the repository, we therefore focus on 135Cs and 237Np.
By the electro-chemical reduction process, amount of actinides in the waste is reduced. InFigure 5.1, even if the exposure dose rates of 237Np and 229Th are reduced, the effects on the overallperformance is small because of the highest peak due to 135Cs.
Another measure suggested for repository performance is the total radiotoxicity index of theradionuclides in the environment that has been released from the repository [13]. The radiotoxicityof a single radionuclide in the environment is obtained by
λNA
3.7 × 1010 · CMPC
∫ t
0e−λτ JNx ,Ny (t − τ)dτ , (5.1)
where λ [sec−1], NA, and CMPC [Ci/m3] are the radioactive-decay constant, Avogadro’s number, andthe maximum permissible concentration [15] of the radionuclide, respectively. JNx ,Ny (t) [mol/yr]is the release rate of the radionuclide from the repository that consists of Nx × Ny array of wastecanisters.
The main advantages of this measure are that (1) it has less uncertainty because, unlike theexposure dose rate, analysis on migration to the biosphere is not necessary, (2) it reflects the isolationmechanisms of the repository because it is based on the release from the repository, (3) it is sensitiveto the initial inventory of the radionuclides in the repository, and (4) it is not affected much bytemporal instability and uncertainty in migration properties because it is obtained by integration ofthe radionuclide release rate [38].
Figure 5.2 shows the radiotoxicity index of the radionuclides for the same reference repository[39]. It is observed that the main radionuclides that determine the toxicity in the environment are135Cs and 239Np. Based on the discussions in Chapter 4, that the toxicity index is sensitively affectedby the initial inventory of 237Np and that the toxicity is effectively reduced by reduction of 237Npinventory.
If we reduce the initial inventory of 237Np, the peak of the total toxicity is also reduced. How-ever, further reduction of 237Np inventory has little effect on the total toxicity when contribution by135Cs becomes predominant. The inventory of fission products including 135Cs remains approxi-mately the same as long as the amount of energy production is the same, and therefore, the toxicityof 135Cs remains the same. Thus, it is not necessary to aim at extremely good recovery efficiency for237Np by the electro-chemical reduction process.2 The repository can contain the initial inventoryof 237Np equivalent to the toxicity of 135Cs in the environment without affecting the overall impact.
In the present study, the release rate of the radiotoxicity from the repository is used as theperformance measure instead of integrated toxicity in the environment defined by Eq. (5.1). The
2Other important criteria are the regulatory standards that the repository is required to meet. Further reduction ofradionuclides released from the repository is not necessary when the repository meets safety standards. In this chapter,the effect of 237Np reduction in sodalite waste is examined as one of the aspects of fuel cycle with electro-chemicalreduction process.
73
Table 5.1 Parameter Values for the Repository
Symbol Description Value
TL Duration for the waste-matrix dissolution 104, 105, 106 yrF Volumetric flow rate of groundwater through the interface
between two adjacent compartments in a row0.0217 m3/yr
εb Porosity in the buffer region 0.41εr Effective porosity in the near-field rock (NFR) region 0.02ρb Density of the solid material in the buffer region 2100 kg/m3
ρr Density of the solid material in the NFR region 2600 kg/m3
V Volume of the NFR region in a compartment 10.85 m3
v Pore velocity of groundwater 1 m/yrS Surface area of a single waste matrix 2.17 m2
d Distance between waste canisters 10 mL Thickness of the buffer region 0.98 m
Table 5.2 Parameter Values for the Repository
Symbol Description Value (135Cs) Value (237Np)
Kdb Soprtion coefficient in the buffer 0.01 m2/yr 1.0 m2/yrKdr Soprtion coefficient in the near-field rock (NFR) region 0.05 m2/yr 1.0 m2/yr
Half-life 2.3 × 106 yr 2.14 × 106 yrCMPC Maximum permissible concentration 1 × 10−5 Ci/m3 2 × 10−8 Ci/m3
C∗ Solubility in groundwater – 2 × 10−5 mol/m3
M◦ Initial inventory per canisterSodalite waste 4.84 mol 4, 0.4, 0.04 molH12 waste 3.19 mol 4, 0.4, 0.04 mol
peak radiotoxicity release rate I peakNx ,Ny
[m3/yr] of a radionuclide from an Nx × Ny array of wastecanisters is obtained by
I peakNx ,Ny
=λNA
3.7 × 1010 · CMPCJ peak
Nx ,Ny, (5.2)
where J peakNx ,Ny
[mol/yr] is the peak release rate of the radionuclide from the canister array. Ground-water is assumed to flow parallel to the rows of Nx canisters.
The radiotoxicity release rate due to 135Cs initially contained in sodalite waste repository iscalculated based on Eq. (5.2). The maximum inventory of 237Np in the repository that does notexceed the toxicity release rate of 135Cs is investigated in the present study.
5.3 Methods and Input Data
5.3.1 Conditions on the Comparison
The environmental impacts of the sodalite waste repository is compared to H12 waste repositoryunder the condition of the same repository capacity. Here, the repository capacity is measured by
74
the electricity generation that produced the waste.In the H12 waste repository, 40000 waste canisters are placed. Since 1.25 waste canisters are
produced from 1 tHM of spent fuel, 40000 canisters of waste is equivalent to 32000 t of spent fuel.With the burnup of 45 GWd/tHM, the entire fuel is used to produce (32000 tHM)×(45 GWd/tHM) =
1440000 GWd (or 3900 GWyr) of thermal energy. Assuming the conversion efficiency of 33%, therepository capacity is equivalent to approximately 1300 GWyr of electricity generation.
The same repository layout design is assumed for both H12 waste and sodalite waste, such asdimensions of the waste canisters and EBS. The parameters for the repository are shown in Table5.1 and Table 5.2.
5.3.2 Peak Radiotoxicity Release Rate
As mentioned earlier, the peak radiotoxicity release rates are evaluated for 135Cs and 237Np,which are the major contributors to the environmental impact. Since the solubility of cesium ingroundwater is sufficiently high, 135Cs is released congruently with degradation of the waste matrix.On the other hand, the release of 237Np from waste forms is considered to be limited by the solubilityof neptunium in the groundwater.
The peak release rate J peakNx ,Ny
of 135Cs from the repository of Nx × Ny array of waste canisters isobtained from Eqs. (3.24), (3.25), and (3.26), and is given as
J peakNx ,Ny
=
Nx Ny M◦
TL, Nx Tres < TL ,
Ny M◦
Tres, Nx Tres ≥ TL ,
(5.3)
where M◦ is the initial inventory of the radionuclide in a single canister, TL is the leach time ofthe waste forms, Tres is the mean residence time of the radionuclide in a single compartment and isgiven as
Tres =RrεrV + RbεbSL
F. (5.4)
The retardation factor Rr in the host rock in the near field is defined as
Rr ≡ 1 +1 − εr
εrρr Kdr , (5.5)
and Rb in the buffer is defined similarly. See Table 5.1 and Table 5.2 for the other parameters.Details of the derivation of formula (5.3) are described in Chapter 3.
For 237Np, the peak release rates are given as
J peakNx ,Ny
=
F NyC∗β
ζ·(1 − γ Nx
), Nx Tres < T ,
F NyC∗β
ζ·
[1 − exp
(−
ζ TRrεrV
)],
Nx Tres > T ,
(5.6)
75
where α, β, γ , and ζ are defined as follows:
α ≡
√Rbλ
D, β ≡
Sεb Dα
sinh αL, γ ≡
FF + ζ
, (5.7)
ζ ≡ λRrεrV + β cosh αL . (5.8)
The first formula in Eq. (5.6) is derived from Eqs. (4.22) and (4.23). The second formula in Eq.(5.6) applies when the radionuclide completely depletes in the last waste canister in the array beforethe advection front of the radionuclide from the first waste canister reaches the repository exit. Thetime T is obtained by solving
M◦e−λT−
β
λ
[1 − e−λT
] [C∗ cosh αL − C
]= 0 , (5.9)
where
C ≡C∗β
ζ·
[1 − exp
(−
ζ TRrεrV
)]. (5.10)
Details of the derivation of formula (5.6) are described in [13].
5.3.3 Sodalite Waste Repository
5.3.3.1 Number of Waste Forms
The highest allowable temperature of the sodalite waste is assumed to be 800 ◦C. The mass ofradionuclides per canister is determined so that the temperature in the sodalite waste is kept below800 ◦C. In the temperature calculation, the cylindrical shape of a waste form is approximated withthe sphere of the same volume. The surface of the waste form is assumed to be cooled and main-tained at 100 ◦C. Steady-state heat transfer is considered. The relationship between the maximumtemperature in the waste form (at the center of the sphere) and the heat generation rate is given as[40]
A0a2
6K= 1T , (5.11)
where 1T = 700 [K] is the temperature difference between the center and the surface of the sphere,K [W/Km] is the thermal conductivity, A0 [W/m3] is the heat density, and a [m] is the radius of thesphere. The radius a is obtained by
a =
(34
r2h)1/3
, (5.12)
where r [m] and h [m] are, respectively, the radius and the height of the original cylindrical shapeof the waste.
The heat generation rate 43πa3 A0 [W] per waste form is obtained as 2575 W, where K =
0.4 W/Km, r = 0.22 m, and h = 1.35 m has been assumed.Table 5.3 shows the amount of fission products in the waste from a high-burnup (48 GWd/tHM)
PWR equivalent to 1 t of the fuel and their decay heat. The heat generation in the waste due to thefission products is 2.43 kW/tHM as shown in Table 5.3. It is assumed that the waste forms are
76
Table 5.3 Fission Products in the Waste from High-Burnup (48 GWd/t) PWR
Heat generation Mass Valencerate [W/tHM] [mol/tHM] [mol.equiv.]
Li 0.00 × 10+0 9.21 × 10−5 1 9.21 × 10−5
Be 5.07 × 10−9 2.41 × 10−5 2 4.82 × 10−5
Zn 0.00 × 10+0 8.64 × 10−7 2 1.73 × 10−6
Ga 0.00 × 10+0 2.42 × 10−5 3 7.27 × 10−5
Ge 0.00 × 10+0 1.13 × 10−2 4 4.51 × 10−2
As 0.00 × 10+0 3.71 × 10−3−3
Se 1.26 × 10−4 1.06 × 10+0−2
Br 0.00 × 10+0 3.80 × 10−1−1
Rb 1.48 × 10−8 6.62 × 10+0 1 6.62 × 10+0
Sr 1.05 × 10+2 1.43 × 10+1 2 2.85 × 10+1
Y 5.60 × 10+2 7.75 × 10+0 3 2.33 × 10+1
Tc 9.36 × 10−3 1.12 × 10+1 4 4.48 × 10+1
Ru 2.45 × 10+0 3.13 × 10+1 2 6.25 × 10+1
Rh 3.94 × 10+2 5.75 × 10+0 3 1.73 × 10+1
Sn 6.70 × 10−3 6.87 × 10−1 4 2.75 × 10+0
Sb 1.21 × 10+1 1.38 × 10−1−3
Te 7.72 × 10−1 5.83 × 10+0−2
I 2.15 × 10−5 2.46 × 10+0−1
Cs 8.38 × 10+2 2.91 × 10+1 1 2.91 × 10+1
Ba 5.18 × 10+2 1.73 × 10+1 2 3.45 × 10+1
Total 2.43 × 10+3 1.33 × 10+2 2.49 × 10+2
fabricated 4 years after discharge of the fuels. The heat generation rate due to fission productsdecreases with time, and hence it takes the maximum value at the time of waste form fabrication.The mass of the fuel equivalent to maximum allowable heat generation rate 2575 W for a singlesodalite waste is obtained as Mcanister = 2575/2430 = 1.06 tHM.
For comparison with H12 waste repository, the capacity of the sodalite waste repository is as-sumed to be 1300 GWyr equivalent. The number of waste canisters in the repository is obtainedby
number of canisters = Mtotal/Mcanister , (5.13)
where Mtotal is the mass of the total fuel used for 1300 GWyr electricity generation. Mtotal is obtainedby
Mtotal × (burnup) × (conversion efficiency) = 1300 GWyr . (5.14)
Assuming the conversion efficiency of 33% and the burnup of 48 GWd/tHM, the fuel dischargedfrom the high-burnup PWR is obtained to be 3.0×104 tHM. The number of sodalite waste canistersin the repository is obtained to be 28300.
Note that the size of a sodalite waste form is assumed to be the same as that of a H12 vitrifiedwaste. Considering that the maximum heat generation rates for H12 waste (2.33 kW) and sodalite
77
waste (2.575 kW) are similar to each other, the specification of the EBS and the design used in theH12 repository can be applied to sodalite waste repository.
5.3.3.2 Fraction of the Fission Products in the Waste
For the integrity of the sodalite waste form, it is considered that the fraction of fission productsmust be less than 30% of the total cation in the waste form.
As shown in Table 5.3, the total amount of FP cations in the waste is 249 mol.equiv./tHM. Asodalite waste obtained in the previous section contains 1.06 tHM equivalent of fission products, andthere for it contains 249 mol.equiv./tHM × 1.06 tHM = 264 mol.equiv. of cation fission products.
Sodalite is produced by
Na12(AlSiO4)12 + 1.9CaCl2 → 1.9Na4Ca2(AlSiO4)6Cl2 ,
and it is known that 1705 g of Na12(AlSiO4)12 and 3.8 mol.equiv. of chloride produce the sodalitewaste form of the volume 837 cm3. The total amount of cation in a waste form is obtained as
(volume of a waste) × (3.8 mol.equiv.)/(837 cm3)
= 932 mol.equiv. (5.15)
Therefore the fraction of cation fission products in a sodalite waste is obtained as
264 mol.equiv.
932 mol.equiv.= 0.283 . (5.16)
Thus, the fraction of the fission products in the sodalite waste (28.3%) is confirmed to be less thanthe suggested limit (30%).
5.4 Results
The peak toxicity release rates are plotted in Figure 5.3 as functions of Nx , the number ofcanisters along the groundwater flow.
The toxicity release rate due to 135Cs from H12 waste repository is shown as case (a). Thisis observed similar to that from the sodalite wastes (case (b)) where the leach time of the sodalitewaste is assumed to be the same as the vitrified waste (TL = 104 yr). It confirms that there is nota significant difference in toxicity release rate due to fission products if the amount of electricitygeneration and the waste form performance are common.
Cases (e), (f), and (g) show the contribution of 237Np in the sodalite wastes, assuming that eachcanister initially contains 4 mol, 0.4 mol, and 0.04 mol of 237Np, respectively. The value 4 mol isapproximately equal to the initial inventory of 237Np in a single H12 waste canister. Little differenceis observed between the peak toxicity release rates of cases (e) and (f). The contribution of 237Npin (e) and (f) is greater than that of 135Cs in (b) for all values of Nx , whereas it becomes smallerin (g) than (b). The peak toxicity release rate of 135Cs and 237Np becomes approximately equal toeach other when there is 0.1 mol of 237Np in each canister. It follows that even if a few percent of
78
237Np is contained in the waste stream during the electro-chemical reduction process, the overalltoxicity release rate would not be affected because the toxicity release rate of 135Cs would be higherthan that of 237Np. In such cases, the further reduction of 237Np in the waste would not reduce thetoxicity release rate.
Results (c) and (d) show the cases that the leach time of sodalite wastes becomes 105 yr and106 yr, respectively. As the leach time becomes longer, the toxicity release rate for small valuesof Nx becomes lower. In case Nx = 4 and the leach time is improved to 106 yr, the peak toxicityrelease rate can be reduced by a factor of 10 by reducing the 237Np inventory to 0.04 mol per canister.This implies the possibility of having a repository of 10 times greater capacity and having the sametoxicity release rate as H12 waste repository.
Figure 5.4 shows the results for which the repository capacity is expanded by the factor of 10.Case (e) is plotted together for comparison. Case (e) is considered to be the toxicity release ratefrom the original H12 repository.
With 10-times greater power generation, the inventory of 135Cs in the repository becomes 10-times greater, and thus the number of canisters increases proportionally. Cases (h) through (m)correspond to cases (b) through (g), respectively. The difference between (h) through (m) and (b)through (g) is that the number of canisters is 10-times greater in (h) through (m). The toxicity
101
102
103
104
105
106
100 101 102 103 104
Peak
toxi
city
rele
ase
rate
[m3 /y
r]
Nx
(a) 135Cs (40000 canisters)(b) 135Cs, TL = 104 yr
(c) 135Cs, TL = 105 yr(d) 135Cs, TL = 106 yr
(e) 237Np, 4 mol(f) 237Np, 0.4 mol
(g) 237Np, 0.04 mol
Figure 5.3 Peak radiotoxicity release rate from the repository. (a) 135Cs from the repository of40000 H12 vitrified wastes containing 3.19 mol/canister of 135Cs, TL = 104 yr. (b)(c)(d): 135Csfrom the repository of 28300 sodalite wastes containing 4.84 mol/canister of 135Cs. The assumedleach time is (b) TL = 104 yr, (c) TL = 105 yr, and (d) TL = 106 yr. (e)(f)(g): 237Np from therepository of 28300 sodalite wastes. The assumed initial inventory of 237Np is (e) 4 mol/canister,(f) 0.4 mol/canister, and (g) 0.04 mol/canister.
79
release rates for cases (h) through (m) are 10-times greater than those for cases (b) through (g).Because of the 10-times greater inventory of 135Cs, toxicity release rate in case (h) is higher than
case (e). However, if the leach time is 106 yr [case (j)] and 237Np content is 0.04 mol/canister [case(m)], it is possible to have a repository of 10-times greater capacity with the level of toxicity releaserate same as the original H12 waste repository.
5.5 Limitations of the Analysis
Because many simplifications and assumptions have been introduced to the present study, inter-pretation of the results requires caution.
By the assumption of steady-state temperature distribution and the spherical-body approxima-tion used in calculation of the allowable heat generation rate in a waste form, the allowable heatgeneration rate is evaluated smaller than the case without these assumptions. Comparison betweenthe H12 waste and the sodalite waste has been performed with the same interval d between can-isters and the same EBS specifications. In order to minimize the number of waste canisters, theseparameters must be optimized for each waste type, and the transient heat transfer analysis must beperformed.
The leach time for sodalite waste is assumed within the range between 104 yr and 106 yr basedon that for the vitrified waste in the present study. Since the leach time can affect the toxicity release
101
102
103
104
105
106
100 101 102 103 104
Peak
toxi
city
rele
ase
rate
[m3 /y
r]
Nx
(h) 135Cs, TL = 104 yr
(i) 135Cs, TL = 105 yr(j) 135Cs, TL = 106 yr
(k) 237Np, 4 mol(l) 237Np, 0.4 mol
(m) 237Np, 0.04 mol(e)
Figure 5.4 Peak radiotoxicity release rate from the repository with extended capacity (10 times).(h)(i)(j): 135Cs from the repository of 283000 sodalite wastes (other assumptions in (b)(c)(d) ap-ply). (k)(l)(m): 237Np from the repository of 283000 sodalite wastes (other assumptions in (e)(f)(g)apply).
80
rate of 135Cs, further knowledge on degradation characteristics of sodalite waste in the groundwateris required.
The peak release rate formula (5.6) for 237Np takes into account the attenuation of the radionu-clide due to its radioactive decay, whereas the formula (5.3) for 135Cs neglects the radioactive decay.In case the leach time becomes greater than 106 yr, Eq. (5.3) should be modified to take into accountthe effects of radioactive decay.
5.6 Conclusion
The radiotoxicity release rates from the sodalite waste repository containing 28300 canisters areobserved in comparison with H12 waste repository of 40000 canisters.
Since the electro-chemical reduction process recovers 237Np, the toxicity release rate from thesodalite waste repository due to 237Np can be reduced significantly if the initial mass of 237Npbecomes less than 0.4 mol/canister. On the other hand, the toxicity release rates due to 135Cs forboth two repositories are the same because the 135Cs inventory is the same.
The radiotoxicity release rate of 237Np becomes equal to that of 135Cs when the initial contentof 237Np in a single sodalite waste form is approximately 0.1 mol. Therefore, the overall peakradiotoxicity release rate would not decrease if 237Np contained in a waste form is reduced below0.1 mol/canister.
As the leach time for sodalite waste form becomes longer, the toxicity release rate for smallvalues of Nx becomes lower. In case the leach time is improved to 106 yr, for a repository withsmall number of connected canisters (i.e., small Nx ), the peak toxicity release rate can be reducedby a factor up to 10 by reducing the 237Np inventory to 0.04 mol/canister. In this case, there is thepossibility of having a repository of 10 times greater capacity and having the same toxicity releaserate as H12 waste repository.
Thus, options of the processes in the fuel cycle can be compared in terms of the performanceof the waste disposal system. The design of the fuel cycle should be considered with the canisterarray configuration in the repository because their effect on the repository performance appears in adifferent manner for a different canister array configuration.
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CHAPTER 6
MARKOV CHAIN MODEL IN A TWO-DIMENSIONAL
ARRAY OF COMPARTMENTS
6.1 Introduction
Analyses of groundwater flow and radionuclide transport at the scale of entire waste reposi-tory are important for the performance assessment and designing of high-level waste repository.In Chapters 3 and 4, significant effects of array configuration of waste canisters in the repositorywere observed because multiple waste canisters affect mass transport from other canisters in thesame groundwater stream and the spatial extent of the canister array affects the travel time in therepository.
In Chapters 3 and 4, a uniform flow of groundwater in homogeneous medium was assumed.However, near-field rock is characterized as heterogeneous media. Radionuclide migration in frac-tured rock around a single waste package was modeled (FFDF model) and studied in [41]. In FFDFmodel, the randomly generated fractures intersect with each other forming a cluster of connectedfractures. The cluster of fractures that connect to waste canisters becomes the main migration pathof the nuclide released from failed canisters. Thus, the migration path of the released nuclide largelydepends on how the clusters of connected fractures are formed. The transport analysis of radionu-clide particles was performed [41] using a random-walk tracking method in such a heterogeneousdomain.
In order to incorporate both the heterogeneity as in FFDF model and the waste array configura-tion, the domain in the FFDF model should be extended to the scale of the repository. A large-scalesimulation of nuclide transport through the rock fracture network is being performed in such a wayas reported in [42] using FFDF model and Earth Simulator [43]. In [42], more than ten thousandfractures are generated for one realization of fractures in a rock domain that contains 109 canistersto analyze the groundwater flow.
However, extending the detailed model of a single-package scale to the repository scale wouldrequire significant amount of computing resources [42]. It is preferable to have a radionuclide mi-gration model at a repository scale that makes use of the results obtained from a separate model(such as FFDF) with a detailed geometry in a small domain. For this purpose, a new compartmentmodel for particle migration in the geologic media is developed, by describing the particle migra-tion by Markov chains. The goal of the new model is to enable efficient analyses of radionuclidemigration at a large scale based on the information obtained from a smaller-scale detailed analysis.
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6.2 Model
6.2.1 Transition Probabilities for a Compartment
A compartment in the present model represents a square region of geologic media. Two-dimensional array of compartments is considered.1 The sizes of the domain and a compartmentmay be assigned arbitrarily.2
The main input data for the present model are the transition probabilities for particles, whichare used to determine the position of the particle after time interval 1t when we know its positionbefore the time interval. These transition probabilities are illustrated in Figure 6.1 as the probabilityof particle migration from one compartment to another. P(1nx ,1ny) denotes the probability thatthe particle in the shaded compartment moves after time interval 1t by 1nx compartments in xdirection and 1ny compartments in y direction. P(0,0) is the probability that the particle will still bein the shaded compartment after 1t .
With a separate model such as FFDF, the particle migration within the compartment and itsvicinity can be analyzed to obtain the spatial distribution of the particles at time 1t . Provided thatthe particles do not interfere with each other, the transition probabilities are obtained as the fractionalnumber of particles in the region corresponding to each neighboring compartment. The obtainedtransition probabilities represent the properties of particle migration within a compartment. Becausethe information of the structure inside a compartment is also contained in the transition probabilities,it is not explicitly considered in this model as opposed to the compartments considered in Chapters3 and 4.
If the spatial distribution of the particles is not deterministically obtained by the separate anal-ysis, the transition probabilities can be assigned randomly based on the statistics obtained by theanalysis. For example, in FFDF model, the migration of particles is greatly affected by the connec-tivity of the fractures through which the particles flow. Since the fractures are randomly generatedin FFDF, the obtained transition probabilities are also random variables with distributions.
6.2.2 Migration of a Single Particle through an Array of Compartments
In the present model, the geologic domain consists of an Nx × Ny array of aforementionedcompartments embedded in the environment. Figure 6.2 shows the array of compartments and theirindices. Migration of a particle is represented by its transition from one compartment to another,which is assumed to occur at each time interval 1t . The detailed trajectory of the particle withineach compartment is not considered in this model.
1A two-dimensional array of compartments has been also considered in the model in Chapters 3 and 4. Radionuclidesin the NFR region, however, has been assumed to migrate along a row of canisters, and the mathematical model formigration in the NFR regions is one-dimensional. In the model in this chapter, radionuclide movements and groundwaterflow in various directions are considered as opposed to unidirectional flow in the NFR assumed in Chapters 3 and 4. Thepresent model also applies to one- and three-dimensional systems although it is illustrated in a two-dimensional systemhere.
2The scale of the domain represented by the compartment array may be various. For example, in Section 6.3, migrationof particles is illustrated in a domain that contains a 10×10 array of waste canisters in a repository. The domain in Section6.5, on the other hand, represents a single waste canister and the geologic medium in its vicinity, in which case the sizeof a compartment is smaller than the size of a waste canister.
83
Figure 6.1 The transition probabilities for a particle migrating from the shaded compartment. Thesubscript coordinates denote the direction of the position after the transition relative to the positionbefore the transition.
Figure 6.2 An Nx × Ny array of compartments. Compartments are specified by n = 1, . . . , N(N ≡ Nx Ny) and by coordinates (nx , ny).
84
Let Pmn denote the transition probability from compartment m to compartment n in time interval1t , i.e., the probability that a particle is in compartment n at time t + 1t , given that it was incompartment m at time t . The transition probabilities Pmn are the input data to be obtained throughthe separate analysis. In this notation, Pm,m+1 corresponds to P(0,1) in Figure 6.1, Pmm to P(0,0),and so forth. It is assumed that Pmn only depends on m and n, but not on time t . The transition isindependent of the past positions of the particle and only depends on the present position. Thus, thesequence of particle positions (i.e., the compartment indices) forms a Markov chain with discretetime and discrete states [44].
Since Pmn is a probability, it must satisfy 0 ≤ Pmn ≤ 1. Since the particle must exist in one ofthe compartments in the array or in the environment after a transition,
Pm,env +
N∑n=1
Pmn = 1 , m = 1, 2, . . . , N , (6.1)
where ‘env’ denotes the index for the environment, and thus Pm,env is the transition probabilityfrom compartment m to the environment in one time step. N ≡ Nx Ny is the total number ofcompartments in the array.
We assume that the particle stays in the environment once it enters the region, i.e., Penv,env = 1and Penv,n = 0 for n = 1, 2, . . . , N .
Given the particle position m at time t , the probability that it is in compartment l at time t + 1tis denoted as Pml by definition. Similarly, given the particle position l at time t +1t , the probabilitythat it is in compartment n at time t + 21t is denoted as Pln . It follows that, given the particlesposition m at time t , the probability that it makes a transition to compartment l at time t + 1t andthen to compartment n at time t + 21t is Pml Pln .
Thus, given the particle position m at time t , the probability P (2)mn that the particle is in compart-
ment n at time t + 21t , regardless of the position at t + 1t , is
P (2)mn =
∑l
Pml Pln . (6.2)
The summation on the right side is taken for all possible positions l at time t + 1t . P (k)mn is referred
to as the k-step transition probability and denotes the probability that a particle in compartment mwill be in compartment n after k time steps.
Eq. (6.2) is generally described by Chapman-Kolmogorov equations [44],
P (k1+k2)mn =
∑l
P (k1)ml P (k2)
ln . (6.3)
The matrix form of Eq. (6.3) is written as
P(k1+k2) = P(k1) · P(k2) , (6.4)
85
where P(k) is the k-step transition probability matrix whose elements are P (k)mn , i.e.,
P(k)≡
P (k)
11 P (k)12 · · · P (k)
1,N P (k)1,env
P (k)21 P (k)
22 · · · P (k)2,N P (k)
2,env...
.... . .
......
P (k)N ,1 P (k)
N ,2 · · · P (k)N ,N P (k)
N ,env
P (k)env,1 P (k)
env,2 · · · P (k)env,N P (k)
env,env
. (6.5)
It follows that P(k)= Pk , where P is the one-step transition probability matrix which consists of
Pmn .Suppose that a particle is in compartment n at t = 0 with probability a(0)
n . The probabilitydistribution of the particle position at t = k1t is obtained by
a(k)= a(0)
· Pk , (6.6)
where a(0) is a row vector that consists of the initial position probabilities a(0)n , i.e.,
a(0)≡
[a(0)
1 , a(0)2 , . . . , a(0)
N , a(0)env
]. (6.7)
It is required that a(0)n satisfies
a(0)env +
N∑n=1
a(0)n = 1 . (6.8)
6.2.3 Migration of Multiple Particles
The aforementioned process is applied independently to each of multiple particles. Let Y i(k)n
denote the indicator random variable that describes existence of the particle i in compartment n attime t = k1t ;
Y i(k)n =
{1 if particle i is in compartment n at time k1t ,
0 otherwise .
The probability of having Y i(k)n = 1, i.e., the probability of finding the particle in compartment
n at time k1t , is P{Y i(k)n = 1} = a(k)
n . Since Y i(k)n is a Bernoulli random variable, its mean is equal
to the probability P{Y i(k)n = 1}, and therefore
E[Y i(k)
n
]= a(k)
n , (6.9)
where E[·] denotes the expectation of the expression.Consider M◦ particles of a nuclide in the compartment array. The initial location and migration
of each particle are determined by the probabilities specified by a(0) and P, respectively. Commonvalues for a(0) and P are used for all particles. Each particle is assumed to independently migratethrough the compartments.
86
The number of particles in compartment n at time t can be obtained by summation of the indi-cator variables Y i(k)
n over all M◦ particles:
M (k)n =
M◦∑i=1
Y i(k)n , (6.10)
where M (k)n is the number of particles in compartment n at time k1t . Assuming that the initial
number of particles M◦ is the order of the magnitude of Avogadro’s number, by Eqs. (6.10) and(6.9),
M (k)n
M◦=
1M◦
M◦∑i=1
Y i(k)n ≈ E
[Y i(k)
n
]= a(k)
n . (6.11)
The weak law of large numbers has been applied to the approximation between the sample meanof Y i(k)
n and its true mean in Eq. (6.11). The weak law of large numbers applied for Y i(k)n is formally
written as [45]
limm→∞
P
{∣∣∣∣∣ 1m
m∑i=1
Y i(k)n − E
[Y i(k)
n
]∣∣∣∣∣ ≥ c
}= 0 , c > 0 , (6.12)
or, informally,1m
m∑i=1
Y i(k)n ≈ E
[Y i(k)
n
]for m � 1 , (6.13)
where m is the total number of independent particles in the system. The approximation in Eq. (6.11)is obtained by substituting m with M◦ in Eq. (6.13).
Thus, by assuming independence of each particle and from the law of large numbers, the frac-tional mass M (k)
n /M◦ of the particles in compartment n at t = k1t is described by the probabilitya(k)
n . The spatial distribution of the particles is described by the vector a(k) which is obtained throughEq. (6.6).
The input parameters to the present model are a(0), the initial probability distribution of theparticles, and P, the one-step transition probability matrix. Values of P should be determined byother studies on particle migration within a single compartment. Value of a(0) is determined byarrangement of radionuclide sources in the repository.
6.3 Illustration 1: Migration at the Repository Scale
6.3.1 Overview of Illustration
For a demonstration, we apply the present model to nuclide migration by dispersion and advec-tion in a hypothetical waste repository which consists of 10 × 10 array of waste packages.
Because of heterogeneous formation of the cluster of interconnected fractures, the compart-ments in the array are not necessarily connected to each other by the fracture cluster. In this il-lustration, the compartment connectivity is randomly determined for each interface based on theprobability Pconnect. Although hypothetical values for the transition probability are used in the
87
Table 6.1 Transition Probabilities for Illustration 1Symbol Value Symbol Value Symbol Value
P(−1,1) 0 P(0,1) 0.18 P(1,1) 0P(−1,0) 0.075 P(0,0) 0.19 P(1,0) 0.375P(−1,−1) 0 P(0,−1) 0.18 P(1,−1) 0
∗ Other transition probabilities P(1nx ,1ny) not shown in thetable are set to zero.
demonstration, the statistics of the connectivity between adjacent compartments should, instead,be obtained based on smaller-scale FFDF analyses and be converted to transition probabilities.
First, we apply the present model to a homogeneous domain assuming that all the compart-ments are connected to their neighboring compartments (i.e., Pconnect = 1). Second, we simulatethe heterogeneous fractured rock domain by randomly assigning connectivity between neighboringcompartments.
6.3.2 Homogeneous Medium with All Compartments Connected
The hypothetical repository consists of a 30×30 array of compartments. Figure 6.3(a) shows theinitial distribution of the nuclide in the repository. Each black square indicates the compartment thatcontains a waste package. The gray-scale colors of compartments indicate the values of probabilitya(k)
n , which is also the mass fraction of the nuclide in the compartment.Particles are released into the geologic media at time t = 0.3 The values for transition proba-
3In a real waste repository, release of radionuclides from waste packages is considered to occur gradually insteadof instantaneously because of the gradual failure of the engineered barrier system and the waste forms. Migration of a
Figure 6.3 Probability a(k)n of the particle existence in each compartment in a 30 × 30 array; (a)
the initial distribution (k = 0), and (b) at t = 601t (k = 60) when Pconnect = 1.
88
bilities shown in Table 6.1 are used (see Figure 6.1 for the symbols). These values are arbitrarilyassigned for the illustration purpose. We assume the same transition probabilities for all compart-ments in the array, i.e., the transition probabilities depend only on the direction of the transition andnot on the current compartment of the particle. The values of transition probabilities in Table 6.1simulates advective migration of particles in positive x direction and dispersion effect both in x andy directions.
Figure 6.3(b) shows the concentration profile at time t = 601t . It is observed that the nuclidemoves in the x direction while it disperses.
The fraction of the nuclide that exists in the environment (i.e., a(k)env) is plotted as a function of
time in Figure 6.4. This is also interpreted as the cumulative distribution function for the time atwhich particle is released into the environment. In this illustration, particles may be released intothe environment not only through the right edge of the compartment array but also through the edgesat the top, the bottom, and the left. It is observed that practically all particles are released from thearray by the time t = 1501t .
6.3.3 Heterogeneous Medium with Random Connectivity
Next, nuclide transport through the heterogeneous fractured rock domain is simulated by ran-domly assigning connectivity between neighboring compartments.
gradually released nuclide can be simulated with the present model by superposing the concentration profiles separatelyobtained for the nuclide released at each time step.
0.0
0.2
0.4
0.6
0.8
1.0
0 50 100 150 200 250 300 350 400
Frac
tion
in th
e en
viro
nmen
t, a(k
)en
v
Time, t/∆t
Pconnect = 1
Pconnect = 0.8
Figure 6.4 Fraction of the nuclide in the environment as a function of time. The curve forPconnect = 0.8 is specific to the connectivity realization shown in Figure 6.5.
89
A compartment is ‘connected’ to each of its neighboring compartment (or to the environmentif it is on the edge of the array) with probability Pconnect = 0.8. The transition probabilities areassigned as follows:4 (1) If the two neighboring compartments are connected, the value shown inTable 6.1 is used for the corresponding transition. (2) If the two neighboring compartments are notconnected, the probabilities for transition between the two compartments are zero. (3) The transitionprobability P(0,0) is determined by
P(0,0) = 1 − P(1,0) − P(0,1) − P(−1,0) − P(0,−1) . (6.14)
The initial distribution shown in Figure 6.3(a) is assumed.Figure 6.5 shows the concentration profile at time t = 601t in the domain with randomly as-
signed connectivity. The lines on the compartment boundaries indicate that the two compartmentsare not connected to each other across the boundary. Since particles cannot move across these lines,the path of a particle migration becomes crooked. The nuclide is observed to accumulate in a com-partment (or a group of isolated compartments) that is only connected to the compartment on its leftbecause the nuclide in the left compartment flows into such a compartment with a high probabilityeven though the path is a dead end. The nuclide slowly moves out of such a compartment by going
4This scheme of probability assignment is highly abstracted. Instead, the transition probabilities could be assignedbased on groundwater flow analyses. In addition, in this demonstration, the connectivity of a pair of neighboring com-partments is assigned independent of the connectivity of other pairs. In the reality, the connectivity of compartments maybe correlated with that of other compartments in the vicinity.
Figure 6.5 Nuclide distribution at t = 601t for one realization of randomly generated connectiv-ity between compartments (Pconnect = 0.8). The lines on the compartment boundaries indicate thatthe two compartments are not connected to each other across the boundary. Nuclide does not moveacross these lines.
90
back to the left compartment. This accumulation is less likely to occur in the reality because therewould be little advective flow from the left compartment into such a dead-end compartment, and thecorresponding transition probability would be smaller.
The fraction of the particles in the environment for this particular case is plotted in Figure6.4. It is observed that, with Pconnect = 0.8, the nuclide spends a longer time for migration withinthe array, and that a(k)
env reaches unity significantly later because of the accumulation in dead-endcompartments.
6.4 Illustration 2: Comparison with a Continuum Model
6.4.1 Continuum Model
In order to compare the results obtained from the present model with previous conventionalmodels, we consider the following problem.
At time t = 0, M◦ [mol] of a nuclide is released at the origin. We shall observe the concentrationof the nuclide at (x, y) = (L , 0). The uniform, steady flow of water in the positive x direction isassumed. The nuclide moves by advection and molecular diffusion. The governing equation in acontinuous domain is written as
∂C∂t
= D(
∂2
∂x2+
∂2
∂y2
)C − v
∂C∂x
, (6.15)
where D and v are the effective diffusion coefficient and the mean velocity of the nuclide, respec-tively. For an infinite domain, the concentration of the nuclide is given as
χ(ξ, ϑ, τ ) =Pe
4πτexp
[−
Pe4τ
((ξ − τ)2
+ ϑ2)] , τ > 0 , (6.16)
where the following nondimensionalization has been introduced:
ξ ≡ x/L , ϑ ≡ y/L , τ ≡ vt/L , χ ≡ C L2h/M◦ , Pe ≡ vL/D , (6.17)
where h is the height of the medium in the direction perpendicular to xy plane.Let 1ξ and 1ϑ denote the normalized distance in x and y direction, respectively, that a particle
moves over a time interval 1τ . In the dimensionless system, the mean and the standard deviationof 1ξ and 1ϑ are
µ1ξ = 1τ , µ1ϑ = 0 , and σ1ξ = σ1ϑ =√
21τ/Pe . (6.18)
6.4.2 Application of the Markov Chain Model
The present Markov chain model is applied to the above problem by preserving the values ofµ1ξ , µ1ϑ , σ1ξ , and σ1ϑ . Instead of infinite domain, finite number of compartments are used. Thedimension l of a compartment is set by l = L/n0, where n0 is an arbitrarily assigned number ofcompartments. The size of the compartment array is set to (4n0 + 1)× (4n0 + 1). The origin, which
91
is the initial position of the nuclide, is located at the center of the array. The purpose of assigningthe compartment array to cover the regions beyond the observation point is to reduce the effect ofthe finite size of the domain used in the Markov chain model.
The position of a compartment can be represented by the coordinates of the point in the midst ofthe compartment. Position of all particles in the compartment is also represented by this mid point.Let point A (ξ0, ϑ0) be the mid point of the compartment that a particle resides before a transition(see Figure 6.6). Then point B (ξ0 +µ1ξ , ϑ0 +µ1ϑ) is the mean coordinates of the particle after thetransition. Let jx denote the difference between nx indices of the compartment containing point Aand the compartment containing point B. Similarly, let jy denote the difference between ny indicesof these compartments.
Assume that the particle makes a transition into either the compartment containing point B orone of the 8 compartments that surround point B. Assuming 1nx and 1ny are independent, thetransition probability can be obtained in a form of
P(1nx ,1ny) = P(1nx ,∗) P(∗,1ny) , (6.19)
where P(1nx ,∗) is the marginal probability that the particle moves by 1nx compartments in x direc-tion regardless of movement in y direction, and P(∗,1ny) is the marginal probability for 1ny .
The transition probabilities P(1nx ,1ny) are determined so that the mean and the standard devia-tion of movement over time interval 1τ are preserved. With the mid point representing the locationof each compartment, the mean and the variance of 1ξ in the Markov chain representation arewritten in terms of transition probabilities by
µ1ξ =1n0
(jx − P( jx −1,∗) + P( jx +1,∗)
), (6.20)
Figure 6.6 Points A and B, and the non-zero transition probabilities. Particle location before atransition is represented by point A (ξ0, ϑ0). Point B (ξ0 + µ1ξ , ϑ0 + µ1ϑ) is the mean coordinatesof the particle after the transition. Transition probabilities are assigned to the 9 compartments aroundpoint B.
92
σ 21ξ =
1n0
[j2x + 2 jx
(P( jx +1,∗) − P( jx −1,∗)
)+ P( jx +1,∗) + P( jx −1,∗) − µ2
σx
](6.21)
The mean and the standard deviation must preserve the values obtained by Eq. (6.18). With thecondition for the sum of the probabilities P( jx +1,∗) + P( jx ,∗) + P( jx −1,∗) = 1, and with Eqs. (6.20)and (6.21), the marginal probabilities P(1nx ,∗) are determined by µ1ξ and σ1ξ as
P( jx +1,∗) =12
[n2
0σ21ξ + (n0µ1ξ − jx)
2+ n0µ1ξ − jx
],
P( jx −1,∗) =12
[n2
0σ21ξ + (n0µ1ξ − jx)
2− n0µ1ξ + jx
],
P( jx ,∗) = 1 − n20σ
21ξ − (n0µ1ξ − jx)
2 ,
P(1nx ,∗) = 0 for 1nx < jx − 1 , 1nx > jx + 1 . (6.22)
With Eqs. (6.18) and (6.22), P(1nx ,∗) is obtained in terms of Pe, n0, and 1τ .The marginal probabilities P(∗,1ϑ) can be determined similarly by
P(∗, jy+1) =12
[n2
0σ21ϑ + (n0µ1ϑ − jy)
2+ n0µ1ϑ − jy
],
P(∗, jy−1) =12
[n2
0σ21ϑ + (n0µ1ϑ − jy)
2− n0µ1ϑ + jy
],
P(∗, jy) = 1 − n20σ
21ϑ − (n0µ1ϑ − jy)
2 ,
P(∗,1ny) = 0 for 1ny < jy − 1 , 1ny > jy + 1 . (6.23)
In the present illustration, groundwater flows in the x direction, and therefore jy = 0 and µ1ϑ = 0.In order to keep the transition probabilities nonnegative, we assign 1τ = Pe/8n2
0. The rangesof the parameters are discussed in the next section.
The probabilities for initial nuclide distribution is assigned as a(0)n = 1 if compartment n is at
the origin, and a(0)n = 0 otherwise. The dimensionless concentration in a compartment after k time
steps can be obtained by
χ =L2hM◦
·M◦
l2ha(k)
n = n20a(k)
n . (6.24)
Figure 6.7 shows the dimensionless concentration at (x, y) = (L , 0) as a function of time forn0 = 5 and n0 = 10. The concentration obtained by Eq. (6.16) is also plotted in together. The valuePe = 25 has been used for all curves. Better agreement with the continuum model is observed forn0 = 10.
6.4.3 Ranges of the Parameters
The marginal probabilities P(1nx ,∗) obtained by Eq. (6.22) must be nonnegative and at mostunity. Applying this condition to Eq. (6.22), we obtain
14
≤ n20σ
21ξ +
(n0µ1ξ − jx +
12
)2
≤94
, (6.25)
93
14
≤ n20σ
21ξ +
(n0µ1ξ − jx −
12
)2
≤94
, (6.26)
0 ≤ n20σ
21ξ + (n0µ1ξ − jx)
2≤ 1 . (6.27)
The possible ranges for n0µ1ξ − jx and n0σ1ξ based on these inequalities are depicted in Figure6.8. The circles indicated as ‘a’ correspond to the range boundaries of inequality (6.25). Thecircles indicated as ‘b’ correspond to the boundaries of inequality (6.26). The circle indicated as’c’ corresponds to the second inequality in (6.27). The shaded area including its boundaries is thepossible range of n0µ1ξ − jx and n0σ1ξ . Because of the definition of jx , the difference betweenn0µ1ξ and jx is at most 1/2.
It is shown in Figure 6.8 that n0σ1ξ can become zero only if n0µ1ξ − jx = 0. This meansthat, with the assumed scheme of transition probability assignment, the present model can simulatean advective migration without diffusion or dispersion only if the advective movement over onetimestep is equal to the length of a compartment.
In the present illustration, the uniform velocity of water is assumed, and therefore n0µ1ξ − jx
has the same value for all particle positions. If the water flow is heterogeneous as in Illustration3, the value of n0µ1ξ − jx varies depending on the location of the particle. In order to satisfyconditions (6.25), (6.26), and (6.27) regardless of the value of n0µ1ξ − jx , n0σ1ξ must be in thefollowing range:
12
≤ n0σ1ξ ≤
√3
2. (6.28)
0.0
0.5
1.0
1.5
2.0
2.5
0 1 2 3 4
Dim
ensi
onle
ss c
once
ntra
tion
C′
Dimensionless time, t′
Continuum modeln0 = 5
n0 = 10
Figure 6.7 Dimensionless concentration χ at (x, y) = (L , 0). The concentration for continuummodel is obtained by Eq. (6.16).
94
By substituting Eq. (6.18) into Eq. (6.28), n0 and 1τ must satisfy
Pe8
≤ n201τ ≤
3Pe8
. (6.29)
Once the size of a compartment l = L/n0 is assigned, the possible range of the time interval 1τ isthus determined.
The same discussion applies to n0µ1ϑ − jy and n0σ1ϑ .
6.5 Illustration 3: Transport around a Waste Cylinder
In the previous illustration, the transition probabilities have been determined by the water ve-locity and the diffusion coefficient. In this section, the same approach is applied to simulate particletransport around a single waste canister. It is also an example of applications to a relatively smalldomain as opposed to the greater scale treated in Illustration 1.
6.5.1 Groundwater Flow and Nuclide Migration around an Infinite Cylinder
Consider a waste solid imbedded in a porous medium through which water is flowing steadilyin accordance with Darcy’s law. The waste solid is treated as a cylinder of infinite length. The flowis taken normal to the axis of the cylinder. Figure 6.9 depicts the streamlines of the flow around thecylinder. The potential head h [m] is governed by
∇2h =
1r
∂
∂r
(r∂h∂r
)+
1r2
∂2h∂θ2
= 0 . (6.30)
The velocity vector Ev [m/yr] of water is, by Darcy’s law,
Ev = −K E∇h , (6.31)
Figure 6.8 Range of the parameters. The circles indicated as ‘a’ correspond to the range bound-aries of inequality (6.25), ‘b’ to (6.26), and ‘c’ to the second inequality in (6.27). The shaded areaincluding its boundaries is the possible range of n0µ1ξ − jx and n0σ1ξ .
95
where K [m/s] is the hydraulic conductivity. Assuming the isotropic, homogeneous medium, K isconstant.
There is no water flowing into the waste solid. Boundary conditions are
vr (r0, θ) = 0 , 0 ≤ θ < 2π , (6.32)
limr→∞
vr (r, θ) = v0 cos θ , limr→∞
vθ (r, θ) = −v0 sin θ , 0 ≤ θ < 2π , (6.33)
where vr [m/yr] and vθ [m/yr] are radial and tangential components of Ev, respectively, and v0 [m/yr]is the pore velocity of the water far away from the cylinder.
The water velocity is obtained from the above equations as [46]
vr (r, θ) = v0
(1 −
r20
r2
)cos θ , vθ (r, θ) = −v0
(1 +
r20
r2
)sin θ , (6.34)
or, in the Cartesian coordinate system,
vx(x, y) = v0
[1 +
r20 (−x2
+ y2)
(x2 + y2)2
], vy(x, y) = 2v0
r20 xy
(x2 + y2)2. (6.35)
Consider diffusive and advective migration of a nuclide in such water streams. The Markovchain model is applied in the same manner shown in Illustration 2 except that the velocity of thenuclide is dependent on the location in this illustration. Nuclide retardation due to adsorption isneglected here.
The system is nondimensionalized in a manner similar to Eq. (6.17). The radius of the cylinder,r0, is used as the characteristic length instead of L .
x ′≡ x/r0 , y′
≡ y/r0 , t ′≡ v0t/r0 , Pe ≡ v0r0/D . (6.36)
Figure 6.9 Streamlines of steady flow of water around an infinite cylinder of radius r0.
96
Over a time interval 1t , the means and the standard deviations of the particle movement in xand y directions are
µ1ξ (x ′
0, y′
0) ≡vx1t
r0= 1τ
[1 +
−x ′20 + y′2
0
(x ′20 + y′2
0 )2
], (6.37)
µ1ϑ(x ′
0, y′
0) ≡vy1t
r0= −21τ ·
x ′
0 y′
0
(x ′20 + y′2
0 )2, (6.38)
σ1ξ = σ1ϑ =√
21τ/Pe , (6.39)
where (x ′20 , y′2
0 ) is the location of the particle at the beginning of the time interval (see point A inFigure 6.6).
As shown in Figure 6.6, the transition probabilities for the Markov chain model are assignedto the 9 compartments around point B as P(1nx ,1ny) = P(1nx ,∗) P(∗,1ny), where P(1nx ,∗) and P(∗,1ny)
are obtained by Eqs. (6.22) and (6.23), respectively. In the present illustration, n0 is defined byn0 ≡ r0/ l where l [m] is the dimension of a single compartment.
6.5.2 Release of Nuclide from the Waste Solid
The nuclide initially exists uniformly in the waste cylinder. As the waste matrix degrades, thenuclide is gradually released into the water through the surface of the waste solid. This gradualrelease of the nuclide into the groundwater is roughly simulated in this illustration by assumingfictitious diffusion of the nuclide within the waste solid. A compartment is considered to representa part of a waste solid if the mid point of the compartment is inside the waste cylinder.
In the dimensionless system, the mean µw1ξ and the standard deviation σ w
1ξ of the particle move-ment 1ξ over a time interval 1τ in the waste cylinder are
µw1ξ = 0 , σ w
1ξ = σ1ξ
√Dw
D, (6.40)
where Dw is the fictitious diffusion coefficient of the particle in the waste cylinder. Similarly forµw
1ξ and σ w1ξ ,
µw1ϑ = 0 , σ w
1ϑ = σ1ϑ
√Dw
D. (6.41)
The transition probabilities for the particles in the waste cylinder are determined by Eqs. (6.19),(6.22), and (6.23).
6.5.3 Results
Figure 6.10 shows the distribution of the nuclide in a 30 × 30 array of compartments at timeτ = 0, 2, 4, and 6. The parameter values used are Pe = 40, n0 = 5, Dw/D = 0.1, and 1τ = 0.2.
Initially the nuclide is contained in the waste cylinder. As it is released into the surroundingmedium where water flows, the nuclide is transported in the positive x direction. At time τ = 6,the nuclide distribution in the vicinity of the waste cylinder reaches a quasi-steady state because thetransport outside the cylinder is significantly faster than change in the release rate from the cylinder.
97
Figure 6.10 Distribution of a nuclide around a single waste cylinder. The parameter values usedare Pe = 40, n0 = 5, Dw/D = 0.1, and 1τ = 0.2.
98
6.6 Illustration 4: Tilted Water Stream and Exit Concentration
As our last illustration, the concentrations at the downstream end of the canister array are ob-served. In Chapter 3, the peak exit concentration has been observed to be constant at an upperbound regardless of Nx , the number of canisters in the flow direction, if Nx is greater than a thresh-old value. The similar observation is made with the Markov chain model in two-dimensional arrayof compartments in this chapter. Of particular interest is profile of the exit concentration when thegroundwater flow direction is tilted with respect to the canister array.
6.6.1 Application of the Model
Consider a 30 × 30 array of compartments and 10 × 10 waste canisters arranged at positionsspecified by black compartments in Figure 6.11. The nuclide is released into the groundwater inthose black compartments. The groundwater steadily and uniformly flows with velocity v. Thenuclide released in water migrates by advection. Retardation of the nuclide due to sorption is notconsidered. The angle formed by the canister array (x direction) and the water flow direction isdenoted by θ .
The transition probabilities are determined by Eqs. (6.19), (6.22), and (6.23), where n0 is definedas
n0 ≡ L/ l = 3 ,
The representative length L [m] and l [m] are the interval between two adjacent waste canisters and
Figure 6.11 Array of compartments considered in Illustration 4. Each black compartment con-tains a waste canister. The concentrations are observed in compartments specified by A and B. Thearrows show the stream of groundwater that passes through compartment A or B.
99
the length of a compartment, respectively.1ξ and 1ϑ denote the normalized distance in x and y direction, respectively, that a particle
moves over a time interval 1τ . With the nondimensionalization (6.17), the mean and the standarddeviations of 1ξ and 1ϑ are
µ1ξ =v1t cos θ
L= 1τ cos θ , (6.42)
µ1ϑ =v1t sin θ
L= 1τ sin θ , (6.43)
σ1ξ = σ1ϑ =
√2D1t
L=√
21τ/Pe . (6.44)
In order to meet the condition (6.29), we assign 1τ = Pe/8n20.
For nuclide released at the initial time step by the amount m [mol/canister], the concentrationin a compartment after k time steps can be obtained by
C =Nx Nym
l2ha(k)
n , (6.45)
where Nx Ny is the total number of waste canisters in the array. For the case that nuclide is graduallyreleased from waste canisters over a duration of time, TL [yr], the overall concentration is obtainedby superposing the concentration profiles separately obtained for the nuclide released at each timestep. The concentration C at time step k is obtained as
C =
k∑κ=0
Nx Nyml2h
a(k−κ)n , k < kL ,
kL−1∑κ=0
Nx Nyml2h
a(k−κ)n , k ≥ kL ,
(6.46)
wherem =
M◦
kLand kL =
TL
1t=
τL
1τ.
In Eq. (6.46), κ can be interpreted as the time step that mass m of the nuclide is released from eachcanister, and kL the duration of release in terms of time steps. The dimensionless concentration χ
corresponding to Eq. (6.46) is
χ =
L2
M◦
k∑κ=0
Nx Nyml2
a(k−κ)n , k < kL ,
L2
M◦
kL−1∑κ=0
Nx Nyml2
a(k−κ)n , k ≥ kL .
(6.47)
100
6.6.2 Observation of Exit Concentration
Figures 6.12, 6.13, and 6.14 show the dimensionless concentration χ in the compartments Aand B specified in Figure 6.11 as functions of dimensionless time τ . Figure 6.12 shows the resultswhen the groundwater is flowing exactly parallel to the x axis (θ = 0◦). Figures 6.13 and 6.14 showthe concentration at A and B, respectively, when θ = 30◦. For all calculations, parameter values1τ = 1/6, Pe = 12, n0 = 3, and τL = 5 are used. The concentration profiles are plotted for variousvalues of Nx , while the value of Ny is fixed to 10. When Nx = 2, 4, 6, and 8, the waste canistersare placed only in the Nx black compartments on the right part of the 30 × 30 array in Figure 6.11.
For the case with groundwater flow direction parallel to x axis, the concentration profiles ob-served at compartment A and B are the same because of geometric symmetry. In Figure 6.12,the concentration increases in the early times as the nuclide is released from waste canisters. ForNx = 2, the concentration levels off at τ = 2 when the advection front of water reaches the down-stream exit of the array.5 The concentration for Nx = 2 rapidly decreases after the release hasstopped at τ = τL = 5. The similar profile is observed for Nx = 4 with a higher peak concentra-tion. These peak concentrations for Nx < τL correspond to the steady-state concentration observedand formulated as Eq. (3.24).
For Nx = 8, and 10, concentration plateau is observed after the nuclide release stops at τL .Because Nx > τL for these cases, nuclide release from canisters completes before the advectionfront reaches the repository exit. The dimensionless plateau concentration is χ = 1, which isthe concentration when the initial mass M◦ of the nuclide in a single canister is spread uniformlyover the surrounding 9 compartments. This plateau concentration corresponds to the upper-boundconcentration formulated as Eq. (3.25).
The sharp peaks observed at τ = τL are due to the fact that the concentration is not uniformnear the canister releasing the nuclide.
When groundwater flow direction is tilted from the canister array (θ = 30◦), the length of thegroundwater stream within the repository changes.
As illustrated in Figure 6.11, the water stream that passes compartment A only intersects 4 or5 rows of canisters. Since the left half of the canister rows have little effect on the stream, nosignificant difference is observed among profiles for nx = 6, 8, and 10 in Figure 6.13.
On the other hand, the streamline that passes compartment B intersects all 10 rows of canisters.The plateau concentrations for nx = 6, 8, and 10 are observed in Figure 6.14. Because this stream-line is longer than that for θ = 0 by factor of 1/ cos 30◦
= 2/√
3, the migration time for this streamis also longer by the same factor. This is observed as the length of the plateau period (see Figure6.15) since the end of the plateau is determined by the migration time within the array.
The above observations imply that the peak exit concentration of a stream that runs throughthe canister array is determined by the migration time along the stream and the duration TL ofnuclide release from canisters. If the migration time is longer than TL , the peak exit concentrationof that specific stream becomes the upper-bound concentration regardless of the direction of theflow relative to canister array.
Another effect of the tilted water flow is potential increase in the cross-sectional area of the water
5In this dimensionless system, the time spent for migration over length L [m] in the flow direction is 1. Thus, time Nxis spent for the nuclide to pass by Nx waste canisters, when groundwater is flowing parallel to the array.
101
0.0
0.2
0.4
0.6
0.8
1.0
1.2
0 5 10 15 20
Dim
ensi
onle
ss c
once
ntra
tion,
χ
Dimensionless time, τ
Compartment A and Bθ = 0°
Nx=2
4 6 8 Nx=10
τL
Figure 6.12 Dimensionless concentration at compartment A and B when water flow direction isparallel to x axis (θ = 0◦). The concentration profiles in compartments A and B are same becauseof symmetry.
0.0
0.2
0.4
0.6
0.8
1.0
1.2
0 5 10 15 20
Dim
ensi
onle
ss c
once
ntra
tion,
χ
Dimensionless time, τ
Compartment Aθ = 30°
Nx=2
4 Nx=6,8,10
τL
Figure 6.13 Dimensionless concentration atcompartment A when water flow direction istilted by θ = 30◦.
0.0
0.2
0.4
0.6
0.8
1.0
1.2
0 5 10 15 20
Dim
ensi
onle
ss c
once
ntra
tion,
χ
Dimensionless time, τ
Compartment Bθ = 30°
Nx=2
4 6 8 Nx=10
τL
Figure 6.14 Dimensionless concentration atcompartment B when water flow direction istilted by θ = 30◦.
102
stream running through the repository. The overall release rate of a nuclide from the repository mayincrease because of the increased cross-section.
6.7 Conclusion
The model for particle migration at multiple scales has been developed using the Markov chainprobability model. The goal of the present model is to enable analyses of radionuclide migration atthe repository scale based on the information obtained from a smaller-scale detailed analysis.
In Illustration 1, nuclide transport in a repository of 10 × 10 waste packages has been demon-strated. The transition probabilities based on random connectivities were assigned to simulate thenuclide transport through the heterogeneous medium with clusters of fractures.
In the comparison with the analytical continuum model of mass transport, the results from thepresent model showed a good agreement.
The concept of this model is not limited to two-dimensional arrays but can also be applied toone-dimensional and three-dimensional arrays of compartments. It is also applicable to systems atvarious scales. For example, the present model has been applied to the scale of 10 × 10 array ofcanisters in a repository in Illustration 1, while it has been applied to the scale of domain around asingle waste canister in Illustration 3.
The upper-bound concentration discussed in Chapter 3 has been reobserved with the presentmodel in two-dimensional array. The groundwater flow direction tilted with respect to the canisterarray would affect the length of streamlines and hence the migration time within the canister array.
0.0
0.2
0.4
0.6
0.8
1.0
1.2
0 5 10 15 20
Dim
ensi
onle
ss c
once
ntra
tion,
χ
Dimensionless time, τ
Compartment BNx=10
θ=0° θ=30°
Figure 6.15 Comparison of dimensionless concentrations at compartment B between θ = 0◦ andθ = 30◦.
103
As the result, the peak exit concentration varies depending on the position of the compartment.In the future work, the transition probabilities should be obtained from the results of analyses by
other models, such as FFDF, at a smaller-scale medium. In order to acquire meaningful results withthe present model, it is essential to properly convert from the results obtained in such smaller-scaleanalyses by other models into the transition probabilities.
The size of a compartment and the time step interval are closely related to and constrained by theeffects of diffusion and dispersion of the nuclide (e.g., Eq. (6.29)). This means that a compartmentsize and a time step used for transport analysis of one nuclide may not be suitable for anothernuclide with a different diffusion coefficient. One way of avoiding this constraint is to consider amodel based on a distribution of residence time of each nuclide in each compartment as discussedin the next chapter.
104
CHAPTER 7
DISTRIBUTIONS OF RESIDENCE TIMES IN THE
COMPARTMENT MODEL
7.1 Introduction
In previous chapters, the effects of the canister array configuration and the footprint size of therepository were observed with compartment models. Mean residence times in compartments werefound to be the key parameters for characterizing migration of radionuclides and their release rates.For example, in Chapter 2, effects of vault array configuration on the radionuclide release rate at theGBI (geosphere/biosphere interface) was discussed in conjunction with the mean residence time inthe unsaturated zone, the mean residence time in an aquifer compartment, and the mean lifetime ofthe radionuclide. In Chapter 3, the release of a radionuclide from the array of compartments wascharacterized by the waste matrix leach time and the mean migration time across the length of therepository.
In this chapter, further observations of the residence times and waste multiplicity are made bydescribing the particle migration with a continuous-time stochastic process. Each compartment isassigned a residence time distribution (RTD) of a radionuclide. The release rate of the nuclide fromthe array of compartments is obtained in terms of the distribution of the total time spent in the array.
In Chapter 2, the compartment size in the aquifer was determined by the dispersivity of theadvective flow. This was due to the dispersion effect produced by the instantaneous mixing ofradionuclide in a compartment and the sequence of such compartments. In the model developedin this chapter, the dispersion effect is included in the RTD, and hence the compartment size isarbitrarily assigned without constraints.
7.2 Framework of the Stochastic Model
Migration of particles is considered in a single array of N compartments arranged parallel to thegroundwater flow direction. The waste form in each compartment initially contains M◦ particles.Each particle represents an atom of a radionuclide. Although the intended application of this studyis the analysis of radionuclide migration, the radioactive decay is neglected and the particle is treatedas stable.
Migration of a particle is described by location of the particle as a function of time. By consid-ering migration of a number of particles, the release rate of the nuclide is obtained.
105
Figure 7.1 Array of compartments and the migration of a single particle. Migration of the particlestarts in compartment n = 1. The particle migrates through N ′ compartments and is releasedfrom the end of the array. By assumption, the particle does not migrate through the upstreamcompartments (dashed lines).
Figure 7.2 Relationship among the particle position index X (t), the time of transition Tn , and theresidence time Un .
106
7.2.1 Migration of a Single Particle
First, migration of a single particle is described in this section. The migration of a particle in thecompartment array is depicted in Figure 7.1. Compartment indices are assigned in such a mannerthat index n = 1 is assigned to the compartment that initially contains the particle, and n = N ′
to the compartment at the downstream end of the array. The compartments in the upstream side ofcompartment n = 1 (drawn in dashed lines in Figure 7.1) have no effect on the migration becausethe particle does not migrate through these compartments by assumption.
The compartment in which the particle resides at time t is denoted by index n = X (t). Theparticle is initially contained in compartment n = 1, and therefore X (0) = 1. At time t = 0, thewaste forms start to degrade because of water contact. After t = 0, the particle spends some time inthe waste form according to the gradual degradation of the waste form. Once it is released from thewaste form, it migrates through the buffer region and the NFR region, and eventually leaves the firstcompartment and enters the second compartment. In the second compartment and thereafter, theparticle moves in the NFR region and the buffer region. The particle will not reenter waste regionsonce it is released from a waste form. It is assumed that the particle movement from compartment nis always made to the adjacent downstream compartment n+1. Thus, X (t) increments by one everytime the particle makes a transition from one compartment to another. Therefore, {X (t) − 1, t ≥ 0}
forms a counting process that represents the number of transitions made by time t . Figure 7.2illustrates X (t) as a function of time.
Let Tn denote the time at which the particle moves from compartment n to n + 1, and Un thetime that the particle spends in compartment n. The quantity Un is referred to as the residence timeof the particle in compartment n. The relationship between Tn and Un is written as
T1 = U1 and Tn = Tn−1 + Un , n = 2, 3, . . . , N ′ , (7.1)
or
Tn =
n∑n′=1
Un′ , n = 1, 2, . . . , N ′ . (7.2)
The relationship among X (t), Tn , and Un is shown in Figure 7.2.Let residence time Un be a positive continuous random variable and its distribution be prescribed
by a PDF fUn . It is assumed that the residence times U1, U2, . . . , UN ′ are mutually independent. Itfollows that Tn−1 (= U1 + · · · + Un−1) and Un are also independent (n = 2, 3, . . . , N ′).
Since we are interested in the release of the particle from the array, focus is put on obtaining thedistribution of TN ′ , i.e., the time that the particle is released from the array of compartments. Therelationship between the distribution of TN ′ and the release rate of the nuclide from the array willbe discussed in Section 7.2.2.
By the first equality in Eq. (7.1),
fT1(t) = fU1(t) . (7.3)
Considering the second equality of Eq. (7.1) and the mutual independence of Tn−1 and Un , the PDFfTn for Tn is obtained by
fTn (t) = ( fTn−1 ∗ fUn )(t) =
∫ t
0fTn−1(t − u) fUn (u)du , t > 0 , n = 2, 3, . . . , N ′ . (7.4)
107
where operator ∗ denotes convolution of the two PDFs. Considering Eq. (7.2), PDF fTn can berewritten in terms of the prescribed PDFs fU1 , fU2 , . . . , fUn as
fTn (t) = ( fU1 ∗ fU2 ∗ · · · ∗ fUn )(t) , t > 0 , n = 1, 2, . . . , N ′ . (7.5)
For compartments n = 2, 3, . . . , N ′, the particle enter each compartment through the upstreamboundary and leave the compartment through the downstream boundary. There is no time spent inthe waste form in those compartments. Since all the compartments are identical, one can assumethat the residence times U2, U3, . . . , and Un are identically distributed. On the other hand, U1 has adifferent distribution because of the time spent in the waste form in the original compartment. Letg0(u) denote the PDF for U1 and g(u) the PDF for U2, U3, . . . , and UN ′ , i.e.,
fU1(u) = g0(u) and fU2(u) = fU3(u) = · · · = fUN (u) = g(u) . (7.6)
Equation (7.5) is rewritten in terms of g0 and g as
fTn (t) = (g0 ∗ g ∗ · · · ∗ g︸ ︷︷ ︸n−1
)(t) , t > 0 , n = 1, 2, . . . , N ′ . (7.7)
Let Yn(t) denote the indicator random variable that describes existence of the particle in com-partments 1 through n at time t ; Yn(t) = 1 if the particle is located in either one of compartments 1through n at time t , and Yn(t) = 0 otherwise. The particle position X (t) at time t is at most n if andonly if the transition from compartment n to n + 1 occurs after time t (i.e., Tn > t). Therefore,
Yn(t) =
{1 if X (t) ≤ n (i.e., Tn > t) ,
0 if X (t) > n (i.e., Tn ≤ t) .(7.8)
The probability of having Yn(t) = 1, i.e., the probability of finding the particle in either one ofcompartments 1 through n at time t , is
P{Yn(t) = 1} = P{Tn > t} = 1 − P{Tn ≤ t} = 1 − FTn (t) , (7.9)
where FTn is the cumulative distribution function (CDF) of Tn and is obtained as
FTn (t) =
∫ t
0fTn (t
′)dt ′ , t ≥ 0 . (7.10)
Since Yn(t) is a Bernoulli random variable, its mean is given as the probability P{Yn(t) = 1}, andtherefore
E [Yn(t)] = 1 − FTn (t) , (7.11)
where E[·] denotes the expectation of the expression.
108
7.2.2 Migration of Multiple Particles
Consider M◦ particles of a nuclide initially contained in compartment n = 1. Each particle isassumed to independently migrate through the compartments according to the process discussed inSection 7.2.1. Thus, there are separate values of Un and Tn for each particle. It is assumed thatPDFs fUn and fTn are common for all particles.
It is assumed that migration process of a particle is not affected by existence of other particlesin the system. Therefore, the residence time of a particle is independent of the residence times ofthe other particles. While this assumption is indispensable for the further development of the modeldescribed below, it limits the applicability of the model. For example, this model is not applicableto solubility-limited release of the nuclide because the solubility-limited release of particles fromthe waste form is controlled by the particles near the surface of the waste form. Precipitation of thenuclide cannot be taken into account because it is based on the interaction between particles. Thepresent model is applicable to the system where the nuclide concentration is kept lower than thesolubility and interaction between particles is negligible.
Let an indicator variable Y (i)n (t) correspond to Yn(t) defined by Eq. (7.8) with superscript (i)
denoting the i’th particle. The total number of particles in compartments 1 through n at time t canbe obtained by summation of the indicator variables Y (i)
n over all M◦ particles:
n∑n′=1
Mn′(t) =
M◦∑i=1
Y (i)n (t) , (7.12)
where Mn′(t) is the number of particles in compartment n′ at time t . Assuming that the initialnumber of particles M◦ is the order of the magnitude of Avogadro’s number, by Eqs. (7.12) and(7.11),
1M◦
n∑n′=1
Mn′(t) =1
M◦
M◦∑i=1
Y (i)n (t) ≈ E
[Y (i)
n (t)]
= 1 − FTn (t) . (7.13)
The weak law of large numbers has been applied to the approximation between the sample meanof Y (i)
n (t) and the true mean of Y (i)n (t) in Eq. (7.13). The weak law of large numbers applied for
Y (i)n (t) is formally written as [45]
limm→∞
P
{∣∣∣∣∣ 1m
m∑i=1
Y (i)n (t) − E
[Y (i)
n (t)]∣∣∣∣∣ ≥ c
}= 0 , c > 0 , (7.14)
or, informally,1m
m∑i=1
Y (i)n (t) ≈ E
[Y (i)
n (t)]
for m � 1 , (7.15)
where m is the total number of independent particles in the system. The approximation in Eq. (7.13)is obtained by substituting m with M◦ in Eq. (7.15).
109
The fractional release rate of the particles at the downstream end of the array is obtained as thefractional rate of decrease in the number of particles in the array;
Fractional release rate = −1
M◦
ddt
N ′∑n=1
Mn(t)
= −ddt
[1 − FTN ′ (t)] = fTN ′ (t) . (7.16)
Thus, for the particles initially contained in compartment n = 1, the fractional release rate at theend of the array is given by fTN ′ , i.e., the PDF for the time of release from the last compartment N ′.
The number of particles in the single compartment n is derived from Eq. (7.13);
Mn(t)M◦
=1
M◦
[n∑
n′=1
Mn′(t) −
n−1∑n′=1
Mn′(t)
]= FTn−1 − FTn . (7.17)
7.2.3 Release from Multiple Waste Forms
For migration of particles from multiple waste forms in the array, the nuclide streams fromsingle waste forms are superposed on each other. The superposition is depicted in Figure 7.3. Forthe array of N compartments, the number N ′ of compartments that particles migrate through variesfrom 1 to N depending on the initial position of the particle.
Figure 7.3(a) depicts the migration of particles from the first waste form in the array. For theseparticles, N ′
= N . For the particles initially contained in the second waste form of the array,N ′
= N − 1 (Figure 7.3(b)). The PDF g0 is used for the compartment n = 1, and g for the N ′− 1
downstream compartments. For particles from each waste form, the release rate at the end of thearray is written as M◦ fTN ′ (t).
The release rate of the particles from all the waste forms in the array is obtained as the sum ofthese release rates of particles from individual waste forms, i.e.,
N∑N ′=1
M◦ fTN ′ (t) . (7.18)
The release rate φ◦
N (t) normalized to the initial number M◦ of particles per canister is
φ◦
N (t) =
N∑N ′=1
fTN ′ (t) =
N∑N ′=1
(g0 ∗ g ∗ · · · ∗ g︸ ︷︷ ︸N ′−1
)(t) , (7.19)
where Eq. (7.7) has been applied.Given the PDF g0 for the residence time in compartment n = 1 and g for the residence times in
the downstream compartments, the normalized release rate φ◦
N (t) can be thus obtained. As shownlater in the illustrations, g0 and g can be assigned as functions or can be obtained from other studieson nuclide migration within a single compartment.
Other studies on nuclide migration inside a single compartment can be used to obtain g0 and g.In such studies, the multiplicity of compartments need not be considered to obtain g0 and g.
On the other hand, once g0 and g are obtained, the present model focuses on the multiplicity ofthe compartments. All the necessary information about the compartments is contained in g0 and g.
110
7.3 Illustration 1
In this section, the release rate of cesium from an array of waste canisters is calculated byassigning RTDs to compartments. Groundwater flow parallel to the array of the canisters is assumed.The effect of the canister array configuration is observed again in this illustration. The results arecompared with the one obtained from VR calculation discussed in Chapter 3.
7.3.1 RTD for the Downstream Compartments
In Chapter 2, it was assumed that a radionuclide in a compartment is immediately mixed becauseof dispersion effect and that the concentration in each compartment is maintained uniform. For sucha compartment, the residence time is known to be exponentially distributed [47] (see Appendix A).
Consider a compartment that contains the NFR region and the buffer region such as the onesin Chapter 3. While the uniform concentration is assumed in the NFR region, the concentration is
Figure 7.3 Concept of superposition of nuclide streams from multiple waste forms. Migration ofparticles from each waste canister is depicted in (a) through (e). The total release rate of particlesfrom all the waste forms is obtained as the sum of the release rates of particles from individual wasteforms (shown as (f)). The filled circles represent the waste forms that initially contains the nuclide.Note that the compartments (dashed-line boxes) on the upstream side of the original compartmenthave no effect on particle migration because particles are assumed to migrate to the downstreamcompartments.
111
not necessarily uniform in the buffer region. However, since the majority of the time in the com-partment is spent in the NFR region, one can assume that the RTD is approximately the exponentialdistribution with the mean residence time defined in Eq. (3.19). In case the time spent in the bufferbecomes longer, the residence time becomes more spread out.
Considering the above discussions, the gamma distribution with rate parameter 3 [yr−1] andshape parameter α is assigned to the residence times in compartments n = 2, 3, . . . , N ′;
g(u) =3αuα−1
0(α)e−3u , u > 0 . (7.20)
The rate parameter 3 is assigned so that the mean is equal to the mean residence time in a singlecompartment defined by Eq. (3.19) in Chapter 3. With parameter values used for cesium in Chapter3,
E [Un] =α
3= 1370 yr , n = 2, 3, . . . , N ′ . (7.21)
The exponential distribution with rate parameter 3 is a special case of the gamma distributionobtained by assigning α = 1.
7.3.2 RTD for the First Compartment
In the first compartment, the residence time U1 is considered as the sum of the residence timesU1,a in the waste form and U1,b in the rest of the compartment including the buffer region and theNFR region;
U1 = U1,a + U1,b . (7.22)
The uniform distribution on time interval 0 < u ≤ TL is assigned to U1,a .
fU1,a (u) =
{1/TL , 0 < u ≤ TL
0 , u > TL .(7.23)
This corresponds to the uniform degradation rate of the waste matrix assumed in Section 3.2.2 withleach time TL [yr].
The gamma distribution with rate parameter 3 and shape parameter α is assigned to the resi-dence time U1,b in the buffer and the NFR.
fU1,b(u) =3αuα−1
0(α)e−3u , t > 0 . (7.24)
The same rate constant 3 as defined for U2, U3, etc. is assumed.Assuming that U1,a and U1,b are mutually independent, the PDF of U1 is obtained by convolution
integral of fU1,a and fU1,b as
fU1(u) = g0(u) = ( fU1,a ∗ fU1,b)(u) . (7.25)
112
7.3.3 Fractional Release Rate from a Single Waste Form
Substituting g0 in Eq. (7.7) with formula (7.25), and considering that fU1,b = g,
fTN ′ (t) = ( fU1,a ∗ fU1,b ∗ g ∗ · · · ∗ g︸ ︷︷ ︸N ′−1
)(t)
= ( fU1,a ∗ g ∗ · · · ∗ g︸ ︷︷ ︸N ′
)(t) , t > 0 , N ′= 1, 2, . . . , N . (7.26)
Since U1,b, U2, . . . , UN ′ are independent and have the gamma distribution with the common rateparameter 3, the sum U1,b + U2 + · · · + UN ′ has the gamma distribution with rate parameter 3 andshape parameter N ′α, and g ∗ · · · ∗ g is obtained as
fU1,b+···+UN ′ (u) = (g ∗ · · · ∗ g︸ ︷︷ ︸N ′
)(u) =3N ′αuN ′α−1
0(N ′α)e−3u , u > 0 , N ′
= 1, 2, . . . , N . (7.27)
Substituting Eqs. (7.27) and (7.23) into Eq. (7.26), the fractional release rate at the end of the array
0.0000
0.0001
0.0002
0 10000 20000 30000
Frac
tiona
l rel
ease
rat
e, f T
N′(t
) [1
/yr]
Time, t [yr]
1 2 3 … 16N′ =
Figure 7.4 Fractional release rate fTN ′ (t) ofcesium initially contained in a single compart-ment based on Eq. (7.28) for N ′
= 1, 2, . . . , 16.Parameters TL = 104 yr, 3 = 7.3 × 10−4 yr−1,and α = 1 are used.
10-5
10-4
10-3
102 103 104 105 106
Nor
mal
ized
rel
ease
rat
e, φ
o N(t
) [1
/yr]
Time, t [yr]
N = 1
2
4
816 32 64
Present ModelVR
Figure 7.5 Normalized release rate φ◦
N (t) ofcesium from the entire array. The normalizedrelease rates based on Eq. (7.31) (solid lines)and that obtained from VR calculation (dashedlines) are shown together. N = 1, 2, 4, 8, 16,32, and 64. Parameters TL = 104 yr, 3 = 7.3 ×
10−4 yr−1, and α = 1 are used.
113
is obtained as
fTN ′ (t) =
∫ t
0fT1,a (t − u)
(3N ′αuN ′α−1
0(N ′α)e−3u
)du
=
1TL
P(N ′α, 3t) , 0 < t ≤ TL ,
1TL
[P(N ′α, 3t) − P(N ′α, 3(t − TL))
], t > TL ,
(7.28)
where P(a, b) is the regularized gamma function and is defined as [48]
P(a, b) ≡γ (a, b)
0(a), γ (a, b) ≡
∫ b
0τ a−1e−τ dτ . (7.29)
When a is a positive integer,
P(a, b) = 1 −
a∑n=1
bn−1
(n − 1)!e−b . (7.30)
Function γ (a, b) is called the lower incomplete gamma function.Figure 7.4 shows the fractional release rate fTN ′ (t) for N ′
= 1, 2, . . . , 16. The shape parameterα = 1, the rate parameter 3 = 7.3 × 10−4 yr−1, and the leach time TL = 104 yr is assumed in thefigure. Thus, mean residence times in compartments 2, 3, . . . , N ′ are α/3 = 1370 yr.
The effect of the canister position on the migration time in the repository is clearly observed.For a greater value of N ′, the release is observed later and more spread out over time. At 14000 yr,most of the cesium from the canister at the downstream end of the repository (N ′
= 1) has beenreleased from the repository, while only a little fraction of cesium has been released for N ′
= 16.Since the total release rate from the repository is the superposition of these individual release rates,the peak release rate would increase with the number of canisters in the array. However, the it stopsincreasing because the time frame of release is different for the nuclide from different canisters.
7.3.4 Release Rate from the Entire Array
By substituting Eq. (7.28) into Eq. (7.19), the normalized release rate of particles from the entirearray with N waste forms is obtained as
φ◦
N (t) =
1TL
N∑N ′=1
P(N ′α, 3t) , 0 < t ≤ TL ,
1TL
N∑N ′=1
[P(N ′α, 3t) − P(N ′α, 3(t − TL))
], t > TL .
(7.31)
Figure 7.5 shows the normalized release rates φ◦
N (t) for N = 1, 2, 4, 8, 16, 32, and 64. Thenormalized release rates obtained from VR calculation in Chapter 3 are plotted together in the figure
114
for comparison. In the VR calculation, parameter values for cesium shown in Table 3.1 are used.The cesium is assumed to be a stable nuclide in the VR calculation.
In VR model, the normalized release rate is defined as
φ◦
N (t) = JNx ,1(t)/M◦ , (7.32)
where JNx ,1(t) [mol/yr] is the release rate of the nuclide from a single array of Nx compartments(defined by Eq. (3.17)), M◦ [mol] is the initial mass of cesium in a single waste form.
For all N values, the release rate φ◦
N (t) overlaps with the VR result for t > 104 yr. The plateauthat is observed in the VR calculation is observed in the release rate in the present model as well.
Before 104 yr, however, some discrepancy between the present model and the VR model isobserved. Greater release rate is observed for the present model before 104 yr. This greater releaserate in the early times is due to the assumed RTD for U1,b, the residence time in the buffer and inthe NFR immediately after the particle is released from a waste form. The residence time U1,b is thetime spent for the migration through the buffer and then for the NFR region. However, the assumedprofile for PDF fU1,b(t) is the time spent after the particle once enter the NFR region. Therefore,U1,b is estimated somewhat shorter, and the overall RTD in the original compartment is shifted.
7.4 Illustration 2
In Section 7.3, discrepancy between the VR results and the result from the present model hasbeen observed because of the assumed profile g0. In the second illustration of the stochastic modelframework, the numerical data of the fractional release rate obtained directly from the VR calcula-tion result is used as PDF g0.
7.4.1 Assignment of Residence Time Distributions
The fractional release rate of cesium from a single compartment is obtained from VR calculation(discussed in Chapter 3) with the parameter values shown in Table 3.1. By discussion of Eq. (7.16),the fractional release rate function is equivalent to the PDF g0 for the residence time U1 in thecompartment that the particle is initially contained. Therefore, this fractional release rate functionis used as the PDF g0. The profile obtained from VR is shown as fT1(t) in Figure 7.6.
As in the previous section, gamma distribution is used as the RTD for the downstream compart-ments n = 2, 3, . . . , N ′. The sum U2 + · · · + UN ′ has the gamma distribution with shape parameterN ′α − 1 and rate parameter 3. The fractional release rate fTN ′ (t) is obtained as the convolution ofg0 and the PDF for this sum, and thus
fTN ′ (t) = (g0 ∗ g ∗ · · · ∗ g︸ ︷︷ ︸N ′−1
)(t)
=
∫ t
0g0(t − u)
(3N ′αuN ′α−1
0(N ′α)e−3u
)du , t > 0 , N ′
= 1, 2, . . . , N . (7.33)
Since g0 is obtained from VR as tabulated data, the integration above is performed numericallyusing the trapezoidal rule [49].
115
7.4.2 Numerical Results
Figure 7.6 shows the fractional release rate fTN ′ (t) for N ′= 1, 2, . . . , 16. The rate parameter
3 = 7.3 × 10−4 yr−1 and the shape parameter α = 1 is used. As mentioned earlier, the release rateprofile for N ′
= 1 is obtained from VR calculation. Difference between this profile for N ′= 1 and
that in Figure 7.4 is observed during the early times (t < 1000 yr). As observed in Figure 7.4, therelease for a greater value of N ′ is observed later and spread out over time.
Figure 7.7 shows the normalized release rate φ◦
N (t) from the entire array for N = 1, 2, 4, 8, 16,32, and 64. The normalized release rates obtained from VR calculation are also plotted in the figure.For all values of N , the release rate profile obtained with the present stochastic model overlaps withthe profile obtained with VR model.
This confirms that the RTD for compartments n = 2, 3, . . . , N ′ can be represented by thegamma distribution with parameters α = 1 and 3 = 7.3 × 10−4 yr−1. This means that the NFR andthe buffer in such a compartment can be treated as a single compartment with uniform concentration.The significance of the buffer region is to increase the residence time for the particles migratingthrough the compartment.
Figure 7.8 and Figure 7.9 show the profiles of fTN ′ (t) and φ◦
N (t), respectively, when the pa-rameters α = 16 and 3 = 1.2 × 10−2 yr−1 are used. With these parameter values, the mean
0.0000
0.0001
0.0002
0 10000 20000 30000
Frac
tiona
l rel
ease
rate
, fT N
′(t) [1
/yr]
Time, t [yr]
1 2 3 … 16N′ =
Figure 7.6 Fractional release rate fTN ′ (t) ofcesium initially contained in a single compart-ment based on Eq. (7.33) for N ′
= 1, 2, . . . , 16.Parameters 3 = 7.3 × 10−4 yr−1 and α = 1 areused.
10-5
10-4
10-3
102 103 104 105 106
Nor
mal
ized
rel
ease
rat
e, φ
o N(t
) [1
/yr]
Time, t [yr]
N = 1
2
4
816 32 64
Present ModelVR
Figure 7.7 Normalized release rate φ◦
N (t) ofcesium from the entire array. The normal-ized release rates based on the stochastic model(solid lines) and that obtained from VR calcula-tion (dashed lines) are shown together. N = 1,2, 4, 8, 16, 32, and 64. Parameters 3 =
7.3 × 10−4 yr−1 and α = 1 are used.
116
residence time in a downstream compartment is the same as that for Figure 7.6 and Figure 7.7 (i.e.,α/3 = 1730 yr) while its standard deviation is smaller by the factor of 4 (
√α/3 = 342 yr).
Because of the smaller value of the standard deviation, the fractional release rate fT16(t) forN ′
= 16 in Figure 7.8, for example, is not as spread out over time as that shown in Figure 7.6. It isalso observed in Figure 7.8 that the peak of fT16 is as high as that of fT1 , whereas in Figure 7.6, thepeak level decreases with N ′.
In contrast to fTN ′ , the peak levels of normalized release rate φ◦
N (t) from the entire array is stillobserved to match those obtained by VR. Because of the smaller standard deviation for the RTD,the release rate is observed to decrease more rapidly after the plateau.
7.5 Discussion
In Section 7.4, PDF g0 has been obtained by VR calculation, and an gamma distribution hasbeen assigned to g. However, the PDF g is not limited to gamma distributions. The PDF g forthe downstream compartments can be obtained by a separate analysis on nuclide migration within asingle compartment and be applied to the present framework. In such a case, the interior structureof the compartment may be more complex than what has been considered in Chapter 2 and Chapter
0.0000
0.0001
0.0002
0 10000 20000 30000
Frac
tiona
l rel
ease
rate
, fT N
′(t) [1
/yr]
Time, t [yr]
1 2 3 … 16N′ =
Figure 7.8 Fractional release rate fTN ′ (t) ofcesium initially contained in a single compart-ment based on Eq. (7.33) for N ′
= 1, 2, . . . , 16.Parameters 3 = 1.2×10−2 yr−1 and α = 16 areused.
10-5
10-4
10-3
102 103 104 105 106
Nor
mal
ized
rel
ease
rat
e, φ
o N(t
) [1
/yr]
Time, t [yr]
N = 1
2
4
816 32 64
Present ModelVR
Figure 7.9 Normalized release rate φ◦
N (t) ofcesium from the entire array. The normal-ized release rates based on the stochastic model(solid lines) and that obtained from VR calcula-tion (dashed lines) are shown together. N = 1,2, 4, 8, 16, 32, and 64. Parameters 3 =
1.2 × 10−2 yr−1 and α = 16 are used.
117
3. In order to obtain g0 and g, we only need such separate analyses for the single-compartmentconfiguration. In such an analysis, the multiplicity of the compartments need not be considered.Once the PDFs g0 and g are obtained from the separate analysis, the same methodology of thepresent model can be applied regardless of the complexity inside a compartment. All necessaryinformation of the compartments is confined in g0 and g.
In Chapter 2 and Chapter 3, the size of a compartment determines the mean residence time andthe magnitude of dispersion effect. In the model discussed in Chapter 2, in order to reproduce thedispersion effects, the length of an aquifer compartment was determined according to the disper-sivity. In Chapter 3, the size of a compartment has been determined by the distance between twoadjacent waste canisters in the array.
For the model in this chapter, the size of a compartment is not restricted in such a way be-cause the dispersion effect is contained in the assigned RTDs for each compartment. The size of acompartment can be assigned independently of the dispersion property.
As assumed for the weak law of large numbers in Section 7.2.2, it is required that particlesdo not interfere with each other. It is safe to make such an assumption for congruently releasednuclides. On the other hand, in case of solubility-limited release, the release rate of a nuclide isbased on existence of atoms interfering with each other around the waste surface. Therefore thepresent model may not be applicable to migration of the nuclides with low solubility, whose releasefrom waste forms is limited by the solubility.
7.6 Conclusion
A compartment model for particle migration has been developed in this chapter by consideringa residence time distribution (RTD) for each compartment.
Fractional release rate of cesium from a compartment array is calculated and compared to theresults obtained in Chapter 3. The following observations are made regarding the compartmentmodels.
With the present stochastic model, the plateau observed in the release rate of cesium obtainedby VR calculation is also observed and showed a good agreement.
The assumption of the instantaneous mixing and the uniform concentration in a compartment inChapter 2 is equivalent to the exponential distribution of the residence time. It was also observed thatthe RTD in the downstream compartments in VR model (Chapter 3) is approximately exponentialdistribution. With the parameter values used for cesium, the existence of the buffer region in VRmodel has relatively small effects on the assumption of the exponential distribution.
In the present model, the information about migration of particles within a compartment witha complex structure is confined in the RTDs. If the interior structure of a compartment is changed,the RTD for the compartment changes accordingly. The dispersion effect is also contained in theRTDs. Therefore, the size of a compartment is not constrained by the dispersion property.
The present stochastic approach is applied to independent particles because of the usage of theweak law of large numbers. The model is not applicable if particles interfere with each other. Forthis reason, the solubility-limited release of radionuclides cannot be incorporated by the presentmodel.
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CHAPTER 8
APPLICABILITY AND LIMITATIONS OF THE
MODELS
Number of assumptions have been made in the analyses in this dissertation. Some of the as-sumptions, such as the uniform degradation of waste matrices and straight groundwater flow streamalong canister arrays, have been made in order to focus on the effects of canister array configurationand the repository footprint. The hypothetical situations considered in the analyses, however, canalso be found in a real repository. In this chapter, applicability and the limitations of the modelsdeveloped in this dissertation are discussed.
8.1 Streamline of Groundwater in the Repository
In Chapter 2, the water in the aquifer has been assumed to flow exactly parallel to or perpen-dicular to the vault array. In Chapters 3 and 4, groundwater in the NFR has been assumed to flowexactly parallel to the row of waste canisters. Although it is considered that the groundwater flowspreferentially through the disturbed zone of the host rock due to excavation of disposal tunnels, de-viation from the straight streamline parallel to the canister array would occur because the mediumis heterogeneous.
To investigate the effects of deviation of the water streamline from the canister array, the Markovchain model in Chapter 6 in the two dimensional domain has been developed (see Section 6.6).If a groundwater streamline on the repository plane forms an angle with the canister arrays, theupper-bound concentration of congruently released nuclide is still observed, and the peak exit con-centration can be characterized by the leach time and the length of the streamline in the repository.The upper-bound concentration does not depend on the direction of the groundwater flow. Thisobservation suggests that the peak concentration at the end of the each streamline can be formulatedby Eqs. (3.24), (3.25), and (4.22) even when the water flow direction is not parallel to the canisterarrays, by replacing Nx with the ratio between the length of the streamline and the length of a singlecompartment.
If the streamline is tilted vertically off the plane of the canister array1 as shown in Figure 8.1,the similar discussion applies. In the figure, the groundwater flow direction is indicated by an arrow,which intersects with the repository by angle θ . With the assumed height h of the repository, the
1This situation can occur if the repository is built in a sedimentary rock region, which is considered to have multiplelayers with different properties. Groundwater tend to flow through the layers with higher permeability. Because this layerstructure is frequently tilted from the horizontal plane, groundwater also tend to flow in a tilted direction.
119
length l of the streamline that intersects with the repository is
l =h
sin θ. (8.1)
In such a case, the concentration at the end of the stream is approximately formulated by Eqs. (3.24),(3.25), and (4.22), with Nx being replaced with
l/d =h
d sin θ, (8.2)
where d is the length of a single compartment. When the height of the repository is defined ash ≡ d, Eq. (8.2) becomes 1/ sin θ . In such a case, when θ = 90◦, the configuration becomes thesame as Nx = 1 (i.e., configuration B in Table 1.1).
This discussion can be further extended to a curved water stream. Since the exit concentrationis determined by the length of the streamline that intersects the repository, the direction of the waterflow at each point of the stream should not affect the exit concentration.
8.2 Failure Time of Canisters and Matrix Degradation Rate
Geologic media are heterogeneous by nature, and their properties may vary from place to placewithin a repository. The issue of heterogeneity should be stated especially when we consider theradionuclide migration at the scale of a repository and discuss on the effects of such repositoryfootprint.
It has been assumed in Chapters 3, 4, and 5 that all the waste canisters in the array fail at the sametime and the waste forms degrade at the same rate. In a real repository, the individual waste canisterscan fail at various times because of possible initial faults in a canister or because of heterogeneityof the physical and geochemical properties of the surrounding environment. The leach rate of thewaste matrix, which directly affects the congruent release rate of radionuclides from the waste, canvary for individual waste forms for the same reasons.
For an analysis of a congruently released nuclide with such heterogeneity of waste canisters,the stochastic model developed in Chapter 7 can be useful. Both the failure time and the matrixdegradation rate can be represented by a distribution of the particle residence time in the waste form.In the illustration in Section 7.3, for example, the density function (7.23) for the residence time can
Figure 8.1 Groundwater stream tilted vertically off the plane of canister array by angle θ .
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be changed for each waste canister. The release rate of the nuclide from the entire repository isobtained by superposing the release rates from individual canisters.
In order to account for the variations of failure time and matrix degradation rate in solubility-limited release of a nuclide, the model in Chapter 4 needs to be modified to reflect different failuretimes and different leach times to individual waste canisters. If radionuclide release from a wasteform is limited by the solubility, the matrix degradation rate does not directly affect the release ratefrom the waste. Therefore, it is considered that variety in the matrix degradation rate has little effecton the exit concentration as long as the release from waste forms is governed by the solubility-limiting mechanism.
8.3 Infiltration Rate of Water
The water infiltration rate evolving with time has been represented by three time domains withdifferent infiltration rates in Chapter 2. Due to its stepwise change between the time domains, asteep increase in the exposure dose rate has been observed. In order to account for a more realistictime-evolution of infiltration rate, an analysis on the water infiltration through the waste vault isnecessary. Once the infiltration rate curve is obtained, the gradually changing infiltration rate canbe represented in the present model by a piecewise step function with a greater number of shortertime domains.
8.4 Determination of the Release Mode
In Chapters 3 and 4, the peak concentration at the repository exit has been formulated as Eqs.(3.24) and (3.25) for congruent release and Eq. (4.22) for solubility-limited release of radionuclides.While the formulas are useful for repository performance evaluation, it is necessary to know before-hand whether release of the given nuclide would be congruent or solubility-limited for the givendesign of a repository. The formulas have been applied in Chapter 5, with the assumption that 237Npis always released by the solubility-limiting mechanism. The observation of numerical calculationshown in Figure 4.1 has been the only justification for this assumption. For an effective applicationof the formulas, however, it is necessary to be able to identify the release mode.
If the initial inventory M◦ is increased from a low value such that a nuclide is congruently re-leased in every compartment, the concentration at the matrix/buffer boundary reaches the solubilityat a certain value of M◦. This threshold can be roughly estimated as follows. As observed in Figure3.5, the concentration profile in the buffer is a straight line when the radionuclide is being congru-ently released. The slope of the straight line is related to the congruent release rate from the wastematrix:
−Sεb D∂Cb
∂ξ= q(t) , 0 ≤ ξ ≤ L , 0 < t < TL , (8.3)
where q is the congruent release rate of the nuclide, i.e.,
q(t) = M◦e−λt/TL , 0 < t < TL . (8.4)
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See Chapter 3 for definitions of the rest of the symbols. Neglecting the radioactive decay (i.e.,assuming λ = 0), and integrating Eq. (8.3) over the buffer thickness, the concentration Cb(0, t) atthe waste/buffer boundary is expressed as
Cb(0, t) = Cb(L , t) +L M◦
Sεb DTL, 0 < t < TL . (8.5)
As long as this boundary concentration Cb(0, t) is lower than the solubility, i.e., Cb(0, t) < C∗,the radionuclide is released congruently. Substituting Cb(0, t) with the right side of Eq. (8.5) andassuming Cb(L , t) = 0 for simplicity, this inequality is written as
M◦ <Sεb DTLC∗
L. (8.6)
With the parameter values for 237Np shown in Table 4.1, the right side of inequality (8.6) is obtainedas 0.0033 mol. See Appendix C for more details.
8.5 Application to Yucca Mountain Repository
It is planned that 70,000 metric tons of waste, which consists of multiple types of waste pack-ages, is put into Yucca Mountain Repository (YMR) [50]. The repository tunnels will be locatedabout 1000 ft above the water table. Radionuclides released from the waste packages would migratedownward in the unsaturated zone, and horizontally after they reach the water table.
The model developed for an LLIW repository in Chapter 2 can be used for YMR, althoughthe size of YMR is greater than the LLIW repository, with some modification and appropriateinterpretation of parameters in the context of the YMR conditions.
The unsaturated compartments in the model of Chapter 2 represent the waste forms and themultiple rock layers of different properties. Each waste vault shown in Chapter 2 can be used torepresent a group of waste packages or a single waste package of YMR. By having multiple wastecompartments, different types of waste packages can be individually specified.
The model assumption that the radionuclides instantaneously dissolve in the water in the repos-itory should be modified to account for the detailed mechanisms of radionuclide release from wastepackages. However, if the details of the release rate profile of a radionuclide from a waste packageis known from another analysis, i.e., if the residence time distribution is known, the mean residencetime 1/µ in the waste package obtained from the profile can be used as a parameter value for themodel in Chapter 2. This approximates the detailed residence time distribution with the exponentialdistribution with the same mean residence time.
Multiple rock layers in the unsaturated zone can be represented by the multiple unsaturatedcompartments with different properties. Migration over 20 km under water table to Amargosa Valleycan be simulated by the migration through aquifer compartments.
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CHAPTER 9
CONCLUSIONS
In order to observe the effects of repository footprint and canister array configurations on arepository performance, compartment models for radionuclide migration in the repository and ingeologic media have been developed in the present dissertation.
The following effects are observed by considering canister array configuration in the compart-ment models developed in this dissertation: (1) the various lengths of migration pathways and thevarious times spent for migration depending on the canister position, (2) radioactive decay duringthe various migration time, (3) various scales of dispersion effect during the migration, and (4) theeffect of nuclide plume from upstream canisters upon nuclide release from a downstream canister.
Analyses have been made for various conditions. Radionuclide release from a water-unsaturatedLILW repository and migration in the far field has been analyzed in Chapter 2. This compartmentmodel can also be applied to an HLW repository located in unsaturated media, such as Yucca Moun-tain Repository. The models developed in Chapters 3 and 4 can be applied to a repository in thesaturated media. Release of a radionuclide congruent to matrix dissolution and solubility-limitedrelease are taken into account. Migration in a two-dimensional system has been illustrated with themodel in Chapter 6. This Markov chain model can also be applied to three-dimensional system.Migration through heterogeneous media at a repository scale and migration at a smaller scale in thevicinity of a single waste canister have been illustrated. Thus, radionuclide release from varioustypes of repositories can be analyzed with the compartment models developed in this dissertation.
The findings of this dissertation include the followings:
1. The performance of far field as a natural barrier can be characterized by comparing the life-time of a radionuclide and the migration time in the far field. If the lifetime of a nuclide issignificantly shorter than the migration time, the nuclide decays out before it reaches bio-sphere. If the lifetime of a nuclide is significantly longer than the migration time, the nuclidewill survive and is mostly released into the biosphere.
2. For a water-saturated repository where the groundwater flows horizontally along the arrayof waste canisters, the peak exit concentrations have been analytically formulated for bothcongruent release and solubility-limited release of radionuclides. The peak exit concentrationis characterized by the duration of radionuclide release from waste canisters and the migra-tion time through in the repository. A theoretical upper-bound concentration has been found,which the concentration does not exceed regardless of the canister array configuration. Withthis upper-bound concentration, theoretical possibility has been shown for designing a repos-itory of a greater capacity without increasing the peak release rate of radioactivity into theenvironment. The discussions of the peak exit concentration are applicable even when thegroundwater flow is not aligned to the canister array.
123
3. The peak release rate of a radionuclide is affected by the initial mass loading in the repositoryfor both congruent-release case and solubility-limited-release case. This effect appears in adifferent manner for a different canister array configuration.
The repository-scale models that take into account the canister array configuration are importantfor better understanding of the repository performance and for the purpose of repository designing.With these compartment models, further studies can be carried out for radionuclide migration inheterogenous media at a greater scale. The models in Chapters 6 and 7 may be suitable for suchapplications since they efficiently make use of the results from a detailed analysis.
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APPENDIX A
DISPERSION EFFECT IN COMPARTMENT MODELS
In Chapter 2, it was assumed that a radionuclide in a compartment is immediately mixed becauseof dispersion effect and that the concentration in each compartment is maintained uniform. Thedispersion effect introduced by this assumption and the series of compartments is discussed here.
A.1 Dispersion in an Array of Compartments
Consider an array of compartments which are connected by a groundwater stream as shown inFigure A.1. Mass M◦ of a stable nuclide is initially contained in the compartment n = 1 at timet = 0. The nuclide migrates by advection along the array of compartments. The nuclide enteringa compartment is instantaneously mixed, and the nuclide concentration inside each compartment isuniform as assumed in Chapter 2.1
The mass balance equation and the initial condition in compartment 1 are written as
dM1
dt= −µM1(t) , t > 0 , and M1(0) = M◦ ,
where M1 is the mass of the nuclide in the first compartment, and µM1 is the release rate of thenuclide from the compartment. The mass M1(t) is obtained as
M1(t) = M◦e−µt , t ≥ 0 . (A.1)
For the downstream compartments (n = 2, 3, . . . , N ), the mass balance equations are writtenas
dMn
dt= µMn−1(t) − µMn(t) , t > 0 , n = 2, 3, . . . , N , (A.2)
1The series of compartments can be considered as a series of stirred tanks of an equal volume connected by steadyflow of water in chemical engineering field. Same discussion applies to such series of tanks [51].
Figure A.1 Array of compartments connected by the groundwater flow. The arrows represent theflow direction of the groundwater that connects the compartments.
125
subject toMn(0) = 0 , t ≥ 0 .
As the solution to the above equations, with Eq. (A.1), the fractional mass of the nuclide in com-partment n is expressed as
Mn(t)M◦
=(µt)n−1
(n − 1)!e−µt , t ≥ 0 . (A.3)
The fractional release rate fn(t) of the nuclide is
fn(t) =µMn(t)
M◦=
µntn−1
(n − 1)!e−µt , t ≥ 0 , n = 1, 2, . . . , N . (A.4)
Particularly for n = 1,f1(t) = µe−µt .
This fractional release rate corresponds to the probability density function (PDF) of the residencetime in a single compartment as shown in Eq. (7.16). Thus, for a single compartment with instan-taneous mixing, the residence time is exponentially distributed with rate parameter µ [47]. Themean of the residence time is 1/µ as mentioned in Chapter 2. Similarly, the residence time in nconsecutive compartments has the gamma distribution with shape parameter n and rate parameterµ.
This can be interpreted as the process in which each independent particle moves through thecompartment array, spending an independent, exponentially distributed residence time in each com-partment with mean 1/µ. The position of each particle forms a Poisson process, and the number n′
of transitions made by time t has the Poisson distribution with mean mpoisson = µt . The probabilitymass function, fpoisson(n′), of the Poisson distribution is
fpoisson(n′) =mpoisson
n′
n′!e−mpoisson . (A.5)
Note the similarity between Eqs. (A.3) and (A.5), where n′ corresponds to n − 1. The spatial massdistribution of the nuclide is described by the Poisson distribution. The mean distance of the nuclidefrom the upstream end of the array is
Lmpoisson = Lµt , (A.6)
where L is the length of a compartment. The extent of longitudinal dispersion is characterized bythe standard deviation σpoisson of the Poisson distribution as
Lσpoisson = L√
mpoisson = L√
µt . (A.7)
A.2 Dispersion in a Continuous Medium
Now, we consider the advection-dispersion problem in one dimensional infinite medium:
R∂C∂t
= D∂2C∂x2
− Rv∂C∂x
, t > 0 ,
126
where C(x, t) is the concentration of the nuclide in the pore water, R the retardation factor, D thedispersion coefficient, and v the flow velocity of groundwater. We consider a situation similar to thecompartment counterpart. We assume a pulsewise injection of a unit mass of the stable nuclide atx = 0, t = 0. The solution is written as
C(x, t) =1
2√
π Dt/Rexp
(−
(x − vt/R)2
4Dt/R
). (A.8)
This is mathematically in the same form as the probability density function, fnormal, of the normaldistribution with mean mnormal and standard deviation σnormal:
fnormal(x) =1
√2πσnormal
exp(
−(x − mnormal)
2
2σ 2normal
), (A.9)
wheremnormal = vt/R , and σnormal =
√2Dt/R . (A.10)
The magnitude of nuclide dispersion at time t can be characterized by the standard deviation σnormal.
A.3 Size of a Compartment and Dispersivity
Since the aforementioned compartment model and the continuum model should represent thesame physical phenomenon, values of the extent of the dispersion should be the same. By equatingLmpoisson to mnormal, and Lσpoisson to σnormal, we obtain
µ =v
RLand L =
2Dv
. (A.11)
The first equation is consistent with Eqs. (2.6) and (2.16). Since the dispersion coefficient D isassociated with the longitudinal dispersivity α as D = αv [52], we obtain from the second equationin Eq. (A.11)
L = 2α . (A.12)
Thus, the length of a compartment in the compartment model represents twice the value of thelongitudinal dispersivity.
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APPENDIX B
TRUE PEAK VALUE OF THE EXIT
CONCENTRATION WITH RADIOACTIVE DECAY
EFFECT
The true peak of the exit concentration is analytically formulated here with radioactive decaytaken into account.
With radioactive decay, the peak exit concentration in the dimensionless system does not neces-sarily correspond to the peak value of the exit concentration with dimension. As shown in Section3.4.4, if the half-life of the radionuclide is significantly short, the true peak may appear before theleach time TL. In order to obtain the peak concentration observed before the leach time TL, the exitconcentration is analytically estimated.
In the dimensionless system, the mass balance equation in the region including the entire bufferand the NFR in the last compartment Nx is obtained by integrating Eq. (3.7′) over 0 < θ < 1 andadding Eq. (3.12′):
ddτ
[(1 − β)
∫ 1
0χb
Nx(θ, τ )dθ + βχNx (τ )
]= ϕw(τ ) + χNx −1(τ ) − χNx (τ ) , τ > 0 , (B.1)
where Eq. (3.15′) has been applied to eliminate ϕb. The first term in the brackets on the left siderepresents the dimensionless mass in the entire buffer region, the the second term represents thedimensionless mass in the NFR.
During the early times τ < τL and τ < Nx , the release rate from the waste matrix is
ϕw(τ ) = 1/τL , 0 < τ < τL , (B.2)
and the NFR concentration in the downstream compartments n > τ have the same value, includingχNx −1 and χNx , because the fresh water that has entered compartment 1 at time 0 has not yet reachedthese compartments, and therefore χNx −1(τ )−χNx (τ ) = 0. We assume that the concentration in thebuffer is uniformly kept at the same level as in the NFR.1 With these conditions and the assumption,Eq. (B.1) is rewritten as
dχNx
dτ=
1τL
, 0 < τ < min(τL, Nx) . (B.3)
1This assumption is not justified from the observation. As observed in Figure 3.5, the concentration in the bufferincreases toward the waste-matrix surface while the nuclide is being released (0 < τ < τL). However, the effect ofthis assumption on the NFR concentration is relatively small, and it is used as an approximation to estimate the NFRconcentration in the last compartment.
128
With the initial condition χNx (0) = 0, the solution to the above equation is
χNx (τ ) =τ
τL, 0 < τ < min(τL, Nx) . (B.4)
The corresponding concentration CNx with dimension is
CNx (t) =t
TLC◦e−λt , 0 < t < min(TL, TNx ) . (B.5)
If 1/λ < min(TL, TNx ), the peak of the concentration CNx appears at t = 1/λ. Let Tdecay ≡ 1/λ.The peak concentration is obtained as
CpeakNx
=Tdecay
TLC◦e−1 . (B.6)
If Tdecay ≥ min(TL, TNx ), the exit concentration increases until it reaches the plateau at t =
min(TL, TNx ). In such cases, the dimensionless peak exit concentration is formulated by Eqs. (3.24)and (3.25). Since the real concentration at plateau decreases exponentially with time because ofradioactive decay, the peak concentration is observed at the beginning of the plateau, i.e., t =
min(TL, TNx ).The peak exit concentration Cpeak
Nxin the system with dimensions is summarized as follows:
CpeakNx
=
Tdecay
TLC◦e−1 , if Tdecay ≤ TL , Tdecay ≤ TNx ,
C◦e−λTL , if TL ≤ Tdecay , TL ≤ TNx ,TNx
TLC◦e−λTNx , if TNx ≤ Tdecay , TNx ≤ TL .
(B.7)
Thus, the peak exit concentration CpeakNx
can be characterized by the mean lifetime of the radionu-clide, Tdecay ≡ 1/λ, the matrix leach time TL, and the migration time over Nx compartments, TNx .
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APPENDIX C
DETERMINATION OF RELEASE MODE
As discussed in Section 8.4, it is important to know whether a radionuclide is released fromwaste forms congruently throughout the release period or solubility-limited release takes place. Inthis section, the concentration at the waste/buffer boundary is analytically formulated in order todetermine the release mode.
C.1 Concentration in the Buffer
During the congruent release of a radionuclide from a waste matrix, the dimensionless con-centration of the radionuclide in the buffer is observed to have linear profiles, being highest at thewaste/buffer interface and the lowest at the buffer/NFR interface (see Figure 3.5). After the releasestops, the concentration profile levels and becomes the same concentration as in the NFR. Once theconcentration at the waste/buffer boundary reaches the solubility during the release, the solubility-limited release takes place. The concentration at the waste/buffer boundary during 0 < t ≤ TL isformulated here.
In the dimensionless system for a congruently released radionuclide (see Section 3.2.6), therelease rate ϕw from the waste form during 0 < τ < τL is
ϕw(τ ) =1τL
, 0 < τ < τL . (C.1)
Since the concentration profile is observed to be linear, we assume a quasi-steady state of diffusionin the buffer. The flux at at any point in the buffer is therefore equal to the release rate ϕw(τ ) fromthe waste matrix:
−(1 − β)αdχb
n
dθ= ϕw(τ ) , 0 < θ < 1 , 0 < τ < τL , (C.2)
where α and β are defined as
α ≡T1 DRbL2
, β ≡RrεrV
RrεrV + RbεbSL. (C.3)
The concentration in the buffer still changes with time because the boundary condition is given bya function of time,
χbn (1, τ ) = χn(τ ) , τ > 0 , (C.4)
where χn is the dimensionless concentration in the NFR. Integrating Eq. (C.2) over θ ∈ (0, 1) andapplying Eqs. (C.1) and (C.4) to substitute ϕw(τ ) and χb
n (1, τ ), respectively, we obtain
χbn (0, τ ) = χn(τ ) +
1τL(1 − β)α
, 0 < τ ≤ τL . (C.5)
130
The corresponding physical concentration Cbn(0, t) is written as
Cbn(0, t) = Cn(t) +
C◦e−λt
τL(1 − β)α, 0 < t ≤ TL . (C.6)
This formulation of Cbn(0, t) becomes Eq. (8.5) when we assume no radioactive decay (λ = 0).
If the concentration Cn in the NFR is known, the concentration Cbn(0, t) can be calculated from
formula (C.6).
C.2 Peak Concentration at the Waste/Buffer Boundary
Whether the concentration at the waste/buffer boundary reaches the solubility can be deter-mined by comparing the peak value of the concentration and the solubility. According to Eq. (C.6),concentration Cb
n(0, t) is at a peak when Cn(t) is at its peak value for congruently released radionu-clides. The peak concentration in the NFR is formulated as Eq. (B.7). Although Eq. (B.7) has beenformulated for the last compartment Nx , it can be applied to any compartment n (= 1, 2, . . . , Nx )in the array by replacing Nx in the formula with n. Substituting Eq. (B.7) into Eq. (C.6), the peakvalue Cb,peak
n and the peak time are obtained as1
Cb,peakn =
C◦e−1[
Tdecay
TL+
1τL(1 − β)α
]at t = Tdecay if Tdecay ≤ TL , Tdecay ≤ Tn ,
C◦e−λTL
[1 +
1τL(1 − β)α
]at t = TL if TL ≤ Tdecay , TL ≤ Tn ,
C◦e−λTn
[Tn
TL+
1τL(1 − β)α
]at t = Tn if Tn ≤ Tdecay , Tn ≤ TL ,
(C.7)where Tn is defined as
Tn ≡ nT1 , T1 ≡RrεrV + RbεbSL
F. (C.8)
C.3 Determination of Release Mode
The peak concentration in the NFR and that at the waste/buffer boundary increase with compart-ment number n if n is smaller than Tdecay/T1 and TL/T1. If n is greater than either one of the values,the peak concentration becomes independent of n. Therefore, the peak concentration Cb,peak
Nxin the
1This formulation of Cb,peakn is an approximation and is not exact particularly for the cases where the first or second
formula in Eq. (C.7) applies. For the derivation of the first formula in Eq. (B.7), uniform concentration in the buffer isassumed in order to approximate the concentration in the NFR. Also for the derivation of the second formula in Eq. (B.7),uniform concentration in the buffer and the NFR is assumed because the concentration in the NFR becomes its peak valueafter release from the waste stops and the concentration profile in the buffer levels. For the derivation of Eq. (C.6), on theother hand, the concentration profile with a constant gradient in the buffer is assumed. Thus, Eq. (C.7) is derived withinconsistent assumptions.
131
last compartment Nx takes the highest concentration among all the compartments. If the peak con-centration Cb,peak
Nxin the last compartment is lower than the solubility C∗, the peak concentrations at
the waste/buffer boundary in all compartments are lower than the solubility. Thus, if
Cb,peakNx
< C∗ (C.9)
is satisfied, the radionuclide is released congruently with matrix dissolution in all compartments inthe array throughout the time period of release.
If Cb,peakn obtained by formula (C.7) is higher than the solubility, the release mode switches to
solubility-limited release when the concentration Cbn(0, t) reaches the solubility. Since the peak
concentrations Cb,peakn+1 , Cb,peak
n+2 , . . . are greater than or equal to Cb,peakn , it implies that the concen-
tration in all the downstream compartments also reach the solubility. On the other hand, the peakconcentrations Cb,peak
1 , . . . , Cb,peakn−1 are less than or equal to Cb,peak
n . Concentrations in the upstreamcompartments can still be lower than the solubility all times even if the concentration in compart-ment n reaches the solubility. If Cb,peak
1 in the first compartment is higher than or equal to thesolubility, i.e.,
Cb,peak1 ≥ C∗ (C.10)
then the boundary concentrations reach the solubility in all compartments in the array.With the parameter values for 237Np shown in Table 4.1,
Tdecay = 3.06 × 106 yr , TL = 1 × 104 yr , T1 = 3.18 × 104 yr . (C.11)
Therefore, the second formula in (C.7) applies to all compartments. By substituting
C◦≡
M◦
RrεrV + RbεbSL(C.12)
into the second formula of (C.6), the inequality condition (C.9) can be rewritten for 237Np as
M◦ < C∗(RrεrV + RbεbSL) ·eλTL
1 +1
τL(1 − β)α
= 3.3 × 10−3 mol . (C.13)
Thus, if the initial inventory of 237Np is less than 3.3 × 10−3 mol, 237Np is released congruentlythroughout the time period of release in all compartments in the array, regardless of the value of Nx .
132
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