NUMERICAL STUDY OF NUCLIDE MIGRATION IN A NONUNIFORM HORIZONTAL FLOW FIELD OF A HIGH-LEVEL RADIOACTIVE WASTE REPOSITORY WITH MULTIPLE CANISTERS

Transcription

1 NUMERICAL STUDY OF NUCLIDE MIGRATION IN A NONUNIFORM HORIZONTAL FLOW FIELD OF A HIGH-LEVEL RADIOACTIVE WASTE REPOSITORY WITH MULTIPLE CANISTERS RADIOACTIVE WASTE MANAGEMENT AND DISPOSAL KEYWORDS: geological radioactive waste disposal, repository performance assessment, flow and transport DOO-HYUN LIM* Japan Atomic Energy Agency 4-33 Muramatsu, Tokai, Ibaraki , Japan Received November 18, 2005 Accepted for Publication February 22, 2006 Migration of nuclides in a water-saturated highlevel radioactive waste repository is analyzed by a newly developed two-dimensional numerical model incorporating a multiple-canister configuration and a nonuniform horizontal flow field of the host rock. The nonuniform flow field is established numerically by obtaining spacedependent groundwater flow velocity vectors using the finite element method. Transport of nuclides is simulated for the instantaneous-pulse-input source condition using the random-walk method. The current study for advectiondominant host rock shows quantitatively that the migration of nuclides in a repository adopting the disposal-pit vertical-emplacement concept is influenced not only by the canister configuration but also by flow boundary conditions, where groundwater flow is considered to be horizontal to the repository plane. The effects of applied hydraulic gradient direction u h on nuclide migration become more significant as the number of canisters increases, while the effects are negligible for the singlecanister configuration. As the number of canisters increases, the results of nuclide migration with respect to u h range more widely and are bounded by two extreme cases. The u h orthogonal to the orientation of the disposal tunnel is observed as most advantageous in terms of the isolation of the radionuclide. The single-canister configuration yields conservative results compared with the multiple-canister configuration. I. INTRODUCTION *Current address: Golder Associates, Inc., Redmond, Washington dlim@golder.com, doohyunlim@cal.berkeley.edu This paper presents two-dimensional ~2-D! numerical results of the migration of nuclides released instantaneously from the failed waste canisters through a hypothetical water-saturated high-level radioactive waste ~HLW! repository with multiple canisters emplaced vertically in a nonuniform horizontal flow field of the host rock. A potential water-saturated HLW repository is considered to contain thousands of waste canisters ~e.g., canisters 1 and 4500 canisters 2! emplaced in a deep geologic formation below the water table. The analyses of groundwater flow in deep underground and nuclide migration with flowing groundwater from the repository to the environment are important basic elements in the safety assessment of HLW repositories. 1 Previous repository performance assessments 1 3 stand on conservative assumptions and simplifications in order to ensure enough safety margin. The current study focuses on the canister configuration and the flow conditions of the host rock for modeling of nuclide migration through a watersaturated HLW repository. Previous repository performance assessment models 1 5 assumed an independent single-canister configuration and0or a uniform flow of groundwater in the water-conducting host rock. In the independent canister models, 1 3 by assuming that migration of radionuclides released from one canister is independent of that released from other canisters, the results from the entire repository were obtained by 222 NUCLEAR TECHNOLOGY VOL. 156 NOV. 2006

2 multiplying the total number of failed canisters by the results from a single-canister configuration. Important effects of the number of canisters and interactions among multiple canisters on the release of radionuclides from the repository have been neglected in the independent canister models by regarding the repository as a collection of independent single canisters. 4 See Ref. 4 for more details on the independent canister model. The connected-canister models 4,5 were developed for the solubility-limit release 4 and the congruent release 5 to investigate effects of multiple canisters on the radionuclide migration in a water-saturated hypothetical repository. For each compartment consisting of a waste matrix, a buffer, and a homogenized near-field rock ~NFR! in the connected-canister models, 4,5 radionuclides are assumed to be transported through the pore water in the buffer region by molecular diffusion and then released into the NFR. For the case where a single array of connected compartments is perpendicular to the uniform groundwater flow, the connected-canister models 4,5 become equivalent to the single-independent canister models 1 3 because nuclides released from each compartment migrate without interactions among other compartments. 4 If a single array of connected compartments is parallel to a groundwater flow, however, migration of nuclides released from each compartment is influenced by other compartments as described in Ref. 4. If the water flow is tilted with respect to the 2-D compartment array, the effects of tilted water flow on the nuclide migration through the connected canisters may be taken into account in the connected-canister models 4,5 to a certain degree, as mentioned in footnote e of Ref. 5, by applying the basic concept of the connected-canister models to each flow component obtained by decomposing water velocity into x and y components. For such cases, except for perpendicular flow to the array, it can be considered that the connected-canister models 4,5 are based on an assumption of perfectly dependent canisters ~or compartments! in the sense that all radionuclides released from one compartment must be transported into the next compartment through the homogenized NFR, characterized by a constant, uniform water velocity. This is mainly because nonuniform transport paths in the NFR, which can play an important role for transport between multiple canisters, 4 was not taken into account explicitly in the connected-canister models. 4,5 The dependency of canisters for nuclide migration has not been treated explicitly with respect to the nonuniform flow condition of the host rock in the previous repository performance assessments. 1 5 Because the dependency of canisters for nuclide migration could vary due to the locations of failed canisters and the groundwater flow conditions of the host rock, a model incorporating both a multiple-canister configuration and a nonuniform flow of the host rock is needed for better characterization of the nuclide migration through a HLW repository. Objectives of the current study are ~a! to develop a numerical model incorporating not only a multiplecanister configuration but also a nonuniform horizontal flow field of the host rock for groundwater flow and radionuclide transport analyses in 2-D space and ~b! to investigate effects of a nonuniform flow field of the host rock on radionuclide migration through a hypothetical water-saturated repository for different canister configurations. In this paper, a 2-D numerical model incorporating a multiple-canister configuration for groundwater flow and radionuclide transport analyses is developed based on the Japanese HLW repository concept 1 ~Sec. II.A!. A hard rock system is considered in the current study. Heterogeneity of the host rock, caused mainly by the fractures, is not taken into account in the current study by assuming the host rock as a homogeneous isotropic water-conducting porous medium. A nonuniform flow field is established numerically using the finite element method for the repository area with respect to variable directions of applied hydraulic gradient for a given canister configuration ~Sec. II.B!. Radionuclide migration is analyzed based on the obtained nonuniform flow field using the random-walk method ~Sec. II.C!. A 2-D numerical code is developed for the disposal-pit verticalemplacement concept 1 ~Sec. III!. Numerical simulations for groundwater flow and radionuclide migration through a water-saturated hypothetical repository are performed for advection-dominant host rock using the newly developed numerical code for different configurations of multiple canisters and variable flow boundary conditions in terms of direction of applied hydraulic gradient ~Secs. IV and V!. II. DEVELOPMENT OF A 2-D MULTIPLE-CANISTER FLOW AND TRANSPORT MODEL II.A. Physical Processes and Conceptual Model The Japanese HLW repository concepts 1 proposed two types of emplacement concepts for waste packages, such as the disposal-pit vertical-emplacement concept and the disposal-tunnel horizontal-emplacement concept. The disposal-pit vertical-emplacement concept shown in Fig. 1 is considered in the current study because of ~a! robust and flexible design for better postclosure safety 2 and ~b! the least complicated emplacement operations. 2 Awaste package includes an overpack and a bentonitefilled buffer surrounding the overpack. The overpack is called a canister hereafter. The bentonite-filled buffer is considered as a mixture of 70 wt% bentonite and 30 wt% sand. 1 The canister, which has a minimum design lifetime of 1000 yr for the reference case of the Japanese concept, 1 contains the waste matrix confining radionuclides. After NUCLEAR TECHNOLOGY VOL. 156 NOV

3 Fig. 1. ~a! Disposal-pit vertical-emplacement concept and ~b! its schematic diagram in 2-D space ~from Ref. 1!, where x D is the disposal tunnel spacing, y D is the waste package pitch in the disposal tunnel, and t D is the diameter of the disposal tunnel. canister failure, the waste matrix, together with radionuclides contained in the matrix, begins to be dissolved by pore water in the buffer. 4 Radionuclides dissolved by pore water migrate through the pore water in the buffer by molecular diffusion due to an extremely small hydraulic conductivity K of the bentonite mixture, such as K m0s ~Ref. 1!. Radionuclides migrate from the buffer to the surrounding host rock. For a repository containing multiple waste packages, emplaced by the disposal-pit vertical-emplacement method, the migration of radionuclides in the repository could interfere among the waste packages to a certain degree. When groundwater flow through the repository is likely to be horizontal, 1,4 which is parallel to the repository plane, some radionuclides released from an upstream canister could migrate into the bentonitefilled buffer surrounding the downstream canister. Such an interference process could affect the migration of radionuclides due to the strong sorption and retardation capabilities of the bentonite-filled buffer. Because the interference process could be strongly dependent on the transport paths of groundwater and radionuclides in the repository, the nonuniform flow field of the repository containing multiple canisters needs to be established for a given repository system and to be used for nuclide migration analysis in a consistent manner. In the current study, a hypothetical water-saturated HLW repository with horizontal groundwater flow is modeled based on the disposal-pit vertical-emplacement method in 2-D space. As shown in Fig. 2, a hypothetical repository consists of N disposal tunnels containing N y packages in each tunnel. The total number of canisters is N{N y for the repository with an area of L x L y, where L x ~N{x D!, L y ~N y {y D 2a D!, x D is the disposal tunnel spacing, y D is the waste package pitch in the disposal tunnel, a D is assumed to be ~x D t D!02, and t D is the diameter of the disposal tunnel. The NFR is conservatively assumed to be a homogeneous isotropic water-conducting porous medium without waterstagnant regions. Since the rock zone around the tunnel could be altered by the excavation, the area of ~N y {y D!{t D for each tunnel, which is considered to be a horizontal cross-sectional area of a disturbed zone below the tunnel, is conservatively assumed to be an excavationdisturbed zone ~EDZ!. The EDZ is assumed to be more permeable than the NFR. For radionuclide transport analysis, all N{N y canisters are assumed to fail simultaneously 1000 yr after emplacement. Release of radionuclides starts instantaneously at the surface of each of the canisters by assuming that mass transport inside the canister and mass transfer between the canister and the surrounding buffer are neglected. Temperature effects are neglected by assuming that the repository will reach a constant ambient temperature by the time the radionuclide release 608C after 1000 yr ~Ref. 1!#. The swelling of the bentonite is not taken into account in the current study. II.B. Groundwater Flow Analysis The domain shown in Fig. 2 for groundwater flow and radionuclide transport can be subdivided into three different regions: buffer, EDZ, and NFR. Because the host rock including the NFR and the EDZ is considered to be a homogeneous isotropic water-conducting medium, a water-stagnant region is not taken into account in the current model. Each of the canisters is assumed to be impermeable. 224 NUCLEAR TECHNOLOGY VOL. 156 NOV. 2006

4 Fig. 2. A 2-D conceptual model of multiple canisters for groundwater flow and radionuclide transport based on the disposal-pit vertical-emplacement concept shown in Fig. 1b. The model includes N disposal tunnels containing N y waste packages in each tunnel. The governing equation 6 for the steady-state spacedependent hydraulic head h~x, y! is given in 2-D Cartesian coordinates for time-independent, incompressible Darcy flow with space-dependent hydraulic conductivity K~x, y! as ] y! ]x K~x, ]h~x, y! ] ]h~x, y! y! 0. ]x ]y K~x, ]y ~1! With assumptions that hydraulic properties for each region of buffer, EDZ, and NFR are homogeneous isotropic, K~x, y! is constant for each region; e.g., K~x, y! K B for buffer, K~x, y! K E for EDZ, and K~x, y! K N for NFR. Two different kinds of flow boundary conditions are prescribed. First, Darcy velocity normal to the surface of the canister is prescribed as zero for each of the canisters. Second, hydraulic head h~x, y! is prescribed at the outer repository boundaries of the L x L y domain with respect to an arbitrary direction of applied hydraulic gradient u h ~0 u h p02! shown in Fig. 3: h x L x 2, y h H { ~L y 02 y! L u sin u h, L y 2 y L y 2, 0 u h p 2, ~2! h x, y L y 2 h H { 1 ~L x 02 x! L u cos u h, L x 2 x L x 2, 0 u h p 2, ~3! h x L x 2, y h H { 1 ~L y 02 y! L u sin u h, and L y 2 y L y 2, 0 u h p 2, ~4! h x, y L y h H { ~L x 02 x! cos u 2 L u h, L x 2 x L x 2, 0 u h p 2, ~5! where h H ¹h{L u, L u L 0 cos~u h u 0!, L 0 ML x 2 L y 2, u 0 tan 1 L y L x, 0 u h p 2. ~6! NUCLEAR TECHNOLOGY VOL. 156 NOV

5 Fig. 3. Flow boundary conditions at the outer rectangular boundaries with respect to an arbitrary direction of applied hydraulic gradient u h ~0 u h p02!. Dotted lines are the equipotential lines for arbitrary u h. The maximum value of hydraulic head h H is calculated based on a given magnitude ¹h and direction u h of hydraulic gradient shown in Eq. ~6!. The minimum value of hydraulic head h L is set to be zero for simplicity as h L 0. The governing Eq. ~1! is solved 7,8 numerically using the finite element method 9 for the given boundary conditions. Triangular finite elements are used to discretize the domain. Hydraulic head at the internal nodes and Darcy velocity at the center of weight for each of the elements are obtained numerically. As a result, a nonuniform flow field of the repository with multiple canisters is established over the entire model space by obtaining a space-dependent flow velocity field with respect to u h. II.C. Radionuclide Transport Analysis Radionuclide migration is simulated based on the established groundwater flow velocity field. See v~x! in Eq. ~7!. Radionuclides released from each of the canisters migrate through the pore water in the buffer by molecular diffusion and then move by advection and0or diffusion in the EDZ and the NFR. Some of the radionuclides released from the upstream canisters could migrate into the buffer regions surrounding the downstream canisters. II.C.1. Transport Equation for Buffer, EDZ, and NFR The transport equation 6 for advection, molecular diffusion, sorption, and radioactive decay in a homogeneous isotropic medium j ~i.e., buffer, EDZ, and NFR! for a single radionuclide is given as R j ]C j ~X, t! ]t where D p, j ¹{~¹C j ~X, t!! ¹{~v~X!C j ~X, t!! R j lc j ~X, t!, t 0, ~7! R j 1 ~1 «j!r s, j «j K d, j, ~8! X position vector of the radionuclide at time t C concentration of the radionuclide in pore space ~mol0m 3! j region that contains the radionuclide at position X R@{# retardation factor for the radionuclide v~x! pore velocity vector at position X obtained from water flow analysis in the previous section ~m0yr! «@{# medium porosity r s density of the solid materials ~kg0m 3! 226 NUCLEAR TECHNOLOGY VOL. 156 NOV. 2006

6 K d distribution coefficient of the radionuclide ~m 3 0kg! l radioactive decay constant ~10yr! D p pore diffusion coefficient for the radionuclide and is assumed to be homogeneous isotropic for each region ~m 2 0yr!, i.e., D p, j D p,b for buffer, D p, j D p,e for EDZ, and D p, j D p,n for NFR. Precursors of a radioactive decay chain and effects of other isotopes of the element are neglected. 4,5 Transport Eq. ~7! can be written in terms of concentration neglecting radioactive decay process C ' as ]C ' j ~X, t! R j D p, j ¹{~¹C ' j ~X, t!! ]t where ¹{~v~X!C j ' ~X, t!!, t 0, ~9! C j ' ~X, t! [C j ~X, t!{e lt. ~10! II.C.2. Standard Random-Walk Scheme Radionuclide transport can be simulated using the random-walk method. 10 A random-walk method is used in the current model because it can be easily combined with any flow model and does not show the numerical dispersion. 10 A random-walk equation, which is equivalent to Eq. ~9! ~Refs. 10 and 11!, can be written in vector notation as X n X n 1 v~xn 1! R j where Dt Z 2 D p, j R j Dt, ~11! n numerical step number X n position vector of the radionuclide at the n numerical step v~x n 1! water ~pore! velocity vector at X n 1 j region that contains the radionuclide at X n 1 Z vector of normally distributed independent random numbers with the mean of 0 and the variance of 1. Random-walk Eq. ~11!, which obeys transport Eq. ~9!, does not take into account the radioactive decay process. For a single-member decay without precursors, the mass reduction by the radioactive decay process 10 can be incorporated in Eq. ~11!, especially for the constant time step Dt, using Eq. ~10! for a certain time t; i.e., the concentration is changed in time t according to the radioactive decay term C j ~X, t! C j ' ~X, t!{e lt. Thus, random-walk Eq. ~11! with a mass reduction by the Eq. ~10! radioactive decay process obeys transport Eq. ~7! taking into account advection, diffusion, sorption, and radioactive decay for a single member without precursors. II.C.3. Extended Random-Walk Reflection Scheme: Local Mass Conservation Random-walk simulations using Eq. ~11! without proper treatments of discontinuity in diffusivity and porosity at the interfaces between different materials cause a local mass conservation ~LMC! error, which is a physically infeasible gathering of mass in the low-diffusivity zone. 10,12,13 Discontinuity a exists at the buffer-edz interface and the EDZ-NFR interface in the current model. Because the LMC error is caused by the drift term that involves the derivative of the diffusion coefficient in space, 10 in the current model, the LMC error is significant at the buffer-edz interface, where the molecular diffusion is the dominant transport mechanism in the buffer region. Since the advection is considered to be dominant both in the EDZ and the NFR in the current study, the LMC error is negligible at the EDZ-NFR interface. Thus, the LMC error is taken into account at the buffer-edz interface in the current random-walk scheme by adopting the previous scheme 14 with a newly obtained transfer ~12!#, called extended random-walk reflection scheme in the current study. In the extended random-walk reflection scheme, different from the standard random-walk scheme shown in Sec. II.C.2, displacements of a particle over the time step Dt are determined by two separate steps such as diffusive transport given in the third term on the right side of Eq. ~11! and advective transport given in the second term on the right side of Eq. ~11!. Because the LMC error occurs due to a random force such as molecular diffusion, a special treatment is required at the buffer-edz interface when the particle crosses the interface over Dt by diffusive transport. For a case in which the particle passes the interface by diffusive transport, a time step is subdivided into two time steps ~i.e., Dt Dt 1 Dt 2!. First, the particle moves to the interface over Dt 1. Second, the particle displaces over Dt 2 from the interface either to the buffer region with the transfer probability P or to the EDZ region with a probability 1 P. It should be applied regardless of whether a particle crosses the interface from one side of the interface or the other. See Appendix B for detailed simulation procedures of the extended random-walk reflection scheme for onedimensional ~1-D! problems. a The canister-buffer interface is not considered for the treatment of the LMC error because the canister is assumed to be impermeable in the current study. NUCLEAR TECHNOLOGY VOL. 156 NOV

7 The transfer probability P at the buffer-edz interface is determined in Appendix A from the analytical solution 15 for diffusion in a composite porous medium by neglecting contact resistance at the interface as P «B D p, B R B «B D p, B R B «E D p, E R E, ~12! where subscripts B and E represent the buffer and the EDZ, respectively. The extended random-walk reflection scheme with transfer probability ~12! is partially verified in Appendix B for 1-D problems. Numerical experiments performed in Appendix B show good agreements between the numerical results obtained using the extended random-walk reflection scheme with transfer probability ~12! and the analytical solutions ~A.1!, ~A.2!, and ~A.3! in Appendix A. Using the extended random-walk reflection scheme, the effect of the LMC error to radionuclide migration in a repository is investigated numerically in Appendix C. The numerical experiment in Appendix C shows that the LMC error can significantly affect the migration of radionuclides in a repository adopting a multiple-canister configuration. III. NUMERICAL CODE AND INPUT DATA A 2-D numerical code, Multiple-Canister Flow and Transport code in 2-Dimensional space ~MCFT2D!,isdeveloped based on the disposal-pit vertical-emplacement method for a water-saturated HLW repository. Variable flow boundary conditions in terms of the magnitude ~¹h 0! and the applied hydraulic gradient direction ~0 u h p02! are incorporated in the current MCFT2D code. The finite element method described in Sec. II.B is used for analysis of time-independent and incompressible Darcy flow. The extended random-walk reflection scheme shown in Sec. II.C.3 is incorporated in MCFT2D for analysis of nuclide migration by advection, diffusion, sorption, and radioactive decay. Using MCFT2D, a hypothetical repository can be modeled for any combination of the number of disposal tunnels N and the number of waste packages for each tunnel N y, depending on the computing resources available, i.e., $N N y % canister configuration. Specifications and input parameters for the repository system are determined mainly based on the Japanese HLW repository concept. 1 The radius of the canister is prescribed as R c 0.41 m. The thickness of the buffer surrounding the canister is prescribed as 0.7 m. Thus, the outer radius of the buffer is R b 1.11 m. For the hard rock system, the disposal tunnel spacing x D is 10 m, the waste package pitch in the disposal tunnel y D is 4.44 m, and the diameter of the disposal tunnel t D is 5 m. For numerical experiments in the current study, 135 Cs, a half-life of yr, is selected because it is considered to be a major contributor to the peak exposure dose rate. 1 Input parameters for hydraulic and transport properties of 135 Cs are given in Table I. Hydraulic conductivity of the NFR is selected conservatively as K N 10 6 m0s from the experimentally obtained average hydraulic conductivities ~e.g., to 10 6 m0s for rock mass, 10 8 to 10 3 m0s for fault and fracture zones! in Ref. 1. Hydraulic conductivity of the EDZ is chosen as K E 10 5 m0s by assuming that the EDZ is more permeable than the NFR. The effective porosity of NFR is chosen as 0.02 from the reference case of Ref. 1. For EDZ, the effective porosity is assumed two times larger than that for NFR. The hydraulic gradient of 0.01, which was used for the reference case in Ref. 1, is used in the current simulations if it is not otherwise specified. To investigate solely the effects of a nonuniform flow field on the transport of long-lived nuclides, such as 135 Cs, mass reduction by radioactive decay process is not taken into account in the current simulations in Secs. IV and V because the radioactive decay process is independent of groundwater flow conditions. IV. PRELIMINARY NUMERICAL EXPERIMENTS IV.A. Migration Interference with Waste Packages Migration of radionuclides in a repository containing multiple waste canisters, emplaced by the disposalpit vertical method, could interfere with the waste packages to a certain degree due to a process in which some of the radionuclides released from the upstream canisters could migrate b into the buffer regions surrounding the downstream canisters, called canister interference hereafter. The canister interference can affect migration of nuclides due to the strong sorption and retardation capabilities of the bentonite-filled buffer. 1 If the mass of nuclides contributing to the canister interference increases ~i.e., the number of particles migrating into the downstream buffer regions increases!, the degree of canister interference is considered to be increased in the current study. See Fig. 5 and footnote d for some numerical results related to the number of particles contributing to the canister interference and the degree of canister interference. Note that the number of contributing particles is different from canister to canister for a given multiple-canister configuration because the number of contributing particles is mainly determined by the location of failed canisters and groundwater flow ~and transport! conditions. b In the current numerical model, it is assumed that nuclides can migrate from the surrounding rock ~e.g., EDZ region! to the buffer region only by the diffusion process based on porosity, diffusivity, and distribution coefficient for both buffer and EDZ regions as shown in Eq. ~12!. Flow rate in the buffer is negligible due to the extremely low hydraulic conductivity of the buffer. 228 NUCLEAR TECHNOLOGY VOL. 156 NOV. 2006

8 TABLE I Input Parameters of Hydraulic and Transport ~ 135 Cs! Properties 1 Symbol Description Value «B Effective porosity in the buffer region 0.41 «N Effective porosity in the NFR region 0.02 «E Effective porosity in the EDZ region 0.04 a K B Hydraulic conductivity in the buffer region m0s K N Hydraulic conductivity in the NFR region 10 6 m0s K E Hydraulic conductivity in the EDZ region 10 5 m0s a D p,b Pore diffusion coefficient of 135 Cs in the buffer region m 2 0yr D p,n Pore diffusion coefficient of 135 Cs in the NFR region m 2 0yr D p,e Pore diffusion coefficient of 135 Cs in the EDZ region m 2 0yr a K d,b Distribution coefficient of 135 Cs in the buffer region 0.01 m 3 0kg K d,n Distribution coefficient of 135 Cs in the NFR region 0.05 m 3 0kg K d,e Distribution coefficient of 135 Cs in the EDZ region 0.05 m 3 0kg a r s,b Density of the solid material in the buffer region 1600 kg0m 3 r s,n Density of the solid material in the NFR region 2640 kg0m 3 r s,e Density of the solid material in the EDZ region 2640 kg0m 3a R B Retardation factor of 135 Cs in the buffer region 24 b R N Retardation factor of 135 Cs in the NFR region 6469 b R E Retardation factor of 135 Cs in the EDZ region 3169 b a Assumed values. b Calculated values based on given «, r, and K for each region using Eq. ~8!. In the current study, the dependency of canisters for nuclide migration can be determined from the degree of canister interference, which is dependent on the number of particles contributing to canister interference. For a given particular case, the dependency of canisters can be considered as independent if there is no canister interference, which occurs when no particles migrate into the downstream buffer regions. As the number of contributing particles increases, the degree of canister interference ~and the dependency of canisters for nuclide migration! increases. Thus, maximum canister interference refers to maximum dependency of canisters in this paper. Transport conditions in the water-flowing rock such as NFR and EDZ can be determined in terms of transport mechanisms using Peclet number, Pe ~i.e., Pe v{l0d!, which defines 6,16 the ratio between the advective transport to the diffusive transport, where v is pore water velocity, D is the molecular diffusion coefficient, and l is the characteristic length. Transport regimes can be classified in terms of Pe based on Ref. 16 as ~a! a diffusiondominant regime ~Pe, 0.3!, c ~b! a transition regime c The diffusion-dominant case ~Pe, 0.3! is of no interest in the current study for the effects of nonuniform flow to the radionuclide transport. This is because radionuclide transport in the diffusion-dominant region is determined by molecular diffusion, which is independent of the groundwater flow conditions. where advection and diffusion are competitive ~0.3, Pe, 300!, and ~c! an advection-dominant regime ~300, Pe, 10 5!. Parameters for NFR given in Table I ~i.e., K N, «N, and D p,n! yield Pe ; 400 ~i.e., advection dominant! for the hydraulic gradient of 0.01, while Pe ; 20 ~i.e., advection-diffusion competitive! is achieved for a smaller hydraulic gradient of In the current study for the evaluation of Pe, the characteristic length l 0.14 m is used assuming that a solute particle should pass through the buffer region with at least five steps for transport simulation, i.e., l ~R b R c!05. A numerical experiment is performed using MCFT2D to show fundamental features of canister interference for different transport conditions, such as Pe ; 400 and Pe ; 20. A single-canister $1 1% configuration is considered for simplicity. An instantaneous-pulse-input source condition is prescribed at an upstream point such that 10 4 particles of 135 Cs are initially located at a point ~x 0.25, y 0.0!, while the center of the canister is located at a point ~x 0, y 0! in Fig. 4. As a result, simulated transport paths shown in Fig. 4 and cumulative distribution functions ~CDFs! of residence time shown in Fig. 5 are obtained for Pe ; 400 and Pe ; 20 for the selected values of u h, i.e., u h 0, p, and 0.03p. For the advection-dominant case ~e.g., Pe ; 400!, numerical results for u h 0 show that all 10 4 particles migrate into the buffer region, and thus, all particles contribute to the canister interference. As a result, the NUCLEAR TECHNOLOGY VOL. 156 NOV

9 Fig. 4. Simulated transport paths of 135 Cs for the advection-dominant case of Pe ; 400 ~¹h 0.01!, and for the advectiondiffusion competitive case of Pe ; 20 ~¹h ! for u h 0, p, and 0.03p. Fig. 5. The CDFs of 135 Cs residence time for the advection-dominant case of Pe ; 400 and the advection-diffusion competitive case of Pe ; 20 for u h 0, p, and 0.03p. Total number of particles N 0 for the simulations is CDFs of residence time are widely distributed for u h 0 as shown in Fig. 5a. For u h 0.03p, however, it is observed that no particle migrates to the buffer, and thus, all particles move by advection and pass the outer repository boundaries without the canister-interference in a short time interval between 1450 and 1670 yr, as shown in Fig. 5a. For u h p, because 45% of particles pass the repository, which is modeled by the single-canister configuration as shown in Fig. 4, without contribution to canister interference, a sharp change is observed around the CDF value of 0.45 in Fig. 5a. 230 NUCLEAR TECHNOLOGY VOL. 156 NOV. 2006

10 For the advection-diffusion competitive case ~e.g., Pe ; 20!, it is observed that 96.7, 61.4, and 24.8% of 10 4 particles contribute to the canister interference for u h 0, p, and 0.03p, respectively. Because of the spreading of transport paths for Pe;20 shown in Figs. 4d, 4e, and 4f, the maximum contribution to canister interference decreases from 100 to 96.7% at u h 0, while the minimum contribution increases from 0 to 24.8% at u h 0.03p. The travel time increases for Pe ; 20 due to a smaller water velocity than that for Pe ; 400. Numerical results in Figs. 4 and 5 show that the degree of canister interference, which determines the dependency of canisters for nuclide migration in the current model, is strongly influenced by the flow and transport paths. Effects of nonuniform flow in the host rock to radionuclide transport are observed to be more significant for the advection-dominant case ~e.g., Pe ; 400! than the advection-diffusion competitive case ~e.g., Pe ; 20! for the cases considered in this section because CDFs for Pe ; 400 range more widely than those for Pe ; 20. IV.B. Nonuniform Flow for Variable u h Nonuniform flow fields in a repository adopting a multiple-canister configuration are established using MCFT2D for various directions of applied hydraulic gradient u h ~0 u h p02!. To illustrate, a nine-canister model with a $3 3% configuration is selected arbitrarily ~see Fig. 6a!. Discretization of the model by triangular finite elements is shown in Fig. 7a. Contour plots of the hydraulic head for an arbitrarily selected $3 3% canister configuration are obtained and shown in Fig. 6 using data in Table I. Figures 6b through 6g show contour plots for variable u h ranging from u h 0 to u h p02 with an increment of 0.10p. Darcy velocity vectors for variable u h are shown in Fig. 7. Darcy velocity, calculated at the center of weight for each of the triangular elements, is uniform for each element but different from element to element. Water flow in the buffer region is negligibly small due to the extremely low hydraulic conductivity of the buffer K B. For instance, Darcy velocity is ;10 7 m0yr for K B m0s. For advection-dominant host rock such as Pe ; 400, which is the case of Table I with a hydraulic gradient of 0.01, as shown in Fig. 7, flow paths are closely related to the radionuclide transport paths. Thus, it is needed to select representative values of u h for each canister configuration in order to effectively take into account variable degrees of canister interference from minimum interference to maximum interference. IV.C. Representative u h One of the methods to determine a representative u h, related to the degree of canister interference and thus the dependency of canisters for each canister configuration, is to use the angle parallel to the projected lines connecting the centers of canisters in different ways. In the current study, u h 0 and u h p02 are selected for any configuration of canisters because u h 0 is the angle parallel to a projected line connecting the centers of canisters orthogonal to the tunnel, while u h p02 is the angle parallel to a projected line connecting the centers of canisters along the tunnel. For 0,u h p02, representative values of u h are obtained either from a projected line connecting canisters $1,1% and $N,2% as u h tan 1 ~ y D 0~~N 1!{x D!! for N 2 and 0,u h u h ', or from a projected line connecting canisters $1,1% and $2, N y % as u h p02 tan 1 ~x D 0~~N y 1!{y D!!for N y 2 and u h ' u h p02. The value u h ' is the angle parallel to a projected line connecting canisters $1,1% and $2,2%, i.e., u h ' tan 1 ~ y D 0x D!. The value x D is the disposal tunnel spacing and y D is the waste pitch in the disposal tunnel. As an example, representative values of u h for different degrees of canister interference are selected for a $2 2% canister configuration as follows: u h 0, u h p02, u h tan 1 ~ y D 02x D! 0.07p, and u h tan 1 ~ y D 0x D! 0.133p. Figures 8a through 8d show the contour plots of the hydraulic head showing streamlines surrounding the four canisters for u h 0, 0.07p, 0.133p, and p02, respectively. Figure 9 shows the particular comparison between the flow paths ~or streamlines! and the simulated transport paths of radionuclides released from canister $1,1% for u h 0.07p. According to the results in Figs. 8 and 9, the dependency of canisters for a repository with a $2 2% configuration could be characterized qualitatively for different u h as ~a! two independent arrays of two horizontally dependent canisters for u h 0 shown in Fig. 8a, ~b! four independent canisters for u h 0.07p shown in Fig. 8b, ~c! two independent canisters and an array of two diagonally dependent canisters ~i.e., $1,1% and $2,2%! for u h 0.133p, and ~d! two independent arrays of two vertically dependent canisters for u h p02 shown in Fig. 8d. As described in footnote d, for the $4 4% canister configuration, even for the dependent canisters, not all nuclides released from the upstream canister interfere with the downstream canister. By taking into account representative values of u h, such as u h 0, 0.07p, 0.133p, and p02 for $2 2%, for groundwater flow and radionuclide transport analyses for a given canister configuration, results can be obtained d For u h 0, the dependency of 16 canisters with a $4 4% configuration can be characterized qualitatively from Fig. 10a as four independent arrays of 4 horizontally dependent canisters. For u h 0.047p, the dependency could be characterized from Fig. 10b as four independent canisters and six independent arrays of two dependent canisters. For a given array of dependent canisters, however, not all nuclides released from the upstream canister interfere with the downstream canisters. Numerical results show that 80 and 22% of nuclides released from canister $1,1% contribute to canister interference for u h 0 and u h 0.047p, respectively. Also, hydraulic and transport conditions such as water velocity, transport paths, and transport distance for each array in Fig. 10b are not necessarily identical. NUCLEAR TECHNOLOGY VOL. 156 NOV

11 Fig. 6. Contour plots of hydraulic head obtained from MCFT2D using data in Table I for an arbitrarily selected $3 3% canister configuration with various directions from u h 0tou h p02 with an increment of 0.10p. 232 NUCLEAR TECHNOLOGY VOL. 156 NOV. 2006

12 Fig. 7. Darcy velocity vectors obtained from MCFT2D using data from Table I for a $3 3% canister configuration for various u h ranging from 0 to p02 with an increment of 0.10p. NUCLEAR TECHNOLOGY VOL. 156 NOV

13 Fig. 8. Contour plots of hydraulic head and streamlines surrounding the four canisters of a $2 2% canister configuration with representative u h using data from Table I. Fig. 9. ~a! Streamline around canister $1,1% for a $2 2% configuration at an arbitrarily selected u h 0.07p and ~b! simulated transport paths of radionuclides released from canister $1,1% for u h 0.07p. for various degrees of canister interference. The value u h 0.07p shown in Fig. 9b is one of several possible values for zero canister interference for $2 2% canister configuration. Similar to the $2 2% configuration, the dependency of canisters for a repository with a $4 4% configuration d is considered as shown in Fig. 10 for u h 0 and u h 0.047p, where u h 0.047p is obtained from 234 NUCLEAR TECHNOLOGY VOL. 156 NOV. 2006

14 Fig. 10. Simulated transport paths for a $4 4% canister configuration with u h 0 and u h 0.047p determined from u h tan 1 ~ y D 03x D!. tan 1 ~ y D 03x D!. As discussed, the dependency of canisters for a repository with an $N N y % configuration could be characterized qualitatively in terms of a set of independent arrays of connected canisters for various u h. V. EFFECTS OF u h ON 135 CS TRANSPORT IN A REPOSITORY WITH MULTIPLE CANISTERS Preliminary numerical experiments discussed in Sec. IV were performed ~a! to investigate fundamental features of canister interference with respect to Pe for the host rock and ~b! to determine representative values of u h related to the various degrees of canister interference for a given $N N y % canister configuration. The results from the preliminary experiments in Sec. IV are utilized in this section for the study on the effects of a nonuniform flow field to radionuclide transport in a repository with multiple canisters. In the current simulations, the host rock, including EDZ and NFR, is considered to be advection dominant as described in Sec. IV.A. Representative u h related to various degrees of canister interference is determined for each canister configuration as shown in Table II ~see Secs. IV.B and IV.C!. V.A. Numerical Simulation A hypothetical water-saturated HLW repository can be modeled by $N N y % canister configurations for variable u h. Parameters to determine the flow condition of the NFR ~e.g., hydraulic conductivity, hydraulic gradient! given in Table I are conservatively assumed from the experimentally obtained data in Ref. 1 as described in Sec. III. Five sets of $N N y % canister configurations are considered for investigating the effects of the nonuniform flow fields on radionuclide transport, i.e., $1 1%, $2 2%, $3 3%, $4 4%, and $6 6%. Note that the number of canisters for $N N y % canister configurations is N{N y. For example, the number of canisters for the $6 6% configuration is 36. For each canister configuration, different values of u h are selected and given in Table II TABLE II Representative Values of u h ~0 u h p02! for Different $N N y % Canister Configurations u h $1 1% $2 2% $3 3% $4 4% $6 6% Description 0 a Parallel to horizontally connected canisters 0.035p b u h tan 1 ~ y D 04{x D! c 0.047p u h tan 1 ~ y D 03{x D! c 0.07p u h tan 1 ~ y D 02{x D! c 0.133p u h tan 1 ~ y D 0x D! c 0.35p Minimum canister interference d 0.5p Parallel to vertically connected canisters a Selected for simulations. b Not selected for simulations. c See Sec. IV.C for u h, x D 10 m, y D 4.44 m. d Determined from preliminary numerical experiments. NUCLEAR TECHNOLOGY VOL. 156 NOV

15 to represent effectively the various degrees of canister interference. In addition, $N 1% configuration is considered for N 1, 2, 4, 8, 12, 16, and 20 for u h orthogonal to the orientation of the disposal tunnel ~u h 0!. A relatively small number of canisters is considered in order to reduce the computational efforts in the current simulations. In the current simulations, the number of canisters is considered up to 36. Analyses for groundwater flow and migration of 135 Cs are performed using MCFT2D for selected canister configurations with representative values of u h given in Table II. For transport analysis, it is assumed that radionuclides release instantaneously from the surface of each of N{N y number of failed canisters simultaneously 1000 yr after emplacement. For simulations, N c particles are initially located uniformly between R c and R c DR for each of the canisters, where R c is the radius of the surface of canister, R c 0.41 m, and DR is a prescribed distance from the surface of the canister, DR 0.01 m. To compare the release of a fixed number of radionuclides from the repository with different canister configurations, N c is determined from the relationship of N c {~N{N y! N 0 Fig. 11. The CDFs of 135 Cs residence time with respect to u h for $1 1%, $2 2%, $3 3%, and $4 4% configurations. The total number of particles for each simulation is A constant time step of Dt 0.5 yr is used for all simulations. A residence time of zero represents the time of instantaneous release of radionuclides after failure of the canisters, which is assumed to be 1000 yr after emplacement. 236 NUCLEAR TECHNOLOGY VOL. 156 NOV. 2006

16 by fixing the total number of particles N 0 for given $N N y % configurations. N is used in the current simulations. An absorbing boundary condition is prescribed at the outer boundaries of the repository, such as top, bottom, left, and right boundaries, while the reflecting boundary condition is prescribed at the surface of each of the canisters, i.e., the impermeable surface of the waste canister. V.B. Numerical Results and Discussion Figures 11 and 12 show the CDFs of 135 Cs residence time for different canister configurations of $1 1%, $2 2%, $3 3%, and $4 4% and different flow boundary conditions in terms of variable u h given in Table II. For a single-canister $1 1% configuration as shown in Fig. 11a, we observe that CDFs of 135 Cs residence time are almost identical for different u h. This is because the migration of 135 Cs through the diffusion-dominant buffer is equivalent for different u h, and the travel time through the prescribed advection-dominant EDZ and NFR regions is negligibly short compared with that through the diffusion-dominant buffer region to differentiate the cases for different u h. Thus, the numerical results in Fig. 11a show that the effects of the applied hydraulic gradient direction u h on 135 Cs migration are negligible for the single-canister configuration. Furthermore, the dependency of canisters for nuclide migration is independent for the single-canister configuration. For multiple-canister configurations such as $2 2%, $3 3%, and $4 4% configurations, however, since canister interference occurs to a certain degree, CDFs vary with respect to u h as shown in Fig. 11. Numerical results in Fig. 11 show that CDFs of 135 Cs residence time are distributed widely as the number of canisters increases, and the CDFs are bounded by two extreme cases, maximum canister interference and minimum canister interference. In the current study, because the canister interference causes a delay of radionuclide travel time due to the strong retardation capability of the bentonite-filled buffer as described in Sec. IV.A, the CDF shifts to the right as the degree of canister interference increases. Thus, the CDF located farthest to the left in Fig. 11 is considered to be caused by the minimum canister interference observed as u h 0.35p, while the CDF located farthest to the right in Fig. 11 is due to the maximum canister interference observed at u h 0. It shows that the dependency of canisters is maximized at u h 0, while the dependency is minimized at u h 0.35p for the cases considered in Fig. 11. Figure 12 shows that the effects of the canister configuration on the migration of 135 Cs vary with respect to the applied hydraulic gradient direction u h. The effects of canister configurations on the migration of 135 Cs are significant where u h is parallel to a projected line connecting the centers of canisters either orthogonal to the tunnel ~Fig. 12a for u h 0! or along the tunnel ~Fig. 12f for u h p02!. The most significant effects are observed for u h 0 as shown in Fig. 12a, while the effects are negligible for u h 0.35p as shown in Fig. 12e. Figure 13 shows the CDF of 135 Cs residence time and the fractional release rate of 135 Cs for a $6 6% canister configuration for u h 0 and u h 0.35p, respectively. The values u h 0 and u h 0.35p are selected from the results obtained in Fig. 11 as the maximum canister interference and as the minimum canister interference, respectively. Figure 14 shows the ranges of the median residence time and peak fractional release rate with respect to u h for different canister configurations based on the results obtained in Figs. 11, 12, and 13. The results of median residence time and peak fractional release rate are distributed widely as the number of canisters increases. For a $6 6% configuration, as an example, the maximum canister interference observed for u h 0 results in ~a!;3.3 times longer median residence time, such as yr for u h 0 and yr for u h 0.35p, and ~b!;3.3 times smaller peak fractional release rate, such as yr 1 for u h 0 and yr 1 for u h 0.35p, than that for the minimum canister interference observed for u h 0.35p. Numerical results in Figs. 11 and 14 show that the maximum canister interference occurs at u h 0, which is the angle orthogonal to the orientation of the disposal tunnel. Because the angle for the maximum canister interference yields the lower limit of the peak fractional release rate as shown in Fig. 14b and the upper limit of median residence time as shown in Fig. 14a for each canister configuration, the angle orthogonal to the orientation of the disposal tunnel is most advantageous among the cases considered in the current study in terms of the isolation of the radionuclide in the repository. Figure 15 shows the results of peak fractional release rate at the maximum canister interference observed at u h 0 obtained by the $N 1% configuration for N 1, 2, 4, 8, 12, 16, and 20. Symbol ~! in Fig. 15 represents the results obtained from the current numerical simulation. Figure 15 shows that the peak fractional release rate decreases exponentially e as the number of dependent canisters increases. For the maximum canister interference case ~u h 0!, the fractional release rate for the $N N y % configuration e Peak fractional release rate at the outer boundary is equivalent to the mode 17 of the probability density function of the residence time of particles in the current study. It is observed for the case of the maximum canister interference that the residence time distributes more widely, and thus, peak fractional release rate decreases exponentially as the number of dependent canister increases. This occurs because most particles released from the upstream canisters should experience additional delay due to the molecular diffusion ~or random movement! through the buffer regions surrounding the downstream canisters. NUCLEAR TECHNOLOGY VOL. 156 NOV

17 Fig. 12. The CDF of 135 Cs residence time with respect to the canister configurations for various u h. 238 NUCLEAR TECHNOLOGY VOL. 156 NOV. 2006

18 Fig. 13. The CDF of 135 Cs residence time for a $6 6% configuration for u h 0 ~maximum canister interference! and for u h 0.35p ~minimum canister interference!. Fig. 14. Median residence time and peak fractional release rate of 135 Cs for $N N % canister configurations. is equivalent to that for the $N 1% because the $N N y % configuration can be characterized qualitatively as N y independent arrays of N horizontally dependent canisters such as $N 1%, as discussed in Sec. IV.C ~see Fig. 10a!. According to the results in Fig. 14, the peak fractional release rate obtained for $N 1% with u h 0, which is equivalent to that for $N N y % with u h 0, can be considered as a lower extreme of the peak fractional release rate for the $N N y % configuration. The current study shows that the single-canister configuration yields conservative results, such as overestimation of release rate and underestimation of residence time, compared with the multiple-canister configuration for any u h considered in the current study. For instance, the results in Fig. 14b show that the singlecanister $1 1% model yields about four times greater peak fractional release for the maximum canister interference case observed for u h 0 and 1.3 times greater NUCLEAR TECHNOLOGY VOL. 156 NOV

19 Fig. 15. Peak fractional release rate for $N 1% with maximum canister interference ~u h 0!. The symbol ~! in the graph is the numerically obtained value in the current simulations. peak fractional release for the minimum canister interference case observed for u h 0.35p compared with those for a 36-canister model with a $6 6% configuration. In addition, the single-canister model yields a maximum of ;12 times greater peak fractional release rate than that for the $20 N y % configuration. Peak fractional release rates of 135 Cs occur relatively early, e.g., ; yr for the $20 1% configuration and ;10 3 yr for the $1 1% configuration. Because of the long half-life of 135 Cs with a half-life of yr, the mass reduction of 135 Cs due to its radioactive decay is ;1.5% of its initial mass at a time of yr. Thus, the effects of mass reduction by radioactive decay are not significant in the current numerical results, such as peak fractional release rate and median residence time. VI. DISCUSSION, LIMITATIONS, AND FUTURE WORK The host rock, including NFR and EDZ, for a watersaturated HLW repository is assumed to be a homogeneous isotropic water-conducting porous medium, while a water-stagnant region could exist not only in the EDZ but also in the NFR in a potential HLW repository. By assuming the EDZ and the NFR as a homogeneous waterconducting medium, as one of possible influences on transport processes, the retardation of the nuclide migration due to the water-stagnant region is not taken into account in the current model. Thus, the homogeneous medium assumption for the EDZ and the NFR can cause faster migration of nuclides than the model taking into account the water-stagnant regions. Generally, the host rock is considered to be fractured. 1 Fractures or faults will be the most significant pathways for groundwater flow and radionuclide transport. 1 For more detailed analysis, because a potential HLW repository could be located in a deep geologic fractured formation with highly channelized heterogeneous flow fields, the heterogeneity due to the fractures in the host rock needs to be incorporated explicitly in the model and to be used for transport analysis in a consistent manner. The model incorporating a discrete-fracture network to represent the heterogeneity of the fractured host rock was developed for a single-canister configuration 7,8 and is being developed for a multiple-canister configuration by the author. Canister interference, defined in Sec. IV.A, is a major process for interference of canisters for the migration of nuclides in the current model and can be influenced by the mass transfer rate at the buffer-edz interface. In the current model, since the extended random-walk reflection scheme has been developed for isotropic media neglecting contact resistance at the interface, the scheme is limited for zero contact resistance at the interface of composite isotropic media. In the current simulations, the host rock ~i.e., NFR and EDZ! has been conservatively assumed to be advection dominant with a high Peclet number of Pe ; 400. The high Peclet number has been used mainly due to the hydraulic gradient ~¹h 0.01! of the host rock and relatively large hydraulic conductivity of the host rock ~i.e., K N 10 6 m0s!, which has been chosen from the experimentally obtained average hydraulic conductivity data 1 for rock mass ranging from to 10 6 m0s ~Sec. III!. Thus, numerical results obtained in the current study for the advection-dominant host rock could be limited depending on the condition of a rock of interest. Results in Sec. IV.A imply that if diffusion becomes a more important transport mechanism in a certain host rock ~i.e., if Pe number decreases!, due to the spreading on transport paths as shown in Fig. 4, the region influenced by the upstream canisters for canister interference could be changed, depending on the flow and transport conditions of site-specific repository environments. For a sufficiently small Pe ~e.g., Pe, 0.3!, because the molecular diffusion, which is independent of groundwater velocity, is the dominant transport mechanism, the effects of u h on nuclide migration should not be significant for a given canister configuration. In the current transport model, to investigate solely the effects of a nonuniform horizontal flow field of the host rock on the 135 Cs transport in a repository with multiple canisters, ~a! the instantaneous-pulse-input source condition is applied at the surface of each canister, ~b! the effect of the angle between the repository plane and groundwater flow is not taken into account, and ~c! a radioactive decay process is incorporated for a 240 NUCLEAR TECHNOLOGY VOL. 156 NOV. 2006

20 single member decay without precursors of the decay chain. Additional detailed transport processes for various nuclides such as glass dissolution, congruent release, and solubility limit release involved in nuclide release from the waste canister are needed to be incorporated in the model for the purpose of safety assessment of an HLW repository. VII. CONCLUSIONS Migration of 135 Cs in a water-saturated HLW repository has been analyzed numerically by a newly developed 2-D numerical model ~MCFT2D! incorporating both a multiple-canister configuration and a nonuniform horizontal flow field of the host rock. Major accomplishments of the current model compared with previous models 1 5 are ~a! more rigorous ~geometric! representation of the waste package consisting of a waste canister ~or overpack! and a bentonite-filled buffer in a 2-D model space, ~b! incorporation of nonuniform flow and transport paths in the host rock, and ~c! integration of models for groundwater flow and radionuclide transport. The current model enables one to take into account the various degrees of the dependency of canisters for nuclide migration in a water-saturated HLW repository in a more rigorous manner, while the dependency has not been treated explicitly with respect to the nonuniform flow condition of the host rock in previous studies. 1 5 Effects of a nonuniform horizontal flow field of the host rock on 135 Cs migration through a hypothetical water-saturated repository have been investigated using MCFT2D for different canister configurations. The current numerical simulations have been performed for the cases where ~a! the ~nonuniform! groundwater flow is horizontal to the repository plane, ~b! the host rock consisting of the EDZ and NFR has been treated as a homogeneous water-conducting porous medium, and ~c! the host rock has been conservatively assumed to be an advection-dominant region represented by a high Peclet number. The following points have been observed from the current numerical results of the migration of 135 Cs released instantaneously from the failed canisters through the hypothetical repository. The migration of 135 Cs in a repository with multiple canisters is significantly influenced not only by the configuration of canisters but also by the groundwater flow boundary conditions. The effects of applied hydraulic gradient direction u h on the transport results of residence time and release rate of 135 Cs migration become more significant as the number of canisters increases, while the effects are negligible for the singlecanister configuration. Note that the dependency of canisters for nuclide migration is independent for the single-canister configuration. As the number of canisters increases, the results of 135 Cs migration with respect to u h range more widely and are bounded by two extreme cases, i.e., maximum and minimum dependency of canisters. The maximum dependency of canisters for 135 Cs migration has been observed at u h orthogonal to the orientation of the disposal tunnel ~i.e., u h 0!, while u h for the minimum dependency varies depending on the repository system parameters, such as canister configuration, pitch, and tunnel spacing. The direction of an applied hydraulic gradient orthogonal to the orientation of the disposal tunnel is observed quantitatively as most advantageous in terms of the isolation of the radionuclide. For the case of the maximum dependency of canisters, the peak fractional release rate decreases exponentially as the number of canisters increases. The single-canister configuration yields conservative results, such as overestimation of release rate or underestimation of residence time, compared with the multiple-canister configuration for variable flow boundary conditions in terms of u h. The current numerical model ~MCFT2D! could be used to help, in terms of release of nuclides from the repository, ~a! to optimize the repository design parameters such as canister configuration, orientation of disposal tunnel, tunnel spacing, and waste package pitch, and ~b! to quantify the degree of conservatism for the transport models adopting conservative assumptions related to the canister configuration and flow condition in the host rock of an HLW repository. APPENDIX A DETERMINATION OF EQ. ~12! TRANSFER PROBABILITY AT BUFFER-EDZ INTERFACE In previous random-walk reflection schemes, 14,18 transfer probabilities at the interface between different materials have been obtained differently from study to study based on their own hypotheses. In addition, medium porosity was not considered in the previous schemes. 14,18 Thus, it is necessary to determine the transfer probability taking into account the porosity as well as the diffusivity at the interface between different materials. In the current study, the transfer probability is determined from the analytical solution for diffusion in composite porous media. A model for 1-D diffusion in two semi-infinite composite porous media with different porosities ~«1, «2! as well as with different diffusivities ~D 1, D 2! is considered as shown in Fig. A.1. An instantaneous-pulse-input source condition is prescribed at x 0 in region 1 ~i.e., x x b!, where x b is an interface between two regions. The analytical solution for the 1-D composite media neglecting the contact resistance at the interface x b is given 15 for x b 0as NUCLEAR TECHNOLOGY VOL. 156 NOV

21 Fig. A.1. A conceptual model for 1-D diffusion in two semiinfinite composite porous media with different porosity ~«1, «2! as well as with different diffusivity ~D 1, D 2!. An instantaneous input source condition is prescribed at x x 0 in region 1; x b represents an interface between the two regions. C 1 ~x, t! and C 0 C 2 ~x, t! C 0 where 2 e ~x x0! 04D 1 t ~F 1MD 2 F 2MD 1!! M4pD 1 t ~F 1MD 2 F 2MD 1 e ~x x 0! 2 04D 1 t M4pD 1 t F 1 D 2 MD 1 ~F 1MD 2 F 2MD 1! e ~x x 0MD 2 0D 1! 2 04D 2 t MpD 2 t, x x b ~A.1!, x x b, ~A.2! C 0 [ M 0 0«1, F 1 [ D 1 {«1, F 2 [ D 2 {«2, ~A.3! and where C solute concentration in pore space D diffusivity ~m 2 0yr! M 0 total mass injected initially at x x 0 «medium porosity. Transfer probability, which takes into account the medium porosity as well as the diffusivity, can be determined from Eq. ~A.1! as follows. Constants in the brackets of the second term on the right side of Eq. ~A.1! are defined as a new constant G. Constant G can be simplified by dividing the numerator and the denominator by MD 1 D 2 as where G «1MD 1 «2MD 2 «1MD 1 «2MD 2, ~A.4! 0 6G6 1. ~A.5! General solutions of transfer probabilities P 1 and P 2 are defined by assuming a linear relationship with the constant G as P 1 [ a{g b ~A.6! and P 2 [ c{g d, ~A.7! where P 1 probability of a particle displaced from the interface into region 1 containing the source initially P 2 probability of a particle displaced from the interface to region 2 a, b, c, d unknown constants. The transfer probabilities must have the following properties. First, if the diffusivity and porosity are the same for both media, the constant G should be zero by Eq. ~A.4!. Then, the probability for a particle to move in either the negative direction P 1 or the positive direction P 2 is the same as P 1 P 2 0.5, called standard Brownian motion. 19,20 Second, if region 2 is impermeable, i.e., «2 MD 2 0, then G should be one by Eq. ~A.4!. In such a case, particles that reach the interface are displaced into region 1 with a probability of unity ~P 1 1!, while no particles can enter region 2 ~P 2 0!, called reflecting Brownian motion. 19,20 Thus, four unknowns a, b, c, and d in Eqs. ~A.6! and ~A.7! are calculated as a b d 1 2 _ and c 1 2 _. Substituting these values of a through d and G from Eq. ~A.4! into Eqs. ~A.6! and ~A.7! yields the transfer probabilities with respect to region 1 ~P 1! and region 2 ~P 2!, respectively: and P 1 [ 1 R 2 P 2 [ 1 R 2 «1MD 1 «1MD 1 «2MD 2 ~A.8! «2MD 2 «1MD 1 «2MD 2 1 P 1. ~A.9! In the current study, since the bentonite-filled buffer and the EDZ refer to regions 1 and 2, respectively, the transfer probability P can be written from Eq. ~A.8! for P 1 P, «1 «B, «2 «E, D 1 D p,b 0R B, and D 2 D p,e 0R E as P «B D p, B R B «B D p, B R B «E D p, E R E, ~A.10! where subscripts B and E represent the buffer and EDZ, respectively. 242 NUCLEAR TECHNOLOGY VOL. 156 NOV. 2006

22 APPENDIX B PARTIAL VERIFICATION OF EXTENDED RANDOM-WALK REFLECTION SCHEME The extended random-walk reflection scheme, described in Sec. II.C.3, is compared with the analytical solutions from Eqs. ~A.1!, ~A.2!, and ~A.3! for the 1-D diffusion in the composite media shown in Fig. A.1. In the extended random-walk reflection scheme, working equations can be written in two different ways based on Eq. ~11!, depending on whether or not a particle crosses the interface over time step Dt. In cases where a particle moves only within the same region over Dt, such as region 1 ~V 1! or region 2 ~V 2! in Fig. A.1, the working equation for particle displacement only by the molecular diffusion, i.e., v~x! 0inEq.~11!, over Dt is x t Dt x t ZM2D i Dt, ~B.1! where x t Dt position of a particle at time t Dt x t position of a particle at time t Z normally distributed independent random number with a mean of 0 and a variance of 1. In Eq. ~B.1!, D i D 1 if x t V 1 and D i D 2 if x t V 2. If a particle crosses the interface ~x b! over Dt, the particle moves to the interface over Dt 1, and displaces over Dt 2 ~i.e., Dt 2 Dt Dt 1! from the interface either to V 1 ~i.e., ZM2D 1 Dt 2! with a probability P 1 in Eq. ~A.8! or to V 2 ~i.e., ZM2D 2 Dt 2! with a probability P 2 in Eq. ~A.9!. Thus, for the case where a particle crosses the interface over Dt, the working equation for a particle displacement can be written as x t Dt x b 6ZM2D 1 Dt 2 6, G P 1 ~B.2! and where and x t Dt x b 6ZM2D 2 Dt 2 6, G P 1, ~B.3! P 1 «1MD 1 «1MD 1 «2MD 2 ~B.4! Dt 2 Dt Dt 1, Dt 1 ~x b x t! 2. ~B.5! 2D i {Z 2 In Eqs. ~B.2! through ~B.5!, x b is the interface between V 1 and V 2, and G is the random number between 0 and 1. For G P 1, the particle is displaced from the interface by 6ZM2D 1 Dt 2 6 to the V 1. For G P 1, however, the particle is displaced from the interface to V 2 by 6ZM2D 2 Dt 2 6. Equations ~B.2! through ~B.5! should be applied regardless of whether a particle crosses the interface from one side of the interface or the other. It is noted that working Eqs. ~B.2! through ~B.5! are for the model in Fig. A.1; i.e., region 1 ~V 1!, which contains the source initially, is located on the left side of the interface ~x b!. For the case where region 1 is located Fig. B.1. Comparison of numerical results 13 ~symbols! using the extended random-walk reflection scheme Eqs. ~B.2! through ~B.5! with analytical solutions ~lines! Eqs. ~A.1!, ~A.2!, and ~A.3!,atT 1 yr for the model in Fig. A.1 with x 0 10 and x b 0: ~a! case I: N , Dt 10 4 yr, D 1 0D 2 1 ~i.e., D 1 D 2 50 m 2 0yr!, and «1 0« , 0.5, 1, 2, and 8 and ~b! case II: N , Dt 10 4 yr, D 1 0D 2 50 ~i.e., D 1 50 m 2 0yr, D 2 1m 2 0yr!, «1 0« and 8. NUCLEAR TECHNOLOGY VOL. 156 NOV

23 interfaces between different materials cause an LMC error as discussed in Sec. II.C.3. The LMC error can be treated using the extended random-walk reflection scheme with Eq. ~12! transfer probability. See Appendix B for the partial verification for 1-D problems. In Appendix C, in order to investigate solely the effects of the LMC error on the radionuclide transport in Fig. B.2. Comparison of numerical results ~symbols! using the extended random-walk reflection scheme with analytical solutions ~lines! at T 10 4 and T 10 5 yr for case III with x 0 1 and x b 0. Region 1 represents the buffer region. Region 2 represents the EDZ region. N , Dt 0.5 yr, D 1 0D ~i.e., D 1 D p,b 0R B m 2 0yr, D 2 D p,e 0R E m 2 0yr!, and «1 0« ~i.e., «1 «B 0.41, «2 «E 0.04!, where values of «and D are determined from Table I and Eq. ~12!. to the right of the interface in a 1-D system, signs between x b and 6ZM2D i Dt 2 6 in Eqs. ~B.2! and ~B.3! should be changed to and, respectively. For verification, first, two cases 13 are considered for comparison of the numerical results using the extended reflection scheme from Eqs. ~B.2! through ~B.5! with the analytical solution from Eqs. ~A.1!, ~A.2!, and ~A.3! for the composite media with ~case I! constant diffusivity and discontinuous porosity and ~case II! discontinuous diffusivity and discontinuous porosity. Second, ~case III! a case for the two semi-infinite composite media consisting of the bentonite-filled buffer and the EDZ is considered. Input parameters for the numerical simulation for each of the cases are shown in each caption of the figures. As a result, Figs. B.1a, B.1b, and B.2 show good agreements between the numerical results obtained using the extended reflection scheme and the analytical solutions for cases I, II, and III, respectively. APPENDIX C EFFECT OF LMC ERROR ON NUCLIDE TRANSPORT IN A REPOSITORY Random-walk simulations without proper treatments of discontinuity in diffusivity and porosity at the Fig. C.1. The CDFs of the 135 Cs residence time obtained from random-walk schemes with and without treating the LMC error. For a flow boundary condition, the magnitude and direction of the applied hydraulic gradient are prescribed as ¹h 0.01 and u h 0, respectively. For each simulation, 10 4 particles are released instantaneously at the surface of the waste canister. A time step for both random-walk schemes is prescribed as 0.5 yr. 244 NUCLEAR TECHNOLOGY VOL. 156 NOV. 2006