Keywords

1 Introduction

The accident at the Fukushima Daiichi Nuclear Power Station in March 2011 generated damaged fuel in three crippled reactors, containing nearly 250 metric tons of uranium and plutonium along with fission products, minor actinides, and other materials such as fuel cladding, assemblies, and in-core structural material [1]. The damaged fuels will have to be disposed of in a deep geological repository. For a prospective repository, a criticality safety assessment (CSA) should be performed to ensure that the repository system including the engineered barriers and far-field geological formations remains sub-critical for tens of thousands to millions of years. For various repository concepts, CSA is considered to include three major stages in a chronological order: (1) the stage before package failure, (2) the stage after package failure, while fissile nuclides remain within the engineered barriers, and (3) the stage in which fissile nuclides originated from multiple packages deposit in far-field host rocks.

This paper summarizes our previous works [2, 3] on neutronics analysis for the disposal of damaged fuels from Fukushima Daiichi reactors, during the three stages in CSA. Current understanding about the conditions of the damaged fuel is very limited, and the location and design of the repository have not been determined. Therefore, the primary objective of our study is to establish a consistent methodology to evaluate the criticality safety for certain repository conditions and engineered barriers designs. The methodology could be further improved and utilized to assist the repository system design and criticality safety assessment in the future. For stages (1) and (2), neutronics analysis for the engineered barrier region consisting of a single waste package containing damaged fuel debris, failed overpack and the buffer materials [2] will be reported in Sect. 2. For stage (3), our study on the criticality conditions for uranium depositions in geological formations resulting from geological disposal of damaged fuel [3] will be reported in Sect. 3.

2 Neutronics Analysis on Engineered Barrier System Containing Damaged Fuel Debris

2.1 Model and Assumptions

The repository is assumed to be in a water-saturated reducing environment. The neutronics model consists of a canister containing fuel debris from Fukushima Daiichi Unit 1 reactor and the buffer surrounding the canister. Because there is no current design for the disposal system for the damaged fuels, the composition and dimension of the canister and buffer are assumed based on the design for spent fuel disposal [4]. The damaged fuel is assumed to be disposed of after 50 years of cooling. The fuel composition after the accident was calculated by burnup code ORIGEN, which was reported in [1]. Gaseous, soluble, and volatile neutron absorbing nuclides in the fission products (such as Xe and Cs) might have been separated from the fuel and released during and after the accident [5]. Therefore, in this study, only physically and chemically stable, and strongly neutron absorbing nuclides in fission products are considered, which include Gd, Nd, Sm, Rh, and Eu isotopes.

The present work considers six nominal time steps for neutronics analysis: the emplacement time (t = 0), the canister failure time (t = T f ), and four steps during the dissolution of debris particles (t = T f  + 0.2 T l , t = T f  + 0.4 T l , t = T f  + 0.6 T l , and t = T f  + 0.8 T l ). At t = 0, the canister only contains fuel debris. The failure time (T f ) of the carbon steel canister is assumed to be 1000 years. After canister failure, water fills the canister, and the canister is modeled as a porous medium with porosity of 0.3. The geometry of neutronics model for the damaged fuel debris at different time steps have been built based on our literature review on defueling process for the Three Mile Island (TMI) accident [6]. A hexagonal lattice of spherical fuel particles is assumed. The pitch distance between particles is assumed to be either (1) make particles contact each other or (2) make the particles lattice fully fill the canister. In the leaching steps, the released materials from the damaged fuel particles is assumed to be either (a) removed from the canister-buffer system, or (b) be homogeneously mixed with the corroded canister. Combinations of the above variations makes four cases: case 1a, case 1b, case 2a, and case 2b. The schematic layout of the MCNP model is shown in Fig. 1. The engineered barriers consist of a carbon-steel canister surrounded by buffer (a mixture of bentonite and silica sand). The canister is filled with spherical fuel particles in a hexagonal lattice. The unit cell of the lattice is shown in the right bottom of Fig. 1. More detailed descriptions about the model parameters can be found in [3].

Fig. 1
figure 1

Schematic layout of the neutronics model of the engineered barrier system containing damaged fuel debris

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2.2 Summary of Numerical Results

The numerical results were calculated by a Monte-Carlo code MCNP [7], and are shown in Fig. 2, where the neutron multiplication factor k eff is plotted against the nominal time steps for various combinations of cases and initial loadings. Note that the time axis only represents the order of the time steps and does not represent the actual time. The failure time (1000 years) should be several orders of magnitude smaller than the leach time. The green points in three figures in Fig. 2 represent cases assuming the canister is filled with water at time zero.

Fig. 2
figure 2

Calculated k eff for various cases versus time. Solid points represent case (1), hollow points represent case (2). Red points represent case (a), and blue points represent case (b). Squares, circles, and triangles represent initial loading of 500 kg, 1000 kg, and 1500 kg, respectively

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The major findings from the numerical results include, (a) the calculated neutron multiplication factor (k eff) is sensitively dependent on assumptions related to moderation, (b) the carbon steel canister plays an important role in reducing the potential for criticality, (c) the maximum k eff of the canister-buffer system could be achieved after a fraction of fissile nuclides been released from the canister, and (d) under several assumptions, the maximum k eff of the canister-buffer system could be principally determined by the dimension and composition of the canister, not by the initial fuel loading.

3 Conditions for Criticality by Uranium Deposition in Water-Saturated Geological Formations

3.1 Model and Assumptions

From previous discussions, in stage (3), a plume of uranium-bearing groundwater originated from multiple canisters containing damaged fuels could form a uranium deposition in geological formations in the far-field. The deposition could locate in either porous or fractured rock. Because the size of the uranium deposition in porous rock is of the order of the grain size of those rock is much smaller than typical neutron mean-free-path, we can consider that uranium deposition in porous rock is homogeneously mixed with rock and water in a neutronics model. For deposition in the fractured rock, two different configurations have been considered for the mixing between uranium deposition and water in the fracture (i.e. fully mixed or fully separated).

Figure 3 shows the schematic of the MCNP model, in which the spherical core is filled with one of the three different geometries (shown right), surrounded by the one-meter-thick rock as reflector. The combination of rock, water, and heavy metal is expressed by two independent variables: void volume fraction (VVF) and heavy-metal volume fraction (HMVF). For the heterogeneous systems, the VVF is given by b/d, representing the averaged fracture volume fraction, or the fracture porosity in rock. For the homogeneous system, VVF represents the void space fraction that is filled with water and heavy metal precipitations, equivalent to the porosity of a porous rock. The HMVF is defined in a similar way, representing the volume fraction of heavy metal precipitations in the entire core. The volume fraction of the solid-phase of the rock then equals to (1-VVF), and the water volume fraction is given by (VVF-HMVF). By definition, the HMVF must be smaller than VVF, because the volume of precipitation cannot exceed the available void space in the rock.

Fig. 3
figure 3

Three geometries for the MCNP simulations: (1) fractured system I, (2) fractured system II, and (3) homogeneous system

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Two types of host rocks are considered in the present study: average sandstone and magnetite-hematite-bearing pelitic gneiss containing 15% iron. For the heterogeneous systems, the fracture aperture takes values of 0.1, 0.2, 0.5, 1.0, 2.0, 3.0, 4.0, 5.0, and 10.0 cm. For given compositions and geometry for rock and heavy metal, calculations have been first performed for various VVF and HMVF parameters, assuming that the mass of heavy metal in the core is 250 MT, which is the total mass of the damaged fuels form three reactor cores. The discrete k eff results have been used to generate a k eff contour plot by interpolation. By defining a nominal sub-criticality criterion k eff < 0.98, the super-critical region can be determined in the parametric space. Within the super-critical parameter range, MCNP calculations have been conducted to obtain the critical mass of heavy metal deposition. More detailed descriptions about the model parameters can be found in [2].

3.2 Summary of Numerical Results

The numerical results for the effective neutron multiplication factor k eff for the deposition containing 250 metric tons of uranium are shown in Fig. 4 (a) and (b) for two types of host rocks. The contour line in red color, referred to as the critical contour line, indicates the nominal criticality criterion, k eff = 0.98. The triangular region results from the fact that the HMVF cannot be greater than VVF. In either case of rock, the k eff value tends to be greater for a greater value of VVF (i.e., to the right along the horizontal axis). A maximum k eff is observed as HMVF increases for a fixed VVF. If the VVF is 0.094 or smaller for sandstone Fig. 4 (a) and 0.265 for iron-rich rock Fig. 4 (b), then the uranium deposition is always subcritical. We call this threshold VVF as the minimum critical VVF hereafter. The comparison between sandstone and iron-rich rock shows importance of rock compositions. For the iron-rich rock, the likelihood of criticality event would be significantly smaller because iron strongly absorbs neutron.

Fig. 4
figure 4

a k eff contour plot for fractured sandstone with fractured geometry I. b k eff contour plot for iron-rich rock with fractured geometry I

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The minimum critical VVF can be found similarly for every combination of rock type, geometry, and a certain mass of uranium deposition. For both rock types with fractured I geometry, the minimum critical VVF becomes the smallest at aperture b = 1.0 cm. Figure 5(a) and (b) show the contour plots for the critical mass for sandstone comparing two geometries as indicated within the figures. The boundary of the plot is extracted from the red contour line from the k eff results Fig. 4 (a) and (b). The values for critical masses are shown in a logarithm scale in unit of metric tons of uranium, and the contour lines for 1, 10, and 100 metric ton are shown in black, red, and blue, respectively.

Fig. 5
figure 5

a Critical mass contour plot for fractured sandstone. b Critical mass contour plot for homogeneous sandstone. The values in the figure and in the side-bar scale are logarithm of MT of uranium included in the system

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From the numerical results, we can conclude that, the k eff for the deposition become greater with (1) smaller concentrations of neutron-absorbing materials in the host rock, (2) larger porosity of the host rock, (3) heterogeneous geometry of the deposition, and (4) greater mass of uranium in the deposition.

4 Conclusions

This paper summarizes our previous works on neutronics analysis for the disposal of damaged fuels from Fukushima Daiichi reactors. Three major stages have been identified for the criticality safety assessment after disposal.

For stages when fissile nuclides in the damaged fuels remains in the vicinity of the engineered barriers, the k eff for a canister containing fuel debris surrounded by buffer was considered over the leaching time. Based on literature review, the fuel debris has been modeled as a hexagonal lattice of spherical fuel particles. Based on the numerical results, the following key observations can be made: (a) the calculated neutron multiplication factor (k eff) is sensitively dependent on assumptions related to moderation, (b) the carbon steel canister plays an important role in reducing the potential for criticality, (c) the maximum k eff of the canister-buffer system could be achieved after a fraction of fissile nuclides been released from the canister, and (d) under several assumptions, the maximum k eff of the canister-buffer system could be principally determined by the dimension and composition of the canister, not by the initial fuel loading. Future works in this area are planned to apply the present approach for damaged fuels from Unit 2 and Unit 3, to consider more modes for release from the canister, such as leaching of the damaged fuels by reducing the radius of each fuel particle, to consider buffer swelling or collapsing due to degradations, and to develop detailed models to connect the models for single canister with models for the deposition from multiple canisters. The dependence on model parameters, such as fuel particle radius, need to be further examined. We will also investigate the option of using backfilling materials to control criticality in the engineered barrier design.

For the stage when fissile nuclides originated from multiple packages deposit in far-field host rocks, the critical masses for uranium depositions were studied for various rock types and geometries. The analysis has been made for two kinds of rocks by considering a finite system with three different geometries, containing various masses of uranium. The three different geometries include heterogeneous (fractured I and II) and homogeneous systems. The exploration was performed to find optimized combinations of geometry, fracture aperture and the model parameter HMVF, to give the minimum rock porosity (VVF) for criticality. The numerical results show that: the k eff for the deposition become greater with (1) smaller concentrations of neutron-absorbing materials in the host rock, (2) larger porosity of the host rock, (3) heterogeneous geometry of the deposition, and (4) greater mass of uranium in the deposition. After the present analysis, we conclude that various far-field critical configurations are conceivable for given conditions of materials and geological formations. Whether any of such critical configurations would occur in actual geological conditions remains unanswered. To answer this question, we need to extend the present study into the following directions. First, from the neutronics point of view, a more “realistic” fractured system with both the fracture orientation and size randomly distributed is suggested. Second, we need to perform the mass transport analysis to explore whether such a configuration obtained by neutronics analysis is likely to be occurred in geological formations.