Modelling uranium dioxide corrosion under repository conditions: A pore-scale study of the chemical and thermal processes

Journal Pre-proof Modelling Uranium Dioxide Corrosion Under Repository Conditions: A Pore-Scale Study of the Chemical and Thermal Processes Min Liu (Conceptualization) (Data curation) (Formal analysis) (Methodology) (Writing - original draft), Qinjun Kang (Conceptualization) (Project administration)review and editing), Hongwu Xu (Conceptualization) (Project administration)review and editing)

PII:

S0010-938X(19)31659-2

DOI:

https://doi.org/10.1016/j.corsci.2020.108530

Reference:

CS 108530

To appear in:

Corrosion Science

Received Date:

10 August 2019

Revised Date:

30 January 2020

Accepted Date:

8 February 2020

Please cite this article as: Min L, Qinjun K, Hongwu X, Modelling Uranium Dioxide Corrosion Under Repository Conditions: A Pore-Scale Study of the Chemical and Thermal Processes, Corrosion Science (2020), doi: https://doi.org/10.1016/j.corsci.2020.108530

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Modelling

Uranium

Dioxide

Corrosion

Under

Repository

Conditions: A Pore-Scale Study of the Chemical and Thermal Processes Min Liu, Qinjun Kang, Hongwu Xu Earth and Environmental Sciences Division, Los Alamos National Laboratory, Los Alamos, USA

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Email addresses: [email protected] Highlights

Reactive transport simulations are performed on UO2 corrosion in defective fuel rod with

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

different orientations of breach

The dependence of UO2 corrosion on flow rates is investigated and optimized flow

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

conditions are suggested for safe storage of spent nuclear waste. Higher pH can limit the corrosion of UO2 by reducing the reaction rate



Dependence of reaction rate on specific surface area is illustrated



A thermal-chemical model is developed to assess the impact of temperature on the process

Abstract

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of UO2 corrosion.

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

This paper investigates the corrosion behavior of spent uranium dioxide (UO2) fuel when placed in geological repositories. We performed pore-scale reactive transport simulations on the corrosion process of UO2 in a defective fuel rod assembly with different orientations of breach or fracture

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on its clad. It is found that the corrosion rate has strong dependency on the breach orientation. The highest corrosion rate of UO2 is measured when the angle between the groundwater flow direction and the clad fracture reaches 180. UO2 corrosion simulations with different flow rates are compared. The results show the higher flow rate accelerates the corrosion of UO2 fuel. The effect of pH on the corrosion process is also determined. It is indicated that higher pH can limit the corrosion of UO2 by reducing the reaction rate. The dependence of reaction rate and time on reactive surface area is explored. Spent fuel with lower surface area demonstrates longer lifetime

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under corrosion conditions. A numerical model coupling thermal conduction and chemical reactions is developed to assess the impact of temperature on the process of UO2 corrosion. The results show that higher temperature leads to larger corrosion rates for UO2. The predicted reaction rates are higher in comparison with the isothermal result. Hence, our results can help improve the fundamental understanding of UO2 corrosion in geological repositories for long-term storage of spent nuclear fuels and provide guidance for the safe operations and selection of appropriate

Keywords

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Uranium dioxide; corrosion; reactive transport; heat and mass transfer

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repositories.

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1. Introduction

Uranium dioxide (UO2) is the major fuel used in nuclear reactors. The spent nuclear fuel is a

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potential waste to be disposed in a geological repository. In the event of canister or cladding failure associated with an engineered barrier system, the spent fuel would be exposed to groundwater in

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the repository (King, 2017). The spent fuel will then be corroded or dissolved leading to release of soluble radionuclides into groundwater (Gray et al., 1992, Sunder et al., 2004, Guo et al., 2019, Migdisov et al., 2019). This would potentially result in great hazards to the environment and thus

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human health. Thus, it is crucial to study the corrosion behavior of spent nuclear fuel to assess its long-term impact on the environment.

The study of UO2 corrosion under waste disposal conditions has attracted great attentions (Gray et al., 1992, Peper et al., 2004, Sunder and Shoesmith, 1991, Torrero et al., 1997, Williamson et al.,

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2012, Cera et al., 2001). Within the failed container, the corrosion of UO2 relies mostly on the redox conditions of the system. During geological storage of irradiated spent fuel, beta gamma irradiation will be predominant over a few decades while alpha irradiation will predominate in the long term. The irradiation field will cause the radiolysis of water. The oxidizing species including H2O2 will be produced as a result of water radiolysis. These processes can thus affect the oxidizing conditions in a geological repository (Sunder et al., 2004, Sunder et al., 1997).

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The oxidative dissolution of un-irradiated UO2 in the presence of aerated water has been broadly investigated (Clarens et al., 2005, De Windt et al., 2003, Shoesmith et al., 1989, de Pablo et al., 2003). However, in these studies, the effect of irradiation field and water radiolysis is ignored for an irradiated fuel under the conditions of long-term storage. To study the corrosion of irradiated UO2, many studies have been performed to simulate the radiolysis products by including H2O2 into the water and introduced modified kinetic laws for UO2 dissolution (Gimenez et al., 1996, Shoesmith, 2000, Sunder et al., 2004, Ekeroth et al., 2006). Under the condition of oxidative

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dissolution in UO2, the dissolution rate is determined based on the concentrations of oxidants. As the amount of H2O2 produced by water radiolysis in a fuel rod is difficult to be measured, a kinetic law is generally used which includes the concentration of H2O2. Another approach is to study

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irradiated UO2 fuel doped with alpha emitters (Muzeau et al., 2009, Cobos et al., 2002, Jegou et al., 2005). This enables the simulations of the formation of oxidants by alpha radiolysis of water

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in the long term. The dissolution rate can fluctuate by several orders of magnitude depending on the oxidizing conditions (Liu et al., 2017b).

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Recently, several numerical models including chemical reactions and solute transport have been developed to simulate the UO2 corrosion (Odorowski et al., 2017, Liu et al., 2017b, De Windt and

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Spycher, 2019). In the numerical model developed by De Windt et al. (2003) and Spycher et al. (2003), the dissolution of non-irradiated UO2 was considered in the studies during spent fuel disposal. They coupled the transport with UO2 reactions and simulated the uranium migration in

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subsurface. Numerical studies by Liu et al. (2016) and Odorowski et al. (2017) have illustrated that the metallic iron and Fe(II) can prevent the oxidative corrosion of spent UO2 fuel. They also indicated that the metallic irons produced by the corrosion of steel waste packages should be considered in geological repositories for UO2. Liu et al. (2017b) applied a numerical model to predict the corrosion rate of UO2 in a failed nuclear waste container. They compared the

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simulations results with the reaction rates experimentally measured for α-doped UO2 specimens. The results show that the UO2 corrosion rate is dependent on the oxidants hydrogen peroxide. The oxygen produced by hydrogen peroxide can increase the corrosion rate in a closed system while hydrogen can significantly reduce the rate to a minimal value. However, they only considered the diffusion process for mass transfer.

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Among these models, the essential processes of fluid flow and solute transport are considered in only few models. Under the condition of leakage at repository, the contact with groundwater will result in the corrosion of the UO2 pellets. The produced radionuclides will potentially be released due to diffusion processes by controlling the source and transport. In fractured rocks, fluid flow will take place accelerating the release of contaminants into groundwater (Ewing et al., 2016, De Windt and Spycher, 2019). It is thus important to identify the optimized flow conditions of groundwater for minimizing radionuclide leakage and contamination. Development of accurate

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models incorporating fluid flow is required to predict the corrosion of spent nuclear fuel. Besides, the heat transfer is not considered in most previous models. However, the spent nuclear

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fuel still produces heat after several dozens of thousands of years. The temperature of the fuel would be about 70 °C even after ten thousand years (Ewing et al., 2016, Ewing, 2015). This will

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increase the repository temperature near the spent fuel. This in turn will affect the oxidative processes of UO2 fuel. Experiments by De Pablo et al. (1999) have shown that the dissolution of UO2 has strong dependency on temperature. Incorporation of thermal conduction into the model

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is important for accurate prediction of the corrosion processes.

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In the geological storage of spent nuclear fuel, clad failure of a fuel rod may occur under longterm stress loading in repositories even if satisfied design and material are applied in manufacture (Sunder et al., 2004). Figure 1 shows a cross-section of a spent fuel with failed clad. The white

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ring refers to the clad while the grey area inside the ring is the UO2 fuel. Parts of the fuel inside the clad are missing. A fracture or defect is shown on the clad. Reactants can enter the fuel rod through the clad defect. This could result in the contact among UO2, air and groundwater (Jernkvist et al., 2004, Sofu, 2015). Corrosion of UO2 will then take place. Soluble radionuclide contaminant produced in the corrosion processes would be released into groundwater systems via the breach

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and cause environmental and health hazards. In current study, the non-irradiated UO2 fuel (pure UO2) within the defective fuel rod is assumed for simplicity which is a rather simple material compared to the complexity of true irradiated spent fuel structure (Ewing, 2015).

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1mm

Figure 1 Cross-section of a fuel rod with a crack on the clad (reproduced from Williamson et al. (2012); the

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white ring refers to the clad and grey area inside the ring is UO2 fuel).

The rapid development of computational resources and modern imaging techniques facilitates

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direct numerical simulations based on detailed microstructures at the pore scale. Pore-scale simulation of fluid flow and chemical reactions has become increasingly popular to study mineral

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reactions and fluid flow (Pereira Nunes et al., 2016b, Mostaghimi et al., 2016, Kang et al., 2006, Shabro et al., 2012, Tartakovsky and Meakin, 2006, Liu et al., 2017a, Liu and Mostaghimi, 2017a,

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Liu and Mostaghimi, 2017d, Chen et al., 2014, Chen et al., 2013, Liu and Mostaghimi, 2017c). For example, Chen et al. (2015) used a lattice Boltzmann method to model multiphase reactions of minerals with fluid flow directly on 2D synthetic geometry. They included dissolution,

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decomposition and precipitation into the reactive model and explored the impact of these reactions on reactive flow. Pereira Nunes et al. (2016b) developed a reactive model based on the particlebased method to simulate chemical dissolutions of carbonate minerals on micro-CT (computed tomography) images of carbonates. They compared their results with experimental observations and predicted the average reaction rate for calcite dissolution. Liu and Mostaghimi (2017b) and

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Liu and Mostaghimi (2018) recently developed a numerical model applying lattice Boltzmann and finite volume methods to model mineral dissolution on micro-CT images of rocks under the condition of fluid flow. In this paper, we apply the reactive transport model developed by Liu and Mostaghimi (2017b) to simulate UO2 corrosion on fuel pellets with different clad defect orientations. Then, we develop a thermal-chemical model by coupling the thermal advection and conduction with chemical reactions. The new model is used to study the effect of temperature on UO2 corrosion. Our aim is 5

to quantitatively determine the effects of various factors on UO2 corrosion behavior and provide insight into the mechanisms of UO2 corrosion under condition of fluid flow.

2. Numerical models In the simulations, the corrosion of spent fuel under long-term disposal conditions is assumed. The flow of groundwater, solute transport of chemical species, chemical reactions and thermal advection and conduction are included in the reactive transport modelling. The mathematical

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formulations and numerical solution for these processes are briefly introduced in this section. 2.1. Flow of groundwater

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The incompressible fluid flow of groundwater at low-Reynolds number is assumed in the

𝛻 ∙ 𝑣 = 0,

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𝛻p = µ𝛻 2 𝑣,

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repository. Thus, the continuity and the Stokes equations are considered (Sahimi, 2011), (1) (2)

2.2. Chemical reactions

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where 𝑣 is fluid velocity vector, p is pressure and µ is viscosity of fluid.

The corrosion of non-irradiated UO2 caused by the intrusion of oxygen from the environment is

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assumed. The effect of alpha radiolysis on the corrosion process is not included in the study due to the availability of the reaction parameters and computational cost. The redox condition is assumed in a failed container flooded with groundwater. In the long-term disposal conditions, the carbonates in groundwater water can also affect the rates

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(Röllin et al., 2001). However, in this study, such effect is not considered. We assume the concentration of carbonate in the system under consideration is sufficiently low so that its effect on UO2 corrosion is negligible. The effect of carbonate on UO2 reaction will be investigated in future work.

With these assumptions, the corrosion of UO2 is simulated in the presence of aerated water. UO2 can be oxidized to UO2 2+ in solution, leading to the increased solubility and transport of uranium

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contaminant (Peper et al., 2004). The species considered in this study are presented in the reaction equation which can be expressed as (Shoesmith et al., 1994), UO2 (𝑠) + O2 + 2H2 O + 2e− → UO2 2+ + 4OH −

(3)

The transport of chemical solute including reactant oxygen and product UO2 2+ is governed by the advection-diffusion equation, 𝜕𝐶𝑂2 + (𝑣 ∙ 𝛻)𝐶𝑂2 = 𝛻 ∙ (𝐷𝑂2 𝛻𝐶𝑂2 ), 𝜕𝑡

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𝜕𝐶𝑈 + (𝑣 ∙ 𝛻)𝐶𝑈 = 𝛻 ∙ (𝐷𝑈 𝛻𝐶𝑈 ), 𝜕𝑡

(4)

where 𝐶𝑂2 and 𝐶𝑈 represent the local solute concentrations of oxygen and UO2 2+ in water,

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respectively; 𝐷𝑂2 and 𝐷𝑈 are the diffusion coefficients for O2 and UO2 2+ , respectively.

𝜕𝐶 = 𝑘𝑟 (𝐶 − 𝐶𝑠 ), 𝜕𝑛

(6)

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𝐷

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At the surface of UO2, a linear reaction kinetics is applied via

where 𝑘𝑟 is the reaction rate constant, 𝑛̅ is the unit normal vector towards the fluid phase, 𝐶 is the local concentration and 𝐶𝑠 is the saturated concentration at chemical equilibrium. When the local surface.

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concentration 𝐶 is lower than the saturated concentration Cs, the dissolution will occur at the solid

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The mineral volume is updated at each time step via 𝜕𝛽 = 𝑉𝑚 𝐴𝑘𝑟 (𝐶 − 𝐶𝑠 ), 𝜕𝑡

(7)

where 𝛽, Vm, and A are the volume fraction, molar volume, and specific surface area of solid UO2, respectively.

2.3. Thermal advection and conduction For the UO2 fuel, the equation of heat conduction inside a fuel pellet is expressed via,

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𝜌𝑈 𝜎𝑈

𝜕𝑇 = 𝛻 ∙ (𝜆𝑈 𝛻𝑇) + 𝑞𝑣 , 𝜕𝑡

(8)

where 𝜌𝑈 is the density of UO2, 𝜎𝑈 is the specific heat capacity of UO2, 𝑇 is the local temperature, 𝜆𝑈 is the thermal conductivity of UO2 and 𝑞𝑣 is the heat generation rate per unit volume of UO2. The heat transfer in groundwater around the fuel is described by advection-diffusion equation, 𝜕𝑇 + (𝑣 ∙ 𝛻)𝑇] = 𝛻 ∙ (𝜆𝑤 𝛻𝑇), 𝜕𝑡

(9)

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𝜌𝑤 𝜎𝑤 [

where 𝜌𝑤 is the density of groundwater, 𝜎𝑤 is the specific heat capacity of groundwater, 𝑇 is the

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local temperature and 𝜆𝑤 is the thermal conductivity of the groundwater.

In the simulations, each pixel of the image is considered as a grid node in the model. Initially, the

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solid node is assigned with a value of 1 and void node is 0. As corrosion occurs, the value of the solid node decreases. When it reaches 0, the solid node converts into fluid node. Then, it will be

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assigned with zero velocity on all the grid node faces for mass conservation. An implicit finite volume method is applied to discretize these governing equation (Mostaghimi

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et al., 2016). The SIMPLE algorithm (Semi Implicit Method for Pressure Linked Equations) is applied for the solution of fluid flow. The system is initially considered at equilibrium status. The

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Dirichlet boundary condition is considered at the inlet for the concentration equation. The constant temperature is enforced at the boundaries of the domain for the equations of heat transfer. More details about the model and validation are described in Mostaghimi et al. (2016) and Liu and Mostaghimi (2017b).

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3. Results and discussions

The numerical simulations are performed directly on the 2D images of UO2 fuel. Figure 2 shows a representative image of the cross-section for a breached UO2 fuel rod. The image size is 500×500 pixels. The resolution is 60 𝜇𝑚. The red color refers to the cross-section of UO2 fuel. The green ring denotes the clad which isolates the UO2 from fluid. The blue color refers to the fluid. There is a breach on the clad. The reactant can enter the clad through the defect and cause corrosion of the UO2 fuel.

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Breach

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5mm

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Clad

UO2 fuel

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Figure 2 The image of UO2 fuel (red-UO2, green-clad, blue-fluid)

3.1. Isothermal modelling of UO2 corrosion

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3.1.1. The impact of breach orientation on corrosion processes

In this section, we apply the isothermal reactive transport model developed by Liu and Mostaghimi

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(2017b) to investigate the impact of breach orientation on UO2 corrosion. Though the materials for an engineered barrier system, especially the canister or clad, are well designed, in certain circumstances, the clad may fail due to long-term stress loading or other factors (Cox, 1990). This

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will result in the wetting of UO2 fuel through contact with groundwater in the event of water intrusion. In a defective spent fuel assembly, the breach on the clad serves as the inlet of reactant and also the outlet of produced contaminant. The orientation of the breach on the clad can control the rate of mass transfer into the clad affecting the corrosion process. Simulations of UO2 corrosion with different breach orientations are compared. The reaction rate of UO2 pellets was

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experimentally measured in orders of 10-10~-11 𝑚𝑜𝑙 ∙ 𝑚−2 ∙ 𝑠 −1 at pH = 8.2 and 25 °C (Ollila, 1999, Cera et al., 2001). Here we set the reaction rate as 5.0×10-11 𝑚𝑜𝑙 ∙ 𝑚−2 ∙ 𝑠 −1 . This rate is used to calculate the dissolution rate constant in Equation (6), by 𝑘𝑟 = 𝜋𝑟/𝑛 where 𝑛 is the number of moles per unit volume of UO2. We calculate 𝑛 as 4.06×104 𝑚𝑜𝑙 ∙ 𝑚−3. Thus, 𝑘𝑟 is calculated as 3.86×10-7 𝑚 ∙ 𝑠 −1. The diffusion coefficient of O2 and UO22+ are 1.97×10-9 𝑚2 ∙ 𝑠 −1 and 7.6×1010

𝑚2 ∙ 𝑠 −1, respectively, at 25 °C (Kerisit and Liu, 2010, Wilke and Chang, 1955). The average

flow velocity is set as 3.5×10-6m ∙ 𝑠 −1 to represent the flow in subsurface (Dagan and Neuman,

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2005). The saturated concentration of O2 and UO22+ in groundwater used in the simulations are 0.24 𝑚𝑜𝑙 ∙ 𝑚−3 and 1 × 10−2 𝑚𝑜𝑙 ∙ 𝑚−3 , respectively (Rounds et al., 2013, Krestou and Panias, 2004). The surface of the UO2 per unit mass considered in the simulations is 0.2 𝑚2 𝑔−1 corresponding to the measurement of realistic UO2 sample (Gómez et al., 2008, Kubota et al., 1965). The diameter of the UO2 fuel is 8.6 mm. The height of spent fuel pellet is usually in the range from 5 mm to 10 mm (Frost, 2013). Only a thin slice of UO2 pellet with thickness (h) of 60 𝜇𝑚 is considered in the simulations for the sake of computational costs. The density of UO2 is 10.97×103 𝑘𝑔 ∙ 𝑚−3 (Wilson, 1996). The total mass and surface area of the UO2 sample are

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5.17×10-2 g and 1.03×10-2 𝑚2 , respectively. The molar mass of UO2 is 270.03𝑔 ∙ 𝑚𝑜𝑙 −1 . Thus, the reaction rate can be expressed as 4.26×10-1 𝑔 ∙ 𝑚−2 ∙ 𝑦𝑟 −1. By multiplying the surface area, the

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dissolved mass of UO2 per year is computed as 4.39×10-3 𝑔. Thereby, the estimated time for the total mass to dissolve is calculated as 11.78 years with the assumption of a constant oxygen input

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that maximizes the corrosion.

The images of UO2 with different breach orientations are shown in Figure 3. These orientations

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are identified by the angle  between flow direction and the breach (as indicated in Figure 3b).

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From left to right, the angle increases from 0 to 180. The breach on the clad is the only entrance for fluid into the fuel rod. In the simulations, the flow direction is fixed from left to right and the

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flow rate at the inlet remains the same.

Angle 

10mm

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Flow

(a)

(b)

(c)

(d)

(e)

Figure 3 Images of fuel rod for the different breach orientations with breach-flow angle of: (a) 0; (b) 45; (c) 90; (d) 135; (e) 180 (red-UO2, green-clad, blue-fluid)

Five simulations of UO2 corrosion corresponding to the five different breach orientations are carried out with the identical reaction parameters in the model. The reaction time for fully dissolution of the UO2 fuel is calculated and demonstrated in Figure 4. As the angle α between the

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breach and flow direction increases, the reaction time for full dissolution of UO2 decreases. The longest time is observed when the angle between breach and flow direction is 0. By contrast, for the location with breach-flow angle of 180, it takes the least time to fully dissolve the UO2 fuel. This can be explained by fast mass transfer of reactant into the clad. As the breach faces the direction that the fluid is flowing from, the transport of reactants into the clad for reaction is faster than those with the other breach orientations. This leads to higher reactant concentration at the surface of UO2 and consequently results in higher reaction rates.

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The oxygen concentration distributions at the orientations with breach-flow angles of 180 and 0 are shown in Figure 5. The oxygen concentration inside the clad at the orientation with a breach-flow

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angle of 180 (Figure 5a) is much higher due to fast mass transfer. When the angle between the breach and fluid flow is 180, it is easier for oxygen to be transported into the clad through the

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breach on the clad. This leads to higher concentration during the same time period. In contrast, the oxygen concentration inside the clad is much lower where the angle is 0. This is because the

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diffusion direction of oxygen at this position is opposite to the flow direction. A large fraction of reactant is taken away by the fluid flow. The amount of oxygen transported into the clad is less

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than that with angle of 180. Consequently, the concentration of oxygen is lower leading to weak

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corrosion of UO2 with the angle of 0.

Figure 4 The reaction time for fully dissolution of UO2 at different orientations

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Flow

(a)

(b)

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Figure 5 O2 concentration distribution for orientations with (a) 180 and (b) 0 at a reaction time of eight

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years

The variations of effective reaction rate with time are also computed. The effective reaction rate ∆∅

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for the UO2 fuel can be measured from the change of mineral phase via, 𝜂𝑒𝑓𝑓 = Δ𝑡𝑉

𝑚𝐴

where ∆∅

is the changes of mineral volume fraction and Δ𝑡 is the reaction time interval which is one year in

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in this calculation (Pereira Nunes et al., 2016a).

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Figure 6 Variations of average reaction rates of UO2 with time

In Figure 6, the effective reaction rates during UO2 corrosion at orientations with different breachflow angles are compared. The simulations of UO2 corrosion at orientation with larger breach-flow angles have comparably higher rates at the start of reaction. This is a result of fast mass transfer into the fuel rod as discussed above. The variations of the reaction rates can be categorized into two stages. At the first stage, the reaction rate experiences a great increase with time. This reflects the process of oxygen transport. Initially, the concentration of oxygen is low and the reaction is

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slow. As oxygen is transported to the UO2 fuel surface, the reaction rates increase due to the growth of oxygen concentration. When the reaction rate reaches a peak, the reaction rate starts to

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decrease which is the second stage. This is due to the limited space inside the clad. Figure 7 shows the concentration distribution of UO22+ at breach-flow angle of 90 after two different

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reaction times. As the reaction of UO2 occurs, the concentration of the products increases. The breach on the clad is not only the inlet of oxygen and water but also the outlet of product. The transport of product out of the clad is slow. As a result, the concentration of reaction products

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inside the clad accumulates, increases and consequently limits the corrosion process of UO2. In

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addition, with the consumption of UO2 fuel, the reaction finally stops due to the lack of reactant.

(a)

(b)

Figure 7 UO22+ concentration distribution at breach-flow angle of 90 at a reaction time of (a) 20 years and (b) 40 years.

These results indicate that the orientation of breach with respect to the groundwater flow direction plays an important role in the mass transfer of reactant into the clad. It can significantly affect the

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corrosion rates of UO2. Hence, it is necessary to consider the existence of different breach orientations and their effect on UO2 dissolution. The breach orientation with highest reaction rate can also be used as the worst case scenario when estimating the UO2 corrosion rate and contaminant leakage. 3.1.2. UO2 corrosion under different flow rates The fluid flow of groundwater brings the reactant into the repository and also takes the corroded radionuclides into the subsurface system. The flow rate is imperative in the processes of UO2

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corrosion by controlling the mass transfer and transport of UO2 2+ . In this section, we investigate the UO2 corrosion behavior under conditions of different flow velocities. The velocity of fluid

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flow in subsurface is distinctive depending on the depth of the geologic repository. The subsurface flow velocity usually varies in the range of 3.15× 10−4 m ∙ 𝑦𝑟 −1 to 3.15 × 103 m ∙ 𝑦𝑟 −1 (Giles,

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1987). Thus, we compare three simulations with different average flow velocities for the UO2 corrosion processes including 11 m ∙ 𝑦𝑟 −1 , 1.1×102 m ∙ 𝑦𝑟 −1 and 1.1×103 m ∙ 𝑦𝑟 −1 . The other

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parameters used in the simulations are identical to those applied in Section 3.1.1. The simulations are performed on the UO2 images with the breach-flow angle of 90.

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Figure 8 shows the image of UO2 after a reaction time of 20 years. For the simulations with a low flow velocity of 11m ∙ 𝑦𝑟 −1, around 40% of the UO2 fuel is dissolved. With medium flow velocity, over 60% of fuel is corroded. The most dissolution is measured as more than 80% in simulations

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with a high flow velocity at 1.1×103m ∙ 𝑦𝑟 −1. The results demonstrate a higher flow velocity leads to a faster reaction. This can be explained by the faster mass transfer. At a higher flow velocity, the mass transfer of reactants at the surface of UO2 fuel is faster. This results in a higher reactant

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concentration as well as a larger reaction rate.

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(b)

(c)

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(a)

Figure 8 The corroded UO2 after a reaction time of 20 years with (a) low flow velocity (11m ∙ 𝑦𝑟 −1 ), (b)

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medium flow velocity (1.1×102m ∙ 𝑦𝑟 −1 ) and (c) high flow velocity (1.1×103m ∙ 𝑦𝑟 −1 ).

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During the corrosion of UO2, the contaminant UO2 2+ is produced. This product is highly soluble in water and can be easily released into groundwater. The accumulative amounts of the produced

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UO2 2+ are calculated and compared in Figure 9. As the UO2 fuel corrosion progresses, the amount of produced UO2 2+ increases until all the fuel is fully dissolved. The reaction time for simulations

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of UO2 complete dissolution with the low, medium and high flow rate is 61, 48 and 41years, respectively. This reflects a faster reaction rate with a larger velocity due to faster mass transfer of reactant to the inlet of fuel rod. In addition, with the same reaction time, the total produced

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contaminant UO2 2+ is higher at higher flow rates. For example, when the reaction time is 20 years, the amount of produced UO2 2+ is about 1.6 × 10−4 mol in the simulations with high flow rate. By contrast, only 0.7 × 10−4 mol of UO2 2+ is produced when the flow rate is the lowest. This is because of the faster reaction rate at higher flow rates. Thus, the production rate of UO2 2+ is also

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higher as a result of faster corrosion for UO2. It is concluded that higher flow rate can result in faster production of contaminant due to UO2 corrosion. The high flow rate would also cause potential rapid leak into the groundwater system. Therefore, identifying a geologic location with optimized flow condition is necessary for the selection of appropriate repository of spent nuclear waste.

15

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Figure 9 The accumulative amount of the produced UO2 2+ with time

3.1.3. The effect of pH on UO2 corrosion process

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The value of pH is usually used to characterize the acidity or basicity of a solution. For a given subsurface environment, the pH of the groundwater system is distinctive. Previous experimental studies (Sunder et al., 1991, Pierce et al., 2005) have shown the reaction between UO2 and water

processes.

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is affected by the pH of the solution. In this section, we explore the effect of pH on UO2 reaction

In the model, we assume the supply of groundwater is infinite and the pH is constant in the groundwater. The concentration of produced OH − is sufficiently low and its effect on pH of the groundwater is negligible. For the simulations, the reaction rate constant of UO2 can be calculated

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via the equation (Torrero et al., 1997), 𝑘𝑟 = 𝑘𝑟0 100.36(𝑝𝐻0 −𝑝𝐻) ,

(10)

Where 𝑘𝑟0 is the reaction rate constant at a referenced pH0 = 8. The reaction rate constant 𝑘𝑟 at a specific pH can be calculated based on equation (10). The flow velocity and diffusion coefficients used in the simulation are identical to those in Section 3.1.1.

16

The pH values of most groundwater systems range between 6 and 9 (Limited and Gascoyne, 1988, Sunder and Shoesmith, 1991). To investigate the effect of pH on the corrosion processes, simulations for pH = 6, 7, 8 and 9 are carried out on the image of UO2 fuel with the breach-flow angle of 90. The reaction times for fully dissolving the UO2 at these pH conditions are compared in Figure 10. The least time is observed at pH = 6. With the increase of pH, it takes longer to dissolve the fuels. At pH = 6, it takes about 25 years to fully dissolve the fuel while the reaction time is over 70 years at pH = 9. The reaction rate constant decreases with the increase of pH. This

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leads to a lower effective reaction rate at high pH value. The results indicate that the pH of the groundwater at the location of repository plays an important

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role in UO2 corrosion especially in the event of canister or container failure. A repository environment with higher pH will contribute to the safety and integrity of the storage system of

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spent nuclear fuel. The current model is demonstrated to be able to quantify the pH effect on UO2 dissolution and predict the average reaction rate for specific pH.

re

80

70

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50 40 30

20 10

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0

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Reaction time (years)

60

pH=6

pH=7

pH=8

pH=9

Figure 10 The reaction time for fully dissolution of UO2 at different pH

3.1.4. Dependence of corrosion rate on specific surface area of UO2 fuel The reactive surface of UO2 fuel plays an important role in determining the corrosion rate and reaction time (Gómez et al., 2008, Kubota et al., 1965, Chazel et al., 2000). The specific surface

17

area of nuclear fuel under different fabrication and working conditions varies. In this section, we compare simulations with different specific surface areas including 0.0001 m2/g, 0.001 m2/g, 0.01 m2/g and 0.2 m2/g. The dependence of corrosion rate on the specific surface area is investigated. In these simulations, the UO2 reaction rate constant kr, pH value and diffusion coefficient of chemical species are set identical to those in Section 3.1.1. The only different parameter in these simulations is the specific surface area. We performed the simulations on the UO2 images with the breach-flow angle of 90° (Figure 3c).

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Due to the high computational cost of simulation at low surface area 0.0001 m2/g, 0.001 m2/g and 0.01 m2/g, the average reaction rate for the first 100 years is computed and demonstrated in Figure

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11a. Then, it is used to estimate the whole lifetime of the fuel under corrosion conditions (Figure 11b).

Predicted fuel life time (years)

10000

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1E-3

1E-4

1E-5

1E-6

1E-7 0.0001

0.001

0.01

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Average reaction rate (g/year)

1E-2

0.1

1

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Specific surface area (m2/g)

(a)

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100000

1E-1

1000

100

10

1 0.0001

0.001

0.01

0.2

Specific surface area (m2/g)

(b)

Figure 11 Predicted (a) average reaction rate and (b) fuel lifetime for different specific surface areas

As shown in Figure 11a, the average reaction rate (𝑟𝑎𝑣𝑔 ) increases linearly with the specific surface

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area (𝐴𝑠𝑝 ) when the logarithmic scales are used on both axes. This correlation is fitted as 𝑟𝑎𝑣𝑔 = 5.3 × 10−3 𝐴𝑠𝑝 − 3 × 10−6 . The lowest average reaction rate is calculated as 5.49×10-7 g/yr for the fuel with surface area of 0.0001 m2/g. As the specific surface area increases, the reaction rate presents higher values. When the specific surface area is 0.01 m2/g, the predicted average reaction rate is 4.38 × 10−5 g/yr. The highest reaction rate is computed as 1.06×10-3 g/yr at the surface area of 0.2 m2/g. The fuel lifetime estimated by using the average reaction rate is demonstrated in Figure 11b. When the specific surface area is 0.2 m2/g, the fuel lifetime is calculated as 49 years, which 18

is the shortest among all the simulations. It takes longer to fully dissolve the fuels with the decrease of surface area. For the fuel with surface area of 0.001 m2/g, the lifetime is evaluated as 9895 years. The longest lifetime of the fuel pellet is computed as 94240 years at the minimum surface area of 0.0001 m2/g. The results demonstrate that the predicted reaction rate and time strongly depend on the specific surface area of the fuel. Spent fuel with lower surface area presents longer lifetime under corrosion conditions. It should be noted that the specific surface area of fuel material varies with fabrication

of reaction rate with given surface area.

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3.2.Thermal-chemical study on the significance of temperature

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and external conditions. The current modelling approach is able to provide quantitative predictions

The temperature in the repository is important for safe storage of spent UO2. The used nuclear fuel

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still produces heat even after thousands of years’ storage (Ewing et al., 2016, Ewing, 2015). The generated heat will affect the temperature in the repository by heat conduction. Experimental

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results have shown high temperature can enhance the corrosion rate of UO2 in aqueous solutions (De Pablo et al., 1999). In this section, the effect of temperature on UO2 corrosion process under

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flow conditions is explored. We consider the heat generation, advection and conduction during UO2 fuel corrosion. A new model coupling thermal and chemical processes is developed based on the reactive transport model of Liu and Mostaghimi (2017b). For the heat conduction in the UO2

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fuel, Equation (8) is solved for the temperature distribution. The advection-diffusion equation (Equation 9) for the heat transfer in groundwater is solved with same method used for concentration Equation (4) and (5), which is developed in Liu and Mostaghimi (2017b). The impact of temperature on the UO2 corrosion is also investigated by using the thermal-chemical

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model in this section. The allowable temperature of the geological repositories is in the range from 20 to 90 °C (Ikonen, 2003). We perform simulations on the images of UO2 fuel rod with a breachflow angle of 90 (Figure 3c). In the simulations, the temperature distribution in the UO2 fuel is calculated. Each grid node is assigned with one specific temperature value. The initial temperature of groundwater is 25 °C. The thermal conductivity of UO2 is set as 7.1 W/m·K (Bobkov et al., 2008). The volumetric heat rate

19

𝑞𝑣 is 7.6×107 W/m3 (Hedin, 1997). The reaction rate constant and diffusion coefficient for each grid at a given temperature T are calculated via the Arrhenius equations (Kang et al., 2010): 𝐸𝑘 1 1 ( − )], 𝑅 𝑇 𝑇0

(11)

𝐸𝐷 1 1 ( − )], 𝑅 𝑇 𝑇0

(12)

𝑘𝑟 (𝑇) = 𝑘𝑟0 exp[−

𝐷(𝑇) = 𝐷0 exp[−

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Where 𝑘𝑟0 and 𝐷0 are the reaction rate constant and diffusion coefficient at temperature, 𝑇0 ; here 25 °C is used as the reference temperature, 𝑅 is the gas constant and ∆𝐸 is the activation energy.

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The reaction rate constant 𝑘𝑟 can be calculated through Equation (8) based on the reaction rate constant at 25 °C and ∆𝐸𝑘 = 68.4 kJ/mol (Delegard and Schmidt, 2008). Diffusion coefficients 𝐷𝑂2

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and 𝐷𝑃 are computed via Equation (9) using the values at 25 °C and ∆𝐸𝐷 = 19.3 kJ/mol (Mrowec,

Table 1 The parameters used in the simulations Parameters (25 °C)

Symbol

Specific surface area of UO2

A

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1980). The parameters used in the simulations are also summarized in Table 1.

SI units

0.2

𝑚2 𝑔−1

𝑉𝑚

2.46×10-5

𝑚3 ∙ 𝑚𝑜𝑙 −1

𝐷𝑂2

1.97×10-9

𝑚2 ∙ 𝑠 −1

Diffusion coefficient of UO22+

𝐷𝑈

7.6×10-10

𝑚2 ∙ 𝑠 −1

thermal conductivity of UO2

𝜆𝑈

7.1

𝑊 ∙ 𝑚−1 ∙ 𝐾 −1

Specific heat capacity of UO2

𝜎𝑈

270

𝐽 ∙ 𝐾𝑔−1 ∙ 𝐾 −1

Density of UO2

𝜌𝑈

10.97×103

kg ∙ 𝑚−3

thermal conductivity of water

𝜆𝑤

0.6

𝑊 ∙ 𝑚−1 ∙ 𝐾 −1

Specific heat capacity of water

𝜎𝑤

4130

𝐽 ∙ 𝐾𝑔−1 ∙ 𝐾 −1

Volumetric heat rate

𝑞𝑣

7.6×107

𝑊 ∙ 𝑚−3

Pellet thickness

h

60

𝜇𝑚

Molar volume of UO2

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Diffusion coefficient of O2

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Value

20

𝑘𝑟

Reaction rate constant

𝑚 ∙ 𝑠 −1

3.86×10-7

Figure 12 shows the temperature distribution during UO2 corrosion at different reaction times. With heat generation and conduction, the temperature changes in the simulation domain as the reaction occurs. Due to the consumption of UO2 by corrosion, the source of heat decays and the average temperature in the domain decrease. In comparison with the isothermal simulation in Section 3.1.1, faster corrosion is observed when heat conduction is considered. The predicted

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reaction time for fully dissolution of UO2 is 42 years, which is 7 years shorter than the isothermal simulation results in Section 3.1.1. This is because in the thermal-chemical model, the temperature

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at the UO2 surface is higher due to heat conduction. For example, in Figure 12(a), the temperature at the UO2 surface reaches up to 45 °C while the temperature remains at 25 °C in isothermal

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models. The higher temperature results in larger reaction rate constant and diffusion coefficient. They can lead to faster reaction and mass transfer and consequently causes more local corrosion

(a)

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lP

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at the same time.

(b)

(c)

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Figure 12 Evolution of temperature distribution during UO2 corrosion at various reaction times: (a) t = 0; (b) t = 20 years and (c) t = 40 years

The heat in a repository is dependent on the irradiation of spent fuel and radioactive decay. The irradiated spent fuel produces heat which improves the temperature in the repository. However, the flow of groundwater can enhance the heat transfer but increase the mass transfer. In addition, the size of disposal cells and engineered barrier can also affect the heat transfer in repository (Spycher et al., 2003, De Windt and Spycher, 2019). Hence, it is essential to achieve a balance

21

between heat dissipation, the size of the disposal cells and the engineered barriers, as well as minimizing water flow when designing systems for the geological storage of spent nuclear fuel. In future work, these factors will be incorporated into the studies.

4. Conclusions We have applied pore-scale reactive transport model to simulate UO2 corrosion under flow conditions. Simulations of UO2 corrosion were carried out directly on the images of the UO2 fuel rod with defected clad. The impact of the breach orientation on UO2 corrosion processes is studied.

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When the angle between the flow direction and breach reaches up to 180 degrees, the results show the highest reaction rate due to strong reactant transport. Identification of the breach orientation is

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important for accurate predication of leakage and contamination due to UO2 corrosion.

Simulations of UO2 corrosion in repositories with different flow rates have been compared. The

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results illustrate that the higher flow rate causes faster formation of UO2 2+ . The high flow rate would also cause potential rapid leak into the groundwater system. Optimized flow conditions with

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a low flow rate is suggested for selection of appropriate repositories. The impact of pH on UO2 corrosion processes is also investigated. The lower pH can lead to higher reaction rates. The results

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indicate that the subsurface environment with higher pH can limit the UO2 corrosion process and benefit the safe storage of spent nuclear waste. The dependence of reaction rate and time on specific surface area is studied. In the fuel with larger surface area, the predicted reaction rate

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presents higher values. The fuel lifetime has great dependency on the surface area. We have also developed a novel model accounting for the complex coupled processes including chemical reactions, solute transport and heat transfer and studied the dependence of corrosion on temperature. The simulations were compared with the isothermal modelling. The results show the

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inclusion of heat transfer can improve the temperature of the repository and thus result in higher reaction rates due to higher reaction rate constant and diffusion coefficient. As a result, the rise of temperature due to heat production and conduction in repository will increase the corrosion rate of UO2 and could accelerate the release of contaminant into groundwater. It is necessary to consider the heat transfer and achieve a balance between heat dissipation, the size of the disposal cells and the engineered barriers, as well as minimizing water flow when designing systems environment for the geologic storage of spent nuclear fuel.

22

Our developed model can capture the complex processes including fluid flow, chemical reactions and heat transfer as shown in our results. Though the current simulations are performed on single pellet with 2D, this numerical framework is applicable for more complex boundary conditions and 3D pellets. In future works, we will apply the model to predict UO2 reactions for a range of applications in more realistic and complicated fuel pellet samples or assemblies. This numerical method can provide a useful tool to quantitatively predict the reaction rate and help improve the fundamental understanding of the complex processes during UO2 corrosion in geological

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repositories. The numerical method developed may also be used for modelling metallic/nonmetallic corrosion and the dissolution behavior of waste form phases containing other radionuclides and fission products (Balmer et al., 2001, Xu et al., 2001, Xu et al., 2000) with input

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parameters of specific elements/phases of interest.

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Data availability

The data required to reproduce the findings within this manuscript are available upon the interest

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of the readers. Please contact Min Liu ([email protected]).

Author Statement

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The contribution from Min Liu: Conceptualization; Data curation; Formal analysis; Methodology; original draft; Writing

The contribution from Qinjun Kang: Conceptualization; Project administration, review and editing

The contribution from Hongwu Xu: Conceptualization; Project administration, review and

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editing

Conflict of interest

The authors declare that there is no conflict of interest associated with this article.

Acknowledgments

23

Research presented in this article was supported by the Laboratory Directed Research and Development (LDRD) program of Los Alamos National Laboratory (LANL) under project number 20180007 DR. LANL, an affirmative action/equal opportunity employer, is managed by Triad National Security, LLC, for the National Nuclear Security Administration of the U.S. Department of Energy under contract 89233218CNA000001.

Appendix. Numerical model validation

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To validate the model, we present a comparison between our simulation results with analytical solutions. The 2D steady-state heat transfer in a round plate is studied. The radius of the plate is 5

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mm. The image of the round plate is demonstrated in Figure A1(a). The heat generation and thermal conductivity is considered as uniform and constant through the round plate. The

(a)

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2mm

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temperature at the boundary of the plate is fixed as 20 °C.

(b)

(c)

Figure A1 (a) The image of round plate, (b) the simulation results and (c) analytical solution for temperature

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distribution in the plate.

Under steady-state conditions, there is no change in temperature (𝜕𝑇⁄𝜕𝑡 = 0) such that the heat conduction in the round plate can be described by the following equation, 𝛻 ∙ (𝜆𝑑 𝛻𝑇) + 𝑞𝑑 = 0

(A1)

where 𝜆𝑑 is the thermal conductivity, 6.5W/m•K (corresponding to UO2 at 125°C) (Bobkov et al., 2008) and the volumetric heat rate 𝑞𝑑 is 1.38×108 W/m3.

24

Equation (A1) can be solved both analytically and numerically. For numerical solutions, we apply an implicit finite volume method to discretize this equation. A Dirichlet boundary condition (constant temperature of 20°C) is applied at the surface of plate. The simulation results and analytical solutions for temperature distribution in the plate are shown in Figures A1 (b) and (c). The different colors refer to various values of temperatures in the domain. The highest temperature is observed in the center of the plate. The temperature decreases as the distance to the center of the plate increases. We compare the simulation results for temperature predictions with analytical

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solutions in Figure A2. The difference in the average temperature estimated is less than 1%.

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180

Analytical solution 160

Numerical simulations

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140

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100 80 60

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Temperature( °C )

120

40

0 0

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20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Distance across the plate

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Figure A2 Comparison between numerical simulation results with analytical solutions

25

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