# Molecule Diffusion Behavior of Tritium and Selenium in Mongolia Clay Rock by Numerical Analysis of the Spatial and Temporal Variation

^{1}

^{2}

^{3}

^{*}

^{†}

## Abstract

**:**

_{a}) decreased with the increases in the compacted density. In fact, there was retardation of Se diffusion in intact TMS clay rock. A two-site sorption model for Se was applied to simulate fast and slow sorption behavior quantitatively.

## 1. Introduction

_{1/2}= 3.56 × 10

^{5}a) and high radioactivity, and it poses the main potential risks from HLW disposal repositories [16]. Effectively, selenium speciation, which occurs in various valence states (selenium (0), selenide (-II), selenite (IV), and selenate (VI)), depends on various pH, dissolved oxygen (DO), and redox potential conditions in solution. In fact, the mobility of selenate (VI) and selenite (IV) in groundwater is obviously much better than that of the others (selenium (0) and selenide (-II)). Therefore, according to a safety assessment for evaluating radionuclide release to the environment from an HLW repository, faster transport of selenate (Se(VI)) and selenite (Se(IV)) from HLW repositories to the geological environment is considered a major issue and concern. Several previous studies have been performed to obtain the sorption and diffusion parameters, K

_{d}values, and the diffusion coefficients under various conditions [17,18]. In those previous works, the apparent diffusion coefficient values obtained from through-diffusion experiments in crushed mudrock and granite were two orders of magnitude different: 1.04 × 10

^{−12}and 1.40 × 10

^{−10}m

^{2}s

^{−1}in synthetic groundwater, respectively. In fact, the distribution coefficient (K

_{d}) for selenite (Se(IV)) was determined with the sorption of various Se species on a mineral surface, and the results basically suggested that the K

_{d}values of selenite (Se(IV)) on the mudrock appeared to increase with the decrease in Se(IV) equilibrium concentration in the solution.

## 2. Materials and Methods

#### 2.1. Theory of Through-Diffusion

_{a}and D

_{e}are apparent and effective diffusion coefficients, respectively; $\alpha $ depends on the porosity of compacted samples ($\theta $), the bulk density of the dry material (${\rho}_{b}$), and the distribution coefficient K

_{d}; and C is the solute concentration in the liquid phase. Both initial and boundary conditions limit the through-diffusion method. Each boundary condition can be expressed as follows:

_{0}), the concentration in the opposing reservoir is kept close to zero, and L is the overall length of the compacted samples.

_{a}is the apparent diffusion coefficient that takes the rock capacity factor ($\alpha $) into consideration, and $S$ is the cross-section area of the compacted samples. For a sufficient TD experimental time, the diffusion process will reach a steady state; subsequently, in Equations (3)–(5), the exponential term tends to zero, and the concentration profile curve, C(x.t) (or M or CR(t)), shows a linear relationship with time (t). Moreover, a numerical analysis was developed for the spatial and temporal variability C(x.t) of HTO and Se(IV) in this study, and those figures were compared and discussed with the diffusion time for the steady state. Here, two different algorithms were implemented, namely the trust-region reflective algorithm and the Levenberg–Marquardt algorithm. Using both provides an effective and important tool in safety assessment for future clay rock repositories when based on concurrent experimental and numerical results of HTO and Se(IV).

#### 2.2. Experiments

#### 2.2.1. Clay Rocks and Liquids

#### 2.2.2. Batch Tests

_{0}) providing stable isotope tracers and selenium dioxide (SeO

_{2}, Sigma-Aldrich, Darmstadt, Germany) was added to the GW solution prior to batch tests. All batch tests were conducted with a solid/liquid ratio of 0.04 g/20 mL.

#### 2.2.3. Sorption Kinetic Experiments

_{t}) were measured using inductively coupled plasma optical emission spectrometry (ICP-OES, iCAP 7000, Thermo, Waltham, MA, USA).

_{t}) and the initial concentration (C

_{0}) as a function of time follows:

_{t}/C

_{0}= C

_{e}+ (1 − C

_{e})exp^(−λ

_{1}t)

_{t}/C

_{0}= C

_{e}+ (1 − C

_{e}) exp^(−λ

_{1}t) + (1 − f)(1 − C

_{e})exp^(−λ

_{2}t)

_{1}and λ

_{2}represent the one- and two-site decay constants, respectively, and f is the proportionality constant between the amount of one specific site. When the reaction reached equilibrium (i.e., length of time is sufficient), the ratio of sorption C

_{t}/C

_{0}was an exponential decay function with an equilibrium concentration C

_{e}.

#### 2.2.4. Sorption Isotherm Experiments

^{−3}to 10

^{−6}M. Generally, the equilibrium concentration of Se(IV) that was adsorbed on the TMS clay reached a constant and stable value with increasing Se(IV) concentrations. A Langmuir-type sorption equation consistently suggested analysis of the sorption capacity of the Se(IV) or other radionuclides in clay rocks. The Langmuir isotherm model is expressed as follows:

^{3}/mol) and the maximum sorption (M, mol/g), were used to describe and understand the affinity and the sorption capacity of the TMS clay rocks, respectively.

#### 2.2.5. Microanalysis and Elemental Analysis (SEM–EDS Experiments)

^{−2}M. After 7 days, the samples were removed and rinsed quickly with DIW to remove excess Se(IV) solution on the sample surface. After drying, the major sorption of Se(IV) on TMS clay was identified and compared using SEM–EDS with an accelerating voltage of 20 kV and a current of 10 μA. EDS was used to analyze the corresponding Se elemental composition of the clay samples.

#### 2.3. Through-Diffusion Experiments (TD): Column Tests

^{®}(PTFE) units and connectors. The PP columns consisted of pressure-resistant polypropene (<10 MPa) with a length of 13.6 cm and an inner diameter of 5 cm, which was filled with crushed clay rock powders having a total porosity of 0.1 to 0.4 and a bulk density of 1.6 to 2.4 g/cm

^{3}.

#### 2.3.1. Water Saturation

^{3}. In order to make sure each pore space in the compacted TMS clay was filled and saturated with water before the TD experiments, the multichannel switch valves were only connected to the GW reservoir without the HTO radiotracer. During the water saturation period (7 to 30 days), approximately 3 to 5 mL effluent from 5 columns was sampled every 7 days. The concentrations of Na, Mg, Ca, and K versus time (C

_{t}) in the effluent were measured using inductively coupled plasma optical emission spectrometry (ICP-OES, iCAP 7000, Thermo). It was evaluated and determined that water saturation had been reached once the Na, Mg, Ca, and K values were within 5% of the corresponding liquid phase concentrations.

#### 2.3.2. Nonreactive Tests—HTO

#### 2.3.3. Reactive Tests—Selenium (Se(IV))

## 3. Results

#### 3.1. Batch Sorption Tests: One- and Two-Site Kinetic Fitting and Langmuir Isotherm

_{1}, and λ

_{2}), while there is only one decay constant for the other by fitting the normalized concentration (i.e., C

_{t}/C

_{0}). Table 2, which compares the least-squares errors (LSEs), suggests that the two-site fitting curves may be more suitable than those with only one site in describing the sorption kinetics of Se on TMS clay in GW. After a trial-and-error fitting process, a Langmuir model was found to obtain numerical results matching our experimental data at various initial Se(IV) concentrations. Figure 2b and Table 3 list the fitting parameters of the Langmuir model.

#### 3.2. Microanalysis and Elemental Analysis for Se Sorption on TMS Clay Rocks

#### 3.3. Diffusion Coefficients of HTO and Se for TMS Clay Rocks

^{3}/g) and intact TMS clay rocks, and the accumulative concentration curves (CR(t)) of HTO and Se obtained by the TD experiments are shown in Figure 4. This indicates that the time lag between HTO and Se in TMS clay rock to diffuse out is approximately 1 and 30 days, respectively, and it reached a diffusion steady state after about 7 and 80 days due to constant diffusing flux. The dimensionless parameter t

_{d}= (D

_{a}·t

_{f}/L

^{2}), an important factor, is introduced here to determine if the diffusion reached equilibrium. Crank (1975) stated and suggested that diffusion steady state will be achieved when t

_{d}> 0.45. The TD results showed that the td values of all columns were higher than 0.45 and good R-squared values (R

^{2}> 0.9) were obtained in five columns, and Se(IV) exhibited obvious retardation behavior in intact TMS clay rocks. Moreover, it showed the lowest diffusion coefficients (D

_{a}= 1.10 × 10

^{−12}m

^{2}/s and D

_{e}= 3.24 × 10

^{−12}m

^{2}/s) in Se(IV) rather than in HTO in all columns (Table 4).

#### 3.4. Spatial and Temporal Variation with Various Diffusion Coefficients by Numerical Analysis

^{−10}to 10

^{−12}m

^{2}/s, and it required more time to reach steady-state diffusion in the Se(IV) TD experiments. In this study, a numerical analysis was applied to assess the spatial and temporal variation of C(x.t) for the various diffusion coefficients by using Equation (3). Figure 5 and Table 5 show the spatial and temporal variation C(x,t) of D

_{a}= 1.00 × 10

^{−10}, 1.00 × 10

^{−11}, and 1.00 × 10

^{−12}m

^{2}/s through a 0.3 cm thick sample and unit cross-sectional area at different times. However, there are also some limitations to TD experiments, such as detection uncertainty or errors. The numerical analysis of spatial variation for the concentration profiles depended on the diffusion flux (C/C

_{0}> 0.001) at various distances (x), set from 0.295 to 0.299 cm, and is given by Equation (3). According to the numerical analysis of spatial and temporal variation C(x,t), shown in Figure 5a,b, at D

_{a}value = 1.00 × 10

^{−10}m

^{2}/s, the straight line reveals a certain time at which the steady-state conditions reached were entirely different, ranging from 0.25 to 0.46 days (6 to 11 h). Moreover, it shows results (steady state) in agreement with the TD experimental results of HTO in compacted TMS clay rocks after 12 h. In contrast, a longer time, from 18 to 46.8 days, is shown in Figure 5e,f for the spatial variation C(x,t) with a decreasing D

_{a}value, as determined by the Se(IV) TD experiments.

## 4. Discussion

^{2}> 0.9), a set of parameters for the maximum capacity (1.75 × 10

^{−4}mol/g) was determined, with which the batch experimental results of Se(IV) in GW could be described adequately. Compared with TMS clay, responsible for the sorption of Se, the study also showed that clay minerals contain iron as a major component, and several studies have also reported different material analyses [27,28,29,30]. We found that Se(IV) has higher retardation than HTO by comparison between HTO and Se(IV) diffusion coefficients for TMS clay rock. In addition to the microporous composition (i.e., porosity), we also recognized that the key sorption (or retardation) of Se(IV) in TMS clay rock depended on the clay mineral composition (iron content), in agreement with the batch sorption and SEM–EDS experiments. For assessing spatial and temporal variability with various diffusion coefficients by numerical analysis, a good method would be to calculate and realize TD experiments by applying different D

_{a}values in the spatial and temporal variation concentration profile C(x,t).

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 3.**SEM–EDS analysis image of TMS clay rock: (

**a**) SEM image of TMS clay particle; (

**b**) EDS mapping analysis; (

**c**) EDS spectra.

**Figure 5.**Numerical analysis of concentration profiles C(x,t): (

**a**) temporal: C(x,t) at D

_{a}= 1 × 10

^{−10}m

^{2}/s and (

**b**) spatial: C(x,t) at D

_{a}= 1 × 10

^{−10}m

^{2}/s; (

**c**) temporal: C(x,t) at D

_{a}= 1 × 10

^{−11}m

^{2}/s and (

**d**) spatial: C(x,t) at D

_{a}= 1 × 10

^{−11}m

^{2}/s; (

**e**) temporal: C(x,t) at D

_{a}= 1 × 10

^{−12}m

^{2}/s and (

**f**) spatial: C(x,t) at D

_{a}= 1 × 10

^{−12}m

^{2}/s.

**Table 1.**The through-diffusion conditions for the HTO and Se(IV) in compacted and intact clay rocks.

Item. Type RN | 1 | 2 | |||||
---|---|---|---|---|---|---|---|

Compacted Powder | Intact Rocks | ||||||

HTO | Se(IV) | ||||||

Column | No. 1 | No. 2 | No. 3 | No. 4 | No. 5 | No.6 | |

Bulk density | 1.6 | 1.8 | 2.0 | 2.2 | 2.4 | - | |

Initial HTO activity A_{0} (dpm/mL)/Se Conc. Co (ppm) | HTO: 40 dpm/mL (V_{0} = 5000 mL) | Se(IV): 3000 ppm |

Parameters | Sorption | |
---|---|---|

1-Site | 2-Site | |

C_{e} | 7.85 × 10^{−1} | 7.76 × 10^{−1} |

λ1 | 4.88 × 10^{0} | 1.02 × 10^{1} |

λ2 | - | 1.86 × 10^{−1} |

f | - | 7.67 × 10^{−1} |

LSE | 3.48 × 10^{−3} | 3.29 × 10^{−4} |

Parameter | K | M (mol/g) | R-Squared |
---|---|---|---|

Se(IV) | 3.67 × 10^{2} | 1.75 × 10^{−4} | 0.9261 |

Item | HTO | Se | ||||
---|---|---|---|---|---|---|

No. 1 (1.6) | No. 2 (1.8) | No. 3 (2.0) | No. 4 (2.2) | No. 5 (2.4) | No. 6 (-) | |

α | 0.4074 | 0.3330 | 0.2593 | 0.1852 | 0.1111 | 2.9546 |

D_{a} × 10^{−10} (m^{2}/s) | 1.92 | 1.77 | 2.49 | 2.84 | 3.53 | 0.011 |

D_{e} × 10^{−11} (m^{2}/s) | 7.81 | 5.89 | 6.47 | 5.25 | 3.92 | 0.324 |

K_{d} (mL/g) | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 1.10 |

t_{d} | 16.67 | 15.29 | 21.55 | 24.51 | 40.50 | 1.34 |

R-squared | 0.99 | 0.99 | 0.99 | 0.99 | 0.99 | 0.97 |

_{b}× K

_{d}.

D_{a} (m^{2}/s) | T (Days) in Temporal Variation | T (Days) in Spatial Variation (x = 0.299 cm) |
---|---|---|

1.00 × 10^{−10} | 0.25 | 0.46 |

1.00 × 10^{−11} | 2.25 | 4.70 |

1.00 × 10^{−12} | 18.0 | 46.8 |

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**MDPI and ACS Style**

Lee, C.-P.; Hu, Y.; Tien, N.-C.; Tsai, S.-C.; Shi, Y.; Liu, W.; Kong, J.; Sun, Y. Molecule Diffusion Behavior of Tritium and Selenium in Mongolia Clay Rock by Numerical Analysis of the Spatial and Temporal Variation. *Minerals* **2021**, *11*, 875.
https://doi.org/10.3390/min11080875

**AMA Style**

Lee C-P, Hu Y, Tien N-C, Tsai S-C, Shi Y, Liu W, Kong J, Sun Y. Molecule Diffusion Behavior of Tritium and Selenium in Mongolia Clay Rock by Numerical Analysis of the Spatial and Temporal Variation. *Minerals*. 2021; 11(8):875.
https://doi.org/10.3390/min11080875

**Chicago/Turabian Style**

Lee, Chuan-Pin, Yanqin Hu, Neng-Chuan Tien, Shih-Chin Tsai, Yunfeng Shi, Weigang Liu, Jie Kong, and Yuzhen Sun. 2021. "Molecule Diffusion Behavior of Tritium and Selenium in Mongolia Clay Rock by Numerical Analysis of the Spatial and Temporal Variation" *Minerals* 11, no. 8: 875.
https://doi.org/10.3390/min11080875