Materials

GMZ and Anji bentonite powders were obtained from Gaomiaozi, Inner Mongolia, and Anji, Zhejiang Province, respectively. The GMZ bentonite has a grain density of 2660 kg/m³, particle size (d50) of 7.1 μm, cation exchange capacity of 77.3 meq/100 g, and external surface area of 25.6 m²/g. The mineral composition is 74.5% montmorillonite, 12 wt% quartz, 7 wt% cristobalite, 4 wt% feldspar, 1 wt% calcite, and 1 wt% kaolinite [38]. In contrast, the Anji bentonite has a particle size (d50) of 11.6 μm, cation exchange capacity of 76 meq/100 g, and external surface area of 60.3 m²/g. The mineral composition is 46 wt% montmorillonite, 33 wt% quartz, 10 wt% orthoclase, 8 wt% microcline, and 3 wt% calcite [27].

Stock solutions of $\mathrm{Re}{\text{O}}_4^{-}$, ${\text{HCrO}}_4^{-}$, and I^- were prepared by weighing certain amounts of KReO₄, K₂Cr₂O₇, and NaI, and then dissolving them in 200 mL of NaCl solution. The initial concentrations of $\mathrm{Re}{\text{O}}_4^{-}$, ${\text{HCrO}}_4^{-}$, and I^- were 1.12 × 10mol/L, (0.26-2.14)×10^-3 mol/L, and 0.04 × 10^-3 mol/L, respectively. An Optima 7000DV inductively coupled plasma optical emission spectrometer (ICP-OES,PerkinElmer, USA) was used to measure the concentrations. All reagents used in this study were of analytical grade.

Diffusion method

A through-diffusion method, which measures the diffusion parameters of ions through a specific thickness of porous materials, was applied to investigate the anion ($\mathrm{Re}{\text{O}}_4^{-}$, ${\text{HCrO}}_4^{-}$, and I^-) diffusion in compacted bentonite. The experiments were performed using 0.10–0.50 mol/L NaCl solution, with the compacted dry density ranging from 1300 to 1700 kg/m³, pH of 5.6±0.1, and a temperature of 15 ± 3 ℃.

The bentonite powder was compacted into blocks (Φ 2.54 cm × 1.2 cm). Two stainless-steel filters ($\text{Φ}$ 2.54 cm × 0.1 cm) were used to sandwich the blocks. Then, the entire assembly was inserted into a cylindrical cell. After the bentonite blocks were saturated with 0.10–0.50 mol/L NaCl solution for one month, a reservoir connected to one side of the diffusion cells (x = 0) was replaced with 200 mL of the prepared stock solution containing $\mathrm{Re}{\text{O}}_4^{-}$, ${\text{HCrO}}_4^{-}$, and I^-. The other side of the diffusion cell (x=L) was connected to a target reservoir filled with 10 mL of NaCl solution. The target reservoir was replaced at given intervals to maintain a low anion concentration gradient, ensuring that it remained at less than 5% of the concentration at x = 0. A detailed description of the equipment and experimental procedure can be found in the literature [12].

The self-programmed Fitting for Diffusion Parameters software was used to calculate the rock capacity factor and effective diffusion coefficient by analyzing the accumulated mass as a function of time. The reliability of the two parameters was evaluated by examining the consistency between the calculated and experimental flux results.

Multi-porosity model

A multi-porosity model was established for the microstructure of montmorillonite because montmorillonite is the predominant mineral in bentonite. This model considers only the through-pores of compacted bentonite, where the total porosity (ε_tot) is subdivided into three components: diffuse double-layer porosity (ε_ddl), interlayer porosity (ε_il), and free-layer porosity (ε_free) [27, 39]. When compacted bentonite is saturated with an aqueous solution, the diffuse double-layer pores form transition zones from the surface of the bentonite particles to free pore water, containing a deficit of anions, water molecules, and an excess of cations. The interlayer pores contain cations and water molecules. Excess cations compensate for the charge deficit of the tetrahedral-octahedral-tetrahedral layers. By contrast, water molecules are arranged in layers [7]. Free-layer pores are spaces that comprise charge-balanced anions, cations, and water molecules.

Owing to the anionic exclusion effect, anionic radionuclides can barely enter the interlayer pores of bentonite. Therefore, the model assumes that the free-layer pores are the predominant diffusion paths, and the accessible porosity ε_acc is defined as ${\epsilon }_{\text{acc}}\approx {\epsilon }_{\text{free}}={\epsilon }_{\text{tot}}-{\epsilon }_{\text{ddl}}-{\epsilon }_{\text{il}}\mathrm{.}$ (1) The diffuse double-layer porosity ${\epsilon }_{\text{ddl}}$, which depends on the ionic strength, external surface area, mass ratio of montmorillonite, and compacted dry density, can be estimated as ${\epsilon }_{\text{ddl}}=\frac{3.09\times {10}^{-10}}{\sqrt{I}}{A}_{\text{ext}}\cdot m\cdot {\rho }_{\text{d}}\mathrm{.}$ (2) The interlayer porosity depends on the compacted dry density, water layer fraction, and the mass ratio of montmorillonite. The interlayer water is related to the degree of compaction, namely, one water layer ranged between 12.2–12.7 Å at a compacted dry density of 1600–2000 kg/m³, two water layers ranged between 15.2–15.7 Å at a compacted dry density of 1300–1600 kg/m³, and three water layers ranged between 18.4–19 Å at compacted dry densities below 1300 kg/m³ [40]. The interlayer porosity ${\epsilon }_{\text{ddl}}$ is approximately given by [39]:

(1) At ρ_d≤1300 kg/m³, ${\epsilon }_{\text{il}}=\frac{m\cdot I_{\text{f}}\cdot 1300\cdot w_3}{1000}.$ (3) (2) At ρd > 1300 kg/m³, ${\epsilon }_{\text{il}}=\frac{m\cdot I_{\text{f}}\cdot {\rho }_{\text{d}}\cdot {\displaystyle \sum }{x}_i\cdot w_i}{1000}\mathrm{,}$ (4) where the amount of interlayer water w_i is given by

• 0.119 kg H ₂O/kg clay for one water layer at 1600 kg/m³≤ρ_d≤ 2000 kg/m³,

• 0.238 kg H₂O/kg clay for two water layers at 1300 kg/m³ ≤ρ_d≤ 1600 kg/m³,

• 0.357 kg H₂O/kg clay for three water layers at ρ_d≤1300 kg/m³.

The layer faction, x_i, is approximately calculated as follows, where the subscript i denotes one, two, or three water layers [27, 39].

• At 1300kg/m³≤ρ_d≤1600kg/m³, $x_1=\mathrm{0,}{x}_2=\frac{{\rho }_{\text{d}}-{\rho }_{\text{d,3WL}\to \text{2WL}}}{{\rho }_{\text{d,2WL}\to \text{1WL}}-{\rho }_{\text{d,3WL}\to \text{2WL}}}\mathrm{,}{x}_3=1-x_2\mathrm{.}$ (5) • At 1600kg/m³≤ρ_d≤2000kg/m³, $x_1=\frac{{\rho }_{\text{d}}-{\rho }_{\text{d,2WL}\to \text{1WL}}}{{\rho }_{\text{d,1WL}}-{\rho }_{\text{d,2WL}\to \text{1WL}}}\mathrm{,}{x}_2=1-x_1\mathrm{,}{x}_3=0.$ (6) The effective diffusion coefficient is estimated by combining Eq. (1) with Archie’s equation, as follows: $D_{\text{e}}=D_{\text{w}}\cdot {\epsilon }_{\text{acc}}{}^n=D_{\text{w}}\cdot {\left({\epsilon }_{\text{tot}}-{\epsilon }_{\text{ddl}}-{\epsilon }_{\text{il}}\right)}^n\mathrm{.}$ (7) where D_w denotes the species diffusion coefficient in water. $\mathrm{Re}{\text{O}}_4^{-}$ is 1.46×10^-9 m²/s, ${\text{HCrO}}_4^{-}$ is 1.13×10^-9 m²/s, and I^- is 2.0×10^-9 m²/s [41]. $n(-)$ denotes the cementation factor.

Database description and analysis

The training dataset was obtained from two sources. One is a dataset containing experimental instances from the JAEA-DDB (223 instances, 1989–2005) [42] and 15 publications (99 instances, 2006–2024), which are listed in Table S1 in the Supporting Information. The other database contained pseudo-instances produced using the multi-porosity model (15,000 instances) [27]. Table 1 summarizes the statistical information for the datasets. For the dataset collected from JAEA-DDB and the literature, only instances of anion diffusion in bentonite were chosen.

Data pre-processing was performed using the Mahalanobis distance (MD) to remove outliers. MD is a distance measure used extensively in multivariate spaces. This accounts for the mean and covariance of the data. The cutoff point (di) is defined as [37] $d_i=\sqrt{\left(X_i-\overline{X}\right)\cdot C^{-1}\cdot \left(X_i-\overline{X}\right)}\mathrm{,}$ (8) where C, Xi, and $\overline{X}$ are the covariance matrix of the sample, the object vector, and the arithmetic mean vector, respectively. In this study, the cutoff point was set to five, resulting in 29 instances as outliers. Therefore, the dataset contained 197 instances from JAEA-DDB and 96 instances from publications on machine-learning models.

The input features were the rock capacity factor, compacted dry density, mass ratio of montmorillonite, species diffusion coefficient in water, ionic strength, and temperature. The input features for the multi-porosity model dataset were the external surface area, mass ratio of montmorillonite, ionic strength, accessible porosity, compacted dry density, cementation factor, fitting parameter, and species diffusion coefficient in water. Among these features, the rock capacity factor indicates the ability of the bentonite barrier to impede radionuclide diffusion into the granite rock. If the rock capacity factor is less than the total porosity, it is equal to the accessible porosity. The external surface area, accessible porosity, and cementation factor indicate the bentonite characteristics, while the species diffusion coefficient in water indicates the radionuclide properties. The remaining features, such as the temperature, ionic strength, and compacted dry density, are parameters related to the experimental conditions. The effective diffusion coefficient is the only output feature.

The test dataset consisted of eight instances obtained from the diffusion of $\mathrm{Re}{\text{O}}_4^{-}$, ${\text{HCrO}}_4^{-}$, and I^- using the through-diffusion method. Given that both the effective diffusion coefficient and the species diffusion coefficient in water were in the range of 10^-13 to 10^-9m²/s, a logarithmic conversion was applied to maintain consistency with the range of other features, which spanned values from 0 to 2000. This data pre-processing improves the performance [37].

Performance evaluation of the machine-learning model

The predictive accuracy was evaluated using R² and MSE. These parameters were respectively calculated as follows: $R^2=1-\frac{{{\displaystyle \sum _{i=1}^N\left(\log D_{\text{e}\mathrm{,}i}^{\text{exp}}-\log D_{\text{e}\mathrm{,}i}^{\text{pred}}\right)}}^2}{{{\displaystyle \sum _{i=1}^N\left(\log D_{\text{e}\mathrm{,}i}^{\text{exp}}-\log D_{\text{e,ave}}^{\text{exp}}\right)}}^2}\mathrm{,}$ (9) $MSE=\frac{1}{N}{\displaystyle \sum _{i=1}^N{\left(\log D_{\text{e}\mathrm{,}i}^{\text{exp}}-\log D_{\text{e}\mathrm{,}i}^{\text{pred}}\right)}^2}\mathrm{,}$ (10) where N is the number of instances. $\log D_{\text{e}\mathrm{,}i}^{\text{exp}}$ and $\log D_{\text{e}\mathrm{,}i}^{\text{pred}}$ represent the experimental and predicted output values, respectively. $\log D_{\text{e,ave}}^{\text{exp}}$ denotes the average of experimental instances. Increased predictive accuracy is associated with an increase in R² and a decrease in MSE.

Five-fold cross-validation (CV) was employed to mitigate overfitting, a situation characterized by high predictive performance in the training or validation datasets, but low accuracy in the test dataset, resulting in poor generalization and reduced robustness of the machine-learning model. In this approach, the dataset was randomly divided into five equally sized subsamples, with four subsamples used for training and one used for testing.

Database distribution and characteristics

Figures 1a-f show the dependence of the effective diffusion coefficient on each input feature for the JAEA-DDB/publications dataset. The dependence of the multi-porosity model can be found in a previous study [27]. The histograms and kernel curves displayed on the top and right sides of each plot correspond to the distribution of the input features and effective diffusion coefficient. The shape of the curves is determined by the data point concentration; a high data point concentration results in a higher peak amplitude.

Fig. 1Full-screen View

Distribution and characteristics of the input features and output variable

The rock capacity factor can be obtained directly using the through-diffusion method, or calculated as follows [27, 39]: $\alpha =\epsilon +{\rho }_{\text{d}}\cdot K_{\text{d}}\mathrm{,}$ (11) where ε (-), ρ_d (kg/m³), and K_d (m³/kg) denote the porosity, compacted dry density, and distribution coefficient, respectively. Specifically, the total porosity of compacted bentonite is equivalent to that of neutral molecules, such as HTO [10, 11, 17]. In contrast, the accessible porosity was assumed to be the porosity of anionic radionuclides such as ³⁶Cl^- and ¹²⁵I^- [10-12]. This assumption is based on the ionic exclusion effect in which radionuclides are hindered from accessing the negatively charged bentonite surface [17, 43]. The rock capacity factor ranged from 0.01 to 19.08 (Table 1). Most data points are concentrated below two. Specifically, 13.3% of the data points exceeded two, 23.2% ranged from unity to two, and 63.5% were less than unity. Notably, only nine data points were higher than ten (Fig. 1a). Oscarson et al. [44] reported that the rock capacity factors of ${}^{99}{\text{TcO}}_4^{-}$ and ¹²⁵I^- were greater than five, accounting for 4.9% of the high values. This abnormal observation may be attributed to the calculation method using Eq. (10). The distribution coefficient may have been overestimated, because it was measured through sorption experiments with powdered bentonite.

The effective diffusion coefficient increased with a decrease in the compacted dry density (Fig. 1b), which is consistent with previous experimental results [13-18]. It was not surprising that the effective diffusion coefficient increased with increasing species diffusion coefficient in water and temperature (Figs. 1c and d). This observed behavior can be attributed to adherence to two diffusion laws: one is known as Archie’s law, which can be expressed as $D_{\text{e}}=D_{\text{w}}\cdot {\epsilon }^n$ [39, 43]. The second is represented by the Arrhenius equation given by $D_{\text{e}}=A\cdot {\text{e}}^{-Ea/RT}$ [45]. Figures 1e and 1f show that the data point distribution is smeared out for concentrated data within a limited range. This can be explained by the fact that there is no strict one-to-one dependency of the effective diffusion coefficient on the ionic mass ratio of montmorillonite and ionic strength.

Measurement of diffusion parameters using the through-diffusion method

The through-diffusion method was used to determine the diffusion parameters of $\mathrm{Re}{\text{O}}_4^{-}$, ${\text{HCrO}}_4^{-}$, and I^- in compacted bentonite. Figure 2 shows the breakthrough curves under various salinity and compaction conditions. The impact of salinity on diffusion in GMZ bentonite is shown in Figs. 2a-2e, while the effect of compacted dry density in Anji bentonite is presented in Figs. 2f-h. The red dots and lines represent the flux results, while the blue dots and lines represent the accumulated mass results. The solid dots represent the experimental data, the lines represent the calculated results for the relationship between the accumulated mass or flux over time, and the shaded area indicates the calculated upper and lower limits, which consider the uncertainties of the rock capacity factor and the effective diffusion coefficient. These uncertainties are associated with various factors, such as the sample weight, volume of bentonite block and stainless-steel filters, dead volume of the diffusion cells, and ICP-OES measurements. The ionic strength I (mol/L) was calculated as follows: $I=\frac{1}{2}{\displaystyle \sum _{i=0}^n{C}_i{z}_i^2}\mathrm{,}$ (12) where C_i is the total concentration of each species i in a solution, including Na⁺, K⁺, Cl^-, $\mathrm{Re}{\text{O}}_4^{-}$, ${\text{HCrO}}_4^{-}$, and I^-. zi is the charge number of species i.

Fig. 2Full-screen View

Flux, $J(L\mathrm{,}t)$, and accumulated mass, A_cum, as a function of time. pH = 5.6 ± 0.1; C₀(Re) = 1.12 × 10^-3 mol/L; C₀(Cr) = (0.26-2.14) × 10^-3 mol/L; C₀(I)=0.04×10^-3 mol/L; T =15±3℃

Table 2 lists the diffusion parameters of $\mathrm{Re}{\text{O}}_4^{-}$, ${\text{HCrO}}_4^{-}$, and I^- in compacted bentonite. Both the rock capacity factor and effective diffusion coefficient in compacted bentonite increase as the ionic strength increase and the compacted dry density decrease. These trends are consistent with those reported in previous experimental studies [9, 16-19, 24]. However, when comparing these results with those of previous studies on GMZ bentonite [24], it was observed that $\mathrm{Re}{\text{O}}_4^{-}$ had a higher effective diffusion coefficient, which can be explained by the fact that a lower compacted dry density was investigated in this study. In comparison with previous studies [9, 18], higher effective diffusion coefficient values for ${\text{HCrO}}_4^{-}$ were observed. This could also be explained by the higher ionic strength and lower mass ratio of montmorillonite in the ${\text{HCrO}}_4^{-}$ diffusion experiments and the higher ionic strength and compacted dry density in the I^- diffusion experiments. The experimental results comply with the diffusion rules for anions and fall within the training dataset range. Therefore, the test dataset was deemed suitable for evaluation purposes. It is noteworthy that the rock capacity factors of the measured $\mathrm{Re}{\text{O}}_4^{-}$, ${\text{HCrO}}_4^{-}$, and I^- are less than the total porosity, indicating that they cannot be adsorbed onto the bentonite surface. The rock capacity factors are equivalent to the accessible porosity.

Prediction by the machine-learning algorithms

Six machine-learning algorithms, namely LightGBM, XGBoost, Catboost, ANN, RF, and SVM, were employed to predict the effective diffusion coefficient using two training datasets. One dataset comprised eight input features and 15,000 pseudo-instances produced by the multi-porosity model. The other dataset included six input features and 293 instances sourced from JAEA-DDB and 15 publications (Table 1). The datasets were divided into training and validation sets at a ratio of 4:1. The test dataset for the machine-learning models consisted of the experimental results listed in Table 2. LightGBM, XGBoost, Catboost, and RF are ensemble-learning algorithms, while ANN and SVM are traditional learning algorithms. Table 3 lists the mean values of the two performance metrics for the test datasets of the six machine-learning models using the five-fold cross-validation technique. LightGBM outperformed the other machine-learning models in terms of predictive performance, achieving the highest $R_{\text{CV}}^2$ of 0.87 and the lowest $MS{E}_{\text{CV}}$ of 0.01.

Hyperparameters, which are an integral part of machine-learning models, cannot be learned from the dataset. They were set prior to model training to control the models’ learning process. The grid search (GS) method was used to tune the hyperparameters. Reasonable settings for each hyperparameter were manually predefined. The model was iterated through each combination of the specified values. For the training datasets, the cross-validation method was used for guidance. After evaluating all combinations, the parameter combination with the best model performance was obtained. Table 4 summarizes the tuned hyperparameters for each machine-learning model.

A comparison between the experimental and predicted effective diffusion coefficients is presented in Fig. 3, where the dots indicate the experimental data, the red lines represent the linear fit of the experimental data, and the shaded areas represent the 95% confidence interval. For the multi-porosity model, the predictive accuracy is ranked in descending order as SVM > XGBoost > RF > LightGBM > CatBoost > ANN (Figs. 3a-3f). The SVM outperformed the other machine-learning models in terms of predictive performance, with an MSE of 0.01 and R² of 0.83. By contrast, when using the JAEA-DDB/publications dataset, the predictive accuracy is ranked in descending order as LightGBM > CatBoost > XGBoost > SVM > RF > ANN (Figs. 3g-3l). All gradient-boosting algorithms exhibited high performance, with R² values above 0.88. LightGBM and XGBoost achieved similar predictive accuracies, with an MSE of 0.01 and R² of 0.91. The JAEA-DDB/publications outperformed the multi-porosity model. This can be attributed to the complexity of the predictive tasks that involve predicting multiple species ($\mathrm{Re}{\text{O}}_4^{-}$, ${\text{HCrO}}_4^{-}$, and I^-) under different salinity and compaction conditions. This complexity poses a significant challenge for effectively training machine-learning models using the dataset generated from the multi-porosity model, as the predictive accuracy is notably influenced by the quality of the model. In general, boosting models that combine weak learners using weight-based aggregation exhibit stronger prediction capabilities. This finding is consistent with the results of previous studies [27, 46]. Notably, LightGBM achieves a higher predictive accuracy among boosting models because it utilizes two innovative techniques: gradient-based one-side sampling and exclusive feature bundling [27, 47].

Fig. 3Full-screen View

(Color online) Comparison between the experimental and predicted effective diffusion coefficient results based on (a-f) the multi-porosity (MP) model dataset and (g-l) the diffusion database system from the Japan Atomic Energy Agency (JAEA-DDB) and 15 publications using the (a, g) Light Gradient-Boosting, (b, h) Extreme Gradient-Boosting, (c, i) Categorical Gradient-Boosting, (d, j) Artificial Neural Network, (e, k) Random Forest, and (f, l) Support Vector Machine models

Shapley Additive Explanation and Feature Importance analyses

The Shapley Additive Explanation (SHAP) and Feature Importance (FI) methods are two widely used feature attribution methods that can identify the weight or significance of input features driving the predictions [34]. Although SHAP and FI analyses employ distinct techniques to characterize their importance, they can reflect the influence on the predicted output by ranking the importance of the input features [48]. In this study, they were applied to the LightGBM model using the JAEA-DDB/publications dataset, which yielded the highest predictive accuracy among the six machine-learning models. Higher SHAP and FI values for a feature indicate a greater impact on the effective diffusion coefficient. As can be seen in Fig. 4, the rock capacity factor and the compacted dry density are the top-two important input features for effective diffusion coefficient prediction. For the remaining four features, the FI analysis is ranked in descending order as follows: T >logD_w > I > m, while the SHAP analysis ranked them as: ogD_w≈T>m>I. The difference in the montmorillonite mass ratio ranking between the two analyses can be attributed to the underlying principles and assumptions of the two analysis technologies.

Fig. 4Full-screen View

Feature Importance and absolute mean Shapley Additive Explanations values for each feature using the Light Gradient-Boosting model

Ionic strength is closely associated with the electrical double layer located at the bentonite interface [9]. Although ionic strength had a limited effect on the effective diffusion coefficient prediction (Fig. 4), its influence on radionuclide diffusion has been investigated in previous experimental diffusion studies [9, 19, 24]. The effective diffusion coefficient increases in solutions with high salinity until the ionic strength exceeds 0.5 mol/L. This observation is explained by the minimum thickness of the electrical double layer, which results in negligible diffuse double-layer pores and a maximum width of free layer pores [9, 19, 24]. In addition, there is an ongoing debate on the effect of the electrical double layer on radionuclide diffusion [9, 49]. This can be explained by the small porosity proportion in the diffused double-layer pores [28]. It is worth noting that the weight of a feature relies on input features, instances, and algorithms. Further research is needed to clarify the importance of ionic strength in radionuclide diffusion.

Partial Dependence Plot analysis

Partial Dependence Plot (PDP) analysis indicates the ability to analyze the relationship between each input and output features [34]. These plots provide a quantitative assessment of the positive and negative effects of the six input features on the effective diffusion coefficient (Fig. 5). A feature with a strong impact on the output variable exhibits significant changes in the PDP curves, indicating a significant contribution to the model’s prediction. By contrast, a feature with little impact results in flat or nearly constant PDP curves.

Fig. 5Full-screen View

Partial Dependence Plot analysis of the effect of input features on the effective diffusion coefficient. The blue lines represent the partial dependence value, while the gray columns represent the data point distribution for each input feature at a certain value

The rock capacity factor, species diffusion coefficient in water, ionic strength, and temperature positively impacted the effective diffusion coefficient, whereas the compacted dry density and montmorillonite mass ratio negatively impacted it. In other words, the effective diffusion coefficient increases with increasing species diffusion coefficient in water, temperature, and ionic strength, which is consistent with Archie’s law [39, 43], the Arrhenius equation [45], and previous experimental results [10, 19-24]. Conversely, the effective diffusion coefficient decreases in compacted bentonite with a high compacted dry density and high montmorillonite mass ratio, which is also consistent with previous experimental results [9, 13-15].

Among the input features, the rock capacity factor had the most significant influence on the effective diffusion coefficient, which is in agreement with the SHAP and FI analyses. An increase from 0.01 to 19.08 in the rock capacity factor resulted in a significant increase in the PDP value from -11.16 to -9.90, representing a substantial increase of approximately 11.3% (Fig. 5a). It is worth noting that the rock capacity factor of radionuclide anions should be lower than the total porosity if the anionic exclusion effect is considered [39, 43], indicating that some anionic instances with a rock capacity factor above the total porosity threshold should be removed from JAEA-DDB. Nonetheless, these instances were retained in this study for database integrity. The percentage increase in the PDP value is ranked in descending order as follows: α (11.3%) > T (8.8%) > $\log D_{\text{w}}$ (6.5%) > I (1.3%).

The compacted dry density had a negative impact, as an increase from 400 to 1700 kg/m³ led to a decrease in the PDP value from -9.85 to -10.74, corresponding to a decrease of approximately 9.0% (Fig. 5b). This finding is consistent with the results of previous studies [13-18]. Additionally, the montmorillonite mass ratio had a negative impact; an increase from 0.33 to 1.0 led to a decrease in the PDP value from -10.02 to -10.31, corresponding to a decrease of approximately 2.8% (Fig. 5e). This indicates that bentonite has a low montmorillonite mass ratio, such as the illite/smectite mixed-layer (I/S) (m = 0.33) and Kunigel V1 (m = 0.46-0.49) bentonites, and exhibits a higher effective diffusion coefficient, which is in agreement with the findings of previous studies [7, 9, 13, 16, 25]. Generally, bentonite barriers with a higher montmorillonite mass ratio exhibit better blocking abilities against radionuclides [5]. As can be seen in Figs. 5d and 5f, the predicted effective diffusion coefficient increases with increasing ionic strength and temperature. The effect becomes significant when the ionic strength and temperature range from 0.01 to 0.6 mol/L and from 22 to 60 ℃, respectively, which is consistent with the findings of previous studies [20, 26]. This indicates that the PDP analysis provides interpretability of the diffusion law and mechanism.