1. Introduction

Spent fuels from nuclear reactors can be disposed of in high-level radioactive waste (HLW) repositories consisting of natural (NBS) and engineered barrier systems (EBS), located at least 500 m underground. A natural barrier implies that it acts as a barrier in its natural state, and an engineered barrier is an artificial barrier consisting of a canister, buffer, backfill, and a near-field rock. Hydraulic conductivity is critical in designing buffer that fills the space between the canister and the near-field rock. The buffer must prevent the release of nuclides to the extent possible. Groundwater infiltration through the buffer into a canister containing spent fuels should be minimized, and the nuclides must move by not advection but diffusion in buffer. Thus, the hydraulic conductivity of the buffer must be lower than that of the near-field rock at ~10⁻¹² m·s⁻¹ [1]. Buffer materials are primarily used by compressing bentonite which is the most suitable natural buffer material, into blocks [2-4]; the compacted bentonite has a low hydraulic conductivity that satisfies the required hydraulic conductivity criterion. Several models for predicting the hydraulic conductivity of buffer materials have been suggested [5, 6]. Hydraulic conductivity varies with the type, dry density, temperature, porosity, and void distribution of the buffer materials [1-6]. Thus, the hydraulic conductivity must be measured under various conditions considering the aforementioned terms to predict the hydraulic conductivity of buffer materials. Cho et al. [5] suggested a prediction model composed of an independent variable of the dry density for the hydraulic conductivity of compacted bentonite produced from Gyeongju in 1999, but this model is applicable only at 20°C. In 2020, Park et al. [6] suggested a prediction model with a high decision coefficient for the hydraulic conductivity of KJ-Ⅱ; this model which is composed of two independent variables of the dry density and temperature is valid at less than 90°C. The temperature of the buffer materials in an HLW repository should be less than 100°C [7] to prevent the bentonite of buffer material from being illitizated because illitizated bentonite has less swelling and higher hydraulic conductivity, which degrades their performance [8-9]. In addition, the swelling of bentonite buffer materials usually decreases with increase in temperature owing to dehydration [8]. However, confirming the performance of the buffer materials at approximately 100°C is critical, considering their safety when designing HLW repositories. In addition, most of the hydraulic conductivity models for buffer materials were derived from one type of bentonite [5, 6]. Thus, a hydraulic conductivity model covering several types of bentonite above 100°C is needed. Hence, in this study, a prediction model for the hydraulic conductivity of various bentonite buffer materials that is valid even at temperatures in range of 100–125°C is proposed based on various test results and literature values.

2. Hydraulic Conductivity Measurement

2.1 Sample Preparation

The KJ-II bentonite produced in Gyeongju, Republic of Korea, and selected Bentonil-WRK bentonite (selected WRK) having a montmorillonite content larger than 90% among the Bentonil-WRK bentonite produced by Clariant in Korea were used for this research. Both bentonites were Ca-type. The basic properties of KJ-II [10] and Bentonil- WRK are listed in Table 1. The KJ-II is composed of 61.9% montmorillonite, 20.9% albite, 5.3% quartz, 4.1% cristobalite, 7.4% calcite, and 3% heulandite [10]. The selected WRK is composed of approximately 94.7% montmorillonite, 2.2% opal, 2% muscovite, 1% quartz, and less than 1% feldspar and zeolite. The particle size distributions for the KJ-Ⅱ and Bentonil-WRK are shown in Fig. 1. Block-type buffer materials with a dry density of ~1.6 g·cm⁻³ were made by compacting the KJ-Ⅱ and selected WRK. The samples had a cylindrical shape with a diameter of 5 cm and thickness of 1 cm, and the initial water content was 12%.

Table 1

Basic properties of KJ-Ⅱ [10] and Bentonil-WRK

Fig. 1

Particle size distribution curves of KJ-Ⅱ and Bentonil-WRK.

2.2 Test Method

The sample was placed in a confined cell in the furnace. A porous disk was installed on the top and bottom of the cell (Fig. 2). The base pressure pump injected deionized (DI) water at a pressure of 1.2 MPa to the bottom of the sample, and the water that permeated the sample flowed into the back pressure pump applying a pressure of 0.2 MPa to the top of the sample. Before measuring the hydraulic conductivity, the sample was left at a hydraulic head difference of 1 MPa until saturation. The sample was considered fully saturated when the water injected into the bottom of the sample came out upward. Approximately seven days were required for the sample to be fully saturated. The hydraulic conductivity (K) was calculated using Darcy’s law as follows:

Fig. 2

Setup of the hydraulic conductivity measurement for KJ-Ⅱ. (a) setup picture, (b) schematic diagram.

where Q is the flow rate, Δh is the hydraulic head difference, L is the thickness of the sample, and A is the crosssectional area of the sample. The sample temperature was adjusted by varying the furnace temperature to a specific value.

2.3 Test Results

The flow rate of the water flowing out of the base pressure pump and that entering the back pressure pump was the same (Fig. 3). The hydraulic conductivity of the samples increased with temperature (Fig. 4). As the temperature increases, the lattice contraction occurs in the compacted bentonite as the adsorbed water turns into free water, so that the hydration weakens [11-15], and the swelling caused by the hydration of bentonite decreases, allowing water to penetrate the bentonite better. For this reason, the slope of the hydraulic conductivity with temperature was larger for the selected WRK than for KJ-Ⅱ. The selected WRK, which has a 1.5 times higher montmorillonite content than KJ-Ⅱ, which is a material with high swelling property depending on the temperature, exhibited a ~1.2 times higher rate of increase in hydraulic conductivity with temperature than KJ-Ⅱ. Thus, it is judged that the change in hydraulic conductivity with temperature is greater in the selected WRK than in KJ-II.

Fig. 3

Change in injection and inflow volume measured by base and back pressure pump, respectively, at 60℃.

Fig. 4

Hydraulic conductivity with temperature for KJ-Ⅱ and selected WRK.

3. Prediction Model for Hydraulic Conductivity

3.1 Collection of Hydraulic Conductivity Data

To increase the validity of the hydraulic conductivity prediction model, the results for hydraulic conductivity under various buffer material conditions were extracted from literature [16-19], as shown in Table 2. For developing the hydraulic conductivity prediction model, 56 data points, of which 10 were from experiments in this study and 46 extracted from the literature, were used.

Table 2

Hydraulic conductivities for various bentonite buffer materials from literature

3.2 Decision of Independent Variables

Before deriving the hydraulic conductivity prediction model, the correlation with hydraulic conductivity was analyzed for several variables to confirm which factors had the greatest influence on the hydraulic conductivity of bentonite buffer materials using Python. The correlation analysis result between the variables was shown as a heatmap that outputs various information that can be expressed in color as a visual graphic in the form of heat distribution on a certain image (Fig. 5). The p-value is in a range from 0 to 1, which is a probability that the null hypothesis is true. In this analysis, a p-value less than 0.05 means that the independent variable has a significant impact on the dependent variable. Temperature and dry density were confirmed to show a relevant correlation with hydraulic conductivity by the analysis (Table 3). However, the variables did not correlate with each other. The temperature variable showed a higher correlation with the hydraulic conductivity than the dry density variable; temperature had a positive correlation, and dry density had a negative correlation. This indicates that temperature has a greater effect on the hydraulic conductivity than dry density, and the higher the temperature, the higher is the hydraulic conductivity. Conversely, the higher the dry density, the lower is the hydraulic conductivity. Consequently, the hydraulic conductivity prediction model had independent temperature and dry density variables.

Fig. 5

Heatmap of the correlation analysis between the variables related to the hydraulic conductivity of bentonite buffer materials.

Table 3

P-values of variables for hydraulic conductivity of bentonite buffer materials

3.3 Derived Hydraulic Conductivity Prediction Models

Multiple linear regression analysis was conducted to derive a regression model using Python. The regression model comprises two independent variables, dry density and temperature, and a constant (Eq. 2, Table 4). The Durbin–Watson value was 1.205, indicating that the independence of the residuals was recognized to some extent. The regression model had a determination coefficient (R²) of 0.665 and an adjusted R² of 0.65, with an R² value less than that of the model proposed by Park et al. [6] (Fig. 6).

Table 4

Coefficients of a multiple linear regression model

Fig. 6

Comparison of the hydraulic conductivities between the measured values and those predicted by the multiple linear regression model.

To enhance the prediction model, a new variable, DT, was derived as follows:

The power type prediction model (DT model) for hydraulic conductivity with R² of 0.778 was derived with the variable DT as follows:

4. Comparison of Hydraulic Conductivity Prediction Models

To compare the accuracy of the finally derived DT model, new hydraulic conductivity values, which were not used in deriving the DT model and the model from Park et al. [6], were extracted from the literature [20-22], and an additional test on the other WRK sample was conducted. The two models were compared with the new data (Table 5). When comparing the model and the measured values for the new data, the DT and Park et al. (2020) models were less accurate (Fig. 8). Since the hydraulic conductivity is a very small value of 10 to the −12th or −13th power, if the first digit of the measured value and the predicted value differ even slightly, it moves away from the 1:1 line in red between the measured value and the predicted value. In the case of the new data, the dry density condition of 1.9 g·cm⁻³ that was not used to derive the models was hypothesized to have influenced the results. In addition, in deriving the two models, there was insufficient hydraulic conductivity data for samples with a dry density of 1.8 g·cm⁻³ or more. Thus, both the DT and Park et al. (2020) models have limitations with relatively lower accuracy than the results for samples with a dry density of less than 1.8 g·cm⁻³ when compared with the measured values for samples with a dry density of 1.8 g·cm⁻³ or more. However, the DT model values were confirmed to be closer to the measured value than those of the Park et al. model [6].

Fig. 7

Hydraulic conductivity with variable DT.

Table 5

Data used for the comparison of the two hydraulic conductivity prediction models

Fig. 8

Comparison of the DT model and the Park et al. [6] model.

5. Conclusions

To develop a predictive model, data were obtained through hydraulic conductivity tests and from literature at 125°C or lower, and the following conclusions were obtained through data analysis.