A review and evaluation of thermal conductivity model of compacted bentonite and its mixture

The thermal conductivity of bentonite plays a crucial role in analyzing the heat transfer process and determining the temperature field distribution within deep geological repositories. Despite considerable efforts in modeling the thermal conductivity of compacted bentonite and its mixtures, a comprehensive synthesis of these studies has not been previously undertaken. This research aimed to thoroughly review predictive models for the thermal conductivity of compacted bentonite and its mixtures, assessing their performance against a substantial dataset comprising 495 measurements of GMZ and MX80 bentonite. Through a systematic compilation and evaluation of seven models for compacted bentonite and three models for bentonite mixtures, the study identified TC2008 and LC2016 as the most accurate models for GMZ and MX80 compacted bentonite, respectively, whereas PT2021 emerged as the superior predictor for GMZ and MX80 bentonite mixtures. This exploration revealed the absence of a single, universally accurate model capable of predicting the thermal conductivities across all bentonite variants, highlighting the necessity for researchers to judiciously select the most fitting model for predicting the thermal conductivity of bentonite. Furthermore, we expressed the inherent limitations in current thermal conductivity models for compacted bentonite and its mixtures, and proposed directions for future inquiry in this domain.


Introduction
In deep geological repositories, compacted bentonite or its mixtures serve as optimal buffer materials, crucial for mitigating groundwater infiltration, preventing radionuclide migration, and dissipating decay heat from radioactive waste [1].For the operational integrity of the repository, the buffer material should efficiently conduct the heat from radionuclide decay to the host rock for maintaining the maximum design temperature of the compacted bentonite below 100 °C and assure the safe functioning of the repository [2].Thus, understanding the thermal conductivities of compacted bentonite and its mixtures is both theoretically and practically significant.
Thermal conductivity stands as a key determinant of the heat transfer capabilities of the bentonite.Existing research [3,4] reports the trend of variations in the thermal conductivity of the compacted bentonite under different conditions, including mineral composition, particle arrangement, dry density, water content, saturation level, temperature, and chemical makeup, alongside examining the effects of quartz sand and other additives on the thermal conductivity of bentonite [5,6].Although several models 1330 (2024) 012060 IOP Publishing doi:10.1088/1755-1315/1330/1/012060 2 have been developed to predict the thermal conductivity models of unfrozen soil, frozen soil, and sandy soil, the thermal conductivity models of compacted bentonite and its mixtures have not been comprehensively evaluated, especially for compacted bentonite and its mixtures.
This study aims to fill this gap by extensively reviewing thermal conductivity models for various compacted bentonite/bentonite mixtures, assembling a significant dataset for GMZ and MX80 bentonite across varied physical properties (e.g., dry density and water content).This dataset facilitates an indepth evaluation of the thermal conductivity models to understand their strengths and weaknesses.Such an analysis is instrumental in assessing the long-term engineering performance of deep geological repositories within the intricately interlinked thermo-hydro-mechanical-chemical environments.

A review of thermal conductivity models of bentonite
An exhaustive review of the literature revealed seven thermal conductivity models for compacted bentonite and three models for bentonite mixtures.These models of compacted bentonite fall into three distinct categories: (1) empirical, (2) normalized, and (3) theoretical.Each model is identified by its initial date and year of publication to facilitate clearer scientific discourse.
(2) SK1998 model Sakashita and Kumada [10] introduced a thermal conductivity model for compacted bentonite, incorporating variables such as porosity.where n denotes the porosity and Sr indicates the degree of saturation.
(3) TC2008 model Tang et al. [11] derived a model indicating a linear relationship between the thermal conductivity of compacted bentonite and the air volume fraction g with coefficients A and B were obtained by linear fitting of the experimental data. =   + . (3)

Normalized model
(1) KS1983 model Johansen [12] introduced a normalized approach for assessing thermal conductivity, using the thermal conductivities of dry and saturated soil as boundaries and defining the Kersten number Ke = ( − dry)/(sat − dry) to estimate thermal conductivity under varying saturation levels.Expanding on this normalization concept, Knutsson [13] proposed a formula to calculate the thermal conductivity of MX80 compacted bentonite, particularly when Sr  10%,  = 0.034n −2.1 . = (1.0 +  10   )(2 1− 0.56  − 0.034 −2.1 ) + 0.034 −2.1 .(4) (2) ZZ2021 model Zhang et al. [14] considered the air volume fraction (g) as a crucial determinant of compacted bentonite's thermal conductivity, proposing a novel formula based on the KS1983 model.This model considers the thermal conductivities of quartz: where Q and o represent the thermal conductivities of quartz and other minerals, respectively, w and a are the thermal conductivities of water and air, respectively, and q refers to the mass fraction (%) of quartz,  and  indicate the fitting parameters.

Theoretical models
(1) LC2016 model Börgesson et al. [15] conceptualized solids, water, and air in compacted bentonite as rectangular blocks, assumed random combinations between the blocks, and derived a weighted geometric average model to calculate the thermal conductivity.Lee et al. [16] considered the inauthenticity of the geometric mean model, and proposed the following formula: where s denotes the thermal conductivity of the soil skeleton, and m and p denote the parameters considering the inauthenticity of the geometric mean model, which are generally obtained by fitting the experimental data.
(2) CZ2015 model Chen et al. [17] employed homogenization techniques to solve the ellipsoidal matrix inclusion problem and developed a thermal conductivity model for compacted bentonite.
where  denotes a geometric parameter and  represents the ratio of the length to width of the pore.The symbol p represents the type of fluid in the pore; p = w represents water, and p = represents gas.

Thermal conductivity model of bentonite mixture
(1) CL2011 model Cho et al. [18] proposed an improved geometric mean model to predict the thermal conductivity of a bentonite-sand mixture: where m denotes the thermal conductivity of the mixture and nm and ns represent the volume fractions of the mixture and matrix, respectively.a, b, and p indicate the parameters considering the inauthenticity of the geometric mean model.In this paper, (2) WC2015 model Wang et al. [19] proposed a thermal conductivity model for a bentonite-sand mixture based on the CZ2015 model: where  represents w (water), a (gas), or m (mixture); w = a (related to ); w = a and m obtained by fitting.In this study, w = a = 0.0362 and m = 1.0965 for the GMZ bentonite and w = a = 0.0280 and m = 0.5995 for the MX80 bentonite.
(3) PT2021 model Peng et al. [20] proposed the following formula for the thermal conductivity of a bentonite-graphite mixture.

Results and discussion
The datasets employed for evaluating the thermal conductivity models were sourced from scientific peer-reviewed journals and research reports on GMZ and MX80 bentonite.These measurements were rigorously selected based on several key criteria: (1) the thermal conductivity was determined at room temperature and atmospheric pressure using transient methods, independent of the temperature effects; (2) the sample preparation process and comprehensive physical properties of the bentonite, including particle density, dry density, mineral composition (notably quartz content), water content, and degree of saturation, were detailed extensively; (3) The focus was on pure bentonite or mixtures where bentonite constituted over 50% of the material.
To assess the performance of various thermal conductivity models, four statistical indices were utilized: (1) Root-mean-square error (RMSE): Xp represents the predicted value, Xm indicates the measurement, Xave denotes the average value of the measurement, and n indicates the number of measurements.
These indices quantify the discrepancies between model predictions and actual measurements, with RMSE and AD indicating the magnitude of deviation, NSE reflecting the relative accuracy of residuals (closer to 1 signifies higher accuracy); PBIAS measures the average percentage difference, where positive values indicate overestimation and negative values suggest underestimation by the model.
As observed, the TC2008 model could yield zero or negative values in scenarios of low water content, suggesting the relationship between thermal conductivity and air volume fraction may not be linear and may necessitate extensive data across the full suction range for correction.Similar issues were noted with other models, such as the KM1982 model producing negative, zero, or unreasonably low values for low water content scenarios.In addition, because of the functional form of Ke (e.g., a logarithmic function), the KS1983 model fails to calculate the thermal conductivity with Sr = 0 because 0 is an invalid input to the logarithmic function.This phenomenon is common in thermal conductivity modeling IOP Publishing doi:10.1088/1755-1315/1330/1/0120606 [30].Therefore, the author suggests that a closed equation with wide applications should be established in the future.

Performance of thermal conductivity model of bentonite mixture
A compilation of 102 measured values from four studies [23,[31][32][33] on the GMZ bentonite mixture and 129 measurements from a single study [26] on the MX80 bentonite mixture were utilized to assess the accuracy of three bentonite mixture thermal conductivity models.
The analysis of these models is presented in table 3 and figure 2, with the most accurate model underscored in bold.For the GMZ bentonite mixture, the PT2021 model outperformed others (RMSE = 0.16 W m −1 °C−1 , AD = 0.03 W m −1 °C−1 , NSE = 0.72, PBIAS = −2.66%),correctly predicting 86.27% of the measurements with less than 20% error.Conversely, the WC2015 model tended to underestimate measured values by 7.22%, highlighting the diverse effectiveness of different models and the nuanced nature of predicting the thermal conductivity in bentonite mixtures.

Discussion
The theoretical models follow various assumptions with each parameter possessing a defined physical significance, which often feature complex expressions challenging practical applications.Conversely, the empirical model fits the measurements of a certain type of bentonite, and the fitting parameters increase the degrees of freedom, which statistically increases the fitting capability of the models.
Although the empirical models tailored to specific bentonite types with adjustable parameters can enhance model adaptability, their broader applicability may be limited.Despite their simplicity and ease of integration into numerical simulations [30], a universal model that precisely forecasts the thermal conductivities across all bentonite types remains elusive.
For instance, the SK1998 model based on Kunigel bentonite overestimated the thermal conductivities for GMZ and MX80 bentonites by 6.35% and 35.82%, respectively.The accuracy of the model tends to correlate with the dataset size, suggesting that models developed and calibrated with extensive datasets could offer broader, more accurate applications [29].
The evaluation results indicated that the SK1998 model performed markedly better with the GMZ dataset than with the MX80 dataset, whereas the CZ2015 model displayed enhanced accuracy for MX80.This variation underscores that a model might excel with one dataset but falter with another, highlighting the importance of selecting the most suitable model for predicting the thermal conductivity of bentonite in engineering practices.
Extensive research has probed the impacts of factors like porosity (dry density), degree of saturation (water content), mineral content, and additives on the thermal conductivity of bentonite [2,[34][35][36].Nonetheless, recent studies examining temperature effects, heating-cooling cycles, aging time, and ion type and concentration on the thermal conductivity of bentonite lack sufficient data for model development.Additionally, the saturation process of bentonite can induce swelling or shrinkage, potentially creating cracks that disrupt heat transfer connectivity, thereby affecting thermal conductivity measurements [37,38].Current models do not fully account for the dynamic alterations in the thermal conductivity of bentonite due to changes in pore geometry during saturation, pointing to a significant avenue for future research.
Modeling and predicting the thermal conductivity of bentonite mixtures present greater challenges compared to compacted bentonite due to the introduction of additives like sand, which introduce complex interactions within the bentonite matrix.Factors such as particle size distribution, shape, the mode of bonding between the particles and the matrix, and the thermal conductivity of the additive significantly influence the thermal conductivity of the mixture.The intricacies of the interaction mechanisms between the bentonite matrix and additives remain elusive, requiring extensive future research.

Conclusions
The thermal conductivity of bentonite is pivotal for designing and evaluating the performance of deep geological repositories intended for high-level radioactive waste disposal.This study reviewed and assessed seven models for predicting the thermal conductivity of compacted bentonite and three models for assessing bentonite mixtures, utilizing datasets for GMZ and MX80 bentonite.The present findings indicated that the TC2008 model was most effective for GMZ compacted bentonite, while the LC2016 model excelled for MX80 compacted bentonite.Among bentonite mixtures, the PT2021 model demonstrated superior accuracy for both GMZ and MX80 datasets.
A universal model capable of precisely predicting thermal conductivity across all bentonite types has yet to emerge, with models varying in performance across different datasets.Thus, selecting the most fitting model for estimating bentonite's thermal conductivity in engineering applications requires meticulous consideration.Future efforts should focus on developing models that account for additional influences such as temperature variations, heating-cooling cycles, aging, ion type and concentration, and alterations in pore structure during saturation.This investigation provides information on the performance and constraints of existing thermal conductivity models for compacted bentonite and its mixtures, offering insights to ensure the long-term operational safety of deep geological repositories.

Figure 1 .
Figure 1.Prediction results of thermal conductivity model of compacted bentonite Table2.Evaluation results of thermal conductivity models for GMZ and MX80 compacted bentonite.

Figure 2 .
Figure 2. Prediction results of thermal conductivity model of bentonite mixture.Table 3. Evaluation results of thermal conductivity models for GMZ and MX80 bentonite mixture.Bentonite type Amount of data CL2011 WC2015 PT2021

Table 1 .
The fitting parameters of the models.

Table 2 .
Evaluation results of thermal conductivity models for GMZ and MX80 compacted bentonite.

Table 3 .
Evaluation results of thermal conductivity models for GMZ and MX80 bentonite mixture.