A review on thermal conductivity of unsaturated bentonite

As unsaturated bentonite is used a buffer/backfill material in the construction of engineered barriers for high-level nuclear waste disposal, understanding its thermal properties is crucial for maintaining the operational stability of the repository. This review paper synthesizes research on the thermal conductivity of unsaturated bentonite, encompassing aspects of heat transfer mechanisms, measurement techniques, influencing factors, and prediction models. The results highlight that the transient and steady-state methods have emerged as the predominant techniques for measuring the thermal conductivity of unsaturated bentonite owing to their efficiency and the prevention of water redistribution within the sample throughout the testing phase. Factors such as dry density, degree of saturation, additive content, and temperature were observed to positively influence the thermal conductivity of unsaturated bentonite, whereas an increase in the ion concentration of the pore fluid decreased the thermal conductivity. The prediction models for the thermal conductivity of unsaturated bentonite are categorized into empirical, normalized, and theoretical models. Although empirical and normalized models provide some insight, they suffer from a lack of theoretical foundation, and the parameters they incorporate often lack clear physical significance, complicating their practical application despite their capacity to represent the multifield coupling characteristics of thermal conductivity. The advancements in microscopic testing methods and computational technology herald the potential of mesoscale models and machine learning as formidable tools for predicting the thermal conductivity of unsaturated bentonite, suggesting a promising direction for future research.


Introduction
Deep geological repositories designed for the disposal of high-level radioactive waste (HLW) in hard rock formations integrate natural geological barriers with engineered ones [1].Bentonite serves as the final engineered barrier between the host rock and the waste canister, thereby mitigating groundwater infiltration, hindering radionuclide migration, dispersing decay heat from radioactive waste, and upholding the structural integrity of the repository [2,3].Because of its low permeability, excellent swelling properties, and superior radionuclide adsorption capacity, bentonite is recognized as a leading candidate for engineered buffer materials in the waste management strategies of numerous countries [4,5].
In the operational phase of a repository, the buffer material is responsible for diffusing the heat generated by radioactive decay into the surrounding host rock.The effectiveness of this heat dissipation, dictated by the thermal conductivity of the bentonite buffer, is crucial for determining the temperature distribution within the repository.Consequently, extensive research has been conducted on the thermal IOP Publishing doi:10.1088/1755-1315/1330/1/012057 2 conductivity of unsaturated bentonite [6,7], aiming to elucidate its heat transfer mechanisms, establish the interplay between thermal conductivity and factors like dry density, saturation, and additives, and develop predictive models for its thermal conductivity [8].
This research compiles and discusses the findings and recent advancements in understanding the heat transfer mechanism, measurement techniques, influencing factors, prediction models, and their practical applicability for assessing the thermal conductivity of unsaturated bentonite.It aspires to furnish a comprehensive reference for evaluating thermal conductivity in the context of deep geological repositories for HLW disposal.

Heat transfer mechanism of unsaturated bentonite
Unsaturated soil is a tri-phase porous material that comprises solid (soil particles), liquid (pore water), and gas (pore air) phases, which are characterized by a complex pore structure nestled within the particle skeletons, wherein both water and air occupy the interstitial spaces.The heat transfer within unsaturated soil predominantly occurs via interactions at the soil particle-particle, particle-water (through water films or bridges), and particle-air interfaces, along with the conveyance of latent heat at elevated temperatures (figure 1) [9].The efficacy of contact heat transfer hinges on the thermal conductivity of this tri-phase medium, its dry density (or porosity), and the degree of saturation (or water content), whereas the efficiency of the latent heat transfer is principally governed by the temperature gradient.Among these, soil particles exhibit the highest thermal conductivity, succeeded by water and air [10].The degree of saturation effects on thermal conductivity for double structure in compacted bentonite [10,11].
Compacted bentonite exhibits a dual structure comprising microstructures (inter-layer and intraaggregate pores) and macrostructures (inter-aggregate pores) (figure 2) [10].This variation in thermal conductivity is intimately linked to water retention mechanisms across different saturation levels.Employing the intersection of two tangent lines, as depicted in figure 2, exhibits three distinct saturation zones: i) low saturation zone I, ii) medium saturation zone II, and iii) high saturation zone III.In zone I, water molecules predominantly adhere to mineral crystal structures and clay particle surfaces without forming free water within the macrostructure, leading to a gradual increase in thermal conductivity with saturation.
As saturation progresses into zone II, the macrostructure progressively accommodates capillary water, leading to the formation of water films and discrete water bridges.This enhancement in thermal transfer path connectivity significantly elevates thermal conductivity.
In zone III, despite higher saturation levels, the additional pore water does not substantially modify heat flow preference through the soil skeleton nor augment the connectivity of the heat transfer pathway, resulting in a moderated increase in thermal conductivity, which eventually stabilizes.

Measurement technique of thermal conductivity of unsaturated bentonite
The thermal conductivity of soil can be determined using thermal probes, hot disks, heat flow meters, and hot boxes (figure 3) [12,13].The thermal probe and hot disk techniques are categorized as transient methods, whereas the heat flow meter and hot box approaches are steady-state methods.The measurements conducted via the heat flow meter and hot box methods tend to yield more reliable outcomes but are hampered by the need for bulky apparatus and the inefficiency caused by the prolonged testing durations.Conversely, the thermal probe and hot-disk techniques demonstrate enhanced testing efficiency and provide precise results, effectively mitigating the limitations associated with the traditional steady-state methods, especially the issue of water redistribution during testing.Additionally, the equipment is portable and user-friendly, rendering the thermal probe and hot-disk methods favored choices for laboratory assessments of the thermal conductivity of unsaturated bentonite.Although the technology for measuring the thermal conductivity of unsaturated bentonite has matured, the ongoing structural transformations during bentonite swelling require further exploration into methodologies capable of dynamically tracking changes in thermal conductivity amidst conditions of unrestricted expansion, finite deformation, and intricate hydromechanical loading paths.

Factors affecting thermal conductivity of unsaturated bentonite
Multiple factors influence the thermal conductivity of unsaturated bentonite, including dry density, degree of saturation, additive content, temperature, pore water chemistry, and other environmental conditions [14,15].

Dry density
The impact of dry density on thermal conductivity, as depicted in figure 4 [14,[16][17][18][19], reveals nearly linear correlations between thermal conductivity and dry density.Nonetheless, these relationships and the corresponding formulas vary with water content, complicating their application in the management of HLW repositories.A notable increase in thermal conductivity-over two-fold-is observed as dry density escalates from 1.2-1.8g/cm 3 at constant water content.This enhancement is attributed to the increased contact area among clay particles with denser compaction, facilitating superior heat conduction and thereby elevating thermal conductivity.

Degree of saturation
The impact of saturation degree on the thermal conductivity of various unsaturated bentonite types is portrayed in figure 5, wherein the thermal conductivity increased with the degree of saturation, thereby maintaining a constant dry density [16][17][18][19].Specifically, thermal conductivity may enhance two to threefold from a dry state to full saturation.This augmentation is attributed to the expansion and amalgamation of liquid bridges between adjacent particles, culminating in the formation of a comprehensive water film encircling soil particles, thereby establishing a pore-water network.The heat predominantly traverses through the solid skeleton, bolstered by this pore-water network at particle contacts, leading to a gradual increment in thermal conductivity until it reaches its apex.

Additive
Research indicates that the inclusion of sand in bentonite significantly elevates its thermal conductivity, as showcased in figure 6 [20,21].A GMZ bentonite-sand mixture with 50% sand content exhibited a thermal conductivity 1.9 times higher than that of compacted bentonite at a 10% water content.Moreover, the thermal conductivity of mixtures with varying water content appears to change linearly with sand inclusion.Given that sand's thermal conductivity substantially surpasses that of bentonite [22], augmenting sand content naturally enhances heat transfer through the sand, thus increasing the overall thermal conductivity.Nonetheless, beyond a 50% sand inclusion, the thermal conductivity of the bentonite-sand mixture shows a decline with further increases in sand content due to the sand skeleton effect, which disrupts heat transfer between sand particles [23].
The addition of materials such as sand, crushed granite, and graphite can improve the thermal conductivity of the bentonite.When integrating additives, their effects on buffering capabilities of the bentonite need to be considered comprehensively.Furthermore, the processing and transportation costs associated with these additives should also be considered.

Temperature and pore water chemistry
Figure 7 illustrates the impact of the temperature on the thermal conductivities of two compacted bentonites (GMZ and MX80).For a set dry density, thermal conductivity augments with an increase in temperature.Specifically, when the temperature ascends from 20 °C to 90 °C, the thermal conductivity of GMZ and MX-80 bentonites can surge by up to 21% and 40%, respectively.This increment is attributed to enhanced water vapor movement and latent heat transfer due to the temperature rise and thermal gradients, facilitating the expansion of water bridge menisci (liquid islands) and thus improving heat transfer in unsaturated bentonites.
The alteration in thermal conductivity with temperature becomes more pronounced at higher water contents.Moreover, as dry density escalates, the effect of the temperature on thermal conductivity markedly diminishes (figure 7b).This phenomenon is associated with the necessity of water for latent heat transfer, forming liquid islands, thereby explaining the minimal thermal conductivity changes in dry bentonite with temperature elevation.However, a higher dry density leads to a reduction in pore air pathways between bentonite particles or aggregates, diminishing latent heat transfer.Consequently, the influence of the temperature on thermal conductivity lessens with the increasing dry density.
Research focusing on the effect of pore water chemistry on the thermal conductivity of unsaturated bentonites remains limited.Siddiqua et al. [25] observed a decrease in thermal conductivity for a bentonite-sand mixture with increasing ion concentration in the pore fluid, potentially due to the diminished thermal conductivity of the pore fluid as ion concentration rises [26].
The exploration of thermal conductivity of the unsaturated bentonite is extensive.During the operational phase of a deep geological repository, bentonite undergoes simultaneous exposure to saline (alkaline) groundwater saturation and high-temperature drying.Investigating the variation of the thermal conductivity under combined temperature-salt (alkali) effects, alongside wetting-drying, salinizationdesalinization, and heating-cooling processes, presents a valuable avenue for future research.

Thermal conductivity model of unsaturated bentonite
Prediction models for the thermal conductivity of unsaturated bentonite are broadly classified into empirical, normalized, and theoretical categories.

Empirical models
The empirical models were formulated based on the limited experimental datasets to establish the correlations between thermal conductivity and soil parameters such as dry density, degree of saturation, water content, and volume fraction of air through linear, polynomial, or power exponent functions [12,27,28].Yoon et al. [24] utilized regression analysis on 147 datasets to predict the thermal conductivity of Gyeongju bentonite, accounting for initial dry density, water content, and temperature variations.As outlined in table 1, these models exhibited straightforward and computationally efficient expressions but lack a robust theoretical underpinning.The coefficients within these empirical equations are highly material-specific, rendering each model applicable to only a select array of bentonite types.
where Sr denotes the degree of saturation and n indicates the porosity.Issues arise when it is applied to a low degree of saturation (Sr < 0.1), owing to the logarithmic form of Ke.Enhancements to this model include Côté and Konrad [31] transitioning from a logarithmic to a hyperbolic equation for the Ke-Sr relationship, introducing a material-specific parameter  Lu et al. [32] developed an updated normalized thermal conductivity model, extending its applicability across 12 soil types, based on the Côté model framework.Subsequent refinements were made by researchers like Chen [33] and Peng [34].Furthermore, Lu and Dong [11] incorporated the soil-water retention mechanism into the model and established a quantitative relation between pore size distribution, soil-water interaction mechanisms, and thermal conductivity variation based on the soilwater retention curve (SWRC).Zhang et al. [10] posited that the volume fraction of air (a) exerts a more substantial influence on the thermal conductivity of the bentonite than the degree of saturation (Sr), employing an inverse sigmoid function to model the Ke-a relationship.
Normalized models account for various factors including bentonite type, porosity, saturation degree, mineral content, pore structure, and the soil-water retention mechanism, achieving high predictive accuracy.Nonetheless, these models often rely on empirical formulas to calculate the thermal conductivity of dry bentonite, with the fitting parameters frequently lacking explicit physical significance.

Theoretical models
Theoretical models for predicting the thermal conductivity of unsaturated bentonite encompass mixing and mathematical frameworks.The mixing model treats the solid, water, and air components of the multi-phase soil system as cubic cells, calculating the thermal conductivity of the bulk medium by blending these elements in specific series and parallel combinations [35].However, the Wiener bounds, which set the upper and lower limits of thermal conductivity of the unsaturated soil (i.e., Wiener bounds, figures 8a and d) within the series and parallel models, generally yield poor predictive accuracy [36].To address this, Börgesson et al. [37] introduced a geometric mean model that presupposes random combinations between the constituent blocks.Mathematical models draw on predictive approaches from other physical domains like dielectric permittivity, magnetic permeability, and electrical conductivity, utilizing mathematical algorithms to compute thermal conductivity based on the conductivity and volume fractions of the components (table 2).Initial mathematical models prominently include Maxwell's equation [38], the Hashin-Shtrikman model (H-S model, figures 8b, d) [39], and the effective medium theory (EMT, figures 8c, d) [40].Notably, the thermal conductivity of unsaturated bentonite aligns more closely with the EMT prediction as it transitions to saturation from the lower H-S bounds during the drying process (figure 8d).
Tong et al. [41] and Chen et al. [42] proposed a model that integrates the thermal conductivity of unsaturated bentonite within a thermo-hydro-mechanical context, employing Wiener and H-S bounds to amalgamate the three-phase medium in series and parallel formats.The model was further refined to account for the impact of the double structure on thermal conductivity by comparing the pores of the soil to random ellipsoids of varied shapes and sizes, utilizing homogenization techniques [43,44].Additionally, the thermal conductivity of bentonite-sand mixtures was calculated using Maxwell's equation, based on the premise of sand particles (suspended matter) was randomly distributed within the bentonite matrix (continuum) [23,45].
Theoretical models for thermal conductivity offer a well-defined physical basis, accommodating the influences of porosity, saturation degree, three-phase medium thermal conductivity, temperature, and the double structure, while adeptly representing the multifaceted coupling characteristics of thermal conductivity and demonstrating superior performance overall.Nonetheless, these models fall short in capturing the anisotropy inherent in thermal conductivity and present certain limitations.
In conclusion, the ongoing evolution of thermal conductivity models underscores the absence of a universally applicable, unified framework.Future research should aim at devising novel models for the thermal conductivity of unsaturated bentonite that integrate multifield (thermo-chemo-hydromechanical) couplings.Advances in microscopic testing methods and computational technologies hold promise for mesoscale models and machine learning as potent methodologies for predicting the thermal conductivity of the unsaturated bentonite.

Conclusions
The thermal conductivity of unsaturated bentonite plays a pivotal role in the design and efficacy evaluation of deep geological repositories for HLW disposal.This review encompasses investigations into heat transfer mechanisms, measurement approaches, influencing variables, prediction models, and their relevance to thermal conductivity of the unsaturated bentonite, yielding the following key insights: Unsaturated bentonite exhibits primary heat transfer mechanisms through direct contact between soil particle interfaces, including particle-water (via water films or bridges) and particle-air interactions, alongside latent heat transfer at elevated temperatures.The unique double structure significantly influences the thermal conductivity of the unsaturated bentonite.
In laboratory settings, the thermal conductivity of unsaturated bentonite is predominantly assessed using the thermal probe or hot disk method.It is observed to increase with the rise in dry density, degree of saturation, additive content, and temperature, whereas a surge in the ion concentration of the pore fluid leads to a decrease in thermal conductivity.
The available prediction models for the thermal conductivity of the unsaturated bentonite are categorized into empirical, normalized, and theoretical frameworks, with the latter encompassing both mixing and mathematical models.Despite these developments, a comprehensive model that addresses the thermal conductivity of unsaturated bentonite through a multifaceted (thermo-chemo-hydromechanical) coupling perspective remains elusive.
Extensive research on unsaturated bentonite's thermal conductivity has been documented both domestically and internationally.Nonetheless, the real-world operational complexities of deep geological repositories for HLW disposal present multifield coupled conditions that have not been sufficiently explored in existing studies.Given the crucial role of bentonite as a buffer/backfill material, the dynamic evolution of the thermal conductivity of the bentonite pellet mixtures under such multifield coupled scenarios warrants additional investigation.

Figure 2 .
Figure 2.The degree of saturation effects on thermal conductivity for double structure in compacted bentonite[10,11].
Key factors:  and d denote the natural density and dry density of compacted bentonite (g/cm 3 ), respectively; w denotes the water content; Sr indicates the degree of saturation; n denotes the porosity; a indicates the volume fraction of air; T denotes the temperature (C); sat indicates the thermal conductivity of saturated bentonite (Wm −1 C− 1 );  denotes the fitting parameter.

Figure 8 .
Figure 8. Schematic of theoretical models of thermal conductivity.