Modeling thermal conductivity of clays: A review and evaluation of 28 predictive models

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Highlights

  • A total of 28 thermal conductivity models for clay were reviewed

  • These models were evaluated with 1250 measurements on 65 clays from 21 studies

  • The DF1979 model performed the best among the 28 models but not satisfactory

  • Model performance dataset/soil dependent and care should be taken for modelling modespecific soil types

  • It demonstrates an urgent call for developing new model with higher accuracy and wider applications

Abstract

Effective thermal conductivity of clay (λeff) has essential applications in disciplines such as environmental and earth science, agriculture and engineering. Much work has been done pertaining to model thermal conductivity of clay, but no work was found to collate and synthesize these works. This study aimed to conduct an extensive review of the predictive models for thermal conductivity of clay and evaluate their performance with a large compiled dataset. A total of 28 models were collated and categorized and their performance was evaluated with a large dataset consisting of 1250 measurements made on 65 clays from 21 studies. The result showed that the DF1979 was the best performing model but not satisfactory, with root-mean-square-error (RMSE) = 0.35 W m−1 °C−1, average deviation (AD) = −0.04 W m−1 °C−1 and Nash-Sutcliff Efficiency (NSE) = 0.54. Further investigations showed that these models are dataset dependent and fairly good performance might be found on certain soils/dataset. Therefore, cares should be taken when selecting the most appropriate model for predicting λeff in practice. Limitations of the thermal conductivity models for clays have been stated and perspectives on future studies were also presented. This work would provide the novice or expert alike information on the advantage and limitations on modeling thermal conductivity of clays for various purposes.

Introduction

Clays differ from other soils in many aspects including particle size and shape, specific surface area and mineralogy (Farrar and Coleman, 1967; Hepper et al., 2006; Maček et al., 2013; Paterson, 2018), which make it special in the thermal, hydraulic and mechanical characteristics (Kang et al., 2015; María and Villar, 2004; Mozumder and Laskar, 2015; Tang et al., 2011; Zhang et al., 2017a). With respect to this, clay has been widely used in disciplines such as environmental and earth science, agriculture and geotechnical/geological engineering. These applications include water treatment (Chang and Juang, 2004), backfilling material for buried high-voltage power cable and oil, gas or hot-water pipes (Newson et al., 2002; Reno and Winterkorn, 1967; Ye et al., 2010) or borehole heat exchanger (Luo et al., 2019; Yu et al., 2015), natural geological engineering barrier for hazardous waste disposal including radioactive nuclear waste (Gens et al., 2002; Madsen, 1998; Mügler et al., 2006; Tang and Cui, 2007; Villar et al., 2005; Yoon et al., 2018) and landfill (Ghorbel-Abid and Trabelsi-Ayadi, 2015; Kaya and Durukan, 2004).

Effective thermal conductivity of clay (λeff), reciprocal of thermal resistivity, is of critical importance and interest among these thermo-hydro-mechanical characteristics. Because λeff of clay is required to determines heat transmission through soil in order to optimize engineering design (He et al., 2021; Reno and Winterkorn, 1967; Wang et al., 2016; Xu et al., 2019b). Although great progress has been made in techniques of λeff measurement (Bristow et al., 1994; Campbell et al., 1991; He et al., 2018a; He et al., 2015; He et al., 2018b; Ren et al., 2003; Wang et al., 2020a; Zhang et al., 2017b), λeff data are still rare and sparse compared to that of soil hydraulic and mechanical properties. Therefore, many efforts have been made to model λeff of clays (Abuel-Naga et al., 2009; Akinyemi et al., 2011; Barry-Macaulay et al., 2011; Caridad et al., 2014; Dao et al., 2014; de Zárate et al., 2010; Garnier et al., 2010; Jougnot and Revil, 2010; Kang et al., 2015; Sun et al., 2020; Tang et al., 2008; Xu et al., 2019d). However, it is interesting to note that there is a lack of study that tends to collate and synthesize these works.

It is also noted that many of these thermal conductivity models for clay were commonly developed or tested on a small dataset composing 2 to less than 200 λeff measurements (Cai et al., 2015; de Zárate et al., 2010; Xu et al., 2019d). In addition, there is a lack of study assessing these models with a large dataset consisting of clays with wide range soil physic properties (e.g. water content, mineralogy, and bulk density). Previous studies (de Vries, 1952a; He et al., 2015; He et al., 2018b; Wang et al., 2020a) have shown that soil thermal conductivity measured with transient methods (e.g., heat pulse probe, thermal needle, and hot-wire method) is more accurate for soils from dryness to saturation compared to steady-state method (e.g., guarded hot plate, divided bar method, and heat flux meter). The steady-state methods result in phase change of ice for frozen soils or water redistribution in unfrozen soils under temperature gradient, especially in unsaturated soils. Therefore, λeff measured with the transient methods are recommended for development and calibration of predictive models for soil thermal conductivity (He et al., 2018b; Zhang et al., 2017b).

The objectives of this study, therefore, were threefold: (1) to conduct an extensive review of the predictive models for thermal conductivity of clay; (2) to compile a large dataset consisting of clays of various types from dry to full saturation states and at different bulk densities; (3) to evaluate the performance of the thermal conductivity models for clay with the compiled dataset. It is hoped that this work would provide the novice or expert alike information on the advantage and limitations on modeling thermal conductivity of clays for various purposes.

Section snippets

A review of thermal conductivity models for clays

The extensive literature search returned 28 thermal conductivity models for clays. These models were categorized into three groups: (1) theoretical/semi-theoretical models (N = 2) and mixing models (N = 4); (2) normalized models (N = 6); and (3) linear and non-linear regression models (N = 14). The initials and year of publication were used to represent the model for better science communications. It should note that only models developed for clays were presented and evaluated in this study.

Published datasets

To evaluate the various models to estimate λeff of clay, experimental data were collated by following several important criteria: (1) reliable and reproducible experimental techniques/setup. λeff were measured on soil samples with transient method (e.g., heat pulse method, hot-wire method and thermal probe) at room temperature under atmosphere pressure condition and thereby temperature effects on the λeff will not be considered in this study; (2) measurements were taken on pure clays or clay

Theoretical/semi-theoretical and mixing models

For the four theoretical/semi-theoretical models, the SK1998 model performed better than the JR2010 model (Fig. 1c and b, Table 2). Although the de Vries (1963) model has been widely applied, it cannot be used to well predict thermal conductivity of clay (Fig. 1a, Table 2). The performance of its simplified from proposed by Tian et al. (2016) is slightly better but still not satisfactory (Fig. 1b, Table 2). Theoretical models attract attention modeling soil physical properties including thermal

Conclusion

In this study, 28 thermal conductivity models for clays were reviewed and evaluated with a large compiled dataset from 21 studies. The resulted showed that the DF1979 model had the best performance on the whole compiled dataset but still not satisfactory. Performance of the thermal conductivity models may be dataset or soil dependent and cares should be taken when choosing the most appropriate model for prediction purpose in practice. Future efforts focusing on the development of more accurate

Declaration of Competing Interest

The author declares no conflict of interest.

Acknowledgements

Datasets used in this study were digitalized from the published literature.

Funding for this research was provided in part by the National Natural Science Foundation of China [Grant No. 41877015, 42077135], Natural Science Foundation of Shaanxi Province [2020JM-169], Young science and technology star of Shaanxi Province, China Postdoctoral Science Foundation [Grant No. 2018M641024], the Training Program Foundation for the Young Talents of Northwest A&F University, and the 111 project [Grant No.

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