The paper discusses existing models used to estimate the thermal conductivity of the soil medium. The considerations are divided into three general sections. In the first section of the paper, we focus on the presentation of empirical models. Here, in the case of Johansen method, different relations for Kersten number are also presented. In the next part, theoretical models are considered. In the following part, selected models were used to predict measured thermal conductivities of coarse- and fine-grained soils, at different water contents. Based on these predictions as well as on the authors’ experience, a critical assessment of the existing models is provided. The remarks as well as advantages and disadvantages of those models are summarized in a tabular form. The latter is important from a practical point of view; based on the table content, one can simply choose a model that is suitable for the particular problem.

#### Keywords

- Heat flow
- thermal conductivity
- soil
- Kersten number

The use of modern geotechnical solutions, such as geothermal piles, requires a new approach to design from engineers. In order to correctly design a thermal pile that combines the classical foundation features with the idea of recovering heat from the ground, one must determine the deformation state implied not only by mechanical loads, but also by the thermal loads resulting from the cyclical fluctuation of temperature in the pile and in the surrounding soil medium. Therefore, to determine the overall state of pile deformation, one must know not only the mechanical parameters of the concrete and the soil, but also their thermal characteristics, especially regarding their thermal conductivity. Thus, the correct determination of the thermal conductivity characteristics of the soil medium is of primary importance for the design of structures for which the surrounding ground constitutes a source of thermal energy.

There exists a large number of models that can be used to predict soil thermal conductivity. In the most general way, they can be divided into two groups (Różański, 2018): empirical and theoretical models. The former are models whose parameters are determined based on the results of laboratory tests (e.g., Kersten, 1949; Johansen, 1975; Donazzi et al., 1979; Côté & Konrad, 2005b; Lu et al., 2007; Chen, 2008; Lu et al., 2014). Their main disadvantage is that the scope of their applicability is usually very limited and gives the possibility of determining the thermal conductivity only for a particular type of soil, e.g., a certain model can work very well for sand, whereas when applied to clay, it produces enormous error. Hence, the empirical models usually are not ‘general’ ones. The main defect of the models included in the second group (theoretical or semi-theoretical ones, e.g., Mickley, 1951; Gemant, 1952; de Vries, 1963; Gori, 1983; Tong et al., 2009; Haigh, 2012; He et al., 2017) is their assumption of the ‘grain-water-air’ microstructure – in general, the geometry of the soil microstructure (‘grain-air-water’ mixture) is rather very simple. As a result, they are rarely able to correctly describe the complex thermal characteristics of partially saturated soils, that is, the change of thermal conductivity with the change in water content. This problem mainly affects fine-grained soils, which are characterized by a specific dependence of thermal conductivity on very low water contents, that is, to a certain limit of water content in the pores (usually quite small), the addition of water does not result in a significant increase in thermal conductivity (e.g., Johansen, 1975).

The current paper is a classical review of existing empirical and theoretical models used for the prediction of soil thermal conductivity. In the first part, selected models are briefly described. In particular, five empirical and six theoretical models are presented. Special attention is paid to the Johansen model (Johansen, 1975), which uses the concept of normalized conductivity, referred to as the Kersten number. Recent modifications of Kersten number are discussed. In Section 4, a critical assessment of thermal conductivity models is provided. Selected models were used to predict measured thermal conductivities of coarse- and fine-grained soils, at different water contents (results are taken from Lu et al., 2007). Based on these predictions as well as on the authors’ experience critical remarks with respect to individual models are formulated. At the end of this section, a tabular summary of the research is provided. The summary contains remarks, advantages and disadvantages of each considered model. The latter is important from a practical point of view, that is, based on the table content, one can simply choose a model that is suitable for the particular problem.

On the basis of the results of thermal conductivity measurements of 19 soils, Kersten (1949) proposed empirical equations to determine the thermal conductivity of the soil medium on the basis of moisture content _{d}^{3}]. For fine-grained soils (silt, clay, etc.) the thermal conductivity of the soil medium can be estimated using the following expression:

The analogous formula for coarse-grained soils is as follows:

Johansen (1975) proposed the determination of thermal conductivity of partially saturated soil medium on the basis of thermal conductivity of dry soil ^{dry}^{sat}_{e}

Based on the data contained in Kersten’s work (1949), Johansen proposed the correlation formulas between the Kersten number _{e}_{r}^{sat}

Different approaches used for evaluation of the Kersten number _{e}

Johansen (1975) | Coarse-grained soil:_{e}_{r}_{e}_{r} | ||

Côté and Konrad (2005a; 2005b) | ^{dry}^{−ηn} | - | |

Lu et al. (2007) | ^{dry} | - | |

He at al. (2017) | - | - |

where:

_{w}

_{s}

The Johansen method is recognized in the literature as providing one of the best predictions of the thermal conductivity of soil medium (Farouki, 1981; Dong et al. 2015; Różański, 2018; Zhang & Wang, 2017). In the following years, various researchers modified the Johansen method. The changes mainly concern the introduction of a slightly improved form of the function describing the _{e} − S_{r}

Côté and Konrad (2005a; 2005b) modified the Kersten number by introducing an additional parameter ^{dry}^{dry}_{r} is the one proposed in the work of He et al. (2017). All formulas for the Johansen method and its subsequent modifications are collected in Table 1. The sign ‘−’ means that the authors did not modify the parameter or they did not refer to it at all in their work.

Donazzi et al. (1979) studied the thermal and hydrological characteristics of soils surrounded by underground installations. Based on the results of the laboratory tests, they proposed the following correlation formula to determine the thermal conductivity of the soil, which can be applied to all types of soil:

Chen (2008) carried out tests on the thermal conductivity of quartz sands in different saturation states. On this basis, the empirical equation of thermal conductivity as a function of porosity and degree of saturation was proposed:

Based on the results of thermal conductivity measurements, the model parameters were determined as:

In the work of Lu et al. (2014), the following empirical equation for thermal conductivity at various saturation states was proposed:

The conductivity of dry soil is evaluated using the approach proposed by Lu et al. (2007) – see Table 1. Parameters _{Cl}_{Sa}_{d}^{3}
]. Note, that this model gives the possibility to estimate the thermal conductivity of the same soils, but in different states of compaction.

In 1912, Wiener (Wiener, 1912) proposed an upper and lower bound for the thermal conductivity of a porous medium consisting of three separate phases: solid, water and air (Fig. 1).

The lower (^{L}^{U}_{i}_{i}

In the work of Mickley (1951), it was assumed that elementary volume of a soil is a unit cube with separated ‘channels’, filled with air and/or water (Fig. 2). In the case of dry soil (_{r}^{2} and unit length, (2) the soil skeleton between the outer walls of a cube of sectional area (1−^{2}) and unit length, and (3) and (4) two columns (air and soil skeleton) in series, each with sectional area ^{2}) and length

The thermal conductivities at different states of saturation are evaluated using the following relations (Mickley, 1951):

dry soil_{a}

fully saturated soil

partially saturated soil

The geometric variables _{r}

Gemant (1952) proposed a model based on an idealized grain shape – a cube with a side

Gemant (1952) proposed the following formula for the conductivity of the soil
_{s}_{w}^{−1}^{−1}
] are the thermal conductivities of the soil skeleton and water, respectively. Remaining parameters are expressed by following relations:

In equations above, _{0}_{0} decreases as the temperature increases following the theory of water adsorption of glass (Farouki, 1981), for example: for _{0}=0.01. Analogically, namely using a separate nomogram, the values of ^{2}/

De Vries model (de Vries, 1963) is based on the Maxwell equation defining the effective electrical conductivity of a mixture of spherical particles dispersed in a continuous fluid (Eucken, 1932). De Vries assumed that the soil medium could be modelled as a mixture of _{i}_{a}_{a}

Gori (1983) proposed a theoretical model to determine the thermal conductivity of soils in different saturation states. Gori assumed that the soil (as a multi-phase medium) consists of periodic cubic cells with a ‘centrally’ located skeleton (solid) surrounded by air (dry soil) or water (saturated soil) or both air and water (partially saturated soil). A geometry of the representative unit cell, at various states of saturation, is presented in Fig. 4.

Due to the fact that the calculation procedure using this model, compared to most other proposals, is relatively difficult, the use of this model in engineering practice is quite limited. In addition, the disadvantage of the model is that while the formulation itself results directly from theoretical considerations, the water content values determining the geometry of the periodicity cell (Fig. 4) were determined empirically depending on the type of soil. In the computational sense, determination of the thermal conductivity of dry soils seems to be relatively simple (Fig. 4a). This prediction depends only on the porosity of the soil and the thermal conductivities of the air and soil skeleton and is given by the following equation:

In the work of Tong et al. (2009), the thermal conductivity of a multicomponent porous medium was considered. A new model based on Wiener as well as Hashin-Shtrikmann

For a macroscopically isotropic medium the bounds of Hashin-Shtrikman provide the narrowest range of estimates, which are expressed only in terms of volume fractions of constituents (e.g., Łydżba, 2011).

bounds was proposed. The value of thermal conductivity was shown to be affected by such parameters as mineral composition, temperature, saturation degree and porosity. As proposed by Tong et al. (2009), the thermal conductivity of a three-component porous medium (solid, water, air) is determined by the following expression:_{1}is the coefficient dependent on soil porosity, while the value of parameter

_{2}depends on porosity, saturation and temperature. Both parameters are in the range from 0 to 1 which is a direct consequence of fulfilling the Wiener bounds. Determination of the coefficients

_{1}and

_{2}requires a series of laboratory tests for samples of porous material with different porosities and saturation degrees and subjected to several different temperatures.

Haigh (2012) model is used to determine the thermal conductivity of sandy soils in different water saturation states. The equation describing thermal conductivity was determined on the basis of the assumed simplified geometry of the soil microstructure. A problem of 1D heat flow between two identical grains of spherical shape and radius

The equation describing the thermal conductivity of the soil is expressed by the following formula:
_{w}_{w}_{s}_{a}_{a}_{s}

Fig. 6 shows the predictions obtained using the considered models for coarse-grained soil, that is, sand with following soil separates: 92% of sand, 7% of silt and 1% of clay. The dry density and the specific density are _{d}^{3}] and _{s}^{3}], respectively. The results of thermal conductivity at different saturation states are taken from the work of Lu et al. (2007). The value of thermal conductivity is assumed analogically as in the work of Lu et al. (2007), that is, _{s}^{−1} K^{−1}].

Due to the high contrast between the conductivity of soil skeleton and air/water, Wiener bounds were omitted in the analysis since they provide a very wide range of possible values of

Kersten model (1949) quite well describes, in a qualitative way, the relationship _{r}_{r}_{r}

Mickley (1951) and de Vries (1963) models do not reflect correctly, neither qualitatively nor quantitatively, the thermal characteristic of the analysed soil. Relatively high porosity (about 40%) may be a problem for Mickley model. De Vries model (1963), due to the assumption of no interaction between the particles, underestimates the thermal conductivity of sand for small values of _{r}

Gemant model (1952) overestimates the thermal conductivity of the analysed soil. This is probably a consequence of the fact that originally Gemant proposed a formula for determining the thermal conductivity of the soil skeleton based only on the clay fraction. Using Gemant proposal, the thermal conductivity of the soil skeleton would be 5.81[^{−1}^{−1}]. This value would improve the prediction _{r}^{−1}^{−1}], which was adopted for the analysis.

Donazzi model (1979) significantly overestimates thermal conductivity for low water contents. Moreover, the convexity of the curve _{r}

Weak prediction was obtained by Lu et al. (2014) model. Almost within the entire range of _{r}

Similar predictions, qualitatively and quantitatively, were obtained using models of Johansen (1975), Côté and Konrad (2005b) and Lu et al. (2007). This is not surprising, as compared to Johansen’s proposal, the other two formulations introduce only slight modifications to the original relationship defining the Kersten number (see Table 1).

Similar analysis was also conducted for fine-grained soil, that is, clay with following content of soil separates: 32%, 38% and 30% respectively for sand, silt and clay; the dry density _{d}^{3}] and the specific density _{s}^{3}]. The results of thermal conductivity measurements and thermal conductivity of the soil skeleton (_{s}^{−1}^{−1}]) for the soil under consideration were taken from Lu et al. (2007). Fig. 7 presents the obtained predictions and measurement data. The analysis omitted those models that are dedicated to sandy soils, that is, Chen (2008) and Haigh (2012) models.

Again, the prediction based on Lu et al. model (2014) is overestimating the measurement data almost in the entire range of _{r}_{r}_{e}_{r}

Based on the predictions presented above as well as on the authors’ experience, some critical remarks with respect to individual models are formulated and presented in a tabular form (Table 2 and Table 3). Tables contain remarks, advantages and disadvantages of each considered model.

Advantages, disadvantages and comments on considered empirical models (Różański, 2018).

Model | Advantages | Disadvantages | Comments |

Simple formula; for every type of soil | Equations do not take into account the quartz content which has the largest contribution in overall value of | Thermal conductivity in dry state cannot be determined | |

Can be used for frozen soils; relatively high quality of prediction; for every type of soil | Possible inaccuracy for dry soil (±20%) | Empirical relations valid for _{r}_{r} | |

Simple formula; for every type of soil | Weak prediction for soils with low water content | The shape of the _{r} | |

For every type of soil; includes the type of soil and the shape of grains | The course of the _{e}_{r} | Modification of the Johansen method (1975) with respect to the Kersten number and dry soil conductivity | |

For every type of soil; very good reflection of thermal conductivity for fine-grained soils with very low water content; the type of soil is taken into account | Unknown influence of the type of soil on the conductivity in the dry state | Modification of the Johansen method with respect to the Kersten number and dry soil conductivity | |

Simple formula; good quality of prediction | Limited applicability | Only for sands with a high quartz content | |

Simple formula; for every type of soil; includes the effect of the dry density on thermal conductivity | For the analysed soils, the model clearly overestimated the values of | Dry soil conductivity should be computed using empirical relation proposed in Lu et al. (2007); possible weaker prediction for soils with high content of sand separate | |

Simple formula; for every type of soil; good quality of prediction | Lack of correlation formulas for determining model parameters | Modification of the Johansen method with respect to the Kersten number |

Advantages, disadvantages and comments on the considered theoretical models (Różański, 2018).

Determination of the range of possible thermal conductivity values of porous media (soils); simple formula | Rough estimate | For coarse soils, due to the contrast between the thermal conductivity of the components, these bounds are very wide | |

For every type of soil | Weak prediction for dry soils or with low water content | Should not be applied to the soils with relatively high porosity | |

For every type of soil | Complicated formula; need to use nomograms | Not applicable to dry soils; possible overestimation of thermal conductivity results if Gemant formula is not used to determine the thermal conductivity of the soil skeleton _{s} | |

For every type of soil; can be used for partially or fully frozen soils | Need to assume values of shape factors _{a}_{r} | For good predictions, one should incorporate in Eq. (22) heterogeneity of solid phase and at least five minerals should be taken into account (Tarnawski & Wagner, 1992, 1993); do not use if the volume fraction of water is less than 0.03 (coarse-grained soil) or 0.05–0.10 (fine-grained soil) | |

For every type of soil; can be used for different temperatures | Very complex formula; Certain parameters should be determined on the basis of laboratory tests’ results | Possible underestimation of thermal conductivity for dry soils | |

Includes the impact of many factors on the thermal conductivity of the porous media | Complex formula; a series of laboratory tests have to be performed | Lack of formulas from which model parameters can be computed | |

High accuracy of prediction | Complex formula; Underestimation of results by a constant factor, about 1.58 | Only for sandy soils with a porosity higher than 0.333 |

The paper presented a classical review of the existing empirical and theoretical models used for prediction of soil thermal conductivity. Five empirical and six theoretical models were widely discussed. Special attention was paid to the Johansen model, which uses the concept of the Kersten number. Selected modifications of Kersten number, presented in different papers, were discussed. Critical assessment of thermal conductivity models was provided in Section 4. Based on model predictions, measurement results (from Lu et al., 2007) for two types of soils, as well as on the authors’ experience, the critical remarks with respect to individual models were formulated. These were presented in a tabular form. From the practical point of view, the discussion provided in Section 4 is very important. Based on the tables’ content, one can simply choose a model that is suitable for the particular engineering problem.

#### Advantages, disadvantages and comments on considered empirical models (Różański, 2018).

Model | Advantages | Disadvantages | Comments |

Simple formula; for every type of soil | Equations do not take into account the quartz content which has the largest contribution in overall value of | Thermal conductivity in dry state cannot be determined | |

Can be used for frozen soils; relatively high quality of prediction; for every type of soil | Possible inaccuracy for dry soil (±20%) | Empirical relations valid for _{r}_{r} | |

Simple formula; for every type of soil | Weak prediction for soils with low water content | The shape of the _{r} | |

For every type of soil; includes the type of soil and the shape of grains | The course of the _{e}_{r} | Modification of the | |

For every type of soil; very good reflection of thermal conductivity for fine-grained soils with very low water content; the type of soil is taken into account | Unknown influence of the type of soil on the conductivity in the dry state | Modification of the Johansen method with respect to the Kersten number and dry soil conductivity | |

Simple formula; good quality of prediction | Limited applicability | Only for sands with a high quartz content | |

Simple formula; for every type of soil; includes the effect of the dry density on thermal conductivity | For the analysed soils, the model clearly overestimated the values of | Dry soil conductivity should be computed using empirical relation proposed in | |

Simple formula; for every type of soil; good quality of prediction | Lack of correlation formulas for determining model parameters | Modification of the Johansen method with respect to the Kersten number |

#### Different approaches used for evaluation of the Kersten number Ke and the conductivity of dry and saturated soil.

Coarse-grained soil:_{e}_{r}_{e}_{r} | |||

^{dry}^{−ηn} | - | ||

^{dry} | - | ||

- | - |

#### Advantages, disadvantages and comments on the considered theoretical models (Różański, 2018).

Determination of the range of possible thermal conductivity values of porous media (soils); simple formula | Rough estimate | For coarse soils, due to the contrast between the thermal conductivity of the components, these bounds are very wide | |

For every type of soil | Weak prediction for dry soils or with low water content | Should not be applied to the soils with relatively high porosity | |

For every type of soil | Complicated formula; need to use nomograms | Not applicable to dry soils; possible overestimation of thermal conductivity results if Gemant formula is not used to determine the thermal conductivity of the soil skeleton _{s} | |

For every type of soil; can be used for partially or fully frozen soils | Need to assume values of shape factors _{a}_{r} | For good predictions, one should incorporate in | |

For every type of soil; can be used for different temperatures | Very complex formula; Certain parameters should be determined on the basis of laboratory tests’ results | Possible underestimation of thermal conductivity for dry soils | |

Includes the impact of many factors on the thermal conductivity of the porous media | Complex formula; a series of laboratory tests have to be performed | Lack of formulas from which model parameters can be computed | |

High accuracy of prediction | Complex formula; Underestimation of results by a constant factor, about 1.58 | Only for sandy soils with a porosity higher than 0.333 |

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