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Empirical and theoretical models for prediction of soil thermal conductivity: a review and critical assessment

Adrian Różański
Adrian Różański
 and 
Natalia Kaczmarek
Natalia Kaczmarek
   | Jul 09, 2020
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Article Category: Original Study
Published Online: Jul 09, 2020
Page range: 330 - 340
Received: Jun 06, 2020
Accepted: Jun 17, 2020
DOI: https://doi.org/10.2478/sgem-2019-0053
Keywords
© 2020 Adrian Różański, et al., published by Sciendo
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

Figure 1

Wiener model scheme (Wiener, 1912): (a) series model - lower limit; (b) parallel model - upper limit.
Wiener model scheme (Wiener, 1912): (a) series model - lower limit; (b) parallel model - upper limit.

Figure 2

The unit cube considered by Mickley (Farouki, 1981): (a) isometric view; (b) lateral view showing the water layer.
The unit cube considered by Mickley (Farouki, 1981): (a) isometric view; (b) lateral view showing the water layer.

Figure 3

An idealized grain model according to Gemant (Farouki, 1981).
An idealized grain model according to Gemant (Farouki, 1981).

Figure 4

Periodic cell according to the Gori model (Gori, 1983): (a) dry soil; (b) low water content – thin film around the grain; (c) partially saturated soil – formed water bridges; (d) the state close to full saturation.
Periodic cell according to the Gori model (Gori, 1983): (a) dry soil; (b) low water content – thin film around the grain; (c) partially saturated soil – formed water bridges; (d) the state close to full saturation.

Figure 5

Assumed geometry of a three-phase medium (Haigh, 2012).
Assumed geometry of a three-phase medium (Haigh, 2012).

Figure 6

Prediction of thermal conductivity for coarse-grained soil at different saturation states (expressed as saturation degree Sr) and measurement results (Lu et al., 2007)
Prediction of thermal conductivity for coarse-grained soil at different saturation states (expressed as saturation degree Sr) and measurement results (Lu et al., 2007)

Figure 7

Prediction of thermal conductivity for fine-grained soil at different saturation states (expressed as saturation degree Sr) and measurement results (from Lu et al., 2007).
Prediction of thermal conductivity for fine-grained soil at different saturation states (expressed as saturation degree Sr) and measurement results (from Lu et al., 2007).

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

Empirical models
ModelAdvantagesDisadvantagesComments
Kersten (1949)Simple formula; for every type of soilEquations do not take into account the quartz content which has the largest contribution in overall value of λThermal conductivity in dry state cannot be determined
Johansen (1975)Can be used for frozen soils; relatively high quality of prediction; for every type of soilPossible inaccuracy for dry soil (±20%)Empirical relations valid for Sr>0.05 (coarse-grained soil) and Sr>0.01 (fine-grained soil)
Donazzi et al. (1979)Simple formula; for every type of soilWeak prediction for soils with low water contentThe shape of the λ − Sr curve does not fully comply with the test results and with most other models presented in literature
Côté and Konrad (2005b)For every type of soil; includes the type of soil and the shape of grainsThe course of the Ke − Sr curve is not entirely consistent with common knowledge for fine-grained soils with low water contentModification of the Johansen method (1975) with respect to the Kersten number and dry soil conductivity
Lu et al. (2007)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 accountUnknown influence of the type of soil on the conductivity in the dry stateModification of the Johansen method with respect to the Kersten number and dry soil conductivity
Chen (2008)Simple formula; good quality of predictionLimited applicabilityOnly for sands with a high quartz content
Lu et al. (2014)Simple formula; for every type of soil; includes the effect of the dry density on thermal conductivityFor the analysed soils, the model clearly overestimated the values of λ in the entire range of water contentDry 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
He at al. (2017)Simple formula; for every type of soil; good quality of predictionLack of correlation formulas for determining model parametersModification 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.

ModelKersten numberThermal conductivity of
dry soilsaturated soil
Johansen (1975)Coarse-grained soil:Ke ≅ 0.7 log Sr + 1.0Fine-grained soil:Ke ≅ log Sr + 1.0λdry=0.135ρd+64.72700−0.947ρd{\lambda ^{{dry}}} = \frac{{0.135{\rho _d} + 64.7}}{{2700 - 0.947{\rho _d}}}λsat=λwnλs(1−n){\lambda ^{{sat}}} = \lambda _w^n\lambda _s^{(1 - n)}
Côté and Konrad (2005a; 2005b)Ke=κSr1+(κ−1)Sr{K_e} = \frac{{\kappa {S_r}}}{{1 + (\kappa - 1){{\text{S}}_{\text{r}}}}}λdry = χ10−ηn-
Lu et al. (2007)Ke=exp{α[1−Srα−1.33]}{K_e} = {exp}\left\{ {\alpha \left[ {1 - S_r^{\alpha - 1.33}} \right]} \right\}λdry = 0.51 − 0.56n-
He at al. (2017)Ke={0Sr=01A exp [(Srn)−B]Sr>0{K_e} = \left\{ {\begin{array}{*{20}{l}} 0&{{S_r} = 0} \\ {\frac{1}{{A\,{exp}\,\left[ {{{({S_r}n)}^{ - B}}} \right]}}}&{{S_r} > 0} \end{array}} \right.--

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

Theoretical models
ModelAdvantagesDisadvantagesComments
Wiener (1912)Determination of the range of possible thermal conductivity values of porous media (soils); simple formulaRough estimateFor coarse soils, due to the contrast between the thermal conductivity of the components, these bounds are very wide
Mickley (1951)For every type of soilWeak prediction for dry soils or with low water contentShould not be applied to the soils with relatively high porosity
Gemant (1952)For every type of soilComplicated formula; need to use nomogramsNot 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 λs
de Vries (1963)For every type of soil; can be used for partially or fully frozen soilsNeed to assume values of shape factors ga; weak prediction for dry soils; weak reflection of real λ − Sr characteristicFor 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)
Gori (1983)For every type of soil; can be used for different temperaturesVery complex formula; Certain parameters should be determined on the basis of laboratory tests’ resultsPossible underestimation of thermal conductivity for dry soils
Tong et al. (2009)Includes the impact of many factors on the thermal conductivity of the porous mediaComplex formula; a series of laboratory tests have to be performedLack of formulas from which model parameters can be computed
Haigh (2012)High accuracy of predictionComplex formula; Underestimation of results by a constant factor, about 1.58Only for sandy soils with a porosity higher than 0.333
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