Advanced Geometric Mean Model for Predicting Thermal Conductivity of Unsaturated Soils

Abstract

An advanced geometric mean model for predicting the effective thermal conductivity ($\lambda$) of unsaturated soils has been developed and successfully verified against an experimental $\lambda$ database consisting of 40 Canadian soils, 15 American soils, 10 Chinese soils, four Japanese soils, three standard sands, and one pyroclastic soil (Pozzolana) from Italy (a total of 667 experimental $\lambda$ entries). Three soil structure-based parameters were used in the model, namely an inter-particle thermal contact resistance factor ($\alpha$), the degree of saturation of a miniscule pore space $(s_{\mathrm{r}})$, and the bulk thermal conductivity of soil solids $(\lambda _{\mathrm{s}})$. The $\alpha$ factor strongly depended on the ratio of $\lambda _{\mathrm{s}}$ to $\lambda _{\mathrm{f}}$ (where $\lambda _{\mathrm{f}}$ is the thermal conductivity of interfacial fluid) and an inter-particle contact coefficient ($\varepsilon$) whose value was obtained by reverse modeling of experimental $\lambda$ data of 40 Canadian soils; the average values of $\varepsilon$ varied between 0.988 and 0.994 for coarse and fine soils, respectively. In general, $\varepsilon$ depends on soil compaction, soil specific surface area, and grain size distribution. The use of $\alpha$ was essential for close $\lambda$ estimates of experimental data at a low range of degree of saturation $(S_{\mathrm{r}})$. For $\lambda _{\mathrm{s}}$ estimates obtained from measured $\lambda$ at soil saturation or a complete soil mineral composition data or experimental quartz content, 69 % of $\lambda$ predictions were less than $0.08\, \hbox {W} {\cdot } \hbox {m}^{-1} {\cdot } \hbox {K}^{-1}$, 15 % were between $0.08\,\hbox {W} {\cdot } \hbox {m}^{-1} {\cdot } \hbox {K}^{-1}$ and $0.13\,\hbox {W} {\cdot } \hbox {m}^{-1} {\cdot } \hbox {K}^{-1}$, and 13 % were between $0.13\,\hbox {W} {\cdot } \hbox {m}^{-1} {\cdot } \hbox {K}^{-1}$ and $0.24\,\hbox {W} {\cdot } \hbox {m}^{-1} {\cdot } \hbox {K}^{-1}$ with respect to experimental data $(\lambda _{\mathrm{exp}})$. The model gives close $\lambda$ estimates with an average root-mean-square error (RMSE) of $0.051\,\hbox {W} {\cdot } \hbox {m}^{-1} {\cdot } \hbox {K}^{-1}$ for 22 Canadian fine soils and an average RMSE of $0.092\,\hbox {W} {\cdot } \hbox {m}^{-1} {\cdot } \hbox {K}^{-1}$ for 18 Canadian coarse soils. In general, better $\lambda$ estimates were obtained for soils containing less content of quartz. Overall, the model estimates were good for all soils at dry state ($\hbox {RMSE} = 0.050\, \hbox {W} {\cdot } \hbox {m}^{-1} {\cdot } \hbox {K}^{-1}$; 22 % of the average $\lambda _{\mathrm{exp}}$), saturated state ($\hbox {RMSE} = 0.090\,\hbox {W} {\cdot } \hbox {m}^{-1} {\cdot } \hbox {K}^{-1}$; 5 % of the average $\lambda _{\mathrm{exp}}$), soil field capacity ($\hbox {RMSE} = 0.105\,\hbox {W} {\cdot } \hbox {m}^{-1} {\cdot } \hbox {K}^{-1}$; 9 % of the average $\lambda _{\mathrm{exp}}$), and satisfactory near a critical degree of saturation, $S_{\mathrm{r-cr}}$ ($\hbox {RMSE} = 0.162\,\hbox {W} {\cdot } \hbox {m}^{-1} {\cdot } \hbox {K}^{-1}$; 26 % of the average $\lambda _{\mathrm{exp}}$).

Appendices

Appendix 1: Physical Characteristics and Thermal Conductivities of 40 Canadian Soils [8]

See Appendix Tables 12, 13.

Table 12 Density, thermal conductivity, grain size distribution of soil solids, and quartz content

Table 13 Porosity and thermal conductivities at various degrees of saturation [8]

Appendix 2: Physical Characteristics and Thermal Conductivities of Ten Chinese Soils [28]

See Appendix Tables 14, 15.

Table 14 Grain size distribution, porosity, bulk density, thermal conductivity of soil solids, and quartz content

Table 15 Thermal conductivities at various degrees of saturation [28]

Appendix 3: Physical Characteristics and Thermal Conductivities of Four Japanese Soils [30, 31]

See Appendix Tables 16, 17.

Table 16 Grain size distribution, porosity, thermal conductivity of soil solids, and quartz content

Table 17 Thermal conductivities at various degrees of saturation [30, 31]

Appendix 4: Physical Characteristics and Thermal Conductivities of 15 American Soils

1.1 Five Soils from Eastern Washington (USA) [18]

See Appendix Tables 18, 19.

Table 18 Grain size distribution, porosity, thermal conductivity of soil solids, and estimated quartz content

Table 19 Thermal conductivities at various water contents [18]

1.2 Nine Soils from Eastern Washington and Alaska (USA) [32]

See Appendix Tables 20, 21.

Table 20 Grain size distribution, porosity, bulk density, thermal conductivity of soil solids, and quartz content

Table 21 Thermal conductivities at various degrees of saturation [32]

1.3 Thermal Conductivity of Norfolk [33]

See Appendix Tables 22, 23.

Table 22 Grain size distribution, porosity, bulk density, thermal conductivity of soil solids, and quartz content

Table 23 Thermal conductivities at various degrees of saturation [33]

Appendix 5: Physical Characteristics and Thermal Conductivity of Pozzolana (Italy) [34]

See Appendix Tables 24, 25 and 26.

Table 24 Grain size distribution, porosity, thermal conductivity of soil solids, and quartz content

Table 25 Mineralogical contents of Pozzolana and their thermal conductivities [34]

Table 26 Thermal conductivities at various degrees of saturation [34]

Appendix 6: Physical Characteristics and Thermal Conductivities of Three Standard Sands [35]

See Appendix Tables 27, 28.

Table 27 Grain size distribution, thermal conductivity of soil solids, and quartz content

Table 28 Thermal conductivities at various degrees of saturation [35]