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 $$\lambda _{\mathrm{exp}}$$ ).