Advanced Geometric Mean Model for Predicting Thermal Conductivity of Unsaturated Soils

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