# A Numerical Model to Estimate the Soil Thermal Conductivity Using Field Experimental Data
## 3 Results and Discussions
Furthermore, the applied least square method (LSM) is adequat  
Figure 4: _Cont._  
Figure 5 shows the daily average cycle of the \(G_{exp}\) values obtained by the methods \(M4\) and \(M5\), with the values of \(\lambda\) from Table 1 used for these methods. The \(M5\) represent the daily average cycle better than \(M4\) for all hours, especially at night. The greatest difference between the experimental data and \(M4\) and \(M5\), during the nighttime, occurred at 05:30 and was 2.4 Wm\({}^{-2}\) and 0.3 Wm\({}^{-2}\), respectively; and during the daytime occurred at 14:30 and was 3.1 Wm\({}^{-2}\) and 0.9 Wm\({}^{-2}\), respectively. The diurnal peak of \(G_{z}\) was better represented by \(M5\), which had a difference of 0.4 Wm\({}^{-2}\) at 12:30, while \(M4\) showed a difference of 2.5 Wm\({}^{-2}\) at the same hour. The best results obtained by \(M5\) prove that the addition of the \(\varepsilon\) parameter includes corrections that interfere in the estimation of the soil heat flux, i.e., \(G_{z}\) is not simply proportional to the soil temperature gradient. Thus, the inclusion of \(\varepsilon\) also influences the soil thermal conductivity estimation, although the model still considers \(\varepsilon\) constant with depth. Further analysis of the influence of the soil water content on the estimation of \(\lambda\) is shown in Figure 6. All experimental data were used to present the relationship between the soil thermal  
Figure 4: Dispersion diagrams between the values of \(G_{exp}\) (experimental) and the estimated values obtained by the different methods using validation data: (**a**) \(G_{exp}\) and \(G_{M1}\); (**b**) \(G_{exp}\) and \(G_{M2}\); (**c**) \(G_{exp}\) and \(G_{M3}\); (**d**) \(G_{exp}\) and \(G_{M4}\); I \(G_{exp}\) and \(G_{M5}\). The dashed red line represents the linear fit of the data. For reference, the 1:1 line is also shown (solid black line). Figure 5: Annual average daily cycle for soil heat flux at \(-0.10\)\(m\), experimental (\(G_{exp}\)) and estimated by \(M4\) (\(G_{M4}\)) and \(M5\) (\(G_{M5}\)) method. conductivity (\(\lambda\)) and the soil water content (\(\theta\)). In this case, we used only the LSM to solve Equations (2) and (3) for each day, i.e., \(M4\) and \(M5\), obtaining \(\lambda\) daily values. Then, these values were grouped by intervals of 0.01 of the soil water content daily average. With the grouped data, the \(\lambda\) mean value and the respective standard deviation of each cluster were calculated, as shown in Figure 6a. Note that the use of different equation results in a different behavior between the soil thermal conductivity and the soil water content. In general, \(M4\) presented higher mean values and standard deviations than \(M5\). Using another approach, the experimental half-hour data of the days presenting similar soil water content daily averages (0.01 intervals) were grouped. With the grouped data, a single value of \(\lambda\) was estimated for each soil water content range, using \(M4\) and \(M5\), as shown in Figure 6b. For low and high values of soil water content, there are few experimental data; however, the models present the similar values of \(\lambda\) in the intervals of low soil water content and differ considerably for intervals of high values soil water content. For the rest of the soil water content values, both equations showed similar behaviors. The results present in Figure 6a,b show that for high values of soil moisture, both methods generate a greater variability on the \(\lambda\) dataset, concluding that other variables commanding \(\lambda\) values can exist when the soil water content is high, i.e., close to the field capacity.