# A Numerical Model to Estimate the Soil Thermal Conductivity Using Field Experimental Data
## 2 Materials and Methods
### Proposed Model  
The relationship between \(G_{z}\) and \(\frac{\Delta T}{\Delta z}\), represented by Equation (2), is described by a linear function of the variables, with \(\lambda\) being the proportionality constant. However, experimentally, we do not obtain a relation defined by a linear function described by \(y=ax\); instead, we obtain a relation defined by a function described by \(y=ax+b\), as shown in Figure 1, independent of the soil water content. This figure illustrates the relationship between the \(G_{z}\) measured at \(z=-0.10\) m and the variation rate between the soil temperature and the depth \(\left[\frac{T_{2}-T_{1}}{z_{2}-z_{1}}\right]\), with \(z_{1}=-0.05\) m and \(z_{2}=-0.15\) m, does not satisfy Equation (2); since the graph \(G_{z}\times\frac{\Delta T}{\Delta z}\) does not pass through the origin, there will be an underestimate or overestimate of \(G_{z}\) with respect to \(\left[\frac{T_{2}-T_{1}}{z_{2}-z_{1}}\right]\) for values negative or positive of \(G_{z}\), respectively. We assume that this behavior may be related to the finite difference method and the experimental measurements of \(T\) and/or \(G_{z}\). Therefore, to estimate \(\lambda\) from the experimental data on the soil heat flux (\(G_{exp}\)) and the soil temperature at different depths, we propose to include an empirical parameter (\(\varepsilon\)) in Equation (2). The new model for \(G_{z}\) can be expressed as:  
\[G_{z}~{}=-\lambda\left[\frac{T_{2}-T_{1}}{z_{2}-z_{1}}\right]+\varepsilon \tag{3}\]  
where \(\varepsilon\) is a constant empirical parameter to be determined, representing a correction factor of an energy flux that is not accounted for by the field experimental measurements. The addition of \(\varepsilon\) becomes the model able to fit the observed behavior in the experimental data (Figure 1). In analytical solutions of the Fourier Law, although not explicitly, this correction is also present, for example in [17; 33; 34]. In these cases, the estimation of this correction is quite complex, requiring a greater number of parameters obtained, in general, from adjustments to experimental data of soil heat flux and soil temperature. Data Fitting: Least Squares Approximation  
Figure 2 shows the daily cycle for the soil heat flux (\(G_{z}\)) at \(z=-0.10\) m and the temperatures at \(z_{1}=-0.05\) m and \(z_{2}=-0.15\) m for four days in different seasons (Summer DOY 21; Autumn DOY 111; Winter DOY 202; and Spring, DOY 325). Precipitation of 0.1 mm occurred in DOY 325 at 01:00 AM (local time). These days were chosen to emphasize the variability in the amplitude of \(G_{z}\) and soil temperature values and the hours when the soil temperature gradient is zero (\(T_{z_{1}}\approx T_{z_{2}}\)), in different seasons. It may be noted that at hours when the soil temperatures approach each other at different depths, the \(\lambda\) obtained from Equation (2), by the division of \(G_{z}\) by \(\left[\frac{T_{2}-T_{1}}{z_{2}-z_{1}}\right]\), can result in nonfinite values or lead to numerical divergences resulting in a divergent solution. Figure 1: Relationship between \(G_{exp}\) at \(z=-0.10\)\(m\) and the ratio of the soil temperature in relation to the depth (\(\Delta T/\Delta z\)) with \(z_{1}=-0.05\)\(m\) and \(z_{2}=-0.15\)\(m\). Data obtained at the Pedras Altas-RS experimental site for the period from 1 –31 July 2015. Figure 2: _Cont._  
The daily average of \(G_{exp}\) and \(T_{soil}\) for the different depths can show variability throughout the year, as shown in Figure 3. The existence of days that have low-temperature variability at different depths or values of \(G_{exp}\) near zero can also cause numerical divergences, especially when isolating \(\lambda\) in Equation (2) [19; 30]. In Figure 3, the daily soil water content (\(\theta\)) is also shown, presenting a high variability throughout the year, with small values in summer and autumn seasons.