## 2 Materials and Methods
### Impact of Compaction on the Thermal Properties  
After knowing the thermal conductivity curve as a function of \(S_{e}\) in an uncompacted porous media, e.g., a relationship such as that presented in Equation (9), it is of interest to characterize the impact of the compaction in the thermal conductivity curve. We proposed to use different theoretical approaches combined with empirical information to include the impact of compaction on the thermal conductivity curve. The [PERSON] [39] model can be used to represent \(\lambda_{0}\). This relationship depends on \(S_{e}\) and on the bulk density (\(\rho\)) of the media [40]:  
\[\lambda_{0}(S_{e})= (\lambda_{0s}-\lambda_{0d})\lambda_{e}+\lambda_{0d}, \tag{22}\]  
where \(\lambda_{0s}\) and \(\lambda_{0d}\) are the thermal conductivities in the saturated and the dry states, respectively; and \(\lambda_{e}\) is the Kersten number, defined as [39; 41]:  
\[\lambda_{e}=\frac{\kappa S_{e}}{1+(\kappa-1)S_{e}}, \tag{23}\]  
where \(\kappa\) is an empirical parameter that depends on the soil type and does not change with \(\rho\)[42]. The \(\lambda_{0s}\) is estimated using the following expression [39]:  
\[\lambda_{0s}=\lambda_{sol}{}^{(1-\theta_{s})}\lambda_{w}{}^{\theta_{s}}, \tag{24}\]  
where \(\lambda_{w}\) is the water thermal conductivity and \(\lambda_{sol}\) is the solid particles thermal conductivity. Equation (24) assumes that the porosity of the porous media is similar to its saturated volumetric water content. The \(\lambda_{0d}\) is determined using the Ball and Arp model [40], which is suitable for any type of soil:  
\[\lambda_{0d}=\frac{(a\lambda_{sol}-\lambda_{air})\rho+\lambda_{air}\rho_{s}}{ \rho_{s}-(1-a)\rho}, \tag{25}\]where \(a\) is an empirical parameter, \(\rho\) is the bulk density, \(\rho_{s}\) is the particle density of the porous media, and \(\lambda_{air}\) is the thermal conductivity of air. The thermal conductivity curve in absence of fluid flow for the compacted porous media, \(\lambda_{c}(\theta)\), needs some inputs from the hydrodynamic properties: first, to determine \(\lambda_{0s}\), the \(\theta_{s}\) is required, as shown in Equation (24); and second, the lower and upper bounds of the thermal conductivity curve are related to the \(\theta_{r}\) and \(\theta_{s}\) that also change as compaction occurs. Also, note that the impact of compaction on the \(\lambda_{c}(\theta)\) is represented through the bulk density that is used to determine \(\lambda_{0d}\), but also through the change in the \(\theta_{s}\) described in Equation (11), which modifies the \(\lambda_{0s}\). The compaction methodology proposed in this investigation required the use of empirical data to determine how the thermal conductivity changed as compaction occurred. First, the thermal conductivity curve must be known for a specific \(\rho\), i.e., the values of \(\lambda_{0}(S_{e})\), \(\lambda_{0s}\), and \(\lambda_{0d}\) were known. Combining this information with Equation (22) allowed the Kersten number (\(\lambda_{e}\)) to be found for different saturation degrees. Then, Equation (23) was used to find the value of \(\kappa\) that represents the best the \(\lambda_{\mathbf{\mathrm{e}}}\) for each porous media. Recall that \(\kappa\) does not change with \(\rho\)[42]. According to [PERSON] and [PERSON] [41] the value of \(\kappa\) depends on soil type; well-graded gravels and coarse sands have \(\kappa\approx 4.6\), silts and clays have \(\kappa\approx 1.9\), and peat has a \(\kappa\approx 0.6\). In this study, we used the \(\kappa\) value as a fitting parameter to obtain a \(\lambda_{e}\) (\(S_{e}\)) for each substrate that agreed with the experimental information. Later, Equations (24) and (25) were used to find the value of \(\lambda_{sol}\) that matched better the thermal conductivity curve of each porous medium, and to find the value of a that represented the behavior of all the porous media.