## 4 Discussion
The lower values obtained for the \(\lambda_{sol}\) of each substrate, compared to natural soils, could be justified by the presence of organic matter and inorganic materials. The \(\lambda_{sol}\) values were fitted so the simulated \(\lambda_{0s}\) and \(\lambda_{0d}\) were similar to the experimental measures. This agreement between the simulated and experimental values was achieved in all the substrates except for substrate S2, where the fitted \(\lambda_{0d}\) (0.047 W m\({}^{-1}\) K\({}^{-1}\)) was lower than the measured \(\lambda_{0d}\) (0.101 W m\({}^{-1}\) K\({}^{-1}\)). However, the fitted and measured \(\lambda_{0s}\) of this substrate agreed with a difference of \(\sim\)1%. In general, for a specific moisture level, as compaction occurs the thermal conductivity of the substrates increase. This behavior is consistent with the findings of other investigations [18; 20] and occurs because as the substrate is compacted, there is more physical contact between the solid particles, which increase thermal conduction [20; 24; 25; 35]. Also, the magnitude of the thermal conductivity is similar to that presented in the literature [15; 20]. Figure 5a-e also shows that substrates S2, S3, and S5 present a linear increase of thermal conductivity with moisture, contrary to the non-linear behavior observed in substrates S1 and S4. In general, the impact of compaction on the thermal conductivity curve is more relevant in the saturated zone of the curve, i.e., when \(S_{e}\) approaches 1. For example, in substrate S2 the \(\lambda_{0s}\) increased by 0.17 W m\({}^{-1}\) K\({}^{-1}\) (\(\sim\)30% increase). In the dry zone of the curve, i.e., when \(S_{e}\) approaches 0, the substrates' \(\lambda_{0d}\) presented slight increases, except for substrate S1, which exhibited an increment of 0.15 W m\({}^{-1}\) K\({}^{-1}\). Similar effects are reported in other investigations, where the thermal conductivity curves of different soils shows increases of 0.15 and of 0.4 W m\({}^{-1}\) K\({}^{-1}\) for \(\lambda_{0d}\) and \(\lambda_{0s}\) when the soil is compacted [18]. However, previous investigations do not account for the hydrodynamic effects of compaction on the conductivity curves. The addition of the hydrodynamic properties on the compaction method could be clearly seen in substrates S1 and S2, as shown in Figure 5a,b. The highest change between the uncompacted and compacted thermal conductivity curves occurred in substrates S1 and S2. In S1, the increase in thermal conductivity was similar in all the range of the curve; contrary to what was observed for substrate S2, where the increment near the saturated part was too large while the increment near the dry part of the curve was too small. The small increment of the dry thermal conductivity in substrate S2 was due to its very low bulk density (Equation (25)). In addition, the large change of density caused a large increment in \(\theta_{r}\) (0.23 to 0.40 cm\({}^{3}\) cm\({}^{-3}\)) (Equation (12)). As a result, the compacted and uncompacted curves intersected. Regarding the hydrodynamic conditions through the hypothetical green roof, our results showed that compaction reduced the amplitude of the fluctuations in the volumetric water content daily cycles (see Appendix B). Also, the cumulative water that exits the roof (Figure 6) suggested that the breakthrough time occurred earlier in the compacted substrates, most likely due to a reduction in the pore spaces. After compaction, the reduction in the breakthrough time of the substrates ranged between one hour (S3) and three days (S4). This result highlights the nonlinearity effect that compaction has on water fluxes. In all the substrates the amount of water that drains the roof increased, especially in the S3 substrate where the cumulative water flux increased by more than three times. This significant increase in the cumulative water flux was the result of a decrease in the pore spaces and the changes in the response of each substrate to the atmospheric conditions.