## 4 Discussion  
To truly quantify the urbanization negative impacts that a green roof mitigates, a quantitative assessment of each of its components must be performed, and the relevant physical processes must be included in this assessment. One aspect that has been typically neglected is the impact that substrate compaction has on the coupled water and heat fluxes across the roof. In this work, we developed a parametric methodology that used empirical substrate data to consider how compaction changed the substrate's hydrodynamic and thermal properties. By analyzing the changes in the water retention curve when the substrate was compacted (Figure 3), it was clear that the pores decreased their size, diminishing the \(\theta_{s}\) and increasing the  
Figure 7: (**a-e**) Heating cumulative thermal energy required to maintain a reference room temperature of 23 \({}^{\circ}\)C below the roof system. (**f-j**) Cooling cumulative thermal energy required to maintain a reference room temperature of 23 \({}^{\circ}\)C below the roof system. air-entry pressure (\(h_{b}=\alpha^{-1}\)) due to an increase in the capillary forces [37]. These changes in the porous media water retention capacity are consistent with the observations published in the scientific literature [2; 15; 24; 46; 47]. For instance, in the S2 compacted water retention curve, the \(h_{b}\) increased from 17 to 526 cm, while the \(h_{b}\) of the S1 substrate increased from 9 to 53 cm. The increases in the \(h_{b}\) values were on the same order of magnitude as those reported by [PERSON] [37]. However, the impact of compaction on the \(\theta_{r}\) was not the same for all the substrates. S1, S2 and S5 showed an increase between 0.05 and 0.07 cm\({}^{3}\)cm\({}^{-3}\) in \(\theta_{r}\) as compaction occurred while S3 and S4 showed an insignificant change. The reason of this change is that the original density values of \(\theta_{r}\) of S3 and S4 were near to zero and the Equation (12) used the original density to determine the compacted values of \(\theta_{r}\). In addition, the changes in \(n\) and small changes in \(\theta_{r}\) drove unexpected retention curves as shown in Figure 3c: the compacted water retention curve of S3 (\(\rho=1\) g cm\({}^{-3}\)) intercepted the uncompacted curve (\(\rho=1.2\) g cm\({}^{-3}\)) at \(\sim\)0.15 cm\({}^{3}\)cm\({}^{-3}\), meaning that below this value the water retention decreased. However, as the pore size decreases, an increase in the capillary forces is expected and thus one would expect a general increase in water retention [21]. This phenomenon most likely occurs because the properties of these artificial substrates are very different than those of natural soils, which are the media used to develop the algorithms that predict the effect of compaction on the hydrodynamic properties [37]. Therefore, for substrates S3 and S4 the proposed model may not be the best representation of how compaction influences the hydrodynamic properties. From the results obtained for substrates S1-S5 (Figure 3), we concluded that the proposed method resulted in physically possible values for volumetric water contents that are greater than 0.2 cm\({}^{3}\) cm\({}^{-3}\). Regarding the thermal properties, the values of \(\kappa\) were within the range of values reported for natural soils, and were typically in the range reported for silt, clay, and peat [41]. The difference between the \(\kappa\) values obtained for the substrates used in this study and those reported for natural soils are most likely explained by the artificial nature of the substrates. The \(\lambda_{sol}\) values depend on the fraction of gravel, sand, silt, clay, and percentage of quartz of each substrate. As a reference, soils with high organic matter have \(\lambda_{sol}\sim\)0.25 W m\({}^{-1}\) K\({}^{-1}\), while silty foam soils have \(\lambda_{sol}\sim\)-2.05 W m\({}^{-1}\) K\({}^{-1}\), and crushed quartz has \(\lambda_{sol}\sim\)7.2 W m\({}^{-1}\) K\({}^{-1}\)[42]. The \(\lambda_{sol}\) presented in Table 3 ranges between 1.103 and 2.124 W m\({}^{-1}\) K\({}^{-1}\).