## 1 Introduction  
Green roofs have been extensively adopted over the last few decades as a technological solution for sustainable development [1; 2; 3], as they integrate vegetation into buildings to minimize some negative impacts of urbanization [2; 3; 4; 5]. A properly designed green roof can reduce pollution, noise levels, building energy consumption, and stormwater runoff [2; 6; 7]. At the same time, green roofs increase biodiversity and provide ecological services in urban environments [8; 9]. Green roof overall performance strongly relies on the interaction between environmental conditions and the green roof's biophysical components [2; 5]. A systematic comprehension of the processes occurring in every component of a green roof, under site-specific conditions, is required to understand the benefits described above. This systematic comprehension is extremely important in arid and semi-arid areas, where construction materials may be different and meteorological conditionssignificantly differ from humid climates, where majority of green roof investigations have been carried out to date [1; 2; 10; 11; 12]. The vegetation and substrate layers are the central components that determine water and heat fluxes on green roofs [2; 3; 12; 13]. This work focuses on green roof substrates, even when it is known that vegetation is relevant on green roof hydraulic and thermal performance. Substrates are engineered media with an improved performance compared to natural soils [14]. Substrates have been typically used in many civil engineering applications. The substrate provides resources that are critical for the vegetation, such as a support media, nutrients and water [2; 15]. The thermal and water retention properties of a green roof greatly rest on the substrate's physical properties [16]. The heat flux through a green roof and the substrate's moisture content affect the buildings' heat gains and losses, which impacts the consumption of energy in the building. Also, substrate's moisture content is key for the survival of the vegetation, especially in arid and semi-arid areas [1; 2]. Therefore, it is important to comprehend the physical, thermal, and hydrodynamic substrate properties and how they change through time. Even when many investigations addressing green roofs thermal performance consider the impacts of moisture on the substrate thermal properties, they commonly neglect how water flow affects the energy balance [17; 18]. [PERSON] [17] developed a widely used model that represents heat fluxes across the substrate and includes the atmosphere-vegetation-substrate energy exchange. [PERSON]'s [17] model assumes that moisture in the substrate diffuses at a constant rate. However, it is widely known that there are highly non-linear processes occurring in this system [2; 19]. [PERSON] and [PERSON] [18] and [PERSON] et al. [20] incorporated how compaction affects the substrate thermal properties. However, their model did not account for the effect of water flow on the energy balance. Soil compaction plays a key role in solute transport, nutrient availability, crop productivity, and water movement [21]. Soil compaction has been typically investigated without considering its impact on the coupled water and heat transport in porous media [22; 23]. There are many investigations that deal with the impact of compaction on water fluxes [21; 24; 25; 26], as well as many studies that investigate how compaction affects heat fluxes in soils [18; 20]. Nonetheless, there is a lack of studies that have investigated the effect of substrate compaction on the combined water and heat transport in green roofs. Hence, substrate compaction in green roofs has been typically neglected. However, studies demonstrate that through time, the substrate settles and changes the thermal and hydraulic behavior of green roofs [7; 12; 27]. The objective of this study is to develop a methodology that can be used to represent the effect of compaction on the hydrodynamic and thermal properties of a porous medium.