## 3 Conceptual Model and Numerical Implementation  
### Conceptual Model  
The formulation used for balance equations, field equations and the structure of state functions is based on the conceptual model described in EN 15026. For the simulation of WAC tests, an isothermal and one-dimensional flow has been assumed. In addition, the PBM is considered non-deformable. The water mass balance, without considering water sources or sinks in the domain, is defined by the expression  
\[\frac{\partial w}{\partial t}+\
abla g=0 \tag{1}\]  
where \(w\) is the water content per total volume unit (kg m\({}^{-3}\)), \(g\) is the water flow rate (kg m\({}^{-2}\) s\({}^{-1}\)) and \(\
abla\cdot\) is the divergence differential operator. The matric suction (defined as the difference between gas pressure and liquid pressure, \(s=P_{G}-P_{L}\)) has been selected as the state variable. In the water content term, the contributions of the liquid, \(w_{\rm w}\), and vapor, \(w_{\rm v}\), phases have been considered according to  
\[w=w_{\rm w}+w_{\rm v}=\phi\;Sr\;\rho_{\rm w}+\phi\;(1-Sr)\rho_{\rm v}=\theta \;\rho_{\rm w}+(\theta_{\rm sat}-\theta)\;\rho_{\rm v} \tag{2}\]  
where \(\phi\) is the porosity (m\({}^{3}\) m\({}^{-3}\)), \(Sr\) is the degree of saturation (m\({}^{3}\) m\({}^{-3}\)), \(\theta\) is the volumetric water content (m\({}^{3}\) m\({}^{-3}\)), \(\theta_{\rm sat}\) is the volumetric water content in saturated conditions (m\({}^{3}\) m\({}^{-3}\)), \(\rho_{\rm w}\) is the liquid water density (kg m\({}^{-3}\)) and \(\rho_{\rm v}\) is the water vapor density (kg m\({}^{-3}\)), estimated by the psychrometric equation (Equation (A2)). The water flow rate, \(g\), is obtained from the contributions of the liquid, \(g_{\rm w}\), and vapor, \(g_{\rm v}\), phases:  
\[g=g_{\rm w}+g_{\rm v} \tag{3}\]  
The liquid and vapor fluxes were defined by adopting the approaches from [PERSON] et al. [22]. The term corresponding to the liquid phase is defined as  
\[g_{\rm w}=k\;\
abla s \tag{4}\]  
where \(k\) is the hydraulic conductivity (s) and \(\
abla s\) is the suction gradient (Pa m\({}^{-1}\)). The hydraulic conductivity has been calculated as  
\[k=\frac{\kappa\,K_{\rm sat}}{\gamma_{\rm w}}\,\rho_{\rm w} \tag{5}\]where \(\gamma_{\rm w}\) is the specific weight of the water (N m\({}^{-3}\)), \(\kappa\) is the relative permeability (dimensionless) and \(K_{\rm sat}\) is the saturated conductivity (m s\({}^{-1}\)), defined as  
\[K_{\rm sat}=\frac{K}{\mu_{\rm w}}\gamma_{\rm w} \tag{6}\]  
where \(K\) is the intrinsic permeability (m\({}^{2}\)). The relative permeability has been determined by the van Genuchten-Mualem [23] approach:  
\[\kappa=Sr^{\frac{1}{2}}\Big{[}1-\Big{(}1-Sr^{\frac{1}{m}}\Big{)}^{m}\Big{]}^{2} \tag{7}\]  
where \(m\) (dimensionless) is a fitting parameter used in the van Genuchten [24] water retention curve:  
\[Sr=\Big{(}1+(\alpha\ s)^{n}\Big{)}^{-m} \tag{8}\]  
where \(\alpha\) (Pa\({}^{-1}\)) and \(n\) (dimensionless) are also van Genuchten model parameters. The water vapor flow rate is calculated as  
\[\mathbf{g}_{\rm v}=\delta_{\rm p}\
abla P_{\rm v} \tag{9}\]  
where \(\delta_{\rm p}\) is the vapor permeability (kg Pa\({}^{-1}\) m\({}^{-1}\) s\({}^{-1}\)) and \(\
abla P_{\rm v}\) is the water vapor pressure gradient (Pa m\({}^{-1}\)). The vapor permeability is calculated by  
\[\delta_{\rm p}=\frac{\delta_{0}}{\mu} \tag{10}\]  
where \(\delta_{0}\) (kg Pa\({}^{-1}\) m\({}^{-1}\) s\({}^{-1}\)) is the vapor permeability of the still air and \(\mu\) (dimensionless) is the diffusional resistance factor.