## 3 Conceptual Model and Numerical Implementation
### Numerical Model and System Abstraction  
To solve the boundary value problem, an explicit finite difference scheme has been used. A detailed description of the spatial and temporal discretization applied is included in Appendix C, but Figure 1 provides a scheme of the discretization of the domain used. The lower grid point (\(i=1\)) corresponds to the base of the sample in contact with the saturated permeable layer. Consequently, a Dirichlet condition of suction equal to 0 Pa has been applied in this boundary. The boundary condition applied to the upper grid point (\(i=n\)) is estimated using the formulation described in EN 15026 for moisture transport across the interfaces between the material and environment. In the WAC tests, the specimens are kept in a desiccator with a controlled relative humidity (\(RH\)) of 32% before being tested. Therefore, the initial condition set is the suction corresponding to this \(RH\), which is calculated using the psychometric law (Equation (A2)). The numerical scheme described in Appendix C has been programmed using the Visual Basic for Application (VBA) language and implemented as a Microsoft Excel(r) function called FLoW1D. Figure 2 shows a snapshot of the tool interface. The code offers the possibility of calculating the relative permeability by means of the van Genuchten [24] or the Brooks and Corey [26] approaches. The van Genuchten model is the default option. If the user wants to use the Brooks and Corey model, the value of \(\lambda\) (empirical parameter of the model) must be included in the input of variables (cell C3 in Figure 2a). The arguments of FLoW1D (**Par**, Tini, Tfin, **So**) are entered in the same way as in a predefined Excel function, as shown in Figure 2b. The **Par** argument is the vector (vectors are identified by using bold font throughout the paper) of material parameters and geometric dimensions (cells C1 to C8 in Figure 2a). The \"Tini\" and \"Tfin\" arguments correspond to the initial and final time of the time increment modeled. The **So** argument is the vector corresponding to the suction at the \"Tini\" time. Given that a match is made between the spreadsheet cells and the grid points of the discretization, the first value for all components of **So** (the initial condition of the boundary value problem) must be defined in cells H3 to H103 in Figure 2b (if the domain is discretized in 101 grid points). By default, the value computed in cell C12, obtained from the initial \(RH\) after applying the psychometric law (Equation (A2)), is assumed as the first value of So. For the following computational times (defined in cells I2, J2, K2 and so on), **So** is the result of the previous time step. The boundary conditions are defined as indicated in Figure 1. For all computational times, a Dirichlet boundary condition is assumed at the bottom grid point. Therefore, cells H3, J3, K3 and so on are equal to the value \"s1\" defined in cell C14 (Figure 2a). If the psychometric law (Equation (A2)) is applied, s1 is a function of the \(RH\) value consigned in cell C13 and consequently is sufficient to change that value to modify the suction value at grid point 1 throughout the computational process. Although a Dirichlet condition has been assumed by default in grid point 1, FLoW1D has the option to activate a von Neumann boundary condition, such as that indicated for the grid point n in Figure 1. If the user activates this option--and since the term corresponding to the liquid phase is assumed to equal to zero--the flow \(g_{\text{B0}}\) applied to grid point 1 is given, according to the EN 15026 standard, by the expression  
\[g_{\text{B0}}=\frac{\delta_{0}}{s_{\text{d,0}}}(P_{\text{v,0}}-P_{\text{v,1}}) \tag{14}\]  
where \(s_{\text{d,0}}\) is the equivalent vapor diffusion thickness of bottom interfaces, \(P_{\text{v,0}}\) is the vapor pressure obtained by the relative humidity at the environment in the bottom interface and \(P_{\text{v,1}}\) is the vapor pressure obtained by the psychometric law (Equation (A2)) at grid point 1.