## 4 Model Application  
In order to illustrate the coupled use of FLoW1D with optimization modules in the Microsoft Excel(r) environment, WAC tests of the limestone used in the construction of the Cathedral of Santa Maria and San Julian in Cuenca, Spain were analyzed. Two different lithotypes of this limestone have been identified: the first lithotype is used in ornamental elements and named ornamental porous stone (OPS), and the second lithotype is a stone used in structural elements, denominated as structural porous stone (SPS). The chemical analysis (see Table S1) performed by X-ray diffraction (see Figure S1) shows that calcite is the major material in both lithotypes, along with a significant fraction of quartz (Q peak in Figure S1b) in SPS. The real and apparent densities [32] are 2746 and 1498 kg m\({}^{-3}\), respectively, for OPS. In the case of SPS, these densities are 2623 and 1886 kg m\({}^{-3}\), respectively. Figure 5(a1,a2) shows samples of the two tested materials. In the image of the optical microscope (Figure 5(b2)), ooliths of an approximate size of 10 \(\upmu\)m are observed with a rounded shape and low crystallinity index, with small regular calcite envelopes whose nuclei are usually formed by quartz clasts, characterizing SPS. Furthermore, the distribution of pore sizes (Figure S2) obtained in the Mercury Intrusion Porosimetry (MIP) test shows that the mean pore size ranges between 1 and 20 \(\upmu\)m for OPS and 0.1 and 10 \(\upmu\)m for SPS [33]. Finally, regarding experimental characterization, Table 2 includes the WAC coefficients obtained according to EN 15801 [14].  
In order to carry out the parameter estimation, FLoW1D was used with the predefined complement _Solver_ of Microsoft Excel(r). The Generalized Reduced Gradient (GRG) was the selected optimization option of _Solver_. This algorithm, developed by [PERSON] and [PERSON] [34], is an extension of the  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Sample** & **WAC Coefficient (kg m\({}^{-2}\) s\({}^{-1/2}\))** & **R\({}^{2}\)** \\ \hline OPS & 0.9157 & 0.976 \\ SPS & 0.1655 & 0.998 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Water absorption by capillarity (WAC) coefficients for samples.  
Figure 5: Ornamental porous stone (OPS) (1) and structural porous stone (SPS) (2). Sample (**a**) and picture from an optical microscope (**b**).  
[PERSON] and [PERSON] [35] reduced gradient method. Its mathematical structure has undergone continuous development [36; 37; 38; 39], but the GRG2 version of this algorithm is the one implemented in _Solver_[40; 41].  
A least squares objective function was adopted. The intrinsic permeability, \(K\), and the parameters of the van Genuchten retention curve, \(n\) and \(m\), were the estimated material parameters. The porosity of the two samples tested was determined by ISO 17892-1 [42], obtaining results of 0.38 and 0.23 for OPS and SPS, respectively. A tortuosity value equal to 1 was assumed, in agreement with the work of [PERSON] and [PERSON] [43].  
The parameter \(\alpha\) of the van Genuchten retention curve can be obtained by the relationship  
\[\alpha=\frac{1}{s_{0}}\left[\left(\frac{1}{Sr_{\mathrm{o}}}\right)^{\frac{1}{n }}-1\right]^{\frac{1}{n}} \tag{15}\]  
where the \(s_{0}\) is the initial suction and \(Sr_{\mathrm{o}}\) is the degree of saturation at the beginning of the WAC test obtained from the water mass in Equation (2):  
\[Sr_{\mathrm{o}}=\frac{w_{\mathrm{o}}-\phi\;\rho_{\mathrm{v}}}{\phi\;(\rho_{ \mathrm{w}}-\rho_{\mathrm{v}})} \tag{16}\]  
where \(w_{\mathrm{o}}\) is the water content at the begging of the WAC test.  
Figure 6 shows the results of two WAC tests carried out with OPS and SPS, respectively, and the simulation developed with the tool FLoW1D.  
As can be observed in Figure 6, a satisfactory fit was obtained for both tests. The estimated values of K, \(n\) and \(m\), are indicated in Table 3. This table also presents the parameter \(\alpha\), which is calculated using Equation (15) using estimated values of \(n\) and \(m\).