## 2 Overview of Important Elements of Critiqued Article  
The general isothermal moisture transfer equation adopted in [1] is:  
\[\frac{\mathrm{d}w}{\mathrm{d}t}+\
abla\mathrm{g}=\frac{\mathrm{d}(w_{\mathrm{ w}}+w_{\mathrm{v}})}{\mathrm{d}t}+\
abla(\mathrm{g}_{\mathrm{w}}+\mathrm{g}_{ \mathrm{v}})=\frac{\mathrm{d}(\mathrm{g}\mathrm{S}_{\mathrm{r}}\mathrm{\rho} _{\mathrm{w}}+\mathrm{\phi}(1-\mathrm{S}_{\mathrm{r}})\mathrm{\rho}_{\mathrm{ v}})}{\mathrm{d}t}+\
abla\left(-\mathrm{k}\
abla\mathrm{P}_{\mathrm{c}}-\mathrm{\delta} _{\mathrm{p}}\
abla\mathrm{P}_{\mathrm{v}}\right)=0 \tag{1}\]  
where \(\mathrm{w}\) (kg/m\({}^{3}\)) is moisture content, \(\mathrm{g}\) (kg/m\({}^{2}\)s) is moisture flow, \(\mathrm{t}\) (s) is time, \(\mathrm{\phi}\) (m\({}^{3}\)/m\({}^{3}\)) is porosity, \(\mathrm{S}_{\mathrm{r}}\) (-) is degree of saturation, \(\mathrm{\rho}\) (kg/m\({}^{3}\)) is density, \(\mathrm{k}\) (kg/msPa) is liquid permeability, \(\mathrm{\delta}_{\mathrm{p}}\) (kg/msPa) is vapour permeability, \(\mathrm{p}\) (Pa) is pressure, and subscripts w, v, and c refer to water, vapour, and capillary, respectively. It has to be noted that the (negative) capillary pressure is used here instead of the (positive) suction in [1], as the former is more physically accurate. Flows are typically presumed to go from high to low flow potential values, along the negative of the potential gradient, whereas the inverse is true for suction. In that respect, it should be remarked that [1]'s Equation (9) is incorrect, so a negative sign was e added at the right hand side. In complement to Equation (1), the liquid storage and transport properties of the material are described with the unimodal van Genuchten and Mualem [2] equations in [1]:  
\[\mathrm{S}_{\mathrm{r}}\left(\mathrm{p}_{\mathrm{c}}\right)=\left(1+\left(- \mathrm{\alpha}\cdot\mathrm{P}_{\mathrm{c}}\right)^{n}\right)^{-\mathrm{m}} \tag{2}\]  
\[\mathrm{k}(\mathrm{S}_{\mathrm{r}})=\mathrm{K}\frac{\mathrm{\rho}_{\mathrm{w}} }{\mathrm{\mu}_{\mathrm{w}}}\mathrm{\kappa}(\mathrm{S}_{\mathrm{r}})=\mathrm{ K}\frac{\mathrm{\rho}_{\mathrm{w}}}{\mathrm{\mu}_{\mathrm{w}}}\mathrm{S}_{ \mathrm{r}}^{0.5}\left(1-\left(1-\mathrm{S}_{\mathrm{r}}\frac{1}{n}\right)^{ \mathrm{m}}\right)^{2} \tag{3}\]  
completed with this equation for the vapour permeability:  
\[\mathrm{\delta}_{\mathrm{p}}\left(\mathrm{S}_{\mathrm{r}}\right)=\mathrm{ \delta}_{0}\mathrm{\tau}\mathrm{\phi}(1-\mathrm{S}_{\mathrm{r}}) \tag{4}\]  
where \(\mathrm{\mu}_{\mathrm{w}}\) (Pas) is liquid water viscosity, \(\mathrm{K}\) (m\({}^{2}\), is intrinsic permeability, \(\mathrm{\delta}_{0}\) (kg/msPa) is the vapour permeability of still air, \(\mathrm{\tau}\) (-) is tortuosity correction (assumed equal to 1), and \(\mathrm{\alpha}\) (1/Pa), \(\mathrm{n}\) (-), and \(\mathrm{m}\) (-) are fitting parameters. The parameterisation of the moisture storage and transport properties hence comprises five parameters: porosity \(\phi\), intrinsic permeability K, and fitting parameters \(\alpha\), n, and m.  
In FLoW1D, the Excel-based simulation tool, Equation (1) is solved with finite differences for the spatial discretisation and forward Euler for the temporal discretisation, and the final implementation is verified by means of three qualification exercises, for all of which decent but not perfect agreements with the reference solutions are obtained; see [1]'s Figures 3 and 4. As an exemplary application of FLoW1D, the model is used to characterize the moisture storage and transport properties of two limestones employed in the Cathedral of Sancta Maria and San Juan in Cuenca, Spain. For this, the parameters K, n, m, and \(\alpha\) are inversely identified by fitting simulations to measurements of capillary absorption, while the parameter \(\phi\) is independently determined from porosity measurements.