## Materials and Methods
### Model Analysis and Selection for Saturated Water Vapour Pressure  
The drift of each saturated water vapour pressure model presented in Section 2.1.1 with respect to the reference of [PERSON] and [PERSON] [4] can be evaluated with its relative error \(RE_{\mathrm{PV0,i}}\)  
\[RE_{\mathrm{PV0,i}}=\Big{(}p_{\mathrm{r0,i}}-p_{\mathrm{r0,(1)}}\Big{)}\Big{/} p_{\mathrm{r0,(1)}} \tag{9}\]  
where \(i\) denotes each of the formulations in Table 1, and \(p_{\mathrm{r0,(1)}}\) is the reference saturated water pressure of [PERSON] and [PERSON] [4], Equation (1).  
However, in model selection, in addition to the similarity to a reference, the number of parameters used is relevant, and parsimony has to be taken into account. Model selection is then a balance between under- and overfitted models [21]. Two of the most usual  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Reference** & \(c^{\mathrm{L}}\) **(J/kg/K)** & \(c^{\mathrm{V}}\) **(J/kg/K)** \\ \hline [PERSON] and [PERSON] [7] & 4180 P & 1870 P \\ [PERSON] et al. [14] & 4180 P & 1900 P \\ [PERSON] and [PERSON] [15] & 4180 P & 1870 P \\ [PERSON] [16] & 4180 P, vol & 1890 P, 1428 vol \\ [PERSON] et al. [17] & 4202 ns & 1620 ns \\ [PERSON] and [PERSON] [12] & 4180 ns & 1900 ns \\ \hline \hline \end{tabular}  
* Superscripts p: isobaric, vol: isochoric, ns: not specified.  
\end{table}
Table 2: Specific heat capacity of water in liquid and vapour forms given in the literature for modelling unsaturated soils and clay-based engineered barriers.  
selection criteria are the Akaike information criterion (AIC) and the Bayesian information criterion (BIC).  
The AIC [18] is an estimator of the expected Kullback-Leibler information loss for each model. In the case of assuming least squares fitting with normally distributed errors for the models, the AIC can be written as [21]  
\[\text{AIC}=n\log\left(\frac{\sum\varepsilon^{2}}{n}\right)+2k \tag{10}\]  
where \(n\) is the sample size, \(k\) is the number of parameters of the model, and \(\varepsilon\) are the residuals from the model. The value of the AIC for each model should be interpreted in terms relative to the values of the other models compared. From a set of \(S\) models, the one obtaining the minimum AIC will define \(\text{AIC}_{\text{min}}\). A normalised measure of the relative likelihood of each model \(i\) is given by the Akaike weights \(Aw\)[21]  
\[Aw_{i}=\frac{\exp[(\text{AIC}_{\text{min}}-\text{AIC}_{i})/2]}{\sum_{s=1}^{S} \exp[(\text{AIC}_{\text{min}}-\text{AIC}_{s})/2]} \tag{11}\]  
\(Aw_{i}\) can be interpreted as the probability that model \(i\) is the best model for the data in terms of Kullback-Leibler information among the \(S\) models analysed. The BIC [19] is based on Bayes factors. In the case of assuming least squares fitting with normally distributed errors for the models, the BIC can be written as  
\[\text{BIC}=n\log\left(\frac{\sum\varepsilon^{2}}{n}\right)+k\;\log(n) \tag{12}\]  
As with the AIC, BIC values should be interpreted in relative terms. From a set of \(S\) models, posterior model probabilities \(Bp\) can be computed for each model \(i\) as [21]  
\[Bp_{i}=\frac{\exp[(\text{BIC}_{\text{min}}-\text{BIC}_{i})/2]}{\sum_{s=1}^{S} \exp[(\text{BIC}_{\text{min}}-\text{BIC}_{s})/2]} \tag{13}\]  
where \(\text{BIC}_{\text{min}}\) is the minimum BIC obtained among the models in the set. \(Bp_{i}\) is the inferred probability that model \(i\) is the most parsimonious model closest to the reference in the model set.  
The model obtaining the \(Aw\) and \(Bp\) values closest to 1 will be selected following AIC and BIC model selection criteria.