## Materials and Methods  
### Reference Information from the Literature  
#### 2.1.1. Saturated Water Vapour Pressure  
Seven expressions from the literature for the saturated water vapour pressure \(p_{\
u 0}\) as a function of temperature were inspected. All the equations and their reference work are included in Table 1. The equation of [PERSON] and [PERSON] [4], Equation (1), is taken as a reference. Given its high accuracy, it was used to develop the IAPWS-95 formulation [2]. The first inspected expression, Equation (2), is taken from [PERSON] [5]. From among [PERSON]'s expressions, equation \(e_{\
u 6}\) is used in the present work, since it is the one with the smallest relative error in the temperature interval 0 to 100 \({}^{\circ}\)C. It involves temperature fractions as an argument of an exponential expression, and includes four parameters. Next, [PERSON]'s [6] expression, Equation (3), includes temperature fractions multiplying and as an argument of an exponential expression, and five parameters.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Reference** & **Expression** & **Coefficients and** \\  & & **Auxiliary Expressions** & **Equation No.** \\ \hline  & & \(c_{1}=-7.85951783\), \\  & & \(c_{2}=1.84408259\), \\  & & \(c_{3}=-11.7866497\), \\  & & \(c_{4}=22.6807411\), \\  & & \(c_{5}=-15.9618719\), \\  & & \(c_{6}=1.80122502\). \\  & & \(\theta=1-\frac{T}{T_{c}}\), \\  & & \(T_{c}=647.096\) K, \\  & & \(p_{c}=22064\) kPa. \\  & & \(c_{1}=0.61121\) kPa, \\  & & \(c_{2}=18.564\), \\  & & \(c_{3}=254.4\)\({}^{\circ}\)C, \\  & & \(c_{4}=255.57\)\({}^{\circ}\)C. \\  & & \(c_{1}=105\), \\  & & \(c_{2}=1.57\), \\  & & \(c_{3}=34.494\), \\  & & \(c_{4}=-4924.99\)\({}^{\circ}\)C, \\  & & \(c_{5}=237.1\)\({}^{\circ}\)C. \\  & & \(T\) in \({}^{\circ}\)C. \\  & & \(c_{1}=\frac{1}{1944}\) kg/m\({}^{3}\), \\  & & \(c_{2}=0.06374\) K\({}^{-1}\), \\  & & \(c_{3}=-1.634\times 10^{-4}\) K\({}^{-2}\). \\ \multirow{2}{*}{[PERSON] et al. [8]} & \multirow{2}{*}{\(p_{\
u 0}=c_{1}\,exp\left(\frac{c_{3}}{T+273}\,C\right)\)} & \(c_{1}=1.36075\times 10^{8}\) kPa, \\  & & \(c_{2}=-5239.7\)\({}^{\circ}\)C. \\ \multirow{2}{*}{[PERSON] et al. [9]} & \multirow{2}{*}{\(p_{\
u 0}=c_{1}\,exp\left(\frac{c_{3}}{T}\right)\)} & \(c_{1}=1.12659\times 10^{8}\) kPa, \\  & & \(c_{2}=-5192.74\) K. \\ \multirow{2}{*}{[PERSON] et al. [10]} & \multirow{2}{*}{\(p_{\
u 0}=10^{-3}\)kg/m\({}^{3}\frac{RT}{MW}exp\left(c_{1}+\frac{c_{3}}{T}\right)\)} & \(c_{1}=19.84\), \\  & & \(c_{2}=-4975.9\) K. \\ \multirow{2}{*}{[PERSON] and [PERSON] [12]} & \multirow{2}{*}{\(p_{\
u 0}=10^{-3}\)kg/m\({}^{3}\frac{RT}{MW}exp\left(c_{1}+\frac{c_{3}}{T}\right)\)} & \(c_{1}=19.891\), \\  & & \(c_{2}=-4974.0\) K. \\ \hline \hline \end{tabular}
\end{table}
Table 1: Expressions for saturated water vapour pressure as a function of temperature \(T\) from the literature. \(R\): universal gas constant (8.3144 J/mol/K), \(MW\): molar mass of water (0.018016 kg/mol).  
Then, expressions used to model thermal problems in unsaturated soils and clay-based engineered barriers were inspected. First, [PERSON] and [PERSON] [7] use an expression for \(p_{\mathrm{V0}}\), Equation (4), that includes a quadratic polynomial of temperature \(T\) inside an exponential function, and three parameters. [PERSON] et al. [8], Equation (5), and [PERSON] et al. [9], Equation (6), use the same expression structure for \(p_{\mathrm{V0}}\), an exponential of the inverse of temperature with two parameters, although their values are slightly different. Finally, [PERSON] et al. [10] and [PERSON] et al. [11], Equation (7), and [PERSON] and [PERSON] [12], Equation (8), use the same expression structure for \(p_{\mathrm{V0}}\), an exponential function involving the inverse of temperature with three parameters. The two former works use exactly the same parameters, while the latter uses somewhat different figures. The expressions and their reference work are included in Table 1.