## Materials and Methods
### Thermodynamic Basis for Analysing the Consistency of the Formulations  
The energy balance equation for the thermal problem in soils can be written in terms of the specific internal energy \(u\) of each phase \(i\) (solid, liquid, gas) and species \(j\) (soil, water, air) composing the mixture of the soil [8; 11; 12; 16; 22; 23]  
\[\sum_{i,j}\frac{\partial}{\partial t}\left(\rho^{ij}\theta^{ij}u^{ij}\right)= \sum_{i,j}-\
abla\cdot\left(-\mathbf{K}\cdot\
abla T+\rho^{ij}\theta^{ij}u^{ ij}\mathbf{v}^{ij}\right)+g \tag{14}\]  
where \(t\) is time, \(\
abla\) is the gradient operator, \(\
abla\cdot\) is the divergence operator, \(\mathbf{K}\) is the thermal conductivity tensor, \(g\) is an energy source term, and \(\rho^{ij}\), \(\theta^{ij}\) and \(\mathbf{v}^{ij}\) are, respectively, the density, the specific volume and the velocity of component (\(i\), \(j\)). However, it is usual to write this balance equation in terms of the specific enthalpies \(h^{ij}\) of each component [7; 9; 10; 14; 17; 23]  
\[\sum_{i,j}\frac{\partial}{\partial t}\left(\rho^{ij}\theta^{ij}h^{ij}\right)= \sum_{i,j}-\
abla\cdot\left(-\mathbf{K}\cdot\
abla T+\rho^{ij}\theta^{ij}h^{ ij}\mathbf{v}^{ij}\right)+g \tag{15}\]The total differentials \(\mathrm{d}u\) and \(\mathrm{d}h\) can, in a general case, be written in terms of the isochoric \(c_{\mathrm{v}}\) and isobaric \(c_{\mathrm{p}}\) specific heat capacities as [20]  
\[\mathrm{d}u=c_{\mathrm{v}}\ \mathrm{d}T+\left[T\left(\frac{\partial P}{ \partial T}\right)_{\mathrm{v}}-P\right]\mathrm{d}\mathrm{v} \tag{16}\]  
\[\mathrm{d}h=c_{\mathrm{p}}\ \mathrm{d}T+\left[-T\left(\frac{\partial\
u}{ \partial T}\right)_{\mathrm{p}}+\mathrm{v}\right]\mathrm{d}P \tag{17}\]  
where \(P\) is pressure, \(\mathrm{v}\) is volume, and ()\(\mathrm{v}\) and ()\(\mathrm{p}\) indicate derivative under constant volume and pressure, respectively. Although specific heat capacities are functions of temperature and pressure, they are usually taken as constants in the modelling of engineering geomaterials. In addition, in such modelling, the differentials \(\mathrm{d}u\) and \(\mathrm{d}h\) are usually formulated as a function of a constant specific heat capacity for each component and temperature only, see [7; 8; 9; 10; 11; 12; 14; 16; 17; 23]. Then, for the components water vapour V and liquid water L, the internal energy and enthalpy terms can be written as  
\[u_{\mathrm{V}}-u_{\mathrm{V0}} =c_{\mathrm{v}}^{\mathrm{V}}(T-T_{0}) \tag{18}\] \[u_{\mathrm{L}}-u_{\mathrm{L0}} =c_{\mathrm{v}}^{\mathrm{L}}(T-T_{0})\] (19) \[h_{\mathrm{V}}-h_{\mathrm{V0}} =c_{\mathrm{p}}^{\mathrm{V}}(T-T_{0})\] (20) \[h_{\mathrm{L}}-h_{\mathrm{L0}} =c_{\mathrm{p}}^{\mathrm{L}}(T-T_{0}) \tag{21}\]  
where \(u_{\mathrm{V0}}\) and \(u_{\mathrm{L0}}\) are the internal energies and \(h_{\mathrm{V0}}\) and \(h_{\mathrm{L0}}\) are the enthalpies of vapour and liquid water, respectively, at reference temperature \(T_{0}\). The enthalpy of vaporisation is an energy sink in the energy balance equation in soils. At a given temperature, the enthalpy of vaporisation \(\Delta h_{\mathrm{LV}}\) is equal to the difference between water vapour and liquid water enthalpies, which following Equations (20) and (21) gives  
\[\Delta h_{\mathrm{LV}}=h_{\mathrm{V}}-h_{\mathrm{L}}=\left[\Delta h_{\mathrm{ LV,T0}}-\left(c_{\mathrm{p}}^{\mathrm{V}}-c_{\mathrm{p}}^{\mathrm{L}}\right)T_{0} \right]+\left(c_{\mathrm{p}}^{\mathrm{V}}-c_{\mathrm{p}}^{\mathrm{L}}\right)T \tag{22}\]  
where \(\Delta h_{\mathrm{LV,T0}}\) is the enthalpy of vaporisation at \(T_{0}\). Given that \(\Delta h_{\mathrm{LV,T0}}\) and \(T_{0}\) are constant, if \(c^{\mathrm{V}}\)p and \(c^{\mathrm{L}}\)p are constant too, the enthalpy of vaporisation should respond to a linear equation  
\[\Delta h_{\mathrm{LV}}=a+b\ T \tag{23}\]  
where \(a\) and \(b\) are constant parameters. An equation such as Equation (23) can be fitted to reference data of the enthalpy of vaporisation to find these parameters.