# Artificial Neural Network-Based Model for Prediction of Frost Heave Behavior of Silty Soil Specimen
## 2 Materials and Methods
### Mass Balance Equation  
The volume fraction (\(\theta\)) of a fully saturated soil can be expressed as the sum of the volumetric water content (\(\theta_{w}\)) and the volumetric ice content (\(\theta_{i}\)). It is also expressed as a function of the void ratio (\(e\)) and ice saturation (\(S_{i}\)). Ice saturation is a temperature-dependent function that can be estimated from an empirical function, such as that shown in Equation (2) [36].  
\[\theta=\theta_{w}+\theta_{i}=\frac{e}{1+e}(1-S_{i})+\frac{e}{1+e}S_{i} \tag{1}\]  
\[S_{i}=\left\{\begin{array}{cc}1-\left[1-\left(T-T_{0}\right)\right]^{\alpha}&T \leq T_{0}\\ 0&T\leq T_{0}\end{array}\right. \tag{2}\]  
where \(T\) is the temperature (K), \(T_{0}\) is the freezing temperature of pore water (\(T_{0}\) = 273.15 K), and \(\alpha\) is an empirical parameter.  
The temperature gradient in a freezing soil causes the movement of pore water toward lower temperature regions under uniform pressure fields [37; 38]. To simulate this phenomenon numerically, the mathematical formulation below is required. First, if a thermodynamic equilibrium is satisfied at the interface between the ice and pore water, the following relation is established between the temperature, ice pressure, and pore water pressure according to the Clapeyron equation, which is based on thermodynamics [39; 40; 41; 42].  
\[\frac{u_{w}}{\rho_{w}}-\frac{u_{i}}{\rho_{i}}=L\ln\frac{T}{T_{0}} \tag{3}\]  
where \(\rho_{w}\) is the water density (kg/m\({}^{3}\)), \(\rho_{i}\) is the ice density (kg/m\({}^{3}\)), \(u_{a}\) is the atmospheric pressure (\(u_{a}\) = 101.3 kPa), and \(L\) is the latent heat of fusion (\(L\) = 334.5 kJ/kg).  
The mass balance equation for the freezing behavior can be expressed as  
\[\frac{\partial(\rho_{w}\theta_{w})}{\partial t}dV+\frac{\partial(\rho_{i} \theta_{i})}{\partial t}dV+\rho_{w}\
abla vdV=0 \tag{4}\]  
where \(t\) is the time and \(dV\) refers to the volume element of the soil. Assuming the pressure and temperature to be independent driving forces for the pore water flow, the velocity of the pore water \(v\) can be calculated as follows [43].  
\[v=-k\
abla\psi=-k\
abla(z+\frac{u_{w}}{\gamma_{w}}+\frac{L}{g}\frac{\Delta T }{T_{0}}) \tag{5}\]  
where \(\psi\) is the hydraulic head, which is defined as the sum of the elevation and pressure heads. \(k\) is the hydraulic conductivity (m/s), which is expressed as a function of temperature [12; 13].  
\[\frac{k}{k_{0}}=\left\{\begin{array}{cc}\left[1-\left(T-T_{0}\right)\right] ^{\beta}&T\leq T_{0}\\ 1&T>T_{0}\end{array}\right. \tag{6}\]  
where \(k_{0}\) is the hydraulic conductivity in the unfrozen zone. \(\beta\) is an experimental parameter that varies with the size and structure of the pore, and its values are in the range of \(-8\) to \(-40\).  
The volume element of the soil can be expressed with respect to the solid volume of the soil, and Equation (4) can be expanded as Equation (8).  
\[dV=(1+e)dV_{s} \tag{7}\]  
\[\frac{\partial}{\partial t}[\rho_{w}e(1-S_{i})+\rho_{i}eS_{i}]+\rho_{w}(1+e) \
abla(-k\
abla\psi)=0 \tag{8}\]  
Rearrangement of these equations yields the final formula  
\[\rho_{w}(\frac{1}{1+e}\frac{\partial e}{\partial t}(1-S_{i})-\frac{\partial S _{i}}{\partial t}\frac{e}{1+e})+\rho_{i}(\frac{1}{1+e}\frac{\partial e}{ \partial t}S_{i}\frac{\partial S_{i}}{\partial t}\frac{e}{1+e})+\rho_{w}\
abla v=0 \tag{9}\]