## 4 Evolution Mechanism of Unfrozen Water Content in Pore
### Governing Equation of Water Ice Transition in Pore  
When describing the fluid motion process within the pore space, [PERSON]'s law is usually used to characterize the state of energy conservation during fluid flow. Therefore, the governing equation of water-ice phase change flow in coal was derived based on the equation of continuous motion of water and [PERSON]'s law. It is assumed that the side length of the micro-element body in any space in the pore is \(\Delta x\), \(\Delta y\) and \(\Delta z\), and the inflow mass and outflow mass of the water in the coal micro-element body moving along the X-axis are \(q_{ux}-\frac{1}{2}\times\triangle x\times\frac{\delta q_{ux}}{\delta x}\) and \(q_{ux}+\frac{1}{2}\times\triangle x\times\frac{\delta q_{ux}}{\delta x}\). Suppose that the unfrozen water fluxes at the center of the cell are \(q_{ux}\), \(q_{uy}\), \(q_{uz}\). Therefore, in the X-axis direction, unfrozen water flux per unit time (m/s) can be expressed as:  
\[q_{ux}+\frac{1}{2}\Delta x\frac{\partial q_{ux}}{\partial x}-(q_{ux}-\frac{1} {2}\Delta x\frac{\partial q_{ux}}{\partial x})=\Delta x\frac{\partial q_{ux} }{\partial x} \tag{1}\]  
Then, the volume change of unfrozen water in coal cell within unit time can be expressed as:  
\[\Delta x\frac{\partial q_{ux}}{\partial x}\times\Delta y\Delta z=\frac{ \partial q_{ux}}{\partial x}\Delta x\Delta y\Delta z \tag{2}\]  
Assume that the unit time is \(\triangle t\) and the density of unfrozen water is \(\rho_{u}\) (kg\(\cdot\)m\({}^{-3}\)). Therefore, the change value of unfrozen water mass along the X-axis is:  
\[\frac{\partial(\rho_{u}q_{ux})}{\partial x}\Delta x\Delta y\Delta z\Delta t \tag{3}\]  
Then, the total change of unfrozen water mass in three directions \(xyz\) can be expressed as:  
\[-\bigg{[}\frac{\partial(\rho_{u}q_{ux})}{\partial x}+\frac{\partial(\rho_{u} q_{uy})}{\partial y}+\frac{\partial(\rho_{u}q_{uz})}{\partial z}\bigg{]}\Delta x \Delta y\Delta z\Delta t \tag{4}\]  
At the same time, the mass of unfrozen water in coal microelement is \(\rho_{u}\theta_{u}\bigtriangleup x\bigtriangleup y\bigtriangleup z\); \(\theta_{u}\) is the volume content of unfrozen water (%). By the same token, the mass of ice in the cell is \(\rho_{i}\theta_{i}\bigtriangleup x\bigtriangleup y\bigtriangleup z\), where \(\rho_{i}\) (kg\(\cdot\)m\({}^{-3}\)) and \(\theta_{i}\) (%) are the density and volume content of ice, respectively. Therefore, the variation of total moisture content of the microelement per unit time is defined as:  
\[\frac{\partial(\rho_{u}\theta_{u}+\rho_{i}\theta_{i})}{\partial t}\Delta x \Delta y\Delta z\Delta t \tag{5}\]  
According to the law of fluid mass conservation, the following can be obtained:  
\[\frac{\partial(\rho_{u}\theta_{u}+\rho_{i}\theta_{i})}{\partial t}\Delta x \Delta y\Delta z\Delta t=-\bigg{[}\frac{\partial(\rho_{u}q_{ux})}{\partial x} +\frac{\partial(\rho_{u}q_{uy})}{\partial y}+\frac{\partial(\rho_{u}q_{uz})}{ \partial z}\bigg{]}\Delta x\Delta y\Delta z\Delta t \tag{6}\]  
By eliminating homogeneity and simplifying, we can obtain:  
\[\frac{\partial(\rho_{u}\theta_{u}+\rho_{i}\theta_{i})}{\partial t}=-\
abla( \rho_{u}q) \tag{7}\]  
Combined with fluid flux defined in [PERSON]'s law of fluid flow in unsaturated porous media:  
\[q_{u}=-K\
abla\psi=-K\
abla(\psi_{m}+\psi_{g}) \tag{8}\]where \(q_{\rm u}\) represents the flux of unfrozen water in coal, m/s; \(K\) represents the permeability coefficient of unfrozen water in coal. It can also be expressed as the influence coefficient of unfrozen water gravity, which is a function of \(\theta_{\rm u}\). \(\psi\) represents the potential energy involved in the unfrozen water flow process, including the matrix potential energy \(\psi_{\rm m}\) and gravitational potential energy \(\psi_{\rm g}\).