## 2 Materials and Methods
### Thermophysical Modeling
\[\frac{\rho}{\epsilon_{p}^{2}}(\mathbf{u}\cdot\
abla)\mathbf{u}=\
abla\cdot \left[-p\mathbf{I}+\frac{\mu}{\epsilon_{p}}(\
abla\mathbf{u}+(\
abla\mathbf{u })^{T})\right]-\frac{\mu}{k}\mathbf{u}-\frac{\rho\epsilon_{p}C}{\sqrt{k}} \mathbf{u}|\mathbf{u}| \tag{2}\]  
where \(\mathbf{u}\) is the water velocity in the channel and adjacent rock (m/s), \(\rho\) is the water density (kg/m\({}^{3}\)), \(p\) is the pressure (Pa), \(\epsilon_{p}\) is the porosity of the medium (1/1), \(\mu\) is the dynamic viscosity (Pa-s), \(k\) is the permeability of the porous medium (m\({}^{2}\)), and \(C_{f}=\frac{1.75}{\sqrt{150\epsilon_{p}^{3}}}\) is the dimensionless coefficient of friction [22]. At the start of the model, the water is considered to be motionless (\(\mathbf{u}=0\)) at atmospheric pressure (relative pressure is zero). At the entrance to the channel, the speed of water inflow is 100 m/day and the exit from the channel is at atmospheric pressure. At large distances from the channel, there is no water flow and a constant temperature, \(Ti\). Figure 5: The structure of wells t1 and t3, adopted for numerical modeling of thawing by circulating water. It is assumed that the physical properties of the water and rock media are not temperature-dependent; the values assigned are water density of 1000 kg/m\({}^{3}\), water dynamic viscosity of 10\({}^{-3}\) Pa-s, medium porosity of 0.4, and medium permeability of 10\({}^{-12}\) m\({}^{2}\) (\(k\)). The thermal properties of media are given in Table 1.  
The resulting distribution of velocities (**u**) is substituted into the heat conduction equation with a convective term describing the water flow and is solved in temporary settings with water temperature (both at the rock surface and the channel inlet) changing during the year, according to the graph shown in Figure 4.  
\[(\sigma c)_{eff}\frac{\partial T}{\partial t}+\rho_{f}c_{f}\textbf{u}\cdot \
abla T=\
abla\cdot(\lambda_{eff}\
abla T) \tag{3}\]  
where \(\left(\sigma c\right)_{eff}=(1-\varepsilon_{p})\rho_{s}c_{s}+\varepsilon_{p} \rho_{f}c_{f}\) and \(\lambda_{eff}=(1-\varepsilon_{p})\lambda_{s}+\varepsilon_{p}\lambda_{f}\) are the effective volumetric heat capacity and thermal conductivity of the medium, respectively; \(\rho\), \(c\), \(\lambda\)--density, specific heat capacity, and thermal conductivity of the medium, respectively. Index \(s\) is a solid matrix of the medium and \(f\) is the fluid (water). The temperature of the medium at the start of the simulation and at large distances from the filtration channel (\(Ti\)) are assumed to be \(-4\)\({}^{\circ}\)C for frozen rock and \(4\)\({}^{\circ}\)C for thawed rock. We assume that there is no heat flux at large distances from the channel. The thermal properties of the rock are presented in Table 1. Calculations were performed using the COMSOL Multiphysics software package [23]. In addition to calculating the change in the temperature distribution in thawed and frozen rock heating by filtration channel with warm water, the change over time in heat flow through the wall of such a cylindrical channel in thawed and frozen rock, respectively, was calculated. By integrating these heat flows over time, the amount of heat transferred by warm water to the thawed and frozen rock, respectively, was calculated. Additionally, the average amount of heat transferred by the channel to the rock per year, the average power of the channel, and the average specific power of the channel were calculated.