## 2 Materials and Methods
### Thermophysical Modeling  
Heat transfer and rock thawing due to physical heating by filtering circulating water with positive temperatures (from 2 to 15 \({}^{\circ}\)C, see Figure 4) through the rock were calculated. The circulating water moves along sub-vertical cracks in the rock, which were, therefore, modeled as sub-vertical cylindrical channels. Model Description  
The thawing of two geological columns of rocks was simulated, one in the thawed zone (well t1, Figure 2) and one in the frozen zone (well t3, Figure 2). Filtering circulating water through sub-vertical cracks in the rock was modeled as water flowing through a cylindrical channel with a 15 cm diameter and 66 m length. Water was modeled as entering the channel at a speed of 100 m/day. The outlet of the channel is at atmospheric pressure. The average temperature of circulating water at the surface of the rock and entering into the channel changes throughout the year according to the graph shown in Figure 4. At an initial time and at a large distance from the channel, the temperature is assumed to be a constant, _Ti_ (Table 1). Geological columns in thawed and frozen zones:  
1. Well t1: total depth 75 m. Structure according to Figure 5: 0-17 m--technogenic deposits with known properties (t 1/1, Table 1); 17-66 m--fractured limestones (t 1/2, Table 1); 66-75 m--weakly fractured marl (impermeable, not modeled). The initial temperature of the well t1 rocks and the temperature at large distances from the filtration channel are assumed to be constant and time-invariant (_Ti_ = 4 \({}^{\circ}\)C, Table 1). 2. Well t3: total depth 70 m. Structure according to Figure 5: 0-8 m--technogenic deposits with known properties (t 3/1, Table 1); 8-68 m--fractured limestones (t 3/2, Table 1); 68 m and below--marl (impermeable, not modeled). The initial temperature of the well t3 rocks (units 3/1 and 3/2) and the temperature at large distances from the filtration channel are assumed to be constant (_Ti_ = -4 \({}^{\circ}\)C, Table 1). \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Characteristic** & **t 1/1** & **t 1/2** & **t 3/1** & **t 3/2** \\ \hline
**Mineralogical Composition:** & **Technogenic Psephite, Magnetic, Pyrite** & **Limestone with an Admixture of Pyrite and Gypsum** & **Technogenic Psephite, Magnetic, Pyrite** & **Limestone with an Admixture of Pyrite and Gypsum** \\ \hline \(\lambda\), W/m/K & 1.94 & 2.10 & 2.37 & 2.91 \\ \hline \(C\), kJ/(m\({}^{3}\cdot\)K) & 2490 & 2381 & 1996 & 1946 \\ \hline \(c\) = \(C/\rho\), J/kg/K & 1406 & 972 & 1018 & 828 \\ \hline \(\rho\), g/cm\({}^{3}\) & 1.77 & 2.45 & 1.96 & 2.35 \\ \hline \(k\), m\({}^{2}\) & \(10^{-12}\) & \(10^{-12}\) & \(10^{-12}\) & \(10^{-12}\) \\ \hline _Ti\({}_{o}\)_\({}^{\circ}\)C & +4 & +4 & –4 & –4 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Thermophysical characteristics of the wells’ units adopted for physical modeling. Figure 4: Annual changes in average temperature of water entering the crack. Table 1 shows the thermal conductivity (\(\lambda\)), volumetric heat capacity (\(C\)), specific heat capacity (\(c=C/|\rho\)), density (\(\rho\)), permeability (\(k\)), initial temperature (\(Ti\)), and composition of the rock units in wells t1 and t3. The distribution of water velocity in the channel was calculated for the motion of an incompressible flow in a free channel by solving the Navier-Stokes equations in a stationary setting:  
\[\rho(\mathbf{u}\cdot\
abla)\mathbf{u}=\
abla\cdot[-p\mathbf{I}+\mu(\
abla \mathbf{u}+(\
abla\mathbf{u})^{T})], \tag{1}\]  
The distribution of water velocity in the rock adjacent to the channel is calculated in a stationary setting by solving the Brinkman equation with a Forchheimer correction [21].