## 2 Theory of Modelling
### Physical Parameters  
The relevant thermal physical parameters change with temperature and pressure. This study is based on shallow low-rank coal with a buried depth of 450 m, which is relatively shallow. Therefore, the temperature range is 280-310 K and the pressure is 0-10 MPa. Refprop software has been used for physical parameters evaluation (Figures 1 and 2). It can be seen that all the thermal physical parameters change nonlinearly with temperature and pressure. Among them, the physical parameters of CO\({}_{2}\) show a sudden change trend under certain temperatures and pressure. Therefore, it is very unreasonable to regard the thermal physical parameters as constants as previous studies. This paper uses Matlab to call Refprop software, then uses polynomial function fitting in Matlab. It is worth mentioning that the Refprop software is based on the most accurate pure fluid and mixturemodels at present. It implements three models for the thermodynamic properties of pure fluids: equations of state explicit in Helmholtz energy, the modified Benedict-Webb-Rubin equation of state, and an extended corresponding states (ECS) model [44]. For CO\({}_{2}\) and CH\({}_{4}\), high accuracy Helmholtz energy equations of state is used. The relationship between density, specific heat capacity, thermal conductivity, dynamic viscosity and pressure/temperature is fitted as follows. These equations are used for numerical solutions. For CH\({}_{4}\): \(T\in[280,310]\)\({}^{\circ}\)C \(P\in[0,10]\) MPa  
\[\rho_{1} =p00+p10\times P_{1}+p01\times T+p20\times P_{1}^{2}+p11\times P_ {1}\times T+p30\times{P_{1}}^{3}+p21\times P_{1}^{2}\times T \tag{18}\] \[C_{g1} =p00+p10\times P_{1}+p01\times T+p20\times P_{1}^{2}+p11\times P_ {1}\times T+p30\times{P_{1}}^{2}+p21\times P_{1}^{2}\times T\] (19) \[\lambda_{g1} =p00+p10\times P_{1}+p01\times T+p20\times P_{1}^{2}+p11\times P_ {1}\times T+p30\times P_{1}^{3}+p21\times P_{1}^{2}\times T\] (20) \[\mu_{1} =p00+p10\times P_{1}+p01\times T+p20\times P_{1}^{2}+p11\times T \times P_{1} \tag{21}\]  
For CO\({}_{2}\): \(T\in[280,310]\)\({}^{\circ}\)C \(P\in[0,10]\) MPa  
Figure 1: CH\({}_{4}\) (**a**) density, (**b**) specific heat capacity, (**c**) thermal conductivity and (**d**) dynamic viscosity as a function of pressure/temperature. \[\rho_{2} =p00+p10\times P_{2}+p20\times T+p20\times P_{2}\times T \tag{22}\] \[C_{g2} =p00+p10\times P_{2}+p01\times T+p20\times{P_{2}}^{2}+p11\times P_{2 }\times T+p30\times{P_{2}}^{3}+p21\times{P_{2}}^{2}\times T\] (23) \[\lambda_{g2} =p00+p10\times P_{2}+p01\times T+p20\times{P_{2}}^{2}+p11\times P_{ 2}\times T+p30\times{P_{2}}^{3}+p21\times{P_{2}}^{2}\times T\] (24) \[\mu_{2} =p00+p10\times P_{2}+p01\times T+p20\times{P_{2}}^{2}+p11\times T \times P_{2}+p30\times{P_{2}}^{3}+p21\times{P_{2}}^{2}\times T \tag{25}\]  
where, \(T\) is temperature, K; \(P_{i}\) is pressure, Pa; \(\mu_{i}\) is the dynamic viscosity, Pa-s; \(C_{gi}\) is the specific heat capacity, J/(kg\(\cdot\)K); \(\rho_{i}\) is the density, kg/m\({}^{3}\); \(\lambda_{gi}\) is the thermal conductivity, W/(m\(\cdot\)K); \(i=1\), CH\({}_{4}\), \(i=2\), CO\({}_{2}\). Tables 2 and 3 list the corresponding coefficients in the equation.