# The Thermodynamic Change Laws of CO2-Coupled Fractured Rock
## 3 Results
### Simulation of CO\({}_{2}\) Heat Transfer in Pores
#### 3.4.4 Coupled Model Control Equations  
The stress field is affected by the temperature and seepage, as well as mechanical and chemical damage during CO\({}_{2}\)-coupled fracturing, which can cause rock deformation. The governing equation in the stress field can be expressed as follows [10]:  
\[G(u_{i}+\frac{1}{1-2u})-K(T-T_{tr})-\alpha_{p}p_{f}=0 \tag{7}\]  
where \(G=E^{*}/2(1+v)\) is the shear modulus; \(\alpha_{T}\) denotes the thermal expansion coefficient of rock; K represents the bulk modulus; \(T\) is the rock temperature; \(T_{tr}\) denotes the rock temperature in its natural state; \(\alpha_{p}=1-(K/K_{s})\) represents the Biot number;\(K_{s}\) is the modulus of rock; and \(p_{f}\) denotes the fluid pressure within the pore.  
During seepage of the liquid CO\({}_{2}\), heat transfer affects the temperature distribution within the rock mass. The total specific heat capacity of the rock mass is linearly correlated to its mass and the specific heat capacity of liquid CO\({}_{2}\). This can be mathematically expressed as follows [11]:  
\[(\rho C)_{t}=\Big{(}1-\phi_{m}-\phi_{f}\Big{)}\rho_{s}C_{s}+s_{w}\phi_{f}\rho_ {w}C_{w} \tag{8}\]  
where \(C_{s}\) and \(C_{w}\) are the specific heat capacity of the rock mass and CO\({}_{2}\), respectively.  
Figure 12: Pore model.  
Figure 13: Fluid–solid coupling model.  
During rock fracturing, deformation is related to the physical properties of the rock mass and its volumetric strain, which can be expressed in the form below [11]:  
\[W_{T}=T\alpha_{s}K_{v}\frac{\partial\varepsilon_{v}}{\partial t} \tag{9}\]  
where \(\alpha_{s}\) is the thermal expansion coefficient of rock; \(K_{v}\) is the bulk modulus; \(\varepsilon_{v}\) denotes the volumetric strain; and \(T\) is the rock temperature.  
The convective heat exchanged between liquid CO\({}_{2}\) and the rock mass is  
\[Q_{tr}=\
abla\cdot\left[-(\frac{K_{f}k_{w}}{\mu_{w}}\
abla p_{w}\cdot\rho_{w} C_{w})\Delta T\right] \tag{10}\]  
where \(\mu_{w}\) and \(p_{w}\) are the viscosity and pressure of the liquid CO\({}_{2}\), respectively.  
Finally, the energy conservation law in the CO\({}_{2}\)-coupled fracturing fluid can be summarized as follows:  
\[\frac{\partial[(\rho C)_{t}\Delta T]}{\partial t}+T\alpha_{s}K\frac{\partial \varepsilon_{v}}{\partial t}+Q_{te}+\
abla\cdot(K_{t}\
abla T)+\Delta Q_{m}=Q_{ T}. \tag{11}\]  
The coupling between temperature and stress fields is initially manifested through heat exchange. Subsequently, it is obtained using fluid seepage and changes in the thermophysical properties of CO\({}_{2}\).  
The mass conservation equation can be expressed as follows [11]:  
\[\frac{\partial(\rho\mu)}{\partial t}+\
abla\cdot(\rho\
u)=-\Delta \tag{12}\]  
where \(\rho\) is the density of supercritical CO\({}_{2}\); \(\mu\) is the rock porosity; \(t\) denotes time; \(\Delta\) represents the seepage process source; and \(\
u\) is the fluid flow rate.  
The temperature gradient between the liquid CO\({}_{2}\) and the rock induces convective heat transfer. According to [PERSON]'s law, the heat transfer in the rock mass is proportional to the temperature difference between liquid CO\({}_{2}\) and the rock [11]:  
\[q_{c}=-\lambda\Delta t \tag{13}\]  
where \(\lambda\) and \(t\) are the thermal conductivity and temperature of liquid CO\({}_{2}\), respectively.  
By coupling the stress and temperature field equations, the governing equations for heat transfer in CO\({}_{2}\)-coupled fractured rocks can be expressed in the form below:  
\[u_{ij}=(1-\Delta Q_{m})\alpha_{p}p_{f}+Q_{w}. \tag{14}\]  
The geometric model involving flow and rock mass was constructed and meshed, and the initial boundary conditions were set. The \(k\)-epsilon model was employed to approximate turbulent stresses.