# The Thermodynamic Change Laws of CO2-Coupled Fractured Rock
## 3 Results
### Characterization of Fracture Surface Roughness of Rock Samples before and after CO2 Coupling  
To characterize the features of rock fracture surfaces before and after CO2 coupling, a three-dimensional electron microscope scanner was employed for scanning, as depicted in Figure 9. The surface roughness of fractures reflects the micro-convex bodies' fluctuations on the rock fracture surface, which has a size effect and is a key indicator of fracture mechanics characteristics.  
Utilizing the Joint Roughness Coefficient method (JRC), the surface of cracks undergoes digitization. In Figure 10a, the rough surface of fractures in the natural state of the rock sample exhibits a stepped shape. Due to significant differences in relief angles, conductivity is poor, resulting in a poor fracturing effect. The rough surface of fractures at the edge of the rock sample is uneven, with a maximum roughness of 0.82 mm. Figure 10b shows that the fracture roughness surface of MTS-coupled rock samples is planar, with the highest roughness reaching 1.13 mm. The fluctuation angle of the rock sample edge is relatively small, indicating good conductivity and a positive fracturing eff  
Figure 8: SEM images of rock samples under MTS coupling (the red circle indicates the hole). (**a**) \(\times\)500; (**b**) \(\times\)2000.  
Figure 9: Three-dimensional electron microscope scanning instrument.  
it is evident that stepped rough surfaces exhibit significant size effects compared to planar rough surfaces.  
To quantify the roughness of surface cracks in rock samples, the roughness of the crack surface was quantified using an empirical expression that combines characterization statistical parameters with JRC [25].  
\[JRC=32.2+32.47\log h_{2}, \tag{2}\]  
\[h_{2}=\left[\frac{1}{(n-1)(\Delta x)^{2}}\sum_{i=1}^{n}{(h_{i+1}-h_{i})^{2}} \right]^{\frac{1}{2}} \tag{3}\]  
where \(x\) represents the root mean square of the fracture profile wall slope; n is the number of data points; and \(h\) denotes the axial distance between points \(i+1\) and \(i\). The height of peak roughness can be obtained from the following expression:  
\[\xi=h_{\text{max}}-h_{\text{min}} \tag{4}\]  
where \(\xi\) represents the height of peak roughness, mm, and \(h_{\text{max}}\) and \(h_{\text{min}}\) are the maximum and minimum roughness heights, mm, respectively.  
\[R_{s}=\sqrt{\frac{1}{n}\sum_{i=1}^{n}{(h_{i}-h_{a})^{2}}} \tag{5}\]  
\[R_{n}=\frac{1}{n}\sum_{i=1}^{n}{|h_{i}-h_{a}|} \tag{6}\]  
Figure 10: Characterization diagram of crack roughness: (**a**) natural samples and (**b**) rock samples under MTS coupling.  
where \(R_{n}\) and \(R_{s}\) represent the average and the root mean square of the roughness height, respectively, and \(h_{i}\) and \(h_{a}\) are the roughness height of point \(i\) and the average height of the elevation line, mm, respectively.  
The roughness parameters of the fractured sample can be calculated using Equations (2)-(6), as shown in Table 3.  
Table 3 indicates that compared to the sample under natural conditions, the CO\({}_{2}\)-coupled rock sample has an average roughness 0.15 mm higher, with a difference of 0.16 mm in the root mean square of the roughness height. The JRC is less than that of the natural state samples, and the difference is small. Therefore, the rough surface of the coupled rock sample exhibits a planar shape. This indicates that the heat transfer effect generated by the rock matrix and CO\({}_{2}\) in the cracks is conducive to crack expansion, resulting in good fracture connectivity, strong conductivity, and a significant fracturing effect.