## 3 Results and Discussion
### Mechanical Property Analysis
#### 3.2.2 Determination and Analysis of Crack Initiation Stress and Damage Stress  
In this study, the crack volume strain method is utilized to identify characteristic stresses in sandstone under different treatment conditions during uniaxial compression [27]. The rock undergoes volumetric, axial, and lateral strains, which are related as follows:  
\[\varepsilon_{y}+2\varepsilon_{x}=\varepsilon_{y} \tag{1}\]  
where \(\varepsilon_{v}\) is the volumetric strain, \(\varepsilon_{x}\) is the lateral strain, and \(\varepsilon_{y}\) is the axial strain. It is assumed that the volumetric strain of the rock \(\varepsilon_{v}\) consists of an elastic volumetric strain \(\varepsilon_{ev}\) and a crack volume strain \(\varepsilon_{cv}\), namely:  
\[\varepsilon_{ev}+\varepsilon_{cv}=\varepsilon_{v} \tag{2}\]  
The elastic volumetric strain \(\varepsilon_{v}\) is computed as:  
\[\varepsilon_{v}=\varepsilon_{1}+\varepsilon_{2}+\varepsilon_{3} \tag{3}\]  
where \(\varepsilon_{1}\) is the strain in the direction of the maximum principal stress \(\sigma_{1}\), \(\varepsilon_{2}\) is the strain in the direction of the intermediate principal stress \(\sigma_{2}\), and \(\varepsilon_{3}\) is the strain in the direction of the minimum principal stress \(\sigma_{3}\). Based on the generalized Hooke law, the formula for the elastic volumetric strain \(\varepsilon_{ev}\) is obtained as:  
\[\varepsilon_{ev}=\frac{1-2\mu}{E}(\sigma_{1}+\sigma_{2}+\sigma_{3}) \tag{4}\]  
where \(\mu\) is the Poisson ratio and \(E\) is the elastic modulus, determined from the elastic deformation stage. For uniaxial compression, where \(\sigma_{2}=\sigma_{3}=0\), this further simplifies to:  
\[\varepsilon_{ev}=\frac{(1-2\mu)\sigma_{1}}{E} \tag{5}\]  
Figure 6: Uniaxial stress–strain curves of sandstone under different treatment conditions. From Equations (3) and (5), the crack-related crack volume strain \(\varepsilon_{cv}\) is derived:  
\[\varepsilon_{cv}=\varepsilon_{v}-\varepsilon_{ev}=\varepsilon_{1}+2\varepsilon_ {2}-\frac{(1-2\mu)\sigma_{1}}{E} \tag{6}\]  
The volumetric strain \(\varepsilon_{v}\), elastic volumetric strain \(\varepsilon_{cv}\), and cracking volumetric strain \(\varepsilon_{cv}\) are calculated by means of Equations (1), (2), (3), (4), (5), and (6), respectively, as illustrated in Figure 7, which results in the appearance of two peaks in the \(\varepsilon_{v}\) and \(\varepsilon_{cv}\) curves. The peak point is vertically upward according to the stress-strain curve, and points B and C are obtained. Point B is the crack initiation stress of the specimen \(\sigma_{ci}\), while point C is the damage stress of the specimen \(\sigma_{cd}\). Finally, point D, which is the peak stress \(\sigma_{c}\), is obtained according to the peak point of the stress-strain curve graph. Through the uniaxial compression experiments, the relationship between crack initiation stress and damage stress of the sandstone under acidification-supercritical CO\({}_{2}\) compounding is summarized, as shown in Figures 8 and 9. The trend of damage stress of the specimens under three processing conditions is consistent with the trend of peak strength, both showing a monotonic decrease. Under supercritical CO\({}_{2}\) soaking conditions, as the soaking pressure increases, the initial stress of the sandstone gradually decreases, reaching a maximum value of 13 MPa at a soaking pressure of 8 MPa, which is close to the original sandstone initial stress of 14.3 MPa. When the soaking pressure is 16 MPa, the initial stress decreases to a minimum value of 11.4 MPa. Under the condition of sandstone acidization treatment, the initial stresses of sandstone soaked in low- and high-concentration hydrochloric acid are 12.5 MPa and 13.7 MPa, respectively, both lower than the original sandstone initial stress. It is evident that acidization treatment also has a significant impact on the initial stress of the sandstone, and within a certain concentration range, the higher the concentration of hydrochloric acid, the more significant the deterioration of the sandstone.