# Experimental Study on Permeability Evolution of Sandstone during Triaxial Compression Damage
## 3 Analysis of Test Results
### Analysis of Rock Damage Evolution  
Throughout the failure process, a considerable number of newly generated fractures exist within the rock, accompanied by evolving damage within the sample. The introduction of the damage variable \(D\) is paramount. Assuming that the microstructural strength of the rock follows the Weibull distribution, the amalgamation of statistical theory and damage mechanics gives rise to a statistical damage expression:  
\[D=1-exp\left[1-\left(\frac{\epsilon}{\epsilon_{0}}\right)^{s}\right] \tag{10}\]  
where \(D\) represents the damage variable, \(\epsilon\) is the strain, and \(s\) and \(\epsilon_{0}\) are parameters of the Weibull distribution.  
The Weibull distribution parameters are predicated on the distinctive features exhibited by the peak of the deviatoric stress-axial strain curve:  
\[s=\frac{1}{ln\left(\frac{E\epsilon_{v}}{\epsilon_{v}-2\mu\alpha_{0}}\right)} \tag{11}\]  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Item** & **Confining Pressure** & **Seepage Pressure** & **Volumetric Compression Stage** & **Volumetric Expansion Stage** \\ \hline
1 & 15 & 2 & \(-121.11\epsilon_{v}^{3}+84.59\epsilon_{v}^{2}-12.83\epsilon_{v}+1\) & \(\exp\left(0.03\epsilon_{v}^{2}+0.08\epsilon_{v}+0.74\right)\) \\  & & & (R\({}^{2}\) = 0.992) & (R\({}^{2}\) = 0.963) \\
2 & 15 & 4 & \(\exp\left(13.37\epsilon_{v}^{2}-3.79\epsilon_{v}+0.62\right)\) & \(\exp\left(-0.01\epsilon_{v}^{2}-0.27\epsilon_{v}+1.13\right)\) \\  & & & (R\({}^{2}\) = 0.986) & (R\({}^{2}\) = 0.945) \\
3 & 15 & 8 & \(\exp\left(16.71\epsilon_{v}^{2}-3.92\epsilon_{v}+1.22\right)\) & \(\exp\left(-0.006\epsilon_{v}^{2}-0.29\epsilon_{v}+1.55\right)\) \\  & & & (R\({}^{2}\) = 0.981) & (R\({}^{2}\) = 0.980) \\ \hline \hline \end{tabular}
\end{table}
Table 6: The fitting parameters for the relationship between permeability and volumetric strain of sand-stone under different seepage pressure.  
Figure 13: The relationship between permeability and volumetric strain in different stages (confining pressure of 15 MPa).  
\[\varepsilon_{0}=\varepsilon_{c}s^{\frac{1}{2}} \tag{12}\]  
where \(E\) is the elastic modulus, \(\mu\) is the Poisson's ratio of the sandstone, \(\sigma_{3}\) is the confining pressure, and \(\sigma_{c}\) and \(\varepsilon_{p}\) are the peak stress and peak strain, respectively.  
The evolution curves of axial strain, the damage variable, and permeability under different conditions are shown in Figure 13(a)-f. During the stages of micro-fracture closure and linear elastic deformation, where new voids and micro-fractures have yet to form, and due to the increased circumferential confinement experienced by the rock sample, both permeability and damage variable changes are relatively gradual, with the damage variable nearly approaching zero. In the phase of crack propagation, with increasing loads, newly generated fractures emerge, causing the damage variable to increase slowly, leading to a corresponding increase in permeability. The change rate of the damage variable demonstrates a strong correlation with the corresponding alteration in permeability, revealing that the variation in the damage variable mirrors the development of internal fractures within the rock specimen. The evolution of rock permeability is closely linked to the development of its internal cracks and fractures. Hence, a higher damage rate within the specimen corresponds to a swifter alteration in permeability.  
Figure 14: Relationship between permeability and the damage variable under different conditions.