# Experimental Study on Permeability Evolution of Sandstone during Triaxial Compression Damage
## 3 Analysis of Test Results
### Permeability Characteristics
#### 3.3.2 Relationship between Permeability and Volumetric Strain  
Throughout the experimental procedure, the evolution of volumetric strain in sandstone unfolds in two discernible stages: volumetric compaction and volumetric expansion. The transitional periods of permeability closely align with these volumetric strain stages. Figure 12 presents the permeability-volumetric strain curves spanning the entirety of the stress-strain progression under varied conditions. Throughout the deformation process of sandstone, the permeability exhibits a distinct pattern of gradual decrease, steady development, gradual increase, and rapid growth, which correlates well with the four stages of sandstone deformation: slow volumetric compression, rapid volumetric compression, rapid volumetric expansion, and slow volumetric expansion. For instance, under conditions of a pore pressure of 2 MPa and a confining pressure of 15 MPa, during the stage of volumetric compaction, due to the weaker radial expansion of the sandstone compared to axial compression, the sandstone undergoes micro-fracture closure, linear elastic deformation, and stable fracture expansion. The predominant deformation in this stage is axial compression, leading to a gradual transition of internal pores and micro-fractures from slow compression to slow expansion. Consequently, the fluid flow pathways within the sandstone expand slowly, yielding a progressive increase in permeability. Interestingly, the rock sample's minimum permeability does not coincide with the moment of peak volumetric compaction (which is the inflection point between the compression stage and the expansion stage), but rather precedes the inflection point of volumetric strain. Under the same pore pressure, with an increase in confining pressure, the minimum permeability \(K_{min}\) and the permeability at the maximum volumetric compression point \(K_{c}\) gradually approach each other. After the point of compaction (in the stage of rapid volumetric expansion), as the sandstone undergoes rapid volumetric expansion, the aperture of newly formed fractures within the rock increases, resulting in an augmented number of fluid flow pathways and a rapid increase in permeability. It is evident from Figure 12 that the maximum permeability \(K_{max}\) of the sandstone is achieved only subsequent to the rock reaching its peak strength, displaying a distinct lag effect. Indeed, the preceding discussion highlights the intricate interrelationship between rock permeability and volumetric strain. To further advance the analysis of fluid flow-stress coupling, the variation in permeability with volumetric strain during the stages of volumetric compression and expansion, with a confining pressure of 15 MPa and differing pore pressure differentials of 2, 4, and 8 MPa, was subjected to fitting. The volumetric strain during the expansion stage is considered as a relative value of volumetric expansion:  
\[\varepsilon_{v1}=\varepsilon_{vr}-\varepsilon_{v0} \tag{7}\] \begin{table}
\begin{tabular}{c c c c} \hline
**Sepage Pressure/MPa** & **Confining Pressure/MPa** & \(K_{cc}\)/(10-17/m2) & \(K_{ci}\)/(10-27/m2) \\ \hline  & 5 & 0.53 & 3.22 \\
2 & 10 & 1.17 & 1.32 \\ \hline
2 & & 0.49 & 1.46 \\
4 & 15 & 1.37 & 1.73 \\
8 & & 2.64 & 3.11 \\ \hline
2 & 20 & 0.78 & 0.99 \\ \hline \end{tabular}
\end{table} 
Table 5: Two key permeability values of sandstone during whole stress–strain process under different conditions. where \(\varepsilon_{v1}\) is the relative value of the volumetric expansion strain, \(\varepsilon_{vr}\) is the maximum volumetric compression strain, and \(\varepsilon_{v0}\) is the volumetric expansion. In the volumetric compression stage, the evolution of sandstone permeability is depicted in Figure 13a. At a pore pressure of 2 MPa, the relationship between permeability and volumetric strain is described using a cubic polynomial:  
\[K=A\varepsilon_{v}^{3}+B\varepsilon_{v}^{2}+C\varepsilon_{v}+1 \tag{8}\]  
where, \(A\), \(B\), and \(C\) are fitting parameters.