# Analysis of Damage and Permeability Evolution of Sandstone under Compression Deformation
## 2 Coupling Analysis of Rock Damage and Permeability
### Permeability and Deformation  
The permeability \(k\) is related to the pore size and crack aperture of the rock and is an important indicator to measure the connectivity of voids. Compared with porosity, there is no precise defined expression for permeability [26], but it can be obtained indirectly by its relationship with porosity. Figure 4 shows the permeability-strain and stress-strain curves of sandstone obtained from permeability tests under conventional triaxial compression. With the increase in the differential stress, the permeability gradually decreased because of pore and crack closure. When the differential stress increased to a critical value, the permeability began to increase, and the growth rate was faster. The test results show that the critical value was close to the dilatancy stress \(\sigma_{d}\). The main reason for the change in the permeability trend was closely related to the unstable crack propagation because the propagation and penetration of cracks caused the rock volume to expand and then induced the permeability to surge. Interestingly, the trend of the permeability-strain curve is very similar to that of the volumetric strain curve, and the turning points are the dilatancy stress. It can be seen that the permeability and volumetric strain of the rock were both affected by the stress state, but the root cause of the induced permeability surge and volume expansion was damage and crack propagation. Based on the theoretical analysis and test results, the relationship between the permeability and deformation of rock could not be obtained temporarily, but it was certain that the permeability was positively correlated with the volumetric strain because of their consistent changing trends induced by damage. The evolution equation of permeability can be determined indirectly through the relationship between porosity and rock deformation. According to the definition of rock porosity, the porosity increment can be expressed as:  
\[\Delta\phi=\frac{\Delta V_{v}}{V}=\frac{\Delta V-\Delta V_{m}}{V}=\epsilon_{V} -\epsilon_{Vm}(1-\phi_{0}) \tag{4}\]  
where \(V\), \(V_{m}\), and \(V_{v}\) are the overall volume, matrix volume, and void volume of the rock, respectively, and \(\epsilon_{V}\) and \(\epsilon_{Vm}\) are the volumetric strain of the rock and the rock matrix. Figure 3: PDF and CDF of Weibull distribution with different shape parameters. Then, the evolution equation of porosity can be obtained. \[\phi=\phi_{0}+\varepsilon_{V}+\varepsilon_{Vm}(\phi_{0}-1) \tag{5}\]  
In the compaction stage of the rock volume, the fluid seepage is dominated by porous media flow. The relationship between permeability and porosity can be described by the cubic law [31; 32]:  
\[k=k_{0}(\phi/\phi_{0})^{3} \tag{6}\]  
In the dilatancy stage of the rock volume, the fluid seepage is dominated by fracture flow because the crack has entered unstable growth. Based on the concept of an equivalent hydraulic radius (Figure 5), the relationship between permeability and porosity can be described as [13; 33]:  
\[k=\frac{e^{2}}{b}\frac{\phi}{\tau^{2}} \tag{7}\]  
\[e\propto\phi^{\chi} \tag{8}\]  
where \(e\) denotes the equivalent hydraulic radius, \(\tau\) is the tortuosity, \(\chi\) is a parameter that depends on the area per unit volume and aperture of the crack, and \(b\) is a constant that depends on the shape of the flow channel (it is equal to two for a circular cross-section and three for a slot or crack cross-section). The tortuosity is the ratio of the actual seepage distance to the apparent distance, which is a function of the porosity:  
\[\tau^{2}=\phi^{1-s} \tag{9}\]  
where \(s\) is a constant whose value varies from one to three (this value for most rocks is about two). Figure 4: Permeability\(-\)strain and stress\(-\)strain curves of sandstone at a confining pressure of 10 MPa; \(k\) denotes permeability. Figure 5: Schematic of the equivalent hydraulic radius in fracture flow.