# Analysis of Damage and Permeability Evolution of Sandstone under Compression Deformation
## 4 Model Validation and Results Analysis
### Permeability Evolution Analysis  
Figure 12 shows the analytical predictions of the permeability evolution of the sandstone specimen under loading deformation at a confining pressure of 5 MPa. The permeability evolution of the rock under compression can be divided into two stages, decreasing and increasing, corresponding to the compaction and dilatation of the rock deformation, respectively. The evolution trend is consistent with experimental results. The influence of the seepage pressure on the permeability evolution is mainly concentrated in the increasing stage. With the increase in the seepage pressure, the growth rate of the permeability rapidly increases. Taking the seepage pressure of 1.5 MPa as an example, the permeability considering the effect of the fluid-solid coupling is about 10 times higher than that without the fluid-solid coupling. During the period from rock dilatation to failure, the permeability surge phenomenon can be clearly observed. This phenomenon becomes more obvious with the increase in the seepage pressure. Thereafter, the permeability enters slow growth until it stabilizes.  
The turning point at which the permeability changes from decreasing to increasing moves forward with the increase in the seepage pressure (Figure 12), the corresponding permeability surge comes earlier. This reflects that the presence of seepage pressure (pore pressure) under the same boundary conditions reduces the strength and aggravates the damage to the rock. This conclusion is consistent with the existing geotechnical theories and engineering practices. Based on the relationship between porosity and permeability (Equation (10)), the trend of the porosity evolution is the same as that of the permeability.  
The permeability contours and velocity streamline in the sandstone specimen in the case of failure are shown in Figure 13. One can see that the permeability gradually decreases from the bottom to the top of the sandstone specimen due to the influence of damage and deformation. Interestingly, the spatial variation in the permeability is not uniform in the height direction but has a deflection angle (the black dashed line in Figure 13). The direction and angle of this deflection largely coincide with the shear failure of the rock, which further validates the coupling relationship between damage and permeability and the reliability of the model.  
Figure 11: Damage evolution of sandstone specimen under triaxial compression conditions with different inlet seepage pressures \(p_{in}\).  
According to the numerical and experimental results, it is found that the permeability of rock in the compaction stage has an exponential function relationship with the effective confining pressure (\(\sigma_{3}\)-\(p\)). However, the relationship in the dilatancy stage is extremely complicated due to the permeability surge phenomenon and cannot be expressed by a single function for the time being.  
\[k=k_{0}\exp(-\beta\cdot p_{ce}) \tag{23}\]  
where \(p_{ce}\) denotes the effective confining pressure, and \(\beta\) is the equation coefficient.  
Figure 14 shows the variation curves and contours of the pore pressure in the height direction. It is noticed that the pore pressure decreases with the increase in the flow distance, but the pressure gradient increases with the increase in the seepage distance. Therefore, the seepage velocity depends on the competitive influences of the permeability evolution and pressure gradient. For the proposed flow-solid coupling analysis model, the analysis results indicate that there is no significant difference in the seepage velocity in the horizontal direction. However, the outlet seepage velocity is higher than the inlet seepage velocity in the vertical direction, and the seepage velocity in the deflection direction is higher than that of the nearby part.  
Figure 12: Permeability evolution of sandstone specimen under triaxial compression conditions with different inlet seepage pressures pin.  
Figure 13: Permeability contours and velocity streamline with different inlet seepage pressures.