# Analysis of Damage and Permeability Evolution of Sandstone under Compression Deformation
## 4 Model Validation and Results Analysis
### Fluid-Solid Coupling Analysis Model  
In order to further investigate the coupling mechanism of the damage-permeability of the rock, a simulation model of conventional triaxial compression tests considering the fluid-solid coupling effect was established and solved. As shown in Figure 8, the cylindrical specimen used in the triaxial compression test could be simplified by considering the two-dimensional axial symmetry. The selected specimen for the test was sandstone with a diameter of 50 mm and a height of 100 mm. The seepage medium was nitrogen. The bottom of the model adopted a fixed constraint, the center line served as the axis of symmetry, and a prescribed displacement was applied at the top in accordance with the triaxial compression test method. During the fluid-solid coupling simulation, the bottom and top of the specimen acted as the seepage inlet and outlet, respectively. To ensure the accuracy and representativeness of the calculation results, a uniform tetrahedral mesh was used in the two-dimensional axisymmetric model.  
It was necessary to perform isotropic compression (\(\sigma_{1}=\sigma_{2}=\sigma_{3}\)) before axial compression (\(\sigma_{1}>\sigma_{2}=\sigma_{3}\)) and then apply differential stress and fluid pressure. The relationship between the prescribed displacement and differential stress can be described by the reaction force in the vertical direction.  
\[\sigma_{diff}=\frac{F_{re}}{\pi r^{2}}-\sigma_{3} \tag{20}\]  
where \(\sigma_{diff}\) denotes the differential stress in the axial compression stage, \(r\) is the radius of the specimen, and \(F_{re}\) denotes the reaction force.  
The basic calculation parameters of sandstone and nitrogen are listed in Table 1. These parameters were obtained from the experimental results, such as the mechanical and permeability tests. The scale parameter and shape parameter were obtained from the above-derived equations and varied with the confining pressure. In order to ensure the comparability of the calculation results, five levels of confining pressures of 5 MPa, 10 MPa, 20 MPa, 30 MPa, and 40 MPa were set in this analytic simulation.  
Figure 8: Simulation model for evolution analysis of rock damage and permeability.  
The compressibility and density of nitrogen are the function of gas pressure and can be derived by the ideal gas law.  
\[\rho_{g}=\frac{pM}{RT} \tag{21}\]  
\[\chi=\frac{1}{\rho_{g}}\left(\frac{\partial\rho_{g}}{\partial p}\right) \tag{22}\]  
where \(\rho_{g}\) and \(M\) are the density and molar mass of nitrogen, \(R\) and \(T\) are the universal gas constant and absolute temperature, and \(\chi\) denotes the compressibility of nitrogen.  
The flow-solid coupling analysis used [PERSON]'s law as the governing equation for fluid flow in the sandstone. The primary reasons were as follows: (1) the test medium (nitrogen) was a Newtonian fluid; (2) the initial permeability of the sandstone was 3.99 \(\times\) 10\({}^{-16}\) m\({}^{2}\), which was tested using the steady-state method and fell within the applicable range of [PERSON]'s law.