## 2 Numerical Study  
### Determination of Sandstone Parameters  
In this study, the numerical simulation is carried out based on the finite difference method, and the research object is divided into different sub-elements after grid division. The acceleration in the X and Y directions can be expressed as follows.  
\[a_{x}=\frac{F_{x}}{m} \tag{1}\]  
\[a_{y}=\frac{F_{y}}{m} \tag{2}\]  
where \(F_{x}\) and \(F_{y}\) are the nodal forces in the \(x\) and \(y\) directions, respectively, and \(m\) is the mass.  
The stress component can be expressed in the following form:  
\[\sigma_{i}=P_{i}+S_{i},\ (i=x,y) \tag{3}\]  
\[\tau_{xy}=S_{xy} \tag{4}\]  
where \(P\) is the external pressure, \(S_{i}\) is the deviatoric stress in the \(i\) direction, and there is a calculation relationship as follows  
\[S_{i}^{n+1}=S_{i}^{n}+2G\left(\dot{e}_{i}-\frac{1}{3}\dot{e}\right)^{n+\frac{ 1}{2}}\Delta t,\ (i=x,y) \tag{5}\]  
\[S_{xy}^{n+1}=S_{xy}^{n}+2G\epsilon_{xy}^{n+\frac{1}{2}}\Delta t \tag{6}\]  
\[\dot{e}=\dot{e}_{x}+\dot{e}_{y} \tag{7}\]  
Here, \(G\) is the shear modulus of the material and \(\dot{e}\) is the volume rate of the material.  
For materials, the relationship between the pressure \(P\) and density described in the linear equation of state is  
\[P=k\cdot(\frac{\rho}{\rho_{0}}-1) \tag{8}\]  
where \(P\) is pressure, \(k\) is the bulk modulus, \(k\) = 4.04 GPa, and \(\rho\) and \(\rho_{0}\) are the initial and present densities. Here, as sandstone is a brittle rock material, the relationship between stress and strain during deformation can be studied using an elastic strength model.  
At the same time, the experimental research is based on to the first strength theory and the third strength theory. If either the principal stress \(\sigma_{1}\) reaches its maximum (\(\sigma\)), or the maximum shear stress \(\tau_{max}\) is greater than the rock dynamic shear strength (\(\tau\)), the material fails.  
Here, a sound wave velocimeter is used to measure the dynamic material parameters of sandstone, as shown in Figure 2. It has two piezoelectric ceramic terminals, the transmitting end can emit waveform signals, and the receiving end can receive signals. According to the geometric configuration of the S-wave and P-wave, the relevant parameters of the material can be calculated by the time when the P-wave and S-wave pass through it.  
In this paper, the dynamic elastic modulus and dynamic Poisson's ratio of the rock materials are calculated by testing the expansion wave velocity \(V_{p}\) and shear wave velocity \(V_{s}\) of the material, as shown in Table 1. The calculation formula is shown in Equations (9) and (10). This paper adopts a rock sample ultrasonic tester to obtain the measurements of \(V_{p}\) and \(V_{s}\), and the working principle of the instrument is shown in Figure 2, which consists of a high-voltage pulse transmitter, a receiver, and a converter. The test of the two wave speeds of the material is obtained in accordance with the provisions of the International Society of Rock Mechanics for 100 mm \(\times\) 50 mm cylindrical specimens, and a total of five tests are prepared to obtain their average values.  
If the velocities \(V_{p}\) and \(V_{s}\) of the P-wave and S-wave are taken into the following equation, the elastic modulus and [PERSON]'s ratio of the material can be acquired.  
\[V_{p}=\sqrt{\frac{E_{d}(1-\mu_{d})}{(1+\mu_{d})(1-2\mu_{d})\rho}} \tag{9}\]  
\[V_{s}=\sqrt{\frac{E_{d}}{2(1+\mu_{d})\rho}} \tag{10}\]  
\[E=\frac{\rho V_{p}{{}^{2}}({{N_{s}}^{2}}-4{{V_{p}}^{2}})}{{{V_{s}}^{2}}-{{V_{p }}^{2}}} \tag{11}\]  
\[v=\frac{{{V_{s}}^{2}}-2{{V_{p}}^{2}}}{2{{V_{s}}^{2}}-2{{V_{p}}^{2}}} \tag{12}\]  
where \(\rho\) is the density of sandstone, \(E\) is the dynamic elastic modulus, and \(\mu\) is the Poisson ratio.