## 3 Safety Evaluation Method
### Judgment Criteria for Tension Fracture of Perforated String  
Similarly to the static analysis of the pipe string, it can be assumed that the position above the neutral point of the pipe string is mainly subjected to tension, and the pipe string is subjected to an axial tensile load under the constraints of the casing, and it is easy to deform and fracture when the tensile load exceeds the strength range that can be withstood by the pipe string itself. In order to simplify the model of the pipe string, first ignore the joints and other components; then, the pipe string can be regarded as a whole structure since assuming that the material of the pipe string is isotropic, the axis of the pipe string and the centerline of the casing is in the same line; the pipe string is straight and stable, and the internal and external pressure is balanced; the bottom of the pipe string is subjected to the axial dynamic shock load. Figure 7a shows the schematic diagram of the behavior of the pipe string without tensile fracture, and Figure 7b,c show the schematic diagram of the fracture behavior of the pipe string in axial tension and the fracture behavior of the center string of the packer, respectively. In the process of the propagation of the shock wave in the fluid medium, it will interact with the structure in the wellbore, and when the shock wave propagates to the rubber wall of the packer, reflection and refraction will occur, the rubber wall of the packer will be in the fluid medium and there will be two interfaces above and below. From the basic theory of shock wave, it can be seen that after the shock load is reflected and transmitted by the packer, the pressure on the lower interface of the packer is the sum of the incident pressure on the lower interface and the reflected pressure on the lower interface. The expression for the pressure at the lower interface of the packer is  
\[P_{LI}=P_{i}+P_{r} \tag{16}\]  
where \(P_{i}\) and \(P_{r}\) are the incident and reflected pressures at the interface under the packer, respectively, MPa. In an enclosed space, the shock wave not only travels forward but also reflects when it encounters the packer. That is, the original shock wave and the reflected shock wave may meet at the same location and superposition, resulting in a pressure at that point that is twice the pressure of the original shock wave. In order to simplify the analysis process, the lower packer interface is considered to be the location where dual pressure is applied. The formula for calculating the axial force of pipe string under shock loading is as follows:  
\[ \begin{array}{l}F_{z}=(P_{i}+P_{r})\frac{\pi}{4}\left({{D_{o}}^{2}-{d_{o}}^{ 2}}\right)=\frac{\pi}{2}P_{p}\left({{D_{o}}^{2}-{d_{o}}^{2}}\right)\\ =\frac{\pi}{2}\left(\rho gH_{0}+1341.12\frac{\left({a{m_{e}}(H_{1}-H_{2})} \right)^{1.00}}{\left({\frac{\pi}{4}\left({(H_{0}-H_{3})(D_{c}-2\delta_{c})}^{2 }-(H_{2}-H_{3})D_{b}{}^{2}-(H_{1}-H_{2})D_{s}{}^{2}\right)}\right)^{0.39}} \right)\left({{D_{o}}^{2}-{d_{o}}^{2}}\right)\end{array}  \tag{17}\]  
Under shock loading, the effective axial tensile force of the pipe string is composed of two parts, including the axial tensile force caused by the self-weight and the instantaneous tensile force caused by the shock load, and the tensile force at the upper end of the pipe string of the deep well long tubing is larger. Based on the safety coefficient method, comparing the effective axial tensile force of the pipe string under shock loading with the magnitude of the tensile strength of the pipe string can be used as a criterion for the judgment of fracture of the pipe string under tension. The following judgment criteria are used to ensure that the pipe string does not break.