## 3 Safety Evaluation Method
### Judgment Criteria for Perforated String Buckling Under Pressure  
Similarly to the static analysis of the pipe string, it can be considered that the pipe string will be subjected to axial shock load under the constraint of the casing, and the pipe string buckling behavior is mainly divided into dynamic sinusoidal buckling behavior and dynamic spiral buckling behavior. In order to simplify the model of pipe string, first the joints and other components should be ignored; 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 6a shows the schematic diagram of the pipe string without buckling behavior, and Figure 6b,c show the schematic diagram of the dynamic sinusoidal buckling behavior and the dynamic spiral buckling behavior of the pipe string, respectively.  
Figure 6: Schematic diagram of pipe string buckling behavior: (**a**) the schematic diagram of the pipe string without buckling behavior; (**b**) the schematic diagram of the dynamic sinusoidal buckling behavior of the pipe string; (**c**) the schematic diagram of the dynamic spiral buckling behavior of the pipe string.  
The magnitude of the shock load pressure on the pipe string at the instant of shock loading is  
\[F_{c}=\frac{\pi}{4}P_{p}\left({D_{o}}^{2}-{d_{o}}^{2}\right) \tag{12}\]  
where \(F_{c}\) is the impact pressure on the pipe string, N; \(d_{o}\) is the inner diameter of the oil pipe in the shot hole section, m.  
The empirical formula for calculating the shock load obtained by fitting can be obtained by substituting into the above formula:  
\[F_{c}=\frac{\pi}{4}\left(\rho gH_{0}+1341.12\frac{\left(am_{e}(H_{1}-H_{2}) \right)^{1.03}}{\left(\frac{\pi}{4}\left((H_{0}-H_{3})(D_{c}-2d_{c})^{2}-(H_{2} -H_{3}){D_{o}}^{2}-(H_{1}-H_{2}){D_{g}}^{2}\right)\right)^{0.98}}\right)\left({D _{o}}^{2}-{d_{o}}^{2}\right) \tag{13}\]  
Considering only the pipe string subjected to axial shock load, when the impact pressure acting on the bottom of the pipe string exceeds the range that can be withstood by the pipe string itself, the pipe string will be subjected to a buckling behavior. Based on the safety coefficient method, comparing the impact pressure at the bottom of the pipe string under shock loading with the size of the critical load of the pipe string under buckling can be used as a criterion for determining the buckling of the pipe string under pressure.  
The criterion for judging the pipe string without buckling is  
\[F_{c}n_{1}\leq F_{cb} \tag{14}\]  
\[F_{cb}=5.66\sqrt{\frac{q_{p}E_{p}I_{p}}{r}} \tag{15}\]  
where \(n_{1}\) is the buckling safety factor of the pipe string; \(F_{cb}\) is the critical buckling load of the pipe string, N; \(E_{p}\) is the modulus of elasticity, Pa; \(I_{p}\) is the cross-section moment of inertia, m\({}^{4}\); \(q_{p}\) is the floating weight of the line, N/m; \(r\) is the inner diameter of the pipe string, m.