## 3 Theoretical Force Analysis of Casing Damaged Wells
### Stress Analysis of Casing Damaged Wells under Non-Uniform In-Situ Stress  
Usually, the casing is in a non-uniform in-situ stress state, i.e., \(\sigma_{\rm H}=\sigma_{\rm h}\). The original in-situ stress in a non-uniform state can be decomposed into the sum of the average stress component and the deviation stress component. Therefore, solving the external extrusion force and stress of the casing can be divided into two steps: firstly, the stress on the casing under the action of average stress; the second is the stress on the casing under deviating  
Figure 5: Distribution of in-situ stress at horizontal section.  
stress, and the superposition of the two stress parts is the stress situation of the casing under the non-uniform original in-situ stress.  
The external extrusion force of the casing under non-uniform in-situ stress is shown in Equation (14):  
\[\left\{\begin{aligned} s_{1}=-\frac{(1-\mu_{k})(\sigma_{H}+ \sigma_{h})}{1+\frac{1}{1+\left(\frac{\eta}{h}\right)^{2}}\frac{1+\mu_{d}}{1+ \mu_{g}}\frac{E_{d}}{E_{d}}\left[1-2\mu_{c}+\left(\frac{\tau_{2}}{\tau_{3}} \right)^{2}\right]}\\ s_{2}=-\frac{\text{C}_{21}+\text{C}_{12}}{\text{C}_{11}\text{C}_{22}- \text{C}_{12}}\text{C}_{21}\frac{2\left(1-\mu_{f}^{2}\right)E_{c}}{1+\mu_{d}E _{f}}\left(1-m_{cl}^{2}\right)^{3}(\sigma_{H}-\sigma_{h})\\ s_{3}=-\frac{\text{C}_{21}+\text{C}_{11}}{\text{C}_{12}\text{C}_{22}- \text{C}_{12}}\text{C}_{21}\frac{2\left(1-\mu_{f}^{2}\right)E_{c}}{1+\mu_{d}E _{f}}\left(1-m_{cl}^{2}\right)^{3}(\sigma_{H}-\sigma_{h})\\ \end{aligned}\right. \tag{14}\]  
\[\left\{\begin{aligned} c_{11}&=A+\frac{1+ \mu_{f}}{1+\mu_{d}}\frac{E_{d}}{E_{f}}\big{(}\frac{5}{3}-2\mu_{cl}\big{)} \big{(}1-m_{cl}^{2}\big{)}^{3}\\ c_{12}&=B-\frac{1+\mu_{f}}{1+\mu_{d}}\frac{E_{d}}{E_{f}} \big{(}\frac{4}{3}-2\mu_{cl}\big{)}\big{(}1-m_{cl}^{2}\big{)}^{3}\\ c_{21}&=C-\frac{1+\mu_{f}}{1+\mu_{d}}\frac{E_{d}}{E_{f}} \big{(}\frac{4}{3}-2\mu_{cl}\big{)}\big{(}1-m_{cl}^{2}\big{)}^{3}\\ c_{22}&=D+\frac{1+\mu_{f}}{1+\mu_{d}}\frac{E_{d}}{E_{f}} \big{(}\frac{5}{3}-2\mu_{cl}\big{)}\big{(}1-m_{cl}^{2}\big{)}^{3}\\ R&=-\frac{4\left(1-\mu_{f}^{2}\right)}{1+\mu_{d}}\frac{E_{d}} {E_{f}}\left(1-m_{cl}^{2}\right)^{3}\end{aligned}\right. \tag{15}\]  
\[\left\{\begin{aligned} A=\big{(}1-\frac{2}{3}\mu_{cl} \big{)}+(5-6\mu_{cl})m_{cl}^{2}+(3-2\mu_{cl})m_{cl}^{4}+\big{(}\frac{5}{3} -2\mu_{cl}\big{)}m_{cl}^{6}\\ B=-\frac{2}{3}\mu_{cl}+2\mu_{cl}m_{cl}^{2}-2(2-\mu_{cl})m_{cl}^{4}- \big{(}\frac{4}{3}-2\mu_{cl}\big{)}m_{cl}^{6}\\ C=-\frac{2}{3}\mu_{cl}-2\mu_{cl}m_{cl}^{2}+2(2-\mu_{cl})m_{cl}^{4}- \big{(}\frac{4}{3}-2\mu_{cl}\big{)}m_{cl}^{6}\\ D=\big{(}1-\frac{2}{3}\mu_{cl}\big{)}+(3-2)m_{cl}^{2}+(3-2\mu_{cl})m_{ cl}^{4}+\big{(}\frac{5}{3}-2\mu_{cl}\big{)}m_{cl}^{6}\\ \end{aligned}\right. \tag{16}\]  
In the equation, c, A, B, C, and D are dimensionless parameters. Under simple and ideal conditions, the stress of the casing can be solved using the above analytical formula, but the calculation is complex and the stress conditions of the casing are complex and variable. In addition, the coupling problem of the casing, cement sheath, and formation makes it difficult to calculate using the analytical formula. Therefore, it is necessary to use finite element software for analysis and research.