# A Study on the Stress Field Around an Elliptical Wellbore with an Eccentric Casing in Deep Wells
## 3 Result Analysis and Discussion
### Stress Distribution of Casing Under Different Eccentricities
#### 3.1.2 Circumferential Stress Distribution  
Figure 6 displays the contour plot of circumferential stress distribution in the casing at different eccentricities. By observing the plots, we can conclude that as the eccentricity increases, the non-uniformity of circumferential stress distribution and the occurrence of stress concentration become more pronounced. The increase in eccentricity leads to an increase in the non-uniformity of circumferential stress distribution in the casing, accompanied by stress concentration. The area of the low-stress region continuously increases and shifts towards the position of the thinner cement ring, while the area of the high-stress region decreases. The specific changes in cementing are as follows. When eccentricity is 0.0, the low-stress area, 80\({}^{\circ}\) to 100\({}^{\circ}\), is on the inner wall of the casing (thinner cement sheath), mostly near the inner wall. The high-stress zone, 0\({}^{\circ}\) to 20\({}^{\circ}\) and 160\({}^{\circ}\) to 180\({}^{\circ}\), diverges from the inner wall to the outer wall. Both zones are symmetrical about the T-Z plane. When eccentricity is 0.3, the circumferential stress distribution is similar to that at an eccentricity of 0.0. The low-stress zone is no longer symmetrical, with an increased area. When eccentricity is 0.6, the low-stress zone expands and shifts towards the thicker position of the cement sheath, radiating from the inner wall to the outer wall. The high-stress zone decreases; the area on the thick cement ring side decreases the most, resulting in stress concentration. When eccentricity is 0.9, the low-stress zone expands to half of the casing area. The high-stress zone on the thick cement ring side further decreases, indicating a significant stress concentration state. \begin{table}
\begin{tabular}{c c c c c} \hline  & \(\epsilon\)**=0.0** & \(\epsilon\)**=0.3** & \(\epsilon\)**=0.6** & \(\epsilon\)**=0.9** \\ \hline \(\sigma_{\text{Max},r}/\text{MPa}\) & 40.17 & 46.46 & 84.54 & 146.93 \\ \(\sigma_{\text{Min},r}/\text{MPa}\) & 27.75 & 30.76 & 35.48 & 36.51 \\ \(K\) & 1.44 & 1.51 & 2.38 & 4.024 \\ \hline \end{tabular}  
* Footnotes: \(\sigma_{\text{Max},r}\) is the maximum radial stress value, \(\sigma_{\text{Min},r}\) is the minimum radial stress value, and \(K\) is the stress non-uniformity coefficient.  
\end{table} 
Table 4: Maximum radial stress and stress non-uniformity coefficient under different eccentricity ratios. Figure 5: Radial stress distribution on the outer wall of the casing under different eccentricity ratios. Figure 7 is a dotted line plot depicting the circumferential stress on the outer wall of the casing at different eccentricities. In the graph, the curves exhibit the following characteristics:  
1. The increase in eccentricity leads to an increase in the non-uniformity of stress distribution and the probability of stress concentration. Comparing Tables 4 and 5, we found that compared to the non-uniformity coefficient of radial stress, the non-uniformity coefficient of circumferential stress is lower. Unlike radial stress, the reason for the non-uniformity of circumferential stress is that the minimum value of circumferential stress is constantly decreasing. 2. The increase in eccentricity reduces the value of circumferential stress on the outer wall of the casing. The maximum stress values under each eccentricity are almost the same, and the orientation of the maximum stress values under each eccentricity is 180 \({}^{\circ}\), which is the thickest position of the cement sheath. The minimum stress values under different eccentricities vary greatly, and the orientation of the minimum stress values under each eccentricity is close to the side of the thick cement sheath. Figure 6: Circumferential stress distribution contour plot of casing at eccentricity: (**a**) \(\varepsilon=0.0\), (**b**) \(\varepsilon=0.3\), (**c**) \(\varepsilon=0.6\), and (**d**) \(\varepsilon=0.9\). Figure 7: Circumferential stress distribution on the outer wall of the casing under different eccentricity ratios.