# 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  
#### 3.1.1 Radial Stress Distribution  
In Figure 4, a contour plot of the radial stresses in the casing is shown for different eccentricities when the model reaches equilibrium. Observing the cloud map can lead to the conclusion that an increase in eccentricity leads to an uneven geometric shape of the cement sheath, which in turn leads to a complex stress distribution in the casing. The low-stress area moves towards the thinner position of the cement sheath and gradually increases its area, and the high-stress zones experience stress concentration at higher levels of eccentricity.  
The specific changes in cementing are as follows: when eccentricity is 0.0, the radial stress distribution is symmetrical, with a clear transition in stress values. The low-stress zone is distributed from 0\({}^{\circ}\) to 180\({}^{\circ}\) on the inner wall of the casing. The high-stress zone is distributed between 0\({}^{\circ}\) to 45\({}^{\circ}\) and 135\({}^{\circ}\) to 180\({}^{\circ}\) on the outer wall of the casing.  
When eccentricity is 0.3, the stress distribution becomes non-uniform, and the area of the low-stress zone decreases. The low-stress zone is now between 0\({}^{\circ}\) to 110\({}^{\circ}\) and 150\({}^{\circ}\) to 180\({}^{\circ}\) on the inner wall of the casing. The high-stress zone increases and is concentrated between 100\({}^{\circ}\) and 170\({}^{\circ}\) on the outer wall of the casing, particularly where the cement sheath is thicker.  
When eccentricity is 0.6, the area of the low-stress region increases and is concentrated between 0\({}^{\circ}\) to 50\({}^{\circ}\) and 170\({}^{\circ}\) to 180\({}^{\circ}\) on the inner wall of the casing. The high-stress zone expands from the outer wall to the inner wall and is concentrated between 120\({}^{\circ}\) and 160\({}^{\circ}\).  
When eccentricity is 0.9, the area of the low-stress region expands and occupies most of the casing from 0\({}^{\circ}\) to 90\({}^{\circ}\). The high-stress zone shrinks, leading to a stress concentration between 140\({}^{\circ}\) and 160\({}^{\circ}\) on the inner wall of the casing, expanding outward in a fan shape.  
Figure 5 is a scatter plot obtained by extracting the radial stresses on the outer surface of the casing for each eccentricity. The overall distribution of the curves has the following characteristics:  
1. Increasing eccentricity leads to a non-uniform distribution of radial stresses on the outer surface of the casing. Table 4 provides numerical values for the maximum and minimum radial stress values on the outer surface of the casing and the stress non-uniformity coefficient for each eccentricity. Eccentricity has a significant impact on the uniformity of radial stress distribution. At high eccentricities, the non-uniformity of radial stress is higher, and stress concentration is more likely to occur.  
Figure 4: Radial stress distribution diagrams for different casing eccentricity values. Radial stress diagram for eccentricity: (**a**) \(\varepsilon=0.0\), (**b**) \(\varepsilon=0.3\), (**c**) \(\varepsilon=0.6\), and (**d**) \(\varepsilon=0.9\).  
2. The location of the maximum radial stress varies with the change in eccentricity as follows: the location of the maximum stress is generally located where the cement ring is relatively thicker. As the eccentricity increases, the probability of stress concentration occurring also increases.