# A Study on the Stress Field Around an Elliptical Wellbore with an Eccentric Casing in Deep Wells
## 2 Finite Element Model of Eccentric Casing in an Elliptical Wellbore  
### Establishment of the Finite Element Model  
According to the theory of elasticity, based on the geometric and force characteristics of the composite model, it can be simplified as a plane strain model. To simplify the calculation process, the following assumptions are made:  
1. The casing, cement annulus, and formation are linear elastic materials. The plastic behavior of the materials is not considered, and the influence of pore pressure on stress distribution is neglected. 2. Assuming small displacements, the changes in casing inner and outer diameters caused by loads, as well as the changes in cement annulus thickness, are neglected. 3. Assuming tight adhesion between all interfaces without any gaps, no debonding occurs during the deformation process. 4. We assume that the model parts are concentric and maintain their concentric status during the deformation process. According to the model symmetry, we selected half of the model for modeling and set the eccentricity of the wellbore. Based on the data in Table 1, we established a combined model under different casing eccentricities. Firstly, we created the casing, formation, and cement annulus as separate parts and applied the TIE contact attribute at their contact interfaces (the TIE contact property is mainly used in finite element analysis to bind two regions together so that there is no relative motion between them), as per Assumption 3. The Saint-Venant principle states that if the actual distributed load is replaced by an equivalent load, the stress and strain only change near the load application location, while the stress values are basically the same far away from the load application location. According to the principle of Saint-Venant, the stress in the surrounding formation is released after drilling, but the stress far away from the wellbore remains unaffected. To eliminate the influence of boundary effects on the results, we made the size of the formation in the combined finite element model 20 times larger than the casing size. In finite element analysis, a model size that is too small can lead to significant boundary effects, affecting the accuracy of the results. A model size 20 times larger than the wellbore diameter provides a sufficient buffer zone, ensuring that the stress distribution near the wellbore is not significantly influenced by boundary conditions. Although larger models can further reduce boundary effects, they also increase computation time and resource consumption. A 20-fold size strikes a good balance between computational efficiency and accuracy. On the bottom side of the model, we set symmetric displacement boundary conditions. On the top side of the model, we applied a maximum horizontal stress of 32 MPa, while on the left and right sides, a minimum horizontal stress of 20 MPa was applied. With the above settings, we established the physical model as shown in Figure 1.  
The model utilizes a structured meshing technique and employs four-node quadrilateral elements (CPE4 elements). Grid refinement has been applied to the casing and cement annulus sections. The entire combined physical model consists of a total of 15,488 nodes and 15,240 elements, as shown in Figure 2.  
Table 1 is a study on the determination of grid accuracy. By comparing the peak stress of casing with different numbers of elements, it was found that when the number of elements was 15,240, the calculation of peak stress tended to be stable, and the calculation time and resource consumption were relatively low. The data in this study are obtained from actual data of a well in Block P13 of the Daqing Oilfield [18], including the geometric and material parameters of the casing, cement annulus, and formation, as well as the numerical values of the maximum and minimum horizontal stresses. Please refer to Table 2 for specific values.