# Study on the Vertical Stability of Drilling Wellbore under Optimized Constraints
## 4 Numerical Calculation Verification  
### Principles of Stability Numerical Calculation and Numerical Models  
Euler stability was determined to be the cause of the wellbore's stability issue. The generalized eigenvalue solution of Euler stability in numerical analysis and the incremental solution of extreme stability were used to determine this stability. Literature [36] provides a detailed explanation of the numerical calculation process for the stability of wellbore structures. Based on numerical calculation principles, the finite element software ABAQUS can be used to calculate numerical simulation eigenvalue and analyze the structural stability. When the eigenvalue \(\lambda\) satisfies \(\lambda<0\) or \(\lambda>1\), the wellbore is vertically stable, and when \(0<\lambda<1\), the wellbore will be unstable.  
The numerical simulation section of this paper is based on the Banji coal mine auxiliary wellbore as the engineering basis and establishes a physical model of the second stage of the wellbore construction process, which is the stage where the wellbore floated to the bottom but is not filled and fixed. Since a wellbore is cylindrical in shape, with constant inner and outer diameters, it can be regarded as an axisymmetric model. The establishment of the entire wellbore structure model is done through the input file. According to the characteristics of the wellbore structure, the entire wellbore is divided into two layers: the lower layer is the bottom of wellbore, and the upper layer is the joint of wellbore. The establishment process of the wellbore is shown below.  
1. Definition of nodes and node sets. The node types are defined by *Node and Nset. A total of 15,433 nodes are defined in the entire wellbore model, and the height of wellbore model is 638.084 m. The nodes at the bottom of the wellbore are named \"BOTOM\", and the nodes at the top of the wellbore are named \"TOP\".  
Figure 6: The sensitivity coefficient of the critical depth of wellbore to each influencing factor.  
2. Definition of elements and elements sets. In this structure, the elements in the bottommost of the wellbore are defined using three-node triangular linear film strain linear shell elements, referred to as S3R. All other elements are specified to use four-node quadrilateral linear reduced integration shell elements, referred to as S4R.
3. Define surfaces. Surfaces are defined by the *Surface command, which describes the faces composed of elements. For all elements, their inner surfaces are named \"In\", and their outer surfaces are named Out.
4. Define materials. Materials are defined by the Materials command. The wellbore structure is made of reinforced concrete. The concrete is graded C70. The elastic modulus of concrete is 38.15 GPa, and its Poisson's ratio is 0.285. The reinforcing steel has an elastic modulus of 210 GPa, and its Poisson's ratio is also 0.285.
5. Define shell parameters. Defined by the Shell section command, the thickness of the shell is 0.85 m.
6. Definition of loads. The height of counterweight water is 333.3 m, and the height of mud is 638.084 m.
7. Define boundaries. The *Boundary command is used to define boundary conditions. We define the corresponding constraints on BOTTOM and TOP.
8. Define buckling. Buckling is defined by the Buckle Step command. We set the number of eigenvalues requested to five.  
The finite element calculation model of the equal cross-sectional wellbore, established according to above definitions, is shown in Figure 7a. The types of elements used in the finite element model are depicted in Figure 7b,c.