# Study on the Vertical Stability of Drilling Wellbore under Optimized Constraints
## 2 Theoretical Methodology  
### Force Analysis of Wellbore under Optimized Construction Method  
Before the wellbore is suspended and sinks to the bottom, the grouting pipe is used to fill the cement slurry on the bedrock surface so that the bottom of wellbore can fall on the bedrock surface. The wellbore's external side is wrapped by cement slurry, and the bottom of the wellbore is in contact with the bedrock surface. It can be assumed that the wellbore is regarded as a slender pressure rod with an upper hinge support and a lower end fixed support. For the convenience of this discussion, the wellbore is regarded as the whole of the continuity of a homogeneous nature, and we made the following basic assumptions:  
1. The material of the wellbore is completely elastic, obeying the law of Huke;
2. The displacement and deformation of the wellbore during flexion is small;
3. The effects of friction and bonding between the wall and mud are not taken into account. The drilling wellbore is regarded as a slender rod hinged at the upper end and solidly supported at the lower end, so the deflection curve function of the drilling wellbore \(y\) can be expressed as [26]  
\[y=\sum_{n=1}^{\infty}C_{n}\bigg{(}\cos\frac{2n+1}{2H}\pi x-\cos\frac{2n-1}{2H }\pi x\bigg{)} \tag{1}\]  
where \(n\) is the number of the half wave number of the flexural curve function; \(C_{n}\) is a coefficient to be determined; \(x\) is the height at any point on the wellbore; and \(H\) is the \begin{table}
\begin{tabular}{c c c c} \hline
**Symbols** & **Meanings** & **Symbols** & **Meanings** \\ \hline \(y\) & The deflection curve function of wellbore & \(\Pi\) & The total potential energy of wellbore \\ \(y^{\prime}\) & The first derivative of \(y\) & \(U\) & Elastic strain energy \\ \(y^{\prime\prime}\) & The second derivative of \(y\) & \(V\) & Total external potential energy \\ \(\delta\) & The maximum deformation value of the deflection curve & \(E\) & Elastic modulus of the material used for wellbore \\ \(D\) & The outer diameter of wellbore & \(I\) & Sectional moment of inertia of wellbore \\ \(d\) & The inner diameter of wellbore & \(A_{1}\) & The coefficient of \(\delta\) \\ \(k\) & Radius of curvature & \(A_{2}\) & The coefficient of \(\delta^{2}\) \\ \(M\) & The in-plane bending moment & \(A_{4}\) & The coefficient of \(\delta^{4}\) \\ \(\rho\) & The ratio of reinforcement & \(x\) & The state variable of the system \\ \(q_{c}\) & Self-weight of wellbore & \(m\) & A control variable of the system \\ \(P_{m}\) & Lateral pressure of mud on the external surface & \(n\) & A control variable of the system \\ \(P_{w}\) & Lateral pressure of counterweight water on the internal surface of wellbore & \(\Pi^{\prime}(x)\) & The first derivative of \(\Pi\) \\ \(R_{A}\) & Counterforce on the bottom of wellbore & \(\Pi^{\prime\prime}(x)\) & The second derivative of \(\Pi\) \\ \(R_{B}\) & The counterforce of wellhead & \(W_{c}\) & The external potential energy change due to self-weight \\ \(R_{B}\) & support & \(W_{m}\) & The change in external potential energy due to the lateral pressure of mud \\ \(H\) & Depth of wellbore & \(W_{m}\) & The change in external potential energy under the lateral pressure of counterweight water \\ \(\gamma_{c}\) & The weight of concrete & \(W_{w}\) & The horizontal component of \(W_{m}\) \\ \(\gamma_{s}\) & The weight of steel & \((W_{m})_{x}\) & The vertical component of \(W_{m}\) \\ \(\gamma_{m}\) & The weight of mud & \((W_{m})_{y}\) & The vertical component of \(W_{m}\) \\  & Gravity of mud per unit length on the outside of & \((W_{w})_{x}\) & The horizontal component of \(W_{w}\) \\ \(q_{m}\) & wellbore. & \((W_{w})_{y}\) & The vertical component of \(W_{w}\) \\ \(\gamma_{w}\) & The weight of counterweight water per unit length & \(\Delta\) & Stability criterion \\ \(q_{w}\) & on the inside of wellbore. & \(\triangle\) & Thickness of wellbore \\ \(H_{w}\) & Height of counterweight water & \(\triangle\) & Thickness of wellbore \\ \hline \end{tabular}
\end{table} 
Table 1: Glossary of symbols and their meanings.