# A New Prediction Model of Annular Pressure Buildup for Offshore Wells
## 3 Wellbore Annulus Volume Model
### Deformation of Casing-Free Section  
The deformation of the casing-free section consists of two parts: one is caused by thermal expansion, and the other is caused by stress. Assuming the total radial strain of the free section of casing is \(\varepsilon_{r}\), then the strain due to the stress part is \(\varepsilon_{r}-\alpha T\). Therefore, the following occurs:  
\[\left\{ \begin{aligned} &\varepsilon_{r}-\alpha T=\frac{1}{E}[ \sigma_{r}-\
u(\sigma_{\theta}+\sigma_{z})]\\ &\varepsilon_{\theta}-\alpha T=\frac{1}{E}[\sigma_{\theta}-\
u( \sigma_{z}+\sigma_{r})]\\ &\varepsilon_{z}-\alpha T=\frac{1}{E}[\sigma_{z}-\
u(\sigma_{r}+ \sigma_{\theta})]\end{aligned} \right. \tag{1}\]  
under planar conditions, \(\varepsilon_{z}=0\);  
Where \(\sigma_{r}\) is radial stress, MPa; \(\sigma_{\theta}\) is circumferential stress, MPa; \(\sigma_{z}\) is axial stress, MPa; \(\varepsilon_{r}\) is the radial strain, m; \(\varepsilon_{\theta}\) is circumferential stress, m; \(\varepsilon_{z}\) is the axial stress, m; \(E\) is the elastic modulus of elastomer, MPa; \(\
u\) is Poisson's ratio of an elastic body, dimensionless; \(\alpha\) is the expansion coefficient of the elastomer, \({}^{\circ}\)C\({}^{-1}\); \(T\) is the temperature change, \({}^{\circ}\)C.  
Due to the difference between the subsea wellhead and the land wellhead, the deformation ability of the deepwater casing in the longitudinal direction is limited after cementing. According to the theory of thermoelasticity, the wellbore formation system can be regarded as the two-dimensional plane strain problem of a multi-layer composite thick-walled barrel for displacement and stress analysis and solution. The geometric equation of the plane strain problem in polar coordinates is as follows:  
\[\left\{ \begin{aligned} &\varepsilon_{r}=\frac{\partial_{\alpha}}{ \partial_{r}}\\ &\varepsilon_{\theta}=\frac{v}{r}+\frac{1}{r}\frac{\partial_{\alpha }}{\partial_{\theta}}\\ &\varepsilon_{z}=\frac{1}{r}\frac{\partial_{\alpha}}{\partial_{ \theta}}+\frac{\partial_{\alpha}}{\partial_{r}}-\frac{v}{r}\end{aligned} \right. \tag{2}\]  
Figure 2: Schematic diagram of wellbore structure. The wellbore formation system belongs to the axisymmetric plane strain problem, which only considers the deformation along the radial direction, so the geometric equation is simplified as follows:  
\[\left\{ \begin{aligned} &\varepsilon_{r}=\frac{d\omega_{r}}{d_{r}}\\ &\varepsilon_{\theta}=\frac{v}{r}\\ &\varepsilon_{z}=0\end{aligned} \right. \tag{3}\]  
In polar coordinates, the physical equation of plane strain considering temperature change is as follows:  
\[\left\{ \begin{aligned} &\varepsilon_{r}=\frac{1+v}{E}[(1-v) \sigma_{r}-v\sigma_{\theta}]+(1+v)\alpha T\\ &\varepsilon_{\theta}=\frac{1+v}{E}[(1-v)\sigma_{r\theta}-v \sigma_{r}]+(1+v)\alpha T\\ \end{aligned} \right. \tag{4}\]  
Therefore, the state equation of stress and strain derived from the formula is as follows:  
\[\left\{ \begin{aligned} &\sigma_{r}=\frac{E}{(1+v)(1-2v)}[(1-v) \varepsilon_{r}+v\varepsilon_{\theta}-(1+v)\alpha T]\\ &\sigma_{\theta}=\frac{E}{(1+v)(1-2v)}[(1-v)\varepsilon_{r}+v \varepsilon_{r}-(1+v)\alpha T]\\ \end{aligned} \right. \tag{5}\]  
Because the wellbore formation system belongs to axisymmetric plane strain problem, the influence of physical force can be ignored. At the same time, the tangential stress is zero, and the magnitude of the circumferential stress is independent of the rotation Angle, so the formula can be simplified as follows:  
The plane stress satisfies the equilibrium equation:  
\[\frac{d\sigma_{r}}{dr}+\frac{\sigma_{r}-\sigma_{\theta}}{r}=0 \tag{6}\]  
The boundary conditions are as follows:  
\[\left\{ \begin{aligned} &\sigma_{r}(a)=-q_{1}\\ &\sigma_{r}(b)=q_{2}\\ \end{aligned} \right.