# Borehole Condition and Limit Pressure Differential Analysis in Carbonate Reservoirs
## 2 Stress Distribution around Borehole in Open-Hole Completion  
During the test, pore pressure around the borehole and stress on the rock skeleton are redistributed. Assume that the original in situ stress state is the following: the overburden pressure \(\sigma_{V}\); the maximum horizontal stress \(\sigma_{H}\); and the minimum horizontal stress \(\sigma_{h}\). For any directional well with inclination angle \(\alpha\) and azimuth angle \(\beta\), the far-field stress with the borehole axis as z-axis can be calculated by the following formula [7]:  
\[\left\{ \begin{array}{l}\sigma_{xx}=\sigma_{H}\cos^{2}\alpha\cos^{2}\beta+ \sigma_{h}\cos^{2}\alpha\sin^{2}\beta+\sigma_{V}\sin^{2}\alpha\\ \sigma_{yy}=\sigma_{H}\sin^{2}\beta+\sigma_{h}\cos^{2}\beta\\ \sigma_{zz}=\sigma_{H}\sin^{2}\alpha\cos^{2}\beta+\sigma_{h}\sin^{2}\alpha \sin^{2}\beta+\sigma_{V}\cos^{2}\alpha\\ \tau_{xy}=-\sigma_{H}\cos\alpha\cos\beta\sin\beta+\sigma_{h}\cos\alpha\cos \beta\sin\beta\\ \tau_{yz}=-\sigma_{H}\sin\alpha\cos\beta\sin\beta+\sigma_{h}\sin\alpha\cos \beta\sin\beta\\ \tau_{zx}=\sigma_{H}\cos\alpha\sin\alpha\cos^{2}\beta+\sigma_{h}\cos\alpha \sin\alpha\sin^{2}\beta-\sigma_{V}\sin\alpha\cos\alpha\end{array} \right. \tag{1}\]  
Based on the plane strain assumption, the stress component around the wellbore can be expressed as follows:  
\[\left\{ \begin{array}{l}\sigma_{\tau}=\frac{\tau_{xx}}{r}p_{i}+\frac{\sigma_{ xx}+\sigma_{yy}}{2}(1-\frac{\tau_{xx}^{2}}{r^{2}})+\frac{\sigma_{xx}-\sigma_{yy}}{2}(1 +\frac{3\tau_{xx}^{4}}{r^{4}}-\frac{4\tau_{xy}^{2}}{r^{2}})\cos 2\theta+ \tau_{xy}(1+\frac{3\tau_{xy}^{4}}{r^{4}}-\frac{4\tau_{xy}^{2}}{r^{2}})\sin 2 \theta\\ \sigma_{\theta}=-\frac{\tau_{xy}}{r}p_{i}+\frac{\sigma_{xx}+\sigma_{yy}}{2}(1 +\frac{\tau_{xy}^{2}}{r^{2}})-\frac{\sigma_{xx}-\sigma_{yy}}{2}(1+\frac{3\tau _{xy}^{4}}{r^{4}})\cos 2\theta-\tau_{xy}(1+\frac{3\tau_{xy}^{4}}{r^{4}})\sin 2\theta\\ \sigma_{z}=\sigma_{zz}-\mu[2(\sigma_{xx}-\sigma_{yy})\frac{\tau_{xy}^{2}}{r^{2 }}\cos 2\theta+4\tau_{xy}\frac{\tau_{xy}^{2}}{r^{2}}\sin 2\theta]\\ \tau_{r\theta}=-\frac{\sigma_{xx}-\sigma_{yy}}{2}(1-\frac{3\tau_{x}^{4}}{r^{4}} +\frac{2\tau_{xy}^{2}}{r^{2}})\sin 2\theta+\tau_{xy}(1-\frac{3\tau_{xy}^{4}}{r^{4}}+ \frac{2\tau_{xy}^{2}}{r^{2}})\cos 2\theta\\ \tau_{\theta z}=\tau_{yz}(1+\frac{\tau_{xy}^{2}}{r^{2}})\cos\theta-\tau_{zx}(1 +\frac{\tau_{xy}^{2}}{r^{2}})\sin\theta\\ \tau_{zr}=\tau_{xz}(1-\frac{\tau_{xy}^{2}}{r^{2}})\cos\theta+\tau_{yz}(1-\frac{ \tau_{xy}^{2}}{r^{2}})\sin\theta\end{array} \right. \tag{2}\]  
where \(\sigma_{xx}\), \(\sigma_{yy}\), \(\sigma_{zz}\), \(\tau_{xy}\), \(\tau_{yz}\), and \(\tau_{zx}\) are stress components in borehole rectangular coordinates, MPa; \(\sigma_{r}\), \(\sigma_{\theta}\), \(\sigma_{z}\), and \(\tau_{\theta z}\) are stress components in borehole cylindrical coordinates, MPa; \(p_{i}\) is wellbore fluid column pressure, MPa; \(\theta\) is the relative angle of any point in borehole wall, radian; \(\mu\) is Poisson's ratio; \(r_{w}\) and \(r\) are the borehole radius and polar radius of a point in formation, m.  
According to the basic principle of stress transformation [8], the stress matrix at any point in formation near the borehole wall can be transformed into the principal stress matrix. \[\left[ \begin{array}{ccc}\sigma_{r}&\tau_{r\theta}&\tau_{zr}\\ \tau_{r\theta}&\sigma_{\theta}&\tau_{\theta z}\\ \tau_{zr}&\tau_{\theta z}&\sigma_{z}\end{array} \right]\rightarrow\left[ \begin{array}{ccc}\sigma_{1}&&\\ &\sigma_{2}&\\ &&\sigma_{3}\end{array} \right] \tag{3}\]  
where \(\sigma_{1}\), \(\sigma_{2}\) and \(\sigma_{3}\) are three principal stresses, MPa.