# Borehole Condition and Limit Pressure Differential Analysis in Carbonate Reservoirs
## 3 Evaluation of Open-Hole Wellbore Condition in Carbonate Rock
### Description of Wellbore Conditions  
At present, the criteria for describing the shear failure of rock in a plane strain state mainly include the Mohr-Coulomb criterion, Drucker-Prager criterion, and Hoek-Brown criterion. The yield surface of the Mohr-Coulomb criterion in stress space constitutes an irregular hexagonal prismatic cone, exhibiting a complex geometry that aptly depicts the shearing behavior of many rock masses. Its independent accommodation of cohesion and internal friction angle renders it widely applicable in the analysis of granular materials such as soils, gravel, and numerous rocks, particularly when accounting for the concurrent effects of shear and normal stresses. The standard expression of the Mohr-Coulomb criterion is as follows:  
\[{\sigma^{\prime}}_{1}={\sigma^{\prime}}_{3}\tan^{2}\left(\frac{\pi}{4}+\frac{ \phi}{2}\right)+2C\tan\left(\frac{\pi}{4}+\frac{\phi}{2}\right) \tag{6}\]  
where, \(\sigma^{\prime}_{1}\) and \(\sigma^{\prime}_{3}\) is the maximum and minimum effective principal stress, respectively. \(C\) is cohesion; \(\phi\) is internal friction angle. The Drucker-Prager criterion, whose yield surface in stress space forms a conical section, simplifies computations and is commonly employed in analyzing the plastic behavior of metals and other materials with superior continuity. It is especially suited to materials where cohesion is relatively minor, and friction dominates; however, in addressing the intricate deformation behavior of rocks, especially within the realm of elastic-plastic mechanics, the D-P criterion lacks the precision of the Mohr-Coulomb criterion in depicting complex rock behaviors. The standard expression of the Drucker-Prager criterion is as follows:  
\[\sqrt{J_{2}}+\alpha I_{1}-k=0 \tag{7}\]  
where \(I_{1}\) and \(J_{2}\) denote the first and second invariants of the stress deviator tensor, respectively; \(\alpha\) and \(k\) are constants related to the cohesion \(C\) and the internal friction angle \(\phi\). The Hoek-Brown criterion captures the inherent nonlinear failure characteristics of rocks and rock masses, along with the influence of structural features and stress states on strength. Nonetheless, limitations include neglect of the intermediate principal stress effect, challenges in accurately determining the criterion parameters, and reduced suitability for rocks with pronounced anisotropy due to prominent joints. \[\sigma_{1}=\sigma_{3}+\sigma_{c}\left(m_{\mathrm{i}}\frac{\sigma_{3}}{\sigma _{c}}+1\right)^{0.5} \tag{8}\]  
where \(\sigma_{1}\) and \(\sigma_{3}\) is the maximum and minimum effective principal stress, respectively. \(\sigma_{c}\) represents the uniaxial compressive strength of the rock; \(m_{\mathrm{i}}\) is an empirical parameter with dimensions reflecting the hardness of the rock, whose value ranges from 0.001 to 25.0. Owing to its broad applicability and comprehensive consideration of physical parameters, the Mohr-Coulomb criterion is adopted herein to characterize the plastic and residual regions of carbonate rocks. Let \(F_{y}=\frac{\sigma^{\prime}_{1}}{\sigma^{\prime}_{3}K^{2}_{y}+2C_{y}K_{y}}\), \(F_{p}=\frac{\sigma^{\prime}_{1}}{\sigma^{\prime}_{3}K^{2}_{y}+2C_{p}K_{p}}\), respectively, where \(K_{y}=\tan\left(\frac{\pi}{4}+\frac{\phi_{p}}{2}\right)\); \(K_{p}=\tan\left(\frac{\pi}{4}+\frac{\phi_{p}}{2}\right)\)pressure differential), and the wellbore is safe. When the surrounding rock of the wellbore is in a plastic state, it will not immediately collapse and the wellbore is safe, but there may be some damage inside the rock, which could lead to collapse when it is immersed in completion fluid for a long time. When the surrounding rock of the wellbore enters the post-peak residual state, spalling and loss of blocks will occur in the wellbore.