# Borehole Condition and Limit Pressure Differential Analysis in Carbonate Reservoirs
## 2 Stress Distribution around Borehole in Open-Hole Completion
For fractured carbonate formations, when fractures are highly developed, pressure changes around the borehole during open-hole testing are similar to those in porous media, as shown below [9]:  
\[p_{(r)}=p_{p}-\frac{\mu Q}{4\pi K_{f}h}[E_{i}(-\frac{\mu C_{f}r^{2}}{4K_{f}t})] \tag{4}\]  
where \(p_{(r)}\) is the pore pressure around the wellbore from the axis \(r\), MPa; \(p_{p}\) is the original pore pressure of the formation, MPa; \(Q\) is the test yield, m\({}^{3}\)/s; \(K_{f}\) is the permeability of fracture system, \(\mu\)m\({}^{2}\); \(h\) is reservoir thickness, m; \(\mu\) is the underground viscosity of crude oil, mPa-s; \(C_{f}\) is the compression coefficient of fractured reservoir, MPa\({}^{-1}\); \(t\) is the test time, s; \(E_{i}\) is a power integral function. According to the principle of effective stress, the effective stress acting on the rock skeleton controls the rock deformation and failure, namely,  
\[\sigma^{\prime}=\sigma-\alpha p_{(r)} \tag{5}\]  
According to Equations (1) to (5), the effective stress distribution around the wellbore can be calculated. Taking a deep carbonate reservoir as an example, its basic characteristic parameters are shown in Table 1.  
When the test pressure differential is 20 MPa, the stress distribution around the borehole is shown in Figure 1. Due to the difference in the redistribution of stress at different well angles after its formation, there are \"safest\" and \"most dangerous\" angles around the well. At the safest angle, the difference between the maximum and minimum effective principal stresses is relatively small, making the well less prone to failure. Conversely, at the most dangerous angle, the difference between the maximum and minimum effective principal stresses is large, which makes it easy to cause wellbore collapse and instability. According to the calculation results, it can also be seen that the stress distribution around the borehole in different directions is also different. When drilling in the direction of the maximum horizontal principal stress, the stress difference on the safest angle of the wellbore is small, only about 94 MPa, whereas, at the most dangerous angle, the stress difference reaches approximately 309 MPa. When drilling in the direction of horizontal minimum principal stress, the differential stress on the wellbore at the safest angle is about 201 MPa, and the differential stress on the wellbore at the most dangerous angle is about 247 MPa. The difference between the two is not significant, which is related to the relative size of the three principal stresses. \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline Vertical depth/m & 7735 \\ Pore pressure/MPa & 89.0 \\ Overburden pressure/MPa & 185.6 \\ Max horizontal principal stress/MPa & 193.4 \\ Min horizontal principal stress/MPa & 135.4 \\ Effective stress coefficient & 0.6 \\ Poisson’s ratio of formation & 0.20 \\ Viscosity/mPa-s & 3.06 \\ Fracture permeability/mD & 1.44 \\ Reservoir thickness/m & 100 \\ Coefficient of compressibility/MPa\({}^{-1}\) & \(0.41\times 10^{-4}\) \\ Reservoir porosity/\% & 0.6 \\ Testing time/h & 72 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Characteristics and test parameters of a carbonate reservoir.