# Research on Pore Pressure Detection While Drilling Based on Mechanical Specific Energy
## 2 Theory
### Downhole Parameters  
In the model of MSE, WOB and torque are the main variables. For conventional drilling, there is generally a lack of the downhole WOB and torque measurements, so they are considered to be computed from surface measurements (hook load, wellhead torque). In some high-angle wells, especially directional and horizontal wells, the friction between the tubular and wellbore wall is high, and there might be beds of cuttings. Therefore, the influence of friction during the transmission of WOB and torque should be considered.  
The following assumptions can be made for obtaining a simplified soft rod model for estimating the friction: (1) the force and deformation of the drill string are within the elastic range; (2) the borehole curvature of the calculation unit is a constant; (3) the drill string touches the upper or lower side of the well wall with the same curvature; (4) the shear forces on the cross-section of the drill string are neglected, and the effect of the drill string stiffness is disregarded, but the drill string can withstand axial pressure; (5) the influence of the dynamic effect of the drill string is neglected. The downhole WOB and torque are calculated by Equation (22) and Equation (23), respectively [42]. Here, in the Figure 1, the models for both WOB and torque are transcendental equations that need to be solved using numerical calculations. A trial value is first taken at the bit, and the WOB and torque are calculated from the bit to the wellhead by unit. Then, the trial results are compared with the actual torque and hook load at the wellhead, respectively. Based on the error, the trial value at the bit is adjusted for iteration.  
Where \(F_{i}\) is the axial force at the lower end of the unit (N); \(F_{i-1}\) is the axial force at the upper end of the unit (N); \(q_{\rm m}\) is the floating weight of the unit in the drilling fluid (N/m); \(L_{i}\) is the length of the \(i\)-th unit (m); \(\beta_{i}\) is the overall angle change rate of the \(i\)-th unit (rad); \(T_{i}\) is the torque at the lower end of the unit (N-m); and \(T_{i-1}\) is the torque at the upper end of the unit (N-m).  
\[F_{i-1}=F_{i}+\frac{q_{\rm m}L_{i}\cos\overline{a}_{i}-\mu_{i}|N_{i}|}{\cos \frac{\beta_{i}}{2}\cos\frac{\Delta{\kappa_{i}}}{2}} \tag{22}\]  
\[T_{i-1}=T_{i}+\mu_{i}r_{i}|N_{i}| \tag{23}\]  
Figure 1: Force analysis for drill string unit [42].  
\[\beta_{i}=\cos^{-1}(\cos\alpha_{i-1}\cos\alpha_{i}+\sin\alpha_{i-1}\sin\alpha_{i} \cos\Delta\phi) \tag{24}\]  
\[N_{i}=\sqrt{\left(F_{i}\Delta\phi\sin\overline{\alpha}_{i}\right)^{2}+\left(F_{ i}\Delta\alpha_{i}+q_{\rm m}L_{i}\sin\overline{\alpha}_{i}\right)^{2}} \tag{25}\]  
where \(\overline{\alpha}_{i}\) is the average of the well slope angles at the ends of the \(i\)-th unit (rad); \(\mu_{i}\) is the friction coefficient of the \(i\)-th unit; \(N_{i}\) is the radial support force on the \(i\)-th unit (N); \(\Delta\alpha_{i}\) is the incremental well slope angle of the \(i\)-th unit (rad); \(\mu_{t}\) is the circumferential friction coefficient; \(r_{i}\) is the radius of curvature of the \(i\)-th unit (m); \(\alpha_{i}\) is the well slope angle at the lower end of the unit (rad); \(\alpha_{i-1}\) is the well slope angle at the upper end of the unit (rad); \(\Delta\phi\) is the azimuth increment (rad); and \(\Delta\alpha_{i}\) is the incremental well slope angle of the \(i\)-th unit (rad).