# Review on the Influence of Complex Stratum on the Drilling Trajectory of the Drilling Robot
## 4 Stratum Force Theory and Method of Complex Coal Seams  
### Quantitative Analysis of Formation Force  
On the basis of qualitative analysis of the influence of complex coal seams considering geological factors on drilling trajectory, quantitative analysis of formation forces must be carried out in order to predict and control well trajectory [87]. The formation force refers to the force effect of the deviation of the formation on the bit, which includes the formation variable well deviation force and the formation variable azimuth force. The theory and method of formation force is to quantitatively analyze and calculate the formation force by comprehensively considering the influence of formation anisotropy and bit anisotropy, formation dip and strike, drilling direction and weight on bit and other factors. In directional drilling, the drilling pressure PB is not generally located in the formation profile, and there is an angle between the drilling direction and the formation updip direction, so the formation force has both the variable well deviation effect and the variable azimuth effect, that is, the formation variable well deviation force Fa and the formation variable azimuth force \(F\)\(\varphi\)[88]. The drilling pressure cone shown in Figure 15 has a half apex angle of well deviation \(\alpha\). The drilling pressure \(\underset{\text{P}_{\text{B}}}{\rightarrow}\) is a generatrix of the cone, and the inclined angle between the plumb plane (P plane) and the formation profile is \(\Delta\varphi\). Then, \(\underset{\text{P}_{\text{B}}}{\rightarrow}\underset{\text{P}_{\text{B}}}{ \rightarrow}\) + \(\underset{\text{P}_{\text{B}}}{\rightarrow}\). The formation force caused by the component \(\underset{\text{P}_{\text{B}}}{\rightarrow}\) is as follows:  
\[\underset{\text{P}_{\text{F}}}{\rightarrow}=\frac{\text{Htg}(\beta-\alpha) \text{cos}\,\alpha/\text{cos}\,\alpha^{\prime}\text{P}_{\text{B}}}{1-\text{ Htg}^{2}(\beta-\alpha)} \tag{5}\]  
in which \(\alpha^{\prime}=\text{arctg}(\text{tg}\alpha\cdot\text{cos}\,\Delta\varphi)\), \(\Delta\phi=\phi_{\text{W}}-\phi_{\text{S}}\). \(\phi_{\text{W}}\) is the azimuth of the well bore, \(\phi_{\text{S}}\) is the azimuth of the formation tilt up, H is the comprehensive cutting anisotropy index, and \(\beta\) is the dip angle of the stratum. By further decomposing \(\underset{\text{F}_{\text{F}}}{\rightarrow}\) into \(\underset{\text{F}_{\text{F}}}{\rightarrow}\underset{\text{F}_{\text{A}}}{ \rightarrow}\) + \(\underset{\text{F}_{\text{\phi}}}{\rightarrow}\), the follows can be deduced:  
\[\text{F}_{\alpha}=\text{F}_{\text{f}}^{\prime}\text{cos}\,\Delta\phi^{\prime },\text{F}_{\Phi  
From this formula, it is easy to derive [PERSON]'s full angle formula. Because of \(\alpha^{\prime}<\alpha\), \(\Delta\Phi^{\prime}\approx\Delta\phi\) can be deduced when \(\alpha\) is small. Then F\({}_{\alpha}\) and F\({}_{\phi}\) can be simplified to the following approximate formula:  
\[\mathrm{F}_{\alpha}=\mathrm{F}_{\mathrm{f}}^{\prime}\mathrm{cos}\,\Delta\phi \tag{8}\]  
\[\mathrm{F}_{\phi}=-\mathrm{F}_{\mathrm{f}}^{\prime}\mathrm{sin}\,\Delta\phi \tag{9}\]  
The symbol of F\({}_{\mathrm{f}}^{\prime}\) specified as pointing upward is positive and pointing downward is negative. The relationship between the symbols, properties and quadrants of the formation oblique force F\({}_{\alpha}\) and the formation azimuthal force F\({}_{\phi}\) is shown in Tables 1 and 2 [88]. Figure 16: Analysis of formation variable well deviation force F\({}_{\alpha}\) and formation variable azimuth force F\({}_{\phi}\). Figure 15: Breakdown of drilling pressure cone and drilling pressure. To sum up, it can be seen that the formation variable well deviation force can be the deflecting force or the declination force, and the formation variable azimuth force can be the the increasing azimuth force or the decreasing azimuth force.