## 2 Data and Methods
### The \"Hard-String\" Drag Calculation Method  
The classic drilling string drag torque models are the \"soft-string\" model and the \"hard-string\" model. The \"soft-string\" model used earlier is simple and has certain accuracy, but it does not take into account the rigidity of the drilling string and is suitable for straight wells and wells with small slope angles. The \"hard-string\" model makes up for this shortcoming. The intelligent analysis method of drag and torque proposed by [PERSON] [29] uses the hard-string model to calculate the drag and has achieved good results. Therefore, this study adopts [PERSON]'s \"hard-string\" model (Equations (5)-(13)) [30] for the real-time calculation of the drilling string drag.  
The differential equation for the force of the overall drill string is:  
\[\frac{d(-F)}{ds}=-EIk_{b}\frac{dk_{b}}{ds}-qcosa\mp\mu_{1}\cdot\mu_{t} \tag{5}\]  
\[\frac{dM}{ds}=\mu_{1}\cdot\mu_{t}\cdot\frac{D_{o}}{2} \tag{6}\]  
where in the \"\(\mp\)\", the \"\(-\)\" stands for pulling up the string, the \"+\" stands for pressing down the string; \(F\) is the axial pressure on the drill string N, s is the depth of the well, m; \(q\) is the gravity of the drill string per unit length, N/m; \(a\) is the inclination angle of the well, rad; \(EI\) is the bending stiffness of the drill string, N m\({}^{2}\); \(n_{t}\) is the contact drag force between the drill string and the well wall, N/m; \(\mu_{1}\) is the axial drag coefficient of the drill string; \(k_{b}\) is the wellbore axis curvature, m\({}^{-1}\).  
The borehole curvature \(k_{b}\) is calculated as:  
\[K_{b}=\sqrt{\left(\frac{d\alpha}{ds}\right)^{2}+sin^{2}\alpha\left(\frac{d \varphi}{ds}\right)^{2}} \tag{7}\]  
where \(\varphi\) is the azimuth angle of the well.  
The contact drag force \(n_{t}\) on the drill column is calculated by:  
\[n_{t}=\sqrt{\frac{A^{2}+B^{2}}{1+\mu_{1}^{2}}} \tag{8}\]  
Figure 4: BP neural network algorithm.  
\[A=EI\frac{d^{2}k_{b}}{ds^{2}}+k_{b}F-k_{n}(-k_{b}M_{T}+EI\cdot k_{b}k_{n})+\frac{q} {k_{b}}\frac{d\alpha}{ds}sin\alpha \tag{9}\]  
\[B=\frac{d}{ds}(-k_{b}M_{T}+EI\cdot k_{b}k_{n})+EI\cdot k_{n}\frac{dk_{b}}{ds}- \frac{q}{k_{b}}\frac{d\varphi}{ds}sin^{2}\alpha \tag{10}\]  
\[k_{n}=\frac{sin\alpha}{k_{b}^{2}}\left(\frac{d\alpha}{ds}\frac{d^{2}\varphi}{ds ^{2}}-\frac{d\varphi}{ds}\frac{d^{2}\alpha}{ds^{2}}\right)+cos\alpha\cdot \left(\frac{1}{k_{b}^{2}}\left(\frac{d\alpha}{ds}\right)^{2}+1\right)\frac{d \varphi}{ds} \tag{11}\]  
We use the finite difference method to solve for the forces on the entire drill string. The drill string can be divided into several small drill string sections, as shown in Figure 5. The forces on the small drill string sections can be obtained by the following equations, with the pressure as positive:  
\[F_{i}=F_{i+1}+\frac{1}{2}EI_{i}(K_{bi}^{2}-K_{bi+1}^{2})+(-q_{i}cos\alpha_{i} \mp\mu_{1}\cdot\mu_{ii})\Delta s_{i} \tag{12}\]  
\[M_{Ti}=\frac{1}{2}\mu_{1}\cdot n_{ti}\cdot D_{bi}\Delta s_{i}+M_{Ti+1} \tag{13}\]  
Among them, \(F_{i}\) and \(F_{i+1}\) are the axial force at the end of the i-th drill string near the ground and near the bit, respectively, \(N\); \(M_{Ti}\) and \(M_{Ti+1}\) are the torque at both ends of the i-th drill string, respectively, in N-m; \(K_{bi}^{2}\) and \(K_{bi+1}^{2}\) are the borehole curvature at both ends of the i-th section, respectively, m-1; \(q_{i}\), \(EI_{i}\), \(n_{ti}\), \(\Delta s_{i}\), and \(D_{bi}\) are the line weight (N/m) and bending stiffness (N-m\({}^{2}\)) of the drill string in the \(i\)-th section, respectively, and the contact with the borehole wall force (N/m), length (m), and outer diameter (m) of small drill string sections.