## 2 Analysis of Factors Affecting the Stability of the Skid-and-Mud-Return Line
### Analysis of Factors Affecting the Stability of the Skid-and-Mud-Return Line  
With the suction action of the subsea mud lift pump, the drilling fluid is drawn from the lower outlet of the suction module. The drilling fluid is returned to the subsea mud lift pump through the subsea mud return line. It is then transported to the drilling vessel through the upper mud return pipeline and recycled after treatment. Therefore, there are three main factors affecting the stability of the skid-and-mud-return line:  
1. The effects of drilling vessels on the skid-and-mud-return line due to wind and sea currents on the sea surface;
2. The effects of sea current generated by the flow of seawater on the skid-and-mud-return line;
3. The effects of the reverse torque and vibration generated by the operation of the mud lift pump on the skid-and-mud-return line. For Factor 1, according to \"Mooring Equipment Guidelines\" released by Oil Companies International Marine Forum (OCIMF) [35], the specification provides wind load calculation formulas for very large oil tankers, as shown in the following equations, where \(F_{xw}\) is the longitudinal (bow-stern direction) wind load, \(F_{yw}\) is the transverse (port-starboard direction) wind load, \(C_{xw}\) is the longitudinal wind resistance coefficient, \(C_{yw}\) is the trans  
Figure 2: The model of the skid-and-mud return pipeline: (a) mechanical model and (b) analytical model. verse wind resistance coefficient, \(\rho_{w}\) is air density, \(V_{w}\) is wind speed, \(A_{T}\) is the transverse windward area, and \(A_{L}\) is the transverse windward area. \[ \begin{array}{l}F_{xw}=\frac{1}{2}C_{xw}\rho_{w}{V_{w}}^{2}A_{T}\\ F_{yw}=\frac{1}{2}C_{yw}\rho_{w}{V_{w}}^{2}A_{L}\end{array}  \tag{1}\]  
Moreover, the calculation formula of water flow load on ultra-large oil tankers is shown as follows, where \(F_{xc}\) is the longitudinal (bow-stern direction) water flow load, \(F_{yc}\) is the transverse (port-stapoard direction) water flow load, \(C_{xc}\) is the longitudinal flow resistance coefficient, \(C_{yc}\) is the transverse flow resistance coefficient, \(\rho_{c}\) is seawater density, \(V_{c}\) is flow velocity, \(L_{BP}\) is vertical line spacing, \(T\) is the average draft depth. \[ \begin{array}{l}F_{xc}=\frac{1}{2}C_{xc}\rho_{c}{V_{c}}^{2}L_{BP}T\\ F_{yc}=\frac{1}{2}C_{yc}\rho_{c}{V_{c}}^{2}L_{BP}T\end{array}  \tag{2}\]  
When these calculation formulas are used, it is important to refer to the corresponding charts in the OCIMF guidelines. The wind resistance coefficients and flow resistance coefficients for different wind angles and water flow angles can be obtained. Then, these coefficients are substituted into the calculation formulas to obtain the longitudinal wind load (\(F_{xw}\)), transverse wind load (\(F_{yw}\)), longitudinal water flow load (\(F_{xc}\)), and transverse water flow load (\(F_{yc}\)). It can be observed that the OCIMF calculation method treats all force structures as a whole. For Factor 2, calculations can be made based on the wave-induced load theory, which can be divided into the following two methods. (1) Flow-induced drag load  
When a steady and uniform water flow passes around a circular structure, the force on the circular structure in the direction of flow is called flow-induced drag force. Flow-induced drag force is generally composed of two force components: friction drag force and pressure drag force. Friction drag force is generated due to the viscosity of the fluid, which forms a boundary layer on the structure's surface. The sum of the frictional shear stress is the friction drag force on the circular structure. Pressure drag force is caused by the separation of the boundary layer at a certain point on the structure's surface. The sum of the normal pressure stress is the pressure drag force on the cylinder.