## 2 Analysis of the Reverse Torque and Vibration in the Mud Pump
### Working Principle and Structure of the Mud Pump Module  
The riserless mud recovery (RMR) closed-loop drilling system generally consists of a drilling unit, a mud return system unit, a mud treatment unit, and a power monitoring unit.  
Figure 1: Analysis flowchart.  
The operational mode of the RMR closed-loop system is shown in Figure 2. The mud pump module is located within the mud return system unit and is installed using a mid-hanging configuration on the mud-lifting pipeline. The main function of the mud pump module is to transport the cuttings generated by the suction module through the mud return pipeline to the surface mud treatment equipment. With the suction action of the subsea mud pump, drilling fluid is drawn from the lower outlet of the suction module, enters the subsea mud pump through the subsea drilling fluid delivery pipeline, and is then lifted to the drilling vessel for processing and subsequent reuse.  
The mud pump module is used to circulate a mixture of drilling fluid, cuttings, and even gas. The external view of the mud pump module is shown in Figure 3. The mud pump is powered by the platform and all equipment is only subjected to the hydrostatic pressure of seawater. The mud pump module is a critical component of the riserless mud recovery closed-loop system. It controls the pressure balance inside the suction module using the subsea pump and facilitates the return of drilling mud. The subsea pump module is fixed on the mud return pipeline and not only needs to ensure the reliability of the entire pipeline structure but also to maintain the smooth flow of the mud return pipeline. To prevent a collision between the lower subsea pump module and the suction module during operation, a subsea anchor is used to secure the lower end of the mud return pipeline, and a tension force is applied to restrict the horizontal movement of the mud return pipeline.  
The lifting module of the mud pump unit mainly consists of drill pipe joints, corrective blocks, ball valves, centrifugal pumps, a four-way connector, a skid assembly, a ball valve control device, a skid, a trolley adapter, a supporting mechanism, and other components. The skid is the foundation of the mud pump module and serves mainly for connection and support. During drilling operation, the skid is installed on the mud lifting pipeline using a center-hanging arrangement, providing support for the internal components of the mud pump module. The skid remains stationary during the mud-lifting process and does not undergo significant vibration. The centrifugal pump used in the module is a multistage centrifugal pump, which increases the power of the pump module, enabling it to meet the requirements of deepwater operation. The mud return riser provides a conduit for drilling mud to flow from the seabed wellhead to the drilling rig. It also serves to fix and support two lifting pump module units, bearing the weight of the seabed anchor and resisting the force applied by sea currents and the movement of the drilling rig.  
Figure 2: Diagram of the riserless mud recovery closed-loop operating mode [37].  
Validation of the Theoretical Calculations and the Numerical Simulation Results for the Reverse Torque of the Single-Stage Mud Pump