Nonlinear Characteristics of Axial-Torsional Coupled Vibrations in Deep Well Drill Strings Affected by Positive Displacement Motor

In deep-well drilling, bottom-hole conditions are complex. The drill string, which has a high length-to-diameter ratio, rotates at high speed, driven by power tools such as the top drive and a positive-displacement motor (PDM), while in contact with the rock. These intermittent contacts and the associated friction lead to coupled axial–torsional vibrations, resulting in more complex motion responses of the drill string and exacerbating drill-bit and drill-string failures. Previous studies on axial–torsional coupling vibrations of drill strings have overlooked the influence of PDM, and the effects of PDM on axial–torsional coupling vibrations in deep wells therefore remain unclear. This paper addresses this issue by developing a deep-well drill-string axial–torsional coupling vibration model and deriving the corresponding differential equations of motion. The model describes the frictional contact and cutting processes during interaction between the drill bit and the rock. Coupling between axial and torsional motion is achieved through the frictional torque and axial force at the drill bit. The model also incorporates the effects of the PDM, taking into account factors such as bit-nozzle pressure drop, tool dimensions, and drilling-fluid flow rate when calculating the PDM output parameters. Additionally, evaluation indices for axial and torsional vibrations are proposed. Field-data validation demonstrates that the developed dynamic model can accurately reproduce the motion characteristics of the drill string. Based on the new axial–torsional coupling vibration model, the nonlinear vibration characteristics of axial–torsional coupling and the effects of PDM speed, torque, and axial force on these vibrations are analyzed. A case study on a deep well in the Shunbei block of the Tarim Basin clarifies the characteristics and influencing factors of axial–torsional coupling vibrations in deep-well drill strings. The axial–torsional coupled-vibration model proposed in this study addresses the limitations of previous research that overlooked the PDM. Considering that PDM has become a standard configuration in deep and ultra-deep-well drilling, this model holds significant practical value for studying drill-string vibrations and for further clarifying the common characteristics of axial–torsional coupled vibrations.