Acknowledgements

Qslip Mud slip flow through the PDM (gal/min)

RPMs bit rotary speed provided by surface (rpm)

ROP rate of penetration (ft/h) RPM bit rotating speed (rpm)

158 Drilling

T torque at bit (ft-lbf) ΔT torque loss (ft-lbf)

RPMideal ideal PDM rotary speed (rpm)

RPMm PDM output rotary speed (rpm)

Tideal ideal PDM output torque (ft-lbf)

Tm PDM output torque (ft-lbf)

Ts torque at bit provided by surface (ft-lbf)

UCS unconfined compressive strength (psi) V volume of rock drilled in one hour (ft-in<sup>2</sup>

WOB weight on bit of surface measurement (lbf)

WOBb bottom hole actual weight on bit (lbf)

y contact semi-width (in)

μ viscosity of mud (cp) γ well inclination (rad)

α helix angle of seal line (degree)

β coefficient of hydraulic horsepower

μ<sup>b</sup> bit-specific coefficient of sliding friction

δ clearance of the slip passage (in)

μ<sup>s</sup> coefficient of friction of drill string

Δγ additional well inclination (rad)

Γ<sup>i</sup> configuration correction factor

η efficiency of PDM

γ<sup>b</sup> inclination of the bottom hole (rad) ϕ rock internal angle of friction (degree)

Wt total mechanical work done by the bit in one hour (ft-lbf)

WV mechanical work required to break the rock drilled in one hour (ft-lbf)

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This work was financially supported by Science Foundation of China University of Petroleum, Beijing (No.2462017YJRC050), National Key Research and Development Project (Grant numbers: 2017ZX05009-003; 2016YFC0303303), National Natural Science Foundation of China (Grant numbers: 51434009; 51521063; 51774301).
