# Mechanical Behaviors of Bottom Hole Assembly with Bent-Housing Positive Displacement Motor Under Rotary Drilling

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## Abstract

Abundant drilling activities had confirmed that the fully rotary drilling can improve the rate of penetration effectively. However, the fully rotary drilling has brought some challenges for the trajectory control ability of the bottom hole assembly (BHA). One of the reasons is the effect of drill-string rotation was ignored in the existing methods, where the bent-housing positive displacement motor (PDM) was regarded as the prebending beam. According to the D’Alembert principle, the dynamical centrifugal force, generated by drill-string rotation, was equivalent to a quasi-static problem. The mechanical model of BHA with bent-housing PDM was established based on the Timoshenko beam theory. The calculated formula of bit side force (BSF) and resultant steering force (RSF) was deduced. The influences of inclination, rotational speed of drill-string, bend angle, eccentricity, stabilizer, weight on bit (WOB) and elbow position on the average BSF and RSF were investigated. The results show that the rotational speed of drill-string has a significant influence on the steering ability. The average BSF increases with the rotational speed of drill-string, while the RSF increases firstly and decreases subsequently. The controlling factor is the transverse component of drill-string gravity in a low rotational speed, while it is the centrifugal force in a high rotational speed. The BSF climbs up and then declines with WOB and rotational speed of drill-string. When the rotational speed of drill-string exceeds 100 RPM or WOB is higher than 80 kN, the BSF will decrease, resulting in a decline in angle buildup. The present method can be utilized to optimize the drilling parameters, BHA configuration and structure of bent-housing PDM.

## Keywords

Rotary drilling Bottom hole assembly Positive displacement motor Mechanical behavior Steering force## Abbreviations

- ABPV
Arrow back pressure valve

- AKO
Adjustable kick-off

- BHA
Bottom hole assembly

- BSF
Bit side force

- CA
Closured azimuth

- CD
Closured displacement

- DC
Drill collar

- DKF
Double kick-off

- DTU
Double-tilted universal

- FAB
Fixed angle build

- HWDP
Heavy weight drill pipe

- MD
Measured depth

- MWD
Measurement while drilling

- NMDC
Non-magnetic drill collar

- NPT
Nonproductive time

- PDM
Positive displacement motor

- PDC
Polycrystalline diamond compact

- ROP
Rate of penetration

- RPM
Revolutions per minute

- RSF
Resultant steering force

- RSDS
Rotary steerable drilling system

- TVD
True vertical depth

- WOB
Weight on bit

## List of symbols

- \(\gamma \)
Bending angle \(({^{\circ }})\)

- \(\alpha \)
Inclination angle of well \(({^{\circ }})\)

- \(\theta \)
The angle between the drill-string and borehole axis \(({^{\circ }})\)

*c*The length between the elbow point and first stabilizer (m)

*e*Eccentricity (mm)

*F*Equivalent concentrated force of bent-housing PDM (N)

*b*The distance between the load

*F*and first stabilizer (m)*E*Young’s modulus (GPa)

- \(F_\mathrm{cen}\)
Concentrated centrifugal force (N)

*y*Deflection of drill-string (mm)

- \(P_{1}\)
Axial force of the first stabilizer (N)

- \(P_{2}\)
Axial force of the second stabilizer (N)

- \(F_{1}\), \(F_{2}\)
Support reaction of the 1st and 2nd stabilizer (N)

- \(m_{1}\), \(m_{2}\), \(m_{3}\)
Linear density of the 1st, 2nd and 3rd span (kg/m)

*w*Rotational speed (rpm)

- \(l_{1}\), \(l_{2}\), \(l_{3}\)
Length of the 1st, 2nd and 3rd span (m)

- \(P_{b}\)
WOB (kN)

- \(q_{2}\)
Uniform transverse load of drill-string (N/m)

*M*(*x*)Bending moment of any position \((\hbox {N}\cdot \hbox {m})\)

- \(X(u_{i})\)
Transcendental function about \(u_{i}\)

- \(Y(u_{i})\)
Transcendental function about \(u_{i}\)

- \(Z(u_{i})\)
Transcendental function about \(u_{i}\)

- \(u_{i}\)
Stability coefficient of drill-string

*i*The order of the span

- \(M_{0}\), \(M_{3}\)
Bending moment on the drill bit and upper tangent point (N \(\cdot \) m)

- \(M_{1}\), \(M_{2}\)
Internal moment acted on the 1st and 2nd stabilizer (N \(\cdot \) m)

- \(D_{w}, \, D_{b}\),
Diameter of the wellbore and drill bit (mm)

- \(D_{s1}\), \(D_{s2}\)
Diameter of the 1st and 2nd stabilizer (mm)

- \(D_\mathrm{dc}\)
Diameter of the drill collar (mm)

- \(e_{0}\), \(e_{3}\)
Eccentricity locates at the drill bit and drill collar (mm)

- \(e_{1}\), \(e_{2}\)
Eccentricity locates at the 1st and 2nd stabilizer (mm)

- \(I_{1}\), \(I_{2}\), \(I_{3}\)
Inertia moment of the 1st, 2nd and 3rd span \((\hbox {m}^{4})\)

- \(P_{a}\)
BSF value (N)

- \(\beta \)
The angle between BSF and

*n*-direction \(({^{\circ }})\)- \(M_{o1}^n , M_{o1}^v \)
Component of bending moment (N \(\cdot \) m)

- \(P_a ^{n}, P_a^v \)
Component of BSF (N)

- \(\bar{{P}}_a \)
Average BSF (N)

- \(F_{a}\)
RSF value (N)

- \(L_{1}\)
The distance between bent-sub and first stabilizer (m)

- \(L_{2}\)
The distance between bent-sub and second stabilizer (m)

- \(R_{1}\)
The radius of drill-string between bent-sub and first stabilizer (m)

- \(R_{2}\)
The radius of drill-string between bent-sub and second stabilizer (m)

- \(d_{1}\)
Maximum deflection between bent-sub and first stabilizer (m)

- \(d_{2}\)
Maximum deflection between bent-sub and second stabilizer (m)

*N*Node number of a circle

- \(\theta _{1}\)
The angle of the segment between bent-sub and first stabilizer \(({^{\circ }})\)

- \(\theta _{2}\)
The angle of the segment between bent-sub and second stabilizer \(({^{\circ }})\)

*k*Buildup rate (\({^{\circ }}\)/30m)

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## Notes

### Acknowledgements

This work was financially supported by the Major National Science and Technology Projects of China (Grant No. 2016ZX05022-001), the 973 Program of China (Grant No. 2013CB228003), the Science and Technology Support Program of Sichuan Province (Grant No. 2015SZ0003), and the Funds of State Key Laboratory of Oil and Gas Reservoir Geology and Exploitation of Southwest Petroleum University (Grant Nos. PLN201611 and G201604).

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