Delay System Modeling of Rotary Drilling Vibrations

Abstract

Vibrations in rotary drilling systems are oscillations occurring without being intentionally provoked. They often have detrimental effects on the system performance and are important source of economic losses; drill bit wear, pipes disconnection, borehole disruption and prolonged drilling time. By this chapter, we provide an improved modeling for the rotary drilling system . Among others, the proposed modeling takes into account; the infinite dimensional settings of problem as well as the nonlinear interconnected dynamics.

Appendix: Notations table

Variable	Signification
$L_p$	Pipe length
$L_b$	Bor Hole Assemble length
L	$= L_p + L_b$
$U_p$, $U_b$	Pipe, drill collar traction/compression deformation
$\varPhi _p$, $\varPhi _b$	Pipe, drill collar torsional deformation
$\varepsilon _{U_p}^i$, $\varepsilon _{\varPhi _p}^i$	Internal damping coefficients
$\gamma _{U_p}^v$, $\gamma _{\varPhi _p}^v$	Viscous damping coefficients
$\rho$	Steel density
E, G	Young’s, shear modulus of drillstring steel
$A_p$, $J_p$	Cross-section and polar inertia moment of one pipe section
$A_b$, $J_b$	Cross-section and polar inertia moment of one drill collar section
$r_{po}$, $r_{pi}$	Outer, inner pipe radius
$r_{bo}$, $r_{bi}$	Outer, inner drill collar radius
$\varPsi _{\varPhi d}$, $\varPsi _{U d}$	D component of rotary table (torsion) induction motor flux
$L_{\varPhi m}$, $L_{U m}$	Torsion, traction/compression induction motor mutual inductance
$I_{\varPhi d}$, $I_{\varPhi q}$	D, Q component of stator current in torsion induction motor
$I_{U d}$, $I_{U q}$	D, Q component of stator current in traction/compression induction motor
$J_{top}$	Top drive inertia
$u_T$	Rotary table motor torque, taken as a control input
H	Force acting in the top hole device
$\zeta _{rg_1}$	Accounts for vibrations in all drilling rig elements except the drilling string, BHA, cables, drawworks, travelling and crown blocks
$\zeta _{rg_2}$	Accounts for elasticity in cables, crown and travelling blocks
$k_{rg_{01}}\zeta _{rg_{ini}}$	Ground reaction force
$u_F (t)$	$= k_{rg_{01}} (\zeta _{rg_1}(t) -\zeta _{rg_0}(t) )$, tension force in the cable at the drawworks level, taken as a control input
$M_{rg_i}$, $\gamma _{rg_1}$, $k_{rg_{ij}}$	Equivalent masses, damping coefficients and stiffness coefficients
$M_{top}$	Top drive mass
$U_b$, $\varPhi _b$	axial, torsional vibrations
${T_{bit}}$	Bit reaction torque
$M_{bit}$	Bit’s mass
$W_{bit}(t)$	Reaction force at the bit
$T_c$, $W_c$	Bottom hole cutting torque and force
$T_f$, $W_f$	Bottom hole friction torque and force
a	Bit radius
l	Length of the wearflat
$\sigma$	Contact stress
$\gamma$	accounts for the distribution and orientation of the frictional forces acting at the wearflat/rock interface
$\mu$	Ratio between the horizontal and the vertical components of the frictional force
$V_b$	$=(\partial _t U_p,\partial _t \varPhi _p)$
$\text {sgn}(V_b)$	designate the orientation of $V_b$ with respect to the horizontal plane
$\mathscr {F}$	Adimensional friction function
d	Depth of cut
$\varepsilon$	Intrinsic specific energy
$\zeta$	Ratio of the vertical to the horizontal force for a sharp cutter
n	Bit blade number