Bit-sticking Phenomena in a Multidegree-of-freedom Controlled Drillstring
ϕr ..
ϕ1 ..
ϕj ..
ϕp ..
ϕl ..
ϕb ..
= – (ϕr
ct Jr
= – (2ϕ1
ct Jp
= – (2ϕj
ct Jp
= – (ϕp
ct Jp
= – (ϕl
ctl Jl
= – (ϕb
Jb ctb
. .. – ϕ1 ) – (ϕr
kt Jr
– ϕr .. – ϕ2
. .. – ϕj+1
– ϕ1
) – (2ϕ1 ...
kt Jp
– ϕj-1
. .. . – ϕp-1
) – (ϕp
kt Jp
. .. . – ϕp
) – (ϕl
ktl Jl
.. – ϕl) – (ϕb
ktb Jb
– ϕp – ϕl
) – (ϕl .
ctb Jl
) –
Tb(ϕb) Jb
– ϕp-1 ) – (ϕp
ctl Jp
– ϕb
) – (2ϕj ...
kt Jp
) +
Tm – Tar (ϕr) Jr
. – ϕr – ϕ2 – ϕj+1 ) – ϕj-1 – ϕl ) ) – (ϕp
ktl Jp
) – (ϕl
ktb Jl
– ϕl – ϕb ) (1)
with ϕr, ϕj (with j=1,…,p), ϕl; ϕb the angular displacements of the top-rotary system, the drill pipes, the drill collars and the bit,
respectively. • ϕr,• ϕj, (with j=l,…,p), • ϕl,• ϕb are the angular velocities of the top-rotary system, the drill pipes, the drill collars and the bit,
respectively. Tm is the drive torque coming from the electrical motor at the surface. It is assumed that arbitrary torques can be applied without taking into account the actuator dynamics generating this torque, then
Tm=u with u>0 one of the system control inputs. The other control input is the Wob. Tar is the viscous damping torque associated with Jr, and corresponds to the lubrication of the mechanical elements of the
top-rotary system with Tar=Cr • coefficient. Tb ( •
ϕb)=Cb• ϕb+Tfb( • ϕr, and cr the viscous damping ϕb) is the TOB such that: Tfb (ϕb ..) W obRb μcb + (μsb – μcb) exp – ϒb ) = sign(ϕb νf (2) with Rb the bit radius; µsb ,µcb with values in (0,1) the static and Coulomb
friction coefficients associated with Jb; 00. In addition, the Coulomb and static friction torque is Tcb and Tsb, respectively, with
Tcb=WobRbµcb sign(φb ;Tsb=WobRbµsb
.. . ) = φb
/|φb | if φb . . The sign function is considered as: = 0, sign(φb . ) C –1,1 if φb .
10 12
= 0. (3)
From a mathematical viewpoint, this sign function is the discontinuity of the system and the cause of stick-slip oscillations.
A particular case of the n-DOF model with 3 DOF presented in 31 captures the main bit-sticking phenomena and drill pipe oscillations:
φr = – (φr .. φr = (φr .. φr = (φp ..
ctb Jb
ct Jp
ct Jr
..
.. – φp
.. – φp ) – (φr
kt Jr
) + (φr
kt Jp
– φb ) + (φp
ktb Jb
– φp – φp ) + ) – – φb Tm
ctb Jp
– Tar Jr
(φp ) – Tb (φr . .. (φb)/Jb . (4)
where Jp is the inertia of the drill pipes. The two bit-sticking situations for this system are shown in Figure 3. The following parameters were used:
Jr = 2122 kg m2 kt , Jb = 471,9698 kg m2 = 698,06314 N m/rad, ktb , Jp = 750 kg m2 = 907,482089 N m/rad, EXPLORATION & PRODUCTION – VOLUME 8 ISSUE 2 – φb ) ) –
ktb Jp
(φp – φb ) ct
= 139,612629 N m s/rad, ctb cr = 425 N m s/rad, cb μcb = 0,5 μcb
= 0,8 υf = 1 γb = 181,49641 N m s/rad
= 50 N m s/rad = 0,9 Rb
= 0,155575 m (5)
As expected, if the Wob is high enough, the bit is stuck permanently. If the torque u is not big enough to overcome the effects of the
bit-rock friction and the Wob is not small enough, the BHA exhibits stick-slip behaviour.
73
2 4 6 8
0 0 204060 Wob (kN)
A. Weight of bit (Wob), desired rotary velocity (Ω) for a fixed torque at the top-rotary system (u) = 8,138Nm; B. u, Ω for a fixed Wob = 74,386N; C. Wob, u at which a Hopf bifurcation (HB) can appear. See reference 31 for more details. Graphics were obtained using XPPAUT.52
HB2 branch C 14 ϕb . B
1 2 3 4 5 6 7 8 9
0 7,000 ) Figure 4: Bifurcation Diagrams with the Hopf Bifurcations of the System A
1 2 3 4 5 6 7 8 9
0 55 60
HB1 LP HB2 65 Periodic orbits branch 70 75 Wob (kN) Stable branch Unstable branch 80 85 90
HB1 LP HB2 7,500 8,000 8,500 u (Nm) Periodic orbits branch Stable branch 9,000 9,500 10,000
HB1 branch
80
100
120
u (kNm)
Ω (radians per second)
Ω (radians per second)
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