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Control de Vibraciones en
Sarta de Perforacion
Miguel Angel Corchero, Francisco Rodrıguez RubioEn colaboracion con Carlos Canudas-de-Wit
Dpto. Ingenierıa de Sistemas y AutomaticaEscuela Superior de Ingenieros
Universidad de Sevilla
JIC’06 - Pamplona - Marzo - 2006
D-OSKIL: A New Me hanism for Suppressing Sti k-Slip
OUTLINE
� Introdu tion.� System Des ription.� System Model.� Velo ity Control.� Sti k-Slip Predi tion based on DF-Analysis.� Drilling Os illation Killer Me hanism.� Con lusions and Future Work.
CDC-ECC'05 - Sevilla - De ember 2005 2
D-OSKIL: A New Me hanism for Suppressing Sti k-Slip
INTRODUCTION� Ri h dynami al behavior.� Display sti k-slip self-ex itedos illations.� Redu ing os illations yield substantial ost savings.� Paper fo used in torsional sti k-slipos illations.Rotatory speed of utting devi e � [0; 6!d℄� Phenomenon mainly due to fri tion.
CDC-ECC'05 - Sevilla - De ember 2005 3
D-OSKIL: A New Me hanism for Suppressing Sti k-Slip
Contributions On Reje ting Os illations� Varying di�erent drilling parameters (rotary speed, WoB, mud, ...)[Sananikone 1992, Pavone 1994 ℄.{ Not formal analysis.{ Good performan e results.� New ontrols to ompensate sti k-slip.{ Soft Torque Rotatory System [Halsey 1988 ℄.{ Top vibration absorber [Jansen 1995 ℄.{ Linear H1 ontrol [Serrarens 1998 ℄.{ ...� Use WoB as a new ontrol variable.CDC-ECC'05 - Sevilla - De ember 2005 4
D-OSKIL: A New Me hanism for Suppressing Sti k-Slip
SYSTEM DESCRIPTION� Rotatory Table moved by DCmotor at onstant velo ity.� Bit oupled by a elasti shaft alledDrillstring.{ Upper part: Sequen e of tubes.{ Lower part: BHA.� Mud pumped to lubri atethe onta t.� Total normal for e alled WoB.� Fri tion torque at Bit alled ToB.
motor
rotatory table
drillstring
drillcollar
bit
mud
transmision
CDC-ECC'05 - Sevilla - De ember 2005 5
D-OSKIL: A New Me hanism for Suppressing Sti k-Slip
SYSTEM MODEL
� Lumped parameter model.� Torsional behavior model.� Two degrees of freedom.� Drilling Rig modelled by an elasti shaft.� ToB = normal for e � fri tion.
Jr �'r + ( _'r � _'b) + k('r � 'b) + dr _'r = vJb �'b + ( _'b � _'r) + k('b � 'r) + db _'b = ��(�)uCDC-ECC'05 - Sevilla - De ember 2005 6
D-OSKIL: A New Me hanism for Suppressing Sti k-Slip
Fri tion Torque and State-Spa e modelFri tion at the bit (LuGre Model)� u = ~u+ u0: Total WoB.� u0: Nominal WoB.� ~u: Control signal.� �( _'b; z): Normalized torque.ToB = �( _'b; z) � u�( _'b; z) = �0z + �1 _z_z = _'b � �0 j _'bjg( _'b)z;g( _'b) = �C + (�S � �C)e�( _'b=vs)2
State-Spa e Model
_x = Ax+Bv +H�(x; z)u_z = f(x; z)A = 0BB� 0 1 �1�kJr � (dr+ )Jr JrkJb Jb � ( +db)Jb
1CCA
B = h 0 1Jr 0 iTH = h 0 0 �1Jb iT
CDC-ECC'05 - Sevilla - De ember 2005 7
D-OSKIL: A New Me hanism for Suppressing Sti k-Slip
VELOCITY CONTROL� Rotary velo ity regulated todesired value ( _'r ! !d)� Standard ontrol form[Christoforou 2003 ℄� Gains designed using 2-time-s aleseparation method.� Equations in losed-loop:_x = A lx+B l!d +H l�(x; z)u_z = f(x; z)
Qualitative Behavior
0 5 10 15 20 25 30 35 40 45 500
5
10
15
20
25
30Stick−Slip simulation
Time [s]
Rot
ator
y S
peed
[rad
/s]
surface
downhole
CDC-ECC'05 - Sevilla - De ember 2005 8
D-OSKIL: A New Me hanism for Suppressing Sti k-Slip
Di�erent Behaviors(lo al stability - lo al attra tive limit y le)
0 20 40 60 80 100 120 140 160 180 2000
5
10
15
Time [s]
Rot
ator
y S
peed
[rad
/s]
Unstable limit cycle produced by perturbation on WoB
0 20 40 60 80 100 120 140 160 180 2000
5
10
15
Time [s]
Rot
ator
y S
peed
[rad
/s]
Stick−Slip cycle produced by perturbation on WoB
surfacedownhole
surfacedownhole
Change on WoB
Change on WoB
Equilibrium Domain
Stable Orbit CDC-ECC'05 - Sevilla - De ember 2005 9
D-OSKIL: A New Me hanism for Suppressing Sti k-Slip
STICK-SLIP PREDICTION BASED ON DF-ANALYSIS� Use of SBDF to predi t limit y le.� Steady-state hara teristi of LuGre fri tion model.Harmoni balan e:G(j!0) = �1N1(A0; !0; y0)y0 [1 +G(0)N0(A0; y0)℄ = W (0)!dy0 = W (0)[1 +G(0)N0(A0; y0)℄!d = !dIm fG(j!0)g = 0Re fG(j!0)g = � 1N1(A;!d)CDC-ECC'05 - Sevilla - De ember 2005 10
D-OSKIL: A New Me hanism for Suppressing Sti k-Slip
Des ribing Fun tion Analysis� Points jp0j; jp1j � 1WoB� Two sets of limit y les predi ted.� Control by modifying !n:{ !n upper bounded by noise and power onstraints.{ !n lower by poor performan e.� Control by keeping WoB small:{ High-performan e drilling requires WoBlarge.{ Knowledge of map [!n;WoB℄.CDC-ECC'05 - Sevilla - De ember 2005 11
D-OSKIL: A New Me hanism for Suppressing Sti k-Slip
The Sti k-Slip Os illations� High-performan e drilling intoos illation zone.� System traje tories at equilibrium.� Perturbations may ause limit y les.� Os illation eliminated by de reasingWoB.� Lo al attra tion domain is enlarged.� WoB returns slowly to nominal valueagain. 0 10 20 30 40 50 60 700
1
2
3
4
5
6
7x 10
4 Stable WoB − wn domain
wn [rad/s]
WoB
[N]
Oscillations zone
prediction
simulation
CDC-ECC'05 - Sevilla - De ember 2005 12
D-OSKIL: A New Me hanism for Suppressing Sti k-Slip
Possible System Traje tories
CDC-ECC'05 - Sevilla - De ember 2005 13
D-OSKIL: A New Me hanism for Suppressing Sti k-Slip
DRILLING OSCILLATION KILLER MECHANISM� Output re e ting the sti k-slip.� Filter F (s) extra ts bias and mainos illatory omponent.� Compute RMS value.� Control law for es u(t) 2 [0; u0℄.� Os illation frequen y estimator.
Oil Well
Drillstring
RMS
value
External
disturbances
yu0
Band-pass
filter
~
Drilling Oscillator Killer
D-OSKIL
Frequency
Estimator
u
u
Adaptation
with
saturation
yf = F (s) � Cx = F (s) � _'b 1� _~u = �~u� satu00 (KI�); ~u(0) = 0� = Z t+Tt y2f (�)d� u = u0 + ~u
CDC-ECC'05 - Sevilla - De ember 2005 14
D-OSKIL: A New Me hanism for Suppressing Sti k-Slip
DOSKIL Typi al Operation (Simulation)
0 20 40 60 80 100 120 140 160 180 200−20
0
20
40
y(t)
Operation under OSKIL
0 20 40 60 80 100 120 140 160 180 200−40
−20
0
20
40
y f and
ζ
0 20 40 60 80 100 120 140 160 180 2001
2
3
4x 10
4
u 0 − u
Time (sec)
Activation of theOSKIL mechanism
Os illation vanishes. WoB re overs nominal value.
CDC-ECC'05 - Sevilla - De ember 2005 15
D-OSKIL: A New Me hanism for Suppressing Sti k-Slip
CONCLUSIONS AND FUTURE WORK
� WoB as additional ontrol variable.� Sti k-slip behavior insight using SBDF.� Simulations show os illations eliminated.� Redesign of velo ity ontrol notne essary.� Stability analysis under investigation.� S aled Drillstring platform is being built. �����M1
M2 �CDC-ECC'05 - Sevilla - De ember 2005 16
Os ila iones en una sarta de perfora i�onEstabilidadModelo de error: _e = A le+H l [�(e3)~u+ ~�(e3)u0℄_~u = ��~u+�(�) on ~�(e3) = �(e3)� ��.
G : � = �~u1 � ~u2 7! y = e3
A : y 7! ~u1 = �(e3)~u
� : y 7! ~u2 = ~�(e3)u020 de Enero 2005 13
Os ila iones en una sarta de perfora i�on
Sistema Lineal Perturbado� De�ni i�on: _e = A le+ g(t; e)g(t; e) = H l(~u1 + ~u2)� Perturba i�on a otada: jj~u1jj1 = supt�0 j~u1j = supt�0 j�(�)j � j~uj � u0 <1jj~u2jj1 = supt�0 j~u2j = supt�0 j~�(�)j � ju0j � 2 � u0 <1jjg(t; e)jj � jjH ljj � [jj~u1jj+ jj~u2jj � 3jjH ljju0℄� Los estados est�an a otados: jje(t)jj � Æ Æ > 0; 8t > 0
20 de Enero 2005 14
Os ila iones en una sarta de perfora i�on
Fun i�on de Transferen ia G(s)
� De�ni i�on: G(s) = �C(sI �A l)�1H l� Expresi�on: G(s) = s(a2s2+a1s+a0)b4s4+b3s3+b2s2+b1s+b0� Cara ter��sti a: Fun i�on SPR.{ 9P; Q > 0 s:t AT lP + PA l = �Q{ PH l = �C{ G : �! y es pasivo onV (e) = eTPe{ G : �! y tiene ganan iaL2 finita 2(G) = supt�0 jG(j!)j−120
−110
−100
−90
−80
−70
−60
−50
−40
−30
−20
Mag
nitu
de (
dB)
10−1
100
101
102
103
−90
−45
0
45
90
Pha
se (
deg)
Bode Diagram
Frequency (rad/sec)
0 0.02 0.04 0.06 0.08 0.1
−0.06
−0.04
−0.02
0
0.02
0.04
System: untitled1Real: 0.00779Imag: 0.0235Frequency (rad/sec): 1.05
Nyquist Diagram
Real Axis
Imag
inar
y A
xis
20 de Enero 2005 15
Os ila iones en una sarta de perfora i�on
Bloque Perturba i�on �(y)� La perturba i�on pertene e al intervalo[a; b℄:{ �(0) = 0{ ay2 � y�(y) � by2 , 8y 2 <{ a = ��S��C!d u0{ b = �S+�C!d u0� La perturba i�on tiene ganan ia L2 �nita:{ 2(�) � max[jaj; jbj℄� La perturba i�on modi� ada��(y) = �(y) + "y es pasiva:{ Si " � jaj ) R t0 y��(y) � 020 de Enero 2005 16
Os ila iones en una sarta de perfora i�on
Bloque de Adapta i�on A(y). Op i�on 1� Se toma �(y) = �y � sign[�(y)℄:{ Se puede asumir sign[�(y)℄ � sign[ _'b℄� Proposi i�on 1: El mapa A�(y) : y 7! (�(y)~u� "y)es pasivo si se toma:�� � "�C� Proposi i�on 2: El equilibrio (e�; ~u�) = (0; 0) esglobalmente asint�oti amente estable, on:V (e; ~u) = 12eTPe+ Z t0 y��(y) ++ �Z t0 y(�(�)~u� "y) + 2u0ymax� �20 de Enero 2005 17
Os ila iones en una sarta de perfora i�on
Bloque de Adapta i�on A(y). Op i�on 2� Se toma �(y) = ��jyj:{ Para �� < u0�ymax la se~nal de ontrol se mantiene a otada ~u 2 [�u0; 0℄.� Proposi i�on 1: La siguiente expresi�on es ierta:Z t0 y�(y)~u+ Z t0 �� y2 + 2u0ymax� � 0� Proposi i�on 2: El equilibrio (e�; ~u�) = (0; 0) es globalmente asint�oti amente estable, on:V (t; e) = 12 eTPe+ Z t0 y("y +�(y)) ++ Z t0 y�(y)~u+ Z t0 �� y2 + 2u0ymax�si tomamos: �� � 12#min(Q)� "
20 de Enero 2005 18
Os ila iones en una sarta de perfora i�on
Respuestas Obtenidas� Diferentes velo idades de respuesta.� Posibles limita iones de poten ia.� Per�les m�as o menos suave.0 50 100 150
0
10
20
30
40
Time [s]
Rot
atio
n V
eloc
ity [r
ad/s
]
Operation under DOSKIL
surface downhole
0 50 100 1502.6
2.8
3
3.2
3.4
3.6
3.8
4
x 104
Time [s]
Wei
ght O
n B
it [N
]
� Disponibilidad de medidas.� Limita i�on de valores de ~u.� Cara ter��sti as adaptativas.
0 50 100 1503.3
3.4
3.5
3.6
3.7
3.8
3.9
4
4.1x 10
4
Time [s]W
eigh
t On
Bit
[N]
0 50 100 1500
10
20
30
40
Time [s]
Rot
atio
n V
eloc
ity [r
ad/s
]
Operation under DOSKIL
surface downhole
20 de Enero 2005 19
Os ila iones en una sarta de perfora i�on
Algunos trabajos en urso y futuros
� Todas las pegas anteriores ......� Compila i�on de todo el trabajo en un solo do umento.� B�usqueda �nan ia i�on.� Amplia i�on del estudio de estabilidad a otras posibles estru turas de ontrol.� Estudio del a oplamiento de os ila iones laterales y torsionales.� Impli a iones de DOSKIL es este a oplamiento.� Futuras pruebas de laboratorio on equipo experimental.20 de Enero 2005 20
Os ila iones en una sarta de perfora i�on
Encoder Incremental salida seno/coseno
Masa ~ 12 kg
Masa ~ 2 kg
Acoplamiento Elastico Constante de torsion
0,4<k<1 N.m/rad
Motor brushless o CC + encoder + reductora 1:10
0-100 tr/mn - par nominal entre 0,5 y 1 Nm ������Estructura
Pieza a taladrar
Útil de perforación
Masa adicional (hasta 15 Kg)
Captor de fuerza
Motor brushless o CC + Captor de posicion +
reductora 1:10 par nominal ~2 Nm
Desplazamiento max. 20 cm
Rail de guía
M1
M2
F = 300 N max �Placas de fijacion a las guías
Masa Total
ap. 30 Kg
20 de Enero 2005 21
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