Analysis, Modeling and Control of an Oilwell Drilling System · 2013. 11. 27. · Oilwell drilling...

Analysis, Modeling and Control ofan Oilwell Drilling System

Dr. Sabine Mondié CuzangeM. Sc. Belem Saldivar Márquez

[email protected] / [email protected]

Centro de Investigación y de Estudios Avanzadosdel Instituto Politécnico Nacional,CINVESTAV IPN, México, D.F.

October, 2012.

CINVESTAV

October 2012S. Mondie & B. Saldivar (CINVESTAV) Drilling system October 2012 1 / 39

Contents

1 Introduction

2 Drilling system modeling

3 Model validation

4 Control strategies to suppress undesirable behaviors:Neutral- type time-delay model approachWave equation approach

S. Mondie & B. Saldivar (CINVESTAV) Drilling system October 2012 2 / 39

IntroductionObjectives

Modeling

The rst objective is to nd a model to describe the behavior of the rod duringdrilling operation. The model must accurately describe the most importantphenomena arising in real wells and it has to be simple enough for analysis andcontrol purposes.

Model validation

In order to validate the model, we test the main strategies used in practice toreduce unwanted behaviors.

Control design

The second objective of our research is the control design for the stabilization ofthe drilling system through Lyapunov theory based on two di¤erent approaches:neutral-type time-delay model and wave equation model.S. Mondie & B. Saldivar (CINVESTAV) Drilling system October 2012 3 / 39

IntroductionProblem statement

Oilwell drillstrings are mechanisms that play a key role in the petroleum extractionindustry. Failures in drillstrings can be signicant in the total cost of theperforation process.

Vertical oilwell drilling system.


ModelingOilwell drilling system

The main components of the drilling system are:

Drillstring, it is composed by pipes sections screwed end to end each other,they are added as the bore hole depth does.

Drill collars, heavy, sti¤ steel tubulars used at the bottom to provide weighton bit and rigidity.

Bit, rock cutting device to create the borehole.

The drilling mud or uid, it has the function of cleaning, cooling andlubricating the bit.


ModelingNon-desired oscillations

Three main types of vibrations can be distinguished:

torsional (stick-slip oscillations),axial (bit bouncing phenomenon) andlateral (whirl motion due to the out-of-balance of the drillstring).


ModelingDistributed parameter model

The mechanical system is described by the following partial di¤erential equation[Challamel, 2000]:

GJ∂2θ

∂ξ2 (ξ, t) I∂2θ

∂t2 (ξ, t) β∂θ

∂t(ξ, t) = 0, ξ 2 (0, L), t > 0,

with boundary conditions:

GJ∂θ

∂ξ(0, t) = ca

∂θ

∂t(0, t)Ω(t)

; GJ

∂θ

∂ξ(L, t) + IB

∂2θ

∂t2 (L, t) = T

∂θ

∂t(L, t)

where:

θ(ξ, t) is the angle of rotation,Ω is the angular velocity at the surface,T is the torque on the bit, L is the length of the rod,IB is a lumped inertia (it represents the assembly at the bottom hole)β 0 is the damping (viscous and structural)I is the inertia, G is the shear modulus and J is the geometrical moment ofinertia.


ModelingNeutral type time delay model

Let the damping β = 0. The distributed parameter model reduces to theunidimensional wave equation: ∂2θ

∂ξ2 (ξ, t) = p2 ∂2θ∂t2 (ξ, t), ξ 2 (0, L), where

p =p

I/GJ. The general solution of the wave equation can be written asθ(ξ, t) = φ(t+ pξ) + ψ(t pξ). The drilling behavior is described by thefollowing neutral-type time-delay equation:

z(t) Υz(t 2Γ) = Ψz(t) ΥΨz(t 2Γ) 1IB

T (z(t)) (1)

+1IB

ΥT (z(t 2Γ)) +ΠΩ(t)(t Γ),

where z(t) is the angular velocity at the bottom end, and

Π =2Ψca

ca +p

IGJΥ =

ca p

IGJca +

pIGJ

, Ψ =

pIGJIB

, Γ =

sI

GJL.

The angular velocity coming from the rotary table Ω(t) is usually taken as controlinput.S. Mondie & B. Saldivar (CINVESTAV) Drilling system October 2012 8 / 39

ModelingTorsional dynamics

The following nonlinear equation was introduced in [Navarro & Cortés, 2007], itallows to approximate the physical phenomenon at the bottom hole:

T (z(t)) = cbz(t)+WobRbµb (z(t)) sign (z(t)) .

the term cbz(t) is a viscous damping torque at the bit which approximates theinuence of the mud drilling and the term WobRbµb (z(t))sign(z(t)) is a dryfriction torque modelling the bit-rock contact. Rb > 0 is the bit radius andWob > 0 the weight on the bit. The bit dry friction coe¢ cient µb(z(t)) isdescribed by:

µb (z(t)) = µcb + (µsb µcb)e γb

vfjz(t)j,

where µsb , µcb 2 (0, 1) are the static and Coulomb friction coe¢ cients and0 < γb < 1 is a constant dening the velocity decrease rate.


ModelingAxial dynamics

The damped harmonic oscillator model describes the axial dynamics of thedrillstring [Challamel, 2000]:

m0Y(t) + c0Y(t) + k0(Y(t) ROPt) = µ1 (T (z(t)) T (z(t h))) ,

where:Y axial positionY axial velocityY axial accelerationROP rate of penetrationm0 massc0 dampingk0 spring constant


Oilwell drilling systemSimulations of the open loop drilling system

The parameters used in the following simulations are shown below, they representa typical case in oilwell drilling operations.

G = 79.3x109N m2, I = 0.095Kg m, L = 1172m,

J = 1.19x105m4, Rb = 0.155575m, vf = 1,

Wob = 97347N, IB = 89Kg m2, ca = 2000N m s,

µcb = 0.5, µsb = 0.8, γb = 0.9,

cb = 0.03N m s rad1, m0 = 37278Kg, c0 = 16100kg s1,

k0 = 1.55 106Kg s2, ROP = 0.01m s1, µ1 = 257m1.

According to the above parameters, the time delay is Γ = 0.3719.


SimulationsStick-slip behavior

Angular velocity at the bottom extremity z(t) for: (a) Ω(t) = 10rad s1

(stick-slip), (b) Ω(t) = 40rad s1.


SimulationsStick-slip behavior

Torque on the bit T(z(t)) for: (a) Ω(t) = 10rad s1, (b) Ω(t) =40rad s1.S. Mondie & B. Saldivar (CINVESTAV) Drilling system October 2012 13 / 39

SimulationBit-bounce behavior

Variable y = YROPt for: (a) Ω(t) = 10rad s1 (bit-bouncing), (b)Ω(t) = 40rad s1.


Practical strategies to reduce the stick-slip phenomenonDecreasing the weight on the bit

Reduction of the stick-slip by decreasing Wob.S. Mondie & B. Saldivar (CINVESTAV) Drilling system October 2012 15 / 39

Practical strategies to reduce the stick-slip phenomenonIncreasing the surface angular velocity

Reduction of the stick-slip by increasing Ω.S. Mondie & B. Saldivar (CINVESTAV) Drilling system October 2012 16 / 39

Practical strategies to reduce the stick-slip phenomenonIntroducing a variation law of the weight on the bit

Using the variation law: Wob(z(t)) = Kw jz(t)j+Wob0.S. Mondie & B. Saldivar (CINVESTAV) Drilling system October 2012 17 / 39

Practical strategies to reduce the stick-slip phenomenonIncreasing the damping at the down end

Velocity at the bottom extremity for diferent values ofcb (N m s rad1): (a) 0,8, (b) 15, (c) 65, (d) 150.


Control strategies

Wave equation approach

The proposal of an energy function for the distributed parameter model of thedrilling system provides a control law Ω(t) that ensures the energy dissipationduring the drilling process. This control law is shown to be e¤ective in suppressingtorsional oscillations in the drilling process.

Neutral-type time-delay model approach

We address the problem of control design for the suppression of axial-torsionalcoupled drilling vibrations via attractive ellipsoid method. Based on a combinationof the Lyapunov method and the principle of attractive sets, we develop ane¤ective methodology allowing the control synthesis for the drilling systemstabilization through the solution of an optimization problem subject to bilinearmatrix constraints.


Dissipativity analysisWave equation

Consider the normalized oilwell drilling model:

ztt(σ, t) = azσσ(σ, t) dzt(σ, t)t > 0, 0 < σ < 1 (2)

where a = GJIL2 and d = β

I coupled to the mixed boundary conditions

zσ(0, t) = g (zt(0, t)Ω(t)) , σ 2 (0, 1), t > 0,zσ(1, t) = kzt(1, t) qµb(zt(1, t))sign(zt(1, t)) hztt(1, t), (3)

where g = caLGJ , k = cbL

GJ , q = WobRbLGJ and h = IBL

GJ .Let the energy function:

E(t) =Z 1

0az2

σ(σ, t)dσ+Z 1

0z2

t (σ, t)dσ+ ah zt(1, t)2. (4)



Di¤erentiation of E(t) along (2), yields

ddt

E(t) = 2aZ 1

0zσ(σ, t)ztσ(σ, t)dσ

+2azt(1, t)(kzt(1, t) qµb(zt(1, t))sign(zt(1, t))hztt(1, t)) 2agzt(0, t) (zt(0, t)Ω(t))

2aZ 1

0zσ(σ, t)ztσ(σ, t)dσ

2dZ 1

0zt(σ, t)zt(σ, t)dσ+ 2ahzt(1, t)ztt(1, t),

sinceµb(zt(1, t))sign(zt(1, t))zt(1, t) = µb(zt(1, t)) jzt(1, t)j ,

we have

ddt E(t) = 2aqµb(zt(1, t)) jzt(1, t)j 2d

R 10 z2

t (σ, t)dσ 2agzt(0, t) (zt(0, t)Ω(t)) 2akz2

t (1, t).(5)



In order to ensure the dissipativity of the system, the control law Ω(t) shouldallow the negativity of (5). Choosing the following control law:

Ω(t) = (1 c1)zt(0, t) + 2c1zreft (1, t) c1

zreft (1, t)

2

zt(0, t)(6)

where zreft (1, t) is the angular velocity to be achieved and c1 > 0 is a free designparameter we obtain that

ddt E(t) = 2aqµb(zt(1, t)) jzt(1, t)j 2d

R 10 z2

t (σ, t)dσ

2agc1

zreft (1, t) zt(0, t)

2

2akz2t (1, t).

Taking into account that µb(zt(1, t)) > 0 and a, q, k, d and g are positiveconstants we nd that d

dt E(t) 0.


Dissipativity analysisNumerical results

The non-growth of the energy of the drilling system (which reects the oscillatorybehavior of the system) is established:

PropositionFor all solutions of (2) under the boundary conditions (3), the energy given by (4)does not grow if the control law (6) is applied.

In order to test the performance of the controller (6), we obtain a delay-equationdescribing the angular velocity at the top end zt(0, t) via the DAlemberttransformation and we simulate the neutral-type model (1) where the velocity atthe bottom extremity z(t) corresponds to zt(1, t).


Dissipativity analysisSimulation result

The stick-slip vibrations are reduced by means of the application of the control law(6) for c1 =0.3. The reference velocity considered in this simulation is 10rad/s.


Attractive ellipsoid methodPreliminaries

Consider the following time-delay system of neutral-type:

x(t) + Dx(t h) = A0x(t) + A1x(t h) + n(t), (7)

x(θ) = φ(θ), θ 2 [h, 0] ,

where h 0 is the time delay, D 2 Rnn is Schur stable, A0, A1 2 Rnn, n(t)satises kn(t)k ζ, for ζ > 0, t 0.

Denition (Ellipsoid)An ellipsoid centered at the origin is a set in Rn such that

EM =n

x 2 Rn : xT Mx 1o

, 0 < M = MT.



Denition (Invariant ellipsoid)

An ellipsoid E is positive invariant for the system (7) if φ(θ) 2 E , θ 2 [h, 0]implies that x(t, φ) 2 E , t 0 for every trajectory of the system.

Denition (Attractive ellipsoid)An ellipsoid E is an attractive domain for the system (7) if1) φ(θ) 2 E , θ 2 [h, 0] implies that x(t, φ) 2 E , t 02)φ(θ) 2 RnnE , for some θ 2 [h, 0] implies that there exists Ta, 0 Ta < ∞,such that x(t, φ) 2 E , t 0.



Lemma

Let the functional V(xt) satisfying:

xT(t)Px(t) V(xt) α kxtk2h , 0 < P = PT ,

d

dtV(xt) + σV(xt) β, 8t 0, σ > 0, β > 0.

Then for any initial function φ 2 PC([h, 0],Rn), the solution x(t, φ) belongs tothe ellipsoid

EP =n

x 2 Rn : xT Px 1o

, P =

σ

β+ ς

P ς > 0

for t Ta(φ, ς) > 0 where

Ta(φ, ς) =1σ

ln

α kφk2

hς

β

ςσ

!S. Mondie & B. Saldivar (CINVESTAV) Drilling system October 2012 27 / 39

Attractive ellipsoid methodProblem statement

Let a neutral-type time-delay system of the form:

x(t) + Dx(t h) = A0x(t) + A1x(t h) + B1u1(t τ) + B2u2(t) (8)

+C0 f (x(t)) + C1 f (x(t h)) +ω,

where h, τ are constant time delays, D 2 Rnn is Schur stable, A0, A1 2 Rnn,B 2 Rnm, f satises k f ()k ζ, ζ > 0, t 0, and ω:

kωkKω= ωTKωω 1, t 0.

where Kω 2 Rnn is a given constant matrix such that Kω = KTω > 0.

Let us consider the following structure of u1:

u1(t τ) = K0 x(t h) + K1x(t h), (9)

for u2(t) consider:u2(t) = K2x(t). (10)


Attractive ellipsoid methodProblem statement

Our aim is to nd the conditions such that the ellipsoid

EP =n

x 2 Rn : xT Px 1o

,

denes an attractive set for the trajectories of the system (8) in closed loop with(9)-(10), i.e.,

lımt!∞

xT(t)Px(t) 1.

Furthermore, the choice of the matrices K0, K1, K2 2 Rmn and P mustguarantee the minimality of the ellipsoid EP. Given that the trace of the matrix Pis inversely related to the axes of the ellipsoid EP, the following optimizationproblem arises:

mın tr(P1)

subject to P 2 Σ1, K0, K1, K2 2 Σ2

where Σ1, Σ2, dene the set of admissible matrices of dimension n n and m nrespectively guaranteeing the invariance property of the ellipsoid EP.S. Mondie & B. Saldivar (CINVESTAV) Drilling system October 2012 29 / 39

Attractive ellipsoid methodMain result

Theorem

Let the optimization problemmın tr(H)

subject to 8>>>><>>>>:Λ := fH, P, S, R, K0, K1, K2, Pk, βg , k = 1, ..., 14.Φ < 0,

H :=

H InIn P

> 0,

P > 0, S > 0, R > 0, β > 0, σ > 0

with optimal solution Λ :=

H, P, S, R, K0, K1, K2, Pk, β

, k = 1, ..., 14. Theellipsoid E(0, P) determined by the matrix σ

βP is a minimum attractive ellipsoid

for system (8).



For the proof we use the Lyapunov functional:

V(xt) = xT(t)Px(t) +Z t

theσ(st)xT(s)Sx(s)ds

+hZ 0

h

Z t

t+θeσ(st) xT(s)Rx(s)dsdθ,

α = λmax (P) + hλmax (S) + h2λmax (R) .

Following the descriptor approach we consider the null terms:

0 = [x(t) Dx(t h) + A0x(t) + A1x(t h) + B1u1(t τ)

+B2u2(t) + C0 f (x(t)) + C1 f (x(t h)) +ω],

0 = [u1(t τ) + K0 x(t h) + K1x(t h)],

0 = [u2(t) + K2x(t)]



We obtain after symmetrization of cross terms:

dV(xt)

dt+ σV(xt) β ηTΦη,

where η = (x(t) x(t h) x(t) x(t h) u1(t τ) u2(t) f (x(t)) f (x(t h))ω)T , Φ is a bilinear symmetric matrix.In order to computationally minimize the trace of P1 it is necessary to includethe following inequality condition

H :=

H InIn P

> 0,

with H > 0, which implies P1 < H (Schur complement).Then, the minimization problem mın tr(H) implies the minimization of mıntr(P1).


Attractive ellipsoid methodTorsional-axial coupled dynamics

Dening the new variables:

x1(t) = z(t)Ω0, x2(t) = Y(t) ROPt, x3(t) = x2(t),

and considering the controller structure:

Ω(t) = u1(t) +Ω0,

the torsional-axial coupled dynamics of the drillstring can be described by:

x(t) + Dx(t 2Γ) = A0x(t) + A1x(t 2Γ) + B1u1(t Γ)+ B2u2(t) + C0 f (x1(t) +Ω0) + C1 f (x1(t 2Γ) +Ω0) +ω

x =

x1 x2 x3T ,

where:

D=

24 Υ 0 00 0 00 0 0

35 , A0 =

264 Ψ cbIB

0 00 0 1

µ1cbm0

k0m0

c0m0

375 ,


Attractive ellipsoid methodTorsional-axial coupled dynamics

A1 =

264ΥcbIB ΥΨ 0 00 0 0

µ1cbm0

0 0

375 , B1 =

24 Π00

35 , B2 =

24 00 c0

m0

35 ,

C0 =

264 1IB

0 µ1

m0

375 , C1 =

264 ΥIB0µ1m0

375 , ω =

24 ω100

35 ,

the time delay is:

Γ =

sI

GJL,

The nonlinear function is dened by:

f (x1(t) +Ω0) = Tnl (x1(t) +Ω0)

= WobRb

µcb+(µsb µcb)e

γbvfjx1(t)+Ω0j

sign (x1(t) +Ω0) .


Attractive ellipsoid methodNumerical results

Using the above result, it is possible to obtain the synthesis of the controllers (9),(10).For σ = 1.5 we get:

P =

24 7.4288 2.2274 0.45332.2274 4.5511 0.70330.4533 0.7033 0.2943

35 , eig(P) =

8<: 0.17973.38868.7060

λmax (S) = 155.40825, λmax (R) = 4.0950, β = 1.6311, α = 126.55

K0 =0.1285 0 0

, K1 =

0.5091 0.1185 0.0566

,

K2 =

53.9090 157.5892 80.9510

.


Attractive ellipsoid methodSimulations of the closed loop drilling system


Attractive ellipsoid methodNumerical results

For ς = 0.1, the trajectories of the drilling system are attracted to the ellipsoid

EP =n

x 2 Rn : xT Px 1o

where

P = 1.0196

24 7.4288 2.2274 0.45332.2274 4.5511 0.70330.4533 0.7033 0.2943

35for t Ta(φ, ς) > 0 where

Ta(φ, ς) = 0.6667 ln

1265.5 kφk2h 10.8740

= 5.6852 s.


Attractive ellipsoid methodSimulations of the closed loop drilling system

x1(t) vs. x2(t) vs. x3(t) converging to the ellipsoid E(0, P).


References

Fridman E., Mondié S., Saldivar B., Bounds on the response of a drilling pipemodel, Special issue on Time-Delay Systems in IMA Journal of MathematicalControl. & Information (2010) 27(4): 513-526.

Saldivar B., Mondié S., Loiseau J.J., Rasvan V., Stick-slip oscillations inoillwell drilstrings: distributed parameter and neutral type retarded modelapproaches. 18th IFAC World Congress, Milano, Italy, (2011), 284-289.

Challamel N., Rock destruction e¤ect on the stability of a drilling structure,Journal of sound and vibration, (2000), 233 (2), 235-254.

E. Fridman, New LyapunovKrasovskii functionals for stability of linearretarded and neutral type systems. Systems & Control Letters 43 (2001)309-319.

Navarro E., Cortés D., Sliding-mode of a multi-DOF oilwell drillstring withstick-slip oscillations, Proceedings of the 2007 American Control Conference,3837-3842, New York City, USA, 2007.


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