Avoiding stick slip vibrations in drilling through startup trajectory design

Journal of Process Control 70 (2018) 24–35

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Journal of Process Control journal homepage: www.elsevier.com/locate/jprocont

Avoiding stick slip vibrations in drilling through startup trajectory design Ulf Jakob F. Aarsnes a,b,∗ , Florent Di Meglio c , Roman J. Shor d a

International Research Institute of Stavanger (IRIS), Oslo, Norway DrillWell – Drilling and Well Centre for Improved Recovery, Stavanger, Norway Centre Automatique et Systèmes, MINES ParisTech, Paris, France d University of Calgary, Department of Chemical and Petroleum Engineering, Calgary, Canada b c

a r t i c l e

i n f o

Article history: Received 24 January 2018 Received in revised form 17 July 2018 Accepted 28 July 2018 Keywords: Drill-string vibrations Stick-slip Distributed systems Differential flatness Delay equations

a b s t r a c t A distributed model of a drill string with a collars section is presented with Coulomb friction as a distributed source term. This model is capable of replicating stick slip oscillations as caused by the reduction in friction from static to dynamic. We design a feed-forward startup trajectory for initiating rotation of the drill string that effectively avoids the stick slip limit cycle. The trajectory design is performed using the differential flatness of the bit angular velocity, and by treating the reduction from static to dynamic friction as an estimated disturbance to be canceled, thus conforming to the canonical 3-DOF controller design for tracking and disturbance rejection. A simulation study illustrates the feasibility of the approach. © 2018 Elsevier Ltd. All rights reserved.

1. Introduction Exploration and production of oil and gas in the deep subsurface, where hydrocarbon reservoirs are found at depths between 2,000 and 20,000 feet, requires that a narrow borehole, between 4 and 24 inches in diameter, be drilled using a slender drill-string through a varied downhole environment and along an often snaking wellpath. Drill string vibrations, and their negative consequences on Rate Of Penetration (ROP) and equipment, is a well known phenomenon when drilling for hydrocarbons. In particular, the torsional oscillations known as stick slip, which are considered to be the most destructive vibrations, are to be avoided. Significant literature exists which seeks to explain the incidence of stick-slip through various models of bit-rock interaction and various complexities of drill-string dynamics. The simplest models assume that the bit-rock interaction law takes the form of a discontinuous frictional force at the bit and abstract the drill-string as a lumped mass, representing the bottom hole assembly (BHA) inertia, and a torsional spring, representing the drill-string stiffness [7,12]. These models may be confounded by introducing higher complexity dynamics at the bit-rock interaction or through higher order

∗ Corresponding author at: International Research Institute of Stavanger (IRIS), Oslo, Norway. E-mail address: [email protected] (U.J. F. Aarsnes). https://doi.org/10.1016/j.jprocont.2018.07.019 0959-1524/© 2018 Elsevier Ltd. All rights reserved.

models along the drill-string [20,26], but still assume that stick slip stems from the non-linearity of the frictional force at the bit. All these models have been used to demonstrate the occurrence of the limit cycle which exhibits itself as stick-slip and may be used to design various types of stick-slip mitigation controllers, including simple tuned PID controllers [19,29], impedance matching controllers [14], H-infinity controllers [33], sliding mode controllers [27], and others [8,31]. Despite this significant research, the vibration mitigating controllers currently applied in the field are mainly PI controllers following the SoftSpeed and SoftTorque approach of tuning the proportional and integral gains to obtain a certain reduction in the proximal (i.e. topside) reflection coefficient over a limited frequency range [18]. One of the reasons other approaches have failed to see a wider degree of adoption is a fundamental limitation of the feedback approach to this problem. Specifically, the dynamics are described by a lightly damped wave equation, the transfer function of which is known to non-proper with a high supremum at high frequencies [11]. These kind of systems, although easily controlled in theory by canceling the reflection coefficient, are very hard to control in practice due to the vanishing delay-robustness margins [22,6]. However, this limitation can to a large degree be overcome by using a topside torque measurement, as is done in some versions of the impedance matching controller [14,2]. In the present paper, we argue that the problem of avoiding entering a stick slip limit cycle when starting up drill string rota-

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25

Fig. 2. Infinitesimal drill string element.

Fig. 1. Schematic indicating the distributed drill string lying in deviate borehole.

tion can be viewed as a classical linear disturbance rejection and tracking problem [9]. Specifically, we want to cancel the impact of the Stribeck-like effect of the torque acting on the BHA as rotation is initiated. Consequently, the main contribution of this paper is an add-on to the industrial state-of-the-art feedback controller taking the form of a feed-forward controller, which comprises two terms. The first feed-forward term handles the reduction between static and dynamic Coulomb friction as a disturbance that is estimated from previous startups and then canceled. The second term is a trajectory planner which uses the systems differential flatness to compute the actuation trend which achieves changes in set-point without exciting new oscillations. This addition, which requires little implementation effort for practitioners, avoids the stick-slip limit cycle at startup. To illustrate the relevance of this approach, it is employed on a distributed model of the drill string with the Coulomb friction given as a distributed source term. As such, the model can effectively replicated the torsional behavior of wells with significant lateral sections, where the dynamics are dominated by the (distributed) drill-string–borehole interaction, which is a particularly challenging scenario. The paper is organized as follows. In Section 2, we derive the simulation model. Then we present a simplified model for control design in 3 and show that it is differentially flat. This is then used to obtain the proposed control architecture (Section 4.1) and feedforward controllers (Sections 4.3 and 4.4). Finally, Section 5 contains numerical simulations on a relevant case study.

Fig. 3. Collar-pipe transition.

J is the polar moment for inertia and G is the shear modulus. Hence the equations for the angular motion are given by

∂(t, x) ∂ω(t, x) + JG =0 ∂t ∂x J

∂ω(t, x) ∂(t, x) + = S(ω, x), ∂t ∂x

(1)

(2)

where the source term is due to frictional contact with the borehole and is modeled as S(ω, x) = −kt Jω(t, x) − F(ω, x),

(3)

where kt is a damping constant representing the viscous shear stresses between the pipe and drilling mud, and F(ω) is a differential inclusion, to be described, representing the Coulomb friction between the drill string and the borehole. 2.2. Discontinuities of a multiple sectioned drill string

2. Model 2.1. Torsional dynamics of drill string We use a distributed model, similar to [1,4,16], except that in this case we consider only the torsional dynamics. That is, for the angular motion, we denote the angular velocity and torque as ω(t, x), (t, x), respectively, with (t, x) ∈ [0, ∞) × [0, L], see Fig. 1. The torque is found from shear strain, given as twist per unit length, and letting  denote the angular displacement in the string s.t. ∂(t,x) = ∂t ω(t, x), we have (t, x) = JG((t, x) − (t, x + dx))/dx, see (Fig. 2). Here

The lowermost section of the drill string is typically made up of drill collars which may have a great impact on the drill string dynamic due to their added inertia. In particular, the transition from the pipes to collars in the drill string will cause reflections in the traveling waves due to the change in characteristic line impedance [1]. We split the drill string into a pipe section with polar moment of inertia and lengths Jp , Lp and a collar section with the same parameters given as Jc , Lc . We use  + , ω+ to denote the strain and velocity at the top of the drill collar and  − , ω− at the bottom of the pipe,

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Fig. 4. Recorded and simulated drill-string response at a bit depth of 1733 m in a well with the survey shown right, using the friction parameters:  = 0.34, frat = 0.55, ωc = 19 (RPM).

see Fig. 3. The boundary conditions at the transition are given by the following continuity constraints ω+ = ω− ,

+ = −

(4)

2.3. Coulomb friction as an inclusion

⎪ ⎩

|ω| < ωc ,

F(ω, x) = −Fc (x)frat ,

ω < ωc ,

(5)

where ωc is the threshold on the angular velocity where the Coulomb friction transitions from static to dynamic, frat ∈ [0, 1] is the ratio between the static and dynamics Coulomb friction, and F(ω) ∈ [−Fc , Fc ] denotes the inclusion where F(ω, x) = −

∂(t, x) − kt Jω(t, x) ∈ [−Fc (x), Fc (x)], ∂x

(6)

and take the boundary values ±Fc (x) if this relation does not hold. To obtain the maximum Coulomb torque function Fc (x), we employ the classic Coulomb friction law, which states that the friction opposing a motion horizontal to the plane is proportional to the normal force with the coefficient . Thus we obtain Fc (x) =  sin((x))gA(x)ro (x),

(8)

and finally, the angular velocity at the top of the drill string is equal the top drive velocity ω(t, x = 0) = ω0 .

ω > ωc ,

F(ω, x) ∈ [−Fc (x), Fc (x)],

At the topside boundary, the top drive is actuated by a motor delivering a torque  m which we assume to be the control input. The topdrive has the inertia ITD and hence satisfies the dynamics

∂ω0 1 (m − (t, x = 0)), = ITD ∂t

The Coulomb friction is modeled as an inclusion

⎧ F(ω, x) = Fc (x)frat , ⎪ ⎨

2.4. Boundary condition

(7)

where (x) is the wellbore inclination, g is the acceleration of gravity, A(x) is the cross sectional area of the drill string, ro (x) is the outer radius of the drill string. The friction factor  is dependent on the wellbore roughness, mud properties, etc. Note that the relation (7) is simplistic in that the normal force is affected by other effects than just gravity, with tortuosity being a particularly important parameter [23]. Consequently, to compensate for such un-modeled effects, the friction factor  may be tuned (typically increased). For the case studied in [32], tortuosity was found to increase torque progressively with MD with a 28% increase reported at 17,000 ft when drilled with a RSS with an unwanted dog leg severity of 0.45 and 0.41 deg/100 ft in the curve section and the slant section, respectively.

2.5. Model validity The effectiveness of this modeling approach is explored in some detail in [3]. Here we illustrate this by briefly considering the openloop fit of the model to full scale field data shown in Figs. 4 and 5. Details of the numerical implementation is given in Appendix A. In both the cases of Figs. 4 and 5, the model accurately replicates the stick slip oscillations of the field data. The cases covers two qualitatively different behavior of the stick slip, demonstrating a certain degree of versatility of the model. In particular we note that, for Fig. 4 where down-hole data is available, a good replication of the angular BHA velocity is also achieved. Down-hole data for the well in Fig. 5 is not available. 2.6. Control problem statement In the following sections, we design a control law  m (t, ω0 ) that drives the bottom velocity ω(t, Lp + Lc ) to a constant set point value ωsp from an initial condition corresponding to full rest, i.e. ω(0, x) ≡ 0. As detailed in Section 4.1, the controller comprises an industry-standard PI controller, to which we add feed forward terms, whose main goal is to avoid entering the basin of attraction of a limit cycle while transitioning from the rest position to the desired set point. To achieve this, we rely on a simplified model for control, described in the next section. 3. Model approximation and flatness In this section we derive a model for control design, relying on two simplifying assumptions: first, we propose a lumped approximation of the drill collar section. Second, we model the torque

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Fig. 5. Recorded and simulated drill-string response at a bit depth of 22,506 m in a well with the survey shown right, using the friction parameters:  = 0.43, frat = 0.43, ωc = 17 (RPM).

due to Coulomb friction as a constant disturbance, whose rejection during the startup avoids entering the stick-slip limit cycle. We now detail these assumptions before rewriting the model as a time-delay system with a flat output.

∂ωL 1 ((t, x = L) − d(t)). = IBHA ∂t

(15)

To evaluate the error introduced by the lumped approximation, transfer functions can readily be derived, see Appendix B.

3.1. Lumped BHA To simplify analysis and facilitate the flat formulation, it is an amenable approximation to represent the BHA section of the drill string, including the collar section, as a single lumped inertial element. This approximation entails lumping the effect of the source term (19) into the lumped dynamics of the BHA. This is a reasonable approximation for most drill-strings as much of the torque acting on the drill string will come from stabilizers located in, or close to, the BHA [3]. The inertia of the lumped BHA is IBHA = Lc Jc ,

(9)

and hence the BHA’s angular velocity is governed by

∂ωL 1 ((t, x = L) − d(t)), = IBHA ∂t

(10)

where d(t) accounts for the now lumped effect of the distributed source term, i.e.:



d(t) ≈

L

S(ω, x).

(11)

0

Here, (11) is meant for illustration and is not used directly. When we later employ the flat formulation, facilitated by this approximation, for control d(t) will be treated as an uncertain disturbance. Using this lumped approximation of BHA, we obtain what we will refer to as the semi-lumped formulation, given by the distributed wave-equation

3.1.1. Control problem statement in approximate coordinates To solve the control problem defined in Section 2.6, we solve the following two sub-problems 1. design a feedfoward controller to reject the step disturbance d. 2. solve the trajectory planning problem for ωL for a trajectory going from zero to the setpoint ωsp Problem 1 is solved in Section 4.3 while Problem 2 is solved in Section 4.4. Both solutions rely on the existence of a flat output for (12)–(15), which we prove in the next two sections. 3.2. Derivation of Riemann invariants The Riemann invariants of a Hyperbolic PDE is the states corresponding to a transformation of the system which has a diagonalized transport matrix, i.e. the system can be written as a series of tranport equations only coupled in the source terms [21]. Define the Riemann invariants ˛=ω+

ct , JG

where ct =



ˇ=ω−  J

ct , JG

(16)

is the velocity of the torsional wave. This trans-

formation enables us to rewrite (1) and (2) in variables ˛, ˇ as the diagonalized PDE system

∂(t, x) ∂ω(t, x) + JG =0 ∂t ∂x

(12)

(17)

∂ω(t, x) ∂(t, x) J + = 0, ∂t ∂x

∂˛ ∂˛ + ct = −S ∂t ∂x

(13)

∂ˇ ∂ˇ − ct = −S. ∂t ∂x

(18)

with the boundary conditions ω(t, x = 0) = ω0 , ω(t, x = L) = ωL governed by

∂ω0 1 (m − (t, x = 0)), = ITD ∂t

(14)

with the source term S=

S 1 = kt (˛ + ˇ) + F. J J

(19)

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For the fully distributed model, the boundary conditions (4) rewrite, in the Riemann coordinates, as follows ˇ+ = ˛− =

1 1 + Z¯ 1 1 + Z¯

¯ + 2Zˇ ¯ −) (˛+ (1 − Z)

(20)

¯ − ), (2˛+ − (1 − Z)ˇ

(21)

where we have denoted the relative magnitude of the impedance as Z¯ =

 c −  c + t t JG

/

JG

.

(22)

We note that for the case of the same material being used at both sides of the discontinuity, the only change is in the polar moment of inertia. That is, for a pipe-collar sections of e.g. steel, we have, following Fig. 3 Z¯ =

Jc . Jp

(23)

Note the meaning of (20) and (21) as reflections of incoming waves from both sides, as they are split into an upward and a downward traveling wave. 3.3. Semi-lumped delay equation The semi-lumped system, (12)–(15), rewrites, with the Riemann Variables as states:

∂˛ ∂˛ + ct =0 ∂t ∂x

(24)

∂ˇ ∂ˇ − ct = 0. ∂t ∂x

(25)

˛(t, x = L) = ˛(t − tD , x = 0)

(26)

ˇ(t, x = 0) = ˇL (t − tD , x = L),

(27)

where tD = Lp /ct . p ITD

and aL =

p IBHA

,

both with dimensions seconds−1 , representing the inertia of the top-drive and BHA, respectively, relative to the line impedance of the drill string p . We will also write ˇL : = ˇ(t, x = L), ˛0 : = ˛(t, x = 0). The semi-lumped approximation of the dynamics then writes as ˇL = 2ωL − ˛0 (t − tD )

(28)

˛0 = 2ω0 − ˇL (t − tD )

(29)

ω˙ 0 = a0 (−ω0 + ˇL (t − tD )) + ω˙ L = aL (−ωL + ˛0 (t − tD )) +

1

m

(30)

1 d. IBHA

(31)

ITD

the flat output) and all its derivatives. This property implies controllability as, using the flat output and its derivatives as a new system of coordinates, the trajectory planning problem becomes trivial, as will appear in Section 4.4. We now show that z(t) = ωL (t) is a flat output for (28)–(31), similarly to [30]. Indeed, the Riemann invariants write as follows as functions of z(t) ˙ + tD )/aL ˛0 = z(t + tD ) + z(t

(32)

˙ − tD )/aL ˇ0 = z(t − tD ) − z(t

(33)

˙ ˛L = z(t) + z(t)/a L

(34)

˙ ˇL = z(t) − z(t)/a L.

(35)

Hence, we can write the top-drive angular velocity as a function of z(t) ω0 =

The solution to this PDE can be written as the delay equations:

Now, we define the frequency constants a0 =

Fig. 6. Block diagram showing the three components of the control architecture conforming to the canonical feedforward formulation of [5]: the feedback block C, the feed-forward disturbance rejection block D and the feed-forward tracking block F.

We note that this system is characterized by the three time constants • : Top drive time constant. • : BHA time constant. • : Drill string travel time. 3.4. System flatness Flatness is a systems property known, for linear finitedimensional systems, as Brunovsky’s form. It has been introduced in [15] for nonlinear finite-dimensional systems and in [24,25] for PDEs. A system has the flatness property if the state and the control input can be parametrized as functions of one output (called

˙ + tD ) − z(t ˙ − tD ) z(t + tD ) + z(t − tD ) z(t , + 2 2aL

(36)

which again allows us to write the input: motor torque,  m , as function of the flat output 2 a0 ˙ − tD ) − a0 z(t − tD ). m = ω˙ 0 + a0 ωf + z(t ITD aL

(37)

4. Tracking and control 4.1. Control architecture The control signal is composed of three terms: m = uc + uf + ud ,

(38)

where uc (t) = − (C ∗ (ωSP − ω0 )) (t) is a feedback term, uf is a ˆ a disturfeed-forward term to ensure tracking and ud = (D ∗ d) bance canceling term, with the controller impulses C(t), D(t) to be designed. This conforms to a canonical 3DOF controller architecture [5], see Fig. 6. The set-point for the top drive velocity ωSP is generated as the sum of the disturbance and the tracking feedforward terms ωd and ωf . The disturbance feed-forward term ωd is needed since the disturbance canceling imposes a trajectory on the top drive velocity. The feed-forward tracking term is design so as to avoid sustained oscillations of the BHA by exploiting the model’s differential flatness. Since these feed-forward terms are initiated as the drill string breaks free from the static friction, the system dynamics are linear and the two terms can be superpositioned. 4.2. Baseline feedback control: SoftSpeed / SoftTorque The current industry standard in handling torsional vibrations are the two products NOV’s SoftSpeed [19,17] and Shell’s SoftTorque [14,29]. The essential approach of all solutions is to reduce

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29

where fc is the frequency (in Hertz) where the minimum of R(ω) is achieved (Fig. 7). The general, for the PID controller, minimum of the reflection coefficient is obtained at



arg minR(ω) = ω

Ki ≡ fc 2. ITD + Kd

(47)

In the following we will use the SoftTorque-like controller, (45) and (46) as our baseline controller, as this type of controller is the most widely used of the stick-slip mitigating controllers in the field. It is worth noting that the industry standard controller that is most often used is a high gain PI control to ensure rapid tracking of the top drive set-point [18]. We will also consider this kind of controller for comparisons and will in this case use the gains. Kp = 100 p ,

Ki = 5ITD .

(48)

These gains were obtained from the field scenario and are similar to those presented by [18,13,28]. Fig. 7. Topside reflection coefficient for Kp = 4 p , fc = 0.2 Hz.

4.3. Disturbance cancellation

the reflection coefficient at the top drive in a certain key frequency range [18]. Assuming for the moment a constant set-point, and defining the m controller transfer function C(s) ≡ ω we obtain the relation: 0

0 (s) ¯ = C(s) + ITD s ≡ C(s), ωTD (s)

Note from (28) to (31) that the down-hole velocity ωL , which is what we want to control and keep constant, is the signal ˛0 delayed and low-pass filtered. Hence, ωL can be controlled by controlling ˛0 . We have ˛0 = 2ω0 − ˇ0

(39) =

2 ud s − a0 + ˇ0 s + a0 ITD s + a0

(50)

=

2 ud s − a0 2e−st D d + . s + a0 ITD s + a0 s + aL IBHA

(51)

while the topside reflection coefficient is given as [19]



C(s) ¯ − p

R(ω) =

¯ C(s) + p

.

(40)

s=jω

Typically, a PI or PID controller is employed, that is, on the form

 m = Kp e + Ki

e( )d + Kd e˙

(41)

0

e = ωTD − ωSP ,

(42)

This results in the feedback relation K ¯ C(s) = Kp + i + (Kd + ITD )s, s

(43)



Kp − p + Ksi + (Kd + ITD )s



R(ω) =

K

Kp + p + si + (Kd + ITD )s s=jω

(44)

where-from we see that impedance matching (R ≈ 0) is obtained with the tuning: 1. Kp = p to match drill string impedance. 2. Ki small but non-zero (to achieve tracking). 3. Kd =− ITD to counter-act the effect of the top drive inertia. The problem with this approach is that −Kd > ITD leads to instability, while −Kd < ITD rapidly degrades performance. Furthermore, the high Kd term leads to excessive noise sensitivity. The approach of Softspeed [19] is to remove the negative integral-action altogether, sets the proportional action to Kp = 4 p ,

(45)

and then tunes the integral gain according to 2

(46)

s − a0 aL −st D ˆ d(t), e s + aL a0

(52)

where dˆ is an estimate of the disturbance. This disturbance canceling term results in the following contribution to the top drive velocity set-point ωd =

that is

2 , Ki = (2fc ) ITD

Hence, to try to cancel the effect of the disturbance, we will use the disturbance canceling term ud = −

t

(49)

1 e−st D ˆ d(t). s + aL IBHA

(53)

4.3.1. Estimating the disturbance magnitude The disturbance will be assumed to take the form of a Heaviside step function acting the instant the BHA releases from the stick phase. Hence, the task of estimating the disturbance equates to estimating the point in time of the release and magnitude of this function. After the release, the resulting change in torque on the BHA affects the top-drive after a delay of tD , see Fig. 8, at which point in time we want to cancel this effect. Considering the case of a stick slip limit cycle being initiated for a industry standard high-gain PI controller, the controller keeps the topdrive angular velocity ω0 approximately constant, while the changes in motor torque  m are due to the disturbance, see Figs. 4 and 5. We have m = −

aL 2e−st D d(t). s + aL

(54)

Hence, considering the field data of Figs. 4 and 5 as an example, we can easily estimate the disturbance magnitudes to approximately 7.5 kNm and 19 kNm, respectively. Alternatively, if the dry and sliding friction coefficient of the drill-string–wellbore interface is known, a disturbance magnitude estimate could be computed directly from a torque and drag model of the well [3]. Consequently, we can parametrize the uncertainty of our estimate of the disturbance in these two parameters: the disturbance

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Fig. 8. Timeline showing the various timing constants of the procedure.

ˆ and the time after release t at which we magnitude estimate |d|, initiate the disturbance cancellation. For a given case of a distributed model startup simulation, there is a range of these two ˆ t , for which the stick slip oscillations are avoided. parameters, |d|, We say that the size of this range gives an indication of the robustness of the procedure, and will investigate this range in Section 5.4. 4.4. Tracking In this Section, we take advantage of the flatness property derived in Section 3.4 to solve the trajectory planning problem. Writing the flat output z(t) as the feed-forward contribution to the (presumed) linear response of the BHA angular velocity ωL (t), the feed-forward tracking contribution to the top drive set-point becomes, from (36), ωf =

˙ + tD ) − z(t ˙ − tD ) z(t + tD ) + z(t − tD ) z(t , + 2 2aL

(55)

Fig. 9. The mollifier step function approximation of (57) and its two first derivatives used by the feed-forward flatness controller.

while the actuation contribution term is, from (37), a0 2 ˙ − tD ) − a0 z(t − tD ). u = ω˙ f + a0 ωf + z(t ITD f aL

(56)

Plugging a reference trajectory zref (t) into (55), (56) yields the associated feed-forward control law F. We will use a mollifier (semi-analytical function) to construct transition trajectories that are booth smooth, and have vanishing derivatives at the end and start points. Specifically we use the integral of the “bump” function as a smooth approximation of the step function with vanishing derivatives:



t

um (t) =

( − 1)d

(57)

0

where the bump function is given as



⎧ 1 ⎪ exp − ⎪ 2 ⎪ 1−t ⎨ for t ∈ (−1, 1)   1 (t) = 1 exp − d

⎪ ⎪ 1 − 2 ⎪ ⎩ −1

Fig. 10. The resulting feed-forward terms of (55) and (56) for z(t) = um (t/3).

(58) 5. Simulations

otherwise

0

This mollifier step function and its derivatives are illustrated in Fig. 9. It can be used to construct reference trajectories for the downhole RPM, by choosing an amplitude, switching time, and switching duration which we denote as Am , tsd , tsr , respectively. Hence, z(t) = Am um

t − t sd

tsr

.

The resulting feed-forward terms are illustrated in Fig. 10.

(59)

We will argue for the benefit for each of the three components of the control design by considering a series of simulations showing their impact on the dynamics. 5.1. Free drill string To highlight certain features of the model dynamics, we initially consider the case of a drill string spinning freely with a uniform velocity of 60RPM and then change the velocity set-point. We will consider the three cases

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31

Fig. 11. Comparison of response to velocity set-point changes for: (a) the industry standard aggressive PI controller, (b) SoftTorque/speed feedback controller, (c) SoftTorque/speed feedback with the proposed differential flatness trajectory planning feed-forward controller. Set-point changes uses the mollifier (57) with a tsr = 5 s switching duration.

• The industry standard high gain PI controller (48). • The baseline SoftTorque/torque PI controller (45),(46). • The baseline SoftTorque/torque PI controller with the proposed differential flatness trajectory planning feed-forward controller, described in Section 4.4. For case (a) and (b), the velocity set-point is changed in two approximate steps according to ωSP = 60 + 90um

t − 20 5

− 60um

t − 40 5

,

(60)

while for case (c) Eq. (60) is used as the desired trajectory zref to generate the set-point and actuation contributions ωf and uf according to (55) and (56). The simulation results are shown in Fig. 11. We see that without the trajectory feed-forward controller the set-point changes induces oscillations on the BHA velocity ωL . These oscillations are sustained with the high-gain PI controller, see Fig. 11(a), while the SoftTorque/torque controller dampens the oscillations over time, see Fig. 11(b). Such oscillations are avoided with the feed-forward flatness controller, see Fig. 11(c). 5.2. Stick slip Next we consider a rotation startup such as is required after each pipe connection procedure while drilling a well. In this scenario the stationary drill string is initially kept in place by the Coulomb friction until enough torque is built up to overcome it. At which point, pipe-rotation is initiated and the Coulomb friction is reduced as it changes from static to dynamic. The resulting release of the stored

energy potentially pushes the drill string into a destructive stick slip limit cycle. Field data examples of this is shown in Figs. 4 and 5. We refer to [3] for a more detailed description of this phenomena. A replication of this effect, with the industry standard high-gain PI controller (48), is shown in Fig. 12(a). This stick slip limit cycle behavior is considered a significant problem in drilling and was the chief motivation for the development of the SoftSpeed/Torque controllers. Indeed, the SoftSpeed/Torque controllers work for certain cases, and it is not difficult to construct parameter sets where using only this feedback is sufficient for avoiding stick slip. However, the reason this topic is still an area receiving significant ongoing research interest is that the SoftSpeed/Torque controllers are often insufficient to avoid stick slip, which is indeed the case here as well, see Fig. 12 b) where the baseline SoftTorque/torque PI controller (45), (46), is used. The novel approach proposed in the present paper is to handle the reduction between static and dynamic Coulomb friction as a disturbance that is estimated from previous startups and then canceled with the feed-forward disturbance canceling term derived in Section 4.3. The timing of this approach is summarized in Fig. 8 . The result of this approach is shown in Fig. 12(c), with the torque trend shown in Fig 13 and the topside Riemann invariants shown in Fig. 14. The disturbance cancellation is initiated when an increasing ˇ0 is estimated, at which point the disturbance feed-forward terms ωd , ud are updated according to (52) and (53). From the torque trend in Fig. 13 case a) we set the estimate dˆ = 10 kNm. The goal is to cancel the effect in the reflecting ˛0 , as shown in Fig. 14. This figure shows that the added actuation of the feed-forward term

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Fig. 12. Comparison of start-up procedure for a pipe subject to Coulomb friction potentially causing stick slip for: (a) the industry standard aggressive PI controller, (b) SoftTorque/speed feedback controller, (c) SoftTorque/speed feedback with the proposed feed-forward disturbance rejection and differential flatness trajectory planning terms, (d) same as in (c) but with a 50 kNm torque limitation, and, (e) counteracting the torque limitation by increasing robustness through increasing the time the ωL set-point RPM is kept high. The corresponding actuation torques are shown in Fig. 13.

successfully removes most of the reflection compared to the case without using the feed-forward term D. Simultaneous to this, the feed-forward trajectory is updated to bring the drill string velocity back down to the desired 60 RPM without inducing additional oscillations. 5.3. Effect of torque constraints In field scenarios, the amount of power available to the rig – due to a constraint in on-site generating capacity – or presence of torque limiters in the top drive controller will set a maximum torque constraint. On typical AC top drives, torque is proportional to electric current, and a large increase in current may cause a power overload. Similarly, a large increase in torque may also exceed the torque

limits on gearing or other components in the top drive or nearsurface rotary equipment. This constraint on motor torque can limit the effectiveness of the feedforward disturbance canceling part of the controller. This is illustrated in Fig. 12 d), where a 50 kNm max torque constraint is used on the actuation. In this case, torsional wave is only partly canceled, and the remaining reflection is sufficient to initiate a stick slip limit cycle. To deal with issues such as this, the robustness of the presented approach can be improved by increasing the time at which the RPM is kept at a high setpoint before being lowered to the desired final RPM setpoint. This is illustrated in Fig. 12 e) where the setpoint is kept at a high RPM for 4 seconds before being lowered to the desired value of 60 RPM. This enables sufficient damping of the reflections and stick slip is avoided.

U.J. F. Aarsnes et al. / Journal of Process Control 70 (2018) 24–35

33

Fig. 13. Comparison of the actuation torque  m for the five cases shown in Fig. 12.

5.4. Robustness analysis

Fig. 14. Comparison of the topside Riemann variables with and without the disturbance cancelling controller D(s), corresponding to Fig. 12(c) and (b), respectively.

Since the approach is based on employing the estimate of the ˆ which is not a-priori known, it is of interest the disturbance, d, to gauge to what degree uncertainty in this estimate can be tolerated. Towards this end, a comprehensive sensitivity study has been ˆ of the discarried out where various timings t and magnitudes |d| turbance estimate was repeatedly used for the same start-up to find the required total transition time t + tsd , see Fig. 8, needed to avoid entering a stick slip limit cycle. This study is summarized in Fig. 15, which shows the contours corresponding to the required total transition time for a given magnitude and timing ˆ t . pair |d|, We see that the least amount of total transition time is required ˆ = 12 kNm, t = tD = 0.81, which corresponds to the around the |d| magnitude and timing that was computed a priori. Further, we note that a great deal of uncertainty is allowed by the approach when the transition time is allowed to be sufficiently large, in this case ttt ≈ 30–35 s of total transition time enables us to avoid stick slip even for a very-wide range of disturbance estimates.

ˆ given in kNm, and actuation timing t , given as difference from the calculated time. The Fig. 15. Robustness parametrized in the magnitude of the disturbance estimate, d, ˆ t pair. contour lines denotes the total transition time, in seconds, of the startup procedure required to avoid initiating a stick slip limit cycle for the given d,

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U.J. F. Aarsnes et al. / Journal of Process Control 70 (2018) 24–35

Appendix A. Numerical implementation In the numerical implementation of the model the wave equation (1), (2) is transformed into transport equations discretized using a first order upwind scheme. This choice is made to ensure numerical robustness and to avoid spurious oscillations, as higher order schemes perform poorly due to the temporal discontinuities introduced by the distributed differential inclusions which are used to represent the Coloumb friction. Numerical accuracy can then be ensured by having a sufficiently fine spatial grid, and this is an amenable approach due to the fact that simulation speed is not of critical importance for the present study. In all simulations a spatial grid of 500 cells is used for the drill string and the time-step is chosen to enforce the Courant–Friedrichs–Lewy (CFL) condition [10]. In the numerical treatment of the model, F is implemented as follows. For cell size x and and time step t, and at cell # j and time step # k t 1 (˛ − ct (˛ − ˛j−1 )+ 2 t j x j

F˜jk = Fig. 16. Comparison of the distributed and semi-lumped approximation transfer functions of drill string torque with an angular velocity input.

ˇj + ct

(A.1)

t − ˇj ) + tkt (˛kj + ˇjk )), (ˇ x j+1

(A.2)

and limited by We conclude from the robustness analysis that there is a tradeoff between performance, parametrized in total transition-time ttt , ˆ And, and robustness, parametrized in allowable uncertainty in d. that if a sufficient lenient total transition-time ttt is chosen, the proposed approach is robust to the aforementioned uncertainties.

Fjk =

⎧ ⎪ ⎪ ⎨ sgn(F˜jk ) min(|F˜jk |, Fc ),

|˛kj + ˇjk | 2

≤ ωc

k k ⎪ ⎪ ⎩ sgn(F˜ k ) min(|F˜ k |, f F ), |˛j + ˇj | > ω rat c c j j

(A.3)

2

The model is updated with an upwind scheme according to 6. Conclusion

˛k+1 = ˛kj − ct j

t k (˛ − ˛kj−1 ) − tkt (˛kj + ˇjk ) − Fjk x j

(A.4)

We have presented a strategy designed to avoid torsional stickslip oscillations at the start-up of a drilling operation, e.g. after a connection. It consists in a feedforward controller, which can easily be added to the standard industry feedback controllers, without disturbing their closed-loop behavior. The approach has been validated on a model shown to reproduce, in open-loop simulations, the dynamics of a drilling system during start-up. We show that this method is somewhat robust to uncertain design parameters in the controller, provided performance (in terms of the start-up duration) is relaxed enough. Future works include the on-line adaptation of the uncertain parameters, based on topside measurements. More precisely, we believe that the disturbance cancellation time t and magnitude dˆ can be effectively adjusted from the measurement of topside torque before the complete release of the drillstring. Besides, the fidelity of the simulation model described in Section 2 to field data makes it amenable to estimate in real time bottom velocity from topside measurements. This requires the design of an observer for a complex system composed of coupled Partial Differential Equations and Ordinary Differential Equations with nonlinear source terms, which, to our best knowledge, would be a novel result. This will also be the topic of future work.

ˇjk+1 = ˇjk − ct

t k k (ˇ − ˇj+1 ) − tkt (˛kj + ˇjk ) − Fjk . x j

(A.5)

Acknowledgment This research was financially supported by ConocoPhillips, AkerBP, Statoil, Wintershall and the RCN grant (203525/O30) DrillWell, by European Union’s Seventh Framework Programme for research, technological development and demonstration under Marie Curie grant agreement no [608695], and the FRIPRO Mobility Grant Fellowship Programme (FRICON).

Appendix B. Error of lumped approximation We will derive the input–output description of the drill string between the input ω0 and the output  0 , without the top-drive (as it acts as a lowpass filter masking the approximation error). Denote this transfer function T0 (s) ≡ g0 (s).

0

(B.1)

For a two-section drill string, ignoring the non-linear part of the source term, it can be found as [4]: gc (s) = Zc

ZL + Zc tanh c Zc + ZL tanh c

(B.2)

g0 (s) = Zp

gc + Zp tanh p . Zp + gc tanh p

(B.3)

with Jp Gp Zp := ct Zc :=

Jc Gc ct

 

k 1+ , s 1+

k , s

Lp p := s ct

 1+

k , s

(B.4)

1+

k , s

(B.5)



c := s

Lc ct

and with the load impedance ZL = 0 for the case of bit off bottom. Now consider the following approximation of the two-section drill string, where the collar section has been lumped into a single inertia element: g˜ 0 (s) = Zp

ZBHA + Zp tanh p , Zp + ZBHA tanh p

(B.6)

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ZBHA = Jc Lc s(1 + k/s) ≡ IBHA (s + 1/k).

(B.7)

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