Constrained MPC Design for Heave DisturbanceAttenuation in Offshore Drilling Systems

Amirhossein Nikoofard∗, Tor Arne Johansen, Hessam Mahdianfar, and Alexey Pavlov

Abstract—This paper presents a constrained model predictivecontrol scheme for regulation of the annular pressure in a wellduring managed pressure drilling from a floating rig subjectto heave motion. The results show that closed-loop simulationwithout disturbance has a fast regulation response and withoutany overshoot. The robustness of controller to deal with heavedisturbances is investigated. The constrained MPC shows gooddisturbance rejection capabilities. The simulation results showthat this controller has better performance than a PID controllerand is also capable of handling constraints of the system withthe heave disturbance.

Index Terms—Managed pressure drilling, heave Compensationand model predictive control.

I. INTRODUCTION

In drilling operations, a fluid called mud is pumped downthrough the drill string and flows through the drill bit at thebottom of the well (see Figure 1). Then the mud flows upthe well annulus carrying cuttings out of the well. To avoidfracturing, collapse of the well, or influx of formation fluidssurrounding the well, it is crucial to control the pressure in theopen part of the annulus within a certain operating window.In conventional drilling, this is done by mixing a mud ofappropriate density and adjusting mud pump flow-rates. Inmanaged pressure drilling (MPD), the annulus is sealed andthe mud exits through a controlled choke, allowing for fasterand more precise control of the annular pressure. In automaticMPD systems, the choke is controlled by an automatic controlsystem which manages the annular mud pressure to be withinspecified upper and lower limits. Different aspects of modelingfor MPD have been studied in the literature [1]–[5]. Estimationand control design in MPD has been investigated by severalresearchers [5]–[10], focusing mainly on pressure controlduring drilling from a fixed platform.

When designing MPD control systems, one should takeinto account various operational procedures and disturbancesthat affect the pressure inside the well. There is a specificdisturbance occurring during drilling from floaters thatsignificantly affects MPD operations. In this case, the rigmoves vertically with the waves, referred to as heave motion.As drilling proceeds, the drill string needs to be extendedwith new sections. Thus, every couple of hours or so, drilling

Amirhossein Nikoofard, Tor Arne Johansen and Hessam Mahdianfar arewith the Center for Autonomous Marine Operations and Systems (AMOS),Department of Engineering Cybernetics, Norwegian University of Science andTechnology, Trondheim, Norway.

Alexey Pavlov is a principal researcher at Statoil Research Centre, Depart-ment of Intelligent Well Construction, Porsgrunn, 3910, Norway.

*Corresponding author :amirhossein.nikoofard@itk.ntnu.no

Figure 1. Schematic of an MPD system (Courtesy of Glenn-Ole Kaasa,Statoil Research Centre.)

is stopped to add a new segment of about 27 meters to thedrill string. During drilling, a heave compensation mechanismis active that isolates the drill string from the heave motionof the rig. However, during connections, the pump is stoppedand the string is disconnected from the heave compensationmechanism and rigidly connected to the rig. The drill stringthen moves vertically with the heave motion of the floatingrig, and acts like a piston on the mud in the well. The heavemotion may be more than 3 meters in amplitude and typicallyhas a period of 10-20 seconds, which causes a severe pressurefluctuations at the bottom of the well. Pressure fluctuationshave been observed to be an order of magnitude higher thanthe standard limits for pressure regulation accuracy in MPD,which is about ±2.5 bar. Downward movement of the drillstring into the well increases pressure(surging), and upward

movement decreases pressure (swabbing). Excessive surgeand swab pressures can lead to mud loss resulting fromhigh pressure fracturing the formation or a kick-sequence(uncontrolled influx from the reservoir) that can potentiallygrow into a blowout as a consequence of low pressure.

Rasmussen et al. [11], compared and evaluated differentMPD methods for compensation of surge and swab pressure.Pavlov et al. [12], presented two nonlinear control algorithmsfor handling heave disturbances in MPD operations.Mahdianfar et al. [13], [14], designed an infinite-dimensionalobserver that estimates the heave disturbance. This estimationis used in a controller to reject the effect of the disturbanceon the down-hole pressure.

Model predictive control (MPC) is one of the popular con-trollers for complex constrained multivariable control problemin industry and has been the subject of many studies sincethe 1970s (e.g. see [15]–[18]). At each sampling time, aMPC control action is acquired by the on-line solution of afinite horizon open-loop optimal control problem. Althoughmore than one control action is obtained, only the first oneis implemented to the plant. At the next sampling time, thecomputation is repeated with new measurements obtained fromthe system. The purpose of this paper is to design a constrainedMPC scheme for controlling the pressure during MPD oilwell drilling. One of the criteria for evaluating the controllerperformance is its ability to handle heave disturbances.Thisscheme is compared with a standard PID-control scheme.

In the following sections, a model based on mass andmomentum balances that provides the governing equations forpressure and flow in the annulus is given. A stochastic mod-eling of waves in the North Sea is used and heave disturbanceinduced by elevation motion of sea surface is modeled. Thedesign of a constrained MPC scheme is presented and appliedon Managed Pressure Drilling to show that this controller hasrobust performance with a comparison with PID controller.

II. MATHEMATICAL MODELING

A. Annulus flow dynamics

The governing equations for flow in an annulus is derivedfrom mass and momentum balances [2]

∂p

∂t= − β

A

∂q

∂x(1)

∂q

∂t= − A

ρ0

∂p

∂x− F

ρ0+Ag cos(α(x)) (2)

where p(x, t) and q(x, t) are the pressure and volumetric flowrate at location x and time t, respectively. The bulk modulusof the mud is denoted by β. A(x) is the cross section area, ρis the (constant) mass density, F is the friction force per unitlength, g is the gravitational constant and α(x) is the anglebetween gravity and the positive flow direction at location xin the well (Figure 2). To derive a set of ordinary differentialequations describing the dynamics of the pressures and flowsat different positions in the well, equations (1) and (2) are

discretized by using a finite volumes method. To solve thisproblem, the annulus is divided into a number of controlvolumes, as shown in Figure 2, and integrating (1) and (2)over each control volume.

volume. This derivation is done similar to the one done in[10].

To incorporate the important pressure dynamics createdby the drill string movement, there are two things to bearin mind: First, the volume of the annulus will continuouslychange by vd(t)Ad, where Ad is the drill string cross sectionarea, due to the top of the drill string moving in and out ofthe well with velocity vd. Second, one also has to considerthe fact that the cross section of the drill bit will in generalbe larger than that of the rest of the drill string. This willcause the generated flow to be “squeezed” in over a smallercross section, increasing the velocity and thus also the frictionaround the drilling bit. (See also Figure 2.) Taking all theseissues into account, the finite volume method results in thefollowing set of equations

p1 =�1

A1l1(�q1 � vdAd)

pi =�i

Aili(qi�1 � qi) , i = 2, 3, .., N � 1

pN =�N

AN lN(q(N�1) � qc + qbpp) (3)

qi =Ai

li⇢i(pi � pi+1) �

Fi (qi) Ai

⇢ili� Aig

�hi

li,

i = 1, 2, ..N � 1 (4)

Here, the numbers 1...N refer to control volume number,with 1 being the lower most control volume representingthe down hole pressure (p1 = pbit), and N being the uppermost volume representing the choke pressure (pN = pc). Thelength of each control volume is denoted l, and the heightdifference is �h. Notice that since the well may be non-vertical, l and �h may in general differ from each other. Toour disposal for control are the back pressure pump flow qbpp

and the choke flow qc. The flow from the back pressure pumpqbpp is linearly related to the pump frequency and cannot bechanged fast enough to compensate for the heave-inducedpressure fluctuations. Therefore, it is the choke flow thatis used primarily for control. It is modelled by an orificeequation:

qc = Kc

ppc � p0G (u) (5)

Here, Kc is the choke constant corresponding to the areaof the choke and the density of the drilling fluid. p0 is the(atmospheric) pressure down stream the choke and G(u) is astrictly increasing and invertible function relating the controlsignal to the actual choke opening, taking its values on theinterval [0, 1].

B. Friction Model

To model the friction force acting on each control volume,we propose to use standard, Newtonian friction factor correla-tions (see e.g. [12] for details). This is due to the observationsfrom measurement data from full scale tests [1] that suggestthe friction force in the annulus is a linear function of the

A

qN-1

qN-2

qc

p1

q1

β,ρ

pN-1

pN

qbpp

α

g

x

l=Δh

DrillString

DrillingBit

Figure 2. Control volumes of annulus hydraulic model.

flow rate, at least for the modest flow rates that can beexpected without externally forced flow (zero flow from themain pump). This corresponds well to Newtonian frictionfactors for laminar flow. See Figure 3 for the measured steadystate friction drops.

0 0.005 0.01 0.015 0.02 0.0250

5

10

15

20

25

30

35

40

45

50

q (m3/s)

p (

bar)

drill stringannulus

Figure 3. Friction losses from observed, full scale testing data

Taking the special geometry of the annulus region intoaccount, it has been proposed [13] to use a modified Reynoldsnumber to calculate the friction factor. However, to keepthe detail level manageable, we will in this paper lump theentire friction model into a constant (or, more likely, slowlyvarying) friction coefficient kfric, giving the resulting friction

Figure 2. Control volumes of annulus hydraulic model [2]

Landent et al. found that [2] a five control volumes modelcould capture the main dynamics of the system in case ofheave disturbance for a well from Ullrigg1 test facility with aparticular length of about 2000m and with water based mud.We will use parameters corresponding to that well as a basecase throughout the paper. The set of nine ordinary differentialequations describing five control volumes in the annulus areas follows [1] and [19]

p1 =β1A1l1

(−q1 − vdAd) (3)

p2 =β2A2l2

(q1 − q2) (4)

p3 =β3A3l3

(q2 − q3) (5)

p4 =β4A4l4

(q3 − q4) (6)

p5 =β5A5l5

(q4 − qc + qbpp) (7)

qi =Ailiρi

(pi − pi+1)− Fi(qi)Ailiρi

−Aig∆hili

(8)

qc = Kc

√pc − p0G(u) (9)

where, i = 1, ..., 4, and the numbers 1, ..., 5 refer to controlvolume number, with 1 being the lower most control volumerepresenting the down hole pressure (p1 = pbit), and 5being the upper most volume representing the choke pressure

1Ullrigg is a full scale drilling test facility located at International ResearchInstitute of Stavanger (IRIS).

(p5 = pc). vd is the heave vertical velocity due to ocean waves.The length of each control volume is denoted by l, and theheight difference is ∆h. Notice since the well may be non-vertical, li and ∆hi may in general differ from each other. Themeans for pressure control are the back pressure pump flowqbpp and the choke flow qc. The flow from the back pressurepump qbpp is linearly related to the pump frequency and cannotbe changed fast enough to compensate for the heave-inducedpressure fluctuations. Therefore, it is the choke flow that isused primarily for control, which is modeled by nonlinearorifice equation. Kc is the choke constant corresponding to thearea of the choke and the density of the drilling fluid. p0 is the(atmospheric) pressure downstream the choke and G(u) is astrictly increasing and invertible function relating the controlsignal to the actual choke opening, taking its values on theinterval [0, 1].Based on experimental results from full scale tests at Ullrigg,the friction force in the annulus is considered to be a linearfunction of the flow rate [2]. Friction force on the ith controlvolume is

Fi(qi) =kfricqiAi

(10)

where kfric is constant (or, more likely, slowly varying)friction coefficient.

B. Waves Response Modeling

Environmental forces due to waves, wind and ocean currentsare considered disturbances to the motion control systemof floating vessels. These forces, which can be describedin stochastic terms, are conceptually separated into low-frequency (LF) and wave-frequency (WF) components, [20].

Ocean waves are random in terms of both time and space.Therefore, a stochastic modeling description seems to bethe most appropriate approach to describe them. The modelpresented in this part is for purposes of controller design andverification. For control system design it is common to assumethe principle of superposition when considering wind and wavedisturbances. For most marine control applications this is agood approximation [20].

During normal drilling operations the WF part of the drill-string motion is compensated for by the heave control system,[21]–[23]. However, during connections the drill-string isdisconnected from the heave compensation mechanism andrigidly connected to the rig. Thus, it moves vertically with theheave motion of the floating rig, and causes severe down-holepressure fluctuations.

1) Linear Approximation for WF Position: When simulat-ing and testing feedback control systems, it is useful to havea simple and effective way of representing the wave forces.Here the motion Response Amplitude Operators (RAO) arerepresented as a state-space model where the wave spectrumis approximated by a linear filter. In this setting RAO vesselmodel is represented in Figure 3,where Hrao(s) is the waveamplitude-to-force transfer function and Hv(s) is the force-to-motion transfer function. In addition to this, the response of

the motion RAOs and the linear vessel dynamics in cascadeis modeled as constant tunable gains, [20]:

K = diag{K1,K2,K3,K4,K5,K6} (11)

This means that the RAO vessel model is approximated as(Figure 3)

Hrao(s)Hv(s) ≈ K (12)

The fixed-gain approximation (equation 12) produces goodresults in a closed-loop system where the purpose is to testrobustness and performance of a feedback control system inthe presence of waves.

Ak τwave1Hv(s)Hrao(s)Hs(s)

w ηw

Figure 3. Linear approximation for computation of wave-induced positions.

If the fixed gain approximation (equation 12) is applied, thegeneralized WF position vector ηw in Figure 3 becomes

ηw = KHs(s)w(s) (13)

where Hs(s) is a diagonal matrix containing linear approxi-mations of the wave spectrum S(ω). The WF position for thedegree of freedom related to heave motion becomes

ηhw = Khξh (14)ξh(s) = hh(s)wh(s) (15)

where hh(s) is a linear approximation of the wave spectraldensity function S(ω) and wh(s) is a zero-mean Gaussianwhite noise process with unity power across the spectrum:

Phww(ω) = 1.0 (16)

Hence, the power spectral density (PSD) function for ξh(s)can be computed as

Phξξ(ω) = |hh(jω)|2Phww(ω) = |hh(jω)|2 (17)

Here we approximate S(ω) with Phξξ(ω), for instance bymeans of nonlinear regression, such that Phξξ(ω) reflects thepower distribution of S(ω) in the relevant frequency range.

2) JONSWAP Spectrum: The JONSWAP spectrum is rep-resentative for wind-generated waves under the assumption offinite water depth and limited fetch, [20], [24]. The spectraldensity function is written

S(ω) = 155H2s

T 41

ω−5 exp(−944

T 41

ω−4)γY (18)

where Hs is the significant wave height, T1 is the averagewave period, γ = 3.3 and

Y = exp

[−(0.191ωT1 − 1√

2σ

)2](19)

where

σ =

{0.07 for ω ≤ 5.24/T1

0.09 for ω > 5.24/T1(20)

The modal period, T0, is related to the average wave periodthrough T1 = 0.834T0, [20].

Figure 4, produced using MSS Toolbox2, shows the JON-SWAP spectrum power distribution curve. The parametervalues for Hs and T0 are taken from [25]. From Figure 4we can see that the JONSWAP spectrum is a narrow bandspectrum, and its energy is mainly focused on 0.5−1.5 rad/s,and the peak frequency is ω0 = 0.7222 rad/s.

0.5 1 1.5 2 2.50

0.5

1

1.5

2

2.5

3

3.5

4JONSWAP Spectrum

[rad/s]

S() [

m2 s

]

JONSWAP spectrumLinear approximation

Figure 4. JONSWAP spectrum and its approximation.

3) Second-Order Wave Transfer Function Approximation:As discussed earlier a linear wave response approximation forHs(s) is usually preferred by ship control systems engineers,because of its simplicity and applicability:

hh(s) =2λω0σs

s2 + 2λω0s+ ω20

(21)

where λ = 0.1017, σ = 1.9528, Hs = 4.70, T0 = 8.70,ω0 = 0.7222 are typical parametrs. The transfer functionapproximation is shown in Figure 4.

III. CONTROLLER DESIGN

The model described by equations (3)-(9) is in form of anonlinear strict feedback system, with an unmatched stochasticdisturbance. By considering aj =

βj

Aj lj, bj =

Aj

ljρj, cj =

Kfric

ρj lj,

the model in state-space form would be

{X = AX +Bua +B1 + Edy = CX

(22)

2MSS. Marine Systems Simulator (2010), version 3.3. Viewed 26.03.2011,http://www.marinecontrol.org.

where

X =[p1 q1 p2 q2 p3 q3 p4 q4 p5

]T

A =

0 −a1 0 0 0 0 0 0 0b1 −c1 −b1 0 0 0 0 0 00 a2 0 −a2 0 0 0 0 00 0 b2 −c2 −b2 0 0 0 00 0 0 a3 0 −a3 0 0 00 0 0 0 b3 −c3 −b3 0 00 0 0 0 0 a4 0 −a4 00 0 0 0 0 0 b4 −c4 −b40 0 0 0 0 0 0 a5 0

B =

[0 0 0 0 0 0 0 0 a

]TB1 = −263.7814

[0 1 0 1 0 1 0 1 0

]TE =

[−22.0857 0 0 0 0 0 0 0 0

]TC =

[1 0 0 0 0 0 0 0 0

](23)

andua = qbpp − qc (24)

The heave disturbance vd in equation (3) will be compensatedby using constrained MPC designed in the next part. Note thatthe hydrostatic pressures in equation (8) are included in thestates pi in (3)-(7). p1 = pbit is the output of the process.

A. Constrained MPC design

Consider the discrete-time linear time-invariant input-affinesystem{

x(k + 1) = Ax(k) +Bu(k) +B1 + Ed(k)y(k) = Cx(k),

(25)

while fulfilling the constraints

ymin ≤ y(k) ≤ ymax, umin ≤ u(k) ≤ umax (26)

at all time instants k ≥ 0.In (25)-(26), n , p and m are the number of states, outputs andinputs respectively, and x(k) ∈ <n , y(k) ∈ <p , d(k) ∈ <nand u(k) ∈ <m are the state, output, disturbance and inputvectors respectively.The constrained MPC solves a constrained optimal regulationproblem at each time k. The following optimization problem

minU

∆={uk,···,uk+N}

{J(u, y, r) =

N∑i=1

[(u(k + i|k)TRu(k + i|k)+

(y(k + i|k)− r(k + i|k))TQ(y(k + i|k)− r(k + i|k))]}

s.t ymin ≤ yi+k|k ≤ ymax i = 1, · · · , N,umin ≤ ut+k|k ≤ ymax i = 1, · · · , N,xk|k = x(k) (27)xi+k+1|k = Axi+k|k +Bui+k|k +B1 + Edi+k|k,

yi+k|k = Cxi+k|k

is solved on-line at each sample time, where N, J and rare the finite horizon, cost function and reference trajectory

respectively, the subscript ”(k + i|k)” denotes the valuepredicted for time k + i, we assume that Q and R are thepositive definite matrices.As the states x(k) is not directly measurable, prediction arecomputed from estimation of states. The state observer isdesigned to provide estimation of states x(k), since the pair(C,A) is detectable. The controller computes the optimalsolution U by solving the quadratic programing (QP). Ifthe future value of disturbances and/or measurement ofdisturbances are not known then disturbances are assumed tobe zero in MPC predictions.

Controller parameters such as weight of inputs, inputs rateand outputs and control horizon must be tuned to achieve thegood performance and stability in this problem. The predictionhorizon should be tuned properly to ensure the closed-loopstability of the control system.

IV. SIMULATION RESULTS

The parameters for simulations, identified from the IRIS-Drill simulator, are given in Table I.

Table IPARAMETER VALUES

Parameter Value Parameter Valuea 2.254× 108 [Pa/m3] g 9.806 [m/s2]b 4.276× 10−8 [m4/Kg] A 0.0269 [m2]Kf 5.725× 105 [sPa/m3] e 0.2638 [m3/s2]c 14.4982 [1/sm2] Ad 0.0291 [m2]KG 0.0650 Kc 2.32qbpp 369.2464 [m3/s] p0 101325 [pa]

The time-step used for discretizing the dynamic optimiza-tion model was 0.1 s. The input weight (R), input rate weight(Rδu), output weight (Q) and prediction horizon (N ) are cho-sen 150, 0, 17 and 100 respectively. In this problem, predictionhorizon is set large compared with the settling time to ensurethe closed-loop stability of the control system. The weightsspecify trade-offs in the controller design. Choosing largeroutput weight or smaller input weight results in overshootin closed-loop response and sometimes broken constraints. Inother words, if you choose larger input weight or smalleroutput weight then the closed loop response is slower orsometimes unstable.

To compare the impact of MPC on the drilling system withother controllers, a PID controller was applied to the systemas well. A PID controller is chosen due to its popularity inthe industry. Proportional, integral and derivative gain arechosen 0.75, 0.002 and -1 respectively. The Bode plot of theloop transfer function with the PID is shown in Figure 5.Bandwidth with PID is less than 1.3 rad/sec, and the phasedrops very quickly. So, it is not realistic to get a bandwidth ofabout 5 rad/sec which would be desirable for this disturbance.

Three different simulations are performed. The firstsimulation is shown in Figures 6 and 7, where the nominal

−500

−400

−300

−200

−100

0

100

Magnitude (

dB

)

10−4

10−3

10−2

10−1

100

101

102

103

−900

−720

−540

−360

−180

0

Phase (

deg)

Bode DiagramGm = 11.8 dB (at 1.23 rad/s) , Pm = 59.7 deg (at 0.32 rad/s)

Frequency (rad/s)

Figure 5. Bode plot of the loop transfer function with the PID.

model is used for generating the measurements. A softconstraint of 2.5bar (compared to the reference pressure) anda constraint of choke opening taking its values on the interval[0, 1] are included in the constrained MPC optimization.Figure 6 compares the responses of the PID controllerand constrained MPC to regulate set point trajectory. Inthe proposed MPC controller, the bottom-hole pressureapproaches to set point fast without any overshoot. Incomparison to MPC controller, PID controller has someovershoot and somewhat slower response. The choke controlsignal in constrained MPC is illustrated in Figure 7.

0 10 20 30 40 50 60257

258

259

260

261

262

263

264

265

266

Time (s)

Pre

ssure

(bar)

Bottom−hole pressure

PID

MPC

Figure 6. Bottom-hole pressure without disturbance.

The second simulation is shown in Figures 8, 9 and 10,where the nominal model with heave disturbance is usedfor generating the measurements. The same constraints as inprevious simulation are enforced to the controller. Figure 8compares the responses of constant input (qbpp = qc) andconstrained MPC to track the set point reference with existingheave disturbance. A constant input couldn’t reduce the effectof heave disturbance and track the set point reference. Figure9 compares the responses of PID controller and constrainedMPC to track the set point reference with a heave disturbance.It is found that the MPC controller is capable of maintainingthe constraints whereas the PID controller is not. The choke

0 10 20 30 40 50 600.26

0.28

0.3

0.32

0.34

0.36

0.38

0.4

0.42

0.44

Time (s)

Choke o

penin

g

Choke Input

Figure 7. MPC control signal to the choke without disturbance.

control signal is illustrated in Figure 10. As indicated in thisfigure, the constrained MPC shows good disturbance rejec-tion capabilities. Figure 11 shows heave disturbance pressurevariations.

0 20 40 60 80 100 120 140 160 180 200250

252

254

256

258

260

262

264

266

268

270

Time(s)

Pre

ssu

re(b

ar)

Bottom−hole Pressure

Constant Input

Constrained MPC

Figure 8. Bottom-hole pressure

0 20 40 60 80 100 120 140 160 180 200256

258

260

262

264

266

268

270

Time (s)

Pre

ssure

(bar)

Bottom−hole pressure

MPC

PID

Figure 9. Bottom-hole pressure with heave disturbance.

The last simulation is shown in Figure 12 where the nominalmodel with heave disturbance is used for generating the mea-surements. The same constraints as in previous simulation areenforced to controller. In this simulation, the heave disturbanceis assumed to be predictable. The heave disturbance is givenby vd = cos(2πt/12)[m], where 2π/12 corresponds closelyto the most dominant wave frequency in the North Atlantic,with reference to the JONSWAP spectrum [2], [19]. The inputweight for MPC with future knowledge of heave disturbanceis chosen R = 85. Figure 12 compares the responses of MPCcontroller without future knowledge of heave disturbance and

0 20 40 60 80 100 120 140 160 180 2000.26

0.28

0.3

0.32

0.34

0.36

0.38

0.4

0.42

0.44

0.46

Time (s)

Choke o

penin

g

Choke input

Figure 10. MPC control signal to the choke with heave disturbance.

0 20 40 60 80 100 120 140 160 180 200−15

−10

−5

0

5

10

15

Time(s)P

ressure

(bar)

Figure 11. Heave disturbance

MPC with future knowledge of heave disturbance to track theset point reference. It is found that the MPC controller withfuture knowledge of heave disturbance reduces the effect ofheave disturbance more than MPC controller without futureknowledge of heave disturbance. This may motivate furtherresearch on short-term heave motion prediction based onforward-looking sensors such as ocean wave radar.

0 20 40 60 80 100 120 140 160 180 200254

256

258

260

262

264

266

268

Time (s)

Pre

ssu

re (

ba

r )

Bottom−hole pressure

MPC without future knowledge of disturbance R=150

MPC with future knowledge of disturbance R=85

Figure 12. Bottom-hole pressure with predictable heave disturbance.

V. CONCLUSIONS

In this paper a dynamical model describing the flow andpressure in the annulus is used. The model was based on ahydraulic transmission line, and is discretized through a finitevolumes method. A stochastic model describing sea waves inthe North Sea was given.

A constrained MPC for controlling bottom hole pressureduring oil well drilling was designed. It was found that

the constrained MPC scheme is able to successfully controlthe downhole pressure. It was also found that a constrainedMPC shows improved attenuation of the heave disturbance.Comparing the PID controller results with MPC shows thatthe MPC controller has a better performance than the PIDcontroller.

ACKNOWLEDGMENT

The authors gratefully acknowledge the financial supportprovided to this project through the Norwegian ResearchCouncil and Statoil ASA (NFR project 210432/E30 IntelligentDrilling).

REFERENCES

[1] I. S. Landet, H. Mahdianfar, U. J. F. Aarsnes, A. Pavlov, and O. M.Aamo, “Modeling for mpd operations with experimental validation,” inIADC/SPE Drilling Conference and Exhibition, no. SPE-150461. SanDiego, California: Society of Petroleum Engineers, March 2012.

[2] I. S. Landet, A. Pavlov, and O. M. Aamo, “Modeling and control ofheave-induced pressure fluctuations in managed pressure drilling,” IEEETransactions on Control Systems Technology, 2013.

[3] J. Petersen, R. Rommetveit, K. S. Bjørkevoll, and J. Frøyen, “A generaldynamic model for single and multi-phase flow operations duringdrilling, completion, well control and intervention,” in IADC/SPE AsiaPacific Drilling Technology Conference and Exhibition, no. 114688-MS.Jakarta, Indonesia: Society of Petroleum Engineers, August 2008.

[4] H. Mahdianfar, A. Pavlov, and O. M. Aamo, “Joint unscented kalmanfilter for state and parameter estimation in managed pressure drilling,” inEuropean Control Conference. Zurich, Switzerland: IEEE, July 2013.

[5] G.-O. Kaasa, Ø. N. Stamnes, O. M. Aamo, and L. S. Imsland, “Sim-plified hydraulics model used for intelligent estimation of downholepressure for a managed-pressure-drilling control system,” SPE Drillingand Completion, vol. 27, no. 1, pp. 127–138, March 2012.

[6] G. Nygaard, L. Imsland, and E. A. Johannessen, “Using nmpc based ona low-order model for control pressure druing oil well drilling,” in 8thInternational Symposium on Dynamics and Control of Process Systems,vol. 8, no. 1. IFAC, 2007.

[7] Ø. Breyholtz, G. Nygaard, H. Siahaan, and M. Nikolaou, “Managedpressure drilling: A multi-level control approach,” in SPE IntelligentEnergy Conference and Exhibition, no. 128151-MS. Utrecht, TheNetherlands: Society of Petroleum Engineers, March 2010.

[8] J. Zhou, Ø. N. Stamnes, O. M. Aamo, and G.-O. Kaasa, “Switchedcontrol for pressure regulation and kick attenuation in a managed pres-sure drilling system,” IEEE Transactions on Control Systems Technology,vol. 19, no. 2, pp. 337–350, 2011.

[9] J. Zhou and G. Nygaard, “Automatic model-based control scheme forstabilizing pressure during dual-gradient drilling,” Journal of ProcessControl, vol. 21, no. 8, pp. 1138–1147, September 2011.

[10] J.-M. Godhavn, A. Pavlov, G.-O. Kaasa, and N. L. Rolland, “Drillingseeking automatic control solutions,” in Proceedings of the 18th WorldCongress, vol. 18, no. 1, The International Federation of AutomaticControl. Milano, Italy: IFAC, September 2011, pp. 10 842–10 850.

[11] O. S. Rasmussen and S. Sangesland, “Evaluation of mpd methodsfor compensation of surge-and-swab pressures in floating drilling op-erations,” no. 108346-MS, IADC/SPE Managed Pressure Drilling &Underbalanced Operations. Texas, U.S.A.: IADC/SPE, March 2007.

[12] A. Pavlov, G.-O. Kaasa, and L. Imsland, “Experimental disturbance re-jection on a full-scale drilling rig,” in 8th IFAC Symposium on NonlinearControl Systems. University of Bologna, Italy: IFAC, September 2010,pp. 1338–1343.

[13] H. Mahdianfar, O. M. Aamo, and A. Pavlov, “Attenuation of heave-induced pressure oscillations in offshore drilling systems,” in AmericanControl Conference (ACC). Montreal, Canada: IEEE, June 2012, pp.4915–4920.

[14] ——, “Suppression of heave-induced pressure fluctuations in mpd,”in Proceedings of the 2012 IFAC Workshop on Automatic Control inOffshore Oil and Gas Production, vol. 1. Trondheim, Norway: IFAC,May 2012, pp. 239–244.

[15] D. Q. Mayne, J. B. Rawlings, and P. O. Rao, C. V.and Scokaert,“Constrained model predictive control: Stability and optimality,” AU-TOMATICA, vol. 36, no. 6, pp. 789–814, 2000.

[16] M. Morari and J. H. Lee, “Model predictive control: past, present andfuture,” AUTOMATICA, vol. 23, no. 4-5, pp. 667–682, May 1999.

[17] C. E. Garcıa, D. M. Prett, and M. Morari, “Model predictive control:Theory and practice—a survey,” AUTOMATICA, vol. 25, no. 3, p.335˙348, May 1989.

[18] J. M. Maciejowski, Predictive Control with Constraints. Prentice Hall,2002.

[19] I. S. Landet, A. Pavlov, O. M. Aamo, and H. Mahdianfar, “Control ofheave-induced pressure fluctuations in managed pressure drilling,” inAmerican Control Conference (ACC). Montreal, Canada: IEEE, June2012, pp. 2270–2275.

[20] T. I. Fossen, Handbook of Marine Craft Hydrodynamics and MotionControl. John Wiley & Sons, April 2011.

[21] U. A. Korde, “Active heave compensation on drill-ships in irregularwaves,” Ocean Engineering, vol. 25, no. 7, pp. 541–561, July 1998.

[22] K. Do and J. Pan, “Nonlinear control of an active heave compensationsystem,” Ocean Engineering, vol. 35, no. 5–6, pp. 558–571, April 2008.

[23] S. Kuchler, T. Mahl, J. Neupert, K. Schneider, and O. Sawodny, “Activecontrol for an offshore crane using prediction of the vessel’s motion,”IEEE/ASME Transactions on Mechatronics, vol. 16, no. 2, pp. 297–309,April 2011.

[24] M. K. Ochi, Ocean Waves - The Stochastic Approach. CambridgeUniversity Press, 2005.

[25] W. H. Michel, “Sea spectra revisited,” Marine Technology, vol. 36, no. 4,pp. 211–227, 1999.