Modelling Two-Phase Flow for Control Design in Oil Well Drilling

Gerhard Nygaard and Geir Nævdal

Abstract— Today, marginal oil wells are being drilled and theoperating margins for the bottom-hole well pressures duringdrilling are becoming narrower. This requires an improvedcontrol of the pressure balance between the reservoir porepressure and the well bottom-hole pressure. In oil well drillingapplications, the pressure is typically controlled manually byadjusting the choke valve. This paper proposes a simple feed-back PI-control scheme with feed-forward compensation of theknown disturbances. A low-dimensional dynamic state modelfor two-phase flow has been developed to be able to tune thecontrol parameters. The proposed method is presented andevaluated using a detailed oil well drilling simulator. The resultsshow that the proposed control design keeps the bottom-holepressure within the operating margins.

I. INTRODUCTION

Controlling the bottom-hole pressure in an oil well duringdrilling can be a challenging task, especially when using agas-liquid mixture as circulation fluid. The pressure balancebetween the pressure in the reservoir and the pressure inthe well during the drilling phase has great impact on thefluid rates from the reservoir when the oil well is set intoproduction at a later stage.

During oil well drilling, a drill fluid is pumped intothe drill string. This drill fluid is flowing down the drillpipe, through the drill bit, and upwards through the annulusbetween the drill string and the sidewall of the well. Oneof the purposes of the drill fluid is to transport the cuttingsfrom the drilling process up to the surface. Another importantscope of the drill fluid is to maintain a certain pressuregradient along the length of the well.

Drilling the oil well when the pressure in the well isbelow the reservoir pore pressure have several benefits. Themost important benefit is that the porous formation is lessdamaged, since the particles from the drilling process donot penetrate into the formation. This leads to a higherproduction rate when the oil well is set into production.

The well pressure is managed during the drilling processby adjusting the density and the flow rate of the drillingfluids. In case the reservoir pore pressure is lower thanthe hydrostatic pressure caused by the drill string liquids,gas has to be injected to reduce the well pressure. Thecomplex behaviour of the resulting two-phase flow resultsin challenges regarding the effort of maintaining a correctwell pressure gradient along the well. In addition, migrationof reservoir fluids (gas and/or liquids) from the reservoirformation makes the task even more difficult. A schematicof an oil well drilling system is shown in Fig. 1.

G. Nygaard is with RF-Petroleum, RF-Rogaland Research, Bergen,Norway Gerhard.Nygaard@rf.no

G. Nævdal is with RF-Petroleum, RF-Rogaland Research, Bergen, Nor-way Geir.Naevdal@rf.no

Fig. 1. Schematic layout of an oil well drilling system

During drilling, disturbances that cause fluctuations in thewell pressure might occur. The operator has to make properactions to avoid variations in the well pressure. Disturbancesarise from several sources. One source is the hydrostaticpressure of the well. The well length is increasing, andhence the well pressure is increased. Another source is thatmore of the reservoir is exposed to the well pressures,as drilling progresses. The reservoir conditions, such asreservoir pressure and permeability results in influx from thereservoir into the well. Due to this, the well flow rate anddensity of the well fluid mixture is changed. A third sourceof disturbance is caused by the pipe connection procedurewhich is performed at equal time intervals during drilling.

The drill string consists of several pipe segments whichare jointed together. As the well is becoming longer, newpipe segments are added to the drill string using the pipeconnection procedure. The pipe connection procedure mainlyconsists of five operations. First the rotation of the drillstring is stopped. Secondly the pumping of the drill fluidinto the drill string is stopped. Then a new pipe segment ismounted to the drill string. Action number four is to restartthe drill fluid pumps, and finally the drill string rotation is re-started. This procedure, and especially stopping and startingof the drill fluid cause severe fluctuations in the well fluidsflow rates, and influence the well pressure. This paper isfocusing on how to avoid such fluctuations a simple PI-control scheme.

To compensate for variations in the well pressure, theoperator might modify the fluid composition and flow rates

Proceedings of the2005 IEEE Conference on Control ApplicationsToronto, Canada, August 28-31, 2005

TA4.2

0-7803-9354-6/05/$20.00 ©2005 IEEE 675

into the drill string. This will change the density of the fluidmixture in the well, affecting the well pressure. However,well pressure is not modified instantly, since the new fluidcomposition need some time to be filled into the whole well.Another way the operator might modify the well pressure, isto adjust the choke valve on top of the annulus part of thewell. Changing the valve opening causes a rapid response inthe bottom-hole pressure.

Both methods are used to compensate the bottom-holepressure, but during pipe connections the well pressure isnormally maintained using the choke valve. One of themain problems for controlling the well pressure during pipeconnections is that the transmission of the signal is usuallybased on a mud pulse telemetry system. This system issending various data from the bit system, but the system isonly operating while the drill fluid is circulating. This meansthat the well pressure at the bit is not available. The controlsystem must then rely on a model of the well system.

II. CONTROL SCHEME

Today, in normal drilling operations the choke valve isadjusted manually by a trained drilling engineer. The fluidcomposition and pressures are evaluated based on steady-state values, and the choke is adjusted accordingly. Re-cently, new procedures for manually adjusting the flow ratesand choke opening prior and during pipe connections aresuggested [1]. These procedures are based on a detailedmechanistic dynamic two-phase flow model. The model isused to evaluate the well conditions and to plan the pipeconnection prior to the actual action. The choke valve isthen adjusted manually according to the calculated set-points.Difficulties might arise if the pipe connection procedure isnot performing as planned.

A different approach for avoiding the pipe connectionpressure fluctuations is described in [2]. A prototype me-chanical system which is using various seals and valves hasbeen developed to be able to continue to pump the drill fluidseven during the pipe connections. The mechanical systemincreases the complexity of the drilling system.

Under-balanced drilling has some similarities with gas-lifted wells, and a control system utilizing a low-ordermodel is described in [3]. A fourth approach is suggestedby [4], where an automatic control system is operating thechoke on-line during the pipe connection. The control systemis utilizing a non-linear model predictive control schemecombined with first-principles model. The model is used foron-line evaluation of the well pressures and fluid flows, andpredictions are made to find the most optimal choke openingduring pipe connections.

This paper presents and evaluates a simple PI-controlscheme for controlling the bottom-hole pressure in the wellduring drilling and pipe connections procedures.

Since it is not possible to tune the parameters in the PIcontrol using experiments and the Ziegler-Nichols closed-loop method [5] in our case, the control setting can be foundby developing a simple model of the process. In [6], a low-order dynamic model is developed for controlling slugging

flow in a gas-lifted production well. An equivalent model forthe under-balanced drilling case will be developed and thecontrol parameters is found based on experiments performedon the low-order model.

III. MODELLING TWO-PHASE FLOW

Several methods for modelling the dynamic two-phaseflow in the well can be used. One method is to focus on themajor effects in the well, and look at certain phenomena. Thistype of model has only a few states, such as flow rates of thefluid components. Another approach is to model the variouseffects more detailed, and use spatial discretization of thewell system. This type of detailed mechanistic modelling isvery time-consuming to perform. The main purpose for bothmethods is to be able to calculate the bottom-hole pressuresufficiently accurate to be able to control the actual pressure.

In control engineering, modelling is used to improve thecontrol system ability to follow the given reference values.This type of models is often linear, or is being linearizedaround stable conditions. In 2-phase fluid flow systems, theinteraction between the gas and the liquid are causing non-linear behaviour. In addition, the actuators and disturbancesmight easily bring the fluid flow outside the validity envelopeof a linearized model.

The low order model is set up in a similar way as in [6],modeling the well as a two-tank system. A mass balance anda pressure balance is set up for the drill string and annulus,respectively. The model is based on the assumption the gasis evenly distributed within the liquid in the whole well.The density of the mixture of gas and liquid ρmix is givenby ρmix = mg+ml

V where mg is the gas mass, ml is theliquid mass, and V is the volume. The additional densitydue to particles from the drilling process is not taken intoaccount. This simplified oil well system is modelled usinga combination of the mass balance of the fluids and thepressure balance [7].

From now on subscript l denotes liquid and subscript gdenotes gas.

A. Mass balance

Fig. 2 is a schematic of the well, showing where themass balances of the drill string and the mass balance ofthe annulus. The mass balance is divided into two systems,the drill string and the annulus between the well and the drillstring.

The mass balances for the fluids in the drill string aregiven by dmg,d

dt = wg,pump − wg,bit, mg,d(0) = m0,g,d,

and dml,d

dt = wl,pump − wl,bit, ml,d(0) = m0,l,d wherem·,d is the mass in the drill string, w·,pump is the mass flowat the pump, and w·,bit is the mass flow at the drill bit.

The mass balance equations for the annulus are given bydmg,a

dt = wg,bit + wg,res − wg,choke, mg,a(0) = m0,g,adml,a

dt = wl,bit +wl,res−wl,choke, ml,a(0) = m0,l,a wherem·,a is the mass in the annulus and w·,choke is the mass flowat the exiting choke valve.

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Fig. 2. Mass balance of simplified oil well geometry

Fig. 3. Pressure balance of simplified oil well geometry

B. Pressure balance

In addition to the mass balance, the pressure balance inthe system is important due to the frictional pressure inducedby the velocity of the liquid. When gas in injected into thewell, the gas volume is changed due to gas compression.The hydrostatic pressure also varies due to variation in themixture density. The fluid flows through the restriction atthe drill bit at the bottom of the well and at the choke valveat the top of the well. In Fig. 3 the various pressures areindicated.

The pressure balance is evaluated at two points andgiven as mass acceleration at the bottom of the drillstring and as mass acceleration at the top of the annulus.The pressure balance equations are given by dwmix,bit

dt =pd,c+pd,g−pd,f−∆pbit−pa,c−pa,g−pa,f

Ad, wmix,bit(0) = 0 and

dwmix,choke

dt = pa,c−∆pchoke−patm

Aa, wmix,choke(0) = 0

where A is the cross sectional area of the drill stringand annulus, the subscript c is the compression pressure,subscript h is the hydrostatic pressure and subscript f is thefrictional pressure, wmix,· is the mixture flow acceleration

before the drill bit flow restriction and before the choke valveand patm is the atmospheric pressure.

C. Well length

When drilling the oil well, the length of the well isincreasing according to the drilling rate. The length of thewell has substantial influence of the well pressure. The welllength L is chosen as a state in the dynamic systems andgiven by dL

dt = vd, L(0) = L0 where vd is the drillingrate, and L0 is the initial well length.

To solve these balance equations, the relations betweenthe pressures and flows are given below.

D. Gas entrainment

If the void fraction of liquid in the mixture, αm, is givenas αm = ρmix

ρlthen this void fraction should be used to

calculate the gas mass rate and liquid mass rate. However,when the velocity is reduced, the friction pressure is reducedand the gas is expanding. This effect causes the liquid toflow out of the well and the gas to be contained in thewell. The gas mass rate at low mixture velocities shouldthen be modified to αe = αm +β (1 − αm)

(1

1+e−n(vt−vm)

)

where β is a factor for entrainment at low velocities, ncorresponds to the slope of the entrainment, vt is a constantreferring to the mixture velocity at the transition betweenfull gas entrainment and minimum gas entrainment and vm

is the current velocity of the mixture. To calculate the massflow rates of gas and liquid, the liquid void fraction αe

is used. This gives the relations wl = αewmix and wg =(1 − αe)wmix.

E. Pump mass rates

The pump mass rates are stable during drilling, but arestopped when pipe connection are performed. The set-pointof the pumps are the total flow rate, wp, and to deliver theserates, the pump pressure is adjusted accordingly. The voidfraction between the gas mass rate and the liquid mass rateare fixed using αp. The resulting pump rates are found usingwl,p = αpwp and wg,p = (1 − αp) wp.

F. Reservoir mass rates

To model the flow from the reservoir into the well, asimple relation called the productivity index PI can be used.This is a linear relation of the pressure difference between thereservoir and the well. The mass rate from the reservoir, wr

can be calculated using the relation wr = PI (pa,bot − pres)where the mass flow from the reservoir is wl,r = αrwmix

and wg,r = (1 − αr)wmix.

G. Valve equations

The mass rate, w, through a restriction is given by thesimple valve equation from [8], w = Cdz

√ρ∆p, where Cd

is the discharge coefficient of the restriction, z is the area ofthe restriction, and ∆p is the differential pressure across therestriction. This relation is used both at the drill bit and thechoke valve.

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H. Gas compression pressure

The gas is compressible and the gas volume is dependenton the pressure conditions. The relation between the gasmass and the compression pressure is based to the perfectgas law in a pressurized tank pc,tank = ρg

ΛMg

T, whereΛ is the gas constant, Mg is the molecular weight of thegas, and T is the average temperature. In the model, themixture of the gas and liquid in the well cause an averagecompression pressure along the depth of the well. Hence,we model the compression pressure as pc,well = patm +K1 (pc,tank − patm) , where and K1 is a compression factor.

I. Hydrostatic pressure

The hydrostatic pressure ph in the well is calculated basedupon the relation between the mixture density ρmix, gravityg and well depth h and is given as ph = ρmixgh.

J. Frictional pressure

The frictional pressure loss is caused by the frictionbetween the fluid and the walls of the well and the drill string.The friction pressure loss is calculated by [9] pf = ρfLv2

2Dwhere v is the fluid velocity and D is the hydraulic diameter.To calculate the friction factor f , the Haaland equation [9]

1√f≈ −1.8 log10

[6.9Re +

(ε/D3.7

)1.11]

is used. Here ε/D is

the relative roughness of the pipe. The Reynolds number Reis calculated using [9] Re = ρvD

µ where µ is the viscosityof the fluid.

K. Calculation scheme

When setting up the model, a total of 7 states havebeen used to describe the system. The state vector is �x =[mg,d,ml,d,mg,a,ml,d, wmix,d, wmix,d, L]. The choke areais defined to be �u = [zchoke]. The inflow from the pumps andthe drilling rate are treated as a disturbances of the system�v = [wg,pump, wg,pump, vd] .

An explicit scheme for calculating the parameters is givenby d�x

dt = f (�x, �u,�v) , �x0 = 0, and �y = g (�x) where y is thesensors values.

Several of the parameters in this model is not easily found,and the model has to be tuned to describe the well systemmore accurate. The need for experimental tuning is thereforerequired for a specific case.

IV. CASE DESCRIPTION

A test case is defined to evaluate the PI-control scheme.Initially, the well is 2000 m deep, and the well is drilled 100m into a reservoir. Well data and reservoir data is given inTable I. Fig. 4 shows the simulation of the case using the de-tailed model. Initially the fluid flow in the well is in a steady-state condition, and the drilling is initiated. After 10 minutes,the first pipe connection procedure is started. The rotation ofthe drill string and the circulation of fluids are stopped for10 minutes. Then the circulation pumps are re-started, andthe drill string starts to rotate. The second pipe connectionprocedure is initiated after 52 minutes, and is completedafter 64 minutes. In this simulation, no adjustments of the

TABLE I

WELL AND RESERVOIR DATA

Parameter ValueInitial well length, hw,i 2000 m

Liquid circulation rate, wl 24 kg/sGas circulation rate, wg 2 kg/s

Reservoir height, hr 100 meterDrilling rate, vd 0.01 m/s

Reservoir pore pressure, pr 215 barWell set-point pressure, pw 205 bar

Reservoir collapse pressure, pc 185 bar

0

10

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30

Mas

s F

low

[kg/

s]

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0

0.5

1

Cho

ke In

dex

[0−

1]

Choke Opening Index

0

50

100

Pre

ssur

e [b

ar]

Choke Differential Pressure

0 20 40 60 80 100 120140

160

180

200

220

Minutes

Pre

ssur

e [b

ar]

Bottomhole Pressure

Well pressureReservoir pressureCollapse Pressure

Fig. 4. Simulated drilling case with no control actions

choke opening are performed. During each pipe connection,the bottom hole pressure is falling from about 205 bar anddown towards 145 bar. As the reservoir collapse pressure isat 185 bar, actions must be taken to avoid that the pressure isfalling below this limit. After the pipe connection procedureis completed, the pressure slowly increases towards the 205bar set-point. However, the pressure raises above the set-point, due to a slugging flow regime in the well. The sluggingflow is caused by a segregation of the gas-liquid mixtureduring the pipe connection.

The data from the simulation of the detailed model is usedto tune the low-dimensional state model. The fluid mass ratesinto the drill string are the same in the low-order model asthe detailed model. During pumping the liquid rate is 24 kg/sand gas rate is 2 kg/s, and during pipe connections both ratesare set to 0 kg/s.

Fig. 5 shows the flow from the annulus section for bothmodels. The inaccuracies in this case are due to the limitedmodelling of the gas entrainment, and the deviation betweenthe detailed model and the low-order model has increased.

Comparison of the models with respect to fluid flow ratefrom the reservoir are given in Fig. 6. The detailed modelshows increased fluid flow from the reservoir as more ofthe reservoir has contact with the well. The low-order modelcould be improved if a different method of modelling thanthe Production Index method was used.

When comparing the pressures between the low-ordermodel and detailed mechanistic model, the low-order model

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0 20 40 60 80 100 1200

5

10

15

20

25

30

35Annulus Liquid Flow Out

Minutes

Mas

s F

low

[kg/

s]

Low−order modelDetailed model

0 20 40 60 80 100 1200

0.5

1

1.5

2

2.5

3Annulus Gas Flow Out

Minutes

Mas

s F

low

[kg/

s]

Low−order modelDetailed model

Fig. 5. Mass flow rates out of annulus

0 20 40 60 80 100 1200

2

4

6

8

10Reservoir Liquid Flow Out

Minutes

Mas

s F

low

[kg/

s]

Low−order modelDetailed model

0 20 40 60 80 100 1200

0.05

0.1

0.15

0.2

0.25

Reservoir Gas Flow Out

Minutes

Mas

s F

low

[kg/

s]

Low−order modelDetailed model

Fig. 6. Mass flow rates out of reservoir

has to be adjusted with respect to the friction pressurelosses in the drill-string and the annulus, in addition to thecompression factor of the gas and liquid mixture. In Fig. 7the pressures at the top of the drillstring and the bottom ofthe drill-string are compared.

In Fig. 8 the pressures in the annulus section are shown.The low-order model is less accurate. This is due to theinaccuracies in the low-order model with regards to themixture flow rates.

However, the low-order model gives relatively accuratevalues for the flow rates and pressures in the well duringdrilling. This low-order model can now be used for designingthe control scheme for the drilling process.

V. CONTROL SCHEME TUNING AND EVALUATION

The low-order model is used to define the control parame-ters for the PI-control scheme with feed-forward compensa-tion of the pump flow rate. The method used for designingthe parameters is the Ziegler-Nichols method for closed loopssystems [5]. The feed-forward compensation are selectedto be Kf = 0.6. The closed loop-system are brought toa critical conditions, marginally stable, by setting the PI-control parameters to Kp,critical = 285 and Ti = ∞. Theresulting fluctuations are shown in Fig. 9. From these sim-

0 20 40 60 80 100 1200

50

100

150

200

250

300Drillstring Top Pressure

Minutes

Pre

ssur

e [b

ar]

Low−order modelDetailed model

0 20 40 60 80 100 120100

150

200

250

300Drillstring Bottom Pressure

Minutes

Pre

ssur

e [b

ar]

Low−order modelDetailed model

Fig. 7. Drill string pressures

0 20 40 60 80 100 1200

5

10

15

20

25

30Annulus Top Pressure

Minutes

Pre

ssur

e [b

ar]

Low−order modelDetailed model

0 20 40 60 80 100 120100

120

140

160

180

200

220

Annulus Bottom Pressure

Minutes

Pre

ssur

e [b

ar]

Low−order modelDetailed model

Fig. 8. Annulus pressures

ulations, the critical time period is found, Tcritical = 76.2.Based on the Ziegler-Nichols rules, the control parameterscan be calculated. According to these rules, the proportionalgain Kp = 0.45Kp,critical = 128.25 and the integral timeconstant should be Ti = Tcritical

1.2 = 63.5. The controlleris then updated using these parameters and the results areshown in Fig. 10.

To evaluate the control scheme, the PI-controller is com-pared with manual adjustment of the choke valve. Whenevaluating the controller, the detailed mechanistic model areused. In Fig. 11 the manual control method are used, wherethe choke is set to a value defined by the operator. As canbe seen, this manual method is working quite good, but itcan be observed that the bottom-hole pressure is increasingabove reservoir pore pressure after the pipe connection isfinished.

In Fig. 12 the PI-control scheme is evaluated using thedetailed mechanistic model is used. As can be seen, the fluc-tuations in the bottom-hole pressure are a bit higher relativeto the fluctuations in Fig. 10. However, when comparing withthe case where the choke valve is manually controlled, thereis a significant improvement. The bottom-hole pressure arekept within the margins during the whole operation.

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0

10

20

30M

ass

Flo

w [k

g/s]

Mixture Flow Rate In

0

0.5

1

Cho

ke In

dex

[0−

1]

Choke Opening Index

0

50

100

Pre

ssur

e [b

ar]

Choke Differential Pressure

0 20 40 60 80 100 120140

160

180

200

220

Minutes

Pre

ssur

e [b

ar]

Bottomhole Pressure

Well pressureReservoir pressureCollapse Pressure

Fig. 9. Well data using PI-control with critical parameters

0

10

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30

Mas

s F

low

[kg/

s]

Mixture Flow Rate In

0

0.5

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Cho

ke In

dex

[0−

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Choke Opening Index

0

50

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Pre

ssur

e [b

ar]

Choke Differential Pressure

0 20 40 60 80 100 120140

160

180

200

220

Minutes

Pre

ssur

e [b

ar]

Bottomhole Pressure

Well pressureReservoir pressureCollapse Pressure

Fig. 10. Well data using PI-control with adjusted parameters

0

10

20

30

Mas

s F

low

[kg/

s]

Mixture Flow Rate In

0

0.5

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Cho

ke In

dex

[0−

1]

Choke Opening Index

0

50

100

Pre

ssur

e [b

ar]

Choke Differential Pressure

0 20 40 60 80 100 120140

160

180

200

220

Minutes

Pre

ssur

e [b

ar]

Bottomhole Pressure

Well pressureReservoir pressureCollapse Pressure

Fig. 11. Simulating manual control with detailed model

0

10

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30

Mas

s F

low

[kg/

s]

Mixture Flow Rate In

0

0.5

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Cho

ke In

dex

[0−

1]

Choke Opening Index

0

50

100

Pre

ssur

e [b

ar]

Choke Differential Pressure

0 20 40 60 80 100 120140

160

180

200

220

Minutes

Pre

ssur

e [b

ar]

Bottomhole Pressure

Well pressureReservoir pressureCollapse Pressure

Fig. 12. Simulating PI-control with detailed model

VI. CONCLUSION

Using a PI-control scheme for adjusting the choke valveduring oil well drilling, improves the stability of the bottom-hole pressure during the whole drilling operations, includ-ing during pipe connections procedures. By using a low-dimensional state model a set of efficient control parameterscan be found.

However, if the circulation flow rates are being modified,or the inflow from the reservoir is changing, then the simplelow-order model is not describing the real process sufficientlyaccurately. The low-order model must then be corrected, andnew control parameters must be found.

ACKNOWLEDGMENT

The authors would like to thank RF-Rogaland Researchfor the permission to publish this work. The project is co-sponsored by the Research Council of Norway.

REFERENCES

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[4] G. H. Nygaard, E. H. Vefring, K. K. Fjelde, G. Nævdal, R. J.Lorentzen, and S. Mylvaganam, “Bottomhole pressure control duringpipe connection in gas dominant wells,” in SPE/IADC UnderbalancedTechnology Conference and Exhibition, The Woodlands, Texas, USA,Oct.11–12, 2004, paper SPE 91578.

[5] G. Stephanopoulos, Chemical Process Control. Englewood Cliffs, NewJersey: Prentice-Hall Inc., 1984.

[6] E. Storkaas, S. Skogestad, and J. M. Godhavn, “A low-dimensionaldynamic model of severe slugging for control design and analysis,” inMultiphase’03, San Remo, Italy, June11–13, 2003.

[7] J. G. Balchen and K. I. Mumme, Process Control. Structures andapplications. New York, USA: VanNostrand Reinhold, 1988.

[8] O. Egeland and J. T. Gravdahl, Modelling and Simulation for AutomaticControl. Trondheim, Norway: MarineCybernetics, 2002.

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