Slug handling with a virtual harp based onnonlinear predictive control for a gravity

separator ?

Christoph Josef Backi ∗,1 Dinesh Krishnamoorthy ∗

Sigurd Skogestad ∗

∗ Department of Chemical Engineering, Norwegian University ofScience and Technology (NTNU), 7491 Trondheim, Norway (e-mail:{christoph.backi,dinesh.krishnamoorthy,sigurd.skogestad}@ntnu.no).

Abstract: This paper presents a nonlinear model predictive control approach for a three-phase gravity separator model. The aim of the controller is to dampen slug-induced, oscillatorydisturbances in the inflow to the gravity separator. This means that, despite the disturbance, thelevels of water and oil as well as the pressure should be held at constant operational setpoints.Additionally, a second objective is to dampen the outflows of water, oil and gas and hencekeep too large oscillations in flows from downstream equipment. Several constraints are addedsuch as the height of the weir, maximal and minimal allowable levels and pressures as well asconstraints on the outflows.

Keywords: Gravity separator, nonlinear model predictive control, slug handling, virtual harp

1. INTRODUCTION

In the oil- and gas-industry, the separation of phases suchas gas, water and hydrocarbons (oil and condensate) froma well stream is crucial to obtain as pure single phasestreams as possible. Thereby, often gravity separators areutilized as a first rough separation stage followed by anumber of refining stages, for example hydrocyclones, gasscrubbers, gas flotation units and in special cases alsomembranes. After all these separation stages, the phasesshould be pure enough to preferably distribute them fur-ther by standard single phase pumps and compressors (oiland gas). In addition, disposal into the sea (water) or rein-jection back into the reservoir (water and gas) are furtheroptions. This paper focuses on gravity separation devices,which are based on separation by gravitational forces anddensity differences between the respective dispersed andcontinuous phases (Arntzen, 2001; Bothamley, 2013). Thismeans that water droplets dispersed in the oil-continuousphase will settle towards their bulk water-continuous phasewhereas oil droplets dispersed in this water-continuousphase will rise to the oil-continuous phase.

Literature regarding control-oriented modeling of gravityseparators is rather sparse. The main modeling efforts aretowards CFD-based models investigating the flow patternsinside the separator and are mostly used for the purposesof design (optimization) or validation (Hansen, 2001; Lalehet al., 2012, 2013; Kharoua et al., 2013).

Gravity separators must be operated within specified lim-its making steady control of the state variables to nominalvalues necessary. These include for example limits on the

? This work was supported by the Norwegian Research Councilunder the project SUBPRO (Subsea production and processing)1 corresponding author

gas pressure as well as on the water level, which should re-main below the weir such that the water-continuous phasedoes not enter the outflow of the oil-continuous phase.Disturbances to the process can cause the state variablesto leave these specified limits making steady control actionnecessary. Such disturbances can for instance be slug in-flows, which express themselves in alternating oscillationsin the inflows of gas and liquid. Riser-induced slugging hasattracted the attention of researchers in recent decadesand can be controlled by a large variety of (nonlinear)control methods (Jahanshahi and Skogestad, 2017; Ja-hanshahi et al., 2017; Jahanshahi and Skogestad, 2013).Furthermore, the installation of large volumes in the formof harp-style slug catchers can avoid excessive oscillatorybehavior. In this work we concentrate on the case, wherethe slug controllers are either not implemented or only ableto dampen the slug-induced oscillations insufficiently, andwhen harp-style slug catchers are not installed.

As the title says, we design a ”virtual harp”, whichprovides a similar effect like a harp-style slug catcher. It isbased on the model predictive control (MPC) methodologyto reduce oscillations in the outflows of gas, water and oiland at the same time keep the state variables (total liquidlevel, water level and pressure) at certain setpoints insidedefined bounds. However, there exists a trade-off betweendampening the slug-induced oscillations in the outflowsand dampening the oscillations in the state variables.Therefore, the exact objective and hence tuning are veryimportant tasks.

The remainder of this paper is structured as follows:Section 2 introduces the mathematical mode, which isthe basis for controller design, while the controller itselfis designed in Section 3. Section 4 presents simulation

Proceedings of the 3rd IFAC Workshop on AutomaticControl in Offshore Oil and Gas Production, Esbjerg,Denmark, May 30 - June 1, 2018

Th_A_Regular_Talk.2

Copyright © 2018, IFAC 120

results, whereas in Section 5 the paper is closed with someconcluding remarks.

2. MATHEMATICAL MODEL

A schematic of the gravity separator is shown in Fig. 1.Basically, it is a long cylinder, where the multiphase inletstream is fed into the separator at the left side. Due toturbulence, a first rough separation occurs in the inletzone. For simplicity, we assume that the gas is completelyflashed out of the liquid phases, meaning that no gasbubbles exist therein. Additionally, we assume that noliquid droplets can be found in the continuous gas phase.The active separation zone of length L follows the inletzone, in which dispersed droplets settle or rise to theirrespective bulk phases due to density differences. Phe-nomena like for example breakage (big droplets break intosmaller droplets) and coalescence (small droplets form abig droplet) are neglected, but are likely to occur. The gas,water and oil phases leave the separator in the outlet zoneon the right hand size of the separator. More simplifyingassumptions include an average velocity for the respectiveoil- and water-continuous layers including the disperseddroplets (no slip between droplets and continuous phase)as well as the absence of an emulsion (dense-packed) layerbetween these layers. Typically, in order to get a moreeven flow in gravity separators, flow-distribution baffles areadded along the x-direction (not specifically shown here).

The focus in this paper is not on overall separation of thethree phases, but for completeness our model also includesdroplet calculations for oil in water and water in oil.

hW

hL

qWoutqOout

qLin

qGin

L

qGout

AW

AL

y

x

Fig. 1. Simplified schematic of the gravity separator witha cross-sectional view on the left-hand side

2.1 Differential part of the model

The mathematical model that is the basis for controllerdesign in Section 3 was introduced in detail in Backiand Skogestad (2017a). The differential part of the modelconsists of three states, the overall liquid level hL (oil pluswater), the water level hW and the gas pressure p.

Based on in- and outflow dynamics, the following differen-tial equations are obtained for the water and liquid levels(Backi and Skogestad, 2017b)

dhLdt

=dVLdt

1

2L√hL(2r − hL)

, (1)

dhWdt

=dVWdt

1

2L√hW (2r − hW )

, (2)

with radius of the gravity separator vessel r and length ofthe active separation zone L . The changes of volumes forthe liquid and water phases are given by the volumetricin- and outflow dynamics

dVLdt

= qL,in − qL,out = qL,in − qW,out − qO,out, (3)

dVWdt

= qW,in − qW,out= qL,in [αφw + (1− α)(1− φo)]− qW,out,

(4)

where α is the water cut, qi denote in- or outflows ofeither liquid (L), water (W) or oil (O), φw represents thefraction of inflowing water going into the water-continuousphase and φo defines the fraction of inflowing oil going intothe oil-continuous phase. In (4), αφw denotes the fractionof total inflowing water and (1 − α)(1 − φo) representsthe fraction of total inflowing oil, both into the water-continuous phase.

The pressure dynamics are derived from the ideal gas lawassuming constant temperature and are scaled here for theunit [bar]

dp

dt=10−5

[RT ρG

MG(qG,in− qG,out) + 105p (qL,in− qL,out)

VSep − VL

](5)

with volume of the active separation zone VSep, liquidvolume

VL=r2L

2

[2 cos−1

(r − hLr

)− sin

(2 cos−1

(r − hLr

))],

universal gas constant R, temperature T , density of gasρG, molar mass of gas MG and the in- and outflows of gas(G) and liquid (L), respectively.

The manipulated variables are the outflows of oil, water

and gas, hence u = [qO,out qW,out qG,out]T

. Disturbancevariables are the inflow of liquid qL,in (sum of the inflowsof water and oil) and the inflow of gas qG,in.

2.2 Algebraic part of the model

In addition to the differential part described above, themodel consists of an algebraic part, which is constitutedby droplet distribution calculations in the oil- and water-continuous phases in the active separation zone. Therefore,we define 10 droplet size classes and assume their diam-eters di = {50, 100, 150, . . . , 500}µm with initial distribu-tions nO/W = 108 [1 5 10 50 100 100 50 10 5 1], whichis weighted by factors FW (water droplets) and FO (oildroplets) to regard for the fractions of inflowing waterand oil to the respective phases. These assumptions im-ply that the droplet volumes for oil and water are alike

V Oi = VWi =π

6d3i .

For each droplet class i the vertical residence time tvi is

compared to its horizontal residence time tW/Oh . Latter

is calculated by the length L divided by the horizontalvelocities of the oil and water phases, which are given bythe inflows of oil and water divided by the respective cross-

sectional areas, vW/Oh =

qO/W,in

AO/W, hence

Copyright © 2018, IFAC 121

tW/Oh =

AO/WL

qO/W,in(6)

with AO = AL−AW and qO,in = qL,in−qW,in. The verticalresidence time is based on Stokes’ law giving a verticalvelocity for each droplet size class i

vvi =gd2i (ρd − ρc)

18µc, (7)

where g is the gravitational acceleration, ρd and ρc arethe densities of the dispersed and continuous phases,respectively, and µc indicates the dynamic viscosity ofthe continuous phase. If a droplet reaches the oil-water-interface, its respective volume is added to its bulk phaseand subtracted from the phase it was dispersed in.

For oil droplets dispersed in the water-continuous phasethe removed volume is calculated via Algorithm 1.

Algorithm 1 Calculation of droplet class positions andseparated volume for oil droplets

for i = 1:10 doif tOh < tOvi then

zOi = tOh vOvi

elsezOi = hW

end if

V OoWi = nOi VOi

zOihW

end for

with∑10i=1 V

OoWi = V OoW and tOvi =

hWvOvi

.

In addition, for water droplets dispersed in the oil-continuous phase, the removed volume is calculated ac-cordingly via Algorithm 2.

Algorithm 2 Calculation of droplet class positions andseparated volume for water droplets

for i = 1:10 doif tWh < tWvi then

zWi = tWh vWvi

elsezWi = hO

end if

VWoOi = nWi V

Wi

zWihO

end for

where∑10i=1 V

WoOi = VWoO and tWvi =

hOvWvi

with hO =

hL − hW .

The positions of droplet classes z =[zO zW

]represents

the vector of algebraic variables.

The factors FW and FO are calculated as follows

FW = tOh,inlet [α (1− φw)] qL,in1

VW,

FO = tWh,inlet [(1− α) (1− φo)] qL,in1

VO,

where tOh,inlet and tWh,inlet are the residence times of dropletsin the inlet zone of the separator in the respective water-

and oil-continuous phases, and VO =∑10i=1 niV

Oi as well

as VW =∑10i=1 n

Wi V

Wi .

The algorithm as listed above is not implementable assuch. In fact, the if-else-end statements switch betweentwo cases, namely if the horizontal residence time issmaller or equal/larger than the vertical residence timesfor each droplet class i. The optimization algorithm relieson a continuous formulation of the algorithm and itsimplementation is achieved by applying approximationsbased on an arctan-function with a steep gradient arounda switching argument ∆t representing the difference inresidence times. Details are presented in Appendix A.

2.3 Maximization of separation efficiency

An investigation for maximizing the separation efficiencyin the given separator model had the objective to maximizethe outflows of oil from the water-continuous phase andwater from the oil-continuous phase. Thereby, the waterlevel and hence the horizontal and vertical residence timesof droplets in the respective phases were subject to changesat a constantly held overall liquid level. Since both the ini-tial distributions and the horizontal and vertical velocitiesare depending on the very same variable, i.e. levels of waterand oil, the optimizer depending on their weights in thecost function will prefer either cleaner water or cleaneroil. This means that either the water level will be as highas possible or zero. In the following investigation we willtherefore concentrate on the case favoring cleaner waterand define the setpoint of water just below the weir.

3. CONTROLLER DESIGN

In order to dampen out slugging inflows and reduce theireffect on the state variables as well as on the outflowsof the gravity separator, we design a nonlinear modelpredictive controller. Thereby, the state variables shouldbe kept at or close to desired setpoints and between safety-related limits. In addition, the variances of the inflowsshould not propagate through to the outflows, meaningthat the gravity separator acts a slug tank and protectsdownstream equipment from too large oscillations of gas,water and oil. This is important since e.g. hydrocyclonesdo not tolerate large fluctuations in inflows with respect toseparation efficiency and optimal operation. There existsa trade-off between oscillation reduction in the state andthe manipulated variables.

In this work, we use a nonlinear model predictive controlstrategy to achieve the objective of balancing the fluctu-ations between the state and the manipulated variables.Before the MPC problem can be formulated, the optimalcontrol problem is first discretized into a finite dimensionalnonlinear optimization problem divided into N elementssuch that each interval is in [tk, tk+1] for all k ∈ {1, . . . , N}.A low order direct collocation scheme is used to providea polynomial approximation of the system trajectoriesfor each time interval [tk, tk+1]. In this work we use athird order Radau collocation scheme for the polynomialapproximation. The resulting discrete version of the DAEmodel in sections 2.1 and 2.2 is represented as

xk+1 = f(xk, zk,uk)

zk = g(xk,uk)(8)

Copyright © 2018, IFAC 122

where xk represents the differential states (from Sec-tion 2.1) at time step k, zk represents the algebraic states(from Section 2.2) and uk denotes the control inputs (ma-nipulated variables). Once the system is discretized, thenonlinear economic MPC problem can be formulated as

min Jstates + Joutflows (9a)

s.t. hWk≤ hweir (9b)

xk+1 = f(xk, zk,uk) (9c)

zk = g(xk,uk) (9d)

xmin ≤ xk ≤ xmax (9e)

umin ≤ uk ≤ umax (9f)

x0 = xinit (9g)

∀k ∈ {1, . . . , N}

with the state setpoint tracking term

Jstates =

N∑k=1

ωhL

∥∥hLk− hspLk

∥∥2+ ωhW

∥∥hWk− hspWk

∥∥2

+ ωp ‖pk − pspk ‖2

and the term penalizing changes in the manipulated vari-ables

Joutflows =

N∑k=1

ωqO,out

∥∥qO,outk − qO,outk−1

∥∥2

+ ωqW,out

∥∥qW,outk − qW,outk−1

∥∥2

+ ωqG,out

∥∥qG,outk − qG,outk−1

∥∥2.

Furthermore, the objective function comprises of the con-straint (9b), which enforces the water level to remain belowthe weir plate hweir. This prevents the water-continuousphase to enter the oil-continuous outflow. In addition,upper and lower bounds are also enforced for the statesand the control inputs in (9e) and (9f), respectively. Weassume that the liquid level, water level and the separa-tor pressures are measured. At each iteration, the initialconditions for the states are enforced in (9g).

The dynamic optimization problem is setup as a nonlinearprogramming problem in CasADi v3.0.1-rc1 (Andersson,2013). The resulting NLP problem is then solved usingIPOPT version 3.12.2 (Wachter and Biegler, 2006) run-ning with a mumps linear solver. The plant simulatoris solved with an ode15s solver. We simulate 600 MPCiterations with a sample time of ∆t = 1 s. The predictionhorizon of the NMPC controller is set to 20 s. The weightsin the objective function are chosen as shown in Table 1.The NMPC problem is initialized with steady state valuesfor liquid inflow of qL,in = 0.59 m3 s−1 and gas inflowqG,in = 0.456 m3 s−1 and the corresponding states withhW = 1.9 m, hL = 2.3 m and p = 68.7 bar. The weir heightis set to hweir = 2 m. The setpoint for water is hspW = 1.9 m,that for the overall liquid level hspL = 2.5 m (to have somereasonable buffer volume above the liquid level), and forthe pressure psp = 68.7 bar. Most parameters are takenfrom Laleh et al. (2012) and the ones not listed here aresummarized in Appendix B.

4. SIMULATIONS

This Section presents simulation results for two cases ofsevere slugging flows entering the gravity separator:

Table 1. Weights used in the NMPC problem.

weight valueωhL

1ωhW

1ωp 1

weight valueωqO,out

50ωqW,out

10ωqG,out

50

1.) Production from one well with only one sinusoidalslugging frequency, which oscillates around a nom-inal value of qL,in = 0.59 m3 s−1 and qG,in =0.456 m3 s−1.

2.) Production from three wells with three sinusoidalslugging frequencies, where each is weighted differ-ently. Here, the main weight has been put on thehigher-frequent oscillations, where all sinusoidals os-cillated around the same nominal values as mentionedunder point 1.) above

The two different cases are demonstrated in Fig. 2, wherethe two different characters of the slugging inflows to thegravity separator become apparent.

0 100 200 300 400 500 600

0

0.5

1

1.5

Fig. 2. Slugging inflows of liquid and gas to the separator

In the subsequent studies, we compare two different waysof MPC implementation. The first one assumed a constantdisturbance in the prediction horizon, whereas for thesecond one the expected future disturbances were includedin the prediction horizon.

4.1 Case 1 - Production from one well

Fig. 3 presents the simulation results for production fromone well with constant disturbance in the prediction hori-zon. It can be seen that stabilization of the pressure andwater level is quite good with the cost that the respectiveoutflows on the right hand side oscillate quite heavily.In contrary, the oil level oscillates more heavily aroundits nominal value compared to the water level and gaspressure, however, its outflow (black plot) is less noisycompared to the the outflows of water and gas.

In Fig. 4, the same simulation case as presented aboveis demonstrated, however, this time with a time-varyingdisturbance anticipating the slug also in the predictionhorizon. What becomes clear directly is the reduced oscil-latory behavior for the outflows, whereas the performance

Copyright © 2018, IFAC 123

Fig. 3. Case 1 - state variables and manipulated variablesfor constant disturbance in the prediction horizon

Fig. 4. Case 1 - state variables and manipulated variablesfor time-varying disturbance in the prediction horizon

with respect to setpoint tracking does not change muchcompared to before.

4.2 Case 2 - Production from three wells

In Fig. 5, simulations for the production from three wellswith constant disturbance in the prediction horizon areshown. Comparing this case to the cases presented inSection 4.1, the heavier oscillations in the disturbancecan clearly be seen in the water and oil levels, howeverperformance for pressure setpoint tracking is the same asbefore. The outflows show a more noisy behavior and seemmore influenced by low-frequent rather than high-frequentslugs.

Fig. 6 presents the same case as above, but again with atime-varying disturbance in the prediction horizon. Likefor the case in Fig. 4, a clear reduction in oscillations canbe seen for all outflows. Setpoint tracking for the statevariables shows about the same performance as in Fig. 5and hence the same trend as in Section 4.1 is identifiable.

Fig. 5. Case 2 - state variables and manipulated variablesfor constant disturbance in the prediction horizon

Fig. 6. Case 2 - state variables and manipulated variablesfor time-varying disturbance in the prediction horizon

4.3 Comparison

In Fig. 7, the cost functions for all presented cases areshown. The top plot shows the case for production fromone well, whereas the bottom plot demonstrates produc-tion from three wells. What is apparent is the betterperformance for simulations with anticipated sinusoidaldisturbances in the predictions horizon (blue plots) com-pared to the cases with constant disturbances (red plots).

5. CONCLUDING REMARKS

In this paper we presented a nonlinear model predictivecontrol scheme for the minimization of the effect of slug-ging inflow disturbances on the state and the manipulatedvariables. Thereby, there exists a trade-off between thereduction in oscillations for the state variables and the ma-nipulated variables. The model that was used for controllerdesign is a differential algebraic equation model with 3differential states defining the dynamic variables liquidlevel, water level and gas pressure as well as 20 algebraic

Copyright © 2018, IFAC 124

constant disturbance in prediction horizon

sinusoidal disturbance in prediction horizon

0 100 200 300 400 500 6000

2

4

6

8

constant disturbance in prediction horizon

sinusoidal disturbance in prediction horizon

Fig. 7. Comparison of the cost function values for simula-tion cases 1 and 2

states characterizing the positions of droplet size classes atthe end of the active separation zone. We defined 10 sizeclasses for oil droplets and water droplets, respectively.

The results show that the controller is able to maintainthe pressure as well as the oil and water levels at theirdesired setpoints and reduce the effect of the disturbanceson the outflows as well. We presented two different kinds ofdisturbances as well as two solution strategies. From thesimulations it became apparent that for both cases thesolution strategy with non-constant, time-varying distur-bances over the prediction horizon was superior.

This work merely presents a proof of concept and hencecomputational times are not reported. However, we areaware that the aspect of real-time applicability is of highsignificance for MPC implementations. For completeness,we used a quite detailed model with droplet calculationsfor oil in water and water in oil. However, in order toreduce computational time, using simpler, linear modelscould be an option.

REFERENCES

Andersson, J. (2013). A general-purpose software frame-work for dynamic optimization. PhD thesis, KU Leuven.

Arntzen, R. (2001). Gravity Separator Revamping. Ph.D.thesis, Norwegian University of Science and Technology,Trondheim, Norway.

Backi, C.J. and Skogestad, S. (2017a). A simple dy-namic gravity separator model for separation efficiencyevaluation incorporating level and pressure control. InProceedings of the 2017 American Control Conference.Seattle, WA, USA.

Backi, C.J. and Skogestad, S. (2017b). Virtual inflowmonitoring for a three phase gravity separator. InProceedings of the 1st IEEE Conference on ControlTechnology and Applications. Kohala Coast, HI, USA.

Bothamley, M. (2013). Gas/Liquid Separators - Quanti-fying Separation Performance - Parts 1–3. Oil and GasFacilities, 2(4,5,6), 21–29, 35–47, 34–47.

Hansen, E.W.M. (2001). Phenomelogical Modeling andSimulation of Fluid Flow and Separation Behaviour inOffshore Gravity Separators. In Proceedings of the 2001

Pressure Vessel and Piping Conference, volume 431, 23–29. Atlanta, GA, USA.

Jahanshahi, E., Backi, C.J., and Skogestad, S. (2017).Anti-slug control based on a virtual flow measurement.Flow Measurement and Instrumentation, 53, 299–307.

Jahanshahi, E. and Skogestad, S. (2013). Closed-loopmodel identification and PID/PI tuning for robust anti-slug control. IFAC Proceedings Volumes, 46(32), 233–240.

Jahanshahi, E. and Skogestad, S. (2017). Nonlinear controlsolutions to prevent slugging flow in offshore oil produc-tion. Journal of Process Control, 54, 138–151.

Kharoua, N., Khezzar, L., and Saadawi, H. (2013). CFDModelling of a Horizontal Three-Phase Separator: APopulation Balance Approach. American Journal ofFluid Dynamics, 3(4), 101–118.

Laleh, A.P., Svrcek, W.Y., and Monnery, W.D. (2012).Computational Fluid Dynamics-Based Study of an Oil-field Separator - Part I: A Realistic Simulation. Oil andGas Facilities, 1(6), 57–68.

Laleh, A.P., Svrcek, W.Y., and Monnery, W.D. (2013).Computational Fluid Dynamics-Based Study of an Oil-field Separator - Part II: An Optimum Design. Oil andGas Facilities, 2(1), 52–59.

Wachter, A. and Biegler, L.T. (2006). On the implemen-tation of an interior-point filter line-search algorithmfor large-scale nonlinear programming. MathematicalProgramming, 106(1), 25–57.

Appendix A. ARCTAN APPROXIMATIONS

The switching argument ∆ti together with the arctan-function ζi(∆ti)

∆ti = tW/Oh − tO/Wvi ,

ζi(∆ti) =1

π

(arctan (1000π∆ti) +

π

2

),

are used to calculate the positions zO/Wi of oil and water

droplets at the end of the active separation zone

zO/Wi = ζi(∆ti)hW/O + (1− ζi(∆ti))

(LvO/Wvi

vW/Oh

).

The function switches between the cases if a droplet sizeclass i hits the interface, meaning ζi(∆ti) = 1 or if itremains dispersed, as for ζi(∆ti) = 0.

Appendix B. PARAMETERS

g gravitational acceleration ≈ 9.8 m s−1

L Length (active separation zone) 10 m

MG Molar mass of the gas 0.01604 kg mol−1

r Radius of the separator 1.65 m

R Universal gas constant 8.314 kg m2

s2 mol KT Temperature 328.5 K

VSep Volume (active separation zone) 85.53 m3

α Water cut of the liquid inflow 0.135

µO Dynamic viscosity oil 0.001 kg (s m)−1

µW Dynamic viscosity water 0.0005 kg (s m)−1

ρG Density of gas 49.7 kg m−3

ρO Density of oil 831.5 kg m−3

ρW Density of water 1030 kg m−3

φw Initial water to water fraction 0.1

φo Initial oil to oil fraction 0.9

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