Control-Oriented Drift-Flux Modeling of Single and Two...

CONTROL-ORIENTED DRIFT-FLUX MODELING OF SINGLE AND TWO-PHASEFLOW FOR DRILLING

Ulf Jakob F. Aarsnes∗

Department of Engineering CyberneticsNorwegian University of Science and Technology

Trondheim 7491Norway

Email: [email protected]

Florent Di Meglio

Centre Automatique et SystemesMINES ParisTech

Paris 75006France

Steinar Evje

Department of Petroleum EngineeringUniversity of Stavanger

Stavanger 4036Norway

Ole Morten Aamo

Department of Engineering CyberneticsNorwegian University of Science and Technology

Trondheim 7491Norway

ABSTRACTWe present a simplified drift-flux model for gas-liquid flow

in pipes. The model is able to handle single and two-phase flowthanks to a particular choice of empirical slip law. A presentedimplicit numerical scheme can be used to rapidly solve the equa-tions with good accuracy. Besides, it remains simple enoughto be amenable to mathematical and control-oriented analysis.In particular, we present an analysis of the steady-states of themodel that yields important considerations for drilling practi-tioners. This includes the identification of 4 distinct operatingregimes of the system, and a discussion on the occurrence ofslugging in underbalanced drilling.

NOMENCLATUREG, L Indices denoting gas and liquidα Volume fraction

∗Address all correspondence to this author.

ρ DensityA Area of flowcL Velocity of sound, liquidcG(T ) Velocity of sound, gasK Slip parameterS Slip parameterCv Choke constantD Hydraulic diameterg Gravity constantf Friction factorkG Gas inflow parameterkL Liquid inflow parameterm Liquid mass variablen Gas mass variablePres Reservoir pressurePs Separator pressurep0 Reference liquid pressureT (s) Temperature

Proceedings of the ASME 2014 Dynamic Systems and Control Conference DSCC2014

October 22-24, 2014, San Antonio, TX, USA

DSCC2014-6121

1 Copyright © 2014 by ASME

Downloaded From: http://proceedings.asmedigitalcollection.asme.org/ on 01/16/2015 Terms of Use: http://asme.org/terms

Y Choke correction factorχL,G Flux fraction, liquid, gasρL,0 Reference liquid densityφ(s) Inclination

1 INTRODUCTIONThe process of drilling for hydrocarbons consists in creating

a borehole, sometimes extending several thousand meters intothe ground, until it reaches an oil or gas reservoir. There is anincreasing drive for automation in many aspects of drilling, to in-crease efficiency and reduce the inherent risk of the operation. Asthe world reaches the end of ’easy oil’ and the world’s reservoirsstart depleting, increasingly challenging wells are considered fordrilling.

Following the demand of the drilling industry, advanced,high fidelity simulators of the drilling process have been devel-oped. Applications of these include training of drilling personneland real time decision support. At the same time, automated con-trol systems for controlling various aspects of the drilling processhave been developed and are gradually being accepted by the in-dustry. In particular, model-based estimation and control strate-gies exist for Managed Pressure Drilling (MPD) [1, 2]. Theseare based on a simple hydraulic model of the pressure and flowdynamics of the drilling system. Indeed, in conventional over-balanced drilling and MPD, the borehole pressure is maintainedabove the reservoir pressure by circulating liquid (usually mudor water) into the well. Conversely, in Under-Balanced Drilling(UBD), oil and gas are produced while drilling. The introductionof a gaseous phase makes modelling the fluid dynamics signif-icantly more complicated. The main contribution of this paperis to present a simple and reliable model for two-phase flow,amenable to the development of model-based control and esti-mation techniques for UBD.

In [1] it is stated that the goal of that paper is to ”at-tract researchers from the control community to problems withindrilling automation”. We submit to that goal but will focus onUnder-Balanced Drilling and the unique challenges posed by theparticulars of this operation, such as the coupling with the reser-voir and the complex dynamics of a distributed multiphase flowsystem.

The layout of the paper is as follows: In section 2 we explainthe basics of the underbalanced drilling process. In section 3 wediscuss how this process can be modelled using the Drift Fluxmodel which can be solved with an implicit numerical schemeas suggested in Section 4. Section 5 suggests a simple proce-dure to match this model to more detailed models or experimen-tal data. In Section 6, we use the model to derive a novel analysisof the steady-states and dynamics encountered in underbalanceddrilling operations. Finally, in Section 7, we propose some inter-esting control and estimation problems from UBD, one of whichis a novel idea derived from the preceding analysis.

FIGURE 1. Drilling process schematic for UBD.

2 UNDERBALANCED DRILLINGConsider the drilling system schematically depicted in Fig.

1. It consists of a circulation system: The drilling fluid is pumpedinto the top of the drill string, and circulated out at the bottom atthe drilling bit. It then flows up trough the annular section aroundthe drill string, transporting the drilled formation particles, calledcuttings, out of the well. The cuttings and any produced fluids arethen separated from the drilling fluid before it is injected into thedrill string again.

One of the main concerns of the driller is to control the pres-sure in the lower sections of the well. Too high pressure resultsin low ROP (rate of penetration), loss of drilling fluid (which insome cases can be excessively expensive) and degrade the fu-ture production rate of the well. Low pressure in the well causesinflux of gas and/or other reservoir fluids, and excessively lowpressure may result in wellbore collapse and a stuck drill string.The biggest danger in drilling is that of a blowout: this happenswhen the gas influx goes unnoticed and starts to displace themuch heavier drilling fluid. Hence this causes a further reduc-tion in downhole pressure and yet more gas influx, i.e. a positivefeedback loop which can spiral out of control. For these reasonsit is important to keep the downhole pressure within a specifiedpressure window at all times, and to closely monitor the dynam-ics in the well.

In UBD operations the downhole pressure is deliberately



kept below the reservoir pore pressure1, causing continuous in-flow of produced fluids from the reservoir. The reservoir inflowis related to the downhole pressure by the Production Index (PI)and pore pressure. It is highly desirable to estimate the Produc-tion Index and pore pressure of the reservoir since these param-eters have a large influence on the dynamics of the system andplay an important role in determining the systems operationalconstraints.

Before starting an UBD operation, initial estimates of thereservoir properties are combined with various requirementsposed by equipment constraints and operational considerationsto develop the so-called operating envelope. The operating en-velope defines the allowable range of the controlled and ma-nipulated variables in the process. Staying within the operat-ing envelope is paramount to minimizing risk and optimizingdrilling performance. To achieve this, a good understanding ofthe multiphase flow and pressure dynamics of the coupled well-reservoir system is required. This has led to the developmentof several advanced, high-fidelity simulators of both over- andunder-balanced drilling. The application of these, however, havelargely been limited to training for and planning of operations.

3 MODELLINGChoosing Model Fidelity

Underbalanced drilling operations is an inherently multi-phase process. Drilling mud, and in some cases injection gas,is pumped into the well through the drill string. In addition,both oil, gas and water may be produced from the reservoir, andcuttings from the drilled formation constitutes a phase of solidparticles. In the literature, the most used model of multiphaseflow in drilling is the Drift Flux Model (DFM) which requiresone distributed state for each phase to model the mass balancewhile the momentum of the mixture is lumped into one equa-tion (a seminal report on the DFM is [3], relevant referencesfor the use of DFM in drilling is [4, 5]). Including an energyequation, this results in a total of n+2 distributed states, where nis the number of phases. The resulting complexity makes thesemodels ill suited for control and estimator design. Instead it hasbeen proposed that a good compromise with sufficient accuracyis achieved by lumping fluids into a gas and a liquid phase, [6,7], and ignoring the energy equation thereby reducing the num-ber of distributed states down to 3. Further simplification can beachieved by using a static momentum balance which reduces thenumber of dynamic distributed states down to 2. The downsideof this reduction is that the pressure waves propagate instanta-neously, reducing the model accuracy during fast transients, [8,9].

Low order lumped (i.e. ODE) models have been suggestedfor one-phase drilling [1, 10, 11], but these are unsuitable for

1I.e. the pressure in the fluids in the reservoir surrounding the borehole.

two-phase flow. Low order lumped models have seen successfuluse in control of severe slugging in two-phase production risers[12–15], and has also been considered for application in UBD[16], but are not able to reproduce the operating envelope whichis essential for UBD operations.

The rationale behind reducing model complexity is to enableestimation and control techniques. Besides, the simplificationshould give insight into the dynamics of the model. Any reduc-tion in complexity from that of the high fidelity models, how-ever, will require calibration or compensation by feedback frommeasurements of some sort. In the following we will focus on amechanistic drift flux model which gives a reasonable compro-mise between complexity and accuracy, see Table 1. We stressthat this model on its own have limited quantitative predictivepower, but produces excellent predictions after being calibratedto actual data or a high fidelity model.

3.1 The drift flux model:The model is developed by expressing the mass conservation

law for the gas and the liquid separately, while combining themomentum equation to obtain a third order hyperbolic PDE. Indeveloping the model, we use the following mass variables

m = αLρL, n = αGρG (1)

where for k = L,G denoting liquid or gas, ρk is the phase density,and αk the volume fraction satisfying

αL +αG = 1. (2)

Further vk denotes the velocities, and P the pressure. All of thesevariables are functions of time and space. We denote t ≥ 0 thetime variable, and s∈ [0,L] the space variable, corresponding to acurvilinear abscissa with s = 0 corresponding to the bottom holeand s = L to the outlet choke position (see Fig. 1). The equationsare as follows,

∂m∂ t

+∂mvL

∂ s= 0, (3)

∂n∂ t

+∂nvG

∂ s= 0, (4)

∂mvL +nvG

∂ t+

∂P+mv2L +nv2

G∂ s

=−(m+n)gsinφ(s)− 2 f (m+n)vm|vm|D

. (5)

In the momentum equation (5), the term (m+n)gsinφ representsthe gravitational source term, while − 2 f (m+n)vm|vm|

D accounts for



TABLE 1. A coarse taxonomy of unsteady two-phase models for UBD

Model Category Trade-off

High fidelity simulation models Very high complexity and accuracy. Includes flow regime predictions. Slow and illsuited for analysis and controller design.

Mechanistic Drift-flux model High complexity and reasonable to good fidelity when using tuning procedures.

DF with static mom.eq. Little reduction in complexity. Reduced bandwidth.

Low Order Lumped models Desired form for analysis and design. Non-acceptable reduction of accuracy w.r.t.classifying UBD operating points.

frictional losses. The mixtures velocity is given as

vm = αGvG +αLvL. (6)

Along with these distributed equations, algebraic relations areneeded to close the system.

Closure Relations: Both the liquid and gas phase areassumed compressible. This is required for the model to handlethe transition from two-phase to single-phase flow. The densitiesare thus given as functions of the pressure as follows

ρG =P

c2G(T )

, ρL = ρL,0 +Pc2

L, (7)

where ck is the velocity of sound and ρL,0 is the reference densityof the liquid phase given at vacuum. Notice that the velocity ofsound in the gas phase cG depends on the temperature as sug-gested by the ideal gas law. The temperature profile is assumedto be known and fed into the model.

Combining (7) with (2) we obtain the following relations forfinding volume fractions from the mass variables:

αG =12−

c2G

c2L

n+m+√

∆

2ρL,0, (8)

∆ =(

ρL,0−c2

G

c2L

n−m)2

+4c2

G

c2L

nρL,0 (9)

Then the pressure can be found where we use a modified expres-sion to ensure pressure is defined when the gas vanishes

P =

(m

1−αG−ρL,0

)c2

L, if αG ≤ α∗Gn

αGc2

G, otherwise.(10)

Because the momentum equation (5) was written for the gas-liquid mixture, a so-called slip law is needed to empirically re-late the velocities of gas and liquid. Traditionally, the Zuber-Findlay [17] slip law is used

vG =C0vm + v∞ (11)

where C0 and v∞ are constant parameters for a given flow regime.However, for smooth transition between single and two-phaseflow, a relation with state–dependent parameters is needed [18,19]. More precisely, we use the following slip law

vG = (K− (K−1)αG)vm +αLS (12)

where K ≥ 1 and S≥ 0 are constant parameters.

3.2 Boundary ConditionsBoundary conditions on the downhole boundary are given

by the mass-rates of gas and liquid injected from the drilling rigand flowing in from the reservoir. Denoting the cross sectionalflow area by A the boundary fluxes are given as:

mvL|s=0 =1A

(WL,res(t)+WL,in j(t)

), (13)

nvG|s=0 =1A

(WG,res(t)+WG,in j(t)

). (14)

The injection mass-rates of gas and liquid, WG,in j,WL,in j, arespecified by the driller and can, within some constraints, be con-sidered as manipulated variables. The inflow from the reservoiris dependent on the pressure at the downhole boundary, usuallygiven by a Vogel-Type Inflow performance relationship (IPR)[20], but within the operational range of a typical UBD opera-tion an affine approximation should suffice, i.e.

WL,res = kL max(Pres−P(0),0) (15)WG,res = kG max(Pres−P(0),0) (16)



Here Pres is the reservoir pore pressure and kG,kL are the produc-tion index (PI) of the gas and liquid respectively.

The topside boundary condition is given by a choke equationrelating topside pressure to mass flow rates

mvL√ρL

+nvG

Y√

ρG

∣∣∣s=L

=Cv(Z(t)

)A

√max(P(s=L, t)−Ps,0),

(17)

where Cv is the choke opening given by the manipulated variableZ, Y ∈ [0,1] is a gas expansion factor for the gas flow and Ps isthe separator pressure, i.e. the pressure downstream the choke.Changing the choke opening is the primary control actuation forthe drilling system.

4 NUMERICAL SCHEMEModel (3)–(17) rewrites as the following nonlinear 3-state

hyperbolic system of conservation laws [9]

∂q1

∂ t+

∂ f1(q1,q2,q3)

∂ s= 0 (18)

∂q2

∂ t+

∂ f2(q1,q2,q3)

∂ s= 0 (19)

∂q3

∂ t+

∂ f3(q1,q2,q3)

∂ s= FW (q1,q2,q3)+FG(q1,q2) (20)

where q = (q1,q2,q3) = (n,m,nvG +mvL) is the set of conser-vative variables. Traditionally, explicit numerical schemes arefavored for such systems, because they preserve shocks and limitnumerical diffusion [6, 21]. However, to ensure their stability,the time and space steps ∆t and ∆s are required to satisfy, at alltimes, Courant–Friedrichs–Lewy (CFL) types of conditions, ofthe form

∣∣∣∣λmax(u)∆t∆s

∣∣∣∣≤ 1 (21)

where λmax(u) is the largest (in absolute value) characteristicvelocity of the problem, i.e. the largest eigenvalue of the ma-trix ∂ f

∂u (u). In the case of two-phase flow, the largest eigenvalue,which corresponds to the propagation of pressure waves in thegas, is of the order of 300 m.s−1. In the case of single-phaseliquid flow, the order of magnitude jumps to around 1000 m.s−1.For, e.g., a 3000 meter-long well with 100 space steps, this im-poses to a time step of the order of ∆t < 0.1s.

For this reason, we choose an unconditionally stable im-plicit scheme to numerically solve the equations, therefore not

subject to CFL conditions. More precisely, consider a time-space grid t ∈ 0,∆t, ..., s ∈ 0,∆s, ...,(P−1)∆s. Denot-ing q(p∆s,n∆t) = qn(p), we consider the following approximateequations

qn1,2(p)−qn−1

1,2 (p)

∆t+

f1,2(qn(p+1))− f1,2(qn(p−1))2∆s

= 0

(22)

qn3(p)−qn−1

3 (p)∆t

+f3(qn(p+1))− f3(qn(p−1))

2∆s= FW (qn(p))+FG(qn(p)) (23)

These equations are valid for p= 1, ...,P−2, yielding 3×(P−2)implicit nonlinear equations to be solved at each time step. Theboundary conditions (13)–(17) yield 3 more equations, that canbe written in the following implicit form

hbottom,1(qn(0)) = hbottom,2(qn(0)) = htop(qn(P)) = 0 (24)

The last 3 equations are given by using a de-centered second-order discretization of the spatial derivative at the boundaries

∂ f1,2(q)∂ s

((P−1)∆s,n∆t)≈

3 f1,2(qn(P−1))−4 f1,2(qn(P−2))+ f1,2(qn(P−3))2∆s

(25)

∂ f3(q)∂ s

(0,n∆t) ≈ −3 f3(qn(0))+4 f3(qn(1))− f3(qn(2))2∆s

(26)

This yields a set of 3P equations with 3P unknowns to be solvedat each time step. To increase computational speed, the Jacobianof these equations is computed analytically, and a simple Newtonalgorithm is used to solve them. More precisely, the algorithmtakes the following steps

1. At t = 0, pick a suitable initial condition, e.g. an equilibriumprofile.

2. At t = n∆t, considering that qn−1(p) is known for all p =0, ...,P− 1 from the previous iteration, do the following tocompute qn(p)

(a) Use qn−1(·) as the initial guess of qn(·).



(b) Rewriting Equations (22)–(26) as F(qn) = 0,compute F(qn).

(c) If the norm of F is low enough (given a pre-definedthreshold value), the Newton algorithm has converged:go to step 3. The norm can be chosen, e.g., as aweighted 2-norm, i.e. FTWF , where W is a suitablychosen weighing matrix.

(d) Compute the 3P×3P Jacobian matrix

J =∂F∂qn =

(∂F

∂qn1(1)· · · ∂F

∂qn1(P−1)

∂F∂qn

2(1)· · · ∂F

∂qn3(1)· · ·)

(27)

(e) Update qn

qn := qn− J−1F(qn) (28)

(f) Repeat steps (b)–(e) until the norm of F is low enough.

3. Increment n and go to step 1.

Remark: The choice of the norm in the stopping criterion for theNewton algorithm is critical to the algorithm performance androbustness. Particular attention should be given to scaling theequations to maximize the conditioning of matrix J.

5 MODEL CALIBRATIONSince the model described in Section 3.1 is of a significant

reduced complexity compared to the high fidelity models, certainparameters need to be adjusted in order to quantitatively repro-duce the behavior of a given system. This can be done by formu-lating the fitting of the model response to measured or simulateddata as an optimization problem, similar to what is done in [22].

5.1 Calibrated parametersThe number of calibrated parameters may vary according to

the considered case, depending on how well known the param-eters are in the given scenario. Here we use the the slip lawparameters K, S, and the friction factor f which are empiricalparameters. In high fidelity models these are found from sophis-ticated relations based on flow regime predictions which intro-duces significant extra complexity to the model. In addition itmay be beneficial to tune the choke coefficient Cv and the gasexpansion factor Y in (17), to ensure good performance of themodel as small changes in these parameters can have a large ef-fect on the model dynamics.

The calibration algorithm is a simple optimization routine.Given a set of n measurements at steady state zi, i = 1, . . . ,n, anda vector of p parameters to calibrate θ ∈ Rp, we consider the

FIGURE 2. System calibrated with OLGA data. The choke openingis incremented at 2,4,6,9,11 hours.

following minimization problem

minθ∈Ω

z1(θ)− z1...

zn(θ)− zn

T

M

z1(θ)− z1...

zn(θ)− zn

(29)

where zi(θ) is the output of the model corresponding to the mea-surement zi, for the vector of parameters θ , M is a weightingmatrix used to normalize the outputs, and Ω is the set of allow-able values which θ can take.

The vector of parameters, θ , which satisfies (29) is foundusing a gradient based optimization procedure in MATLAB.The result of using this procedure on a data set generated withOLGA can be seen in Fig. 2. In this example only BHCP andWHP from each steady state were used to tune the four vari-ables θ = [K,S, f ,Cv,Y ]T . The resulting calibration of the sim-ple model has a good performance in the operating range coveredby this dataset, but may deteriorate when the system moves to adistinctly different operating regime.

6 APPLICATION OF MODEL TO UBDIn this section, to be more clear, P(0) will be referred to

as Bottom-Hole Circulating Pressure (BHCP) and P(L) as Well-Head Pressure (WHP). The current approach to controlling pres-sure in UBD is manual control of the outlet choke. Often, oper-ators monitor the WHP and try to stabilize it around a constant



FIGURE 3. UBD operating envelope. Taken from [23].

value2. Their a priori knowledge of potential steady-state oper-ating points is given by graphs similar to Fig. 3, taken from [23].These graphs determine, for a constant WHP, the correspondencebetween BHCP and gas flow rate, and determine a set of operat-ing constraints referred to as the operating envelope [24].

We now perform a steady-state analysis of the model. Thekey difference of this approach compared with state-of-the-artsteady-state analyses software such as the one used in [23, 24], isto include the choke model, thus ”classifying” operating pointsin terms of choke opening rather than WHP. This allows us toidentify a broader variety of operating points, as well as clearlydefining control challenges.

6.1 Classifying operating pointsTo understand the need for a characterization of steady-

states, consider the simulation depicted in Fig. 4, correspondingto a UBD operation in a dry gas well (i.e. WL,res = 0). The chokeopening is decremented after 3, 6 and 9 hours, leaving enoughtime between each decrement for the system to reach equilib-rium. Between t = 0 and t = 6h, the system exhibits an intuitivebehavior: both BHCP and WHP increase as the choke openingis decreased. However, during the next step, between t = 6h andt = 9h, the WHP increases before settling at a lower value thanthe previous equilibrium. The counter-intuitive notion that low-ering the choke opening can lower the equilibrium WHP compli-cates in itself the manual control of the operation. This difficultyis increased by the inverse response of the WHP, which seemsto indicate the presence of unstable zeros in the transfer func-tion between this output and the choke opening. When the chokeopening is decreased further after t = 9h, the system drifts to-wards overbalanced conditions. We now use the model describedin Section 3.1 to get insight into these mechanisms.

2The reason for this is that WHP determines the flow rates through the choke,thus the amount of gas and liquid that the separator has to handle at once. This isone of the main constraints in UBD.

FIGURE 4. Transient simulation with decrementing choke opening.

6.2 Steady-state analysisAt steady states of the system (3)-(12), the mass-rates are

constant throughout the well, while the pressure gradient is givenby the three terms: acceleration due to the expanding gas, frictionand hydrostatic pressure. Given a value of the BHCP, integrat-ing these equations yields the corresponding equilibrium valueof the WHP, as depicted by the solid line in Fig 5. However, thiscurve does not take into account the choke model on the topsideboundary s = L. Indeed, to each value of the BHCP correspondsunique values of the inflow rates of gas and liquid, through Equa-tions (13)-(17). Only the WHPs which yield a total out flow rate(given by (17)) equal to the sum of the in flow rates can be steady-states. This is represented by the dotted lines in Fig. 5, which,for various values of the choke opening, are the WHP requiredfor the in flow and out flow rates to match. The intersections ofthe solid and each dotted line is the steady-state equilibrium forthe choke opening corresponding to the dotted line.

6.2.1 Steady-state taxonomy Depending on the op-erating point, the system exhibits significantly different dynam-ics. Specifically, we identify the four distinct operating regimesof the UBD system, ordered by increasing equilibrium BHCP,and illustrated on Fig. 6.

Intuitive regime This regime corresponds to the first 6 hours ofthe transient simulation shown in Fig. 4. It is stable andminimum phase.

Non-intuitive regime A small increase in BHCP yields a lowerinflux of gas into the system, which increases the hydrostaticpressure and eventually decreases the WHP. This regime isstable for a constant choke opening but with a inverse re-



FIGURE 5. Alternative operating envelope showing operating pointsfor different choke openings.

FIGURE 6. Using the choke as actuation, the system can have 4 dis-tinct operating regimes.

sponse in the WHP. Consequently, if a constant WHP is en-forced as a boundary condition, steady-states in this regionbecomes unstable.

Unstable regime For larger values of the equilibrium BHCP,there is no stable equilibrium. The system will then eitherenter a limit cycle of severe slugging, or, drift towards thesteady-state in overbalanced conditions.

Stable, overbalanced regime The system only contains the liq-

uid phase and the difference; BHCP-WHP, is constant.

6.3 Control EnvelopeThe behaviour of the system w.r.t. to changing choke open-

ing can perhaps more clearly be seen in Fig. 7. In this so-calledcontrol envelope the steady state values of BHCP and WHP (i.e.the green and red dots in Fig. 5) is plotted over a range of chokeopenings given on the x-axis.

Consider a well initially in the overbalanced regime. Clos-ing the choke will cause the system to move along the red lineuntil a choke opening of 0.38. This corresponds to a choke open-ing where the red dotted line in Fig. 5 is below the WHP minimaoccurring at the transition to underbalanced conditions. The sys-tem will then move to the stable steady states given by the bluecurve in Fig. 7. Reducing choke opening when the system is inthis state will make the system move along the blue curve. Theend of the blue curve, moving towards left, in Fig 7, is the limitto the unstable regime. Closing the choke past this point will ei-ther cause the system to go to the overbalanced regime (as in thesimulation shown in Fig. 4) or enter a severe slugging limit cycleas shown in Fig. 8.

6.4 SluggingBy a slight modification to the parameters used to obtain

the simulation shown in Fig 4 (increasing Pres and decreasingY ), the system becomes capable of severe slugging. Specifically,instead of moving to the overbalance steady-state when the chokeis closed sufficiently, it enters a slugging limit-cycle (see Fig.8and 9). We note that the limit-cycle takes the system into theoverbalanced regime for a brief period, which requires the modelto be capable to handle the transition to liquid only flow.

The possible occurrence and control of severe slugging inproduction risers have been studied extensively (see e.g.[25–27]), but slugging in underbalanced drilling operations have notyet gotten significant attention. Due to the different boundaryconditions the causes for slugging in production risers, such asdensity waves and casing heading, cannot necessarily explainslugging during UBD. Hence, the causes for severe slugging inUBD is not yet well understood. To move forwards toward in-creased automation of UBD this must be amended, and as such,the investigation of the causes for severe slugging in UBD wasmajor motivation for the implementation of the current model.

7 FURTHER CONTROL AND ESTIMATION PROB-LEMS IN UBDThis section contains suggestions for how applied control

theory can make an impact in UBD.



FIGURE 7. Control envelope showing steady state points plottedagainst choke opening. The system shows a hysteresis-like behaviourin that it will converge to different steady-states depending on whetherthe system is over- or under-balanced.

7.1 Estimation of PI and pore pressureEstimation of the Production Index (PI) and pore pressure

amounts to identifying the amount of produced fluids and relat-ing it to the measured BHCP as by (13)-(14). In typical UBDoperation available real-time measurements of significance areBHCP and WHP. Additionally, measurements of the amount ofproduced gas is available after it has been separated from thedrilling fluid with a delay of 2 - 12 hours. When drilling, thePI and pore pressure will remain more or less constant until afracture in the reservoir is encountered, at which time kG,kL willincrease. Rapid estimation of the PI using the immediately avail-

FIGURE 8. Decrementing choke opening into slugging instability.

FIGURE 9. Void fraction (αG) profiles at various stages of a sluggingcycle.

able measurements of BHCP and WHP has considerable benefitsas it enables the drillers to optimize the process on-line as wellas pinpointing the exact position of the fracture based on the po-sition of the drilling bit when the fracture was encountered [28].

7.2 Connection scenarioFor every 30 meters drilled, the drill string has to be ex-

tended in a procedure called a drill string connection. During thisoperation the main pump is shut down, stopping the circulationof drilling fluid. This causes a separation between the phases as



the gas travels to the topmost part of the well when gravity pullsthe liquid down to the bottom. The gas expands as it travels up-wards increasing the pressure in the well. During this transientprocess, it is desirable to keep the BHCP at a specified set-pointby dynamically controlling the choke opening Z. Designing acontroller then amounts to finding a dynamic mapping from themeasured BHCP and WHP to the choke opening Z which keepsBHCP within some predefined constraint for the duration of theconnection[29].

7.3 Extending the Operating Envelope the UnstableRegime

Often during UBD operations, produced gas is flared afterseparation from the drilling fluid. In these cases it can be desir-able to minimize the amount of gas produced due to economicor environmental considerations of the gas flaring. As discussedin Section 6.2 and shown in Fig. 7, the operating points corre-sponding to the least amount of produced gas (i.e. closest to thereservoir pressure) are in the unstable regime. Hence, these po-tential operating points are currently not part of the operating en-velope and drilling currently does not occur in this regime. Usingfeedback control, however, these steady-states can be stabilizedthereby extending the operating envelope, potentially enablingmore cost effective and environmentally friendly operations.

ACKNOWLEDGMENTThis work was supported by Statoil and the Norwegian Re-

search Council (NFR project 210432/E30 Intelligent Drilling).

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