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Adaptive PI Control of Bottom Hole Pressure during Oil Well Drilling
Proceedings of the 3rd IFAC Conference on Proceedings of the 3rd IFAC Conference on Advances in Proportional-Integral-Derivative Proceedings of on Proceedings of the the 3rd 3rd IFAC IFAC Conference Conference on Control Advances in Proportional-Integral-Derivative Control Available online at www.sciencedirect.com Ghent, Belgium, May 9-11, 2018 Advances in Proportional-Integral-Derivative Proportional-Integral-Derivative Control Proceedings of the 3rd IFAC Conference on Control Advances in Ghent, Belgium, May 9-11, 2018 Ghent, Belgium, May 9-11, 2018 Advances in Proportional-Integral-Derivative Control Ghent, Belgium, May 9-11, 2018 Ghent, Belgium, May 9-11, 2018
ScienceDirect
IFAC PapersOnLine 51-4 (2018) 166–171
Adaptive Adaptive PI PI Control Control of of Bottom Bottom Hole Hole Adaptive PI Control of Bottom Hole Pressure during Oil Well Drilling Adaptive Control Bottom Hole PressurePI during Oil of Well Drilling Pressure during Oil Well Drilling Pressure during Oil ∗∗Well Drilling Jing Zhou
Jing Zhou ∗∗ Jing Jing Zhou Zhou Jing Zhou ∗ University of Agder, 4898 Department of Engineering Department of Engineering Sciences, Sciences, University of Agder, 4898 Department of Engineering Engineering Sciences, University of Agder, Agder, 4898 4898 Grimstad, Norway jing.zhou@ uia.no). Department of Sciences, University of Grimstad, Norway (e-mail: (e-mail: jing.zhou@ uia.no). Department of Engineering Sciences, University of Agder, 4898 Grimstad, Norway (e-mail: (e-mail: jing.zhou@ uia.no). Grimstad, Norway jing.zhou@ uia.no). Grimstad, Norway (e-mail: jing.zhou@ uia.no). Abstract: In this paper, we studied the bottom hole pressure (BHP) control in an oil well during Abstract: In this paper, we the hole (BHP) control an well Abstract: In this this paper, wells we studied studied the bottom bottom hole pressure pressure (BHP) control in inbeing an oil oildrilled. well during during drilling. Today marginal with narrow pressure windows(BHP) are frequently This Abstract: In paper, we studied the bottom hole pressure control in an oil well during drilling. Today marginal wells with narrow pressure windows are frequently being drilled. This Abstract: In this paper, we studied the hole pressure (BHP) control inbeing an oil well during drilling. Today marginal wells with narrow pressure windows are frequently drilled. This requires accurate and precise control to bottom balance the bottom hole pressure between the pore and drilling. Today marginal wells with narrow pressure windows are frequently being drilled. This requires accurate and precise to the bottom hole between pore and drilling. Today marginal wellscontrol with narrow pressure windows arepressure frequently being the drilled. This requires accurate and precise control to balance balance the hole pressure between the pore and fracture of the reservoir. This presents three schemes to the requires accurate and control balance the bottom bottom hole pressure the pore and fracture pressure pressure of theprecise reservoir. Thistopaper paper presents three control control schemesbetween to stabilize stabilize the BHP BHP requires accurateproportional-integral(PI) and control balance the hole pressure between pore and fracture pressure of theprecise reservoir. Thistopaper paper presents threefeed-forward control schemes to stabilize stabilize the BHP BHP prole, including control, PIbottom with control andthe adaptive PI fracture pressure of the reservoir. This presents three control schemes to the PI prole, including control, PI with control and fracture pressureproportional-integral(PI) ofcontrol. the reservoir. This paper presents threefeed-forward control schemes to stabilize BHP prole, including proportional-integral(PI) control, PI carried with feed-forward control and adaptive adaptive PI with feed-forward The proposed schemes are out through simulations onthe a highprole, including proportional-integral(PI) control, PI with feed-forward control and adaptive PI with feed-forward control. The proposed schemes are carried out through simulations on a highprole, including control, PI with feed-forward control and adaptive PI with feed-forward control. The proposed schemes carried out through simulations on aaInhighfidelity hydraulicproportional-integral(PI) drilling simulator for flow rate are changes and BHP set-point changes. fast with feed-forward control. The proposed schemes are carried out through simulations on highfidelity hydraulic drilling simulator for flow rate changes and BHP set-point changes. In fast with feed-forward control. The proposed schemes are carried out through simulations on a highfidelity hydraulic drilling simulator for flow rate changes and BHP set-point changes. In fast set-point changes and flow rate changes, the adaptive PI controller exhibits less tracking error fidelity for flow changes and BHPexhibits set-point changes. fast set-pointhydraulic changes drilling and flowsimulator rate changes, the rate adaptive PI controller less trackingInerror fidelity for flow rate changes and simulation BHPexhibits set-point changes. Inerror fast set-point changes drilling andthan flowsimulator rateconventional changes, the PI adaptive PI The controller exhibits less tracking and lesshydraulic oscillations the solution. results illustrate the set-point changes and flow rate changes, the adaptive PI controller less tracking simulation results illustrateerror the and less oscillations than the conventional PI solution. The set-point changes andthan flow rateconventional changes, adaptive PI The controller exhibits less tracking and less oscillations oscillations than the conventional PI solution. The simulation results illustrateerror the effectiveness of proposed control schemes.the PI and less the solution. simulation results illustrate the effectiveness of proposed control schemes. and less oscillations thancontrol the conventional effectiveness of schemes. effectiveness of proposed proposed control schemes. PI solution. The simulation results illustrate the © 2018, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. effectiveness proposed control schemes. Keywords: PIofcontrol, adaptive control, oil well drilling, pressure control. Keywords: PI control, adaptive control, Keywords: PI PI control, control, adaptive adaptive control, control, oil oil well well drilling, drilling, pressure pressure control. control. Keywords: oil well drilling, pressure control. Keywords: PI control, adaptive control, oil well drilling, pressure control. 1. INTRODUCTION 1. INTRODUCTION 1. 1. INTRODUCTION INTRODUCTION Main pump 1. INTRODUCTION q Main pump Controlling the bottom hole pressure in an oil well is one pump q Main pump pump pump Controlling the bottom hole pressure in an oil well is one q Main Controlling the bottom hole pressure in an oil well is one pump of the critical tasks during drilling. During well drilling, Controlling thetasks bottom hole drilling. pressureDuring in an oil well is one pump Main pumpq of the critical during well drilling, q Controlling thetasks bottom hole pressure instring an oil well is and one pump of the during drilling. During well drilling, a uid pumped into the topside qchoke of the critical critical during During drilling, a drilling drilling uid is istasks pumped intodrilling. the drill drill stringwell topside and qchoke of the critical tasks during drilling. During drilling, aa drilling uid is pumped the drill string topside and through the drill bit at theinto bottom hole of thewell well Merritt drilling uid is pumped into the drill string topside and choke through the drill bit at the bottom hole of the well Merritt qchoke a drillingWhite uiddrill is (1999); pumped into the (2006). drill string topside and choke through the bit at the bottom hole of the Merritt (1967); Nygaard Thewell mud then through the drill bit at the bottom hole of the well Merritt (1967); White (1999); Nygaard (2006). The mud then through thecuttings drill(1999); bit at bottom hole the well Merritt (1967); White Nygaard (2006). The mud then transports in the the annulus sideof of the well (i.e. (1967); White (1999); Nygaard (2006). The mud then q transports cuttings in the annulus side of the well (i.e. qback (1967); White (1999); Nygaard (2006). The mud then transports cuttings in annulus side of the well (i.e. in outside the string) the drill rig, transports cuttings in the the annulus sideup qback Annulus pump in the the well well bore bore outside the drill drill string) upofto tothe thewell drill (i.e. rig, back Annulus pump transports cuttings in annulus sideup ofto the in the well bore outside the drill string) the drill rig, where a choke valve andthe a back pressure pump iswell used(i.e. to qback in the well bore outside the drill string) up to the drill rig, Annulus pump where a choke valve and a back pressure pump is used to back Annulus pump in the well bore outside drill to the where aathe choke valve and aa back pressure is used to control annular pressure, seestring) Figureup 1pump for a schematic where choke valve andthe back pressure pump isdrill usedrig, to Annulus pump control the annular pressure, see Figure 1 for a schematic where athe choke and a back is used to control pressure, see Figure for overview of annular thevalve system. control the annular pressure, seepressure Figure 11pump for aa schematic schematic overview of the system. control the annular pressure, see Figure 1 for a schematic overview of the system. Today marginal wells with narrow pressure windows overview of the system. Today marginal wells with narrow pressure windows are are overview ofbeing the system. Today marginal wells with are frequently drilled. Thisnarrow requirespressure accuratewindows and precise Today marginal wells with narrow pressure windows are frequently being drilled. This requires accurate and precise Today wells with narrow are frequently being This requires accurate and precise control marginal to balance the BHP between the pore windows and fracture frequently being drilled. drilled. Thisbetween requirespressure accurate andfracture precise control to balance the BHP the pore and frequently being drilled. This requires accurate and precise control to balance the BHP between the pore and fracture pressure of the reservoir. A stable well bore promotes control toofbalance the BHP A between the pore andpromotes fracture Non pressure the reservoir. stable well bore Non control toof the BHP A between the pore andpromotes fracture pressure the reservoir. stable well bore efcient personnel safety. A well pressure ofbalance theand reservoir. A stable bore promotes return Non efcient drilling drilling and personnel safety. well A destabilized destabilized well Non return pressure of theand reservoir. A production. stable well borelow promotes efcient drilling personnel safety. A destabilized well bore can reduce or eliminate Too a mud efcient drilling and personnel safety. A destabilized well valve return Non return bore can reduce or eliminate production. Too low a mud valve efcient drilling and personnel safety. A destabilized well bore can reduce Too aa mud pressure can leador toeliminate a kick orproduction. well bore collapse and too bore can can reduce orto eliminate production. Too low lowand mud valve return valve pressure lead a kick or well bore collapse too bore can reduce or eliminate production. Too low a mud pressure can lead to a kick or well bore collapse and too high a mud pressure can create well bore fracturing and pressure can lead to a kick or well bore collapse and too valve high a mud pressure can create well bore fracturing and pressure can pressure lead to acan kick or stability well collapse and and too high mud create well bore fracturing losses.aa Preventing these costly problems requires high mud pressure can create wellbore bore fracturing and losses. Preventing these costly stability problems requires high a Preventing mud pressure pressure cancostly create well control bore fracturing and losses. these stability problems requires an accurate control. Pressure is a challenglosses. Preventing these costly stability problems requires an accurate pressure control. Pressure control is a challenglosses. costly stability problems requires an pressure control. Pressure is aadynamics challengingaccurate taskPreventing during wellthese drilling, due to thecontrol complex an accurate pressure control. Pressure control is challenging task during well drilling, due to thecontrol complex an accurate pressure control. Pressure is adynamics challenging task during well due to dynamics of the multiphase owdrilling, potentially consisting of drilling mud, ing task during well drilling, dueconsisting to the the complex complex dynamics of the multiphase ow potentially of drilling mud, ing task during well drilling, dueconsisting to the complex dynamics of ow potentially of drilling mud, oil,the gasmultiphase and cuttings. Fig. 1. of the multiphase ow potentially consisting of drilling mud, oil, gas and cuttings. 1. A A simplified simplified schematic schematic drawing drawing of of the the drilling drilling of the multiphase ow potentially consisting drilling mud, Fig. oil, gas and cuttings. The main objective is to precisely bottomofhole pressure Fig. 1. A simplified schematic drawing of the drilling drilling system. oil, gas and cuttings. Fig. 1. A simplified schematic drawing of the The main objective is to precisely bottom hole pressure system. oil, gas and cuttings. The main objective is to precisely bottom hole pressure prolemain throughout the iswell bore continuously while drilling, Fig. system. 1. A simplified schematic drawing of the drilling system. The objective to precisely bottom hole pressure prole throughout the continuously while drilling, The main objective iswell to bore precisely bottom holeabove pressure prole throughout the well bore continuously while drilling, i.e. maintain pressure in the pressure prole theannular well bore continuously while drilling, i.e. to tothroughout maintain the the annular pressure in the the well well above the hole hole system. pressure is is indirectly indirectly stabilized stabilized by by applying applying feedfeedprole throughout theannular welland bore continuously while drilling, i.e. maintain pressure in the well above the hole pressure pressure is indirectly indirectly stabilized by applying feedporeto or collapse the pressure below the fracture or sticking back control to stabilize the topside annulus pressure i.e. to maintain the annular pressure in the well above the hole is stabilized by applying feedback control to stabilize the topside annulus pressure pore or collapse pressure and below the fracture or sticking i.e. toor maintain the annular pressure in the wellor above the back hole istheindirectly stabilized by applying feedpore collapse pressure below fracture sticking back pressure control to stabilize the topsidecorresponding annulus pressure pressure. Basically, this and amounts tothe stabilizing the down instead, where pressure set-point pore or collapse pressure and below the fracture or sticking control to stabilize the topside annulus pressure instead, where the pressure set-point corresponding to to a a pressure. Basically, this amounts to stabilizing the down pore or collapse pressure below the fracture or sticking back control tothe stabilize the topside annulus pressure pressure. Basically, this amounts to stabilizing the down instead, where pressure set-point corresponding to hole annular pressure at aand critical depth within its margins, desired bottom hole pressure is calculated online using pressure. Basically, this amounts to stabilizing the down instead, where the pressure set-point corresponding to a a desired bottom hole pressure is calculated online using hole annular pressure at a critical depth within its margins, pressure. Basically, this amounts to stabilizing down instead, where model. the set-point corresponding to a hole annular pressure at critical depth within its margins, desired bottom hole pressure is online using i.e. at particular where the pressure margins This is common hole annular at aadepth critical depth within itsthe margins, bottom holepressure pressure is calculated calculated online using i.e. either either at a apressure particular depth where the pressure margins adesired a steady-state steady-state model. This strategy strategy is the the most most common hole annular at abit critical depth within its the margins, desired bottommodel. holemainly pressure is calculated online using i.e. at depth where the pressure margins a steady-state steady-state model. This strategy strategy is availability the most most common are either small, or aapressure atparticular the drill where conditions are most aand due of i.e. either at particular depth where the pressure margins This is the common and straightforward straightforward mainly due to to the the availability of high high are small, or the drill bit where conditions are the most i.e. either at aat particular depth where the pressure margins aand steady-state model.mainly This strategy ismeasurements. the most common are small, or at the drill bit where conditions are the most straightforward mainly due to the the availability of high high uncertain. frequency and robust topside pressure are small, or at the drill bit where conditions are the most and straightforward due to availability of frequency and robust topside pressure measurements. The The uncertain. are small, at the drill where conditions arethe thechoke most direct and straightforward mainly due to theatmeasurements. availability ofdepth high uncertain. frequency and robust topside pressure The Basic two or strategies for bit closed-loop control of BHP control is that the BHP the critical uncertain. frequency and robust topside pressure measurements. The direct BHP control is that the BHP at the critical depth Basic two strategies for closed-loop control of the choke uncertain. frequency and robust topside pressure measurements. The Basic two strategies for closed-loop control of the choke direct BHP control is that the BHP at the critical depth are used: indirect topside control and direct bottom hole is stabilized at a desired set-point directly. Even though a Basic twoindirect strategies for closed-loop the choke BHP control is that the BHP at theEven critical depth are used: topside control andcontrol direct of bottom hole direct is stabilized at a desired set-point directly. though a Basic two strategies for closed-loop control thebottom choke direct BHP control is that the BHP at theEven critical depth are used: indirect topside control and bottom hole is stabilized stabilized at aa desired desired set-point directly. Even though a control. The indirect topside control isdirect thatof the BHP measurement usually exists, an estimate of the presare used: indirect topside control and direct bottom hole is at set-point directly. though BHP measurement usually exists, an estimate of the pres-a control. The indirect topside control is that the bottom are used:The indirect topside control and is hole is stabilized at a desired set-point directly. Even control. indirect topside control that the BHP measurement usually exists, an an estimate of though the prespres-a control. The indirect topside control isdirect that bottom the bottom bottom BHP measurement usually exists, estimate of the control. indirect topside control is that the bottom BHP measurement an estimate of the pres2405-8963 The © 2018 2018, IFAC (International Federation of Automatic Control) by Elsevier Ltd.usually All rightsexists, reserved. Copyright © IFAC 166 Hosting ∗ ∗ ∗ ∗ ∗
Copyright 2018 IFAC 166 Control. Peer review© responsibility of International Federation of Automatic Copyright © under 2018 IFAC IFAC 166 Copyright © 2018 166 10.1016/j.ifacol.2018.06.060 Copyright © 2018 IFAC 166
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1.1 Managed Pressure Drilling Recent experience indicates that in order to optimize the drilling operation the entire drilling system, not just the mechanics or software, needs to be designed from a control system point of view. Automatic control of drilling operations in a well can be a challenging task, due to the very complex dynamics of the multiphase flow potentially consisting of drilling mud, oil, gas and cuttings. Managed Pressure Drilling (MPD) Hannegan et al. (2004) is a technology that enables precisely control of the annular pressure during drilling and aims to prevent drilling related problems. By allowing manipulation of the topside choke and pumps, MPD provides a means of quickly affecting pressure to counteract disturbances, and several control schemes are found in the literature. This is typically achieved through a closed, pressurized fluid system in which flow rate, mud density, and back pressure on the fluid returns (choke manifold) are used to set and control the BHP under both static and dynamic conditions. Control of BHP during well drilling is a challenging task when there are disturbances and uncertainties in the drilling systems. 1.2 Control Solutions Manual control of the choke valve is commonly applied in todays drilling operations. The control systems are operated by the drilling crew, where the various inputs to the drilling system are adjusted independently. Therefore, it is low reaction to changes in set-point and disturbances. State-of-the-art solutions typically employ conventional PID control applied to the choke, using one of the above strategies PI controllers are relatively standard. PID control is a powerful control method because of the simple parameter tuning and limited requirements for knowledge about the process. PID control in the drilling process has been studied in Godhavn (2009); Zhou and Nygaard (2010); Siahaan et al. (2014); Zhou et al. (2016). The model-based control for drilling operation has been studied in Nygaard et al. (2007); Calsen et al. (2008); Zhou et al. (2008, 2009); Stamnes et al. (2009); Breyholtz et al. (2009); Zhou and Nygaard (2010); Zhou et al. (2011); Zhou and Nygaard (2011); Kaasa et al. (2012). However, the modelbased control method depends on the accuracy of the developed drilling model and the complexity is increased by the fact that a large set of parameters in such models are uncertain or unknown and possibly changing. For MPD, gain-scheduled PI control with feed forward for the choke to control the BHP is a high performance controller in MPD operations. There are signicant drawbacks with both strategies. In both cases, the PI controller relies heavily on integral action to balance the pressure drop caused by friction, which is signicant, and the proportional feedback gain must be low to prevent generating pressure pulses by fast changes in the control input. As a result, the control system based on conventional PI control will react slowly to fast pressure changes, which results from 167
167
movements of the drill string. Another drawback, is the uncertainty in the modeled bottom hole pressure, due to uncertainties in the friction and mud compressibility parameters in both the drill string and annulus. Typically, the model is calibrated by tuning these parameters to t the measured BHP. This is typically a computation routine that is initiated manually. There is signicant potential to improve existing PI control algorithm. In this paper, we investigate three types of controllers for BHP control in face of pipe connection and set-point changes. First control is standard PI control. The second is the combination of PI control and feed-forward control. Then a methodology for adaptive PI is presented. The corresponding designs are based on using only the tracking error, its derivative, its integral, and the current value of the adaptive gains in order to update the PI gains. The conventional independent parallel realization, which most existing adaptive designs have used, yields a linearly parametrized adaptive control problem. Case study simulations are provided to demonstrate the capabilities of the proposed algorithms. 2. PRSSURE CONTROL DESIGN In this section, the automatic control method is described, where a PI control, a PI with feed-forward control, and an adaptive PI with feed-forward control are applied. By using back pressure MPD, the choke valve opening is controlled using the proposed control methods while the main pump flow and the back pressure pump flow are manually operated. The proposed control schemes can be described by the structure in Figure 2, where the feedback controller calculates an error value as the difference between a measured process variable and a desired set-point. The controller attempts to minimize the error by adjusting the process by use of a manipulated variable and compensate the effects of the disturbance. The disturbance consists of known and unknown disturbances, for example, the flow change through the main pump during pipe connection is regarded as the measured disturbance to the process. Distrubance
Set-point ݕ௦௧ -
sure is needed between samples because that the transfer rate of the measurement is usually slow, or for additional safety because the sensor itself may be unreliable.
ٔ
Feedforward Controller
݁ሺݐሻ
+
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Feedback Controller
+
+
ٔ
Process
ݕሺݐሻ
Fig. 2. Control structure 2.1 PI Control PI control is a powerful control method because of the simple parameter tuning and limited requirements for knowledge about the process. In this section, a conventional PI control is used.
IFAC PID 2018 168 Belgium, May 9-11, 2018 Ghent,
u = uP I = −KP e − KI
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∫
edτ
(1)
where KP > 0 and KI > 0 are tuning gains for PI control. The variable e is defined as the mismatch between the controlled BHP y and the desired set-point yset , such as e(t) = y(t) − yset . The PI controller can drive the control variable to approach its set-point without any model needed in the computation in the controller provided the PI parameters KP and KI are tuned well. 2.2 PI and Feed-Forward Control Scheme Combination of feed-forward and feedback control gives a better performance compared to individual use of feedback control. Such a control can be expressed as u = uP I + uf f
(2)
u f f = Kf f ω f f (3) where Kf f > 0 is a tuning gain for feed-forward control and ωf f denotes a feed-forward function. 2.3 Adaptive PI and Feed-Forward Control Adaptive PI control is one approach to improve the robustness and autonomy of PI controllers as well as capture the essence of adaptive control theory within a simple architecture. Numerous publications in the control community have considered this problem. In this section, an adaptive PI control uaP I is expressed as ∫ ˆ ˆ (4) uaP I = KP e + KI edτ
ˆ I are the adaptive proportional gain and ˆ P and K where K adaptive integral gain. These adaptive gains are updated by using the following adaptation laws: ˆ˙ P = −γP e2 K ∫ ˆ˙ I = −γI e edτ K
(5) (6)
where γP > 0 and γI > 0 are the adaptation gains for proportional and integral gains. The adaptive PI control is achieved by utilizing only the feedback tracking error and its integral as driving signals as well as the current gain values to adjust the adaptive gains. The combination of adaptive PI and feed-forward control can be expressed as u = uP I + uf f + uaP I
(7)
2.4 Stability analysis Consider the following first order system ay˙ = −by + u (8) where y is the output, u is the input, a > 0 is an unknown parameter and b > 0 is a known parameter. In several articles Godhavn (2009); Zhou et al. (2008); Kaasa et al. (2012), the pressure dynamics was modeled by a first-order differential equation as (8). From (4) and (7), the derivative of e = y − yset is given as 168
ae˙ = −by + uP I + uf f + uaP I ∫ = −by − KP e − Ki edτ + Kf f ωf f ∫ ˆ ˆ +KP e + KI edτ ∫ = −KP e − Ki edτ − be − byset ∫ T ˆ ˆ +Kf f ωf f + KP e + KI edτ ∫ ∫ ˆP e + K ˆ I edτ (9) = −be − KP e − Ki edτ + K
where Kf f = b, and ωf f = yset . Consider the following Lyapunov function:
(∫ )2 1 2 1 edτ V = ae + Ki 2 2 1 ˆ2 1 ˆ2 + (10) K + K 2γP p 2γI I Using (5) and (6), its derivative is obtained as ∫ ∫ ˆ ˆ ˙ V = e(−KP e − Ki edτ − be + KP e + KI edτ ) (∫ ) 1 ˆ ˆ˙ 1 ˆ ˆ˙ +Ki e edτ + Kp K P + K I KI γP γI 1 ˆ ˆ˙ = −(KP + b)e2 + Kp (K P + γP e2 ) γP ∫ 1 ˆ ˆ˙ + KI (K I + γI e edτ ) γI = −(KP + b)e2 ≤ 0 (11) which shows ∫ that V is globally bounded. Thus the sigˆ I are bounded. It further ˆ P , and K nals e(t), edτ , K implies that the tracking error e(t) converges to zero as limt→∞ e(t) = 0 by using LaSalle-Yoshizawa theorem in Khalil (2002). 3. CASE STUDIES In this section, two case studies will be used to demonstrate the proposed methodologies in face of uncertainties and disturbances. The control objective is to control the bottom hole pressure at the desired set point when the main pump is shuttled down and then started up during the pipe connection and set-point changes. The control variable u is the choke opening. The tuning rules for PI control gains are MIT in Rifai (2009). Large value of (KP , KI ) gives the fast speed of response and good disturbance rejection and small value gives good robustness to time delay and uncertainties. The gain for feed-forward control is chosen by trial and error. The adaptive gains are updated by using the following adaptation laws. Simulations are carried out using a high fidelity drilling simulator developed by the International Research Institute of Stavanger Lage et al. (2000). Throughout the simulations, the aim of controller is to maintain the BHP around the desired BHP in two cases. The first case is the BHP control during pipe connection. The second case is the BHP stabilization during set-point changes. This can be a challenge for drilling in formation with very
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tight margin between pore pressure and fracture pressure. The proposed PI control, PI with feed-forward control and adaptive PI and feed-forward control are evaluated for two cases in face of time-delay in the BHP measurement. The cases are related to a well being drilled recently in the North Sea. The well bore configuration is given in Table 1. Parameter
Value
Well Length True vertical depth (TVD) Drillpipe outer diameter Drillpipe inner diameter Casing inner diameter Water density
2270 (m) 1951 (m) 5 (inch) 5 4 32 (inch) 8 58 (m) 1000 (kg/m3 )
Table 1: Well bore Configuration in Drilling Simulator
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4. CONCLUSION In this paper, we investigate the robustness of three types of controllers for BHP control during well drilling, including PI control, PI with feed-forward control and adaptive PI with feed-forward control. The proposed control schemes are designed to stabilize the bottom hole pressure and achieve the asymptotic tracking. The simulation results are evaluated in a high-fidelity drilling simulator and illustrate the effectiveness of proposed control schemes. Case study simulations show that the adaptive PI with feed-forward control are able to improve control performance in face of uncertainties and time-delays. REFERENCES
3.1 Case 1- BHP control during pipe connection In the first case, the objective is to control BHP at the desire set-point 280 bar during pipe connection, while the main pump is shuttled down and opened again. Case 1a is when there is 5 seconds time-delay in the BHP measurement and case 1b is when there is 15 seconds timedelay in the BHP measurement. Figures 3-5 and Figures 6-8 show the BHP with PI control, PI with feed-forward control, and adaptive PI with feed-forward control for case 1a and case 1b. separately. In this case, the control parameters are set as KP = −2.5×10−3 , KI = −8.3×10−5 and Kf f = 7.3−5 and the adaptation gains are set as γp = 1 × 10−9 and γI = 3 × 10−12 , the disturbance is the change of the flow rate through the main pump qpump . 3.2 Case 2- BHP control during set-point changes In the second case, the desired set-point for BHP is changed and the aim of controller is to maintain the BHP at the desired set-point. Figures 9-11 show the bottom hole pressure with PI control, PI and feed-forward, and adaptive PI with feed-forward control when there is 10 seconds time-delay in the BHP measurement. In this case, the control parameters are set as KP = −2.5 × 10−3 , KI = −8.3 × 10−5 and Kf f = 5.0−6 and the adaptation gains are set as γp = 1.0 × 10−9 and γI = 3.0 × 10−12 , the disturbance is the change of the set-point pressure. 3.3 Results In conclusion of simulation results, PI control is not good when the measurement has time delays. PI+feed-forward control improves the performance when there is 5 second delay in the BHP, but it is not good when the delay is 15 second. Adaptive PI+feed-forward control gives the best performance when there is time-delay in the BHP measurement. The amplitude and frequency of oscillation due to delay in BHP are reduced within the acceptance region. The simulation results show that the combination of adaptive PI and feed-forward is robust to the delays in BHP measurements and gives the best performance when there is time-delay in the measurement. 169
Breyholtz, Ø., Nygaard, G., and Nikolaou, M. (2009). Advanced automatic control or dual-gradient drilling. In SPE Annual Technical Conference and Exhibition, SPE 124631. New Orleans, Louisiana, USA. Calsen, L.A., Gerhard, N., Gravdal, J., Nikolaou, M., and Schubert, J. (2008). Performing the dynamic shut-in procedure because of a kick incident when using automatic coordinate control of pump rates and choke-valve opening. In SPE/IADC Managed Pressure Drilling and Underbalanced Operations Conference and Exhibition, SPE/IADC-113693MS. Abu Dhabi, UAE. Godhavn, J.M. (2009). Control requirements for highend automatic mpd operations. In SPE/IADC Drilling Conference and Exhibition, SPE 119442. Amsterdam, Netherlands. Hannegan, D., Todd, R.J., Pritchard, D.M., and Jonasson, B. (2004). MPD uniquely applicable to methane hydrate drilling. In SPE/IADC Underbalanced Technology Conference and Exhibition, SPE91560. Houston, Texas, U.S.A. Kaasa, G.O., Stamnes, O.N., Aamo, O., and Imsland, L. (2012). Simplified hydraulics model used for intelligent estimation of downhole pressure for a managed-pressuredrilling control system. SPE Drilling and Completion, SPE-143097-PA, 27, 127–138. Khalil, H.K. (2002). Nonlinear Systems. 3rd edition, Prentice Hall, U.S. Lage, A., Fjelde, K., and Time, R. (2000). Underbalanced drilling dynamics: Two-phase flow modeling and experiments. In IADC/SPE Asia Pacific Drilling Technology, SPE -62743-MS. Kuala Lumpur, Malaysia. Merritt, H.E. (1967). Hydraulic Control Systems. John Wiley & Sons, U.S. Nygaard, G. (2006). Multivariable process control in high temperature and high pressure environment using non-intrusive multi sensor data fusion. PhD Thesis, Norwegian University of Science and Technology. Nygaard, G., Johannessen, E., Gravdal, J.E., and Iversen, F. (2007). Automatic coordinated control of pump rates and choke valve for compensating pressure fluctuations during surge and swab operations. In IADC/SPE Managed Pressure Drilling and Underbalanced Operations Conference and Exhibition, SPE 108344. Galveston, Texas, U.S.A. Rifai, K. (2009). Nonlinearly parameterized adaptive pid control for parallel and series realizations. In American Control Conference, 5150–5155. St. Louis, USA.
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Case 1: 15 Seconds delay in sensor
Fig. 3. PI control for case 1a
Fig. 6. PI control for case 1b
Fig. 4. PI + Feed-forward control for case 1a
Fig. 7. PI + Feed-forward control for case 1b
Fig. 5. Adaptive PI + Feed-forward control for case 1a
Fig. 8. Adaptive PI + Feed-forward control for case 1b
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Decision and Control. White, F.M. (1999). Fluid Mechanics, 4th edition. McGraw-Hill Int, Boston, USA. Zhou, J., Gravdal, J., Strand, P., and Hovland, S. (2016). Automated kick control procedure for an influx in managed pressure drilling operations by utilizing pwd. Journal of Modeling, Identification and Control, 37, 3140. Zhou, J. and Nygaard, G. (2010). Control and estimation of downhole pressure in managed pressure drilling operations. In Proceedings of the 4th International Symposium on Communications, Control and Signal Processing. Limassal, Cyrpus. Zhou, J., Stamnes, O.N., Aamo, O.M., and Kaasa, G.O. (2008). Adaptive output feedback control of a managed pressure drilling system. In Proceedings of the 47th IEEE Conference on Decision and Control, 3008–3013. Cancun, Mexico. Zhou, J., Stamnes, Ø.N., Aamo, O.M., and Kaasa, G.O. (2009). Pressure regulation with kick attenuation in a managed pressure drilling system. In Proceedings of the 48th IEEE Conference on Decision and Control. Shanghai, China. Zhou, J., Stamnes, Ø.N., Aamo, O.M., and Kaasa, G.O. (2011). Switched control for pressure regulation and kick attenuation in a managed pressure drilling system. IEEE Transactions on Control Systems Technology, 19, 337–350. Zhou, J. and Nygaard, G. (2011). Automatic modelbased control scheme for stabilizing pressure during dual gradient drilling. Journal of Process Control, 21, 1138– 1147.
Fig. 9. PI control for case 2
Fig. 10. PI + Feed-forward control for case 2
Fig. 11. Adaptive PI + Feed-forward control for case 2 Siahaan, H.B., Bjrkevoll, K.S., and Gravdal, J.E. (2014). Possibilities of using wired drill pipe telemetry during managed pressure drilling in extended reach wells. In SPE Intelligent Energy Conference and Exhibition, SPE 167856. Utrecht, Netherlands. Stamnes, Ø.N., Zhou, J., Kaasa, G.O., and Aamo, O.M. (2009). Adaptive observer design for nonlinear systems with parametric uncertainties in unmeasured state dynamics. In Proceedings of the 48th IEEE Conference on 171
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Adaptive Adaptive PI PI Control Control of of Bottom Bottom Hole Hole Adaptive PI Control of Bottom Hole Pressure during Oil Well Drilling Adaptive Control Bottom Hole PressurePI during Oil of Well Drilling Pressure during Oil Well Drilling Pressure during Oil ∗∗Well Drilling Jing Zhou
Jing Zhou ∗∗ Jing Jing Zhou Zhou Jing Zhou ∗ University of Agder, 4898 Department of Engineering Department of Engineering Sciences, Sciences, University of Agder, 4898 Department of Engineering Engineering Sciences, University of Agder, Agder, 4898 4898 Grimstad, Norway jing.zhou@ uia.no). Department of Sciences, University of Grimstad, Norway (e-mail: (e-mail: jing.zhou@ uia.no). Department of Engineering Sciences, University of Agder, 4898 Grimstad, Norway (e-mail: (e-mail: jing.zhou@ uia.no). Grimstad, Norway jing.zhou@ uia.no). Grimstad, Norway (e-mail: jing.zhou@ uia.no). Abstract: In this paper, we studied the bottom hole pressure (BHP) control in an oil well during Abstract: In this paper, we the hole (BHP) control an well Abstract: In this this paper, wells we studied studied the bottom bottom hole pressure pressure (BHP) control in inbeing an oil oildrilled. well during during drilling. Today marginal with narrow pressure windows(BHP) are frequently This Abstract: In paper, we studied the bottom hole pressure control in an oil well during drilling. Today marginal wells with narrow pressure windows are frequently being drilled. This Abstract: In this paper, we studied the hole pressure (BHP) control inbeing an oil well during drilling. Today marginal wells with narrow pressure windows are frequently drilled. This requires accurate and precise control to bottom balance the bottom hole pressure between the pore and drilling. Today marginal wells with narrow pressure windows are frequently being drilled. This requires accurate and precise to the bottom hole between pore and drilling. Today marginal wellscontrol with narrow pressure windows arepressure frequently being the drilled. This requires accurate and precise control to balance balance the hole pressure between the pore and fracture of the reservoir. This presents three schemes to the requires accurate and control balance the bottom bottom hole pressure the pore and fracture pressure pressure of theprecise reservoir. Thistopaper paper presents three control control schemesbetween to stabilize stabilize the BHP BHP requires accurateproportional-integral(PI) and control balance the hole pressure between pore and fracture pressure of theprecise reservoir. Thistopaper paper presents threefeed-forward control schemes to stabilize stabilize the BHP BHP prole, including control, PIbottom with control andthe adaptive PI fracture pressure of the reservoir. This presents three control schemes to the PI prole, including control, PI with control and fracture pressureproportional-integral(PI) ofcontrol. the reservoir. This paper presents threefeed-forward control schemes to stabilize BHP prole, including proportional-integral(PI) control, PI carried with feed-forward control and adaptive adaptive PI with feed-forward The proposed schemes are out through simulations onthe a highprole, including proportional-integral(PI) control, PI with feed-forward control and adaptive PI with feed-forward control. The proposed schemes are carried out through simulations on a highprole, including control, PI with feed-forward control and adaptive PI with feed-forward control. The proposed schemes carried out through simulations on aaInhighfidelity hydraulicproportional-integral(PI) drilling simulator for flow rate are changes and BHP set-point changes. fast with feed-forward control. The proposed schemes are carried out through simulations on highfidelity hydraulic drilling simulator for flow rate changes and BHP set-point changes. In fast with feed-forward control. The proposed schemes are carried out through simulations on a highfidelity hydraulic drilling simulator for flow rate changes and BHP set-point changes. In fast set-point changes and flow rate changes, the adaptive PI controller exhibits less tracking error fidelity for flow changes and BHPexhibits set-point changes. fast set-pointhydraulic changes drilling and flowsimulator rate changes, the rate adaptive PI controller less trackingInerror fidelity for flow rate changes and simulation BHPexhibits set-point changes. Inerror fast set-point changes drilling andthan flowsimulator rateconventional changes, the PI adaptive PI The controller exhibits less tracking and lesshydraulic oscillations the solution. results illustrate the set-point changes and flow rate changes, the adaptive PI controller less tracking simulation results illustrateerror the and less oscillations than the conventional PI solution. The set-point changes andthan flow rateconventional changes, adaptive PI The controller exhibits less tracking and less oscillations oscillations than the conventional PI solution. The simulation results illustrateerror the effectiveness of proposed control schemes.the PI and less the solution. simulation results illustrate the effectiveness of proposed control schemes. and less oscillations thancontrol the conventional effectiveness of schemes. effectiveness of proposed proposed control schemes. PI solution. The simulation results illustrate the © 2018, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. effectiveness proposed control schemes. Keywords: PIofcontrol, adaptive control, oil well drilling, pressure control. Keywords: PI control, adaptive control, Keywords: PI PI control, control, adaptive adaptive control, control, oil oil well well drilling, drilling, pressure pressure control. control. Keywords: oil well drilling, pressure control. Keywords: PI control, adaptive control, oil well drilling, pressure control. 1. INTRODUCTION 1. INTRODUCTION 1. 1. INTRODUCTION INTRODUCTION Main pump 1. INTRODUCTION q Main pump Controlling the bottom hole pressure in an oil well is one pump q Main pump pump pump Controlling the bottom hole pressure in an oil well is one q Main Controlling the bottom hole pressure in an oil well is one pump of the critical tasks during drilling. During well drilling, Controlling thetasks bottom hole drilling. pressureDuring in an oil well is one pump Main pumpq of the critical during well drilling, q Controlling thetasks bottom hole pressure instring an oil well is and one pump of the during drilling. During well drilling, a uid pumped into the topside qchoke of the critical critical during During drilling, a drilling drilling uid is istasks pumped intodrilling. the drill drill stringwell topside and qchoke of the critical tasks during drilling. During drilling, aa drilling uid is pumped the drill string topside and through the drill bit at theinto bottom hole of thewell well Merritt drilling uid is pumped into the drill string topside and choke through the drill bit at the bottom hole of the well Merritt qchoke a drillingWhite uiddrill is (1999); pumped into the (2006). drill string topside and choke through the bit at the bottom hole of the Merritt (1967); Nygaard Thewell mud then through the drill bit at the bottom hole of the well Merritt (1967); White (1999); Nygaard (2006). The mud then through thecuttings drill(1999); bit at bottom hole the well Merritt (1967); White Nygaard (2006). The mud then transports in the the annulus sideof of the well (i.e. (1967); White (1999); Nygaard (2006). The mud then q transports cuttings in the annulus side of the well (i.e. qback (1967); White (1999); Nygaard (2006). The mud then transports cuttings in annulus side of the well (i.e. in outside the string) the drill rig, transports cuttings in the the annulus sideup qback Annulus pump in the the well well bore bore outside the drill drill string) upofto tothe thewell drill (i.e. rig, back Annulus pump transports cuttings in annulus sideup ofto the in the well bore outside the drill string) the drill rig, where a choke valve andthe a back pressure pump iswell used(i.e. to qback in the well bore outside the drill string) up to the drill rig, Annulus pump where a choke valve and a back pressure pump is used to back Annulus pump in the well bore outside drill to the where aathe choke valve and aa back pressure is used to control annular pressure, seestring) Figureup 1pump for a schematic where choke valve andthe back pressure pump isdrill usedrig, to Annulus pump control the annular pressure, see Figure 1 for a schematic where athe choke and a back is used to control pressure, see Figure for overview of annular thevalve system. control the annular pressure, seepressure Figure 11pump for aa schematic schematic overview of the system. control the annular pressure, see Figure 1 for a schematic overview of the system. Today marginal wells with narrow pressure windows overview of the system. Today marginal wells with narrow pressure windows are are overview ofbeing the system. Today marginal wells with are frequently drilled. Thisnarrow requirespressure accuratewindows and precise Today marginal wells with narrow pressure windows are frequently being drilled. This requires accurate and precise Today wells with narrow are frequently being This requires accurate and precise control marginal to balance the BHP between the pore windows and fracture frequently being drilled. drilled. Thisbetween requirespressure accurate andfracture precise control to balance the BHP the pore and frequently being drilled. This requires accurate and precise control to balance the BHP between the pore and fracture pressure of the reservoir. A stable well bore promotes control toofbalance the BHP A between the pore andpromotes fracture Non pressure the reservoir. stable well bore Non control toof the BHP A between the pore andpromotes fracture pressure the reservoir. stable well bore efcient personnel safety. A well pressure ofbalance theand reservoir. A stable bore promotes return Non efcient drilling drilling and personnel safety. well A destabilized destabilized well Non return pressure of theand reservoir. A production. stable well borelow promotes efcient drilling personnel safety. A destabilized well bore can reduce or eliminate Too a mud efcient drilling and personnel safety. A destabilized well valve return Non return bore can reduce or eliminate production. Too low a mud valve efcient drilling and personnel safety. A destabilized well bore can reduce Too aa mud pressure can leador toeliminate a kick orproduction. well bore collapse and too bore can can reduce orto eliminate production. Too low lowand mud valve return valve pressure lead a kick or well bore collapse too bore can reduce or eliminate production. Too low a mud pressure can lead to a kick or well bore collapse and too high a mud pressure can create well bore fracturing and pressure can lead to a kick or well bore collapse and too valve high a mud pressure can create well bore fracturing and pressure can pressure lead to acan kick or stability well collapse and and too high mud create well bore fracturing losses.aa Preventing these costly problems requires high mud pressure can create wellbore bore fracturing and losses. Preventing these costly stability problems requires high a Preventing mud pressure pressure cancostly create well control bore fracturing and losses. these stability problems requires an accurate control. Pressure is a challenglosses. Preventing these costly stability problems requires an accurate pressure control. Pressure control is a challenglosses. costly stability problems requires an pressure control. Pressure is aadynamics challengingaccurate taskPreventing during wellthese drilling, due to thecontrol complex an accurate pressure control. Pressure control is challenging task during well drilling, due to thecontrol complex an accurate pressure control. Pressure is adynamics challenging task during well due to dynamics of the multiphase owdrilling, potentially consisting of drilling mud, ing task during well drilling, dueconsisting to the the complex complex dynamics of the multiphase ow potentially of drilling mud, ing task during well drilling, dueconsisting to the complex dynamics of ow potentially of drilling mud, oil,the gasmultiphase and cuttings. Fig. 1. of the multiphase ow potentially consisting of drilling mud, oil, gas and cuttings. 1. A A simplified simplified schematic schematic drawing drawing of of the the drilling drilling of the multiphase ow potentially consisting drilling mud, Fig. oil, gas and cuttings. The main objective is to precisely bottomofhole pressure Fig. 1. A simplified schematic drawing of the drilling drilling system. oil, gas and cuttings. Fig. 1. A simplified schematic drawing of the The main objective is to precisely bottom hole pressure system. oil, gas and cuttings. The main objective is to precisely bottom hole pressure prolemain throughout the iswell bore continuously while drilling, Fig. system. 1. A simplified schematic drawing of the drilling system. The objective to precisely bottom hole pressure prole throughout the continuously while drilling, The main objective iswell to bore precisely bottom holeabove pressure prole throughout the well bore continuously while drilling, i.e. maintain pressure in the pressure prole theannular well bore continuously while drilling, i.e. to tothroughout maintain the the annular pressure in the the well well above the hole hole system. pressure is is indirectly indirectly stabilized stabilized by by applying applying feedfeedprole throughout theannular welland bore continuously while drilling, i.e. maintain pressure in the well above the hole pressure pressure is indirectly indirectly stabilized by applying feedporeto or collapse the pressure below the fracture or sticking back control to stabilize the topside annulus pressure i.e. to maintain the annular pressure in the well above the hole is stabilized by applying feedback control to stabilize the topside annulus pressure pore or collapse pressure and below the fracture or sticking i.e. toor maintain the annular pressure in the wellor above the back hole istheindirectly stabilized by applying feedpore collapse pressure below fracture sticking back pressure control to stabilize the topsidecorresponding annulus pressure pressure. Basically, this and amounts tothe stabilizing the down instead, where pressure set-point pore or collapse pressure and below the fracture or sticking control to stabilize the topside annulus pressure instead, where the pressure set-point corresponding to to a a pressure. Basically, this amounts to stabilizing the down pore or collapse pressure below the fracture or sticking back control tothe stabilize the topside annulus pressure pressure. Basically, this amounts to stabilizing the down instead, where pressure set-point corresponding to hole annular pressure at aand critical depth within its margins, desired bottom hole pressure is calculated online using pressure. Basically, this amounts to stabilizing the down instead, where the pressure set-point corresponding to a a desired bottom hole pressure is calculated online using hole annular pressure at a critical depth within its margins, pressure. Basically, this amounts to stabilizing down instead, where model. the set-point corresponding to a hole annular pressure at critical depth within its margins, desired bottom hole pressure is online using i.e. at particular where the pressure margins This is common hole annular at aadepth critical depth within itsthe margins, bottom holepressure pressure is calculated calculated online using i.e. either either at a apressure particular depth where the pressure margins adesired a steady-state steady-state model. This strategy strategy is the the most most common hole annular at abit critical depth within its the margins, desired bottommodel. holemainly pressure is calculated online using i.e. at depth where the pressure margins a steady-state steady-state model. This strategy strategy is availability the most most common are either small, or aapressure atparticular the drill where conditions are most aand due of i.e. either at particular depth where the pressure margins This is the common and straightforward straightforward mainly due to to the the availability of high high are small, or the drill bit where conditions are the most i.e. either at aat particular depth where the pressure margins aand steady-state model.mainly This strategy ismeasurements. the most common are small, or at the drill bit where conditions are the most straightforward mainly due to the the availability of high high uncertain. frequency and robust topside pressure are small, or at the drill bit where conditions are the most and straightforward due to availability of frequency and robust topside pressure measurements. The The uncertain. are small, at the drill where conditions arethe thechoke most direct and straightforward mainly due to theatmeasurements. availability ofdepth high uncertain. frequency and robust topside pressure The Basic two or strategies for bit closed-loop control of BHP control is that the BHP the critical uncertain. frequency and robust topside pressure measurements. The direct BHP control is that the BHP at the critical depth Basic two strategies for closed-loop control of the choke uncertain. frequency and robust topside pressure measurements. The Basic two strategies for closed-loop control of the choke direct BHP control is that the BHP at the critical depth are used: indirect topside control and direct bottom hole is stabilized at a desired set-point directly. Even though a Basic twoindirect strategies for closed-loop the choke BHP control is that the BHP at theEven critical depth are used: topside control andcontrol direct of bottom hole direct is stabilized at a desired set-point directly. though a Basic two strategies for closed-loop control thebottom choke direct BHP control is that the BHP at theEven critical depth are used: indirect topside control and bottom hole is stabilized stabilized at aa desired desired set-point directly. Even though a control. The indirect topside control isdirect thatof the BHP measurement usually exists, an estimate of the presare used: indirect topside control and direct bottom hole is at set-point directly. though BHP measurement usually exists, an estimate of the pres-a control. The indirect topside control is that the bottom are used:The indirect topside control and is hole is stabilized at a desired set-point directly. Even control. indirect topside control that the BHP measurement usually exists, an an estimate of though the prespres-a control. The indirect topside control isdirect that bottom the bottom bottom BHP measurement usually exists, estimate of the control. indirect topside control is that the bottom BHP measurement an estimate of the pres2405-8963 The © 2018 2018, IFAC (International Federation of Automatic Control) by Elsevier Ltd.usually All rightsexists, reserved. Copyright © IFAC 166 Hosting ∗ ∗ ∗ ∗ ∗
Copyright 2018 IFAC 166 Control. Peer review© responsibility of International Federation of Automatic Copyright © under 2018 IFAC IFAC 166 Copyright © 2018 166 10.1016/j.ifacol.2018.06.060 Copyright © 2018 IFAC 166
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1.1 Managed Pressure Drilling Recent experience indicates that in order to optimize the drilling operation the entire drilling system, not just the mechanics or software, needs to be designed from a control system point of view. Automatic control of drilling operations in a well can be a challenging task, due to the very complex dynamics of the multiphase flow potentially consisting of drilling mud, oil, gas and cuttings. Managed Pressure Drilling (MPD) Hannegan et al. (2004) is a technology that enables precisely control of the annular pressure during drilling and aims to prevent drilling related problems. By allowing manipulation of the topside choke and pumps, MPD provides a means of quickly affecting pressure to counteract disturbances, and several control schemes are found in the literature. This is typically achieved through a closed, pressurized fluid system in which flow rate, mud density, and back pressure on the fluid returns (choke manifold) are used to set and control the BHP under both static and dynamic conditions. Control of BHP during well drilling is a challenging task when there are disturbances and uncertainties in the drilling systems. 1.2 Control Solutions Manual control of the choke valve is commonly applied in todays drilling operations. The control systems are operated by the drilling crew, where the various inputs to the drilling system are adjusted independently. Therefore, it is low reaction to changes in set-point and disturbances. State-of-the-art solutions typically employ conventional PID control applied to the choke, using one of the above strategies PI controllers are relatively standard. PID control is a powerful control method because of the simple parameter tuning and limited requirements for knowledge about the process. PID control in the drilling process has been studied in Godhavn (2009); Zhou and Nygaard (2010); Siahaan et al. (2014); Zhou et al. (2016). The model-based control for drilling operation has been studied in Nygaard et al. (2007); Calsen et al. (2008); Zhou et al. (2008, 2009); Stamnes et al. (2009); Breyholtz et al. (2009); Zhou and Nygaard (2010); Zhou et al. (2011); Zhou and Nygaard (2011); Kaasa et al. (2012). However, the modelbased control method depends on the accuracy of the developed drilling model and the complexity is increased by the fact that a large set of parameters in such models are uncertain or unknown and possibly changing. For MPD, gain-scheduled PI control with feed forward for the choke to control the BHP is a high performance controller in MPD operations. There are signicant drawbacks with both strategies. In both cases, the PI controller relies heavily on integral action to balance the pressure drop caused by friction, which is signicant, and the proportional feedback gain must be low to prevent generating pressure pulses by fast changes in the control input. As a result, the control system based on conventional PI control will react slowly to fast pressure changes, which results from 167
167
movements of the drill string. Another drawback, is the uncertainty in the modeled bottom hole pressure, due to uncertainties in the friction and mud compressibility parameters in both the drill string and annulus. Typically, the model is calibrated by tuning these parameters to t the measured BHP. This is typically a computation routine that is initiated manually. There is signicant potential to improve existing PI control algorithm. In this paper, we investigate three types of controllers for BHP control in face of pipe connection and set-point changes. First control is standard PI control. The second is the combination of PI control and feed-forward control. Then a methodology for adaptive PI is presented. The corresponding designs are based on using only the tracking error, its derivative, its integral, and the current value of the adaptive gains in order to update the PI gains. The conventional independent parallel realization, which most existing adaptive designs have used, yields a linearly parametrized adaptive control problem. Case study simulations are provided to demonstrate the capabilities of the proposed algorithms. 2. PRSSURE CONTROL DESIGN In this section, the automatic control method is described, where a PI control, a PI with feed-forward control, and an adaptive PI with feed-forward control are applied. By using back pressure MPD, the choke valve opening is controlled using the proposed control methods while the main pump flow and the back pressure pump flow are manually operated. The proposed control schemes can be described by the structure in Figure 2, where the feedback controller calculates an error value as the difference between a measured process variable and a desired set-point. The controller attempts to minimize the error by adjusting the process by use of a manipulated variable and compensate the effects of the disturbance. The disturbance consists of known and unknown disturbances, for example, the flow change through the main pump during pipe connection is regarded as the measured disturbance to the process. Distrubance
Set-point ݕ௦௧ -
sure is needed between samples because that the transfer rate of the measurement is usually slow, or for additional safety because the sensor itself may be unreliable.
ٔ
Feedforward Controller
݁ሺݐሻ
+
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Feedback Controller
+
+
ٔ
Process
ݕሺݐሻ
Fig. 2. Control structure 2.1 PI Control PI control is a powerful control method because of the simple parameter tuning and limited requirements for knowledge about the process. In this section, a conventional PI control is used.
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u = uP I = −KP e − KI
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∫
edτ
(1)
where KP > 0 and KI > 0 are tuning gains for PI control. The variable e is defined as the mismatch between the controlled BHP y and the desired set-point yset , such as e(t) = y(t) − yset . The PI controller can drive the control variable to approach its set-point without any model needed in the computation in the controller provided the PI parameters KP and KI are tuned well. 2.2 PI and Feed-Forward Control Scheme Combination of feed-forward and feedback control gives a better performance compared to individual use of feedback control. Such a control can be expressed as u = uP I + uf f
(2)
u f f = Kf f ω f f (3) where Kf f > 0 is a tuning gain for feed-forward control and ωf f denotes a feed-forward function. 2.3 Adaptive PI and Feed-Forward Control Adaptive PI control is one approach to improve the robustness and autonomy of PI controllers as well as capture the essence of adaptive control theory within a simple architecture. Numerous publications in the control community have considered this problem. In this section, an adaptive PI control uaP I is expressed as ∫ ˆ ˆ (4) uaP I = KP e + KI edτ
ˆ I are the adaptive proportional gain and ˆ P and K where K adaptive integral gain. These adaptive gains are updated by using the following adaptation laws: ˆ˙ P = −γP e2 K ∫ ˆ˙ I = −γI e edτ K
(5) (6)
where γP > 0 and γI > 0 are the adaptation gains for proportional and integral gains. The adaptive PI control is achieved by utilizing only the feedback tracking error and its integral as driving signals as well as the current gain values to adjust the adaptive gains. The combination of adaptive PI and feed-forward control can be expressed as u = uP I + uf f + uaP I
(7)
2.4 Stability analysis Consider the following first order system ay˙ = −by + u (8) where y is the output, u is the input, a > 0 is an unknown parameter and b > 0 is a known parameter. In several articles Godhavn (2009); Zhou et al. (2008); Kaasa et al. (2012), the pressure dynamics was modeled by a first-order differential equation as (8). From (4) and (7), the derivative of e = y − yset is given as 168
ae˙ = −by + uP I + uf f + uaP I ∫ = −by − KP e − Ki edτ + Kf f ωf f ∫ ˆ ˆ +KP e + KI edτ ∫ = −KP e − Ki edτ − be − byset ∫ T ˆ ˆ +Kf f ωf f + KP e + KI edτ ∫ ∫ ˆP e + K ˆ I edτ (9) = −be − KP e − Ki edτ + K
where Kf f = b, and ωf f = yset . Consider the following Lyapunov function:
(∫ )2 1 2 1 edτ V = ae + Ki 2 2 1 ˆ2 1 ˆ2 + (10) K + K 2γP p 2γI I Using (5) and (6), its derivative is obtained as ∫ ∫ ˆ ˆ ˙ V = e(−KP e − Ki edτ − be + KP e + KI edτ ) (∫ ) 1 ˆ ˆ˙ 1 ˆ ˆ˙ +Ki e edτ + Kp K P + K I KI γP γI 1 ˆ ˆ˙ = −(KP + b)e2 + Kp (K P + γP e2 ) γP ∫ 1 ˆ ˆ˙ + KI (K I + γI e edτ ) γI = −(KP + b)e2 ≤ 0 (11) which shows ∫ that V is globally bounded. Thus the sigˆ I are bounded. It further ˆ P , and K nals e(t), edτ , K implies that the tracking error e(t) converges to zero as limt→∞ e(t) = 0 by using LaSalle-Yoshizawa theorem in Khalil (2002). 3. CASE STUDIES In this section, two case studies will be used to demonstrate the proposed methodologies in face of uncertainties and disturbances. The control objective is to control the bottom hole pressure at the desired set point when the main pump is shuttled down and then started up during the pipe connection and set-point changes. The control variable u is the choke opening. The tuning rules for PI control gains are MIT in Rifai (2009). Large value of (KP , KI ) gives the fast speed of response and good disturbance rejection and small value gives good robustness to time delay and uncertainties. The gain for feed-forward control is chosen by trial and error. The adaptive gains are updated by using the following adaptation laws. Simulations are carried out using a high fidelity drilling simulator developed by the International Research Institute of Stavanger Lage et al. (2000). Throughout the simulations, the aim of controller is to maintain the BHP around the desired BHP in two cases. The first case is the BHP control during pipe connection. The second case is the BHP stabilization during set-point changes. This can be a challenge for drilling in formation with very
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tight margin between pore pressure and fracture pressure. The proposed PI control, PI with feed-forward control and adaptive PI and feed-forward control are evaluated for two cases in face of time-delay in the BHP measurement. The cases are related to a well being drilled recently in the North Sea. The well bore configuration is given in Table 1. Parameter
Value
Well Length True vertical depth (TVD) Drillpipe outer diameter Drillpipe inner diameter Casing inner diameter Water density
2270 (m) 1951 (m) 5 (inch) 5 4 32 (inch) 8 58 (m) 1000 (kg/m3 )
Table 1: Well bore Configuration in Drilling Simulator
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4. CONCLUSION In this paper, we investigate the robustness of three types of controllers for BHP control during well drilling, including PI control, PI with feed-forward control and adaptive PI with feed-forward control. The proposed control schemes are designed to stabilize the bottom hole pressure and achieve the asymptotic tracking. The simulation results are evaluated in a high-fidelity drilling simulator and illustrate the effectiveness of proposed control schemes. Case study simulations show that the adaptive PI with feed-forward control are able to improve control performance in face of uncertainties and time-delays. REFERENCES
3.1 Case 1- BHP control during pipe connection In the first case, the objective is to control BHP at the desire set-point 280 bar during pipe connection, while the main pump is shuttled down and opened again. Case 1a is when there is 5 seconds time-delay in the BHP measurement and case 1b is when there is 15 seconds timedelay in the BHP measurement. Figures 3-5 and Figures 6-8 show the BHP with PI control, PI with feed-forward control, and adaptive PI with feed-forward control for case 1a and case 1b. separately. In this case, the control parameters are set as KP = −2.5×10−3 , KI = −8.3×10−5 and Kf f = 7.3−5 and the adaptation gains are set as γp = 1 × 10−9 and γI = 3 × 10−12 , the disturbance is the change of the flow rate through the main pump qpump . 3.2 Case 2- BHP control during set-point changes In the second case, the desired set-point for BHP is changed and the aim of controller is to maintain the BHP at the desired set-point. Figures 9-11 show the bottom hole pressure with PI control, PI and feed-forward, and adaptive PI with feed-forward control when there is 10 seconds time-delay in the BHP measurement. In this case, the control parameters are set as KP = −2.5 × 10−3 , KI = −8.3 × 10−5 and Kf f = 5.0−6 and the adaptation gains are set as γp = 1.0 × 10−9 and γI = 3.0 × 10−12 , the disturbance is the change of the set-point pressure. 3.3 Results In conclusion of simulation results, PI control is not good when the measurement has time delays. PI+feed-forward control improves the performance when there is 5 second delay in the BHP, but it is not good when the delay is 15 second. Adaptive PI+feed-forward control gives the best performance when there is time-delay in the BHP measurement. The amplitude and frequency of oscillation due to delay in BHP are reduced within the acceptance region. The simulation results show that the combination of adaptive PI and feed-forward is robust to the delays in BHP measurements and gives the best performance when there is time-delay in the measurement. 169
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Case 1: 15 Seconds delay in sensor
Fig. 3. PI control for case 1a
Fig. 6. PI control for case 1b
Fig. 4. PI + Feed-forward control for case 1a
Fig. 7. PI + Feed-forward control for case 1b
Fig. 5. Adaptive PI + Feed-forward control for case 1a
Fig. 8. Adaptive PI + Feed-forward control for case 1b
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Fig. 9. PI control for case 2
Fig. 10. PI + Feed-forward control for case 2
Fig. 11. Adaptive PI + Feed-forward control for case 2 Siahaan, H.B., Bjrkevoll, K.S., and Gravdal, J.E. (2014). Possibilities of using wired drill pipe telemetry during managed pressure drilling in extended reach wells. In SPE Intelligent Energy Conference and Exhibition, SPE 167856. Utrecht, Netherlands. Stamnes, Ø.N., Zhou, J., Kaasa, G.O., and Aamo, O.M. (2009). Adaptive observer design for nonlinear systems with parametric uncertainties in unmeasured state dynamics. In Proceedings of the 48th IEEE Conference on 171