Nonlinear Model Predictive Controller for Kick Attenuation in Managed Pressure Drilling*

11th IFAC Symposium on Dynamics and Control of 11th Symposium on and Process including Biosystems 11th IFAC IFACSystems, Symposium on Dynamics Dynamics and Control Control of of 11th IFACSystems, Symposium on Dynamics and Control of Process including Biosystems June 6-8,Systems, 2016. NTNU, Trondheim, Norway Available online at www.sciencedirect.com Process including Biosystems Process Systems, including Biosystems June June 6-8, 6-8, 2016. 2016. NTNU, NTNU, Trondheim, Trondheim, Norway Norway June 6-8, 2016. NTNU, Trondheim, Norway

ScienceDirect

IFAC-PapersOnLine 49-7 (2016) 248–253

Nonlinear Model Predictive Controller for Nonlinear Model Predictive Controller for Nonlinear Model Predictive Controller for Nonlinear Model Predictive Controller for Kick Attenuation in Managed Pressure Kick Attenuation in Managed Pressure Kick Attenuation in Managed Pressure  Kick Attenuation in Managed Pressure Drilling  Drilling Drilling Drilling ∗ ∗∗

Anirudh Nandan ∗ Syed Imtiaz ∗∗ Anirudh Nandan ∗∗ Syed Imtiaz ∗∗ Anirudh Anirudh Nandan Nandan Syed Syed Imtiaz Imtiaz ∗∗ ∗ Faculty of Engineering and Applied Science, Memorial University of ∗ ∗ Faculty of Engineering and Applied Science, Memorial University of of Engineering Applied Science, University ∗ Faculty Newfoundland, St. and John’s, NL A1B 3X5Memorial Canada (e-mail: Faculty of Engineering and Applied Science, Memorial University of of Newfoundland, St. John’s, NL A1B 3X5 Canada (e-mail: Newfoundland, St. John’s, NL A1B 3X5 Canada (e-mail: [email protected]) Newfoundland, St. John’s, NL A1B 3X5 Canada (e-mail: [email protected]) ∗∗ [email protected]) and Applied Science, Memorial University of ∗∗ Faculty of Engineering [email protected]) ∗∗ Faculty of Engineering and Applied Science, Memorial University of of Engineering and Applied Science, University ∗∗ Faculty Newfoundland, St. John’s, NL A1B 3X5 Memorial Canada (e-mail: Faculty of Engineering Applied Science, Memorial University of of Newfoundland, St. and John’s, NL A1B 3X5 Canada (e-mail: Newfoundland, NL A1B 3X5 Canada (e-mail: [email protected]) Newfoundland, St. St. John’s, John’s, NL A1B 3X5 Canada (e-mail: [email protected]) [email protected]) [email protected]) Abstract: We propose a new design of nonlinear model predictive controller (NMPC) for Abstract: We propose a new pressure design ofdrilling nonlinear model predictive controller (NMPC) for Abstract: We a design nonlinear model predictive controller (NMPC) for automatic control of managed (MPD) system. The proposed Abstract: We propose propose a new new pressure design of ofdrilling nonlinear model predictive controllercontroller (NMPC)acts for automatic control of managed (MPD) system. The proposed controller acts automatic control of managed pressure drilling (MPD) system. The proposed controller acts in a pressure control mode and tracks bottom-hole-pressure (BHP) set point during normal automatic control of managed pressure drilling (MPD) system. The proposed controller acts in a pressureand control mode andswitches tracks bottom-hole-pressure (BHP) normal in a control mode tracks (BHP) set point during normal operations, automatically to a flow control mode in set thepoint eventduring of abnormal in a pressure pressureand control mode and andswitches tracks bottom-hole-pressure bottom-hole-pressure (BHP) set point during normal operations, automatically to a flow control mode in the event of abnormal operations, and automatically switches to a flow control mode in the event of abnormal situations, i.e. gasautomatically kick. It contains kick, when it flow occurs, withinmode certain threshold byoftheabnormal deft use operations, and switches to a control in the event situations, i.e. gas kick. It contains kick, when it occurs, within certain threshold by the deft use situations, gas kick. kick, when it within certain by deft of nonlineari.e. state We use output control architecture and employ offset-free situations, i.e. gasconstraints. kick. It It contains contains kick, whenfeedback it occurs, occurs, within certain threshold threshold by the the deft use use of nonlinear state constraints. We use output feedback control architecture and employ offset-free of nonlinear state constraints. We use output feedback control architecture and employ offset-free NMPC algorithm which utilizes recursive-discretization for discretization of the model and use of nonlinear state constraints. We use output feedback control architecture and employ offset-free NMPC algorithm which utilizes recursive-discretization for discretization of the model and use NMPC algorithm which utilizes recursive-discretization for of and active method for optimal control calculation. We demonstrate that the proposed controller is NMPCset algorithm which utilizes recursive-discretization for discretization discretization of the the model model and use use active set method for optimal control calculation. We demonstrate that the proposed controller is active set method for optimal control calculation. We demonstrate that the proposed controller is able toset track a bottom hole pressure set point andWe contain influx inthat the the presence of measurement active method for optimal control calculation. demonstrate proposed controller is able to track a bottom hole pressure set point and contain influx in the presence of measurement able to track a bottom hole pressure set point and contain influx in the presence of measurement noise and plant model mismatches. able to track a bottom hole pressure set point and contain influx in the presence of measurement noise and plant model mismatches. noise and plant model noise and plant model mismatches. mismatches. © 2016, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: Predictive control, Pressure control, Flow control, Constraints, Output feedback, Keywords: Predictive control, Pressure control, Flow Flow control, Constraints, Constraints, Output feedback, feedback, Keywords: Predictive Pressure Disturbance rejection,control, Tracking, Drillingcontrol, Keywords: Predictive control, Pressure control, Flow control, control, Constraints, Output Output feedback, Disturbance rejection, Tracking, Drilling Disturbance Disturbance rejection, rejection, Tracking, Tracking, Drilling Drilling 1. INTRODUCTION controller was used to track a choke pressure setpoint and 1. INTRODUCTION INTRODUCTION controller was used to track aa choke pressure and 1. controller was used to track setpoint and simulation results sequencesetpoint was shown 1. INTRODUCTION controller was usedfor to pipe trackextension a choke choke pressure pressure setpoint and simulation results for pipe extension sequence was shown simulation results for pipe extension sequence was shown it was suggested that gain scheduling can be used Managed pressure drilling (MPD) is an overbalanced and simulation results for pipe extension sequence was shown and it was nonlinearities. suggested thatInstead gain scheduling can be PID, used Managed pressure(Malloy drilling (MPD) is an overbalanced overbalanced and it suggested gain can used tackling of gain scheduled Managed pressure drilling (MPD) an drilling technique et al., 2009) is in which the bottom for and it was was nonlinearities. suggested that thatInstead gain scheduling scheduling can be be PID, used Managed pressure(Malloy drilling (MPD) is an overbalanced for tackling of gain scheduled drilling technique et al., 2009) in which the bottom for tackling nonlinearities. Instead of gain scheduled PID, a nonlinear controller for BHP regulation was designed drilling technique (Malloy et al., 2009) in which the bottom hole pressure is regulated by employing an automated for tackling nonlinearities. Instead of gain scheduled PID, drilling technique (Malloy et by al., employing 2009) in which the bottom a nonlinear controller for BHP regulation was designed hole pressure is regulated an automated a nonlinear for designed (Godhavncontroller et al., 2011) usingregulation feedback was linearization hole pressure regulated by an automated choke manifold.is advantages of MPD are the in nonlinear controller for BHP BHP regulation was designed hole isThe regulated by employing employing anenabling automated (Godhavn et al., 2011) using feedback linearization chokepressure manifold. The advantages of MPD MPD are enabling the ain in (Godhavn et al., 2011) using feedback linearization technique and tracking of BHP setpoint was enabled by choke manifold. The advantages of are enabling the drilling of so-called undrillable wells in which pressure winin (Godhavn et al., 2011) using feedback linearization choke manifold. The advantages of MPD are enabling the technique and tracking tracking of kicks BHP still setpoint wasthat enabled by drilling of so-called so-called undrillable wells inhandling which pressure pressure win- technique and of BHP setpoint was enabled by a BHP estimator. Large require drilling drilling of undrillable wells in which window is very narrow and ensures safer of reservoir and tracking of kicks BHP still setpoint wasthat enabled by drilling of so-called undrillable wells inhandling which pressure win- technique abe BHP estimator. Large require drilling dow is very narrow and ensures safer of reservoir a BHP estimator. Large kicks still require that drilling stopped, so that kick can be handled by conventional dow very ensures handling reservoir influx. will beand an influx of safer reservoir fluidsof kick abeBHP estimator. Large kicks still require that drilling dow is is There very narrow narrow and ensures safer handling ofcalled reservoir stopped, so that kick can be handled by conventional influx. There will be an influx of reservoir fluids called kick be stopped, that can be handled by conventional handlingso order improve during influx. There will be influx reservoir fluids kick is less than thecalled reservoir when the bottom be stopped, somethod. that kick kickIn can be to handled by safety conventional influx. There will hole be an anpressure influx of ofp reservoir fluids called kick kick kick handling method. In order to improve safety during less than the reservoir when the bottom hole pressure pbh bh is kick handling method. In order to improve safety during such operations, in (Carlsen et al., 2013) PI, IMC, and is less than the reservoir when the bottom hole pressure p pressure p . If a kick is unmitigated, large quantities of bh res kick handling method. In order to improve safety during than quantities the reservoir when the pbottom hole pressure pbh is lesslarge such operations, in (Carlsen et al., 2013) PI, IMC, and pressure . If a kick is unmitigated, of res such operations, in (Carlsen et al., 2013) PI, IMC, and MPC pressure controllers were designed to automate kick pressure p . If a kick is unmitigated, large quantities of reservoir fluids may flow to the surface endangering people res such operations, in (Carlsen et al., 2013) PI, IMC, and pressure pfluids a kick istounmitigated, large quantities of MPC pressure controllers were designed to automate kick res . Ifmay reservoir flow the surface endangering people MPC pressure controllers were designed to automate kick handling sequence. In (Nandan et al., 2014), a robust H reservoir fluids may flow to the surface endangering people ∞ and the environment. The traditional method to handle a MPC pressure controllers were designed to automate kick reservoir fluids may flow to the surface method endangering people sequence. In (Nandan et al., aa robust H ∞ and environment. to handle aa handling handling sequence. In (Nandan et al., 2014), robust H loop shaping controller was deigned for2014), handling variations ∞ and the environment. The traditional to handle kick the requires drilling toThe be traditional stopped andmethod the annulus needs handling sequence. In (Nandan et al., 2014), a robust H ∞ and the environment. The traditional method to handle a loop shaping controller was deigned for handling variations kick requires drilling to be stopped and the annulus needs loop shaping controller was deigned for handling variations in mud density, well length, and mud flow rate. For severe kick requires drilling to be stopped and the annulus needs to be shut-off and that contributes to the non-productive loop shaping controller was deigned for handling variations kick requires drilling to be stopped and the annulus needs in mud density, well length, and mud flow rate. For severe to be shut-off and that to the non-productive mud well length, mud flow severe changes in the flow and and choke opening, gainFor switching to be shut-off and contributes to the non-productive time has contributes the potential NPT and in in mud density, density, wellrate length, and mud flow rate. rate. For severe to be (NPT). shut-offMPD and that that contributes to to thereduce non-productive changes in the flow rate and choke opening, gain switching time (NPT). MPD has the potential to reduce NPT and changes in the flow rate and choke opening, gain switching robust controller was suggested. The advantage offered time (NPT). MPD has the potential to reduce NPT and enhance safety. It was reported in (Vieira et al., 2008) changes in the flow rate and choke opening, gain switching time (NPT). MPD has the potential to reduce NPT2008) and robust controller was suggested. The advantage offered by by enhance safety. It was reported in (Vieira et al., robust controller The offered by control iswas its suggested. ability to track a BHP setpoint enhance safety. It was reported in (Vieira et al., 2008) that without MPD it took 65 days to drill a particular robust controller was suggested. The advantage advantage offeredbut by enhance safety. It was reported in (Vieira et al., 2008) pressure pressure control is its ability to track a BHP setpoint but that without MPD it took 65 days to drill a particular pressure control is its ability to track a BHP setpoint but during a kick, continued pressure setpoint tracking will not that without MPD it took 65 days to drill a particular well while using MPD it took only 45 days. There is popressure control is its ability to track a BHP setpoint but that without MPD it took 65 days to drill a particular during a kick, continued pressure setpoint tracking will not well while using MPD it took through only 45 automatic days. There is po- during aa kick, continued pressure setpoint tracking will attenuate a kick (Zhou et al., 2011). For kick attenuation well while using it only days. is tential to reduce NPT further pressure kick, continued pressure setpoint tracking will not not well while using MPD MPD it took took through only 45 45 automatic days. There There is popo- during attenuate aa kick (Zhou et al., 2011). For kick attenuation tential to reduce NPT further pressure attenuate kick (Zhou et al., 2011). For kick attenuation flow controllers have been designed. Feedback linearised tential to reduce NPT further through automatic pressure management and automation usually leads to enhanced attenuate a kick (Zhou et al., 2011). For kick attenuation tential to reduce NPT further through automatic pressure flow controllers have been designed. Feedback linearised management and automation usually enhanced controllers have designed. Feedback linearised were been presented in (Hauge et al., 2012) management and usually leads to enhanced safety. Automated MPD solutions rangeleads fromto flow controllers have been designed. Feedback linearised management and automation automation usually leads toautomating enhanced flow flow controllers were presented in (Hauge et al., 2012) safety. Automated MPD solutions range from automating flow controllers were presented in (Hauge et al., 2012) and (Hauge et al., 2013). The choke opening was used to safety. Automated MPD solutions range from automating conventional well control methods to model based control flow (Hauge controllers were presented in (Hauge etwas al.,used 2012) safety. Automated MPD solutions range frombased automating and et al., 2013). The choke opening to conventional well control methods to model control and (Hauge et al., 2013). The choke opening was used to regulate the exit flow rate and thereby the in/out flux. An conventional well control methods to model based control of pressure and flow rate. An extensive review of comand (Hauge et al., 2013). The choke opening was used to conventional well control methods to model based control regulate the exit flow rate and thereby the in/out flux. An of pressure and flow rate. An extensive review of comregulate the exit flow rate and thereby the in/out flux. An in/out flux and estimator was also presented in (Hauge of pressure and flow rate. An extensive review of computer controland in managed pressure drilling review can be found in regulate the exit flow rate and thereby the in/out flux. An of pressure flow rate. An extensive of comin/out flux and presented (Hauge puter control in managed drilling can be found in in/out flux estimator was also presented in (Hauge al., 2012) andestimator (Hauge etwas al., also 2013) they alsoin puter control in pressure drilling can be in (Nikolaou, 2013). Controlpressure requirements MPD system in/out flux and and estimator was also presented inestimate (Hauge puter control in managed managed pressure drillingfor can be found found in et et al., 2012) and (Hauge et al., 2013) they also estimate (Nikolaou, 2013). Control requirements for MPD system et al., 2012) and (Hauge et al., 2013) they also estimate bit flow rate. In (Santos et al., 2003), a well control method (Nikolaou, 2013). Control requirements for MPD system was discussed in (Godhavn et al., 2010), a simple PID et al., 2012) and (Hauge et al., 2013) they also estimate (Nikolaou, 2013). Control requirements for MPD system bit flow rate. In (Santos et al., 2003), a well control method was discussed in (Godhavn et al., 2010), a simple PID bit flow rate. (Santos al., well control which in/outaaflow formethod detectwas discussed flowinvolves rate. In Incomparing (Santos et et the al., 2003), 2003), wellrates control method was discussed in in (Godhavn (Godhavn et et al., al., 2010), 2010), a a simple simple PID PID bit which involves comparing the in/out flow rates for detect This which involves comparing the in/out flow rates for detectwork was supported by Natural Science and Engineering Reing kicks and remedying by increasing the back pressure which involves comparing the in/out flow rates for detect This work was supported by Natural Science and Engineering Reing kicks and remedying by increasing the back pressure  search of supported Canada (NSERC) andScience Research Development This Council work was by Natural andand Engineering Reing kicks and remedying by increasing the back pressure was presented. Flow control is an effective strategy for  This Council work was supported by Natural Science andand Engineering Reing kicks and remedying by increasing the back pressure search of Canada (NSERC) and Research Development was presented. presented. Flow Flow control control is is an an effective effective strategy strategy for for Corporation of of Newfoundland and Labrador (RDC). search Council Canada (NSERC) and Research and Development was search Council of Canada (NSERC) and Research and Development was presented. Flow control is an effective strategy for Corporation Corporation of of Newfoundland Newfoundland and and Labrador Labrador (RDC). (RDC).

Corporation of Newfoundland and Labrador (RDC). Copyright © 2016, 2016 IFAC 248Hosting by Elsevier Ltd. All rights reserved. 2405-8963 © IFAC (International Federation of Automatic Control) Copyright © 2016 IFAC 248 Copyright ©under 2016 responsibility IFAC 248Control. Peer review of International Federation of Automatic Copyright © 2016 IFAC 248 10.1016/j.ifacol.2016.07.268

IFAC DYCOPS-CAB, 2016 June 6-8, 2016. NTNU, Trondheim, NorwayAnirudh Nandan et al. / IFAC-PapersOnLine 49-7 (2016) 248–253

suppressing kicks but typically in an MPD operation BHP must track a set point. In (Zhou et al., 2011) a switching controller which works as a pressure controller during normal operation and as a flow controller while handling kicks was presented. In order to perform overbalanced drilling, reservoir pressure estimates are required, in (Zhou et al., 2011) a nonlinear passivity based observer for reservoir pressure and kick estimation was developed. A nonlinear pressure/flow switching controller was designed for dual gradient drilling (DGD) in (Zhou and Nygaard, 2011). DGD is a variant of MPD in which muds of varying densities are used and as a result the hydrostatic pressure is piece-wise linear. Nonlinear controllers depend heavily on tuning and necessary expertise might not be available for upstream operations. Model predictive controller (MPC) and nonlinear MPC design have also been considered for MPD, they are well suited for MPD because their ability to handle constraints and nonlinearity respectively. An NMPC scheme for control of underbalanced drilling (UBD) was developed in (Nygaard and Nævdal, 2006). BHP was regulated by computing optimal choke opening in receding horizon fashion. A two phase model of drilling well was used as an UBD well produces hydrocarbons while drilling. In (Breyholtz et al., 2009) NMPC was used to coordinate pump flow rate and choke opening in order to control BHP. The control was evaluated for pressure regulation during pipe extension sequence but control of kicks were not treated. In (Breyholtz et al., 2010), linear MPC control of DGD was considered. The bottom hole pressure and hook position were controlled by manipulating main pump and sub sea pump flow rates as well as the drill string velocity. The focus of the study was on optimal movement of drill string in order to minimize pressure variations. Work done in (Breyholtz et al., 2010) was further expanded in (Breyholtz et al., 2011), robustness and results in presence of noise was analysed. Use of linear MPC was considered in (Møgster et al., 2013) which utilized step response models between the inputs, namely the mud flow rate and choke opening, and the outputs namely the bottom hole pressure and the pressure at the casing shoe. The controller was implemented using Statoil’s in-house MPC software SEPTIC. The innovation was in manipulating pressure at two points, BHP and pressure at the casing shoe, but its effectiveness in dealing with kicks and severe drop in pumping rate was not studied. In (Pedersen et al., 2013) UBD control was performed by using First Order Plus Time Delay (FOPTD) models. The bottom hole pressure and return flow rate were regulated by manipulating the choke opening and mud pump flow rate. Regulating outlet flow will be useful in UBD as it produces during drilling. But an MPD well, the kind of well considered in this work, produces only during a kick and at other times a BHP set point has to be tracked hence that requires setting the NMPC problem differently from that in (Pedersen et al., 2013). In (Mahdianfar et al., 2013) a joint unscented Kalman filter was developed for state and parameter estimation as frictional loses and annular geometry are uncertain and estimation of those parameters will improve control solutions. Thus for kick rejection switching pressure/flow controller offers the best solution and NMPC is very well suited for MPD because of its ability to handle constraints and nonlinearities. We make use of NMPC technique to implement the switched 249

249

pressure/flow control philosophy. The controller presented in this paper has the following features: The NMPC operates as a pressure controller which tracks BHP under normal drilling conditions. The controller acts more like a flow controller when a kick occurs and contains a kick within a tunable threshold. The controller is able to work under different mud flow rates and choke opening without any deterioration in performance. The designed NMPC controller is able to perform well under persistent disturbance, plant model mismatch, and noisy measurements. The controller is able to regulate bottom hole pressure during pipe extension sequence as well. 2. SYSTEM DESCRIPTION qp Mud Pit Pp qb

Pc

qc uc

Degasser & Shaker

pbh

Fig. 1. Schematic representation of managed pressure drilling The MPD process consists of two control volumes, the drill string and the annulus. The schematic representation of MPD process is shown in Figure 1. The drilling mud is pumped into the drill string under pump pressure pp and at flow rate qp . The mud exits the drill string through the drill bit at a flow rate qbit . The drilling mud then flows through the annulus control volume and exists it through a choke at pressure pc and flow rate qc . The pump pressure, choke pressure, and bit flow rate are given by Equations (1), (2), and (3) respectively. βd and βa are bulk moduli of mud in drill string and annulus respectively. ρd and ρa are the mud densities in the drill string and annulus respectively. Vd and Va are the volumes of the drill string and the annulus respectively. M is a mass like property. The pressure at the bottom hole pbh is given by Equation (4). The flow through the choke is given by Equation (5) where uc ∈ [0, 1] is the choke opening. The kick flow rate qk is given by Equation (6). fd and fa are frictional loss coefficients in the drill string and the annulus respectively which relate volumetric flow rate and frictional pressure drop. Due to the addition of reservoir fluids and cuttings in the annulus,

IFAC DYCOPS-CAB, 2016 250 June 6-8, 2016. NTNU, Trondheim, NorwayAnirudh Nandan et al. / IFAC-PapersOnLine 49-7 (2016) 248–253

generally mud density changes when mud flows from the drill string into the annulus and that induces pressure changes equal to (ρa − ρd )ghtv . Frictional loss and mud density are major sources of uncertainty and when there is a reservoir flow it acts as a persistent disturbance. The drilling model which is considered here is based on the detailed model presented in (Kaasa et al., 2012). βd p˙p = (qp − qbit ) (1) Vd βa (qbit − qc + qb + qk ) (2) p˙c = Va 1 (pp − pc − pf d − pf a q˙bit = M − (ρa − ρd )ghtv ) (3) (4) pbh = pc + pf a + ρa ghtv  (pc − po ) qc = uc Cd Ao (5) ρa qk = Kpi (pres − pbh ) (6) 32ρd fd |qp |qp Ld (7) pf d = π 2 Dd 5 32ρa fa |qbit |qbit La (8) pf a = 2 π 2 (Da − Dd )(Da 2 − Dd 2 ) The controlled output y is given by Equation (9), the manipulated variable u is given by Equation (10), the state vector x is represented by Equation (11), and the disturbance d is given by Equation (12). y = pbh (9) u = uc (10) T

x(k + T ) = x(k) +



k+T

f (x(τ ), u)dτ,

(15)

k

where x(k) is the current state and T is the sampling time. In order to design an offset free NMPC we utilize the results presented in (Morari and Maeder, 2012) and (Rawlings and Mayne, 2009). As a first step, disturbance model is incorporated in the nominal model resulting in an augmented model represented by faug . During state prediction disturbance is held constant, given by Equation (17). The prediction model is numerically integrated using explicit Runge-Kutta 4,5 method. The predicted state is given by Equation (16).  k+T faug (x(τ ), d(k))dτ, (16) x(k + T ) = x(k) + k

d(k + T ) = d(k)

(17)

3.2 Equilibrium state and input targets According to (Morari and Maeder, 2012) incorporating disturbance model and an integrator is not always sufficient for offset free tracking. The augmented model must be used to compute equilibrium state targets to achieve certain output setpoint. In this work, our objective is to track an output reference r(k) = pref , therefore we have to compute relevant state targets. The state and input targets are denoted by x ¯ and u ¯ respectively and they are computed by solving the equilibrium Equations (18) and (19) they are implemented as equality constraints in the optimization problem. ˆ x, u ¯ , d(k)), (18) x ¯ = faug (¯

x = [pp , pc , qbit ] (11) (12) d = qk The nominal state space equation of the system is given by Equation (13), where f is the system of equations described by Equations (1) - (3) with qk = 0. The output is given by Equation (14), where g is given by Equation (4). x˙ = f (x, u) (13) y = g(x) (14)

ˆ r(k) = gaug (¯ x, d(k)), (19) where gaug is the augmented output model. The augmented output model is chosen depending on the control objective. If the objective is to track BHP setpoint the augmented output is given by Equation (20), where g1 aug is given by Equation (4). If the objective is to track choke pressure setpoint the augmented output model is given by Equation (21), where g2 aug is given by Equation (2).

3. CONTROLLER DESIGN

y1 (k + T ) = g1 aug (x(τ ), d(k)), y2 (k + T ) = g2 aug (x(τ ), d(k))

Nonlinear model predictive controller (NMPC) selects optimal control actions by using an optimization algorithm such that the error between the reference and predicted states of the system is minimized. NMPC utilizes a nonlinear numerical model of the system to predict the future states of the system for a finite horizon and computes a sequence of optimal control actions for that finite horizon but applies only the first control action and discards the rest. This process of prediction, control computation, and control application is repeated again. Since the controller predicts and computes over a horizon which is forever moving, this is called moving horizon control. Therefore, NMPC is an optimal moving horizon controller. The core elements of an NMPC are the cost function, prediction model, state constraints, and input constraints. In this section the design of each of those elements is explained. 3.1 Prediction model The predicted states are given by the Equation (15). 250

(20) (21)

3.3 Cost function The objective is to track x ¯ and u ¯ , the state and input equilibrium targets respectively, resulting in the offset free tracking of the reference r(k) = pref . J = min u

k+m  κ=k

(ˆ x(κ) − x ¯(κ))T λ1 (ˆ x(κ) − x ¯(κ))

¯ (κ))2 , + λ2 (u(κ) − u

(22)

and λ2 ∈ R are cost function weights where λ1 ∈ R and m is the prediction horizon. The optimization problem is constrained by the following constraints (23) x ˆ ∈ X, x ˆ ∈ Xnl , u ∈ U, x ¯ ∈ X, u ¯ ∈ U, (24) where the state and input constraint sets X and U are given by 3×3

IFAC DYCOPS-CAB, 2016 June 6-8, 2016. NTNU, Trondheim, NorwayAnirudh Nandan et al. / IFAC-PapersOnLine 49-7 (2016) 248–253

40

495 p bh p estimate res p setpoint

38

bh

36

pres p

490

frac

34 32

u c [%]

Pressure [bar]

251

485

30 28 26

480

24 22 475

0

50

100

150

200

250

300

350

400

20

450

0

50

100

150

200

250

300

350

400

450

400

450

Time [s]

Time [s]

Fig. 2. Kick attenuation: Bottom hole pressure   pp min ≤ pp ≤ pp max X :=  pc min ≤ pc ≤ pc max  , (25) qbit min ≤ qbit ≤ qbit max   U := uc min ≤ uc ≤ uc max . (26) The control objective is not only to track a bottom hole setpoint but also to contain the reservoir influx within certain threshold and in order to achieve it we include a nonlinear state constraint given by following equation Xnl = [0 ≤ (qc − q¯bit ) ≤ ] (27) where q¯bit is the equilibrium state target for the bit flow rate. By adding this nonlinear constraint the annular discharge is constrained within a tunable threshold . The controller uses recursive-discretization for discretization of the model and uses active set method for optimal control calculation.

Fig. 3. Kick attenuation: Choke opening

3.4 Observer

Fig. 4. Kick attenuation: Kick estimate

35 30

q k [LPM]

25 20 15 10 5 0

0

50

100

150

200

250

350

100 Constraint value Threshold

90 80 70 60 50 Kick handling 40 30

4. SIMULATION

300

Time [s]

(qc ! q7bit ) [LP M]

The designed NMPC scheme depends on the estimates of kick and the bit flow rate for predicting the states of the MPD system. The bit flow rate and kick flow rate are estimated using the observer developed in (Zhou et al., 2011). In order to perform overbalanced drilling, the estimate of the reservoir pressure is required. The reservoir pressure is estimated using the observer developed in (Zhou et al., 2011). Observers are not presented here and readers are referred to the cited articles for the details.

40

Setpoint revision

20

The performance of the controller is evaluated by simulating a kick and the effect of measurement noise on the outlet flow rate constraint is also studied. The simulation model consists of Equations (1) - (6). It is assumed that BHP measurements are available without time-delay. 4.1 Outlet flow constrained pressure regulation The initial bottom hole pressure setpoint is pref = 480 bar. In this simulation mud is pumped at the rate of 1200 LP M . A kick is encountered at 120s, and that leads to violation of the flow constraint threshold of  = 10, as 251

10 0 0

120

240

360

480

Time [s]

Fig. 5. Kick attenuation: Value of flow constraint shown in Figure 2. The controller responds by closing the choke as shown in Figure 3 and that causes an increase in pbh . Due to the increase in pressures, the reservoir pressure estimator is able to estimate the new reservoir pressure as shown in Figure 2. It is to be noted that the controller is not tracking the reservoir pressure which will

IFAC DYCOPS-CAB, 2016 June 6-8, 2016. NTNU, Trondheim, NorwayAnirudh Nandan et al. / IFAC-PapersOnLine 49-7 (2016) 248–253 252

495

40 pbh pres estimate p setpoint bh p res pfrac

30 25

q k [LPM]

Pressure [bar]

490

35

485

20 15

480

10 5

475

0

50

100

150

200

250

300

350

400

0

450

Time [s]

100

150

200

250

300

350

400

450

Fig. 8. Effect of noise: Kick estimate 100

38

90

36

80

34

70

(qc ! q7bit ) [LP M]

40

32 30 28

Constraint value Threshold

60 Kick handling

50 40 30

26

20

24

10

22 20

50

Time [s]

Fig. 6. Effect of noise: Bottom hole pressure

u c [%]

0

0

0

50

100

150

200

250

300

350

400

0

450

120

240

360

480

Time [s]

Time [s]

Fig. 9. Effect of noise: Value of flow constraint

Fig. 7. Effect of noise: Choke opening be possible only by resorting to complete flow control, instead it gives up pressure tracking in order to satisfy the outlet flow constraint. Using the new reservoir estimate, pref is revised to 475 bar at 252s. Eventually, kick is completely rejected by restoring overbalanced condition. 4.2 Effect of measurement noise on outlet flow constraint In this case the effect of measurement noise and plantmodel mismatch on the flow constraint is tested. A measurement noise of 0.1 bar is added to pressure measurements and plant-model mismatch is introduced by augmenting state equations with random processes. The setpoint is pref = 480 bar and mud is pumped at the rate of 1200 LP M . A kick is encountered at 120s due to a 5 bar step change in the reservoir pressure. The controller responds by slightly closing the choke, as shown in Figure 7, in order to increase pbh . Due to the increase in pbh , shown in Figure 6, the kick flow rate decreases as shown in Figure 8. It can be seen in Figure 9, the constraint value oscillates around the threshold value  = 10. Due to the measurement noise and plant model mismatch the flow intermittently exceeds the threshold marginally and the controller brings it back into the acceptable region. 252

5. CONCLUSION An NMPC based outlet flow constrained pressure regulator was designed for managed pressure drilling system. A nonlinear model of MPD well augmented with an input disturbance model was used to design the controller. An observer was used to estimate unmeasured state and disturbance. The controller presented in this article is able to perform well in the presence of a persistent disturbance. It was found that the controller is able to attenuate kicks and track a BHP setpoint. But the uncertainties in the frictional loss, choke model, and well geometry were not considered. Estimates of the uncertain parameters can be incorporated in the prediction model in order to improve performance and ensure stability. Robust NMPC algorithms can also be utilized to address model uncertainties. The above mentioned limitations will be addressed in a future article. REFERENCES Breyholtz, Ø., Nygaard, G., Godhavn, J.M., and Vefring, E.H. (2009). Evaluating control designs for coordinating pump rates and choke valve during managed pressure drilling operations. In Control Applica-

IFAC DYCOPS-CAB, 2016 June 6-8, 2016. NTNU, Trondheim, NorwayAnirudh Nandan et al. / IFAC-PapersOnLine 49-7 (2016) 248–253

Table 1. Values of well parameters used in simulations Parameter Va Vd TV D M βa βd ρa ρd fd fa Cd Ao po Kpi

Value 89.9456 25.5960 3500 8.04 × 108 2.3 × 109 2.3 × 109 1300 1300 1.65 × 1010 2.08 × 109 0.6 2 × 10−3 1.013 × 105 6.133 × 10−9

Unit m3 m3 m kg/m3 Pa Pa kg/m3 kg/m3 s2 /m6 s2 /m6 − m2 Pa m3 /(s Pa)

Table 2. Controller tuning parameters Parameter λ1 λ2 m T

Value diag[0,1,0] 1000 4 6

Unit − − − s

Table 3. State and input constraints Parameter pp min pp max pc min pc max qbit min qbit max uc min uc max

Value 8 × 105 150 × 105 8 × 105 50 × 105 −0.002 0.0283 0 100

Unit Pa Pa Pa Pa m3 /s m3 /s % %

tions,(CCA) & Intelligent Control,(ISIC), 2009 IEEE, 731–738. IEEE. Breyholtz, Ø., Nygaard, G., and Nikolaou, M. (2010). Automatic control of managed pressure drilling. In American Control Conference (ACC), 2010, 442–447. IEEE. Breyholtz, Ø., Nygaard, G., Nikolaou, M., et al. (2011). Managed-pressure drilling: Using model predictive control to improve pressure control for during dual-gradient drilling. SPE Drilling & Completion, 26(02), 182–197. Carlsen, L.A., Nygaard, G., and Nikolaou, M. (2013). Evaluation of control methods for drilling operations with unexpected gas influx. Journal of Process Control, 23(3), 306–316. Godhavn, J.M., Pavlov, A., Kaasa, G.O., and Rolland, N.L. (2011). Drilling seeking automatic control solutions. In Proceedings of the 18th World Congress, volume 18, 10. Godhavn, J.M. et al. (2010). Control requirements for automatic managed pressure drilling system. SPE Drilling & Completion, 25, 336–345. Hauge, E., Aamo, O., Godhavn, J.M., and Nygaard, G. (2013). A novel model-based scheme for kick and loss mitigation during drilling. Journal of Process Control, 23(4), 463–472. Hauge, E., Aamo, O.M., and Godhavn, J.M. (2012). Model-based estimation and control of in/out-flux during drilling. In American Control Conference (ACC), 2012, 4909–4914. IEEE. 253

253

Kaasa, G.O., Stamnes, Ø.N., Aamo, O.M., Imsland, L.S., et al. (2012). Simplified hydraulics model used for intelligent estimation of downhole pressure for a managedpressure-drilling control system. SPE Drilling & Completion, 27(01), 127–138. Mahdianfar, H., Pavlov, A., and Aamo, O.M. (2013). Joint unscented kalman filter for state and parameter estimation in managed pressure drilling. In Control Conference (ECC), 2013 European, 1645–1650. IEEE. Malloy, K.P., Stone, R., Medley, G.H., Hannegan, D.M., Coker, O.D., Reitsma, D., Santos, H.M., Kinder, J.I., Eck-Olsen, J., McCaskill, J.W., et al. (2009). Managedpressure drilling: What it is and what it is not. In IADC/SPE Managed Pressure Drilling and Underbalanced Operations Conference & Exhibition. Society of Petroleum Engineers. Møgster, J., Godhavn, J.M., and Imsland, L. (2013). Using mpc for managed pressure drilling. Modeling, Identification, and Control. Morari, M. and Maeder, U. (2012). Nonlinear offset-free model predictive control. Automatica, 48(9), 2059–2067. Nandan, A., Imtiaz, S., and Butt, S. (2014). Robust control of managed pressure drilling. In Oceans-St. John’s, 2014, 1–8. IEEE. Nikolaou, M. (2013). Computer-aided process engineering in oil and gas production. Computers & Chemical Engineering, 51, 96–101. Nygaard, G. and Nævdal, G. (2006). Nonlinear model predictive control scheme for stabilizing annulus pressure during oil well drilling. Journal of Process Control, 16(7), 719–732. Pedersen, T., Godhavn, J.M., et al. (2013). Model predictive control of flow and pressure in underbalanced drilling. In Dynamics and Control of Process Systems, volume 10, 307–312. Rawlings, J.B. and Mayne, D.Q. (2009). Model predictive control: theory and design. Nob Hill Publishing. Santos, H., Leuchtenberg, C., Shayegi, S., et al. (2003). Micro-flux control: the next generation in drilling process for ultra-deepwater. In Offshore Technology Conference. Offshore Technology Conference. Vieira, P., Arnone, M.A., Cook, I., Moyse, K., Haojie, H.W., Qutob, H.H., Yuesheng, C., Qing, C., et al. (2008). Constant bottomhole pressure: Managedpressure drilling technique applied in an exploratory well in saudi arabia. In SPE/IADC Managed Pressure Drilling and Underbalanced Operations Conference and Exhibition. Society of Petroleum Engineers. Zhou, J., Aamo, O.M., Kaasa, G.O., et al. (2011). Switched control for pressure regulation and kick attenuation in a managed pressure drilling system. Control Systems Technology, IEEE Transactions on, 19(2), 337– 350. Zhou, J. and Nygaard, G. (2011). Automatic modelbased control scheme for stabilizing pressure during dual-gradient drilling. Journal of Process Control, 21(8), 1138–1147.