Nonlinear model predictive control of managed pressure drilling

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Research article

Nonlinear model predictive control of managed pressure drilling Anirudh Nandan, Syed Imtiaz n Faculty of Engineering and Applied Science, Memorial University of Newfoundland, St. John's, NL, Canada A1B 3X5

art ic l e i nf o

a b s t r a c t

Article history: Received 29 February 2016 Received in revised form 20 January 2017 Accepted 21 March 2017

A new design of nonlinear model predictive controller (NMPC) is proposed for managed pressure drilling (MPD) system. The NMPC is based on output feedback control architecture and employs offset-free formulation proposed in [1]. NMPC uses active set method for computing control inputs. The controller implements an automatic switching from constant bottom hole pressure (CBHP) regulation to flow control mode in the event of a reservoir kick. In the flow control mode the controller automatically raises the bottom hole pressure setpoint, and thereby keeps the reservoir fluid flow to the surface within a tunable threshold. This is achieved by exploiting constraint handling capability of NMPC. In addition to kick mitigation the controller demonstrated good performance in containing the bottom hole pressure (BHP) during the pipe connection sequence. The controller also delivered satisfactory performance in the presence of measurement noise and uncertainty in the system. & 2017 ISA. Published by Elsevier Ltd. All rights reserved.

Keywords: Managed pressure drilling (MPD) Nonlinear model predictive controller (NMPC) Gas kick

1. Introduction Managed pressure drilling (MPD) is a marginally overbalanced drilling technique [2] where the bottom hole pressure is regulated by employing an automated choke manifold. MPD enables drilling of so-called undrillable wells where pressure window is very narrow. One of the major safety issues in drilling is influx of reservoir fluids or commonly known as ‘reservoir kick’. Reservoir kick happens if for some reason the bottom hole pressure (BHP), (pbh) drops below the reservoir pressure (pres). If a kick is unmitigated, large quantity of reservoir fluid may flow to the rig surface which may lead to catastrophic accidents. MPD can prevent and recover the system quickly from such abnormal situation. MPD can also expedite the drilling process, in [3] it was reported that without MPD it took 65 days to drill a particular well, while using MPD the drilling period was brought down to 45 days. Automated MPD solutions range from simple proportional integral derivative (PID) controller to model based control of pressure at different points, and control of drilling fluid flow rate. A review of automatic control in MPD can be found in [4]. Early on constant bottom hole pressure regulators were usually implemented for MPD systems. A simple PID controller to track choke pressure setpoint was developed in [5]. The controller demonstrated good performance for pipe extension sequence. A nonlinear controller for BHP regulation was designed in [6] using n

Corresponding author. E-mail addresses: [email protected] (A. Nandan), [email protected] (S. Imtiaz).

feedback linearization technique. These controllers, however were not configured for kick mitigation. In order to improve safety during such operations, in [7] PI, IMC, and MPC pressure controllers were designed to automatically mitigate kicks while drilling. In [8], a robust H∞ loop shaping controller was designed for handling variations in mud density, well length, and mud flow rates. For large changes in the flow rate and choke opening, gain switching robust controller was suggested. The advantage of a pressure controller is its ability to track a BHP setpoint. However, in the event of a ‘reservoir influx’ continued pressure setpoint tracking will not attenuate a kick [9]. In the event of reservoir fluid influx into the bottom hole region, flow controllers are better suited for handling such reservoir kicks. Several flow control strategies have been used successfully to mitigate reservoir kick. Feedback linearised flow controllers were presented in [10] and [11]. In combination with a bit flow estimator the controllers manipulate the choke opening to regulate the exit flow rate and thereby the in/out flux. In [12] a well control method was proposed that compares the in/out flow rates for detecting kicks and subsequently kick is mitigated by manipulating the back pressure. Flow control is an effective strategy for suppressing kicks, but does not provide optimal performance in tracking the bottom hole pressure trajectory. In [9] a switching controller was implemented which works as a pressure controller during normal operation, and as a flow controller in the event of a reservoir kick. A nonlinear passivity based observer was developed to estimate kick magnitude and reservoir pressure. A similar nonlinear pressure/flow switching controller was designed for dual gradient drilling (DGD) in [13].

http://dx.doi.org/10.1016/j.isatra.2017.03.013 0019-0578/& 2017 ISA. Published by Elsevier Ltd. All rights reserved.

Please cite this article as: Nandan A, Imtiaz S. Nonlinear model predictive control of managed pressure drilling. ISA Transactions (2017), http://dx.doi.org/10.1016/j.isatra.2017.03.013i

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Linear and nonlinear model predictive controller (LMPC/NMPC) have also been considered for MPD. In [14,15] linear MPC was designed for DGD, the focus of the study was on optimal movement of drill string in order to minimize pressure variations. The hook position and bottom hole pressure were controlled by manipulating the drill string velocity and main pump and sub-sea pump flow rates. Controller performance was demonstrated in presence of noise and uncertainty. [16] implemented linear MPC to control the bottom hole pressure and the pressure at the casing shoe by manipulating the mud flow rate and choke opening. The controller was implemented using Statoil's in-house MPC software SEPTIC. The controller regulated BHP and casing shoe pressure, however effectiveness of the controller in dealing with kicks and severe drop in pumping rate was not studied. [17] implemented MPC on UBD system 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 is useful in UBD as it allows hydrocarbons to come to the surface during drilling. NMPCs are well suited for UBD because of their ability to handle nonlinearity. An NMPC scheme was used for control of underbalanced drilling (UBD) [18]. BHP was regulated by computing optimal choke opening in a receding horizon fashion. A two phase model of drilling well was used to model UBD well drilling. [19] used NMPC to coordinate pump flow rate and choke opening in order to control BHP. The controller was evaluated for pressure regulation during pipe extension sequence, however mitigation of kicks were not considered. The above literature review clearly shows MPC/ NMPC have been used successfully to UBD and DGD systems. The application of MPC/NMPC to MPD system is very limited. Besides, one of the major strength of MPC is handling of constraints, which was not fully exploited in these applications. In this paper, we present a new design of NMPC for MPD application which implements the philosophy of switched pressure/ flow control by cleverly employing the constraints of NMPC. 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 the kick within a tunable threshold. The rest of the paper is organized as follows: a brief description of MPD is furnished in Section 2, followed by the design of the controller and optimization scheme and details on the implemented observer to estimate bit flow rate, kick flow rate, and reservoir pressure are provided in Section 3. The simulation results are presented in Section 4 with concluding remarks in Section 5.

qb qp

pp pc pbh

NMPC

uc

qp Mud Pit pp qb

pc

qc uc

Degasser & Shaker

Ocean

Rock formation Drill string

Annulus

Rock formation

Bit Reservoir pbh

Fig. 1. Schematic representation of managed pressure drilling.

Table 1 Values of well parameters used in simulations. Parameter

Value

Unit

Va

89.9456

m3

Vd

25.5960

TVD M

3500

m3 m

βa

2.3 × 109

βd ρa

2.3 × 109 1300

ρd

1300

kg/m3

fd

1.65 × 1010

s2/m6

fa

2. System description

Cd Ao

2.08 × 109 0.6 2  10 3

s2/m6 −

The MPD system consists of two control volumes, the drill string and the annular mud return section. The schematic representation of MPD system is shown in Fig. 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 annular control volume and exits through a choke at pressure pc, and flow rate qc. The pump pressure, choke pressure, and bit flow rate are given by Eqs. (1), (2), and (3) respectively; βd and βa are bulk moduli of mud in drill string and annulus respectively. Similarly, ρd and ρa are the mud densities, fd and fa are frictional loss coefficients, Vd and Va are the volumes of the drill string and the annulus respectively, and M is a mass like property. The pressure at the bottom hole pbh is

po

1.013 × 105

m2 Pa

Kpi

6.133 × 10−9

m3/(s Pa)

8.04 × 108

kg/m3 Pa

Pa kg/m3

given by Eq. (4). The flow through the choke is given by Eq. (5) where uc ∈ [0, 1] is the choke opening. The kick flow rate qk is given by Eq. (6) where Kpi is a parameter characteristic to reservoir properties. Frictional loss and mud density are major sources of uncertainty, a reservoir kick it acts as a persistent disturbance. Detailed derivation of the model can be found in [20].

pṗ =

βd Vd

(qp − qbit )

(1)

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Fig. 2. Kick handling and pressure setpoint revision (nominal case). Table 2 Values of observer parameters used in simulations. Parameter

Value

l1 l2 l3 γ1 γ2 γ3 Ko

1  10 0.2 0.2 2  10 0.005 5  106

pċ =

7

6

4.9066 × 10−9

βa Va

̇ = qbit

(qbit − qc + qkick )

⎞ 1⎛ 2 ⎜ p − pc − fd qp2 − fa qbit − (ρa − ρd )gh TVD ⎟ ⎠ M⎝ p

Table 3 Controller tuning parameters. Unit

Parameter

Value

Unit

− − − − − −

λ1 λ2 m T

diag[0,1,0] 1000 4 6

− − −

m3/(s Pa)

(2)

2 pbh = pc + fa qbit + ρa gh TVD

qc = uc Cd A o

(4)

2(pc − po ) ρa

qkick = Kpi(pres − pbh ) (3)

s

(5)

(6)

The above system is considered our ‘real MPD system’ and can

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be represented in the following standard state space form

Table 4 State and input constraints. Parameter

Value

Unit

ppmin

8 × 105

Pa

ppmax

150 × 105

Pa

pcmin

8 × 105

Pa

pcmax

50 × 10 −0.002

Pa

max qbit

0.0283

ucmin

0

m3/s %

ucmax

100

%

min qbit

5

ẋ = fϕ (x, d, u)

(7)

y = gϕ(x, d)

(8)

where x = [pp pc qbit ] ; y = pbh ; u = [uc qp] and d = qkick . The ‘reservoir kick’ qkick is the only persistent disturbance d.

m3/s

3. Controller design The NMPC uses a model similar to the system model described in the previous section. However, the controller relies on observers

Fig. 3. Kick handling and pressure setpoint revision with constraint softening.

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to estimate the unmeasured state, qbit, and disturbance qkick. The prediction model is given by the following set of equations:

^ y^ = gaug (x^ , d )

⎞ β ⎛ ̇ p^p = d ⎜ qp − q^bit + l1(pp − p^p )⎟ Vd ⎝ ⎠

⎡ ⎤ ^ where x^ = ⎢ p^p p^c q^bit ⎥ ; y^ = p^bh ; u = [uc qp]and d = q^kick , and faug and ⎣ ⎦ gaug are the augmented state transition model and output model ^ both augmented with the estimated disturbance, d (i.e. ‘reservoir influx’, q^ ).

⎞ β ⎛ ̇ p^c = a ⎜ q^bit − qc + q^kick⎟ Va ⎝ ⎠

(

̇ q^bit = − γ1 pp − p^p

(9)

(10)

)

(11)

2 p^bh = p^c + fa q^bit + ρa gh TVD

(12)

Or compactly the prediction model is given by

^ ̇ x^ = faug (x, d , u)

(14)

kick

The ‘reservoir kick’ can be estimated by detecting difference between the in/out flow flux. We use a formulation based on the observer proposed by [9] to estimate these parameters. When there is no kick the in/out flow must be zero. As the reservoir fluid enters into the system, there would be a flow difference between the in/out flux, and that error can be used for estimating kick flow rate. In order to estimate q^ a new variable q1 is introduced given kick

(13)

by Eq. (15). The time derivative of q1 is given by Eq. (16) obtained using Eqs. (1) and (10).

Fig. 4. Kick handling and pressure setpoint revision under plant-model mismatch and measurement noise.

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q1 =

|qc − q¯bit | ≤ ϵ

Va V p + d pp βa c βd

(15)

q1̇ = qp + q^kick − qc

(16)

Using the derived measurement (q1), reservoir flow estimator is formed (given by Eqs. (17) and (18))and it is driven by the dynamics of topside pressure measurements pp and pc.

̇ q^1 = qp + q^res − qc + l2(q1 − q^1)

(17)

̇ q^kick = γ2(q1 − q^1)

(18)

The NMPC was designed with an objective to track the bottom hole pressure setpoint (i.e. r = pref ), and to manage ‘reservoir kick’ in case of influx of reservoir fluids in the system. We design our NMPC based on the results presented in [1] and [21]. The formulation implements an offset free design by introducing the concept target equilibrium. A target equilibrium is defined as the solution to the following equilibrium problem

(

)

^ x¯ = faug x¯ , u¯ , d (k ) ,

(19)

(27)

where q¯bit is the equilibrium state target for the bit flow rate. By adding this nonlinear constraint the discharge of reservoir fluid through the choke is constrained within a tunable threshold ϵ.

4. Results and discussions The NMPC controller was implemented in MATLAB on the simulated MPD system described in Section 2. The developed NMPC scheme uses sequential discretization technique for solving finite horizon optimization problem and active set method for computing optimal control actions. Sampling time of 6s and a prediction and a control horizon of 4 sampling intervals were chosen for all simulations. Longer prediction horizons did not improve performance and they can potentially increase numerical error [22]. MPD system parameters used for simulation are given in Table 1. The controller tuning parameters, and the state and input constraints used for all simulations are provided in Tables 3 and 4 respectively. Several scenarios were simulated to demonstrate controller performance and robustness under noise and system uncertainty. 4.1. Outlet flow constrained pressure regulation

(

)

^ r (k ) = gaug x¯ , d (k ) .

(20)

where x¯ and u¯ are the equilibrium target based on reference r(k) ^ and estimated disturbance d (k ). The targets are integrated into the standard MPC cost function as equilibrium constraints (given by Eqs. (24) and (25)). The cost function minimizes the error between the equilibrium state targets x¯ and the system states x(k); the equilibrium input target u¯ and the current input u(k). k+m

J = min u

∑

T

( x^(κ ) − x¯ ) λ ( x^(κ ) − x¯ ) + λ ( u(κ ) − u¯ ) ) 1

2

2

κ= k

subject to x^(k + T ) = x(k ) +

∫k

k+T

^ f (x(τ ), d (τ ), u(τ ))dτ ,

(21)

(22)

^ ^ d (k + T ) = d (k )

(23)

^ x¯ = faug (x¯ , u¯ , d (k ))

(24)

^ r (k ) = gaug (x¯ , d (k ))

(25)

x min ≤ x ≤ x max ; umin ≤ u ≤ umax

(26)

The initial bottom hole pressure setpoint is pref = 480bar . In this simulation mud is pumped at the rate of 1200lpm. A kick is encountered at 120 s , and that leads to violation of the flow constraint threshold of ϵ = 10, as shown in Fig. 2d. The controller responds by constricting the choke as shown in Fig. 2b 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 Fig. 2a. It is to be noted that the controller is not tracking the reservoir pressure which can be possible only by resorting to complete flow control, instead it gives up pressure tracking in order to satisfy flow constraints. In order to reset the pressure setpoint it is necessary to estimate the reservoir pressure. In the event of ‘reservoir kick’, because of the flow of the reservoir fluid into the system it is possible to estimate reservoir pressure under those conditions. The reservoir pressure is estimated by using a parameter update law and assuming a reservoir model. In order to estimate pres, Eq. (16) is

1230 qbit qbit estimate

1220

1210

where λ1 ∈ R and λ2 ∈ R are cost function weights and m is the prediction horizon. One of the important addition to the design of the controller is management of ‘reservoir kick’. The controller serves dual objectives, under normal condition the controller is configured to track predetermined bottom hole setpoint tightly, however as soon reservoir fluid enters into the system, and upsets the in/out flow balance, the controller gives up on the pressure setpoint and tries to contain the reservoir fluid influx into the drilling system within the specified threshold. Thus the controller acts in ‘pressure control mode’ under normal condition and switches to ‘flow control mode’ in case of ‘reservoir kick’. In order to achieve such behavior we include a nonlinear state constraint given by following equation

q bit [LPM]

3×3

1200

1190

1180

1170 0

50

100

150

200

250

300

350

400

450

Time [s] Fig. 5. Bit flow rate estimate under plant-model mismatch.

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modified using Eq. (6) resulting in Eq. (28) as in [9]. q2 is estimated by injecting error in the derived measurement q1 and given by Eq. (29), and the reservoir pressure pres is estimated using Eq. (30). Generally, only inaccurate estimates of the productivity index Kpi will be available and hence it is replaced by a tuning parameter Ko.

q2̇ = qp + Kpi(p^res − pbh ) − qc

(28)

̇ q^2 = qp + Ko(p^res − pbh ) − qc + l3(q1 − q^2)

(29)

̇ p^res = γ3(q1 − q^2)

(30)

Using the new reservoir pressure estimate, bottom hole pressure reference pref was revised to 475bar at 252s . Eventually due to overbalanced conditions the kick is completely rejected. The parameters used for tuning the observer is provided in Table 2.

7

4.2. Effect of constraint softening: outlet flow constrained pressure regulation Here we test the ability of the designed controller to track a bottom hole setpoint and to contain the reservoir influx within a threshold. In this simulation the initial bottom hole pressure setpoint is pref = 470bar and mud is pumped at the rate of 1500 lpm . A kick is encountered at 120 s , and that leads to violation of the flow constraint as shown in Fig. 3d, initially a threshold of ϵ = 10 lpm was chosen. The controller responds by closing down the choke as shown in Fig. 3b and that causes an increase in bottom hole pressure, pbh shown in Fig. 3a. Using the new reservoir estimate, pref is revised to 475bar at 252 s . The flow constraints are relaxed ( ϵ = 100) during the setpoint revision for faster tracking of setpoint as shown in Fig. 3d. Due to the constraint softening, the setpoint could be revised in less than 30 s .

Fig. 6. Bottom hole pressure tracking during pipe connection sequence.

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4.3. Robustness under plant-model mismatch A measurement noise of 0.1bar is added to pressure measurements and plant-model mismatch is introduced by augmenting state equations with random noise. The robustness of controller is tested by tracking a higher pressure setpoint with lower mud flow rate, forcing the controller to work at lower choke opening. The initial bottom hole pressure setpoint is pref = 480bar and mud is pumped at the rate of

1200 lpm leading to a lower choke opening. A kick is encountered at 120 s , leading to violation of flow constraint. Unlike the previous case the constraint does not settle at that threshold value due to noise. It can be seen in Fig. 4d the differential flow (qc − q^bit ) occasionally violates the threshold during kick handling but the controller acts to nudge it back to the acceptable region. Therefore, reasonable noise and plant uncertainty does not affect the flow constraint handling considerably. During normal setpoint tracking the noise and plant uncertainty does not affect the controller as shown in (i.e., 0 − 120 s ) Fig. 4d. With the help of the new reservoir pressure estimate, the setpoint is revised. Flow constraint is again relaxed during setpoint revision. Bottom hole pressure and drilling pressure window are shown in Fig. 4a. Choke opening is shown in Fig. 4b. Kick flow rate is shown in Fig. 4c. This NMPC being a state feedback controller the estimate of the bit flow rate is required and it is shown in Fig. 5. 4.4. Controller performance during pipe connection sequence We test the ability of the controller to track bottom hole pressure pbh set point pref during pipe connection sequence. In order to get the best performance during pipe connection the flow constraint must be switched off as the objective is solely to regulate BHP. Typically in pipe connection sequence the mud pump flow rate is ramped down from a nominal value to 0 lpm in approximately 60 s to 120 s . While performing pipe extension there will be no mud flow, then mud pump is ramped up from no flow to a nominal value. To test the NMPC we used a similar sequence as shown in Fig. 6d, the mud pump flow rate is ramped down at 60 s from 1500 lpm to 0 lpm in 60 s ; between 120 s and 180 s there is no flow in the system; starting at 180 s mud pump is ramped back to 1500 lpm in 60 s . The setpoint to be tracked is pref = 470bar . A measurement noise of 0.1bar is added to topside pressure measurements. The bottom hole pressure pbh is shown in Fig. 6a. The controller responds by closing down the choke in order to trap the pressure as shown in Fig. 6b. The initial overshoot is because of back flow, whenever there is a negative change in pump flow rate there will be momentary increase in pressure but eventually pressure will subside. In order to maintain a constant pbh, the choke pressure pc has to increase (shown in Fig. 6c) to compensate for the loss in frictional pressure drop.

5. Conclusions In this article a nonlinear model predictive controller for pressure regulation and reservoir flow containment was presented. The control objectives were achieved by penalizing the deviation of BHP from the setpoint and enforcing hard constraints on the in/out flow rate flux. The controller was designed as an output feedback controller which regulates bottom hole pressure by manipulating the choke opening. Equilibrium state references were generated by using the dynamic model of MPD and disturbance model was incorporated in the prediction model for offset free output tracking. It was shown that in the event of a kick, reservoir influx was contained by the controller within the

allowed threshold. It was also shown that the controller is able to perform well in presence of measurement and model uncertainty. The flow constraint is not severely affected due to measurement noise and it was shown constraint softening lead to considerably improved performance. The controller is also able to regulate bottom hole pressure during severe loss in mud flow rate which typically occurs during the pipe extension sequence. The controller is able to work under different mud flow rates and choke opening without any deterioration in performance. For further progressing the work we recommend treating frictional losses as uncertain parameters and utilizing parameter estimation for updating the model of the plant. Also we plan to test the proposed controller tested on an experimental set up which is under construction.

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