Pressure Regulation with Kick Attenuation in a Managed Pressure ...
Joint 48th IEEE Conference on Decision and Control and 28th Chinese Control Conference Shanghai, P.R. China, December 16-18, 2009
ThC10.3
Pressure Regulation with Kick Attenuation in a Managed Pressure Drilling System Jing Zhou, Øyvind Nistad Stamnes, Ole Morten Aamo and Glenn-Ole Kaasa Abstract— This paper presents a switched control scheme for regulation of the annular pressure in a well during drilling. The main feature is automatic kick attenuation while drilling reservoir sections. A simple mathematical model is presented that captures the dominant phenomena of the drilling system and forms the basis for observer and control design. A switched pressure control algorithm for feedback control of the choke valve opening and the back-pressure pump is proposed, and asymptotic stability properties are established. Kick attenuation is guaranteed for a set of reservoir models, including the most common ones. As part of the control scheme, novel observers for estimation of the flow rate through the bit and kick/loss detection are developed. Simulation results obtained with a high fidelity drilling simulator are presented to demonstrate the effectiveness of the proposed switched control scheme. Index Terms— Managed pressure drilling, adaptive observer, pressure control, kick detection.
I. I NTRODUCTION A. Automatic pressure control Controlling the annulus pressure in an oil well during drilling can be a challenging task, due to the very complex dynamics of the multiphase flow potentially consisting of drilling mud, oil, gas and cuttings. By allowing manipulation of the topside choke and pumps, Managed Pressure Drilling (MPD) provides a means of quickly affecting pressure to counteract disturbances, and several control schemes are found in the literature. Since downhole pressure measurements are at best unreliable due to slow sampling or transmission delays, the core ingredient in the control schemes is an estimator for the downhole pressure. In [1], [2] a sophisticated two-phase flow simulator is used to predict the downhole pressure based on the drift-flux formulation. In [3], an unscented Kalman filter exploiting downhole measurements is used to tune the predicted pressure loss due to friction in both the drill string and the annulus. To facilitate model-based observer and controller design, low order models for the downhole pressure have recently been developed [4]. In [5], [6], nonlinear adaptive observers were developed based on the simple dynamic model from [4], consisting of only three ordinary differential equations. In [7], nonlinear model predictive control in combination with an unscented Kalman filter was used to control the Jing Zhou is with Petroleum Department, International Research Institute of Stavanger, No-5008, Bergen, Norway. [email protected] Øyvind Nistad Stamnes and Ole Morten Aamo are with Department of Engineering Cybernetics, Norwegian University of Science and Technology, No-7491, Trondheim, Norway. [email protected],
bottomhole pressure based on a two-phase flow model in [1], [2]. There is a significant potential to improve existing control and estimation schemes. B. Kick detection and attenuation While drilling in the reservoir section, one may drill into a reservoir section with an unexpected high pore pressure, such as a high-pressure gas pocket. The resulting intrusion of formation fluids into the well bore is termed a kick. If the formation fluid is gas, it will expand on its way up the well bore, pushing mud out of the well. The average density of the fluid column therefore decreases, which in turn decreases the bottomhole pressure, causing the influx of gas to increase. If not counteracted, this unstable effect can escalate into a blow out causing severe financial losses, environmental contamination and potentially loss of human lives. It is of great importance to be able to detect a kick in its early phase, so that it can be attenuated in a controlled manner. In conventional well control procedures, the first response to detecting a kick is to shut down the mud pumps and isolate the well by closing the blow-out preventer (BOP). Stopping the mud pumps will remove the frictional pressure drop, decreasing the downhole pressure and increasing the influx. However, since the BOP is closed, the pressure will start to increase as more fluids enter the well until the downhole pressure balances the reservoir pressure. Several methods for kick detection have been presented, see [8], [9], [10], [11], [12], [13], [14], [15], however, there is a significant potential to improve existing control, estimation schemes, and kick detection methods. It is of particular interest to improve performance of the schemes during critical drilling operations such as pipe connections, as well as to add kick attenuation capabilities. The latter is the main topic of this paper. II. M ATHEMATICAL M ODELLING The modelling is based on dividing the system into two control volumes: one for the drill string and one for the annulus. Figure 1 shows a schematic of the two control volumes considered. The dynamics of the mud pump pressure pp in the drillstring and the choke pressure pc in the annulus, is given as p˙ p
=
p˙ c
=
[email protected] Glenn-Ole Kaasa is with StatoilHydro ASA, Research Centre, No-3908, Porsgrunn, Norway. [email protected]
978-1-4244-3872-3/09/$25.00 ©2009 IEEE
5586
βd (qpump − qbit ) , Vd βa (t) qbit + qres + u(t) + V˙ a , Va (t)
(1) (2)
ThC10.3 Mud Pump
p0 u p ,Zp
Control Choke
pc
q pump
qchoke
•
p0
pp qback
Flow rate estimation: .
uc , z c Backpressure Pump
qˆbit
=
pˆ˙ p
=
−γ1 (ˆ pp − pp ), βd qpump − qˆbit + l1 (pp − pˆp ) , Vd
ub , Zb lbit
Drillstring
lw
Annulus
• h lbit
h lw
pbit qbit
qres
Fig. 1.
A simplified schematical drawing of the drilling system. •
Both pp and pc are measured. The downhole pressure pbit , which is to be regulated and computed in two ways. Either by considering the pressure difference through the annulus, obtaining 2 , pbit (t) = pc + ρa (t)gh + Fa qbit
(4)
or the pressure difference through the drillstring, obtaining 2 . pbit (t) = pp + ρd gh − Fd qpump
(5)
ρa is the density of the fluid in the annulus, ρd is the density of the mud, g is the acceleration of gravity, h is the vertical depth at the bit, and Fa and Fd are friction coefficients for the annulus and drillstring, respectively. While ρd and Fd are considered known, ρa and Fa are unknown due to the complex nature of the fluid properties in the annulus caused by unknown reservoir influx, qres , as well as unknown properties of cuttings and the well wall. Therefore, equation (5) can be considered a direct measurement of pbit , while some parameter estimation scheme is needed to obtain pbit from equation (4). Equation (5) is not valid when mud circulation stops since the bit does not allow backflow from the annulus to the drillstring. III. C ONTROLLER DESIGN We propose the following switched control scheme for regulation of the annular pressure and automatic kick attenuation in the managed pressure drilling system (1)–(2), (4). • Physical measurements: pp (t), pc (t), and qpump (t). • Derived measurements: pbit (t)
=
p1 (t)
=
2 (t) + ρd gh, pp (t) − Fd qpump Vd Va pc (t) + pp (t). βa βd
t˙s
=
σ(t)
=
1, ts (0) = 0, (10) 1 ts ≥ 0 , (11) 0 ts < 0 −kP σ (t) (pbit (t) − pref ) − qˆbit , (12)
•
= =
γ2 (p1 − pˆ1 ), (13) qpump + qˆres + u + l2 (p1 − pˆ1 ), (14)
where γ2 and l2 are positive design constants. Reservoir pore pressure estimation: pˆ˙ res = γ3 (p1 − pˆ2 ) (15) ˙pˆ2 = qpump + k(ˆ pres − pbit ) + u + l3 (p1 − pˆ2 ) (16)
where γ3 , k, and l3 are positive design constants. Kick detection: When σ(t)=1 and qˆres ≥ q¯res , set ts = −Tw . q¯res > 0 and Tw > 0 are tuning constants. • Kick attenuation: While σ(t)=0, let pref (t) = p ˆres (t)+ ∆pref . ∆pref > 0 is a tuning constant. As the switching signal suggests, the controller has two modes of operation. In the normal mode, the switching signal is chosen as 1 and kP is selected as a suitable positive constant to regulate the annular pressure pbit to the desired set-point pref . If one unexpectedly drills into a pocket of gas that has a pressure above pref , a kick incident occurs and the controller is set in kick handling mode by toggling the switching signal to 0. In this mode of operation the controller reduces to a pure flow controller, giving the closed loop a self-regulating property with respect to attenuating the kick. Remark 1: While stability of the switched system is not analyzed, only the individual modes will be given in Sections IV-A and IV-B. The switching logic contains a positive constant dwell time, Tw , which prevents instability caused by frequent switching [16]. •
A. Flow rate estimation We assume that qbit can be treated as an unknown constant, which is reasonable during well drilling operation with nearly constant mud pump flow, and propose an observer based on (1) and a parameter update law for qˆbit given as (8)–(9). Defining the error variables p˜p = pp − pˆp and q˜bit = qbit − qˆbit , the error dynamics becomes
(6) (7)
where kP is a positive design constant. Reservoir influx estimation: qˆ˙res pˆ˙ 1
where βd and Vd are the bulk modulus and volume of the drill string, βa and Va are the bulk modulus and volume of the annulus, qpump is the mud pump rate, qbit is the flow rate through the drill bit, and qres is the reservoir influx, u is the control input, which is physically applied by distributing the desired u to the control choke and backpressure pump such that u(t) = qback (t) − qchoke (t). (3)
(9)
where γ1 and l1 are positive design constants. Control input:
u =
Drill Bit
(8)
Vd . p˜ βd p
=
−l1 p˜p − q˜bit ,
(17)
q˜bit
=
γ1 p˜p ,
(18)
.
which is linear with an exponentially stable origin.
5587
ThC10.3 Lemma 1: The adaptive observer (8)–(9) achieves exponential parameter convergence, that is limt→∞ qˆbit = qbit . B. Kick detection Consider the derived measurement (7). Using (1)–(2), the time derivative of p1 is p˙ 1
=
qpump + qres + u.
= −γ2 p˜1 , = q˜res − l2 p˜1 ,
(20) (21)
which is linear and time invariant. By inspecting eigenvalues, we get the following result. Lemma 2: Under the stated assumptions, the adaptive observer (13)–(14) achieves exponential parameter convergence, that is lim qˆres = qres . (22) t→∞
C. Reservoir pore pressure estimation Assume that the reservoir may be described by the reservoir model (23) qres (t) = k0 pres − pbit (t) , where k0 is the so-called production index, and pres is the reservoir pore pressure. Then the derivative of p1 in (19) is given as p˙ 1
= qpump + k0 (pres − pbit ) + u.
(24)
Consider the adaptive observer (15)–(16). We cannot use the production index k0 in the observer, since it is generally unknown. Forming the error dynamics by defining p˜2 = p1 − pˆ2 and p˜res = pres − pˆres , we have p˜˙ 2
=
−l3 p˜2 + k p˜res + (k0 − k) (pres − pbit ) (25)
p˜˙ res
=
−γ3 p˜2 ,
(26)
which is linear and time invariant, and driven by pres − pbit which converges to zero as analyzed in Section IV-B. Lemma 3: Under the stated assumptions, the adaptive observer (15)–(16) achieves parameter convergence, that is lim pˆres = pres .
t→∞
A. Normal operation with pressure control In normal operation, we have that σ (t) = 1, and assume that qres = 0, ρ˙ a = 0, βa and Va are constant. Under these simplifications, the dynamics of system (28) becomes
(19)
Assuming that qres is an unknown constant, Va is slowly varying, and βa , Va are known prior to kick, an observer is designed for kick detection based on the dynamics of p1 as given in (13)–(14). Forming the error dynamics by defining p˜1 = p1 − pˆ1 and q˜res = qres − qˆres , we have q˜˙res p˜˙ 1
where q˜bit = qbit − qˆbit .
e˙ bit
We analysis the stability properties of the closed-loop in two modes of operation: normal operation with pressure control and kick attenuation operation. With the controller (12), the dynamic model for pbit (t) is obtained by differentiating (4) with respect to time and inserting (2) to obtain βa (t) p˙ bit = − kP σ (t) (pbit (t) − pref ) + q˜bit + qres Va +ρ˙ a (t)gh, (28)
(29)
where ebit = pbit − pref is the regulation error. Clearly, (29) is a linear system with an exponentially stable origin, driven by the flow estimation error. Since the flow estimation error converges exponentially fast to zero, so does the regulation error pbit − pref . Thus, we have the following result. Lemma 4: Under the stated assumptions, the controller (12) and the adaptive observer (8)–(9) in closed loop with system (28) achieves exponential parameter convergence and set-point regulation, that is lim qˆbit = qbit , lim pbit = pref .
t→∞
t→∞
(30)
B. Kick attenuation When a kick is detected, we set σ (t) = 0 and the closed loop dynamics becomes p˙ bit =
βa (t) (qres + u ˜ (t)) + ρ˙ a (t) gh, Va
(31)
where u ˜ accounts for inaccuracies in the flow control. Assuming that in closed loop, the net mass flow into the annulus is proportional to qres (t) + u ˜ (t), a mass balance in the annulus can be written as m ˙ = ρ¯ (t) qres (t) + u ˜ (t) , (32)
where ρ¯ (t) is some effective density of the inflowing fluid. For slowly varying Va , we get Va ρ˙ a = ρ¯ (t) qres (t) + u ˜ (t) , (33)
and therefore (31) gives p˙ bit
=
where γ (t)
γ (t) qres (t) + u ˜ (t) , =
βa (t) + ρ¯ (t) gh . Va
(34)
(35)
Assume that the reservoir dynamics can be modelled as x˙ =
(27)
IV. C LOSED - LOOP A NALYSIS
βa (˜ qbit − kP ebit ) , Va
=
qres
=
f (x, ξ) ,
(36)
h (x, ξ) ,
(37)
with f (0, 0) = 0, h (0, 0) = 0, ξ denotes the pressure driving force ξ (t) = pres − pbit , and pres is the reservoir pressure and is assumed constant (slowly varying). The dynamics of ξ can be written as ξ˙ (t) = −p˙ bit (t) = −γ (t) (qres + u ˜ (t)) .
(38)
Suppose now that γ (t) is bounded below by γmin > 0. Assuming u ˜ (t) = 0, the dynamics of the system can be represented by a (negative) feedback connection consisting
5588
ThC10.3 uniformly asymptotically stable.
of the system ξ˙ (t) = −γmin h (x (t) , ξ (t)) + e1 (t) , x˙ (t) = f (x (t) , ξ (t)) , qres (t)
=
h (x (t) , ξ (t)) ,
(39) (40) (41)
and the memoryless function φ (t) = (γ (t) − γmin ) e2 (t) ,
(42)
by setting e1 (t) = −φ (t) and e2 (t) = qres (t). Clearly, (42) is passive since φ (t) e2 (t) = (γ (t) − γmin ) e22 (t) ≥ 0 (due to γmin < γ (t)). By Theorem 6.4 in [17], we have the following result. Lemma 5: Suppose γ (t) > γmin . If system (39)–(41) is strictly passive with storage function V , then the origin of the interconnection (39)–(42) is uniformly asymptotically stable. If V is radially unbounded, the origin is globally uniformly asymptotically stable. Suppose now that γ (t), in addition to the lower bound, is bounded above by γmax . The dynamics of the system can then be represented by a (negative) feedback connection consisting of the system ξ˙ (t) x˙ (t)
= −γmin h (x (t) , ξ (t)) + e1 (t) , = f (x (t) , ξ (t)) ,
y (t)
=
(43) (44)
(γmax − γmin ) h (x (t) , ξ (t)) + e1 (t) , (45)
and the memoryless function φ (t) =
γ (t) − γmin e2 (t) , γmax − γ (t)
(46)
by setting e1 (t) = −φ (t) and e2 (t) = y (t). Clearly, (46) γ(t)−γmin is passive since φ (t) e2 (t) = γmax −γ(t) e22 (t) ≥ 0. By Theorem 6.4 in [17], we have the following result. Lemma 6: Suppose γmin < γ (t) < γmax . If system (43)–(45) is strictly passive with storage function V , then the origin of the interconnection (43)–(46) is uniformly asymptotically stable. If V is radially unbounded, the origin is globally uniformly asymptotically stable. Corollary 1: Suppose γmin < γ (t) , and assume that the reservoir model (36)–(37) is linear and minimal with transfer function R (s) = qres (s)/ξ(s). If R (s) G1 (s) = s + γmin R (s)
Remark 2: Lemmas 5–6 and their corollaries imply in particular that qres → 0 and pbit → pres as t → ∞. The above results characterize a set of reservoir models for which kick attenuation is guaranteed under the proposed control. Next, we present two examples showing that common reservoir models are covered by the results. Example 1: Let the reservoir model be the common production index model, i.e. the simplest and probably the most commonly used reservoir inflow model, qres = k (pres − pbit ) .
(48)
is strictly positive real, then the origin of (43)–(46) is globally
(49)
In this case, R (s) = k, so G1 (s) = k/ (s + γmin k) , which is strictly positive real for k > 0. Thus, by Corollary 1, qres → 0 and pbit → pres . Example 2: Consider the following simple distributed parameter model for the pressure distribution in a thin reservoir layer filled with gas. The model is based on the ideal gas law and Fick’s law. ∂ 2 p¯ ∂ p¯ = α 2 , (x, t) ∈ (0, L) × [0, ∞), ∂t ∂x with boundary conditions
(50)
α ∂ p¯ (0, t) , (51) A ∂x and initial condition p¯ (x, 0) = p¯0 (x). The flow into the annulus is assumed to be given by the pressure difference between the reservoir pressure and the well pressure, p¯ (L, t) = pres , qres (t) =
qres (t) = k (¯ p (0, t) − pbit ) .
(52)
The coordinate change p (x, t) = p¯ (x, t) − pres , gives
(47)
is strictly positive real, then the origin of (39)–(42) is globally uniformly asymptotically stable. Proof: The system (39)–(41) becomes (47) in this case. Since G1 (s) is strictly positive real, it is strictly passive so the result follows from Lemma 5. Corollary 2: Suppose γmin < γ (t) < γmax , and assume that the reservoir model (36)–(37) is linear and minimal with transfer function R (s) = qres (s)/ξ(s). If s + γmax R (s) G2 (s) = s + γmin R (s)
Proof: The system (43)–(45) becomes (48) in this case. Since G2 (s) is strictly positive real, it is strictly passive so the result follows from Lemma 6.
∂p ∂t
=
p (0, t)
=
p (L, t)
=
∂2p , (x, t) ∈ (0, L) × [0, ∞), (53) ∂x2 α ∂p (0, t) − (pres − pbit ), t ∈ [0, ∞)(54) kA ∂x 0, t ∈ [0, ∞). (55) α
System (43)–(45) becomes ξ˙ (t) ∂p ∂t
=
−γmin k (p (0, t) + ξ) + e1 (t) , (56) 2 ∂ p = α 2, (57) ∂x α ∂p p (0, t) = (0, t) − ξ, p (L, t) = 0, (58) kA ∂x qres (t) = (γmax − γmin ) k (p (0, t) + ξ) + e1 (t) .(59) It can be shown that if γmax < 3γmin /2, the storage function Z kγmin (γmax − γmin ) L 2 1 V = p dx + k (γmax − γmin ) ξ 2 4A 2 0 (60)
5589
ThC10.3 satisfies qres (t) e1 (t)
≥
L
1 1 V˙ + p2 dx + 2 λL 0 λ +δ1 ξ 2 + δ2 p2 (0, t) , Z
Z
L 0
∂p ∂x
2
dx
(61)
for strictly positive constants λ, δ1 , and δ2 . Thus (56)–(59) is strictly passive, so qres → 0, pbit → pres by Lemma 6. V. S IMULATION R ESULTS
pressure regulation is on, while it is quickly attenuated when only flow control is applied. These effects are even more evident in Figures 3 which shows the same information for pref = 460(bar). A lower initial pressure set-point causes a more severe kick incident because the pressure difference between the set point and the pore pressure of the gas pocket is larger. The simulation results verify our theoretical results of Section IV-B that state that flow control will have a selfregulating capability for attenuation of a kick.
In this section we test our proposed switched control scheme on WeMod, which is a high fidelity drilling simulator developed by IRIS. The control signal relates to the choke opening, zc , and the flow rate of the backpressure pump, qback , through u = qback − qchoke (zc ), where the exact form of the valve equation qchoke (zc ) is generally unknown. We will therefore employ a fast flow controller to achieve the desired flow through the choke. Assuming that the backpressure pump is operated manually and that both qback and qchoke are measured, we apply the PI controller
zc (k)
=
∆t ec (k), zc (k − 1) + kcp ec (k) − ec (k − 1) + τci
set (k) = qback (k)−u(k). The physical parameters where qchoke used in the simulations are summarized in Table 1, and were found by fitting the model (1)–(2) to steady state data from WeMod. The controller design parameters are chosen as: kP = 3 × 10−4 , kcp = 0.5,τci = 5, q¯res = 1 × 10−4 (pref = 480)/5 × 10−4 (pref = 460), ∆pref = 4(bar), Tw = 5(min), l1 = 0.05, γ1 = 0.01, l2 = 0.2, γ2 = 0.01, l3 = 0.2, γ3 = 50. The controller is turned on at t = 30sec, with observer initials pˆp (0) = 0, pˆc (0) = 0, qˆbit (0) = 5/600, qˆres (0) = 0. Next, we present two simulation cases.
pbit Gas mas rate
5
0
30
35
0
5
10 5 0 −5
15 20 Time [min]
25
30
35
3
Estimated reservoir gas flow rate [m /s]
x 10
0
10
5
10
15 20 Time [min]
25
Pressure 30 Control 35 Flow Control
Fig. 2. Annular Bottomhole Pressure pbit , approx gas mass rate and estimated gas flow rate qˆres
pref = 460 Annulus Bottomhole Pressure [bar]
500 480 460 0
qˆres
Description Volume drill string (m3 ) Bulk modulus drill string (bar) Volume drill string (m3 ) Bulk modulus drill string (bar) kg Density drill string ( m 3) s2 Friction factor drill string ( bar m6 ) Vertical depth of bit (m) Pump flow rate ( liters min )
25
0.2
−4
Table 1: Parameter values with WeMod Value 26.7 13050 100 7317 0.0125 145620 3600 1000
15 20 Time [min] WeMod Gas mas rate [kg/s]
5
10
5
10
15 20 25 Time [min] WeMod Gas mas rate [kg/s]
30
35
30
35
2 1 0
0 −3
Para. Vd βd Va βa ρd Fd hbit qpump
10
0.4
qˆres
set (k) − qchoke (k) qchoke
0
pbit
=
Annulus Bottomhole Pressure [bar]
500 480 460
Gas mas rate
ec (k)
pref = 480
6 4 2 0 −2
x 10
0
5
15 20 25 Time [min] Estimated reservoir gas flow rate [m3/s]
10
15 20 Time [min]
25
Pressure 30 Control 35 Flow Control
Fig. 3. Annular Bottomhole Pressure pbit , Approx gas mass rate and estimated gas flow rate qˆres
B. Switched pressure regulation
A. Effect of pressure regulation after kick In this case, we demonstrate that pressure regulation needs to be switched off when a kick incident occurs. Figure 2 compares the annular pressure for continued pressure regulation and only flow control. The kick occurs at t = 15min, at which point continued pressure control clearly leads to increased flow through the choke, while switching the pressure regulation off allows the bottomhole pressure to increase. The effect of this on the reservoir influx is shown in Figure 2. The influx is allowed to continue when
When the pressure regulation is switched off, the downhole pressure will drift, as clearly shown in Figure 3. Thus, as soon as a new pressure setpoint that is higher than the pore pressure of the gas pocket can be computed, pressure regulation should be switched back on. In this simulation case, the switched pressure control is tested for pref = 460(bar). When the kick is detected, we switch to the flow control for a prescribed period of time Tw , and then we switch back to the pressure regulation, setting the new pressure set-point based on the pore pressure estimate. Figure 4 shows the flow rate through the drill bit, qbit , and its estimate, qˆbit , and the estimated reservoir pore pressure, pˆres . The estimated bit flow rate follows that of WeMod quite
5590
ThC10.3
q
lation results obtained with high fidelity drilling model are presented to demonstrate the effectiveness of the proposed switched control scheme.
Drill bit flow rate [m3/s]
0.0165 WeMod Estimated q
0.016
bit
[m3/s]
1
pref = 460 0.017
0.0155 0.015
ACKNOWLEDGEMENTS
pump
0
5
500
10 15 20 25 Time [min] Estimated Reservoir pore pressure [bar]
30
The main part of this work was conducted as a part a research project at Norwegian University of Science and Technology (NTNU), with a direct grant funded by StatoilHydro ASA. The work has also been supported by the MaxWells project at International Research Institute of Stavanger (IRIS). The MaxWells project is funded by grants from the Research Council of Norway and StatoilHydro ASA.
35
pˆres [bar]
490 480 470
Estimated p
460
Annular Pressure pbit
450
res
5
10
15
20 Time [min]
25
30
35
Drill flow rate qbit & qˆbit and estimated pore pressure pˆres
Fig. 4.
Annulus Bottomhole Pressure [bar]
500
R EFERENCES
490
pbit
480 470
WeMod Estimated
460 450
0
5
10
15 20 Time [min] Control Valve Openning
25
30
35
0
5
10
15 20 Time [min]
25
30
35
0.2
zc
0.15 0.1 0.05
Annular Bottomhole Pressure pbit & pˆbit and Choke Valve zc WeMod Gas mas rate [kg/s]
Gas mas rate
2 1.5 1 0.5 0
0
5
10
5
10
−3
qˆres [m3 /s]
6
x 10
15 20 25 Time [min] 3 Estimated gas flow rate [m /s]
30
35
30
35
4 2 0 −2
Fig. 6.
1
Fig. 5.
0
15 20 Time [min]
25
[1] G. Nygaard, Multivariable process control in high temperature and high pressure environment using non-intrusive multi sensor data fusio. PhD Thesis, Norwegian University of Science and Technology, 2006. [2] B. Fossli and S. Sangesland, “Managed pressure drilling for subsea applications; well control challenges in deep waters,” in SPE/IADC Underbalanced Technology Conference and Exhibition, SPE 91633, 2004. [3] J. E. Gravdal, R. Lorentzen, K. Fjelde, and E. Vefring, “Tuning of computer model parameters in managed-pressure drilling applications using an unscented kalman filter technique,” in SPE Annual Technical Conference and Exhibition, SPE 97028, 2005. [4] G.-O. kaasa, “A simple dynamic model of drilling for control,” Technical Report, StatoilHydro Resarch Centre, Porsgrunn, Norway, 2007. [5] J. Zhou, O. N. Stamnes, O. M. Aamo, and G.-O. Kaasa, “Adaptive output feedback control of a managed pressure drilling system,” in Proceedings of the 47th IEEE Conference on Decision and Control, Cancun, Mexico, 2008, pp. 3308–3313. [6] O. N. Stamnes, J. Zhou, G.-O. Kaasa, and O. M. Aamo, “Adaptive observer design for the bottomhole pressure of a managed pressure drilling system,” in Proceedings of the 47th IEEE Conference on Decision and Control, Cancun, Mexico, 2008, pp. 2961–2966. [7] G. Nygaard, L. Imsland, and E. A. Johannessen, “Using nmpc based on a low-order model for controlling pressure during oilwell drilling,” in 8th International Symposium on Dynamics and Control of Process Systems, June 6-8 2007. [8] J. J. Azar and G. R. Samuel, Drilling Engineering. Penwell Corporation, 2007. [9] J. Speers and G. Gehrig, “Delta flow: An accurate, reliable system for detecting kicks and loss of circulation during drilling,” SPE Drilling Engineering, pp. 359–363, 1987. [10] J. F. S. Stokka, J. O. Andersend and J. Welde, “Gas kick warner an early gas influx detection method,” in IADC/SPE Drilling Conference, Amsterdam, February 1993., Amsterdam, February 1993, p. SPE/IADC 25713. [11] B. W. Swanson, A. G. Gardner, N. P. Brown, and P. J. Murray, “Slimhole early kick detection by real-time drilling analysis,” SPE Drilling and Completion, pp. 27–32, March, 1997. [12] M. Doria and C. Morooka, “Kick detection in floating drilling rigs,” in SPE/IADC 39004. IADC/SPE Drilling Conference, Amsterdam, August, 1997. [13] C. L. Helio Santos and S. Shayegi, “Micro-flux control:the next generation in drilling process for ultra-deepwater,” in SPE Latin American and Caribbean Petroleum Engineering Conference, ser. SPE8113, Port-of-Spain, Trinildad, West Indies, April 2003. [14] H. Santos, E. Catak, J. Kinder, and P. Sonnemann, “Kick detection and control in oil-based mud: Real well-test results using microflux control equipment,” in SPE/IADC Drilling Conference, Proceedings, vol. 1. Society of Petroleum Engineers (SPE), Richardson, TX 75083-3836, United States, 2007, pp. 429–438. [15] D. Hargreaves, S. Jardine, and B. Jeffryes, “Early kick detection for deepwater drilling: New probabilistic methods applied in the field,” in SPE Annual Technical Conference and Exhibition, 2001. [16] D. Liberzon, Switching in Systems and Control. Birkhauser Boston, 2003. [17] H. K. Khalil, Nonlinear Systems. U.S.: 3rd ed, Prentice-Hall, 2002.
Approx gas mass rate in well and estimated gas flow rate qˆres
accurately. Figure 5 shows the annular downhole pressure pbit and its estimate pˆbit and the choke opening zc . The downhole pressure is estimated accurately. Figure 6 shows the reservoir gas mass flow and the estimated reservoir influx qˆres . The simulation results show that the estimators work well compared to WeMod, and that the proposed control algorithm is able to detect and attenuate the kick during drilling. VI. C ONCLUSIONS In this paper, a switched control scheme is presented for regulation of the annular pressure in the well during drilling and attenuation of kick incidents while drilling a reservoir. Novel observers for estimation the flow rate through the bit and kick/loss detection are developed. Global uniformly asymptotically stabilization of the dynamic system is established for pressure regulation and kick attenuation. Simu-
5591
ThC10.3
Pressure Regulation with Kick Attenuation in a Managed Pressure Drilling System Jing Zhou, Øyvind Nistad Stamnes, Ole Morten Aamo and Glenn-Ole Kaasa Abstract— This paper presents a switched control scheme for regulation of the annular pressure in a well during drilling. The main feature is automatic kick attenuation while drilling reservoir sections. A simple mathematical model is presented that captures the dominant phenomena of the drilling system and forms the basis for observer and control design. A switched pressure control algorithm for feedback control of the choke valve opening and the back-pressure pump is proposed, and asymptotic stability properties are established. Kick attenuation is guaranteed for a set of reservoir models, including the most common ones. As part of the control scheme, novel observers for estimation of the flow rate through the bit and kick/loss detection are developed. Simulation results obtained with a high fidelity drilling simulator are presented to demonstrate the effectiveness of the proposed switched control scheme. Index Terms— Managed pressure drilling, adaptive observer, pressure control, kick detection.
I. I NTRODUCTION A. Automatic pressure control Controlling the annulus pressure in an oil well during drilling can be a challenging task, due to the very complex dynamics of the multiphase flow potentially consisting of drilling mud, oil, gas and cuttings. By allowing manipulation of the topside choke and pumps, Managed Pressure Drilling (MPD) provides a means of quickly affecting pressure to counteract disturbances, and several control schemes are found in the literature. Since downhole pressure measurements are at best unreliable due to slow sampling or transmission delays, the core ingredient in the control schemes is an estimator for the downhole pressure. In [1], [2] a sophisticated two-phase flow simulator is used to predict the downhole pressure based on the drift-flux formulation. In [3], an unscented Kalman filter exploiting downhole measurements is used to tune the predicted pressure loss due to friction in both the drill string and the annulus. To facilitate model-based observer and controller design, low order models for the downhole pressure have recently been developed [4]. In [5], [6], nonlinear adaptive observers were developed based on the simple dynamic model from [4], consisting of only three ordinary differential equations. In [7], nonlinear model predictive control in combination with an unscented Kalman filter was used to control the Jing Zhou is with Petroleum Department, International Research Institute of Stavanger, No-5008, Bergen, Norway. [email protected] Øyvind Nistad Stamnes and Ole Morten Aamo are with Department of Engineering Cybernetics, Norwegian University of Science and Technology, No-7491, Trondheim, Norway. [email protected],
bottomhole pressure based on a two-phase flow model in [1], [2]. There is a significant potential to improve existing control and estimation schemes. B. Kick detection and attenuation While drilling in the reservoir section, one may drill into a reservoir section with an unexpected high pore pressure, such as a high-pressure gas pocket. The resulting intrusion of formation fluids into the well bore is termed a kick. If the formation fluid is gas, it will expand on its way up the well bore, pushing mud out of the well. The average density of the fluid column therefore decreases, which in turn decreases the bottomhole pressure, causing the influx of gas to increase. If not counteracted, this unstable effect can escalate into a blow out causing severe financial losses, environmental contamination and potentially loss of human lives. It is of great importance to be able to detect a kick in its early phase, so that it can be attenuated in a controlled manner. In conventional well control procedures, the first response to detecting a kick is to shut down the mud pumps and isolate the well by closing the blow-out preventer (BOP). Stopping the mud pumps will remove the frictional pressure drop, decreasing the downhole pressure and increasing the influx. However, since the BOP is closed, the pressure will start to increase as more fluids enter the well until the downhole pressure balances the reservoir pressure. Several methods for kick detection have been presented, see [8], [9], [10], [11], [12], [13], [14], [15], however, there is a significant potential to improve existing control, estimation schemes, and kick detection methods. It is of particular interest to improve performance of the schemes during critical drilling operations such as pipe connections, as well as to add kick attenuation capabilities. The latter is the main topic of this paper. II. M ATHEMATICAL M ODELLING The modelling is based on dividing the system into two control volumes: one for the drill string and one for the annulus. Figure 1 shows a schematic of the two control volumes considered. The dynamics of the mud pump pressure pp in the drillstring and the choke pressure pc in the annulus, is given as p˙ p
=
p˙ c
=
[email protected] Glenn-Ole Kaasa is with StatoilHydro ASA, Research Centre, No-3908, Porsgrunn, Norway. [email protected]
978-1-4244-3872-3/09/$25.00 ©2009 IEEE
5586
βd (qpump − qbit ) , Vd βa (t) qbit + qres + u(t) + V˙ a , Va (t)
(1) (2)
ThC10.3 Mud Pump
p0 u p ,Zp
Control Choke
pc
q pump
qchoke
•
p0
pp qback
Flow rate estimation: .
uc , z c Backpressure Pump
qˆbit
=
pˆ˙ p
=
−γ1 (ˆ pp − pp ), βd qpump − qˆbit + l1 (pp − pˆp ) , Vd
ub , Zb lbit
Drillstring
lw
Annulus
• h lbit
h lw
pbit qbit
qres
Fig. 1.
A simplified schematical drawing of the drilling system. •
Both pp and pc are measured. The downhole pressure pbit , which is to be regulated and computed in two ways. Either by considering the pressure difference through the annulus, obtaining 2 , pbit (t) = pc + ρa (t)gh + Fa qbit
(4)
or the pressure difference through the drillstring, obtaining 2 . pbit (t) = pp + ρd gh − Fd qpump
(5)
ρa is the density of the fluid in the annulus, ρd is the density of the mud, g is the acceleration of gravity, h is the vertical depth at the bit, and Fa and Fd are friction coefficients for the annulus and drillstring, respectively. While ρd and Fd are considered known, ρa and Fa are unknown due to the complex nature of the fluid properties in the annulus caused by unknown reservoir influx, qres , as well as unknown properties of cuttings and the well wall. Therefore, equation (5) can be considered a direct measurement of pbit , while some parameter estimation scheme is needed to obtain pbit from equation (4). Equation (5) is not valid when mud circulation stops since the bit does not allow backflow from the annulus to the drillstring. III. C ONTROLLER DESIGN We propose the following switched control scheme for regulation of the annular pressure and automatic kick attenuation in the managed pressure drilling system (1)–(2), (4). • Physical measurements: pp (t), pc (t), and qpump (t). • Derived measurements: pbit (t)
=
p1 (t)
=
2 (t) + ρd gh, pp (t) − Fd qpump Vd Va pc (t) + pp (t). βa βd
t˙s
=
σ(t)
=
1, ts (0) = 0, (10) 1 ts ≥ 0 , (11) 0 ts < 0 −kP σ (t) (pbit (t) − pref ) − qˆbit , (12)
•
= =
γ2 (p1 − pˆ1 ), (13) qpump + qˆres + u + l2 (p1 − pˆ1 ), (14)
where γ2 and l2 are positive design constants. Reservoir pore pressure estimation: pˆ˙ res = γ3 (p1 − pˆ2 ) (15) ˙pˆ2 = qpump + k(ˆ pres − pbit ) + u + l3 (p1 − pˆ2 ) (16)
where γ3 , k, and l3 are positive design constants. Kick detection: When σ(t)=1 and qˆres ≥ q¯res , set ts = −Tw . q¯res > 0 and Tw > 0 are tuning constants. • Kick attenuation: While σ(t)=0, let pref (t) = p ˆres (t)+ ∆pref . ∆pref > 0 is a tuning constant. As the switching signal suggests, the controller has two modes of operation. In the normal mode, the switching signal is chosen as 1 and kP is selected as a suitable positive constant to regulate the annular pressure pbit to the desired set-point pref . If one unexpectedly drills into a pocket of gas that has a pressure above pref , a kick incident occurs and the controller is set in kick handling mode by toggling the switching signal to 0. In this mode of operation the controller reduces to a pure flow controller, giving the closed loop a self-regulating property with respect to attenuating the kick. Remark 1: While stability of the switched system is not analyzed, only the individual modes will be given in Sections IV-A and IV-B. The switching logic contains a positive constant dwell time, Tw , which prevents instability caused by frequent switching [16]. •
A. Flow rate estimation We assume that qbit can be treated as an unknown constant, which is reasonable during well drilling operation with nearly constant mud pump flow, and propose an observer based on (1) and a parameter update law for qˆbit given as (8)–(9). Defining the error variables p˜p = pp − pˆp and q˜bit = qbit − qˆbit , the error dynamics becomes
(6) (7)
where kP is a positive design constant. Reservoir influx estimation: qˆ˙res pˆ˙ 1
where βd and Vd are the bulk modulus and volume of the drill string, βa and Va are the bulk modulus and volume of the annulus, qpump is the mud pump rate, qbit is the flow rate through the drill bit, and qres is the reservoir influx, u is the control input, which is physically applied by distributing the desired u to the control choke and backpressure pump such that u(t) = qback (t) − qchoke (t). (3)
(9)
where γ1 and l1 are positive design constants. Control input:
u =
Drill Bit
(8)
Vd . p˜ βd p
=
−l1 p˜p − q˜bit ,
(17)
q˜bit
=
γ1 p˜p ,
(18)
.
which is linear with an exponentially stable origin.
5587
ThC10.3 Lemma 1: The adaptive observer (8)–(9) achieves exponential parameter convergence, that is limt→∞ qˆbit = qbit . B. Kick detection Consider the derived measurement (7). Using (1)–(2), the time derivative of p1 is p˙ 1
=
qpump + qres + u.
= −γ2 p˜1 , = q˜res − l2 p˜1 ,
(20) (21)
which is linear and time invariant. By inspecting eigenvalues, we get the following result. Lemma 2: Under the stated assumptions, the adaptive observer (13)–(14) achieves exponential parameter convergence, that is lim qˆres = qres . (22) t→∞
C. Reservoir pore pressure estimation Assume that the reservoir may be described by the reservoir model (23) qres (t) = k0 pres − pbit (t) , where k0 is the so-called production index, and pres is the reservoir pore pressure. Then the derivative of p1 in (19) is given as p˙ 1
= qpump + k0 (pres − pbit ) + u.
(24)
Consider the adaptive observer (15)–(16). We cannot use the production index k0 in the observer, since it is generally unknown. Forming the error dynamics by defining p˜2 = p1 − pˆ2 and p˜res = pres − pˆres , we have p˜˙ 2
=
−l3 p˜2 + k p˜res + (k0 − k) (pres − pbit ) (25)
p˜˙ res
=
−γ3 p˜2 ,
(26)
which is linear and time invariant, and driven by pres − pbit which converges to zero as analyzed in Section IV-B. Lemma 3: Under the stated assumptions, the adaptive observer (15)–(16) achieves parameter convergence, that is lim pˆres = pres .
t→∞
A. Normal operation with pressure control In normal operation, we have that σ (t) = 1, and assume that qres = 0, ρ˙ a = 0, βa and Va are constant. Under these simplifications, the dynamics of system (28) becomes
(19)
Assuming that qres is an unknown constant, Va is slowly varying, and βa , Va are known prior to kick, an observer is designed for kick detection based on the dynamics of p1 as given in (13)–(14). Forming the error dynamics by defining p˜1 = p1 − pˆ1 and q˜res = qres − qˆres , we have q˜˙res p˜˙ 1
where q˜bit = qbit − qˆbit .
e˙ bit
We analysis the stability properties of the closed-loop in two modes of operation: normal operation with pressure control and kick attenuation operation. With the controller (12), the dynamic model for pbit (t) is obtained by differentiating (4) with respect to time and inserting (2) to obtain βa (t) p˙ bit = − kP σ (t) (pbit (t) − pref ) + q˜bit + qres Va +ρ˙ a (t)gh, (28)
(29)
where ebit = pbit − pref is the regulation error. Clearly, (29) is a linear system with an exponentially stable origin, driven by the flow estimation error. Since the flow estimation error converges exponentially fast to zero, so does the regulation error pbit − pref . Thus, we have the following result. Lemma 4: Under the stated assumptions, the controller (12) and the adaptive observer (8)–(9) in closed loop with system (28) achieves exponential parameter convergence and set-point regulation, that is lim qˆbit = qbit , lim pbit = pref .
t→∞
t→∞
(30)
B. Kick attenuation When a kick is detected, we set σ (t) = 0 and the closed loop dynamics becomes p˙ bit =
βa (t) (qres + u ˜ (t)) + ρ˙ a (t) gh, Va
(31)
where u ˜ accounts for inaccuracies in the flow control. Assuming that in closed loop, the net mass flow into the annulus is proportional to qres (t) + u ˜ (t), a mass balance in the annulus can be written as m ˙ = ρ¯ (t) qres (t) + u ˜ (t) , (32)
where ρ¯ (t) is some effective density of the inflowing fluid. For slowly varying Va , we get Va ρ˙ a = ρ¯ (t) qres (t) + u ˜ (t) , (33)
and therefore (31) gives p˙ bit
=
where γ (t)
γ (t) qres (t) + u ˜ (t) , =
βa (t) + ρ¯ (t) gh . Va
(34)
(35)
Assume that the reservoir dynamics can be modelled as x˙ =
(27)
IV. C LOSED - LOOP A NALYSIS
βa (˜ qbit − kP ebit ) , Va
=
qres
=
f (x, ξ) ,
(36)
h (x, ξ) ,
(37)
with f (0, 0) = 0, h (0, 0) = 0, ξ denotes the pressure driving force ξ (t) = pres − pbit , and pres is the reservoir pressure and is assumed constant (slowly varying). The dynamics of ξ can be written as ξ˙ (t) = −p˙ bit (t) = −γ (t) (qres + u ˜ (t)) .
(38)
Suppose now that γ (t) is bounded below by γmin > 0. Assuming u ˜ (t) = 0, the dynamics of the system can be represented by a (negative) feedback connection consisting
5588
ThC10.3 uniformly asymptotically stable.
of the system ξ˙ (t) = −γmin h (x (t) , ξ (t)) + e1 (t) , x˙ (t) = f (x (t) , ξ (t)) , qres (t)
=
h (x (t) , ξ (t)) ,
(39) (40) (41)
and the memoryless function φ (t) = (γ (t) − γmin ) e2 (t) ,
(42)
by setting e1 (t) = −φ (t) and e2 (t) = qres (t). Clearly, (42) is passive since φ (t) e2 (t) = (γ (t) − γmin ) e22 (t) ≥ 0 (due to γmin < γ (t)). By Theorem 6.4 in [17], we have the following result. Lemma 5: Suppose γ (t) > γmin . If system (39)–(41) is strictly passive with storage function V , then the origin of the interconnection (39)–(42) is uniformly asymptotically stable. If V is radially unbounded, the origin is globally uniformly asymptotically stable. Suppose now that γ (t), in addition to the lower bound, is bounded above by γmax . The dynamics of the system can then be represented by a (negative) feedback connection consisting of the system ξ˙ (t) x˙ (t)
= −γmin h (x (t) , ξ (t)) + e1 (t) , = f (x (t) , ξ (t)) ,
y (t)
=
(43) (44)
(γmax − γmin ) h (x (t) , ξ (t)) + e1 (t) , (45)
and the memoryless function φ (t) =
γ (t) − γmin e2 (t) , γmax − γ (t)
(46)
by setting e1 (t) = −φ (t) and e2 (t) = y (t). Clearly, (46) γ(t)−γmin is passive since φ (t) e2 (t) = γmax −γ(t) e22 (t) ≥ 0. By Theorem 6.4 in [17], we have the following result. Lemma 6: Suppose γmin < γ (t) < γmax . If system (43)–(45) is strictly passive with storage function V , then the origin of the interconnection (43)–(46) is uniformly asymptotically stable. If V is radially unbounded, the origin is globally uniformly asymptotically stable. Corollary 1: Suppose γmin < γ (t) , and assume that the reservoir model (36)–(37) is linear and minimal with transfer function R (s) = qres (s)/ξ(s). If R (s) G1 (s) = s + γmin R (s)
Remark 2: Lemmas 5–6 and their corollaries imply in particular that qres → 0 and pbit → pres as t → ∞. The above results characterize a set of reservoir models for which kick attenuation is guaranteed under the proposed control. Next, we present two examples showing that common reservoir models are covered by the results. Example 1: Let the reservoir model be the common production index model, i.e. the simplest and probably the most commonly used reservoir inflow model, qres = k (pres − pbit ) .
(48)
is strictly positive real, then the origin of (43)–(46) is globally
(49)
In this case, R (s) = k, so G1 (s) = k/ (s + γmin k) , which is strictly positive real for k > 0. Thus, by Corollary 1, qres → 0 and pbit → pres . Example 2: Consider the following simple distributed parameter model for the pressure distribution in a thin reservoir layer filled with gas. The model is based on the ideal gas law and Fick’s law. ∂ 2 p¯ ∂ p¯ = α 2 , (x, t) ∈ (0, L) × [0, ∞), ∂t ∂x with boundary conditions
(50)
α ∂ p¯ (0, t) , (51) A ∂x and initial condition p¯ (x, 0) = p¯0 (x). The flow into the annulus is assumed to be given by the pressure difference between the reservoir pressure and the well pressure, p¯ (L, t) = pres , qres (t) =
qres (t) = k (¯ p (0, t) − pbit ) .
(52)
The coordinate change p (x, t) = p¯ (x, t) − pres , gives
(47)
is strictly positive real, then the origin of (39)–(42) is globally uniformly asymptotically stable. Proof: The system (39)–(41) becomes (47) in this case. Since G1 (s) is strictly positive real, it is strictly passive so the result follows from Lemma 5. Corollary 2: Suppose γmin < γ (t) < γmax , and assume that the reservoir model (36)–(37) is linear and minimal with transfer function R (s) = qres (s)/ξ(s). If s + γmax R (s) G2 (s) = s + γmin R (s)
Proof: The system (43)–(45) becomes (48) in this case. Since G2 (s) is strictly positive real, it is strictly passive so the result follows from Lemma 6.
∂p ∂t
=
p (0, t)
=
p (L, t)
=
∂2p , (x, t) ∈ (0, L) × [0, ∞), (53) ∂x2 α ∂p (0, t) − (pres − pbit ), t ∈ [0, ∞)(54) kA ∂x 0, t ∈ [0, ∞). (55) α
System (43)–(45) becomes ξ˙ (t) ∂p ∂t
=
−γmin k (p (0, t) + ξ) + e1 (t) , (56) 2 ∂ p = α 2, (57) ∂x α ∂p p (0, t) = (0, t) − ξ, p (L, t) = 0, (58) kA ∂x qres (t) = (γmax − γmin ) k (p (0, t) + ξ) + e1 (t) .(59) It can be shown that if γmax < 3γmin /2, the storage function Z kγmin (γmax − γmin ) L 2 1 V = p dx + k (γmax − γmin ) ξ 2 4A 2 0 (60)
5589
ThC10.3 satisfies qres (t) e1 (t)
≥
L
1 1 V˙ + p2 dx + 2 λL 0 λ +δ1 ξ 2 + δ2 p2 (0, t) , Z
Z
L 0
∂p ∂x
2
dx
(61)
for strictly positive constants λ, δ1 , and δ2 . Thus (56)–(59) is strictly passive, so qres → 0, pbit → pres by Lemma 6. V. S IMULATION R ESULTS
pressure regulation is on, while it is quickly attenuated when only flow control is applied. These effects are even more evident in Figures 3 which shows the same information for pref = 460(bar). A lower initial pressure set-point causes a more severe kick incident because the pressure difference between the set point and the pore pressure of the gas pocket is larger. The simulation results verify our theoretical results of Section IV-B that state that flow control will have a selfregulating capability for attenuation of a kick.
In this section we test our proposed switched control scheme on WeMod, which is a high fidelity drilling simulator developed by IRIS. The control signal relates to the choke opening, zc , and the flow rate of the backpressure pump, qback , through u = qback − qchoke (zc ), where the exact form of the valve equation qchoke (zc ) is generally unknown. We will therefore employ a fast flow controller to achieve the desired flow through the choke. Assuming that the backpressure pump is operated manually and that both qback and qchoke are measured, we apply the PI controller
zc (k)
=
∆t ec (k), zc (k − 1) + kcp ec (k) − ec (k − 1) + τci
set (k) = qback (k)−u(k). The physical parameters where qchoke used in the simulations are summarized in Table 1, and were found by fitting the model (1)–(2) to steady state data from WeMod. The controller design parameters are chosen as: kP = 3 × 10−4 , kcp = 0.5,τci = 5, q¯res = 1 × 10−4 (pref = 480)/5 × 10−4 (pref = 460), ∆pref = 4(bar), Tw = 5(min), l1 = 0.05, γ1 = 0.01, l2 = 0.2, γ2 = 0.01, l3 = 0.2, γ3 = 50. The controller is turned on at t = 30sec, with observer initials pˆp (0) = 0, pˆc (0) = 0, qˆbit (0) = 5/600, qˆres (0) = 0. Next, we present two simulation cases.
pbit Gas mas rate
5
0
30
35
0
5
10 5 0 −5
15 20 Time [min]
25
30
35
3
Estimated reservoir gas flow rate [m /s]
x 10
0
10
5
10
15 20 Time [min]
25
Pressure 30 Control 35 Flow Control
Fig. 2. Annular Bottomhole Pressure pbit , approx gas mass rate and estimated gas flow rate qˆres
pref = 460 Annulus Bottomhole Pressure [bar]
500 480 460 0
qˆres
Description Volume drill string (m3 ) Bulk modulus drill string (bar) Volume drill string (m3 ) Bulk modulus drill string (bar) kg Density drill string ( m 3) s2 Friction factor drill string ( bar m6 ) Vertical depth of bit (m) Pump flow rate ( liters min )
25
0.2
−4
Table 1: Parameter values with WeMod Value 26.7 13050 100 7317 0.0125 145620 3600 1000
15 20 Time [min] WeMod Gas mas rate [kg/s]
5
10
5
10
15 20 25 Time [min] WeMod Gas mas rate [kg/s]
30
35
30
35
2 1 0
0 −3
Para. Vd βd Va βa ρd Fd hbit qpump
10
0.4
qˆres
set (k) − qchoke (k) qchoke
0
pbit
=
Annulus Bottomhole Pressure [bar]
500 480 460
Gas mas rate
ec (k)
pref = 480
6 4 2 0 −2
x 10
0
5
15 20 25 Time [min] Estimated reservoir gas flow rate [m3/s]
10
15 20 Time [min]
25
Pressure 30 Control 35 Flow Control
Fig. 3. Annular Bottomhole Pressure pbit , Approx gas mass rate and estimated gas flow rate qˆres
B. Switched pressure regulation
A. Effect of pressure regulation after kick In this case, we demonstrate that pressure regulation needs to be switched off when a kick incident occurs. Figure 2 compares the annular pressure for continued pressure regulation and only flow control. The kick occurs at t = 15min, at which point continued pressure control clearly leads to increased flow through the choke, while switching the pressure regulation off allows the bottomhole pressure to increase. The effect of this on the reservoir influx is shown in Figure 2. The influx is allowed to continue when
When the pressure regulation is switched off, the downhole pressure will drift, as clearly shown in Figure 3. Thus, as soon as a new pressure setpoint that is higher than the pore pressure of the gas pocket can be computed, pressure regulation should be switched back on. In this simulation case, the switched pressure control is tested for pref = 460(bar). When the kick is detected, we switch to the flow control for a prescribed period of time Tw , and then we switch back to the pressure regulation, setting the new pressure set-point based on the pore pressure estimate. Figure 4 shows the flow rate through the drill bit, qbit , and its estimate, qˆbit , and the estimated reservoir pore pressure, pˆres . The estimated bit flow rate follows that of WeMod quite
5590
ThC10.3
q
lation results obtained with high fidelity drilling model are presented to demonstrate the effectiveness of the proposed switched control scheme.
Drill bit flow rate [m3/s]
0.0165 WeMod Estimated q
0.016
bit
[m3/s]
1
pref = 460 0.017
0.0155 0.015
ACKNOWLEDGEMENTS
pump
0
5
500
10 15 20 25 Time [min] Estimated Reservoir pore pressure [bar]
30
The main part of this work was conducted as a part a research project at Norwegian University of Science and Technology (NTNU), with a direct grant funded by StatoilHydro ASA. The work has also been supported by the MaxWells project at International Research Institute of Stavanger (IRIS). The MaxWells project is funded by grants from the Research Council of Norway and StatoilHydro ASA.
35
pˆres [bar]
490 480 470
Estimated p
460
Annular Pressure pbit
450
res
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10
15
20 Time [min]
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Drill flow rate qbit & qˆbit and estimated pore pressure pˆres
Fig. 4.
Annulus Bottomhole Pressure [bar]
500
R EFERENCES
490
pbit
480 470
WeMod Estimated
460 450
0
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10
15 20 Time [min] Control Valve Openning
25
30
35
0
5
10
15 20 Time [min]
25
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0.2
zc
0.15 0.1 0.05
Annular Bottomhole Pressure pbit & pˆbit and Choke Valve zc WeMod Gas mas rate [kg/s]
Gas mas rate
2 1.5 1 0.5 0
0
5
10
5
10
−3
qˆres [m3 /s]
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x 10
15 20 25 Time [min] 3 Estimated gas flow rate [m /s]
30
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4 2 0 −2
Fig. 6.
1
Fig. 5.
0
15 20 Time [min]
25
[1] G. Nygaard, Multivariable process control in high temperature and high pressure environment using non-intrusive multi sensor data fusio. PhD Thesis, Norwegian University of Science and Technology, 2006. [2] B. Fossli and S. Sangesland, “Managed pressure drilling for subsea applications; well control challenges in deep waters,” in SPE/IADC Underbalanced Technology Conference and Exhibition, SPE 91633, 2004. [3] J. E. Gravdal, R. Lorentzen, K. Fjelde, and E. Vefring, “Tuning of computer model parameters in managed-pressure drilling applications using an unscented kalman filter technique,” in SPE Annual Technical Conference and Exhibition, SPE 97028, 2005. [4] G.-O. kaasa, “A simple dynamic model of drilling for control,” Technical Report, StatoilHydro Resarch Centre, Porsgrunn, Norway, 2007. [5] J. Zhou, O. N. Stamnes, O. M. Aamo, and G.-O. Kaasa, “Adaptive output feedback control of a managed pressure drilling system,” in Proceedings of the 47th IEEE Conference on Decision and Control, Cancun, Mexico, 2008, pp. 3308–3313. [6] O. N. Stamnes, J. Zhou, G.-O. Kaasa, and O. M. Aamo, “Adaptive observer design for the bottomhole pressure of a managed pressure drilling system,” in Proceedings of the 47th IEEE Conference on Decision and Control, Cancun, Mexico, 2008, pp. 2961–2966. [7] G. Nygaard, L. Imsland, and E. A. Johannessen, “Using nmpc based on a low-order model for controlling pressure during oilwell drilling,” in 8th International Symposium on Dynamics and Control of Process Systems, June 6-8 2007. [8] J. J. Azar and G. R. Samuel, Drilling Engineering. Penwell Corporation, 2007. [9] J. Speers and G. Gehrig, “Delta flow: An accurate, reliable system for detecting kicks and loss of circulation during drilling,” SPE Drilling Engineering, pp. 359–363, 1987. [10] J. F. S. Stokka, J. O. Andersend and J. Welde, “Gas kick warner an early gas influx detection method,” in IADC/SPE Drilling Conference, Amsterdam, February 1993., Amsterdam, February 1993, p. SPE/IADC 25713. [11] B. W. Swanson, A. G. Gardner, N. P. Brown, and P. J. Murray, “Slimhole early kick detection by real-time drilling analysis,” SPE Drilling and Completion, pp. 27–32, March, 1997. [12] M. Doria and C. Morooka, “Kick detection in floating drilling rigs,” in SPE/IADC 39004. IADC/SPE Drilling Conference, Amsterdam, August, 1997. [13] C. L. Helio Santos and S. Shayegi, “Micro-flux control:the next generation in drilling process for ultra-deepwater,” in SPE Latin American and Caribbean Petroleum Engineering Conference, ser. SPE8113, Port-of-Spain, Trinildad, West Indies, April 2003. [14] H. Santos, E. Catak, J. Kinder, and P. Sonnemann, “Kick detection and control in oil-based mud: Real well-test results using microflux control equipment,” in SPE/IADC Drilling Conference, Proceedings, vol. 1. Society of Petroleum Engineers (SPE), Richardson, TX 75083-3836, United States, 2007, pp. 429–438. [15] D. Hargreaves, S. Jardine, and B. Jeffryes, “Early kick detection for deepwater drilling: New probabilistic methods applied in the field,” in SPE Annual Technical Conference and Exhibition, 2001. [16] D. Liberzon, Switching in Systems and Control. Birkhauser Boston, 2003. [17] H. K. Khalil, Nonlinear Systems. U.S.: 3rd ed, Prentice-Hall, 2002.
Approx gas mass rate in well and estimated gas flow rate qˆres
accurately. Figure 5 shows the annular downhole pressure pbit and its estimate pˆbit and the choke opening zc . The downhole pressure is estimated accurately. Figure 6 shows the reservoir gas mass flow and the estimated reservoir influx qˆres . The simulation results show that the estimators work well compared to WeMod, and that the proposed control algorithm is able to detect and attenuate the kick during drilling. VI. C ONCLUSIONS In this paper, a switched control scheme is presented for regulation of the annular pressure in the well during drilling and attenuation of kick incidents while drilling a reservoir. Novel observers for estimation the flow rate through the bit and kick/loss detection are developed. Global uniformly asymptotically stabilization of the dynamic system is established for pressure regulation and kick attenuation. Simu-
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