2012 American Control Conference Fairmont Queen Elizabeth, Montréal, Canada June 27-June 29, 2012
Control of Heave-Induced Pressure Fluctuations in Managed Pressure Drilling Ingar Skyberg Landet, Alexey Pavlov, Ole Morten Aamo, Hessam Mahdianfar*
Abstract—Managed pressure drilling is an advanced pressure control method which is intended to meet increasingly high demands in drilling operations in the oil and gas industry. In this method, the circulating drilling fluid, which takes cuttings out of the well, is released at the surface through a controlled choke. This choke is used for active control of the fluid pressure in the well. The corresponding automatic control system keeps the pressure at the bottom of the well at a specified set-point despite various disturbances. One of such disturbances, vertical motion of the drill string, causes severe pressure fluctuations which need to be actively attenuated. In this paper we present two different disturbance rejection strategies based on discretized partial differential equations for the well hydraulic system. The performance of the controllers is shown through simulations both under idealized conditions as well as by simulations on a high fidelity drilling simulator.
I. I NTRODUCTION In drilling operations performed in the oil and gas industry, one of the most important challenges is to control the pressure of the drilling fluid, often called drilling mud. This drilling fluid is pumped at high pressure into the drill string at the top of the well, flows through the drill bit into the well, and continues up the well annulus carrying cuttings out to the surface (See Figure 1 taken from [1]). In addition to transporting cuttings, the mud is used to control the pressure in the annulus. This pressure control is crucial, as the pressure has to be above the pore pressure to prevent unwanted inflow of hydrocarbons from the surrounding formations into the well. Also, it should not exceed the strength of the rock of the surrounding formations to prevent the well from fracturing. Without proper control, these issues can result in time consuming, expensive and dangerous consequences including loss of mud, a stuck drill string, loss of well and catastrophes due to gas kicks and oil blowouts. Conventionally, pressure control is done by changing the mud density whenever the pressure needs to be changed, i.e. when the borehole reaches an area with different pore or fracture pressures, as specified by geophysical data. For example, to increase the down hole pressure, one could increase the density of the mud. However, this conventional method Ingar Skyberg Landet is a graduate student at the Department of Engineering Cybernetics at the Norwegian University of Science and Technology (NTNU). Email:
[email protected]. Alexey Pavlov is a principal researcher at Statoil Research Centre, Department of Intelligent Well Construction, Porsgrunn, 3910, Norway. Email:
[email protected]. Ole Morten Aamo is a professor at the Department of Engineering Cybernetics at NTNU. Email:
[email protected]. Hessam Mahdianfar is a PhD student at the Department of Engineering Cybernetics at NTNU. Email:
[email protected]
978-1-4577-1094-0/12/$26.00 ©2012 AACC
Figure 1. system.
Well and hardware configuration in a managed pressure drilling
provides only slow and inaccurate pressure control, which is insufficient for certain demanding drilling operations. A relatively new method that provides faster and more accurate pressure control is called Managed Pressure Drilling (MPD). With this method, the well annulus is sealed off and a control choke is installed to release the mud at the top of the well. By manipulating the choke opening, it is possible to significantly influence the annulus pressure. To maintain the controllability in case of a shut down of the main pump, a back pressure pump is installed at the control choke, see Figure 1. This hardware allows the active control of the well pressure, and an automatic control system for the choke makes it fast and accurate. This active control enables the drilling of wells that would not have been possible using the conventional methods. When designing such a control system for MPD, several disturbances affecting the pressure need to be accounted for. One such disturbance is the vertical motion of the drill string, which can cause significant fluctuations of the pressure in the well. This motion corresponds to several operational procedures, but in particular, this is an issue when drilling from a floating rig. In this case, waves will affect the entire rig causing it to move vertically, so called heave motion. Nor-
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mally, heave compensation systems on the rig will decouple this motion from the motion of the drill string, thus keeping the pressure unaffected. However, during connections, when the drill string is extended with another section, the heave compensation mechanisms are deactivated and the drill string is rigidly connected to the rig. This in turn results in the drill string acting like a piston down into the well, causing large pressure fluctuations. Design of an automatic control system for MPD with active compensation of such heave-induced pressure fluctuations is the main subject of this paper. Results on MPD control systems design and implementation can be found in a number of papers, where such aspects as bottom hole pressure observer design [2], pressure control [3], [4], gas kick attenuation [5], [6] and implementation aspects [7], [8], [9] are considered. Still, there are not a lot of published results on heave generated pressure fluctuations. An attempt at making an automatic control system for this scenario together with experimental results can be found in [1]. Here, a simple model of the well hydraulics developed in [10] was used for controller design. This worked well in simulations, but failed in full scale testing under realistic conditions. To the best of our knowledge, the problem of compensation of heave-induced pressure fluctuations in managed pressure drilling is still an open problem. The hypothesis made and defended in this paper is that these fluctuations can be compensated by designing an automatic control system based on a high order model of the well hydraulics and using advanced nonlinear control techniques. In this paper we firstly present a model of the heave-induced pressure fluctuations that would match experimental data with sufficient accuracy. Secondly, we present two control strategies for compensation or attenuation of the pressure oscillations. Thirdly, we evaluate the efficiency of these controllers in simulations against the developed model under idealistic conditions, as well as against a high fidelity simulator of the well hydraulics. The goal is to show the efficiency of the proposed controllers by using simulations and comparison to existing measurement data from earlier full scale tests presented in [1]. This paper is organized in the following way: Section 2 presents modeling of the well hydraulics in case of heaveinduced pressure fluctuations. Section 3 presents two control strategies for attenuation of the pressure fluctuations. Section 4 presents simulation results and Section 5 contains conclusions.
following subsections we will present the hydraulic model of the annulus and a model describing the oscillatory motion of the drill string. A. Hydraulic Transmission Line We will model the well annulus with the drilling mud as a hydraulic transmission line, and a common way of doing this is by the following partial differential equations (see e.g. [11], Chapter 11) @p @q = (1) @t A @x @q A @p F = + Agcos (↵ (x)) (2) @t ⇢ @x ⇢ Here, p(x, t) and q(x, t) are pressure and volumetric flow rate at location x ant time t, respectively. The bulk modulus of the mud is denoted by . A(x) is the cross section area, ⇢ is the (constant) mass density, F is the friction force per unit length, g is the gravitational constant and ↵ (x) is the angle between gravity and the positive flow direction at location x in the well. (See Figure 2.) We discretize (1) and (2) using a finite volumes method to get a set of ordinary differential equations describing the dynamics of the pressures and flows at different positions in the well. This is done by dividing the annulus into a number of control volumes, as shown in Figure 2, and integrating (1) and (2) over each control volume. This derivation is done similar to the one done in [10]. To incorporate the important pressure dynamics created by the drill string movement, there are two things to bear in mind: First, the volume of the annulus will continuously change by vd (t)Ad , where Ad is the drill string cross section area, due to the top of the drill string moving in and out of the well with velocity vd . Second, one also has to consider the fact that the cross section of the drill bit will in general be larger than that of the rest of the drill string. This will cause the generated flow to be “squeezed” in over a smaller cross section, increasing the velocity and thus also the friction around the drilling bit. (See also Figure 2.) Taking all these issues into account, the finite volume method results in the following set of equations p˙1
=
p˙ i
=
In this paper we consider the case of controlling the pressure at the bottom of the well under the disturbance from the vertical motion of the drill string following the heave motion of the floating rig. Since this situation occurs during drill string connections, the main pump is disconnected and there is no flow through the drill string (the drill bit is equipped with a one-way valve which prevents back flow from the annulus into the drill string). In this case the main dynamics of interest is the hydraulic dynamics in the annulus, which is affected by the vertical motion of the drill string, the controlled choke opening and the flow from the back pressure pump. In the
i
Ai l i N
( q1
v d Ad )
(qi
qi ) , i = 2, 3, .., N
1
1
(q(N 1) qc + qbpp ) (3) AN l N Ai Fi (qi ) Ai hi q˙i = (pi pi+1 ) Ai g , l i ⇢i ⇢i l i li i = 1, 2, ..N 1 (4) Here, the numbers 1...N refer to control volume number, with 1 being the lower most control volume representing the down hole pressure (p1 = pbit ), and N being the upper most volume representing the choke pressure (pN = pc ). The length of each control volume is denoted l, and the height difference is h. Notice that since the well may be non-vertical, l and h may in general differ from each other. To our disposal for control are the back pressure pump flow qbpp and the p˙ N
II. P ROBLEM D ESCRIPTION AND M ODELING
1
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Figure 2.
Control volumes of annulus hydraulic model.
Figure 4. Simulation results compared to measured data, pbit , different number of control volumes.
50 drill string annulus
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N=5 N=15 N=50 Measured
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12 25 20
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Figure 3.
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Friction losses from observed, full scale testing data
2 0
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time [s] choke flow qc . The flow from the back pressure pump qbpp cannot be changed fast enough to compensate for the heave- Figure 5. Simulation results compared to measured data, p , different c induced pressure fluctuations. Therefore, it is the choke flow number of control volumes. that is used primarily for control. It is modelled by an orifice equation: p qc = Kc pc p0 G (u) (5) friction model into a constant (or, more likely, slowly varying) Here, Kc is the choke constant corresponding to the area friction coefficient kf ric , giving the resulting friction force on of the choke and the density of the drilling fluid. p0 is the control volume i kf ric qi Fi (qi ) = (6) (atmospheric) pressure down stream the choke and G(u) is a Ai strictly increasing and invertible function relating the control signal to the actual choke opening, taking its values on the To see the validity of this model for the scenario of interest, consider Figure 4 and Figure 5. Here, we have compared interval [0, 1]. the pressure as estimated by our model for different number of control volumes to measured pressures taken from full B. Friction Model scale tests at Ullrigg1 . Most of the parameters, like the bulk To model the friction force acting on each control volume, modulus, mass density and geometry were either available we propose to use standard, Newtonian friction factor correla- from specifications of the well or measured during the tests. tions (see e.g. [12] for details). This is due to the observations The friction coefficients were identified from measurement from measurement data from full scale tests [1] that suggest data. The motion of the drill string can be seen in Figure 6. the friction force in the annulus is a linear function of the As we can see, the main dynamics of the pressures are flow rate, at least for the modest flow rates that can be reproduced. There might be some large, instantaneous errors, expected without externally forced flow (zero flow from the but these are quickly corrected and are mostly due to the main pump). This corresponds well to Newtonian friction fact that the models are not of high enough order to capture factors for laminar flow. See Figure 3 for the measured steady the steepest pressure fronts. The error in the amplitude of state friction drops. the oscillations is never more than a couple of bar. Also, Taking the special geometry of the annulus region into there seems to be little to be gained by changing the number account, it has been proposed [13] to use a modified Reynolds 1 Ullrigg is a full scale drilling test facility located at International Research number to calculate the friction factor. However, to keep the detail level manageable, we will in this paper lump the entire Institute of Stavanger (IRIS)
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for some small ✏.
18 17.5
III. D ISTURBANCE R EJECTION /ATTENUATION C ONTROLLERS
17
In this section we consider two controllers, nonlinear and linear, for solving the disturbance rejection/attenuation problem stated in the previous section. Both controllers have their roots in the output regulation theory, see e.g. [15].
xd [m]
16.5 16 15.5 15
A. Nonlinear Output Regulation Controller
14.5 14
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Figure 6.
Heave motion of drill string, relative position xd .
of control volumes within the interval used here. Notice that the presented modeling is an achievement by itself. So far there has been no other works on modeling of heaveinduced pressure fluctuations in MPD with comparison to experimental data resulting in such a good match. C. Heave Model We will model the heave motion of the drill string as an oscillatory motion driven by the waves on the open sea. To this end, we have consulted the JONSWAP spectrum (see e.g. [14] for details), which is the result of a large measurement project designed to find the frequency spectrum of the waves in the North Atlantic. It seems that the peak of this spectrum rad is at the frequency !waves = 2⇡ 12 s , so we choose to consider the speed of the drill string vd to be the output of the harmonic oscillator with this frequency w˙1 = w2 2 w˙2 = !waves w1 (7) vd = w2 If we wish to extend our wave model, it is quite simple to generate a signal vd spanning more of the frequency range of the JONSWAP spectrum by adding several such harmonic oscillators with different frequencies and amplitudes according to the spectrum. (See for instance [14]).
When we consider the total system model (3) with linear friction, it is clear that the only nonlinearity is in the choke flow. Thus it should be simple to find a feedback linearizing control and then apply some linear control strategy to the linearized input-output map. However, we will try to utilize the nonlinear output regulation theory on the full, nonlinear system. This theory considers a general nonlinear system of the form x˙ = f (x, u, w) (10) e = hr (x, w) (11) y = hm (x, w) (12) Here, x 2 Rn is the state vector, u 2 Rk is the control input, e 2 Rlr is the regulated output and y 2 Rlm is the measured output. The external input w 2 Rm is generated by a so called exosystem of the form w˙ = Sw
(13)
This exosystem is written here as a linear system, but could in general be nonlinear as well. It could include any number of harmonic disturbances of the form (7) and also a large range of reference signals for the regulated output. Here we clearly see that the problem presented in the previous section can be considered as a particular case of the output regulation problem. More specifically, the right-hand side in (10) can be chosen according to equations (3)-(6), the regulated output can be chosen equal to e = pbit pref and the measured output y can be chosen depending on the available measurements in the system. In this section, for simplicity, we assume that all states of the system and the exosystem are available for measurements2 . We will discuss in the simulations section how one can lift the requirement D. Problem Statement of measuring the system states. Under these assumptions, an With the available models, the control problem can be output regulation controller can be found in the following formalized as follows. Given the annulus hydraulic model (3) form: with the choke flow model (5) and friction model (6), find a u = c (w) + K (x ⇡ (w)) (14) controller such that Here K is an appropriate matrix which ensures quadratic |pbit (t) pref (t)| ! 0 (8) stability of a certain matrix function on the system matrices (see [15] for details), and c (w) , ⇡ (w) are the solutions to where pref is a set-point specified by the operator. the so-called regulator equations This control goal should be achieved regardless of the drill d ⇡(w(t)) = f (⇡ (w (t)) , c(w (t)), w (t)) (15) string motion vd generated by (7). In reality, (8) can be dt substituted by the requirement of practical regulation, i.e. that 0 = ⇡1 (w(t)) pref (16) after transients |pbit (t)
pref (t)| < ✏
2 In
fact, the vertical position and acceleration of the drill string can be
(9) available for measurement from exisiting heave compensation systems. 2273
It is a straightforward task to solve the regulator equations by back substitution from (16) for any number of control volumes. Although we assumed the heave model to consist of a single frequency (7), it can be extended to include an arbitrary number of harmonics. As follows from [15], the controller (14) with the computed ⇡(w) and c(w) guarantees boundedness of solutions of the closed-loop system and makes pbit pref bit ! 0 as t ! +1.
Output Regulation Internal Model pref
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p1 [bar]
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B. Linear Internal Model Controller Even though the drillstring may move following a simple sinusoidal wave of one frequency, the steady-state output of the controller presented in the previous subsection will contain a large number of harmonics due to the nonlinear nature of the control strategy (the steady-state control output c(w) is, in general, a nonlinear function of w(t)). In practice however, only some dominant harmonics can be the most important. In this section we propose a linear controller that is capable of generating the first harmonic of the wave frequency to compensate for the heave-induced pressure fluctuations. This is a controller based on the internal model principle, a popular linear control design tool for rejecting harmonic disturbances, see e.g. [16]. The idea is that if one can find the so-called disturbance generating polynomial and include it in a stable closed loop error feedback control system, the disturbance will vanish asymptotically (for the case of a linear system). In our case we apply this design strategy to a nonlinear system, which will not lead to exact disturbance rejection, but rather result in attenuation of the corresponding pressure fluctuations. If we consider the same harmonic disturbance model (7), one can readily find the disturbance generating polynomial as 2 (s) = s2 + !waves (17) A very simple way of creating such a controller is then Q1 Q2 s C (s) = Q0 + + 2 (18) 2 s s + !waves which includes a PI controller in addition to the internal model based on the disturbance generating polynomial. Here, Qi are appropriate constants and could for instance be tuned by classical frequency domain methods. Again, this method can easily be extended to any number of frequencies. The paper [17] proves passivity of the controller (18) and the closed loop stability of this controller and a passive nonlinear system. The hydraulic transmission line is quite clearly dissipative due to the friction force, but to ensure the correct control action applied to the choke, we must take one more step. Let us name the output signal from the controller (18) v(t) (that is, C(s) is the transfer function from the error signal e(t) to the signal v(t)) and suggest the following control signal to the choke: u(t) = G 1 ( v(t)) (19) The polarity v must be enforced to ensure that the transmission line is passive with input v and output p1 3 . Thus we 3 The proof of this is omitted for space requirements. It can be shown by the use of a positive real transfer function with qc as input and p1 as output, and then showing passivity from v to p1 by using the memoryless mapping from v to qc and assuming pc > p0 .
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Figure 7. Simulated performance of the proposed controllers under perfect information.
actually have a controller that is totally independent on the system parameters, and the only information needed is the frequency of the wave disturbance. Of course, the guarantees on rejection are weaker than for the output regulation controller, but this might be offset by the low demand for information about the system. IV. S IMULATION R ESULTS U NDER P ERFECT C ONDITIONS To demonstrate the effectiveness of the proposed controllers, we will show simulation results for the two controllers with the described disturbance (7) assuming we can measure both the full state of the model (3) and both states of the oscillator (7). The output regulation controller was developed for a model with 5 control volumes, but, in general, there is no limit to the number of control volumes one can use. Simulation results can be seen in Figure 7. In this scenario, the well model is also made up of 5 control volumes due to the assumption of perfect information. The harmonic disturbance is such that the relative drill string position is given by 2⇡ xd = cos 2⇡ 12 t [m], where 12 corresponds closely to the most dominant wave frequency in the North Atlantic, with reference to the JONSWAP spectrum. We can see that the output regulation controller achieves the promised asymptotic rejection of disturbances in the bottomhole pressure, while the linear internal model controller is stable and confines the pressure fluctuations to a quite narrow band around the pressure set point. In Figure 7, this is also compared to the uncontrolled case, where one would simply keep the choke at 50% open. V. S IMULATIONS W ITH A H IGH F IDELITY D RILLING S IMULATOR To assess the performance of the proposed controllers under more realistic and non-ideal conditions, we will present simulation results using a high fidelity drilling simulator program named “IRISDrill for MATLAB”, developed by IRIS. This simulator is based on a fine discretization of a PDE describing the hydraulics in the annulus similar to PDE (1)-(2) but taking more hydraulic effects into account and having a more advanced nonlinear friction model. Therefore it is much closer
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Parameter Al A l⇢
⇢g h kf ric
Value
VI. C ONCLUSION In this paper we have considered the problem of compensation of heave-induced pressure fluctuations in MPD operations. Firstly, we have presented a model accurately describing the dynamics of this process. The accuracy of the model has been verified against data from full scale tests. Secondly, based on this model we have presented two controllers for rejection/attenuation of these pressure fluctuations. These controllers have demonstrated good attenuation of the heaveinduced pressure fluctuations sufficient from a practical point of view. This conclusion has been supported by simulations with a commercial high fidelity drilling simulator. Future work on this subject should go towards reducing the need to measure some of the signals, specifically the down hole pressure pbit .
Pa 1.91 ⇤ 108 m 3 4 5.1477 ⇤ 10 8 m kg 6.1569 ⇤ 106 P a a 9.1030 ⇤ 106 sP m3
Table I PARAMETERS IDENTIFIED FROM THE IRISD RILL SIMULATOR . 275
Output Regulation Internal Model p ref
Uncontrolled case 270
p1 [bar]
265
260
VII. ACKNOWLEDGMENTS The authors wish to thank Statoil ASA for providing the test data from full scale MPD tests. Also, we wish to thank 250 Gerhard Nygaard and Lars Næsheim at the International 0 20 40 60 80 100 120 time [s] Research Institute of Stavanger for the access and guidance Figure 8. Simulated performance of the two controllers on the IRISDrill to the IRISDrill for MATLAB simulator. 255
simulator.
R EFERENCES
to reality than the simplified model (3) used for controller design purposes. The same controllers as in the last section were used, but this time the output regulation controller had to rely on an observer for the unmeasured states of the hydraulic model. The state estimation used in this case was a straight forward observer based on the hydraulic model of the form x ˆ˙ = f (ˆ x, u, w) + ko (y yˆ) (20) ⇥ ⇤T p 1 pN y = (21) where we assume the states of the harmonic disturbance to be known. The necessary parameters had to be identified through initial tests with the simulator program, and their values can be found in Table V. One of the main limitations in using a lower order model for controller design, is that its pressure changes are not as fast as those of the real system. This was compensated by increasing the bulk modulus in the model (3) to a larger value than what is likely the case. Still, the low order model is not quite as fast as the IRISDrill simulator. The simulator was run with the same disturbance as in the previous section, and simulation results can be seen in Figure 8. We can see that in the uncontrolled case the heave motion of the drill string results in pressure fluctuations of ±10 bar. The nonlinear output regulation controller has a very short transient period, and the resulting pressure oscillations are confined to ±2bar, while the linear internal model-based controller, after the transients, confines the oscillations to ±4bar. From a practical point of view, the attenuation of pressure fluctuations in both cases is significant, especially in the case of the nonlinear controller (a standard accuracy requirement for an automatic MPD control system is ±2.5 bar).
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