Adaptive Output Feedback Control of a Managed Pressure Drilling System

Transcription

1 Proceedings of the 47th IEEE Conference on Decision and Control Cancun, Mexico, Dec 9-, 8 WeB5 Adaptive Output Feedback Control of a Managed Pressure Drilling Syste Jing Zhou, Øyvind Nistad Stanes, Ole Morten Aao and Glenn-Ole Kaasa Abstract This paper presents a nonlinear observer-based control schee to stabilize the annular pressure profile throughout the well bore continuously while drilling A siple echanistic odel is presented that captures the doinant phenoena of the drilling syste and fors the basis for odelbased observer and control design A new nonlinear adaptive observer is developed for state estiations A new adaptive controller is designed to stabilize the annular pressure and achieve asyptotic tracking by applying feedback control of the choke valve opening and the ain pups Index Ters Drilling, nonlinear observer, adaptive control, stabilization, tracking I INTRODUCTION During well drilling, a drilling fluid ud is puped into the drill string topside and through the drill bit at the bottohole of the well [], [] The ud then transports cuttings in the annulus side of the well ie in the well bore outside the drill string up to the drill rig, where a choke valve and a backpressure pup are used to control the annular pressure A ore elaborate description of the drilling process is given in [3] The ain objective is to precisely control the annular pressure profile throughout the well bore continuously while drilling, ie to aintain the annular pressure in the well above the pore or collapse pressure and below the fracture or sticking pressure Usually, this aounts to stabilizing the downhole annular pressure at a critical depth within its argins, ie either at a particular depth where the pressure argins are sall, or at the drill bit where conditions are the ost uncertain Basically, two strategies for closed-loop control of the choke are used: indirect topside control and direct bottohole control Indirect topside control is to stabilize the bottohole pressure indirectly by applying feedback control to stabilize the topside annulus pressure instead, where the pressure setpoint corresponding to a desired bottohole pressure is calculated online using a steady-state odel This strategy is the ost coon and straightforward ainly due to the availability of high-frequency and robust topside pressure easureents Direct bottohole control is to stabilize the bottohole pressure at the critical depth at a desired setpoint directly Even though a bottohole easureent usually Jing Zhou, Øyvind Nistad Stanes and Ole Morten Aao are with Departent of Engineering Cybernetics, Norwegian University of Science and Technology, No-749, Trondhei, Norway jingzhou@itkntnuno, oyvista@itkntnuno, aao@ntnuno Glenn-Ole Kaasa is with StatoilHydro ASA, Research Centre, No-398, Porsgrunn, Norway GKAA@StatoilHydroco exists, an estiate of the pressure is needed between saples because the transfer rate of the easureent usually is slow, or for additional safety because the sensor itself ay be unreliable State-of-the-art solutions typically eploy conventional PI control applied to the choke, using one of the above strategies There are significant drawbacks with both strategies One is that the control syste based on conventional PI control will react slowly to fast pressure changes, which results fro oveents of the drill string Another drawback, is the uncertainty in the odelled bottohole pressure, due to uncertainties in the friction and ud copressibility paraeters in both the drill string and annulus There is significant potential to iprove existing algoriths, either the control law itself, or the observer used to estiate the critical downhole pressure Model-based control enables iproved copensation of pressure fluctuations during particularly critical drilling operations Also, by using odelbased copensation with adaptation of uncertain paraeters rather than integral action in the controller, one typically enable faster reaction to changes in setpoints and disturbances In the absence of full-state easureent, observer design is an effective way to control systes, such as in [4], [5], [6], [7] In this paper, we will address nonlinear adaptive observer-based control of a drilling syste in the presence of unknown paraeters and uneasured downhole pressure A siple dynaic odel developed for the observer and odel-based control design, is further developed to better describe the liquid fluid flow behavior A new nonlinear observer is developed by using Lyapunov techniques to estiate the uneasured downhole pressure Precise and robust estiation of the annular pressure during drilling allows for reduced pressure argins Online adaptation of unknown odel paraeters can extract ore inforation fro the syste The adaptive controller is designed by using Lyapunov techniques and paraeter estiation to stabilize the annular pressure at the desired setpoint The stabilization of the dynaic syste is deonstrated by the proposed control Itis shown that the proposed controller can guarantee asyptotic tracking Siulation results are presented to illustrate the effectiveness of the proposed control schee II MODEL In this section, we present a odel developed in [8], which captures the doinant phenoena of the drilling syste and fors the basis for odel-based observer and control design The odel only considers fluid phase flow and the well is divided into two separate copartents Figure shows /8/$5 8 IEEE 38

2 47th IEEE CDC, Cancun, Mexico, Dec 9-, 8 WeB5 Mud Control Pup q p c pup Choke p q choke p u, p p p p q back u, z c c Backpressure Pup and a 4 = ρ d ρ ag M, the syste 3 is rewritten as ṗ p = a q bit + b u p 4 Fig Drillstring h l bit lbit q bit Drill Bit l w p bit q res Annulus h l w u, A siplified scheatical drawing of the drilling syste b b q bit = a p p p c a 3 qbit θ q bit + q res + a 4 v 3 5 ṗ c = a 5 q bit + q res + u + v 6 v The ain variable of interest is the annular downhole pressure p bit given by p bit = p c + M a q bit + F a q bit + q res + ρ a gv 3 7 Our objective is to design a control law for the control input u which stabilizes p bit at the desired set-point p ref the two control volues considered, one control volue for the drill string and one for the annulus The volues are connected through the drill bit The detailed derivation of the odel is given in [8], where the dynaics of the drilling syste is described by ṗ p = a q bit + b u p q bit = a p p p c F d M q bit q bit F a M q bit + q res q bit + q res + ρ d ρ a g M v 3 ṗ c = a 5 v q bit + q res + u + v 3 The states p p and p c are the inlet ud pup pressure and outlet choke pressure bar, q bit is the flow rate through the drill bit 3 /s, u = q back q choke is control input, u p, q back and q choke are the flow rates through the ud pup, the back pressure pup and the choke valve, and q res is the reservoir influx, v, v and v 3 are the annulus volue, rate of change of the annulus volue and vertical depth of the bit, respectively The rest of the quantities in 3 are constant paraeters and can be explained as a = β d V d, a = M, a 5 = β a, b = β d V d, M = M a + M d ; V d : volue of the drill string; β d and β a : bulk odulus of the drill string and the annulus; M d and M a : density per eter of the drill string and the annulus; F d and F a : friction factor of the drill string and the annulus; ρ d and ρ a : density in the drill string and the annulus; g: gravity The paraeters are known except as stated in the following assuptions Assuption : The reservoir influx q res is an unknown constant Assuption : θ = Fa M > is an unknown constant Assuption 3: The flow rate q bit > and q bit + q res Using these assuptions and the notations a 3 = F d M A Observer III NONLINEAR ADAPTIVE OBSERVER Consider that p p and p c are easured and q bit is uneasured, where the paraeter θ and q res are unknown constants The following change of coordinates is defined ξ q bit + l p p, 8 where l is a positive constant This gives the dynaics ξ = q bit + l ṗ p = l a q bit + l b u p + a p p p c a 3 q bit θ q bit + q res + a 4 v 3 9 Defining Θ = [θ,θ,θ 3 ] T,θ = θ,θ = θq res, and θ 3 = θq res, the equation 9 can be written as ξ = l a q bit + l b u p + a p p p c a 3 q bit Θ T φq bit + a 4 v 3, where φq bit = [q bit,q bit,] T and Θ will be estiated in the observer design An adaptive observer for q bit is developed as follows ˆξ = l a ˆq bit + l b u p + a p p p c a 3ˆq bit ˆΘ T φˆq bit + a 4 v 3, ˆq bit = ˆξ l p p, where ˆΘ = [ˆθ, ˆθ, ˆθ 3 ] T and φˆq bit = [ˆq bit,ˆq bit,] T Firstly, we obtain the following error ters θ q bit ˆθ ˆq bit = θ q bit + θ ˆq bit 3 θ q bit ˆθ ˆq bit = θ q bit θ θ ˆq bit = θ q bit ˆq bit + θ ˆq bit 4 q bit ˆq bit = q bit + ˆq bit q bit, 5 where θ i = θ i ˆθ i and q bit = q bit ˆq bit, the error dynaics of ξ becoes ξ = l a q bit a 3 + θ q bit ˆq bit θ ˆq bit θ q bit θ ˆq bit θ 3, 6 39

3 47th IEEE CDC, Cancun, Mexico, Dec 9-, 8 WeB5 and since ξ = ξ ˆξ = q bit, we get ξ = l a ξ a3 + θ q bit + ˆq bit ξ θ ξ B Lyapunov Analysis Θ T φˆq bit 7 Consider the Lyapunov function U ξ, Θ = ξ + Θ T Γ Θ, 8 where Γ is the adaptation gain Using 7, the tie derivative of U is U = l a ξ a 3 + θ q bit + ˆq bit ξ θ ξ + Θ T Γ Θ Γφˆqbit ξ 9 This suggests that we should choose an adaptation law satisfying Θ = Γφˆqbit ξ, giving U = l a ξ a 3 + θ q bit + ˆq bit ξ θ q res ξ If q bit >, ˆq bit >, and q bit > q res, the tie-derivative of U satisfies U l a ξ Thus, ξ as t Consider the following dynaic equation ˆq bit = a 3ˆq bit ˆθ ˆq bit ˆθ ˆq bit ˆθ 3 +a p p p c + a 4 v 3 + l a q bit ˆq bit 3 It can be shown that ˆq bit > if ˆq bit >, and C Adaptation Law p p > p c a a 4 v 3 + a ax{ˆθ 3 } 4 Note that cannot be used for paraeter estiation because ξ is unavailable We introduce a new variable σ Θ + η p p, ˆξ, 5 where η is a vector function to be designed to assign σ a desired dynaics Differentiating σ with respect to tie, gives σ = η ṗ p + η ˆξ 6 p p ˆξ Let an estiate ˆθ of the paraeter vector be given by ˆσ = η a ˆq bit + b u p + η ˆξ, 7 p p ˆξ ˆΘ = ˆσ η p p, ˆξ 8 The resulting estiation error is then governed by Θ = Θ = σ η ˆΘ p p, ˆξ ˆσ η p p, ˆξ = a η p p ξ 9 Copared with, this suggests that η should be selected such that ˆq η bit a Γφˆq bit = Γ ˆq bit 3 p p A solution η can be found by integrating 3 η = 3l a ˆξ l p p 3 Γφˆξ l p p dp p = Γ a l a ˆξ l p p 3 a p p The resulting partial derivatives becoe η = Γ ˆξ l p p ˆξ l p p a p p, 3 η l a ˆξ l p p ˆξ = Γ l a ˆξ l p p 33 Lea : With the application of the adaptive nonlinear observer, and the paraeter update law 7 8, in the set { A = p p, ˆq bit,p c : q bit > q res, ˆq bit >, p p > p c a a 4 v 3 + a ax{ˆθ 3 }, }, 34 the signals ξ and Θ are bounded and the observation error converges to zero, ie, li [q bit ˆq bit ] = IV ADAPTIVE CONTROLLER DESIGN A Annular pressure profile The annular downhole pressure p bit, which fro 5 and 7, can be written as p bit = M a a p p + M d a p c M a a 3 q bit +M d Θ T φq bit + f, 35 where f = M a a 4 + ρ a gv 3 B Controller design Based on the observer, the syste 4 6 is rewritten as ṗ p = a ˆq bit + b u p a ξ ˆq bit = a p p p c a 3ˆq bit ˆΘ T φˆq bit 36 +a 4 v 3 + l a ξ 37 ṗ c = a 5 ˆq bit + q res + u + v + v ξ, 38 3

4 47th IEEE CDC, Cancun, Mexico, Dec 9-, 8 WeB5 and the output y p p,p c, ˆq bit = ˆp bit p p,p c, ˆq bit = M a a p p + M d a p c M a a 3ˆq bit +M d ˆΘT φˆq bit + f 39 Hence the syste with 39 has relative degree one Our objective is to design a control law for the control input u which stabilizes the annular down-hole pressure p bit at the desired set-point p ref Define the set point error as the following e = y p p,p c, ˆq bit p ref 4 Coputing the derivative of e fro 36 4 gives ė = ẏ p p,p c, ˆq bit = M da a 5 u + M a a ˆpp a ξ v + M da a 5 ˆpc + v ξ + q res + M dˆθˆq bit + M dˆθ M a a 3ˆq bit ˆq + l a ξ where M d φ T ˆq bit Γφˆq bit ξ + f, 4 ˆp p = a ˆq bit + b u p 4 ˆp c = ˆq bit + v 43 ˆq = a p p p c a 3ˆq bit ˆΘ T φˆq bit + a 4 v 3 44 Thus the control law is designed as u = ˆq bit + v + ˆq res + v C e M a a ˆpp f M d a a k B e 5 M dˆθˆq bit + M dˆθ M a a 3ˆq bit ˆq, 45 where C and k are positive constants, and B = M a a a a a 5 M d M a φ T ˆq bit Γφˆq bit v +l a M dˆθˆq bit + M dˆθ M a a 3ˆq bit 46 k > 4kl a, 47 and the paraeter adaptive law for q res is given by ˆq res = γm da a 5 v e, 48 where γ is a positive adaptation gain C Lyapunov analysis Consider the control Lyapunov function V = ku + e + γ q res 49 Using, 45 and 48, the derivative of V is Let V C e + B e ξ k B e kl a ξ γ q res ˆq res γ M da a 5 e v C e kl a ǫ l ξ kl a ǫ l ξ +B e ξ k B e 5 ǫ l = 4kk l a 5 Note that ǫ l, due to 47 and the fact that k,k,l and a are strictly positive constants We obtain that V C e kl a ǫ l ξ 4k ξ + B e ξ k B e C e kl a ǫ l ξ ξ k B e k C e kl a ǫ l ξ, 5 where Young s inequality was used Since V is positive definite and V is negative seidefinite in A, it proves that signals e, ξ, Θ, q res are bounded Fro the LaSalle- Yoshizawa Theore in [9], it further follows that e, ξ as t The signals y, ξ, ˆΘ, ˆq res are bounded and asyptotic tracking is achieved, ie, li [ˆp bit p ref ] = 53 Note that u in 45 includes the signals p p,p c and ˆq bit To ake the control input u bounded and the syste stable, we will ensure that p p and ˆq bit are bounded Towards that end, we consider the dynaics of p p and ˆq bit, given as ˆq bit = a p p a 3ˆq bit + M d y + M a a p p M a a 3ˆq bit + M d ˆΘT φˆq bit + f ˆΘ T φˆq bit + a 4 v 3 + l a ξ = p p M a 3ˆq bit y + l a ξ + dt M d M d M d 54 ṗ p = a q bit + b u p, 55 where dt = Maa4 M d + ρag M d + a 4 v 3 t is a bounded ter Assuing that p p is bounded, we obtain the boundedness of ˆq bit fro the boundedness of y, ξ,v 3,p p in 54 and ˆq bit > Therefore, u is bounded fro 45 Now we have a conclusion that the signals in the closed-loop syste can be shown to be bounded, as stated in the following theore Reark : In practice the pup will not be able to drive the pressure to infinity Thus, we can assue that the pup speed signal will be such that p p stays bounded Theore : With the application of the adaptive nonlinear observer, the control law 45, and the paraeter update law 7 8, in the set { A = p p, ˆq bit,p c : q bit > q res, ˆq bit >, p p > p c a 4 v 3 + } ax{ˆθ 3 }, 56 a a 3

5 47th IEEE CDC, Cancun, Mexico, Dec 9-, 8 WeB5 all signals y,p p, ˆq bit, ˆΘ, and ˆq res are bounded and asyptotic tracking is achieved given as D Tracking perforance li [ˆp bit p ref ] = 57 We now introduce the following useful Lea and proposition as in [], [], [] Lea : Micaelli and Sason 993 [], Lea Let f : R + R be any differentiable function If ft converges to zero as t and its derivative satisfies ft = f t + ηt t where f is a uniforly continuous function and ηt tends to zero as t, then ft and f t tend to zero as t Proposition : Suppose f is differentiable on a,b and f is bounded on a,b Then f t is uniforly continuous on a,b Considering the error dynaics 7, we have fro Lea and Theore that li ξ = 58 li l a ξ a3 + θ q bit + ˆq bit ξ θ ξ = 59 Define f t = Θ T φˆq bit So f is differentiable and f t = φ T Γφ ξ + ˆq bit θt + θ a p p p c a 3ˆq bit ˆΘ T φˆq bit + a 4 v 3 + l a ξ 6 Fro the boundedness of p p,p c, ˆq bit, ξ, Θ,v 3 in A, we have f t is bounded in A Therefore f t is uniforly continuous in A and li ΘT φˆq bit = by using Lea and Proposition Fro 35 and 39, we have p bit ˆp bit = M a a 3 q bit ˆq bit +M d Θ T φq bit ˆΘ T φˆq bit = M a a 3 qbit ˆq bit + M ΘT d φˆq bit q +M d Θ T bit ˆq bit q bit ˆq bit 6 Since li [q bit ˆq bit ] =, li [qbit ˆq bit ] =, and li ΘT φˆq bit = in A, it follows that li [p bit ˆp bit ] =, which in conjunction with Theore gives Theore : With the application of the adaptive nonlinear observer, the control law 45, and the paraeter update laws 7 8, in the set { A = p p, ˆq bit,p c : q bit > q res, ˆq bit >, p p > p c a 4 v 3 + } ax{ˆθ 3 }, 6 a a all signals y,p p, ˆq bit,p c, ˆΘ, and ˆq res are bounded and asyptotic tracking is achieved given as li [p bit p ref ] = 63 V SIMULATION RESULTS In this section we test our proposed controller on odel 3 When doing so, we need to distribute the control signal u fro 45 to the two physical actuation devices, the backpressure pup and the choke opening, according to u = q back q choke We assue that the backpressure pup is set at a constant rate, while the choke opening is related to the choke flow by the standard valve equation q choke = K c ρ a p c p z c 64 For siulation studies, the following values are selected for the syste: β a = β d = 4, V d = 83, V a = 96, M d = 57, M a = 7, F d = 65, F a = 8, ρ a = ρ d = 5 5, h bit =, g = 98, p =, K c = 466, q res =, Va =, q pup =, q back = 3 The paraeters F a and q res need not be known in the controller design The design objective is to stabilize p bit at the desired set point p ref = 3bar With the proposed adaptive observer and controller, we take the following set of design paraeters: l = 5,C =,k =,Γ = diag{695 3,6,},γ = 5 The initials are set as p p =,p c = 7,q bit = 4, ˆq bit =, ˆq res = q res and ˆF a = 6F a, respectively Figure shows the annular downhole pressure p bit, ˆp bit and p ref and the choke opening z c Figure 3 shows the paraeter estiations Clearly, the annular pressure asyptotically tracks the pressure reference and paraeter convergence is achieved The proposed nonlinear observer-based controller has been tested on WeMod, a siulator based on a distributed paraeter odel of the fluid dynaics in the well [3] The odel 3 was fitted to steady state data resulting in the paraeter values in Table I and the initials V a = and V a = We turn on the observer at t = 5in with initials ˆq bit = /6, ˆΘ = [8 Fa /M;;] and design paraeters l = 4 and Γ = diag,, The controller starts at t = in with design paraeters C = 5, k = 3 7 and γ = 6 The set point is changed fro 34 to 3bar at t = 5in The pup is changed fro 5 to 5l/in at t = 3in Fro t = 5in to t = 5in3s approxiately 6 of the drill pipe is pulled out of the bore hole Figure 4 shows the pup pressure p p, the choke pressure p c and the pup flow u p, the annular downhole pressure p bit and ˆp bit, the flow through the bit q bit and ˆq bit, and the actual and desired choke opening z c Fro Figure 4 we can see that the controller is able to suppress the changes in downhole pressure with axiu deviation fro the desired set point 3bar of approxiately 5bar The desired choke opening z c is calculated fro 64, but there is a difference between the desired choke opening and the actual choke opening due to the additional actuator dynaics in WeMod The siulation results show that the presented odel can fit the data and the annular pressure can track the pressure reference well with the proposed controller 3

6 47th IEEE CDC, Cancun, Mexico, Dec 9-, 8 WeB5 Annular pressure p bit Choke control input z c [%] t [s] t [s] Fig Siulations of observed-based stabilization with adaptive controller p bit solid, ˆp bit dashed-dot and p ref dashed Estiation of θ Estiation of θ Estiation of θ 3 Fig tsec 5 3 x tsec 4 x tsec Paraeter estiations with adaptive nonlinear observer TABLE I PARAMETER VALUES WITH WEMOD Paraeter Value Description V d 67 Volue drill string 3 β d 3 Bulk odulus drill string bar β a 73 Bulk odulus drill string bar ρ a 5 Density annulus 3 ρ d 5 Density drill string 3 F d bar s 7 Friction factor drill string 6 bar s F a 6 Friction factor annulus 6 M a 6 Density per eter of annulus 4 M d 6 Density per eter of drill string 4 h bit Vertical depth of bit VI CONCLUSIONS During well drilling, the annular downhole pressure should be precisely controlled throughout the well bore continuously while drilling A choke valve and a back pressure pup is used to control the annular pressure This paper presents a nonlinear adaptive observer control applied to stabilize the annular pressure A siple odel is used to capture the doinant phenoena of the drilling syste and for the observer and odel-based control design A new nonlinear adaptive observer-based control is developed to stabilize the annular pressure and achieve asyptotic tracking The 5 5 p p[barg] p c[barg] u p[ l/in] p bit[barg] 3 ˆp bit[barg] q bit[ l/in] ˆq bit[ l/in] Fig 4 Actual z c Desired z c in Siulations of observed-based control with WeMod siulation results are presented to illustrate the effectiveness of the proposed control schee REFERENCES [] G Nygaard, Multivariable process control in high teperature and high pressure environent using non-intrusive ulti sensor data fusio PhD Thesis, Norwegian University of Science and Technology, 6 [] G Nygaard and G Nævdal, Modelling two-phase flow for control design in oil well drilling, in Proceedings of the 5 IEEE Conference on Control Applications, Toronto, Canada,, 5, pp [3] Ø N Stanes, J Zhou, G-O Kaasa, and O M Aao, Adaptive observer design for the bottohole pressure of a anaged pressure drilling syste, in Proceedings of the 47th IEEE Conference on Decision and Control, 8 [4] Y Tan, I Kanellakopoulos, and Z P Jiang, Nonlinear observer/controller design for a class of nonlinear systes, in Proceedings of the 37th IEEE Conference on Decision and Control, Florida US, 998, pp [5] B Yao and L Xu, Observer-based adaptive robust control of a class of nonlinear systes with dynaic uncertainties, International Journal of Robust and Nonlinear Control, vol, pp , [6] J Zhou, G-O Kaasa, and O M Aao, Nonlinear adaptive observer control for a riser slugging syste in unstable wells, in Proceedings of Aerican Control Conference, USA, 8, pp [7] G-O Kaasa, V Alstad, J Zhou, and O M Aao, Attenuation of slugging in unstable oil wells by nonlinear control, in Proceedings of the 7th IFAC World Congress, Seoul, Korea, 8, pp [8] G-O Kaasa, A Siple Dynaic Model of Drilling for Control, Technical Report, StatoilHydro Resarch Centre Porsgrunn, 7 [9] H K Khalil, Nonlinear Systes US: 3rd ed, Prentice-Hall, [] A Micaelli and C Sason, Trajectory tracking for unicycle-type and two-steering-wheels obile robots, Rapport de recherche 97, INRIA, Sophia-Antipolis, France, 993 [] A A J Lefeber, Tracking control of nonlinear echanical systes PhD Thesis, Proefschrift Universiteit Twente, Enschede, [] E Panteley, M A Lorła, and A Teel, Relaxed persistency of excitation for unifor asyptotic stability, IEEE Transactiosn on Autoatic Control, vol 46, pp , [3] G Nygaard and J E Gravdal, Weod for atlab user s guide, International Research Institute of Stavanger, Bergen, Norway, Tech Rep Publ nr 7/34, 7 33