Modelling and Identification of a Sucker-Rod Pumping ... · PDF fileModelling and...

6
Modelling and Identification of a Sucker-Rod Pumping System of Oil Wells LUIZ H. S. TORRES, LEIZER SCHNITMAN Universidade Federal da Bahia Centro de Capacitac ¸˜ ao Tecnol´ ogica em Automac ¸˜ ao Industrial (CTAI) Rua Aristides Novis, n 02, Escola Polit´ ecnica segundo andar, 40.210-630, Salvador, Bahia Brazil [email protected], [email protected] Abstract: The aim of this paper is present an application of modelling and identification techniques in a sucker-rod pump system. The results show a dynamic system model and its relationship with the real process variables in this kind of artificial lift oil method. The model obtained from experiments with real data is representative and can be used for applications, such as for instance, a control system design. Key–Words: Modelling and identification, Polynomial models, Oil industry, Process automation, Production mon- itoring, Sucker-Rod pump systems. 1 Introduction The sucker-rod pump system is the artificial lift method most used in the current on-shore petroleum industry due to the simplicity of its equipments and facilities [1]. This method is also considered as the first technique used to lift oil up from wells. Studies have shown that its popularity is related to low cost of investments and maintenance, deep and outflow flex- ibility, good energy efficiency and the possibility for operating in different fluid compositions and viscosi- ties in a wide range of temperatures [2]. Although this lift method is already well-known and widely used, there are still some circumstances in which improvements in the operational conditions are still possible, especially when dealing with produc- tion control strategies of the pump unit for increasing the system productivity. The development of low cost sensors turned possible the measurement of bottom- hole variables that assists the production monitoring, application of new control strategies, and enhance- ment of the process automation [3, 4, 5]. Simulation softwares have assisted the studies about the sucker-rod pump system in modelling and design of control systems, especially into controlling the dynamic fluid level of the annular well. The val- idation of these softwares increases the possibilities in a successful implementation of a control system, for instance. However, such simulators, speaking gen- erally, are limited to theoretical models (mostly phe- nomenological) and therefore in lacking of evidences in situations close to the real production fields. On the other hand, a modelled system mathemat- ically does not always need be represented by phe- nomenological models by using physical laws (ma- terial balance, energy, etc.). The inputs and outputs of the system may be grouped by some interdepen- dence, in order to extract a relation of cause and effect in phenomena that occur with the elements of a set of sampled data. The modelled system does not need be limited to a physical modelling (white-box), but also apply to systems with observable data (black-box), or systems that include some additional information (gray-box) [6]. In recent years the application of empirical modelling techniques or, as is also known, modelling and identification techniques of dynamic systems (e.g.: ARX, ARMAX, NARMAX, etc.) in the oil and gas industry, has experienced a significant growth [7, 8, 9, 10, 11, 12, 13]. The white-box models in this application may be- come arduous and hard to be obtained since the pro- cesses involved in the oil production may not be well understood (e.g: synergy among various dynamic events and the bottom of the well and the surface). New research, therefore, by using modelling and iden- tification techniques, especially in artificial lift sys- tems, may contribute to a better understanding of dy- namic models in the analysis and design of automatic control, for instance. Recent Advances in Mathematical Methods, Intelligent Systems and Materials ISBN: 978-1-61804-168-5 19

Transcript of Modelling and Identification of a Sucker-Rod Pumping ... · PDF fileModelling and...

Modelling and Identification of a Sucker-Rod Pumping System ofOil Wells

LUIZ H. S. TORRES, LEIZER SCHNITMANUniversidade Federal da Bahia

Centro de Capacitacao Tecnologicaem Automacao Industrial (CTAI)

Rua Aristides Novis, n 02, Escola Politecnicasegundo andar, 40.210-630, Salvador, Bahia

[email protected], [email protected]

Abstract: The aim of this paper is present an application of modelling and identification techniques in a sucker-rodpump system. The results show a dynamic system model and its relationship with the real process variables in thiskind of artificial lift oil method. The model obtained from experiments with real data is representative and can beused for applications, such as for instance, a control system design.

Key–Words: Modelling and identification, Polynomial models, Oil industry, Process automation, Production mon-itoring, Sucker-Rod pump systems.

1 IntroductionThe sucker-rod pump system is the artificial liftmethod most used in the current on-shore petroleumindustry due to the simplicity of its equipments andfacilities [1]. This method is also considered as thefirst technique used to lift oil up from wells. Studieshave shown that its popularity is related to low cost ofinvestments and maintenance, deep and outflow flex-ibility, good energy efficiency and the possibility foroperating in different fluid compositions and viscosi-ties in a wide range of temperatures [2].

Although this lift method is already well-knownand widely used, there are still some circumstances inwhich improvements in the operational conditions arestill possible, especially when dealing with produc-tion control strategies of the pump unit for increasingthe system productivity. The development of low costsensors turned possible the measurement of bottom-hole variables that assists the production monitoring,application of new control strategies, and enhance-ment of the process automation [3, 4, 5].

Simulation softwares have assisted the studiesabout the sucker-rod pump system in modelling anddesign of control systems, especially into controllingthe dynamic fluid level of the annular well. The val-idation of these softwares increases the possibilitiesin a successful implementation of a control system,for instance. However, such simulators, speaking gen-erally, are limited to theoretical models (mostly phe-nomenological) and therefore in lacking of evidences

in situations close to the real production fields.

On the other hand, a modelled system mathemat-ically does not always need be represented by phe-nomenological models by using physical laws (ma-terial balance, energy, etc.). The inputs and outputsof the system may be grouped by some interdepen-dence, in order to extract a relation of cause and effectin phenomena that occur with the elements of a set ofsampled data. The modelled system does not need belimited to a physical modelling (white-box), but alsoapply to systems with observable data (black-box),or systems that include some additional information(gray-box) [6]. In recent years the application ofempirical modelling techniques or, as is also known,modelling and identification techniques of dynamicsystems (e.g.: ARX, ARMAX, NARMAX, etc.) inthe oil and gas industry, has experienced a significantgrowth [7, 8, 9, 10, 11, 12, 13].

The white-box models in this application may be-come arduous and hard to be obtained since the pro-cesses involved in the oil production may not be wellunderstood (e.g: synergy among various dynamicevents and the bottom of the well and the surface).New research, therefore, by using modelling and iden-tification techniques, especially in artificial lift sys-tems, may contribute to a better understanding of dy-namic models in the analysis and design of automaticcontrol, for instance.

Recent Advances in Mathematical Methods, Intelligent Systems and Materials

ISBN: 978-1-61804-168-5 19

2 The Sucker-Rod Pump SystemIn this artificial lift method a rotary movement ofprime mover (either an electric or a combustion mo-tor) localized on the surface of the pump unit is con-verted in alternative movement of the rod string. Thissame column transmits the an alternative movementto the pump components that are located at the bot-tom of the well, that are responsible to elevate the fluidfrom reservoir up to the surface. The sucker-rod pumpsystem could be divided in downhole and surface ele-ments (see Fig.(1)).

Figure 1: Components of a sucker-rod pump system.

The rod string is the link between the pump unitlocalized on the surface and the bottomhole pump.The bottomhole pump is a kind of alternative pump ofpositive displacement of simple effect, in other words,the fluid is displaced in a one way direction of the al-ternative movement. The function of the bottomholepump is providing energy to (increasing the pressureof) the fluid from reservoir [1]. In Fig.(2) the bottom-hole scheme is presented. The annular well and pumpinlet level are also shown.

The pumping cycle generated by the relativemovement of the valves have repercussions on the bot-tomhole pressure. The oil production is controlled byvarying the prime mover velocity, which implies inthe manipulation of the pumping speed, measured incycles per minute (CPM). In this control strategy thevariable speed drive (VSD) technique is used. Thatallows to adjust the pumping speed through a fre-quency inverter device [14, 15]. It is important toremark that the production performance is associatedwith the annular fluid level, and the operation with theminimum possible annular level (minimum bottom-hole pressure) the reservoir oil outflow is maximized[16]. In therms of a control system design to increase

Figure 2: Bottomhole scheme with sucker-rod pumpsystem.

the oil production, a dynamic model of a sucker-rodpump system may relate the pumping speed of the unitand the fluid level in the annular well. This dynamicmodel may also reveal the relationship of its parame-ters with the real process. According to the literature[17, 16], these parameters are normally related to fluidcharacteristics in the well, environmental properties atthe bottom of the hole, and mechanical assembly, forexample.

3 Modelling and Identification

3.1 Polynomial modelsBy considering that it is impossible to control andmonitoring the systems which are unknown, the mod-elling and identification techniques are importanttools to understanding the behaviour of a process.These techniques aim to obtain mathematical modelsby using a set of sampled data from a system. In otherwords, the models describe the cause and effect rela-tionships through the output and input signals.

The polynomial models are structures frequentlyused in modelling and identification procedures. Suchmodels are parametric input-output structures able torepresent dynamic behaviour of a wide class of linearand non-linear systems.

Consider below the general expression of thesemodels,

y(k) =q−dB(q−1)

A(q−1)u(k−1)+

C(q−1)

D(q−1)A(q−1)e(k) ,

(1)where,

• y(k) is the process output;

Recent Advances in Mathematical Methods, Intelligent Systems and Materials

ISBN: 978-1-61804-168-5 20

• u(k) is the process input;

• q−1 is the unit delay operator and u(k)q−1 =u(k − 1);

• d is the dead time, in multiples of the sample time(d ≥ 0);

• e(k) is a possible noise. It may be considered awhite noise, for instance.

The polynomials A(q−1), B(q−1), C(q−1), andD(q−1) are defined as,

A(q−1) = 1 + a1q−1 + a2q

−2 + . . .+ anaq−na

B(q−1) = b0 + b1q−1 + b2q

−2 + . . .+ bnbq−nb

C(q−1) = 1 + c1q−1 + c2q

−2 + . . .+ cncq−nc

D(q−1) = 1 + d1q−1 + d2q

−2 + . . .+ dndq−nd ,

(2)where na, nb, nc e nd are the degrees of the polynomi-als A(q−1), B(q−1), C(q−1) e D(q−1), respectively.

There is a unit natural delay of sample time be-tween the input and the output of the system. This factis due to discrete models use the zero-order holder insampling the input. The zero-order holder keeps thevalue of the last input in k − 1 up to the process has aresponse in k. To illustrate the model above, one mayconsider the block diagram in Fig.(3).

Figure 3: Block diagram of a parametric input-outputmodel. Two models can be identified in this structure:a deterministic model and a noise model.

3.2 Practical ConsiderationsIn this section some considerations are presentedabout the sucker-rod pump systems and the real plantused in this work.

In systems that are assembled with sucker-rodpumps, often it is desired that an operation range is

very close to the pump inlet level. This operationrange is characterized by the complete pump fillingwith the least bottomhole pressure possible. That pro-vides the minimum back pressure on the productionzone of the reservoir and, in turn, it increases the oilproduction [16]. In the Laboratorio de Elevacao Ar-tificial - LEA (in english Artificial Lift Lab) at theUniversidade Federal da Bahia, there is a real plantof a sucker-rod pump with an artificial well of 32m ofheight, fully instrumented, with full access and visiblebottomhole. All components of this equipment are in-dustrial and the plant also has a supervisory system todata acquisition and control. A scheme of this well ispresented in Fig(4).

Figure 4: Well scheme with sucker-rod pump system.

In modelling and identification procedures it of-ten used a deviation variables (input and output sig-nals of the process) from a reference or a nominaloperation point. In this work, a reference is chosenby considering the desired operation range and thepump inlet level. A scheme of this well with the ref-erence chosen and this operation range is also shownin Fig(4). It is important to remark that the productionperformance is associated with the annular fluid level.Therefore a dynamic model of a sucker-rod pump maybe developed by relating the pumping speed of the unitand the fluid level in the annular well in order to as-sist the system monitoring or for control purposes, forinstance. Hence in this model the level in the annularwell (measured in meters) is considered as the pro-cess output y(k) , and the pumping speed (measuredin CPM) as the process input u(k).

In [17] some relationships between the pumpingspeed and the fluid level in the annular well (and otherpossible variables and parameters of the whole sys-tem) based in theoretical studies and phenomenologi-cal models of a sucker-rod pump are presented. How-ever, these studies indicate a flow rate from the annu-

Recent Advances in Mathematical Methods, Intelligent Systems and Materials

ISBN: 978-1-61804-168-5 21

lar well to the production tubing is linear in the head(level), which would indicate laminar flow by consid-ering a single-phase and incompressible fluid. In or-der to verify this property in the real plant used in thiswork some tests are performed and a step signal isapplied to the process input. The process output isshown in Fig.(5).

Figure 5: Process output yielded by a step input.

By observing the Fig.(5), the process responsewith no dead time and it seems that can be well fittedby a linear first order model. This sounds reasonablebecause this process can be seen as a simplified fluidlevel system in a SISO tank [18]. From Eq.(1) a struc-ture of the ARX model may be proposed as following,

y(k) =B(q−1)

A(q−1)u(k − 1) +

1

A(q−1)e(k) , (3)

A(q−1) = 1 + a1q−1

B(q−1) = b0 ,(4)

where, na = n, nb ≤ n, and nc = nd = 0.

4 Experimental ResultsA recursive least square (RLS) is used to estimate theparameters. The results are shown as following,

a1 = −9, 307× 10−1

b0 = 3, 500× 10−3 ,(5)

A(q−1) = 1− 9, 307× 10−1q−1

B(q−1) = 3, 500× 10−3 .(6)

An application of a pseudo random binary signal(PRBS) in the process input yields in Fig.(6) a com-parison between the estimated output and the processoutput.

Figure 6: Comparison between the estimated outputand the process output

Figure 7: Block diagram of the identified model

The input-output ARX structure may be proposedin the block diagram in Fig.(7)

By using another set of sampled data from a un-correlated and new PRBS, in Fig.(8) a visual valida-tion between the ARX model output and the real pro-cess output.

An analysis of the result in Fig (8) shows theidentified model clearly follows the process response.However, it can be seen in some sections in Fig (8)there are differences between the estimated outputand process output due to unmodelled dynamics, forexample. Despite of these differences, the obtainedmodel may meet the main expectations in a controlsystem design. In other words, the model incorporatesthe main features of actual system dynamics and it issufficiently representative.

Another important point to consider is that the pa-rameters of the plant a1 and b0 are associated withsome physical quantities of the sucker-rod pump sys-

Recent Advances in Mathematical Methods, Intelligent Systems and Materials

ISBN: 978-1-61804-168-5 22

Figure 8: Visual validation between the ARX modeloutput and the real process output

tem. According to the literature [17, 16], these pa-rameters are normally related to fluid characteristicsin the well, environmental properties at the bottom ofthe hole, and mechanical assembly. For a linear firstorder model the physical parameters are:

• productivity index (PI), related to the flow fromreservoir (production zone) and the bottomholepressure;

• specific weight (γf ) related to the pumped fluidcomposition (water, oil and/or gas);

• transverse internal area of the annular well(AAN ), related to the mechanical design of thewell;

The identified model in Eq.(6) provides a straight-forward design for controller algorithms, for instance,since the parameters are directly associated with phys-ical quantities. These parameters, in turn, representthe relationships among the physical attributes of thesystem that have more easily known uncertainties.

5 ConclusionIn this paper the results presented a dynamic model forsucker-rod pump systems that is developed by usingmodelling and identification techniques. It should beremembered that one of main challenges that has beenrepeatedly found is the validation of models (mostlyphenomenological) for these systems. Once the com-ponents of pumping, such as rod string and bottom-hole pump are located in the deep subsurface andwithout the possibility of easy access. It does nothappen with the plant used in this work, whose sys-tem is fully instrumented and the bottom is accessi-ble and visible. An contribution of this paper is that

the results were not limited to theoretical models (orsimulators) and therefore in lacking of evidences insituations close to the real production fields. The sup-port of the LEA laboratory may confirm the modeldeveloped within the desired operation range is rep-resentative since incorporates the main features of thedynamics of real plant. An accurate model is an im-portant key in a successful design of a control system,for example. The identified model also exhibited thecause and effect phenomena that occur with the inputand output signals and some physical parameters ofthis equipment. Is is crucial to a better understandingof dynamic properties of the sucker-rod pumping sys-tems. These results also can further contribute to theimprovement of existing or a new simulation software.

Acknowledgements: The research was supported bythe CTAI (facilities and infrastructure) at the Universi-dade Federal da Bahia and CAPES (financial support).

References:

[1] W.L. Lake, Petroleum Engineering Handbook,2006, Society of Petroleum Engineers, Richard-son, USA.

[2] G. Takcs, Sucker-Rod Pumping Manual, 2002,Penn–Well Books, Tulsa, USA.

[3] G.V. Moises, T.A. Rolim and J.M. Formigli,Gedig: Petrobras corporate program for digitalintegrated field management, in Proc. of SPEIntelligent Energy Conference and Exhibition,2008, Amsterdam, The Netherlands, ISBN: 978-1-55563-166-6.

[4] B. Smith, M. Hall, A. Franklin, E.S. Jo-hansen and H. Nalmis, Field-wide deploy-ment of in-well optical flow meters and pres-sure/temperature gauges at buzzard field, inProc. of SPE Intelligent Energy Conference andExhibition, 2008, Amsterdam, The Netherlands,ISBN: 978-1-55563-166-6.

[5] M.A.D. Bezerra, L. Schnitman, M.A. BarretoFilho and J.A.M. Felippe de Souza, Patternrecognition for downhole dynamometer card inoil rod pump system using artificial neural net-works, in Proc. of 11th International Conferenceon Enterprise Information Systems, 2009, Milan,Italy, pp. 351-355.

[6] L.A. Aguirre, Introduction to System Identifica-tion - Linear and Non-Linear Techniques Ap-plied to Real Systems. (in portuguese), v. 1,3rd ed., 2007, Editora UFMG, Belo Horizonte,Brazil, 730 p..

[7] J.M. Galvao, L. Schnitman and H. Lepkison,Dynamic system identification by using AR-MAX and NARMAX applied to oil industry, in

Recent Advances in Mathematical Methods, Intelligent Systems and Materials

ISBN: 978-1-61804-168-5 23

Proc. of 5th National Congress of MechanicalEngineering (CONEM), 2008, Salvador, Brazil.

[8] D.J. Pagano, V.A. Dallagnol Filho andA. Plucenio, Identification of polinomialNARMAX models for an oil well operating bycontinuous gas-lift., in Proc. of InternationalSymposium on Advanced Control of ChemicalProcess (ADCHEM), v. 2, 2006, Gramado,Brazil, pp. 1113-1118.

[9] A. Plucenio, D.J. Pagano and E. Camponogara,Gas-lift optimization and control with non-linearMPC., in Proc. of IFAC International Sympo-sium on Advanced Control of Chemical Pro-cesses (ADCHEM), 2009, Istanbul, Turkey.

[10] E. Camponogara, A. Plucenio and A.F. Teixeira,An automation system for gas-lifted oil wells:Model identification, control, and optimization,Journal of Petroleum Science and Engineering,2010, v. 70, pp. 157-167.

[11] E.B. Cajueiro, L. Schnitman and R.A. Kalid, In-ferring Polished Rod Position from Torque Cur-rent of the Motor., In: Recent Advances of Ap-plied & Biomedical Informatics and Computa-tional Engineering in Systems Applications, v.1, 1rd ed., 2011, WSEAS Press, Florence, Italy,pp. 283-287.

[12] E.B. Cajueiro, L. Schnitman and R.A. Kalid, Us-ing NARX model with wavelet network to in-ferring the polished rod position. InternationalJournal of Mathematics and Computers in Sim-ulation, 2012, v. 6, pp. 66-73.

[13] L.H.S. Torres, Modelling, Identification andAdaptive Control of Rod Pumping Systems forOil Wells (in portuguese), 2012, M.Sc. Disserta-tion, Universidade Federal da Bahia, Salvador.

[14] R. Peterson, T. Smigura, C. Brunings, W. Qui-jada, and A.Gomez, A production increases atPDVSA using a improved SRP control, in Proc.of SPE Annual Technical Conference and Exhi-bition, 2006, San Antonio, USA, ISBN: 978-1-55563-149-9.

[15] B. Ordonez, A. Codas and U.F. Moreno, Sucker-Rod Pumping System: Simulator and Dy-namic Level Control Using Downhole Pressure,in Proc. of IEEE International Conference onEmerging Technologies and Factory Automa-tion, 2008, Hamburg, Germany, pp. 282-289.

[16] B. Ordonez, A. Codas and U.F. Moreno, Improv-ing the operational conditions for the sucker-rodpumping system, in Proc. of IEEE InternationalConference on Control Applications, 2009, SaintPetersburg, Russia, pp. 1259-1264.

[17] M. de A. Barreto Filho, Estimation of avaragereservoir pressure and completion skin factorsof wells that produce using sucker rod pumping.,2001, P.h.D Thesis, The University of Texas atAustin, Austin.

[18] K. Ogata, Modern Control Engineering, 2000,Prentice Hall, USA.

Recent Advances in Mathematical Methods, Intelligent Systems and Materials

ISBN: 978-1-61804-168-5 24