# Modelling and Identiﬁcation of a Sucker-Rod Pumping ... and Identiﬁcation of a Sucker-Rod...

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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

Brazilluizhenrique@ieee.org, leizer@ufba.br

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.

KeyWords: 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) =qdB(q1)

A(q1)u(k1)+ C(q

1)

D(q1)A(q1)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;

q1 is the unit delay operator and u(k)q1 =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(q1), B(q1), C(q1), andD(q1) are defined as,

A(q1) = 1 + a1q1 + a2q

2 + . . .+ anaqna

B(q1) = b0 + b1q1 + b2q

2 + . . .+ bnbqnb

C(q1) = 1 + c1q1 + c2q

2 + . . .+ cncqnc

D(q1) = 1 + d1q1 + d2q

2 + . . .+ dndqnd ,

(2)where na, nb, nc e nd are the degrees of the polynomi-als A(q1), B(q1), C(q1) e D(q1), 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 sucke

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