DETERMINING THE SURFACE DYNAMOMETER CARD OF A PUMPING

SYSTEM FROM THE TORQUE CURVE OF A THREE-PHASE INDUCTION

MOTOR.

Wilton Lacerda Silva∗, Daniel Goncalves de Souza Neto†, Manuel de Almeida Barreto

Filho‡, Antonio Marcus Nogueira Lima§, Amauri Oliveira†

∗Federal Institute of Bahia - IFBAAv. Amazonas, no 3150, Zabele, Vitoria da Conquista BA,Brazil

†Department of Electrical Engineering, Federal University of Bahia - UFBAAristides Novis Street, no 02, Federacao, Salvador BA, CEP 40210-630, Brazil

‡Department of Materials Science and Technology, Federal University of Bahia - UFBAand Petroleo Brasileiro S.A - PETROBRAS

§Department of Electrical Engineering, Federal University of Campina Grande -UFCGCampina Grande - PB, Brazil

Emails: wiltonlacerda@ifba.edu.br, daniel.onda@gmail.com,

barretofilho@petrobras.com.br, amnlima@dee.ufcg.edu.br, amauri@ufba.br

Abstract— This paper proposes to obtain the surface dynamometer card, in a sucker-rod pumping system,from the torque curve of the three-phase induction motor that drives the beam pumping unit. This will lead toa minimally invasive, easy installation and maintenance method, without using the load cell, to plot the surfacedynamometer card of a pumping system. It has been shown the mathematical basis which allows us to infer thepolished rod load by using the torque of the induction motor. A simulation has been performed and a surfacedynamometer card has been plotted from data of the motor torque. The torque of the induction motor of abeam pumping unit with API designation C640D-305-144, that operates in a well of 1280 m deep, has been alsoobtained by using the methodology of rotor slot harmonics, and the surface dynamometer card of a pumpingsystem has been plotted.

Keywords— Sucker-rod pumping, surface dynamometer card, three-phase induction motor, torque curve,rotor slot harmonics.

1 Introduction

The sucker-rod pumping (RP) is the most ap-plied artificial lift technology of oil in the world(Thomas, 2001). Figure 1 presents a RP sys-tem. In particular, it highlights the downholeequipments such as: subsurface pump, tubing androd string, besides surface equipments, here rep-resented by a conventional beam pumping unit.Evaluating operation conditions of a well is essen-tial, so one of the most applied tools by profession-als in the oil industry is the surface dynamometercard of a pumping system. It provides informationabout the rod string load versus its displacementfor each pumping cycle (Mantecon, 1989). The useof dynamometers to measure the polished rod loadis quite usual and they have been improved in re-cent decades (Takacs, 2003). This paper proposesto obtain the surface dynamometer card from thetorque curve of the three-phase induction motor(TIM) that drives the beam pumping unit, ie,without using the load cell.

One of the methods to obtain the surfacedynamometer card without using a dynamome-ter is to use the TIM torque curve as well asdata from the manufacturer of the beam pumpingunit (BPU) (Gibbs, 1987). The work of (Rowlanet al., 2005) introduced a methodology to obtainnetwork torque data from the power of three-

phase induction motor and its speed. In thiscontext, it is essential to know the efficiency ofthe transmission system, ie, motor, gearbox andsheaves/belts. Similarly, this paper proposes touse the rotor slot harmonics, which are presentin the line current of the induction motor, to ob-tain the motor speed (Aiello et al., 2005; Nandiet al., 2002) and from this information infers mo-tor torque. This methodology has advantages overthe previous one since the number of sensors re-quired to measure parameters to estimate the mo-tor torque and speed is much smaller. However,when it is compared with the traditional use of theload cell to obtain polished rod load, both method-ologies are less intrusive to the RP system, thenthe installation and mainly the maintenance costsare reduced.

2 Mathematical basis to obtain the

surface dynamometer card of a

pumping system from the torque curve

of the induction motor

The map of the loads that operates on the pol-ished rod in function of its displacement (surfacedynamometer card) plays a key role in the analysisand diagnostic process of the sucker-rod pumping.Moreover, its use as a production optimizationtool, through the automation of wells and pump-

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engex

API - 25 E

Rod string

Production casing

Conventional beam pumping unit

Subsurface pump

Polishedrod

Figure 1: Schematic representation of a sucker-rod pump-ing system.

off control, is a natural tendency. The polishedrod load and counterweights generates torque onthe gearbox shaft of the beam pumping unit. Thetorque factor method, equation (1), approved byAmerican Petroleum Institute (API) is one of theways to obtain the torque which is submitted thegearbox shaft of the BPU.

TR(θ) = TF (θ)[QPR(θ)−QSU ]−Msin(θ+τ) (1)

For:

TR - gearbox torque of the BPU, in (N ·m).

TF - torque factor, in (m).

QPR - polished rod load, in (N).

QSU - structural unbalance of the BPU, in(N).

M - counterbalance moment, in (N ·m).

θ - rotational angle of the crank, in (rad).

τ - phase angle between the crank arm andcounterweight line, in (rad).

The torque of the induction motor can be con-verted to the BPU gearbox torque using equation(2):

TR(θ) = υTM (θ) (2)

In which υ is a constant of proportionalitythat includes the ratio between the average speedof the motor and crank, and the efficiency of: themotor, the transmission system (shaves/belts),the gearbox and the rotational and articulatedparts of the BPU.

The equation (1) can be rewritten as a func-tion of the induction motor torque, in which thepolished rod load is isolated:

βC

χψ A

ρ

P

J

KH

Φ

PRαR

θ

I

Downstroke

G

Figure 2: Geometry of a conventional beam pumping unit.

QPR(θ) =1

TF (θ)[υTM (θ) +Msin(θ + τ)] +QSU

(3)The rotational angle of the crank, figure 2,

can be obtained from the equation (4):

θ(t) = ω(t)t+ ε (4)

The instantaneous rotational speed of thecrank ω, in rad/s, can be obtained by the productof the instantaneous rotational speed of the mo-tor and the ratio between the average speed of themotor and crank. The average speed of the crankcan be estimated from the pulse train provided bya Hall effect sensor mounted on the BPU, whichis triggered by a magnet fastened on the crank.

The angle ε can be inferred from the BPU ge-ometry, figure 2, through the following equations:

ε = φ− γ (5)

In which the angles φ and γ are:

φ = arcsin(I/K) (6)

γ = arccos

[

(P +R)2 +K2 − C2

2(P +R)K

]

(7)

The crank arm R is the solution of the bi-quadratic equation (8):

ER4 + FR2 + L = 0 (8)

In which:

E = sin2(SPR/2A) (9)

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F =2(P 2 + C2 +K2)cos2(

SPR

2A

)

+

2(P 2− C2

−K2) (10)

L =

[

(C2 +K2 − P 2)sin

(

SPR

2A

)]2

−

[

CKsin

(

SPR

A

)]2

(11)

A, C, P , K, H , I , G e J are dimensions ofthe conventional BPU geometry and SPR is thecourse of the polished rod. The dimension J varieswith the angle θ and it can be obtained from theequation (12):

J =√

K2 +R2 − 2KRcos(θ − φ) (12)

The torque factor TF is a function of the an-gle θ, crank arm R and BPU geometry and it canbe inferred from the equation (13):

TF (θ) =AR

C

sinα

sinβ(13)

For:

α = β + ψ − (θ − φ) (14)

β = arccos

[

P 2 + C2 − J2

2CP

]

(15)

Finally, one can check that:

ψ = χ− ρ (16)

In which:

χ = arccos

(

C2 + J2 − P 2

2CJ

)

(17)

ρ = arcsin

(

Rsin(θ − φ)

J

)

(18)

Therefore, if data from the counterbalancemoment M , structural unbalance QSU , polishedrod course SPR and BPU geometry, that are pro-vided by manufacturers of beam pumping units,besides the torque and speed versus time curves ofthe induction motor are available, one can deter-mine the polished rod load QPR along the pump-ing cycle, from the equation (1). Moreover, to plot(QPR vs. S(θ)) it is necessary to know the posi-tion of the polished rod S(θ), which is a functionof the angle θ for each load value. The equation(19) allows us to obtain the values of polished rodposition as a function of the rotational angle ofthe crank:

S(θ) = A(ψb − ψ) (19)

0 1 2 3 4 5 6 7 8-50

0

50

100

150

200

250

300

350

Time (s)

Tor

que

(Nm

)

Simulated torque curve

Figure 3: Simulated torque curve.

0 0.5 1 1.5 2 2.5 3

2

3

4

5

6

7

8

x 104

Rod displacement (m)

Rod

load

(N)

Simulated surface dynamometer cardRecovered surface dynamometer card

Figure 4: Simulated surface dynamometer cards.

In which the angle ψb is the minimal value ofψ that occurs at the beginning of the downstrokeand it is given by:

ψb = arccos

(

C2 +K2 − (P +R)2

2CK

)

(20)

2.1 Simulation to obtain surface dynamometercard

The work of (Barreto Filho, 2001) developed aprogram that simulates the movement of the rodstring, its deformation and interaction with theproduced fluid, besides the acceleration and de-celeration of the same within the tubing alongthe pumping cycle, using the downhole conditions,the rod string design, the characteristics of thepumped fluid and the pumping unit data. Thelinearization of the induction motor torque versusspeed curve, that will be described in this paper,has been also employed in his simulator.

The induction motor torque versus timecurve, figure 3, has been obtained in the simulatordeveloped by Barreto Filho and it has been em-ployed to recover the surface dynamometer card,figure 4, by means of the torque factor methodpreviously described.

In figure 4 one can note a slight difference

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between the simulated and recovered surface dy-namometer cards. This is due to the contributionof the inertial components of BPU, that have beendescribed by (Gibbs, 1975) and they have beentaken into account only in the simulator.

2.2 Detection of the motor speed and torque bymeans of the rotor slot harmonics methodol-ogy

The rotor slot harmonics methodology (Nandiet al., 2001) can be employed to estimate the rota-tional speed of the axis of a three phase inductionmotor. The equation of rotor slot harmonics fre-quencies can be written as:

fsh =

[

λZ1− s

p± δ

]

f1 (21)

For:fsh - rotor slot harmonic frequency, in (Hz).Z - number of rotor slots.s - slip.p - number of fundamental poles pairs.λ - any positive integer number.f1 - fundamental supply frequency, in (Hz).δ - order of the stator time harmonic (δ = 1,

3, 5, etc.).

When equation (21) is employed to estimatethe rotational speed of the motor, it can be sim-plified as follows: the λ can be taken equal to one,since the first set of components exhibits good am-plitude to detect rotor slot harmonics. Thus, theslip of the motor can be expressed by:

s = 1−p

Z

[

fshf1

∓ δ

]

(22)

It is possible to determine the rotational speedof the induction motor shaft, in (Hz), by meansof the motor slip (Krause et al., 2002; Nasar andBoldea, 2002), through the following equation:

n =60

Z(fsh ± δf1) (23)

It is known that the generation of the rotorslot harmonics, which are present in the line cur-rent of the induction motor, depends strongly onthe number of rotor slots and the number of fun-damental poles pairs (Silva and Olivera, 2012a).So, if these parameters are known, it is possibleto identify the frequency spectral components f1and fsh.

The methodology of rotor slot harmonics en-ables us to estimate the slip and speed of inductionmotors with squirrel cage rotor, which drive thebeam pumping units for artificial lift of oil (Silvaand Oliveira, 2013). If the instantaneous motorspeed is known, one can infer the instantaneousspeed of any part of the BPU and indirectly the

0 200 400 600 800 1000 1200 1400 1600 18000

1

2

3

4

5

6

7

8

9

Rotational speed (rpm)

Tor

que

(N.m

)

Rated Torque

Figure 5: TIM torque vs. speed curve (solid line) and linearapproximation between null and rated torque (dashed line).

gearbox and motor torque. The linear relation-ship between the TIM torque and slip has beenproposed to obtain the torque of the inductionmotor. Figure 5 shows the TIM torque curve thathas been obtained from a motor equivalent circuitsimulation. In this moment two points which willbe employed in this method can be highlighted,the rated torque of the TIM Tn, which happensto rated speed and slip sn, and torque for syn-chronous speed for which the slip is zero. Thus,a linear equation can be constructed by using thepoints (0,0) and (sn,Tn), and it is represented bythe dashed line in figure 5. The TIM torque canbe estimated from this linear equation that is afunction of the slip and it can be expressed by:

TM = Tns

sn(24)

For:TM - motor torque, in (N ·m).Tn - rated motor torque, in (N ·m).s - slip.sn - rated slip.

Data such as: slip, speed and rated torque canbe obtained from the TIM nameplate. The slip orspeed can be estimated by using the methodologyof rotor slot harmonics. Thus, if these data areavailable it is possible to estimate the torque onthe motor shaft.

3 Experimental results

The methodology of rotor slot harmonics has beenemployed to infer the surface dynamometer cardof an oil well equipped with sucker-rod pumping.The subsurface pump of the well is at a depth of1280 m and it is driven by a conventional beampumping unit with API designation C640D-305-144, figure 6. The geometrical dimensions of thispumping unit in inches are: A = 180, C = 120.08,I = 120, P = 144.50, R = 47 and structural un-balance -520 kgf . From these values and the pre-vious equations, it is possible to calculate someparameters such as: φ = 36.91 and ψb = 68.4.

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Figure 6: Photo of a conventional beam pumping unit withAPI designation C640D-305-144.

One can determine that this pumping unit op-erates with 189.6 degrees of upstroke and 170.4degrees of downstroke.

The TIM power, that drives this BPU, is 75cv, its rated speed is 1190 rpm, it has 3 poles pairs,58 rotor slots, its rated current and torque are re-spectively 95 A and 443 N ·m and it has been di-rectly supplied by the power grid. The schematicfor current acquisition in one of the motor phasesis shown in figure 7. The current signal, thathas been acquired by a current probe which hasa Hall effect sensor, has been acquired and condi-tioned in the analog to digital converter DAQ6009of National Instruments with a sampling rate of 10ks/s. The current signal is shown in figure 8. Inthis figure, one can observe that variations of theload directly affect the current signal amplitude.A magnification of this signal has been performedto allow us a better viewing of the current wave-form and the result is shown in figure 9.

An algorithm has been used to identify f1and fsh, since these parameters are critical todetermine the motor speed and slip, equations(22) and (23). Several digital signal processingtools have been employed to identify these param-eters, but two techniques should be highlighted:the Chirp Z-Transform (CZT) and Short TimeFourier Transform (STFT), which provide betteraccuracy in the identification process (Silva andOlivera, 2012b).

Once the slip and speed are determined, thetorques can be estimated in the sucker-rod pump-ing system. Figure 10 shows the estimated torquecurves. One can note that both the current sig-

Currentprobe

Signalconditioning

Computer -Digital signalprocessing

Pha

seA

Pha

se B

Pha

se C

Inductionmotor

GearBox

Sheavesandbelts

Controller

Figure 7: Schematic representation of the method em-ployed to acquire the current signal in the field.

0 5 10 15 20 25−100

−80

−60

−40

−20

0

20

40

60

80

100

Time (s)

Am

plitu

de (

A)

Induction motor currentCrank position

Figure 8: The current signal acquired in one of the motorphases. The vertical dashed lines are pulses provided by aHall effect sensor mounted on the BPU, which is triggeredby a magnet fastened on the crank. Each pulse indicatesthat the mechanical horse’s head is on the highest position.

5.45 5.5 5.55 5.6 5.65

−50

−40

−30

−20

−10

0

10

20

30

40

50

Time (s)

Am

plitu

de (

A)

Induction motor currentCrank position

Figure 9: Magnification of a current signal segment.

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0 50 100 150 200 250 300 350 400−1.5

−1

−0.5

0

0.5

1

1.5

2x 10

6

Crank angle (degrees)

Tor

que

(lbf−

in)

Counterbalance torqueGearbox maximum torqueNet torqueRod load torque

Figure 10: Torque curves of the beam pumping unit withAPI designation 640D-305-144.

0 50 100 1501

1.2

1.4

1.6

1.8

2

2.2

2.4x 10

4

Rod displacement (in)

Rod

load

(lb

f)

Estimated surface dynamometer cardRecovered surface dynamometer card

Figure 11: Estimated and reference surface dynamometercards.

nal waveform (figure 9) and gearbox torque curve(figure 10) indicate that the BPU is not com-pletely balanced. However, it appears that thegearbox torque peak is less than the maximumgearbox torque allowable. Using the polished rodtorque curve and its position, it is possible to es-timate the surface dynamometer card of the oilwell equipped with RP. The estimated surface dy-namometer card is shown in figure 11. It has beencompared with a reference surface dynamometercard that has been obtained by using a load celland positioning encoder. One can note many sim-ilarities between them.

4 Conclusion

The surface dynamometer card of a pumping sys-tem is a very important tool in the analysis and di-agnostic process of the sucker-rod pumping. Thus,the availability of other tools and techniques toobtain such card is relevant in this context. Thispaper presents the theoretical and practical basisto obtain the surface dynamometer card by usingthe methodology of rotor slot harmonics, whichare present in the line current of the three-phaseinduction motor.

Positive results have been achieved in the re-covery process of the surface dynamometer card.The rotor slot harmonics methodology indicatesthat it is possible to determine the gearbox torqueon the BPU. Thus, this paper shows that thismethodology to obtain the surface dynamome-ter card is feasible and it is an auxiliary tool forprofessionals in the oil industry that work withsucker-rod pumping systems which are driven bythree-phase induction motors.

5 Acknowledgment

The authors thank FAPESB and CNPq for thefinancial support throughout the development ofthis project and PETROBRAS.

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Nasar, S. A. and Boldea, I. (2002). The InductionMachine Handbook, 1 edn, CRC Press.

Rowlan, O. L., Mccoy, J. N. and Podio, A. L.(2005). Best method to balance torque load-ings on a pumping unit gearbox, Journalof Canadian Petroleum Technology - JCPT44(7): 27–33.

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