Application of Fourier Descriptors and PearsonCorrelation for Fault Detection in Sucker Rod

Pumping SystemFabio Soares de Lima

PETROBRASPetroleo Brasileiro S.A.

Aracaju, SE, BrazilEmail: fslima@petrobras.com.br

Luiz Affonso H. GuedesUFRN

Universidade Federal do Rio G. do NorteNatal, RN, Brazil

Email: affonso@dca.ufrn.br

Diego R. SilvaIF-RN

Instituto Federal de EducacaoNatal, RN, Brazil

Email: diego@dca.ufrn.br

Abstract—This paper presents a study of a new proposal fordown-hole dynamometer cards patterns for the automatic faultdiagnosis of Sucker Rod Pumping System. The accuracy andquick identification of down-hole problems is essential to therisk decrease and production improvement of the petroleumindustry. The main idea is a card recognition through a digitalimage processing (Fourier Descriptors) and statistical (PearsonCorrelation) tools. Successful results were reached when thisproposal was applied in simulated cards and PETROBRAS realcards.

I. INTRODUCTION

The Sucker Rod pumping is the most common form ofartificial lift in the world [1], [2]. It is estimated that 90% ofartificially lifted wells use Sucker Rod pumping systems [3],[4]. In Brazil, 64% of the whole production of wells is on rodpump [5]. In practice, the analysis of the Sucker Rod pumpingsystem is through a card. This card is called dynamometercard. With this card it is possible to get the down-holecondition of the rod-pumped wells. The dynamometer card is aplot of load versus position that reflects the present pumpingconditions [6], [7]. While the system is operating, the cardcan assume several shapes that might be a normal operationor fault situations. The Sucker Rod System fault diagnosis is avisual interpretation process [8]. However, this approach canbe influenced by several factors, such as complex behaviorof the system, diversity of the dynamometer card shapes, andexperience and skill of the engineer.

Nowadays, the main petroleum engineer of the productionfield is responsible for more than a hundred wells. In this case,the traditional process of interpretation is not suitable for theprompt diagnosis of the down-hole conditions. The accuracyand quick identification of down-hole problems is essential tothe risk decrease and production improvement of petroleumindustry. Because the fault diagnosis of sucker rod pumpingsystem is a process of pattern recognition of dynamometercards, several works using artificial neural network in recogni-tion and classification have been made to improve the accuracyand efficiency of sucker rod pumping system fault diagnosis.

The recognition and classification of objects based on theirvisual similarity has become a central task in current industrial

imaging systems. This research suggests a new proposal to an-alyze and to diagnose any Sucker Rod Pumping System basedin a better description model and a statistical recognition. Thegreat works using a neural networks.

Therefore, the effective shape description is essential inretrieval systems. The Fourier descriptor (FD) is probablythe best-known boundary-based shape descriptor. It has beenproven to outperform most other boundary-based methods interms of retrieval accuracy and efficiency. [9] In addition, therecognition is being done through a correlation, a simple sta-tistical principle, that suggests a cause-effect relation betweentwo variables, in other words, the relation between the wellcard and the pattern card.

This paper is divided in five sections. In the next, the suckerrod system is explained and some card patterns are presentedwith their meanings. The third section explains the boundarydescriptors, emphasizing the Fourier Descriptors and, afterthat, showing a correlation principle that was used. The modelproposed is shown in the fourth section with the obtainedresults. Finally, the conclusions are commented in the fifthsection.

II. SUCKER ROD PUMPING

In the history, the first artificial lift method, that appeared inthe oil industry, was the Sucker Rod Pumping, appearing afterthat the oil industry birth. The importance of this method isshowed in the number of installations that are operating in theworld. The Figure 1 presented a Sucker Rod Pumping Unit.

The great success of the Sucker Rod Pumping System islinked with low cost in investment, maintenance, flow anddeep flexibility, good energetic efficiency and the possibility tooperate several down-hole conditions. But, the main advantageis the lowest cost/production relationship during the productionfield life.

A. Components of the Sucker Rod Pumping

1) Downhole Pump: The Downhole Pump is a positivedisplacement pump, in other words, when the fluid gets insuction, it does not return.

978-1-4244-2728-4/09/$25.00 ©2009 IEEE

Fig. 1. Sucker Rod Pumping Unit

2) Rod String: The string rod is responsible for providingthe surface energy to the downhole pump.

3) Sucker Rod Pumping Unit: The pumping unit changesthe rotation movement of the electric motor in the appliedreciprocate movement for polish rod, while the reduction boxdecreases the rotational speed of the electric motor to allowthe pumping speed.

4) Dynamometer Card: A dynamometer card is a graph ofeffects originated to active charge in the pump during a pumpcycle.

B. Reference Card Patterns

There are two types of dynamometer cards: the surface andthe down-hole card. The charges are recorded in the surfacethrough the dynamometers and in the down-hole throughespecial devices or mathematical models. The dynamometercard is one of main tools to analyze and condition the reviewof the system. This card is a record of the charges along therod string path. It is possible to view several conditions ofpumping through the dynamometer card.

In this subsection, some Sucker Rod Pumping System cardsare presented. Each card showed was chosen based on the mainproblems of oil fields and they can be found in other previousworks.[10]

These presented cards (2) are some reference patterns forthe proposed model in the forward section.

1) Normal Operation: The normal pumping pattern isassociated with the follow characteristics:

• High volumetric Efficiency;• Low interference of gas;• Low or medium suction pressure.2) Fluid Pound: These patterns are associated with the

follow characteristics:• Low suction pressure;• Low interference of gas;

Fig. 2. Cards Patterns

• Blocked Pump Suction.3) Valve Leak: This patterns that happen when there is a

leak in the down-hole valves.4) Pump Hitting: There are some problems about the down-

hole pump. These problems are pump hitting in the side ofsleeve, in the top or at the bottom.

III. THEORETICAL BASE

A. Boundary Descriptors

The shape descriptors are mathematical methods that repre-sent the object or the region shape. The descriptors are sepa-rated in Contour-Based Descriptors and Regional Descriptors[11].

The boundary descriptors describe the object shape andare based on the contour. The internal shape is described byregional descriptors.

The ideal descriptors may have the invariant characteristicsof: Translation, Rotation, Scale and Begin point.

In this work, has been used a shape descriptor and it iscalled Fourier Descriptors.

1) Fourier Descriptors: In addition to the good retrievaland classification performance, the main advantages of theFourier Descriptors are that they are compact and computa-tionally light, they are easy to implement, their matching isstraightforward, and their sensitivity to noise is low.

The contour-based shape description is based on one-dimensional boundary function (shape signature). Let (xk, yk),where k = 0, 1, 2, ..., N − 1 represents the object boundarycoordinates, in which N is the boundary length. Complexcoordinate function z(k) (Ref. 2) expresses the boundarypoints in an object centered coordinate system:

z(k) = (xk − xc) + j(yk − yc)(1)

in which (xc, yc) is the object centroid.

Fourier descriptors can be formed by the boundary functionz(k) using the Discrete Fourier Transform (DFT):

Fn =1N

N−1∑k=0

z(k)e−j2πnk

N (2)

for n = 0, 1, 2, ..., N −1 and Fn are the transform coefficientsof z(k). The descriptors can be made by the rotation invariantusing the magnitudes of the transform coefficients, |Fn|. Thescale can be normalized by dividing the magnitudes of thecoefficients by |F1|.B. Correlation Analysis

In probability and stochastic processes, the correlation showthe force degree and direction of linear relationship betweentwo random variables. In general statistical usage, correlationrefers to the departure of two random variables from inde-pendence. In this broad sense there are several coefficients,measuring the degree of correlation, adapted to the nature ofthe data [12].

A number of different coefficients is used in several situa-tions. The most known is the Pearson correlation coefficient,or simply Pearson correlation, which is obtained by dividingthe covariance of the two variables by the product of theirstandard deviations.

1) Pearson Correlation: The Pearson correlation (or“product-moment correlation coefficient”, or also “r of Pear-son”) measure the correlation degree and the direction betweentwo variables of metric scale. This coefficient is representedby r and can be between −1 and 1. So, r can be analyzed inthe following manners:

+1: It means a perfect correlation and the variables arein the same direction;

−1: It means a perfect correlation too, but, in this analy-sis, the variables direction is opposite.

0: In this case, the variables does not have a lineardependence.

In other words, the signal of the result correlation showsif the correlation is positive or negative and the proportionvariable shows the correlation force.

The Pearson correlation coefficient is calculated accordingto the next formula:

r =∑n

i=1 (xi − x) (yi − y)√∑ni=1 (xi − x)2 ·

√∑ni=1 (yi − y)2

Where x1, x2, . . ., xn e y1, y2, . . ., yn are measure valuesboth of variables. Moreover, xn can be written:

x =1n·

n∑i=1

xi

And yn can be:

y =1n·

n∑i=1

yi

These variables (xn and yn) are arithmetics meanings, bothof variables x and y.

Fig. 3. Proposed Model Flow

Fig. 4. The Real Field Card Example

IV. PROPOSED MODEL AND RESULT

The model proposes using processing image tools to recog-nize patterns. In this article, these tools are used in equipmentfault detection of the petroleum industry.

A. Proposed Model

The proposed model is based on frequency correlation ofimages. Pattern cards of faults are selected and processed,generating a Fourier descriptor for each pattern card. And thisstep is repeated for each field card to be analyzed. Later, theFourier descriptor of the field card is correlated with eachFourier descriptor of the pattern cards. The result of eachcorrelation is compared in a maximum function. The highervalue is linked to the closest pattern of the field card.

The Figure 3 showed a information flow in the proposedmodel.

1) Data Acquisition: The data is obtained through theSISAL� software [13]. The SISAL� is the company softwarethat overlooks the majority of PETROBRAS wells. Thissupervisory software gathers the field variables like: current,force, horse head position, head pressure, down-hole pressure,among others.

The dynamometer cards are a two dimension graph of force(ordinate axis) versus horse head position (abscissa axis). Thiscard consists of one hundred points as you can see in Figure4.

2) Patterns Selection: The pattern cards are selected by anexpert engineer. After it, the selected cards are available forsystem to process. From this moment, the treatment is similarto the field card as the pattern cards.

3) Fourier Descriptors Generator: After the data acquisi-tion and patterns selection, the cards are processed using themathematical tool showed in the Section III-A1.

In the Figure 5 it is presented the processed frequencies andthe behavior of frequencies that compose the figure which canbe observed. In this way, each field card can be correlatedwith a pattern card that has an own identity.

Fig. 5. Fourier Descriptor Example of a Real Card

Fig. 6. The Intern Structure of Correlation Analysis

4) Correlation Analysis: Using the Pearson Correlation ispossible to compare the Fourier descriptor of the field cardwith each Fourier descriptor of the pattern cards. In this way,in each correlation of the field card with a pattern card isgenerated a value that is the degree of close between them.In the end of this analysis, a coefficient table is generated toeach performed correlation. In the Figure 6, it is possible tosee the intern structure of this step.

5) Maximum Classifier and Fault Recognition: It was nec-essary using a classifier to recognize what pattern is closer tothe field card. In this work, a simple classifier was used andit is just a maximum function. This function is applied in thegenerated table in Section IV-A3. After it, the correlation thathas the higher value represents the fault.

B. Results

The proposed model above was applied to Sucker-RodPumping System Fault Diagnosis of 102 dynamometer cardsfrom PETROBRAS fields. The final model in this work hassix pattern cards. The results were very good and it obtain acorrect percentage of 89%, when compared with other worksthat used Neural Network tool [1], [8], [4], [10], [6].

Some tests using different conditions of Fourier Descriptorsnormalization had been done, but what has been observed is

the maintenance of analysis correct percentage.

V. CONCLUSION AND FUTURE WORKS

Nowadays, the quantity of onshore wells using Sucker-RodPumping System is higher and it stickles the engineer work.In addition, the difficulty in recognizing a specific card shapeaugment as the amount of noise increase mainly as a functionof well depth.

This study, using the processing image and the correlationtools, is suitable for fault diagnosis of Sucker-Rod PumpingSystem and could help to interpret the down-hole condition ofoil well promptly and correctly.

The proposed model presented a high efficiency for pro-cessed field cards and seemed to be very robust to inherentproblems in the processing images, like rotation, translationand scale. The great advantage this model in relation withother methods is not require big quantity of data to train. Just,the specialist draw the pattern card of fault.

In the future, it is desired to test other correlations and todevelop an intermediate step selecting the main frequenciesof cards. In this new step, it is wished to test another math-ematical tool, like the Principal Component Analysis (PCA),to simplify and decrease the processed data. In this form, itwill be possible to develop a prediction tool of the Sucker-RodSystem faults, based on the frequency analysis of each patterncard. It will permit the prediction and the planning of themaintenance, so the field engineer manages his equipments.

REFERENCES

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