Tarea 19_metodo API Rp11l

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New API Design Calculations for Sucker rodPumping Systems?FRED D. GRIFFIN*

ABSTRACTThe recently published APZ R P 11 L: Recom mende d

Practice for D esign C alculations for Sucker-ro d Pump ingSys tems takes into account many more of the wellvariables than had been previously considered in sizingpumping equipment. This paper compares the new APImethod with the more conventional, simplified methodwhich has been used for so many years. Direct compari-sons are made between maximum polished-rod load,minimum polished-rod load, and peak torque on the gearreducer. Both the API and the older method are thencompared with measured results taken from 77 wells,covering a wide range of conditions.

INTRODUCTIONThe new API recommended practice was first pub-

lished in March 1967. It was the final result of athorough and time-consuming effort that began in 1954.In that year, Sucker Rod Pumping Research, Inc. wascreated by a group of users and manufacturers ofsucker-rod pumping equipment to make a study in dep thof the many complex problems associated with artificial-lift mechanisms. To perform the necessary work, Mid-west Research Institute of Kansas City was retained.

Midwest Research Institute, in the beginning, at-tempted to study sucker-rod pumping systems throughuse of a mechanical model designed to incorporate all ofthe important well variables. Although this approach didwork, it was not entirely satisfactory. The mechanicalmodel proved difficult to use and control, and the dataobtained could no t be easily correlated. It did, however,provide a better understanding of the many complexvariables encountered in a sucker-rod pumping system.

Midwest then set up an electric analog which was ableto produce a large amount of data that could be moreeasily correlated. Having completed the project, a finalreport was submitted t o all participating members ofSucker Rod Pumping Research as well as to the API.*D Schropp (Phillips Petroleum Co., Santa Barbara,Calif.) made an outstanding contribution to the industryby condensing and re-plotting the mass of data con-tained in Midwest s final report into six families ofcurves. These were made available to API.

Lufkin Foundry and M achine Co. , Lufkin, Texas?Presented at the spring meeting of the Rocky MountainDistrict. API Division o f Production, ay 1968SA portion of this report is reproduced beginning on p 232 o fthis edit ion of Drill ing and P roduction Pract ice.

The API Committee on Standardization of ProductionEquipment, at that time headed by Max Halderson,Phillips Petroleum Co., then created a task force tostudy the final report. Ed Pierce, Sunray DX Oil Co.,was chairman of this task group, which included topengineering and production talent from many majorusers and manufacturers. It was hoped that, afterstudying Midwest s report , the task group would be in aposition to recommend t o t he Productiqn EquipmentCommittee that an API recommended practice beadopted. Th e task force did an outstanding job in takingthe mass of data from the report and presenting it in amore usable and convenient form. The curves preparedby Schropp, previously mentioned, form the basis of allcalculations in APZ RP 1 I L .

SCOPE OF API RP 111In any sucker-rod pumping system, polished-rod loads

and, consequently, counterbalance and peak torque arecomplex functions of many well variables. Some of theseare

1. Polished-rod stroke2 Pumping speed3 . Physical characteristics of the rod string4. Fluid-column load5. Polished-rod acceleration pattern6. Mechanical friction7. Compressibility8. Pump submergence9. Dynamics of the sucker-rod stringNeither conventional calculations nor API RP I l L

consider items 6. or 7., because the magnitude of thesevalues is almost always unknown. Further, the conven-tional method usually does not take into considerationpump submergence, although this is not always the case.

The API calculation method does consider pumpsubmergence (when it is known) and also one other veryimportant variable not considered in the conventionalmethod, viz., rod dynamics. This is the primary differ-ence between the new API method and the conventionalmethod of calculation.

Sucker-rod string dynamics take into account: aviscous damping; b unit geometry; c, spring constant ofrods and tubing; d ratio of pumping speed to thenatural frequency of the rod string; e ratio of rodstretch to polished-rod stroke; f variation of angularvelocity of the cranks; g motor slippage, if electricallydriven; and h system inertia.


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NEW API DESIGN CALCULATIONS FOR SUCKER ROD PUMPING SYSTEMS 22

In the API method some consideration is given to allthese dynamic factors, except unit geometry and systeminertia. The unit geometry for API RP 11L is assumed tobe of the conventional type, i. e., the equalizer bearing islocated directly over the gear-reducer crank shaft. Thus,the conventional crankbalanced and air-balanced unitsboth would fall into this category.

SIGNIFICANCE OF NON-DIMENSIONALPARAMETERS

Although this paper does not go into the highlytechnical aspects of API RP I lL , it is important toconsider briefly two independent variables that appearthroughout this publication. These appear as the abscissain Fig. 1 through 6 herein (p. 225 through 230).

1.N / N O 1 , ratio of pumping speed to the naturalfrequency of the combination rod string.2 . Fo/Skr , ratio of rod stretch to polished-rod stroke.

These variables, or parameters, are the two mostimportant in well-load behavior.

The value of Not will usually be of the magnitude of30 vibrations per minute or greater. Seldom will pum-ping speed exceed 20 spm. Therefore, this value will varybetween 0 and 0.6. Because the pumping speed willnever exceed the natural frequency of vibration of therod string, the higher harmonics ( 2 N O t ,3No: 4NO1 , tc.)will never be involved. The lower-order harmonics( 1 2 N O 1 ,1 3 ~ , , 1 4N0: etc.) will be in the operatingrange.

It is convenient to think of N / N O 1 as pumpingspeed , even though it is, in reali ty, demensionless.Assume, for example, that a tapered rod string with aparticular size pump has a natural f requency of vibrationof 4 0 per minute. Then, when we say that N/N, 0 .25 ,we are merely saying tha t the pumping speed is 1 0 spmand the rod string is operating at a 4th-order frequency,i. e., N / N o l 0 . 2 5 = N / 4 0 or N 10 spm

Similarly, the dimensionless parameter, F,/Sk,, repre-sents rod stretch. The actual rod stretch induced bythe fluid load Fo is Folkr , where kr is the springconstant of the rod string. Thus (Fo/kr ) /S is the rodstretch expressed as a fraction of the polished-rodstroke. A sucker-rod system having a value of Fo/Skr0 . 4 is one that has 40 percent of the polirhed-rod stroketaken up in rod stretch. When operating at very lowspeeds, where overtravel is nil, this means that the netstroke at the plunger is 60 percent of the polished-rodstroke. Once again, it is convenient to think of Fo/S kr asrod stretch although it, too, is dimensionless. We shouldnot try to picture the value of Fo/S kr as a measure ofthe size of the pumping unit. For example, two unitsshown on Fig. 7 (p . 23 1) (repor t numbers 303 and 435)each have an Fo/Skr value of 0.269. The unit for reportnumber 303 is a 74-in. stroke unit pumping from 4,300ft within a 2-in. pump, 718- to 314-in. rods, and requiresa 320 API reducer size.

The unit for report number 435 is a 120-in. strokeunit pumping from 9,425 f t with a 1-114-in. pump, 1-in.

718-in. 314-in. rods, and requires a 64 0 API gear-reducer size.It is seen that one value of the non-dimensionalparameter Fo/S kr can cover a whole series of pumpingunits, rod strings, and plunger sizes. Thus, thes,e twoimportant non-dimensional parameters, N / N o andFo/SKr , allow us to correlate a whole group of pumpinginstallations without having to consider an infinitenumber of cases on an individual basis.

NOMENCLATUREIn comparing the formulas for calculating peak

.polished-rod load, minimum polished-rod load, and peaktorque for both the conventional and API methods, thefollowing nomenclature will apply.PPRL = peak polished-rod load, poundsMPRL = minimum polished-rod load, pounds

PT = peak torque, inch-poundsWra = weight of rod string in air, poundsWrf = weight of rod string in fluid, poundsL = pump depth, feetS = polished-rod stroke, inchesN = pumping speed, strokes per minuteA p = full plunder area, square inchesA , = average area of rod string, square inchesF = 0 . 4 3 3 L ( A p )= total weight of fluid based on

full plunger area, poundskr = spring constant of the rod string, pounds perinchG = modifying factor for conventional units to

correct for the deviation from simple har-monic motion-usually has a magnitude ofabout 1.05 for conventional crank-balancedunits and a value of 1 for air-balanced units.G I =same as G , except that it applies to theupstroke on special geometry units only.Usually has a value of about 0 93G2 =same as G , except that it applies to thedownstroke of special geometry units only.Usually has a value of about 1.2.

F l = peak polished-rod load less the weight of therod string in fluid, poundsF2 = weight of the rod string in fluid less theminimum polished-rod load, poundsT = crank torque without correction factors,

inch-poundsT , = torque adjustment constantPEAK POLISHED-ROD LOAD

The most widely used conventional method of pre-dicting peak polished-rod load is shown in two forms inEquations ( l ) , (2), and (3) for conventional units.

For those who prefer to express fluid load as afunction of net plunger area:PPRL 0 .433L (A p-A , ) r a r a ( ~ ~ 2 / 7 0 , 5 0 0 )1 )


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222 FRED D GRIFFIN

Another approach, which gives identical results, de-fines fluid load as a function of full plunger area:

PPRL 0 .4 33 L ( A p ) W r f + Wra ( s N ~ / ~ o , ~ o o )2)

P PR L = F o + W r f + ra ( ~ ~ 2 / 7 0 , 5 0 0 ) 3)For units with special geometry:

For air-balanced units:PPR L Fo W rf 0 .7 Wra ( ~ ~ 2 / 7 0 , 5 0 0 )3b)

By inspection, i t is obvious tha t although the conven-tional method of predicting peak polished-rod load doesconsider the acceleration of the rod string, it does nottake into account the harmonic effects of avibrating rodstring.

The API method for predicting peak polished-rod loadis as follows:

PPR L Wrf [ (F l /Skr) (Skr) ] 4)The term (F1 /S kr ) s a non-dimensional parameter takenfrom a curve in APZ RP ~ I L hich plots (F1/Skr)against ( N / N o ) for a series of values of (Fo /S kr ) . Thesecurves take into account the effect of rod-string har-monics, as well as the normal acceleration effects. TheAPI method does not introduce any modifying factorsto take into account units with special geometry.

MINIMUM POL ISHED-RO D LOA DThe conventional formula for minimum polished-rodload for conventional geometry units is:

MPRL Wrf Wra ( ~ ~ 2 / 7 0 , 5 0 0 ) 5)For units with special geometry:

For air-balanced units:MPRL r f 1 .3 Wra ( ~ ~ 2 / 7 0 , 5 0 0 ) 5b)

Here again the deceleration of the rod string is con-sidered, but the dynamic effects are not.

The API method for predicting minimum polished-rodload is:

The term (FzISk,) is a non-dimensional parameter takenfrom a curve in APZ RP II L which plots (F2/Skr)against ( N / N o ) for a series of values of (F0/Skr) . Thesecurves do consider the normal deceleration effects, plusthe effects of rod harmonics.

PEAK TORQUEThe conventional method for calculating peak torque

for units with conventional geometry is:PT [ (PPRL) ( f i fPRL) ] S /2 G 7 )

For units with special geometry:PT [(PPR L x ( G I ) M P R L ) ] x ( G 2 ) r S / 2 8)

The API method for calculating peak torque is:PT (2T / s ~ ~ ) S kr x Ta 9)

The factor ( 2 ~ / ~ 2 k ~ )s taken from a curve in APZ RPI I L which plots ( 2 ~ / ~ 2 k ~ )gainst ( N / N o ) for variousvalues of (Fo /S kr ) .Ta is a torque adjustment factor.

Both the conventional and API methods of calculatingpeak torque assume that the peak and minimumpolished-rod loads occur at a crank position whichresults in an optimum mechanical advantage. Thus, theassumption is that the peak and minimum polished-rodloads for conventional units occur at the 75-deg and the285-deg crank positions. As is commonly known, this isnot always the case. When these loads do come atpositions on the crank cycle other than these assumedpositions, then errors of considerable magnitude mayresult no matter which method of calculation is used.

The assumption is also made that there is no fluidpound or gas interference. This is not the usual case, butit would be difficult to include these factors in anymathematical formulation.

APZ RP 1IL does not include a peak-torque predic-tion for units with special geometry.Another assumption the API method makes is that

the mechanical efficiency of the pumping unit is 100percent. Some conventional methods of calculating peaktorque make this same assumption, although at least onemajor manufacturer uses a mechanical efficiency of 93percent. Values tabulated in this paper for the conven-tional method, however, assume 100 percent mechanicalefficiency so they will be on a comparable basis with theAPI method. It could be argued at length as to just whatthe mechanical efficiency of a pumping unit from thepolished rod to the slow-speed gear should be;however, it definitely is not 100 percent. An efficiencyof 93 percent seems reasonable when comparing themechanism of a pumping unit with similar types ofmachinery. It is recommended that the API consider amechanical efficiency factor in future revisions of APZR P I ? L .

Still another assumption the API method, as well asthe conventional method, makes in predicting peaktorque s that the pumping unit is always perfectly


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N W API DESIGN CALCULATIONS FOR SUCKER ROD PUMPING SYSTEMS 223

counterbalanced. This is a very naive assumption, to saythe least Of the 77 wells listed in Fig. 7, the average&crease in actual torque on the gear reducer over what itwould have been with perfect counterbalance was 19.7percent. Some units were overloaded well over 50percent with respect to gear-reducer torque because ofan out-of-counterbalance condition. It is regrettably truethat many operators do not appreciate fully the realimportance of correct counterbalance as a most impor-tant influence on the torque imposed on a gear reducer.An out-of-counterbalance condition should always beconsidered, if we are to be realistic, in determining thesize of the pumping-unit, reducer n o matter whichmethod of calculation is used. It is, therefore, recom-mended that a multiplying factor of 1.2 always beapplied t o the calculated required torque before the finalselection of the pumping-unit reducer is made. Thisrecommendation will not necessarily result in a larger

.gear reducer than would have been selected previously;but it may do so, depending upon where the calculatedtorque, as modified, falls.

COMPARISON OF PREDICTED AND MEASUREDLOADS FOR 77 WELLS

In comparing the API method with the conventionalmethod of calculating peak polished-rod load, minimumpolished-rod load, and peak torque, it is meaningful tocompare both methods with measured results takenfrom dynamometer studies recorded over a period ofseveral years. In these studies, the peak and minimumloads are taken directly from the dynamometer cards.The counterbalance effect is calculated from thereported position and size of the counterweights. Thepeak torque on the gear reducer in all cases had beencalculated by API Std 1 1 E using accurate torque factorsfor the polished-rod positions taken at every 15 deg ofcrank rotation.

An effort was made in selecting the dynamometerstudies, results of which are recorded in Fig. 7 (p. 231) ,to cover a wide range of condit ions consideringproduction, depth, pumping speed, and pumping unitsize. Dynamometer cards with obvious well abnormali-ties were not considered.Peak Polished-rod Loads

Referring to Fig. and 2 (p . 225 and 226), it isnoted that strictly from a standpoint of averages, theAPI method fo r predicting peak polished-rod load isvery accurate, predicting loads that average only 1.4percent greater than measured. The conventionalmethod predicts peak polished-rod loads that averageonly 3.43 percent less than measured. Although the"average" deviation for both methods is certainlyaccurate enough, not too .much can be said for therange of predicted loads. For the API method(excluding the very extreme deviations), 95 percent of

the wells fall in a range of from -22 percent to +32percent. Similarly, the conventional method predictsloads that are measured t o be -30 percent to +25percent.

It is disappointing in both methods that the range of'deviation of calculated from measured is so great. Itshould not be so surprising, perhaps, if we stop toconsider that so many of the variables that affect wellloads are either too complex for mathematical treatmentor are variables that depend upon well data that is nearlyalways unavailable at the time of computation.Minimum Polished-rod Loads

In the calculations for minimum polished-rod loads,results show that the API method is definitely superiorto the conventional one, see Fig. 3 and 4 (p. 227 and228). The API method made predictions that averagedonly 28 lb above measured; whereas, the conventionalmethod predicted minimum loads that averaged 1,299 lbabove. Once again, the range of deviation in bothmethods was wider than would have been desired.

Peak TorqueConsidering the fact that both methods, on an average

basis, predict peak loads that deviate from measuredonly a small amount; and considering the fact that theAPI method was shown t o be considerably moreaccurate with respect to minimum load, then we couldexpect the API method to be more accurate on a torquecomparison. This proves to be the case.

Referring to Fig. 5 and 6 (p. 229 and 230), the APImethod predicts torque values that average 7.26 percentabove measured. The conventional method calculatestorques that are 18.82 percent below measured results.Thus, the API method is more conservative than theolder method of calculation. The wide range of deviationfrom measured in the torque calculations reflects, ofcourse, the wide range of deviations in both the peakand minimum polished-rod loads discussed previously.

DEGREE OF DIFFICULTY OF CALCULATIONSBecause the API method takes into account many

more variables than the conventional method, it is moredifficult and time-consuming to make the necessarycalculations. For one not too familiar with API RP I I Lor for one who uses it very infrequently, it will take atleast three times as long to make the necessary calcula-tions for designing a pumping-unit-sucker-rod systemthan does the old conventional way. Although it is moreaccurate, at least to some degree, the increased difficultyin the calculations will limit its universal use unless acomputer program is written which will tabulate all thewell loads, reducer torques, rod stresses, etc. withinsome set limits which have been found to be practical.


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224 FRE D D. GRIFFINFor example, a computer program could be writtenaround PZ RP 11L which would print out for, say,every 500 ft of depth and in production increments of,say, 100 BID all conceivable pumping-unit-sucker-rodcombinations that would satisfy all the limiting condi-

tions of the program. Some of these conditions might beto limit pumping speed to a minimum of 2 or 3 spm anda maximum of perhaps 20 or 25 spm. Other limitationswould have to be a limiting rod stress, a minimum figurefor minimum load, etc.It is believed that the computer approach whichwould provide the user and the manufacturer withprinted data in book form is the best course for the APIto pursue in obtaining the maximum benefit from PIR P 1IL Some oil companies have done this with theirown method of computation and have found that thedata books are widely used and widely accepted by fieldpersonnel.

CONCLUSIONS AN D RECOMMENDATIONSUse of PZ RP 11L makes it possible to:1. Predict values for peak polished-rod load which areslightly more accurate than the more conventionalmethods of calculation.2. Predict values for minimum load which are muchmore accurate than those determined by conventionalmethods.

3 Predict gear-reducer torques more accurately, pri-marily because minimum loads can be more accuratelydetermined.PZ RP 11L makes some broad assumptions inpredicting peak torque which can cause a considerable

error in unit selection. These are:a. That the maximum and minimum loads occur atthe 75-deg and 285-deg crank position, where thepumping-unit torque factors are optimum. Inas-much as any calculation method for sizing equip-ment would almost always be made in the absenceof a dynamometer card, this assumption seemsvalid.b. That pumping-unit efficiency is 100 percent. Amore realistic value in the range of 90 to 95 percentshould be introduced.c. That the pumping unit is perfectly counterbal-anced. Years of experience have shown that this is

unrealistic. Torque calculations for t he gear reducershould include a multiplying factor of 1.2 inaddition to the efficiency factor mentioned in b.)to compensate for out-of-counterbalance condition.TO make PI RP IIL more useful, a computer

program should be written which will tabulate in bookform many sucker-rod pumping systems for a given setof conditions, from which the user can make a selectionbased on his own judgment.


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NEW API DESIGN CALCULATIONS FOR SUCKER ROD PUMPING SYSTEMS 225maso-w4 30

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PEAK POLISHED RODLOADo API METHOD

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2 6 FRED D GRIFFIN

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PEAK POLISHED ROD LOADCONVENTIONAL ME THOD


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N W API DESIGN CALCULATIONS FOR SUCKER -ROD PUMPING SYSTEMS 227

MINIMUM POLISHEDROD LOADCONVENTIONAL METHOD

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MINIMUM POLISHED ROD LOADAPI METHOD


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228 FRED D GRIFFIN

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6ooo-MINIMUM POLISHED RODAPI METHOD


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23 FRED D GRIFFIN

PEAKTORQUEAPI METHOD

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I I~ t d 4m PEAK TORQUEa CONVENTIONAL METHOD


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Fig. 7 Comparison of Predicted and Measured Loads for 7 7 Wells

Tarea 19_metodo API Rp11l

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