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  1. MATHEMATICAL MODEL ENHANCES PUMPING -UNIT DESIGN

    J.P. Byrd Consultant Lufkin, Tex. A performance effectiveness (PE) model has been developed to assist determining the optimal and most economical performance of different pumping-unit geometries. This new, modified approach, called the PE model, recognizes rod and structural loading; rod loading alone; torsional loading; lift efficiency; surface efficiency of the prime mover, belts, drive train, and structural bearings; prime-mover size requirement; and power consumption. The limitations include not considering cost of the unit, prime mover, or rod string; assuming the prime mover to be a standard Nema D, oil field motor; no signaling for overloading of the system components; and assuming pumping a full barrel of incompressible fluid, off bottom with each stroke. PUMPING-UNIT DESIGN One of the most helpful and convenient aids in the successful application of conventional beam-pumping units is the American Petroleum Institute bulletin, 11L-3, Sucker Rod Pumping System Design Book. Expanding the work of Sucker Rod Pumping Research Inc. and the Midwest Research Institute, the API produced this set of tables (11L-3) containing thousands of different precalculated pumping options, or modes. The designs are generated by using a model of the wave equation applied to sucker-rod pumping with conventional beam units. These API tables have been widely accepted, and though certain sections have been questioned and in some cases revised, in general they have made a substantial contribution to the petroleum industry in facilitating the application of conventional pumping units. According to these tables, there are 20 API approved sucker-rod sizes, 18 approved stroke lengths, and 10 different API plunger diameters. Thus, in lifting a given amount of fluid from a particular depth, with a conventional pumping unit, theoretically there could be about 3,600 beam and sucker-rod options, or pumping modes, for a single, artificial-lift application. Not considered is variation in pumping speed. Obviously, some of these thousands of pumping modes are either impractical, uneconomical, or both. But even the elimination of 90% of them still leaves over 350 pumping modes to consider. In the API 11L-3 design tables, for a single application, i.e., lifting 400 b/d from 3,500 ft, requires the theoretical evaluation of nearly 400 different pumping modes. Other applications require the consideration of even more than 400. But which of these hundreds of pumping modes is the most effective as regards to economy, longevity, and efficiency? Theoretically, any of them can do the job. But which one is best? Perhaps the "best pumping mode" is in the eye of the operator. Often consideration is given to the pumping mode having the lowest torque, or lowest rod or structural loading, or highest efficiency, etc. But the pumping mode having the lowest torque, might not be the most efficient, nor the mode resulting in the lowest rod loading might not afford maximum economy, etc. Thus, the question arises, "What is the best and most effective pumping mode, when considering all, or most, of the dominant factors involved?" PAST RATING CONCEPTS A significant approach to optimum pumping mode selection was made in 1975 by Manuel Estrada.1 Included in his study was a section on an economic index (EI). According to the study the EI gives the most economical pumping combination when considering torsional, structural, prime mover, and lifting requirements. By selecting the lowest EI number, the most economical pumping system is defined. Estrada's equation for the economic index is: [SEE FORMULA] where: WMAX = Peak polished rod load, lb TP = Peak torque, in-lb (inbalance) PPR Polished rod horsepower LE Lift efficiency This is a simple and direct equation, relating some of the important variables of the beam pumping system, and casting them into a series of index numbers, each associated with a particular pumping mode (Table 1). Later in 1980, Luis Valera M. included an extension and refinement of the Estrada pumping mode index.2 Valera entitled his rating number, the comparative economic index (CEI). This index assumes the overall economy of a beam and sucker-rod pumping system is a direct function of PPRL, PT, nameplate horsepower (HPNP), the cyclic load factor, and an inverse function of lift efficiency. Combined into a simple mathematical expression the correlation is: CEI = 10-9 x PPRL x PT x HPNP x CLF LE where: PPRL = Peak polished rod load, lb PT = Peak torque, in.-lb (in-balance) HPNP = Nameplate horsepower CLF = Cyclic load factor LE = Lift efficiency The correlation assumes a weight of 1.0 for each of the five variables used to calculate CEI values. Wherever experience dictates, the weighting can be done empirically. For instance, in an area where power costs are excessively high, the CLF could be weighted greater than one. For a given situation, the selection of the lowest CEI assures maximum economy (Table 2). In 1982, Solomon D. Lekia,3 not only expanded, but refined the work of Estrada and Valera. The designation that Lekia used for indexing the various pumping modes was called the performance index (PIX) which is an important number in evaluating the overall economy of a beam and sucker-rod pumping system. It is given conceptually as follows: PIX = 10-8 PPRL x PT x HPNP x CLF LE x ITE where: PPRL = Peak polished rod load, lb PT = Peak torque, in.-lb (in-balance) HPNP = Nameplate horsepower CLF = Cyclic load factor LE = Lift Efficiency ITE = Index of torsional effectiveness A power of one is given to each of the six variables to weight them equally. Peak polished-rod load, peak torque, nameplate horsepower, and cyclic load factor appear in the numerator to keep their value as low as possible on various installations. Conversely, lift efficiency (LE) and index of torsional effectiveness (ITE) appear in the denominator because high lift efficiencies (LE) and indices of torsional effectiveness (ITE) are indicative of good pumping operations. For a given design application, selecting the lowest PIX value assures maximum economy (Table 3). The work of Estrada, Valera, and Lekia are important concepts, expansions, and refinements for developing a valid procedure for selecting the optimum pumping mode for a beam and sucker-rod pumping application. Often, in the past, pumping unit design considered one, or perhaps two major variables, such as peak torque or peak polished-rod load, etc. One virtue of indexing pumping modes is that most of the important variables can be considered in the formulation-not just one or two. Although there are many different modes for the venerable conventional pumping unit, it was not until the 1920s that a significantly different beam unit geometry, the air-balance unit, became popular. This unit has its own spectrum of pumping modes. With the advent of the Mark II pumping unit in the mid-1950s, a third menu of pumping mode possibilities was added. As the Mark II patents expired in the late 1970s, other beam pumping geometries appeared with their own unique series of modes, further adding to the vast number of pumping mode possibilities to be reckoned with. Each pumping mode would have a different kinematic or performance output, and the most desirable pumping mode for one beam unit might be different from the optimum mode of another type of geometry. Thus, evaluating the possible pumping modes for a single application, considering two or three different geometries, could become a sizeable task. For instance, in comparing two different unit geometries for an application, one might be superior in reducing structural load, rod-load range, and lift efficiency, while a second might lower torque peaks, the cyclic-load factor, and surface efficiency. Which is the more effective pumping mode? Obviously, the substantial number of physical constraints on the typical well often makes the number of pumping mode options manageable. But which one is best? PERFORMANCE DISPARITY To further illustrate the performance disparity of conventional-unit pumping-mode options, some practical, others impractical, reference is made to API bulletin 11L-3 for several dramatic examples which underscore the need for some kind of rating index in addition to the regular, comprehensive, predictive survey. EXAMPLE NO. 1 On p. 370 of these tables, a pumping mode using API-75 rods and pumping 8.0-300 in. strokes/min (spm) with a 1 1/2-in. plunger, the peak polished-rod load is given as 20,366 lb. To handle this same pumping application (p. 373), a conventional unit using API-98 rods and pumping 11-100 in. spm with a 2.75 in. plunger, will develop a peak polished-rod load of 40,403 lb. This is almost exactly twice the structural load requirement when using the pumping mode employing API-75 rods. Thus, the conventional unit performs the same amount of work per day in each case, i.e., lifting 600 bbl of fluid/day from 6,500 ft, but by selecting the proper pumping mode, the unit structural load can be cut in half. EXAMPLE NO. 2 On p. 372 of the API bulletin, it can be seen that a pumping mode of 10.4-300 in. spm driving a 1.25-in. pump, with API-97 rods, develops a peak torque of 2,358,000 in.-lb to lift this same application of 600 bbl of fluid/day from 6,500 ft. On the preceding page (371), using API-87 rods, a conventional unit, pumping 19.9-64 in. spm and driving a 2-in. plunger develops a peak torque of but 346,000 in.-lb. Thus, selecting the previous pumping mode requires a speed reducer to accommodate nearly seven times as much peak torque as is required with a second mode to handle the same pumping job. In one case, peak torque slightly overloads an API 320 in.-lb reducer, while in the second case, the same pumping job requires nearly the largest beam pumping speed reducer manufactured, a 2,560,000 in.-lb box. In both cases, the same amount of work is performed per day, i.e., lifting 600 bbl from 6,500 ft. In this conventional unit application, one pumping mode developed a peak torque about 700% greater than that of the second pumping mode. EXAMPLE NO. 3 The desirability of selecting an optimum pumping mode is strikingly demonstrated on p. 148 of the API 11L-3 tables. To lift 400 b/d from 3,500 ft with an API-77 rod string, listed are, among others, two different pumping modes. One requires a polished-rod horsepower of 57.4, another needs but 10.7. Assuming these figures are correct, rod-string losses would be 118 times greater in the former pumping mode compared to the latter, and over five times as much polished-rod power would be consumed by the higher horsepower mode. Furthermore, as regards to only lift efficiency, which is but one of the factors involved in total system efficiency, in the 57.4 hp mode about 20% of the polished-rod input energy is devoted to fluid elevation, while 80% is wasted as heat loss. In the contrasting pumping mode, about 96% of the polished-rod work is devoted to beneficial fluid lift, and only 4% to heat loss. This should be adequate justification for the further understanding and exploration of different beam-pumping modes. EXAMPLE NO. 4 One of the most important aspects of proper pumping mode selection involves prime-mover horsepower requirements. On p. 372 of the API tables, using API-97 rods and driving 10.4-300 in. spm, with a 1 3/4-in. plunger, the resulting polished-rod horsepower is 89.9. This number is a direct function of the size of the prime mover required. With the same API-97 rod string, it can be seen that using a 2.75-in. pump and driving 9-120 in. spm, the polished-rod horsepower required is 30.7 or approximately one third the amount needed in the preceding example. Obviously, if the API 11L-3 figures are correct, this means a prime mover three times as large would be required to perform the same job, when lifting 600 bbl of fluid/day from 6,500 ft with a conventional beam-pumping unit. Although misapplications of the magnitude of the four examples listed above seldom, if ever, occur, such disparity, even theoretical, emphasizes the fact that proper pumping mode selection can significantly increase the effectiveness and the economy of lifting fluid with a beam and sucker-rod pumping system. PERFORMANCE EFFECTIVENESS The performance effectiveness rating (PE) is a simple and direct mathematical model (see box) that seeks to consider balance, and harmonize most of the dominant factors concerned with performance effectiveness in lifting fluid with a beam and sucker-rod system. Following is a list and rationale of the various PE equation components. Px is the structural and rod load factor, relating the dead weight of rods and fluid to the peak polished-rod load. This is the reciprocal of the impulse factor used in earlier peak polished-rod load formulation. Rx simply ratios the weight of rods and fluid to the load range of the system in operation. Tx is a mathematical relationship of the ratio of average torque to peak torque, modified by a constant to account for a fundamental differential in ranges between Px, Rx, and Tx as well as attempting to balance torsional considerations properly to rod and structural factors. Lx is simply the ratio of hydraulic horsepower to polished-rod horsepower, and is the quantity known as lift efficiency (LE). Sx is the surface efficiency of the machinery from the input of the Nema "D" motor to the output of the pumping unit. This equation for surface efficiency not only covers the mechanical efficiency of the pumping unit proper at rated capacity, or thereabouts, but also considers prime mover and belt efficiency as well. Mx is a factor based on lift efficiency, surface efficiency, and the cylic factor, giving appropriate credit to a smaller prime mover adequately handling the required hydraulic work load. Cx is the inverse of the cylic load factor times a 1.5 multiplier, which is a direct index of the power consumed. Although the PE concept is primarily a performance effectiveness index, a prime-mover size factor and a power-consumption factor were arbitrarily added to the equation. No attempt has been made to consider either the first cost of the pumping unit and prime mover or their maintenance costs. The larger the PE index number, the more effective the pumping mode. Several important application functions in the optimizing of beam-pumping modes can be facilitated by using the new performance effectiveness system (PE), or perhaps one of the three earlier optimizing versions. Because of differences in the mathematical models used, a similar, though not exact, correlation should be expected from the various tables. Unfortunately, the PE system tables are still proprietary but hopefully will be available in the future. Pumping-mode optimizing tables can come in at least two different arrangements. The first has an index number included in the regular arrangement of the tables, such as the examples in Tables 1, 2, and 3. Or secondly, the tables can be arranged in either ascending or descending order according to the indexing system used. Pumping-mode indexing tables can perform several useful functions such as: Comparing the existing pumping mode to the optimum pumping mode to see if they are the same or similar Comparing two different pumping unit geometries using the same pumping mode to determine the difference in performance effectiveness Comparing the optimum performance effectiveness mode of one geometry to the same pumping mode of a second nonoptimized geometry Comparing the optimum pumping mode of one geometry to the optimum pumping mode of another geometry. Because the new PE tables are not currently available, in the following examples a combination of the PE model and the Lekia optimizing tables have been substituted. EXAMPLE 1 The problem was to produce 400 b/d from 4,000 ft with API-76 (Grade D) rods employing a Class III beam-pumping unit. The operator, using his own experience, selected a pumping mode of 16.5 - 54 in. spm with a 2 1/2-in. plunger and 76 rods. Was this selection, based on the operator's experience, similar to the optimum pumping mode? EXPERIENCE SELECTION 16.5-54 in. spm x 2 1/2 in. Px = 0.765 Rx = 0.996 Tx = 1.536 Lx = 0.923 Sx = 0.767 Mx = 0,366 Cx = 0.776 PE = 3.476 OPTIMUM SELECTION 13.6-74 in. spm x 2 in. Px = 0.7080 Rx = 0.8780 Tx = 2.296 Lx = 0.773 Sx = 0.750 Mx = 0.407 Cx = 1.053 PE = 3.711 Obviously, the PE's of both pumping modes are reasonably close, showing that the operator's experience has resulted in a good pumping effectiveness mode, and an economical pumping arrangement. Changing the existing pumping mode should provide a 5 or 6% improvement, which is significant enough to be considered. EXAMPLE 2 An operator needs to produce 400 b/d from 8,500 ft with API-86 (Grade D) rods. The pumping mode selected is 12.7 - 120 in. spm with a 1.5-in. plunger. In this case, which pumping unit geometry is the most effective, Unit E or Unit F? UNIT E 12.7 - 120 in. spm x 1.5-in. plunger Px = 0.695 Rx = 1.151 Tx = 1.852 Lx = 0.719 Sx = 0.779 Mx = 0.339 cx = 0.909 PE = 3.319 UNIT F 12.7 - 120 in. spm x 1.5-in. plunger Px = 0.702 Rx = 1.162 Tx = 2.351 Lx = 0.710 Sx = 0.751 Mx = 0.407 Cx = 1.145 PE = 3.798 The PE for unit F appears to be some 13% better than the effectiveness of unit E, both operating with the same pumping mode. Presumably the PE is not optimum for either geometry. EXAMPLE 3 Another operator must produce 400 b/d from 3,500 ft with API-77 rods. The optimum pumping mode for Unit G is 14.6 - 64 in. spm with a 2.0-in. plunger. A comparable Unit H (different geometry) employs essentially the same pumping mode, except it is not known if this mode is optimum for geometry H. UNIT G 14.6 - 64 in. spm x 2.0-in. plunger Px = 0,692 Rx = 0,835 Tx = 3,002 Lx = 0,764 Sx = 0,775 Mx = 0,398 Cx = 1,008 PE = 4,148 (Optimized) UNIT H 14.4 - 64 in. spm x 2.0-in. plunger Px = 0.746 Rx = 0.973 Tx = 1.735 Lx = 0.828 Sx = 0.785 Mx = 0.393 Cx = 0.906 PE = 3.544 (Not optimized) In this comparison, Unit G optimized appears to be about 15% more effective than Unit H, not optimized. EXAMPLE 4 Lifting 500 b/d from 4,500 ft with API-87 rods, optimumly, with Geometry 1, suggests a pumping mode of 17.1 - 54 in. spm x 2.5-in. plunger, with its accompanying PE number. Also a second PE Number is desired for Geometry J when it is operating in the identical (though nonoptimized) pumping mode as Geometry 1. And finally, a third PE number is calculated for Geometry J when it is operating in its optimum pumping mode, and compared to Unit I when also optimumly driven. UNIT I OPTIMIZED 17.1 - 54 in. spm x 2.5-in. plunger Px = 0.749 Rx = 1.008 TX = 1.961 Lx = 0.815 Sx = 0.776 Mx = 0.395 Cx = 0.938 PE = 3.684 UNIT J WITH I PUMPING MODE Px = 0.754 Rx = 1.009 Tx = 1.768 Lx = 0.780 Sx = 0.773 Mx = 0.347 Cx = 0.863 PE = 3.339 UNIT J OPTIMIZED 11.8 - 86 in. spm x 2.25 plunger Px = 0.720 Rx = 0.924 Tx = 2.667 Lx = 0.796 sx = 0.774 mx = 0.418 Cx = 1.017 PE = 4.091 Although Unit I optimized shows a 9.5% greater effectiveness than Unit J in the same pumping mode, when Unit J's mode is optimized, its PE becomes 10% greater than Unit I's optimized mode and 18.5% more effective than its own nonoptimized mode. This example illustrates that a rigorous comparison of the PE for different pumping unit geometries, over a given application, cannot be finally evaluated until the optimum PE for both geometries is determined and compared. ACKNOWLEDGMENTS Grateful acknowledgment is made to Kermit Brown and John Day for their guidance and assistance and to Lufkin Industries Inc. for its support. REFERENCES Estrada, Manuel E., "Design and Optimizing Tables for the Mark II Oil Field Pumping Unit," The University of Tulsa, Graduate Division, 1975. Valera M., Luis A., "A Technique for Determining Optimum Geometry and the Most Economical Pumping Mode for Different Beam and Sucker Rod Systems," The University of Tulsa, Graduate Division, 1980. Lekia, Solomon, and Keteh, D., "An Improved Technique for Evaluating Performance Characteristics and Economy of the Conventional and Mark II Beam and Sucker Rod Pumping Systems," The University of Tulsa, Graduate Division, 1982. BIBLIOGRAPHY Gibbs, S.G., "Predicting the Behavior of Sucker Rod Pumping Systems," Journal of Petroleum Technology, July 1963. Gibbs, S.G., "Computing Gearbox Torque and Motor Loading for Ultra-High Slip Prime Movers," Paper SPE 5149, 49th Annual Fall Meeting of the SPE-AIME, Houston, Oct. 6-9, 1984, also in JPT, September 1975, pp. 1153-1159. Byrd, J.P., and Beasley, W.L., "Predicting Prime Mover Requirements, Power Costs, and Electrical Demands for Beam Pumping Units," Paper No. 374035, 25th Annual Technical Meeting of the Petroleum Society of CIM, Calgary, May 1-11, 1974. Brown, Kermit, Day, John, Byrd, Joe, et al., The Technology Of Artifical Lift Methods, Vol. 2a, PennWell Publishing Co., Tulsa. Copyright 1990 Oil & Gas Journal. All Rights Reserved.

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