PCP Axial Load: Theory and Lab Results

SPE 90153 PCP Axial Load: Theory and Lab Results F.J.S. Alhanati, SPE, C-FER Technologies and P. Skoczylas, SPE, C-FER Technologies

Copyright 2004, Society of Petroleum Engineers Inc. This paper was prepared for presentation at the SPE Annual Technical Conference and Exhibition held in Houston, Texas, U.S.A., 26–29 September 2004. This paper was selected for presentation by an SPE Program Committee following review of information contained in a proposal submitted by the author(s). Contents of the paper, as presented, have not been reviewed by the Society of Petroleum Engineers and are subject to correction by the author(s). The material, as presented, does not necessarily reflect any position of the Society of Petroleum Engineers, its officers, or members. Papers presented at SPE meetings are subject to publication review by Editorial Committees of the Society of Petroleum Engineers. Electronic reproduction, distribution, or storage of any part of this paper for commercial purposes without the written consent of the Society of Petroleum Engineers is prohibited. Permission to reproduce in print is restricted to a proposal of not more than 300 words; illustrations may not be copied. The proposal must contain conspicuous acknowledgment of where and by whom the paper was presented. Write Librarian, SPE, P.O. Box 833836, Richardson, TX 75083-3836, U.S.A., fax 01-972-952-9435. 01-972-952-9435.

Abstract Accurate estimation of pump axial load in Progressing Cavity Pump (PCP) applications is an important aspect of the installation design. design. It affects the calculation of rod stresses, stresses, and therefore the determination of the rod string torque capacity. In deviated deviated wells, wells, it also directly directly affects affects the the calculated rod-tubing contact loads, and therefore the evaluation of wear related issues. This paper discusses the current theory related to the calculation of axial load in PCP applications, in light of laboratory test results obtained with 25 different pump models. The axial load is normally estimated by multiplying the differential pressure across the pump by an effective pump cross sectional sectional area. However, given given the complexity complexity of the  pump geometry, it is not straightforward how to properly calculate this effective area. In the paper, it is shown shown that intuitive choices, such as the circle defined by the rotor minor diameter, the circle defined by the rotor major diameter, or the  pump cavity area, area, do not yield adequate adequate results. A new method is suggested that not only matches quite well the extensive laboratory data for single-lobe pumps but also matches reasonably well the limited data available for multi-lobe pumps. Introduction In a PCP application, the dominant load affecting stresses in the rod string is is torque. Axial load does exist, exist, however, both both due to rod weight and due to a load generated at the downhole  pump, and it can be significant in many cases. Other design considerations affected by the rod string axial load are pump space-out and, in deviated wells, rod-tubing wear and rod fatigue life. life. It is therefore therefore quite important to to be able to accurately calculate the axial load in many PCP applications.

Calculation of Pump Axial Load The axial load generated at the pump is normally estimated  based on the pressure forces acting on the pump rotor, which is connected to the rod string, as illustrated in Figure 1. Note that, for a downhole-driven PCP, the formulation would be slightly different. The pump axial load is therefore determined by multiplying the differential pressure across the pump by an effective pump cross sectional area, and applying a small correction to account for the cross sectional area of the rod string:

 F  = ∆ P × Aeff  − P d  × Arod 

[Eq. 1]

where ∆ P = P  d  −

P i

[Eq. 2]

The geometry of a PCP pump is somehow complex; so, it is not immediately obvious what value for the cross-sectional area ( A  Aeff ) area one should use. A few approaches have been suggested in the literature1,2,3, and the formulas most commonly used in the industry, for single-lobe pumps, are listed below. is the cross-sectional cross-sectional area of the Stator Cavity Area. This is stator cavity when the rotor is not present:

 Aeff 

=

 D 2

π  

4

+ 4eD

[Eq. 3]

Rotor Minor Diameter Area. This is the cross-sectional cross-sectional area of the rotor:

 Aeff 

=

 D 2

π  

4

[Eq. 4]

Rotor Major Diameter Area. This is the area of a circle having a diameter equal to the major diameter of the rotor:

 Aeff 

=

π  

4

( D + 2e )2

[Eq. 5]

Area Open to Flow. This is the area open to flow flow when the rotor is installed inside the stator:

 Aeff 

=

4eD

[Eq. 6]

the area of the smallest Circumscribed Cavity Area. This is the circle which can be drawn completely outside the entire stator cavity:

 Aeff 

=

π  

4

( D + 4e )2

[Eq. 7]

2

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 Note that calculation of these areas requires knowledge of the detailed geometry of the pump, namely the rotor minor diameter ( D)  D) and the eccentricity (e (e), and these values are not readily available from the manufacturers catalogs. Later in the paper, it is shown that most of these intuitive choices do not yield adequate results. In fact, only perhaps the least intuitive one, the circumscribed circumscribed cavity area, does.

New Method When investigating this topic, the authors searched not only for a formulation that would yield adequate results, but also for a formulation that could be justified on a physical basis. The proposed new formulation is:

 Aeff 

=  N S  × A Flow +

A Rotor 

[Eq. 8]

For a single lobe pump ( N  s=2), this yields:

 Aeff 

= 8eD +

π  

4

D2

[Eq. 9]

The physical justification for this formulation is that it is the differential pressure across the cavities that results in the  pump axial load. In a PCP, the total differential pressure is distributed through all the cavities, but there are as many trains of cavities as the number of lobes in the stator. Therefore the  pump pressure differential would apply to the area open to flow, but as many times as the number of lobes in the stator. In addition, the pump differential pressure also applies to the cross sectional area of the rotor. Later in the paper, it is shown that this new formula matches well laboratory data not only for single-lobe pumps,  but also for multi-lobe multi-lobe pumps.

Laboratory Test Program Unlike beam pump aplications, a good measurement of axial load cannot be easily obtained in a PCP application. Therefore, to the authors’ knowledge, there has been little verification of the formulations being used to calculate axial loads to date. As part of a Joint Industry Project (JIP), a laboratory study was undertaken at C-FER Technologies’ laboratories in Edmonton, Alberta, Canada, in 1995 and 1996, to study the  performance of PCPs. PCPs. Twenty-five pumps from different manufacturers were installed in a test well, and run at several levels of speed, intake and discharge pressure. One of the many pump performance parameters measured was axial load, using a load cell specifically designed and constructed for this test program. Figure 2 shows the test well, with the wellhead installed. Also visible in the figure (at the top of the drivehead, immediately below the polish rod clamp) is the load cell which measured torque and axial load. Table 1 lists the pumps which were tested in the study. (Note that the test program took place in 1995-1996, and many of these pumps may no longer be available.) Only one of these pumps (#23) was a multilobe pump (with a 2:3 lobe configuration); all the others were single-lobe pumps. Table 2 lists the levels of speed, intake and discharge  pressure at which these pumps were tested. Note that not every pump was tested at all combinations of intake and

discharge pressure, pressure, for three reasons: reasons: the rated differential  pressure of the pump was not to be exceeded; in some cases (high speeds, high pressures, larger displacement pumps), the  power requirements exceeded the test system’s limits; and in some cases the pump was not able to produice any fluid at higher differential pressures (zero efficiency). All tests were conducted with water. In the test well, the pump was in a vertical position and  just one tubing joint below the wellhead. Measured independent parameters included flow rate, torque, and axial load; however, only axial load results are reported here.

Test Results Figure 3 shows the measured vs. calculated axial load for all the tests conducted with single lobe pumps in water, and using different effective area formulatins. It is evident from the figure that the calculated values for most pump effective area formulations formulations are too small. Only the Circumscribed Cavity Area and the New Area yield acceptable results. Figure 4 plots the errors (calculated force – measured force) for each formulation. Figure 5 is similar, except that it shows a relative error (percent of measured). Note that some data points were excluded from Figure 5 (while still present in Figures 3 and 4). This corresponds to data points where the calculated force was less than zero or the measured force was was less than 5 kN. This was done because small values of force can lead to very large relative errors, even for very small small absolute errors. errors. This is especially true for the Open Flow Area and Minor Diameter Area, for which many points have a calculated force less than zero (as can be seen in Figure 3.) Figures 4 and 5 again show that only the the Circumscribed Cavity Area and the New Area give reasonable results, both in terms of having a small mean error and a small standard deviation. deviation. Of these two, two, the Circumscribed Circumscribed Cavity Cavity Area has a smaller mean error, but the New Area has a smaller standard deviation (both in absolute and relative errors). Multilobe Pumps. As described above, the new method for calculating the pump effective area was developed considering both single and multi lobe pumps. Therefore, if the physical physical basis on which it was developed was correct, it should also yield reasonable results for for the 2:3 multilobe pumps. Figure 6 shows the measured vs. calculated axial load for all the tests conducted with this multilobe pump. pump. The errors are plotted plotted in Figure 7. While the circumscribed cavity area provides a slightly more accurate result for single lobe pumps, the New Area, or “Ns × Flow Area + Rotor Area” (Ns representing the number of stator lobes) yielded better results for multilobe pumps, and can perhaps be considered a better general equation. Conclusion 1. The effective area for use in calculating the axial load on the rotor of a progressing cavity pump subjected to a differential pressure is not immediately obvious. 2. Various areas have been suggested in the past, but many intuitive choices do not yield adequate results, when compared to lab test data.

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3. A new method is suggested, that not only matches quite well extensive laboratory data for single lobe pumps, but also matches reasonably well limited data available for multi-lobe  pumps.

Nomenclature  Aeff  = effective cross-sectional area for the pump (m²)  A flow = area of the the stator stator cavity cavity open open to flow (m²)  Arod  = cross-sectional area of the rod at the top of the pump pump rotor (m²)  Arotor  = rotor cross-sectional area (m)  D = rotor minor diameter (m) e= rotor eccentricity (m)  F  = axial load (N) pump discharge pressure (Pa)  P d  d  =  P i = pump intake pressure (Pa) ∆ P   = differential pressure (Pa) Acknowledgements The authors would like to acknowledge the Participants of C-FER’s Progressing Cavity Pumping System Technology Development JIP, for their contributions to the test program reported in this paper. References 1. Cholet, H.: Progressing H.:  Progressing Cavity Pumps, Pumps, Editions Technip, Paris (1997). 2. Presber, T.: “Lifting Heavy Heavy Oil with with Screw Pumps,” Pumps,”  presented at the 2nd  AOSTRA Can.-China Heavy Oil Technical Symposium, Beijing, China, Oct. 29-Nov. 1, 1990. 3. Klein, S.T. et al : “Well Optimization Package for Progressive Cavity Pumping Systems,” paper SPE 52162  presented at the 1999 SPE Mid-Continent Operations Symposium, Oklahoma City, March 28-31.

3

4

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Pump # 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Pump Manufactur  er BMW Pump RotaliftCorod GriffinLegrand RotaliftCorod BMW Pump RotaliftCorod RotaliftCorod RotaliftCorod GriffinLegrand GriffinLegrand National Oilwell National Oilwell BMW Pump BMW Pump RotaliftCorod GriffinLegrand National Oilwell RotaliftCorod GriffinLegrand National Oilwell Kudu Industries Kudu Industries Kudu Industries Kudu Industries Kudu Industries

Pump Model 28-1200 12-H-10

Nominal Displacement (m³/d/RPM) 0.28 0.1

Pressure Rating (kPa) 11760 11775

40-B-095

0.151

11968

12-E-17

0.17

11772

28-1200 (HS) 12-N-27

0.28

11772

0.27

11772

12-N-27

0.27

11772

12-N-27

0.27

11772

40-A-195

0.31

11968

40-A-195

0.31

11968

14-1200

0.35

9663

14-1200

0.35

9663

83-600 120-600 8-H-75

0.83 1.2 0.75

5886 5886 7848

30-A-400

0.636

8927

18-1500

0.48

12429

5-H-110

1.1

4905

30-600

0.95

8927

14-2100

0.626

9663

400TP900

0.806

8829

600TP900

1.2

8829

900TP1000

1.68

9810

200TP1200

0.392

11772

200TP1200

0.392

11772

Figure 1. Pressure Forces on PCP

Table 1. Pumps Tested in Study Test Speeds 75, 100, 150, 200 RPM RPM 100,200,300,400 RPM

Pumps 1, 2, 3, 3, 4, 6, 7, 7, 8 Pumps 5, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 24, 25 Pump 23

50,100,150,200 RPM Intake Pressures 200, 2000, 4000, 8000 kPa Discharge Pressures 200, 1000, 2000, 3000, 4000, 5000, 6000, 7000, 8000, 9000, 10000, 11000, 12000 kPa Table 2. 2. Test Conditions Conditions

Figure 2. Test Well

SPE 90153

5

40

35

30

25    )    N 20    k    (   e   c   r   o    F 15    d   e   r   u   s   a   e 10    M

5

0 Cavity Area Circumscribed Cavity Area Minor Diameter Area Major Diameter Area Open Area New Area

−5

−10 −10

0

10  x Area ∆P  x

20 − P   x A d 

30

40

50

 (kN)

rod 

Figure 3. Measured vs. Calculated Calculated Axial Load

New Area

New Area

Open Area   e   p   y    T   a   e   r    A   e   v    i    t   c   e    f    f    E

Open Area   e   p   y    T   a   e   r    A   e   v    i    t   c   e    f    f    E

Major Diameter Area

Minor Diameter Area

Major Diameter Area

Minor Diameter Area

Circumscribed Cavity Area

Circumscribed Cavity Area

Cavity Area

Cavity Area

−40

−30

−20

Figure 4. Error in Calculated Calculated Axial Load

−10 Error (kN)

0

10

−100

−50

0 Percent Error

Figure 5. Percent Error in Calculated Calculated Axial Load

50

6

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20

15

   N    k  ,    d10   a   o    L    l   a    i   x    A    d   e   r   u   s   a 5   e    M

0

Circumscribed Cavity Area 3 x Flow Area + Rotor Area

−5 −10

−5

0

5 Calculated Axial Load, kN

Figure 6. Measured vs. Calculated Axial Axial Load for Multilobe PCP

3 x Flow Area + Rotor Area

Circumscribed Cavity Area

−5

−4

−3 −2 −1 Error in Calculated Load, kN

Figure 7. Error in Calculated Load for Multilobe PCP PCP

0

1

10

15

20