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APPLICATIONS OF VISCOUS OIL IN A
ROD SUCKER PUMP SYSTEM
By
Utkarsh Bhargava
Submitted in Partial Fulfillment of the Requir...
ABSTRACT
When producing a well for some amount of time, there always comes a moment when there
is not enough natural energ...
ii
ACKNOWLEDGEMENTS
I would like to express my sincere appreciation to my committee members: Dr. Her-Yuan
Chen, Dr. Guoyin...
iii
TABLE OF CONTENTS
LIST OF TABLES.........................................................................................
iv
LIST OF TABLES
Table 1: Equations for Output Parameters in API-RP 11L: Conventional Units………….13
v
LIST OF FIGURES
Figure 1: Dimensionless Plunger Stroke Factor API-RP 11L plot (Takacs, 2015)……...…4
Figure 2: Dimensionl...
vi
Figure 43: Conventional Units: Effect of Efficiency and Viscosity on MPRL….……...…43
Figure 44: Polished Rod Horsepower ...
vii
LIST OF ABBREVIATIONS AND SYMBOLS
psi Pounds per square inch (unit of pressure)
In-lbf Inch-pound force (unit of torqu...
This thesis is accepted on behalf of the
Faculty of the Institute by the following committee:
Tan Nguyen
Advisor
Her Yuan ...
1
CHAPTER 1
INTRODUCTION
Artificial lift assists in the downhole process of extracting oil by decreasing the bottomhole
pr...
2
stage is marked by the opening of the traveling valve and the closing of the standing valve.
This means no fluid can lea...
3
CHAPTER 2
LITERATURE REVIEW
2.1 API-RP 11L Procedure
The API-RP 11L procedure is a trial and error method based upon man...
4
Figure 1: Dimensionless Plunger Stroke Factor API-RP 11L plot (Takacs, 2015)
Figure 2: Dimensionless PPRL API-RP 11L plo...
5
Figure 3: Dimensionless MPRL API-RP 11L plot (Takacs, 2015)
Figure 4: Dimensionless Peak Torque API-RP 11L plot (Takacs,...
6
Figure 5: Dimensionless PRHP API-RP 11L plot (Takacs, 2015)
2.2 Pumping Units Summary
The three most common sucker rod p...
7
Figure 6: Conventional & Reverse Mark Pumping Units Photo (Lufkin, 2013)
2.2.2 Air Balanced Units
The use of compressed ...
8
Figure 7: Air Balanced Pumping Units Photo (Lufkin, 2013)
2.2.3 Mark II Units
The Mark II units have a very unique geome...
9
Figure 8: Mark II Pumping Units Photo (Lufkin, 2013)
2.3 Efficiency vs. Viscosity
Figure 9: Relationship between pump ef...
10
viscosity changes. High viscosity fluids lead to higher flow resistances, which leads to a
decrease the mechanical effi...
11
2.5 Annular Flow Frictional Pressure Drop
The annular flow frictional pressure drop equation for Newtonian fluids is:
𝑑...
12
CHAPTER 3
OBJECTIVE
The goal of this work is threefold.
First, all of the plots in the API-RP 11L procedure were digiti...
13
CHAPTER 4
DIGITIZATION OF API-RP 11L PLOTS
Digitization is defined as the process of extracting numerical data values f...
14
Figure 10: API-RP 11L Dimensionless Figure 11: Digitization of Dimensionless
Plunger Stroke Plot (Takacs, 2015) Plunger...
15
Figure 12: API-RP 11L Dimensionless Figure 13: Digitization of Dimensionless
PPRL plot (Takacs, 2015) PPRL plot
Eq. (3)...
16
Figure 14: API-RP 11L Dimensionless Figure 15: Digitization of Dimensionless
MPRL plot (Takacs, 2015) MPRL plot
Eq. (4)...
17
The digitization for the minimum polished rod load also, similarly to the peak polished
rod load, came out to be quite ...
18
Figure 18: API-RP 11L Dimensionless Figure 19: Digitization of Dimensionless
PRHP plot (Takacs, 2015) PRHP plot
Eqs. 6 ...
19
The digitization of the polished rod horsepower dimensionless group also presented a
similar challenge to the torque. H...
20
CHAPTER 5
FRICTIONAL PRESSURE LOSSES GRADIENT DEVELOPMENT
Figure 20: Diagram for Rod Movement in Tubular (Li, 2012)
The...
21
Eq. (11) gives the general form of the velocity profile with c1 and c2 as the two integral
constants:
𝑣𝑣 = −
𝑟𝑟2
4𝜇𝜇
𝑑𝑑...
22
First, the equation is solved without the 𝑢𝑢𝑟𝑟 term. Breaking up the integral, the flow rate
equation can be rewritten ...
23
𝑄𝑄 =
1
4
𝜋𝜋
2𝜇𝜇
𝑑𝑑𝑃𝑃
𝑑𝑑𝑑𝑑
(𝑟𝑟1
4
+ 𝑟𝑟2
4
− 2𝑟𝑟1
2
𝑟𝑟2
2) −
𝜋𝜋
2𝜇𝜇
𝑑𝑑𝑑𝑑
𝑑𝑑𝑑𝑑
(𝑟𝑟2
2
− 𝑟𝑟1
2) �
𝑟𝑟2
2−𝑟𝑟1
2(2 ln
𝑟𝑟2
𝑟𝑟1
...
24
𝑄𝑄 =
𝜋𝜋
8𝜇𝜇
𝑑𝑑𝑑𝑑
𝑑𝑑𝑑𝑑
�𝑟𝑟2
4
− 𝑟𝑟1
4
−
�𝑟𝑟2
2−𝑟𝑟1
2�
2
ln
𝑟𝑟2
𝑟𝑟1
� (30)
Now, the 𝑢𝑢𝑟𝑟 term must be added in to solve t...
25
𝑑𝑑𝑃𝑃𝑓𝑓
𝑑𝑑𝑑𝑑
=
𝜋𝜋�𝑟𝑟2
2−𝑟𝑟1
2�(𝑢𝑢�−𝑢𝑢𝑟𝑟)
𝜋𝜋
8𝜇𝜇
�𝑟𝑟2
4−𝑟𝑟1
4−
�𝑟𝑟2
2−𝑟𝑟1
2�
2
ln
𝑟𝑟2
𝑟𝑟1
�
(36)
𝑑𝑑𝑃𝑃𝑓𝑓
𝑑𝑑𝑑𝑑
=
8𝜇𝜇(𝑟𝑟2
2−...
26
CHAPTER 6
RESULTS AND DISCUSSIONS
6.1 Results
Following the digitization plots in Chapter 4, the viscosity effect plots...
27
Figure 22: MPRL: SPE 20152 vs Simulator Comparisons
Figure 23: PT: SPE 20152 vs Simulator Comparisons
28
Figure 24: PRHP: SPE 20152 vs Simulator Comparisons
Figure 25: CBE: SPE 20152 vs Simulator Comparisons
29
6.1.1 Polished Rod Load Results
Figure 26: Conventional Units: Viscosity Effect on Polished Rod Loads at 5000 ft
Fig. 2...
30
Figure 27: Comparison of the PPRL of the Three Units at 5000 ft
Figure 28: Comparison of the MPRL of the Three Units at...
31
The effect viscosity has on the polished rod loads at 5000 ft for both air balanced units and
Mark II units is similar ...
32
Figure 30: Conventional Units: Effect of Viscosity and Length on MPRL
The results from Figs. 26-28 are simply shown for...
33
Figure 31: Conventional Units: Percent Increase of PPRL at 5000 ft due to Viscosity
Figure 32: Conventional Units: Perc...
34
Figs. 31 and 32 show both the magnitudes (bars) of the PPRL and MPRL at 5000 ft on the
left y-axis and the percent incr...
35
Figure 34: Comparisons of MPRL Percent Increase of the Three Units at 5000 ft
Figs. 33 and 34 show the comparisons of t...
36
6.1.2 Peak Torque Results
Figure 35: Conventional Units: Effect of Viscosity on Peak Torque (Upstroke /
Downstroke)
Mov...
37
Figure 36: Air Balanced Units: Effect of Viscosity on Peak Torque (Upstroke /
Downstroke)
Fig. 36 shows the behavior of...
38
Figure 37: Mark II Units: Effect of Viscosity on Peak Torque (Upstroke / Downstroke)
When comparing both Figs. 36 and 3...
39
Figure 38: Comparisons of Peak Torque Upstroke of the Three Units at 5000 ft
Figure 39: Comparisons of Peak Torque Down...
40
geometry as well as the features described in the literature review section for the Mark II
units. This unit not only h...
41
Figure 41: Conventional Units: Percent Increase of Peak Torque at 5000 ft due to
Viscosity
Fig. 41 shows the magnitudes...
42
6.1.3 Efficiency Results
Figure 42: Conventional Units: Effect of Efficiency and Viscosity on PPRL
From the general rel...
43
Figure 43: Conventional Units: Effect of Efficiency and Viscosity on MPRL
Figs. 42 and 43 show the effect of efficiency...
44
Figure 44: Polished Rod Horsepower (HP) vs Efficiency
6.2 Discussions
This study is an attempt to further expand the AP...
45
The next stage of the project was implementing the API-RP 11L procedure along with the
equations generated from the dig...
46
it can be seen, viscous oil production is much more costly and difficult than lighter oil
production.
47
CHAPTER 7
CONCLUDING REMARKS
The goal of this work was to improve upon the API-RP 11L procedure as it is used only
for ...
48
CHAPTER 8
FUTURE WORK
The following are some recommendations for suggested areas of further research:
• Drill string De...
49
REFERENCES
Black Gold Pump and Supply, Inc. Plungers: Monel. (2015). Signal Hill, California.
Heavy Oil Industry Challe...
50
Rohatgi, Ankit. (2015). WebPlotDigitizer User Manual, Version 3.9. Austin, Texas.
<http://arohatgi.info/WebPlotDigitize...
51
APPENDIX
Figure 45: Air Balanced Units: Viscosity Effect on Polished Rod Loads at 5000 ft
Figure 46: Mark II Units: Vis...
52
Figure 47: Air Balanced Units: Effect of Viscosity and Length on PPRL
Figure 48: Air Balanced Units: Effect of Viscosit...
53
Figure 49: Air Balanced Units: % Increase of PPRL at 5000 ft due to Viscosity
Figure 50: Air Balanced Units: % Increase...
54
Figure 51: Mark II Units: Effect of Viscosity and Length on PPRL
Figure 52: Mark II Units: Effect of Viscosity and Leng...
55
Figure 53: Mark II Units: % Increase of PPRL at 5000 ft due to Viscosity
Figure 54: Mark II Units: % Increase of MPRL a...
56
Figure 55: Conventional Units: Effect of Viscosity and Length on Peak Torque
(Upstroke)
Figure 56: Conventional Units: ...
57
Figure 57: Air Balanced Units: Effect of Viscosity and Length on Peak Torque
(Upstroke)
Figure 58: Air Balanced Units: ...
58
Figure 59: Mark II Units: Effect of Viscosity and Length on Peak Torque (Upstroke)
Figure 60: Mark II Units: Effect of ...
59
Figure 61: Air Balanced Units: Effect of Viscosity and Length on Peak Torque
Figure 62: Air Balanced Units: % Increase ...
60
Figure 63: Mark II Units: Effect of Viscosity and Length on Peak Torque
Figure 64: Mark II Units: % Increase of Peak To...
61
Figure 65: Air Balanced Units: Effect of Efficiency and Viscosity on PPRL
Figure 66: Air Balanced Units: Effect of Effi...
62
Figure 67: Mark II Units: Effect of Efficiency and Viscosity on PPRL
Figure 68: Mark II Units: Effect of Efficiency and...
63
COMPUTER SIMULATOR PROGRAM CODE
Sub CommandButton1_Click()
Q = Sheet1.Cells(3, 2)
Bo = Sheet1.Cells(4, 2)
G = Sheet1.Ce...
64
WrfSkr = Wrf / Skr
Sheet1.Cells(7, 6) = WrfSkr
SpS = -288.464 * (NNop) ^ 6 + 464.4255 * (NNop) ^ 5 - 285.941 * (NNop) ^...
65
PPRL = Wrf + (F1Skr) * (Skr)
Sheet1.Cells(3, 9) = PPRL
MPRL = Wrf - (F2Skr) * (Skr)
Sheet1.Cells(4, 9) = MPRL
CBE = 1.0...
66
Sheet1.Cells(24, 9) = PTM
PRHPM = (F3Skr) * (Skr) * (Sp) * (N) * (0.00000253)
Sheet1.Cells(26, 9) = PRHPM
NEMAHPM = (PR...
Applications of Viscous Oil in a Rod Sucker Pump System
By Utkarsh Bhargava
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Utkarsh Bhargava - Thesis

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Utkarsh Bhargava - Thesis

  1. 1. APPLICATIONS OF VISCOUS OIL IN A ROD SUCKER PUMP SYSTEM By Utkarsh Bhargava Submitted in Partial Fulfillment of the Requirements for the Master of Petroleum Engineering New Mexico Institute of Mining and Technology Department of Petroleum Engineering Socorro, New Mexico November 9th , 2015
  2. 2. ABSTRACT When producing a well for some amount of time, there always comes a moment when there is not enough natural energy in the reservoir for oil to rise to the surface. The pressure decreases significantly over time which will, in turn, decrease production. Artificial lift is a process used in production engineering on oil and gas wells that can solve these problems through a variety of pumps and lifts by reducing the bottomhole pressure and thereby increasing the production rate of the well. This stimulation is necessary for about 95% of all oil and gas wells, of which the majority being sucker-rod pumps; beam pumping is the most common artificial lift method used in the industry. There are many aspects of a sucker rod pump that must be designed very carefully to not only avoid failure, but to also ensure safety of the operators. The API-RP 11L procedure is a trial and error method based on correlations of research test data which, in summary, helps the designer to calculate the important parameters of a pumping unit. These include polished rod loads, torque, horsepower, etc. based on the preliminary selection of components through a series of steps. However, there is a limitation of the API-RP 11L procedure; it does not take into account the viscosity of the oil when predicting pump displacement and associated loads. In this work, a computer simulator was developed according to the API-RP 11L procedure to predict the important output parameters of a pumping unit. Many of the operating characteristics used to calculate these output parameters are found in tables and figures, but in order to implement them into a program, these plots had to be digitized. The effect of heavy oil also was investigated by introducing an original equation for the frictional pressure losses gradient. This equation was used in order to predict the magnitude of increase of the peak polished rod load (PPRL), minimum polished rod load (MPRL) and peak torque (PT). Studies were also done to see the effect the length of the rod and viscosity has on the rod loads and torque, respectively. Efficiency was also incorporated in this study, as the effect of the inertial forces and dynamic fluid loads were seen when studying the relationship between both polished rod loads and polished rod horsepower and efficiency. Based on the simulator results and sensitivity analysis, the API-RP 11L procedure is an effective tool to predict pump displacement, polished rod loads, torque requirements, and horsepower within certain bounds of some parameters of inviscid fluids. From the results of the simulator and the equation developed, it has been determined that heavy oil has a major effect on rod loads and torque on a sucker rod pumping unit. For example, oil with a viscosity of 6000 centipoise can add almost 3500 psi onto the polished rod loads that a sucker rod pump can handle and about 130000 in-lbf of torque. Viscosity effect comparisons between three of the most common pumping units: conventional, air balanced, and Mark II are also found in this work. The incorporation of efficiency and its relationship with the polished rod loads and polished rod horsepower (PRHP) is also quite a significant finding: with an increase in the efficiency, a decrease in the PPRL and PRHP are seen, while an increase in the MPRL is observed. Keywords: Artificial Lift Methods, Sucker Rod Pump, API-RP 11L procedure, heavy oil applications
  3. 3. ii ACKNOWLEDGEMENTS I would like to express my sincere appreciation to my committee members: Dr. Her-Yuan Chen, Dr. Guoyin Zhang, and my primary research advisor, Dr. Tan Nguyen, for providing me the opportunity to embark on this project. I would also like to thank all the staff, students and faculty of the Petroleum Engineering department of New Mexico Institute of Mining and Technology, particularly Dr. Thomas Engler and Mrs. Karen Balch. To my family and friends in the United States and in India, thank you all for your unyielding love and support. God bless you all. Your assistance and support has been invaluable throughout the duration of my graduate studies.
  4. 4. iii TABLE OF CONTENTS LIST OF TABLES............................................................................................................. iv LIST OF FIGURES .............................................................................................................v LIST OF ABBREVIATIONS AND SYMBOLS ............................................................. vii 1. INTRODUCTION .........................................................................................................1 2. LITERATURE REVIEW ..............................................................................................3 2.1. API-RP 11L Procedure .........................................................................................3 2.2. Pumping Units Summary......................................................................................6 2.2.1. Conventional Units ......................................................................................6 2.2.2. Air Balanced Units.......................................................................................7 2.2.3. Mark II Units................................................................................................8 2.3. Efficiency vs. Viscosity ........................................................................................9 2.4. Sucker Rod Pump for Viscous Oil......................................................................10 2.5. Annular Flow Frictional Pressure Drop..............................................................11 3. OBJECTIVE ................................................................................................................12 4. DIGITIZATION OF API-RP 11L PLOTS..................................................................13 5. FRICTIONAL PRESSURE LOSSES GRADIENT DEVELOPMENT......................20 6. RESULTS AND DISCUSSIONS................................................................................26 6.1. Results.................................................................................................................26 6.1.1. Polished Rod Load Results........................................................................29 6.1.2. Peak Torque Results ..................................................................................36 6.1.3. Efficiency Results......................................................................................42 6.2. Discussions .........................................................................................................44 7. CONCLUDING REMARKS.......................................................................................47 8. FUTURE WORK.........................................................................................................48 9. REFERENCES ............................................................................................................49 10. APPENDIX..................................................................................................................51
  5. 5. iv LIST OF TABLES Table 1: Equations for Output Parameters in API-RP 11L: Conventional Units………….13
  6. 6. v LIST OF FIGURES Figure 1: Dimensionless Plunger Stroke Factor API-RP 11L plot (Takacs, 2015)……...…4 Figure 2: Dimensionless PPRL API-RP 11L plot (Takacs, 2015)…………….................…4 Figure 3: Dimensionless MPRL API-RP 11L plot (Takacs, 2015)…………………...……5 Figure 4: Dimensionless Peak Torque API-RP 11L plot (Takacs, 2015)……............……..5 Figure 5: Dimensionless PRHP API-RP 11L plot (Takacs, 2015)……….…….…………..6 Figure 6: Conventional & Reverse Mark Pumping Units Photo (Lufkin, 2013)…………...7 Figure 7: Air Balanced Pumping Units Photo (Lufkin, 2013)……………………………...8 Figure 8: Mark II Pumping Units Photo (Lufkin, 2013)…………………………………..9 Figure 9: Relationship between pump efficiency and viscosity (Herzog, 2005)………….9 Figure 10: API-RP 11L Dimensionless Plunger Stroke Plot (Takacs, 2015)…...………...14 Figure 11: Digitization of Dimensionless Plunger Stroke Plot………………….………..14 Figure 12: API-RP 11L Dimensionless PPRL Plot (Takacs, 2015)…...……..…..……….15 Figure 13: Digitization of Dimensionless PPRL Plot…………………………….………15 Figure 14: API-RP 11L Dimensionless MPRL Plot (Takacs, 2015)…………….……..…16 Figure 15: Digitization of Dimensionless MPRL Plot……………………..……………..16 Figure 16: API-RP 11L Dimensionless PT Plot (Takacs, 2015)…...………..…………....17 Figure 17: Digitization of Dimensionless PT Plot…………………………….………….17 Figure 18: API-RP 11L Dimensionless PRHP Plot (Takacs, 2015)……………….……...18 Figure 19: Digitization of Dimensionless PRHP Plot…………..………...………………18 Figure 20: Diagram for Rod Movement in Tubular (Li, 2012)…………………………....20 Figure 21: PPRL: SPE 20152 vs Simulator Comparisons……………………………...…26 Figure 22: MPRL: SPE 20152 vs Simulator Comparisons……………………………….27 Figure 23: PT: SPE 20152 vs Simulator Comparisons……………………………………27 Figure 24: PPHP: SPE 20152 vs Simulator Comparisons………………………………...28 Figure 25: CBE: SPE 20152 vs Simulator Comparisons…………………………………28 Figure 26: Conventional Units: Viscosity Effect on Polished Rod Loads at 5000 ft...……29 Figure 27: Comparison of the PPRL of the Three Units at 5000 ft………………………..30 Figure 28: Comparison of the MPRL of the Three Units at 5000 ft……………………….30 Figure 29: Conventional Units: Effect of Viscosity and Length on PPRL…………..……31 Figure 30: Conventional Units: Effect of Viscosity and Length on MPRL………….……32 Figure 31: Conventional Units: % Increase of PPRL at 5000 ft due to Viscosity…………33 Figure 32: Conventional Units: % Increase of MPRL at 5000 ft due to Viscosity…...……33 Figure 33: Comparisons of PPRL % Increase of the Three Units at 5000 ft………………34 Figure 34: Comparisons of MPRL % Increase of the Three Units at 5000 ft...……………35 Figure 35: Conventional Units: Effect of Viscosity on PT (Upstroke / Downstroke)…….36 Figure 36: Air Balanced Units: Effect of Viscosity on PT (Upstroke / Downstroke)..……37 Figure 37: Mark II Units: Effect of Viscosity on PT (Upstroke / Downstroke)……...……38 Figure 38: Comparisons of Peak Torque Upstroke of the Three Units at 5000 ft…..…..…39 Figure 39: Comparisons of Peak Torque Downstroke of the Three Units at 5000 ft…...…39 Figure 40: Conventional Units: Effect of Viscosity and Length on Peak Torque…………40 Figure 41: Conventional Units: % Increase of Peak Torque at 5000 ft due to Viscosity.…41 Figure 42: Conventional Units: Effect of Efficiency and Viscosity on PPRL…..……...…42
  7. 7. vi Figure 43: Conventional Units: Effect of Efficiency and Viscosity on MPRL….……...…43 Figure 44: Polished Rod Horsepower (HP) vs Efficiency……………………………...…44 Figure 45: Air Balanced Units: Viscosity Effect on Polished Rod Loads at 5000 ft………51 Figure 46: Mark II Units: Viscosity Effect on Polished Rod Loads at 5000 ft…………….51 Figure 47: Air Balanced Units: Effect of Viscosity and Length on PPRL...………………52 Figure 48: Air Balanced Units: Effect of Viscosity and Length on MPRL.………………52 Figure 49: Air Balanced Units: % Increase of PPRL at 5000 ft due to Viscosity.…………53 Figure 50: Air Balanced Units: % Increase of MPRL at 5000 ft due to Viscosity...………53 Figure 51: Mark II Units: Effect of Viscosity and Length on PPRL………………………54 Figure 52: Mark II Units: Effect of Viscosity and Length on MPRL……..………………54 Figure 53: Mark II Units: % Increase of PPRL at 5000 ft due to Viscosity ………………55 Figure 54: Mark II Units: % Increase of MPRL at 5000 ft due to Viscosity………………55 Figure 55: Conventional Units: Effect of Viscosity and Length on PT (Upstroke)…….…56 Figure 56: Conventional Units: Effect of Viscosity and Length on PT (Downstroke)……56 Figure 57: Air Balanced Units: Effect of Viscosity and Length on PT (Upstroke)…..……57 Figure 58: Air Balanced Units: Effect of Viscosity and Length on PT (Downstroke)….…57 Figure 59: Mark II Units: Effect of Viscosity and Length on PT (Upstroke)…...…………58 Figure 60: Mark II Units: Effect of Viscosity and Length on PT (Downstroke)…..………58 Figure 61: Air Balanced Units: Effect of Viscosity and Length on PT……………………59 Figure 62: Air Balanced Units: % Increase of Peak Torque at 5000 ft due to Viscosity..…59 Figure 63: Mark II Units: Effect of Viscosity and Length on Peak Torque.………………60 Figure 64: Mark II Units: % Increase of Peak Torque at 5000 ft due to Viscosity...………60 Figure 65: Air Balanced Units: Effect of Efficiency and Viscosity on PPRL….....………61 Figure 66: Air Balanced Units: Effect of Efficiency and Viscosity on MPRL...……….…61 Figure 67: Mark II Units: Effect of Efficiency and Viscosity on PPRL………......………62 Figure 68: Mark II Units: Effect of Efficiency and Viscosity on MPRL……….....………62
  8. 8. vii LIST OF ABBREVIATIONS AND SYMBOLS psi Pounds per square inch (unit of pressure) In-lbf Inch-pound force (unit of torque) cp Centipoise (unit of viscosity) PPRL Peak Polished Rod Load MPRL Minimum Polished Rod Load PT Peak Torque (in-lbf) PRHP Polished Rod Horsepower (HP) CBE Counterbalance Effect (lbf) PD Pump Displacement N Pumping Speed (spm – strokes per minute) S Surface Stroke Length (in) Sp Plunger Stroke Length (in) kr Spring constant of the rod string Fo/Skr Dimensionless Rod Stretch N/No Dimensionless Pump Speed N/N’o Dimensionless Pump Speed (with rod-tapering) 𝜂𝜂 Efficiency API American Petroleum Institute 𝜏𝜏 Shear Stress 𝜇𝜇 Viscosity γ Shear Rate r1 Inner Radius (Rod) r2 Outer Radius (Rod to Wall) u, v Velocity ur Velocity of Rod 𝑢𝑢� Average Flow Velocity Q Flow Rate dPf / dL Frictional Pressure Drop
  9. 9. This thesis is accepted on behalf of the Faculty of the Institute by the following committee: Tan Nguyen Advisor Her Yuan Chen Committee Member Guoyin Zhang Committee Member 11/9/15 Date I release this document to the New Mexico Institute of Mining and Technology. Utkarsh Bhargava 11/9/15 Student’s Signature Date
  10. 10. 1 CHAPTER 1 INTRODUCTION Artificial lift assists in the downhole process of extracting oil by decreasing the bottomhole pressure in the reservoir. These various methods are very important techniques in the industry and must be used in the majority pressure deficient wells. Without the presence of artificial lift, it would be impossible to meet current demands. As mentioned before, this stimulation is necessary for about 95% of all oil/gas wells (Hyne, 2012). Sucker-rod lifting, or beam pumping, is the most common artificial lift method used in the industry. About 750,000 out of the 2 million oil wells around the world use sucker rod pumps as their choice of artificial method. In the United States alone, sucker rod pumps account for about 350,000 wells which also takes into account the 80% of all US oil wells that produce a maximum of 10 barrels per day (Lea, 2014). Because these pumping systems occupy lots of space and are very heavy, they are primarily found on onshore locations. There are many different parts of a sucker rod pumping system, which include the surface equipment (prime mover, gear reducer and pumping unit) and the downhole equipment (sucker rod string and subsurface pump). The prime mover simply supplies power to the system; these are most often internal combustion engines, but can also be electric motors. The gear reducer stabilizes the speed and torque of the pump. The power supplied by the prime mover is generally very high speed and low torque, while the pumping unit, conversely, requires low speed and high torque (Kelly, 2000). The gear reducer is essential in this alteration. The pumping unit reciprocates the motion of the prime mover and gear reducer. The latter two exhibit rotational behavior, while the pumping unit establishes a vertical motion. The pumping unit also includes the counterweight (for conventional and Mark II units) or the compressed air assembly (for air balanced units) which evenly distribute the weight of the rod string. The pumping unit on the surface is connected to the subsurface pump via a sucker rod string, which also enables the subsurface pump to have a vertical motion from its usual rotational motion. A pump jack is also vital as it is connected to the downhole pump at the bottom of the well by many interconnected sucker rods. Sucker rods maintain about 25-30 feet in length with several choices in diameter ranging from ½ inch to 1 ⅛ inch; these are joined together via couplings. Before detailing how the system works, it should be noted that sucker rod pumps work by positive displacement of the fluid and not by centrifugal force. First, as stated earlier, the prime mover supplies power to the gear reducer; the gear reducer not only reduces the angular velocity, but increases the relative torque. The crank turns counterclockwise and lifts the counterweight, which, in turn pivots the beam and submerges the plunger, which signifies the beginning of the downstroke stage and the end of the upstroke stage. For a closer examination of the pump cycle, these two stages are very important in the functioning of the pump. There are two valves needed: the standing valve and traveling valve. These are located at the bottom of the tubing and at the bottom of the sucker rods, respectively. Fluid flow dictates the opening and closing of these valves. The downstroke
  11. 11. 2 stage is marked by the opening of the traveling valve and the closing of the standing valve. This means no fluid can leave the well, and only flow into the plunger. The standing valve must be closed as there is fluid being carried above it and the fluid is moving upward towards the traveling valve, which is open. Conversely, the upstroke stage is discernable by the traveling valve closing and the standing valve opening. In this stage, while more fluid is pumped into well, fluid not only in the plunger, but also above it is lifted out of the casing. This cycle repeats itself when the top of the stroke is reached by the plunger, as the downstroke process happens again where the traveling valve opens and standing valve closes. These two valves work together to move the fluid up the tubing toward the surface in what is known as the pump cycle. This process works quite well for inviscid fluids or low viscosity fluids, however, there are challenges that lead to pumping oil of heavier crudes. Heavy oil is characterized by very low API grades (10̊ to 22.3̊), low hydrogen to carbon ratios, and high specific gravities. They also contain large amounts of asphaltic content along with sulfur, nitrogen and other heavy metals (Halliburton, 2009). Along with having very little to no flowability due to their high viscosity, heavy oil is also quite difficult and costly to produce. The viscosity of these crudes range from a couple hundred centipoise to 10,000 cp. Most heavy oils are also found in poorly consolidated sands (Halliburton, 2009), which present many challenges in themselves. The reservoirs that contain heavy oil influence both cementing and the wellbore integrity. Maintaining the pressure throughout the system, controlling the heat transfer, and having a means of effective sand control becomes infinitely more important in the production of such reservoirs. It is difficult to recover these oils in their natural state through ordinary production methods; increasing the temperature (adding heat) or dilution may be required for the oil to flow through the pipeline. However, this work focuses on the conventional method of extraction in determining the viability that a regular sucker rod pumping system has for producing heavy oil.
  12. 12. 3 CHAPTER 2 LITERATURE REVIEW 2.1 API-RP 11L Procedure The API-RP 11L procedure is a trial and error method based upon many correlations of research test data. It contains three steps: the preliminary selection of components, calculating the operating characteristics with formulas, tables and figures, and lastly, finding the pump displacement and associated loads from those characteristics. To proceed with the API-RP 11L procedure and calculate all of the output parameters, a minimum amount of information must be known. This information is based on the preliminary components chosen: pump depth, fluid level, pumping speed, surface stroke length, specific gravity, pump plunger diameter, the nominal tubing diameter, and the design and size of the sucker rod. The API-RP 11L procedure adopted and expanded upon work conducted by the Midwest Research Institute in 1964 that ran numerous computer simulations over many different operating conditions (Takacs, 2015). The work conducted is a function of two dimensionless variables: pump speed and rod stretch. The dimensionless pump speed quantity may be expressed as N/No or N/N’o. N/No is defined as the pump speed in strokes per minute (N) divided by (No), the natural frequency of the string, while N/N’o is simply (N/No)/Fc, where Fc is a frequency factor of the rod string; this is based off the rod API grade and plunger diameter. The dimensionless rod stretch is noted as Fo/Skr; this is defined as the weight of the fluid imposed on the rods divided by the product of the polished rod stroke length and the spring constant of the rod string. After these dimensionless qualities are calculated, then the dimensionless variables to help calculate the associated loads and stresses can be calculated. These include the plunger stroke, the pump displacement, the peak and minimum polished rod loads, the polished rod horsepower, the peak torque, and the counterbalance effect. This is normally done with plots and figures, however, these plots have been digitized and equations have been generated for the most important parameters when simulating a pumping system. API-RP 11L plots: With the calculation of both the dimensionless rod stretch and dimensionless pump speed, the following plots enable the operator to calculate the dimensionless groups that will allow the final calculation for the output pumping parameters. For example, in Fig. 1, with a dimensionless pump speed of 0.274 and rod stretch of 0.13, the plot can be used to find the (Sp/S) value of 0.98.
  13. 13. 4 Figure 1: Dimensionless Plunger Stroke Factor API-RP 11L plot (Takacs, 2015) Figure 2: Dimensionless PPRL API-RP 11L plot (Takacs, 2015)
  14. 14. 5 Figure 3: Dimensionless MPRL API-RP 11L plot (Takacs, 2015) Figure 4: Dimensionless Peak Torque API-RP 11L plot (Takacs, 2015)
  15. 15. 6 Figure 5: Dimensionless PRHP API-RP 11L plot (Takacs, 2015) 2.2 Pumping Units Summary The three most common sucker rod pumping units are conventional, air balanced, and Mark II units. A summary of the geometry of these pumping units are described in order to help explain some of the results below. 2.2.1 Conventional Units Conventional units are the most common in the industry and the simplest to operate; they are require the least amount of maintenance and are the standard for all operating pumping units. These systems allow for the widest range of sizes that are available and are the cheapest in terms of operating costs than the other pumping units. Conventional units and Mark II units both employ counterweights in order to balance the load of the fluid and rod with the surface equipment, however, the air balanced units use compressed air instead of counterweights.
  16. 16. 7 Figure 6: Conventional & Reverse Mark Pumping Units Photo (Lufkin, 2013) 2.2.2 Air Balanced Units The use of compressed air as opposed to the counterweights allows for more control of the counterbalance, as adjustments can be made without having to stop the system. This is beneficial as these systems generally are of lower weight than the conventional systems, and thus more portable, allowing their use offshore.
  17. 17. 8 Figure 7: Air Balanced Pumping Units Photo (Lufkin, 2013) 2.2.3 Mark II Units The Mark II units have a very unique geometry; it uses a lever system that is design to decrease upstroke acceleration and decrease the peak rod load (Lufkin, 2013). This unique geometry also is designed to produce a slower upstroke and faster downstroke, which decreases the torque on the gear reducer up to 35%. Common in both the air balanced and Mark II units, as opposed to the conventional units, is the difference in the position of the gear reducer. The gear reducer in the air balanced and Mark II units is located underneath the equalizer and away from the Samson post. In the two former units, the gear reducer is moved towards the Samson posts turning in the preferred direction of rotation. This geometry creates an upstroke that occurs in the crank rotation of 195° and a downstroke of 165°. This change of geometry from the usual 180° reduces the acceleration at the beginning of the upstroke where the load is the greatest, which results in the decrease of the polished rod load. This shift in the gear reducer also creates a scenario that decreases the maximum torque factor on the upstroke and increases the maximum downstroke torque factor. A greater mechanical advantage is created for lifting the heavy load on the upstroke and a lower mechanical advantage for the reduced load on the downstroke (Lufkin, 2013).
  18. 18. 9 Figure 8: Mark II Pumping Units Photo (Lufkin, 2013) 2.3 Efficiency vs. Viscosity Figure 9: Relationship between pump efficiency and viscosity (Herzog, 2005) Fig. 9, from the Machinery Lubrication Publication, shows that as the viscosity increases, there is an associated increase in the volumetric efficiency, however a decrease in the mechanical efficiency. Volumetric efficiency refers to the comparison of the actual fluid flow out of a pump and the flow based on internal leakage and flow losses. The mechanical efficiency deals with the frictional losses and energy required to overcome the drag force and have constant flow (Rudnick, 2013). These parameters are both heavily influenced by
  19. 19. 10 viscosity changes. High viscosity fluids lead to higher flow resistances, which leads to a decrease the mechanical efficiency, however increase in viscosity leads to an increase in the volumetric efficiency as the slip decreases and there is also less leakage with viscous oils than lighter oils. As can also be seen from Fig. 9, there is a range of the viscosity that optimizes both efficiencies along with the overall efficiency. The overall efficiency is defined as the product of the mechanical and volumetric efficiencies. 2.4 Sucker Rod Pump for Viscous Oil Viscous oil is pumped quite frequently in many parts of the world, not only with progressive cavity pumps, but also with sucker rod pumps. The study conducted in SPE 2152 (Juch, 1969), took place in Venezuela, as most of the crude oil pumped in Venezuela is very viscous with 9̊ to 16̊ API gravities and with viscosities that are as high as 10,000 cp. Because of this, there is a growing need to improve the pump efficiencies when dealing with a viscous reservoir. A new line of sucker rod pumps were introduced as a result. The new technology of streamlined pumps have improved the production performance of viscous crude wells in Venezuela (Juch, 1969). Two very important problems arise when dealing with pumping viscous oil. With viscous oils, there is more of a resistance to flow which causes severe pressure variance over time. This can be damaging to the pump performance due to huge energy losses as well as gas leaks, not to mention the pump and equipment wear. Also, issues are seen in the upstroke and downstroke phases of the pump. The valve positions that allow for fluid flow are delayed depending on the API gravity, which affects not only the pump efficiency, but also the pump performance (Juch, 1969). However, there are techniques that aid in pump design that enable pumping of viscous oils. One such method involves the use of a friction ring hold-down mechanism. This involves the use of Monel plungers, as regular plungers are not very efficient in high viscous and corrosive wells. Monel plungers are much sturdier and made from corrosion resistant materials (Black, 2015), which help to get a higher hold-down force than with regular plungers. Monel plungers also enable higher flow capacity with a higher hold-down force. Another method involves the use of ring valves; they reside in rod guides and sit in the rim of the tube connector (Juch, 1969). The installation of a ring valve enables the traveling valve to open earlier on the downstroke, which not only addresses the problem of the valve actions to be delayed for viscous oil, but also allow for the downstroke stage to occur without any problems. Conversely, the standing valves also open earlier on the upstroke, which causes a major decrease in the pressure differential across the standing valve on the downstroke. This decrease in pressure drop greatly improves the volumetric efficiency (Juch, 1969). Also, the use of a ring valve allows for loads during the pump cycle (PPRL, MPRL, etc.) to occur in a more gradual fashion and improve the load distribution.
  20. 20. 11 2.5 Annular Flow Frictional Pressure Drop The annular flow frictional pressure drop equation for Newtonian fluids is: 𝑑𝑑𝑃𝑃𝑓𝑓 𝑑𝑑𝑑𝑑 = 8𝜇𝜇𝑢𝑢� �𝑟𝑟2 2+𝑟𝑟1 2− 𝑟𝑟2 2−𝑟𝑟1 2 ln 𝑟𝑟2 𝑟𝑟1 � (1) It can be seen that Eq. (1) is quite similar to the equation derived (Eq. 38), in chapter 5, with the only difference being the incorporation of the velocity of the rod. The boundary conditions in order to find the two constants for Eq. (1) were different, being: B.C. 1: r = r1  u = 0 B.C. 2: r = r2  u = 0 These two boundary conditions imply the velocity will equal 0 at both r1 and r2, which removes the velocity of the rod and solely bases the frictional pressure drop on the average flow velocity and the viscosity.
  21. 21. 12 CHAPTER 3 OBJECTIVE The goal of this work is threefold. First, all of the plots in the API-RP 11L procedure were digitized in order to get a single equation for each output parameter solely based on the dimensionless rod stretch and dimensionless pump speed. Secondly, these equations along with others from the API-RP 11L procedure were coded into a simulator using Microsoft Excel VBA. This is the working simulator that is used to calculate the output parameters, such as polished rod loads, torque, horsepower etc., for any and all inputs that the user enters. Lastly, the effect of viscosity was studied. Because the API-RP 11L procedure is limited to inviscid flow, there are no heavy oil applications associated with it. A new equation for the frictional pressure losses gradient was developed and was used in conjunction with the output parameters from the simulator to further study the effect viscosity has on pump characteristics and the magnitude to which they are affected.
  22. 22. 13 CHAPTER 4 DIGITIZATION OF API-RP 11L PLOTS Digitization is defined as the process of extracting numerical data values from graphs in scientific publications (Rohatgi, 2015). In the case of the API-RP 11L procedure, to calculate output parameters without the use of plots and instead with equations, the plots had to be digitized. These plots were recreated and manipulated in Excel using the solver macro add-in; the distance between the differences of the squares of the two plots was minimized to get the best results possible. To get accurate results, a 10% error had to be used or else the solver could not complete the iteration process. The plots found in literature to calculate the dimensionless groups also needed to be manipulated and an equation was generated for each of them found. As it can be seen, the plots only allow for a certain bound in these two dimensionless groups, in some cases, (0 - 0.6) for the pump speed and (0.1 - 0.5) for the rod stretch, while an equation would allow for the calculation outside of these parameters. The equations for the various output parameters are below and each of these includes a dimensionless group that can be calculated using the equation generated based on the digitization of each plot in literature. Table 1: Equations for Output Parameters in API-RP 11L: Conventional Units It can be seen from Table 1 that to calculate the PPRL, the dimensionless group of F1/Skr needs to be calculated. The effective weight of the rods in fluid is simply based on the weight of the rods in air and specific gravity of the fluid. This is also seen for the dimensionless group F2/Skr to calculate the MPRL, Sp/S (Ideal and Actual Pump Displacement), 2T/S2 kr (Peak Torque) and F3/Skr (PRHP). All of these dimensionless groups can be found using the equations that were generated based on the digitization of the plots found in literature. The digitization results involve the specific equation generated for each output parameter dependent upon the dimensionless pump speed (N/No or N/N’o) and rod stretch (Fo/Skr), as explained in the literature review section. These are seen below:
  23. 23. 14 Figure 10: API-RP 11L Dimensionless Figure 11: Digitization of Dimensionless Plunger Stroke Plot (Takacs, 2015) Plunger Stroke Plot Eq. (2) generated for the plunger stroke divided by the surface stroke in terms of dimensionless pumping speed and dimensionless rod stretch is: 𝑆𝑆𝑝𝑝 𝑆𝑆 = −288.464 � 𝑁𝑁 𝑁𝑁𝑜𝑜 ′ � 6 + 464.4255 � 𝑁𝑁 𝑁𝑁𝑜𝑜 ′ � 5 − 285.941 � 𝑁𝑁 𝑁𝑁𝑜𝑜 ′ � 4 + 88.94985 � 𝑁𝑁 𝑁𝑁𝑜𝑜 ′ � 3 − 13.189 � 𝑁𝑁 𝑁𝑁𝑜𝑜 ′ � 2 + 1.124451 � 𝑁𝑁 𝑁𝑁𝑜𝑜 ′ � + 5.554806 � 𝐹𝐹𝑜𝑜 𝑆𝑆𝑘𝑘𝑟𝑟 � 4 − 7.0981 � 𝐹𝐹𝑜𝑜 𝑆𝑆𝑘𝑘𝑟𝑟 � 3 + 3.325249 � 𝐹𝐹𝑜𝑜 𝑆𝑆𝑘𝑘𝑟𝑟 � 2 − 1.54031 � 𝐹𝐹𝑜𝑜 𝑆𝑆𝑘𝑘𝑟𝑟 � + 0.994469 (2) Eq. (2) along with the rest have been generated in dimensionless pump speed to the 6th degree polynomial and dimensionless rod stretch to the 4th degree in order to get the most accurate results possible. These equations were also generated in order to exceed the bounds of these plots. Fig. 10 is only valid for the range of the dimensionless pump speed from 0 to 0.6 and the dimensionless rod stretch from 0.05 to 0.5. Eqs. (1-6) using the solver macro add-in allow for the calculation at ranges outside of these limits, if desired. As it can be seen, while the digitization and solver coefficient finder was done to the best of its ability, there is a degree of uncertainty at N/N’o = 0.2 to N/N’o = 0.35 at all of the dimensionless rod stretch values with about a 5% error, however the model was optimized at all other ranges.
  24. 24. 15 Figure 12: API-RP 11L Dimensionless Figure 13: Digitization of Dimensionless PPRL plot (Takacs, 2015) PPRL plot Eq. (3) generated for the peak polished rod load dimensionless group (F1/Skr) in terms of dimensionless pumping speed and dimensionless rod stretch is: 𝐹𝐹1 𝑆𝑆𝑘𝑘𝑟𝑟 = −187.783 � 𝑁𝑁 𝑁𝑁𝑜𝑜 � 6 + 341.1534 � 𝑁𝑁 𝑁𝑁𝑜𝑜 � 5 − 210.786 � 𝑁𝑁 𝑁𝑁𝑜𝑜 � 4 + 55.62392 � 𝑁𝑁 𝑁𝑁𝑜𝑜 � 3 − 5.58467 � 𝑁𝑁 𝑁𝑁𝑜𝑜 � 2 + .748049 � 𝑁𝑁 𝑁𝑁𝑜𝑜 � + 1.252434 � 𝐹𝐹𝑜𝑜 𝑆𝑆𝑘𝑘𝑟𝑟 � 4 − .39606 � 𝐹𝐹𝑜𝑜 𝑆𝑆𝑘𝑘𝑟𝑟 � 3 − .90549 � 𝐹𝐹𝑜𝑜 𝑆𝑆𝑘𝑘𝑟𝑟 � 2 + 1.225137 � 𝐹𝐹𝑜𝑜 𝑆𝑆𝑘𝑘𝑟𝑟 � − 0.13983 (3) The digitization for the peak polished rod load, as can be seen from Fig. 12, turned out quite well in reference to the plot in literature on the left. The only slight outliers may potentially occur with very small Fo/Skr values and high N/No values. However, for all other values of both rod stretch and pump speed the results are deemed quite strong.
  25. 25. 16 Figure 14: API-RP 11L Dimensionless Figure 15: Digitization of Dimensionless MPRL plot (Takacs, 2015) MPRL plot Eq. (4) generated for the minimum polished rod load dimensionless group (F2/Skr) in terms of dimensionless pumping speed and dimensionless rod stretch is: 𝐹𝐹2 𝑆𝑆𝑘𝑘𝑟𝑟 = −21.2101 � 𝑁𝑁 𝑁𝑁𝑜𝑜 � 6 + 86.57893 � 𝑁𝑁 𝑁𝑁𝑜𝑜 � 5 − 82.5498 � 𝑁𝑁 𝑁𝑁𝑜𝑜 � 4 + 29.56686 � 𝑁𝑁 𝑁𝑁𝑜𝑜 � 3 − 3.48565 � 𝑁𝑁 𝑁𝑁𝑜𝑜 � 2 + .63318 � 𝑁𝑁 𝑁𝑁𝑜𝑜 � − 30.5551 � 𝐹𝐹𝑜𝑜 𝑆𝑆𝑘𝑘𝑟𝑟 � 4 + 37.86197 � 𝐹𝐹𝑜𝑜 𝑆𝑆𝑘𝑘𝑟𝑟 � 3 − 16.7337 � 𝐹𝐹𝑜𝑜 𝑆𝑆𝑘𝑘𝑟𝑟 � 2 + 3.10824 � 𝐹𝐹𝑜𝑜 𝑆𝑆𝑘𝑘𝑟𝑟 � − 0.20801 (4)
  26. 26. 17 The digitization for the minimum polished rod load also, similarly to the peak polished rod load, came out to be quite accurate compared with the plot found in literature, with all of the predicted values being less than 5% from the actual values. Figure 16: API-RP 11L Dimensionless Figure 17: Digitization of Dimensionless PT plot (Takacs, 2015) PT plot Eq. (5) generated for the peak torque dimensionless group (2𝑇𝑇/𝑆𝑆2 𝑘𝑘𝑟𝑟) in terms of dimensionless pumping speed and dimensionless rod stretch is: 2𝑇𝑇 𝑆𝑆2 𝑘𝑘𝑟𝑟 = −74.9862 � 𝑁𝑁 𝑁𝑁𝑜𝑜 ′ � 6 + 132.8622 � 𝑁𝑁 𝑁𝑁𝑜𝑜 ′ � 5 − 87.8538 � 𝑁𝑁 𝑁𝑁𝑜𝑜 ′ � 4 + 27.16812 � 𝑁𝑁 𝑁𝑁𝑜𝑜 ′ � 3 − 3.52134 � 𝑁𝑁 𝑁𝑁𝑜𝑜 ′ � 2 + .663331 � 𝑁𝑁 𝑁𝑁𝑜𝑜 ′ � + 5.705248 � 𝐹𝐹𝑜𝑜 𝑆𝑆𝑘𝑘𝑟𝑟 � 4 − 5.62858 � 𝐹𝐹𝑜𝑜 𝑆𝑆𝑘𝑘𝑟𝑟 � 3 + 1.028574 � 𝐹𝐹𝑜𝑜 𝑆𝑆𝑘𝑘𝑟𝑟 � 2 + .573612 � 𝐹𝐹𝑜𝑜 𝑆𝑆𝑘𝑘𝑟𝑟 � − 0.037408 (5) The digitization for the peak torque dimensionless group presented quite a distinct challenge. As can be seen from Fig. 16, the rod stretch value (Fo/Skr) at 0.5 varying the dimensionless pump speed presents very odd behavior for the torque factor needed in order to calculate the peak torque needed for the system. It was handled to the best of the solver’s ability, while keeping in mind the other values, and along with the rest of the rod stretch values gives accurate results anyhow.
  27. 27. 18 Figure 18: API-RP 11L Dimensionless Figure 19: Digitization of Dimensionless PRHP plot (Takacs, 2015) PRHP plot Eqs. 6 and 7 generated for the polished rod horsepower dimensionless group (F3/Skr) in terms of dimensionless pumping speed and dimensionless rod stretch are: 𝐼𝐼𝐼𝐼 𝐹𝐹𝑜𝑜 𝑆𝑆𝑘𝑘𝑟𝑟 < .3 𝐹𝐹3 𝑆𝑆𝑘𝑘𝑟𝑟 = −1.70788 � 𝑁𝑁 𝑁𝑁𝑜𝑜 � 6 − 12.8682 � 𝑁𝑁 𝑁𝑁𝑜𝑜 � 5 + 18.29899 � 𝑁𝑁 𝑁𝑁𝑜𝑜 � 4 − 6.6556 � 𝑁𝑁 𝑁𝑁𝑜𝑜 � 3 + 1.49254 � 𝑁𝑁 𝑁𝑁𝑜𝑜 � 2 + .071709 � 𝑁𝑁 𝑁𝑁𝑜𝑜 � + 11.87353 � 𝐹𝐹𝑜𝑜 𝑆𝑆𝑘𝑘𝑟𝑟 � 4 − 20.4182 � 𝐹𝐹𝑜𝑜 𝑆𝑆𝑘𝑘𝑟𝑟 � 3 + 8.6778 � 𝐹𝐹𝑜𝑜 𝑆𝑆𝑘𝑘𝑟𝑟 � 2 − .58959 � 𝐹𝐹𝑜𝑜 𝑆𝑆𝑘𝑘𝑟𝑟 � + 0.076676 (6) 𝐼𝐼𝐼𝐼 𝐹𝐹𝑜𝑜 𝑆𝑆𝑘𝑘𝑟𝑟 > .3 𝐹𝐹3 𝑆𝑆𝑘𝑘𝑟𝑟 = −4.56469 � 𝑁𝑁 𝑁𝑁𝑜𝑜 � 6 − 15.5761 � 𝑁𝑁 𝑁𝑁𝑜𝑜 � 5 + 11.95129 � 𝑁𝑁 𝑁𝑁𝑜𝑜 � 4 + 3.109074 � 𝑁𝑁 𝑁𝑁𝑜𝑜 � 3 − 1.20497 � 𝑁𝑁 𝑁𝑁𝑜𝑜 � 2 + .303059 � 𝑁𝑁 𝑁𝑁𝑜𝑜 � + 13.80571 � 𝐹𝐹𝑜𝑜 𝑆𝑆𝑘𝑘𝑟𝑟 � 4 − 20.11 � 𝐹𝐹𝑜𝑜 𝑆𝑆𝑘𝑘𝑟𝑟 � 3 + 8.649151 � 𝐹𝐹𝑜𝑜 𝑆𝑆𝑘𝑘𝑟𝑟 � 2 − .40286 � 𝐹𝐹𝐹𝐹 𝑆𝑆𝑘𝑘𝑟𝑟 � − 0.063143 (7)
  28. 28. 19 The digitization of the polished rod horsepower dimensionless group also presented a similar challenge to the torque. However, in this case, the rod stretch values of 0.4 and 0.6 demonstrate equal behavior at low pump speeds, however deviate from each other at higher pump speeds. This behavior, along with the other rod stretch groups, did not allow for simply one equation to model their behavior, as the results compared to the actual API-RP 11L plot were quite inaccurate. A piecewise function was then attributed to the calculation of the dimensionless group to calculate the polished rod horsepower to account for the behavior at the 0.4 and 0.6 Fo/Skr groups.
  29. 29. 20 CHAPTER 5 FRICTIONAL PRESSURE LOSSES GRADIENT DEVELOPMENT Figure 20: Diagram for Rod Movement in Tubular (Li, 2012) The momentum eq. is applied for an incompressible fluid, a 1-D, isothermal, and steady state system, while assuming no convection. Cylindrical coordinates were also used in the derivation. The relationship between τ and 𝑑𝑑𝑝𝑝𝑓𝑓 𝑑𝑑𝑑𝑑 (the frictional pressure drop gradient) can be written as: 𝜏𝜏 = 𝑟𝑟 2 𝑑𝑑𝑝𝑝𝑓𝑓 𝑑𝑑𝑑𝑑 + 𝑐𝑐1 𝑟𝑟 (8) For Newtonian fluids: 𝜏𝜏 = 𝜇𝜇𝜇𝜇 = −𝜇𝜇 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 = 𝑟𝑟 2 𝑑𝑑𝑝𝑝𝑓𝑓 𝑑𝑑𝑑𝑑 + 𝑐𝑐1 𝑟𝑟 (9) Dividing by the viscosity on both sides and multiplying by dr: −𝑑𝑑𝑑𝑑 = � 𝑟𝑟 2𝜇𝜇 𝑑𝑑𝑝𝑝𝑓𝑓 𝑑𝑑𝑑𝑑 + 𝑐𝑐1 𝜇𝜇𝜇𝜇 � 𝑑𝑑𝑑𝑑 (10) Downstroke Friction Force 𝑟𝑟1 𝑟𝑟2
  30. 30. 21 Eq. (11) gives the general form of the velocity profile with c1 and c2 as the two integral constants: 𝑣𝑣 = − 𝑟𝑟2 4𝜇𝜇 𝑑𝑑𝑝𝑝𝑓𝑓 𝑑𝑑𝑑𝑑 − 𝑐𝑐1 𝜇𝜇 ln 𝑟𝑟 + 𝑐𝑐2 (11) To model the velocity profile, two boundary conditions are needed to find c1 and c2. During the upstroke, there are two movements: the upward movement of the rod and the fluid flow in the annulus between the rods and production casing. The boundary conditions can be established based on Fig. 20. When r = r1, the velocity is taken as the velocity of the rod and when r = r2, the velocity at the wall, is 0. B.C. 1: r = r1  u = ur B.C. 2: r = r2  u = 0 Solving these equations with the two boundary conditions and rearranging leads to the velocity profile below: 𝑢𝑢(𝑟𝑟) = 1 4𝜇𝜇 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 �(𝑟𝑟2 2 − 𝑟𝑟2) − (𝑟𝑟2 2 − 𝑟𝑟1 2) ln� 𝑟𝑟2 𝑟𝑟 � ln� 𝑟𝑟2 𝑟𝑟1 � � + 𝑢𝑢𝑟𝑟 (12) The flow rate can be calculated as: 𝑄𝑄 = ∫ 𝑢𝑢(𝑟𝑟)2𝜋𝜋𝜋𝜋 𝑑𝑑𝑑𝑑 (13) Plugging in the velocity profile, the equation can be rewritten as: 𝑢𝑢(𝑟𝑟)2𝜋𝜋𝜋𝜋 𝑑𝑑𝑑𝑑 = �� 1 4𝜇𝜇 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 (𝑟𝑟2 2 − 𝑟𝑟2) + 𝑢𝑢𝑟𝑟 − 1 4𝜇𝜇 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 (𝑟𝑟2 2 − 𝑟𝑟1 2)� ln� 𝑟𝑟2 𝑟𝑟 � ln� 𝑟𝑟2 𝑟𝑟1 � � 2𝜋𝜋𝜋𝜋 𝑑𝑑𝑑𝑑 (14) 𝑄𝑄 = ∫ � 1 4𝜇𝜇 𝑑𝑑𝑃𝑃 𝑑𝑑𝑑𝑑 �(𝑟𝑟2 2 − 𝑟𝑟2) − (𝑟𝑟2 2 − 𝑟𝑟1 2) ln� 𝑟𝑟2 𝑟𝑟 � ln� 𝑟𝑟2 𝑟𝑟1 � � + 𝑢𝑢𝑟𝑟� 2𝜋𝜋𝜋𝜋 𝑑𝑑𝑑𝑑 (15) 𝑄𝑄 = 2𝜋𝜋 ∫ � 1 4𝜇𝜇 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 �(𝑟𝑟2 2 − 𝑟𝑟2) − (𝑟𝑟2 2 − 𝑟𝑟1 2) ln� 𝑟𝑟2 𝑟𝑟 � ln� 𝑟𝑟2 𝑟𝑟1 � � + 𝑢𝑢𝑟𝑟� 𝑟𝑟 𝑑𝑑𝑑𝑑 𝑟𝑟2 𝑟𝑟1 (16)
  31. 31. 22 First, the equation is solved without the 𝑢𝑢𝑟𝑟 term. Breaking up the integral, the flow rate equation can be rewritten as: 𝑄𝑄 = ∫ 𝜋𝜋 2𝜇𝜇 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 (𝑟𝑟2 2 − 𝑟𝑟2)𝑟𝑟 𝑑𝑑𝑑𝑑 𝑟𝑟2 𝑟𝑟1 − ∫ 𝜋𝜋 2𝜇𝜇 𝑟𝑟2 𝑟𝑟1 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 (𝑟𝑟2 2 − 𝑟𝑟1 2) ln� 𝑟𝑟2 𝑟𝑟 � ln� 𝑟𝑟2 𝑟𝑟1 � 𝑟𝑟 𝑑𝑑𝑑𝑑 (17) Simplifying: 𝑄𝑄 = ∫ 𝜋𝜋 2𝜇𝜇 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 𝑟𝑟2 2 𝑟𝑟 𝑑𝑑𝑑𝑑 𝑟𝑟2 𝑟𝑟1 − ∫ 𝜋𝜋 2𝜇𝜇 𝑟𝑟2 𝑟𝑟1 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 𝑟𝑟3 𝑑𝑑𝑑𝑑 − 𝜋𝜋 2𝜇𝜇 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 (𝑟𝑟2 2 − 𝑟𝑟1 2) ∫ ln� 𝑟𝑟2 𝑟𝑟 � ln� 𝑟𝑟2 𝑟𝑟1 � 𝑟𝑟 𝑑𝑑𝑑𝑑 𝑟𝑟2 𝑟𝑟1 (18) Integrating before plugging in the limits of integration: 𝑄𝑄 = � 𝜋𝜋 2𝜇𝜇 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 𝑟𝑟2 2 1 2 𝑟𝑟2 − 𝜋𝜋 2𝜇𝜇 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 1 4 𝑟𝑟4 � � 𝑟𝑟2 𝑟𝑟1 − 𝜋𝜋 2𝜇𝜇 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 (𝑟𝑟2 2 − 𝑟𝑟1 2) � 𝑟𝑟2�2ln 𝑟𝑟2 𝑟𝑟 +1� 4 ln 𝑟𝑟2 𝑟𝑟1 � � 𝑟𝑟2 𝑟𝑟1 (19) Now plugging in the lower and upper bounds of integration, the flow rate equation can be written as such: 𝑄𝑄 = 𝜋𝜋 2𝜇𝜇 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 1 2 𝑟𝑟2 2 𝑟𝑟2 2 − 𝜋𝜋 2𝜇𝜇 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 1 4 𝑟𝑟2 4 − 𝜋𝜋 2𝜇𝜇 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 1 2 𝑟𝑟2 2 𝑟𝑟1 2 + 𝜋𝜋 2𝜇𝜇 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 1 4 𝑟𝑟1 4 − 𝜋𝜋 2𝜇𝜇 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 (𝑟𝑟2 2 − 𝑟𝑟1 2) � 𝑟𝑟2 2�2ln 𝑟𝑟2 𝑟𝑟2 +1� 4 ln 𝑟𝑟2 𝑟𝑟1 − 𝑟𝑟1 2�2ln 𝑟𝑟2 𝑟𝑟1 +1� 4 ln 𝑟𝑟2 𝑟𝑟1 � (20) Combining like terms and simplifying: 𝑄𝑄 = 𝜋𝜋 2𝜇𝜇 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 1 4 𝑟𝑟2 4 − 𝜋𝜋 2𝜇𝜇 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 1 2 𝑟𝑟2 2 𝑟𝑟1 2 + 𝜋𝜋 2𝜇𝜇 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 1 4 𝑟𝑟1 4 − 𝜋𝜋 2𝜇𝜇 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 (𝑟𝑟2 2 − 𝑟𝑟1 2) � 𝑟𝑟2 2−𝑟𝑟1 2(2 ln 𝑟𝑟2 𝑟𝑟1 +1) 4 ln 𝑟𝑟2 𝑟𝑟1 � (21) 𝑄𝑄 = 1 4 𝜋𝜋 2𝜇𝜇 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 (𝑟𝑟1 4 + 𝑟𝑟2 4) − 1 2 𝜋𝜋 2𝜇𝜇 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 𝑟𝑟1 2 𝑟𝑟2 2 − 𝜋𝜋 2𝜇𝜇 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 (𝑟𝑟2 2 − 𝑟𝑟1 2) � 𝑟𝑟2 2−𝑟𝑟1 2(2 ln 𝑟𝑟2 𝑟𝑟1 +1) 4 ln 𝑟𝑟2 𝑟𝑟1 � (22) Combining the first two terms to have 𝜋𝜋 2𝜇𝜇 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 common in both remaining terms:
  32. 32. 23 𝑄𝑄 = 1 4 𝜋𝜋 2𝜇𝜇 𝑑𝑑𝑃𝑃 𝑑𝑑𝑑𝑑 (𝑟𝑟1 4 + 𝑟𝑟2 4 − 2𝑟𝑟1 2 𝑟𝑟2 2) − 𝜋𝜋 2𝜇𝜇 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 (𝑟𝑟2 2 − 𝑟𝑟1 2) � 𝑟𝑟2 2−𝑟𝑟1 2(2 ln 𝑟𝑟2 𝑟𝑟1 +1) 4 ln 𝑟𝑟2 𝑟𝑟1 � (23) Distributing and subtracting the terms: 𝑄𝑄 = 𝜋𝜋 8𝜇𝜇 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 �𝑟𝑟1 4 + 𝑟𝑟2 4 − 2𝑟𝑟1 2 𝑟𝑟2 2 − 4(𝑟𝑟2 2 − 𝑟𝑟1 2) � 𝑟𝑟2 2−𝑟𝑟1 2(2 ln 𝑟𝑟2 𝑟𝑟1 +1) 4 ln 𝑟𝑟2 𝑟𝑟1 �� (24) 𝑄𝑄 = 𝜋𝜋 8𝜇𝜇 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 � 𝑟𝑟1 4 ln 𝑟𝑟2 𝑟𝑟1 +𝑟𝑟2 4 ln 𝑟𝑟2 𝑟𝑟1 −2𝑟𝑟1 2 𝑟𝑟2 2 ln 𝑟𝑟2 𝑟𝑟1 −�𝑟𝑟2 2−𝑟𝑟1 2�� 𝑟𝑟2 2−2𝑟𝑟1 2 ln 𝑟𝑟2 𝑟𝑟1 −𝑟𝑟1 2� ln 𝑟𝑟2 𝑟𝑟1 � (25) 𝑄𝑄 = 𝜋𝜋 8𝜇𝜇 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 � 𝑟𝑟1 4 ln 𝑟𝑟2 𝑟𝑟1 +𝑟𝑟2 4 ln 𝑟𝑟2 𝑟𝑟1 −2𝑟𝑟1 2 𝑟𝑟2 2 ln 𝑟𝑟2 𝑟𝑟1 −𝑟𝑟2 4+2𝑟𝑟1 2 𝑟𝑟2 2 ln 𝑟𝑟2 𝑟𝑟1 +𝑟𝑟1 2 𝑟𝑟2 2+𝑟𝑟1 2 𝑟𝑟2 2−2𝑟𝑟1 4 ln 𝑟𝑟2 𝑟𝑟1 −𝑟𝑟1 4 ln 𝑟𝑟2 𝑟𝑟1 � (26) Combining like terms: 𝑄𝑄 = 𝜋𝜋 8𝜇𝜇 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 � −𝑟𝑟1 4 ln 𝑟𝑟2 𝑟𝑟1 +𝑟𝑟2 4 ln 𝑟𝑟2 𝑟𝑟1 +2𝑟𝑟1 2 𝑟𝑟2 2−𝑟𝑟1 4−𝑟𝑟2 4 ln 𝑟𝑟2 𝑟𝑟1 � (27) Rewriting 2𝑟𝑟1 2 𝑟𝑟2 2 − 𝑟𝑟1 4 − 𝑟𝑟2 4 into −(𝑟𝑟2 2 − 𝑟𝑟1 2)2 : 𝑄𝑄 = 𝜋𝜋 8𝜇𝜇 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 � �𝑟𝑟2 4−𝑟𝑟1 4� ln 𝑟𝑟2 𝑟𝑟1 −�𝑟𝑟2 2−𝑟𝑟1 2� 2 ln 𝑟𝑟2 𝑟𝑟1 � (28) Breaking up the fractional term: 𝑄𝑄 = 𝜋𝜋 8𝜇𝜇 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 � �𝑟𝑟2 4−𝑟𝑟1 4� ln 𝑟𝑟2 𝑟𝑟1 ln 𝑟𝑟2 𝑟𝑟1 − �𝑟𝑟2 2−𝑟𝑟1 2� 2 ln 𝑟𝑟2 𝑟𝑟1 � (29) Cancelling the ln 𝑟𝑟2 𝑟𝑟1 from the first term yields the flow rate equation without the 𝑢𝑢𝑟𝑟 term:
  33. 33. 24 𝑄𝑄 = 𝜋𝜋 8𝜇𝜇 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 �𝑟𝑟2 4 − 𝑟𝑟1 4 − �𝑟𝑟2 2−𝑟𝑟1 2� 2 ln 𝑟𝑟2 𝑟𝑟1 � (30) Now, the 𝑢𝑢𝑟𝑟 term must be added in to solve the new frictional pressure drop: 𝑄𝑄 = 2𝜋𝜋 ∫ � 1 4𝜇𝜇 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 �(𝑟𝑟2 2 − 𝑟𝑟2) − (𝑟𝑟2 2 − 𝑟𝑟1 2) ln� 𝑟𝑟2 𝑟𝑟 � ln� 𝑟𝑟2 𝑟𝑟1 � �� 𝑟𝑟 𝑑𝑑𝑑𝑑 𝑟𝑟2 𝑟𝑟1 + 2𝜋𝜋 ∫ 𝑢𝑢𝑟𝑟 𝑟𝑟2 𝑟𝑟1 𝑟𝑟 𝑑𝑑𝑑𝑑 (31) Substituting Eq. (30) into the first part of Eq. (31): 𝑄𝑄 = 𝜋𝜋 8𝜇𝜇 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 �𝑟𝑟2 4 − 𝑟𝑟1 4 − �𝑟𝑟2 2−𝑟𝑟1 2� 2 ln 𝑟𝑟2 𝑟𝑟1 � + 2𝜋𝜋(𝑢𝑢𝑟𝑟) � 1 2 𝑟𝑟2 � � 𝑟𝑟2 𝑟𝑟1 (32) Integrating and simplifying yields the final flow rate equation: 𝑄𝑄 = 𝜋𝜋 8𝜇𝜇 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 �𝑟𝑟2 4 − 𝑟𝑟1 4 − �𝑟𝑟2 2−𝑟𝑟1 2� 2 ln 𝑟𝑟2 𝑟𝑟1 � + 𝜋𝜋(𝑢𝑢𝑟𝑟)(𝑟𝑟2 2 − 𝑟𝑟1 2) (33) Average velocity can be expressed as: 𝑄𝑄 = 𝜋𝜋(𝑟𝑟2 2 − 𝑟𝑟1 2)𝑢𝑢� (34) Equating Eq. (33) to the average velocity equation will enable the calculation of the 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 . 𝜋𝜋(𝑟𝑟2 2 − 𝑟𝑟1 2)𝑢𝑢� = 𝜋𝜋 8𝜇𝜇 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 �𝑟𝑟2 4 − 𝑟𝑟1 4 − �𝑟𝑟2 2−𝑟𝑟1 2� 2 ln 𝑟𝑟2 𝑟𝑟1 � + 𝜋𝜋(𝑢𝑢𝑟𝑟)(𝑟𝑟2 2 − 𝑟𝑟1 2) (35) Simplifying and isolating the 𝑑𝑑𝑃𝑃𝑓𝑓 𝑑𝑑𝑑𝑑 term:
  34. 34. 25 𝑑𝑑𝑃𝑃𝑓𝑓 𝑑𝑑𝑑𝑑 = 𝜋𝜋�𝑟𝑟2 2−𝑟𝑟1 2�(𝑢𝑢�−𝑢𝑢𝑟𝑟) 𝜋𝜋 8𝜇𝜇 �𝑟𝑟2 4−𝑟𝑟1 4− �𝑟𝑟2 2−𝑟𝑟1 2� 2 ln 𝑟𝑟2 𝑟𝑟1 � (36) 𝑑𝑑𝑃𝑃𝑓𝑓 𝑑𝑑𝑑𝑑 = 8𝜇𝜇(𝑟𝑟2 2−𝑟𝑟1 2)(𝑢𝑢�−𝑢𝑢𝑟𝑟) �𝑟𝑟2 4−𝑟𝑟1 4− �𝑟𝑟2 2−𝑟𝑟1 2� 2 ln 𝑟𝑟2 𝑟𝑟1 � (37) The final equation is seen below: 𝑑𝑑𝑃𝑃𝑓𝑓 𝑑𝑑𝑑𝑑 = 8𝜇𝜇(𝑢𝑢�−𝑢𝑢𝑟𝑟) �𝑟𝑟2 2+𝑟𝑟1 2− 𝑟𝑟2 2−𝑟𝑟1 2 ln 𝑟𝑟2 𝑟𝑟1 � (38) This equation is the final equation used to calculate the effect of viscosity and the additional pressure that must be added on to the predicted loads from the simulator. These are found in the next section of this work, the results and discussion. It can be seen from Eq. (38), that for any constant average flow velocity and velocity of the rod, the frictional pressure drop gradient can be calculated based on the viscosity. This gradient can be multiplied by the length of the rod to calculate the pressure due to the viscous force and added to the polished rod loads (PPRL and MPRL); these loads can then be used to calculate the effect on the torque. Eq. (38) can be compared to Eq. (1) in the literature review section to see the effect that the velocity of the rod has on the frictional pressure drop gradient.
  35. 35. 26 CHAPTER 6 RESULTS AND DISCUSSIONS 6.1 Results Following the digitization plots in Chapter 4, the viscosity effect plots are presented below. As described in the literature review section, the API-RP 11L process works primarily for inviscid or very low viscosity fluid. The digitized equations (Eqs. 1-6) in conjunction with Eq. (38) were imbedded into the simulator to study the effect of fluid viscosity on the pump design parameters such as PPRL, MPRL, and PT. The simulator also was used to study the effect of efficiency on these parameters. Before the viscosity results are shown, the accuracy of the simulator must be established in order to validate the results. When comparing the results of the simulator to those in literature, the results correlate quite well. This was found both with the SPE 20152 paper (Jennings, 1989) and different presentations outlining examples of the API-RP 11L procedure. When comparing the results of the simulator to SPE 20152, the main outliers occur in the PT and the PPRL, with the computer simulator under-predicting these two factors. However, the estimated MPRL, the CBE and the PRHP are quite close to those in literature, being less than 5% off the actual values in the published paper. The differences can be attributed to the digitization; the process using the solver technique is a good estimation tool to try and fit the data, however, exact results were not seen. A graphical comparison of the five important parameters (PPRL, MPRL, PT, PRHP, CBE) between SPE 20152 and the simulator developed of all three units are shown: Figure 21: PPRL: SPE 20152 vs Simulator Comparisons
  36. 36. 27 Figure 22: MPRL: SPE 20152 vs Simulator Comparisons Figure 23: PT: SPE 20152 vs Simulator Comparisons
  37. 37. 28 Figure 24: PRHP: SPE 20152 vs Simulator Comparisons Figure 25: CBE: SPE 20152 vs Simulator Comparisons
  38. 38. 29 6.1.1 Polished Rod Load Results Figure 26: Conventional Units: Viscosity Effect on Polished Rod Loads at 5000 ft Fig. 26 shows quantitatively how the fluid viscosity affects the PPRL and MPRL for a conventional pumping unit. As the fluid viscosity increases, the friction between the fluids and the rods as well as the fluid and production casing increase also. This leads to higher polished rods. A 10.053% increase in the PPRL and a 21.961% increase in the MPRL are observed when the fluid viscosity varies from 0 to 4000 cp. Due to the fact that some heavy oils have viscosities as high as 10,000 cp, this increase is quite significant and must be taken into account when designing a pumping unit. Similar plots to that of Fig. 26 can be seen in the Appendix for air balanced and Mark II units. 10.053% increase 21.961% increase
  39. 39. 30 Figure 27: Comparison of the PPRL of the Three Units at 5000 ft Figure 28: Comparison of the MPRL of the Three Units at 5000 ft
  40. 40. 31 The effect viscosity has on the polished rod loads at 5000 ft for both air balanced units and Mark II units is similar to that seen in the conventional unit (Fig. 26), however with different magnitudes. Figs. 27 and 28 show the magnitudes of the PPRL and MPRL of all three pumping units. It can be seen from both of these plots that conventional units have larger PPRL and MPRL than air balanced and Mark II units. The geometry of both of these units as described in the literature review chapter help to explain why the polished rod loads are lower than those of the conventional unit. The position of the gear reducer in both of these systems allow for more degrees of the crank travel on the upstroke (180° to 195°), and this decreases the loads. Also unique to the Mark II units is a lever system that is also used to further decrease the upstroke acceleration and decrease the rod load (Lufkin, 2013). Figure 29: Conventional Units: Effect of Viscosity and Length on PPRL
  41. 41. 32 Figure 30: Conventional Units: Effect of Viscosity and Length on MPRL The results from Figs. 26-28 are simply shown for a rod that is 5000 ft in length for all three pumping units. In Figs. 29 and 30, the length of the rod is varied in the simulator developed along with the viscosity from 0 to 6000 cp for the conventional unit to see the effect both of these factors have on the PPRL and MPRL. As the length of the rod increases, the loads increase as well due to more friction. The trends are otherwise similar to those seen at only 5000 ft, with a gradual increase of load due to viscosity. However, it can be seen that as the length of the rod increases, the increase of viscosity has a greater impact on the increase of the polished rod loads. This is to say that as the length of the rod increases, the effect of viscosity is much greater. This is because of the “rod heavy” phenomenon where the counterbalance effect (counterweights for the conventional and Mark II units) of the rods is greater than the effect of the weights, which causes more stress on the rods. It should be noted that based on the results for the polished rod loads, the effect of viscosity can be neglected at 500 centipoise or below, with only about a 1.5% increase on the PPRL and a 2.7% increase on the MPRL.
  42. 42. 33 Figure 31: Conventional Units: Percent Increase of PPRL at 5000 ft due to Viscosity Figure 32: Conventional Units: Percent Increase of MPRL at 5000 ft due to Viscosity
  43. 43. 34 Figs. 31 and 32 show both the magnitudes (bars) of the PPRL and MPRL at 5000 ft on the left y-axis and the percent increase (line) due to viscosity of these loads on the right y-axis. It can be seen that for conventional units, when increasing the viscosity from 0 to 6000 cp, the increase of the PPRL is about 15% and the increase of the MPRL is 33%. Again, it should be noted that based on purely the polished rod loads, the effect of viscosity can be neglected at 500 centipoise or below. The results for the effect of viscosity and length as well as the percent increase are quite similar for the air balanced and Mark II units as seen in the appendix. Figure 33: Comparisons of PPRL Percent Increase of the Three Units at 5000 ft
  44. 44. 35 Figure 34: Comparisons of MPRL Percent Increase of the Three Units at 5000 ft Figs. 33 and 34 show the comparisons of the percent increase for both the PPRL and the MPRL for all of the three pumping units at 5000 feet. There is about a 14-16% increase evident in the PPRL for all three pumping units when the viscosity is increased from 0 to 6000 cp and a 32-40% increase in the MPRL for the three pumping units. It can be seen that in both of these plots, the Mark II units have a higher percent increase in the loads due to the viscosity increase than the conventional and air balanced units. Due to the geometry and the lever system, the Mark II units are designed to decrease the rod load. However, with the effect of viscosity affecting all of the pumping units similarly, the lower magnitude initially with the inviscid fluid in the Mark II units will cause the increase to be of a higher rate with high viscosity fluid than the other pumping units. Again, with the increase of the fluid viscosity, the friction between both the fluid and the rods and the fluid and the casing are also higher which causes these increases of the polished rod loads. These percent increases are quite significant to study when designing a pumping system as some oil can have a viscosity of up to 10,000 cp.
  45. 45. 36 6.1.2 Peak Torque Results Figure 35: Conventional Units: Effect of Viscosity on Peak Torque (Upstroke / Downstroke) Moving from the effect viscosity has on the polished rod loads, there is also a sizable impact it has on the peak torque, both on the upstroke and downstroke as well as the overall torque. In Figs. 35-37, the peak torque in both the upstroke and downstroke are presented for all three pumping units. There are similar increases in the peak torque on the upstroke, due to the increased stress put on the rod and more friction with increased viscosity as the fluid is being carried to the surface. There are also similar decreases associated with the peak torque on the downstroke as for all the pumping units, the heavier the oil is, the less torque needed to lower the plunger after the fluid has been unloaded on the surface. The different behavior is discussed beneath each plot. In Fig. 35, for the conventional units, the peak torque on the upstroke demonstrates an increase of 35% over 4000 cp and the peak torque on the downstroke experiences about a 39% decrease over 4000 cp. This is the expected behavior for the conventional unit with more torque needed on the upstroke for an inviscid fluid than on the downstroke. 35.305% increase 39.041% decrease
  46. 46. 37 Figure 36: Air Balanced Units: Effect of Viscosity on Peak Torque (Upstroke / Downstroke) Fig. 36 shows the behavior of the peak torque on the upstroke and downstroke for an air balanced unit. There is about a 36% increase on the upstroke at 4000 cp and a 35% decrease on the downstroke at 4000 cp. This is a distinct difference from Fig. 35 as it can be seen that more torque is needed at low viscosity on the downstroke in air balanced units than conventional units. This phenomenon is also seen in the Mark II units and explained below. 35.463% decrease 36.096% increase
  47. 47. 38 Figure 37: Mark II Units: Effect of Viscosity on Peak Torque (Upstroke / Downstroke) When comparing both Figs. 36 and 37 with Fig. 35, it can be seen that more torque is needed on the downstroke for inviscid fluids in the air balanced and Mark II units than in conventional systems. This is much more evident for the Mark II units in Fig. 37, however. This can be explained from the geometry of both the air balanced and Mark II units. Both of these pumping units employ more degrees of crank travel on the upstroke, which leads to a 195° upstroke. This change with the shifting of the gear reducer, first and foremost, reduces the polished rod loads by decreasing the acceleration on the upstroke stage where the greatest load can be seen. However, the unique position of the gear reducer also has consequences for the torque. Because the gear reducer is shifted from underneath the equalizer (conventional) towards the Samson post, the cross-yoke, located by the horsehead, is forward of the gear reducer (air balanced, Mark II). In lifting the heavier load on the upstroke, a greater mechanical advantage is created, and a lesser mechanical advantage is created for the reduced downstroke load. Because of this, both the maximum torque factor on the upstroke decreases and the maximum torque factor on the downstroke increases (Lufkin, 2013). This shift is more evident in the Mark II units and can be seen in Fig. 37, and to a lesser extent in the air balanced units in Fig. 36 as well. Also, with some heavy oils having viscosity as high as 10,000 cp, this very high increase in torque must be considered in the pumping unit design. 32.798% decrease 40.702% increase
  48. 48. 39 Figure 38: Comparisons of Peak Torque Upstroke of the Three Units at 5000 ft Figure 39: Comparisons of Peak Torque Downstroke of the Three Units at 5000 ft Figs. 38 and 39 simply compare the magnitudes of the peak torque on the upstroke and downstroke for all three pumping systems at 5000 feet. These results agree with the
  49. 49. 40 geometry as well as the features described in the literature review section for the Mark II units. This unit not only helps to lower the power costs and the size of the prime mover, but also reduce the torque requirements on the gear reducer up to 35% (Lufkin, 2013). Figure 40: Conventional Units: Effect of Viscosity and Length on Peak Torque Fig. 40 shows the relationship between the length of the rod and viscosity on the overall peak torque for a conventional unit. This is the expected result: an exponential increase of torque as the length increases with the effect of viscosity being much greater at high lengths. This is again because of the “rod heavy” phenomenon where the counterbalance effect (counterweights for the conventional and Mark II units) of the rods is greater than the effect of the weights, which causes more stress on the rods. This plot also shows that, similarly to the polished rod loads, at fluid viscosities 500 cp and below, the viscosity effects can be neglected, with only a 4.4% increase in the PT.
  50. 50. 41 Figure 41: Conventional Units: Percent Increase of Peak Torque at 5000 ft due to Viscosity Fig. 41 shows the magnitudes of the peak torque at 5000 feet along with the percent increase at the different viscosities from 0 to 6000 cp for a conventional unit. This plot shows that for a 6000 cp fluid, there is approximately a 53% increase in the peak torque required. When compared to the polished rod load percent increase plots (Figs. 31-32), it can be concluded that heavy oil has a greater impact on the torque requirements than the polished rod load requirements. This is even more prominent due to the fact that some heavy oils have viscosities up to 10,000 cp, which may even double the torque needed in a pumping unit. This will shift major emphasis on design considerations.
  51. 51. 42 6.1.3 Efficiency Results Figure 42: Conventional Units: Effect of Efficiency and Viscosity on PPRL From the general relationship of viscosity and efficiency seen in Fig. 9, there are also correlations seen with each of these variables on the polished rod loads, as seen in Figs. 42 and 43. As the efficiency was increased in the simulator, both the stroke speed (N) and the plunger stroke length (Sp) decreased. The equation for the stroke speed is: 𝑁𝑁 = 𝑄𝑄∗𝐵𝐵𝑜𝑜 0.1166∗𝑒𝑒∗𝐷𝐷𝑝𝑝 2∗𝑆𝑆 (39) The plunger stroke length is a function of both the dimensionless rod stretch and dimensionless pump speed as found in the digitization chapter. Because the change in efficiency has a direct relationship with the stroke speed and it is used in the calculation of the dimensionless pump speed, the efficiency has a direct relationship with both of these parameters. The change in the dimensionless pump speed impacts all of the plots involved in the digitization, however, most notably, the polished rod loads. The range of efficiency that was observed was from 0.75 to 1; pump efficiency is ordinarily a fraction around 0.9, and values outside of this range were resulting in abnormal output.
  52. 52. 43 Figure 43: Conventional Units: Effect of Efficiency and Viscosity on MPRL Figs. 42 and 43 show the effect of efficiency on the polished rod loads for conventional units only. These results are similar to those of air balanced and Mark II units and those plots can be found in the Appendix section. From Figs. 42 and 43, it can be seen that as the pump efficiency increases from 0.75 to 1, the PPRL slightly decreases, while the MPRL slightly increases; this is due to fluid inertia effects. Pump efficiency affects the dynamic fluid load on the plunger (Svinos, 2008). Because sucker rod pumps employ high speed prime movers, it is important to consider the rotating moments of inertia when simulating pumping parameters. These moments of inertia refer to those associated with the counterweights, cranks, gear reducer, walking beam, horsehead, and the motor. All of these individual moments of inertia are seen to lower the torque, PPRL, and PRHP and increase the MPRL. This is because as the torque on the gear reducer increases during pumping, the prime mover decreases speed. This then causes the crank and counterweight inertia to release energy which in turn, lowers the torque the gear reducer can supply. The inertia of the walking beam and horsehead, conversely, adds torque to the gear reducer, however, this effect is quite small. As the prime mover slows down with the high torque, the polished rod load slows down as well (Svinos, 2008). This reduction in speed leads to lower dynamic forces which leads to a lower PPRL, PRHP and a higher MPRL. It is known from previous results that as the viscosity increases, an increased load will also be seen. Fig. 44 is the plot between the PRHP and Efficiency: (It should be noted that the Polished Rod Horsepower is the same for all three pumping units as described in API-RP 11L)
  53. 53. 44 Figure 44: Polished Rod Horsepower (HP) vs Efficiency 6.2 Discussions This study is an attempt to further expand the API-RP 11L process to include the application of viscous oil and the effect viscosity has on the pumping characteristics the process helps to predict. The approach involved four stages: the digitization of the API-RP 11L plots to develop equations for each pumping parameter which served as inputs to the second stage – the simulator developed, the development of the frictional pressure losses gradient equation, and lastly, determining the effect of viscosity on several pumping characteristics with the use of the simulator and pressure gradient equation. Each stage of the project presented numerous challenges in both the design and optimization. The first objective of the project was the digitization of the API-RP 11L plots, which proved to be quite the task as the solver data analysis tool in Excel is quite involved. The technique is based upon setting an objective cell equal to a certain value, or maximizing or minimizing it by changing certain variable cells. In the case of this work, the variable cells are the coefficients of each digitization equation based on the plots in Figures 10-19. To subject the objective cell to the appropriate delineation of optimization, constraints must be added. These included subjecting specific cells to maintain their values, most of which being the most important values. These were primarily the first and last points or points of deviation in each plot. This was the most time-consuming part, as the constraints had to be optimized to ensure the best fit plot for the work conducted. This stage also was the backbone of the project as the primary results were based off of the digitization process.
  54. 54. 45 The next stage of the project was implementing the API-RP 11L procedure along with the equations generated from the digitization into a computer simulator. This simulator enables the operator to predict certain pump characteristics without the use of tables and figures found in literature. The simulator is set up so the user can input all of the pertinent data (production rate, specific gravity, pump depth, stroke speed and length, etc...) and the simulator will present all of the API dimensionless groups along with the associated loads, torques, and horsepowers needed in all three pumping units (conventional, air balanced, and Mark II) to model any pumping system. The accuracy of the simulator was validated from SPE 20152, as seen from Figs. 21-25 in the results section. The third stage of the project was solving the equation for the frictional pressure losses gradient. This included going through the general form of the momentum equation with many assumptions (described in chapter 5) in order to find an equation for the gradient dependent on the viscosity, holding the velocities constant. The derivation and simplification of certain integrals proved to be the challenge, especially when calculating the flow rate based upon the velocity profile. In deriving the final pressure gradient equation, there were numerous steps taken. First, the theoretical equation of the pressure drop in annular flow of Newtonian fluids was considered. The velocity of the rod was not considered in this step in order to validate the results of the derivation agreed to that in literature. Then after the work correlated to that in literature, the velocity of the rod was added in order to get the final equation. The result included an equation that stated if the average flow velocity and the velocity of the rod were held constant along with the radii, the viscosity could be varied in order to calculate the pressure drop gradient. This gradient could then be multiplied by the total length to achieve the additional pressure friction would induce based on the viscosity that must be added to the loads calculated in the simulator. This equation was used quite extensively in the last stage of the project. The final stage of the project was numerically finding the effects of viscous oil on the pumping characteristics predicted from the simulator. The results are seen not only from Figs. 26-44 in the results section, but also in the plots seen in the Appendix. It can be seen that based on all of the plots that there is quite a correlation between viscosity and the associated loads and torque on the system as a whole. High viscosity fluids greatly increase the magnitude of the polished rod loads and torque, both on the upstroke and downstroke as well as overall. The effect is seen more so when calculating the torque than the polished rod loads, but is still quite high in both parameters. Of course, the standard gravity of oil or lighter oils can be produced with standard pump designs, however pumping very heavy oil will call for major design changes when designing a pump. Rod centralizers, paraffin scrapers and rod guides will need to be employed as they not only help keep the rods and couplings away from the tubing to decrease the wear, but also aid in the stabilization of the pump. Pump and rod wear occur much more frequently for heavy oil applications. Viscous oil production will also call for a heavier rod string for more support and that leads to a heavier counterbalance effect. As
  55. 55. 46 it can be seen, viscous oil production is much more costly and difficult than lighter oil production.
  56. 56. 47 CHAPTER 7 CONCLUDING REMARKS The goal of this work was to improve upon the API-RP 11L procedure as it is used only for inviscid or low viscosity fluids. Also, to create a simulator where instead of using tables and figures to calculate the parameters needed, equations are used. The following conclusions can be drawn from the study: • The digitization used in this study along with the computer simulator seem to slightly underestimate the peak polished rod load and peak torque. • Heavy oil has a major effect on rod loads and torque, but as seen in the plots in the results section, the effect on the torque is much higher than is seen on the rod loads due to the stress that is put on the gear reducer (>50% vs ~25%). • For conventional units: o 10% increase on PPRL from 0 – 4000 cp o 25% increase on MPRL from 0 – 4000 cp o 35% increase on the Peak Torque Upstroke from 0 – 4000 cp o 40% decrease on the Peak Torque Downstroke from 0 – 4000 cp o 55% increase on Peak Torque from 0 – 6000 cp Based on these conclusions, this work shows the modeling of a pumping system becomes far more complicated when dealing with a heavy oilfield. As stated before, heavy oil reservoirs present a mountain of problems, from cementing and keeping the wellbore integrity to very close pressure maintenance (Halliburton, 2009). Cost is also one of the most, if not the most important factor in the decision to drill a heavy oil reservoir. This research helps to quantify the effect of the viscous oil on the pump system based on the momentum equation as well as give the operator an easier method to predict pump loads and torque requirements. However, with the rod stretch seen downhole with the production of heavy oil as well, there is still much to be understood regarding the effect of viscosity not only on the surface equipment, but also downhole.
  57. 57. 48 CHAPTER 8 FUTURE WORK The following are some recommendations for suggested areas of further research: • Drill string Design / Buckling Applications • Drag and Torque Applications for straight and inclined wellbores • Finding the ideal stabilizer positions as well as the distance required between them.
  58. 58. 49 REFERENCES Black Gold Pump and Supply, Inc. Plungers: Monel. (2015). Signal Hill, California. Heavy Oil Industry Challenges – Halliburton Solutions Summary. (2009). Houston, Texas. <http://www.halliburton.com/public/solutions/contents/Heavy_Oil/related_docs/ H06685.pdf> Herzog, S., Neveu, C., Placek, D. (2005). The Benefits of Maximum Efficiency Hydraulic Fluids. Noria Publication: Machinery Lubrication. Louisville, Kentucky. Hyne, Norman J. (2012). Nontechnical Guide to Petroleum Geology, Exploration, Drilling & Production, 3rd Edition. Tulsa, Oklahoma. Jennings, J.W, SPE. (1989). Design of Sucker-Rod Pump Systems. Society of Petroleum Engineers SPE 20152. Texas A&M University. <https://www.onepetro.org/download/conference-paper/SPE-20152- MS?id=conference-paper%2FSPE-20152-MS> Juch, A.H, Watson, R.J, SPE. (1969). New Concepts in Sucker-Rod Pump Design. Society of Petroleum Engineers SPE 2152. Shell de Venezuela, Ltd. <https://www.onepetro.org/download/journal-paper/SPE-2172-PA?id=journal- paper%2FSPE-2172-PA> Kelly, Michael. (2000). Rod Pumping Overview – Class Notes. New Mexico Institute of Mining and Technology. Lea, J.F., Rowlan, Lynn. (2014). Guidelines & Recommended Practices: Selection of Artificial Lift Systems for Deliquifying Gas Wells. Artificial Lift Research and Development Council. Austin, Texas. Li, J., Han, M., Han, X. (2012). A New Pump Application in Heavy Oil Recovery. Design Innovation Papers. China University of Petroleum. <http://fluidsengineering.asmedigitalcollection.asme.org/article.aspx?articleid=14 40796> Lufkin Conventional & Reverse Mark Pumping Units Operator’s Manual A-82. (2013). Lufkin, Texas. Lufkin Air Balanced Pumping Units Operator’s Manual A-82. (2013). Lufkin, Texas. Lufkin Mark II Pumping Units Operator’s Manual A-82. (2013). Lufkin, Texas.
  59. 59. 50 Rohatgi, Ankit. (2015). WebPlotDigitizer User Manual, Version 3.9. Austin, Texas. <http://arohatgi.info/WebPlotDigitizer/userManual.pdf> Rudnick, Leslie. (2013). Synthetics, Mineral Oils and Bio-Based Lubricants, 2nd Edition. Chemical Industries. Philadelphia, Pennsylvania. Svinos, J.G, Treiberg, Terry. (2008). XROD-V: Advanced Modern Design and Simulation of Rod Pumping Systems for Vertical Wells, 3rd Edition. La Habra, California. <http://www.doverals.com/File%20Library/Theta/Manual%20and%20Brochure/ manual-xrod.pdf> Takacs, Gabor. (2015). Sucker Rod Pumping Handbook: Production Engineering Fundamentals and Long-Stroke Rod Pumping. Hungary.
  60. 60. 51 APPENDIX Figure 45: Air Balanced Units: Viscosity Effect on Polished Rod Loads at 5000 ft Figure 46: Mark II Units: Viscosity Effect on Polished Rod Loads at 5000 ft 10.449% increase 24.605% increase 10.731% increase 26.822% increase
  61. 61. 52 Figure 47: Air Balanced Units: Effect of Viscosity and Length on PPRL Figure 48: Air Balanced Units: Effect of Viscosity and Length on MPRL
  62. 62. 53 Figure 49: Air Balanced Units: % Increase of PPRL at 5000 ft due to Viscosity Figure 50: Air Balanced Units: % Increase of MPRL at 5000 ft due to Viscosity
  63. 63. 54 Figure 51: Mark II Units: Effect of Viscosity and Length on PPRL Figure 52: Mark II Units: Effect of Viscosity and Length on MPRL
  64. 64. 55 Figure 53: Mark II Units: % Increase of PPRL at 5000 ft due to Viscosity Figure 54: Mark II Units: % Increase of MPRL at 5000 ft due to Viscosity
  65. 65. 56 Figure 55: Conventional Units: Effect of Viscosity and Length on Peak Torque (Upstroke) Figure 56: Conventional Units: Effect of Viscosity and Length on Peak Torque (Downstroke)
  66. 66. 57 Figure 57: Air Balanced Units: Effect of Viscosity and Length on Peak Torque (Upstroke) Figure 58: Air Balanced Units: Effect of Viscosity and Length on Peak Torque (Downstroke)
  67. 67. 58 Figure 59: Mark II Units: Effect of Viscosity and Length on Peak Torque (Upstroke) Figure 60: Mark II Units: Effect of Viscosity and Length on Peak Torque (Downstroke)
  68. 68. 59 Figure 61: Air Balanced Units: Effect of Viscosity and Length on Peak Torque Figure 62: Air Balanced Units: % Increase of Peak Torque at 5000 ft due to Viscosity
  69. 69. 60 Figure 63: Mark II Units: Effect of Viscosity and Length on Peak Torque Figure 64: Mark II Units: % Increase of Peak Torque at 5000 ft due to Viscosity
  70. 70. 61 Figure 65: Air Balanced Units: Effect of Efficiency and Viscosity on PPRL Figure 66: Air Balanced Units: Effect of Efficiency and Viscosity on MPRL
  71. 71. 62 Figure 67: Mark II Units: Effect of Efficiency and Viscosity on PPRL Figure 68: Mark II Units: Effect of Efficiency and Viscosity on MPRL
  72. 72. 63 COMPUTER SIMULATOR PROGRAM CODE Sub CommandButton1_Click() Q = Sheet1.Cells(3, 2) Bo = Sheet1.Cells(4, 2) G = Sheet1.Cells(5, 2) Muf = Sheet1.Cells(6, 2) L = Sheet1.Cells(7, 2) H = Sheet1.Cells(8, 2) S = Sheet1.Cells(9, 2) Dp = Sheet1.Cells(10, 2) C = Sheet1.Cells(11, 2) Lp = Sheet1.Cells(12, 2) e = Sheet1.Cells(13, 2) API = Sheet1.Cells(14, 2) RodClass = Sheet1.Cells(15, 2) SinkerBarDia = Sheet1.Cells(16, 2) ServiceFactor = Sheet1.Cells(17, 2) N = Sheet1.Cells(31, 2) Sp = Sheet1.Cells(32, 2) Ta = Sheet1.Cells(11, 6) N = (Q * Bo) / (0.1166 * e * Dp ^ 2 * S) Sheet1.Cells(31, 2) = N IPD = 0.1166 * Sp * N * Dp ^ 2 Sheet1.Cells(20, 2) = IPD Fo = 0.34 * G * Dp ^ 2 * L Sheet1.Cells(22, 2) = Fo Kr = 1 / (L * 0.000000997) Sheet1.Cells(23, 2) = Kr Skr = Sp * Kr Sheet1.Cells(24, 2) = Skr Wrf = 1.566 * L * (1 - 0.128 * G) Sheet1.Cells(25, 2) = Wrf FoSkr = Fo / Skr Sheet1.Cells(3, 6) = FoSkr NNo = (N * H) / 245000 Sheet1.Cells(5, 6) = NNo NNop = (NNo / 1.191) Sheet1.Cells(6, 6) = NNop
  73. 73. 64 WrfSkr = Wrf / Skr Sheet1.Cells(7, 6) = WrfSkr SpS = -288.464 * (NNop) ^ 6 + 464.4255 * (NNop) ^ 5 - 285.941 * (NNop) ^ 4 + 88.94985 * (NNop) ^ 3 - 13.189 * (NNop) ^ 2 + 1.124451 * (NNop) - 8.44032 + 5.554806 * (FoSkr) ^ 4 - 7.0981 * (FoSkr) ^ 3 + 3.325249 * (FoSkr) ^ 2 - 1.54031 * (FoSkr) + 9.434789 Sheet1.Cells(4, 6) = SpS F1Skr = -187.783 * (NNo) ^ 6 + 341.1534 * (NNo) ^ 5 - 210.786 * (NNo) ^ 4 + 55.62392 * (NNo) ^ 3 - 5.58467 * (NNo) ^ 2 + 0.748049 * (NNo) + 3.789787 + 1.252434 * (FoSkr) ^ 4 - 0.39606 * (FoSkr) ^ 3 - 0.90549 * (FoSkr) ^ 2 + 1.225137 * (FoSkr) - 3.80377 Sheet1.Cells(8, 6) = F1Skr F2Skr = -21.2101 * (NNo) ^ 6 + 86.57893 * (NNo) ^ 5 - 82.5498 * (NNo) ^ 4 + 29.56686 * (NNo) ^ 3 - 3.48565 * (NNo) ^ 2 + 0.63318 * (NNo) - 0.0152 - 30.5551 * (FoSkr) ^ 4 + 37.86197 * (FoSkr) ^ 3 - 16.7337 * (FoSkr) ^ 2 + 3.10824 * (FoSkr) - 0.19281 Sheet1.Cells(9, 6) = F2Skr TSkr = -74.9862 * (NNop) ^ 6 + 132.8622 * (NNop) ^ 5 - 87.8538 * (NNop) ^ 4 + 27.16812 * (NNop) ^ 3 - 3.52134 * (NNop) ^ 2 + 0.663331 * (NNop) + 0.039938 + 5.705248 * (FoSkr) ^ 4 - 5.62858 * (FoSkr) ^ 3 + 1.028574 * (FoSkr) ^ 2 + 0.573612 * (FoSkr) - 0.00253 Sheet1.Cells(10, 6) = TSkr If FoSkr < 0.3 Then F3Skr = -1.70788 * (NNo) ^ 6 - 12.8682 * (NNo) ^ 5 + 18.29899 * (NNo) ^ 4 - 6.6556 * (NNo) ^ 3 + 1.49254 * (NNo) ^ 2 + 0.071709 * (NNo) - 0.37493 + 11.87353 * (FoSkr) ^ 4 - 20.4182 * (FoSkr) ^ 3 + 8.6778 * (FoSkr) ^ 2 - 0.58959 * (FoSkr) + 0.451606 Sheet1.Cells(12, 6) = F3Skr ElseIf FoSkr > 0.3 Then F3Skr = -4.56469 * (NNo) ^ 6 - 15.5761 * (NNo) ^ 5 + 11.95129 * (NNo) ^ 4 + 3.109074 * (NNo) ^ 3 - 1.20497 * (NNo) ^ 2 + 0.303059 * (NNo) + 0.031177 + 13.80571 * (FoSkr) ^ 4 - 20.11 * (FoSkr) ^ 3 + 8.649151 * (FoSkr) ^ 2 - 0.40286 * (FoSkr) - 0.09432 Sheet1.Cells(12, 6) = F3Skr End If Sp = (Sps * S) Sheet1.Cells(32, 2) = Sp APD = 0.1166 * Sp * SpS * N * Dp ^ 2 Sheet1.Cells(21, 2) = APD
  74. 74. 65 PPRL = Wrf + (F1Skr) * (Skr) Sheet1.Cells(3, 9) = PPRL MPRL = Wrf - (F2Skr) * (Skr) Sheet1.Cells(4, 9) = MPRL CBE = 1.06 * (Fo / 2 + Wrf) Sheet1.Cells(7, 9) = CBE PT = (TSkr) * (Skr) * (S / 2) * (Ta) Sheet1.Cells(6, 9) = PT PRHP = (F3Skr) * (Skr) * (Sp) * (N) * (0.00000253) Sheet1.Cells(8, 9) = PRHP NEMAHP = (PRHP * 1.375) / (e * 0.7) Sheet1.Cells(10, 9) = NEMAHP PPRLA = Wrf + 0.85 * ((F1Skr * Skr)) + 0.15 * Fo Sheet1.Cells(12, 9) = PPRLA MPRLA = PPRLA - ((F2Skr + F1Skr) * Skr) Sheet1.Cells(13, 9) = MPRLA CBEA = 0.52 * (PPRLA + 1.25 * MPRLA) Sheet1.Cells(16, 9) = CBEA PTA = ((0.233 * PPRLA) - (0.3 * MPRLA)) * S Sheet1.Cells(15, 9) = PTA PRHPA = (F3Skr) * (Skr) * (Sp) * (N) * (0.00000253) Sheet1.Cells(17, 9) = PRHPA NEMAHPA = (PRHP * 1.375) / (e * 0.7) Sheet1.Cells(19, 9) = NEMAHPA PPRLM = Wrf + 0.75 * (F1Skr * Skr) + 0.25 * Fo Sheet1.Cells(21, 9) = PPRLM MPRLM = PPRLM - ((F2Skr + F1Skr) * Skr) Sheet1.Cells(22, 9) = MPRLM CBEM = 0.53 * (PPRLM + MPRLM) Sheet1.Cells(25, 9) = CBEM PTM = 0.48 * (TSkr) * Skr * S * Ta
  75. 75. 66 Sheet1.Cells(24, 9) = PTM PRHPM = (F3Skr) * (Skr) * (Sp) * (N) * (0.00000253) Sheet1.Cells(26, 9) = PRHPM NEMAHPM = (PRHP * 1.375) / (e * 0.7) Sheet1.Cells(28, 9) = NEMAHPM End Sub
  76. 76. Applications of Viscous Oil in a Rod Sucker Pump System By Utkarsh Bhargava Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the last page. To copy otherwise, to republish, to post on servers or to redistribute to lists, requires prior specific permission and may require a fee.

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