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Productivity Index of Horizontal Oil Wells
Nasser AlMolhem
Problem Report Submitted to the
College of Engineering and Mineral Resources
At West Virginia University
In partial fulfillments of the requirements
For the degree of
Masters of Science
In
Petroleum and Natural Gas Engineering
Kashy Aminian, Ph.D., Chair
Sam Ameri, Prof.
Mehrdad Zamirian, Ph.D.
Department of Petroleum and Natural Gas Engineering
Morgantown, West Virginia
2016
Keywords: productivity index, oil horizontal wells
Copyright 2016 Nasser AlMolhem
Abstract
Productivity Index of Horizontal Oil Wells
Nasser AlMolhem
Productivity index is the practical approach to characterize the performance of oil wells. During the evolution of petroleum industry, many productivity index PI solutions for different well types have been developed. Initially, PI values were calculated for oil vertical wells. As the drilling technology progressed, PI solutions were considered for horizontal wells. There are different methods for predicting the PI of both vertical and horizontal wells. The main objective of this study is to compare the PI values generated from those different approaches. Moreover, this research aims to highlight the most influential reservoir properties to the PI. A range of static data was given to perform this sensitivity analysis.
iii
Acknowledgement Firstly, I would like to thank God for giving me the support to be successful in my life.
Also, I would like to thank my parents for their encouragement. I want to express my
deepest gratitude and appreciation to my advisor Dr. Kashy Aminian, throughout his
advising through my research, and for giving me the opportunity to work under his
supervision.
I want to mention the support of the chairman of the Petroleum and Natural Gas
Engineering Department at West Virginia University, Professor Samuel Ameri for his
incomparable personality and his fatherhood to every student, which makes our
department the best environment for study.
Also, I sincerely thank Dr. Zamirian for his guidance, support during my research
work and for being on the committee.
My special thanks to all the faculty and staff at the Department of petroleum and
Natural Gas Engineering. I would like to send my special thanks to all my relatives in
Saudi, and all my friends and colleagues that I have met in Morgantown, West Virginia.
I also would like to acknowledge Saudi Aramco for their support and consultation
throughout the research.
iv
Table of Content
Abstract ..........................................................................................................................................
Acknowledgement .................................................................................................................. iii
List of Tables .............................................................................................................................. 3
List of Figures ............................................................................................................................. 4
CHAPTER I. INTRODUCTION ................................................................................................. 1 1.1 Overview .................................................................................................................................. 1 1.1.1 Background ............................................................................................................................ 1 1.1.2 General Definition ................................................................................................................ 1 1.1.3 Geometry of Horizontal Wells.......................................................................................... 2 1.1.3.1 Build Rate ........................................................................................................................... 2 1.1.3.2 Azimuth ............................................................................................................................... 2 1.1.4 Advantages of Drilling Horizontal Wells ...................................................................... 3 1.1.5 Disadvantages of Drilling Horizontal Wells ................................................................ 4
CHAPTER II. THEORY ............................................................................................................... 5 2.1 Transient State Flow ........................................................................................................... 5 2.2 Pseudo Steady State Flow .................................................................................................. 6 2.3 Late Transient Flow ............................................................................................................. 7 2.4 Steady State Flow .................................................................................................................. 7 Literature Review ............................................................................................................................... 8
CHAPTER III. METHODOLOGY .............................................................................................. 9 3.1 Steady State Methods ....................................................................................................... 10 3.1.1 Vertical Well PI ................................................................................................................... 10 3.1.2 Borisov’s Method ............................................................................................................... 10 3.1.3 Giger-Reiss-Jourdan’s Method ...................................................................................... 10 3.1.4 Joshi’s Method..................................................................................................................... 11 3.1.5 Renard-Dupuy’s Method ................................................................................................. 13 3.2 Pseudo Steady State Methods ....................................................................................... 14 3.2.1 Vertical Well PI ................................................................................................................... 14 3.2.2 Babu-Odeh’s Method ........................................................................................................ 14 3.2.3 Kuchuk’s Method ............................................................................................................... 15 3.2.4 Economides’ Method ........................................................................................................ 16
CHAPTER IV. DISCUSSION AND RESULTS ...................................................................... 17 4.1 PI of Steady State Wells ................................................................................................... 17 4.1.1 Reservoir Radius = 2000 ft ............................................................................................ 17 4.1.2 Reservoir Radius = 5000 ft ............................................................................................ 30 4.2 PI of Pseudo Steady State Wells ................................................................................... 42 4.2.1 Reservoir Radius = 2000 ft ............................................................................................ 42 4.2.2 Reservoir Radius = 5000 ft ............................................................................................ 49
CHAPTER V. CONCLUSION AND FUTURE WORK .......................................................... 55
NOMENCLATURE .................................................................................................................... 57
REFERENCES ............................................................................................................................ 59
v
vi
List of Tables Table 1. Classification of Horizontal Wells 2 Table 2. PI of steady state vertical oil well 18 Table 3. PI of Borisov’s steady state horizontal oil well 19 Table 4. PI of Giger-Reiss-Jourdan’s isotropic steady state horizontal oil well 20 Table 5. PI of Giger-Reiss-Jourdan’s anisotropic steady state horizontal oil well 22 Table 6. PI of Jushi’s isotropic steady state horizontal oil well 23 Table 7. PI of Jushi’s anisotropic steady state horizontal oil well 25 Table 8. PI of Renard-Dupuy’s isotropic steady state horizontal oil well 26 Table 9. PI of Renard-Dupuy’s anisotropic steady state horizontal oil well 28 Table 10. PI of the vertical steady state oil well – Re=5000 ft 30 Table 11. PI of Borisov’s steady state horizontal oil well – Re = 5000 ft 31 Table 12. PI of Giger-Reiss-Jourdan’s isotropic steady state horizontal oil well – Re = 5000 ft 33 Table 13. PI of Giger-Reiss-Jourdan’s anisotropic steady state horizontal oil well 34 Table 14. PI of Jushi’s isotropic steady state horizontal oil well – Re= 5000 ft 36 Table 15. PI of Jushi’s anisotropic steady state horizontal oil well – Re=5000 ft 37 Table 16. PI of Renard-Dupuy’s isotropic steady state horizontal oil well – Re=5000 ft 39 Table 17. PI of Renard-Dupuy’s anisotropic steady state horizontal oil well – Re = 5000 ft 40 Table 18. PI of pseudo steady state vertical oil well 43 Table 19. PI of Babu-Odeh’s pseudo steady state horizontal oil well 44 Table 20. PI of Kuckuk’s pseudo steady state horizontal oil well 46 Table 21. PI of Economides pseudo steady state horizontal oil well 47 Table 22. PI of pseudo steady state vertical oil well 50 Table 23. PI of Babu-Odeh’s pseudo steady state horizontal oil well – Re = 5000 ft 51 Table 24. PI of Economides pseudo steady state horizontal oil well – Re = 5000 ft 52
vii
List of Figures Figure 1. Azimuth of Horizontal Wells. (Directional drilling. 2015. Web. 26 April. 2016) ................................. 3 Figure 2. Transient and pseudo steady state flow. (Reservoir flow. 2014. Web. April 26. 2016) ..................... 6 Figure 3. Impact of the crucial parameters on vertical oil well PI ........................................................................... 18 Figure 4. Impact of the crucial parameters on Borisov’s PI ........................................................................................ 20 Figure 5. Impact of the crucial parameters on Giger-Reiss-Jourdan’s isotropic PI ............................................ 21 Figure 6. Impact of the crucial parameters on Giger-Reiss-Jourdan’s anisotropic PI ....................................... 23 Figure 7. Impact of the crucial parameters on Jushi’s isotropic PI ........................................................................... 24 Figure 8. Impact of the crucial parameters on Jushi’s anisotropic PI ...................................................................... 26 Figure 9. Impact of the crucial parameters on Renard-Dupuy’s isotropic PI ....................................................... 27 Figure 10. Impact of the crucial parameters on Renard-Dupuy’s anisotropic PI ............................................... 28 Figure 11. PIs of Base Case Steady State Oil Wells - Re = 2000 ft .............................................................................. 29 Figure 12. PIs of Maximum Effect (1-D Case) Steady State Oil Wells - Re = 2000 ft .......................................... 29 Figure 13. Impact of the crucial parameters on vertical oil PI .................................................................................. 31 Figure 14. Impact of the crucial parameters on Borisov’s PI at Re = 5000 ft ....................................................... 32 Figure 15. Impact of the crucial parameters on Giger-Reiss-Jourdan’s isotropic PI.......................................... 33 Figure 16. Impact of the crucial parameters on Giger-Reiss-Jourdan’s anisotropic PI .................................... 35 Figure 17. Impact of the crucial parameters on Jushi’s isotropic PI ........................................................................ 36 Figure 18. Impact of the crucial parameters on Jushi’s anisotropic PI ................................................................... 38 Figure 19. Impact of the crucial parameters on Renard-Dupuy’s isotropic PI..................................................... 39 Figure 20. Impact of the crucial parameters on Renard-Dupuy’s anisotropic PI ............................................... 41 Figure 21. PI’s of Base Case Steady State Oil Wells - Re = 5000 ft ............................................................................. 41 Figure 22. PI’s of Maximum Effect (1-D Case) Steady State Oil Wells - Re = 5000 ft ......................................... 42 Figure 23. Impact of the crucial parameters on vertical oil well PI ......................................................................... 43 Figure 24. Impact of the crucial parameters on Babu-Odeh’s PI............................................................................... 45 Figure 25. Impact of the crucial parameters on Kuckuk’s PI ...................................................................................... 46 Figure 26. Impact of the crucial parameters on Economides PI ................................................................................ 48 Figure 27. PIs of Base Case Pseudo Steady State Oil Wells - Re = 2000 ft .............................................................. 48 Figure 28. PIs of Case 1-D Pseudo Steady State Oil Wells - Re = 2000 ft ................................................................. 49 Figure 29. Impact of the crucial parameters on vertical oil well PI ......................................................................... 50 Figure 30. Impact of the crucial parameters on Babu-Odeh’s PI at Re = 5000 ft ................................................ 52 Figure 31. Impact of the crucial parameters on Economides’s PI ............................................................................. 53 Figure 32. PIs of Base Case Pseudo Steady State Oil Wells - Re = 5000 ft .............................................................. 54 Figure 33. PIs of Case 1-D Pseudo Steady State Oil Wells - Re = 5000 ft ................................................................. 54
1
CHAPTER I. INTRODUCTION
1.1 Overview
1.1.1 Background Fluid deliverability is the main measure of the wells performance in the petroleum
industry. Initially, vertical wells were the only type of wells that produce oil
reservoirs. Since the 1920s, drilling technology has been improved to drill wells at
deviated angles. This improvement allowed drilling engineers to geo-steer their wells
horizontally, which will increase the production rates by as much as 20 times more
than drilling vertically. This is true since petroleum prospects are more extensive
aerially compared to their thickness (usually thickness is less than 150 ft). In addition,
directional drilling permits accessing reservoirs that cannot be accessed directly
using vertical wells.
The first oil well drilled in North America was in Oil Springs, Ontario in 1858.
Moreover, production in Santa Barbara County, CA began in the 1890s with the
development of the Summerland Oil Field, which included the world’s first offshore
oil well. Historical records suggest that horizontal drilling dates go back to as early as
1920s, and was first utilized in Pennsylvania in 1944. Nevertheless, in the 1980s,
horizontal drilling became a popular tradition when improved equipment, motor, and
other technologies were developed.
1.1.2 General Definition A horizontal well is a well which is drilled in such a way that the wellbore deviates
laterally to an approximate horizontal orientation within the target formation. The
horizontal components usually extend to at least 100 ft in the targeted reservoir,
measured from the initial point of penetration to the toe of the well. A deviated well
can be categorized as a horizontal well when its inclination exceeds 85°.
2
1.1.3 Geometry of Horizontal Wells
1.1.3.1 Build Rate The build rate is the increase in the inclination of a horizontal well. Generally, it is
expressed in °/100 ft. It is denoted as a decline rate if the inclination is decreasing
(negative). Based on the buildup/decline rate, horizontal wells can be categorized as
short, medium, and long radius (Table 1.1).
Table 1. Classification of Horizontal Wells
Category Build Rate
Long radius 2° to 6°/100 ft
Medium radius 6° to 35°/100 ft
Short radius 1.5° to 3° / 1 ft
1.1.3.2 Azimuth The azimuth of a borehole at a point is the direction of the borehole on the horizontal
plane, measures as a clockwise angle (0° - 360°) from the North reference. All
magnetic tools give readings referenced to magnetic north; however, the final
calculated coordinates are reference to either true north or grid north. Figure 1
shows the azimuth direction of a horizontal well.
3
Figure 1. Azimuth of Horizontal Wells. (Directional drilling. 2015. Web. 26 April. 2016)
1.1.4 Advantages of Drilling Horizontal Wells The advantages of drilling horizontal wells are:
1. In reservoirs with water and gas coning problems, horizontal wells have been
used to minimize coning problems and enhance oil production.
2. In naturally fractured reservoirs, horizontal wells have been used to intersect
fractures and produce from them, which will maximize the cumulative
production.
3. Enables drilling multiple wells with one surface wellbore (multi-lateral).
4. A long horizontal well provides a large reservoir contact area and therefore
enhances the productivity.
5. It provides solution to drilling under inaccessible locations, such as mountains,
riverbeds, and populated cities.
4
6. It can be utilized as a remedial operation to sidetrack around an obstruction
(fish).
7. This technique is applied to relief wells in case of blow-out.
1.1.5 Disadvantages of Drilling Horizontal Wells The disadvantages of drilling horizontal wells are:
1. Higher drilling and completion costs. Typically it costs about 1.4 to 3 times
more than drilling a vertical well.
2. Needs complex drilling and completion technologies.
3. Generally, it is difficult to produce from multiple zones using a single
horizontal well.
5
CHAPTER II. THEORY The productivity index is a measure of the ability of a well to produce. It is the ration
of the total oil flow rate to the pressure drawdown. In other words, it is the
hydrocarbon volume delivered per psi of drawdown at the sand-face (STB/psi/day).
It is mathematically expressed as:
𝐽𝐽 = 𝑞𝑞∆𝑃𝑃
= 𝑞𝑞
(𝑝𝑝𝑝𝑝 − 𝑝𝑝𝑝𝑝𝑝𝑝)
During the production cycle of a reservoir, the producing oil well goes through four
main stages based on the pressure drawdown and boundary conditions. These four
stages are:
• Transient state.
• Pseudo steady state.
• Steady state.
• Late transient.
2.1 Transient State Flow Transient state flow takes place when a well is first put into production. It is also
known as the infinite acting or unsteady state flow in which the pressure disturbance
caused by the production of a well has not reached any reservoir boundary. It is
described as the fluid flowing condition at which the rate of change of pressure with
respect to time at any position in the reservoir is not zero or constant. This is true
since the pressure migrates outward from the well without facing any boundaries.
Mathematically, transient flow is described as: 𝑑𝑑𝑝𝑝𝑑𝑑𝑑𝑑
= 𝑝𝑝(𝑝𝑝, 𝑑𝑑),𝑝𝑝ℎ𝑒𝑒𝑝𝑝𝑒𝑒 𝑝𝑝 𝑖𝑖𝑖𝑖 𝑑𝑑ℎ𝑒𝑒 𝑝𝑝𝑟𝑟𝑑𝑑𝑖𝑖𝑟𝑟𝑖𝑖 𝑟𝑟𝑎𝑎𝑑𝑑 𝑑𝑑 𝑖𝑖𝑖𝑖 𝑑𝑑ℎ𝑒𝑒 𝑑𝑑𝑖𝑖𝑡𝑡𝑒𝑒.
Figure 2 illustrations the progression of the transient flow pattern.
6
Figure 2. Transient and pseudo steady state flow. (Reservoir flow. 2014. Web. April 26. 2016)
2.2 Pseudo Steady State Flow Pseudo steady state flow begins when the pressure disturbance created by the
production well is felt at the boundary of the well’s drainage area. In other words,
when the fluid mass situated at the drainage boundary starts moving towards the
producing well, pseudo steady state begins. Mathematically, it is expressed as: 𝑑𝑑𝑝𝑝𝑑𝑑𝑑𝑑
= 𝑐𝑐𝑐𝑐𝑎𝑎𝑖𝑖𝑑𝑑𝑟𝑟𝑎𝑎𝑑𝑑 𝑝𝑝𝑟𝑟𝑑𝑑𝑒𝑒
Figure 2 shows the behavior of the pseudo steady state flow pattern.
7
2.3 Late Transient Flow This flow regime takes place between the unsteady state and the pseudo steady state
flow regimes. Moreover, it happens when the pressure disturbance caused by the
production of a well has reached some of the reservoir boundaries.
2.4 Steady State Flow Steady state flow occurs when the production of a well does not change the pressure
at any point in the reservoir over time. It is usually due to an aquifer support or gas
cap expansion. Mathematically, it is expressed as: 𝑑𝑑𝑝𝑝𝑑𝑑𝑑𝑑
= 0
8
Literature Review There are two well categories in which any well is classified: vertical or horizontal.
Generally, un-stimulated horizontal oil well produces two to five times to that of a
stimulated vertical well. On the other hand, horizontal wells might produce less in
thicker reservoirs (reservoir with thicknesses higher than 500 ft). In addition, they
are less efficient in low vertical permeability, and in stratified reservoirs. To
overcome these drawbacks, stimulation technology can be utilized.
Many calculations have been computed to evaluate the productivity index of a
horizontal well and many flow models have been employed for this purpose.
Parallelepiped models with no flow/constant pressure boundaries at the top or
bottom, and either no flow or infinite acting boundaries at the sides were extensively
used to approximate the well drainage area.
The first model was presented by Borisov, in which constant pressure drainage
ellipse was assumed. After that, Joshi introduced an equation that accounted for the
vertical to horizontal permeability anisotropy. Then, Economides developed it to be
used in the elliptical coordinates. However, this model did not account for the well
and reservoir configurations, as well as early time or late time phenomena.
Babu and Odeh presented equations that were complicated to calculate the pressure
drawdown at any point by integrating appropriate point source functions in space
and time. The assumption of their solutions is based on that the well is parallel to the
y-axis of the parallelepiped model (Economides, 1996). Additionally, using a
numerical inverter, Goode and Thambynayagam introduced a model for horizontal
well pressure transient response in Laplace space. After that, Kuchuk improved
Goode and Thambynayagam’s equations by including constant pressure at the
boundaries.
Normally, as the horizontal well length increases, the productivity index associated
increases. However, producing high volumes of fluids from long horizontal wells will
result in high-pressure losses along the wellbore. As a result, this will decrease the
productivity of the well.
9
CHAPTER III. METHODOLOGY Reservoir and fluid Properties (oil):
Formation thickness (h), ft. 10 – 50
Horizontal permeability (kh), md. 5 – 50
Viscosity (u), cp. 0.5
Formation volume factor (Bo), bbl/stb 1.5
Depth (d), ft. 6000 - 7000
Drainage radius (Reh), ft. 2000 - 5000
Length of horizontal well (L), ft 1000 – 3000
Wellbore radius (rw), ft 0.5
Vertical permeability (kv), md 0.5 – 5
Temperature (T), °F 120 – 130
Average reservoir radius (re), ft 2000 – 5000
Skin factor (S) 0
Average reservoir pressure (P), psia 3000
Flowing bottom hole pressure (Pwf), psia 500
Drainage area (a*b), ft2 2000 * 2000 – 5000 * 5000
Location of the center of well in the vertical plane (Zo), ft mid-point
Standoff (Zw) 0.25 – 0.75
Porosity, % 10%
To calculate the productivity index of oil wells, there are multiple approaches for each
flow regime.
10
3.1 Steady State Methods There are four major steady state equations to calculate the productivity index of oil
horizontal wells. The resulted PI from these equations is compared to the vertical
well’s PI. These methods are:
1. Vertical well PI.
2. Borisov’s method.
3. Giger-Reiss-Jourdan method.
4. Joshi’s method.
5. Rendard-Dupuy method.
3.1.1 Vertical Well PI The following equation is used to predict the PI of oil vertical well:
𝐽𝐽 = 0.00708 ∗ 𝑘𝑘 ∗ ℎ
𝐵𝐵𝑐𝑐 ∗ 𝜇𝜇 ∗ �ln �𝑝𝑝𝑒𝑒𝑝𝑝𝑝𝑝� + 𝑖𝑖�
3.1.2 Borisov’s Method Borisov proposed the following equation to predict the PI of oil horizontal well in an
isotropic reservoir:
𝐽𝐽 = 0.00708 ℎ 𝑘𝑘ℎ
𝜇𝜇𝑐𝑐 𝐵𝐵𝑐𝑐 �ln �4 𝑝𝑝𝑒𝑒ℎ𝐿𝐿 �+ �𝐿𝐿ℎ� ln � ℎ
2 𝜋𝜋 𝑝𝑝𝑝𝑝��
3.1.3 Giger-Reiss-Jourdan’s Method
11
Giger, Reiss, and Jourdan proposed the following equation to predict the PI of oil
horizontal well in an isotropic reservoir:
𝐽𝐽 = 0.00708 𝐿𝐿 𝐾𝐾ℎ
𝜇𝜇𝑐𝑐 𝐵𝐵𝑐𝑐 ��𝐿𝐿ℎ� ln(𝑋𝑋) + ln � ℎ2 𝑝𝑝𝑝𝑝��
For anisotropic reservoir:
𝐽𝐽 = 0.00708 𝐾𝐾ℎ
𝜇𝜇𝑐𝑐 𝐵𝐵𝑐𝑐 ��1ℎ� ln(𝑋𝑋) + �𝐵𝐵
2
𝐿𝐿 � ln � ℎ2 𝑝𝑝𝑝𝑝��
Where:
𝑋𝑋 = 1 + �1 + � 𝐿𝐿
2 𝑝𝑝𝑒𝑒ℎ�2
𝐿𝐿(2 𝑝𝑝𝑒𝑒ℎ)�
𝐵𝐵 = �𝐾𝐾ℎ𝐾𝐾𝐾𝐾
3.1.4 Joshi’s Method
12
Joshi proposed the following equation to predict the PI of oil horizontal well in an
isotropic reservoir:
𝐽𝐽 = 0.00708 ℎ 𝐾𝐾ℎ
𝜇𝜇𝑐𝑐 𝐵𝐵𝑐𝑐 �ln(𝑅𝑅) + �ℎ𝐿𝐿� ln � ℎ2 𝑝𝑝𝑝𝑝��
For anisotropic reservoir:
𝐽𝐽 = 0.00708 ℎ 𝐾𝐾ℎ
𝜇𝜇𝑐𝑐 𝐵𝐵𝑐𝑐 �ln(𝑅𝑅) + �𝐵𝐵2ℎ𝐿𝐿 � ln � ℎ
2 𝑝𝑝𝑝𝑝��
Where:
𝐵𝐵 = �𝐾𝐾ℎ𝐾𝐾𝐾𝐾
𝑟𝑟 = 𝐿𝐿2 ∗ �0.5 + �0.25 + �
2 𝑝𝑝𝑒𝑒ℎ𝐿𝐿
�4
�
0.5
𝑅𝑅 = 𝑟𝑟 + �𝑟𝑟2 − �𝐿𝐿 2� �
2
�𝐿𝐿 2� �
13
3.1.5 Renard-Dupuy’s Method Renard and Dupuy proposed the following equation to predict the PI of oil horizontal
well in an isotropic reservoir:
𝐽𝐽 = 0.00708 ℎ 𝐾𝐾ℎ
𝜇𝜇𝑐𝑐 𝐵𝐵𝑐𝑐 �cosh−1 �2𝑟𝑟𝐿𝐿 � + �ℎ𝐿𝐿� ln � ℎ
2 𝜋𝜋 𝑝𝑝𝑝𝑝��
For anisotropic reservoir:
𝐽𝐽 = 0.00708 ℎ 𝐾𝐾ℎ
𝜇𝜇𝑐𝑐 𝐵𝐵𝑐𝑐 �cosh−1 �2𝑟𝑟𝐿𝐿 � + �𝐵𝐵ℎ𝐿𝐿 � ln � ℎ
2 𝜋𝜋 𝑝𝑝′𝑝𝑝��
Where:
𝑟𝑟 = 𝐿𝐿2 ∗ �0.5 + �0.25 + �
2 𝑝𝑝𝑒𝑒ℎ𝐿𝐿
�4
�
0.5
𝐵𝐵 = �𝐾𝐾ℎ𝐾𝐾𝐾𝐾
𝑝𝑝′𝑝𝑝 = (1 + 𝐵𝐵)𝑝𝑝𝑝𝑝
2 𝐵𝐵
14
3.2 Pseudo Steady State Methods There are three major pseudo steady state equations to calculate the productivity
index of oil horizontal wells. The resulted PI from these equations is compared to the
vertical well’s PI. These methods are:
1. Vertical Well PI.
2. Babu-Odeh method.
3. Kuchuk method.
4. Economides method.
3.2.1 Vertical Well PI The following equation is used to predict the PI of oil vertical well:
𝐽𝐽 = 𝑘𝑘 ∗ ℎ
141.2 ∗ 𝐵𝐵𝑐𝑐 ∗ 𝜇𝜇 ∗ �ln �𝑝𝑝𝑒𝑒𝑝𝑝𝑝𝑝�+ 𝑖𝑖 − 0.75�
3.2.2 Babu-Odeh’s Method This method is meant to provide an easier model for calculating the PI of a horizontal
well. They presented the following equation:
𝐽𝐽 = 0.00708 𝑏𝑏 √𝑘𝑘𝑘𝑘 𝑘𝑘𝑘𝑘
𝜇𝜇 𝐵𝐵 �ln �𝐶𝐶𝐻𝐻 𝐴𝐴12 𝑝𝑝𝑝𝑝� − 0.75 + 𝑆𝑆𝑅𝑅�
Where:
SR is a function that depends strongly on the well length L. SR = 0 when L = b (the fully
penetrating case).
15
ln(𝐶𝐶𝐻𝐻) = 6.28 ∗ �𝑟𝑟ℎ���
𝑘𝑘𝑘𝑘𝑘𝑘𝑘𝑘� ∗ �
13− 𝑘𝑘𝑐𝑐𝑟𝑟
+ �𝑘𝑘𝑐𝑐𝑟𝑟�2� − ln (sin �
180 𝑘𝑘𝑐𝑐ℎ
�)
− 0.5 ln��𝑟𝑟ℎ��
𝑘𝑘𝑘𝑘𝑘𝑘𝑘𝑘� − 1.088
Here: xo and zo are the coordinates measuring the center of the well in the vertical
plane, (a) is the dimension of the drainage area.
3.2.3 Kuchuk’s Method Kuchuk suggested an approximate infinite conductivity solution to calculate the
productivity index as following:
𝐽𝐽 =7.08 ∗ 10−3 ∗ 𝐾𝐾𝐻𝐻 ∗ ℎ
𝜇𝜇 ∗ 𝐵𝐵𝑐𝑐 ∗ �𝑃𝑃𝑝𝑝𝑃𝑃 + 𝑆𝑆𝑆𝑆∗�
𝑆𝑆𝑆𝑆 ∗= ℎ
2𝐿𝐿 12
�𝐾𝐾𝑘𝑘𝐾𝐾𝑘𝑘 ∗ 𝑆𝑆𝑡𝑡
𝑆𝑆𝑡𝑡 = �2𝜋𝜋 ∗ 𝐿𝐿 1
2 �𝐾𝐾𝐾𝐾 ∗ 𝐾𝐾𝑘𝑘µ ∗ 𝑞𝑞 � ∗ 𝛥𝛥𝑃𝑃𝑖𝑖
Since the skin is zero, both Sm and Sm* should be negligible.
𝐾𝐾𝐻𝐻 = �𝐾𝐾𝑘𝑘 ∗ 𝐾𝐾𝐾𝐾
16
𝑃𝑃𝑝𝑝𝑃𝑃 = ℎ
2𝐿𝐿 12
�𝐾𝐾𝑘𝑘𝐾𝐾𝑘𝑘 �ln �
8ℎ𝜋𝜋 ∗ 𝑝𝑝′𝑝𝑝
𝑐𝑐𝑐𝑐𝑑𝑑 �𝜋𝜋 ∗ 𝑍𝑍𝑝𝑝2ℎ �� +
𝑍𝑍𝑝𝑝 − ℎ
𝐿𝐿 12
�𝐾𝐾𝑘𝑘𝐾𝐾𝑘𝑘 �
3.2.4 Economides’ Method This approach is general, readily reproduce well-known analytical solutions, and can
be used for transient, mixed, and no flow boundary conditions. The PI equation is
given by:
𝐽𝐽 = 𝐾𝐾𝑟𝑟𝐾𝐾𝑒𝑒 ∗ 𝑘𝑘𝑒𝑒
887.22 𝐵𝐵 𝜇𝜇 � 𝑃𝑃𝑃𝑃 + 𝑘𝑘𝑒𝑒2 𝜋𝜋 𝐿𝐿 Σ 𝑖𝑖 �
Where:
𝑃𝑃𝑃𝑃 = 𝑘𝑘𝑒𝑒 𝐶𝐶𝐻𝐻4 𝜋𝜋 ℎ
+ 𝑘𝑘𝑒𝑒
2 𝜋𝜋 𝐿𝐿 𝑆𝑆𝑘𝑘
𝑆𝑆𝑘𝑘 = ln �ℎ
2 𝜋𝜋 𝑝𝑝𝑝𝑝� −
ℎ 6 𝐿𝐿
+ 𝑆𝑆𝑒𝑒
𝑆𝑆𝑒𝑒 = ℎ𝐿𝐿
��2𝑍𝑍𝑝𝑝ℎ
� − 12
�2𝑍𝑍𝑝𝑝ℎ�2
−12
� − ln �𝑖𝑖𝑖𝑖𝑎𝑎 �𝜋𝜋 𝑍𝑍𝑝𝑝ℎ
��
17
CHAPTER IV. DISCUSSION AND RESULTS
Initially, the most influential reservoir properties that greatly impact the productivity
index value were distinguished. To achieve this goal, many sensitivity studies were
performed by testing the lower and upper limits of each reservoir property. As a
result, these crucial properties are reservoir thickness, reservoir permeability, and
reservoir radius (dimension). To capture the effect of these properties, two main
cases were generated.
In the first case, the impact of the reservoir permeability and thickness on the PI value
was examined at the lower limit of the reservoir radius (2000 ft for oil wells). That is,
the PI value was calculated at various limits of thickness and permeability,
independently and collectively. Furthermore, the PI value was computed at the
minimum reservoir values (case 1-base).
Next, the PI value was calculated at the maximum reservoir thickness without
changing the other reservoir parameters (case 1-A). Also, the effect of reservoir
permeability on the PI value was estimated by applying its maximum value (case 1-
B). In order to distinguish between the effect of permeability and thickness on the PI,
case 1-C was created by increasing both parameters at a similar magnitude. That is,
both thickness and permeability were increased by a factor of 5. Case 1-D represents
the maximum influential reservoir properties. In some models, where the length is a
crucial reservoir parameter, additional case (case 1-D-1) was generated to account
for length impact on the PI. Similarly, the impact of these properties on the PI was
tested at the upper limit of the reservoir radius (5000 ft for oil wells).
4.1 PI of Steady State Wells
4.1.1 Reservoir Radius = 2000 ft The PI values of the vertical well are illustrated in Table 2. Clearly, Case 1-A promotes
a PI value that is five times greater than the 1-base case. Similarly, increasing the
permeability up to 50 mD in Case 1-B will result in a PI value that is ten times higher
18
than the PI of Case 1-base. By increasing the reservoir thickness and permeability by
a factor of 5 (Case 1-C), the PI value will increase 25 times compared to the PI of the
1-base Case. In case 1-D, maximizing the influential parameters will promote the
highest PI value among all the cases (50 times greater than the PI of Case 1-base).
Figure 3 below indicates the percentage effect of the crucial parameters on the
productivity index value.
Table 2. PI of steady state vertical oil well
Properties Case 1-base Case 1-A Case 1-B Case 1-C Case 1-D Thickness, ft 10 50 10 50 50
Permeability, md 1.58 1.58 15.81 7.91 15.81
Viscosity, cp 0.5 0.5 0.5 0.5 0.5
Formation volume factor, bbl/stb 1.5 1.5 1.5 1.5 1.5
Reservoir radius, ft 2000 2000 2000 2000 2000
Wellbore radius, ft 0.5 0.5 0.5 0.5 0.5
PI, stb/psi/day 0.018 0.090 0.180 0.450 0.900
Figure 3. Impact of the crucial parameters on vertical oil well PI
19
The PI values of the horizontal oil well using Borisov’s model are illustrated in Table
3. Clearly, Case 1-A promotes a PI value that is 10.3 times greater than the 1-base
case. Similarly, increasing the permeability up to 50 mD in Case 1-B will result in a PI
value that is ten times higher than the PI of Case 1-base. By increasing the reservoir
thickness and permeability by a factor of 5 (Case 1-C), the PI value will increase 51.3
times compared to the PI of the 1-base Case. In case 1-D, maximizing the influential
parameters will promote the highest PI value among all the cases (102.7 times greater
than the PI of Case 1-base). In case 1-D, maximizing the influential parameters will
promote the highest PI value among all the cases. Figure 4 below indicates the
percentage effect of the crucial parameters on the productivity index value.
Table 3. PI of Borisov’s steady state horizontal oil well
Properties Case 1-base Case 1-A Case 1-B Case 1-C Case 1-D Thickness, ft 10 50 10 50 50
Permeability, md 1.58 1.58 15.81 7.91 15.81
Viscosity, cp 0.5 0.5 0.5 0.5 0.5
Formation volume factor, bbl/stb 1.5 1.5 1.5 1.5 1.5
Reservoir radius, ft 2000 2000 2000 2000 2000
Length, ft 1000 1000 1000 1000 1000
Wellbore radius, ft 0.5 0.5 0.5 0.5 0.5
PI, stb/psi/day 0.001 0.013 0.013 0.065 0.130
20
Figure 4. Impact of the crucial parameters on Borisov’s PI
The PI values of the horizontal oil well using isotropic Giger-Reiss-Jourdan’s model
are illustrated in Table 4. Clearly, Case 1-A promotes a PI value that is 4.6 times
greater than the 1-base case. Similarly, increasing the permeability up to 50 mD in
Case 1-B will result in a PI value that is ten times higher than the PI of Case 1-base. By
increasing the reservoir thickness and permeability by a factor of 5 (Case 1-C), the PI
value will increase 23.1 times compared to the PI of the 1-base Case. In case 1-D,
maximizing the influential parameters will promote the highest PI value among all
the cases (46.2 times greater than the PI of Case 1-base). Figure 5 below indicates
the percentage effect of the crucial parameters on the productivity index value.
Table 4. PI of Giger-Reiss-Jourdan’s isotropic steady state horizontal oil well
Properties Case 1-base Case 1-A Case 1-B Case 1-C Case 1-D Thickness, ft 10 50 10 50 50
Horizontal permeability, md 5.00 5.00 50.00 25.00 50.00
Vertical permeability, md 0.50 0.50 5.00 2.50 5.00
Viscosity, cp 0.5 0.5 0.5 0.5 0.5
21
Formation volume factor, bbl/stb 1.5 1.5 1.5 1.5 1.5
Reservoir radius, ft 2000 2000 2000 2000 2000
Length, ft 1000 1000 1000 1000 1000
Wellbore radius, ft 0.5 0.5 0.5 0.5 0.5
B, dimensionless factor 3.162 3.162 3.162 3.162 3.162
X, dimensionless factor 8.123 8.123 8.123 8.123 8.123
PI, stb/psi/day 0.223 1.030 2.229 5.152 10.304
Figure 5. Impact of the crucial parameters on Giger-Reiss-Jourdan’s isotropic PI
The PI values of the horizontal oil well using anisotropic Giger-Reiss-Jourdan’s model
are illustrated in Table 5. Clearly, Case 1-A promotes a PI value that is 2.9 times
greater than the 1-base case. Similarly, increasing the permeability up to 50 mD in
Case 1-B will result in a PI value that is ten times higher than the PI of Case 1-base. By
increasing the reservoir thickness and permeability by a factor of 5 (Case 1-C), the PI
value will increase 14.3 times compared to the PI of the 1-base Case. In case 1-D,
maximizing the influential parameters will promote the highest PI value among all
22
the cases (28.7 times greater than the PI of Case 1-base). Figure 6 below indicates
the percentage effect of the crucial parameters on the productivity index value.
Table 5. PI of Giger-Reiss-Jourdan’s anisotropic steady state horizontal oil well
Properties Case 1-base Case 1-A Case 1-B Case 1-C Case 1-D
Thickness, ft 10 50 10 50 50
Horizontal permeability, md 5.00 5.00 50.00 25.00 50.00
Vertical permeability, md 0.50 0.50 5.00 2.50 5.00
Viscosity, cp 0.5 0.5 0.5 0.5 0.5
Formation volume factor, bbl/stb 1.5 1.5 1.5 1.5 1.5
Reservoir radius, ft 2000 2000 2000 2000 2000
Length, ft 1000 1000 1000 1000 1000
Wellbore radius, ft 0.5 0.5 0.5 0.5 0.5
B, dimensionless factor 3.162 3.162 3.162 3.162 3.162
X, dimensionless factor 8.123 8.123 8.123 8.123 8.123
PI, stb/psi/day 0.203 0.583 2.030 2.913 5.826
23
Figure 6. Impact of the crucial parameters on Giger-Reiss-Jourdan’s anisotropic PI
The PI values of the horizontal oil well using isotropic Jushi’s model are illustrated in
Table 6. Clearly, Case 1-A promotes a PI value that is 4.6 times greater than the 1-
base case. On the other hand, increasing the permeability up to 50 mD in Case 1-B will
result in a PI value that is similar to the PI of Case 1-base. By increasing the reservoir
thickness and permeability by a factor of 5 (Case 1-C), the PI value will increase 23.1
times compared to the PI of the 1-base Case. In case 1-D, maximizing the influential
parameters will promote the highest PI value among all the cases (46.2 times greater
than the PI of Case 1-base). Figure 7 below indicates the percentage effect of the
crucial parameters on the productivity index value.
Table 6. PI of Jushi’s isotropic steady state horizontal oil well
Properties Case 1-base Case 1-A Case 1-B Case 1-C Case 1-D Thickness, ft 10 50 10 50 50
Horizontal permeability, md 5.00 5.00 5.00 25.00 50.00
Vertical permeability, md 0.50 0.50 0.50 2.50 5.00
Viscosity, cp 0.5 0.5 0.5 0.5 0.5
24
Formation volume factor, bbl/stb 1.5 1.5 1.5 1.5 1.5
Reservoir radius, ft 2000 2000 2000 2000 2000
Length, ft 1000 1000 1000 1000 1000
Wellbore radius, ft 0.5 0.5 0.5 0.5 0.5
a, dimensionless factor 2031.49 2031.49 2031.49 2031.49 2031.49
B, dimensionless factor 3.16 3.16 3.16 3.16 3.16
R, dimensionless factor 8.00 8.00 8.00 8.00 8.00
PI, stb/psi/day 0.224 1.037 0.224 5.186 10.373
Figure 7. Impact of the crucial parameters on Jushi’s isotropic PI
The PI values of the horizontal oil well using anisotropic Jushi’s model are illustrated
in Table 7. Clearly, Case 1-A promotes a PI value that is 2.9 times greater than the 1-
base case. On the other hand, increasing the permeability up to 50 mD in Case 1-B will
result in a PI value that is similar to the PI of Case 1-base. By increasing the reservoir
thickness and permeability by a factor of 5 (Case 1-C), the PI value will increase 14.3
times compared to the PI of the 1-base Case. In case 1-D, maximizing the influential
parameters will promote the highest PI value among all the cases (28.6 times greater
25
than the PI of Case 1-base). Figure 8 below indicates the percentage effect of the
crucial parameters on the productivity index value.
Table 7. PI of Jushi’s anisotropic steady state horizontal oil well
Properties Case 1-base Case 1-A Case 1-B Case 1-C Case 1-D
Thickness, ft 10 50 10 50 50
Horizontal permeability, md 5.00 5.00 5.00 25.00 50.00
Vertical permeability, md 0.50 0.50 0.50 2.50 5.00
Viscosity, cp 0.5 0.5 0.5 0.5 0.5
Formation volume factor, bbl/stb 1.5 1.5 1.5 1.5 1.5
Reservoir radius, ft 2000 2000 2000 2000 2000
Length, ft 1000 1000 1000 1000 1000
Wellbore radius, ft 0.5 0.5 0.5 0.5 0.5
a, dimensionless factor 2031.49 2031.49 2031.49 2031.49 2031.49
B, dimensionless factor 3.162 3.162 3.162 3.162 3.162
R, dimensionless factor 8.001 8.001 8.001 8.001 8.001
PI, stb/psi/day 0.204 0.585 0.204 2.924 5.848
26
Figure 8. Impact of the crucial parameters on Jushi’s anisotropic PI
The PI values of the horizontal oil well using isotropic Renard-Dupuy’s model are
illustrated in Table 8. . Clearly, Case 1-A promotes a PI value that is 4.7 times greater
than the 1-base case. On the other hand, increasing the permeability up to 50 mD in
Case 1-B will result in a PI value that is similar to the PI of Case 1-base. By increasing
the reservoir thickness and permeability by a factor of 5 (Case 1-C), the PI value will
increase 23.6 times compared to the PI of the 1-base Case. In case 1-D, maximizing
the influential parameters will promote the highest PI value among all the cases (47.1
times greater than the PI of Case 1-base). Figure 9 below indicates the percentage
effect of the crucial parameters on the productivity index value.
Table 8. PI of Renard-Dupuy’s isotropic steady state horizontal oil well
Properties Case 1-base Case 1-A Case 1-B Case 1-C Case 1-D Thickness, ft 10 50 10 50 50
Horizontal permeability, md 5.00 5.00 5.00 25.00 50.00
Vertical permeability, md 0.50 0.50 0.50 2.50 5.00
Viscosity, cp 0.5 0.5 0.5 0.5 0.5
27
Formation volume factor, bbl/stb 1.5 1.5 1.5 1.5 1.5
Reservoir radius, ft 2000 2000 2000 2000 2000
Length, ft 1000 1000 1000 1000 1000
Wellbore radius, ft 0.5 0.5 0.5 0.5 0.5
a, dimensionless factor 2031.490 2031.490 2031.490 2031.490 2031.490
B, dimensionless factor 3.162 3.162 3.162 3.162 3.162
Effective wellbore radius rw', ft 0.329 0.329 0.329 0.329 0.329
PI, stb/psi/day 0.226 1.064 0.226 5.320 10.640
Figure 9. Impact of the crucial parameters on Renard-Dupuy’s isotropic PI
The PI values of the horizontal oil well using anisotropic Renard-Dupuy’s model are
illustrated in Table 9. Clearly, Case 1-A promotes a PI value that is 4.1 times greater
than the 1-base case. On the other hand, increasing the permeability up to 50 mD in
Case 1-B will result in a PI value that is similar to the PI of Case 1-base. By increasing
the reservoir thickness and permeability by a factor of 5 (Case 1-C), the PI value will
increase 20.6 times compared to the PI of the 1-base Case. In case 1-D, maximizing
the influential parameters will promote the highest PI value among all the cases (41.2
times greater than the PI of Case 1-base). Figure 10 below indicates the percentage
effect of the crucial parameters on the productivity index value.
28
Table 9. PI of Renard-Dupuy’s anisotropic steady state horizontal oil well
Properties Case 1-base Case 1-A Case 1-B Case 1-C Case 1-D Thickness, ft 10 50 10 50 50
Horizontal permeability, md 5.00 5.00 5.00 25.00 50.00
Vertical permeability, md 0.50 0.50 0.50 2.50 5.00
Viscosity, cp 0.5 0.5 0.5 0.5 0.5
Formation volume factor, bbl/stb 1.5 1.5 1.5 1.5 1.5
Reservoir radius, ft 2000 2000 2000 2000 2000
Length, ft 1000 1000 1000 1000 1000
Wellbore radius, ft 0.5 0.5 0.5 0.5 0.5
a, dimensionless factor 2031 2031 2031 2031 2031
B, dimensionless factor 3.162 3.162 3.162 3.162 3.162
Effective wellbore radius rw', ft 0.329 0.329 0.329 0.329 0.329
PI, stb/psi/day 0.222 0.914 0.222 4.568 9.135
Figure 10. Impact of the crucial parameters on Renard-Dupuy’s anisotropic PI
29
Based on Figures 11 and 12 below, Renard-Dupuy model predicts the highest
productivity index (12.5 times higher PI than the PI of a vertical well in 1-base and
11.8 times in 1-D) among all the models. On the other hand, Borisov’s equation will
result in the lowest PI (0.056 of the vertical well’s PI value in 1-base and 0.144 of the
PI in 1-D).
Figure 11. PIs of Base Case Steady State Oil Wells - Re = 2000 ft
Figure 12. PIs of Maximum Effect (1-D Case) Steady State Oil Wells - Re = 2000 ft
30
4.1.2 Reservoir Radius = 5000 ft The PI values of the vertical oil well are illustrated in Table 10. Clearly, Case 1-A
promotes a PI value that is five times greater than the 1-base case. Similarly,
increasing the permeability up to 50 mD in Case 1-B will result in a PI value that is ten
times higher than the PI of Case 1-base. By increasing the reservoir thickness and
permeability by a factor of 5 (Case 1-C), the PI value will increase 25 times compared
to the PI of the 1-base Case. In case 1-D, maximizing the influential parameters will
promote the highest PI value among all the cases (50 times greater than the PI of Case
1-base). Figure 13 below indicates the percentage effect of the crucial parameters on
the productivity index value.
Table 10. PI of the vertical steady state oil well – Re=5000 ft
Properties Case 1-base Case 1-A Case 1-B Case 1-C Case 1-D Thickness, ft 10 50 10 50 50
Permeability, md 1.58 1.58 15.81 7.91 15.81 Viscosity, cp 0.5 0.5 0.5 0.5 0.5
Formation volume factor, bbl/stb 1.5 1.5 1.5 1.5 1.5
Reservoir radius, ft 5000 5000 5000 5000 5000 Wellbore radius, ft 0.5 0.5 0.5 0.5 0.5
PI, stb/psi/day 0.016 0.081 0.162 0.405 0.810
31
Figure 13. Impact of the crucial parameters on vertical oil PI
The PI values of the horizontal oil well using Borisov’s model are illustrated in Table
11. Clearly, Case 1-A promotes a PI value that is 10.2 times greater than the 1-base
case. Similarly, increasing the permeability up to 50 mD in Case 1-B will result in a PI
value that is ten times higher than the PI of Case 1-base. By increasing the reservoir
thickness and permeability by a factor of 5 (Case 1-C), the PI value will increase 50.9
times compared to the PI of the 1-base Case. In case 1-D, maximizing the influential
parameters will promote the highest PI value among all the cases (101.8 times greater
than the PI of Case 1-base). In case 1-D-1, maximizing the influential parameters as
well as the horizontal well length will only promote a PI value that is 35.4 greater than
the vertical well’s PI. Figure 14 below indicates the percentage effect of the crucial
parameters on the productivity index value.
Table 11. PI of Borisov’s steady state horizontal oil well – Re = 5000 ft
Properties Case 1-base Case 1-A Case 1-B Case 1-C Case 1-D Case 1-D-1
Thickness, ft 10 50 10 50 50 50
Permeability, md 1.581 1.581 15.811 7.906 15.811 15.811
32
Viscosity, cp 0.5 0.5 0.5 0.5 0.5 0.5 Formation volume factor,
bbl/stb 1.5 1.5 1.5 1.5 1.5 1.5
Reservoir radius, ft 5000 5000 5000 5000 5000 5000
Length, ft 1000 1000 1000 1000 1000 3000
Wellbore radius, ft 0.5 0.5 0.5 0.5 0.5 0.5
PI, stb/psi/day 0.0013 0.0128 0.0126 0.0639 0.1279 0.0444
Figure 14. Impact of the crucial parameters on Borisov’s PI at Re = 5000 ft
The PI values of the horizontal oil well using isotropic Giger-Reiss-Jourdan’s model
are illustrated in Table 12. Clearly, Case 1-A promotes a PI value that is 4.6 times
greater than the 1-base case. Similarly, increasing the permeability up to 50 mD in
Case 1-B will result in a PI value that is ten times higher than the PI of Case 1-base. By
increasing the reservoir thickness and permeability by a factor of 5 (Case 1-C), the PI
value will increase 23.6 times compared to the PI of the 1-base Case. In case 1-D,
maximizing the influential parameters will promote a PI value 47.3 times greater than
the PI of Case 1-base. In case 1-D-1, maximizing the influential parameters as well as
the horizontal well length will promote the highest PI value among all the cases (76.1
33
times greater than the PI of Case 1-base). Figure 15 below indicates the percentage
effect of the crucial parameters on the productivity index value. Table 12. PI of Giger-Reiss-Jourdan’s isotropic steady state horizontal oil well – Re = 5000 ft
Properties Case 1-base Case 1-A Case 1-B Case 1-C Case 1-D Case 1-D-1 Thickness, ft 10 50 10 50 50 50
Horizontal permeability, md 5.00 5.00 50.00 25.00 50.00 50 Vertical permeability, md 0.50 0.50 5.00 2.50 5.00 5
Viscosity, cp 0.5 0.5 0.5 0.5 0.5 0.5 Formation volume factor,
bbl/stb 1.5 1.5 1.5 1.5 1.5 1.5
Reservoir radius, ft 5000 5000 5000 5000 5000 5000 Length, ft 1000 1000 1000 1000 1000 3000
Wellbore radius, ft 0.5 0.5 0.5 0.5 0.5 0.5
B, dimensionless factor 3.162 3.162 3.162 3.162 3.162 3.162 X, dimensionless factor 20.050 20.050 20.050 20.050 20.050 6.813
PI, stb/psi/day 0.156 0.739 1.562 3.695 7.389 11.895
Figure 15. Impact of the crucial parameters on Giger-Reiss-Jourdan’s isotropic PI
The PI values of the horizontal oil well using anisotropic Giger-Reiss-Jourdan’s model
are illustrated in Table 13. Clearly, Case 1-A promotes a PI value that is 3.3 times
greater than the 1-base case. Similarly, increasing the permeability up to 50 mD in
Case 1-B will result in a PI value that is ten times higher than the PI of Case 1-base. By
34
increasing the reservoir thickness and permeability by a factor of 5 (Case 1-C), the PI
value will increase 16.3 times compared to the PI of the 1-base Case. In case 1-D,
maximizing the influential parameters will promote a PI value 32.6 times greater than
the PI of Case 1-base. In case 1-D-1, maximizing the influential parameters as well as
the horizontal well length will promote the highest PI value among all the cases (62.8
times greater than the PI of Case 1-base). Figure 16 below indicates the percentage
effect of the crucial parameters on the productivity index value.
Table 13. PI of Giger-Reiss-Jourdan’s anisotropic steady state horizontal oil well
Properties Case 1-base Case 1-A Case 1-B Case 1-C Case 1-D Case 1-D-1 Thickness, ft 10 50 10 50 50 50
Horizontal permeability, md 5.00 5.00 50.00 25.00 50.00 50 Vertical permeability, md 0.50 0.50 5.00 2.50 5.00 5
Viscosity, cp 0.5 0.5 0.5 0.5 0.5 0.5 Formation volume factor,
bbl/stb 1.5 1.5 1.5 1.5 1.5 1.5
Reservoir radius, ft 5000 5000 5000 5000 5000 5000 Length, ft 1000 1000 1000 1000 1000 3000
Wellbore radius, ft 0.5 0.5 0.5 0.5 0.5 0.5 B, dimensionless factor 3.162 3.162 3.162 3.162 3.162 3.162 X, dimensionless factor 20.050 20.050 20.050 20.050 20.050 6.813
PI, stb/psi/day 0.146 0.476 1.462 2.382 4.764 9.180
35
Figure 16. Impact of the crucial parameters on Giger-Reiss-Jourdan’s anisotropic PI
The PI values of the horizontal oil well using isotropic Jushi’s model are illustrated in
Table 14. Clearly, Case 1-A promotes a PI value that is 4.7 times greater than the 1-
base case. Similarly, increasing the permeability up to 50 mD in Case 1-B will result
in a PI value that is ten times higher than the PI of Case 1-base. By increasing the
reservoir thickness and permeability by a factor of 5 (Case 1-C), the PI value will
increase 23.6 times compared to the PI of the 1-base Case. In case 1-D, maximizing
the influential parameters will promote a PI value 47.3 times greater than the PI of
Case 1-base. In case 1-D-1, maximizing the influential parameters as well as the
horizontal well length will promote the highest PI value among all the cases (76.9
times greater than the PI of Case 1-base). Figure 17 below indicates the percentage
effect of the crucial parameters on the productivity index value.
36
Table 14. PI of Jushi’s isotropic steady state horizontal oil well – Re= 5000 ft
Properties Case 1-base Case 1-A Case 1-B Case 1-C Case 1-D Case 1-D-1 Thickness, ft 10 50 10 50 50 50
Horizontal permeability, md 5 5 50 25 50 50 Vertical permeability, md 0.5 0.5 5 2.5 5 5
Viscosity, cp 0.5 0.5 0.5 0.5 0.5 0.5 Formation volume factor,
bbl/stb 1.5 1.5 1.5 1.5 1.5 1.5
Reservoir radius, ft 5000 5000 5000 5000 5000 5000 Length, ft 1000 1000 1000 1000 1000 3000
Wellbore radius, ft 0.5 0.5 0.5 0.5 0.5 0.5 a, dimensionless factor 5,012.52 5,012.52 5,012.52 5,012.52 5,012.52 5,113.74 B, dimensionless factor 3.162 3.162 3.162 3.162 3.162 3.162 R, dimensionless factor 20.0 20.0 20.0 20.0 20.0 6.668
PI, stb/psi/day 0.156 0.740 1.564 3.698 7.395 12.025
Figure 17. Impact of the crucial parameters on Jushi’s isotropic PI
37
The PI values of the horizontal oil well using anisotropic Jushi’s model are illustrated
in Table 15. Clearly, Case 1-A promotes a PI value that is 3.3 times greater than the
1-base case. Similarly, increasing the permeability up to 50 mD in Case 1-B will result
in a PI value that is ten times higher than the PI of Case 1-base. By increasing the
reservoir thickness and permeability by a factor of 5 (Case 1-C), the PI value will
increase 16.3 times compared to the PI of the 1-base Case. In case 1-D, maximizing
the influential parameters will promote a PI value 32.6 times greater than the PI of
Case 1-base. In case 1-D-1, maximizing the influential parameters as well as the
horizontal well length will promote the highest PI value among all the cases (63.3
times greater than the PI of Case 1-base). Figure 18 below indicates the percentage
effect of the crucial parameters on the productivity index value.
Table 15. PI of Jushi’s anisotropic steady state horizontal oil well – Re=5000 ft
Properties Case 1-base Case 1-A Case 1-B Case 1-C Case 1-D Case 1-D-1 Thickness, ft 10 50 10 50 50 50
Horizontal permeability, md 5 5 50 25 50 50 Vertical permeability, md 0.5 0.5 5 2.5 5 5
Viscosity, cp 0.5 0.5 0.5 0.5 0.5 0.5 Formation volume factor,
bbl/stb 1.5 1.5 1.5 1.5 1.5 1.5
Reservoir radius, ft 5000 5000 5000 5000 5000 5000 Length, ft 1000 1000 1000 1000 1000 3000
Wellbore radius, ft 0.5 0.5 0.5 0.5 0.5 0.5 a, dimensionless factor 5,012.52 5,012.52 5,012.52 5,012.52 5,012.52 5,113.74 B, dimensionless factor 3.162 3.162 3.162 3.162 3.162 3.162 R, dimensionless factor 20.0 20.0 20.0 20.0 20.0 6.668
PI, stb/psi/day 0.146 0.477 1.463 2.383 4.766 9.257
38
Figure 18. Impact of the crucial parameters on Jushi’s anisotropic PI
The PI values of the horizontal oil well using isotropic Renard-Dupuy’s model are
illustrated in Table 16. Clearly, Case 1-A promotes a PI value that is 4.8 times greater
than the 1-base case. On the other hand, increasing the permeability up to 50 mD in
Case 1-B will not affect the PI (same PI as Case 1-base’s PI). By increasing the
reservoir thickness and permeability by a factor of 5 (Case 1-C), the PI value will
increase 24 times compared to the PI of the 1-base Case. In case 1-D, maximizing the
influential parameters will promote a PI value 48 times greater than the PI of Case 1-
base. In case 1-D-1, maximizing the influential parameters as well as the horizontal
well length will promote the highest PI value among all the cases (77.4 times greater
than the PI of Case 1-base). Figure 19 below indicates the percentage effect of the
crucial parameters on the productivity index value.
39
Table 16. PI of Renard-Dupuy’s isotropic steady state horizontal oil well – Re=5000 ft
Properties Case 1-base Case 1-A Case 1-B Case 1-C Case 1-D Case 1-D-1
Thickness, ft 10 50 10 50 50 50
Horizontal permeability, md 5.00 5.00 5.00 25.00 50.00 50.00
Vertical permeability, md 0.50 0.50 0.50 2.50 5.00 5.00
Viscosity, cp 0.5 0.5 0.5 0.5 0.5 0.5
Formation volume factor, bbl/stb 1.5 1.5 1.5 1.5 1.5 1.5
Reservoir radius, ft 5000 5000 5000 5000 5000 5000
Length, ft 1000 1000 1000 1000 1000 3000
Wellbore radius, ft 0.5 0.5 0.5 0.5 0.5 0.5 a, dimensionless factor 5,012 5,012 5,012 5,012 5,012 5,113 B, dimensionless factor 3.162 3.162 3.162 3.162 3.162 3.162
Effective wellbore radius rw', ft 0.329 0.329 0.329 0.329 0.329 0.329 PI, stb/psi/day 0.157 0.753 0.157 3.765 7.530 12.143
Figure 19. Impact of the crucial parameters on Renard-Dupuy’s isotropic PI
The PI values of the horizontal oil well using anisotropic Renard-Dupuy’s model are
illustrated in Table 17. Clearly, Case 1-A promotes a PI value that is 4.4 times greater
than the 1-base case. On the other hand, increasing the permeability up to 50 mD in
40
Case 1-B will not affect the PI (same PI as Case 1-base’s PI). By increasing the
reservoir thickness and permeability by a factor of 5 (Case 1-C), the PI value will
increase 21.8 times compared to the PI of the 1-base Case. In case 1-D, maximizing
the influential parameters will promote a PI value 43.5 times greater than the PI of
Case 1-base. In case 1-D-1, maximizing the influential parameters as well as the
horizontal well length will promote the highest PI value among all the cases (73.7
times greater than the PI of Case 1-base). Figure 20 below indicates the percentage
effect of the crucial parameters on the productivity index value.
Table 17. PI of Renard-Dupuy’s anisotropic steady state horizontal oil well – Re = 5000 ft
Properties Case 1-base Case 1-A Case 1-B Case 1-C Case 1-D Case 1-D-1 Thickness, ft 10 50 10 50 50 50
Horizontal permeability, md 5.00 5.00 5.00 25.00 50.00 50.00 Vertical permeability, md 0.50 0.50 0.50 2.50 5.00 5.00
Viscosity, cp 0.5 0.5 0.5 0.5 0.5 0.5 Formation volume factor,
bbl/stb 1.5 1.5 1.5 1.5 1.5 1.5
Reservoir radius, ft 5000 5000 5000 5000 5000 5000 Length, ft 1000 1000 1000 1000 1000 3000
Wellbore radius, ft 0.5 0.5 0.5 0.5 0.5 0.5 a, dimensionless factor 5,012.52 5,012.52 5,012.52 5,012.52 5,012.52 5,113.74 B, dimensionless factor 3.162 3.162 3.162 3.162 3.162 3.162
Effective wellbore radius rw', ft 0.329 0.329 0.329 0.329 0.329 0.329
PI, stb/psi/day 0.155 0.674 0.155 3.372 6.744 11.427
41
Figure 20. Impact of the crucial parameters on Renard-Dupuy’s anisotropic PI
Based on Figures 21 and 22 below, Renard-Dupuy model predicts the highest
productivity index (9.8 times higher PI than the PI of a vertical well in 1-base and 15
times in 1-D) among all the models. On the other hand, Borisov’s equation will result
in the lowest PI (0.082 of the vertical well’s PI value in 1-base and 0.055 of the PI in
1-D).
Figure 21. PI’s of Base Case Steady State Oil Wells - Re = 5000 ft
42
Figure 22. PI’s of Maximum Effect (1-D Case) Steady State Oil Wells - Re = 5000 ft
4.2 PI of Pseudo Steady State Wells
4.2.1 Reservoir Radius = 2000 ft
The PI values of the vertical well are illustrated in Table 18. Clearly, Case 1-A
promotes a PI value that is five times greater than the 1-base case. Similarly,
increasing the permeability up to 50 mD in Case 1-B will result in a PI value that is ten
times higher than the PI of Case 1-base. By increasing the reservoir thickness and
permeability by a factor of 5 (Case 1-C), the PI value will increase 25 times compared
to the PI of the 1-base Case. In case 1-D, maximizing the influential parameters will
promote the highest PI value among all the cases (50 times greater than the PI of Case
1-base). Figure 23 below indicates the percentage effect of the crucial parameters
on the productivity index value.
43
Table 18. PI of pseudo steady state vertical oil well
Properties Case 1-base Case 1-A Case 1-B Case 1-C Case 1-D Thickness, ft 10 50 10 50 50
Permeability, md 1.58 1.58 15.81 7.91 15.81 Viscosity, cp 0.5 0.5 0.5 0.5 0.5
Formation volume factor, bbl/stb 1.5 1.5 1.5 1.5 1.5
Reservoir radius, ft 2000 2000 2000 2000 2000 Wellbore radius, ft 0.5 0.5 0.5 0.5 0.5
PI, stb/psi/day 0.020 0.099 0.198 0.495 0.989
Figure 23. Impact of the crucial parameters on vertical oil well PI
The PI values of the horizontal oil well using Babu-Odeh’s model are illustrated in
Table 19. Clearly, Case 1-A promotes a PI value that is 3.1 times greater than the 1-
base case. On the other hand, increasing the permeability up to 50 mD in Case 1-B will
not affect the PI value (same PI as 1-basse Case). By increasing the reservoir thickness
and permeability by a factor of 5 (Case 1-C), the PI value will increase 15.5 times
compared to the PI of the 1-base Case. In case 1-D, maximizing the influential
44
parameters will promote the highest PI value among all the cases (30.9 times greater
than the PI of Case 1-base). Figure 24 below indicates the percentage effect of the
crucial parameters on the productivity index value.
Table 19. PI of Babu-Odeh’s pseudo steady state horizontal oil well
Properties Case 1-base Case 1-A Case 1-B Case 1-C Case 1-D Thickness, ft 10 50 10 50 50
Horizontal permeability, md 5 5 5 25 50 Vertical permeability, md 0.5 0.5 0.5 2.5 5
Viscosity, cp 0.5 0.5 0.5 0.5 0.5 Formation volume factor,
bbl/stb 1.5 1.5 1.5 1.5 1.5
Reservoir radius, ft 2000 2000 2000 2000 2000 Wellbore radius, ft 0.5 0.5 0.5 0.5 0.5
a, dimensionless factor 2000 2000 2000 2000 2000 B, dimensionless factor 2000 2000 2000 2000 2000
Vertical location, ft 5 25 5 25 25 Well location in x-direction, ft 1000 1000 1000 1000 1000
Shape factor CH 6.33E+13 4.48E+2 6.33E+13 4.48E+2 4.48E+2 Skin effect, SR 0 0 0 0 0
Area, sq ft 4,000,000 4,000,000 4,000,000 4,000,000 4,000,000 PI, stb/psi/day 0.787 2.434 0.787 12.172 24.344
45
Figure 24. Impact of the crucial parameters on Babu-Odeh’s PI
The PI values of the horizontal oil well using Kuckuk’s model are illustrated in Table
20. This approach does not depend on the reservoir radius. However, modifying the
well length will greatly impact the reservoir productivity index. Clearly, Case 1-A
promotes a PI value that is lower than 1-base case (PI of Case 1-A is 0.7 of that in base-
case). On the other hand, increasing the permeability up to 50 mD in Case 1-B will
increase the PI ten times than the base-Case’s PI. By increasing the reservoir thickness
and permeability by a factor of 5 (Case 1-C), the PI value will only increase to around
3.5 times higher compared to the PI of the 1-base Case. In case 1-D, maximizing the
influential parameters will promote a PI value 7.1 times greater than the PI of Case 1-
base. In case 1-D-1, maximizing the influential parameters as well as the horizontal
well length will promote the highest PI value among all the cases (21.2 times greater
than the PI of Case 1-base). Figure 25 below indicates the percentage effect of the
crucial parameters on the productivity index value.
46
Table 20. PI of Kuckuk’s pseudo steady state horizontal oil well
Figure 25. Impact of the crucial parameters on Kuckuk’s PI
Properties Case 1-base Case 1-A Case 1-B Case 1-C Case 1-D Case 1-D-1 Thickness, ft 10 50 10 50 50 50
X-direction permeability, md 5 5 50 25 50 50 Y-direction permeability, md 5 5 50 25 50 50
Vertical permeability, md 0.5 0.5 5 2.5 5 5 Average horizontal permeability, md 5 5 50 25 50 50
Formation volume factor, bbl/stb 1.5 1.5 1.5 1.5 1.5 1.5 Viscosity, cp 0.5 0.5 0.5 0.5 0.5 0.5
Length, ft 1000 1000 1000 1000 1000 3000 Wellbore radius, ft 0.5 0.5 0.5 0.5 0.5 0.5
Effective wellbore radius, ft 0.329 0.329 0.329 0.329 0.329 0.329 Vertical well location, ft 5 25 5 25 25 25
Dimensionless pressure PD 0.12 0.86 0.12 0.86 0.86 0.29 PI, stb/psi/day 3.88 2.73 38.78 13.67 27.35 82.04
47
The PI values of the horizontal oil well using Economides’s model are illustrated in
Table 21. Clearly, Case 1-A promotes a PI value that is 3.7 times greater than the 1-
base case. Similarly, increasing the permeability up to 50 mD in Case 1-B will result
in a PI value that is ten times higher than the PI of Case 1-base. By increasing the
reservoir thickness and permeability by a factor of 5 (Case 1-C), the PI value will
increase 18.7 times compared to the PI of the 1-base Case. In case 1-D, maximizing
the influential parameters will promote the highest PI value among all the cases (37.5
times greater than the PI of Case 1-base). Figure 26 below indicates the percentage
effect of the crucial parameters on the productivity index value.
Table 21. PI of Economides pseudo steady state horizontal oil well
Properties Case 1-base Case 1-A Case 1-B Case 1-C Case 1-D
Thickness, ft 10 50 10 50 50
X-direction permeability, md 5 5 50 25 50
Vertical permeability, md 0.5 0.5 5 2.5 5
y-direction permeability, md 5 5 50 25 50
Average permeability, md 2.321 2.321 23.208 11.604 23.208 Viscosity, cp 0.5 0.5 0.5 0.5 0.5
Formation volume factor, bbl/stb 1.5 1.5 1.5 1.5 1.5 Wellbore radius, ft 0.5 0.5 0.5 0.5 0.5
Skin factor 0 0 0 0 0 Reservoir radius, ft 2000 2000 2000 2000 2000
L/Xe 0.5 0.5 0.5 0.5 0.5 Shape factor, CH 1.47 1.47 1.47 1.47 1.47
Stand off, Zw 0 0 0 0 0 Eccentricity effect, Se 0 0 0 0 0
Skin effect, Sx 1.16 2.76 1.16 2.76 2.76 Dimensionless pressure PD 23.78 5.56 23.78 5.56 5.56
PI, stb/psi/day 0.29 1.08 2.89 5.42 10.83
48
Figure 26. Impact of the crucial parameters on Economides PI
Based on Figures 27 and 28 below, Kuckuk model predicts the highest productivity
index (194 times higher PI than the PI of a vertical well in 1-base and 28 times in 1-
D) among all the models. On the other hand, Vertical well equation will result in the
lowest PI.
Figure 27. PIs of Base Case Pseudo Steady State Oil Wells - Re = 2000 ft
49
Figure 28. PIs of Case 1-D Pseudo Steady State Oil Wells - Re = 2000 ft
4.2.2 Reservoir Radius = 5000 ft The PI values of the vertical oil well are illustrated in Table 22. Clearly, Case 1-A
promotes a PI value that is five times greater than the 1-base case. Similarly,
increasing the permeability up to 50 mD in Case 1-B will result in a PI value that is ten
times higher than the PI of Case 1-base. By increasing the reservoir thickness and
permeability by a factor of 5 (Case 1-C), the PI value will increase 25 times compared
to the PI of the 1-base Case. In case 1-D, maximizing the influential parameters will
promote the highest PI value among all the cases (50 times greater than the PI of Case
1-base). Figure 29 below indicates the percentage effect of the crucial parameters on
the productivity index value.
50
Table 22. PI of pseudo steady state vertical oil well
Properties Case 1-base Case 1-A Case 1-B Case 1-C Case 1-D Thickness, ft 10 50 10 50 50
Permeability, md 1.58 1.58 15.81 7.91 15.81 Viscosity, cp 0.5 0.5 0.5 0.5 0.5
Formation volume factor, bbl/stb 1.5 1.5 1.5 1.5 1.5
Reservoir radius, ft 5000 5000 5000 5000 5000 Wellbore radius, ft 0.5 0.5 0.5 0.5 0.5
PI, stb/psi/day 0.018 0.088 0.176 0.441 0.882
Figure 29. Impact of the crucial parameters on vertical oil well PI
The PI values of the horizontal oil well using Babu-Odeh’s model are illustrated in
Table 23. Clearly, Case 1-A promotes a PI value that is 3.1 times greater than the 1-
base case. On the other hand, increasing the permeability up to 50 mD in Case 1-B will
not affect the PI value (same PI as 1-basse Case). By increasing the reservoir thickness
and permeability by a factor of 5 (Case 1-C), the PI value will increase 15.5 times
compared to the PI of the 1-base Case. In case 1-D, maximizing the influential
parameters will promote the highest PI value among all the cases (30.9 times greater
51
than the PI of Case 1-base). Figure 30 below indicates the percentage effect of the
crucial parameters on the productivity index value.
Table 23. PI of Babu-Odeh’s pseudo steady state horizontal oil well – Re = 5000 ft
Properties Case 1-base Case 1-A Case 1-B Case 1-C Case 1-D Thickness, ft 10 50 10 50 50
Horizontal permeability, md 5 5 5 25 50 Vertical permeability, md 0.5 0.5 0.5 2.5 5
Viscosity, cp 0.5 0.5 0.5 0.5 0.5 Formation volume factor,
bbl/stb 1.5 1.5 1.5 1.5 1.5
Reservoir radius, ft 5000 5000 5000 5000 5000 Wellbore radius, ft 0.5 0.5 0.5 0.5 0.5
a, dimensionless factor 2000 2000 2000 2000 2000 B, dimensionless factor 2000 2000 2000 2000 2000 Vertical well location, ft 5 25 5 25 25
Well location in x-direction, ft 1000 1000 1000 1000 1000 ln(CH) 31.77 6.11 31.77 6.11 6.11
Shape factor CH 6.33E+13 4.48E+2 6.33E+13 4.48E+2 4.48E+2 Skin effect, SR 0 0 0 0 0
Area, sq ft 4,000,000 4,000,000 4,000,000 4,000,000 4,000,000 PI, stb/psi/day 0.787 2.434 0.787 12.172 24.344
52
Figure 30. Impact of the crucial parameters on Babu-Odeh’s PI at Re = 5000 ft
The PI values of the horizontal oil well using Economides’s model are illustrated in
Table 24. Clearly, Case 1-A promotes a PI value that is 4.1 times greater than the 1-
base case. Similarly, increasing the permeability up to 50 mD in Case 1-B will result
in a PI value that is ten times higher than the PI of Case 1-base. By increasing the
reservoir thickness and permeability by a factor of 5 (Case 1-C), the PI value will
increase 20.5 times compared to the PI of the 1-base Case. In case 1-D, maximizing
the influential parameters will promote a PI value 40.9 times greater than the PI of
Case 1-base. In case 1-D-1, maximizing the influential parameters as well as the
horizontal well length will promote the highest PI value among all the cases (82.9
times greater than the PI of Case 1-base). Figure 31 below indicates the percentage
effect of the crucial parameters on the productivity index value.
Table 24. PI of Economides pseudo steady state horizontal oil well – Re = 5000 ft
Properties Case 1-base Case 1-A Case 1-B Case 1-C Case 1-D Case 1-D-1 Thickness, ft 10 50 10 50 50 50
X-direction permeability, md 5 5 50 25 50 50 Vertical permeability, md 0.5 0.5 5 2.5 5 5
y-direction permeability, md 5 5 50 25 50 50
53
Average permeability, md 2.321 2.321 23.208 11.604 23.208 23.208 Viscosity, cp 0.5 0.5 0.5 0.5 0.5 0.5
Formation volume factor, bbl/stb 1.5 1.5 1.5 1.5 1.5 1.5
Well Length, ft 1000 1000 1000 1000 1000 3000 Wellbore radius, ft 0.5 0.5 0.5 0.5 0.5 0.5
Skin factor 0 0 0 0 0 0 Reservoir radius, ft 5000 5000 5000 5000 5000 5000
L/Xe 0.2 0.2 0.2 0.2 0.2 0.6 Shape factor CH 2.262 2.262 2.262 2.262 2.262 1.206
Stand off, Zw 0 0 0 0 0 0 Eccentricity effect, Se 0 0 0 0 0 0
Skin effect, Sx 1.16 2.76 1.16 2.76 2.76 2.76 Dimensionless pressure PD 90.97 20.21 90.97 20.21 20.21 10.37
PI, stb/psi/day 0.19 0.78 1.90 3.89 7.78 15.75
Figure 31. Impact of the crucial parameters on Economides’s PI
Based on Figures 32 and 33 below, Babu-Odeh’s model predicts the highest
productivity index among all the models. On the other hand, Vertical well’s equation
54
will result in the lowest PI. This is applicable for both the base case and case 1-D. In
Economides’s model, case 1-D-1 will promote even higher productivity index by
increasing the well length to 3000 ft (not shown since it’s the only model where it’s
possible to increase the well length in Re of 5000 ft).
Figure 32. PIs of Base Case Pseudo Steady State Oil Wells - Re = 5000 ft
Figure 33. PIs of Case 1-D Pseudo Steady State Oil Wells - Re = 5000 ft
55
CHAPTER V. CONCLUSION AND FUTURE WORK The main objective of this research is to compute and compare the productivity index
of vertical and horizontal oil wells using different pseudo-steady and steady state
approaches. Generally, Renard-Dupuy’s model will generate the highest PI value in
steady state flow regime. On the other hand, Borisov’s model results in the lower PI
among all the models.
In pseudo steady state flow, Kuchuk’s model will generate the highest PI value for oil
wells in smaller reservoirs (e.g. re = 2000 ft). The PI in pseudo steady state flow is
independent of reservoir radius except when using Economides and the vertical well
equation. Economides’s PI is proportional to the reservoir radius; whereas the
vertical well’s PI is inversely proportional.
In addition, the most influential reservoir parameters were determined by creating
different case studies in which the upper and lower limits of those parameters were
used as inputs in the PI models. Based on these case studies, the most influential
parameters were the reservoir radius, permeability, thickness, and in some cases,
horizontal well length. That is, these inputs/parameters greatly increase/decrease
the calculated PI in all the steady and pseudo steady state models.
In steady state models, the PI is directly proportional to the reservoir thickness,
permeability, and radius. Except in Borisov’s model, drilling a longer horizontal well
will result in higher PI. Similarly, the productivity index in a pseudo steady state will
increase as the reservoir thickness and/or permeability increases. Reservoir radius
(area) will positively influence the PI in Economides’s model, but not in Kuckuk’s or
Babu-Odeh’s models.
Future work might include using a three-phase/three-dimensional simulator to
compute the PI for the different well configurations. This also might include other
well configurations such as slanted and multilateral wells. Using a sensitivity analysis
in a 3-D simulator will enable creating hundreds of case studies in which we can
56
determine the optimum parameters value to generate the highest PI and capture how
these influential parameters affect the PI value over time. Finally, simulation runs can
test how well completion (hole size, open/cased hole, completion intervals etc.) can
impact the value of PI.
57
NOMENCLATURE
J Productivity index, bbl/psi/day
K Permeability, md
Kh Horizontal permeability, md
Kv Vertical permeability, md
Kx Permeability in x-direction, md
Ky Permeability in y-direction, md
Kz Permeability in z-direction, md
Kave Average permeability, md
Pi Initial reservoir pressure, psi
Pwf Flowing bottom-hole pressure, psi
q Production rate, stb/day
reh Drainage radius, ft
rw Wellbore radius, ft
rw’ Effective wellbore radius, ft
S Skin factor
μo Oil viscosity, cp
Bo Oil formation volume factor, bbl/stb
T Temperature, °F
h Formation thickness, ft
CH Shape factor
L Horizontal well length, ft
Zw Stand off, or distance of well from middle of reservoir, ft
ϕ Porosity, %
A Area, ft2
Xo, Zo Coordinates measuring the center of well in vertical plane
D Depth, ft
a Half major axis of drainage ellipse, ft
X Dimensionless drainage configuration parameter
SR Skin effect
58
PWD Dimensionless pseudo steady state pressure
Sm Van Everdingen mechanical skin
Xe Extent of drainage area in x-direction, ft
PD Dimensionless pressure
Sx Skin effect
Se Eccentricity effect in vertical direction
59
REFERENCES
1. Borisov, J.P.: “Oil Production using Horizontal and Multiple Deviation Wells,” Nedra, Moscow (1964). Translated by J. Strauss, S.D. Josh (ed.), Phillips Petroleum Co., the R&D library translation, Bartlesville, Oklahoma (1984).
2. Joshi, S.D.: “Augmentation of Well Productivity with Slant and Horizontal Wells,” JPT 729-739, (June 1988).
3. Economides, M.J., and Brand, C.W: “Well Configuration in Anisotropic Reservoirs” SPE Formation Evaluation, December 1996.
4. Babu, D.K. and Odeh, A.S.,: “Productivity of a Horizontal Well,” SPERE, 417-421, (November 1989).
5. Goode, P.A. and Thambynayagam, R.K.M.: “Pressure drawdown and Build-up analysis of Horizontal wells in Anisotropic Media,” SPEFE, 683-697 (December 1987).
6. Kuchuk, F.J and Goode, P.A., Brice , B.W., Sherrard, D.W., and Thambynayagam, R.K.M.: “Pressure Transient Analysis and Inflow Performance of Horizontal wells,” paper SPE 18300, (1988).
7. Goode, P. A., and F. J. Kuchuk. "Inflow performance of horizontal wells." SPE Reservoir Engineering 6.03 (1991): 319-323.
8. Escobar, Freddy H., et al. "An Improved Correlation to Estimate Productivity Index in Horizontal Wells." SPE Asia Pacific Oil and Gas Conference and Exhibition. Society of Petroleum Engineers, 2004.
9. Odeh, A.S. and Babu, D.K.: “Transient Flow Behavior of Horizontal Wells: Pressure Drawdown and Buildup Analysis,” SPEFE (March 1990) 7-15.
10. Besson, J.: “Performance of Slanted and Horizontal Wells on an Anisotropic Medium,” paper SPE 20965, 1990.
11. Jushi, S. D. “Horizontal Well Technology”. Penn Well Publishing Co., Tulsa, OK, 1991.