Productivity Index of Horizontal Oil Wells Nasser AlMolhem ...wvuscholar.wvu.edu/reports/AlMolhem_Nasser.pdf · As the drilling technology progressed, PI solutions were considered

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��


𝐽𝐽 = 0.00708 ℎ 𝐾𝐾ℎ

𝜇𝜇𝑐𝑐 𝐵𝐵𝑐𝑐 �ln(𝑅𝑅) + �𝐵𝐵2ℎ𝐿𝐿 � ln � ℎ

2 𝑝𝑝𝑝𝑝��

Where:


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��


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


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


Permeability, md 1.58 1.58 15.81 7.91 15.81

Viscosity, cp 0.5 0.5 0.5 0.5 0.5



Length, ft 1000 1000 1000 1000 1000


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


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



Length, ft 1000 1000 1000 1000 1000


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






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



Viscosity, cp 0.5 0.5 0.5 0.5 0.5



Length, ft 1000 1000 1000 1000 1000



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




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




Viscosity, cp 0.5 0.5 0.5 0.5 0.5

24



Length, ft 1000 1000 1000 1000 1000


a, dimensionless factor 2031.49 2031.49 2031.49 2031.49 2031.49


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-


result in a PI value that is similar to the PI of Case 1-base. By increasing the reservoir




25



Table 7. PI of Jushi’s anisotropic steady state horizontal oil well


Thickness, ft 10 50 10 50 50



Viscosity, cp 0.5 0.5 0.5 0.5 0.5



Length, ft 1000 1000 1000 1000 1000




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




Viscosity, cp 0.5 0.5 0.5 0.5 0.5

27



Length, ft 1000 1000 1000 1000 1000




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


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





28

Table 9. PI of Renard-Dupuy’s anisotropic steady state horizontal oil well




Viscosity, cp 0.5 0.5 0.5 0.5 0.5



Length, ft 1000 1000 1000 1000 1000


a, dimensionless factor 2031 2031 2031 2031 2031


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


Permeability, md 1.58 1.58 15.81 7.91 15.81 Viscosity, cp 0.5 0.5 0.5 0.5 0.5


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




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






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


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


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




34



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



Table 13. PI of Giger-Reiss-Jourdan’s anisotropic steady state horizontal oil well


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


bbl/stb 1.5 1.5 1.5 1.5 1.5 1.5


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


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


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



36

Table 14. PI of Jushi’s isotropic steady state horizontal oil well – Re= 5000 ft


Horizontal permeability, md 5 5 50 25 50 50 Vertical permeability, md 0.5 0.5 5 2.5 5 5


bbl/stb 1.5 1.5 1.5 1.5 1.5 1.5


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









Table 15. PI of Jushi’s anisotropic steady state horizontal oil well – Re=5000 ft


Horizontal permeability, md 5 5 50 25 50 50 Vertical permeability, md 0.5 0.5 5 2.5 5 5


bbl/stb 1.5 1.5 1.5 1.5 1.5 1.5


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



Case 1-B will not affect the PI (same PI as Case 1-base’s PI). By increasing the


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



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



40

Case 1-B will not affect the PI (same PI as Case 1-base’s PI). By increasing the








Table 17. PI of Renard-Dupuy’s anisotropic steady state horizontal oil well – Re = 5000 ft


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


bbl/stb 1.5 1.5 1.5 1.5 1.5 1.5


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







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





PI, stb/psi/day 0.020 0.099 0.198 0.495 0.989


The PI values of the horizontal oil well using Babu-Odeh’s model are illustrated in



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




Table 19. PI of Babu-Odeh’s pseudo steady state horizontal oil well


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


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



46

Table 20. PI of Kuckuk’s pseudo steady state horizontal oil well

Figure 25. Impact of the crucial parameters on Kuckuk’s PI


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. PI of Economides pseudo steady state horizontal oil well


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







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





PI, stb/psi/day 0.018 0.088 0.176 0.441 0.882


The PI values of the horizontal oil well using Babu-Odeh’s model are illustrated in



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


51



Table 23. PI of Babu-Odeh’s pseudo steady state horizontal oil well – Re = 5000 ft


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


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. PI of Economides pseudo steady state horizontal oil well – Re = 5000 ft


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.

Documents

Productivity Index of Horizontal Oil Wells Nasser AlMolhem ...wvuscholar.wvu.edu/reports/AlMolhem_Nasser.pdf · As the drilling technology progressed, PI solutions were considered