Predicting Horizontal Gas Well Deliverability in a...

PREDICTING HORIZONTAL GAS WELL DELIVERABILITY IN A CONVENTIONAL RESERVOIR FROM VERTICAL WELL DATA

Osade O. Edo-Osagie

A Problem Report Submitted to the

College of Engineering and Mineral Resources at

West Virginia University

In Partial Fulfillment of the Requirements of the degree of

Master of Science

In

Petroleum and Natural Gas Engineering

Khashayar Aminian, Ph.D.

Samuel Ameri, M.S.

Ilkin H. Bilgesu, Ph.D.

Department of Petroleum and Natural Gas Engineering

Morgantown, West Virginia

2011

Keywords: Horizontal Well, Vertical Well, Conventional Gas Reservoir, Deliverability, ECLIPSE

Copyright 2011 Osade O. Edo-Osagie

ABSTRACT

PREDICTING HORIZONTAL GAS WELL DELIVERABILITY IN A CONVENTIONAL RESERVOIR FROM

VERTICAL WELL DATA

Osade Edo-Osagie

This research project focuses on predicting the deliverability of horizontal gas wells in a

conventional reservoir based on well test data made available from the already existing vertical

wells. ECLIPSE reservoir simulation software by Schlumberger was employed for this research.

The first objective of the project was to obtain production results from ECLIPSE and compare

the results to the mathematical gas flow equations to check for validity and consistency. The

ultimate objective however was to determine how the deliverability constants (A and B values)

for the horizontal gas well are impacted by various reservoir parameters including well length,

drainage area, shape aspect ratio (xe/ye), permeability etc.

Using ECLIPSE completions tool template, production results were obtained. The deliverability

constants A and B were determined and different scenarios in terms of reservoir and well

parameters were taken into account. Monitoring the effects on deliverability was necessary in

achieving desired results.

Results were summarized in terms of the ratio of the horizontal to vertical deliverability

constants profile. Since the vertical deliverability is supposedly known and the horizontal to

vertical deliverability ratio profiles were obtained for each scenario, the horizontal deliverability

can be determined by identifying the case matching the particular situation.

iii

DEDICATION

I dedicate this paper first to my father Engr. Osato Edo-Osagie for ever willing support in

numerous forms all through my education, and then to my mother Mrs. Helen Edo-Osagie also

for her love, care and support, and finally to my siblings Osato Jr., Ifueko, Eseosa and Osaze

Edo-Osagie whom I’ve always seemed to lean on at one point or another.

iv

ACKNOWLEDGEMENT

I would like to thank God for his continued guidance. I would also express my gratitude to Dr.

Khashayar Aminian for his supervision throughout this research work and for being an excellent

advisor throughout my graduate studies at West Virginia University. Special thanks are also

directed to the rest of my graduate committee: Prof. Sam Ameri and Dr. Ilkin H. Bilgesu, for

their necessary input and overseeing of my research work.

Finally I would thank my colleagues and friends in the department of Petroleum and Natural

Gas Engineering, in the College of Engineering and Mineral Resources at the University and

everyone else that helped me even in the least amount during my research and graduate

studies here.

v

TABLE OF CONTENTS

ABSTRACT .........................................................................................................................................ii

DEDICATION .................................................................................................................................... iii

ACKNOWLEDGEMENT ..................................................................................................................... iv

TABLE OF CONTENTS........................................................................................................................ v

LIST OF FIGURES ............................................................................................................................. vii

LIST OF TABLES ................................................................................................................................ xi

CHAPTER 1....................................................................................................................................... 1

INTRODUCTION ............................................................................................................................... 1

CHAPTER 2....................................................................................................................................... 2

LITERATURE REVIEW ....................................................................................................................... 2

HORIZONTAL/MULTILATERAL WELLS ......................................................................................... 2

ECLIPSE ........................................................................................................................................ 3

GAS FLOW EQUATIONS ............................................................................................................... 4

DELIVERABILITY ........................................................................................................................... 9

CHAPTER 3..................................................................................................................................... 10

METHODOLOGY ............................................................................................................................ 10

REPRESENTATIVE DATA ............................................................................................................ 10

PRODUCTION ESTIMATION ....................................................................................................... 11

NON-DARCY INTEGRATION ....................................................................................................... 17

DELIVERABILITY CONSTANTS .................................................................................................... 19

PERFORMANCE PREDICTION .................................................................................................... 19

CHAPTER 4..................................................................................................................................... 20

vi

RESULTS AND DISCUSSIONS .......................................................................................................... 20

HORIZONTAL WELL LENGTH ..................................................................................................... 25

SHAPE ASPECT RATIO ................................................................................................................ 26

DRAINAGE AREA........................................................................................................................ 27

FORMATION HEIGHT ................................................................................................................. 29

RESERVOIR VERTICAL AND HORIZONTAL PERMEABILITY ......................................................... 30

DIMENSIONLESS WELL LENGTH ................................................................................................ 34

CHAPTER 5..................................................................................................................................... 44

SUMMARY AND CONCLUSIONS .................................................................................................... 44

APPENDIX ...................................................................................................................................... 45

APPENDIX A: USING GAS FLOW EQUATIONS ............................................................................ 45

APPENDIX B: USING ECLIPSE ..................................................................................................... 52

NOMENCLATURE ........................................................................................................................... 72

REFERENCES .................................................................................................................................. 74

vii

LIST OF FIGURES

Figure 3.1: Vertical well mathematical model ................................................................................ 5

Figure 3.2: Horizontal well mathematical model ........................................................................... 5

Figure 3.3: Shape related skin factor sCA,h, for a horizontal well in a rectangular drainage area

(xe/ye=2) .......................................................................................................................................... 8

Figure 3.4: ECLIPSE template model definition ............................................................................ 12

Figure 3.5: ECLIPSE template reservoir description ..................................................................... 12

Figure 3.6: ECLIPSE generated vertical well model ...................................................................... 13

Figure 3.7: ECLIPSE template well configuration .......................................................................... 14

Figure 3.8: ECLIPSE template production specification ................................................................ 14

Figure 3.9: ECLIPSE generated horizontal well model .................................................................. 15

Figure 3.10: Reservoir relative dimensions .................................................................................. 16

Figure 3.11: Vertical well grid allocation ...................................................................................... 16

Figure 3.12: Horizontal well grid allocation .................................................................................. 17

Figure 3.13: ECLIPSE data manager well completions specification ............................................ 18

Figure 4.1: IPR curve for reservoir pressure of 2000psia ............................................................. 21




Figure 4.5: AH/AV vs L/2xe .............................................................................................................. 25

Figure 4.6: AH/AV vs L/2xe at different area aspect ratios ............................................................. 26

Figure 4.7: AH/AV vs L/2xe at different area aspect ratios at twice the drainage area ................. 27

Figure 4.8: AH/AV vs L/2xe at xe/ye =1 for different drainage areas ............................................... 28

viii

Figure 4.9: AH/AV vs L/2xe at xe/ye =2 for different drainage areas ............................................... 28

Figure 4.10: AH/AV vs L/2xe at xe/ye =5 for different drainage areas ............................................. 29

Figure 4.11: AH/AV vs L/2xe at different reservoir thicknesses ...................................................... 30

Figure 4.12: AH/AV vs L/2xe at different reservoir thicknesses for kV/kH=1/10 ............................. 31



Figure 4.15: AH/AV vs L/2xe at 60ft reservoir thickness for at different kV/ kH ratios.................... 33

Figure 4.16: AH/AV vs L/2xe at 120ft reservoir thickness for at different kV/ kH ratios.................. 33

Figure 4.17: AH/AV vs L/2xe at 60ft reservoir thickness for at different kV/ kH ratios.................... 34

Figure 4.18: AH/AV vs L/2xe at different LD values for constant kV/ kH for xe/ye =1 ....................... 35

Figure 4.19: AH/AV vs L/2xe at different LD values for constant h for xe/ye =1 ............................... 35

Figure 4.20: AH/AV vs L/2xe at LD =1 for xe/ye =1 ........................................................................... 36



Figure 4.23: Average AH/AV vs L/2xe at LD =1 for xe/ye =1 ............................................................. 38



Figure 4.26: AH/AV vs L/2xe at different LD values for constant kV/ kH for xe/ye =2 ....................... 39

Figure 4.27: AH/AV vs L/2xe at different LD values for constant h for xe/ye =2 ............................... 40






ix


Figure A.1: Shape related skin factor SCA,h, for horizontal well in a rectangular drainage area

(xe/ye=2) ....................................................................................................................................... 46

Figure A.2: Gas properties .exe program for computing μ and z ................................................. 47

Figure A.3: Shape related skin factor SCA,h, for horizontal well in a square drainage area

(xe/ye=1) ....................................................................................................................................... 49


(xe/ye=1) ....................................................................................................................................... 50


(xe/ye=5) ....................................................................................................................................... 50

Figure B.1: Model Definition ........................................................................................................ 52

Figure B.2: Reservoir description (Layers) .................................................................................... 52

Figure B.3: Reservoir description (Rock properties) ..................................................................... 53

Figure B.4: Reservoir description (Initial conditions) ................................................................... 53

Figure B.5: Reservoir description (Aquifers) ................................................................................. 54

Figure B.6: Wells (Well deviation, vertical well) ........................................................................... 54

Figure B.7: Wells (Well deviation, 1st lateral) .............................................................................. 55

Figure B.8: Wells (Well deviation, 2nd lateral) ............................................................................. 55

Figure B.9: Wells (Casing) ............................................................................................................. 56

Figure B.10: Wells (Tubing) ........................................................................................................... 56

Figure B.11: Production ................................................................................................................ 57

Figure B.12: Fluid properties (PVT correlations) .......................................................................... 57

Figure B.13: Fluid properties (Advanced) ..................................................................................... 58

Figure B.14: Simulation controls (Gridding) ................................................................................. 58

Figure B.15: Simulation controls (Tuning) .................................................................................... 59

x

Figure B.16: Economics ................................................................................................................. 59

Figure B.17: Gas rate profile from ECLIPSE results ....................................................................... 60

Figure B.18: Field pressure profile from ECLIPSE results.............................................................. 61

Figure B.19: Bottomhole pressure profile from ECLIPSE results .................................................. 62

Figure B.20: Bottomhole pressure profile showing straight line decline ..................................... 63

Figure B.21: Field pressure profile showing selected point in the pseudosteady state period ... 64

xi

LIST OF TABLES

Table 3.1: Shape dependent skin factors, sCA, for vertical wells .................................................... 7

Table 3.2: Base case reservoir and wellbore parameters ............................................................ 11

Table 4.1: Vertical and horizontal well production rates at different conditions ........................ 20

Table 4.2: Relative percentage increases in production rate ....................................................... 23

Table 4.3: Vertical well sample runs at different rates and times ............................................... 24

Table 4.4: Corresponding horizontal well runs at the rates and times ........................................ 24

Table A.1: Shape dependent skin factors, sCA, for vertical wells .................................................. 51

Table B.1: Pressure and rate profiles for base case at xe/ye =1 with A values ............................. 65

Table B.2: AH/AV values for base case at different xe/ye values ................................................... 66

Table B.3: AH/AV values for 4000000ft drainage area .................................................................. 67

Table B.4.1: Total horizontal well skin factor for A=2000000ft2 .................................................. 67

Table B.4.2: Total horizontal well skin factor for A=4000000ft2 .................................................. 68

Table B.5: AH/AV values for different h values at kv/kh = 1/10 ..................................................... 68



Table B.8: AH/AV values for different LD values for xe/ye = 1 (kv/kh = constant) ........................... 70

Table B.9: AH/AV values for different LD values for xe/ye = 1 (h = constant) ................................. 70

Table B.10: AH/AV values for different LD values for xe/ye = 2 (kv/kh = constant) ......................... 71

Table B.11: AH/AV values for different LD values for xe/ye = 2 (h = constant) ............................... 71

1

CHAPTER 1

INTRODUCTION

Horizontal wells offer potential for increase in productivity, reduction of water or gas coning,

reduction of turbulence in gas wells, intersection of fractures in naturally fractured reservoirs

hence drain reservoirs more effectively, improvement in drainage area per well and hence

reduction in the number of vertical wells in low permeability reservoirs, increase in injectivity of

an injection well, enhancement of sweep efficiency, etc. Drilling horizontal wells instead of

vertical wells in hydrocarbon formations is therefore important because of many of these

advantages. It is however necessary to predict the performance of a horizontal well to

determine its economic feasibility. Currently, there is limited information on ways to compute

productivity, devise procedures that evaluate the completion efficiency, and predict

performance of the horizontal wells.

Previous studies have been conducted in the area of predicting horizontal well deliverability but

not in the attempt to predict it from that of a vertical well based on certain parameters in

conventional reservoirs. For example, Chase and Steffy (2004) published a paper on predicting

horizontal gas well deliverability using dimensionless IPR curves. Chen et al (2000) developed a

deliverability model to predict the performance of multilateral wells. Wang and Economides

(2009) predicted horizontal well deliverability with turbulence effects. This research, however,

tackles the aforementioned problem by dealing differently with comparing the deliverability of

a horizontal gas well with that of a vertical one for prediction purposes.

The main purpose of this research was to determine how the deliverability constants of a

horizontal gas well differed as compared to that of a vertical well with reservoir and well

parameters. The parameters taken into consideration were the horizontal well length,

reservoir drainage area, shape aspect ratio, formation height and permeability. The

dimensionless length (LD), a combination of certain parameters, which is explained later in this

paper, was also taken into account.

2

CHAPTER 2

LITERATURE REVIEW

HORIZONTAL/MULTILATERAL WELLS

The drilling of horizontal wells started in the 1980s and became common in the early 1990s.

They have since then proliferated and become essential for today’s hydrocarbon production.

Advantages of drilling a horizontal well over a vertical well include to increase productivity,

reduce water or gas coning, reduce turbulence in gas wells, intersect fractures in naturally

fractured reservoirs hence drain reservoirs more effectively, improve drainage area per well

hence reducing the number of vertical wells in low permeability reservoirs, increase injectivity

of an injection well and enhance sweep efficiency.

Multilateral wells offer the potential for substantial improvement in well economics. In

multilateral well completions, two or more horizontal wellbores are drilled from a single parent

wellbore, enabling drainage of multiple reservoir targets. This technology allows the same

magnitude of reservoir exposure with a lesser number of wells on the surface/platform. The

benefits from multilateral well technology include increased production per platform slot,

exploitation of reservoirs with vertical permeability barriers, and production from natural

fracture systems, and others similar to those listed for horizontal wells. The modeling of

multilateral wells may be complicated for certain configurations because of the interplay

between reservoir performance and pressure loss in the wellbore. There have also been limited

information on ways to compute productivity, devise procedures that evaluate the completion

efficiency, and predict performance of the completion.

There are different methods that have been presented to determine productivity indices or skin

values for multilateral well systems.

For this research, horizontal wells were formed in each case from two laterals made from the

vertical well in the reservoir formation facing opposite directions (hence forming one longer

lateral or one horizontal well).

3

ECLIPSE

Reservoir simulation can be defined as an area of reservoir engineering in which computer

models are used to predict fluid flow in a porous media, in this case the reservoir, assumed to

mimic typical conditions. They can help in the development of new fields and can be analyzed

for production forecasts in order to help make important investment decisions. There are

numerous reservoir simulation software used today by professionals, some of which include

CMG, ECLIPSE, PETREL etc. to name a few. ECLIPSE (Office 2010) software by Schlumberger was

employed for this research.

ECLIPSE is an oil and gas reservoir simulator first developed by Exploration Consultants Limited

(ECL) but is currently owned, developed, marketed and maintained by SIS (formerly known as

GeoQuest), a division of Schlumberger. The name ECLIPSE was an acronym for "ECL´s Implicit

Program for Simulation Engineering". Ian Cheshire was one of the main engineers that

developed the program.

ECLIPSE uses the finite differences method to solve material and energy balance equations

modeling a subsurface petroleum reservoir. Versions include ECLIPSE 100 which solves the

black oil equations (a fluid model) on corner-point grids, ECLIPSE 300 that solves the reservoir

flow equations for compositional hydrocarbon descriptions and thermal simulation and

Intersect, a next generation reservoir simulator developed in partnership with Chevron.

ECLIPSE software covers a wide spectrum of reservoir simulation, specializing in blackoil,

compositional and thermal volume-difference reservoir simulation, and streamline reservoir

simulation. By choosing from a wide range of add-on options, like coalbed methane, gas field

operations, calorific value-based controls, reservoir coupling, and surface networks, simulator

capabilities can be tailored for different models, and can help enhance the scope of reservoir

simulation studies.

4

GAS FLOW EQUATIONS

The pseudo-steady state equation for a well on the basis of a rectangular drainage area is given as (Chaudhri, 2003):

gcAmwe

sc

scg

wfR DqcsssACrrkhT

TPzqpp

''ln

10335.50 3

22

where

2

15 '10222.2

pwpwf

ag

hr

kD

1045.1101073.2' ak

vha kkk

and

s => equivalent negative skin factor due to either well stimulation or horizontal well,

dimensionless

sm => mechanical skin factor, dimensionless

sCA => shape-related skin factor, dimensionless

c’ => shape factor conversion constant, dimensionless

k => permeability, millidarcy

h => reservoir height, feet

pR => average reservoir pressure, pounds per square inches

pwf => well flowing pressure, pounds per square inches

qg => gas flow rate, thousand standard cubic feet per day

T => reservoir temperature, Rankin

µ => gas viscosity at well flowing conditions, centipoise

z => gas compressibility factor evaluated at some average pressure between pR and

pwf, dimensionless

β’ => high velocity flow coefficient, per foot

g => gas gravity, dimensionless

rw => wellbore radius, feet

5

hp => perforated interval, feet

ka => permeability in near wellbore region, millidarcy

AC’ => 0.738 for a rectangular drainage area, dimensionless

In order to calculate gas flow rate, the above equation can be rewritten as:

gCAmwesc

wfRsc

gDqcsssrrTPz

ppkhTq

'738.0ln

10019866.0 226

The mathematical model for the vertical well is shown in figure 3.1.

Figure 3.1: Vertical well mathematical model

The mathematical model for the horizontal well is shown in figure 3.2.

Figure 3.2: Horizontal well mathematical model

Also note that:

2, ftArev

evrLa 2/'

a’

2xe

b’ L

2ye

2xe

6

evrb '

ftrrLr eveveh ,2/

5.0

5.04

/225.05.02/

LrL eh

5.0

vh kk

4

' Lrw

0'c for vertical gas wells

386.1'c for horizontal gas wells

0s for vertical gas wells (no stimulation)

ww rrs /ln '

for horizontal gas wells

h

vD

k

k

h

LL

2

sCA is obtained from charts:

For a vertical well, sCA is obtained from table 3.1.

7

Table 3.1: Shape dependent skin factors, sCA, for vertical wells (Fetkovich and Vienot, 1985)

For a horizontal well, sCA is obtained from using LD and 2xe values for the sCA,h charts. An

example is illustrated in figure 3.3.

8

Figure 3.3: Shape related skin factor sCA,h, for a horizontal well in a rectangular drainage area (xe/ye=2)

(Golan and Whitson, 1986)

Mechanical skin, sm, is assumed to be negligible

For a vertical gas well therefore, the flow equation becomes:

gCAwesc

wfRsc

gDqsrrTPz

ppkhTq

738.0ln

10019866.0 226

………………………….................................2.1

And for a horizontal gas well, the flow equation becomes:

gCAwesc

wfRsc

gDqssrrTPz

ppkhTq

386.1738.0ln

10019866.0 226

………………………………………2.2

Production rate values are obtained by substituting for the parameters remaining, the values

aforementioned previously in the data section.

(D = 0 for calculations without non-Darcy or turbulence effect)

9

DELIVERABILITY

The constants A and B can be determined from flow tests for at least two rates in which the

flow rate, qsc, and corresponding well-flowing or bottom-hole pressure, pwf, values are

measured; reservoir pressure, pR, also must be known. A is usually referred to as the laminar

constant while B is referred to as the turbulence constant.

gcAmwe

sc

scg

scscwfRDqcsssACrr

khT

TPzqBqAqpp

''ln

10335.50 3

222

(Chaudhri, 2003)

Using the gas flow equations for a horizontal well therefore,

'738.0ln1422'738.0ln

10337.50 3

cssr

r

kh

Tzcss

r

r

khT

TpzA cA

w

ecA

w

e

sc

sc

HH

sc

sc Dkh

TzD

khT

TpzB

1422

10337.50 3

……………………………………………………………….2.3

Or for a vertical well,

cA

w

ecA

w

e

sc

sc sr

r

kh

Tzs

r

r

khT

TpzA 738.0ln1422738.0ln

10337.50 3

VV

sc

sc Dkh

TzD

khT

TpzB

1422

10337.50 3

……………………………………………………………….2.4

10

CHAPTER 3

METHODOLOGY

It is important to note that for this research, synthetic data which fall within real average limits

have been employed as the sample data for the use as the base case and the results obtained

are relative to the various syntactic situations which were predefined.

The methodology to achieve the results included the following steps:

- Obtaining production results from gas flow equations and from ECLIPSE simulation runs

for the vertical and horizontal wells during pseudosteady state period and then

comparing both to check for consistency and validity.

- Integrating non-Darcy (turbulence) into both methods and analyze similarly accordingly.

- Deriving deliverability constants A and B for both vertical and horizontal wells using the

analytical and simulation approach and testing for validity and consistency.

- Finally determining how the constants A and B vary with respect to change in the

following parameters:

o Horizontal well length

o Reservoir drainage area

o Shape aspect ratio

o Reservoir vertical and horizontal permeability

o Formation height

o Dimensionless horizontal well length

REPRESENTATIVE DATA

Unless otherwise indicated, the reservoir and wellbore parameters were used for the base case

for the project are shown in table 3.2.

11

Table 3.2: Base case reservoir and wellbore parameters

PRODUCTION ESTIMATION

Production information was obtained in two ways:

- Using the pseudo-steady state gas flow mathematical equations.

- Using ECLIPSE software (completions tool template).

Pseudo-Steady State Gas Flow Mathematical Equations:

The gas flow equations used for the wells have been illustrated in the previous section.

Equation 2.1 shows the general equation used in the case of a vertical well while equation 2.2

shows the general equation used in the case of a horizontal well.

Parameter Value Unit

Initial reservoir pressure 2000 psia

Well flowing pressure 500 psia

Reservoir temperature 110 °F

Standard conditions pressure 14.7 psia

Standard conditions temperature 520 R

Horizontal permeability 10 md

Vertical permeability 1 md

Drainage area 2000000 ft2

Shape aspect ratio (xe/ye) 2

Reservoir thickness 60 ft

Horizontal well length 1000 ft

Hole diameter 9.5 in

Reservoir true vertical depth 5000 ft

Porosity 0.1

Gas gravity 0.7

12

Production using ECLIPSE:

Results were obtained using the completions tool model template. The model was defined as a

dry gas model with simulation length and reporting of about 1 year. The reservoir was

described according to the parameters previously mentioned in the data section. Figures 3.4

and 3.5 show some of the model input templates.

Figure 3.4: ECLIPSE template model definition

Figure 3.5: ECLIPSE template reservoir description

13

For vertical well production, a vertical well was created down through the formation and

perforated from the producing reservoir’s top to bottom depth. The vertical well model is

illustrated as follows:

Figure 3.6: ECLIPSE generated vertical well model

For horizontal well production, two equidistant laterals of the same length each were made

from the vertical well in the middle of the reservoir facing opposite directions (hence a

combined horizontal well of length twice the length of one of the laterals). Production was

obtained from the well while only the laterals were perforated.

14

Figure 3.7: ECLIPSE template well configuration

Figure 3.8: ECLIPSE template production specification

15

The horizontal well model generated is illustrated as follows:

Figure 3.9: ECLIPSE generated horizontal well model

The reservoir was rendered transparent in order to show the two laterals from “Well1” forming

the horizontal well. The relative reservoir dimensions can be seen on the next illustration.

16

Figure 3.10: Reservoir relative dimensions

After inputting all the necessary parameters, the simulation was run and results were obtained.

Figure 3.11: Vertical well grid allocation

17

Figure 3.12: Horizontal well grid allocation

NON-DARCY INTEGRATION

Calculation of D-factor:

2

15 '10222.2

pwpwf

ag

hr

kD

1045.1101073.2' ak

vha kkk

For a vertical well, the perforated interval is the height of the formation while the length of the

well is regarded as the perforation interval for a horizontal well.

For the base case of the horizontal well for example,

91045.1101045.110 10655.7)162.3(1073.21073.2'

ak

7

2

915

2

15

1000.5)1000)(2/792.0(0114.0

)10655.7)(60)(162.3)(7.0(10222.2'10222.2

pwpwf

ag

hr

hkD

18

In case of the vertical well,

91045.1101045.110 10655.7)162.3(1073.21073.2'

ak

4

2

915

2

15

1039.1)60)(2/792.0(0114.0

)10655.7)(60)(162.3)(7.0(10222.2'10222.2

pwpwf

ag

hr

hkD

The non-Darcy factor, D, is directly substituted into the gas flow equation for analytical

calculations and the resulting equation is a quadratic one and the gas flow rate can thus be

obtained from solving the quadratic equation.

Using ECLIPSE, D is specified in the Well completions specification data of the Schedule Section

of the Data Manager Module as shown:

Figure 3.13: ECLIPSE data manager well completions specification

19

DELIVERABILITY CONSTANTS

Obtaining the deliverability constants A and B from the gas flow equations have been illustrated

in the previous section. Equation 2.4 shows the general equation used in the case of a vertical

well while equation 2.3 shows the general equation used in the case of a horizontal well.

Determining the constants from ECLIPSE simulation involved obtaining the reservoir and

bottomhole pressure profiles and relative production values. A and B values were derived

during pseudosteady state accordingly.

sc

wfR

q

ppA

22

…….from simulation without non-Darcy integration

Then 2

22

sc

scwfR

q

AqppB

……..from simulation with non-Darcy integration

PERFORMANCE PREDICTION

The results obtained from following the methodology described above were compared and

checked for validity and consistency. After that task was completed, vital analysis was

performed using the ECLIPSE reservoir simulation software. Variations in the values of

constants A and B with respect to change in horizontal well length, reservoir drainage area,

drainage area aspect ratio, reservoir vertical and horizontal permeability, formation height and

finally dimensionless horizontal well length, which is a combination of the well length,

formation height and ratio of vertical to horizontal permeability, are derived for horizontal

wells as compared to vertical wells. Results for each case show a horizontal to vertical

deliverability constant ratio profiles. Since the vertical deliverability is supposedly known and

the horizontal to vertical deliverability ratio profiles were obtained for each case, the horizontal

deliverability can be determined by identifying the case matching the particular.

20

CHAPTER 4

RESULTS AND DISCUSSIONS

Using the pseudo-steady state gas flow rate mathematical equations to obtain production

information, the base case reservoir and wellbore parameters were employed in the necessary

equations to obtain production values for both vertical and horizontal wells. Note that this

result was obtained at different reservoir and bottom hole pressures. Refer to appendix A for

gas flow equations calculations.

Based on the parameters specified, the production rate for vertical and horizontal wells at

different reservoir and bottomhole pressure is summarized in Table 4.1.

Notice that the vertical well is significantly affected by the non-Darcy term but the horizontal

well is relatively unaffected in terms of production as seen in the 1000ft long case.

Table 4.1: Vertical and horizontal well production rates at different conditions

PR

BHP q BHP q BHP q BHP q BHP q BHP q

400 24436.58 400 36201.29865 400 147041.6 400 141279.7286 400 225726.4 400 285753.9

600 23569.9 600 34143.89758 600 138684.9 600 133698.7578 600 212897.9 600 269513.8

800 22177.67 800 31232.44576 800 126859.3 800 122790.2883 800 194744.1 800 246532.4

1000 20148.62 1000 27272.00854 1000 110772.9 1000 107783.1858 1000 170049.5 1000 215270.8

1200 17511.63 1200 22651.52425 1200 92005.47 1200 90012.22987 1200 141239.3 1200 178799.1


400 20813.26 400 29347.82707 400 119204.4 400 115360.8923 400 182992.9 400 231656.2

600 19736.66 600 27150.97399 600 110281.2 600 107081.0183 600 169294.8 600 214315.4

800 18254.14 800 24388.49827 800 99060.67 800 96543.56903 800 152069.9 800 192509.9

1000 16210.63 1000 20821.6305 1000 84572.85 1000 82806.62941 1000 129829.4 1000 164354.9

1200 13367.97 1200 16363.20513 1200 66463.71 1200 65409.67751 1200 102029.7 1200 129162.5


400 17259.89 400 23129.0696 400 93945.15 400 91524.33192 400 144217 400 182568.6

600 16055.96 600 20962.73362 600 85145.98 600 83211.90423 600 130709.2 600 165468.7

800 14323.61 800 18100.64933 800 73520.82 800 72114.86186 800 112863.2 800 142876.9

1000 12074.78 1000 14633.09896 1000 59436.4 1000 58551.90331 1000 91241.95 1000 115505.9

1200 9040.826 1200 10410.81525 1200 42286.42 1200 41853.61248 1200 64914.69 1200 82177.45


400 13802.99 400 17556.57748 400 71310.92 400 69897.61589 400 109470.8 400 138582.3

600 12462.96 600 15419.37973 600 62630.1 600 61569.9074 600 96144.66 600 121712.4

800 10534.58 800 12577.64017 800 51087.59 800 50399.60352 800 78425.52 800 99281.22

1000 7943.22 1000 9050.324663 1000 36760.41 1000 36417.10725 1000 56431.61 1000 71438.46

1200 4533.254 1200 4877.699654 1200 19812.13 1200 19715.22943 1200 30413.98 1200 38501.97

1000' long Horizontal

Well (w/ Non-Darcy)

1500' long

Horizontal Well

2000ft long

Horizontal Well

1600

1400

Vertical Well (With

Non-Darcy)

1000' long

Horizontal Well

2000

1800

Vertical Well (Without

Non-Darcy)

21

The following IPR curves are generated from the results above:

Figure 4.1: IPR curve for reservoir pressure of 2000psia


0

50000

100000

150000

200000

250000

300000

350000

0 200 400 600 800 1000 1200 1400

q, m

scfd

BHP, psia

Pr = 2000psia

1000' HW

1500' HW

2000' HW

VW W/ ND

VW w/o ND

0

50000

100000

150000

200000

250000

0 200 400 600 800 1000 1200 1400

q, m

scfd

BHP, psia

Pr = 1800psia

1000' HW

1500' HW

2000' HW

VW w/ ND

VW w/o ND

22



0

20000

40000

60000

80000

100000

120000

140000

160000

180000

200000

0 200 400 600 800 1000 1200 1400

q, m

scfd

BHP, psia

Pr = 1600psia

1000' HW

1500' HW

2000' HW

VW w/ ND

VW w/o ND

0

20000

40000

60000

80000

100000

120000

140000

160000

0 200 400 600 800 1000 1200 1400

q, m

scfd

BHP, psia

Pr = 1400psia

1000' HW

1500' HW

2000' HW

VW w/ ND

VW w/o ND

23

Here is a table that shows the relative percentage increases in production rate:

Table 4.2: Relative percentage increases in production rate

Since this is a comparative analysis, a known value of production rate was directly employed in

the program as the constant well control property, while the bottomhole pressure (well flowing

pressure) was varied accordingly. This enables the procurement of a bottom hole pressure

profile (pressure versus time). The pseudo-steady state zone was identified from this profile as

evident by a constant change in pressure (straight line decline). A bottom hole pressure was

obtained at a particular time from within the pseudo-steady state zone. The corresponding

reservoir pressure at that same time was noted from the reservoir pressure profile. These

values of reservoir pressure and well flowing pressure were employed back into the gas flow

rate equations and a value of production, in thousand standard cubic feet per day, was

obtained and compared to the original value obtained initially from the pseudo-steady state gas

flow rate equations.

An alternative method employed was keeping the bottomhole pressure constant as the well

control property of the reservoir model and running simulations to obtain reservoir and

bottomhole pressure, and production profiles. Identifying a particular time during the

pseudosteady state region, the pressure values were employed into the gas flow equation and

PR VW w/ ND VW w/o ND 1000' HW 1500' HW 2000' HW

2000 0 40.828346 472.0135 778.1088 1011.624

1800 0 33.605267 442.675 733.0706 954.6091

1600 0 26.3692922 413.284 687.9521 897.4922

1400 0 19.3938134 384.9512 644.4578 842.4314

Average percentage increase in q over vertical well (with non-darcy effects):

PR VW w/ ND VW w/o ND 1000' HW 1500' HW 2000' HW

2000 110.52253 148.317215 148.3172 148.3172 148.3172

1800 73.278284 93.9036094 93.90361 93.90361 93.90361

1600 35.967546 43.9113305 43.91133 43.91133 43.91133

1400 0 0 0 0 0

Avg increase over 1400 psia PR:

24

a production rate value is obtained. Values are compared for validity. Different scenarios were

tried out to test for consistency.

Deliverability constants A and B values were determined and compared in a similar fashion

following the predefined equations shown in the methodology section. Note that B values were

negligible for horizontal wells as compared to that of vertical wells so focus was shifted to only

the deliverability constant A values for the remainder of this research work.

It was discovered at the end of the analysis that the values obtained using the gas flow

equations were very close to those obtained from ECLIPSE simulation runs with a percentage

error of about 6% or less. Refer to appendix B for ECLIPSE calculations and simulation run

results. The following tables show deliverability constant A values of sample runs at different

rates and times for a vertical well and corresponding horizontal well. Notice how the ratio of

horizontal to vertical A value is constant at 0.32.

Table 4.3: Vertical well sample runs at different rates and times

Table 4.4: Corresponding horizontal well runs at the rates and times

Thus illustrated are the variations in the values of the deliverability constant with respect to

change in the different factors. Starting from the base case scenario, all other factors except

t, days q, mscf/d pR, psia pwf, psia A=(pR^2-pwf^2)/q

600 1000 1781.4 1736.7 157

600 2000 1564.1 1467.2 147

400 3000 1561.3 1414.9 145

400 4000 1414.6 1199.4 140

200 5000 1641.4 1402.9 145

200 6000 1569.9 1269.1 142

t, days q, mscf/d pR, psia pwf, psia A=(pR^2-pwf^2)/q

600 1000 1781.4 1766.7 52

600 2000 1564.1 1533.4 48

400 3000 1561.3 1515.7 47

400 4000 1414.6 1349.3 45

200 5000 1641.4 1568.6 47

200 6000 1570 1480 46

25

that in consideration remain constant unless otherwise specified. In all cases, the horizontal

well is aligned in the x-direction in the middle of the producing reservoir.

HORIZONTAL WELL LENGTH

As previously seen, an increase in the length of a horizontal well increases the production rate

of the well (thereby decreasing the value of the deliverability constant A). In considering the

horizontal well length, the ratio of L/2xe is taken into account. The variation is illustrated in the

following chart, with AH/AV referring to the ratio of horizontal to vertical deliverability constant

A value.

Figure 4.5: AH/AV vs L/2xe

Note that a vertical well will result from a horizontal well length of 0ft, making the ratio AH/AV

equal to 1. The decline in that ratio can be seen in figure 4.5. L/2xe is used as the variable axis

for the rest of the analysis.

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

AH/A

V

L/2xe

26

SHAPE ASPECT RATIO

Next, the ratio is investigated for the change in the reservoir shape aspect ratio, xe/ye, which is

basically the ratio of the length to the breadth of the reservoir. Refer to figure 4.6 for the

resulting profile.

Figure 4.6: AH/AV vs L/2xe at different area aspect ratios

The aspect ratio definitely does impact the deliverability constant A value as shown in the chart

above. The trend shows less change however with further increase in the value of this aspect

ratio.

0

0.1

0.2

0.3

0.4

0.5

0.6

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

AH/A

V

L/2Xe

Xe/Ye=1

Xe/Ye=2

Xe/Ye=5

27

DRAINAGE AREA

Doubling the drainage area affects the ratio AH/AV. The profiles may look similar but the value

does drop indicating that the production is increased more so in the horizontal case than in the

vertical.

Figure 4.7: AH/AV vs L/2xe at different area aspect ratios at twice the drainage area

Figures 4.8, 4.9 and 4.10 show individual comparisons for each aspect ratio.

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

AH/A

V

L/2Xe

A=4000000sqft

Xe/Ye=1

Xe/Ye=2

Xe/Ye=5

28

Figure 4.8: AH/AV vs L/2xe at xe/ye =1 for different drainage areas


0

0.1

0.2

0.3

0.4

0.5

0.6

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

AH/A

V

L/2Xe

Xe/Ye=1

A=2000000ft^2

A=4000000ft^2

0

0.1

0.2

0.3

0.4

0.5

0.6

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

AH/A

V

L/2Xe

Xe/Ye=2

A=2000000ft^2

A=4000000ft^2

29


The trends show somewhat less difference in the AH/AV ratios with increasing well length and

aspect ratio.

FORMATION HEIGHT

Investigating the effect of the formation height on the deliverability constant, we realize the

profile illustrated in figure 4.11.

0

0.1

0.2

0.3

0.4

0.5

0.6

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

AH/A

V

L/2Xe

Xe/Ye=5

A=2000000ft^2

A=4000000ft^2

30

Figure 4.11: AH/AV vs L/2xe at different reservoir thicknesses

Remember that all other factors remain constant according to the base case described earlier

except that in consideration, in this case of which is the height of the formation. The trend

shows an increase in the ratio of AH to AV with increasing formation height meaning that the

horizontal well produces less in comparison with the vertical well with greater formation

heights. The difference is somewhat less pronounced with increasing well lengths.

RESERVOIR VERTICAL AND HORIZONTAL PERMEABILITY

In considering permeability, the ratio of vertical to horizontal permeability kV/kH is taken into

account. From the illustration seen in figures 4.12 and 4.13, changing the value of kV or kH has

no effect on the ratio AH/AV if kV/kH remains constant. The reservoir is considered to be

isotropic, meaning that kx = ky = kH, and kz = kV.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

AH/A

V

L/2Xe

h=60

h=100

h=120

31

Figure 4.12: AH/AV vs L/2xe at different reservoir thicknesses for kV/kH=1/10


0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

AH/A

V

L/2Xe

KV/kH = 1/10

h=60

h=100

h=120

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

AH/A

V

L/2Xe

KV/kH =2/20

h=60

h=100

h=120

32

However, a difference in profile is seen as the ratio of kV to kH changes. An increase in KV/kH

leads to a decrease in AH/AV, less pronounced in thinner formations. Figure 4.14 shows the

relationship involved when the ratio KV/kH is doubled.


Figures 4.15, 4.16 and 4.17 directly compare the ratios individually for the different reservoir

thicknesses. It can be seen how much the decrease in AH/AV is less pronounced in thinner

formations.

0

0.1

0.2

0.3

0.4

0.5

0.6

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

AH/A

V

L/2Xe

KV/kH = 2/10

h=60

h=100

h=120

33

Figure 4.15: AH/AV vs L/2xe at 60ft reservoir thickness for at different kV/ kH ratios


0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

AH/A

V

L/2Xe

h=60ft

kV/kH=0.1

kV/kH=0.2

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

AH/A

V

L/2Xe

h=100ft

kV/kH=0.1

kV/kH=0.2

34


DIMENSIONLESS WELL LENGTH

The dimensionless well length, LD, is a combination of certain parameters. It can be used to

estimate the deliverability of a horizontal well with knowledge of that of a vertical well in the

absence of information about the parameters involved. LD is defined as follows:

h

vD

k

k

h

LL

2

It involves the well length, formation height and vertical to horizontal permeability ratio. In

obtaining our deliverability profile, since AH/AV is plotted against L/2xe which contains the well

length, there is left the option of changing both h and kV/kH. Extreme cases of keeping one

factor constant and changing the other are considered. This is illustrated for different aspect

ratios.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

AH/A

V

L/2Xe

h=120ft

kV/kH=0.1

kV/kH=0.2

35

For xe/ye = 1, keeping kV/kH constant (at 0.1) yields figure 4.18.

Figure 4.18: AH/AV vs L/2xe at different LD values for constant kV/ kH for xe/ye =1 Whereas keeping h constant (at 60ft) yields figure 4.19.

Figure 4.19: AH/AV vs L/2xe at different LD values for constant h for xe/ye =1

0

0.1

0.2

0.3

0.4

0.5

0.6

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

AH/A

V

L/2Xe

LD=1

LD=2

LD=3

0

0.1

0.2

0.3

0.4

0.5

0.6

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

AH/A

V

L/2Xe

LD=1

LD=2

LD=3

36

Analyzing both by comparing and contrasting gives figures 4.20, 4.21 and 4.22.

Figure 4.20: AH/AV vs L/2xe at LD =1 for xe/ye =1


0

0.1

0.2

0.3

0.4

0.5

0.6

0 0.2 0.4 0.6 0.8

AH/A

V

L/2Xe

LD=1

h=cons

kV/kH=cons

0

0.1

0.2

0.3

0.4

0.5

0.6

0 0.2 0.4 0.6 0.8

AH/A

V

L/2Xe

LD=2

h=cons

kV/kH=cons

37


Smaller LD values show relatively more scattered trends. General trends can be estimated by

taking the average of both curves in each case as shown in figures 4.23, 4.24 and 4.25.

0

0.1

0.2

0.3

0.4

0.5

0.6

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

AH/A

V

L/2Xe

LD=3

h=cons

kV/kH=cons

38

Figure 4.23: Average AH/AV vs L/2xe at LD =1 for xe/ye =1


0

0.1

0.2

0.3

0.4

0.5

0.6

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

AH/A

V

L/2Xe

LD=1

0

0.1

0.2

0.3

0.4

0.5

0.6

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

AH

/AV

L/2Xe

LD=2

39


For xe/ye = 2, keeping kV/kH constant (at 0.1) yields figure 4.26.

Figure 4.26: AH/AV vs L/2xe at different LD values for constant kV/ kH for xe/ye =2

0

0.1

0.2

0.3

0.4

0.5

0.6

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

AH

/AV

L/2Xe

LD=3

0

0.1

0.2

0.3

0.4

0.5

0.6

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

AH/A

V

L/2Xe

LD=1

LD=2

LD=3

40

Whereas keeping h constant yields figure 4.27.

Figure 4.27: AH/AV vs L/2xe at different LD values for constant h for xe/ye =2

Analyzing by comparing both ends with profiles shown in figures 4.28, 4.29 and 4.30.


0

0.1

0.2

0.3

0.4

0.5

0.6

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

AH/A

V

L/2Xe

LD=1

LD=2

LD=3

0

0.1

0.2

0.3

0.4

0.5

0.6

0 0.2 0.4 0.6 0.8

AH/A

V

L/2Xe

LD=1

h=cons

kV/kH=cons

41



0

0.1

0.2

0.3

0.4

0.5

0.6

0 0.2 0.4 0.6 0.8

AH/A

V

L/2Xe

LD=2

h=cons

kV/kH=cons

0

0.1

0.2

0.3

0.4

0.5

0.6

0 0.2 0.4 0.6 0.8

AH/A

V

L/2Xe

LD=3

h=cons

kV/kH=cons

42

General trends can be estimated by taking the average of both curves in each case as shown in

figures 4.31, 4.32 and 4.33.



0

0.1

0.2

0.3

0.4

0.5

0.6

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

AH/A

V

L/2Xe

LD=1

0

0.1

0.2

0.3

0.4

0.5

0.6

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

AH/A

V

L/2Xe

LD=2

43


0

0.1

0.2

0.3

0.4

0.5

0.6

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

AH/A

V

L/2Xe

LD=3

44

CHAPTER 5

SUMMARY AND CONCLUSIONS

The main purpose of this research was to determine how the deliverability constants of a

horizontal gas well differed as compared to that of a vertical well with reservoir and well

parameters. The parameters taken into consideration were the horizontal well length,

reservoir drainage area, area aspect ratio, formation height and permeability. The

dimensionless length (LD), a combination of the permeability ratio, reservoir thickness and well

length was also considered.

This study has showed that given the known deliverability of a vertical well, the deliverability of

a horizontal well producing in a similar reservoir can be estimated. Horizontal well deliverability

can be determined with availability of information on some of the different individual

parameters such as well length, shape aspect ratio, drainage area, formation height or

formation permeability, or collective parameters as seen in the case of the dimensionless well

length.

The ratio of the horizontal to vertical deliverability is seen to decrease with increasing

horizontal well length, aspect ratio xe/ye, drainage area and vertical to horizontal permeability

ratio. AH/AV increased with increasing formation height. In general, the ratio also decreases

with increasing values of the dimensionless well length. When vertical deliverability is known,

since the horizontal to vertical deliverability ratio profiles have been obtained for each case, the

horizontal deliverability can be determined by identifying the case matching the particular

situation.

Extensive recommendations include using real case study data for the analysis and comparing

the results to the actual production since the data used for this project, although typical, were

synthetic. Furthermore, a broader variation of scenarios and cases can be considered, maybe

extending even to unconventional reservoirs.

45

APPENDIX

APPENDIX A: USING GAS FLOW EQUATIONS

The calculations for the pseudo-steady state gas flow rate equations are done using the

reservoir and wellbore parameters specified in the data section. An example using the base

case is shown as follows:

For the horizontal well,

gCAmwesc

wfRsc

gDqcsssrrTPz

ppkhTq

'738.0ln

10019866.0 223

First, note that:

Mechanical skin (sm) = 0

Turbulence skin (Dqg) = 0

500'

10005002/10002/'

ev

ev

rb

rLa

ftrrLr eveveh 7075005002/10002/5.05.0

8001000/)707(225.05.02/1000/225.05.02/5.0

45.0

4

LrL eh

162.31105.05.0 vh kk

ftL

rw 2504

1000

4

'

386.1'c for horizontal gas wells

45.6)2/12/5.9/(250ln/ln ' ww rrs

64.210

1

)60(2

1000

2

h

v

Dk

k

h

LL

46

2xe = 2000

sCA is obtained from charts using LD and 2xe:

Figure A.1: Shape related skin factor SCA,h, for horizontal well in a rectangular drainage area (xe/ye=2)


From the above charts, we deduce:

SCA = 2.70

Note that:

μ and z values are obtained from the gas properties.exe program

47

Figure A.2: Gas properties .exe program for computing μ and z Therefore, we have that:

dmscf

DqcsssrrTPz

ppkhTq

gCAmwesc

wfRsc

g

/147215

0386.170.2045.6738.0)2/12/5.9(798ln)7.14)(460110)(83.0)(0134.0(

4002000)520)(60)(10(10019866.0

'738.0ln

10019866.0

223

223

Non-Darcy check:

48

g

g

gCAmwesc

wfRsc

g

Dq

Dq

DqcsssrrTPz

ppkhTq

735.1

255400

386.170.2045.6738.0)2/12/5.9(798ln)7.14)(460110)(83.0)(0134.0(

4002000)520)(60)(10(10019866.0

'738.0ln

10019866.0

223

223

91045.1101045.110 10655.7)162.3(1073.21073.2'

ak

7

2

915

2

15

100.5)1000)(24/5.9(0114.0

)10655.7)(60)(162.3)(7.0(10222.2'10222.2

pwpwf

ag

hr

hkD

dmscfqg /141439)100.5(2

)255400)(100.5(4)735.1(735.17

72

For the vertical well,

gCAmwesc

wfRsc

gDqcsssrrTPz

ppkhTq

'738.0ln

10019866.0 223

ftre 7982000000

dmscf

DqcsssrrTPz

ppkhTq

gCAmwesc

wfRsc

g

/36200

738.0185.0)24/5.9(798ln)7.14)(460110)(83.0)(0134.0(

4002000)520)(60)(10(10019866.0

'738.0ln

10019866.0

223

223

Non-Darcy check:

mdkk ha 10

91045.1101045.110 1015.2)10(1073.21073.2'

ak

4

2

915

2

15

1023.1)60)(2/792.0(0114.0

)1015.2)(60)(10)(7.0(10222.2'10222.2

pwpwf

ag

hr

hkD

49

g

g

g

q

qq

4

4

223

1023.1055.7

255400

1023.1738.0185.0)24/5.9(798ln)7.14)(460110)(83.0)(0134.0(

4002000)520)(60)(10(10019866.0

dmscfqg /25162)1023.1(2

)255400)(1023.1(4)055.7(055.74

42

Notes:

Figure A.3: Shape related skin factor SCA,h, for horizontal well in a square drainage area (xe/ye=1)


50





51

Table A.1: Shape dependent skin factors, sCA, for vertical wells (Fetkovich and Vienot, 1985)

52

APPENDIX B: USING ECLIPSE

Figure B.1: Model Definition

Figure B.2: Reservoir description (Layers)

53

Figure B.3: Reservoir description (Rock properties)

Figure B.4: Reservoir description (Initial conditions)

54

Figure B.5: Reservoir description (Aquifers)

Figure B.6: Wells (Well deviation, vertical well)

55

Figure B.7: Wells (Well deviation, 1st lateral)

Figure B.8: Wells (Well deviation, 2nd lateral)

56

Figure B.9: Wells (Casing)

Figure B.10: Wells (Tubing)

57

Figure B.11: Production

Figure B.12: Fluid properties (PVT correlations)

58

Figure B.13: Fluid properties (Advanced)

Figure B.14: Simulation controls (Gridding)

59

Figure B.15: Simulation controls (Tuning)

Figure B.16: Economics

60

Setting the control mode to constant gas production (same as obtained using the pseudo-

steady state gas flow rate equation), the results were obtained.

Figure B.17: Gas rate profile from ECLIPSE results

61

Figure B.18: Field pressure profile from ECLIPSE results

62

Figure B.19: Bottomhole pressure profile from ECLIPSE results

The pseudo-steady state zone was identified from the bottom hole pressure profile as evident

by a constant change in pressure (straight line decline) and is illustrated below:

63

Figure B.20: Bottomhole pressure profile showing straight line decline

64

Figure B.21: Field pressure profile showing selected point in the pseudosteady state period

Picking pressures from both bottom hole and reservoir profiles at a particular time, the gas

production rate was recalculated using the equations. Resulting values were similar with a

percentage error of about 6%.

65

Typical pressures and production rate profiles at constant bottomhole pressure(including

deliverability constant A calculation):

Table B.1: Pressure and rate profiles for base case at xe/ye =1 with A values

Xe/Ye=1

TIME FPR WBHP:WELL1WGPR:WELL1WPI:WELL1A=(pr^2-pwf^2)/q

(DAYS) (PSIA) (PSIA) (MSCF/DAY)

0 2001.814 2001.814 0 0

1 1953.406 500 52404.36 52.49987 68.04389011

2.2 1902.696 500 45891.92 50.18465 73.43892876

3.64 1846.873 500 42313.79 48.81127 74.70236991

5.368 1785.157 500 39193.65 47.54726 74.93014103

7.4416 1717.331 500 36051.41 46.20619 74.87160247

9.92992 1643.508 500 32781.48 44.72971 74.7714794

12.9159 1564.068 500 29392.93 43.10166 74.72237824

16.49908 1479.666 500 25937.96 41.32454 74.771156

20.7989 1391.236 500 22490.21 39.4139 74.94535835

25.89945 1300.907 500 19163.45 37.41547 75.26615191

31 1222.409 500 16442.26 35.64339 75.67600176

37.12066 1142.012 500 13829.04 33.79975 76.23016212

44.46545 1061.213 500 11387.5 31.92183 76.94166444

53.23272 982.0283 500 9180.849 30.06127 77.81192508

62 916.1876 500 7491.19 28.50267 78.67904624

72.52073 852.1014 500 5975.59 26.97933 79.67024811

82.76037 800.6285 500 4852.288 25.75431 80.58176986

93 757.7945 500 3981.943 24.73443 81.43071886

105.2876 716.0877 500 3191.06 23.7425 82.34932635

114.6438 688.9201 500 2705.95 23.09742 83.00631781

124 665.514 500 2307.664 22.54423 83.59487096

135.2275 641.9918 500 1924.649 21.98777 84.25094495

145.1137 624.0888 500 1645.696 21.56626 84.75855646

155 608.7 500 1413.871 21.20457 85.23809923

166.8635 593.1132 500 1186.898 20.83942 85.75569764

176.4318 582.0855 500 1031.369 20.58235 86.12198054

186 572.4742 500 898.846 20.35801 86.47386882

197.4819 562.6401 500 766.3261 20.12895 86.86109774

207.2409 555.3311 500 669.825 19.95904 87.17587348

217 548.9161 500 586.5344 19.81017 87.47800834

228.7109 542.2542 500 501.6149 19.65618 87.79569924

238.3554 537.4266 500 440.9295 19.54471 88.05804724

248 533.1773 500 388.1156 19.4467 88.31915349

259.5735 528.7869 500 334.1596 19.34557 88.62716035

269.2867 525.538 500 294.631 19.27083 88.89157933

279 522.6707 500 260.0284 19.20494 89.16209914

290.6559 519.7005 500 224.4643 19.13676 89.49578134

300.3279 517.5205 500 198.5417 19.08676 89.79180226

310 515.5909 500 175.726 19.04254 90.10628712

321.6065 513.5862 500 152.1478 18.99662 90.50912525

335 511.622 500 129.173 18.95168 91.0183383

66

Pressures and production rate profiles analyses:

xe/ye for base case scenario:

Table B.2: AH/AV values for base case at different xe/ye values

xe/ye for 4000000ft drainage area:

Xe/Ye h A AV AH AH/AV L/2Xe

1 60 2000000 74.77148 39.37909 0.526659 0.25

1 60 2000000 74.77148 24.07294 0.321954 0.5

1 60 2000000 74.77148 16.72308 0.223656 0.75


2 60 2000000 77.11218 33.52636 0.434774 0.25

2 60 2000000 77.11218 18.57235 0.240849 0.5

2 60 2000000 77.11218 11.03715 0.143131 0.75


5 60 2000000 91.38722 36.13112 0.395363 0.25

5 60 2000000 91.38722 18.07394 0.197773 0.5

5 60 2000000 91.38722 7.599936 0.083162 0.75

67

Table B.3: AH/AV values for 4000000ft drainage area

The table below shows how the total skin factor of a horizontal well changes with changes in

well length and drainage area aspect ratio.

Table B.4.1: Total horizontal well skin factor for A=2000000ft2


1 60 2000000 78.83854 35.35144 0.448403 0.25

1 60 2000000 78.83854 21.38375 0.271235 0.5

1 60 2000000 78.83854 14.59482 0.185123 0.75


2 60 2000000 80.63957 29.5891 0.36693 0.25

2 60 2000000 80.63957 16.10817 0.199755 0.5

2 60 2000000 80.63957 9.28085 0.115091 0.75


5 60 2000000 90.43511 29.88688 0.330479 0.25

5 60 2000000 90.43511 14.59054 0.161337 0.5

5 60 2000000 90.43511 5.987876 0.066212 0.75

xe/ye 1 1 1 2 2 2 5 5 5

2xe 1414.214 1414.214 1414.214 2000 2000 2000 3162.278 3162.278 3162.278

L 353.5534 707.1068 1060.66 500 1000 1500 790.5694 1581.139 2371.708

rev 797.8846 797.8846 797.8846 797.8846 797.8846 797.8846 797.8846 797.8846 797.8846

reh 881.8544 958.496 1029.447 914.3801 1017.626 1111.32 975.7107 1125.79 1258.092

a 890.7576 991.634 1099.853 931.6248 1080.777 1244.147 1016.532 1271.952 1560.47

rw' 88.38835 176.7767 265.165 125 250 375 197.6424 395.2848 592.927

s -5.40808 -6.10123 -6.50669 -5.75465 -6.4478 -6.85327 -6.2128 -6.90595 -7.31141

LD 0.931695 1.86339 2.795085 1.317616 2.635231 3.952847 2.083333 4.166667 6.249999

L/2xe 0.25 0.5 0.75 0.25 0.5 0.75 0.25 0.5 0.75

SCA 4 3.2 2.65 3.7 2.7 2.5 3.8 2.75 2.4

S+SCA -1.40808 -2.90123 -3.85669 -2.05465 -3.7478 -4.35327 -2.4128 -4.15595 -4.91141

A=2000000ft2

68

Table B.4.2: Total horizontal well skin factor for A=4000000ft2

h, kV, kH:

Table B.5: AH/AV values for different h values at kv/kh = 1/10

xe/ye 1 1 1 2 2 2 5 5 5

2xe 2000 2000 2000 2828.427 2828.427 2828.427 4472.136 4472.136 4472.136

L 500 1000 1500 707.1068 1414.214 2121.32 1118.034 2236.068 3354.102

rev 797.8846 797.8846 797.8846 797.8846 797.8846 797.8846 797.8846 797.8846 797.8846

reh 914.3801 1017.626 1111.32 958.496 1095.815 1217.745 1040.505 1236.399 1405.245

a 931.6248 1080.777 1244.147 991.634 1215.137 1465.778 1118.087 1508.812 1957.169

rw' 125 250 375 176.7767 353.5535 530.33 279.5085 559.017 838.5255

s -5.75465 -6.4478 -6.85327 -6.10123 -6.79438 -7.19984 -6.55937 -7.25252 -7.65799

LD 1.317616 2.635231 3.952847 1.86339 3.726781 5.590169 2.946278 5.892557 8.838835

L/2xe 0.25 0.5 0.75 0.25 0.5 0.75 0.25 0.5 0.75

SCA 3.25 2.5 2.3 3.05 2.35 2.2 3.4 2.55 2.25

S+SCA -2.50465 -3.9478 -4.55327 -3.05123 -4.44438 -4.99984 -3.15937 -4.70252 -5.40799

A=4000000ft2

kv,kh h A AV AH AH/AV L 2XE L/2XE LD

1,10 60 2000000 77.11218 33.52636 0.434774 500 2000 0.25 1.317616

1,10 60 2000000 77.11218 18.57235 0.240849 1000 2000 0.5 2.635231

1,10 60 2000000 77.11218 11.03715 0.143131 1500 2000 0.75 3.952847


1,10 100 2000000 46.2914 25.6614 0.554345 500 2000 0.25 0.790569

1,10 100 2000000 46.2914 14.49078 0.313034 1000 2000 0.5 1.581139

1,10 100 2000000 46.2914 9.024089 0.194941 1500 2000 0.75 2.371708


1,10 120 2000000 38.58969 23.49269 0.608782 500 2000 0.25 0.658808

1,10 120 2000000 38.58969 13.43441 0.348135 1000 2000 0.5 1.317616

1,10 120 2000000 38.58969 8.511164 0.220555 1500 2000 0.75 1.976424

69




2,20 60 2000000 38.67345 17.09656 0.442075 500 2000 0.25 1.317616

2,20 60 2000000 38.67345 9.611054 0.248518 1000 2000 0.5 2.635231

2,20 60 2000000 38.67345 5.780181 0.149461 1500 2000 0.75 3.952847


2,20 100 2000000 23.22789 13.06058 0.56228 500 2000 0.25 0.790569

2,20 100 2000000 23.22789 7.53128 0.324234 1000 2000 0.5 1.581139

2,20 100 2000000 23.22789 4.781046 0.205832 1500 2000 0.75 2.371708


2,20 120 2000000 19.3709 11.93893 0.616333 500 2000 0.25 0.658808

2,20 120 2000000 19.3709 6.97219 0.359931 1000 2000 0.5 1.317616

2,20 120 2000000 19.3709 4.507652 0.232702 1500 2000 0.75 1.976424


2,10 60 2000000 77.11218 29.77397 0.386112 500 2000 0.25 1.317616

2,10 60 2000000 77.11218 16.40022 0.21268 1000 2000 0.5 2.635231

2,10 60 2000000 77.11218 9.48982 0.123065 1500 2000 0.75 3.952847


2,10 100 2000000 46.2914 22.21986 0.48 500 2000 0.25 0.790569

2,10 100 2000000 46.2914 12.38444 0.267532 1000 2000 0.5 1.581139

2,10 100 2000000 46.2914 7.510256 0.162239 1500 2000 0.75 2.371708


2,10 120 2000000 38.58969 20.21456 0.523833 500 2000 0.25 0.658808

2,10 120 2000000 38.58969 11.35303 0.294199 1000 2000 0.5 1.317616

2,10 120 2000000 38.58969 7.001782 0.181442 1500 2000 0.75 1.976424

70

For LD for xe/ye = 1:

Table B.8: AH/AV values for different LD values for xe/ye = 1 (kv/kh = constant)

Table B.9: AH/AV values for different LD values for xe/ye = 1 (h = constant)

kv/kh h A AV AH AH/AV L 2XE L/2XE LD

0.1 55.90116 2000000 83.75037 45.72022 0.545911 353.55 1414.214 0.25 1

0.1 111.8023 2000000 41.90689 20.11248 0.479933 707.1 1414.214 0.5 1

0.1 167.7035 2000000 27.97355 12.44478 0.444877 1060.65 1414.214 0.75 1


0.1 27.95058 2000000 167.1624 66.50045 0.397819 353.55 1414.214 0.25 2

0.1 55.90116 2000000 83.75037 26.35369 0.31467 707.1 1414.214 0.5 2

0.1 83.85175 2000000 55.81836 14.98783 0.268511 1060.65 1414.214 0.75 2


0.1 18.63372 2000000 251.6255 88.48958 0.351672 353.55 1414.214 0.25 3

0.1 37.26744 2000000 125.4932 33.08806 0.263664 707.1 1414.214 0.5 3

0.1 55.90116 2000000 83.75037 17.97502 0.214626 1060.65 1414.214 0.75 3


0.115202 60 2000000 78.03051 42.89476 0.549718 353.55 1414.214 0.25 1

0.028801 60 2000000 78.03051 34.73364 0.445129 707.1 1414.214 0.5 1

0.0128 60 2000000 78.03051 30.08603 0.385567 1060.65 1414.214 0.75 1


0.460809 60 2000000 78.03051 32.62421 0.418096 353.55 1414.214 0.25 2

0.115202 60 2000000 78.03051 24.70465 0.316602 707.1 1414.214 0.5 2

0.051201 60 2000000 78.03051 20.22076 0.259139 1060.65 1414.214 0.75 2


1.03682 60 2000000 78.03051 29.10804 0.373034 353.55 1414.214 0.25 3

0.259205 60 2000000 78.03051 21.25794 0.272431 707.1 1414.214 0.5 3

0.115202 60 2000000 78.03051 16.84891 0.215927 1060.65 1414.214 0.75 3

71

For LD for xe/ye = 2:

Table B.10: AH/AV values for different LD values for xe/ye = 2 (kv/kh = constant)

Table B.11: AH/AV values for different LD values for xe/ye = 2 (h = constant)


0.1 79.05694 2000000 58.50502 28.8808 0.493647 500 2000 0.25 1

0.1 158.1139 2000000 29.31548 12.09994 0.412749 1000 2000 0.5 1

0.1 237.1708 2000000 19.59463 7.156631 0.365234 1500 2000 0.75 1


0.1 39.52847 2000000 117.1103 43.02205 0.367363 500 2000 0.25 2

0.1 79.05694 2000000 58.50502 16.12481 0.275614 1000 2000 0.5 2

0.1 118.5854 2000000 39.04414 8.541682 0.21877 1500 2000 0.75 2


0.1 26.35231 2000000 175.2055 56.8557 0.324509 500 2000 0.25 3

0.1 52.70463 2000000 87.78705 19.94745 0.227225 1000 2000 0.5 3

0.1 79.05694 2000000 58.50502 9.826241 0.167956 1500 2000 0.75 3


0.0576 60 2000000 77.11218 37.20946 0.482537 500 2000 0.25 1

0.0144 60 2000000 77.11218 28.69255 0.372089 1000 2000 0.5 1

0.0064 60 2000000 77.11218 23.46245 0.304264 1500 2000 0.75 1


0.2304 60 2000000 77.11218 29.1091 0.37749 500 2000 0.25 2

0.0576 60 2000000 77.11218 20.73447 0.268887 1000 2000 0.5 2

0.0256 60 2000000 77.11218 15.56279 0.20182 1500 2000 0.75 2


0.5184 60 2000000 77.11218 26.10432 0.338524 500 2000 0.25 3

0.1296 60 2000000 77.11218 17.69884 0.229521 1000 2000 0.5 3

0.0576 60 2000000 77.11218 12.57746 0.163106 1500 2000 0.75 3

72

NOMENCLATURE

s => equivalent negative skin factor due to either well stimulation or horizontal well,

dimensionless

sm => mechanical skin factor, dimensionless

sCA => shape-related skin factor, dimensionless

c’ => shape factor conversion constant, dimensionless

k => permeability, millidarcy

kh => horizontal permeability, millidarcy

kv => vertical permeability, millidarcy

h => reservoir height, feet

pR => average reservoir pressure, pounds per square inches

pwf => well flowing pressure, pounds per square inches

qg => gas flow rate, thousand standard cubic feet per day

T => reservoir temperature, Rankin

µ => gas viscosity at well flowing conditions, centipoise

z => gas compressibility factor evaluated at some average pressure between pR and pwf,

dimensionless

β’ => high velocity flow coefficient, per foot

g => gas gravity, dimensionless

rw => wellbore radius, feet

hp => perforated interval, feet

ka => permeability in near wellbore region, millidarcy

AC’ => 0.738 for a rectangular drainage area, dimensionless

LD => dimensionless well length

xe = > drainage radius in the x direction

ye => drainage radius in the y direction

DH => horizontal well turbulence factor

73

DV => vertical well turbulence factor

AH => deliverability constant A value for horizontal well

AV => deliverability constant A value for vertical well

74

REFERENCES

Chen, W. and Zhu, D.: “A Comprehensive Model of Multilateral Well Deliverability” SPE 64751,

2000.

Chase, R.W. and Steffy, C. R.: “Predicting Horizontal Gas Well Deliverability Using Dimensionless

IPR Curves” SPE 91101, 2004.

Wang, X. and Economides, M. J.: “Horizontal Well Deliverability with Turbulence Effects” SPE

121382, 2009.

"Natural gas." Encyclopædia Britannica. Encyclopædia Britannica Online. Encyclopædia

Britannica, 2011. Web. 26 Apr. 2011.

<http://www.britannica.com/EBchecked/topic/406163/natural-gas>.

Cho, H. and Shah, S.: “Prediction of Specific Productivity Index for Long Horizontal Wells”, SPE

67237, 2001.

Joshi, S.: “Horizontal Well Technology”, PennWell, 1991.

“ECLIPSE Reservoir Simulation” www.slb.com,

<http://www.slb.com/services/software/reseng.aspx>.

Chaudhri, A. U.: “Gas Well Testing Handbook”, Elsevier Science, 2003.

Fetkovich, M. J., and M. E. Vienot. “Shape Factors, CA, Expressed as a Skin, sCA,” J. Petroleum

Technol. (Feb. 1979) 211-16

Golan, M., and Whitson, C. H., Well Performance, International Human Resources Development

Corporation, Boston, 1986.

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