Simulation Of Hydraulic Fracturing

UNIVERSITY OF OKLAHOMA GRADUATE COLLEGE

FRACTURE FACE INTERFERENCE OF FINITE CONDUCTIVITY FRACTURED WELLS USING NUMERICAL SIMULATION

A THESIS SUBMITTED TO THE GRADUATE FACULTY in partial fulfillment of the requirements for the Degree of MASTER OF SCIENCE IN NATURAL GAS ENGINEERING AND MANAGEMENT

By SVJETLANA LALE Norman, Oklahoma 2008

FRACTURE FACE INTERFERENCE OF FINITE CONDUCTIVITY FRACTURED WELLS USING NUMERICAL SIMULATION

A THESIS APROVED FOR THE MEWBOURNE SCHOOL OF PETROLEUM AND GEOLOGICAL ENGINEERING

BY

_______________________________ Dr. Jeffrey G. Callard – Chair _______________________________ Dr. Djebbar Tiab _______________________________ Dr. Samuel Osisanya _______________________________ Dr. Dean S. Oliver

©Copyright by SVJETLANA LALE 2008 All Rights Reserved.

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Acknowledgments

I would like to express my sincere appreciation to my advisor, Dr. Callard G. Jeffrey for his guidance through my studies. He provided me with excellent environment and opportunity to work and think as petroleum engineer. He impressed me with his brilliant ideas and hard work in academic research. I learned from him about importance of application of new technologies to research and industry and also to recognize the problem, analyze, and solve it. Special thanks to him for giving me opportunity to participate in the Devon Project on fractured shale gas reservoir, which supported my thesis.

Many thanks to Dr. Tiab Djebbar for guiding me through well test analysis problems and providing me with good knowledge in that field.

Especial thanks to Dr. Dean Oliver and Dr. Osisanya Samuel for serving as my committee members and supporting my study.

I am grateful to the group of Dr. Dean Oliver assistants who helped me to find the right direction in Eclipse software usage and make this research study much easier.

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A lot of appreciation to the MPGE faculty and stuff Dr Faruk Civan, Robert A. Hubbard, Dr Chandra Rai, Sonya Grant, Shalli Young, Mona Troxell, Cynthia Willis, to make these two years happy and fruitful.

I give a huge appreciation to my parents Stoja and Milinko Lale and my family Jelenka and Muhamed Kuburic, Hana Kulosman, and friends for their support and unlimited love.

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Table of Contents

Acknowledgments …………………………………………………………….

iv

Table of Content …………….………………………………………………..

vi

List of Tables ………………………………………………………………… xvi List of Figures ………………………..……………………….……………… xvii Abstract ……………………………….……………………………………… xxi 1.

INTRODUCTION…………………………………………..……………. 1 1.1.

From Conventional to Unconventional Reservoirs ……………….

1

1.2.

Tight Gas Reservoirs ………………………..…………………….

2

1.3.

Shale Gas Reservoirs …………………..……………..…………...

4

1.4.

Hydraulic Fracturing Stimulation ……….………………………...

5

1.5.

Fractures Types ……………….……………………………….….

7

1.6.

Fracture Flow Regimes ………….…………………………….….

9

1.7.

Problem Statement ……………….………………..…………..….

10

1.8.

Thesis Organization ……………………………..…………..……

11

2. HISTORICAL BACKGROUND AND TYPE CURVES ……………….

13

2.1.

Historical Background ……………………………………………

13

2.2.

Agarwal Finite Conductivity Type Curves ………………...…….

17

2.3.

Bennett Finite Conductivity Type Curves ………………….……

19

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3. NUMERICAL MODELING – VERTICAL WELL …………..…….…

4.

23

3.1.

Steps For Type Curve Development ..……….…………….…… 23

3.2.

Numerical Model ……..………………………....…………..…. 23 3.2.1.

Reservoir Discretization Into The Blocks ………..

24

3.2.2.

Data Input of Fluid and Reservoir Properties ..…..

30

3.2.3.

Other Data Included in Numerical Model ……..…

30

3.2.4.

Time Steps ……………….….……..……………..

32

3.3.

Numerical Simulation ………………………………….…….… 33

3.4.

Verification of Numerical Model ..……………………….……. 36 3.4.1.

Constant Flow Rate Case ………..………………..

36

3.4.2.

Constant Pressure Case …………..….……………

40

NUMERICAL MODELING – POINT SOURCE (HORIZONTAL WELL) ………………………………………………..

44

4.1.

Numerical Modeling Methodology……………….…………..… 44

4.2.

Numerical Model ……..………………………….………….…. 44 4.2.1. Constant Flow Rate Case for Point Source (Horizontal Well) …………………………………………..

46

4.2.2. Constant Pressure Rate Case for Point Source (Horizontal Well) …………………………………………..

49

5. FRACTURE FACE INTERFERENCE FOR VERTICAL WELL ……..

52

5.1. Fracture Face Interference Definition ………………….…………..

52

5.2. Vertical Well Numerical Model ……………….….…………..…....

57

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5.3. Constant Flow Rate Case ………………………….…….………..….. 58 5.4. Constant Pressure Case ……….…………………………..………….

61

6. FRACTURE FACE INTERFERENCE FOR POINT SOURCE (HORIZONTAL WELL) …………………………………………………

65

6.1. Point Source (Horizontal Well) Numerical Model …………………

65

6.2. Constant Flow Rate Case ……………………….……………...…..

66

6.3. Constant Pressure Case ……………………………………….…....

68

6.4. McAlister Well Data ……………………………………………….

71

7. SENSITIVITY ANALYSIS OF RESEARCH RESULTS ….……………

74

7.1. Constant Flow Rate Case ……………………….……………....…...

74

7.2. Constant Pressure Case ……………………………………………...

76

7.3. Sensitivity Analysis of Change of Fracture Half-Length ……….......

77

7.4. Sensitivity Analysis of Change of Number of Grid Blocks in z Direction for Point Source ..……………………………………

79

8. SUMMARY AND RECOMMENDATIONS …………….………..…..

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8.1. Summary …………………………………………………………….

81

8.2. Recommendations for Future Work …………………………………

82

Reference …………………………………………….………….….………..

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Appendix A ..…………………………………………………………………

87

Appendix A1 – Data File For FCD=100, Constant Rate Case and Vertical Well 88 Appendix A2 – Data File For FCD=100, Constant Pressure Case and Vertical Well ……………………………………………….. Appendix A3 – Data File For FCD=100, Constant Rate Case and Point Source

91 94

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Appendix A4 – Data File For FCD=100, Constant Pressure Case and Point Source …………………..…………………………..

97

Appendix A5 – Data File For FCD=100, Constant Rate Case, xf/y=255 and Two Vertical Wells ……………………………………

100

Appendix A6 – Data File For FCD=100, Constant Pressure Case, xf/y=128 and Two Vertical Wells ……………………………………

103

Appendix A7 – Data File for FCD=100, Constant Rate Case, xf/y=255 and Point Source ……………………………………..…….

106

Appendix A8 – Data File for FCD=100, Constant Pressure Case, xf/y=255 and Point Source …………………………………….….….

109

Appendix A9 – Data File for FCD=100, Constant Rate Case, xf/y=255 and Fracture Half-Length 506[ft] …………………………

112

Appendix B …………………………………………………………………

115

Table 1 – FCD=1 – Results of Numerical Simulation and Dimensionless Time, Pressure and Rate ……………………..……. 116 Table 2 – FCD=5 – Results of Numerical Simulation and Dimensionless Time, Pressure and Rate ………………………..… 122 Table 3 – FCD=10 – Results of Numerical Simulation and Dimensionless Time, Pressure and Rate ……………………….…

128

Table 4 - FCD=25– Results of Numerical Simulation and Dimensionless Time, Pressure and Rate ………………………..…

134

Table 5 – FCD=100 – Results of Numerical Simulation and Dimensionless Time, Pressure and Rate ………………….………

140

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Table 6 – FCD=500 – Results of Numerical Simulation and Dimensionless Time, Pressure and Rate …………………………

147

Table 7 – FCD=100 – Results of Numerical Simulation and Dimensionless Time, Pressure and Rate for Point Source ……………………………....

152

Table 8 – FCD=100 – Results of Numerical Simulation and Dimensionless Time, and Pressure for xf/y=255, Constant Rate Case, Vertical Well …..

160

Table 9 – FCD=100 – Results of Numerical Simulation and Dimensionless Time, and Rate for xf/y=128, Constant Pressure Case, Vertical Well …..

162

Table 10 – Production Data and Dimensionless Time and Flow Rate for McAlister O.H. 16 …………………………..

164

Table 11 – FCD=100 – Results of Numerical Simulation and Dimensionless Time, Pressure and Rate for xf/y=255, Point Source ……………..……... 165 Table 12 – FCD=100 – Results of Numerical Simulation and Dimensionless Time, and Pressure for Constant Rate Case, xf2 =xf1/4 and xf/y= 255 ………………………………………… 168 Appendix C…………………………………………………………………… 169 Figure 1 – Finite conductivity type curve with deviations for fracture face interference for constant flow rate for and FCD=1 – Line source – Vertical well …………………………………………………….

170

Figure 2 – Finite conductivity type curve with deviations for fracture face interference for constant flow rate and FCD=5 – Line source – Vertical well ……………………………………………………… 170 Figure 3 – Finite conductivity type curve with deviations for fracture face

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interference for constant flow rate and FCD=10 – Line source – Vertical well ……………………………………………………..

171

Figure 4 – Finite conductivity type curve with deviations for fracture face interference for constant flow rate and FCD=25 – Line source – Vertical well ……………………………………………………...

171

Figure 5 – Finite conductivity type curve with deviations for fracture face interference for constant flow rate for and FCD=100 – Line source – Vertical well ……………………………………………………..

172

Figure 6 – Finite conductivity type curve with deviations for fracture face interference for constant flow rate and FCD=500 – Line source – Vertical well ………………………………………………………

172

Figure 7 – Finite conductivity type curve with deviations for fracture face interference for constant pressure case and FCD=1 – Line source – Vertical well ……………………………………………………… 173 Figure 7A – Dimensionless rate versus dimensionless time with deviations for fracture face interference for constant pressure case and FCD=1 – Line source – Vertical well ……………………………………..

173

Figure 8 – Finite conductivity type curve with deviations for fracture face interference for constant pressure case and FCD=5 – Line source – Vertical well ………………………………………………………. 174 Figure 8A – Dimensionless rate versus dimensionless time with deviations for fracture face interference for constant pressure case and FCD=5 – Line source – Vertical well ………………………………………. 174

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Figure 9 – Finite conductivity type curve with deviations for fracture face interference for constant pressure case and FCD=10 – Line source – Vertical well ……………………………………………………… 175 Figure 9A – Dimensionless rate versus dimensionless time with deviations for fracture face interference for constant pressure case and FCD=10 – Line source – Vertical well ……………………………………….. 175 Figure 10 – Finite conductivity type curve with deviations for fracture face interference for constant pressure case and FCD=25 – Line source – Vertical well ……………………………………………………… 176 Figure 10A – Dimensionless rate versus dimensionless time with deviations for fracture face interference for constant pressure case and FCD=25 – Line source – Vertical well ………………………………………. 176 Figure 11 – Finite conductivity type curve with deviations for fracture face interference for constant pressure case and FCD=100 – Line source – Vertical well …………………………………………………….. 177 Figure 11A – Dimensionless rate versus dimensionless time with deviations for fracture face interference for constant pressure case and FCD=100 – Line source – Vertical well ………………………………….….. 177 Figure 12 – Finite conductivity type curve with deviations for fracture face interference for constant pressure case and FCD=500 – Line source – Vertical well …………………………………………………….. 178 Figure 12A – Dimensionless rate versus dimensionless time with deviations for fracture face interference for constant pressure case and FCD=500 –

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Line source – Vertical well ………………………………………. 178 Figure 13 – Finite conductivity type curve with deviations for fracture face interference for constant rate case and FCD=1 – Point source – Horizontal well ………………………………………………….. 179 Figure 14 – Finite conductivity type curve with deviations for fracture face interference for constant rate case and FCD=5 – Point source – Horizontal well …………………………………………………..

179

Figure 15 – Finite conductivity type curve with deviations for fracture face interference for constant rate case and FCD=10 – Point source – Horizontal well …………………………………… 180 Figure 16 – Finite conductivity type curve with deviations for fracture face interference for constant rate case and FCD=25 – Point source – Horizontal well …………………………………..

180

Figure 17 – Finite conductivity type curve with deviations for fracture face interference for constant rate case and FCD=100 – Point source – Horizontal well ………………………………….. 181 Figure 18 – Finite conductivity type curve with deviations for fracture face interference for constant rate case and FCD=500 – Point source – Horizontal well ………………………………….. 181 Figure 19 – Finite conductivity type curve with deviations for fracture face interference for constant pressure case and FCD=1 – Point source – Horizontal well ………………………………….. 182 Figure 19A – Dimensionless rate versus dimensionless time with deviations

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for fracture face interference for constant pressure case and FCD=1 – Point source – Horizontal well ………………………………….. 182 Figure 20 – Finite conductivity type curve with deviations for fracture face interference for constant pressure case and FCD=5 – Point source – Horizontal well ………………………………….. 183 Figure 20A – Dimensionless rate versus dimensionless time with deviations for fracture face interference for constant pressure case and FCD=5 – Point source – Horizontal well ………………………………….. 183 Figure 21 – Finite conductivity type curve with deviations for fracture face interference for constant pressure case and FCD=10 – Point source – Horizontal well ………………………………….. 184 Figure 21A – Dimensionless rate versus dimensionless time with deviations for fracture face interference for constant pressure case and FCD=10 – Point source – Horizontal well ……………………….. 184 Figure 22 – Finite conductivity type curve with deviations for fracture face interference for constant pressure case and FCD=25 – Point source – Horizontal well ………………………………….. 185 Figure 22A – Dimensionless rate versus dimensionless time with deviations for fracture face interference for constant pressure case and FCD=25 – Point source – Horizontal well ……………………….. 185 Figure 23 – Finite conductivity type curve with deviations for fracture face interference for constant pressure case and FCD=100 – Point source – Horizontal well ………………………………….. 186

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Figure 23A – Dimensionless rate versus dimensionless time with deviations for fracture face interference for constant pressure case and FCD=100 – Point source – Horizontal well …………………………186 Figure 24 – Finite conductivity type curve with deviations for fracture face interference for constant pressure case and FCD=500 – Point source – Horizontal well ………………………………….. 187 Figure 24A – Dimensionless rate versus dimensionless time with deviations for fracture face interference for constant pressure case and FCD=500 – Point source – Horizontal well …………………….

187

Appendix D …………………………………………………………………... 188 Nomenclature …………………………………………………………………. 189

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List of Tables

Table 3.1. – Bennett (1985) empirical guidelines for design of x and y grids …………………………………………………..…...

24

Table 3.2. – Reservoir, fracture and fluid PVT properties for constant pressure case ……………………………………………………

32

Table 3.3. – Fracture real and equivalent permeability ……………………...

37

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List of Figures

Figure 1.1. – Capillary pressure and relative permeability in conventional and unconventional reservoir (Shanley, 2004)………………..….

3

Figure 1.2. – Process specification in hydraulically fractured wells in tight gas reservoir (Friedel, 2004)……………………………….…….. Figure 1.3. - Fracture flow regimes (Cinco-Ley, 1981) .…………………..….

6 9

Figure 2.1 – Agarwal (1979) constant rate finite conductivity type curve …… 18 Figure 2.2. – Agarwal (1979) constant pressure finite conductivity type cure .

19

Figure 2.3. – Bennett (1985) constant rate finite conductivity type curve …..

20

Figure 2.4. – Bennett (1985) constant pressure finite conductivity type curve

21

Figure 3.1. – Quarter of the reservoir, grid block distribution ……………....

25

Figure 3.2. – Grid block distribution in numerical model ……………………

26

Figure 3.3. – Well and fracture location in square reservoir (Nashawi, 2007) .

27

Figure 3.4. - Reservoir with grid blocks – imported from Eclipse …………..

28

Figure 3.5. - Part of reservoir grid with well and fracture …………….…….

29

Figure 3.6. - Model simulation results (symbols) with Bennett (1985) finite conductivity type curve for constant rate case (lines) ………

40

Figure 3.7.- Dimensionless flow rate qD versus dimensionless time tDxf for constant pressure case - line source (vertical well) ……………………. Figure 3.8. – Model simulation results (symbols) with Bennett (1985) finite

42

xviii

conductivity type curve for constant pressure case (lines)……

43

Figure 4.1. – Fracture position in point source (horizontal well) …………..

45

Figure 4.2. – Bennett (1985) finite conductivity type curve for constant rate case and numerical model results for point source (horizontal well) .

48

Figure 4.3. - Dimensionless flow rate versus dimensionless time in function of fracture half length for constant pressure case and point source (horizontal well) ………………………………………

50

Figure 4.4. – Bennett (1985) finite conductivity type curve for constant pressure case and point source (horizontal well) ……………..

50

Figure 5.1. – Part of the reservoir with two wells and two fractures ……….

52

Figure 5.2 - Depletion in the reservoir after 260 days for case xf/y=8 ………………………………………………….

53

Figure 5.3. – Depletion in the reservoir after 449 days for case of xf/y=8 ……………………………………………...

54

Figure 5.4. – Depletion in the reservoir after 516 days for case of xf/y=8 ………………………………………………

55

Figure 5.5. - Depletion in the reservoir after 1580 days for case of xf/y=8 ……………………………………………..

55

Figure 5.6. - Depletion in the reservoir at the end of the reservoir life …………………………………………….

56

Figure 5.7. – Numerical model with two wells and two fractures ………….

58

Figure 5.8. – Examples of different xf/y ratios ……………………………..

59

xix

Figure 5.9. – Constant rate case - Finite conductivity type curve for family of finite conductivity fractures with deviations for fracture face interference for FCD=100 ……………………………………..

60

Figure 5.10. – Constant pressure case - Finite conductivity type curve for family of finite conductivity fractures with deviations for fracture face interference for FCD=100 (reciprocal rate) ……………….

63

Figure 5.11. – Constant pressure case – Finite conductivity type curve for family of finite conductivity fractures with deviations for fracture face interference for FCD=100 (rate) …………………………..

63

Figure 6.1. – Point sources (horizontal well) with two vertical fractures …...

66

Figure 6.2. – Finite conductivity type curve with deviations for fracture face interference for constant flow rate and FCD=100 – point source (horizontal well) ………………………………………………… 67 Figure 6.3. – Finite conductivity type curve with deviations for fracture face interference for constant pressure and FCD=100 – point source (horizontal well) ……………………………………….

69

Figure 6.4. – Dimensionless rate versus dimensionless time with deviations for fracture face interference for constant pressure case and FCD=100 – Point source (horizontal well) ……………………. Figure 6.5. – McAlister O.H. 16 monthly gas production data …………….

70 71

Figure 6.6. – Finite conductivity type curve with deviations for fracture face interference and McAlister O. H. 16 well data …………..

73

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Figure 7.1. – Sensitivity analysis of change of initial pressure for FCD=5 …………………………………………………….….. 75 Figure 7.2. – Sensitivity analysis of initial pressure change for FCD=100 ……………………………………………………... 75 Figure 7.3. – Sensitivity analysis of the change of productivity index for FCD=5 ………………………………………………… 76 Figure 7.4. – Sensitivity analysis of the productivity index change for FCD=100 ……………………………………………. 77 Figure 7.5. – Sensitivity analysis for different fractures half-lengths ……………………………………………………... 79 Figure 7.6. – Sensitivity analysis for different number of the grid blocks in z direction for point source, constant rate case and FCD=1 ……..

80

xxi

Abstract

Supply and demand of natural gas has allowed economic exploitation from unconventional reservoirs. Tight-gas and shale gas reservoirs depend on hydraulic fracturing technology to achieve economical gas production. This includes multi stage fracture stimulation treatments in horizontal wells. In this work, performance prediction using finite fracture conductivity models for vertical wells has been extended to model the effects of horizontal well penetration into the stimulated finite conductivity fractures as well as the interference effects of multiple fractures created by multistage fracture stimulation treatments.

Investigation in this direction was performed using numerical simulation. A study was conducted using ECLIPSE, version 2007.1, numerical simulator to model reservoir and well performance for a single phase flow system. The start point was to create a numerical model with outcomes that will match previous results presented by Bennett (1985) for a vertical fracture intersected by a vertical well. After model validation, the investigation of a horizontal well penetration and fracture face interference was performed utilizing the validated model and incorporating geometry for a horizontal well penetration into a finite conductivity vertical fracture and the inclusion of a second vertical fracture.

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Fracture face interference for six cases was generated utilizing a dimensionless parameter of fracture half-length to distance between two fractures.

A case history for a tight gas reservoir demonstrating the combined effects of a horizontal well completed with multi stage stimulation is included.

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1. Introduction

1.1.

From Conventional to Unconventional Reservoirs

Conventional reservoirs produce economic volumes of gas and oil at economic flow rates without large stimulation treatment or any other special recovery process. These reservoirs have high to medium permeabilities with vertical wells, and perforated pay interval.

Unconventional reservoirs do not produce enough oil and gas to have economic production flow rate without massive hydraulic stimulation treatments or special recovery processes. Unconventional reservoirs include tight gas, coal-bed methane, and shale.

Development of the unconventional reservoirs is based on higher prices and higher risks than development of the conventional reservoirs. For a long time they have not been very popular among engineers because it was very difficult to evaluate them and the right recovery techniques had to be successfully chosen and carefully applied in order to avoid production problems. New technologies made this kind of reservoirs very perspective in the future. Today, daily gas production from tight and unconventional

2

reservoirs in USA is more than 25% of total gas production- Naik (2004), and with constant increase of gas price, the future of these reservoirs is secure.

1.2.

Tight Gas Reservoirs

The first tight gas production was developed in the Western United States. Tight gas reservoirs can be found in any geological and tectonic setting. They may and may not contain natural

fractures, but cannot be produced economically without hydraulic

fracturing. Tight gas reservoirs are often defined as a gas bearing sandstone or carbonate matrix with in-situ permeability to gas less than 0.1 millidarcies. Most of tight gas reservoirs permeabilities are the function of the pressures. The pores are irregularly distributed through the reservoir and they are poorly connected by very narrow capillaries resulting in very low permeability. Gas flows through these rocks at low rates and it is not generated in the reservoir beds. Source beds sometimes commingle with reservoir. Figure 1.1. presents comparison of traditional reservoir with low-permeability reservoir. In a traditional reservoir, there is relative permeability in excess of 2% to one or both fluid phases across a wide range of water saturation. Critical water saturation and irreducible water saturation often occur at similar values of water saturation in the traditional reservoirs. Under these condition the absence of widespread water production commonly implies that a reservoir system is at, or near, irreducible water

3

saturation. On the other hand, in tight gas reservoir irreducible water saturation and critical water saturation can be dramatically different.

Figure 1.1. – Capillary pressure and relative permeability in conventional and unconventional reservoir (Shanley, 2004)

In traditional reservoir, there is wide range of water saturations at which both water and gas can flow. Situation is opposite in tight gas reservoir. There is a broad range of water saturation in which neither gas nor water can flow in tight gas reservoir. In some

4

extreme cases, there is virtually no mobile water phase even at very high water saturations.

1.3.

Shale Gas Reservoirs

Another unconventional source of natural gas is shale gas. Because of its matrix low permeability, and higher capillary pressure, commercial production may be achieved only with fractures to provide permeability. Shale gas has been produced from shales with natural fractures for a long time, but lately due to the hydraulic fracturing stimulation improvement its production has been increased. Very often shale gas has been produced using horizontal wells technology. Some of the gas is held in natural fractures, some in the pore spaces, and some is adsorbed onto the organic material. The gas in the fractures is produced immediately, and gas adsorbed onto organic material is released as the formation pressure declines. Gas is usually generated in place from shale with high total organic carbon content. The Barnett Shale in Forth Worth Basin is the most active shale gas play in USA. Due to the high gas prices and use of horizontal well technology to increase production, drilling expanded significantly in past few years. The Barnett Shale wells are deep – about 8,000 feet. Most economic wells are between 300 and 500 feet of thickness (Daniels, 2007).

5

1.4. Hydraulic Fracturing Stimulation

One of the stimulation methods for increasing well productivity and developing commercial wells in low-permeability or tight-gas formations is hydraulic fracturing. The purpose of this stimulation technique is to expose a large surface area of the lowpermeability formation to flow into the well bore. To increase reservoir area in direct communication with the well bore, it is necessary to create a highly conductive path some distance away from the well bore. Using this method, a greater volume of fluid can be produced into the well bore per unit of time and result is an increased production rate without drilling another well.

The hydraulic fracturing stimulation has applied to low-permeability gas formation with in-situ permeability of 0.1 md or less, and tight-gas formation with pores irregularly distributed throughout reservoir which have poor connection by very narrow capillaries. Since the permeability in these formation is low, gas flows through these rocks at low rates. The goal of hydraulic fracturing is to increase gas production flow rates.

Hydraulic fracturing treatment is pumping a suitable fluid, usually water, into the formation at a rate faster than fluid can leak off into the rock. When the fluid pressure or stress at the sand-face is higher than earth compressive stress, fracturing of formation matrix has initiated along a plane perpendicular to the minimum compressive stress. Fluid has been injected until the fracture is open wide enough to accept proppant. Then

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proppant has been added to the fracturing fluid and import to the fracture to keep it open. The hydraulic fracturing treatment is applied on a massive scale, which involves the use of at least 50,000 to 500,000 gal of treating fluid and 100,000 to 1 million pounds of proppant. When sufficient proppant has been injected, the pumps are shut down, pressure in the fracture drops, and earth compressive stress closes the fracture. Pressure in the fracture must exceed pore pressure by an amount equal to the minimum effective rock matrix stress to keep the fracture open after hydraulic fracturing. This pressure is fracture closer pressure.

Figure 1.2. – Process Specification in Hydraulically Fractured Wells in Tight Gas Reservoir (Friedel, 2004)

Figure 1.2. presents major physical processes at fractured wells which may imply -

simultaneous flow of three phases

-

hydraulic and mechanical damage close to the fracture

-

filtercake increase or decrease

7

-

damaging proppant pack by gel residue

-

viscous fingering through proppant pack

-

unbroken fracturing fluids within the proppant pack.

Inertial non-Darcy flow, geomechanical effects like stress dependency of reservoir permeability and fracture closure have great influence on the well production.

The fracture orientation depends on the stress distribution in the formation. If the least principal stress in the formation is horizontal, then a vertical fracture is obtained, otherwise the result will be horizontal fracture. Vertical fractures are more common for depths higher than 2,000 ft.

1.5. Fracture Types

Three fracture types occur in hydraulically fractured wells: -

uniform-flux fracture

-

infinite-conductivity fracture

-

finite conductivity fracture

Uniform-flux fractures occur when fluid enters the fracture at a uniform flow rate per unit area of fracture face enabling pressure drop in the fracture.

8

Fractures with infinite permeability and conductivity have little or no pressure drop along its axis. These fractures are referred as infinite-conductivity fractures. They exist in highly propped tight-gas formations. Usually, fractures with dimensionless conductivity FCD > 500 are treated as infinite-conductivity fractures.

Finite-conductivity fractures are the fractures with significant pressure drop along its axis. This model is very common case, unless formation permeability is extremely low – in microdarcy range.

Cinco-Ley (1978) showed that for practical values of dimensionless time the pressure behavior depends on time, and dimensionless fracture conductivity, FCD: FCD =

k f wf kx f

…………………………………………………………… (1)

where kf [md] – fracture permeability wf [ft] – fracture width k [md] – formation permeability xf [ft] – fracture half-length

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1.6. Fracture Flow Regimes

Figure 1.3. - Fracture flow regimes (Cinco-Ley, 1981)

Four flow regimes occur in the fracture and formation around a hydraulically fractured well, Figure 1.3: -

fracture linear flow (a)

-

bilinear flow (b)

-

formation linear flow (c.)

-

pseudo-radial flow (d)

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Fracture linear flow is very short. During this flow period, most of the fluid entering the well bore comes from fluid expansion in the fracture. The flow regime is linear. It may be masked by well bore-storage effects.

Bilinear flow evolves only in finite-conductivity fractures as fluid in the surrounding formation flows linearly into the fracture and before fracture-tip effects begin to influence well behavior. Most of the fluid entering the well bore during this flow period comes from the formation.

Duration of formation linear flow increases with higher fracture conductivities.

Pseudo-radial flow occurs with fractures of all conductivities. After a sufficiently long flow period, the fracture appears to the reservoir as an expanded well bore. If the fracture length is large relative to the drainage area, then boundary effects change or mask the pseudo-radial flow regime.

1.7. Problem Statement

Increasing gas price, declining production in conventional reservoirs and increasing demand for a gas focused attention of the industry onto exploration and development of unconventional gas reservoirs. Production from tight gas reservoir still presents main

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challenge in petroleum industry because there is available only limited knowledge about causes and solutions of the problems concerning gas production from tight gas reservoirs. Generally, very interested topic for research are finite conductivity fractures. Not so many studies have been published about this topic. The main goal of this study is to answer on the question regarding influence of the interference of the finite conductivity fractures on the pressure data for constant rate production or flow rate data for constant pressure production mode. The aim is to extend a current solution from finite conductivity vertical fractured wells to horizontal wells and especially to horizontal wells with multi stage stimulation treatments with a use of numerical solution techniques. It is necessary to develop type curves for constant rate and pressure production mode, with dimensionless fractures conductivities, and different length to distance ratios as parameters. Final goal is to provide sensitivity analysis of the achieved results – developed type curves for different reservoir and well performances.

1.8. Thesis Organization

Chapter 2 contains a complete review of the published studies concerning finite conductivity fractures. The Agarwal finite conductivity type curves for constant pressure and constant rate were analyzed and compared with Bennett’s finite conductivity type curves for constant pressure and constant rate.

12

Chapter 3 presents the development of the numerical model for line source – vertical well, numerical simulation process and verification of the numerical model which was provided by type curve matching with Bennett solutions. Chapter 4 presents numerical model for point source – horizontal well, numerical simulation results and verification of new developed type curves. Chapter 5 describes study of fracture face interference with vertical wells, containing numerical model for simulation, and development of new type curves with length to distance ratios as parameters. Similar to the chapter 5, chapter 6 describes study of fracture face interference, numerical model for simulation and developed new type curves with length to distance ratio as parameters, but for the point source – horizontal wells. At the end real well data were implemented to provide numerical model verification. Chapter 7 analyses sensitivity of developed type curves on pressure, productivity index, fracture half-length, and number of grid blocks change. Chapter 8 presents summary of the complete investigation with recommendations for the future work.

13

2. Historical Background And Type Curves

2.1. Historical Background

The concept of finite flow-capacity fractures was developed by Cinco-Ley (1978). They used semi analytical approach to point out the need to consider fracture to be finite if the dimensionless fracture conductivity, FCD is less than 300 which is the case of very long fractures and low capacity fractures. This is the first step in the technology of evaluation of massive hydraulic fracturing. Limitation of this technique was its application to systems with small, constant compressibility or system with a constant fluid viscosity-compressibility product. Cinco – Ley type curve can be used for postfracture analysis of data from a constant-rate flow test or a pressure-buildup test and it represents the modeling of vertical hydraulic fracture in an infinite-acting reservoir under the following assumptions: •

the fracture has finite conductivity that is uniform throughout the fracture

•

the fracture has two equal-length wings

•

well bore-storage effects are ignored

14

Agarwal, R.G (1979) discussed the limitations of the conventional analysis methods and alternative techniques for determining fracture half-length and fracture flow capacity on MHF wells with finite flow-capacity fractures. Low-permeability gas wells normally produce at a constant well pressure, but if the rate declines smoothly with bottom hole flowing pressure, then constant rate type curve should be used. Agarwal developed a set of constant well rate and constant well pressure type curves for MHF wells using numerical simulation and discussed the type curve matching technique and actual application of new type curves. Agarwal type curve is useful for analyzing flow tests or long-term production data in wells produced at essentially constant bottom hole pressure, or for wells producing at constant flow rates.

Cinco-Ley, H., Samaniego, V.F. (1981) analyzed finite conductivity fractures and defined bilinear flow regime which is the result of two linear flow regimes. One flow regime is linear flow within the fracture and another is linear flow into the fracture from the matrix. The bilinear flow regime is characterized by 0.25 slope on a log-log plot of pressure drop versus time for the early time pressure data. After bilinear flow, the linear flow occurs with 0.5 slope. They analytically defined that bilinear flow exists when most of the fluid entering the well bore comes from the formation and when fracture tip effects have not yet affected the well behavior.

Bennett, C.O. (1985) developed finite conductivity type-curves for constant pressure and constant rate modes of production using analytical solution for multi layered reservoirs. They identified parts of the type-curves with bilinear and linear flow periods,

15

and also part with the straight line. Their study can be applied to cases where the fracture extends above or below the productive interval, and cases where the fracture conductivity is the function of depth.

Bennett, C.O. (1986) incorporated numerical and analytical solutions for performance of finite-conductivity, vertically fractured wells in single layer reservoirs. They concluded that the fracture height and fracture length effects on the well response can be significant for the homogeneous single layer reservoirs if the conductivity of the fracture is the function of the depth or if fracture height is higher than formation height, hf>h. For multi layer reservoirs, vertical gradients may be significant even if fracture height is equal to the formation height, hf=h.

Tiab, D. (1994), (1995) developed Tiab’s direct synthesis (TDS) technique. This method interprets log-log plots of pressure and pressure derivatives versus time for different ratios of xe/xf for a vertically fractured well inside a closed system without using type curve matching. At this time he has developed TDS method for uniform flux fracture and infinite conductivity fracture. However, the finite conductivity fractures have been observed later by Tiab, D. (1995). A log-log plot of pressure and pressure derivative versus time for the well intersected by a finite conductivity hydraulic fracture in a closed system, may have several straight lines which correspondence to the bilinear, linear, infinite-acting radial flow and pseudo-steady state flow. The slopes and intersection points can be used to calculate permeability, skin factor in the absence of the infinite-acting radial flow line, well bore storage coefficient, half-fracture length in

16

the absence of the linear flow regime straight line of slope 0.5, fracture conductivity in the absence of the bi-linear flow line of slope 0.25 and drainage area.

Nashawi, I.S, Qasem, F.H, Gharbi R.(2003), performed comprehensive study of applying the constant-pressure liquid solution to transient rate-decline analysis of gas wells. Pseudopressure, non-Darcy flow effects, and formation damage have been incorporated in the liquid solution theory to simulate actual real gas flow around the well bore. The investigation shows that for constant-pressure gas production, the conventional semilog plot of the reciprocal dimensionless rate versus the dimensionless time used for liquid solution must be modified to account for high velocity flow effects, especially when reservoir permeability is relatively high (>1md) and the well test is affected by non-Darcy flow and formation damage.

Nashawi, I.S., Malallah, A.H. (2007), investigated pressure buildup and draw down tests influenced with well bore storage effect. These effects dominate at the early time enabling good formation characterization of the area surrounding the well bore. Constant bottom hole pressure tests are immune on these adverse effects. Nashawi and Malallah developed technique of analysis of finite conductivity fractured wells producing at constant bottom hole pressure from closed reservoirs without type curve matching. They used log-log plots of the reciprocal rate and derivative of reciprocal rate versus time for analysis of all the dominant flows: bilinear, pseudo-radial, and boundary-dominated flow and calculation of fracture conductivity, formation permeability, skin factor, well drainage area and reservoir shape factor.

17

2.2. Agarwal Finite Conductivity Type Curves

Agarwal has developed finite conductivity type curves for constant pressure and constant rate production modes for low permeability reservoirs with in-situ permeability less than 0.1[md]. These type curves were defined using numerical simulation. Assumptions that has been used: -

constant compressibility-viscosity product in the system

-

uniform fracture flow capacity

-

ignored well bore storage effect

-

ignored damage

-

no well bore cleanup effects

-

neglected confining pressure and turbulence effects

-

insignificant drainage boundary effects for the duration of the test.

Agarwal constant rate finite conductivity type curve, Figure 2.1., is log-log plot of dimensionless pressure, pD, versus dimensionless time in function of the fracture halflength, tDxf, with dimensionless fracture conductivity, FCD, as a parameter. Dimensionless fracture conductivity is in the range from 0.1 to 500, where higher values correspond to the higher fracture flow capacities. Higher values of the dimensionless fracture conductivities may be the consequence of the lower formation permeability or short fracture length.

18

Figure 2.1– Agarwal (1979) constant rate finite conductivity type curve

Dotted line on the Figure 2.1. presents infinite fracture flow capacity. At early times lower values of tDxf, there are deviations among the dimensionless fracture conductivities but they are diminished at later time. Dimensionless time ranges from 105

to the 1. For the time less than 10-5, porosity and compressibility in the fracture have

great influence on the type curve. Besides the pressure draw down data, this type curve may be applied to analysis of the pressure buildup data if producing time, tP, before shut in is significantly large compared with the shut-in time, ∆t. Otherwise the effect of small producing time is the lower fracture flow capacity.

Agarwal constant pressure finite conductivity type curves, Figure 2.2., are used when well produces at a constant well pressure. Instead of the dimensionless pressure, the reciprocal dimensionless flow rate, 1/qD, was plotted on log-log paper versus time in

19

function of fracture half-length, tDxf, and with dimensionless fracture conductivity, FCD, as a parameter.

Figure 2.2. – Agarwal (1979) constant pressure finite conductivity type curve

These type curves have the same tDxf and FCD ranges with similar shape to the constant rate type curves.

2.3. Bennett Finite Conductivity Type Curves

Bennett finite conductivity type curves have been developed for the multi layer and single layer reservoir, using analytical approach. Assumptions that have been used: -

reservoir boundaries are impermeable

20

-

reservoir is uniform and homogeneous

-

fluid is slightly compressible with constant viscosity

-

gravitational effects are negligible

-

flow in the reservoir parallel to the fracture face is negligible

-

reservoir is infinite in the direction perpendicular to the fracture face

-

fracture length is finite

Bennett constant rate finite conductivity type curves, Figure 2.3., are similar to the Agarwal ones. The dimensionless time axis scales from 10-6 to 1, while the Agarwal dimensionless time scale starts at 10-5. Bennett defined time periods corresponding to the different flow regimes and applied them to the finite flow capacity type curves. The part of the curves on the left side of the triangles defines bilinear flow period, that can be presented by the straight line on the Cartesian plot pwD versus tDxf0.25.

Figure 2.3. – Bennett (1985) constant rate finite conductivity type curve

21

The time period between x letters defines the time for which the straight line will exist on the Cartesian plot ∆p versus tDxf0.5. Linear flow period will occur between circles on the curves and this is the time period for which straight lines are defined on log-log plot pwD versus tDxf with slope of 0.5. The square data points define time period with asymptotic expansion which is correspondent to the straight line on the Cartesian plot ∆p versus t0.3. Finally, the dimensionless fracture conductivities are in smaller range in Bennett type curve comparing with Agarwals.

Bennett constant pressure finite conductivity type curves, Figure 2.4., are the similar to the Bennett constant rate finite flow type curves.

Figure 2.4. – Bennett (1985) constant pressure finite conductivity type curve

The points marked with triangles, circles, squares or x letter denote the same definition of the flow regimes and corresponding plots.

22

Difference between Bennett’s finite conductivity type curves and Agarwal finite conductivity type curves for constant pressure is in the time scale, presented number of dimensionless fracture conductivities, flow regimes definition, and the way that they have been developed.

Since the Bennett finite conductivity type curves have flow regimes data, these type curves have been used for further research.

23

3. Numerical Modeling – Vertical Well

3.1. Steps for Type-Curve Development

Three preparation steps were required to provide type-curve development. The first step is the preparation of numerical model for line source – vertical well and point source – horizontal well. The second step is data file development for numerical simulation using synthetic data of reservoir, reservoir geometry, and fluid properties. The third step is converting simulation results – flow rates and pressures to the dimensionless ones. The fourth step is plotting and comparing these results with already developed Bennett finite conductivity type curves.

3.2. Numerical Model

Developing of numerical simulation model was performed for synthetic reservoir and fluid data and it involves three consecutive steps: 1. Discretization of the reservoir into blocks 2. Assumption of synthetic data input of PVT and rock properties 3. Approximation of time integrals

24

3.2.1. Reservoir Discretization Into The Blocks

The single layer reservoir has been discretizated into 103,515 blocks with distribution x:y:z=335:309:1. This huge reservoir has been chosen to avoid boundary effects. Block dimensions are determined using Bennett’s (1985)., recommendations for design of x and y grids given in Table 3.1.

Table 3.1. – The Bennett (1985) empirical guidelines for design of x and y grids A. For All Grid Blocks ∆xi+1/2 ≤ ∆xi ≤ 2∆xi-1,

i = 2…. (Nx-1)

∆yj+1/2 ≤ ∆yj ≤ 2∆yj-1,

j = 2…. (Ny-1)

B. Near the Fracture (x/Lxf ≤ 1.5, y/Lxf ≤ 1) ∆x/Lxf ≤ 10-2

at the well for CfD ≥ 100

∆x/Lxf ≤ 10-3

at the well for CfD < 100

∆x/Lxf ≤ 1.5x10-2

at the fracture tip

max (∆x/Lxf) ≤ 0.15 b/Lxfj = 2∆y1/Lxf ≤ 2x10-3 ∆y1= ∆y2 =∆y3=∆y4 max (∆y/Lxf ) ≤ 0.2 C. Away From the Fracture (x/Lxf > 1.5, y/Lxf > 1) max (∆x/Lxe) ≤ 0.17 max (∆y/Lxe ) ≤ 0.17

25

According to this table, grid blocks dimensions of the model and their uneven distribution are determined.

Fracture’s blocks in x direction have different dimensions increasing to maximum value and than decreasing to the minimal dimension equal to the well grid block. This minimal dimension is the tip of the fracture and at that point, the distance between the well and fracture tip is the half-fracture length in x direction. Adjacent grid block dimensions increase until the maximum. All next grid blocks have the same dimension.

dimension in y direction

In y direction, the minimal dimension of the grid has the block with well.

Well

xf dimension in x direction

Figure 3.1. – Quarter of the reservoir, grid block distribution

26

The dimension of the adjacent grid blocks increases to the maximal value and then they have constant dimension.

Figures 3.1 and 3.2. show the uneven distribution of grid blocks in the reservoir. Since the reservoir is symmetric relative to the well and fracture position, the quarter of the reservoir has been observed.

dx, dy - Length of Block x or y Respectively [ft]

1000

100

10

Fracture Half-Length

dx

dy

1

Well

0

10

20 n- Number of Blocks

Figure 3.2. – Grid block distribution in numerical model

30

40

27

The blue circles (Figure 3.2.) present dimension of the blocks in x direction, while red crosses present the dimension of the blocks in y direction. In z direction all grid blocks have the same dimension and the fracture height is equal to the reservoir height.

Three dimensional aspect of the well and fracture position in the reservoir is presented in Figure 3.3. The vertical well is located in the center of the square reservoir and that grid block has the minimum x and y dimensions. Finite fracture is parallel with x axis and totally intersects the well symmetric to the y axis.

Figure 3.3. – Well and fracture location in square reservoir (Nashawi, 2007)

28

Both fracture half-lengths in x axis direction are equal, Figure 3.3. Total number of grid blocks in x and y direction as well the total reservoir dimension:

Total number of blocks Total reservoir Total number of blocks Total reservoir in x direction dimension in x direction in y direction dimension in y direction

nx

x [ft]

ny

y[ft]

335

100,646

309

100,978

Well is in central block with 2[ft] dimension, while fracture is in direction of x axis. The middle of the fracture starts in block 1 and fracture continues to the adjacent 20 blocks in both directions of x axis to the total fracture half-length of 2,043[ft].

Analyzed part of the reservoir

Figure 3.4. - Reservoir with grid blocks – imported from Eclipse

29

Figure 3.4. is imported from Eclipse shows huge reservoir with 103,515 grid blocks. Since the grid block dimensions are much lower than total dimension of the reservoir, gridding effect is displayed on the Figure 3.4. as blue color of the reservoir. To be able to analyze reservoir gridding, the part of the reservoir in the white square is going to be zoomed out.

Fracture length 2xf Well

Fracture

Figure 3.5. - Part of reservoir grid with well and fracture

Figure 3.5. presents the result of the increased central part of the reservoir. Grid blocks are defined with blue lines where blue thick lines are the effect of fine gridding. Fracture is extending in x direction with fracture half-length xf , defined by green line. Black circle in the center of the picture is the well. According to the scale below, initial reservoir pressure value is correspondent to red color and during numerical simulation pressure decrease is observed by color change from beginning red to final blue.

30

Since this picture presents Day 0, reservoir is presented by red color because the reservoir pressure has maximum value.

3.2.2. Data Input of Fluid and Reservoir Properties

Fluid properties that are needed to model single-phase fluid flow are those that appear in the flow equations. To simplify simulation a single-phase slightly compressible fluid with characteristics given in Table 3.2. has been chosen. Model for numerical Fracture length f simulation is low permeability reservoir 2x with permeability of 0.1[md]. Rock compressibility has been neglected for simplification purposes. Model does not account for well bore storage, skin, frictional losses in the well bore and capillary pressures.

3.2.3. Other Data Included in Numerical Model

Fracture width 0.5[in] was very low and unacceptable for simulation by simulator, because the well bore radius was 0.3[ft] and fracture width had to be higher than this dimension. The most convenient dimension was 2[ft], the dimension of the smallest grid block with well. Since the fracture porosity of 35% corresponds to the fracture width of 0.5[in], the equivalent fracture porosity was calculated using equation:

31

φe =

wφ f we

……………………………….……………………….……… (2)

where: w [ft] – fracture width we [ft] – equivalent fracture width φf [-] – fracture porosity, fraction φe [-] – equivalent fracture porosity, fraction

Fracture permeability is the function of the dimensionless fracture conductivity.

kf =

FCD kx f w

………………………………………………………….. (3)

where: FCD – dimensionless fracture conductivity k [md] – formation permeability xf [ft] – fracture half-length w [ft] – fracture width

Equivalent fracture permeability

k fe =

kfw we

………………………………………………………….…... (4)

where we [ft] – equivalent fracture width

32

Summary of all reservoir, fracture and fluid properties are listed in Table 3.2. Table 3.2. – Reservoir, fracture and fluid PVT properties for constant pressure case Reservoir Properties Initial pressure pi [psi] 5,000 Bottom hole flowing pressure BHFP [psi] 500 Formation porosity, fraction 0.2 φ Formation permeability k [md] 0.1 Formation height (reservoir thickness) h [ft] 100 Rock compressibility c [psi-1] 0 Skin s 0 Well bore radius rw [ft] 0.3 Fracture Properties Fracture half length xf [ft] 2043 Fracture width w [in] 0.5 Fracture porosity 0.35 φf Equivalent Fracture Properties Adjusted for Numerical Simulation Equivalent fracture width we [ft] 2 Equivalent fracture porosity 0.0073 φfe Fluid Properties Compressibility cf [psi-1] 3.00E-06 Viscosity 1 µ [cp] FVF B [RB/stb] 1

3.2.4. Time Steps

The start date of simulation is determined, by the default, to be January 1st, 1997. Initially, the first moment of simulation was chosen to be the first second when production started (1.15741x10-5 days) for both cases – constant flow rate and constant pressure case. But to cut the simulation time in the constant pressure case, the first moment of observation was chosen to be 1.03x10-4 days. For the verification purposes

33

of this model, it was necessary to compare simulation results with Bennett Type Curves which have logarithmic scale. Chosen time steps have geometrical progression with factor 2.

3.3. Numerical Simulation

A single-well and two-well simulation models in 3D reservoir were set up with the Black Oil simulator Eclipse-100 from Geoquest-Schlumberger. Reservoir grid is developed in Cartesian coordinates with block centered geometry. To develop physical model, the following facts are assumed: -

isothermal flow

-

no diffusion nor dispersion process presented

-

no chemical reactions presented

-

thermodinamical equilibrium

-

one phase system

The inflow equation used by Eclipse is defined by the volumetric production rate of each phase at stock tank conditions:

(

q p, j = Twj M p, j P j − Pw − H wj

) …………………….………………..(5)

where: qp,j – volumetric flow rate of phase p in connection j at stock tank conditions. The flow is positive from the formation into the well, and negative from the well into formation Twj – connection transmissibility factor Mp,j – phase mobility at the connection

34

Pj – nodal pressure in the grid block containing the connection Pw – bottom hole pressure of the well Hwj – well bore pressure head between the connection and the well’s bottom hole datum depth. Pw+Hwj is the pressure in the well at the connection j, called “connection pressure”

Connection transmissibility factor in the Cartesian grid:

Twj =

cθKh …………………………………………..……….……. (6)  ro  ln  + s  rw 

where: c – unit conversion factor (0.001127 in field units) θ – the segment angle connecting with the well (2π) for the well located in the center of

the grid block Kh – effective permeability times net thickness of the connection. ro – pressure equivalent radius of the grid block rw – well bore radius s - skin factor

Pressure equivalent radius of the grid block is distance from the well at which the local pressure is equal to the average nodal pressure of the block. Peaceman’s formula has been used in Cartesian grid for rectangular grid blocks in an anisotropic reservoir:

35

ro = 0.28

  Ky  D x2    K x  

1 K 2  + D 2y  x  Ky  

Ky  K  x

1 4  Kx  +  Ky  

1 12 2        

1 4   

……………………………... (7)

where: Dx, Dy – the x- and y- dimensions of the grid block Kx, Ky – x- and y- direction permeabilities

Two cases have been examined: 1. Constant pressure production mode 2. Constant flow rate production mode

In the first case the BHFP is assumed to be 500[psi] and bottom hole flowing rate has been determined as a result of simulation. Data for the case of the constant pressure are given in Table 3.2. All data for constant flow rate case are the same, except the pressure input. Instead of pressure, for constant flow rate case input will be flow rate of 100[stb/day] and pressure will be the result of the numerical simulation.

36

3.4. Verification of Numerical Model

To verify developed model, Bennett finite conductivity type curves for constant flowing rate and constant pressure were used.

3.4.1. Constant Flow Rate Case

For each dimensionless fracture conductivity, FCD, presented in Bennett type curves for constant flowing rate, fracture porosity is calculated using equation (2). Real fracture porosity has been assumed to be 35% and equivalent is calculated in function of the fracture width, and it is 0.73%

For calculating fracture permeabilities, input data that have been used are listed below:

Data for real fracture permeability calculation Formation permeability k [md] Fracture half length xf [ft] Fracture width w [in]

0.1 2043 0.5

Data for equivalent fracture permeability calculation Formation permeability k [md] 0.1 Fracture half length xf [ft] 2043 Equivalent fracture width we [ft] 2

Where equivalent fracture width must be higher than the well bore radius and in this case it was convenient to set it to 2 [ft] because that was dimension of the grid block with well.

37

Calculated real and equivalent fracture permeability using correlations (3) and (4) are given in Table 3.3.

Table 3.3. – Fracture real and equivalent permeability Dimensionless Fracture Conductivity FCD

Real Fracture Permeability

Equivalent Fracture Permeability

kf

kfe

1 5 10 25 100 500

4,903 24,516 49,032 122,580 490,320 2,451,600

102 511 1,022 2,554 10,215 51,075

Six data files were made for six different fracture dimensionless conductivities, FCD = 1, 5, 10, 25, 100, 500 with only difference in equivalent fracture permeability (keyword: EQUALS) which is the function of the FCD.

Data file for FCD=100 is given in Appendix A1. Well diameter is assumed to be 0.6 [ft] and skin is neglected.

Assumed flow rate was 100[stb/d] and it was control mode for constant flow rate case (keyword: WCONPROD).

In SUMMARY section of Data File, output data well BHP and well production rate were requested. Numerical simulation results for all six dimensionless fractures conductivities are presented in Tables 1 to 6 in Appendix B.

38

To be able to compare numerical simulation results to the Bennett type curves, it was necessary to transform time and pressure into dimensionless time in function of fracture half-length and dimensionless pressure. Correlation for dimensionless time in function of fracture half-length:

t Dxf =

0.0063288k

φct µx f 2

t ……………………………………..……..…..……. (8)

where:

k [md] – formation permeability

t [days] – production time

φ [-] – reservoir porosity, fraction ct [psi-1] – total system compressibility

µ [cp] – fluid viscosity

xf [ft] – fracture half-length

Correlation for dimensionless pressure:

pD =

(

kh pi − pwf 141.2qBµ

) ………………………………………….…………... (9)

39

where:

h [ft] – total reservoir thickness

q [stb/day] – surface rate

pi [psi] – initial pressure

pwf [psi] – well bore flowing pressure

B [RB/stb] – liquid formation volume factor FVF

For the reservoir, fracture, and fluid properties given in Table 3.3 dimensionless time and dimensionless pressure can be calculated using time and pressure multipliers from simulation output of time in days and well bore flowing pressure in psi. tM= 2.53E-04 [day-1] pM= 7.08E-04 [psi-1]

Results of numerical simulation of six dimensionless fractures conductivities for dimensionless pressure and dimensionless time are in Table 1 to Table 6 in Appendix C.

Figure 3.6. presents graphical solution of numerical simulation results for constant rate case. These results are colored data points with Bennett type curve results shown in background. Match with Bennett type curves for all six dimensionless fracture conductivities provides verification of the numerical model.

40

Dimensionless Pressure pD

10

1 FCD=1 0.1

FCD=5 FCD=10 FCD=25

0.01

FCD=100 FCD=500

0.001 1.E-06

1.E-05

1.E-04

1.E-03

1.E-02

1.E-01

1.E+00

Dimensionless Time tDxf Figure 3.6. – Model simulation results (symbols) with Bennett (1985) finite conductivity type curve for constant rate case (lines)

3.4.2. Constant Pressure Case

Only difference between these and Data Files for constant flow rate case is the control mode (in keyword: WCONPROD) which is BHP instead of the flow rate. Data File for FCD=100 is in Appendix A2.

Results of numerical simulation for six dimensionless fractures conductivities from 1 to 500 are given in Table 1 to Table 6 in Appendix B.

41

Equation (8) was used to transform time from the model simulation runs into dimensionless time. For flow rate conversion from field units into dimensionless rate, equation (10) is used:

qD =

141.2 Bµq kh pi − pwf

(

)

………………………………….……………….… (10)

where:

B [RB/stb] – liquid formation volume factor FVF

µ [cp] – fluid viscosity

q [stb/day] – surface rate

k [md] – formation permeability

h [ft] – total reservoir thickness

pi [psi] – initial pressure

pwf [psi] – well bore flowing pressure

Rate multiplier calculated using equation (10) and based on numerical model data set:

qM=3.14E-03 [day/STB]

42

Dimensionless Flow Rate qD

1000

100

10

1

0.1

FCD=1 FCD=5 FCD=10 FCD=25 FCD=100 FCD=500

0.01 1.00E-07 1.00E-06 1.00E-05 1.00E-04 1.00E-03 1.00E-02 1.00E-01 1.00E+00

Dimensionless Time tDxf Figure 3.7. – Dimensionless flow rate qD versus dimensionless time tDxf for constant pressure case - line source (vertical well)

Results of numerical simulation for different dimensionless fractures conductivities, as well as dimensionless time and dimensionless flow rates, are presented in Table 1 to 6 in Appendix B. Graphical solution is presented on Figure 3.7.

Figure 3.8. presents numerical results plotted on Bennett type curve with dimensionless time in function of fracture half-length and reciprocal dimensionless flow rate for fractures dimensionless conductivities, FCD, from 1 to 500.

Reciprocal Dimensionless Flow Rate 1/qD

43

10

1

FCD=1 FCD=5

0.1

FCD=10 FCD=25

0.01

0.001 1.00E-06

FCD=100 FCD=500

1.00E-05

1.00E-04

1.00E-03

1.00E-02

1.00E-01

1.00E+00

Dimensionless Time tDxf

Figure 3.8. – Model simulation results (symbols) with Bennett (1985) finite conductivity type curve for constant pressure case (lines)

Colored data points are results of numerical simulation of previous described numerical model. Match with Bennett finite conductivity type curves for all six dimensionless fracture conductivities provides verification of numerical model for this case.

44

4. Numerical Modeling – Point Source (Horizontal Well)

4.1. Numerical Modeling Methodology

Point source – horizontal wells numerical modeling is similar to the previous described numerical modeling of line source – vertical wells. The same model with synthetic data of reservoir, reservoir and fracture geometry and fluid properties. After model development, the numerical simulation results converted into the dimensionless time and dimensionless pressure or dimensionless rate were compared with results of numerical simulation of vertical well finite conductivity for constant rate and constant pressure production.

4.2. Numerical Model

Model reservoir has been discretizated into 931,635 blocks with distribution x:y:z=335:309:9.

Uneven grid block distribution has been respected for the x, y and z layers like in the previous described model.

45

Well is located in the central fifth block in z direction and also in 168th block in x and 155th block in y directions. Fracture’s blocks in x direction have the same distribution as in the previous model - different dimensions increasing to maximum value and than decreasing to the minimal dimension equal to the well grid block.

Total number of grid blocks in x, y and z directions as well as the total reservoir dimension are: Total number of blocks in x direction

Total reservoir dimension in x direction

Total number of blocks in y direction

Total reservoir Total number of Total reservoir dimension in y blocks in z dimension in z direction direction direction

nx

x [ft]

ny

y[ft]

nz

z[ft]

335

100,646

309

100,978

9

100

Point sources are in central blocks with 2[ft] dimension in x, y and z directions, while fracture is in direction of x axis, Figure 4.1.

Figure 4.1. Fracture position in point source (horizontal well)

46

The middle of the fracture starts in block 1 and fracture continues to the adjacent 20 blocks in both directions of x axis to the total fracture half-length of 2,043[ft]. Fluid, fracture and rock properties are the same as in the basic numerical model for Two cases have been examined: 1. Constant pressure production mode 2. Constant flow rate production mode In the first case the BHFP is assumed to be 500 [psi] and bottom hole flowing rate has been determined as a result of simulation. All data for constant flow rate case are the identical to the constant pressure case, except the pressure input. Instead of pressure, for constant flow rate case input will be flow rate of 100 [stb/day] and pressure will be the Fracture length 2x

result of the numerical simulation.

f

4.2.1. Constant Flow Rate Case for Point Source (Horizontal Well)

For each dimensionless fracture conductivity, FCD, presented in Bennett type curves for constant flowing rate, fracture porosity is calculated using equation (2). Data for real and equivalent fracture width are given below:

Real data Fracture porosity Fracture width

φ [%] w [in]

35 0.5

Equivalent data Fracture porosity φe [%] Equivalent fracture width we [ft]

0.73 2

47

Real and equivalent fracture permeabilities in function of the dimensionless fracture conductivities are the same as in the previous model and they are presented in Table 3.3. in Chapter 3.

Six data files were made for six different fracture dimensionless conductivities, FCD = 1, 5, 10, 25, 100, 500 with only difference in equivalent fracture permeability which is the function of the FCD.

Eclipse data input file for point source – horizontal well, constant rate and FCD=100 is given in Appendix A3.

In SUMMARY section of Data File, output data well BHP and well production rate were requested. Numerical simulation results are presented in Table 7 in Appendix B for FCD=100.

Conversion of time and pressure into dimensionless time in function of fracture halflength and dimensionless pressure was done using correlations (8) and (9) and results are the same multipliers as for the vertical well: tM= 2.53E-04 [day-1] pM= 7.08E-04 [psi-1]

Results of numerical simulation for dimensionless fracture conductivity, FCD=100 and also values of dimensionless pressure and dimensionless time in function of the fracture half-length are given in Table 7 in Appendix B.

48

Dimensionless Pressure pD

10

1

FCD=1

0.1

FCD=5 FCD=10 FCD=25 0.01

FCD=100 FCD=500

0.001 1.E-06

1.E-05

1.E-04

1.E-03

1.E-02

1.E-01

1.E+00

Dimensionless Time tDxf

Figure 4.2. – Bennett (1985) finite conductivity type curve for constant rate case and numerical model results for point source (horizontal well)

The point source (horizontal well) is compared to the line source (vertical well) Bennett type curve (Figure 4.2.) Deviation can be seen to be greater for the lower dimensionless fracture conductivities and that for all fracture conductivities the solutions converge to line source (vertical well) at late times. This type curve is developed for the ratio fracture half-length versus reservoir thickness xf/h =2043/100.

49

4.2.2. Constant Pressure Case for Point Source (Horizontal Well)

Only difference between these and Data Files for constant flow rate case is the control mode (in keyword: WCONPROD) which is BHP instead of the flow rate. Data File for FCD=100 is in Appendix A4.

Results of the numerical simulation are given in Table 7 in Appendix B.

To transform time into dimensionless time in function of the fracture half-length and to transform the flow rate into dimensionless one the correlations (8) and (10) have been used respectively.

Time and rate multipliers are the same as for the vertical well: tM= 2.53E-04 [day-1]

qM=3.14E-03 [day/STB]

Results of numerical simulation for dimensionless fracture conductivity FCD=100, as well as dimensionless times in function of fracture half-length and dimensionless flow rates, are presented in Table 7 in Appendix B and plotted in the Figure 4.3. for six different dimensionless fracture’s conductivities.

50

1000

Dimensionless Flow Rate qD

FCD=1 FCD=5 100

FCD=10 FCD=25 FCD=100

10

FCD=500

1

0.1 1.00E-07

1.00E-06

1.00E-05

1.00E-04

1.00E-03

1.00E-02

1.00E-01

Dimensionless Time tDxf

Reciprocal Dimensionless Flow Rate 1/q D

Figure 4.3. – Dimensionless flow rate versus dimensionless time in function of fracture half length for constant pressure case and point source (horizontal well)

10

1

FCD=1 FCD=5

0.1

FCD=10 FCD=25

0.01

0.001 1.00E-06

FCD=100 FCD=500 1.00E-05

1.00E-04

1.00E-03

1.00E-02

1.00E-01

1.00E+00

Dimensionless Time tDxf

Figure 4.4. – Bennett (1985) finite conductivity type curve for constant pressure case and numerical model results for point source (horizontal well)

51

Colored data points on Figure 4.4. presents numerical results converted in dimensionless ones. They are plotted on Bennett type curve with dimensionless time in function of fracture half-length and reciprocal dimensionless flow rate for fractures dimensionless conductivities, FCD, from 1 to 500.

Like in the previous case, the deviations are higher for the lower dimensionless fractures conductivities and they are smaller for the higher dimensionless fracture conductivities. This type curve is developed for the ratio fracture half-length versus reservoir thickness xf/h =2043/100.

52

5. Fracture Face Interference for Vertical Well

5.1. Fracture Face Interference Definition

Visualization of fracture face interference is shown using the images imported from Eclipse simulation runs. The Figure 5.1 presents the day 0 for the reservoir with two wells presented by black circles and two fractures presented by green lines.

Well 1

Fracture 1

Fracture 2

Well 2

Fracture length 2xf

Figure 5.1. – Part of the reservoir with two wells and two fractures

53

Red color of the reservoir present initial reservoir pressure. During the life of the reservoir – reservoir simulation, this color will change according to the color scale below the picture showing pressure depletion. As depletion proceeds a scale change is made to display region being influenced.

One of the simulation case with length to distance ratio of xf/y=8 were imported from Eclipse for different time spots with obvious pressure change.

Fracture 1

Well 2

Fracture 2

Fracture distance, y

Well 1

Fracture length 2xf

Figure 5.2. – Depletion in the reservoir after 260 days for case xf/y=8 Depletion at Figure 5.2. is observed by color change from beginning red – initial pressure to the orange – pressure in the reservoir after some time, 260 days for this example. It starts and continues from the well and fractures in both directions of y axis.

54

For the same case xf/y=8 at day 449 (Figure 5.3.), depletion in the reservoir area between two fractures is higher than outside of the fracture, showing near complete interference, defined by lighter orange color of the part of the reservoir between two fractures.

Fracture 1 Well 2

Fracture 2

Fracture distance, y

Well 1

Fracture length 2xf

Figure 5.3. – Depletion in the reservoir after 449 days for case of xf/y=8

After 516 days (Figure 5.4.), depletion in the reservoir area between two fractures is still higher than outside of the fracture, but depletion outside the well continues.

55

Fracture 1 Well 1

Well 2 Fracture 2

Figure 5.4. – Depletion in the reservoir after 516 days for case of xf/y=8

Fractures Tips

Figure 5.5. - Depletion in the reservoir after 1580 days for case of xf/y=8

56

After 1580 days (Figure5.5.), the radial flow occurs around the wells and fractures.

Figure 5.6. presents total depletion at the end of the life of the reservoir. After 22 years, the area between two fractures is totally depleted due to the fracture face interference. Reservoir will be depleted from both sides of the fractures although the total depletion can be expected near the fractures and between two fractures due to the fracture face interference.

Fractures Tips

Figure 5.6. - Depletion in the reservoir at the end of the reservoir life

57

5.2. Vertical Well Numerical Model

After matching Bennett’s solutions and model verification, the FCD=100 type curve for constant rate was the object of further research. Numerical model that has been used in previous research has been used for this research with few modifications. The y dimension of the reservoir has been doubled by doubling the number of blocks in y direction. Total number of blocks in Total reservoir Total number of Total reservoir x direction dimension in x direction blocks in y direction dimension in y direction

nx

x [ft]

ny

y[ft]

335

100,646

618

201,956

Two vertical wells are located in the center of each of the two halves of the reservoir presented by numerical model, data file in Appendix 5. Two vertical parallel fractures were extending throughout the wells in both direction of x axis (Figure 5.7). For better analysis, fractures have been enlarged and defined by green lines

and

fracture distance, y. This model is developed for two wells presented by black circles, but this could represent two fractures in the same horizontal well.

Fracture length 2x

f

58

Well 1 Enlarged Fracture

y

201,000

100,500

Initial Reservoir

Added Part of Reservoir Well 2 Enlarged Fracture

Figure 5.7. – Numerical model with two wells and two fractures

5.3. Constant Flow Rate Case

Total production flow rate was also doubled to 200 [stb/day]. Reservoir and fracture physical characteristics and fluid PVT properties were constant. The only variable was the distance between two fractures, y which was the maximum at the start of the research. This distance has been decreased by removing grid blocks between fractures.

59

In order to make this observation dimensionless and applicable to the real well data, different cases of length between two fractures are defined by length to distance ratio (xf/y), where xf is fracture half-length and y is distance between two fractures. By decreasing the length y, this length to distance ratio will increase. The length to distance ratios 0.028, 4, 16, 63.8, and 255 were analyzed.

h h

2xf

2xf

y2

y1

If

y1 < y2

then

xf y1

>

xf y2

Figure 5.8. – Examples of different xf/y ratios

Figure 5.8. presents two cases of two fractures with y1 and y2 distance between them and their reflection on the length to distance ratio. For lower distance between two fractures, the length to distance ratio will be higher.

Data file for the case of constant rate, dimensionless fracture conductivity FCD=100 and length to distance ratio xf/y=255 is presented in Appendix A5 and results of numerical simulation are presented in the Table 8 in Appendix B.

60

Input data for simulation FCD, xf and kf were constant: FCD = 100 xf = 2034 [ft] kf = 10,215 [md]

Graphical solutions are given in Figure 5.9. For the higher distance y, the lower length to distance ratio of 0.028, the simulation result fits the type curve of FCD=100. There is no deviation from that base case. Decreasing distance y, length to distance ratio increases and deviation appears earlier. 10

Dimensionless Pressure PD

FCD - dimensionless fracture conductivity xf - fracture half-length y - distance between two fractures

FCD=100

1

0.1

FCD

length to distance ratio

1

xf y

5 10

0.028 4.0

25 0.01

100

16.0 63.8

500

255

0.001 0.000001

0.00001

0.0001

0.001

0.01

0.1

1

Dimensionless Time tDxf

Figure 5.9. – Constant rate case - Finite conductivity type curve for family of finite conductivity fractures with deviations for fracture face interference for FCD=100 The higher length to distance ratio, the earlier the deviation. For the length to distance ratio equal 255 the deviation is the greatest and it starts at early time.

61

For lower length to distance ratio, the dimensionless pressure will be lower, and pressure will decrease slowly. Production time will be longer with constant rate and this case is optimal. However, for the higher deviations from the base case FCD=100 – higher length to distance ratios, the dimensionless pressure will be higher and pressure decrease faster. This will cause shorter production time with constant production rate and this presents the worse production option.

The same analysis was done for the FCD=1, 5, 10, 25, and 500 and graphical results are presented in the Figures 1 to 6 in Appendix C. Difference between data files for different dimensionless conductivities, FCD, is the number of grid blocks between two fractures, the location of the wells and fractures.

5.4. Constant Pressure Case

Research methodology for this case was the same as for the case of the constant flow rate. The reservoir was doubled and its geometry has already been described. Reservoir and fracture physical characteristics and fluid PVT properties were constant. The only variable was the distance between two fractures, y which was the maximum at the start of the research. This distance has been decreased by removing grid blocks between fractures.

62

Data file for length to distance ratio of 128 is given in the Appendix A6. Results of numerical simulation and dimensionless time and flow rate are given in Table 9 in Appendix B. Plot of well production rate versus time with length to distance ratio as parameter is presented in Figure 5.10.

In this case instead of the reciprocal dimensionless flow rate for one fracture and one well, the result that has been plotted is the reciprocal dimensionless flow rate for two fracture system. Results of the simulation are the flow rates for one well but numerical model is two-well two-fracture system. Total flow rate will be equal to the arithmetic average of the flow rate of both of the wells (fractures) – Equation (11), and it represents the secondary axis on the Bennett finite conductivity type curve with deviations for fracture face interference.

2 qDtfs

=

1 qD1 +qD2 2

………………………………………………… (11)

Dimensionless results of the simulation were plotted on the Figure 5.10. and deviations from the base case have been observed similarly like in the case of the constant flow rate. For lower length to distance ratios, the deviations were not so high, but for higher ratios these deviations were more explicit. The reciprocal dimensionless flow rate 1/qD will be lower and production flow rate will be higher.

Reciprocal Dimensionless Flow Rate 1/q D

10

FCD - dimensionless fracture conductivity xf - fracture half-length y - distance between two fractures

FCD=100 1

length to distance ratio xf y

FCD

0.1

0.01

1 5 10

0.028

25

16.0

4.0 63.8

100

128 255

500

0.001 1.00E-06

1.00E-05

1.00E-04

1.00E-03

1.00E-02

1.00E-01

Reciprocal Dimensionless Flow Rate for Two Fracture System 2/qDtfs

63

1.00E+00

Dimensionless Time tDxf

Figure 5.10. – Constant pressure case - Finite conductivity type curve for family of finite conductivity fractures with deviations for fracture face interference for FCD=100 (reciprocal rate)

Dimensionless Flow Rate q D

1000

100

10

FCD=100

length to distance ratio xf y

1

0.1 0.000001

0.028 4.0 16.0 63.8 255 128

0.00001

0.0001

0.001

0.01

0.1

1

Dimensionless Time tDxf

Figure 5.11. – Constant pressure case - Finite conductivity type curve for family of finite conductivity fractures with deviations for fracture face interference for FCD=100 (rate)

64

This will cause the higher production with constant pressure (Figure 5.11). Contrary, the higher length to distance ratio will result with higher reciprocal dimensionless flow rate and lower flow rate. This means, the production will be lower with constant pressure for higher deviations from base case and this is the worse case of the production.

The similar analysis was done for dimensionless fractures conductivities of FCD=1, 5, 10, 25, and 500. Results are presented in the Figures 7 to 12 in Appendix C.

65

6. Fracture Face Interference for Point Source (Horizontal Well)

6.1. Point Source (Horizontal Well) Numerical Model

The FCD=100 type curve for constant rate and for constant pressure cases were the object of analysis of fracture face interference for point source (horizontal well). Numerical model and simulation methodology were the same as in the previous described case. Besides doubled y dimension of the reservoir, the number of layers was increased to 9 blocks keeping total reservoir thickness h=100[ft]. Total number of blocks in x direction

Total reservoir Total number of dimension in x blocks in y direction direction

Total reservoir dimension in y direction

Total number Total reservoir of blocks in z dimension in z direction direction

nx

x [ft]

ny

y[ft]

nz

z[ft]

335

100,646

618

201,956

9

100

The point sources are located in the center of the reservoir, 5th block in z direction and in the same position like vertical wells in previous model. Two vertical parallel fractures were extending throughout the point sources in both direction of x axis. The Figure 6.1. presents the fractures and point sources locations.

66

Figure 6.1. – Point sources (horizontal well) with two vertical fractures

6.2. Constant Flow Rate Case

Production flow rate remains 200 [stb/day]. Reservoir and fracture physical characteristics and fluid PVT properties were constant. The only variable was the distance between two fractures, y which was the maximum at the start of the research. This distance has been decreased by removing grid blocks between fractures. The length to distance ratios 4, 16, 63.8, and 255 were analyzed. Fracture length 2x

f

67

Eclipse simulator data input file for the point source case with constant rate, dimensionless fracture conductivity FCD=100 and length to distance ratio xf/y=255 is presented in Appendix A7 and results of numerical simulation are presented in the Table 11 in Appendix B.

Graphical solutions are shown in Figure 6.2 comparing effects of various distance ratios with FCD =100 to previously developed point source solutions for all FCD curves. Observations are similar to those made for the line source (vertical well). Decreasing distance y, length to distance ratio increases and deviation appears earlier.

10

Dimensionless Pressure PD

FCD - dimensionless fracture conductivity xf - fracture half-length y - distance between two fractures 1

FCD=100

FCD 1 length to distance ratio

0.1

xf y

5 10

4.0

25 0.01

16.0

100

63.8 500

0.001 0.000001

255

0.00001

0.0001

0.001

0.01

0.1

1

Dimensionless Time tDxf

Figure 6.2. – Finite conductivity type curve with deviations for fracture face interference for constant flow rate and FCD=100 – point source (horizontal well) with previous point source solutions for all FCD values

68

This type curve is developed for the ratio fracture half-length versus reservoir thickness xf/h =2043/100 The same analysis was done for the FCD=1, 5, 10, 25, and 500 and graphical results are presented in the Appendix C – Figures 13 to 18 .

6.3. Constant Pressure Case

The research methodology is almost identical to the one described in the case of the constant flow rate. The only difference is the control mode and in this case it is BHP=500 psi. Reservoir geometry has already been described and reservoir and fracture physical characteristics and fluid PVT properties were constant. The only variable was the distance between two fractures, y which was the maximum at the start of the research. Simulation methodology was the same, the distance between fractures has been decreased by removing grid blocks between fractures.

Eclipse simulator data input file for length to distance ratio, xf/y=255 and dimensionless fracture conductivity, FCD=100 is given in the Appendix A8 for point source. Results of numerical simulation and dimensionless time and flow rate are given in Table 11 in Appendix B. Plot of well production rate versus time with length to distance ratio as parameter is presented in Figure 6.3.

69

Reciprocal Dimensionless Flow Rate 1/q

D

FCD - dimensionless fracture conductivity xf - fracture half-length y - distance between two fractures

Reciprocal Dimensionless Flow Rate For Two Fractures System 2/q Dtfs

10

FCD=100

1 FCD 1 length to distance ratio 0.1

0.01

5 10

xf y

25

4.0

100

16.0 63.8

500

0.001 0.000001

128 255

0.00001

0.0001

0.001

0.01

0.1

1

Dimensionless Time tDxf

Figure 6.3. – Finite conductivity type curve with deviations for fracture face interference for constant pressure and FCD=100 – point source (horizontal well) with previous point source solutions for all FCD values In these cases, the match with base case is observed at earlier time for lower length to distance ratios and deviations are established at later times. This is especially obvious for lower length to distance ratios. For the higher length to distance ratios, deviations start earlier and increase earlier comparing with base case FCD=100 and lower length to distance ratios.

For lower length to distance ratio, the reciprocal dimensionless flow rate 1/qD will be lower and production flow rate will be higher (Figure 6.4.). This will cause the higher production with constant pressure. Contrary, the higher length to distance ratio will

70

result with higher reciprocal dimensionless flow rate and lower flow rate. This means, the production will be lower with constant pressure for higher deviations from base case and this is the worse case of the production.

Dimensionless Flow Rate q D

100

FCD=100 10

length to distance ratio xf y

4.0 16.0 1

63.8 255

0.1 1.E-06

1.E-05

1.E-04

1.E-03

1.E-02

1.E-01

1.E+00

Dimensionless Time tDxf

Figure 6.4. – Dimensionless rate versus dimensionless time with deviations for fracture face interference for constant pressure case and FCD=100 – Point source (horizontal well)

This type curve is developed for the ratio fracture half-length versus reservoir thickness xf/h =2043/100 The similar analysis was done for dimensionless fractures conductivities of FCD=1, 5, 10, 25, and 500. Results are presented in the Figures 19 to 24 in Appendix B.

71

6.4. McAlister Well Data

The motivation for predicting of fracture face interference was to determine if observed well performance data could be attributed to this model. Gas well McAlister O.H. 16 in East Newark field has been the subject of investigation. Well data were obtained from Rail Road Commission of Texas, Oil and Gas Division.

Production Rate [Mcf/month]

1,000,000

100,000

10,000

1,000 0

10

20

30

40

50

60

Time [months]

Figure 6.5. – McAlister O.H. 16 monthly gas production data

Monthly gas production for the subject gas well in a semilog format is shown in Figure 6.5. The McAlister O.H. 16 well was completed on December 15, 2002, as a horizontal well in the Barnett Shale formation stimulated with hydraulic fracturing. Cumulative gas production was 2,166 MMcf on October 1, 2007.

72

Time in months and monthly gas rate have been converted into dimensionless parameters using multipliers tM= 2.5E-6 [months-1] qM=1,200 [Mcf/month-1] where t Dxf = t[months]t M  Mcf  qD = q   qM  month 

Monthly gas production data, dimensionless time and dimensionless flow rate are listed in Table 10 in Appendix B.

Results are plotted on Point source finite conductivity type curve for constant pressure with deviations for fracture face interference, Figure 6.6. The FCD=100 type curve was selected based on information supplemented by service company and the operator.

McAlister O. H. 16 well production data are presented as black squares. Comparing production data with developed deviations from the base case of FCD=100, it is apparent that well data can match curve of length to distance ratio xf/y=128. The interpretation would be the fracture’s half-length of this well is equal to 128 times distance between fractures. Other length to distance ratios could also be matched. A unique match would require prior knowledge of fracture length or formation permeability. Alternatively, fracture length to distance ratios could be interpreted from completion data or micro seismic analysis. The finding of this work is simply that production performance of

73

fracture stimulated horizontal wells can be modeled by the effects of fracture face interference.

Reciprocal Dimensionless Flow Rate 1/q

D

FCD - dimensionless fracture conductivity xf - fracture half-length y - distance between two fractures

Reciprocal Dimensionless Flow Rate For Two Fractures System 2/q Dtfs

10

FCD=100

1 FCD 1 0.1

0.01

5 10 25

4.0

100

16.0

500

63.8

0.001 0.000001

128

0.00001

0.0001

0.001

0.01

0.1

1

Dimensionless Time tDxf

Figure 6.6. – Point Source finite conductivity type curve with deviations for fracture face interference and McAlister O. H. 16 well data

74

7. Sensitivity Analysis of Research Results

7.1. Constant Rate Case

Initial pressure for the numerical model based on synthetic data was 5,000[psi]. The aim of the sensitivity analysis in this case is to investigate the possibility of type curve change with initial pressure change. Three pressures values have been chosen for this investigation: pi=1000, 2000, and 5,000 [psi] for the fracture dimensionless conductivity FCD=5 (Figure 7.1.) and FCD=100 (Figure 7.2).

Results of simulation are the flow rates that have been converted in dimensionless ones using equation (10) and time was converted in dimensionless time in function of the fracture half-length using equation (8). According to the Figures 7.1. and 7.2., the type curve matching has obtained providing confirmation of the numerical simulation results and verification of the numerical model. The new finite conductivity type curves for initial pressures 1,000, 2,000 and 5,000 [psi] do not have any deviations from the Bennett finite conductivity type curve for FCD=5 and FCD=100.

75

Dimensionless Pressure p D

10

1

0.1 pi=1,000 pi=2,000

0.01

0.001 1.E-06

pi=5,000

1.E-05

1.E-04

1.E-03

1.E-02

1.E-01

1.E+00

Dimensionless Time tDxf

Figure 7.1. – Sensitivity analysis of change of initial pressure for FCD=5

Dimensionless Pressure pD

10

1

0.1 pi=1,000 pi=2,000

0.01

0.001 1.E-06

pi=5,000

1.E-05

1.E-04

1.E-03

1.E-02

1.E-01

1.E+00

Dimensionless Time tDxf

Figure 7.2. – Sensitivity analysis of initial pressure change for FCD=100

76

7.2. Constant Pressure Case

Additional verification of the numerical model was performed by changing well productivity indexes by adding keyword WELPI in Schedule section of Data File for FCD=5 and FCD=100. The three different well productivity indexes were set up: initial one, twice higher and twice lower than initial ones. Results are given in the Figures 7.3 and 7.4. Results of the analysis are plotted on Bennett finite conductivity type curve for FCD=5 and FCD=100 and they match the numerical simulation solution and provide numerical model verification.

Reciprocal Dimensionless Flow Rate 1/q D

10

1

0.1 Jinitial=1,285 J=2*Jinitial=2,536

0.01

0.001 1.E-06

J=0.5*Jinitial=634

1.E-05

1.E-04

1.E-03

1.E-02

1.E-01

1.E+00

Dimensionless Time tDxf

Figure 7.3. – Sensitivity analysis of the change of productivity index for FCD=5

77

These figures showed that for the different productivity indexes, it is obvious to have type curve matching. There is no deviation from the developed type curve and constant pressure production mode is not sensitive to the productivity index change. Reciprocal Dimensionless Flow Rate 1/q D

10

1

0.1 Jinitial=25,710 J=2*Jinitial=50,730

0.01

0.001 1.E-06

J=0.5*Jinitial=12,682

1.E-05

1.E-04

1.E-03

1.E-02

1.E-01

1.E+00

Dimensionless Time tDxf

Figure 7.4. – Sensitivity analysis of the productivity index change for FCD=100

7.3. Sensitivity Analysis of Change of Fracture Half-Length for Vertical Well

Finite fracture type curves with deviations for fracture face interference have been observed. To get those results, the fracture half-length of 2,043[ft] have been used as input data. To check developed type curves for different length to distance ratios, it was

78

necessary to change fracture half-length. It was chosen to use fracture half-length xf = 506 [ft], with fracture permeability of kf = 2,530 [md].

Data file for length to distance ratio equal to 255 is given in Appendix A9. Since the fracture half length is about four times less than in the previous case, the distance between two fractures were adjusted to get the same ratio, 255. Using this methodology it was possible to compare deviation for fracture face interference of two fractures with different half-lengths.

Time and pressure multipliers are calculated using equations (8) and (9): tM = 4.12E-03 [day-1] pM = 7.08E-04 [psi-1]

Results of numerical simulation and dimensionless time and pressure data are given in Table 12 in Appendix B. These data were plotted on Bennett finite conductivity type curve with deviations for fracture face interference developed in previous research. Results are the same. Colored data point matches the derived curves of fracture face interference for the same length to distance ratios and different fractures half-lengths. Figure 7.5. presents summary of these results.

79

10

FCD=100

Dimensionless Pressure pD

1

0.1 length to distance ratio 506 y

2043 y 0.01

0.001 1.E-06

1.E-05

1.E-04

1.E-03

1.E-02

2

2

8

8

36

36

255

255

1.E-01

1.E+00

Dimensionless Time tDxf

Figure 7.5. – Sensitivity analysis for different fractures half-lengths

According to the previous figure, there are no deviations from base cases. Developed type curves for different length to distance ratios are not sensitive for the fracture halflength change.

7.4. Sensitivity Analysis of Change of Number of Grid Blocks in z Direction for Point Source

The analysis of point source for both cases – constant pressure and constant flow rate was performed for 9 blocks in z direction. The sensitivity analysis aim was to check the

80

simulation results if number of grid blocks increases to the 13. Results are plotted on the Figure 7.6.

According to the figure, the change of the number of grid blocks in z direction do not have influence on the analysis of the point source which will have the same performance independently on the number of grid blocks in the z direction.

Dimensionless Pressure pD

10

FCD=1 1

0.1 9 blocks 0.01

0.001 1.E-06

13 blocks

1.E-05

1.E-04

1.E-03

1.E-02

1.E-01

1.E+00

Dimensionless Time tDxf Figure 7.6. – Sensitivity analysis for different number of the grid blocks in z direction for point source, constant rate case and FCD=1

81

8. Summary and Recommendations

8.1. Summary

1. A single layer numerical model (two-dimensional) was developed for low permeability hydraulically fractured reservoirs from finite conductivity vertical fractures for both constant pressure and constant rate at the well bore. 2. The numerical model was extended to nine layers (three-dimensional) with a connection in the central layer in vertical direction to extend the solution for horizontal wells. 3. The

numerical

models

(two-dimensional

and

three-dimensional)

also

incorporated a second vertical fracture and the results of fracture face interference was determined for both vertical (offsetting) and horizontal wells (multistage completion).

82

8.2. Recommendations for Future Work

1. The future work includes the application of the pressure derivative on the newly developed type curves for constant rate production and rate integral and integralderivative (normalized rate) for a constant pressure production. 2. Numerical model extension for multiple (more than two) hydraulically stimulated fractures. 3. The third goal of the future work should be investigation of the fracture face interference influence of the different ratios of fracture half-length and reservoir thickness for the horizontal well case. 4. Incorporation of micro seismic data for verification of the rate transient analysis using newly developed type curves. 5. The ability to predict future performance of multistage completion of horizontal wells in tight reservoirs to allow economic optimization in field development.

83

Reference

Agarwal, R.G., Carter, R.D., Polloc, C.B., – Evaluation and Performance Prediction of Low-Permeability Gas Wells Stimulated by Massive Hydraulic Fracturing – Journal of Petroleum Technology, March 1979, 362-372, Trans.AIME 267

Bennett, C.O., Camacho, R.G., Reynolds, A.C., Raghavan, R., – Approximate Solutions for Fractured Wells Producing Layered Reservoirs - SPE Journal – October 1985, 729-742

Bennett, C.O., Reynolds, A.C., Raghavan,R., Jacques, L.E., – Performance of Finite Conductivity, Vertically Fractured Wells in Single-Layer Reservoirs – SPE Formation Evaluation – August 1986, 399-412

Bennion, D.B., Thomas, F.B., Bietz, R.F. – Low Permeability Gas Reservoirs: Problems, Opportunities and Solutions for Drilling, Completion, Stimulation and Production - SPE 35577 presented at Gas Technology Conference, Calgary, Canada – May 1996

84

Bennion, D.B., Thomas, F.B. and Ma, T. – Formation Damage Processes Reducing Productivity of Low Permeability Gas Reservoirs – SPE 60325 presented at the SPE Rocky Mountain Regional/Low Permeability Reservoir Symposium and Exhibition, Denver, Colorado – March 2000

Cinco-Ley, H., Samaniego, V.F., Dominguez, N.,– Transient Pressure Behavior for a Well With Finite-Conductivity Fracture – SPE Journal, August 1978, 253-264, Trans. AIME 265

Cinco-Ley, H., Samaniego, V.F., – Transient Pressure Analysis for Fractured Wells – SPE paper 7490, Journal of Petroleum Technology, September 1981, 1749-1766

Daniels, J., Waters, G., LeCalvez, J., Lassek, J., Bentley, D. – Contacting More of Barnett Shale Through an Integration of Real-Time Microseismic Monitoring, Petrophysics and Hydraulic Fracture Design – SPE 110562 – SPE Annual Technical Conference and Exhibition held in Anaheim, October 2007

Friedel, T. - Numerical Simulation of Production From Tight-Gas Reservoirs by Advanced Stimulation Technologies – PhD Dissertation, TUniversity Bergakademie Freiberg, July 2004

Naik, G.C. - Tight Gas Reservoirs – An Unconventional Natural Energy Source for the Future –2006

85

Nashawi, I.S, Qasem, F.H, Gharbi, R, – Transient Pressure Analysis of Gas Wells Producing at Constant Pressure –Journal of Petroleum Science and Engineering – 40 (2003), 89-102

Nashawi, I.S, Malallah, A.H – Well Test Analysis of Finite-Conductivity Fractured Wells Producing at Constant Bottomhole Pressure – Journal of Petroleum Science and Engineering – 57 (2007), 303-320 Nederlof, M. H – The Scope for Natural Gas Supplies from Unconventional Sources – Ann. Rev. Energy. 13, 1988, 95-117

Railroad

Comission

of

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and

Gas

Division

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http://webapps.rrc.state.tx.us/PDQ/quickLeaseReportBuilderAction.do, Downloaded April 2008

Shanley, K.W., Cluff, R.M, and Robinson, J.M. – Factors Controlling Prolific Gas Production From Low Permeability Sandstone Reservoirs: Implications for Resource Assessment, Prospect Development, and Risk Analysis – The American Association of Petroleum Geology Bulletin 88 (8), 2004, 1083-1121

Stevens, S.H., Kuuskraa, J., Kuuskraa, V. – Unconventional Natural Gas in the United States: Production, Reserves and Resource Potential – 1988

86

Tiab, D – Analysis of Pressure Derivative Without Type-Curve Matching: Vertically Fractured Wells in Closed Systems – Journal of Petroleum Science and Engineering, 11 (1994), Paper SPE 26138

Tiab, D – Analysis of Pressure and Pressure Derivative Without Type-Curve Matching: 1-Skin and Well bore Storage – Paper SPE 25423 presented at the Production Operations Symposium held in Oklahoma City, March 1993. Also, Journal of Petroleum Science and Engineering 12 (1995)

Tiab, D – Analysis of Pressure Derivative Data for Hydraulically Fractured Wells by the Tiab’s Direct Synthesis Technique – Journal of Petroleum Science and Engineering, 49 (2005), Paper SPE 52201

Van Golf-Racht, T.D. – Fundamentals of Fractured Reservoir Engineering – 1982

87

APPENDIX

A

88

Appendix A1 – Eclipse Data Input File for FCD=100, Constant Rate Case And Vertical Well – Line Source -- Constant Flow Rate Case q = 100 [STB/DAY] -- Vertical fracture FCD=100

NOECHO

RUNSPEC

=========================================================================

TITLE Vertical fracture model, (Wf)r=0.5 in ==> (Wf)e=2 ft

DIMENS ---- dx 335

dy dz 309 1

/

-- Fluid phases present WATER

-- Units FIELD --length of stack used by linear solver NSTACK 50 / ------------ #wells, WELLDIMS 1

# connections,

#groups,

#wells per group

1

1

1

/

-- Start simulation date START 1 JAN 1997 / -- run to be restarted from unified restart file UNIFIN -- Restart and summary files written are to be unified UNIFOUT GRID

===========================================================================

TOPS 103515*4950 / DXV -- reservoir 139*340 3*256 128 64 32 16 8 4 -- half fracture-1 2 4 8 16 32 64 128 6*256 128 64 32 16 8 4 -- well 2 -- half fracture-1 4 8 16 32 64 128 6*256 128 64 32 16 8 4 2 -- reservoir 4 8 16 32 64 128 3*256 139*340 / DYV 147*340 256 128 64 32 16 8 4 2 4 8 16 32 64 128 256 147*340 /

89

Appendix A1 – Eclipse Data Input File for FCD=100, Constant Rate Case And Vertical Well – Line Source - continued DZ 103515*100 /

EQUALS PERMX PORO PERMX PORO / COPY PERMX PERMX /

0.1 0.2 10215 0.0073

PERMY PERMZ

1 1 149 149

1 1

335 335 187 187

335 335

1 1

1 1 155 155

309 309

309 309 155 155

1 1

1 1 1 1

1 1 1 1

/ / / /

-----

reservoir X permeability reservoir Porosity equivalent fracture X permeability equivalent fracture porosity/

1 / 1 /

INIT GRIDFILE 0 1 /

RPTGRID TRANX TRANY / PROPS

==========================================================================

PVTW -- PREF 4014.7 /

BW(PREF) CW 1.0 3.0D-6

ROCK -- PREF 4014.7 / DENSITY -- OIL 44.09

VW(PREF) 1.0 0

CVW

CR 0

WATER 62.28

GAS 0.066 /

RPTPROPS / SOLUTION

=========================================================================

-DATUM -DEPTH EQUIL 5000

DATUM PRESS

RPTSOL -- Fluid -- in place FIP=1

5000

OWC DEPTH 1*

OWC PCOW

GOC DEPTH

1*

1*

GOC PCOG 1*

RSVD TABLE

1*

RVVD TABLE 1*

SOLN METH 1*

/

Create init Restart file RESTART=2 /

RPTRST BASIC=2 / SUMMARY

==========================================================================

90

Appendix A1 – Eclipse Data Input File for FCD=100, Constant Rate Case And Vertical Well – Line Source - continued -- Well quantities -- Well BHP WBHP / -- Well water production rate WWPR /

RUNSUM EXCEL

SCHEDULE

==========================================================================

RPTRST BASIC=2 / RPTSCHED WELSPECS / WELSPECS -- WELL GROUP -- NAME NAME W1 G /

-LOCATIONI J 168 155

BHP PHASE DRAINAGE FLAG FLAG FLAG PRESS FLAG DEPTH RADIUS GAS SHUT CROSS TABLE DENS 1* WATER 1* STD SHUT NO 1* SEG

/

COMPDAT -- WELL --LOCATION-OPEN/ SAT CONN WELL EFF SKIN D PENETRATION -- NAME I J K1 K2 SHUT TAB FACT DIAM KH FACTOR FACTOR DIRECTION W1 168 155 1 1 OPEN 1* 1* 0.60 1* 0 0 Z / / WCONPROD -- WELL OPEN/ -- NAME SHUT W1 OPEN / TSTEP 0.00001157 3.46E-05 1.03E-04 3.08E-04 9.20E-04 2.75E-03 8.20E-03 2.45E-02 7.31E-02 2.18E-01 6.52E-01 1.95E+00 5.81E+00 1.74E+01 5.18E+01 1.55E+02 4.62E+02 1.38E+03 END

CNTL MODE WRAT

OIL WATER GAS LIQU RATE RATE RATE RATE 1* 100 1* 1*

1.39E-05 4.15E-05 1.24E-04 3.70E-04 1.10E-03 3.30E-03 9.84E-03 2.94E-02 8.78E-02 2.62E-01 7.83E-01 2.34E+00 6.98E+00 2.08E+01 6.22E+01 1.86E+02 5.55E+02 /

1.67E-05 4.98E-05 1.49E-04 4.44E-04 1.32E-03 3.96E-03 1.18E-02 3.53E-02 1.05E-01 3.15E-01 9.39E-01 2.80E+00 8.37E+00 2.50E+01 7.47E+01 2.23E+02 6.66E+02

RES BHP RATE 1* 1*

2.00E-05 5.97E-05 1.78E-04 5.32E-04 1.59E-03 4.75E-03 1.42E-02 4.23E-02 1.26E-01 3.77E-01 1.13E+00 3.37E+00 1.00E+01 3.00E+01 8.96E+01 2.67E+02 7.99E+02

THP 1*

VFP TABLE 1*

ALQ 1* /

2.40E-05 7.17E-05 2.14E-04 6.39E-04 1.91E-03 5.70E-03 1.70E-02 5.08E-02 1.52E-01 4.53E-01 1.35E+00 4.04E+00 1.21E+01 3.60E+01 1.08E+02 3.21E+02 9.58E+02

2.88E-05 8.60E-05 2.57E-04 7.67E-04 2.29E-03 6.84E-03 2.04E-02 6.10E-02 1.82E-01 5.43E-01 1.62E+00 4.85E+00 1.45E+01 4.32E+01 1.29E+02 3.85E+02 1.15E+03

91

Appendix A2 – Eclipse Data Input File for FCD=100, Constant Pressure Case And Vertical Well – Line Source -- Constant Pressure Case BHP=500 [psi] -- Vertical Fracture FCD=100

NOECHO RUNSPEC

=========================================================================

TITLE Vertical hydraulic fracture model, (Wf)r=0.5 in ==> (Wf)e=2 ft

DIMENS ---- dx 335

dy dz 309 1

/

-- Fluid phases present WATER

-- Units FIELD --length of stack used by linear solver NSTACK 50 / ------------ #wells, WELLDIMS 1

# connections,

#groups,

#wells per group

1

1

1

/

-- Start simulation date START 1 JAN 1997 / -- run to be restarted from unified restart file UNIFIN -- Restart and summary files written are to be unified UNIFOUT GRID

===========================================================================

TOPS 103515*4950 / DXV -- reservoir 139*340 3*256 128 64 32 16 8 4 -- half fracture-1 2 4 8 16 32 64 128 6*256 128 64 32 16 8 4 -- well 2 -- half fracture-1 4 8 16 32 64 128 6*256 128 64 32 16 8 4 2 -- reservoir 4 8 16 32 64 128 3*256 139*340 / DYV 147*340 256 128 64 32 16 8 4 2 4 8 16 32 64 128 256 147*340 /

92

Appendix A2 – Eclipse Data Input File for FCD=100, Constant Pressure Case And Vertical Well – Line Source - continued DZ 103515*100 / EQUALS PERMX PORO PERMX PORO / COPY PERMX PERMX /

0.1 0.2 10215 0.0073

PERMY PERMZ

1 1 149 149

1 1

335 335 187 187

335 335

1 1

1 1 155 155

309 309 155 155

309 309

1 1

1 1 1 1

1 1 1 1

/ / / /

-----

reservoir X permeability reservoir Porosity equivalent fracture X permeability equivalent fracture X porosity/

1 / 1 /

INIT GRIDFILE 0 1 /

RPTGRID TRANX TRANY / PROPS

==========================================================================

PVTW -- PREF 4014.7 /

BW(PREF) 1

ROCK -- PREF 4014.7 / DENSITY -- OIL 44.09

CW 3.0D-6

VW(PREF) 1 0

CVW

CR 0

WATER 62.28

GAS 0.066 /

RPTPROPS / SOLUTION

=========================================================================

-DATUM -DEPTH EQUIL 5000

DATUM PRESS

OWC DEPTH

5000

1*

RPTSOL -- Fluid -- in place FIP=1 SUMMARY

OWC PCOW

GOC DEPTH

1*

1*

GOC PCOG 1*

RSVD TABLE

1*

RVVD TABLE 1*

SOLN METH 1*

/

Create init Restart file RESTART=2 / ==========================================================================

-- Well quantities WBHP / WWPR /

93

Appendix A2 – Eclipse Data Input File for FCD=100, Constant Pressure Case And Vertical Well – Line Source - continued SCHEDULE RPTRST BASIC=3

==========================================================================

FREQ=1 /

RPTSCHED WELSPECS / WELSPECS -- WELL GROUP -- NAME NAME W1 G /

-LOCATIONI J 168 155

BHP PHASE DRAINAGE FLAG FLAG FLAG PRESS FLAG DEPTH RADIUS GAS SHUT CROSS TABLE DENS 1* WATER 1* STD SHUT NO 1* SEG

COMPDAT -- WELL --LOCATION-OPEN/ SAT CONN -- NAME I J K1 K2 SHUT TAB FACT W1 168 155 1 1 OPEN 1* 1* / WCONPROD -- WELL OPEN/ -- NAME SHUT W1 OPEN / TSTEP 0.00001157 3.46E-05 1.03E-04 3.08E-04 9.20E-04 2.75E-03 8.20E-03 2.45E-02 7.31E-02 2.18E-01 6.52E-01 1.95E+00 5.81E+00 1.74E+01 5.18E+01 1.55E+02 4.62E+02 1.38E+03 4.12E+03

END

CNTL MODE BHP

WELL DIAM 0.60

OIL WATER GAS LIQU RATE RATE RATE RATE 1* 1* 1* 1*

1.39E-05 4.15E-05 1.24E-04 3.70E-04 1.10E-03 3.30E-03 9.84E-03 2.94E-02 8.78E-02 2.62E-01 7.83E-01 2.34E+00 6.98E+00 2.08E+01 6.22E+01 1.86E+02 5.55E+02 1.66E+03 4.95E+03

1.67E-05 4.98E-05 1.49E-04 4.44E-04 1.32E-03 3.96E-03 1.18E-02 3.53E-02 1.05E-01 3.15E-01 9.39E-01 2.80E+00 8.37E+00 2.50E+01 7.47E+01 2.23E+02 6.66E+02 1.99E+03 5.93E+03

/

EFF SKIN D PENETRATION KH FACTOR FACTOR DIRECTION 1* 0 0 Z /

RES BHP RATE 1* 500

2.00E-05 5.97E-05 1.78E-04 5.32E-04 1.59E-03 4.75E-03 1.42E-02 4.23E-02 1.26E-01 3.77E-01 1.13E+00 3.37E+00 1.00E+01 3.00E+01 8.96E+01 2.67E+02 7.99E+02 2.39E+03 7.12E+03

THP 1*

VFP TABLE 1*

ALQ 1* /

2.40E-05 7.17E-05 2.14E-04 6.39E-04 1.91E-03 5.70E-03 1.70E-02 5.08E-02 1.52E-01 4.53E-01 1.35E+00 4.04E+00 1.21E+01 3.60E+01 1.08E+02 3.21E+02 9.58E+02 2.86E+03 8.55E+03

2.88E-05 8.60E-05 2.57E-04 7.67E-04 2.29E-03 6.84E-03 2.04E-02 6.10E-02 1.82E-01 5.43E-01 1.62E+00 4.85E+00 1.45E+01 4.32E+01 1.29E+02 3.85E+02 1.15E+03 3.43E+03 1.03E+04

/

94

Appendix A3 – Eclipse Data Input File for FCD=100, Constant Rate Case And Horizontal Well - Point Source -- Vertical fracture FCD=100, constant rate and point source NOECHO RUNSPEC

=========================================================================

TITLE Vertical fracture model, (Wf)r=0.5 in ==> (Wf)e=2 ft

DIMENS ---- dx 335

dy dz 309 9

/

-- Fluid phases present WATER

-- Units FIELD --length of stack used by linear solver NSTACK 50 / ------------ #wells, WELLDIMS 1

# connections,

#groups,

#wells per group

1

1

1

/

-- Start simulation date START 1 JAN 1997 / -- run to be restarted from unified restart file UNIFIN -- Restart and summary files written are to be unified UNIFOUT GRID

===========================================================================

TOPS 103515*4950 103515*4971 103515*4987 103515*4995 103515*4999 103515*5001 103515*5005 103515*5013 103515*5029 /

DXV -- reservoir 139*340 3*256 128 64 32 16 8 4 -- half fracture-1 2 4 8 16 32 64 128 6*256 128 64 32 16 8 4 -- well 2 -- half fracture-1 4 8 16 32 64 128 6*256 128 64 32 16 8 4 2 -- reservoir 4 8 16 32 64 128 3*256 139*340 / DYV 147*340 256 128 64 32 16 8 4 2 4 8 16 32 64 128 256 147*340 /

95

Appendix A3 – Eclipse Data Input File for FCD=100, Constant Rate Case And Horizontal Well - Point Source - continued DZ 103515*21 103515*16 103515*8 103515*4 103515*2 103515*4 103515*8 103515*16 103515*21 /

EQUALS PERMX PORO PERMX PORO / COPY PERMX PERMX /

0.1 0.2 10215 0.0073

PERMY PERMZ

1 1 149 149

1 1

335 335 187 187

335 335

1 1

1 1 155 155

309 309

309 309 155 155

1 1

1 1 1 1

9 9 9 9

/ / / /

-----

reservoir X permeability reservoir porosity equivalent fracture X perm. equivalent fracture porosity/

9 / 9 /

INIT GRIDFILE 0 1 / RPTGRID TRANX TRANY / PROPS

====================================================================

PVTW -- PREF 4014.7 /

BW(PREF) CW 1.0 3.0D-6

ROCK -- PREF 4014.7 / DENSITY -- OIL 44.09

VW(PREF) 1.0 0

CVW

CR 0

WATER 62.28

GAS 0.066 /

RPTPROPS / SOLUTION

=========================================================================

-DATUM -DEPTH EQUIL 5000

DATUM PRESS

OWC DEPTH

5000

1*

RPTSOL -- Fluid -- in place FIP=1

OWC PCOW

GOC DEPTH

1*

1*

GOC PCOG 1*

RSVD TABLE

1*

RVVD TABLE 1*

SOLN METH 1*

/

Create init Restart file RESTART=2 /

RPTRST BASIC=2 / SUMMARY

==========================================================================

96

Appendix A3 – Eclipse Data Input File for FCD=100, Constant Rate Case And Horizontal Well - Point Source - continued -- Well BHP WBHP / -- Well water production rate WWPR / RUNSUM EXCEL

SCHEDULE

==========================================================================

RPTRST BASIC=2 / RPTSCHED WELSPECS / WELSPECS -- WELL GROUP -- NAME NAME W1 G /

-LOCATIONI J 168 155

BHP PHASE DRAINAGE FLAG FLAG FLAG PRESS FLAG DEPTH RADIUS GAS SHUT CROSS TABLE DENS 1* WATER 1* STD SHUT NO 1* SEG

/

COMPDAT -- WELL --LOCATION-OPEN/ SAT CONN WELL EFF SKIN D PENETRATION -- NAME I J K1 K2 SHUT TAB FACT DIAM KH FACTOR FACTOR DIRECTION W1 168 155 5 5 OPEN 1* 1* 0.60 1* 0 0 Z / / WCONPROD -- WELL OPEN/ -- NAME SHUT W1 OPEN / TSTEP 0.00001157 3.46E-05 1.03E-04 3.08E-04 9.20E-04 2.75E-03 8.20E-03 2.45E-02 7.31E-02 2.18E-01 6.52E-01 1.95E+00 5.81E+00 1.74E+01 5.18E+01 1.55E+02 4.62E+02 1.38E+03 END

CNTL MODE WRAT

OIL WATER GAS LIQU RATE RATE RATE RATE 1* 100 1* 1*

1.39E-05 4.15E-05 1.24E-04 3.70E-04 1.10E-03 3.30E-03 9.84E-03 2.94E-02 8.78E-02 2.62E-01 7.83E-01 2.34E+00 6.98E+00 2.08E+01 6.22E+01 1.86E+02 5.55E+02 /

1.67E-05 4.98E-05 1.49E-04 4.44E-04 1.32E-03 3.96E-03 1.18E-02 3.53E-02 1.05E-01 3.15E-01 9.39E-01 2.80E+00 8.37E+00 2.50E+01 7.47E+01 2.23E+02 6.66E+02

RES BHP RATE 1* 1*

2.00E-05 5.97E-05 1.78E-04 5.32E-04 1.59E-03 4.75E-03 1.42E-02 4.23E-02 1.26E-01 3.77E-01 1.13E+00 3.37E+00 1.00E+01 3.00E+01 8.96E+01 2.67E+02 7.99E+02

THP 1*

VFP TABLE 1*

ALQ 1* /

2.40E-05 7.17E-05 2.14E-04 6.39E-04 1.91E-03 5.70E-03 1.70E-02 5.08E-02 1.52E-01 4.53E-01 1.35E+00 4.04E+00 1.21E+01 3.60E+01 1.08E+02 3.21E+02 9.58E+02

2.88E-05 8.60E-05 2.57E-04 7.67E-04 2.29E-03 6.84E-03 2.04E-02 6.10E-02 1.82E-01 5.43E-01 1.62E+00 4.85E+00 1.45E+01 4.32E+01 1.29E+02 3.85E+02 1.15E+03

97

Appendix A4 – Eclipse Data Input File for FCD=100, Constant Pressure Case And Horizontal Well - Point Source --- Vertical fracture, FCD=100, constant pressure, point source -NOECHO RUNSPEC

=========================================================================

TITLE Vertical hydraulic fracture model, (Wf)r=0.5 in ==> (Wf)e=2 ft

DIMENS ---- dx 335

dy dz 309 9

/

-- Fluid phases present WATER

-- Units FIELD --length of stack used by linear solver NSTACK 50 / ------------ #wells, WELLDIMS 1

# connections,

#groups,

#wells per group

1

1

1

/

-- Start simulation date START 1 JAN 1997 / -- run to be restarted from unified restart file UNIFIN -- Restart and summary files written are to be unified UNIFOUT GRID

===========================================================================

TOPS 103515*4950 103515*4971 103515*4987 103515*4995 103515*4999 103515*5001 103515*5005 103515*5013 103515*5029 / DXV -- reservoir 139*340 3*256 128 64 32 16 8 4 -- half fracture-1 2 4 8 16 32 64 128 6*256 128 64 32 16 8 4 -- well 2 -- half fracture-1 4 8 16 32 64 128 6*256 128 64 32 16 8 4 2 -- reservoir 4 8 16 32 64 128 3*256 139*340 / DYV 147*340 256 128 64 32 16 8 4 2 4 8 16 32 64 128 256 147*340 /

98

Appendix A4 – Eclipse Data Input File for FCD=100, Constant Pressure Case And Horizontal Well - Point Source - continued DZ 103515*21 103515*16 103515*8 103515*4 103515*2 103515*4 103515*8 103515*16 103515*21 /

EQUALS PERMX PORO PERMX PORO / COPY PERMX PERMX /

0.1 0.2 10215 0.0073

PERMY PERMZ

1 1 149 149

1 1

335 335 187 187

335 335

1 1

1 1 155 155

309 309 155 155

309 309

1 1

1 1 1 1

9 9 9 9

/ / / /

-----

reservoir X permeability reservoir Porosity equivalent fracture X permeability equivalent fracture X porosity/

9 / 9 /

INIT GRIDFILE 0 1 /

RPTGRID TRANX TRANY / PROPS

PVTW -- PREF 5000 /

==========================================================================

BW(PREF) CW 1 3.0D-6 1

ROCK -- PREF 5000 / DENSITY -- OIL 44.09

VW(PREF) 0

CVW

CR 0

WATER 62.28

GAS 0.066 /

RPTPROPS / SOLUTION

=========================================================================

-DATUM -DEPTH EQUIL 5000

DATUM PRESS

RPTSOL -- Fluid -- in place FIP=1 SUMMARY

5000

OWC DEPTH 1*

OWC PCOW

GOC DEPTH

1*

1*

GOC PCOG 1*

RSVD TABLE

1*

RVVD TABLE 1*

SOLN METH 1*

/

Create init Restart file RESTART=2 / ==========================================================================

-- Well quantities WBHP /

99

Appendix A4 – Eclipse Data Input File for FCD=100, Constant Pressure Case And Horizontal Well - Point Source - continued -- Well water production rate WWPR /

SCHEDULE

RPTRST BASIC=3

==========================================================================

FREQ=1 /

RPTSCHED WELSPECS / WELSPECS -- WELL GROUP -- NAME NAME W1 G /

-LOCATIONI J 168 155

BHP PHASE DRAINAGE FLAG FLAG FLAG PRESS FLAG DEPTH RADIUS GAS SHUT CROSS TABLE DENS 1* WATER 1* STD SHUT NO 1* SEG

COMPDAT -- WELL --LOCATION-OPEN/ -- NAME I J K1 K2 SHUT W1 168 155 5 5 OPEN / WCONPROD -- WELL OPEN/ -- NAME SHUT W1 OPEN / TSTEP 0.00001157 3.46E-05 1.03E-04 3.08E-04 9.20E-04 2.75E-03 8.20E-03 2.45E-02 7.31E-02 2.18E-01 6.52E-01 1.95E+00 5.81E+00 1.74E+01 5.18E+01 1.55E+02 4.62E+02 1.38E+03 4.12E+03

END

CNTL MODE BHP

SAT TAB 1*

CONN WELL FACT DIAM 1* 0.60

OIL WATER GAS LIQU RATE RATE RATE RATE 1* 1* 1* 1*

1.39E-05 4.15E-05 1.24E-04 3.70E-04 1.10E-03 3.30E-03 9.84E-03 2.94E-02 8.78E-02 2.62E-01 7.83E-01 2.34E+00 6.98E+00 2.08E+01 6.22E+01 1.86E+02 5.55E+02 1.66E+03 4.95E+03

1.67E-05 4.98E-05 1.49E-04 4.44E-04 1.32E-03 3.96E-03 1.18E-02 3.53E-02 1.05E-01 3.15E-01 9.39E-01 2.80E+00 8.37E+00 2.50E+01 7.47E+01 2.23E+02 6.66E+02 1.99E+03 5.93E+03

/

EFF SKIN D PENETRATION KH FACTOR FACTOR DIRECTION 1* 0 0 Z /

RES BHP RATE 1* 500

2.00E-05 5.97E-05 1.78E-04 5.32E-04 1.59E-03 4.75E-03 1.42E-02 4.23E-02 1.26E-01 3.77E-01 1.13E+00 3.37E+00 1.00E+01 3.00E+01 8.96E+01 2.67E+02 7.99E+02 2.39E+03 7.12E+03

THP 1*

VFP TABLE 1*

ALQ 1* /

2.40E-05 7.17E-05 2.14E-04 6.39E-04 1.91E-03 5.70E-03 1.70E-02 5.08E-02 1.52E-01 4.53E-01 1.35E+00 4.04E+00 1.21E+01 3.60E+01 1.08E+02 3.21E+02 9.58E+02 2.86E+03 8.55E+03

2.88E-05 8.60E-05 2.57E-04 7.67E-04 2.29E-03 6.84E-03 2.04E-02 6.10E-02 1.82E-01 5.43E-01 1.62E+00 4.85E+00 1.45E+01 4.32E+01 1.29E+02 3.85E+02 1.15E+03 3.43E+03 1.03E+04

/

100

Appendix A5 – Eclipse Data Input File for FCD=100, Constant Rate Case, xf/y=255 and Two Vertical Wells – Line Sources -- FCD=100, Constant Rate Case -- Vertical fracture xf/y=255, two vertical well

NOECHO RUNSPEC

=========================================================================

TITLE Vertical fracture model, (Wf)r=0.5 in ==> (Wf)e=2 ft

DIMENS ---- dx 335

dy dz 312 1

/

-- Fluid phases present WATER

-- Units FIELD --length of stack used by linear solver NSTACK 50 / ------------ #wells, WELLDIMS 2

# connections,

#groups,

#wells per group

1

2

2

/

-- Start simulation date START 1 JAN 1997 / -- run to be restarted from unified restart file UNIFIN -- Restart and summary files written are to be unified UNIFOUT GRID

===========================================================================

TOPS 104520*4950 / DXV -- reservoir 139*340 3*256 128 64 32 16 8 4 -- half fracture-1 2 4 8 16 32 64 128 6*256 128 64 32 16 8 4 -- well 2 -- half fracture-1 4 8 16 32 64 128 6*256 128 64 32 16 8 4 2 -- reservoir 4 8 16 32 64 128 3*256 139*340 / DYV 147*340 256 128 64 32 16 8 4 2 4 4 2 4 8 16 32 64 128 256 147*340 /

101

Appendix A5 – Eclipse Data Input File for FCD=100, Constant Rate Case, xf/y=255 and Two Vertical Wells – Line Sources - continued DZ 104520*100 /

EQUALS PERMX PORO PERMX PERMX PORO PORO / COPY PERMX PERMX /

0.1 0.2 10215 10215 0.0073 0.0073

PERMY PERMZ

1 1 149 149 149 149

1 1

335 335 187 187 187 187

335 335

1 1 155 158 155 158

1 1

312 312 155 158 155 158

312 312

1 1

1 1 1 1 1 1

1 1 1 1 1 1

/ / / / / /

-------

reservoir X permeability reservoir Porosity equivalent fracture X permeability equivalent fracture X permeability equivalent fracture porosity equivalent fracture porosity/

1 / 1 /

INIT GRIDFILE 0 1 / RPTGRID TRANX TRANY / PROPS

==========================================================================

PVTW -- PREF 4014.7 /

BW(PREF) CW 1.0 3.0D-6

ROCK -- PREF 4014.7 / DENSITY -- OIL 44.09

VW(PREF) 1.0 0

CVW

CR 0

WATER 62.28

GAS 0.066 /

RPTPROPS / SOLUTION

=========================================================================

-DATUM -DEPTH EQUIL 5000

DATUM PRESS

OWC DEPTH

5000

1*

RPTSOL -- Fluid -- in place FIP=1

OWC PCOW

GOC DEPTH

1*

1*

GOC PCOG 1*

RSVD TABLE

1*

RVVD TABLE 1*

SOLN METH 1*

/

Create init Restart file RESTART=2 /

RPTRST BASIC=2 / SUMMARY

==========================================================================

-- Well quantities

102

Appendix A5 – Eclipse Data Input File for FCD=100, Constant Rate Case, xf/y=255 and Two Vertical Wells – Line Sources - continued -- Well BHP WBHP / -- Well water production rate WWPR /

RUNSUM EXCEL

SCHEDULE

==========================================================================

RPTRST BASIC=2 / RPTSCHED WELSPECS / WELSPECS -- WELL GROUP -- NAME NAME W1 G W2 G /

-LOCATIONI J 168 155 168 158

BHP PHASE DRAINAGE FLAG FLAG FLAG PRESS FLAG DEPTH RADIUS GAS SHUT CROSS TABLE DENS 1* WATER 1* STD SHUT NO 1* SEG 1* WATER 1* STD SHUT NO 1* SEG

/ /

COMPDAT -- WELL --LOCATION-OPEN/ SAT CONN WELL EFF SKIN D PENETRATION -- NAME I J K1 K2 SHUT TAB FACT DIAM KH FACTOR FACTOR DIRECTION W1 168 155 1 1 OPEN 1* 1* 0.60 1* 0 0 Z / W2 168 158 1 1 OPEN 1* 1* 0.60 1* 0 0 Z / / WCONPROD -- WELL -- NAME W1 W2

OPEN/ SHUT OPEN OPEN

CNTL MODE WRAT WRAT

OIL RATE 1* 1*

WATER RATE 100 100

GAS RATE 1* 1*

LIQU RATE 1* 1*

RES BHP RATE 1* 1* 1* 1*

THP 1* 1*

VFP TABLE 1* 1*

ALQ 1* / 1* /

/ TSTEP 1.03E-04 3.08E-04 9.20E-04 2.75E-03 8.20E-03 2.45E-02 7.31E-02 2.18E-01 6.52E-01 1.95E+00 5.81E+00 1.74E+01 5.18E+01 1.55E+02 4.62E+02 / END

1.24E-04 3.70E-04 1.10E-03 3.30E-03 9.84E-03 2.94E-02 8.78E-02 2.62E-01 7.83E-01 2.34E+00 6.98E+00 2.08E+01 6.22E+01 1.86E+02 5.55E+02

1.49E-04 4.44E-04 1.32E-03 3.96E-03 1.18E-02 3.53E-02 1.05E-01 3.15E-01 9.39E-01 2.80E+00 8.37E+00 2.50E+01 7.47E+01 2.23E+02 6.66E+02

1.78E-04 5.32E-04 1.59E-03 4.75E-03 1.42E-02 4.23E-02 1.26E-01 3.77E-01 1.13E+00 3.37E+00 1.00E+01 3.00E+01 8.96E+01 2.67E+02

2.14E-04 6.39E-04 1.91E-03 5.70E-03 1.70E-02 5.08E-02 1.52E-01 4.53E-01 1.35E+00 4.04E+00 1.21E+01 3.60E+01 1.08E+02 3.21E+02

2.57E-04 7.67E-04 2.29E-03 6.84E-03 2.04E-02 6.10E-02 1.82E-01 5.43E-01 1.62E+00 4.85E+00 1.45E+01 4.32E+01 1.29E+02 3.85E+02

103

Appendix A6 – Eclipse Data Input File for FCD=100, Constant Pressure Case, xf/y=128 and Two Vertical Wells – Line Sources -- FCD=100, Constant Pressure Case -- Vertical fracture xf/y=128, two vertical wells NOECHO RUNSPEC

=========================================================================

TITLE Vertical fracture model, (Wf)r=0.5 in ==> (Wf)e=2 ft

DIMENS ---- dx 335

dy dz 314 1

/

-- Fluid phases present WATER

-- Units FIELD --length of stack used by linear solver NSTACK 50 / ------------ #wells, WELLDIMS 2

# connections,

#groups,

#wells per group

1

2

2

/

-- Start simulation date START 1 JAN 1997 / -- run to be restarted from unified restart file UNIFIN -- Restart and summary files written are to be unified UNIFOUT GRID

===========================================================================

TOPS 105190*4950 / DXV -- reservoir 139*340 3*256 128 64 32 16 8 4 -- half fracture-1 2 4 8 16 32 64 128 6*256 128 64 32 16 8 4 -- well 2 -- half fracture-1 4 8 16 32 64 128 6*256 128 64 32 16 8 4 2 -- reservoir 4 8 16 32 64 128 3*256 139*340 / DYV 147*340 256 128 64 32 16 8 4 2 4 8 8 4 2 4 8 16 32 64 128 256 147*340 /

104

Appendix A6 – Eclipse Data Input File for FCD=100, Constant Pressure Case, xf/y=128 and Two Vertical Wells – Line Sources - continued DZ 105190*100 /

EQUALS PERMX PORO PERMX PERMX PORO PORO / COPY PERMX PERMX /

0.1 0.2 10215 10215 0.0073 0.0073

PERMY PERMZ

1 1 149 149 149 149

1 1

335 335 187 187 187 187

335 335

1 1

1 1 155 160 155 160

314 314

314 314 155 160 155 160

1 1

1 1 1 1 1 1

1 1 1 1 1 1

/ / / / / /

-------

reservoir X permeability reservoir Porosity equivalent fracture X permeability equivalent fracture X permeability equivalent fracture porosity equivalent fracture porosity/

1 / 1 /

INIT GRIDFILE 0 1 / RPTGRID TRANX TRANY / PROPS

==========================================================================

PVTW -- PREF 4014.7 /

BW(PREF) CW 1.0 3.0D-6

ROCK -- PREF 4014.7 / DENSITY -- OIL 44.09

VW(PREF) 1.0 0

CVW

CR 0

WATER 62.28

GAS 0.066 /

RPTPROPS / SOLUTION

=========================================================================

-DATUM -DEPTH EQUIL 5000

DATUM PRESS

OWC DEPTH

5000

1*

RPTSOL -- Fluid -- in place FIP=1

OWC PCOW

GOC DEPTH

1*

1*

GOC PCOG 1*

RSVD TABLE

1*

RVVD TABLE 1*

SOLN METH 1*

/

Create init Restart file RESTART=2 /

RPTRST BASIC=2 / SUMMARY

==========================================================================

105

Appendix A6 – Eclipse Data Input File for FCD=100, Constant Pressure Case, xf/y=128 and Two Vertical Wells – Line Sources - continued -- Well BHP WBHP / -- Well water production rate WWPR /

RUNSUM EXCEL

SCHEDULE

==========================================================================

RPTRST BASIC=2 / RPTSCHED WELSPECS / WELSPECS -- WELL GROUP -- NAME NAME W1 G W2 G /

-LOCATIONI J 168 155 168 160

BHP PHASE DRAINAGE FLAG FLAG FLAG PRESS FLAG DEPTH RADIUS GAS SHUT CROSS TABLE DENS 1* WATER 1* STD SHUT NO 1* SEG 1* WATER 1* STD SHUT NO 1* SEG

/ /

COMPDAT -- WELL --LOCATION-OPEN/ SAT CONN WELL EFF SKIN D PENETRATION -- NAME I J K1 K2 SHUT TAB FACT DIAM KH FACTOR FACTOR DIRECTION W1 168 155 1 1 OPEN 1* 1* 0.60 1* 0 0 Z / W2 168 160 1 1 OPEN 1* 1* 0.60 1* 0 0 Z / / WCONPROD -- WELL -- NAME W1 W2

OPEN/ SHUT OPEN OPEN

CNTL MODE BHP BHP

OIL WATER GAS LIQU RATE RATE RATE RATE 1* 1* 1* 1* 1* 1* 1* 1*

RES BHP RATE 1* 500 1* 500

THP 1* 1*

VFP TABLE 1* 1*

ALQ 1* / 1* /

/ TSTEP 1.03E-04 3.08E-04 9.20E-04 2.75E-03 8.20E-03 2.45E-02 7.31E-02 2.18E-01 6.52E-01 1.95E+00 5.81E+00 1.74E+01 5.18E+01 1.55E+02 4.62E+02 / END

1.24E-04 3.70E-04 1.10E-03 3.30E-03 9.84E-03 2.94E-02 8.78E-02 2.62E-01 7.83E-01 2.34E+00 6.98E+00 2.08E+01 6.22E+01 1.86E+02 5.55E+02

1.49E-04 4.44E-04 1.32E-03 3.96E-03 1.18E-02 3.53E-02 1.05E-01 3.15E-01 9.39E-01 2.80E+00 8.37E+00 2.50E+01 7.47E+01 2.23E+02 6.66E+02

1.78E-04 5.32E-04 1.59E-03 4.75E-03 1.42E-02 4.23E-02 1.26E-01 3.77E-01 1.13E+00 3.37E+00 1.00E+01 3.00E+01 8.96E+01 2.67E+02

2.14E-04 6.39E-04 1.91E-03 5.70E-03 1.70E-02 5.08E-02 1.52E-01 4.53E-01 1.35E+00 4.04E+00 1.21E+01 3.60E+01 1.08E+02 3.21E+02

2.57E-04 7.67E-04 2.29E-03 6.84E-03 2.04E-02 6.10E-02 1.82E-01 5.43E-01 1.62E+00 4.85E+00 1.45E+01 4.32E+01 1.29E+02 3.85E+02

106

Appendix A7 – Eclipse Data Input File for FCD=100, Constant Rate Case, xf/y=255 and Horizontal Well - Point Source -- Constant rate case, point source -- FCD=100 -- Two vertical fractures, xf/y=255 NOECHO RUNSPEC

=========================================================================

TITLE Vertical fracture model, (Wf)r=0.5 in ==> (Wf)e=2 ft

DIMENS ---- dx 335

dy dz 312 9

/

-- Fluid phases present WATER -- Units FIELD --length of stack used by linear solver NSTACK 50 / ------------ #wells, WELLDIMS 2

# connections,

#groups,

#wells per group

1

2

2

/

-- Start simulation date START 1 JAN 1997 / -- run to be restarted from unified restart file UNIFIN -- Restart and summary files written are to be unified UNIFOUT GRID

===========================================================================

TOPS 104520*4950 104520*4971 104520*4987 104520*4995 104520*4999 104520*5001 104520*5005 104520*5013 104520*5029 / DXV -- reservoir 139*340 3*256 128 64 32 16 8 4 -- half fracture-1 2 4 8 16 32 64 128 6*256 128 64 32 16 8 4 -- well 2 -- half fracture-1 4 8 16 32 64 128 6*256 128 64 32 16 8 4 2 -- reservoir 4 8 16 32 64 128 3*256 139*340 / DYV 147*340 256 128 64 32 16 8 4 2 4 4 2 4 8 16 32 64 128 256 147*340 /

107

Appendix A7 – Eclipse Data Input File for FCD=100, Constant Rate Case, xf/y=255 and Horizontal Well - Point Source - continued DZ 104520*21 104520*16 104520*8 104520*4 104520*2 104520*4 104520*8 104520*16 104520*21 / EQUALS PERMX PORO PERMX PERMX PORO PORO / COPY PERMX PERMX /

0.1 0.2 10215 10215 0.0073 0.0073

PERMY PERMZ

1 1 149 149 149 149

1 1

335 335 187 187 187 187

335 335

1 1

1 1 155 158 155 158

312 312

312 312 155 158 155 158

1 1

1 1 1 1 1 1

9 9 9 9 9 9

/ / / / / /

-------

reservoir X permeability reservoir Porosity equivalent fracture X permeability equivalent fracture X permeability equivalent fracture porosity equivalent fracture porosity/

9 / 9 /

INIT GRIDFILE 0 1 / RPTGRID TRANX TRANY / PROPS

==========================================================================

PVTW -- PREF 4014.7 /

BW(PREF) CW 1.0 3.0D-6

ROCK -- PREF 4014.7 / DENSITY -- OIL 44.09

VW(PREF) 1.0 0

CVW

CR 0

WATER 62.28

GAS 0.066 /

RPTPROPS / SOLUTION

=========================================================================

-DATUM -DEPTH EQUIL 5000

DATUM PRESS

OWC DEPTH

5000

1*

RPTSOL -- Fluid -- in place FIP=1

OWC PCOW

GOC DEPTH

1*

1*

GOC PCOG 1*

RSVD TABLE

1*

RVVD TABLE 1*

SOLN METH 1*

/

Create init Restart file RESTART=2 /

RPTRST BASIC=2 / SUMMARY

==========================================================================

108

Appendix A7 – Eclipse Data Input File for FCD=100, Constant Rate Case, xf/y=255 and Horizontal Well - Point Source - continued -- Well BHP WBHP / -- Well water production rate WWPR /

RUNSUM EXCEL

SCHEDULE

==========================================================================

RPTRST BASIC=2 / RPTSCHED WELSPECS / WELSPECS -- WELL GROUP -- NAME NAME W1 G W2 G /

-LOCATIONI J 168 155 168 158

BHP PHASE DRAINAGE FLAG FLAG FLAG PRESS FLAG DEPTH RADIUS GAS SHUT CROSS TABLE DENS 1* WATER 1* STD SHUT NO 1* SEG 1* WATER 1* STD SHUT NO 1* SEG

COMPDAT -- WELL --LOCATION-OPEN/ -- NAME I J K1 K2 SHUT W1 168 155 5 5 OPEN W2 168 158 5 5 OPEN

SAT TAB 1* 1*

CONN WELL FACT DIAM 1* 0.60 1* 0.60

/ /

EFF SKIN D PENETRATION KH FACTOR FACTOR DIRECTION 1* 0 0 Z / 1* 0 0 Z /

/ WCONPROD -- WELL -- NAME W1 W2

OPEN/ SHUT OPEN OPEN

CNTL MODE WRAT WRAT

OIL RATE 1* 1*

WATER RATE 100 100

GAS RATE 1* 1*

LIQU RATE 1* 1*

RES BHP RATE 1* 1* 1* 1*

THP 1* 1*

VFP TABLE 1* 1*

ALQ 1* / 1* /

/ TSTEP 1.03E-04 3.08E-04 9.20E-04 2.75E-03 8.20E-03 2.45E-02 7.31E-02 2.18E-01 6.52E-01 1.95E+00 5.81E+00 1.74E+01 5.18E+01 1.55E+02 4.62E+02 / END

1.24E-04 3.70E-04 1.10E-03 3.30E-03 9.84E-03 2.94E-02 8.78E-02 2.62E-01 7.83E-01 2.34E+00 6.98E+00 2.08E+01 6.22E+01 1.86E+02 5.55E+02

1.49E-04 4.44E-04 1.32E-03 3.96E-03 1.18E-02 3.53E-02 1.05E-01 3.15E-01 9.39E-01 2.80E+00 8.37E+00 2.50E+01 7.47E+01 2.23E+02 6.66E+02

1.78E-04 5.32E-04 1.59E-03 4.75E-03 1.42E-02 4.23E-02 1.26E-01 3.77E-01 1.13E+00 3.37E+00 1.00E+01 3.00E+01 8.96E+01 2.67E+02

2.14E-04 6.39E-04 1.91E-03 5.70E-03 1.70E-02 5.08E-02 1.52E-01 4.53E-01 1.35E+00 4.04E+00 1.21E+01 3.60E+01 1.08E+02 3.21E+02

2.57E-04 7.67E-04 2.29E-03 6.84E-03 2.04E-02 6.10E-02 1.82E-01 5.43E-01 1.62E+00 4.85E+00 1.45E+01 4.32E+01 1.29E+02 3.85E+02

109

Appendix A8 – Eclipse Data Input File for FCD=100, Constant Pressure Case, xf/y=255 and Horizontal Well - Point Source -- Constant pressure case, point source -- FCD=100 -- Two vertical fractures, xf/y=255

NOECHO RUNSPEC

=========================================================================

TITLE Vertical fracture model, (Wf)r=0.5 in ==> (Wf)e=2 ft

DIMENS ---- dx 335

dy dz 312 9

/

-- Fluid phases present WATER

-- Units FIELD --length of stack used by linear solver NSTACK 50 / ------------ #wells, WELLDIMS 2

# connections,

#groups,

#wells per group

1

2

2

/

-- Start simulation date START 1 JAN 1997 / -- run to be restarted from unified restart file UNIFIN -- Restart and summary files written are to be unified UNIFOUT GRID

===========================================================================

TOPS 104520*4950 104520*4971 104520*4987 104520*4995 104520*4999 104520*5001 104520*5005 104520*5013 104520*5029 / DXV -- reservoir 139*340 3*256 128 64 32 16 8 4 -- half fracture-1 2 4 8 16 32 64 128 6*256 128 64 32 16 8 4 -- well 2 -- half fracture-1 4 8 16 32 64 128 6*256 128 64 32 16 8 4 2 -- reservoir 4 8 16 32 64 128 3*256 139*340 / DYV 147*340 256 128 64 32 16 8 4 2 4 4 2 4 8 16 32 64 128 256 147*340 /

110

Appendix A8 – Eclipse Data Input File for FCD=100, Constant Pressure Case, xf/y=255 and Horizontal Well - Point Source - continued DZ 104520*21 104520*16 104520*8 104520*4 104520*2 104520*4 104520*8 104520*16 104520*21 / EQUALS PERMX PORO PERMX PERMX PORO PORO / COPY PERMX PERMX /

0.1 0.2 10215 10215 0.0073 0.0073

PERMY PERMZ

1 1 149 149 149 149

1 1

335 335 187 187 187 187

335 335

1 1

1 1 155 158 155 158

312 312

312 312 155 158 155 158

1 1

1 1 1 1 1 1

9 9 9 9 9 9

/ / / / / /

-------

reservoir X permeability reservoir Porosity equivalent fracture X permeability equivalent fracture X permeability equivalent fracture porosity equivalent fracture porosity/

9 / 9 /

INIT GRIDFILE 0 1 / RPTGRID TRANX TRANY / PROPS

==========================================================================

PVTW -- PREF 4014.7 /

BW(PREF) CW 1.0 3.0D-6

ROCK -- PREF 4014.7 / DENSITY -- OIL 44.09

VW(PREF) 1.0 0

CVW

CR 0

WATER 62.28

GAS 0.066 /

RPTPROPS / SOLUTION

=========================================================================

-DATUM -DEPTH EQUIL 5000

DATUM PRESS

OWC DEPTH

5000

1*

RPTSOL -- Fluid -- in place FIP=1

OWC PCOW

GOC DEPTH

1*

1*

GOC PCOG 1*

RSVD TABLE

1*

RVVD TABLE 1*

SOLN METH 1*

/

Create init Restart file RESTART=2 /

RPTRST BASIC=2 / SUMMARY

==========================================================================

111

Appendix A8 – Eclipse Data Input File for FCD=100, Constant Pressure Case, xf/y=255 and Horizontal Well - Point Source - continued -- Well BHP WBHP / -- Well water production rate WWPR /

RUNSUM EXCEL

SCHEDULE

==========================================================================

RPTRST BASIC=2 / RPTSCHED WELSPECS / WELSPECS -- WELL GROUP -- NAME NAME W1 G W2 G /

-LOCATIONI J 168 155 168 158

BHP PHASE DRAINAGE FLAG FLAG FLAG PRESS FLAG DEPTH RADIUS GAS SHUT CROSS TABLE DENS 1* WATER 1* STD SHUT NO 1* SEG 1* WATER 1* STD SHUT NO 1* SEG

COMPDAT -- WELL --LOCATION-OPEN/ -- NAME I J K1 K2 SHUT W1 168 155 5 5 OPEN W2 168 158 5 5 OPEN

SAT TAB 1* 1*

CONN WELL FACT DIAM 1* 0.60 1* 0.60

/ /

EFF SKIN D PENETRATION KH FACTOR FACTOR DIRECTION 1* 0 0 Z / 1* 0 0 Z /

/ WCONPROD -- WELL -- NAME W1 W2

OPEN/ SHUT OPEN OPEN

CNTL OIL MODE RATE BHP 1* BHP 1*

WATER GAS LIQU RES BHP RATE RATE RATE RATE 1* 1* 1* 1* 500 1* 1* 1* 1* 500

THP 1* 1*

VFP TABLE 1* 1*

ALQ 1* / 1* /

/ TSTEP 1.03E-04 3.08E-04 9.20E-04 2.75E-03 8.20E-03 2.45E-02 7.31E-02 2.18E-01 6.52E-01 1.95E+00 5.81E+00 1.74E+01 5.18E+01 1.55E+02 4.62E+02 / END

1.24E-04 3.70E-04 1.10E-03 3.30E-03 9.84E-03 2.94E-02 8.78E-02 2.62E-01 7.83E-01 2.34E+00 6.98E+00 2.08E+01 6.22E+01 1.86E+02 5.55E+02

1.49E-04 4.44E-04 1.32E-03 3.96E-03 1.18E-02 3.53E-02 1.05E-01 3.15E-01 9.39E-01 2.80E+00 8.37E+00 2.50E+01 7.47E+01 2.23E+02 6.66E+02

1.78E-04 5.32E-04 1.59E-03 4.75E-03 1.42E-02 4.23E-02 1.26E-01 3.77E-01 1.13E+00 3.37E+00 1.00E+01 3.00E+01 8.96E+01 2.67E+02

2.14E-04 6.39E-04 1.91E-03 5.70E-03 1.70E-02 5.08E-02 1.52E-01 4.53E-01 1.35E+00 4.04E+00 1.21E+01 3.60E+01 1.08E+02 3.21E+02

2.57E-04 7.67E-04 2.29E-03 6.84E-03 2.04E-02 6.10E-02 1.82E-01 5.43E-01 1.62E+00 4.85E+00 1.45E+01 4.32E+01 1.29E+02 3.85E+02

112

Appendix A9 – Eclipse Data Input File for FCD=100, Constant Rate Case, xf/y=255 and Fracture Half-Length 506[ft] for Vertical Well – Line Sources -- Constant rate case, FCD=100 -- Fracture half-length 506 -- Vertical fracture xf/y=255 NOECHO RUNSPEC

=========================================================================

TITLE Vertical fracture model, (Wf)r=0.5 in ==> (Wf)e=2 ft

DIMENS ---- dx 323

dy dz 311 1

/

-- Fluid phases present WATER

-- Units FIELD --length of stack used by linear solver NSTACK 50 / ------------ #wells, WELLDIMS 2

# connections, 2

#groups,

#wells per group

1

2

/

-- Start simulation date START 1 JAN 1997 / -- run to be restarted from unified restart file UNIFIN -- Restart and summary files written are to be unified UNIFOUT GRID

===========================================================================

TOPS 100453*4950 / DXV -- reservoir 139*340 3*256 128 64 32 16 8 4 -- half fracture-1 2 4 8 16 32 64 128 128 64 32 16 8 4 -- well 2 -- half fracture-1 4 8 16 32 64 128 128 64 32 16 8 4 2 -- reservoir 4 8 16 32 64 128 3*256 139*340 / DYV 147*340 256 128 64 32 16 8 4 2 2 2 4 8 16 32 64 128 256 147*340 /

113

Appendix A9 – Eclipse Data Input File for FCD=100, Constant Rate Case, xf/y=255 and Fracture Half-Length 506[ft] for Vertical Well – Line Sources - continued DZ 100453*100 /

EQUALS PERMX PORO PERMX PERMX PORO PORO / COPY PERMX PERMX /

0.1 0.2 2530 2530 0.0073 0.0073

PERMY PERMZ

1 1 149 149 149 149

1 1

323 323 175 175 175 175

323 323

1 1

1 1 155 157 155 157

311 311

311 311 155 157 155 157

1 1

1 1 1 1 1 1

1 1 1 1 1 1

/ / / / / /

-------

reservoir X permeability reservoir Porosity equivalent fracture X permeability equivalent fracture X permeability equivalent fracture porosity equivalent fracture porosity/

1 / 1 /

INIT GRIDFILE 0 1 / RPTGRID TRANX TRANY / PROPS

==========================================================================

PVTW -- PREF 4014.7 /

BW(PREF) CW 1.0 3.0D-6

ROCK -- PREF 4014.7 / DENSITY -- OIL 44.09

VW(PREF) 1.0 0

CVW

CR 0

WATER 62.28

GAS 0.066 /

RPTPROPS / SOLUTION

=========================================================================

-DATUM -DEPTH EQUIL 5000

DATUM PRESS

OWC DEPTH

5000

1*

RPTSOL -- Fluid -- in place FIP=1

OWC PCOW

GOC DEPTH

1*

1*

GOC PCOG 1*

RSVD TABLE

1*

RVVD TABLE 1*

SOLN METH 1*

/

Create init Restart file RESTART=2 /

RPTRST BASIC=2 / SUMMARY

==========================================================================

114

Appendix A9 – Eclipse Data Input File for FCD=100, Constant Rate Case, xf/y=255 and Fracture Half-Length 506[ft] for Vertical Well – Line Sources - continued -- Well quantities WBHP / -- Well water production rate WWPR /

RUNSUM EXCEL

SCHEDULE

==========================================================================

RPTRST BASIC=2 / RPTSCHED WELSPECS / WELSPECS -- WELL GROUP -- NAME NAME W1 G W2 G /

-LOCATIONI J 162 155 162 157

BHP PHASE DRAINAGE FLAG FLAG FLAG PRESS FLAG DEPTH RADIUS GAS SHUT CROSS TABLE DENS 1* WATER 1* STD SHUT NO 1* SEG 1* WATER 1* STD SHUT NO 1* SEG

COMPDAT -- WELL --LOCATION-OPEN/ SAT CONN WELL EFF SKIN D PENETRATION -- NAME I J K1 K2 SHUT TAB FACT DIAM KH FACTOR FACTOR DIRECTION W1 162 155 1 1 OPEN 1* 1* 0.60 1* 0 0 Z / W2 162 157 1 1 OPEN 1* 1* 0.60 1* 0 0 Z / / WCONPROD -- WELL -- NAME W1 W2

OPEN/ SHUT OPEN OPEN

CNTL MODE WRAT WRAT

OIL RATE 1* 1*

WATER RATE 100 100

GAS RATE 1* 1*

LIQU RATE 1* 1*

RES BHP RATE 1* 1* 1* 1*

THP 1* 1*

VFP TABLE 1* 1*

ALQ 1* / 1* /

/ TSTEP 1.18E-02 2.94E-02 7.31E-02 1.82E-01 4.52E-01 1.13E+00 2.80E+00 6.97E+00 1.74E+01 4.33E+01 1.08E+02 2.68E+02 6.67E+02 END

1.42E-02 3.52E-02 8.77E-02 2.18E-01 5.43E-01 1.35E+00 3.36E+00 8.36E+00 2.09E+01 5.20E+01 1.29E+02 3.22E+02 8.00E+02

1.70E-02 4.23E-02 1.05E-01 2.62E-01 6.51E-01 1.62E+00 4.03E+00 1.00E+01 2.51E+01 6.23E+01 1.55E+02 3.86E+02 9.61E+02

2.04E-02 5.07E-02 1.26E-01 3.14E-01 7.82E-01 1.95E+00 4.84E+00 1.20E+01 3.01E+01 7.48E+01 1.86E+02 4.63E+02 1.15E+03

2.45E-02 6.09E-02 1.52E-01 3.77E-01 9.38E-01 2.33E+00 5.81E+00 1.45E+01 3.61E+01 8.98E+01 2.23E+02 5.56E+02 1.38E+03

/

/ /

115

APPENDIX

B

116

Table 1 – FCD=1 – Results of Numerical Simulation and Dimensionless Time, Pressure and Rate t [days] 0 1.16E-05 2.55E-05 4.22E-05 6.22E-05 8.62E-05 1.15E-04 1.50E-04 1.91E-04 2.41E-04 3.01E-04 3.72E-04 4.58E-04 5.61E-04 6.85E-04 8.34E-04 1.01E-03 1.23E-03 1.48E-03 1.79E-03 2.16E-03 2.61E-03 3.14E-03 3.78E-03 4.54E-03 5.46E-03 6.56E-03 7.88E-03 9.47E-03 1.14E-02 1.37E-02 1.64E-02 1.97E-02 2.37E-02 2.84E-02 3.41E-02 4.10E-02 4.92E-02 5.90E-02 7.08E-02 8.50E-02 1.02E-01 1.22E-01 1.47E-01

tDxf 0 2.92E-09 6.44E-09 1.07E-08 1.57E-08 2.18E-08 2.91E-08 3.78E-08 4.83E-08 6.09E-08 7.60E-08 9.41E-08 1.16E-07 1.42E-07 1.73E-07 2.11E-07 2.56E-07 3.10E-07 3.75E-07 4.53E-07 5.46E-07 6.58E-07 7.93E-07 9.54E-07 1.15E-06 1.38E-06 1.66E-06 1.99E-06 2.39E-06 2.88E-06 3.46E-06 4.15E-06 4.98E-06 5.99E-06 7.19E-06 8.63E-06 1.04E-05 1.24E-05 1.49E-05 1.79E-05 2.15E-05 2.58E-05 3.09E-05 3.71E-05

q [stb/day] 0 11,480 6,451 5,335 4,870 4,635 4,506 4,432 4,387 4,356 4,331 4,308 4,283 4,254 4,221 4,183 4,138 4,085 4,025 3,955 3,875 3,785 3,684 3,573 3,452 3,322 3,186 3,045 2,903 2,763 2,627 2,499 2,379 2,269 2,167 2,072 1,982 1,896 1,814 1,733 1,656 1,582 1,512 1,445

qD 0 3.60E+01 2.02E+01 1.67E+01 1.53E+01 1.45E+01 1.41E+01 1.39E+01 1.38E+01 1.37E+01 1.36E+01 1.35E+01 1.34E+01 1.33E+01 1.32E+01 1.31E+01 1.30E+01 1.28E+01 1.26E+01 1.24E+01 1.22E+01 1.19E+01 1.16E+01 1.12E+01 1.08E+01 1.04E+01 1.00E+01 9.56E+00 9.11E+00 8.67E+00 8.24E+00 7.84E+00 7.47E+00 7.12E+00 6.80E+00 6.50E+00 6.22E+00 5.95E+00 5.69E+00 5.44E+00 5.20E+00 4.96E+00 4.74E+00 4.53E+00

p [psi] 5000 4,961.11 4,942.71 4,929.86 4,920.28 4,913.04 4,907.68 4,903.80 4,901.09 4,899.23 4,897.95 4,897.00 4,896.21 4,895.45 4,894.62 4,893.68 4,892.57 4,891.27 4,889.74 4,887.94 4,885.84 4,883.39 4,880.57 4,877.31 4,873.60 4,869.40 4,864.71 4,859.50 4,853.75 4,847.51 4,840.80 4,833.68 4,826.18 4,818.33 4,810.17 4,801.67 4,792.82 4,783.56 4,773.84 4,763.62 4,752.82 4,741.50 4,729.63 4,717.21

pD n/a 2.75E-02 4.06E-02 4.97E-02 5.65E-02 6.16E-02 6.54E-02 6.81E-02 7.00E-02 7.14E-02 7.23E-02 7.29E-02 7.35E-02 7.40E-02 7.46E-02 7.53E-02 7.61E-02 7.70E-02 7.81E-02 7.94E-02 8.09E-02 8.26E-02 8.46E-02 8.69E-02 8.95E-02 9.25E-02 9.58E-02 9.95E-02 1.04E-01 1.08E-01 1.13E-01 1.18E-01 1.23E-01 1.29E-01 1.34E-01 1.40E-01 1.47E-01 1.53E-01 1.60E-01 1.67E-01 1.75E-01 1.83E-01 1.91E-01 2.00E-01

117

Table 1 – FCD=1 – Results of Numerical Simulation and Dimensionless Time, Pressure and Rate - continued t [days] 1.76E-01 2.12E-01 2.54E-01 3.05E-01 3.66E-01 4.39E-01 5.27E-01 6.32E-01 7.58E-01 9.10E-01 1.09E+00 1.31E+00 1.57E+00 1.89E+00 2.26E+00 2.72E+00 3.26E+00 3.91E+00 4.69E+00 5.63E+00 6.76E+00 8.11E+00 9.73E+00 1.17E+01 1.40E+01 1.68E+01 2.02E+01 2.42E+01 2.91E+01 3.49E+01 4.19E+01 5.02E+01 6.02E+01 7.23E+01 8.68E+01 1.04E+02 1.25E+02 1.50E+02 1.80E+02 2.16E+02 2.59E+02 3.11E+02 3.73E+02 4.48E+02 5.38E+02

tDxf q [stb/day] 4.46E-05 1,381 5.35E-05 1,321 6.42E-05 1,263 7.70E-05 1,208 9.24E-05 1,155 1.11E-04 1,104 1.33E-04 1,056 1.60E-04 1,010 1.91E-04 966 2.30E-04 924 2.76E-04 885 3.31E-04 847 3.97E-04 810 4.77E-04 775 5.72E-04 742 6.87E-04 710 8.24E-04 680 9.89E-04 652 1.19E-03 624 1.42E-03 598 1.71E-03 573 2.05E-03 549 2.46E-03 526 2.95E-03 504 3.54E-03 483 4.25E-03 463 5.10E-03 444 6.12E-03 426 7.35E-03 408 8.82E-03 392 1.06E-02 376 1.27E-02 361 1.52E-02 346 1.83E-02 333 2.19E-02 320 2.63E-02 307 3.16E-02 295 3.79E-02 284 4.55E-02 273 5.46E-02 262 6.55E-02 252 7.86E-02 242 9.43E-02 232 1.13E-01 223 1.36E-01 214

qD 4.33E+00 4.15E+00 3.96E+00 3.79E+00 3.62E+00 3.46E+00 3.31E+00 3.17E+00 3.03E+00 2.90E+00 2.78E+00 2.66E+00 2.54E+00 2.43E+00 2.33E+00 2.23E+00 2.13E+00 2.04E+00 1.96E+00 1.88E+00 1.80E+00 1.72E+00 1.65E+00 1.58E+00 1.52E+00 1.45E+00 1.39E+00 1.34E+00 1.28E+00 1.23E+00 1.18E+00 1.13E+00 1.09E+00 1.04E+00 1.00E+00 9.63E-01 9.26E-01 8.90E-01 8.55E-01 8.22E-01 7.90E-01 7.59E-01 7.29E-01 7.00E-01 6.72E-01

p [psi] 4,704.24 4,690.69 4,676.54 4,661.71 4,646.15 4,629.88 4,612.84 4,595.10 4,576.59 4,557.19 4,537.01 4,515.99 4,493.99 4,472.96 4,449.37 4,424.22 4,399.38 4,372.86 4,345.06 4,316.60 4,286.66 4,255.43 4,222.96 4,188.75 4,153.52 4,116.44 4,077.95 4,037.91 3,996.19 3,952.88 3,907.90 3,861.13 3,812.92 3,762.64 3,710.70 3,656.93 3,601.32 3,543.52 3,483.49 3,421.05 3,356.09 3,288.44 3,217.92 3,144.33 3,067.62

pD 2.09E-01 2.19E-01 2.29E-01 2.40E-01 2.51E-01 2.62E-01 2.74E-01 2.87E-01 3.00E-01 3.14E-01 3.28E-01 3.43E-01 3.58E-01 3.73E-01 3.90E-01 4.08E-01 4.25E-01 4.44E-01 4.64E-01 4.84E-01 5.05E-01 5.27E-01 5.50E-01 5.75E-01 5.99E-01 6.26E-01 6.53E-01 6.81E-01 7.11E-01 7.42E-01 7.73E-01 8.07E-01 8.41E-01 8.76E-01 9.13E-01 9.51E-01 9.91E-01 1.03E+00 1.07E+00 1.12E+00 1.16E+00 1.21E+00 1.26E+00 1.31E+00 1.37E+00

118

Table 1 – FCD=1 – Results of Numerical Simulation and Dimensionless Time, Pressure and Rate - continued t [days] 6.46E+02 7.75E+02 9.30E+02 1.12E+03 1.34E+03 1.61E+03 1.93E+03 2.12E+03 2.31E+03 2.54E+03 2.77E+03 3.05E+03 3.33E+03 3.66E+03 3.99E+03 4.36E+03 4.58E+03 4.79E+03 5.16E+03 5.46E+03 5.75E+03 6.12E+03 6.48E+03 6.69E+03 6.90E+03 7.27E+03 7.63E+03 7.96E+03 8.28E+03 8.65E+03 9.01E+03 9.38E+03 9.66E+03 9.94E+03 1.03E+04 1.07E+04 1.10E+04 1.14E+04 1.17E+04 1.19E+04 1.23E+04 1.27E+04 1.30E+04 1.34E+04 1.38E+04

tDxf q [stb/day] 1.63E-01 206 1.96E-01 197 2.35E-01 190 2.82E-01 182 3.38E-01 175 4.06E-01 168 4.87E-01 162 5.36E-01 158 5.84E-01 155 6.43E-01 152 7.01E-01 149 7.71E-01 146 8.41E-01 144 9.25E-01 141 1.01E+00 139 1.10E+00 136 1.16E+00 135 1.21E+00 134 1.30E+00 132 1.38E+00 130 1.45E+00 129 1.55E+00 128 1.64E+00 126 1.69E+00 126 1.74E+00 125 1.84E+00 124 1.93E+00 123 2.01E+00 122 2.09E+00 121 2.19E+00 120 2.28E+00 119 2.37E+00 118 2.44E+00 118 2.51E+00 117 2.60E+00 116 2.70E+00 116 2.79E+00 115 2.88E+00 114 2.95E+00 114 3.02E+00 113 3.11E+00 113 3.20E+00 112 3.29E+00 112 3.38E+00 111 3.48E+00 111

qD 6.45E-01 6.20E-01 5.95E-01 5.71E-01 5.49E-01 5.28E-01 5.07E-01 4.97E-01 4.87E-01 4.78E-01 4.69E-01 4.60E-01 4.51E-01 4.43E-01 4.35E-01 4.28E-01 4.24E-01 4.20E-01 4.14E-01 4.09E-01 4.05E-01 4.01E-01 3.96E-01 3.94E-01 3.92E-01 3.88E-01 3.85E-01 3.82E-01 3.79E-01 3.76E-01 3.73E-01 3.71E-01 3.69E-01 3.67E-01 3.65E-01 3.62E-01 3.60E-01 3.58E-01 3.57E-01 3.56E-01 3.54E-01 3.52E-01 3.50E-01 3.49E-01 3.47E-01

p [psi] 2,987.31 2,904.00 2,817.14 2,726.80 2,633.10 2,536.18 2,435.83 2,380.51 2,329.27 2,272.74 2,220.41 2,162.64 2,109.18 2,050.27 1,995.80 1,940.64 1,909.28 1,879.26 1,832.10 1,795.71 1,761.10 1,720.87 1,682.76 1,661.47 1,640.83 1,606.65 1,574.02 1,546.09 1,519.21

pD 1.43E+00 1.48E+00 1.55E+00 1.61E+00 1.68E+00 1.74E+00 1.82E+00 1.86E+00 1.89E+00 1.93E+00 1.97E+00 2.01E+00 2.05E+00 2.09E+00 2.13E+00 2.17E+00 2.19E+00 2.21E+00 2.24E+00 2.27E+00 2.29E+00 2.32E+00 2.35E+00 2.36E+00 2.38E+00 2.40E+00 2.43E+00 2.45E+00 2.47E+00

119

Table 1 – FCD=1 – Results of Numerical Simulation and Dimensionless Time, Pressure and Rate - continued t [days] 1.40E+04 1.43E+04 1.47E+04 1.51E+04 1.54E+04 1.58E+04 1.61E+04 1.65E+04 1.68E+04 1.72E+04 1.75E+04 1.79E+04 1.83E+04 1.86E+04 1.90E+04 1.94E+04 1.97E+04 2.01E+04 2.04E+04 2.06E+04 2.10E+04 2.13E+04 2.17E+04 2.21E+04 2.24E+04 2.28E+04 2.32E+04 2.35E+04 2.39E+04 2.43E+04 2.45E+04 2.47E+04 2.51E+04 2.55E+04 2.58E+04 2.62E+04 2.66E+04 2.69E+04 2.73E+04 2.77E+04 2.80E+04 2.84E+04 2.87E+04 2.91E+04 2.94E+04

tDxf q [stb/day] 3.55E+00 110 3.62E+00 110 3.71E+00 109 3.80E+00 109 3.90E+00 109 3.99E+00 108 4.08E+00 108 4.17E+00 107 4.26E+00 107 4.34E+00 107 4.43E+00 106 4.53E+00 106 4.62E+00 106 4.71E+00 105 4.80E+00 105 4.90E+00 105 4.99E+00 104 5.08E+00 104 5.14E+00 104 5.21E+00 104 5.30E+00 103 5.39E+00 103 5.49E+00 103 5.58E+00 103 5.67E+00 102 5.76E+00 102 5.85E+00 102 5.95E+00 102 6.04E+00 101 6.13E+00 101 6.19E+00 101 6.25E+00 101 6.34E+00 101 6.43E+00 100 6.53E+00 100 6.62E+00 100 6.71E+00 100 6.80E+00 100 6.90E+00 99 6.99E+00 99 7.08E+00 99 7.17E+00 99 7.26E+00 99 7.36E+00 98 7.43E+00 98

qD 3.46E-01 3.45E-01 3.44E-01 3.42E-01 3.41E-01 3.40E-01 3.38E-01 3.37E-01 3.36E-01 3.35E-01 3.34E-01 3.33E-01 3.32E-01 3.31E-01 3.30E-01 3.29E-01 3.28E-01 3.27E-01 3.26E-01 3.25E-01 3.25E-01 3.24E-01 3.23E-01 3.22E-01 3.21E-01 3.20E-01 3.20E-01 3.19E-01 3.18E-01 3.17E-01 3.17E-01 3.16E-01 3.16E-01 3.15E-01 3.14E-01 3.14E-01 3.13E-01 3.12E-01 3.12E-01 3.11E-01 3.11E-01 3.10E-01 3.09E-01 3.09E-01 3.08E-01

120

Table 1 – FCD=1 – Results of Numerical Simulation and Dimensionless Time, Pressure and Rate - continued t [days] 2.97E+04 3.00E+04 3.04E+04 3.08E+04 3.11E+04 3.15E+04 3.19E+04 3.22E+04 3.26E+04 3.30E+04 3.33E+04 3.37E+04 3.41E+04 3.44E+04 3.48E+04 3.52E+04 3.54E+04 3.56E+04 3.60E+04 3.63E+04 3.67E+04 3.71E+04 3.74E+04 3.78E+04 3.82E+04 3.85E+04 3.89E+04 3.93E+04 3.96E+04 4.00E+04 4.04E+04 4.07E+04 4.11E+04 4.15E+04 4.18E+04 4.22E+04 4.25E+04 4.27E+04 4.31E+04 4.35E+04 4.38E+04 4.42E+04 4.46E+04 4.49E+04 4.53E+04

tDxf q [stb/day] 7.50E+00 98 7.59E+00 98 7.69E+00 98 7.78E+00 98 7.87E+00 97 7.96E+00 97 8.05E+00 97 8.15E+00 97 8.24E+00 97 8.33E+00 97 8.42E+00 96 8.52E+00 96 8.61E+00 96 8.70E+00 96 8.79E+00 96 8.88E+00 96 8.94E+00 96 9.00E+00 96 9.09E+00 95 9.18E+00 95 9.28E+00 95 9.37E+00 95 9.46E+00 95 9.55E+00 95 9.65E+00 95 9.74E+00 94 9.83E+00 94 9.92E+00 94 1.00E+01 94 1.01E+01 94 1.02E+01 94 1.03E+01 94 1.04E+01 94 1.05E+01 94 1.06E+01 93 1.07E+01 93 1.07E+01 93 1.08E+01 93 1.09E+01 93 1.10E+01 93 1.11E+01 93 1.12E+01 93 1.13E+01 93 1.14E+01 92 1.14E+01 92

qD 3.08E-01 3.07E-01 3.07E-01 3.06E-01 3.06E-01 3.05E-01 3.05E-01 3.04E-01 3.04E-01 3.03E-01 3.03E-01 3.02E-01 3.02E-01 3.01E-01 3.01E-01 3.00E-01 3.00E-01 3.00E-01 2.99E-01 2.99E-01 2.99E-01 2.98E-01 2.98E-01 2.97E-01 2.97E-01 2.96E-01 2.96E-01 2.96E-01 2.95E-01 2.95E-01 2.94E-01 2.94E-01 2.94E-01 2.93E-01 2.93E-01 2.93E-01 2.92E-01 2.92E-01 2.92E-01 2.91E-01 2.91E-01 2.91E-01 2.90E-01 2.90E-01 2.90E-01

121

Table 1 – FCD=1 – Results of Numerical Simulation and Dimensionless Time, Pressure and Rate - continued t [days] 4.57E+04 4.60E+04 4.64E+04 4.67E+04 4.71E+04 4.75E+04 4.78E+04 4.82E+04 4.86E+04 4.89E+04 4.93E+04 4.97E+04 5.00E+04 5.04E+04 5.08E+04 5.10E+04 5.13E+04 5.16E+04 5.20E+04 5.24E+04 5.27E+04 5.31E+04 5.35E+04 5.38E+04 5.42E+04 5.46E+04 5.49E+04 5.53E+04 5.57E+04 5.60E+04 5.64E+04 5.68E+04 5.71E+04 5.75E+04 5.79E+04 5.82E+04 5.86E+04 5.89E+04 5.93E+04 5.97E+04 6.00E+04 6.04E+04 6.08E+04 6.11E+04 6.14E+04 6.16E+04

tDxf q [stb/day] 1.15E+01 92 1.16E+01 92 1.17E+01 92 1.18E+01 92 1.19E+01 92 1.20E+01 92 1.21E+01 92 1.22E+01 92 1.23E+01 91 1.24E+01 91 1.25E+01 91 1.26E+01 91 1.26E+01 91 1.27E+01 91 1.28E+01 91 1.29E+01 91 1.30E+01 91 1.31E+01 91 1.31E+01 91 1.32E+01 91 1.33E+01 90 1.34E+01 90 1.35E+01 90 1.36E+01 90 1.37E+01 90 1.38E+01 90 1.39E+01 90 1.40E+01 90 1.41E+01 90 1.42E+01 90 1.43E+01 90 1.43E+01 90 1.44E+01 89 1.45E+01 89 1.46E+01 89 1.47E+01 89 1.48E+01 89 1.49E+01 89 1.50E+01 89 1.51E+01 89 1.52E+01 89 1.53E+01 89 1.54E+01 89 1.55E+01 89 1.55E+01 89 1.56E+01 89

qD 2.89E-01 2.89E-01 2.89E-01 2.88E-01 2.88E-01 2.88E-01 2.88E-01 2.87E-01 2.87E-01 2.87E-01 2.86E-01 2.86E-01 2.86E-01 2.85E-01 2.85E-01 2.85E-01 2.85E-01 2.85E-01 2.84E-01 2.84E-01 2.84E-01 2.83E-01 2.83E-01 2.83E-01 2.83E-01 2.82E-01 2.82E-01 2.82E-01 2.82E-01 2.81E-01 2.81E-01 2.81E-01 2.81E-01 2.80E-01 2.80E-01 2.80E-01 2.80E-01 2.79E-01 2.79E-01 2.79E-01 2.79E-01 2.79E-01 2.78E-01 2.78E-01 2.78E-01 2.78E-01

122

Table 2 – FCD=5 – Results of Numerical Simulation and Dimensionless Time, Pressure and Rate t [days] 0 1.16E-05 2.55E-05 4.22E-05 6.22E-05 8.62E-05 1.15E-04 1.50E-04 1.91E-04 2.41E-04 3.01E-04 3.72E-04 4.58E-04 5.61E-04 6.85E-04 8.34E-04 1.01E-03 1.23E-03 1.48E-03 1.79E-03 2.16E-03 2.61E-03 3.14E-03 3.78E-03 4.54E-03 5.46E-03 6.56E-03 7.88E-03 9.47E-03 1.14E-02 1.37E-02 1.64E-02 1.97E-02 2.37E-02 2.84E-02 3.41E-02 4.10E-02 4.92E-02 5.90E-02 7.08E-02 8.50E-02 1.02E-01 1.22E-01 1.47E-01

tDxf 0 2.92E-09 6.44E-09 1.07E-08 1.57E-08 2.18E-08 2.91E-08 3.78E-08 4.83E-08 6.09E-08 7.60E-08 9.41E-08 1.16E-07 1.42E-07 1.73E-07 2.11E-07 2.56E-07 3.10E-07 3.75E-07 4.53E-07 5.46E-07 6.58E-07 7.93E-07 9.54E-07 1.15E-06 1.38E-06 1.66E-06 1.99E-06 2.39E-06 2.88E-06 3.46E-06 4.15E-06 4.98E-06 5.99E-06 7.19E-06 8.63E-06 1.04E-05 1.24E-05 1.49E-05 1.79E-05 2.15E-05 2.58E-05 3.09E-05 3.71E-05

q [stb/day] 0 25,920 14,477 11,973 10,930 10,405 10,118 9,952 9,851 9,781 9,725 9,672 9,616 9,552 9,478 9,391 9,290 9,172 9,035 8,877 8,698 8,495 8,268 8,017 7,744 7,451 7,144 6,828 6,508 6,192 5,888 5,600 5,332 5,084 4,856 4,643 4,443 4,252 4,069 3,891 3,719 3,555 3,398 3,250

qD 0 8.13E+01 4.54E+01 3.76E+01 3.43E+01 3.26E+01 3.17E+01 3.12E+01 3.09E+01 3.07E+01 3.05E+01 3.03E+01 3.02E+01 3.00E+01 2.97E+01 2.95E+01 2.91E+01 2.88E+01 2.83E+01 2.79E+01 2.73E+01 2.67E+01 2.59E+01 2.52E+01 2.43E+01 2.34E+01 2.24E+01 2.14E+01 2.04E+01 1.94E+01 1.85E+01 1.76E+01 1.67E+01 1.60E+01 1.52E+01 1.46E+01 1.39E+01 1.33E+01 1.28E+01 1.22E+01 1.17E+01 1.12E+01 1.07E+01 1.02E+01

p [psi] 5,000 4,983 4,975 4,969 4,965 4,961 4,959 4,957 4,956 4,955 4,955 4,954 4,954 4,953 4,953 4,953 4,952 4,952 4,951 4,950 4,949 4,948 4,947 4,945 4,944 4,942 4,940 4,937 4,935 4,932 4,929 4,926 4,923 4,919 4,915 4,912 4,908 4,904 4,899 4,895 4,890 4,885 4,880 4,874

pD n/a 1.22E-02 1.80E-02 2.21E-02 2.51E-02 2.74E-02 2.91E-02 3.03E-02 3.12E-02 3.17E-02 3.22E-02 3.25E-02 3.27E-02 3.29E-02 3.32E-02 3.35E-02 3.39E-02 3.43E-02 3.47E-02 3.53E-02 3.60E-02 3.68E-02 3.76E-02 3.87E-02 3.99E-02 4.12E-02 4.27E-02 4.43E-02 4.61E-02 4.81E-02 5.02E-02 5.25E-02 5.49E-02 5.73E-02 5.99E-02 6.26E-02 6.54E-02 6.83E-02 7.13E-02 7.45E-02 7.79E-02 8.15E-02 8.52E-02 8.91E-02

123

Table 2 – FCD=5 – Results of Numerical Simulation and Dimensionless Time, Pressure and Rate - continued t [days] 1.76E-01 2.12E-01 2.54E-01 3.05E-01 3.66E-01 4.39E-01 5.27E-01 6.32E-01 7.58E-01 9.10E-01 1.09E+00 1.31E+00 1.57E+00 1.89E+00 2.26E+00 2.72E+00 3.26E+00 3.91E+00 4.69E+00 5.63E+00 6.76E+00 8.11E+00 9.73E+00 1.17E+01 1.40E+01 1.68E+01 2.02E+01 2.42E+01 2.91E+01 3.49E+01 4.19E+01 5.02E+01 6.02E+01 7.23E+01 8.68E+01 1.04E+02 1.25E+02 1.50E+02 1.80E+02 2.16E+02 2.59E+02 3.11E+02 3.73E+02 4.48E+02 5.38E+02

tDxf q [stb/day] 4.46E-05 3,109 5.35E-05 2,975 6.42E-05 2,847 7.70E-05 2,723 9.24E-05 2,604 1.11E-04 2,490 1.33E-04 2,382 1.60E-04 2,279 1.91E-04 2,180 2.30E-04 2,086 2.76E-04 1,995 3.31E-04 1,908 3.97E-04 1,825 4.77E-04 1,745 5.72E-04 1,668 6.87E-04 1,595 8.24E-04 1,525 9.89E-04 1,458 1.19E-03 1,393 1.42E-03 1,331 1.71E-03 1,271 2.05E-03 1,213 2.46E-03 1,158 2.95E-03 1,104 3.54E-03 1,053 4.25E-03 1,003 5.10E-03 955 6.12E-03 908 7.35E-03 863 8.82E-03 819 1.06E-02 777 1.27E-02 736 1.52E-02 697 1.83E-02 660 2.19E-02 624 2.63E-02 589 3.16E-02 556 3.79E-02 525 4.55E-02 495 5.46E-02 466 6.55E-02 439 7.86E-02 413 9.43E-02 389 1.13E-01 366 1.36E-01 345

qD 9.76E+00 9.33E+00 8.93E+00 8.54E+00 8.17E+00 7.81E+00 7.47E+00 7.15E+00 6.84E+00 6.54E+00 6.26E+00 5.99E+00 5.73E+00 5.47E+00 5.23E+00 5.00E+00 4.78E+00 4.57E+00 4.37E+00 4.18E+00 3.99E+00 3.81E+00 3.63E+00 3.46E+00 3.30E+00 3.15E+00 3.00E+00 2.85E+00 2.71E+00 2.57E+00 2.44E+00 2.31E+00 2.19E+00 2.07E+00 1.96E+00 1.85E+00 1.75E+00 1.65E+00 1.55E+00 1.46E+00 1.38E+00 1.30E+00 1.22E+00 1.15E+00 1.08E+00

p [psi] 4,869 4,863 4,856 4,850 4,843 4,836 4,828 4,821 4,812 4,805 4,796 4,787 4,778 4,768 4,757 4,746 4,735 4,723 4,710 4,696 4,682 4,667 4,651 4,635 4,617 4,599 4,579 4,559 4,537 4,513 4,488 4,462 4,433 4,403 4,371 4,336 4,299 4,260 4,218 4,173 4,124 4,073 4,018 3,959 3,897

pD 9.31E-02 9.73E-02 1.02E-01 1.06E-01 1.11E-01 1.16E-01 1.22E-01 1.27E-01 1.33E-01 1.38E-01 1.45E-01 1.51E-01 1.57E-01 1.65E-01 1.72E-01 1.80E-01 1.88E-01 1.96E-01 2.06E-01 2.15E-01 2.25E-01 2.36E-01 2.47E-01 2.58E-01 2.71E-01 2.84E-01 2.98E-01 3.13E-01 3.28E-01 3.45E-01 3.63E-01 3.81E-01 4.01E-01 4.23E-01 4.46E-01 4.70E-01 4.96E-01 5.24E-01 5.54E-01 5.86E-01 6.20E-01 6.57E-01 6.96E-01 7.37E-01 7.81E-01

124

Table 2 – FCD=5 – Results of Numerical Simulation and Dimensionless Time, Pressure and Rate - continued t [days] 6.46E+02 7.75E+02 9.30E+02 1.12E+03 1.34E+03 1.61E+03 1.93E+03 2.12E+03 2.31E+03 2.54E+03 2.77E+03 3.05E+03 3.33E+03 3.66E+03 3.99E+03 4.36E+03 4.58E+03 4.79E+03 5.16E+03 5.46E+03 5.75E+03 6.12E+03 6.48E+03 6.69E+03 6.90E+03 7.27E+03 7.63E+03 7.96E+03 8.28E+03 8.65E+03 9.01E+03 9.38E+03 9.66E+03 9.94E+03 1.03E+04 1.07E+04 1.10E+04 1.14E+04 1.17E+04 1.19E+04 1.23E+04 1.27E+04 1.30E+04 1.34E+04 1.38E+04

tDxf q [stb/day] 1.63E-01 325 1.96E-01 306 2.35E-01 289 2.82E-01 273 3.38E-01 258 4.06E-01 244 4.87E-01 231 5.36E-01 225 5.84E-01 219 6.43E-01 213 7.01E-01 208 7.71E-01 203 8.41E-01 198 9.25E-01 193 1.01E+00 189 1.10E+00 185 1.16E+00 182 1.21E+00 180 1.30E+00 177 1.38E+00 175 1.45E+00 172 1.55E+00 170 1.64E+00 168 1.69E+00 166 1.74E+00 165 1.84E+00 163 1.93E+00 161 2.01E+00 160 2.09E+00 158 2.19E+00 157 2.28E+00 156 2.37E+00 154 2.44E+00 153 2.51E+00 152 2.60E+00 151 2.70E+00 150 2.79E+00 149 2.88E+00 148 2.95E+00 147 3.02E+00 146 3.11E+00 145 3.20E+00 145 3.29E+00 144 3.38E+00 143 3.48E+00 142

qD 1.02E+00 9.60E-01 9.06E-01 8.55E-01 8.09E-01 7.66E-01 7.26E-01 7.06E-01 6.88E-01 6.69E-01 6.53E-01 6.36E-01 6.22E-01 6.06E-01 5.93E-01 5.80E-01 5.72E-01 5.66E-01 5.56E-01 5.48E-01 5.41E-01 5.33E-01 5.26E-01 5.22E-01 5.18E-01 5.12E-01 5.06E-01 5.02E-01 4.97E-01 4.92E-01 4.88E-01 4.84E-01 4.81E-01 4.78E-01 4.74E-01 4.70E-01 4.67E-01 4.64E-01 4.62E-01 4.59E-01 4.57E-01 4.54E-01 4.51E-01 4.49E-01 4.46E-01

p [psi] 3,830 3,760 3,686 3,607 3,525 3,438 3,348 3,298 3,251 3,199 3,151 3,097 3,047 2,992 2,941 2,889 2,860 2,831 2,786 2,752 2,719 2,680 2,644 2,624 2,604 2,571 2,540 2,513 2,487

pD 8.28E-01 8.78E-01 9.31E-01 9.87E-01 1.04E+00 1.11E+00 1.17E+00 1.21E+00 1.24E+00 1.28E+00 1.31E+00 1.35E+00 1.38E+00 1.42E+00 1.46E+00 1.49E+00 1.52E+00 1.54E+00 1.57E+00 1.59E+00 1.62E+00 1.64E+00 1.67E+00 1.68E+00 1.70E+00 1.72E+00 1.74E+00 1.76E+00 1.78E+00

125

Table 2 – FCD=5 – Results of Numerical Simulation and Dimensionless Time, Pressure and Rate - continued t [days] 1.40E+04 1.43E+04 1.47E+04 1.51E+04 1.54E+04 1.58E+04 1.61E+04 1.65E+04 1.68E+04 1.72E+04 1.75E+04 1.79E+04 1.83E+04 1.86E+04 1.90E+04 1.94E+04 1.97E+04 2.01E+04 2.04E+04 2.06E+04 2.10E+04 2.13E+04 2.17E+04 2.21E+04 2.24E+04 2.28E+04 2.32E+04 2.35E+04 2.39E+04 2.43E+04 2.45E+04 2.47E+04 2.51E+04 2.55E+04 2.58E+04 2.62E+04 2.66E+04 2.69E+04 2.73E+04 2.77E+04 2.80E+04 2.84E+04 2.87E+04 2.91E+04 2.94E+04

tDxf q [stb/day] 3.55E+00 142 3.62E+00 141 3.71E+00 140 3.80E+00 140 3.90E+00 139 3.99E+00 138 4.08E+00 138 4.17E+00 137 4.26E+00 137 4.34E+00 136 4.43E+00 135 4.53E+00 135 4.62E+00 134 4.71E+00 134 4.80E+00 133 4.90E+00 133 4.99E+00 132 5.08E+00 132 5.14E+00 132 5.21E+00 131 5.30E+00 131 5.39E+00 130 5.49E+00 130 5.58E+00 130 5.67E+00 129 5.76E+00 129 5.85E+00 128 5.95E+00 128 6.04E+00 128 6.13E+00 127 6.19E+00 127 6.25E+00 127 6.34E+00 127 6.43E+00 126 6.53E+00 126 6.62E+00 126 6.71E+00 125 6.80E+00 125 6.90E+00 125 6.99E+00 124 7.08E+00 124 7.17E+00 124 7.26E+00 123 7.36E+00 123 7.43E+00 123

qD 4.44E-01 4.42E-01 4.40E-01 4.38E-01 4.36E-01 4.34E-01 4.32E-01 4.30E-01 4.28E-01 4.27E-01 4.25E-01 4.23E-01 4.22E-01 4.20E-01 4.18E-01 4.17E-01 4.15E-01 4.14E-01 4.13E-01 4.12E-01 4.11E-01 4.09E-01 4.08E-01 4.07E-01 4.05E-01 4.04E-01 4.03E-01 4.02E-01 4.01E-01 4.00E-01 3.99E-01 3.98E-01 3.97E-01 3.96E-01 3.95E-01 3.94E-01 3.93E-01 3.92E-01 3.91E-01 3.90E-01 3.89E-01 3.88E-01 3.87E-01 3.87E-01 3.86E-01

126

Table 2 – FCD=5 – Results of Numerical Simulation and Dimensionless Time, Pressure and Rate - continued t [days] 2.97E+04 3.00E+04 3.04E+04 3.08E+04 3.11E+04 3.15E+04 3.19E+04 3.22E+04 3.26E+04 3.30E+04 3.33E+04 3.37E+04 3.41E+04 3.44E+04 3.48E+04 3.52E+04 3.54E+04 3.56E+04 3.60E+04 3.63E+04 3.67E+04 3.71E+04 3.74E+04 3.78E+04 3.82E+04 3.85E+04 3.89E+04 3.93E+04 3.96E+04 4.00E+04 4.04E+04 4.07E+04 4.11E+04 4.15E+04 4.18E+04 4.22E+04 4.25E+04 4.27E+04 4.31E+04 4.35E+04 4.38E+04 4.42E+04 4.46E+04 4.49E+04 4.53E+04

tDxf q [stb/day] 7.50E+00 123 7.59E+00 122 7.69E+00 122 7.78E+00 122 7.87E+00 122 7.96E+00 121 8.05E+00 121 8.15E+00 121 8.24E+00 121 8.33E+00 120 8.42E+00 120 8.52E+00 120 8.61E+00 120 8.70E+00 120 8.79E+00 119 8.88E+00 119 8.94E+00 119 9.00E+00 119 9.09E+00 119 9.18E+00 118 9.28E+00 118 9.37E+00 118 9.46E+00 118 9.55E+00 118 9.65E+00 117 9.74E+00 117 9.83E+00 117 9.92E+00 117 1.00E+01 117 1.01E+01 117 1.02E+01 116 1.03E+01 116 1.04E+01 116 1.05E+01 116 1.06E+01 116 1.07E+01 115 1.07E+01 115 1.08E+01 115 1.09E+01 115 1.10E+01 115 1.11E+01 115 1.12E+01 115 1.13E+01 114 1.14E+01 114 1.14E+01 114

qD 3.85E-01 3.84E-01 3.83E-01 3.83E-01 3.82E-01 3.81E-01 3.80E-01 3.80E-01 3.79E-01 3.78E-01 3.77E-01 3.77E-01 3.76E-01 3.75E-01 3.74E-01 3.74E-01 3.73E-01 3.73E-01 3.72E-01 3.72E-01 3.71E-01 3.70E-01 3.70E-01 3.69E-01 3.69E-01 3.68E-01 3.67E-01 3.67E-01 3.66E-01 3.66E-01 3.65E-01 3.64E-01 3.64E-01 3.63E-01 3.63E-01 3.62E-01 3.62E-01 3.61E-01 3.61E-01 3.60E-01 3.60E-01 3.59E-01 3.59E-01 3.58E-01 3.58E-01

127

Table 2 – FCD=5 – Results of Numerical Simulation and Dimensionless Time, Pressure and Rate - continued t [days] 4.57E+04 4.60E+04 4.64E+04 4.67E+04 4.71E+04 4.75E+04 4.78E+04 4.82E+04 4.86E+04 4.89E+04 4.93E+04 4.97E+04 5.00E+04 5.04E+04 5.08E+04 5.10E+04 5.13E+04 5.16E+04 5.20E+04 5.24E+04 5.27E+04 5.31E+04 5.35E+04 5.38E+04 5.42E+04 5.46E+04 5.49E+04 5.53E+04 5.57E+04 5.60E+04 5.64E+04 5.68E+04 5.71E+04 5.75E+04 5.79E+04 5.82E+04 5.86E+04 5.89E+04 5.93E+04 5.97E+04 6.00E+04 6.04E+04 6.08E+04 6.11E+04 6.14E+04 6.16E+04

tDxf q [stb/day] 1.15E+01 114 1.16E+01 114 1.17E+01 114 1.18E+01 113 1.19E+01 113 1.20E+01 113 1.21E+01 113 1.22E+01 113 1.23E+01 113 1.24E+01 113 1.25E+01 112 1.26E+01 112 1.26E+01 112 1.27E+01 112 1.28E+01 112 1.29E+01 112 1.30E+01 112 1.31E+01 112 1.31E+01 111 1.32E+01 111 1.33E+01 111 1.34E+01 111 1.35E+01 111 1.36E+01 111 1.37E+01 111 1.38E+01 111 1.39E+01 110 1.40E+01 110 1.41E+01 110 1.42E+01 110 1.43E+01 110 1.43E+01 110 1.44E+01 110 1.45E+01 110 1.46E+01 110 1.47E+01 109 1.48E+01 109 1.49E+01 109 1.50E+01 109 1.51E+01 109 1.52E+01 109 1.53E+01 109 1.54E+01 109 1.55E+01 109 1.55E+01 109 1.56E+01 108

qD 3.57E-01 3.57E-01 3.57E-01 3.56E-01 3.56E-01 3.55E-01 3.55E-01 3.54E-01 3.54E-01 3.53E-01 3.53E-01 3.53E-01 3.52E-01 3.52E-01 3.51E-01 3.51E-01 3.51E-01 3.50E-01 3.50E-01 3.49E-01 3.49E-01 3.49E-01 3.48E-01 3.48E-01 3.47E-01 3.47E-01 3.47E-01 3.46E-01 3.46E-01 3.46E-01 3.45E-01 3.45E-01 3.45E-01 3.44E-01 3.44E-01 3.43E-01 3.43E-01 3.43E-01 3.42E-01 3.42E-01 3.42E-01 3.41E-01 3.41E-01 3.41E-01 3.41E-01 3.40E-01

128

Table 3 - FCD=10 – Results of Numerical Simulation and Dimensionless Time, Pressure and Rate t [days] 0 1.16E-05 2.55E-05 4.22E-05 6.22E-05 8.62E-05 1.15E-04 1.50E-04 1.91E-04 2.41E-04 3.01E-04 3.72E-04 4.58E-04 5.61E-04 6.85E-04 8.34E-04 1.01E-03 1.23E-03 1.48E-03 1.79E-03 2.16E-03 2.61E-03 3.14E-03 3.78E-03 4.54E-03 5.46E-03 6.56E-03 7.88E-03 9.47E-03 1.14E-02 1.37E-02 1.64E-02 1.97E-02 2.37E-02 2.84E-02 3.41E-02 4.10E-02 4.92E-02 5.90E-02 7.08E-02 8.50E-02 1.02E-01 1.22E-01 1.47E-01

tDxf 0 2.92E-09 6.44E-09 1.07E-08 1.57E-08 2.18E-08 2.91E-08 3.78E-08 4.83E-08 6.09E-08 7.60E-08 9.41E-08 1.16E-07 1.42E-07 1.73E-07 2.11E-07 2.56E-07 3.10E-07 3.75E-07 4.53E-07 5.46E-07 6.58E-07 7.93E-07 9.54E-07 1.15E-06 1.38E-06 1.66E-06 1.99E-06 2.39E-06 2.88E-06 3.46E-06 4.15E-06 4.98E-06 5.99E-06 7.19E-06 8.63E-06 1.04E-05 1.24E-05 1.49E-05 1.79E-05 2.15E-05 2.58E-05 3.09E-05 3.71E-05

q [stb/day] 0 36,748 20,490 16,945 15,472 14,731 14,326 14,092 13,949 13,851 13,773 13,698 13,619 13,529 13,424 13,301 13,158 12,991 12,797 12,575 12,321 12,034 11,713 11,358 10,972 10,559 10,126 9,680 9,229 8,784 8,355 7,949 7,572 7,223 6,902 6,602 6,319 6,049 5,790 5,538 5,295 5,061 4,839 4,628

qD 0 1.15E+02 6.43E+01 5.32E+01 4.85E+01 4.62E+01 4.50E+01 4.42E+01 4.38E+01 4.35E+01 4.32E+01 4.30E+01 4.27E+01 4.25E+01 4.21E+01 4.17E+01 4.13E+01 4.08E+01 4.02E+01 3.95E+01 3.87E+01 3.78E+01 3.68E+01 3.56E+01 3.44E+01 3.31E+01 3.18E+01 3.04E+01 2.90E+01 2.76E+01 2.62E+01 2.49E+01 2.38E+01 2.27E+01 2.17E+01 2.07E+01 1.98E+01 1.90E+01 1.82E+01 1.74E+01 1.66E+01 1.59E+01 1.52E+01 1.45E+01

p [psi] 5,000 4,988 4,982 4,978 4,975 4,973 4,971 4,970 4,969 4,968 4,968 4,968 4,967 4,967 4,967 4,967 4,966 4,966 4,965 4,965 4,964 4,963 4,962 4,961 4,960 4,959 4,957 4,956 4,954 4,952 4,950 4,948 4,945 4,943 4,940 4,938 4,935 4,932 4,929 4,926 4,923 4,919 4,915 4,912

pD n/a 8.59E-03 1.27E-02 1.56E-02 1.77E-02 1.93E-02 2.05E-02 2.14E-02 2.20E-02 2.24E-02 2.27E-02 2.29E-02 2.31E-02 2.33E-02 2.34E-02 2.37E-02 2.39E-02 2.42E-02 2.45E-02 2.49E-02 2.54E-02 2.59E-02 2.66E-02 2.73E-02 2.81E-02 2.91E-02 3.01E-02 3.13E-02 3.25E-02 3.39E-02 3.54E-02 3.70E-02 3.87E-02 4.04E-02 4.22E-02 4.41E-02 4.60E-02 4.80E-02 5.02E-02 5.24E-02 5.48E-02 5.73E-02 5.99E-02 6.26E-02

129

Table 3 - FCD=10 – Results of Numerical Simulation and Dimensionless Time, Pressure and Rate - continued t [days] 1.76E-01 2.12E-01 2.54E-01 3.05E-01 3.66E-01 4.39E-01 5.27E-01 6.32E-01 7.58E-01 9.10E-01 1.09E+00 1.31E+00 1.57E+00 1.89E+00 2.26E+00 2.72E+00 3.26E+00 3.91E+00 4.69E+00 5.63E+00 6.76E+00 8.11E+00 9.73E+00 1.17E+01 1.40E+01 1.68E+01 2.02E+01 2.42E+01 2.91E+01 3.49E+01 4.19E+01 5.02E+01 6.02E+01 7.23E+01 8.68E+01 1.04E+02 1.25E+02 1.50E+02 1.80E+02 2.16E+02 2.59E+02 3.11E+02 3.73E+02 4.48E+02 5.38E+02

tDxf q [stb/day] 4.46E-05 4,427 5.35E-05 4,235 6.42E-05 4,052 7.70E-05 3,875 9.24E-05 3,706 1.11E-04 3,543 1.33E-04 3,387 1.60E-04 3,237 1.91E-04 3,095 2.30E-04 2,959 2.76E-04 2,827 3.31E-04 2,701 3.97E-04 2,579 4.77E-04 2,462 5.72E-04 2,348 6.87E-04 2,239 8.24E-04 2,134 9.89E-04 2,032 1.19E-03 1,934 1.42E-03 1,838 1.71E-03 1,745 2.05E-03 1,655 2.46E-03 1,568 2.95E-03 1,483 3.54E-03 1,401 4.25E-03 1,323 5.10E-03 1,246 6.12E-03 1,173 7.35E-03 1,102 8.82E-03 1,034 1.06E-02 969 1.27E-02 908 1.52E-02 850 1.83E-02 795 2.19E-02 744 2.63E-02 696 3.16E-02 650 3.79E-02 608 4.55E-02 568 5.46E-02 530 6.55E-02 496 7.86E-02 463 9.43E-02 433 1.13E-01 405 1.36E-01 380

qD 1.39E+01 1.33E+01 1.27E+01 1.22E+01 1.16E+01 1.11E+01 1.06E+01 1.02E+01 9.71E+00 9.28E+00 8.87E+00 8.48E+00 8.09E+00 7.72E+00 7.37E+00 7.03E+00 6.70E+00 6.38E+00 6.07E+00 5.77E+00 5.48E+00 5.19E+00 4.92E+00 4.65E+00 4.40E+00 4.15E+00 3.91E+00 3.68E+00 3.46E+00 3.24E+00 3.04E+00 2.85E+00 2.67E+00 2.50E+00 2.33E+00 2.18E+00 2.04E+00 1.91E+00 1.78E+00 1.66E+00 1.55E+00 1.45E+00 1.36E+00 1.27E+00 1.19E+00

p [psi] 4,908 4,903 4,899 4,894 4,890 4,885 4,880 4,875 4,869 4,863 4,857 4,850 4,844 4,836 4,829 4,820 4,812 4,803 4,793 4,783 4,772 4,760 4,747 4,734 4,719 4,704 4,687 4,670 4,650 4,630 4,607 4,583 4,557 4,529 4,499 4,467 4,432 4,394 4,354 4,311 4,265 4,215 4,162 4,105 4,045

pD 6.54E-02 6.84E-02 7.15E-02 7.47E-02 7.81E-02 8.17E-02 8.50E-02 8.88E-02 9.29E-02 9.70E-02 1.01E-01 1.06E-01 1.11E-01 1.16E-01 1.21E-01 1.27E-01 1.33E-01 1.40E-01 1.47E-01 1.54E-01 1.62E-01 1.70E-01 1.79E-01 1.89E-01 1.99E-01 2.10E-01 2.21E-01 2.34E-01 2.48E-01 2.62E-01 2.78E-01 2.95E-01 3.14E-01 3.34E-01 3.55E-01 3.78E-01 4.02E-01 4.29E-01 4.57E-01 4.88E-01 5.21E-01 5.56E-01 5.93E-01 6.34E-01 6.76E-01

130

Table 3 - FCD=10 – Results of Numerical Simulation and Dimensionless Time, Pressure and Rate - continued t [days] 6.46E+02 7.75E+02 9.30E+02 1.12E+03 1.34E+03 1.61E+03 1.93E+03 2.12E+03 2.31E+03 2.54E+03 2.77E+03 3.05E+03 3.33E+03 3.66E+03 3.99E+03 4.36E+03 4.58E+03 4.79E+03 5.16E+03 5.46E+03 5.75E+03 6.12E+03 6.48E+03 6.69E+03 6.90E+03 7.27E+03 7.63E+03 7.96E+03 8.28E+03 8.65E+03 9.01E+03 9.38E+03 9.66E+03 9.94E+03 1.03E+04 1.07E+04 1.10E+04 1.14E+04 1.17E+04 1.19E+04 1.23E+04 1.27E+04 1.30E+04 1.34E+04 1.38E+04

tDxf q [stb/day] 1.63E-01 356 1.96E-01 334 2.35E-01 314 2.82E-01 295 3.38E-01 278 4.06E-01 263 4.87E-01 248 5.36E-01 241 5.84E-01 235 6.43E-01 228 7.01E-01 222 7.71E-01 216 8.41E-01 211 9.25E-01 206 1.01E+00 201 1.10E+00 196 1.16E+00 194 1.21E+00 191 1.30E+00 188 1.38E+00 185 1.45E+00 183 1.55E+00 180 1.64E+00 177 1.69E+00 176 1.74E+00 175 1.84E+00 172 1.93E+00 170 2.01E+00 169 2.09E+00 167 2.19E+00 166 2.28E+00 164 2.37E+00 163 2.44E+00 161 2.51E+00 160 2.60E+00 159 2.70E+00 158 2.79E+00 157 2.88E+00 156 2.95E+00 155 3.02E+00 154 3.11E+00 153 3.20E+00 152 3.29E+00 151 3.38E+00 150 3.48E+00 149

qD 1.12E+00 1.05E+00 9.84E-01 9.26E-01 8.73E-01 8.24E-01 7.79E-01 7.56E-01 7.36E-01 7.15E-01 6.97E-01 6.78E-01 6.62E-01 6.45E-01 6.30E-01 6.16E-01 6.08E-01 6.00E-01 5.89E-01 5.81E-01 5.73E-01 5.64E-01 5.56E-01 5.52E-01 5.48E-01 5.41E-01 5.35E-01 5.30E-01 5.25E-01 5.19E-01 5.15E-01 5.10E-01 5.06E-01 5.03E-01 4.99E-01 4.95E-01 4.92E-01 4.88E-01 4.86E-01 4.83E-01 4.80E-01 4.77E-01 4.74E-01 4.71E-01 4.69E-01

p [psi] 3,980 3,912 3,839 3,763 3,682 3,597 3,509 3,459 3,413 3,362 3,315 3,262 3,213 3,158 3,108 3,057 3,027 2,999 2,955 2,921 2,888 2,850 2,814 2,793 2,774 2,741 2,710 2,683 2,658

pD 7.22E-01 7.71E-01 8.22E-01 8.76E-01 9.33E-01 9.93E-01 1.06E+00 1.09E+00 1.12E+00 1.16E+00 1.19E+00 1.23E+00 1.27E+00 1.30E+00 1.34E+00 1.38E+00 1.40E+00 1.42E+00 1.45E+00 1.47E+00 1.50E+00 1.52E+00 1.55E+00 1.56E+00 1.58E+00 1.60E+00 1.62E+00 1.64E+00 1.66E+00

131

Table 3 - FCD=10 – Results of Numerical Simulation and Dimensionless Time, Pressure and Rate - continued t [days] 1.40E+04 1.43E+04 1.47E+04 1.51E+04 1.54E+04 1.58E+04 1.61E+04 1.65E+04 1.68E+04 1.72E+04 1.75E+04 1.79E+04 1.83E+04 1.86E+04 1.90E+04 1.94E+04 1.97E+04 2.01E+04 2.04E+04 2.06E+04 2.10E+04 2.13E+04 2.17E+04 2.21E+04 2.24E+04 2.28E+04 2.32E+04 2.35E+04 2.39E+04 2.43E+04 2.45E+04 2.47E+04 2.51E+04 2.55E+04 2.58E+04 2.62E+04 2.66E+04 2.69E+04 2.73E+04 2.77E+04 2.80E+04 2.84E+04 2.87E+04 2.91E+04 2.94E+04

tDxf q [stb/day] 3.55E+00 149 3.62E+00 148 3.71E+00 147 3.80E+00 147 3.90E+00 146 3.99E+00 145 4.08E+00 144 4.17E+00 144 4.26E+00 143 4.34E+00 143 4.43E+00 142 4.53E+00 141 4.62E+00 141 4.71E+00 140 4.80E+00 140 4.90E+00 139 4.99E+00 139 5.08E+00 138 5.14E+00 138 5.21E+00 138 5.30E+00 137 5.39E+00 137 5.49E+00 136 5.58E+00 136 5.67E+00 135 5.76E+00 135 5.85E+00 134 5.95E+00 134 6.04E+00 134 6.13E+00 133 6.19E+00 133 6.25E+00 133 6.34E+00 132 6.43E+00 132 6.53E+00 132 6.62E+00 131 6.71E+00 131 6.80E+00 131 6.90E+00 130 6.99E+00 130 7.08E+00 130 7.17E+00 129 7.26E+00 129 7.36E+00 129 7.43E+00 129

qD 4.67E-01 4.65E-01 4.62E-01 4.60E-01 4.58E-01 4.56E-01 4.53E-01 4.51E-01 4.49E-01 4.48E-01 4.46E-01 4.44E-01 4.42E-01 4.40E-01 4.39E-01 4.37E-01 4.35E-01 4.34E-01 4.33E-01 4.32E-01 4.30E-01 4.29E-01 4.27E-01 4.26E-01 4.25E-01 4.23E-01 4.22E-01 4.21E-01 4.19E-01 4.18E-01 4.17E-01 4.17E-01 4.16E-01 4.14E-01 4.13E-01 4.12E-01 4.11E-01 4.10E-01 4.09E-01 4.08E-01 4.07E-01 4.06E-01 4.05E-01 4.04E-01 4.03E-01

132

Table 3 - FCD=10 – Results of Numerical Simulation and Dimensionless Time, Pressure and Rate - continued t [days] 2.97E+04 3.00E+04 3.04E+04 3.08E+04 3.11E+04 3.15E+04 3.19E+04 3.22E+04 3.26E+04 3.30E+04 3.33E+04 3.37E+04 3.41E+04 3.44E+04 3.48E+04 3.52E+04 3.54E+04 3.56E+04 3.60E+04 3.63E+04 3.67E+04 3.71E+04 3.74E+04 3.78E+04 3.82E+04 3.85E+04 3.89E+04 3.93E+04 3.96E+04 4.00E+04 4.04E+04 4.07E+04 4.11E+04 4.15E+04 4.18E+04 4.22E+04 4.25E+04 4.27E+04 4.31E+04 4.35E+04 4.38E+04 4.42E+04 4.46E+04 4.49E+04 4.53E+04

tDxf q [stb/day] 7.50E+00 128 7.59E+00 128 7.69E+00 128 7.78E+00 127 7.87E+00 127 7.96E+00 127 8.05E+00 127 8.15E+00 126 8.24E+00 126 8.33E+00 126 8.42E+00 126 8.52E+00 125 8.61E+00 125 8.70E+00 125 8.79E+00 125 8.88E+00 124 8.94E+00 124 9.00E+00 124 9.09E+00 124 9.18E+00 124 9.28E+00 123 9.37E+00 123 9.46E+00 123 9.55E+00 123 9.65E+00 123 9.74E+00 122 9.83E+00 122 9.92E+00 122 1.00E+01 122 1.01E+01 122 1.02E+01 121 1.03E+01 121 1.04E+01 121 1.05E+01 121 1.06E+01 121 1.07E+01 120 1.07E+01 120 1.08E+01 120 1.09E+01 120 1.10E+01 120 1.11E+01 120 1.12E+01 119 1.13E+01 119 1.14E+01 119 1.14E+01 119

qD 4.03E-01 4.02E-01 4.01E-01 4.00E-01 3.99E-01 3.98E-01 3.97E-01 3.97E-01 3.96E-01 3.95E-01 3.94E-01 3.93E-01 3.93E-01 3.92E-01 3.91E-01 3.90E-01 3.90E-01 3.89E-01 3.89E-01 3.88E-01 3.87E-01 3.87E-01 3.86E-01 3.85E-01 3.85E-01 3.84E-01 3.83E-01 3.83E-01 3.82E-01 3.81E-01 3.81E-01 3.80E-01 3.80E-01 3.79E-01 3.78E-01 3.78E-01 3.77E-01 3.77E-01 3.76E-01 3.76E-01 3.75E-01 3.75E-01 3.74E-01 3.74E-01 3.73E-01

133

Table 3 - FCD=10 – Results of Numerical Simulation and Dimensionless Time, Pressure and Rate - continued t [days] 4.57E+04 4.60E+04 4.64E+04 4.67E+04 4.71E+04 4.75E+04 4.78E+04 4.82E+04 4.86E+04 4.89E+04 4.93E+04 4.97E+04 5.00E+04 5.04E+04 5.08E+04 5.10E+04 5.13E+04 5.16E+04 5.20E+04 5.24E+04 5.27E+04 5.31E+04 5.35E+04 5.38E+04 5.42E+04 5.46E+04 5.49E+04 5.53E+04 5.57E+04 5.60E+04 5.64E+04 5.68E+04 5.71E+04 5.75E+04 5.79E+04 5.82E+04 5.86E+04 5.89E+04 5.93E+04 5.97E+04 6.00E+04 6.04E+04 6.08E+04 6.11E+04 6.14E+04 6.16E+04

tDxf q [stb/day] 1.15E+01 119 1.16E+01 119 1.17E+01 118 1.18E+01 118 1.19E+01 118 1.20E+01 118 1.21E+01 118 1.22E+01 118 1.23E+01 118 1.24E+01 117 1.25E+01 117 1.26E+01 117 1.26E+01 117 1.27E+01 117 1.28E+01 117 1.29E+01 117 1.30E+01 116 1.31E+01 116 1.31E+01 116 1.32E+01 116 1.33E+01 116 1.34E+01 116 1.35E+01 116 1.36E+01 115 1.37E+01 115 1.38E+01 115 1.39E+01 115 1.40E+01 115 1.41E+01 115 1.42E+01 115 1.43E+01 115 1.43E+01 114 1.44E+01 114 1.45E+01 114 1.46E+01 114 1.47E+01 114 1.48E+01 114 1.49E+01 114 1.50E+01 114 1.51E+01 113 1.52E+01 113 1.53E+01 113 1.54E+01 113 1.55E+01 113 1.55E+01 113 1.56E+01 113

qD 3.73E-01 3.72E-01 3.72E-01 3.71E-01 3.71E-01 3.70E-01 3.70E-01 3.69E-01 3.69E-01 3.68E-01 3.68E-01 3.67E-01 3.67E-01 3.66E-01 3.66E-01 3.66E-01 3.65E-01 3.65E-01 3.64E-01 3.64E-01 3.64E-01 3.63E-01 3.63E-01 3.62E-01 3.62E-01 3.62E-01 3.61E-01 3.61E-01 3.60E-01 3.60E-01 3.60E-01 3.59E-01 3.59E-01 3.58E-01 3.58E-01 3.58E-01 3.57E-01 3.57E-01 3.56E-01 3.56E-01 3.56E-01 3.55E-01 3.55E-01 3.55E-01 3.54E-01 3.54E-01

134

Table 4 - FCD=25– Results of Numerical Simulation and Dimensionless Time, Pressure and Rate t [days] 0 1.16E-05 2.55E-05 4.22E-05 6.22E-05 8.62E-05 1.15E-04 1.50E-04 1.91E-04 2.41E-04 3.01E-04 3.72E-04 4.58E-04 5.61E-04 6.85E-04 8.34E-04 1.01E-03 1.23E-03 1.48E-03 1.79E-03 2.16E-03 2.61E-03 3.14E-03 3.78E-03 4.54E-03 5.46E-03 6.56E-03 7.88E-03 9.47E-03 1.14E-02 1.37E-02 1.64E-02 1.97E-02 2.37E-02 2.84E-02 3.41E-02 4.10E-02 4.92E-02 5.90E-02 7.08E-02 8.50E-02 1.02E-01 1.22E-01 1.47E-01

tDxf 0 2.92E-09 6.44E-09 1.07E-08 1.57E-08 2.18E-08 2.91E-08 3.78E-08 4.83E-08 6.09E-08 7.60E-08 9.41E-08 1.16E-07 1.42E-07 1.73E-07 2.11E-07 2.56E-07 3.10E-07 3.75E-07 4.53E-07 5.46E-07 6.58E-07 7.93E-07 9.54E-07 1.15E-06 1.38E-06 1.66E-06 1.99E-06 2.39E-06 2.88E-06 3.46E-06 4.15E-06 4.98E-06 5.99E-06 7.19E-06 8.63E-06 1.04E-05 1.24E-05 1.49E-05 1.79E-05 2.15E-05 2.58E-05 3.09E-05 3.71E-05

q [stb/day] 0 58,234 32,424 26,827 24,508 23,350 22,724 22,364 22,143 21,989 21,865 21,747 21,621 21,479 21,314 21,120 20,894 20,631 20,326 19,976 19,576 19,124 18,618 18,060 17,452 16,801 16,118 15,414 14,702 13,999 13,321 12,679 12,080 11,526 11,014 10,535 10,083 9,653 9,236 8,831 8,439 8,062 7,703 7,360

qD 0 1.83E+02 1.02E+02 8.42E+01 7.69E+01 7.33E+01 7.13E+01 7.02E+01 6.95E+01 6.90E+01 6.86E+01 6.82E+01 6.78E+01 6.74E+01 6.69E+01 6.63E+01 6.56E+01 6.47E+01 6.38E+01 6.27E+01 6.14E+01 6.00E+01 5.84E+01 5.67E+01 5.48E+01 5.27E+01 5.06E+01 4.84E+01 4.61E+01 4.39E+01 4.18E+01 3.98E+01 3.79E+01 3.62E+01 3.46E+01 3.31E+01 3.16E+01 3.03E+01 2.90E+01 2.77E+01 2.65E+01 2.53E+01 2.42E+01 2.31E+01

p [psi] 5,000 4,992 4,989 4,986 4,984 4,983 4,982 4,981 4,980 4,980 4,980 4,980 4,979 4,979 4,979 4,979 4,979 4,978 4,978 4,978 4,977 4,977 4,976 4,976 4,975 4,974 4,973 4,972 4,971 4,970 4,969 4,967 4,966 4,964 4,963 4,961 4,959 4,957 4,956 4,954 4,951 4,949 4,947 4,945

pD n/a 5.42E-03 8.02E-03 9.84E-03 1.12E-02 1.22E-02 1.30E-02 1.35E-02 1.39E-02 1.41E-02 1.43E-02 1.44E-02 1.45E-02 1.47E-02 1.48E-02 1.49E-02 1.51E-02 1.52E-02 1.54E-02 1.57E-02 1.60E-02 1.63E-02 1.67E-02 1.72E-02 1.77E-02 1.83E-02 1.89E-02 1.97E-02 2.05E-02 2.13E-02 2.22E-02 2.32E-02 2.43E-02 2.53E-02 2.65E-02 2.76E-02 2.89E-02 3.01E-02 3.15E-02 3.29E-02 3.44E-02 3.59E-02 3.76E-02 3.93E-02

135

Table 4 - FCD=25– Results of Numerical Simulation and Dimensionless Time, Pressure and Rate - continued t [days] 1.76E-01 2.12E-01 2.54E-01 3.05E-01 3.66E-01 4.39E-01 5.27E-01 6.32E-01 7.58E-01 9.10E-01 1.09E+00 1.31E+00 1.57E+00 1.89E+00 2.26E+00 2.72E+00 3.26E+00 3.91E+00 4.69E+00 5.63E+00 6.76E+00 8.11E+00 9.73E+00 1.17E+01 1.40E+01 1.68E+01 2.02E+01 2.42E+01 2.91E+01 3.49E+01 4.19E+01 5.02E+01 6.02E+01 7.23E+01 8.68E+01 1.04E+02 1.25E+02 1.50E+02 1.80E+02 2.16E+02 2.59E+02 3.11E+02 3.73E+02 4.48E+02 5.38E+02

tDxf q [stb/day] 4.46E-05 7,033 5.35E-05 6,720 6.42E-05 6,419 7.70E-05 6,128 9.24E-05 5,844 1.11E-04 5,570 1.33E-04 5,305 1.60E-04 5,050 1.91E-04 4,804 2.30E-04 4,565 2.76E-04 4,333 3.31E-04 4,108 3.97E-04 3,888 4.77E-04 3,674 5.72E-04 3,466 6.87E-04 3,266 8.24E-04 3,073 9.89E-04 2,887 1.19E-03 2,708 1.42E-03 2,535 1.71E-03 2,368 2.05E-03 2,210 2.46E-03 2,059 2.95E-03 1,915 3.54E-03 1,780 4.25E-03 1,653 5.10E-03 1,533 6.12E-03 1,421 7.35E-03 1,315 8.82E-03 1,217 1.06E-02 1,126 1.27E-02 1,043 1.52E-02 966 1.83E-02 895 2.19E-02 830 2.63E-02 770 3.16E-02 715 3.79E-02 664 4.55E-02 617 5.46E-02 573 6.55E-02 533 7.86E-02 496 9.43E-02 463 1.13E-01 431 1.36E-01 403

qD 2.21E+01 2.11E+01 2.01E+01 1.92E+01 1.83E+01 1.75E+01 1.66E+01 1.58E+01 1.51E+01 1.43E+01 1.36E+01 1.29E+01 1.22E+01 1.15E+01 1.09E+01 1.02E+01 9.64E+00 9.06E+00 8.50E+00 7.95E+00 7.43E+00 6.93E+00 6.46E+00 6.01E+00 5.59E+00 5.19E+00 4.81E+00 4.46E+00 4.13E+00 3.82E+00 3.53E+00 3.27E+00 3.03E+00 2.81E+00 2.60E+00 2.42E+00 2.24E+00 2.08E+00 1.93E+00 1.80E+00 1.67E+00 1.56E+00 1.45E+00 1.35E+00 1.26E+00

p [psi] 4,942 4,939 4,937 4,934 4,931 4,928 4,924 4,920 4,917 4,913 4,908 4,903 4,898 4,893 4,887 4,881 4,874 4,867 4,859 4,850 4,841 4,831 4,819 4,807 4,794 4,780 4,765 4,748 4,730 4,711 4,689 4,666 4,642 4,614 4,585 4,554 4,520 4,484 4,445 4,402 4,357 4,309 4,257 4,201 4,142

pD 4.11E-02 4.30E-02 4.50E-02 4.71E-02 4.91E-02 5.13E-02 5.37E-02 5.64E-02 5.91E-02 6.19E-02 6.51E-02 6.84E-02 7.20E-02 7.58E-02 7.99E-02 8.45E-02 8.93E-02 9.45E-02 1.00E-01 1.06E-01 1.13E-01 1.20E-01 1.28E-01 1.36E-01 1.46E-01 1.56E-01 1.66E-01 1.78E-01 1.91E-01 2.05E-01 2.20E-01 2.36E-01 2.54E-01 2.73E-01 2.94E-01 3.16E-01 3.40E-01 3.66E-01 3.93E-01 4.23E-01 4.55E-01 4.90E-01 5.26E-01 5.66E-01 6.08E-01

136

Table 4 - FCD=25– Results of Numerical Simulation and Dimensionless Time, Pressure and Rate - continued t [days] 6.46E+02 7.75E+02 9.30E+02 1.12E+03 1.34E+03 1.61E+03 1.93E+03 2.12E+03 2.31E+03 2.54E+03 2.77E+03 3.05E+03 3.33E+03 3.66E+03 3.99E+03 4.36E+03 4.58E+03 4.79E+03 5.16E+03 5.46E+03 5.75E+03 6.12E+03 6.48E+03 6.69E+03 6.90E+03 7.27E+03 7.63E+03 7.96E+03 8.28E+03 8.65E+03 9.01E+03 9.38E+03 9.66E+03 9.94E+03 1.03E+04 1.07E+04 1.10E+04 1.14E+04 1.17E+04 1.19E+04 1.23E+04 1.27E+04 1.30E+04 1.34E+04 1.38E+04

tDxf q [stb/day] 1.63E-01 377 1.96E-01 353 2.35E-01 331 2.82E-01 310 3.38E-01 292 4.06E-01 275 4.87E-01 260 5.36E-01 252 5.84E-01 245 6.43E-01 238 7.01E-01 232 7.71E-01 226 8.41E-01 220 9.25E-01 214 1.01E+00 209 1.10E+00 204 1.16E+00 201 1.21E+00 199 1.30E+00 195 1.38E+00 192 1.45E+00 190 1.55E+00 187 1.64E+00 184 1.69E+00 183 1.74E+00 181 1.84E+00 179 1.93E+00 177 2.01E+00 175 2.09E+00 173 2.19E+00 172 2.28E+00 170 2.37E+00 168 2.44E+00 167 2.51E+00 166 2.60E+00 165 2.70E+00 163 2.79E+00 162 2.88E+00 161 2.95E+00 160 3.02E+00 159 3.11E+00 158 3.20E+00 157 3.29E+00 156 3.38E+00 155 3.48E+00 154

qD 1.18E+00 1.11E+00 1.04E+00 9.74E-01 9.16E-01 8.64E-01 8.16E-01 7.91E-01 7.69E-01 7.47E-01 7.28E-01 7.08E-01 6.90E-01 6.72E-01 6.56E-01 6.40E-01 6.32E-01 6.24E-01 6.12E-01 6.03E-01 5.95E-01 5.86E-01 5.77E-01 5.73E-01 5.68E-01 5.61E-01 5.55E-01 5.49E-01 5.44E-01 5.38E-01 5.33E-01 5.28E-01 5.25E-01 5.21E-01 5.17E-01 5.13E-01 5.09E-01 5.05E-01 5.02E-01 5.00E-01 4.97E-01 4.93E-01 4.90E-01 4.87E-01 4.85E-01

p [psi] 4,078 4,011 3,940 3,864 3,784 3,701 3,613 3,564 3,519 3,468 3,421 3,369 3,320 3,266 3,216 3,165 3,136 3,108 3,064 3,030 2,998 2,960 2,924 2,904 2,884 2,852 2,821 2,794 2,769

pD 6.53E-01 7.00E-01 7.51E-01 8.04E-01 8.61E-01 9.20E-01 9.82E-01 1.02E+00 1.05E+00 1.08E+00 1.12E+00 1.16E+00 1.19E+00 1.23E+00 1.26E+00 1.30E+00 1.32E+00 1.34E+00 1.37E+00 1.40E+00 1.42E+00 1.44E+00 1.47E+00 1.48E+00 1.50E+00 1.52E+00 1.54E+00 1.56E+00 1.58E+00

137

Table 4 - FCD=25– Results of Numerical Simulation and Dimensionless Time, Pressure and Rate - continued t [days] 1.40E+04 1.43E+04 1.47E+04 1.51E+04 1.54E+04 1.58E+04 1.61E+04 1.65E+04 1.68E+04 1.72E+04 1.75E+04 1.79E+04 1.83E+04 1.86E+04 1.90E+04 1.94E+04 1.97E+04 2.01E+04 2.04E+04 2.06E+04 2.10E+04 2.13E+04 2.17E+04 2.21E+04 2.24E+04 2.28E+04 2.32E+04 2.35E+04 2.39E+04 2.43E+04 2.45E+04 2.47E+04 2.51E+04 2.55E+04 2.58E+04 2.62E+04 2.66E+04 2.69E+04 2.73E+04 2.77E+04 2.80E+04 2.84E+04 2.87E+04 2.91E+04 2.94E+04

tDxf q [stb/day] 3.55E+00 154 3.62E+00 153 3.71E+00 152 3.80E+00 151 3.90E+00 151 3.99E+00 150 4.08E+00 149 4.17E+00 149 4.26E+00 148 4.34E+00 147 4.43E+00 147 4.53E+00 146 4.62E+00 145 4.71E+00 145 4.80E+00 144 4.90E+00 144 4.99E+00 143 5.08E+00 143 5.14E+00 142 5.21E+00 142 5.30E+00 141 5.39E+00 141 5.49E+00 140 5.58E+00 140 5.67E+00 140 5.76E+00 139 5.85E+00 139 5.95E+00 138 6.04E+00 138 6.13E+00 137 6.19E+00 137 6.25E+00 137 6.34E+00 136 6.43E+00 136 6.53E+00 136 6.62E+00 135 6.71E+00 135 6.80E+00 135 6.90E+00 134 6.99E+00 134 7.08E+00 134 7.17E+00 133 7.26E+00 133 7.36E+00 133 7.43E+00 132

qD 4.82E-01 4.80E-01 4.78E-01 4.75E-01 4.73E-01 4.71E-01 4.68E-01 4.66E-01 4.64E-01 4.62E-01 4.60E-01 4.58E-01 4.56E-01 4.55E-01 4.53E-01 4.51E-01 4.49E-01 4.48E-01 4.46E-01 4.45E-01 4.44E-01 4.42E-01 4.41E-01 4.39E-01 4.38E-01 4.36E-01 4.35E-01 4.34E-01 4.32E-01 4.31E-01 4.30E-01 4.30E-01 4.28E-01 4.27E-01 4.26E-01 4.25E-01 4.24E-01 4.23E-01 4.21E-01 4.20E-01 4.19E-01 4.18E-01 4.17E-01 4.16E-01 4.15E-01

138

Table 4 - FCD=25– Results of Numerical Simulation and Dimensionless Time, Pressure and Rate - continued t [days] 2.97E+04 3.00E+04 3.04E+04 3.08E+04 3.11E+04 3.15E+04 3.19E+04 3.22E+04 3.26E+04 3.30E+04 3.33E+04 3.37E+04 3.41E+04 3.44E+04 3.48E+04 3.52E+04 3.54E+04 3.56E+04 3.60E+04 3.63E+04 3.67E+04 3.71E+04 3.74E+04 3.78E+04 3.82E+04 3.85E+04 3.89E+04 3.93E+04 3.96E+04 4.00E+04 4.04E+04 4.07E+04 4.11E+04 4.15E+04 4.18E+04 4.22E+04 4.25E+04 4.27E+04 4.31E+04 4.35E+04 4.38E+04 4.42E+04 4.46E+04 4.49E+04 4.53E+04

tDxf q [stb/day] 7.50E+00 132 7.59E+00 132 7.69E+00 132 7.78E+00 131 7.87E+00 131 7.96E+00 131 8.05E+00 130 8.15E+00 130 8.24E+00 130 8.33E+00 130 8.42E+00 129 8.52E+00 129 8.61E+00 129 8.70E+00 129 8.79E+00 128 8.88E+00 128 8.94E+00 128 9.00E+00 128 9.09E+00 128 9.18E+00 127 9.28E+00 127 9.37E+00 127 9.46E+00 127 9.55E+00 126 9.65E+00 126 9.74E+00 126 9.83E+00 126 9.92E+00 125 1.00E+01 125 1.01E+01 125 1.02E+01 125 1.03E+01 125 1.04E+01 124 1.05E+01 124 1.06E+01 124 1.07E+01 124 1.07E+01 124 1.08E+01 124 1.09E+01 123 1.10E+01 123 1.11E+01 123 1.12E+01 123 1.13E+01 123 1.14E+01 122 1.14E+01 122

qD 4.15E-01 4.14E-01 4.13E-01 4.12E-01 4.11E-01 4.10E-01 4.09E-01 4.08E-01 4.07E-01 4.07E-01 4.06E-01 4.05E-01 4.04E-01 4.03E-01 4.03E-01 4.02E-01 4.01E-01 4.01E-01 4.00E-01 3.99E-01 3.99E-01 3.98E-01 3.97E-01 3.96E-01 3.96E-01 3.95E-01 3.94E-01 3.94E-01 3.93E-01 3.92E-01 3.92E-01 3.91E-01 3.91E-01 3.90E-01 3.89E-01 3.89E-01 3.88E-01 3.88E-01 3.87E-01 3.87E-01 3.86E-01 3.85E-01 3.85E-01 3.84E-01 3.84E-01

139

Table 4 - FCD=25– Results of Numerical Simulation and Dimensionless Time, Pressure and Rate - continued t [days] 4.57E+04 4.60E+04 4.64E+04 4.67E+04 4.71E+04 4.75E+04 4.78E+04 4.82E+04 4.86E+04 4.89E+04 4.93E+04 4.97E+04 5.00E+04 5.04E+04 5.08E+04 5.10E+04 5.13E+04 5.16E+04 5.20E+04 5.24E+04 5.27E+04 5.31E+04 5.35E+04 5.38E+04 5.42E+04 5.46E+04 5.49E+04 5.53E+04 5.57E+04 5.60E+04 5.64E+04 5.68E+04 5.71E+04 5.75E+04 5.79E+04 5.82E+04 5.86E+04 5.89E+04 5.93E+04 5.97E+04 6.00E+04 6.04E+04 6.08E+04 6.11E+04 6.14E+04 6.16E+04

tDxf q [stb/day] 1.15E+01 122 1.16E+01 122 1.17E+01 122 1.18E+01 122 1.19E+01 121 1.20E+01 121 1.21E+01 121 1.22E+01 121 1.23E+01 121 1.24E+01 121 1.25E+01 120 1.26E+01 120 1.26E+01 120 1.27E+01 120 1.28E+01 120 1.29E+01 120 1.30E+01 120 1.31E+01 120 1.31E+01 119 1.32E+01 119 1.33E+01 119 1.34E+01 119 1.35E+01 119 1.36E+01 119 1.37E+01 119 1.38E+01 118 1.39E+01 118 1.40E+01 118 1.41E+01 118 1.42E+01 118 1.43E+01 118 1.43E+01 118 1.44E+01 117 1.45E+01 117 1.46E+01 117 1.47E+01 117 1.48E+01 117 1.49E+01 117 1.50E+01 117 1.51E+01 117 1.52E+01 116 1.53E+01 116 1.54E+01 116 1.55E+01 116 1.55E+01 116 1.56E+01 116

qD 3.83E-01 3.83E-01 3.82E-01 3.82E-01 3.81E-01 3.81E-01 3.80E-01 3.80E-01 3.79E-01 3.79E-01 3.78E-01 3.78E-01 3.77E-01 3.77E-01 3.76E-01 3.76E-01 3.75E-01 3.75E-01 3.75E-01 3.74E-01 3.74E-01 3.73E-01 3.73E-01 3.72E-01 3.72E-01 3.71E-01 3.71E-01 3.71E-01 3.70E-01 3.70E-01 3.69E-01 3.69E-01 3.69E-01 3.68E-01 3.68E-01 3.67E-01 3.67E-01 3.67E-01 3.66E-01 3.66E-01 3.65E-01 3.65E-01 3.65E-01 3.64E-01 3.64E-01 3.64E-01

140

Table 5 – FCD=100 – Results of Numerical Simulation and Dimensionless Time, Pressure and Rate t [days] 0 1.16E-05 2.55E-05 4.22E-05 6.22E-05 8.62E-05 1.15E-04 1.50E-04 1.91E-04 2.41E-04 3.01E-04 3.72E-04 4.58E-04 5.61E-04 6.85E-04 8.34E-04 1.01E-03 1.23E-03 1.48E-03 1.79E-03 2.16E-03 2.61E-03 3.14E-03 3.78E-03 4.54E-03 5.46E-03 6.56E-03 7.88E-03 9.47E-03 1.14E-02 1.37E-02 1.64E-02 1.97E-02 2.37E-02 2.84E-02 3.41E-02 4.10E-02 4.92E-02 5.90E-02 7.08E-02 8.50E-02 1.02E-01 1.22E-01 1.47E-01

tDxf 0 2.92E-09 6.44E-09 1.07E-08 1.57E-08 2.18E-08 2.91E-08 3.78E-08 4.83E-08 6.09E-08 7.60E-08 9.41E-08 1.16E-07 1.42E-07 1.73E-07 2.11E-07 2.56E-07 3.10E-07 3.75E-07 4.53E-07 5.46E-07 6.58E-07 7.93E-07 9.54E-07 1.15E-06 1.38E-06 1.66E-06 1.99E-06 2.39E-06 2.88E-06 3.46E-06 4.15E-06 4.98E-06 5.99E-06 7.19E-06 8.63E-06 1.04E-05 1.24E-05 1.49E-05 1.79E-05 2.15E-05 2.58E-05 3.09E-05 3.71E-05

q [stb/day] 0 116,839 65,262 54,250 49,678 47,341 46,040 45,279 44,809 44,488 44,231 43,991 43,735 43,448 43,113 42,722 42,266 41,734 41,118 40,410 39,600 38,684 37,659 36,526 35,290 33,965 32,572 31,133 29,672 28,223 26,819 25,480 24,223 23,052 21,955 20,917 19,923 18,959 18,015 17,088 16,179 15,296 14,441 13,617

qD 0 3.67E+02 2.05E+02 1.70E+02 1.56E+02 1.49E+02 1.44E+02 1.42E+02 1.41E+02 1.40E+02 1.39E+02 1.38E+02 1.37E+02 1.36E+02 1.35E+02 1.34E+02 1.33E+02 1.31E+02 1.29E+02 1.27E+02 1.24E+02 1.21E+02 1.18E+02 1.15E+02 1.11E+02 1.07E+02 1.02E+02 9.77E+01 9.31E+01 8.86E+01 8.42E+01 8.00E+01 7.60E+01 7.23E+01 6.89E+01 6.56E+01 6.25E+01 5.95E+01 5.65E+01 5.36E+01 5.08E+01 4.80E+01 4.53E+01 4.27E+01

p [psi] 5,000 4,996 4,994 4,993 4,992 4,991 4,991 4,991 4,990 4,990 4,990 4,990 4,990 4,990 4,990 4,990 4,989 4,989 4,989 4,989 4,989 4,989 4,988 4,988 4,988 4,987 4,987 4,986 4,986 4,985 4,984 4,984 4,983 4,982 4,981 4,980 4,980 4,979 4,978 4,976 4,975 4,974 4,973 4,971

pD n/a 2.70E-03 3.99E-03 4.89E-03 5.55E-03 6.04E-03 6.41E-03 6.67E-03 6.86E-03 6.98E-03 7.07E-03 7.14E-03 7.19E-03 7.24E-03 7.30E-03 7.37E-03 7.44E-03 7.53E-03 7.64E-03 7.76E-03 7.91E-03 8.07E-03 8.27E-03 8.49E-03 8.75E-03 9.04E-03 9.36E-03 9.72E-03 1.01E-02 1.06E-02 1.10E-02 1.15E-02 1.21E-02 1.26E-02 1.32E-02 1.38E-02 1.45E-02 1.52E-02 1.59E-02 1.67E-02 1.75E-02 1.84E-02 1.94E-02 2.05E-02

141

Table 5 – FCD=100 – Results of Numerical Simulation and Dimensionless Time, Pressure and Rate - continued t [days] 1.76E-01 2.12E-01 2.54E-01 3.05E-01 3.66E-01 4.39E-01 5.27E-01 6.32E-01 7.58E-01 9.10E-01 1.09E+00 1.31E+00 1.57E+00 1.89E+00 2.26E+00 2.72E+00 3.26E+00 3.91E+00 4.69E+00 5.63E+00 6.76E+00 8.11E+00 9.73E+00 1.17E+01 1.40E+01 1.68E+01 2.02E+01 2.42E+01 2.91E+01 3.49E+01 4.19E+01 5.02E+01 6.02E+01 7.23E+01 8.68E+01 1.04E+02 1.25E+02 1.50E+02 1.80E+02 2.16E+02 2.59E+02 3.11E+02 3.73E+02 4.48E+02 5.38E+02

tDxf q [stb/day] 4.46E-05 12,823 5.35E-05 12,056 6.42E-05 11,316 7.70E-05 10,599 9.24E-05 9,907 1.11E-04 9,242 1.33E-04 8,605 1.60E-04 8,000 1.91E-04 7,429 2.30E-04 6,888 2.76E-04 6,377 3.31E-04 5,897 3.97E-04 5,444 4.77E-04 5,018 5.72E-04 4,621 6.87E-04 4,253 8.24E-04 3,914 9.89E-04 3,600 1.19E-03 3,309 1.42E-03 3,040 1.71E-03 2,790 2.05E-03 2,560 2.46E-03 2,350 2.95E-03 2,157 3.54E-03 1,981 4.25E-03 1,820 5.10E-03 1,672 6.12E-03 1,536 7.35E-03 1,411 8.82E-03 1,297 1.06E-02 1,194 1.27E-02 1,100 1.52E-02 1,016 1.83E-02 938 2.19E-02 868 2.63E-02 803 3.16E-02 744 3.79E-02 690 4.55E-02 640 5.46E-02 594 6.55E-02 551 7.86E-02 513 9.43E-02 477 1.13E-01 445 1.36E-01 415

qD 4.02E+01 3.78E+01 3.55E+01 3.33E+01 3.11E+01 2.90E+01 2.70E+01 2.51E+01 2.33E+01 2.16E+01 2.00E+01 1.85E+01 1.71E+01 1.57E+01 1.45E+01 1.33E+01 1.23E+01 1.13E+01 1.04E+01 9.54E+00 8.75E+00 8.03E+00 7.37E+00 6.77E+00 6.21E+00 5.71E+00 5.25E+00 4.82E+00 4.43E+00 4.07E+00 3.74E+00 3.45E+00 3.19E+00 2.94E+00 2.72E+00 2.52E+00 2.34E+00 2.16E+00 2.01E+00 1.86E+00 1.73E+00 1.61E+00 1.50E+00 1.39E+00 1.30E+00

p [psi] 4,969 4,968 4,966 4,964 4,962 4,959 4,957 4,954 4,951 4,947 4,944 4,940 4,935 4,930 4,925 4,919 4,913 4,906 4,899 4,891 4,882 4,872 4,861 4,850 4,837 4,824 4,809 4,792 4,774 4,755 4,734 4,711 4,687 4,660 4,631 4,600 4,567 4,531 4,492 4,451 4,406 4,358 4,306 4,251 4,193

pD 2.16E-02 2.29E-02 2.42E-02 2.56E-02 2.71E-02 2.88E-02 3.07E-02 3.27E-02 3.49E-02 3.73E-02 3.99E-02 4.28E-02 4.59E-02 4.93E-02 5.30E-02 5.71E-02 6.15E-02 6.64E-02 7.17E-02 7.75E-02 8.38E-02 9.06E-02 9.82E-02 1.06E-01 1.15E-01 1.25E-01 1.36E-01 1.47E-01 1.60E-01 1.73E-01 1.88E-01 2.04E-01 2.22E-01 2.41E-01 2.61E-01 2.83E-01 3.07E-01 3.32E-01 3.60E-01 3.89E-01 4.21E-01 4.55E-01 4.91E-01 5.30E-01 5.72E-01

142

Table 5 – FCD=100 – Results of Numerical Simulation and Dimensionless Time, Pressure and Rate - continued t [days] 6.46E+02 7.75E+02 9.30E+02 1.12E+03 1.34E+03 1.61E+03 1.93E+03 2.12E+03 2.31E+03 2.54E+03 2.77E+03 3.05E+03 3.33E+03 3.66E+03 3.99E+03 4.36E+03 4.58E+03 4.79E+03 5.16E+03 5.46E+03 5.75E+03 6.12E+03 6.48E+03 6.69E+03 6.90E+03 7.27E+03 7.63E+03 7.96E+03 8.28E+03 8.65E+03 9.01E+03 9.38E+03 9.66E+03 9.94E+03 1.03E+04 1.07E+04 1.10E+04 1.14E+04 1.17E+04 1.19E+04 1.23E+04 1.27E+04 1.30E+04 1.34E+04 1.38E+04

tDxf q [stb/day] 1.63E-01 387 1.96E-01 362 2.35E-01 339 2.82E-01 319 3.38E-01 299 4.06E-01 282 4.87E-01 266 5.36E-01 258 5.84E-01 251 6.43E-01 244 7.01E-01 237 7.71E-01 231 8.41E-01 225 9.25E-01 219 1.01E+00 213 1.10E+00 208 1.16E+00 206 1.21E+00 203 1.30E+00 199 1.38E+00 196 1.45E+00 193 1.55E+00 190 1.64E+00 188 1.69E+00 186 1.74E+00 185 1.84E+00 182 1.93E+00 180 2.01E+00 178 2.09E+00 177 2.19E+00 175 2.28E+00 173 2.37E+00 171 2.44E+00 170 2.51E+00 169 2.60E+00 168 2.70E+00 166 2.79E+00 165 2.88E+00 164 2.95E+00 163 3.02E+00 162 3.11E+00 161 3.20E+00 160 3.29E+00 159 3.38E+00 158 3.48E+00 157

qD 1.22E+00 1.14E+00 1.07E+00 9.99E-01 9.40E-01 8.85E-01 8.35E-01 8.10E-01 7.87E-01 7.64E-01 7.44E-01 7.23E-01 7.05E-01 6.86E-01 6.70E-01 6.54E-01 6.45E-01 6.37E-01 6.25E-01 6.16E-01 6.07E-01 5.98E-01 5.89E-01 5.84E-01 5.80E-01 5.72E-01 5.65E-01 5.60E-01 5.54E-01 5.49E-01 5.43E-01 5.38E-01 5.34E-01 5.31E-01 5.26E-01 5.22E-01 5.18E-01 5.14E-01 5.12E-01 5.09E-01 5.06E-01 5.02E-01 4.99E-01 4.96E-01 4.93E-01

p [psi] 4,130 4,063 3,992 3,917 3,838 3,755 3,668 3,620 3,574 3,524 3,477 3,425 3,377 3,323 3,273 3,222 3,193 3,166 3,122 3,088 3,055 3,018 2,982 2,962 2,942 2,910 2,879 2,852 2,827

pD 6.16E-01 6.64E-01 7.14E-01 7.67E-01 8.23E-01 8.82E-01 9.43E-01 9.78E-01 1.01E+00 1.05E+00 1.08E+00 1.12E+00 1.15E+00 1.19E+00 1.22E+00 1.26E+00 1.28E+00 1.30E+00 1.33E+00 1.35E+00 1.38E+00 1.40E+00 1.43E+00 1.44E+00 1.46E+00 1.48E+00 1.50E+00 1.52E+00 1.54E+00

143

Table 5 – FCD=100 – Results of Numerical Simulation and Dimensionless Time, Pressure and Rate - continued t [days] 1.40E+04 1.43E+04 1.47E+04 1.51E+04 1.54E+04 1.58E+04 1.61E+04 1.65E+04 1.68E+04 1.72E+04 1.75E+04 1.79E+04 1.83E+04 1.86E+04 1.90E+04 1.94E+04 1.97E+04 2.01E+04 2.04E+04 2.06E+04 2.10E+04 2.13E+04 2.17E+04 2.21E+04 2.24E+04 2.28E+04 2.32E+04 2.35E+04 2.39E+04 2.43E+04 2.45E+04 2.47E+04 2.51E+04 2.55E+04 2.58E+04 2.62E+04 2.66E+04 2.69E+04 2.73E+04 2.77E+04 2.80E+04 2.84E+04 2.87E+04 2.91E+04 2.94E+04

tDxf q [stb/day] 3.55E+00 156 3.62E+00 156 3.71E+00 155 3.80E+00 154 3.90E+00 153 3.99E+00 153 4.08E+00 152 4.17E+00 151 4.26E+00 150 4.34E+00 150 4.43E+00 149 4.53E+00 149 4.62E+00 148 4.71E+00 147 4.80E+00 147 4.90E+00 146 4.99E+00 146 5.08E+00 145 5.14E+00 145 5.21E+00 144 5.30E+00 144 5.39E+00 143 5.49E+00 143 5.58E+00 142 5.67E+00 142 5.76E+00 141 5.85E+00 141 5.95E+00 141 6.04E+00 140 6.13E+00 140 6.19E+00 139 6.25E+00 139 6.34E+00 139 6.43E+00 138 6.53E+00 138 6.62E+00 138 6.71E+00 137 6.80E+00 137 6.90E+00 136 6.99E+00 136 7.08E+00 136 7.17E+00 135 7.26E+00 135 7.36E+00 135 7.43E+00 135

qD 4.91E-01 4.89E-01 4.86E-01 4.84E-01 4.81E-01 4.79E-01 4.76E-01 4.74E-01 4.72E-01 4.70E-01 4.68E-01 4.66E-01 4.64E-01 4.62E-01 4.60E-01 4.59E-01 4.57E-01 4.55E-01 4.54E-01 4.53E-01 4.51E-01 4.50E-01 4.48E-01 4.47E-01 4.45E-01 4.44E-01 4.42E-01 4.41E-01 4.40E-01 4.38E-01 4.37E-01 4.37E-01 4.35E-01 4.34E-01 4.33E-01 4.32E-01 4.30E-01 4.29E-01 4.28E-01 4.27E-01 4.26E-01 4.25E-01 4.24E-01 4.23E-01 4.22E-01

144

Table 5 – FCD=100 – Results of Numerical Simulation and Dimensionless Time, Pressure and Rate - continued t [days] 2.97E+04 3.00E+04 3.04E+04 3.08E+04 3.11E+04 3.15E+04 3.19E+04 3.22E+04 3.26E+04 3.30E+04 3.33E+04 3.37E+04 3.41E+04 3.44E+04 3.48E+04 3.52E+04 3.54E+04 3.56E+04 3.60E+04 3.63E+04 3.67E+04 3.71E+04 3.74E+04 3.78E+04 3.82E+04 3.85E+04 3.89E+04 3.93E+04 3.96E+04 4.00E+04 4.04E+04 4.07E+04 4.11E+04 4.15E+04 4.18E+04 4.22E+04 4.25E+04 4.27E+04 4.31E+04 4.35E+04 4.38E+04 4.42E+04 4.46E+04 4.49E+04 4.53E+04

tDxf q [stb/day] 7.50E+00 134 7.59E+00 134 7.69E+00 134 7.78E+00 133 7.87E+00 133 7.96E+00 133 8.05E+00 132 8.15E+00 132 8.24E+00 132 8.33E+00 132 8.42E+00 131 8.52E+00 131 8.61E+00 131 8.70E+00 131 8.79E+00 130 8.88E+00 130 8.94E+00 130 9.00E+00 130 9.09E+00 129 9.18E+00 129 9.28E+00 129 9.37E+00 129 9.46E+00 129 9.55E+00 128 9.65E+00 128 9.74E+00 128 9.83E+00 128 9.92E+00 127 1.00E+01 127 1.01E+01 127 1.02E+01 127 1.03E+01 127 1.04E+01 126 1.05E+01 126 1.06E+01 126 1.07E+01 126 1.07E+01 126 1.08E+01 125 1.09E+01 125 1.10E+01 125 1.11E+01 125 1.12E+01 125 1.13E+01 125 1.14E+01 124 1.14E+01 124

qD 4.21E-01 4.20E-01 4.19E-01 4.18E-01 4.17E-01 4.17E-01 4.16E-01 4.15E-01 4.14E-01 4.13E-01 4.12E-01 4.11E-01 4.10E-01 4.10E-01 4.09E-01 4.08E-01 4.08E-01 4.07E-01 4.06E-01 4.05E-01 4.05E-01 4.04E-01 4.03E-01 4.03E-01 4.02E-01 4.01E-01 4.00E-01 4.00E-01 3.99E-01 3.98E-01 3.98E-01 3.97E-01 3.96E-01 3.96E-01 3.95E-01 3.95E-01 3.94E-01 3.94E-01 3.93E-01 3.92E-01 3.92E-01 3.91E-01 3.91E-01 3.90E-01 3.90E-01

145

Table 5 – FCD=100 – Results of Numerical Simulation and Dimensionless Time, Pressure and Rate - continued t [days] 4.57E+04 4.60E+04 4.64E+04 4.67E+04 4.71E+04 4.75E+04 4.78E+04 4.82E+04 4.86E+04 4.89E+04 4.93E+04 4.97E+04 5.00E+04 5.04E+04 5.08E+04 5.10E+04 5.13E+04 5.16E+04 5.20E+04 5.24E+04 5.27E+04 5.31E+04 5.35E+04 5.38E+04 5.42E+04 5.46E+04 5.49E+04 5.53E+04 5.57E+04 5.60E+04 5.64E+04 5.68E+04 5.71E+04 5.75E+04 5.79E+04 5.82E+04 5.86E+04 5.89E+04 5.93E+04 5.97E+04 6.00E+04 6.04E+04 6.08E+04 6.11E+04 6.14E+04 6.16E+04

tDxf q [stb/day] 1.15E+01 124 1.16E+01 124 1.17E+01 124 1.18E+01 123 1.19E+01 123 1.20E+01 123 1.21E+01 123 1.22E+01 123 1.23E+01 123 1.24E+01 122 1.25E+01 122 1.26E+01 122 1.26E+01 122 1.27E+01 122 1.28E+01 122 1.29E+01 122 1.30E+01 121 1.31E+01 121 1.31E+01 121 1.32E+01 121 1.33E+01 121 1.34E+01 121 1.35E+01 121 1.36E+01 120 1.37E+01 120 1.38E+01 120 1.39E+01 120 1.40E+01 120 1.41E+01 120 1.42E+01 120 1.43E+01 119 1.43E+01 119 1.44E+01 119 1.45E+01 119 1.46E+01 119 1.47E+01 119 1.48E+01 119 1.49E+01 119 1.50E+01 118 1.51E+01 118 1.52E+01 118 1.53E+01 118 1.54E+01 118 1.55E+01 118 1.55E+01 118 1.56E+01 118

qD 3.89E-01 3.88E-01 3.88E-01 3.87E-01 3.87E-01 3.86E-01 3.86E-01 3.85E-01 3.85E-01 3.84E-01 3.84E-01 3.83E-01 3.83E-01 3.82E-01 3.82E-01 3.81E-01 3.81E-01 3.81E-01 3.80E-01 3.80E-01 3.79E-01 3.79E-01 3.78E-01 3.78E-01 3.77E-01 3.77E-01 3.76E-01 3.76E-01 3.76E-01 3.75E-01 3.75E-01 3.74E-01 3.74E-01 3.73E-01 3.73E-01 3.73E-01 3.72E-01 3.72E-01 3.71E-01 3.71E-01 3.71E-01 3.70E-01 3.70E-01 3.70E-01 3.69E-01 3.69E-01

146

Table 6 – FCD=500 – Results of Numerical Simulation and Dimensionless Time, Pressure and Rate t [days] 0 1.16E-05 2.55E-05 4.22E-05 6.22E-05 8.62E-05 1.15E-04 1.50E-04 1.91E-04 2.41E-04 3.01E-04 3.72E-04 4.58E-04 5.61E-04 6.85E-04 8.34E-04 1.01E-03 1.23E-03 1.48E-03 1.79E-03 2.16E-03 2.61E-03 3.14E-03 3.78E-03 4.54E-03 5.46E-03 6.56E-03 7.88E-03 9.47E-03 1.14E-02 1.37E-02 1.64E-02 1.97E-02 2.37E-02 2.84E-02 3.41E-02 4.10E-02 4.92E-02 5.90E-02 7.08E-02 8.50E-02 1.02E-01 1.22E-01 1.47E-01

tDxf 0 2.92E-09 6.44E-09 1.07E-08 1.57E-08 2.18E-08 2.91E-08 3.78E-08 4.83E-08 6.09E-08 7.60E-08 9.41E-08 1.16E-07 1.42E-07 1.73E-07 2.11E-07 2.56E-07 3.10E-07 3.75E-07 4.53E-07 5.46E-07 6.58E-07 7.93E-07 9.54E-07 1.15E-06 1.38E-06 1.66E-06 1.99E-06 2.39E-06 2.88E-06 3.46E-06 4.15E-06 4.98E-06 5.99E-06 7.19E-06 8.63E-06 1.04E-05 1.24E-05 1.49E-05 1.79E-05 2.15E-05 2.58E-05 3.09E-05 3.71E-05

q [stb/day] 0 264,200 148,579 122,893 111,672 105,603 101,959 99,657 98,169 97,166 96,421 95,772 95,116 94,389 93,545 92,557 91,405 90,058 88,495 86,694 84,630 82,289 79,662 76,742 73,543 70,093 66,446 62,649 58,762 54,879 51,086 47,450 44,024 40,832 37,874 35,127 32,554 30,132 27,835 25,657 23,595 21,672 19,891 18,254

qD 0 8.29E+02 4.66E+02 3.86E+02 3.50E+02 3.31E+02 3.20E+02 3.13E+02 3.08E+02 3.05E+02 3.03E+02 3.01E+02 2.98E+02 2.96E+02 2.94E+02 2.90E+02 2.87E+02 2.83E+02 2.78E+02 2.72E+02 2.66E+02 2.58E+02 2.50E+02 2.41E+02 2.31E+02 2.20E+02 2.08E+02 1.97E+02 1.84E+02 1.72E+02 1.60E+02 1.49E+02 1.38E+02 1.28E+02 1.19E+02 1.10E+02 1.02E+02 9.45E+01 8.73E+01 8.05E+01 7.40E+01 6.80E+01 6.24E+01 5.73E+01

p [psi] 5,000 4,998 4,998 4,997 4,997 4,996 4,996 4,996 4,996 4,996 4,995 4,995 4,995 4,995 4,995 4,995 4,995 4,995 4,995 4,995 4,995 4,995 4,995 4,994 4,994 4,994 4,994 4,993 4,993 4,993 4,992 4,992 4,991 4,991 4,990 4,989 4,989 4,988 4,987 4,986 4,985 4,984 4,983 4,981

pD n/a 1.20E-03 1.76E-03 2.16E-03 2.45E-03 2.68E-03 2.86E-03 3.00E-03 3.10E-03 3.17E-03 3.23E-03 3.27E-03 3.30E-03 3.33E-03 3.36E-03 3.39E-03 3.43E-03 3.48E-03 3.54E-03 3.61E-03 3.69E-03 3.78E-03 3.89E-03 4.01E-03 4.16E-03 4.32E-03 4.51E-03 4.73E-03 4.97E-03 5.24E-03 5.54E-03 5.87E-03 6.23E-03 6.62E-03 7.05E-03 7.52E-03 8.02E-03 8.57E-03 9.18E-03 9.84E-03 1.06E-02 1.14E-02 1.22E-02 1.32E-02

147

Table 6 – FCD=500 – Results of Numerical Simulation and Dimensionless Time, Pressure and Rate - continued t [days] 1.76E-01 2.12E-01 2.54E-01 3.05E-01 3.66E-01 4.39E-01 5.27E-01 6.32E-01 7.58E-01 9.10E-01 1.09E+00 1.31E+00 1.57E+00 1.89E+00 2.26E+00 2.72E+00 3.26E+00 3.91E+00 4.69E+00 5.63E+00 6.76E+00 8.11E+00 9.73E+00 1.17E+01 1.40E+01 1.68E+01 2.02E+01 2.42E+01 2.91E+01 3.49E+01 4.19E+01 5.02E+01 6.02E+01 7.23E+01 8.68E+01 1.04E+02 1.25E+02 1.50E+02 1.80E+02 2.16E+02 2.59E+02 3.11E+02 3.73E+02 4.48E+02 5.38E+02

tDxf q [stb/day] 4.46E-05 16,753 5.35E-05 15,374 6.42E-05 14,101 7.70E-05 12,921 9.24E-05 11,825 1.11E-04 10,816 1.33E-04 9,890 1.60E-04 9,048 1.91E-04 8,283 2.30E-04 7,581 2.76E-04 6,939 3.31E-04 6,349 3.97E-04 5,805 4.77E-04 5,305 5.72E-04 4,851 6.87E-04 4,437 8.24E-04 4,062 9.89E-04 3,720 1.19E-03 3,406 1.42E-03 3,118 1.71E-03 2,853 2.05E-03 2,612 2.46E-03 2,392 2.95E-03 2,192 3.54E-03 2,010 4.25E-03 1,845 5.10E-03 1,693 6.12E-03 1,554 7.35E-03 1,426 8.82E-03 1,311 1.06E-02 1,206 1.27E-02 1,111 1.52E-02 1,026 1.83E-02 948 2.19E-02 876 2.63E-02 811 3.16E-02 751 3.79E-02 696 4.55E-02 645 5.46E-02 599 6.55E-02 556 7.86E-02 517 9.43E-02 481 1.13E-01 448 1.36E-01 418

qD 5.26E+01 4.82E+01 4.42E+01 4.05E+01 3.71E+01 3.39E+01 3.10E+01 2.84E+01 2.60E+01 2.38E+01 2.18E+01 1.99E+01 1.82E+01 1.66E+01 1.52E+01 1.39E+01 1.27E+01 1.17E+01 1.07E+01 9.78E+00 8.95E+00 8.20E+00 7.51E+00 6.88E+00 6.31E+00 5.79E+00 5.31E+00 4.88E+00 4.48E+00 4.11E+00 3.78E+00 3.49E+00 3.22E+00 2.97E+00 2.75E+00 2.54E+00 2.36E+00 2.18E+00 2.02E+00 1.88E+00 1.74E+00 1.62E+00 1.51E+00 1.41E+00 1.31E+00

p [psi] 4,980 4,978 4,977 4,975 4,972 4,970 4,968 4,965 4,962 4,959 4,955 4,951 4,947 4,942 4,937 4,931 4,925 4,918 4,911 4,902 4,894 4,884 4,873 4,862 4,849 4,836 4,821 4,804 4,787 4,768 4,747 4,724 4,699 4,673 4,644 4,613 4,580 4,544 4,505 4,464 4,419 4,371 4,320 4,265 4,206

pD 1.42E-02 1.54E-02 1.66E-02 1.80E-02 1.95E-02 2.11E-02 2.29E-02 2.49E-02 2.70E-02 2.94E-02 3.19E-02 3.47E-02 3.78E-02 4.12E-02 4.49E-02 4.89E-02 5.33E-02 5.81E-02 6.34E-02 6.91E-02 7.54E-02 8.22E-02 8.97E-02 9.78E-02 1.07E-01 1.16E-01 1.27E-01 1.39E-01 1.51E-01 1.65E-01 1.79E-01 1.96E-01 2.13E-01 2.32E-01 2.52E-01 2.74E-01 2.97E-01 3.23E-01 3.50E-01 3.80E-01 4.11E-01 4.45E-01 4.82E-01 5.20E-01 5.62E-01

148

Table 6 – FCD=500 – Results of Numerical Simulation and Dimensionless Time, Pressure and Rate - continued t [days] 6.46E+02 7.75E+02 9.30E+02 1.12E+03 1.34E+03 1.61E+03 1.93E+03 2.12E+03 2.31E+03 2.54E+03 2.77E+03 3.05E+03 3.33E+03 3.66E+03 3.99E+03 4.36E+03 4.58E+03 4.79E+03 5.16E+03 5.46E+03 5.75E+03 6.12E+03 6.48E+03 6.69E+03 6.90E+03 7.27E+03 7.63E+03 7.96E+03 8.28E+03 8.65E+03 9.01E+03 9.38E+03 9.66E+03 9.94E+03 1.03E+04 1.07E+04 1.10E+04 1.14E+04 1.17E+04 1.19E+04 1.23E+04 1.27E+04 1.30E+04 1.34E+04 1.38E+04

tDxf q [stb/day] 1.63E-01 390 1.96E-01 365 2.35E-01 342 2.82E-01 321 3.38E-01 301 4.06E-01 284 4.87E-01 268 5.36E-01 260 5.84E-01 252 6.43E-01 245 7.01E-01 239 7.71E-01 232 8.41E-01 226 9.25E-01 220 1.01E+00 215 1.10E+00 210 1.16E+00 207 1.21E+00 204 1.30E+00 200 1.38E+00 197 1.45E+00 195 1.55E+00 191 1.64E+00 189 1.69E+00 187 1.74E+00 186 1.84E+00 183 1.93E+00 181 2.01E+00 179 2.09E+00 178 2.19E+00 176 2.28E+00 174 2.37E+00 172 2.44E+00 171 2.51E+00 170 2.60E+00 169 2.70E+00 167 2.79E+00 166 2.88E+00 165 2.95E+00 164 3.02E+00 163 3.11E+00 162 3.20E+00 161 3.29E+00 160 3.38E+00 159 3.48E+00 158

qD 1.22E+00 1.14E+00 1.07E+00 1.01E+00 9.46E-01 8.91E-01 8.41E-01 8.15E-01 7.92E-01 7.69E-01 7.49E-01 7.28E-01 7.09E-01 6.90E-01 6.74E-01 6.58E-01 6.49E-01 6.41E-01 6.28E-01 6.19E-01 6.10E-01 6.01E-01 5.92E-01 5.87E-01 5.83E-01 5.75E-01 5.68E-01 5.63E-01 5.57E-01 5.51E-01 5.46E-01 5.41E-01 5.37E-01 5.33E-01 5.29E-01 5.25E-01 5.21E-01 5.17E-01 5.14E-01 5.12E-01 5.08E-01 5.05E-01 5.02E-01 4.99E-01 4.96E-01

p [psi] 4,144 4,077 4,006 3,932 3,853 3,770 3,683 3,635 3,589 3,539 3,492 3,441 3,392 3,338 3,289 3,238 3,209 3,181 3,137 3,103 3,071 3,033 2,998 2,978 2,958 2,926 2,895 2,868 2,843

pD 6.06E-01 6.54E-01 7.04E-01 7.57E-01 8.12E-01 8.71E-01 9.33E-01 9.67E-01 9.99E-01 1.03E+00 1.07E+00 1.10E+00 1.14E+00 1.18E+00 1.21E+00 1.25E+00 1.27E+00 1.29E+00 1.32E+00 1.34E+00 1.37E+00 1.39E+00 1.42E+00 1.43E+00 1.45E+00 1.47E+00 1.49E+00 1.51E+00 1.53E+00

149

Table 6 – FCD=500 – Results of Numerical Simulation and Dimensionless Time, Pressure and Rate - continued t [days] 1.40E+04 1.43E+04 1.47E+04 1.51E+04 1.54E+04 1.58E+04 1.61E+04 1.65E+04 1.68E+04 1.72E+04 1.75E+04 1.79E+04 1.83E+04 1.86E+04 1.90E+04 1.94E+04 1.97E+04 2.01E+04 2.04E+04 2.06E+04 2.10E+04 2.13E+04 2.17E+04 2.21E+04 2.24E+04 2.28E+04 2.32E+04 2.35E+04 2.39E+04 2.43E+04 2.45E+04 2.47E+04 2.51E+04 2.55E+04 2.58E+04 2.62E+04 2.66E+04 2.69E+04 2.73E+04 2.77E+04 2.80E+04 2.84E+04 2.87E+04 2.91E+04 2.94E+04

tDxf q [stb/day] 3.55E+00 157 3.62E+00 157 3.71E+00 156 3.80E+00 155 3.90E+00 154 3.99E+00 153 4.08E+00 153 4.17E+00 152 4.26E+00 151 4.34E+00 151 4.43E+00 150 4.53E+00 149 4.62E+00 149 4.71E+00 148 4.80E+00 147 4.90E+00 147 4.99E+00 146 5.08E+00 146 5.14E+00 145 5.21E+00 145 5.30E+00 144 5.39E+00 144 5.49E+00 143 5.58E+00 143 5.67E+00 142 5.76E+00 142 5.85E+00 142 5.95E+00 141 6.04E+00 141 6.13E+00 140 6.19E+00 140 6.25E+00 140 6.34E+00 139 6.43E+00 139 6.53E+00 139 6.62E+00 138 6.71E+00 138 6.80E+00 137 6.90E+00 137 6.99E+00 137 7.08E+00 136 7.17E+00 136 7.26E+00 136 7.36E+00 135 7.43E+00 135

qD 4.93E-01 4.91E-01 4.89E-01 4.86E-01 4.83E-01 4.81E-01 4.79E-01 4.76E-01 4.74E-01 4.72E-01 4.70E-01 4.68E-01 4.66E-01 4.64E-01 4.63E-01 4.61E-01 4.59E-01 4.57E-01 4.56E-01 4.55E-01 4.53E-01 4.52E-01 4.50E-01 4.49E-01 4.47E-01 4.46E-01 4.44E-01 4.43E-01 4.41E-01 4.40E-01 4.39E-01 4.38E-01 4.37E-01 4.36E-01 4.35E-01 4.34E-01 4.32E-01 4.31E-01 4.30E-01 4.29E-01 4.28E-01 4.27E-01 4.26E-01 4.25E-01 4.24E-01

150

Table 6 – FCD=500 – Results of Numerical Simulation and Dimensionless Time, Pressure and Rate - continued t [days] 2.97E+04 3.00E+04 3.04E+04 3.08E+04 3.11E+04 3.15E+04 3.19E+04 3.22E+04 3.26E+04 3.30E+04 3.33E+04 3.37E+04 3.41E+04 3.44E+04 3.48E+04 3.52E+04 3.54E+04 3.56E+04 3.60E+04 3.63E+04 3.67E+04 3.71E+04 3.74E+04 3.78E+04 3.82E+04 3.85E+04 3.89E+04 3.93E+04 3.96E+04 4.00E+04 4.04E+04 4.07E+04 4.11E+04 4.15E+04 4.18E+04 4.22E+04 4.25E+04 4.27E+04 4.31E+04 4.35E+04 4.38E+04 4.42E+04 4.46E+04 4.49E+04 4.53E+04

tDxf q [stb/day] 7.50E+00 135 7.59E+00 135 7.69E+00 134 7.78E+00 134 7.87E+00 134 7.96E+00 133 8.05E+00 133 8.15E+00 133 8.24E+00 132 8.33E+00 132 8.42E+00 132 8.52E+00 132 8.61E+00 131 8.70E+00 131 8.79E+00 131 8.88E+00 131 8.94E+00 130 9.00E+00 130 9.09E+00 130 9.18E+00 130 9.28E+00 130 9.37E+00 129 9.46E+00 129 9.55E+00 129 9.65E+00 129 9.74E+00 128 9.83E+00 128 9.92E+00 128 1.00E+01 128 1.01E+01 127 1.02E+01 127 1.03E+01 127 1.04E+01 127 1.05E+01 127 1.06E+01 126 1.07E+01 126 1.07E+01 126 1.08E+01 126 1.09E+01 126 1.10E+01 126 1.11E+01 125 1.12E+01 125 1.13E+01 125 1.14E+01 125 1.14E+01 125

qD 4.23E-01 4.22E-01 4.21E-01 4.20E-01 4.19E-01 4.18E-01 4.17E-01 4.16E-01 4.16E-01 4.15E-01 4.14E-01 4.13E-01 4.12E-01 4.11E-01 4.11E-01 4.10E-01 4.09E-01 4.09E-01 4.08E-01 4.07E-01 4.06E-01 4.06E-01 4.05E-01 4.04E-01 4.04E-01 4.03E-01 4.02E-01 4.01E-01 4.01E-01 4.00E-01 3.99E-01 3.99E-01 3.98E-01 3.97E-01 3.97E-01 3.96E-01 3.96E-01 3.95E-01 3.95E-01 3.94E-01 3.93E-01 3.93E-01 3.92E-01 3.92E-01 3.91E-01

151

Table 6 – FCD=500 – Results of Numerical Simulation and Dimensionless Time, Pressure and Rate - continued t [days] 4.57E+04 4.60E+04 4.64E+04 4.67E+04 4.71E+04 4.75E+04 4.78E+04 4.82E+04 4.86E+04 4.89E+04 4.93E+04 4.97E+04 5.00E+04 5.04E+04 5.08E+04 5.10E+04 5.13E+04 5.16E+04 5.20E+04 5.24E+04 5.27E+04 5.31E+04 5.35E+04 5.38E+04 5.42E+04 5.46E+04 5.49E+04 5.53E+04 5.57E+04 5.60E+04 5.64E+04 5.68E+04 5.71E+04 5.75E+04 5.79E+04 5.82E+04 5.86E+04 5.89E+04 5.93E+04 5.97E+04 6.00E+04 6.04E+04 6.08E+04 6.11E+04 6.14E+04 6.16E+04

tDxf q [stb/day] 1.15E+01 124 1.16E+01 124 1.17E+01 124 1.18E+01 124 1.19E+01 124 1.20E+01 124 1.21E+01 123 1.22E+01 123 1.23E+01 123 1.24E+01 123 1.25E+01 123 1.26E+01 123 1.26E+01 122 1.27E+01 122 1.28E+01 122 1.29E+01 122 1.30E+01 122 1.31E+01 122 1.31E+01 122 1.32E+01 121 1.33E+01 121 1.34E+01 121 1.35E+01 121 1.36E+01 121 1.37E+01 121 1.38E+01 121 1.39E+01 120 1.40E+01 120 1.41E+01 120 1.42E+01 120 1.43E+01 120 1.43E+01 120 1.44E+01 120 1.45E+01 119 1.46E+01 119 1.47E+01 119 1.48E+01 119 1.49E+01 119 1.50E+01 119 1.51E+01 119 1.52E+01 119 1.53E+01 118 1.54E+01 118 1.55E+01 118 1.55E+01 118 1.56E+01 118

qD 3.91E-01 3.90E-01 3.89E-01 3.89E-01 3.88E-01 3.88E-01 3.87E-01 3.87E-01 3.86E-01 3.86E-01 3.85E-01 3.85E-01 3.84E-01 3.84E-01 3.83E-01 3.83E-01 3.83E-01 3.82E-01 3.82E-01 3.81E-01 3.81E-01 3.80E-01 3.80E-01 3.79E-01 3.79E-01 3.78E-01 3.78E-01 3.77E-01 3.77E-01 3.77E-01 3.76E-01 3.76E-01 3.75E-01 3.75E-01 3.75E-01 3.74E-01 3.74E-01 3.73E-01 3.73E-01 3.73E-01 3.72E-01 3.72E-01 3.71E-01 3.71E-01 3.71E-01 3.71E-01

152

Table 7 – FCD=100 – Results of Numerical Simulation and Dimensionless Time, Pressure and Rate for Point Source t [days] 0.00E+00 1.16E-05 2.55E-05 4.22E-05 6.22E-05 8.62E-05 1.15E-04 1.50E-04 1.91E-04 2.41E-04 3.01E-04 3.72E-04 4.58E-04 5.61E-04 6.85E-04 8.34E-04 1.01E-03 1.23E-03 1.48E-03 1.79E-03 2.16E-03 2.61E-03 3.14E-03 3.78E-03 4.54E-03 5.46E-03 6.56E-03 7.88E-03 9.47E-03 1.14E-02 1.37E-02 1.64E-02 1.97E-02 2.37E-02 2.84E-02 3.41E-02 4.10E-02 4.92E-02 5.90E-02 7.08E-02 8.50E-02

tDxf q [stb/day] 0.00E+00 0 2.92E-09 64,317 6.44E-09 48,107 1.07E-08 41,483 1.57E-08 38,149 2.18E-08 2.91E-08 3.78E-08 4.83E-08 6.09E-08 7.60E-08 9.41E-08 1.16E-07 1.42E-07 1.73E-07 2.11E-07 2.56E-07 3.10E-07 3.75E-07 4.53E-07 5.46E-07 6.58E-07 7.93E-07 9.54E-07 1.15E-06 1.38E-06 1.66E-06 1.99E-06 2.39E-06 2.88E-06 3.46E-06 4.15E-06 4.98E-06 5.99E-06 7.19E-06 8.63E-06 1.04E-05 1.24E-05 1.49E-05 1.79E-05 2.15E-05

36,306 35,242 34,614 34,235 33,990 33,812 33,659 33,504 33,334 33,138 32,909 32,641 32,328 31,964 31,543 31,059 30,508 29,888 29,195 28,431 27,601 26,717 25,787 24,825 23,850 22,882 21,937 21,026 20,154 19,322 18,523 17,747 16,988 16,238 15,495 14,759

qD 0.00E+00 2.02E+02 1.51E+02 1.30E+02 1.20E+02

p [psi] 5,000n/a 4,993 4,991 4,990 4,989

1.14E+02 1.11E+02 1.09E+02 1.07E+02 1.07E+02 1.06E+02 1.06E+02 1.05E+02 1.05E+02 1.04E+02 1.03E+02 1.02E+02 1.01E+02 1.00E+02 9.90E+01 9.75E+01 9.57E+01 9.38E+01 9.16E+01 8.92E+01 8.66E+01 8.38E+01 8.09E+01 7.79E+01 7.48E+01 7.18E+01 6.88E+01 6.60E+01 6.32E+01 6.06E+01 5.81E+01 5.57E+01 5.33E+01 5.10E+01 4.86E+01 4.63E+01

4,988 4,988 4,987 4,987 4,987 4,987 4,987 4,987 4,987 4,987 4,986 4,986 4,986 4,986 4,986 4,986 4,985 4,985 4,985 4,985 4,984 4,984 4,983 4,983 4,982 4,981 4,981 4,980 4,979 4,978 4,977 4,976 4,975 4,974 4,973 4,972

pD 4.91E-03 6.21E-03 7.08E-03 7.78E-03 8.25E-03 8.62E-03 8.88E-03 9.07E-03 9.19E-03 9.28E-03 9.35E-03 9.40E-03 9.45E-03 9.51E-03 9.58E-03 9.65E-03 9.74E-03 9.85E-03 9.97E-03 1.01E-02 1.03E-02 1.05E-02 1.07E-02 1.10E-02 1.12E-02 1.16E-02 1.19E-02 1.23E-02 1.28E-02 1.32E-02 1.37E-02 1.43E-02 1.48E-02 1.54E-02 1.60E-02 1.67E-02 1.74E-02 1.81E-02 1.89E-02 1.97E-02

153

Table 7 – FCD=100 – Results of Numerical Simulation and Dimensionless Time, Pressure and Rate for Point Source - continued t [days] 1.02E-01 1.22E-01 1.47E-01 1.76E-01 2.12E-01 2.54E-01 3.05E-01 3.66E-01 4.39E-01 5.27E-01 6.32E-01 7.58E-01 9.10E-01 1.09E+00 1.31E+00 1.57E+00 1.89E+00 2.26E+00 2.72E+00 3.26E+00 3.91E+00 4.69E+00 5.63E+00 6.76E+00 8.11E+00 9.73E+00 1.17E+01 1.40E+01 1.68E+01 2.02E+01 2.42E+01 2.91E+01 3.49E+01 4.19E+01 5.02E+01 6.02E+01 7.23E+01 8.68E+01 1.04E+02 1.25E+02 1.50E+02 1.80E+02 2.16E+02

tDxf q [stb/day] 2.58E-05 14,035 3.09E-05 13,326 3.71E-05 12,633 4.46E-05 11,959 5.35E-05 11,301 6.42E-05 10,660 7.70E-05 10,034 9.24E-05 9,423 1.11E-04 8,830 1.33E-04 8,257 1.60E-04 7,709 1.91E-04 7,186 2.30E-04 6,685 2.76E-04 6,210 3.31E-04 5,759 3.97E-04 5,332 4.77E-04 4,927 5.72E-04 4,548 6.87E-04 4,194 8.24E-04 3,866 9.89E-04 3,561 1.19E-03 3,278 1.42E-03 3,015 1.71E-03 2,770 2.05E-03 2,545 2.46E-03 2,337 2.95E-03 2,147 3.54E-03 1,972 4.25E-03 1,813 5.10E-03 1,666 6.12E-03 1,531 7.35E-03 1,407 8.82E-03 1,294 1.06E-02 1,191 1.27E-02 1,097 1.52E-02 1,013 1.83E-02 936 2.19E-02 866 2.63E-02 802 3.16E-02 743 3.79E-02 688 4.55E-02 638 5.46E-02 593

qD 4.40E+01 4.18E+01 3.96E+01 3.75E+01 3.55E+01 3.35E+01 3.15E+01 2.96E+01 2.77E+01 2.59E+01 2.42E+01 2.25E+01 2.10E+01 1.95E+01 1.81E+01 1.67E+01 1.55E+01 1.43E+01 1.32E+01 1.21E+01 1.12E+01 1.03E+01 9.46E+00 8.69E+00 7.98E+00 7.33E+00 6.74E+00 6.19E+00 5.69E+00 5.23E+00 4.80E+00 4.41E+00 4.06E+00 3.74E+00 3.44E+00 3.18E+00 2.94E+00 2.72E+00 2.52E+00 2.33E+00 2.16E+00 2.00E+00 1.86E+00

p [psi] 4,971 4,969 4,968 4,966 4,965 4,963 4,961 4,959 4,956 4,954 4,951 4,948 4,944 4,940 4,936 4,932 4,927 4,922 4,916 4,910 4,903 4,896 4,888 4,879 4,869 4,858 4,847 4,834 4,820 4,805 4,789 4,771 4,752 4,731 4,708 4,684 4,657 4,628 4,597 4,564 4,528 4,489 4,448

pD 2.06E-02 2.16E-02 2.27E-02 2.38E-02 2.51E-02 2.64E-02 2.78E-02 2.94E-02 3.11E-02 3.29E-02 3.49E-02 3.71E-02 3.95E-02 4.21E-02 4.50E-02 4.81E-02 5.15E-02 5.52E-02 5.93E-02 6.38E-02 6.86E-02 7.39E-02 7.97E-02 8.60E-02 9.29E-02 1.00E-01 1.09E-01 1.17E-01 1.27E-01 1.38E-01 1.49E-01 1.62E-01 1.76E-01 1.90E-01 2.07E-01 2.24E-01 2.43E-01 2.63E-01 2.85E-01 3.09E-01 3.34E-01 3.62E-01 3.91E-01

154

Table 7 – FCD=100 – Results of Numerical Simulation and Dimensionless Time, Pressure and Rate for Point Source - continued t [days] 2.59E+02 3.11E+02 3.73E+02 4.48E+02 5.38E+02 6.46E+02 7.75E+02 9.30E+02 1.12E+03 1.34E+03 1.61E+03 1.93E+03 2.12E+03 2.31E+03 2.54E+03 2.77E+03 3.05E+03 3.33E+03 3.66E+03 3.99E+03 4.36E+03 4.58E+03 4.79E+03 5.16E+03 5.46E+03 5.75E+03 6.12E+03 6.48E+03 6.69E+03 6.90E+03 7.27E+03 7.63E+03 7.96E+03 8.28E+03 8.65E+03 9.01E+03 9.38E+03 9.66E+03 9.94E+03 1.03E+04 1.07E+04 1.10E+04 1.14E+04

tDxf q [stb/day] 6.55E-02 550 7.86E-02 512 9.43E-02 476 1.13E-01 444 1.36E-01 414 1.63E-01 387 1.96E-01 362 2.35E-01 339 2.82E-01 318 3.38E-01 299 4.06E-01 282 4.87E-01 266 5.36E-01 258 5.84E-01 251 6.43E-01 243 7.01E-01 237 7.71E-01 230 8.41E-01 224 9.25E-01 218 1.01E+00 213 1.10E+00 208 1.16E+00 205 1.21E+00 203 1.30E+00 199 1.38E+00 196 1.45E+00 193 1.55E+00 190 1.64E+00 187 1.69E+00 186 1.74E+00 185 1.84E+00 182 1.93E+00 180 2.01E+00 178 2.09E+00 176 2.19E+00 175 2.28E+00 173 2.37E+00 171 2.44E+00 170 2.51E+00 169 2.60E+00 168 2.70E+00 166 2.79E+00 165 2.88E+00 164

qD 1.73E+00 1.61E+00 1.49E+00 1.39E+00 1.30E+00 1.21E+00 1.14E+00 1.06E+00 9.98E-01 9.38E-01 8.84E-01 8.34E-01 8.09E-01 7.86E-01 7.63E-01 7.43E-01 7.23E-01 7.04E-01 6.86E-01 6.69E-01 6.53E-01 6.44E-01 6.36E-01 6.24E-01 6.15E-01 6.06E-01 5.97E-01 5.88E-01 5.84E-01 5.79E-01 5.72E-01 5.65E-01 5.59E-01 5.54E-01 5.48E-01 5.43E-01 5.38E-01 5.34E-01 5.30E-01 5.26E-01 5.22E-01 5.18E-01 5.14E-01

p [psi] 4,403 4,355 4,303 4,248 4,189 4,126 4,060 3,989 3,914 3,835 3,752 3,665 3,616 3,571 3,521 3,474 3,422 3,374 3,320 3,270 3,219 3,190 3,162 3,119 3,085 3,052 3,014 2,979 2,959 2,939 2,907 2,876 2,849 2,824

pD 4.23E-01 4.57E-01 4.93E-01 5.32E-01 5.74E-01 6.19E-01 6.66E-01 7.16E-01 7.69E-01 8.25E-01 8.84E-01 9.46E-01 9.80E-01 1.01E+00 1.05E+00 1.08E+00 1.12E+00 1.15E+00 1.19E+00 1.23E+00 1.26E+00 1.28E+00 1.30E+00 1.33E+00 1.36E+00 1.38E+00 1.41E+00 1.43E+00 1.45E+00 1.46E+00 1.48E+00 1.50E+00 1.52E+00 1.54E+00

155

Table 7 – FCD=100 – Results of Numerical Simulation and Dimensionless Time, Pressure and Rate for Point Source - continued t [days] 1.17E+04 1.19E+04 1.23E+04 1.27E+04 1.30E+04 1.34E+04 1.38E+04 1.40E+04 1.43E+04 1.47E+04 1.51E+04 1.54E+04 1.58E+04 1.61E+04 1.65E+04 1.68E+04 1.72E+04 1.75E+04 1.79E+04 1.83E+04 1.86E+04 1.90E+04 1.94E+04 1.97E+04 2.01E+04 2.04E+04 2.06E+04 2.10E+04 2.13E+04 2.17E+04 2.21E+04 2.24E+04 2.28E+04 2.32E+04 2.35E+04 2.39E+04 2.43E+04 2.45E+04 2.47E+04 2.51E+04 2.55E+04 2.58E+04 2.62E+04

tDxf q [stb/day] 2.95E+00 163 3.02E+00 162 3.11E+00 161 3.20E+00 160 3.29E+00 159 3.38E+00 158 3.48E+00 157 3.55E+00 156 3.62E+00 156 3.71E+00 155 3.80E+00 154 3.90E+00 153 3.99E+00 152 4.08E+00 152 4.17E+00 151 4.26E+00 150 4.34E+00 150 4.43E+00 149 4.53E+00 148 4.62E+00 148 4.71E+00 147 4.80E+00 147 4.90E+00 146 4.99E+00 145 5.08E+00 145 5.14E+00 145 5.21E+00 144 5.30E+00 144 5.39E+00 143 5.49E+00 143 5.58E+00 142 5.67E+00 142 5.76E+00 141 5.85E+00 141 5.95E+00 140 6.04E+00 140 6.13E+00 140 6.19E+00 139 6.25E+00 139 6.34E+00 139 6.43E+00 138 6.53E+00 138 6.62E+00 137

qD 5.11E-01 5.09E-01 5.05E-01 5.02E-01 4.99E-01 4.96E-01 4.93E-01 4.91E-01 4.88E-01 4.86E-01 4.83E-01 4.81E-01 4.78E-01 4.76E-01 4.74E-01 4.72E-01 4.70E-01 4.68E-01 4.66E-01 4.64E-01 4.62E-01 4.60E-01 4.58E-01 4.56E-01 4.55E-01 4.54E-01 4.52E-01 4.51E-01 4.49E-01 4.48E-01 4.46E-01 4.45E-01 4.43E-01 4.42E-01 4.40E-01 4.39E-01 4.38E-01 4.37E-01 4.36E-01 4.35E-01 4.34E-01 4.32E-01 4.31E-01

156

Table 7 – FCD=100 – Results of Numerical Simulation and Dimensionless Time, Pressure and Rate for Point Source - continued t [days] 2.66E+04 2.69E+04 2.73E+04 2.77E+04 2.80E+04 2.84E+04 2.87E+04 2.91E+04 2.94E+04 2.97E+04 3.00E+04 3.04E+04 3.08E+04 3.11E+04 3.15E+04 3.19E+04 3.22E+04 3.26E+04 3.30E+04 3.33E+04 3.37E+04 3.41E+04 3.44E+04 3.48E+04 3.52E+04 3.54E+04 3.56E+04 3.60E+04 3.63E+04 3.67E+04 3.71E+04 3.74E+04 3.78E+04 3.82E+04 3.85E+04 3.89E+04 3.93E+04 3.96E+04 4.00E+04 4.04E+04 4.07E+04 4.11E+04 4.15E+04

tDxf q [stb/day] 6.71E+00 137 6.80E+00 137 6.90E+00 136 6.99E+00 136 7.08E+00 136 7.17E+00 135 7.26E+00 135 7.36E+00 135 7.43E+00 134 7.50E+00 134 7.59E+00 134 7.69E+00 134 7.78E+00 133 7.87E+00 133 7.96E+00 133 8.05E+00 132 8.15E+00 132 8.24E+00 132 8.33E+00 132 8.42E+00 131 8.52E+00 131 8.61E+00 131 8.70E+00 130 8.79E+00 130 8.88E+00 130 8.94E+00 130 9.00E+00 130 9.09E+00 129 9.18E+00 129 9.28E+00 129 9.37E+00 129 9.46E+00 128 9.55E+00 128 9.65E+00 128 9.74E+00 128 9.83E+00 128 9.92E+00 127 1.00E+01 127 1.01E+01 127 1.02E+01 127 1.03E+01 126 1.04E+01 126 1.05E+01 126

qD 4.30E-01 4.29E-01 4.28E-01 4.27E-01 4.26E-01 4.25E-01 4.24E-01 4.23E-01 4.22E-01 4.21E-01 4.20E-01 4.19E-01 4.18E-01 4.17E-01 4.16E-01 4.15E-01 4.14E-01 4.13E-01 4.13E-01 4.12E-01 4.11E-01 4.10E-01 4.09E-01 4.08E-01 4.08E-01 4.07E-01 4.07E-01 4.06E-01 4.05E-01 4.04E-01 4.04E-01 4.03E-01 4.02E-01 4.02E-01 4.01E-01 4.00E-01 3.99E-01 3.99E-01 3.98E-01 3.97E-01 3.97E-01 3.96E-01 3.95E-01

157

Table 7 – FCD=100 – Results of Numerical Simulation and Dimensionless Time, Pressure and Rate for Point Source - continued t [days] 4.18E+04 4.22E+04 4.25E+04 4.27E+04 4.31E+04 4.35E+04 4.38E+04 4.42E+04 4.46E+04 4.49E+04 4.53E+04 4.57E+04 4.60E+04 4.64E+04 4.67E+04 4.71E+04 4.75E+04 4.78E+04 4.82E+04 4.86E+04 4.89E+04 4.93E+04 4.97E+04 5.00E+04 5.04E+04 5.08E+04 5.10E+04 5.13E+04 5.16E+04 5.20E+04 5.24E+04 5.27E+04 5.31E+04 5.35E+04 5.38E+04 5.42E+04 5.46E+04 5.49E+04 5.53E+04 5.57E+04 5.60E+04 5.64E+04 5.68E+04

tDxf q [stb/day] 1.06E+01 126 1.07E+01 126 1.07E+01 125 1.08E+01 125 1.09E+01 125 1.10E+01 125 1.11E+01 125 1.12E+01 125 1.13E+01 124 1.14E+01 124 1.14E+01 124 1.15E+01 124 1.16E+01 124 1.17E+01 124 1.18E+01 123 1.19E+01 123 1.20E+01 123 1.21E+01 123 1.22E+01 123 1.23E+01 123 1.24E+01 122 1.25E+01 122 1.26E+01 122 1.26E+01 122 1.27E+01 122 1.28E+01 122 1.29E+01 121 1.30E+01 121 1.31E+01 121 1.31E+01 121 1.32E+01 121 1.33E+01 121 1.34E+01 121 1.35E+01 120 1.36E+01 120 1.37E+01 120 1.38E+01 120 1.39E+01 120 1.40E+01 120 1.41E+01 120 1.42E+01 119 1.43E+01 119 1.43E+01 119

qD 3.95E-01 3.94E-01 3.94E-01 3.93E-01 3.93E-01 3.92E-01 3.92E-01 3.91E-01 3.90E-01 3.90E-01 3.89E-01 3.89E-01 3.88E-01 3.88E-01 3.87E-01 3.86E-01 3.86E-01 3.85E-01 3.85E-01 3.84E-01 3.84E-01 3.83E-01 3.83E-01 3.82E-01 3.82E-01 3.81E-01 3.81E-01 3.81E-01 3.80E-01 3.80E-01 3.79E-01 3.79E-01 3.78E-01 3.78E-01 3.77E-01 3.77E-01 3.77E-01 3.76E-01 3.76E-01 3.75E-01 3.75E-01 3.74E-01 3.74E-01

158

Table 7 – FCD=100 – Results of Numerical Simulation and Dimensionless Time, Pressure and Rate for Point Source - continued t [days] 5.71E+04 5.75E+04 5.79E+04 5.82E+04 5.86E+04 5.89E+04 5.93E+04 5.97E+04

tDxf q [stb/day] 1.44E+01 119 1.45E+01 119 1.46E+01 119 1.47E+01 119 1.48E+01 119 1.49E+01 118 1.50E+01 118 1.51E+01 118

qD 3.74E-01 3.73E-01 3.73E-01 3.72E-01 3.72E-01 3.72E-01 3.71E-01 3.71E-01

159

160

Table 8 – FCD=100 – Results of Numerical Simulation and Dimensionless Time, and Pressure for xf/y=255, Constant Rate Case, Vertical Well t [days] 0 1.03E-04 2.27E-04 3.76E-04 5.54E-04 7.68E-04 1.03E-03 1.33E-03 1.70E-03 2.15E-03 2.68E-03 3.32E-03 4.09E-03 5.01E-03 6.11E-03 7.43E-03 9.02E-03 1.09E-02 1.32E-02 1.60E-02 1.93E-02 2.32E-02 2.80E-02 3.37E-02 4.05E-02 4.87E-02 5.86E-02 7.04E-02 8.46E-02 1.02E-01 1.22E-01 1.46E-01 1.76E-01 2.11E-01 2.53E-01 3.04E-01 3.65E-01 4.38E-01 5.26E-01 6.31E-01 7.57E-01 9.09E-01

p [psi] 5,000.00 4,992.24 4,990.58 4,990.04 4,989.81 4,989.65 4,989.48 4,989.29 4,989.08 4,988.82 4,988.52 4,988.17 4,987.76 4,987.30 4,986.76 4,986.15 4,985.47 4,984.69 4,983.83 4,982.88 4,981.83 4,980.69 4,979.46 4,978.12 4,976.67 4,975.10 4,973.41 4,971.56 4,969.55 4,967.35 4,964.94 4,962.30 4,959.40 4,956.23 4,952.77 4,948.98 4,944.80 4,940.22 4,935.19 4,929.68 4,923.68 4,917.04

tDxf n/a 2.60E-08 5.74E-08 9.50E-08 1.40E-07 1.94E-07 2.59E-07 3.37E-07 4.30E-07 5.43E-07 6.77E-07 8.39E-07 1.03E-06 1.26E-06 1.54E-06 1.88E-06 2.28E-06 2.76E-06 3.34E-06 4.03E-06 4.87E-06 5.87E-06 7.07E-06 8.51E-06 1.02E-05 1.23E-05 1.48E-05 1.78E-05 2.14E-05 2.57E-05 3.08E-05 3.70E-05 4.44E-05 5.34E-05 6.41E-05 7.69E-05 9.23E-05 1.11E-04 1.33E-04 1.60E-04 1.91E-04 2.30E-04

pD n/a 5.49E-03 6.67E-03 7.05E-03 7.22E-03 7.33E-03 7.45E-03 7.58E-03 7.74E-03 7.92E-03 8.13E-03 8.38E-03 8.67E-03 9.00E-03 9.38E-03 9.81E-03 1.03E-02 1.08E-02 1.15E-02 1.21E-02 1.29E-02 1.37E-02 1.45E-02 1.55E-02 1.65E-02 1.76E-02 1.88E-02 2.01E-02 2.16E-02 2.31E-02 2.48E-02 2.67E-02 2.88E-02 3.10E-02 3.34E-02 3.61E-02 3.91E-02 4.23E-02 4.59E-02 4.98E-02 5.41E-02 5.88E-02

t [days] p [psi] tDxf pD 1.09E+00 4,909.81 2.76E-04 6.39E-02 1.31E+00 1.57E+00 1.89E+00 2.26E+00 2.72E+00 3.26E+00 3.91E+00 4.69E+00 5.63E+00 6.76E+00 8.11E+00 9.73E+00 1.17E+01 1.40E+01 1.68E+01 2.02E+01 2.42E+01 2.91E+01 3.49E+01 4.19E+01 5.02E+01 6.02E+01 7.23E+01 8.68E+01 1.04E+02 1.25E+02 1.50E+02 1.80E+02 2.16E+02 2.59E+02 3.11E+02 3.73E+02 4.48E+02 5.38E+02 6.46E+02 7.75E+02 9.30E+02 1.12E+03 1.34E+03 1.61E+03

4,901.91 4,893.26 4,883.73 4,873.37 4,862.00 4,849.61 4,836.05 4,821.26 4,805.09 4,787.37 4,768.10 4,747.04 4,724.00 4,698.87 4,671.57 4,641.74 4,609.29 4,573.92 4,535.50 4,493.69 4,448.38 4,399.52 4,346.28 4,288.90 4,227.00 4,160.51 4,088.75 4,011.49 3,928.38 3,839.10 3,743.36 3,640.69 3,530.76 3,413.38 3,287.72 3,154.61 3,013.14 2,863.37 2,705.39 2,539.42

3.31E-04 3.97E-04 4.77E-04 5.72E-04 6.86E-04 8.24E-04 9.88E-04 1.19E-03 1.42E-03 1.71E-03 2.05E-03 2.46E-03 2.95E-03 3.54E-03 4.25E-03 5.10E-03 6.12E-03 7.35E-03 8.82E-03 1.06E-02 1.27E-02 1.52E-02 1.83E-02 2.19E-02 2.63E-02 3.16E-02 3.79E-02 4.55E-02 5.46E-02 6.55E-02 7.86E-02 9.43E-02 1.13E-01 1.36E-01 1.63E-01 1.96E-01 2.35E-01 2.82E-01 3.38E-01 4.06E-01

6.95E-02 7.56E-02 8.23E-02 8.97E-02 9.77E-02 1.07E-01 1.16E-01 1.27E-01 1.38E-01 1.51E-01 1.64E-01 1.79E-01 1.95E-01 2.13E-01 2.33E-01 2.54E-01 2.77E-01 3.02E-01 3.29E-01 3.59E-01 3.91E-01 4.25E-01 4.63E-01 5.04E-01 5.47E-01 5.95E-01 6.45E-01 7.00E-01 7.59E-01 8.22E-01 8.90E-01 9.63E-01 1.04E+00 1.12E+00 1.21E+00 1.31E+00 1.41E+00 1.51E+00 1.63E+00 1.74E+00

161

Table 8 – FCD=100 – Results of Numerical Simulation and Dimensionless Time, and Pressure for xf/y=255, Constant Rate Case, Vertical Well continued t [days] 1.93E+03 2.12E+03 2.31E+03 2.54E+03 2.77E+03 3.05E+03 3.33E+03 3.66E+03 3.99E+03

p [psi] 2,364.91 2,267.88 2,177.39 2,076.76 1,982.99 1,878.72 1,781.67 1,674.00 1,573.90

tDxf 4.87E-01 5.36E-01 5.84E-01 6.43E-01 7.01E-01 7.71E-01 8.41E-01 9.25E-01 1.01E+00

pD 1.87E+00 1.93E+00 2.00E+00 2.07E+00 2.14E+00 2.21E+00 2.28E+00 2.36E+00 2.43E+00

162

Table 9 – FCD=100 – Results of Numerical Simulation and Dimensionless Time, and Rate for xf/y=128, Constant Pressure Case, Vertical Well t [days] q [stb/day] 1.03E-04 57,731.64 2.27E-04 45,780.68 3.76E-04 44,300.47 5.54E-04 43,639.16 7.68E-04 43,039.15 1.03E-03 42,374.53 1.33E-03 41,616.37 1.70E-03 40,751.01 2.15E-03 39,772.57 2.68E-03 38,679.54 3.32E-03 37,470.74 4.09E-03 36,153.84 5.01E-03 34,743.22 6.11E-03 33,263.45 7.43E-03 31,736.06 9.02E-03 30,187.47 1.09E-02 28,655.41 1.32E-02 27,173.31 1.60E-02 25,762.59 1.93E-02 24,437.02 2.32E-02 23,200.53 2.80E-02 22,034.77 3.37E-02 20,919.01 4.05E-02 19,829.81 4.87E-02 18,746.77 5.86E-02 17,653.63 7.04E-02 16,544.00 8.46E-02 15,415.49 1.02E-01 14,281.31 1.22E-01 13,151.09 1.46E-01 12,038.30 1.76E-01 10,958.50 2.11E-01 9,924.63 2.53E-01 8,949.95 3.04E-01 8,041.69 3.65E-01 7,205.55 4.38E-01 6,445.61 5.26E-01 5,762.60 6.31E-01 5,158.12 7.57E-01 4,625.07 9.09E-01 4,155.15 1.09E+00 3,743.63

tDxf 2.60E-08 5.74E-08 9.50E-08 1.40E-07 1.94E-07 2.59E-07 3.37E-07 4.30E-07 5.43E-07 6.77E-07 8.39E-07 1.03E-06 1.26E-06 1.54E-06 1.88E-06 2.28E-06 2.76E-06 3.34E-06 4.03E-06 4.87E-06 5.87E-06 7.07E-06 8.51E-06 1.02E-05 1.23E-05 1.48E-05 1.78E-05 2.14E-05 2.57E-05 3.08E-05 3.70E-05 4.44E-05 5.34E-05 6.41E-05 7.69E-05 9.23E-05 1.11E-04 1.33E-04 1.60E-04 1.91E-04 2.30E-04 2.76E-04

qD 181.149 143.650 139.005 136.930 135.047 132.962 130.583 127.868 124.797 121.368 117.575 113.443 109.016 104.373 99.581 94.722 89.914 85.264 80.837 76.678 72.798 69.140 65.639 62.222 58.823 55.393 51.911 48.370 44.812 41.265 37.774 34.385 31.141 28.083 25.233 22.609 20.225 18.082 16.185 14.512 13.038 11.747

t [days] q [stb/day] 1.31E+00 3,381.58 1.57E+00 3,060.39 1.89E+00 2,774.68 2.26E+00 2,519.97 2.72E+00 2,292.29 3.26E+00 2,088.78 3.91E+00 1,905.76 4.69E+00 1,740.14 5.63E+00 1,589.66 6.76E+00 1,452.11 8.11E+00 1,327.37 9.73E+00 1,214.18 1.17E+01 1,111.43 1.40E+01 1,018.41 1.68E+01 934.05 2.02E+01 856.77 2.42E+01 786.07 2.91E+01 721.38 3.49E+01 662.60 4.19E+01 609.34 5.02E+01 561.33 6.02E+01 518.11 7.23E+01 478.61 8.68E+01 442.62 1.04E+02 409.61 1.25E+02 379.35 1.50E+02 351.48 1.80E+02 325.87 2.16E+02 302.37 2.59E+02 280.82 3.11E+02 261.06 3.73E+02 242.92 4.48E+02 226.29 5.38E+02 211.05 6.46E+02 197.05 7.75E+02 184.30 9.30E+02 172.62 1.12E+03 161.96 1.34E+03 152.23 1.61E+03 143.35 1.93E+03 135.22 2.12E+03 131.05

tDxf 3.31E-04 3.97E-04 4.77E-04 5.72E-04 6.86E-04 8.24E-04 9.88E-04 1.19E-03 1.42E-03 1.71E-03 2.05E-03 2.46E-03 2.95E-03 3.54E-03 4.25E-03 5.10E-03 6.12E-03 7.35E-03 8.82E-03 1.06E-02 1.27E-02 1.52E-02 1.83E-02 2.19E-02 2.63E-02 3.16E-02 3.79E-02 4.55E-02 5.46E-02 6.55E-02 7.86E-02 9.43E-02 1.13E-01 1.36E-01 1.63E-01 1.96E-01 2.35E-01 2.82E-01 3.38E-01 4.06E-01 4.87E-01 5.36E-01

qD 10.611 9.603 8.706 7.907 7.193 6.554 5.980 5.460 4.988 4.556 4.165 3.810 3.487 3.196 2.931 2.688 2.467 2.264 2.079 1.912 1.761 1.626 1.502 1.389 1.285 1.190 1.103 1.023 0.949 0.881 0.819 0.762 0.710 0.662 0.618 0.578 0.542 0.508 0.478 0.450 0.424 0.411

163

Table 9 – FCD=100 – Results of Numerical Simulation and Dimensionless Time, and Rate for xf/y=128, Constant Pressure Case, Vertical Well continued t [days] q [stb/day] tDxf 2.31E+03 127.42 5.84E-01 2.54E+03 123.68 6.43E-01 2.77E+03 120.42 7.01E-01 3.05E+03 117.04 7.71E-01 3.33E+03 114.08 8.41E-01 3.66E+03 111.01 9.25E-01 3.99E+03 108.32 1.01E+00

qD 0.400 0.388 0.378 0.367 0.358 0.348 0.340

164

Table 10 – Production Data and Dimensionless Time and Flow Rate for McAlister O.H. 16 t [months] q[Mcf/m] 1 45,816 2 95,857 3 76,889 4 78,556 5 69,757 6 66,602 7 59,948 8 51,841 9 53,079 10 47,095 11 49,919 12 44,319 13 44,979 14 43,737 15 38,344 16 41,482 17 39,781 18 38,887 19 37,787 20 39,142 21 36,193 22 34,558 23 34,100 24 32,084 25 32,300 26 32,002 27 28,602 28 31,073

tDxf 2.70E-06 5.40E-06 8.10E-06 1.08E-05 1.35E-05 1.62E-05 1.89E-05 2.16E-05 2.43E-05 2.70E-05 2.97E-05 3.24E-05 3.51E-05 3.78E-05 4.05E-05 4.32E-05 4.59E-05 4.86E-05 5.13E-05 5.40E-05 5.67E-05 5.94E-05 6.21E-05 6.48E-05 6.75E-05 7.02E-05 7.29E-05 7.56E-05

qD 3.82E+01 7.99E+01 6.41E+01 6.55E+01 5.81E+01 5.55E+01 5.00E+01 4.32E+01 4.42E+01 3.92E+01 4.16E+01 3.69E+01 3.75E+01 3.64E+01 3.20E+01 3.46E+01 3.32E+01 3.24E+01 3.15E+01 3.26E+01 3.02E+01 2.88E+01 2.84E+01 2.67E+01 2.69E+01 2.67E+01 2.38E+01 2.59E+01

t [months] q[Mcf/m] 29 30,127 30 29,846 31 28,822 32 29,362 33 26,080 34 29,420 35 28,784 36 27,173 37 29,166 38 28,066 39 23,200 40 26,474 41 25,212 42 26,565 43 24,300 44 24,601 45 23,552 46 23,236 47 23,874 48 22,521 49 21,305 50 22,362 51 21,997 52 22,933 53 22,379 54 20,673 55 26,054

tDxf 7.83E-05 8.10E-05 8.37E-05 8.64E-05 8.91E-05 9.18E-05 9.45E-05 9.72E-05 9.99E-05 1.03E-04 1.05E-04 1.08E-04 1.11E-04 1.13E-04 1.16E-04 1.19E-04 1.22E-04 1.24E-04 1.27E-04 1.30E-04 1.32E-04 1.35E-04 1.38E-04 1.40E-04 1.43E-04 1.46E-04 1.49E-04

qD 2.51E+01 2.49E+01 2.40E+01 2.45E+01 2.17E+01 2.45E+01 2.40E+01 2.26E+01 2.43E+01 2.34E+01 1.93E+01 2.21E+01 2.10E+01 2.21E+01 2.03E+01 2.05E+01 1.96E+01 1.94E+01 1.99E+01 1.88E+01 1.78E+01 1.86E+01 1.83E+01 1.91E+01 1.86E+01 1.72E+01 2.17E+01

165

Table 11 – FCD=100 – Results of Numerical Simulation and Dimensionless Time, Pressure and Rate for xf/y=255, Point Source t [days] 0.00E+00 1.03E-04 2.27E-04 3.76E-04 5.54E-04 7.68E-04 1.03E-03 1.33E-03 1.70E-03 2.15E-03 2.68E-03 3.32E-03 4.09E-03 5.01E-03 6.11E-03 7.43E-03 9.02E-03 1.09E-02 1.32E-02 1.60E-02 1.93E-02 2.32E-02 2.80E-02 3.37E-02 4.05E-02 4.87E-02 5.86E-02 7.04E-02 8.46E-02 1.02E-01 1.22E-01 1.46E-01 1.76E-01 2.11E-01 2.53E-01 3.04E-01 3.65E-01 4.38E-01 5.26E-01 6.31E-01 7.57E-01

tDxf n/a 2.60E-08 5.74E-08 9.50E-08 1.40E-07 1.94E-07 2.59E-07 3.37E-07 4.30E-07 5.43E-07 6.77E-07 8.39E-07 1.03E-06 1.26E-06 1.54E-06 1.88E-06 2.28E-06 2.76E-06 3.34E-06 4.03E-06 4.87E-06 5.87E-06 7.07E-06 8.51E-06 1.02E-05 1.23E-05 1.48E-05 1.78E-05 2.14E-05 2.57E-05 3.08E-05 3.70E-05 4.44E-05 5.34E-05 6.41E-05 7.69E-05 9.23E-05 1.11E-04 1.33E-04 1.60E-04 1.91E-04

q [stb/day] 0 41,141 35,004 33,913 33,474 33,108 32,708 32,249 31,717 31,107 30,412 29,627 28,749 27,780 26,727 25,597 24,400 23,157 21,892 20,627 19,383 18,181 17,032 15,942 14,911 13,935 13,007 12,121 11,271 10,458 9,684 8,948 8,254 7,602 6,993 6,423 5,892 5,400 4,945 4,528 4,146

qD n/a 1.29E+02 1.10E+02 1.06E+02 1.05E+02 1.04E+02 1.03E+02 1.01E+02 9.95E+01 9.76E+01 9.54E+01 9.30E+01 9.02E+01 8.72E+01 8.39E+01 8.03E+01 7.66E+01 7.27E+01 6.87E+01 6.47E+01 6.08E+01 5.70E+01 5.34E+01 5.00E+01 4.68E+01 4.37E+01 4.08E+01 3.80E+01 3.54E+01 3.28E+01 3.04E+01 2.81E+01 2.59E+01 2.39E+01 2.19E+01 2.02E+01 1.85E+01 1.69E+01 1.55E+01 1.42E+01 1.30E+01

p [psi] 5,000 4,989 4,987 4,987 4,987 4,987 4,986 4,986 4,986 4,986 4,985 4,985 4,985 4,984 4,984 4,983 4,982 4,982 4,981 4,980 4,979 4,978 4,976 4,975 4,974 4,972 4,970 4,968 4,966 4,964 4,962 4,959 4,956 4,953 4,950 4,946 4,942 4,937 4,932 4,927 4,921

pD n/a 7.69E-03 8.88E-03 9.26E-03 9.43E-03 9.54E-03 9.66E-03 9.79E-03 9.95E-03 1.01E-02 1.03E-02 1.06E-02 1.09E-02 1.12E-02 1.16E-02 1.20E-02 1.25E-02 1.30E-02 1.37E-02 1.43E-02 1.51E-02 1.59E-02 1.68E-02 1.77E-02 1.87E-02 1.98E-02 2.10E-02 2.24E-02 2.38E-02 2.53E-02 2.71E-02 2.89E-02 3.10E-02 3.32E-02 3.57E-02 3.83E-02 4.13E-02 4.45E-02 4.81E-02 5.20E-02 5.63E-02

166

Table 11 – FCD=100 – Results of Numerical Simulation and Dimensionless Time, Pressure and Rate for xf/y=255, Point Source continued t [days] 9.09E-01 1.09E+00 1.31E+00 1.57E+00 1.89E+00 2.26E+00 2.72E+00 3.26E+00 3.91E+00 4.69E+00 5.63E+00 6.76E+00 8.11E+00 9.73E+00 1.17E+01 1.40E+01 1.68E+01 2.02E+01 2.42E+01 2.91E+01 3.49E+01 4.19E+01 5.02E+01 6.02E+01 7.23E+01 8.68E+01 1.04E+02 1.25E+02 1.50E+02 1.80E+02 2.16E+02 2.59E+02 3.11E+02 3.73E+02 4.48E+02 5.38E+02 6.46E+02 7.75E+02 9.30E+02 1.12E+03 1.34E+03

tDxf q [stb/day] 2.30E-04 3,794 2.76E-04 3,474 3.31E-04 3,179 3.97E-04 2,908 4.77E-04 2,659 5.72E-04 2,432 6.86E-04 2,225 8.24E-04 2,036 9.88E-04 1,865 1.19E-03 1,708 1.42E-03 1,564 1.71E-03 1,431 2.05E-03 1,310 2.46E-03 1,200 2.95E-03 1,100 3.54E-03 1,008 4.25E-03 925 5.10E-03 849 6.12E-03 780 7.35E-03 716 8.82E-03 658 1.06E-02 605 1.27E-02 557 1.52E-02 514 1.83E-02 475 2.19E-02 440 2.63E-02 407 3.16E-02 377 3.79E-02 349 4.55E-02 324 5.46E-02 300 6.55E-02 279 7.86E-02 259 9.43E-02 241 1.13E-01 225 1.36E-01 210 1.63E-01 196 1.96E-01 183 2.35E-01 172 2.82E-01 161 3.38E-01 151

qD 1.19E+01 1.09E+01 9.98E+00 9.13E+00 8.34E+00 7.63E+00 6.98E+00 6.39E+00 5.85E+00 5.36E+00 4.91E+00 4.49E+00 4.11E+00 3.77E+00 3.45E+00 3.16E+00 2.90E+00 2.67E+00 2.45E+00 2.25E+00 2.06E+00 1.90E+00 1.75E+00 1.61E+00 1.49E+00 1.38E+00 1.28E+00 1.18E+00 1.10E+00 1.02E+00 9.43E-01 8.75E-01 8.14E-01 7.57E-01 7.06E-01 6.58E-01 6.14E-01 5.75E-01 5.38E-01 5.05E-01 4.75E-01

p [psi] 4,914 4,907 4,899 4,890 4,881 4,870 4,859 4,846 4,833 4,818 4,802 4,784 4,765 4,744 4,721 4,696 4,668 4,639 4,606 4,571 4,532 4,491 4,445 4,396 4,343 4,286 4,224 4,157 4,086 4,008 3,925 3,836 3,740 3,638 3,528 3,410 3,284 3,151 3,010 2,860 2,702

pD 6.10E-02 6.61E-02 7.17E-02 7.78E-02 8.46E-02 9.19E-02 9.99E-02 1.09E-01 1.18E-01 1.29E-01 1.40E-01 1.53E-01 1.66E-01 1.81E-01 1.98E-01 2.15E-01 2.35E-01 2.56E-01 2.79E-01 3.04E-01 3.31E-01 3.61E-01 3.93E-01 4.27E-01 4.65E-01 5.06E-01 5.50E-01 5.97E-01 6.48E-01 7.02E-01 7.61E-01 8.24E-01 8.92E-01 9.65E-01 1.04E+00 1.13E+00 1.21E+00 1.31E+00 1.41E+00 1.52E+00 1.63E+00

167

Table 11 – FCD=100 – Results of Numerical Simulation and Dimensionless Time, Pressure and Rate for xf/y=255, Point Source continued t [days] 1.61E+03 1.93E+03 2.12E+03 2.31E+03 2.54E+03 2.77E+03 3.05E+03 3.33E+03 3.66E+03 3.99E+03

tDxf q [stb/day] 4.06E-01 143 4.87E-01 134 5.36E-01 130 5.84E-01 127 6.43E-01 123 7.01E-01 120 7.71E-01 116 8.41E-01 113 9.25E-01 110 1.01E+00 108

qD 4.47E-01 4.22E-01 4.09E-01 3.98E-01 3.86E-01 3.76E-01 3.65E-01 3.56E-01 3.46E-01 3.38E-01

p [psi] 2,536 2,362 2,265 2,174 2,074 1,980 1,876 1,778 1,671 1,571

pD 1.74E+00 1.87E+00 1.94E+00 2.00E+00 2.07E+00 2.14E+00 2.21E+00 2.28E+00 2.36E+00 2.43E+00

168

Table 12 – FCD=100 – Results of Numerical Simulation and Dimensionless Time, and Pressure for Constant Rate Case, xf2 =xf1/4 and xf/y= 255 t [days] 0 1.18E-02 2.60E-02 4.30E-02 6.34E-02 8.79E-02 1.17E-01 1.53E-01 1.95E-01 2.46E-01 3.06E-01 3.80E-01 4.67E-01 5.72E-01 6.98E-01 8.50E-01 1.03E+00 1.25E+00 1.51E+00 1.83E+00 2.20E+00 2.66E+00 3.20E+00 3.85E+00 4.63E+00 5.57E+00 6.70E+00 8.05E+00 9.67E+00 1.16E+01 1.39E+01 1.67E+01 2.01E+01 2.41E+01

p [psi] 5,000.00 4,957.43 4,941.38 4,926.60 4,912.28 4,897.89 4,883.12 4,867.75 4,851.55 4,834.40 4,815.98 4,796.29 4,775.11 4,752.31 4,727.67 4,700.91 4,672.03 4,640.79 4,606.86 4,570.07 4,530.11 4,486.84 4,439.87 4,389.05 4,333.97 4,274.38 4,209.65 4,139.93 4,064.61 3,983.13 3,895.77 3,801.71 3,700.74 3,592.52

tDxf n/a 4.86E-05 1.07E-04 1.77E-04 2.61E-04 3.62E-04 4.83E-04 6.28E-04 8.03E-04 1.01E-03 1.26E-03 1.56E-03 1.92E-03 2.36E-03 2.88E-03 3.50E-03 4.25E-03 5.15E-03 6.23E-03 7.52E-03 9.08E-03 1.09E-02 1.32E-02 1.59E-02 1.91E-02 2.29E-02 2.76E-02 3.32E-02 3.98E-02 4.79E-02 5.75E-02 6.90E-02 8.28E-02 9.95E-02

pD n/a 3.02E-02 4.15E-02 5.20E-02 6.21E-02 7.23E-02 8.28E-02 9.37E-02 1.05E-01 1.17E-01 1.30E-01 1.44E-01 1.59E-01 1.75E-01 1.93E-01 2.12E-01 2.32E-01 2.54E-01 2.78E-01 3.04E-01 3.33E-01 3.63E-01 3.97E-01 4.33E-01 4.72E-01 5.14E-01 5.60E-01 6.09E-01 6.62E-01 7.20E-01 7.82E-01 8.49E-01 9.20E-01 9.97E-01

t [days] 2.90E+01 3.48E+01 4.18E+01 5.01E+01 6.01E+01 7.21E+01 8.66E+01 1.04E+02 1.25E+02 1.50E+02 1.80E+02 2.16E+02 2.60E+02 3.12E+02 3.74E+02 4.49E+02 5.38E+02 6.46E+02 7.75E+02 7.91E+02 8.10E+02 8.13E+02 8.17E+02 8.23E+02 8.24E+02 8.25E+02 8.27E+02 8.30E+02 8.30E+02 8.31E+02 8.32E+02 8.32E+02 8.32E+02

p [psi] 3,476.51 3,352.52 3,220.49 3,080.39 2,932.67 2,776.96 2,612.36 2,440.07 2,260.05 2,072.51 1,878.02 1,676.82 1,469.32 1,255.68 1,036.80 812.67 583.76 350.13 113.66 85.84 51.92 46.64 40.07 29.88 28.29 26.31 23.22 18.41 17.65 16.72 15.25 15.02 14.74

tDxf 1.19E-01 1.43E-01 1.72E-01 2.06E-01 2.48E-01 2.97E-01 3.57E-01 4.29E-01 5.15E-01 6.18E-01 7.42E-01 8.91E-01 1.07E+00 1.28E+00 1.54E+00 1.85E+00 2.22E+00 2.66E+00 3.19E+00 3.26E+00 3.34E+00 3.35E+00 3.37E+00 3.39E+00 3.39E+00 3.40E+00 3.41E+00 3.42E+00 3.42E+00 3.42E+00 3.43E+00 3.43E+00 3.43E+00

pD 1.08E+00 1.17E+00 1.26E+00 1.36E+00 1.46E+00 1.57E+00 1.69E+00 1.81E+00 1.94E+00 2.07E+00 2.21E+00 2.35E+00 2.50E+00 2.65E+00 2.81E+00 2.97E+00 3.13E+00 3.29E+00 3.46E+00 3.48E+00 3.50E+00 3.51E+00 3.51E+00 3.52E+00 3.52E+00 3.52E+00 3.52E+00 3.53E+00 3.53E+00 3.53E+00 3.53E+00 3.53E+00 3.53E+00

169

APPENDIX

C

170

10

Dimensionless Pressure PD

FCD - dimensionless fracture conductivity xf - fracture half-length y - distance between two fractures

FCD=1

1

0.1

FCD

length to distance ratio

1

xf y

5 10

0.028 4.0

25 0.01

100

16.0 63.8

500

255

0.001 0.000001

0.00001

0.0001

0.001

0.01

0.1

1

Dimensionless Time tDxf

Figure 1 – Finite conductivity type curve with deviations for fracture face interference for constant flow rate for and FCD=1 – Line source – Vertical well

10

Dimensionless Pressure PD

FCD - dimensionless fracture conductivity xf - fracture half-length y - distance between two fractures

FCD=5

1

0.1

FCD

length to distance ratio

1

xf y

5 10

0.028 4.0

25 0.01

100

16.0 63.8

500

255

0.001 0.000001

0.00001

0.0001

0.001

0.01

0.1

1

Dimensionless Time tDxf

Figure 2 – Finite conductivity type curve with deviations for fracture face interference for constant flow rate and FCD=5 – Line source – Vertical well

171

10

Dimensionless Pressure PD

FCD - dimensionless fracture conductivity xf - fracture half-length y - distance between two fractures

FCD=10

1

FCD

length to distance ratio

1

xf y

0.1

5 10

0.028 4.0

25 0.01

100

16.0 63.8

500

255

0.001 0.000001

0.00001

0.0001

0.001

0.01

0.1

1

Dimensionless Time tDxf

Figure 3 – Finite conductivity type curve with deviations for fracture face interference for constant flow rate and FCD=10 – Line source – Vertical well 10

Dimensionless Pressure PD

FCD - dimensionless fracture conductivity xf - fracture half-length y - distance between two fractures

FCD=25

1

0.1

FCD

length to distance ratio

1

xf y

5 10

0.028 4.0

25 0.01

100

16.0 63.8

500

255

0.001 0.000001

0.00001

0.0001

0.001

0.01

0.1

1

Dimensionless Time tDxf

Figure 4 – Finite conductivity type curve with deviations for fracture face interference for constant flow rate and FCD=25 – Line source – Vertical well

172 10

Dimensionless Pressure PD

FCD - dimensionless fracture conductivity xf - fracture half-length y - distance between two fractures

FCD=100

1

0.1

FCD

length to distance ratio

1

xf y

5 10

0.028 4.0

25 0.01

100

16.0 63.8

500

255

0.001 0.000001

0.00001

0.0001

0.001

0.01

0.1

1

Dimensionless Time tDxf

Figure 5 – Finite conductivity type curve with deviations for fracture face interference for constant flow rate for and FCD=100 – Line source – Vertical well

10

Dimensionless Pressure PD

FCD - dimensionless fracture conductivity xf - fracture half-length y - distance between two fractures

FCD=500

1

0.1

FCD

length to distance ratio

1

xf y

5 10

0.028 4.0

25 0.01

100

16.0 63.8

500

255

0.001 0.000001

0.00001

0.0001

0.001

0.01

0.1

1

Dimensionless Time tDxf

Figure 6 – Finite conductivity type curve with deviations for fracture face interference for constant flow rate and FCD=500 – Line source – Vertical well

Reciprocal Dimensionless Flow Rate 1/q D

10

FCD - dimensionless fracture conductivity xf - fracture half-length y - distance between two fractures

FCD=1 1

length to distance ratio xf y

FCD 1 0.1 5 10 25 100

0.028 4.0 16

500

63.8

0.01

128 255

0.001 1.00E-06

1.00E-05

1.00E-04

1.00E-03

1.00E-02

1.00E-01

Reciprocal Dimensionless Flow Rate For Two Fracture System 2/qDtfs

173

1.00E+00

Dimensionless Time tDxf

Figure 7 – Finite conductivity type curve with deviations for fracture face interference for constant pressure case and FCD=1 – Line source – Vertical well

Dimensionless Flow Rate q D

100

FCD=1

10

1

length to distance ratio xf y 0.028 4.0

0.1

16 63.8 255

0.01 1.E-06

1.E-05

1.E-04

1.E-03

1.E-02

1.E-01

1.E+00

Dimensionless Time tDxf

Figure 7A – Dimensionless rate versus dimensionless time with deviations for fracture face interference for constant pressure case and FCD=1 – Line source – Vertical well

Reciprocal Dimensionless Flow Rate 1/q D

10

FCD - dimensionless fracture conductivity xf - fracture half-length y - distance between two fractures

FCD=5

1

length to distance ratio xf y

FCD

0.1

1 5 10

0.028

25

16.0

4.0 63.8

0.01 100

128

500

255

0.001 1.00E-06

1.00E-05

1.00E-04

1.00E-03

1.00E-02

1.00E-01

Reciprocal Dimensionless Flow Rate For Two Fracture System 2/qDtfs

174

1.00E+00

Dimensionless Time tDxf

Figure 8 – Finite conductivity type curve with deviations for fracture face interference for constant pressure case and FCD=5 – Line source – Vertical well

Dimensionless Flow Rate q D

100

FCD=5 10

1

length to distance ratio xf y 0.028 4.0 16.0 63.8 255

0.1 1.E-06

1.E-05

1.E-04

1.E-03

1.E-02

1.E-01

1.E+00

Dimensionless Time tDxf

Figure 8A – Dimensionless rate versus dimensionless time with deviations for fracture face interference for constant pressure case and FCD=5 – Line source – Vertical well

Reciprocal Dimensionless Flow Rate 1/q D

10

FCD - dimensionless fracture conductivity xf - fracture half-length y - distance between two fractures

FCD=10

1

length to distance ratio xf y

FCD

0.1

1 5 10

0.028

25

16.0

4.0 63.8

0.01 100

128 255

500

0.001 1.00E-06

1.00E-05

1.00E-04

1.00E-03

1.00E-02

1.00E-01

Reciprocal Dimensionless Flow Rate For Two Fractures System 2/qDtfs

175

1.00E+00

Dimensionless Time tDxf

Figure 9 – Finite conductivity type curve with deviations for fracture face interference for constant pressure case and FCD=10 – Line source – Vertical well

Dimensionless Flow Rate q D

100

FCD=10 10

length to distance ratio xf y 1

0.028 4.0 16.0 63.8 255

0.1 1.E-06

1.E-05

1.E-04

1.E-03

1.E-02

1.E-01

1.E+00

Dimensionless Time tDxf

Figure 9A – Dimensionless rate versus dimensionless time with deviations for fracture face interference for constant pressure case and FCD=10 – Line source – Vertical well

Reciprocal Dimensionless Flow Rate 1/q D

10

FCD - dimensionless fracture conductivity xf - fracture half-length y - distance between two fractures

FCD=25 1

length to distance ratio xf y

FCD

0.1

1 5 10

0.028

25

16.0

4.0 63.8

0.01 100

128 255

500

0.001 1.00E-06

1.00E-05

1.00E-04

1.00E-03

1.00E-02

1.00E-01

Reciprocal Dimensionless Flow Rate For Two Fractures System 2/qDtfs

176

1.00E+00

Dimensionless Time tDxf

Figure 10 – Finite conductivity type curve with deviations for fracture face interference for constant pressure case and FCD=25 – Line source – Vertical well

Dimensionless Flow Rate q D

100

FCD=25 10

length to distance ratio xf y 0.028

1

4.0 16.0 63.8 255

0.1 1.E-06

1.E-05

1.E-04

1.E-03

1.E-02

1.E-01

1.E+00

Dimensionless Time tDxf

Figure 10A – Dimensionless rate versus dimensionless time with deviations for fracture face interference for constant pressure case and FCD=25 – Line source – Vertical well

177

FCD - dimensionless fracture conductivity xf - fracture half-length y - distance between two fractures

Reciprocal Dimensionless Flow Rate for Two Fracture System 2/qDtfs

Reciprocal Dimensionless Flow Rate 1/q D

10

FCD=100 1

length to distance ratio xf y

FCD

0.1

1

0.028

5 10

4.0 16.0

25

0.01

63.8

100

128 255

500

0.001 1.00E-06

1.00E-05

1.00E-04

1.00E-03

1.00E-02

1.00E-01

1.00E+00

Dimensionless Time tDxf

Figure 11 – Finite conductivity type curve with deviations for fracture face interference for constant pressure case and FCD=100 – Line source – Vertical well

Dimensionless Flow Rate q D

1000

100

10

FCD=100

length to distance ratio xf y

1

0.1 0.000001

0.028 4.0 16.0 63.8 255 128

0.00001

0.0001

0.001

0.01

0.1

1

Dimensionless Time tDxf

Figure 11A – Dimensionless rate versus dimensionless time with deviations for fracture face interference for constant pressure case and FCD=100 – Line source – Vertical well

Reciprocal Dimensionless Flow Rate 1/q D

10

FCD - dimensionless fracture conductivity xf - fracture half-length y - distance between two fractures

FCD=500

1

length to distance ratio xf y

FCD

0.1

1 5 10

0.028

25

16.0

4.0 63.8

0.01 100

255 128

500

0.001 1.00E-06

1.00E-05

1.00E-04

1.00E-03

1.00E-02

1.00E-01

Reciprocal Dimensionless Flow Rate For Two Fractures System 2/qDtfs

178

1.00E+00

Dimensionless Time tDxf

Figure 12 – Finite conductivity type curve with deviations for fracture face interference for constant pressure case and FCD=500 – Line source – Vertical well

Dimensionless Flow Rate q D

1000

FCD=500

100

10

length to distance ratio xf y 0.028

1

4.0 16.0 63.8 255

0.1 1.E-06

1.E-05

1.E-04

1.E-03

1.E-02

1.E-01

1.E+00

Dimensionless Time tDxf

Figure 12A – Dimensionless rate versus dimensionless time with deviations for fracture face interference for constant pressure case and FCD=500 – Line source – Vertical well

179

10

Dimensionless Pressure PD

FCD - dimensionless fracture conductivity xf - fracture half-length y - distance between two fractures 1

FCD=1

FCD 1 length to distance ratio

0.1

xf y

5 10

4.0

25 0.01

16.0

100

63.8 500

0.001 0.000001

255

0.00001

0.0001

0.001

0.01

0.1

1

Dimensionless Time tDxf

Figure 13 – Finite conductivity type curve with deviations for fracture face interference for constant rate case and FCD=1 – Point source – Horizontal well 10

Dimensionless Pressure PD

FCD - dimensionless fracture conductivity xf - fracture half-length y - distance between two fractures 1

FCD=5

FCD 1 length to distance ratio

0.1

xf y

5 10

4.0

25 0.01

16.0

100

63.8 500

0.001 0.000001

255

0.00001

0.0001

0.001

0.01

0.1

1

Dimensionless Time tDxf

Figure 14 – Finite conductivity type curve with deviations for fracture face interference for constant rate case and FCD=5 – Point source – Horizontal well

180 10

Dimensionless Pressure PD

FCD - dimensionless fracture conductivity xf - fracture half-length y - distance between two fractures 1

FCD=10

FCD 1 length to distance ratio

0.1

xf y

5 10

4.0

25 0.01

16.0

100

63.8 500

0.001 0.000001

255

0.00001

0.0001

0.001

0.01

0.1

1

Dimensionless Time tDxf

Figure 15 – Finite conductivity type curve with deviations for fracture face interference for constant rate case and FCD=10 – Point source – Horizontal well

10

Dimensionless Pressure PD

FCD - dimensionless fracture conductivity xf - fracture half-length y - distance between two fractures 1

FCD=25

FCD 1 length to distance ratio

0.1

xf y

5 10

4.0

25 0.01

16.0

100

63.8 500

0.001 0.000001

255

0.00001

0.0001

0.001

0.01

0.1

1

Dimensionless Time tDxf

Figure 16 – Finite conductivity type curve with deviations for fracture face interference for constant rate case and FCD=25 – Point source – Horizontal well

181

10

Dimensionless Pressure PD

FCD - dimensionless fracture conductivity xf - fracture half-length y - distance between two fractures 1

FCD=100

FCD 1 length to distance ratio

0.1

xf y

5 10

4.0

25 0.01

16.0

100

63.8 500

0.001 0.000001

255

0.00001

0.0001

0.001

0.01

0.1

1

Dimensionless Time tDxf

Figure 17 – Finite conductivity type curve with deviations for fracture face interference for constant rate case and FCD=100 – Point source – Horizontal well

10

Dimensionless Pressure PD

FCD - dimensionless fracture conductivity xf - fracture half-length y - distance between two fractures 1

FCD=500

FCD 1 length to distance ratio

0.1

xf y

5 10

4.0

25 0.01

16.0

100

63.8 500

0.001 0.000001

255

0.00001

0.0001

0.001

0.01

0.1

1

Dimensionless Time tDxf

Figure 18 – Finite conductivity type curve with deviations for fracture face interference for constant rate case and FCD=500 – Point source – Horizontal well

182

Reciprocal Dimensionless Flow Rate 1/q

D

FCD - dimensionless fracture conductivity xf - fracture half-length y - distance between two fractures

Reciprocal Dimensionless Flow Rate For Two Fractures System 2/q Dtfs

10

FCD=1

1 FCD 1 length to distance ratio 0.1

5 10

xf y

25

0.01

4.0

100

16.0 63.8

500

128 255

0.001 0.000001

0.00001

0.0001

0.001

0.01

0.1

1

Dimensionless Time tDxf

Dimensionless Flow Rate q D

Figure 19 – Finite conductivity type curve with deviations for fracture face interference for constant pressure case and FCD=1 – Point source – Horizontal well

10

FCD=1

1

length to distance ratio xf y

4.0 16 63.8 255 0.1 1.E-06

1.E-05

1.E-04

1.E-03

1.E-02

1.E-01

1.E+00

Dimensionless Time tDxf

Figure 19A – Dimensionless rate versus dimensionless time with deviations for fracture face interference for constant pressure case and FCD=1 – Point source – Horizontal well

183

Reciprocal Dimensionless Flow Rate 1/q

D

FCD - dimensionless fracture conductivity xf - fracture half-length y - distance between two fractures

Reciprocal Dimensionless Flow Rate For Two Fractures System 2/q Dtfs

10

FCD=5

1 FCD 1 length to distance ratio 0.1

0.01

5 10

xf y

25

4.0

100

16.0 63.8

500

128 255

0.001 0.000001

0.00001

0.0001

0.001

0.01

0.1

1

Dimensionless Time tDxf

Figure 20 – Finite conductivity type curve with deviations for fracture face interference for constant pressure case and FCD=5 – Point source – Horizontal well

Dimensionless Flow Rate q D

100

FCD=5 10

length to distance ratio xf y 1

4.0 16.0 63.8 255

0.1 1.E-06

1.E-05

1.E-04

1.E-03

1.E-02

1.E-01

1.E+00

Dimensionless Time tDxf

Figure 20A – Dimensionless rate versus dimensionless time with deviations for fracture face interference for constant pressure case and FCD=5 – Point source – Horizontal well

184

Reciprocal Dimensionless Flow Rate 1/q

D

FCD - dimensionless fracture conductivity xf - fracture half-length y - distance between two fractures

Reciprocal Dimensionless Flow Rate For Two Fractures System 2/q Dtfs

10

FCD=10

1 FCD 1 length to distance ratio 0.1

5 10

xf y

25

0.01

4.0

100

16.0 63.8

500

128 255

0.001 0.000001

0.00001

0.0001

0.001

0.01

0.1

1

Dimensionless Time tDxf

Figure 21 – Finite conductivity type curve with deviations for fracture face interference for constant pressure case and FCD=10 – Point source – Horizontal well

Dimensionless Flow Rate q D

100

FCD=10 10 length to distance ratio xf y

4.0 1

16.0 63.8 255

0.1 1.E-06

1.E-05

1.E-04

1.E-03

1.E-02

1.E-01

1.E+00

Dimensionless Time tDxf

Figure 21A – Dimensionless rate versus dimensionless time with deviations for fracture face interference for constant pressure case and FCD=10 – Point source – Horizontal well

185

Reciprocal Dimensionless Flow Rate 1/q

D

FCD - dimensionless fracture conductivity xf - fracture half-length y - distance between two fractures

Reciprocal Dimensionless Flow Rate For Two Fractures System 2/q Dtfs

10

FCD=25

1 FCD 1 length to distance ratio 0.1

5 10

xf y

25

0.01

4.0

100

16.0 63.8

500

128 255

0.001 0.000001

0.00001

0.0001

0.001

0.01

0.1

1

Dimensionless Time tDxf

Figure 22 – Finite conductivity type curve with deviations for fracture face interference for constant pressure case and FCD=25 – Point source – Horizontal well

Dimensionless Flow Rate q D

100

FCD=25 10 length to distance ratio xf y

4.0 1

16.0 63.8 255

0.1 1.E-06

1.E-05

1.E-04

1.E-03

1.E-02

1.E-01

1.E+00

Dimensionless Time tDxf

Figure 22A – Dimensionless rate versus dimensionless time with deviations for fracture face interference for constant pressure case and FCD=25 – Point source – Horizontal well

186

Reciprocal Dimensionless Flow Rate 1/q

D

FCD - dimensionless fracture conductivity xf - fracture half-length y - distance between two fractures

Reciprocal Dimensionless Flow Rate For Two Fractures System 2/q Dtfs

10

FCD=100

1 FCD 1 length to distance ratio 0.1

0.01

5 10

xf y

25

4.0

100

16.0 63.8

500

128 255

0.001 0.000001

0.00001

0.0001

0.001

0.01

0.1

1

Dimensionless Time tDxf

Figure 23 – Finite conductivity type curve with deviations for fracture face interference for constant pressure case and FCD=100 – Point source – Horizontal well

Dimensionless Flow Rate q D

100

FCD=100 10

length to distance ratio xf y

4.0 16.0 1

63.8 255

0.1 1.E-06

1.E-05

1.E-04

1.E-03

1.E-02

1.E-01

1.E+00

Dimensionless Time tDxf

Figure 23A – Dimensionless rate versus dimensionless time with deviations for fracture face interference for constant pressure case and FCD=100 – Point source – Horizontal well

187

Reciprocal Dimensionless Flow Rate 1/q

D

FCD - dimensionless fracture conductivity xf - fracture half-length y - distance between two fractures

Reciprocal Dimensionless Flow Rate For Two Fractures System 2/q Dtfs

10

FCD=500

1 FCD 1 length to distance ratio 0.1

5 10

xf y

25

0.01

4.0

100

16.0 63.8

500

128 255

0.001 0.000001

0.00001

0.0001

0.001

0.01

0.1

1

Dimensionless Time tDxf

Figure 24 – Finite conductivity type curve with deviations for fracture face interference for constant pressure case and FCD=500 – Point source – Horizontal well

Dimensionless Flow Rate q D

1000

FCD=500

100

10

length to distance ratio xf y

4.0 1

16.0 63.8 255

0.1 1.E-06

1.E-05

1.E-04

1.E-03

1.E-02

1.E-01

1.E+00

Dimensionless Time tDxf

Figure 24A – Dimensionless rate versus dimensionless time with deviations for fracture face interference for constant pressure case and FCD=500 – Point source – Horizontal well

188

APPENDIX

D

189

Nomenclature

pi

- initial formation pressure [psi]

BHFP

- bottom hole flowing pressure [psi]

c

- formation compressibility [psi-1]

cf,w, o, g

- fluid compressibility – water, oil, gas respectively [psi-1]

ct

- total system compressibility [psi-1]

n

- fluid viscosity [cp]

B

- fluid FVF [RB/stb]

D

- depth [ft]

qD

- dimensionless flow rate

pD

- dimensionless pressure

tDxf

- dimensionless time in function of the fracture half-length

qD1

- dimensionless flow rate for well 1

qD2

- dimensionless flow rate for well 2

qDtfs

- dimensionless flow rate for two fracture system

T

- formation temperature [oR]

ppi

- initial pseudopressure

ppwf

- bottom hole pseudopressure

Cf

- fracture flow capacity

kf

- fracture permeability [md]

kfe

- equivalent fracture permeability [md]

190

w

- fracture width [ft]

we

-equivalent fracture width [ft]

C

- formation flow capacity

k

- formation permeability [md]

h

- formation net pay thickness [ft]

xf

- fracture half-length [ft]

rw

- wellbore radius [ft]

s

- skin [-]

FCD

- dimensionless fracture conductivity [-]

φ

- formation porosity [%]

φf

- fracture porosity [%]

φfe

- equivalent fracture porosity [%]

qp,j

- volumetric flow rate of phase p in connection j

Twj

- connection transmissibility factor

Mp,j

- phase mobility at the connection

Pj

- nodal pressure in the grid block containing the connection

Hwj

- well bore pressure head

c

- unit conversion factor (0.001127 in field units)

θ

- the segment angle connecting with the well (2π) for the well located in the center of the grid block

Kh

- effective permeability times net thickness of the connection.

ro

- pressure equivalent radius of the grid block

Dx, Dy

- the x- and y- dimensions of the grid block

191

Kx, Ky

- x- and y- direction permeabilities

t

- production time [days]