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

iv

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.

v

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 …………..…….… 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

4. 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 …………….………..….. 81

8.1. Summary ……………………………………………………………. 81

8.2. Recommendations for Future Work ………………………………… 82

Reference …………………………………………….………….….……….. 83 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 ……………………………………………….. 91

Appendix A3 – Data File For FCD=100, Constant Rate Case and Point Source 94

ix

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


Dimensionless Time, Pressure and Rate ……………………….… 128

Table 4 - FCD=25– Results of Numerical Simulation and

Dimensionless Time, Pressure and Rate ………………………..… 134


Dimensionless Time, Pressure and Rate ………………….……… 140

x


Dimensionless Time, Pressure and Rate ………………………… 147

Table 7 – FCD=100 – Results of Numerical Simulation and Dimensionless Time,

Pressure and Rate for Point Source …………………………….... 152


and Pressure for xf/y=255, Constant Rate Case, Vertical Well ….. 160


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


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


interference for constant flow rate and FCD=5 – Line source –

Vertical well ……………………………………………………… 170


xi


Vertical well …………………………………………………….. 171



Vertical well ……………………………………………………... 171


interference for constant flow rate for and FCD=100 – Line source –

Vertical well …………………………………………………….. 172



Vertical well ……………………………………………………… 172


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



Vertical well ………………………………………………………. 174



Line source – Vertical well ………………………………………. 174

xii



Vertical well ……………………………………………………… 175



Line source – Vertical well ……………………………………….. 175



Vertical well ……………………………………………………… 176






Vertical well …………………………………………………….. 177



Line source – Vertical well ………………………………….….. 177



Vertical well …………………………………………………….. 178



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interference for constant rate case and FCD=1 – Point source –

Horizontal well ………………………………………………….. 179


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








face interference for constant pressure case and FCD=1 –


Figure 19A – Dimensionless rate versus dimensionless time with deviations

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for fracture face interference for constant pressure case and FCD=1 –






for fracture face interference for constant pressure case and FCD=5 –






for fracture face interference for constant pressure case and

FCD=10 – Point source – Horizontal well ……………………….. 184






FCD=25 – Point source – Horizontal well ……………………….. 185




xv



FCD=100 – Point source – Horizontal well …………………………186






FCD=500 – Point source – Horizontal well ……………………. 187

Appendix D …………………………………………………………………... 188

Nomenclature …………………………………………………………………. 189

xvi

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

xvii

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)……………………………….…….. 6

Figure 1.3. - Fracture flow regimes (Cinco-Ley, 1981) .…………………..…. 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) ……………………. 42

Figure 3.8. – Model simulation results (symbols) with Bennett (1985) finite

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


FCD=100 – Point source (horizontal well) ……………………. 70

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

xxii

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.

1

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 low-

permeability 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

6

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:

fkx

fwfkCDF = …………………………………………………………… (1)

where

kf [md] – fracture permeability

wf [ft] – fracture width

k [md] – formation permeability

xf [ft] – fracture half-length

9

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)

10

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

11

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 post-

fracture 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 half-

length, 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 10-

5 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.

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

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

xf

dimension in x direction

dim

ensi

on in

y d

irect

ion

Well

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.

Figure 3.2. – Grid block distribution in numerical model

1

10

100

1000

0 10 20 30 40

n- Number of Blocks

dx,

dy

- L

eng

th o

f B

lock

x o

r y

Res

pec

tive

ly [

ft]

dx dyFracture Half-Length

Well

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 in x direction

Total reservoir dimension in x direction

Total number of blocks in y direction

Total reservoir 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.

Well

Fracture length 2xf

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

simulation is low permeability reservoir 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:

Fracture length 2x

f

31

ew

fwe

φφ = ……………………………….……………………….……… (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.

w

fkxCDFfk = ………………………………………………………….. (3)

where:

FCD – dimensionless fracture conductivity



w [ft] – fracture width

Equivalent fracture permeability

ew

wfkfek = ………………………………………………………….…... (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] 5,000Bottom hole flowing pressure BHFP [psi] 500Formation porosity, fraction φ 0.2Formation permeability k [md] 0.1Formation height (reservoir thickness) h [ft] 100Rock compressibility c [psi-1] 0Skin s 0Well bore radius rw [ft] 0.3

Fracture Properties Fracture half length xf [ft] 2043Fracture width w [in] 0.5Fracture porosity φf 0.35

Equivalent Fracture Properties Adjusted for Numerical Simulation Equivalent fracture width we [ft] 2Equivalent fracture porosity φfe 0.0073

Fluid Properties Compressibility cf [psi-1] 3.00E-06Viscosity µ [cp] 1FVF 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:

( )wjHwPjPjpMwjTjpq −−= ,, …………………….………………..(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:

swror

KhcwjT

+

=

ln

θ …………………………………………..……….……. (6)

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

4

1

4

1

2

1

2

1

22

1

2

28.0

+

+

=

yKxK

xK

yK

yKxK

yDxK

yKxD

or ……………………………... (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

Data for equivalent fracture permeability

calculation Formation permeability k [md] 0.1 Formation permeability k [md] 0.1Fracture half length xf [ft] 2043 Fracture half length xf [ft] 2043Fracture width w [in] 0.5 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

Real Fracture Permeability

Equivalent Fracture Permeability

FCD kf kfe 1 4,903 1025 24,516 511

10 49,032 1,02225 122,580 2,554

100 490,320 10,215500 2,451,600 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:

txc

kt

ftDxf 2

0063288.0

µφ= ……………………………………..……..…..……. (8)

where:


t [days] – production time

φ [-] – reservoir porosity, fraction

ct [psi-1] – total system compressibility

µ [cp] – fluid viscosity


Correlation for dimensionless pressure:

( )µqB

wfpipkhDp

2.141

−= ………………………………………….…………... (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

0.001

0.01

0.1

1

10

1.E-06 1.E-05 1.E-04 1.E-03 1.E-02 1.E-01 1.E+00

Dimensionless Time tDxf

Dim

ensi

on

less

Pre

ssu

re p

D

FCD=1

FCD=5

FCD=10

FCD=25

FCD=100

FCD=500

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:

( )wfpipkh

qBDq

−= µ2.141

………………………………….……………….… (10)

where:

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

µ [cp] – fluid viscosity

q [stb/day] – surface rate


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

0.01

0.1

1

10

100

1000

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


Dim

ensi

on

less

Flo

w R

ate

q D

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

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.

43

0.001

0.01

0.1

1

10

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


Rec

ipro

cal D

imen

sio

nle

ss F

low

Rat

e 1/

q D

FCD=1

FCD=5

FCD=10

FCD=25

FCD=100

FCD=500

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 reservoir dimension in x

direction


Total reservoir dimension in y

direction

Total number of blocks in z direction

Total reservoir dimension in z

direction

nx x [ft] ny y[ft] n z 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

result of the numerical simulation.

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 Equivalent data Fracture porosity φ [%] 35 Fracture porosity φe [%] 0.73Fracture width w [in] 0.5 Equivalent fracture width we [ft] 2

Fracture length 2x

f

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 half-

length 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

0.001

0.01

0.1

1

10

1.E-06 1.E-05 1.E-04 1.E-03 1.E-02 1.E-01 1.E+00


Dim

ensi

on

less

Pre

ssu

re p

D

FCD=1

FCD=5

FCD=10

FCD=25

FCD=100

FCD=500

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

0.1

1

10

100

1000

1.00E-07 1.00E-06 1.00E-05 1.00E-04 1.00E-03 1.00E-02 1.00E-01


Dim

ensi

on

less

Flo

w R

ate

qD

FCD=1

FCD=5

FCD=10

FCD=25

FCD=100

FCD=500

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

0.001

0.01

0.1

1

10

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


Rec

ipro

cal D

imen

sio

nle

ss F

low

Rat

e 1/

qD

FCD=1

FCD=5

FCD=10

FCD=25

FCD=100

FCD=500

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.

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

Well 1

Well 2

Fracture length 2xf

Fracture 1

Fracture 2

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.

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.

Well 1

Well 2

Fracture length 2xf

Fracture 1

Fracture 2

Fra

ctu

re d

ista

nce

, y

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.

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.

Well 1

Well 2

Fracture length 2xf

Fracture 1

Fracture 2

Fra

ctu

re d

ista

nce

, y

55

Well 1

Well 2

Fracture 2

Fracture 1

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

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

Fractures Tips

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 reservoir dimension in x direction


Total reservoir 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

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.

Well 1

Well 2

Initial Reservoir

Added Part of Reservoir

Enlarged Fracture

Enlarged Fracture

201,

000

100,

500

y

Well 1

Well 2

Initial Reservoir


Enlarged Fracture

Enlarged Fracture

201,

000

100,

500

Well 1

Well 2

Initial Reservoir


Enlarged Fracture

Enlarged Fracture

201,

000

100,

500

y

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.

If y1 < y2 then 21 yfx

yfx

>

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.

y22xf

h

y1

2xf

h

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.

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.

0.001

0.01

0.1

1

10

0.000001 0.00001 0.0001 0.001 0.01 0.1 1


Dim

ensi

on

less

Pre

ssu

re P

D

0.028

4.0

16.0

63.8

255

FCD - dimensionless fracture conductivity

xf - fracture half-lengthy - distance between two fractures

length to distance ratio

xf

y5

25

10

FCD

1

100

500

FCD=100

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

12

21 DDDtfsqqq +=

………………………………………………… (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.

63

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

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

0.1

1

10

100

1000

0.000001 0.00001 0.0001 0.001 0.01 0.1 1


Dim

ensi

on

less

Flo

w R

ate

qD

0.0284.016.063.8255128

length to distance ratioxf

y

FCD=100

0.001

0.01

0.1

1

10

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


Rec

ipro

cal D

imen

sio

nle

ss F

low

Rat

e 1/

qD

xf

y




5

FCD

1

500

100

10

25

FCD=100

0.028

4.0

16.0

63.8

128

255

Rec

ipro

cal D

imen

sio

nle

ss F

low

Rat

e fo

r T

wo

Fra

ctu

re S

yste

m 2

/qD

tfs

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 reservoir dimension in x

direction


Total reservoir dimension in y

direction

Total number of blocks in z

direction

Total reservoir dimension in z

direction

nx x [ft] ny y[ft] n z 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.

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

0.001

0.01

0.1

1

10

0.000001 0.00001 0.0001 0.001 0.01 0.1 1


Dim

ensi

on

less

Pre

ssu

re P

D

4.0

16.0

63.8

255




xf

y5

25

10

FCD

1

100

500

FCD=100

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 .


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

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

0.001

0.01

0.1

1

10

0.000001 0.00001 0.0001 0.001 0.01 0.1 1

4.0

16.0

63.8

128

255

Rec

ipro

cal

Dim

ensi

on

less

Flo

w R

ate

1/q

D


5

FCD

1

500

100

10

25


xf

y



FCD=100

Rec

ipro

cal

Dim

ensi

on

less

Flo

w R

ate

Fo

r T

wo

Fra

ctu

res

Sys

tem

2/q

Dtf

s

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.

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.

0.1

1

10

100

1.E-06 1.E-05 1.E-04 1.E-03 1.E-02 1.E-01 1.E+00


Dim

ensi

on

less

Flo

w R

ate

qD

4.0

16.0

63.8

255

FCD=100


y

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.

1,000

10,000

100,000

1,000,000

0 10 20 30 40 50 60

Time [months]

Pro

du

ctio

n R

ate

[Mcf

/mo

nth

]

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

MDxf tmonthstt ][=

MD qmonth

Mcfqq

=

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.

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

0.001

0.01

0.1

1

10

0.000001 0.00001 0.0001 0.001 0.01 0.1 1

4.0

16.0

63.8

128

Rec

ipro

cal

Dim

ensi

on

less

Flo

w R

ate

1/q

D


5

FCD

1

500

100

10

25



FCD=100

Rec

ipro

cal

Dim

ensi

on

less

Flo

w R

ate

Fo

r T

wo

Fra

ctu

res

Sys

tem

2/q

Dtf

s

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

Figure 7.1. – Sensitivity analysis of change of initial pressure for FCD=5 Figure 7.2. – Sensitivity analysis of initial pressure change for FCD=100

0.001

0.01

0.1

1

10

1.E-06 1.E-05 1.E-04 1.E-03 1.E-02 1.E-01 1.E+00


Dim

ensi

on

less

Pre

ssu

re p

D

pi=1,000

pi=2,000

pi=5,000

0.001

0.01

0.1

1

10

1.E-06 1.E-05 1.E-04 1.E-03 1.E-02 1.E-01 1.E+00


Dim

ensi

on

less

Pre

ssu

re p

D

pi=1,000

pi=2,000

pi=5,000

76


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.

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

0.001

0.01

0.1

1

10

1.E-06 1.E-05 1.E-04 1.E-03 1.E-02 1.E-01 1.E+00


Rec

ipro

cal D

imen

sio

nle

ss F

low

Rat

e 1/

qD

Jinitial=1,285

J=2*Jinitial=2,536

J=0.5*Jinitial=634

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.

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

0.001

0.01

0.1

1

10

1.E-06 1.E-05 1.E-04 1.E-03 1.E-02 1.E-01 1.E+00


Rec

ipro

cal D

imen

sio

nle

ss F

low

Rat

e 1/

qD

Jinitial=25,710

J=2*Jinitial=50,730

J=0.5*Jinitial=12,682

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

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 half-

length 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

0.001

0.01

0.1

1

10

1.E-06 1.E-05 1.E-04 1.E-03 1.E-02 1.E-01 1.E+00


Dim

ensi

on

less

Pre

ssu

re p

D

2 2

8 8

36 36

255 255


2043y

506y

FCD=100

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.

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

0.001

0.01

0.1

1

10

1.E-06 1.E-05 1.E-04 1.E-03 1.E-02 1.E-01 1.E+00


Dim

ensi

on

less

Pre

ssu

re p

D

9 blocks

13 blocks

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 integral-

derivative (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

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Rocky Mountain Regional/Low Permeability Reservoir Symposium and Exhibition,

Denver, Colorado – March 2000

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a Well With Finite-Conductivity Fracture – SPE Journal, August 1978, 253-264,

Trans. AIME 265

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– SPE paper 7490, Journal of Petroleum Technology, September 1981, 1749-1766

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the Future –2006

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Producing at Constant Pressure –Journal of Petroleum Science and Engineering – 40

(2003), 89-102

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86

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87

A P P E N D I X 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 dy dz 335 309 1 / -- Fluid phases present WATER -- Units FIELD --length of stack used by linear solver NSTACK 50 / ------------ #wells, # connections, #groups, #wells per group WELLDIMS 1 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 0.1 1 335 1 309 1 1 / -- reservoir X permeability PORO 0.2 1 335 1 309 1 1 / -- reservoir Porosity PERMX 10215 149 187 155 155 1 1 / -- equivalent fracture X permeability PORO 0.0073 149 187 155 155 1 1 / -- equivalent fracture porosity/ / COPY PERMX PERMY 1 335 1 309 1 1 / PERMX PERMZ 1 335 1 309 1 1 / / INIT GRIDFILE 0 1 / RPTGRID TRANX TRANY / PROPS ========================================================================== PVTW -- PREF BW(PREF) CW VW(PREF) CVW 4014.7 1.0 3.0D-6 1.0 0 / ROCK -- PREF CR 4014.7 0 / DENSITY -- OIL WATER GAS 44.09 62.28 0.066 / RPTPROPS / SOLUTION ========================================================================= -- DATUM DATUM OWC OWC GOC GOC RSVD RVVD SOLN -- DEPTH PRESS DEPTH PCOW DEPTH PCOG TABLE TABLE METH EQUIL 5000 5000 1* 1* 1* 1* 1* 1* 1* /

RPTSOL -- Fluid Create init -- in place Restart file FIP=1 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 -LOCATION- BHP PHASE DRAINAGE FLAG FLAG FLAG PRESS FLAG -- NAME NAME I J DEPTH RADIUS GAS SHUT CROSS TABLE DENS W1 G 168 155 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/ CNTL OIL WATER GAS LIQU RES BHP THP VFP ALQ -- NAME SHUT MODE RATE RATE RATE RATE RATE TABLE W1 OPEN WRAT 1* 100 1* 1* 1* 1* 1* 1* 1* / / TSTEP 0.00001157 1.39E-05 1.67E-05 2.00E-05 2.40E-05 2.88E-05 3.46E-05 4.15E-05 4.98E-05 5.97E-05 7.17E-05 8.60E-05 1.03E-04 1.24E-04 1.49E-04 1.78E-04 2.14E-04 2.57E-04 3.08E-04 3.70E-04 4.44E-04 5.32E-04 6.39E-04 7.67E-04 9.20E-04 1.10E-03 1.32E-03 1.59E-03 1.91E-03 2.29E-03 2.75E-03 3.30E-03 3.96E-03 4.75E-03 5.70E-03 6.84E-03 8.20E-03 9.84E-03 1.18E-02 1.42E-02 1.70E-02 2.04E-02 2.45E-02 2.94E-02 3.53E-02 4.23E-02 5.08E-02 6.10E-02 7.31E-02 8.78E-02 1.05E-01 1.26E-01 1.52E-01 1.82E-01 2.18E-01 2.62E-01 3.15E-01 3.77E-01 4.53E-01 5.43E-01 6.52E-01 7.83E-01 9.39E-01 1.13E+00 1.35E+00 1.62E+00 1.95E+00 2.34E+00 2.80E+00 3.37E+00 4.04E+00 4.85E+00 5.81E+00 6.98E+00 8.37E+00 1.00E+01 1.21E+01 1.45E+01 1.74E+01 2.08E+01 2.50E+01 3.00E+01 3.60E+01 4.32E+01 5.18E+01 6.22E+01 7.47E+01 8.96E+01 1.08E+02 1.29E+02 1.55E+02 1.86E+02 2.23E+02 2.67E+02 3.21E+02 3.85E+02 4.62E+02 5.55E+02 6.66E+02 7.99E+02 9.58E+02 1.15E+03 1.38E+03 / END

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 dy dz 335 309 1 / -- Fluid phases present WATER -- Units FIELD --length of stack used by linear solver NSTACK 50 / ------------ #wells, # connections, #groups, #wells per group WELLDIMS 1 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 0.1 1 335 1 309 1 1 / -- reservoir X permeability PORO 0.2 1 335 1 309 1 1 / -- reservoir Porosity PERMX 10215 149 187 155 155 1 1 / -- equivalent fracture X permeability PORO 0.0073 149 187 155 155 1 1 / -- equivalent fracture X porosity/ / COPY PERMX PERMY 1 335 1 309 1 1 / PERMX PERMZ 1 335 1 309 1 1 / / INIT GRIDFILE 0 1 / RPTGRID TRANX TRANY / PROPS ========================================================================== PVTW -- PREF BW(PREF) CW VW(PREF) CVW 4014.7 1 3.0D-6 1 0 / ROCK -- PREF CR 4014.7 0 / DENSITY -- OIL WATER GAS 44.09 62.28 0.066 / RPTPROPS / SOLUTION ========================================================================= -- DATUM DATUM OWC OWC GOC GOC RSVD RVVD SOLN -- DEPTH PRESS DEPTH PCOW DEPTH PCOG TABLE TABLE METH EQUIL 5000 5000 1* 1* 1* 1* 1* 1* 1* / RPTSOL -- Fluid Create init -- in place Restart file FIP=1 RESTART=2 / SUMMARY ========================================================================== -- 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 -LOCATION- BHP PHASE DRAINAGE FLAG FLAG FLAG PRESS FLAG -- NAME NAME I J DEPTH RADIUS GAS SHUT CROSS TABLE DENS W1 G 168 155 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/ CNTL OIL WATER GAS LIQU RES BHP THP VFP ALQ -- NAME SHUT MODE RATE RATE RATE RATE RATE TABLE W1 OPEN BHP 1* 1* 1* 1* 1* 500 1* 1* 1* / / TSTEP 0.00001157 1.39E-05 1.67E-05 2.00E-05 2.40E-05 2.88E-05 3.46E-05 4.15E-05 4.98E-05 5.97E-05 7.17E-05 8.60E-05 1.03E-04 1.24E-04 1.49E-04 1.78E-04 2.14E-04 2.57E-04 3.08E-04 3.70E-04 4.44E-04 5.32E-04 6.39E-04 7.67E-04 9.20E-04 1.10E-03 1.32E-03 1.59E-03 1.91E-03 2.29E-03 2.75E-03 3.30E-03 3.96E-03 4.75E-03 5.70E-03 6.84E-03 8.20E-03 9.84E-03 1.18E-02 1.42E-02 1.70E-02 2.04E-02 2.45E-02 2.94E-02 3.53E-02 4.23E-02 5.08E-02 6.10E-02 7.31E-02 8.78E-02 1.05E-01 1.26E-01 1.52E-01 1.82E-01 2.18E-01 2.62E-01 3.15E-01 3.77E-01 4.53E-01 5.43E-01 6.52E-01 7.83E-01 9.39E-01 1.13E+00 1.35E+00 1.62E+00 1.95E+00 2.34E+00 2.80E+00 3.37E+00 4.04E+00 4.85E+00 5.81E+00 6.98E+00 8.37E+00 1.00E+01 1.21E+01 1.45E+01 1.74E+01 2.08E+01 2.50E+01 3.00E+01 3.60E+01 4.32E+01 5.18E+01 6.22E+01 7.47E+01 8.96E+01 1.08E+02 1.29E+02 1.55E+02 1.86E+02 2.23E+02 2.67E+02 3.21E+02 3.85E+02 4.62E+02 5.55E+02 6.66E+02 7.99E+02 9.58E+02 1.15E+03 1.38E+03 1.66E+03 1.99E+03 2.39E+03 2.86E+03 3.43E+03 4.12E+03 4.95E+03 5.93E+03 7.12E+03 8.55E+03 1.03E+04 / END

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 dy dz 335 309 9 / -- Fluid phases present WATER -- Units FIELD --length of stack used by linear solver NSTACK 50 / ------------ #wells, # connections, #groups, #wells per group WELLDIMS 1 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 0.1 1 335 1 309 1 9 / -- reservoir X permeability PORO 0.2 1 335 1 309 1 9 / -- reservoir porosity PERMX 10215 149 187 155 155 1 9 / -- equivalent fracture X perm. PORO 0.0073 149 187 155 155 1 9 / -- equivalent fracture porosity/ / COPY PERMX PERMY 1 335 1 309 1 9 / PERMX PERMZ 1 335 1 309 1 9 / / INIT GRIDFILE 0 1 / RPTGRID TRANX TRANY / PROPS ==================================================================== PVTW -- PREF BW(PREF) CW VW(PREF) CVW 4014.7 1.0 3.0D-6 1.0 0 / ROCK -- PREF CR 4014.7 0 / DENSITY -- OIL WATER GAS 44.09 62.28 0.066 / RPTPROPS / SOLUTION ========================================================================= -- DATUM DATUM OWC OWC GOC GOC RSVD RVVD SOLN -- DEPTH PRESS DEPTH PCOW DEPTH PCOG TABLE TABLE METH EQUIL 5000 5000 1* 1* 1* 1* 1* 1* 1* / RPTSOL -- Fluid Create init -- in place Restart file FIP=1 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 -LOCATION- BHP PHASE DRAINAGE FLAG FLAG FLAG PRESS FLAG -- NAME NAME I J DEPTH RADIUS GAS SHUT CROSS TABLE DENS W1 G 168 155 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/ CNTL OIL WATER GAS LIQU RES BHP THP VFP ALQ -- NAME SHUT MODE RATE RATE RATE RATE RATE TABLE W1 OPEN WRAT 1* 100 1* 1* 1* 1* 1* 1* 1* / / TSTEP 0.00001157 1.39E-05 1.67E-05 2.00E-05 2.40E-05 2.88E-05 3.46E-05 4.15E-05 4.98E-05 5.97E-05 7.17E-05 8.60E-05 1.03E-04 1.24E-04 1.49E-04 1.78E-04 2.14E-04 2.57E-04 3.08E-04 3.70E-04 4.44E-04 5.32E-04 6.39E-04 7.67E-04 9.20E-04 1.10E-03 1.32E-03 1.59E-03 1.91E-03 2.29E-03 2.75E-03 3.30E-03 3.96E-03 4.75E-03 5.70E-03 6.84E-03 8.20E-03 9.84E-03 1.18E-02 1.42E-02 1.70E-02 2.04E-02 2.45E-02 2.94E-02 3.53E-02 4.23E-02 5.08E-02 6.10E-02 7.31E-02 8.78E-02 1.05E-01 1.26E-01 1.52E-01 1.82E-01 2.18E-01 2.62E-01 3.15E-01 3.77E-01 4.53E-01 5.43E-01 6.52E-01 7.83E-01 9.39E-01 1.13E+00 1.35E+00 1.62E+00 1.95E+00 2.34E+00 2.80E+00 3.37E+00 4.04E+00 4.85E+00 5.81E+00 6.98E+00 8.37E+00 1.00E+01 1.21E+01 1.45E+01 1.74E+01 2.08E+01 2.50E+01 3.00E+01 3.60E+01 4.32E+01 5.18E+01 6.22E+01 7.47E+01 8.96E+01 1.08E+02 1.29E+02 1.55E+02 1.86E+02 2.23E+02 2.67E+02 3.21E+02 3.85E+02 4.62E+02 5.55E+02 6.66E+02 7.99E+02 9.58E+02 1.15E+03 1.38E+03 / END

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 dy dz 335 309 9 / -- Fluid phases present WATER -- Units FIELD --length of stack used by linear solver NSTACK 50 / ------------ #wells, # connections, #groups, #wells per group WELLDIMS 1 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 0.1 1 335 1 309 1 9 / -- reservoir X permeability PORO 0.2 1 335 1 309 1 9 / -- reservoir Porosity PERMX 10215 149 187 155 155 1 9 / -- equivalent fracture X permeability PORO 0.0073 149 187 155 155 1 9 / -- equivalent fracture X porosity/ / COPY PERMX PERMY 1 335 1 309 1 9 / PERMX PERMZ 1 335 1 309 1 9 / / INIT GRIDFILE 0 1 / RPTGRID TRANX TRANY / PROPS ========================================================================== PVTW -- PREF BW(PREF) CW VW(PREF) CVW 5000 1 3.0D-6 1 0 / ROCK -- PREF CR 5000 0 / DENSITY -- OIL WATER GAS 44.09 62.28 0.066 / RPTPROPS / SOLUTION ========================================================================= -- DATUM DATUM OWC OWC GOC GOC RSVD RVVD SOLN -- DEPTH PRESS DEPTH PCOW DEPTH PCOG TABLE TABLE METH EQUIL 5000 5000 1* 1* 1* 1* 1* 1* 1* / RPTSOL -- Fluid Create init -- in place Restart file FIP=1 RESTART=2 / SUMMARY ========================================================================== -- 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 -LOCATION- BHP PHASE DRAINAGE FLAG FLAG FLAG PRESS FLAG -- NAME NAME I J DEPTH RADIUS GAS SHUT CROSS TABLE DENS W1 G 168 155 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/ CNTL OIL WATER GAS LIQU RES BHP THP VFP ALQ -- NAME SHUT MODE RATE RATE RATE RATE RATE TABLE W1 OPEN BHP 1* 1* 1* 1* 1* 500 1* 1* 1* / / TSTEP 0.00001157 1.39E-05 1.67E-05 2.00E-05 2.40E-05 2.88E-05 3.46E-05 4.15E-05 4.98E-05 5.97E-05 7.17E-05 8.60E-05 1.03E-04 1.24E-04 1.49E-04 1.78E-04 2.14E-04 2.57E-04 3.08E-04 3.70E-04 4.44E-04 5.32E-04 6.39E-04 7.67E-04 9.20E-04 1.10E-03 1.32E-03 1.59E-03 1.91E-03 2.29E-03 2.75E-03 3.30E-03 3.96E-03 4.75E-03 5.70E-03 6.84E-03 8.20E-03 9.84E-03 1.18E-02 1.42E-02 1.70E-02 2.04E-02 2.45E-02 2.94E-02 3.53E-02 4.23E-02 5.08E-02 6.10E-02 7.31E-02 8.78E-02 1.05E-01 1.26E-01 1.52E-01 1.82E-01 2.18E-01 2.62E-01 3.15E-01 3.77E-01 4.53E-01 5.43E-01 6.52E-01 7.83E-01 9.39E-01 1.13E+00 1.35E+00 1.62E+00 1.95E+00 2.34E+00 2.80E+00 3.37E+00 4.04E+00 4.85E+00 5.81E+00 6.98E+00 8.37E+00 1.00E+01 1.21E+01 1.45E+01 1.74E+01 2.08E+01 2.50E+01 3.00E+01 3.60E+01 4.32E+01 5.18E+01 6.22E+01 7.47E+01 8.96E+01 1.08E+02 1.29E+02 1.55E+02 1.86E+02 2.23E+02 2.67E+02 3.21E+02 3.85E+02 4.62E+02 5.55E+02 6.66E+02 7.99E+02 9.58E+02 1.15E+03 1.38E+03 1.66E+03 1.99E+03 2.39E+03 2.86E+03 3.43E+03 4.12E+03 4.95E+03 5.93E+03 7.12E+03 8.55E+03 1.03E+04 / END

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 dy dz 335 312 1 / -- Fluid phases present WATER -- Units FIELD --length of stack used by linear solver NSTACK 50 / ------------ #wells, # connections, #groups, #wells per group WELLDIMS 2 2 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 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 0.1 1 335 1 312 1 1 / -- reservoir X permeability PORO 0.2 1 335 1 312 1 1 / -- reservoir Porosity PERMX 10215 149 187 155 155 1 1 / -- equivalent fracture X permeability PERMX 10215 149 187 158 158 1 1 / -- equivalent fracture X permeability PORO 0.0073 149 187 155 155 1 1 / -- equivalent fracture porosity PORO 0.0073 149 187 158 158 1 1 / -- equivalent fracture porosity/ / COPY PERMX PERMY 1 335 1 312 1 1 / PERMX PERMZ 1 335 1 312 1 1 / / INIT GRIDFILE 0 1 / RPTGRID TRANX TRANY / PROPS ========================================================================== PVTW -- PREF BW(PREF) CW VW(PREF) CVW 4014.7 1.0 3.0D-6 1.0 0 / ROCK -- PREF CR 4014.7 0 / DENSITY -- OIL WATER GAS 44.09 62.28 0.066 / RPTPROPS / SOLUTION ========================================================================= -- DATUM DATUM OWC OWC GOC GOC RSVD RVVD SOLN -- DEPTH PRESS DEPTH PCOW DEPTH PCOG TABLE TABLE METH EQUIL 5000 5000 1* 1* 1* 1* 1* 1* 1* / RPTSOL -- Fluid Create init -- in place Restart file FIP=1 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 -LOCATION- BHP PHASE DRAINAGE FLAG FLAG FLAG PRESS FLAG -- NAME NAME I J DEPTH RADIUS GAS SHUT CROSS TABLE DENS W1 G 168 155 1* WATER 1* STD SHUT NO 1* SEG / W2 G 168 158 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 OPEN/ CNTL OIL WATER GAS LIQU RES BHP THP VFP ALQ -- NAME SHUT MODE RATE RATE RATE RATE RATE TABLE W1 OPEN WRAT 1* 100 1* 1* 1* 1* 1* 1* 1* / W2 OPEN WRAT 1* 100 1* 1* 1* 1* 1* 1* 1* / / TSTEP 1.03E-04 1.24E-04 1.49E-04 1.78E-04 2.14E-04 2.57E-04 3.08E-04 3.70E-04 4.44E-04 5.32E-04 6.39E-04 7.67E-04 9.20E-04 1.10E-03 1.32E-03 1.59E-03 1.91E-03 2.29E-03 2.75E-03 3.30E-03 3.96E-03 4.75E-03 5.70E-03 6.84E-03 8.20E-03 9.84E-03 1.18E-02 1.42E-02 1.70E-02 2.04E-02 2.45E-02 2.94E-02 3.53E-02 4.23E-02 5.08E-02 6.10E-02 7.31E-02 8.78E-02 1.05E-01 1.26E-01 1.52E-01 1.82E-01 2.18E-01 2.62E-01 3.15E-01 3.77E-01 4.53E-01 5.43E-01 6.52E-01 7.83E-01 9.39E-01 1.13E+00 1.35E+00 1.62E+00 1.95E+00 2.34E+00 2.80E+00 3.37E+00 4.04E+00 4.85E+00 5.81E+00 6.98E+00 8.37E+00 1.00E+01 1.21E+01 1.45E+01 1.74E+01 2.08E+01 2.50E+01 3.00E+01 3.60E+01 4.32E+01 5.18E+01 6.22E+01 7.47E+01 8.96E+01 1.08E+02 1.29E+02 1.55E+02 1.86E+02 2.23E+02 2.67E+02 3.21E+02 3.85E+02 4.62E+02 5.55E+02 6.66E+02 / END

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 dy dz 335 314 1 / -- Fluid phases present WATER -- Units FIELD --length of stack used by linear solver NSTACK 50 / ------------ #wells, # connections, #groups, #wells per group WELLDIMS 2 2 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 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 0.1 1 335 1 314 1 1 / -- reservoir X permeability PORO 0.2 1 335 1 314 1 1 / -- reservoir Porosity PERMX 10215 149 187 155 155 1 1 / -- equivalent fracture X permeability PERMX 10215 149 187 160 160 1 1 / -- equivalent fracture X permeability PORO 0.0073 149 187 155 155 1 1 / -- equivalent fracture porosity PORO 0.0073 149 187 160 160 1 1 / -- equivalent fracture porosity/ / COPY PERMX PERMY 1 335 1 314 1 1 / PERMX PERMZ 1 335 1 314 1 1 / / INIT GRIDFILE 0 1 / RPTGRID TRANX TRANY / PROPS ========================================================================== PVTW -- PREF BW(PREF) CW VW(PREF) CVW 4014.7 1.0 3.0D-6 1.0 0 / ROCK -- PREF CR 4014.7 0 / DENSITY -- OIL WATER GAS 44.09 62.28 0.066 / RPTPROPS / SOLUTION ========================================================================= -- DATUM DATUM OWC OWC GOC GOC RSVD RVVD SOLN -- DEPTH PRESS DEPTH PCOW DEPTH PCOG TABLE TABLE METH EQUIL 5000 5000 1* 1* 1* 1* 1* 1* 1* / RPTSOL -- Fluid Create init -- in place Restart file FIP=1 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 -LOCATION- BHP PHASE DRAINAGE FLAG FLAG FLAG PRESS FLAG -- NAME NAME I J DEPTH RADIUS GAS SHUT CROSS TABLE DENS W1 G 168 155 1* WATER 1* STD SHUT NO 1* SEG / W2 G 168 160 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 OPEN/ CNTL OIL WATER GAS LIQU RES BHP THP VFP ALQ -- NAME SHUT MODE RATE RATE RATE RATE RATE TABLE W1 OPEN BHP 1* 1* 1* 1* 1* 500 1* 1* 1* / W2 OPEN BHP 1* 1* 1* 1* 1* 500 1* 1* 1* / / TSTEP 1.03E-04 1.24E-04 1.49E-04 1.78E-04 2.14E-04 2.57E-04 3.08E-04 3.70E-04 4.44E-04 5.32E-04 6.39E-04 7.67E-04 9.20E-04 1.10E-03 1.32E-03 1.59E-03 1.91E-03 2.29E-03 2.75E-03 3.30E-03 3.96E-03 4.75E-03 5.70E-03 6.84E-03 8.20E-03 9.84E-03 1.18E-02 1.42E-02 1.70E-02 2.04E-02 2.45E-02 2.94E-02 3.53E-02 4.23E-02 5.08E-02 6.10E-02 7.31E-02 8.78E-02 1.05E-01 1.26E-01 1.52E-01 1.82E-01 2.18E-01 2.62E-01 3.15E-01 3.77E-01 4.53E-01 5.43E-01 6.52E-01 7.83E-01 9.39E-01 1.13E+00 1.35E+00 1.62E+00 1.95E+00 2.34E+00 2.80E+00 3.37E+00 4.04E+00 4.85E+00 5.81E+00 6.98E+00 8.37E+00 1.00E+01 1.21E+01 1.45E+01 1.74E+01 2.08E+01 2.50E+01 3.00E+01 3.60E+01 4.32E+01 5.18E+01 6.22E+01 7.47E+01 8.96E+01 1.08E+02 1.29E+02 1.55E+02 1.86E+02 2.23E+02 2.67E+02 3.21E+02 3.85E+02 4.62E+02 5.55E+02 6.66E+02 / END

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 dy dz 335 312 9 / -- Fluid phases present WATER -- Units FIELD --length of stack used by linear solver NSTACK 50 / ------------ #wells, # connections, #groups, #wells per group WELLDIMS 2 2 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 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 0.1 1 335 1 312 1 9 / -- reservoir X permeability PORO 0.2 1 335 1 312 1 9 / -- reservoir Porosity PERMX 10215 149 187 155 155 1 9 / -- equivalent fracture X permeability PERMX 10215 149 187 158 158 1 9 / -- equivalent fracture X permeability PORO 0.0073 149 187 155 155 1 9 / -- equivalent fracture porosity PORO 0.0073 149 187 158 158 1 9 / -- equivalent fracture porosity/ / COPY PERMX PERMY 1 335 1 312 1 9 / PERMX PERMZ 1 335 1 312 1 9 / / INIT GRIDFILE 0 1 / RPTGRID TRANX TRANY / PROPS ========================================================================== PVTW -- PREF BW(PREF) CW VW(PREF) CVW 4014.7 1.0 3.0D-6 1.0 0 / ROCK -- PREF CR 4014.7 0 / DENSITY -- OIL WATER GAS 44.09 62.28 0.066 / RPTPROPS / SOLUTION ========================================================================= -- DATUM DATUM OWC OWC GOC GOC RSVD RVVD SOLN -- DEPTH PRESS DEPTH PCOW DEPTH PCOG TABLE TABLE METH EQUIL 5000 5000 1* 1* 1* 1* 1* 1* 1* / RPTSOL -- Fluid Create init -- in place Restart file FIP=1 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 -LOCATION- BHP PHASE DRAINAGE FLAG FLAG FLAG PRESS FLAG -- NAME NAME I J DEPTH RADIUS GAS SHUT CROSS TABLE DENS W1 G 168 155 1* WATER 1* STD SHUT NO 1* SEG / W2 G 168 158 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 / W2 168 158 5 5 OPEN 1* 1* 0.60 1* 0 0 Z / / WCONPROD -- WELL OPEN/ CNTL OIL WATER GAS LIQU RES BHP THP VFP ALQ -- NAME SHUT MODE RATE RATE RATE RATE RATE TABLE W1 OPEN WRAT 1* 100 1* 1* 1* 1* 1* 1* 1* / W2 OPEN WRAT 1* 100 1* 1* 1* 1* 1* 1* 1* / / TSTEP 1.03E-04 1.24E-04 1.49E-04 1.78E-04 2.14E-04 2.57E-04 3.08E-04 3.70E-04 4.44E-04 5.32E-04 6.39E-04 7.67E-04 9.20E-04 1.10E-03 1.32E-03 1.59E-03 1.91E-03 2.29E-03 2.75E-03 3.30E-03 3.96E-03 4.75E-03 5.70E-03 6.84E-03 8.20E-03 9.84E-03 1.18E-02 1.42E-02 1.70E-02 2.04E-02 2.45E-02 2.94E-02 3.53E-02 4.23E-02 5.08E-02 6.10E-02 7.31E-02 8.78E-02 1.05E-01 1.26E-01 1.52E-01 1.82E-01 2.18E-01 2.62E-01 3.15E-01 3.77E-01 4.53E-01 5.43E-01 6.52E-01 7.83E-01 9.39E-01 1.13E+00 1.35E+00 1.62E+00 1.95E+00 2.34E+00 2.80E+00 3.37E+00 4.04E+00 4.85E+00 5.81E+00 6.98E+00 8.37E+00 1.00E+01 1.21E+01 1.45E+01 1.74E+01 2.08E+01 2.50E+01 3.00E+01 3.60E+01 4.32E+01 5.18E+01 6.22E+01 7.47E+01 8.96E+01 1.08E+02 1.29E+02 1.55E+02 1.86E+02 2.23E+02 2.67E+02 3.21E+02 3.85E+02 4.62E+02 5.55E+02 6.66E+02 / END

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 dy dz 335 312 9 / -- Fluid phases present WATER -- Units FIELD --length of stack used by linear solver NSTACK 50 / ------------ #wells, # connections, #groups, #wells per group WELLDIMS 2 2 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 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 0.1 1 335 1 312 1 9 / -- reservoir X permeability PORO 0.2 1 335 1 312 1 9 / -- reservoir Porosity PERMX 10215 149 187 155 155 1 9 / -- equivalent fracture X permeability PERMX 10215 149 187 158 158 1 9 / -- equivalent fracture X permeability PORO 0.0073 149 187 155 155 1 9 / -- equivalent fracture porosity PORO 0.0073 149 187 158 158 1 9 / -- equivalent fracture porosity/ / COPY PERMX PERMY 1 335 1 312 1 9 / PERMX PERMZ 1 335 1 312 1 9 / / INIT GRIDFILE 0 1 / RPTGRID TRANX TRANY / PROPS ========================================================================== PVTW -- PREF BW(PREF) CW VW(PREF) CVW 4014.7 1.0 3.0D-6 1.0 0 / ROCK -- PREF CR 4014.7 0 / DENSITY -- OIL WATER GAS 44.09 62.28 0.066 / RPTPROPS / SOLUTION ========================================================================= -- DATUM DATUM OWC OWC GOC GOC RSVD RVVD SOLN -- DEPTH PRESS DEPTH PCOW DEPTH PCOG TABLE TABLE METH EQUIL 5000 5000 1* 1* 1* 1* 1* 1* 1* / RPTSOL -- Fluid Create init -- in place Restart file FIP=1 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 -LOCATION- BHP PHASE DRAINAGE FLAG FLAG FLAG PRESS FLAG -- NAME NAME I J DEPTH RADIUS GAS SHUT CROSS TABLE DENS W1 G 168 155 1* WATER 1* STD SHUT NO 1* SEG / W2 G 168 158 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 / W2 168 158 5 5 OPEN 1* 1* 0.60 1* 0 0 Z / / WCONPROD -- WELL OPEN/ CNTL OIL WATER GAS LIQU RES BHP THP VFP ALQ -- NAME SHUT MODE RATE RATE RATE RATE RATE TABLE W1 OPEN BHP 1* 1* 1* 1* 1* 500 1* 1* 1* / W2 OPEN BHP 1* 1* 1* 1* 1* 500 1* 1* 1* / / TSTEP 1.03E-04 1.24E-04 1.49E-04 1.78E-04 2.14E-04 2.57E-04 3.08E-04 3.70E-04 4.44E-04 5.32E-04 6.39E-04 7.67E-04 9.20E-04 1.10E-03 1.32E-03 1.59E-03 1.91E-03 2.29E-03 2.75E-03 3.30E-03 3.96E-03 4.75E-03 5.70E-03 6.84E-03 8.20E-03 9.84E-03 1.18E-02 1.42E-02 1.70E-02 2.04E-02 2.45E-02 2.94E-02 3.53E-02 4.23E-02 5.08E-02 6.10E-02 7.31E-02 8.78E-02 1.05E-01 1.26E-01 1.52E-01 1.82E-01 2.18E-01 2.62E-01 3.15E-01 3.77E-01 4.53E-01 5.43E-01 6.52E-01 7.83E-01 9.39E-01 1.13E+00 1.35E+00 1.62E+00 1.95E+00 2.34E+00 2.80E+00 3.37E+00 4.04E+00 4.85E+00 5.81E+00 6.98E+00 8.37E+00 1.00E+01 1.21E+01 1.45E+01 1.74E+01 2.08E+01 2.50E+01 3.00E+01 3.60E+01 4.32E+01 5.18E+01 6.22E+01 7.47E+01 8.96E+01 1.08E+02 1.29E+02 1.55E+02 1.86E+02 2.23E+02 2.67E+02 3.21E+02 3.85E+02 4.62E+02 5.55E+02 6.66E+02 / END

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 dy dz 323 311 1 / -- Fluid phases present WATER -- Units FIELD --length of stack used by linear solver NSTACK 50 / ------------ #wells, # connections, #groups, #wells per group WELLDIMS 2 2 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 0.1 1 323 1 311 1 1 / -- reservoir X permeability PORO 0.2 1 323 1 311 1 1 / -- reservoir Porosity PERMX 2530 149 175 155 155 1 1 / -- equivalent fracture X permeability PERMX 2530 149 175 157 157 1 1 / -- equivalent fracture X permeability PORO 0.0073 149 175 155 155 1 1 / -- equivalent fracture porosity PORO 0.0073 149 175 157 157 1 1 / -- equivalent fracture porosity/ / COPY PERMX PERMY 1 323 1 311 1 1 / PERMX PERMZ 1 323 1 311 1 1 / / INIT GRIDFILE 0 1 / RPTGRID TRANX TRANY / PROPS ========================================================================== PVTW -- PREF BW(PREF) CW VW(PREF) CVW 4014.7 1.0 3.0D-6 1.0 0 / ROCK -- PREF CR 4014.7 0 / DENSITY -- OIL WATER GAS 44.09 62.28 0.066 / RPTPROPS / SOLUTION ========================================================================= -- DATUM DATUM OWC OWC GOC GOC RSVD RVVD SOLN -- DEPTH PRESS DEPTH PCOW DEPTH PCOG TABLE TABLE METH EQUIL 5000 5000 1* 1* 1* 1* 1* 1* 1* / RPTSOL -- Fluid Create init -- in place Restart file FIP=1 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 -LOCATION- BHP PHASE DRAINAGE FLAG FLAG FLAG PRESS FLAG -- NAME NAME I J DEPTH RADIUS GAS SHUT CROSS TABLE DENS W1 G 162 155 1* WATER 1* STD SHUT NO 1* SEG / W2 G 162 157 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 OPEN/ CNTL OIL WATER GAS LIQU RES BHP THP VFP ALQ -- NAME SHUT MODE RATE RATE RATE RATE RATE TABLE W1 OPEN WRAT 1* 100 1* 1* 1* 1* 1* 1* 1* / W2 OPEN WRAT 1* 100 1* 1* 1* 1* 1* 1* 1* / / TSTEP 1.18E-02 1.42E-02 1.70E-02 2.04E-02 2.45E-02 2.94E-02 3.52E-02 4.23E-02 5.07E-02 6.09E-02 7.31E-02 8.77E-02 1.05E-01 1.26E-01 1.52E-01 1.82E-01 2.18E-01 2.62E-01 3.14E-01 3.77E-01 4.52E-01 5.43E-01 6.51E-01 7.82E-01 9.38E-01 1.13E+00 1.35E+00 1.62E+00 1.95E+00 2.33E+00 2.80E+00 3.36E+00 4.03E+00 4.84E+00 5.81E+00 6.97E+00 8.36E+00 1.00E+01 1.20E+01 1.45E+01 1.74E+01 2.09E+01 2.51E+01 3.01E+01 3.61E+01 4.33E+01 5.20E+01 6.23E+01 7.48E+01 8.98E+01 1.08E+02 1.29E+02 1.55E+02 1.86E+02 2.23E+02 2.68E+02 3.22E+02 3.86E+02 4.63E+02 5.56E+02 6.67E+02 8.00E+02 9.61E+02 1.15E+03 1.38E+03 / END

115

A P P E N D I X B

116

Table 1 – FCD=1 – Results of Numerical Simulation and Dimensionless Time, Pressure and Rate

t [days] tDxf q [stb/day] qD p [psi] pD 0 0 0 0 5000 n/a

1.16E-05 2.92E-09 11,480 3.60E+01 4,961.11 2.75E-022.55E-05 6.44E-09 6,451 2.02E+01 4,942.71 4.06E-024.22E-05 1.07E-08 5,335 1.67E+01 4,929.86 4.97E-026.22E-05 1.57E-08 4,870 1.53E+01 4,920.28 5.65E-028.62E-05 2.18E-08 4,635 1.45E+01 4,913.04 6.16E-021.15E-04 2.91E-08 4,506 1.41E+01 4,907.68 6.54E-021.50E-04 3.78E-08 4,432 1.39E+01 4,903.80 6.81E-021.91E-04 4.83E-08 4,387 1.38E+01 4,901.09 7.00E-022.41E-04 6.09E-08 4,356 1.37E+01 4,899.23 7.14E-023.01E-04 7.60E-08 4,331 1.36E+01 4,897.95 7.23E-023.72E-04 9.41E-08 4,308 1.35E+01 4,897.00 7.29E-024.58E-04 1.16E-07 4,283 1.34E+01 4,896.21 7.35E-025.61E-04 1.42E-07 4,254 1.33E+01 4,895.45 7.40E-026.85E-04 1.73E-07 4,221 1.32E+01 4,894.62 7.46E-028.34E-04 2.11E-07 4,183 1.31E+01 4,893.68 7.53E-021.01E-03 2.56E-07 4,138 1.30E+01 4,892.57 7.61E-021.23E-03 3.10E-07 4,085 1.28E+01 4,891.27 7.70E-021.48E-03 3.75E-07 4,025 1.26E+01 4,889.74 7.81E-021.79E-03 4.53E-07 3,955 1.24E+01 4,887.94 7.94E-022.16E-03 5.46E-07 3,875 1.22E+01 4,885.84 8.09E-022.61E-03 6.58E-07 3,785 1.19E+01 4,883.39 8.26E-023.14E-03 7.93E-07 3,684 1.16E+01 4,880.57 8.46E-023.78E-03 9.54E-07 3,573 1.12E+01 4,877.31 8.69E-024.54E-03 1.15E-06 3,452 1.08E+01 4,873.60 8.95E-025.46E-03 1.38E-06 3,322 1.04E+01 4,869.40 9.25E-026.56E-03 1.66E-06 3,186 1.00E+01 4,864.71 9.58E-027.88E-03 1.99E-06 3,045 9.56E+00 4,859.50 9.95E-029.47E-03 2.39E-06 2,903 9.11E+00 4,853.75 1.04E-011.14E-02 2.88E-06 2,763 8.67E+00 4,847.51 1.08E-011.37E-02 3.46E-06 2,627 8.24E+00 4,840.80 1.13E-011.64E-02 4.15E-06 2,499 7.84E+00 4,833.68 1.18E-011.97E-02 4.98E-06 2,379 7.47E+00 4,826.18 1.23E-012.37E-02 5.99E-06 2,269 7.12E+00 4,818.33 1.29E-012.84E-02 7.19E-06 2,167 6.80E+00 4,810.17 1.34E-013.41E-02 8.63E-06 2,072 6.50E+00 4,801.67 1.40E-014.10E-02 1.04E-05 1,982 6.22E+00 4,792.82 1.47E-014.92E-02 1.24E-05 1,896 5.95E+00 4,783.56 1.53E-015.90E-02 1.49E-05 1,814 5.69E+00 4,773.84 1.60E-017.08E-02 1.79E-05 1,733 5.44E+00 4,763.62 1.67E-018.50E-02 2.15E-05 1,656 5.20E+00 4,752.82 1.75E-011.02E-01 2.58E-05 1,582 4.96E+00 4,741.50 1.83E-011.22E-01 3.09E-05 1,512 4.74E+00 4,729.63 1.91E-011.47E-01 3.71E-05 1,445 4.53E+00 4,717.21 2.00E-01

117

Table 1 – FCD=1 – Results of Numerical Simulation and Dimensionless Time, Pressure and Rate - continued

t [days] tDxf q [stb/day] qD p [psi] pD 1.76E-01 4.46E-05 1,381 4.33E+00 4,704.24 2.09E-012.12E-01 5.35E-05 1,321 4.15E+00 4,690.69 2.19E-012.54E-01 6.42E-05 1,263 3.96E+00 4,676.54 2.29E-013.05E-01 7.70E-05 1,208 3.79E+00 4,661.71 2.40E-013.66E-01 9.24E-05 1,155 3.62E+00 4,646.15 2.51E-014.39E-01 1.11E-04 1,104 3.46E+00 4,629.88 2.62E-015.27E-01 1.33E-04 1,056 3.31E+00 4,612.84 2.74E-016.32E-01 1.60E-04 1,010 3.17E+00 4,595.10 2.87E-017.58E-01 1.91E-04 966 3.03E+00 4,576.59 3.00E-019.10E-01 2.30E-04 924 2.90E+00 4,557.19 3.14E-011.09E+00 2.76E-04 885 2.78E+00 4,537.01 3.28E-011.31E+00 3.31E-04 847 2.66E+00 4,515.99 3.43E-011.57E+00 3.97E-04 810 2.54E+00 4,493.99 3.58E-011.89E+00 4.77E-04 775 2.43E+00 4,472.96 3.73E-012.26E+00 5.72E-04 742 2.33E+00 4,449.37 3.90E-012.72E+00 6.87E-04 710 2.23E+00 4,424.22 4.08E-013.26E+00 8.24E-04 680 2.13E+00 4,399.38 4.25E-013.91E+00 9.89E-04 652 2.04E+00 4,372.86 4.44E-014.69E+00 1.19E-03 624 1.96E+00 4,345.06 4.64E-015.63E+00 1.42E-03 598 1.88E+00 4,316.60 4.84E-016.76E+00 1.71E-03 573 1.80E+00 4,286.66 5.05E-018.11E+00 2.05E-03 549 1.72E+00 4,255.43 5.27E-019.73E+00 2.46E-03 526 1.65E+00 4,222.96 5.50E-011.17E+01 2.95E-03 504 1.58E+00 4,188.75 5.75E-011.40E+01 3.54E-03 483 1.52E+00 4,153.52 5.99E-011.68E+01 4.25E-03 463 1.45E+00 4,116.44 6.26E-012.02E+01 5.10E-03 444 1.39E+00 4,077.95 6.53E-012.42E+01 6.12E-03 426 1.34E+00 4,037.91 6.81E-012.91E+01 7.35E-03 408 1.28E+00 3,996.19 7.11E-013.49E+01 8.82E-03 392 1.23E+00 3,952.88 7.42E-014.19E+01 1.06E-02 376 1.18E+00 3,907.90 7.73E-015.02E+01 1.27E-02 361 1.13E+00 3,861.13 8.07E-016.02E+01 1.52E-02 346 1.09E+00 3,812.92 8.41E-017.23E+01 1.83E-02 333 1.04E+00 3,762.64 8.76E-018.68E+01 2.19E-02 320 1.00E+00 3,710.70 9.13E-011.04E+02 2.63E-02 307 9.63E-01 3,656.93 9.51E-011.25E+02 3.16E-02 295 9.26E-01 3,601.32 9.91E-011.50E+02 3.79E-02 284 8.90E-01 3,543.52 1.03E+001.80E+02 4.55E-02 273 8.55E-01 3,483.49 1.07E+002.16E+02 5.46E-02 262 8.22E-01 3,421.05 1.12E+002.59E+02 6.55E-02 252 7.90E-01 3,356.09 1.16E+003.11E+02 7.86E-02 242 7.59E-01 3,288.44 1.21E+003.73E+02 9.43E-02 232 7.29E-01 3,217.92 1.26E+004.48E+02 1.13E-01 223 7.00E-01 3,144.33 1.31E+005.38E+02 1.36E-01 214 6.72E-01 3,067.62 1.37E+00

118


t [days] tDxf q [stb/day] qD p [psi] pD 6.46E+02 1.63E-01 206 6.45E-01 2,987.31 1.43E+007.75E+02 1.96E-01 197 6.20E-01 2,904.00 1.48E+009.30E+02 2.35E-01 190 5.95E-01 2,817.14 1.55E+001.12E+03 2.82E-01 182 5.71E-01 2,726.80 1.61E+001.34E+03 3.38E-01 175 5.49E-01 2,633.10 1.68E+001.61E+03 4.06E-01 168 5.28E-01 2,536.18 1.74E+001.93E+03 4.87E-01 162 5.07E-01 2,435.83 1.82E+002.12E+03 5.36E-01 158 4.97E-01 2,380.51 1.86E+002.31E+03 5.84E-01 155 4.87E-01 2,329.27 1.89E+002.54E+03 6.43E-01 152 4.78E-01 2,272.74 1.93E+002.77E+03 7.01E-01 149 4.69E-01 2,220.41 1.97E+003.05E+03 7.71E-01 146 4.60E-01 2,162.64 2.01E+003.33E+03 8.41E-01 144 4.51E-01 2,109.18 2.05E+003.66E+03 9.25E-01 141 4.43E-01 2,050.27 2.09E+003.99E+03 1.01E+00 139 4.35E-01 1,995.80 2.13E+004.36E+03 1.10E+00 136 4.28E-01 1,940.64 2.17E+004.58E+03 1.16E+00 135 4.24E-01 1,909.28 2.19E+004.79E+03 1.21E+00 134 4.20E-01 1,879.26 2.21E+005.16E+03 1.30E+00 132 4.14E-01 1,832.10 2.24E+005.46E+03 1.38E+00 130 4.09E-01 1,795.71 2.27E+005.75E+03 1.45E+00 129 4.05E-01 1,761.10 2.29E+006.12E+03 1.55E+00 128 4.01E-01 1,720.87 2.32E+006.48E+03 1.64E+00 126 3.96E-01 1,682.76 2.35E+006.69E+03 1.69E+00 126 3.94E-01 1,661.47 2.36E+006.90E+03 1.74E+00 125 3.92E-01 1,640.83 2.38E+007.27E+03 1.84E+00 124 3.88E-01 1,606.65 2.40E+007.63E+03 1.93E+00 123 3.85E-01 1,574.02 2.43E+007.96E+03 2.01E+00 122 3.82E-01 1,546.09 2.45E+008.28E+03 2.09E+00 121 3.79E-01 1,519.21 2.47E+008.65E+03 2.19E+00 120 3.76E-01 9.01E+03 2.28E+00 119 3.73E-01 9.38E+03 2.37E+00 118 3.71E-01 9.66E+03 2.44E+00 118 3.69E-01 9.94E+03 2.51E+00 117 3.67E-01 1.03E+04 2.60E+00 116 3.65E-01 1.07E+04 2.70E+00 116 3.62E-01 1.10E+04 2.79E+00 115 3.60E-01 1.14E+04 2.88E+00 114 3.58E-01 1.17E+04 2.95E+00 114 3.57E-01 1.19E+04 3.02E+00 113 3.56E-01 1.23E+04 3.11E+00 113 3.54E-01 1.27E+04 3.20E+00 112 3.52E-01 1.30E+04 3.29E+00 112 3.50E-01 1.34E+04 3.38E+00 111 3.49E-01 1.38E+04 3.48E+00 111 3.47E-01

119


t [days] tDxf q [stb/day] qD 1.40E+04 3.55E+00 110 3.46E-01 1.43E+04 3.62E+00 110 3.45E-01 1.47E+04 3.71E+00 109 3.44E-01 1.51E+04 3.80E+00 109 3.42E-01 1.54E+04 3.90E+00 109 3.41E-01 1.58E+04 3.99E+00 108 3.40E-01 1.61E+04 4.08E+00 108 3.38E-01 1.65E+04 4.17E+00 107 3.37E-01 1.68E+04 4.26E+00 107 3.36E-01 1.72E+04 4.34E+00 107 3.35E-01 1.75E+04 4.43E+00 106 3.34E-01 1.79E+04 4.53E+00 106 3.33E-01 1.83E+04 4.62E+00 106 3.32E-01 1.86E+04 4.71E+00 105 3.31E-01 1.90E+04 4.80E+00 105 3.30E-01 1.94E+04 4.90E+00 105 3.29E-01 1.97E+04 4.99E+00 104 3.28E-01 2.01E+04 5.08E+00 104 3.27E-01 2.04E+04 5.14E+00 104 3.26E-01 2.06E+04 5.21E+00 104 3.25E-01 2.10E+04 5.30E+00 103 3.25E-01 2.13E+04 5.39E+00 103 3.24E-01 2.17E+04 5.49E+00 103 3.23E-01 2.21E+04 5.58E+00 103 3.22E-01 2.24E+04 5.67E+00 102 3.21E-01 2.28E+04 5.76E+00 102 3.20E-01 2.32E+04 5.85E+00 102 3.20E-01 2.35E+04 5.95E+00 102 3.19E-01 2.39E+04 6.04E+00 101 3.18E-01 2.43E+04 6.13E+00 101 3.17E-01 2.45E+04 6.19E+00 101 3.17E-01 2.47E+04 6.25E+00 101 3.16E-01 2.51E+04 6.34E+00 101 3.16E-01 2.55E+04 6.43E+00 100 3.15E-01 2.58E+04 6.53E+00 100 3.14E-01 2.62E+04 6.62E+00 100 3.14E-01 2.66E+04 6.71E+00 100 3.13E-01 2.69E+04 6.80E+00 100 3.12E-01 2.73E+04 6.90E+00 99 3.12E-01 2.77E+04 6.99E+00 99 3.11E-01 2.80E+04 7.08E+00 99 3.11E-01 2.84E+04 7.17E+00 99 3.10E-01 2.87E+04 7.26E+00 99 3.09E-01 2.91E+04 7.36E+00 98 3.09E-01 2.94E+04 7.43E+00 98 3.08E-01

120



121


t [days] tDxf q [stb/day] qD 4.57E+04 1.15E+01 92 2.89E-01 4.60E+04 1.16E+01 92 2.89E-01 4.64E+04 1.17E+01 92 2.89E-01 4.67E+04 1.18E+01 92 2.88E-01 4.71E+04 1.19E+01 92 2.88E-01 4.75E+04 1.20E+01 92 2.88E-01 4.78E+04 1.21E+01 92 2.88E-01 4.82E+04 1.22E+01 92 2.87E-01 4.86E+04 1.23E+01 91 2.87E-01 4.89E+04 1.24E+01 91 2.87E-01 4.93E+04 1.25E+01 91 2.86E-01 4.97E+04 1.26E+01 91 2.86E-01 5.00E+04 1.26E+01 91 2.86E-01 5.04E+04 1.27E+01 91 2.85E-01 5.08E+04 1.28E+01 91 2.85E-01 5.10E+04 1.29E+01 91 2.85E-01 5.13E+04 1.30E+01 91 2.85E-01 5.16E+04 1.31E+01 91 2.85E-01 5.20E+04 1.31E+01 91 2.84E-01 5.24E+04 1.32E+01 91 2.84E-01 5.27E+04 1.33E+01 90 2.84E-01 5.31E+04 1.34E+01 90 2.83E-01 5.35E+04 1.35E+01 90 2.83E-01 5.38E+04 1.36E+01 90 2.83E-01 5.42E+04 1.37E+01 90 2.83E-01 5.46E+04 1.38E+01 90 2.82E-01 5.49E+04 1.39E+01 90 2.82E-01 5.53E+04 1.40E+01 90 2.82E-01 5.57E+04 1.41E+01 90 2.82E-01 5.60E+04 1.42E+01 90 2.81E-01 5.64E+04 1.43E+01 90 2.81E-01 5.68E+04 1.43E+01 90 2.81E-01 5.71E+04 1.44E+01 89 2.81E-01 5.75E+04 1.45E+01 89 2.80E-01 5.79E+04 1.46E+01 89 2.80E-01 5.82E+04 1.47E+01 89 2.80E-01 5.86E+04 1.48E+01 89 2.80E-01 5.89E+04 1.49E+01 89 2.79E-01 5.93E+04 1.50E+01 89 2.79E-01 5.97E+04 1.51E+01 89 2.79E-01 6.00E+04 1.52E+01 89 2.79E-01 6.04E+04 1.53E+01 89 2.79E-01 6.08E+04 1.54E+01 89 2.78E-01 6.11E+04 1.55E+01 89 2.78E-01 6.14E+04 1.55E+01 89 2.78E-01 6.16E+04 1.56E+01 89 2.78E-01

122


t [days] tDxf q [stb/day] qD p [psi] pD 0 0 0 0 5,000 n/a

1.16E-05 2.92E-09 25,920 8.13E+01 4,983 1.22E-022.55E-05 6.44E-09 14,477 4.54E+01 4,975 1.80E-024.22E-05 1.07E-08 11,973 3.76E+01 4,969 2.21E-026.22E-05 1.57E-08 10,930 3.43E+01 4,965 2.51E-028.62E-05 2.18E-08 10,405 3.26E+01 4,961 2.74E-021.15E-04 2.91E-08 10,118 3.17E+01 4,959 2.91E-021.50E-04 3.78E-08 9,952 3.12E+01 4,957 3.03E-021.91E-04 4.83E-08 9,851 3.09E+01 4,956 3.12E-022.41E-04 6.09E-08 9,781 3.07E+01 4,955 3.17E-023.01E-04 7.60E-08 9,725 3.05E+01 4,955 3.22E-023.72E-04 9.41E-08 9,672 3.03E+01 4,954 3.25E-024.58E-04 1.16E-07 9,616 3.02E+01 4,954 3.27E-025.61E-04 1.42E-07 9,552 3.00E+01 4,953 3.29E-026.85E-04 1.73E-07 9,478 2.97E+01 4,953 3.32E-028.34E-04 2.11E-07 9,391 2.95E+01 4,953 3.35E-021.01E-03 2.56E-07 9,290 2.91E+01 4,952 3.39E-021.23E-03 3.10E-07 9,172 2.88E+01 4,952 3.43E-021.48E-03 3.75E-07 9,035 2.83E+01 4,951 3.47E-021.79E-03 4.53E-07 8,877 2.79E+01 4,950 3.53E-022.16E-03 5.46E-07 8,698 2.73E+01 4,949 3.60E-022.61E-03 6.58E-07 8,495 2.67E+01 4,948 3.68E-023.14E-03 7.93E-07 8,268 2.59E+01 4,947 3.76E-023.78E-03 9.54E-07 8,017 2.52E+01 4,945 3.87E-024.54E-03 1.15E-06 7,744 2.43E+01 4,944 3.99E-025.46E-03 1.38E-06 7,451 2.34E+01 4,942 4.12E-026.56E-03 1.66E-06 7,144 2.24E+01 4,940 4.27E-027.88E-03 1.99E-06 6,828 2.14E+01 4,937 4.43E-029.47E-03 2.39E-06 6,508 2.04E+01 4,935 4.61E-021.14E-02 2.88E-06 6,192 1.94E+01 4,932 4.81E-021.37E-02 3.46E-06 5,888 1.85E+01 4,929 5.02E-021.64E-02 4.15E-06 5,600 1.76E+01 4,926 5.25E-021.97E-02 4.98E-06 5,332 1.67E+01 4,923 5.49E-022.37E-02 5.99E-06 5,084 1.60E+01 4,919 5.73E-022.84E-02 7.19E-06 4,856 1.52E+01 4,915 5.99E-023.41E-02 8.63E-06 4,643 1.46E+01 4,912 6.26E-024.10E-02 1.04E-05 4,443 1.39E+01 4,908 6.54E-024.92E-02 1.24E-05 4,252 1.33E+01 4,904 6.83E-025.90E-02 1.49E-05 4,069 1.28E+01 4,899 7.13E-027.08E-02 1.79E-05 3,891 1.22E+01 4,895 7.45E-028.50E-02 2.15E-05 3,719 1.17E+01 4,890 7.79E-021.02E-01 2.58E-05 3,555 1.12E+01 4,885 8.15E-021.22E-01 3.09E-05 3,398 1.07E+01 4,880 8.52E-021.47E-01 3.71E-05 3,250 1.02E+01 4,874 8.91E-02

123


t [days] tDxf q [stb/day] qD p [psi] pD 1.76E-01 4.46E-05 3,109 9.76E+00 4,869 9.31E-022.12E-01 5.35E-05 2,975 9.33E+00 4,863 9.73E-022.54E-01 6.42E-05 2,847 8.93E+00 4,856 1.02E-013.05E-01 7.70E-05 2,723 8.54E+00 4,850 1.06E-013.66E-01 9.24E-05 2,604 8.17E+00 4,843 1.11E-014.39E-01 1.11E-04 2,490 7.81E+00 4,836 1.16E-015.27E-01 1.33E-04 2,382 7.47E+00 4,828 1.22E-016.32E-01 1.60E-04 2,279 7.15E+00 4,821 1.27E-017.58E-01 1.91E-04 2,180 6.84E+00 4,812 1.33E-019.10E-01 2.30E-04 2,086 6.54E+00 4,805 1.38E-011.09E+00 2.76E-04 1,995 6.26E+00 4,796 1.45E-011.31E+00 3.31E-04 1,908 5.99E+00 4,787 1.51E-011.57E+00 3.97E-04 1,825 5.73E+00 4,778 1.57E-011.89E+00 4.77E-04 1,745 5.47E+00 4,768 1.65E-012.26E+00 5.72E-04 1,668 5.23E+00 4,757 1.72E-012.72E+00 6.87E-04 1,595 5.00E+00 4,746 1.80E-013.26E+00 8.24E-04 1,525 4.78E+00 4,735 1.88E-013.91E+00 9.89E-04 1,458 4.57E+00 4,723 1.96E-014.69E+00 1.19E-03 1,393 4.37E+00 4,710 2.06E-015.63E+00 1.42E-03 1,331 4.18E+00 4,696 2.15E-016.76E+00 1.71E-03 1,271 3.99E+00 4,682 2.25E-018.11E+00 2.05E-03 1,213 3.81E+00 4,667 2.36E-019.73E+00 2.46E-03 1,158 3.63E+00 4,651 2.47E-011.17E+01 2.95E-03 1,104 3.46E+00 4,635 2.58E-011.40E+01 3.54E-03 1,053 3.30E+00 4,617 2.71E-011.68E+01 4.25E-03 1,003 3.15E+00 4,599 2.84E-012.02E+01 5.10E-03 955 3.00E+00 4,579 2.98E-012.42E+01 6.12E-03 908 2.85E+00 4,559 3.13E-012.91E+01 7.35E-03 863 2.71E+00 4,537 3.28E-013.49E+01 8.82E-03 819 2.57E+00 4,513 3.45E-014.19E+01 1.06E-02 777 2.44E+00 4,488 3.63E-015.02E+01 1.27E-02 736 2.31E+00 4,462 3.81E-016.02E+01 1.52E-02 697 2.19E+00 4,433 4.01E-017.23E+01 1.83E-02 660 2.07E+00 4,403 4.23E-018.68E+01 2.19E-02 624 1.96E+00 4,371 4.46E-011.04E+02 2.63E-02 589 1.85E+00 4,336 4.70E-011.25E+02 3.16E-02 556 1.75E+00 4,299 4.96E-011.50E+02 3.79E-02 525 1.65E+00 4,260 5.24E-011.80E+02 4.55E-02 495 1.55E+00 4,218 5.54E-012.16E+02 5.46E-02 466 1.46E+00 4,173 5.86E-012.59E+02 6.55E-02 439 1.38E+00 4,124 6.20E-013.11E+02 7.86E-02 413 1.30E+00 4,073 6.57E-013.73E+02 9.43E-02 389 1.22E+00 4,018 6.96E-014.48E+02 1.13E-01 366 1.15E+00 3,959 7.37E-015.38E+02 1.36E-01 345 1.08E+00 3,897 7.81E-01

124


t [days] tDxf q [stb/day] qD p [psi] pD 6.46E+02 1.63E-01 325 1.02E+00 3,830 8.28E-017.75E+02 1.96E-01 306 9.60E-01 3,760 8.78E-019.30E+02 2.35E-01 289 9.06E-01 3,686 9.31E-011.12E+03 2.82E-01 273 8.55E-01 3,607 9.87E-011.34E+03 3.38E-01 258 8.09E-01 3,525 1.04E+001.61E+03 4.06E-01 244 7.66E-01 3,438 1.11E+001.93E+03 4.87E-01 231 7.26E-01 3,348 1.17E+002.12E+03 5.36E-01 225 7.06E-01 3,298 1.21E+002.31E+03 5.84E-01 219 6.88E-01 3,251 1.24E+002.54E+03 6.43E-01 213 6.69E-01 3,199 1.28E+002.77E+03 7.01E-01 208 6.53E-01 3,151 1.31E+003.05E+03 7.71E-01 203 6.36E-01 3,097 1.35E+003.33E+03 8.41E-01 198 6.22E-01 3,047 1.38E+003.66E+03 9.25E-01 193 6.06E-01 2,992 1.42E+003.99E+03 1.01E+00 189 5.93E-01 2,941 1.46E+004.36E+03 1.10E+00 185 5.80E-01 2,889 1.49E+004.58E+03 1.16E+00 182 5.72E-01 2,860 1.52E+004.79E+03 1.21E+00 180 5.66E-01 2,831 1.54E+005.16E+03 1.30E+00 177 5.56E-01 2,786 1.57E+005.46E+03 1.38E+00 175 5.48E-01 2,752 1.59E+005.75E+03 1.45E+00 172 5.41E-01 2,719 1.62E+006.12E+03 1.55E+00 170 5.33E-01 2,680 1.64E+006.48E+03 1.64E+00 168 5.26E-01 2,644 1.67E+006.69E+03 1.69E+00 166 5.22E-01 2,624 1.68E+006.90E+03 1.74E+00 165 5.18E-01 2,604 1.70E+007.27E+03 1.84E+00 163 5.12E-01 2,571 1.72E+007.63E+03 1.93E+00 161 5.06E-01 2,540 1.74E+007.96E+03 2.01E+00 160 5.02E-01 2,513 1.76E+008.28E+03 2.09E+00 158 4.97E-01 2,487 1.78E+008.65E+03 2.19E+00 157 4.92E-01 9.01E+03 2.28E+00 156 4.88E-01 9.38E+03 2.37E+00 154 4.84E-01 9.66E+03 2.44E+00 153 4.81E-01 9.94E+03 2.51E+00 152 4.78E-01 1.03E+04 2.60E+00 151 4.74E-01 1.07E+04 2.70E+00 150 4.70E-01 1.10E+04 2.79E+00 149 4.67E-01 1.14E+04 2.88E+00 148 4.64E-01 1.17E+04 2.95E+00 147 4.62E-01 1.19E+04 3.02E+00 146 4.59E-01 1.23E+04 3.11E+00 145 4.57E-01 1.27E+04 3.20E+00 145 4.54E-01 1.30E+04 3.29E+00 144 4.51E-01 1.34E+04 3.38E+00 143 4.49E-01 1.38E+04 3.48E+00 142 4.46E-01

125



126



127



128

Table 3 - FCD=10 – Results of Numerical Simulation and Dimensionless Time, Pressure and Rate



129

Table 3 - FCD=10 – Results of Numerical Simulation and Dimensionless Time, Pressure and Rate - continued

t [days] tDxf q [stb/day] qD p [psi] pD 1.76E-01 4.46E-05 4,427 1.39E+01 4,908 6.54E-022.12E-01 5.35E-05 4,235 1.33E+01 4,903 6.84E-022.54E-01 6.42E-05 4,052 1.27E+01 4,899 7.15E-023.05E-01 7.70E-05 3,875 1.22E+01 4,894 7.47E-023.66E-01 9.24E-05 3,706 1.16E+01 4,890 7.81E-024.39E-01 1.11E-04 3,543 1.11E+01 4,885 8.17E-025.27E-01 1.33E-04 3,387 1.06E+01 4,880 8.50E-026.32E-01 1.60E-04 3,237 1.02E+01 4,875 8.88E-027.58E-01 1.91E-04 3,095 9.71E+00 4,869 9.29E-029.10E-01 2.30E-04 2,959 9.28E+00 4,863 9.70E-021.09E+00 2.76E-04 2,827 8.87E+00 4,857 1.01E-011.31E+00 3.31E-04 2,701 8.48E+00 4,850 1.06E-011.57E+00 3.97E-04 2,579 8.09E+00 4,844 1.11E-011.89E+00 4.77E-04 2,462 7.72E+00 4,836 1.16E-012.26E+00 5.72E-04 2,348 7.37E+00 4,829 1.21E-012.72E+00 6.87E-04 2,239 7.03E+00 4,820 1.27E-013.26E+00 8.24E-04 2,134 6.70E+00 4,812 1.33E-013.91E+00 9.89E-04 2,032 6.38E+00 4,803 1.40E-014.69E+00 1.19E-03 1,934 6.07E+00 4,793 1.47E-015.63E+00 1.42E-03 1,838 5.77E+00 4,783 1.54E-016.76E+00 1.71E-03 1,745 5.48E+00 4,772 1.62E-018.11E+00 2.05E-03 1,655 5.19E+00 4,760 1.70E-019.73E+00 2.46E-03 1,568 4.92E+00 4,747 1.79E-011.17E+01 2.95E-03 1,483 4.65E+00 4,734 1.89E-011.40E+01 3.54E-03 1,401 4.40E+00 4,719 1.99E-011.68E+01 4.25E-03 1,323 4.15E+00 4,704 2.10E-012.02E+01 5.10E-03 1,246 3.91E+00 4,687 2.21E-012.42E+01 6.12E-03 1,173 3.68E+00 4,670 2.34E-012.91E+01 7.35E-03 1,102 3.46E+00 4,650 2.48E-013.49E+01 8.82E-03 1,034 3.24E+00 4,630 2.62E-014.19E+01 1.06E-02 969 3.04E+00 4,607 2.78E-015.02E+01 1.27E-02 908 2.85E+00 4,583 2.95E-016.02E+01 1.52E-02 850 2.67E+00 4,557 3.14E-017.23E+01 1.83E-02 795 2.50E+00 4,529 3.34E-018.68E+01 2.19E-02 744 2.33E+00 4,499 3.55E-011.04E+02 2.63E-02 696 2.18E+00 4,467 3.78E-011.25E+02 3.16E-02 650 2.04E+00 4,432 4.02E-011.50E+02 3.79E-02 608 1.91E+00 4,394 4.29E-011.80E+02 4.55E-02 568 1.78E+00 4,354 4.57E-012.16E+02 5.46E-02 530 1.66E+00 4,311 4.88E-012.59E+02 6.55E-02 496 1.55E+00 4,265 5.21E-013.11E+02 7.86E-02 463 1.45E+00 4,215 5.56E-013.73E+02 9.43E-02 433 1.36E+00 4,162 5.93E-014.48E+02 1.13E-01 405 1.27E+00 4,105 6.34E-015.38E+02 1.36E-01 380 1.19E+00 4,045 6.76E-01

130


t [days] tDxf q [stb/day] qD p [psi] pD 6.46E+02 1.63E-01 356 1.12E+00 3,980 7.22E-017.75E+02 1.96E-01 334 1.05E+00 3,912 7.71E-019.30E+02 2.35E-01 314 9.84E-01 3,839 8.22E-011.12E+03 2.82E-01 295 9.26E-01 3,763 8.76E-011.34E+03 3.38E-01 278 8.73E-01 3,682 9.33E-011.61E+03 4.06E-01 263 8.24E-01 3,597 9.93E-011.93E+03 4.87E-01 248 7.79E-01 3,509 1.06E+002.12E+03 5.36E-01 241 7.56E-01 3,459 1.09E+002.31E+03 5.84E-01 235 7.36E-01 3,413 1.12E+002.54E+03 6.43E-01 228 7.15E-01 3,362 1.16E+002.77E+03 7.01E-01 222 6.97E-01 3,315 1.19E+003.05E+03 7.71E-01 216 6.78E-01 3,262 1.23E+003.33E+03 8.41E-01 211 6.62E-01 3,213 1.27E+003.66E+03 9.25E-01 206 6.45E-01 3,158 1.30E+003.99E+03 1.01E+00 201 6.30E-01 3,108 1.34E+004.36E+03 1.10E+00 196 6.16E-01 3,057 1.38E+004.58E+03 1.16E+00 194 6.08E-01 3,027 1.40E+004.79E+03 1.21E+00 191 6.00E-01 2,999 1.42E+005.16E+03 1.30E+00 188 5.89E-01 2,955 1.45E+005.46E+03 1.38E+00 185 5.81E-01 2,921 1.47E+005.75E+03 1.45E+00 183 5.73E-01 2,888 1.50E+006.12E+03 1.55E+00 180 5.64E-01 2,850 1.52E+006.48E+03 1.64E+00 177 5.56E-01 2,814 1.55E+006.69E+03 1.69E+00 176 5.52E-01 2,793 1.56E+006.90E+03 1.74E+00 175 5.48E-01 2,774 1.58E+007.27E+03 1.84E+00 172 5.41E-01 2,741 1.60E+007.63E+03 1.93E+00 170 5.35E-01 2,710 1.62E+007.96E+03 2.01E+00 169 5.30E-01 2,683 1.64E+008.28E+03 2.09E+00 167 5.25E-01 2,658 1.66E+008.65E+03 2.19E+00 166 5.19E-01 9.01E+03 2.28E+00 164 5.15E-01 9.38E+03 2.37E+00 163 5.10E-01 9.66E+03 2.44E+00 161 5.06E-01 9.94E+03 2.51E+00 160 5.03E-01 1.03E+04 2.60E+00 159 4.99E-01 1.07E+04 2.70E+00 158 4.95E-01 1.10E+04 2.79E+00 157 4.92E-01 1.14E+04 2.88E+00 156 4.88E-01 1.17E+04 2.95E+00 155 4.86E-01 1.19E+04 3.02E+00 154 4.83E-01 1.23E+04 3.11E+00 153 4.80E-01 1.27E+04 3.20E+00 152 4.77E-01 1.30E+04 3.29E+00 151 4.74E-01 1.34E+04 3.38E+00 150 4.71E-01 1.38E+04 3.48E+00 149 4.69E-01

131



132



133



134

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



135

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

t [days] tDxf q [stb/day] qD p [psi] pD 1.76E-01 4.46E-05 7,033 2.21E+01 4,942 4.11E-022.12E-01 5.35E-05 6,720 2.11E+01 4,939 4.30E-022.54E-01 6.42E-05 6,419 2.01E+01 4,937 4.50E-023.05E-01 7.70E-05 6,128 1.92E+01 4,934 4.71E-023.66E-01 9.24E-05 5,844 1.83E+01 4,931 4.91E-024.39E-01 1.11E-04 5,570 1.75E+01 4,928 5.13E-025.27E-01 1.33E-04 5,305 1.66E+01 4,924 5.37E-026.32E-01 1.60E-04 5,050 1.58E+01 4,920 5.64E-027.58E-01 1.91E-04 4,804 1.51E+01 4,917 5.91E-029.10E-01 2.30E-04 4,565 1.43E+01 4,913 6.19E-021.09E+00 2.76E-04 4,333 1.36E+01 4,908 6.51E-021.31E+00 3.31E-04 4,108 1.29E+01 4,903 6.84E-021.57E+00 3.97E-04 3,888 1.22E+01 4,898 7.20E-021.89E+00 4.77E-04 3,674 1.15E+01 4,893 7.58E-022.26E+00 5.72E-04 3,466 1.09E+01 4,887 7.99E-022.72E+00 6.87E-04 3,266 1.02E+01 4,881 8.45E-023.26E+00 8.24E-04 3,073 9.64E+00 4,874 8.93E-023.91E+00 9.89E-04 2,887 9.06E+00 4,867 9.45E-024.69E+00 1.19E-03 2,708 8.50E+00 4,859 1.00E-015.63E+00 1.42E-03 2,535 7.95E+00 4,850 1.06E-016.76E+00 1.71E-03 2,368 7.43E+00 4,841 1.13E-018.11E+00 2.05E-03 2,210 6.93E+00 4,831 1.20E-019.73E+00 2.46E-03 2,059 6.46E+00 4,819 1.28E-011.17E+01 2.95E-03 1,915 6.01E+00 4,807 1.36E-011.40E+01 3.54E-03 1,780 5.59E+00 4,794 1.46E-011.68E+01 4.25E-03 1,653 5.19E+00 4,780 1.56E-012.02E+01 5.10E-03 1,533 4.81E+00 4,765 1.66E-012.42E+01 6.12E-03 1,421 4.46E+00 4,748 1.78E-012.91E+01 7.35E-03 1,315 4.13E+00 4,730 1.91E-013.49E+01 8.82E-03 1,217 3.82E+00 4,711 2.05E-014.19E+01 1.06E-02 1,126 3.53E+00 4,689 2.20E-015.02E+01 1.27E-02 1,043 3.27E+00 4,666 2.36E-016.02E+01 1.52E-02 966 3.03E+00 4,642 2.54E-017.23E+01 1.83E-02 895 2.81E+00 4,614 2.73E-018.68E+01 2.19E-02 830 2.60E+00 4,585 2.94E-011.04E+02 2.63E-02 770 2.42E+00 4,554 3.16E-011.25E+02 3.16E-02 715 2.24E+00 4,520 3.40E-011.50E+02 3.79E-02 664 2.08E+00 4,484 3.66E-011.80E+02 4.55E-02 617 1.93E+00 4,445 3.93E-012.16E+02 5.46E-02 573 1.80E+00 4,402 4.23E-012.59E+02 6.55E-02 533 1.67E+00 4,357 4.55E-013.11E+02 7.86E-02 496 1.56E+00 4,309 4.90E-013.73E+02 9.43E-02 463 1.45E+00 4,257 5.26E-014.48E+02 1.13E-01 431 1.35E+00 4,201 5.66E-015.38E+02 1.36E-01 403 1.26E+00 4,142 6.08E-01

136


t [days] tDxf q [stb/day] qD p [psi] pD 6.46E+02 1.63E-01 377 1.18E+00 4,078 6.53E-017.75E+02 1.96E-01 353 1.11E+00 4,011 7.00E-019.30E+02 2.35E-01 331 1.04E+00 3,940 7.51E-011.12E+03 2.82E-01 310 9.74E-01 3,864 8.04E-011.34E+03 3.38E-01 292 9.16E-01 3,784 8.61E-011.61E+03 4.06E-01 275 8.64E-01 3,701 9.20E-011.93E+03 4.87E-01 260 8.16E-01 3,613 9.82E-012.12E+03 5.36E-01 252 7.91E-01 3,564 1.02E+002.31E+03 5.84E-01 245 7.69E-01 3,519 1.05E+002.54E+03 6.43E-01 238 7.47E-01 3,468 1.08E+002.77E+03 7.01E-01 232 7.28E-01 3,421 1.12E+003.05E+03 7.71E-01 226 7.08E-01 3,369 1.16E+003.33E+03 8.41E-01 220 6.90E-01 3,320 1.19E+003.66E+03 9.25E-01 214 6.72E-01 3,266 1.23E+003.99E+03 1.01E+00 209 6.56E-01 3,216 1.26E+004.36E+03 1.10E+00 204 6.40E-01 3,165 1.30E+004.58E+03 1.16E+00 201 6.32E-01 3,136 1.32E+004.79E+03 1.21E+00 199 6.24E-01 3,108 1.34E+005.16E+03 1.30E+00 195 6.12E-01 3,064 1.37E+005.46E+03 1.38E+00 192 6.03E-01 3,030 1.40E+005.75E+03 1.45E+00 190 5.95E-01 2,998 1.42E+006.12E+03 1.55E+00 187 5.86E-01 2,960 1.44E+006.48E+03 1.64E+00 184 5.77E-01 2,924 1.47E+006.69E+03 1.69E+00 183 5.73E-01 2,904 1.48E+006.90E+03 1.74E+00 181 5.68E-01 2,884 1.50E+007.27E+03 1.84E+00 179 5.61E-01 2,852 1.52E+007.63E+03 1.93E+00 177 5.55E-01 2,821 1.54E+007.96E+03 2.01E+00 175 5.49E-01 2,794 1.56E+008.28E+03 2.09E+00 173 5.44E-01 2,769 1.58E+008.65E+03 2.19E+00 172 5.38E-01 9.01E+03 2.28E+00 170 5.33E-01 9.38E+03 2.37E+00 168 5.28E-01 9.66E+03 2.44E+00 167 5.25E-01 9.94E+03 2.51E+00 166 5.21E-01 1.03E+04 2.60E+00 165 5.17E-01 1.07E+04 2.70E+00 163 5.13E-01 1.10E+04 2.79E+00 162 5.09E-01 1.14E+04 2.88E+00 161 5.05E-01 1.17E+04 2.95E+00 160 5.02E-01 1.19E+04 3.02E+00 159 5.00E-01 1.23E+04 3.11E+00 158 4.97E-01 1.27E+04 3.20E+00 157 4.93E-01 1.30E+04 3.29E+00 156 4.90E-01 1.34E+04 3.38E+00 155 4.87E-01 1.38E+04 3.48E+00 154 4.85E-01

137



138



139



140




141


t [days] tDxf q [stb/day] qD p [psi] pD 1.76E-01 4.46E-05 12,823 4.02E+01 4,969 2.16E-022.12E-01 5.35E-05 12,056 3.78E+01 4,968 2.29E-022.54E-01 6.42E-05 11,316 3.55E+01 4,966 2.42E-023.05E-01 7.70E-05 10,599 3.33E+01 4,964 2.56E-023.66E-01 9.24E-05 9,907 3.11E+01 4,962 2.71E-024.39E-01 1.11E-04 9,242 2.90E+01 4,959 2.88E-025.27E-01 1.33E-04 8,605 2.70E+01 4,957 3.07E-026.32E-01 1.60E-04 8,000 2.51E+01 4,954 3.27E-027.58E-01 1.91E-04 7,429 2.33E+01 4,951 3.49E-029.10E-01 2.30E-04 6,888 2.16E+01 4,947 3.73E-021.09E+00 2.76E-04 6,377 2.00E+01 4,944 3.99E-021.31E+00 3.31E-04 5,897 1.85E+01 4,940 4.28E-021.57E+00 3.97E-04 5,444 1.71E+01 4,935 4.59E-021.89E+00 4.77E-04 5,018 1.57E+01 4,930 4.93E-022.26E+00 5.72E-04 4,621 1.45E+01 4,925 5.30E-022.72E+00 6.87E-04 4,253 1.33E+01 4,919 5.71E-023.26E+00 8.24E-04 3,914 1.23E+01 4,913 6.15E-023.91E+00 9.89E-04 3,600 1.13E+01 4,906 6.64E-024.69E+00 1.19E-03 3,309 1.04E+01 4,899 7.17E-025.63E+00 1.42E-03 3,040 9.54E+00 4,891 7.75E-026.76E+00 1.71E-03 2,790 8.75E+00 4,882 8.38E-028.11E+00 2.05E-03 2,560 8.03E+00 4,872 9.06E-029.73E+00 2.46E-03 2,350 7.37E+00 4,861 9.82E-021.17E+01 2.95E-03 2,157 6.77E+00 4,850 1.06E-011.40E+01 3.54E-03 1,981 6.21E+00 4,837 1.15E-011.68E+01 4.25E-03 1,820 5.71E+00 4,824 1.25E-012.02E+01 5.10E-03 1,672 5.25E+00 4,809 1.36E-012.42E+01 6.12E-03 1,536 4.82E+00 4,792 1.47E-012.91E+01 7.35E-03 1,411 4.43E+00 4,774 1.60E-013.49E+01 8.82E-03 1,297 4.07E+00 4,755 1.73E-014.19E+01 1.06E-02 1,194 3.74E+00 4,734 1.88E-015.02E+01 1.27E-02 1,100 3.45E+00 4,711 2.04E-016.02E+01 1.52E-02 1,016 3.19E+00 4,687 2.22E-017.23E+01 1.83E-02 938 2.94E+00 4,660 2.41E-018.68E+01 2.19E-02 868 2.72E+00 4,631 2.61E-011.04E+02 2.63E-02 803 2.52E+00 4,600 2.83E-011.25E+02 3.16E-02 744 2.34E+00 4,567 3.07E-011.50E+02 3.79E-02 690 2.16E+00 4,531 3.32E-011.80E+02 4.55E-02 640 2.01E+00 4,492 3.60E-012.16E+02 5.46E-02 594 1.86E+00 4,451 3.89E-012.59E+02 6.55E-02 551 1.73E+00 4,406 4.21E-013.11E+02 7.86E-02 513 1.61E+00 4,358 4.55E-013.73E+02 9.43E-02 477 1.50E+00 4,306 4.91E-014.48E+02 1.13E-01 445 1.39E+00 4,251 5.30E-015.38E+02 1.36E-01 415 1.30E+00 4,193 5.72E-01

142


t [days] tDxf q [stb/day] qD p [psi] pD 6.46E+02 1.63E-01 387 1.22E+00 4,130 6.16E-017.75E+02 1.96E-01 362 1.14E+00 4,063 6.64E-019.30E+02 2.35E-01 339 1.07E+00 3,992 7.14E-011.12E+03 2.82E-01 319 9.99E-01 3,917 7.67E-011.34E+03 3.38E-01 299 9.40E-01 3,838 8.23E-011.61E+03 4.06E-01 282 8.85E-01 3,755 8.82E-011.93E+03 4.87E-01 266 8.35E-01 3,668 9.43E-012.12E+03 5.36E-01 258 8.10E-01 3,620 9.78E-012.31E+03 5.84E-01 251 7.87E-01 3,574 1.01E+002.54E+03 6.43E-01 244 7.64E-01 3,524 1.05E+002.77E+03 7.01E-01 237 7.44E-01 3,477 1.08E+003.05E+03 7.71E-01 231 7.23E-01 3,425 1.12E+003.33E+03 8.41E-01 225 7.05E-01 3,377 1.15E+003.66E+03 9.25E-01 219 6.86E-01 3,323 1.19E+003.99E+03 1.01E+00 213 6.70E-01 3,273 1.22E+004.36E+03 1.10E+00 208 6.54E-01 3,222 1.26E+004.58E+03 1.16E+00 206 6.45E-01 3,193 1.28E+004.79E+03 1.21E+00 203 6.37E-01 3,166 1.30E+005.16E+03 1.30E+00 199 6.25E-01 3,122 1.33E+005.46E+03 1.38E+00 196 6.16E-01 3,088 1.35E+005.75E+03 1.45E+00 193 6.07E-01 3,055 1.38E+006.12E+03 1.55E+00 190 5.98E-01 3,018 1.40E+006.48E+03 1.64E+00 188 5.89E-01 2,982 1.43E+006.69E+03 1.69E+00 186 5.84E-01 2,962 1.44E+006.90E+03 1.74E+00 185 5.80E-01 2,942 1.46E+007.27E+03 1.84E+00 182 5.72E-01 2,910 1.48E+007.63E+03 1.93E+00 180 5.65E-01 2,879 1.50E+007.96E+03 2.01E+00 178 5.60E-01 2,852 1.52E+008.28E+03 2.09E+00 177 5.54E-01 2,827 1.54E+008.65E+03 2.19E+00 175 5.49E-01 9.01E+03 2.28E+00 173 5.43E-01 9.38E+03 2.37E+00 171 5.38E-01 9.66E+03 2.44E+00 170 5.34E-01 9.94E+03 2.51E+00 169 5.31E-01 1.03E+04 2.60E+00 168 5.26E-01 1.07E+04 2.70E+00 166 5.22E-01 1.10E+04 2.79E+00 165 5.18E-01 1.14E+04 2.88E+00 164 5.14E-01 1.17E+04 2.95E+00 163 5.12E-01 1.19E+04 3.02E+00 162 5.09E-01 1.23E+04 3.11E+00 161 5.06E-01 1.27E+04 3.20E+00 160 5.02E-01 1.30E+04 3.29E+00 159 4.99E-01 1.34E+04 3.38E+00 158 4.96E-01 1.38E+04 3.48E+00 157 4.93E-01

143



144



145



146




147


t [days] tDxf q [stb/day] qD p [psi] pD 1.76E-01 4.46E-05 16,753 5.26E+01 4,980 1.42E-022.12E-01 5.35E-05 15,374 4.82E+01 4,978 1.54E-022.54E-01 6.42E-05 14,101 4.42E+01 4,977 1.66E-023.05E-01 7.70E-05 12,921 4.05E+01 4,975 1.80E-023.66E-01 9.24E-05 11,825 3.71E+01 4,972 1.95E-024.39E-01 1.11E-04 10,816 3.39E+01 4,970 2.11E-025.27E-01 1.33E-04 9,890 3.10E+01 4,968 2.29E-026.32E-01 1.60E-04 9,048 2.84E+01 4,965 2.49E-027.58E-01 1.91E-04 8,283 2.60E+01 4,962 2.70E-029.10E-01 2.30E-04 7,581 2.38E+01 4,959 2.94E-021.09E+00 2.76E-04 6,939 2.18E+01 4,955 3.19E-021.31E+00 3.31E-04 6,349 1.99E+01 4,951 3.47E-021.57E+00 3.97E-04 5,805 1.82E+01 4,947 3.78E-021.89E+00 4.77E-04 5,305 1.66E+01 4,942 4.12E-022.26E+00 5.72E-04 4,851 1.52E+01 4,937 4.49E-022.72E+00 6.87E-04 4,437 1.39E+01 4,931 4.89E-023.26E+00 8.24E-04 4,062 1.27E+01 4,925 5.33E-023.91E+00 9.89E-04 3,720 1.17E+01 4,918 5.81E-024.69E+00 1.19E-03 3,406 1.07E+01 4,911 6.34E-025.63E+00 1.42E-03 3,118 9.78E+00 4,902 6.91E-026.76E+00 1.71E-03 2,853 8.95E+00 4,894 7.54E-028.11E+00 2.05E-03 2,612 8.20E+00 4,884 8.22E-029.73E+00 2.46E-03 2,392 7.51E+00 4,873 8.97E-021.17E+01 2.95E-03 2,192 6.88E+00 4,862 9.78E-021.40E+01 3.54E-03 2,010 6.31E+00 4,849 1.07E-011.68E+01 4.25E-03 1,845 5.79E+00 4,836 1.16E-012.02E+01 5.10E-03 1,693 5.31E+00 4,821 1.27E-012.42E+01 6.12E-03 1,554 4.88E+00 4,804 1.39E-012.91E+01 7.35E-03 1,426 4.48E+00 4,787 1.51E-013.49E+01 8.82E-03 1,311 4.11E+00 4,768 1.65E-014.19E+01 1.06E-02 1,206 3.78E+00 4,747 1.79E-015.02E+01 1.27E-02 1,111 3.49E+00 4,724 1.96E-016.02E+01 1.52E-02 1,026 3.22E+00 4,699 2.13E-017.23E+01 1.83E-02 948 2.97E+00 4,673 2.32E-018.68E+01 2.19E-02 876 2.75E+00 4,644 2.52E-011.04E+02 2.63E-02 811 2.54E+00 4,613 2.74E-011.25E+02 3.16E-02 751 2.36E+00 4,580 2.97E-011.50E+02 3.79E-02 696 2.18E+00 4,544 3.23E-011.80E+02 4.55E-02 645 2.02E+00 4,505 3.50E-012.16E+02 5.46E-02 599 1.88E+00 4,464 3.80E-012.59E+02 6.55E-02 556 1.74E+00 4,419 4.11E-013.11E+02 7.86E-02 517 1.62E+00 4,371 4.45E-013.73E+02 9.43E-02 481 1.51E+00 4,320 4.82E-014.48E+02 1.13E-01 448 1.41E+00 4,265 5.20E-015.38E+02 1.36E-01 418 1.31E+00 4,206 5.62E-01

148


t [days] tDxf q [stb/day] qD p [psi] pD 6.46E+02 1.63E-01 390 1.22E+00 4,144 6.06E-017.75E+02 1.96E-01 365 1.14E+00 4,077 6.54E-019.30E+02 2.35E-01 342 1.07E+00 4,006 7.04E-011.12E+03 2.82E-01 321 1.01E+00 3,932 7.57E-011.34E+03 3.38E-01 301 9.46E-01 3,853 8.12E-011.61E+03 4.06E-01 284 8.91E-01 3,770 8.71E-011.93E+03 4.87E-01 268 8.41E-01 3,683 9.33E-012.12E+03 5.36E-01 260 8.15E-01 3,635 9.67E-012.31E+03 5.84E-01 252 7.92E-01 3,589 9.99E-012.54E+03 6.43E-01 245 7.69E-01 3,539 1.03E+002.77E+03 7.01E-01 239 7.49E-01 3,492 1.07E+003.05E+03 7.71E-01 232 7.28E-01 3,441 1.10E+003.33E+03 8.41E-01 226 7.09E-01 3,392 1.14E+003.66E+03 9.25E-01 220 6.90E-01 3,338 1.18E+003.99E+03 1.01E+00 215 6.74E-01 3,289 1.21E+004.36E+03 1.10E+00 210 6.58E-01 3,238 1.25E+004.58E+03 1.16E+00 207 6.49E-01 3,209 1.27E+004.79E+03 1.21E+00 204 6.41E-01 3,181 1.29E+005.16E+03 1.30E+00 200 6.28E-01 3,137 1.32E+005.46E+03 1.38E+00 197 6.19E-01 3,103 1.34E+005.75E+03 1.45E+00 195 6.10E-01 3,071 1.37E+006.12E+03 1.55E+00 191 6.01E-01 3,033 1.39E+006.48E+03 1.64E+00 189 5.92E-01 2,998 1.42E+006.69E+03 1.69E+00 187 5.87E-01 2,978 1.43E+006.90E+03 1.74E+00 186 5.83E-01 2,958 1.45E+007.27E+03 1.84E+00 183 5.75E-01 2,926 1.47E+007.63E+03 1.93E+00 181 5.68E-01 2,895 1.49E+007.96E+03 2.01E+00 179 5.63E-01 2,868 1.51E+008.28E+03 2.09E+00 178 5.57E-01 2,843 1.53E+008.65E+03 2.19E+00 176 5.51E-01 9.01E+03 2.28E+00 174 5.46E-01 9.38E+03 2.37E+00 172 5.41E-01 9.66E+03 2.44E+00 171 5.37E-01 9.94E+03 2.51E+00 170 5.33E-01 1.03E+04 2.60E+00 169 5.29E-01 1.07E+04 2.70E+00 167 5.25E-01 1.10E+04 2.79E+00 166 5.21E-01 1.14E+04 2.88E+00 165 5.17E-01 1.17E+04 2.95E+00 164 5.14E-01 1.19E+04 3.02E+00 163 5.12E-01 1.23E+04 3.11E+00 162 5.08E-01 1.27E+04 3.20E+00 161 5.05E-01 1.30E+04 3.29E+00 160 5.02E-01 1.34E+04 3.38E+00 159 4.99E-01 1.38E+04 3.48E+00 158 4.96E-01

149



150



151



152

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

t [days] tDxf q [stb/day] qD p [psi] pD 0.00E+00 0.00E+00 0 0.00E+00 5,000n/a 1.16E-05 2.92E-09 64,317 2.02E+02 4,993 4.91E-03 2.55E-05 6.44E-09 48,107 1.51E+02 4,991 6.21E-03 4.22E-05 1.07E-08 41,483 1.30E+02 4,990 7.08E-03

6.22E-05 1.57E-08 38,149 1.20E+02 4,989 7.78E-03

8.62E-05 2.18E-08 36,306 1.14E+02 4,988 8.25E-03 1.15E-04 2.91E-08 35,242 1.11E+02 4,988 8.62E-03 1.50E-04 3.78E-08 34,614 1.09E+02 4,987 8.88E-03 1.91E-04 4.83E-08 34,235 1.07E+02 4,987 9.07E-03

2.41E-04 6.09E-08 33,990 1.07E+02 4,987 9.19E-03 3.01E-04 7.60E-08 33,812 1.06E+02 4,987 9.28E-03 3.72E-04 9.41E-08 33,659 1.06E+02 4,987 9.35E-03 4.58E-04 1.16E-07 33,504 1.05E+02 4,987 9.40E-03

5.61E-04 1.42E-07 33,334 1.05E+02 4,987 9.45E-03 6.85E-04 1.73E-07 33,138 1.04E+02 4,987 9.51E-03 8.34E-04 2.11E-07 32,909 1.03E+02 4,986 9.58E-03 1.01E-03 2.56E-07 32,641 1.02E+02 4,986 9.65E-03 1.23E-03 3.10E-07 32,328 1.01E+02 4,986 9.74E-03 1.48E-03 3.75E-07 31,964 1.00E+02 4,986 9.85E-03 1.79E-03 4.53E-07 31,543 9.90E+01 4,986 9.97E-03 2.16E-03 5.46E-07 31,059 9.75E+01 4,986 1.01E-02 2.61E-03 6.58E-07 30,508 9.57E+01 4,985 1.03E-02 3.14E-03 7.93E-07 29,888 9.38E+01 4,985 1.05E-02 3.78E-03 9.54E-07 29,195 9.16E+01 4,985 1.07E-02 4.54E-03 1.15E-06 28,431 8.92E+01 4,985 1.10E-02 5.46E-03 1.38E-06 27,601 8.66E+01 4,984 1.12E-02 6.56E-03 1.66E-06 26,717 8.38E+01 4,984 1.16E-02

7.88E-03 1.99E-06 25,787 8.09E+01 4,983 1.19E-02 9.47E-03 2.39E-06 24,825 7.79E+01 4,983 1.23E-02 1.14E-02 2.88E-06 23,850 7.48E+01 4,982 1.28E-02 1.37E-02 3.46E-06 22,882 7.18E+01 4,981 1.32E-02 1.64E-02 4.15E-06 21,937 6.88E+01 4,981 1.37E-02 1.97E-02 4.98E-06 21,026 6.60E+01 4,980 1.43E-02 2.37E-02 5.99E-06 20,154 6.32E+01 4,979 1.48E-02 2.84E-02 7.19E-06 19,322 6.06E+01 4,978 1.54E-02 3.41E-02 8.63E-06 18,523 5.81E+01 4,977 1.60E-02 4.10E-02 1.04E-05 17,747 5.57E+01 4,976 1.67E-02 4.92E-02 1.24E-05 16,988 5.33E+01 4,975 1.74E-02 5.90E-02 1.49E-05 16,238 5.10E+01 4,974 1.81E-02 7.08E-02 1.79E-05 15,495 4.86E+01 4,973 1.89E-02 8.50E-02 2.15E-05 14,759 4.63E+01 4,972 1.97E-02

153

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

t [days] tDxf q [stb/day] qD p [psi] pD 1.02E-01 2.58E-05 14,035 4.40E+01 4,971 2.06E-02 1.22E-01 3.09E-05 13,326 4.18E+01 4,969 2.16E-02 1.47E-01 3.71E-05 12,633 3.96E+01 4,968 2.27E-02 1.76E-01 4.46E-05 11,959 3.75E+01 4,966 2.38E-02 2.12E-01 5.35E-05 11,301 3.55E+01 4,965 2.51E-02 2.54E-01 6.42E-05 10,660 3.35E+01 4,963 2.64E-02 3.05E-01 7.70E-05 10,034 3.15E+01 4,961 2.78E-02 3.66E-01 9.24E-05 9,423 2.96E+01 4,959 2.94E-02 4.39E-01 1.11E-04 8,830 2.77E+01 4,956 3.11E-02 5.27E-01 1.33E-04 8,257 2.59E+01 4,954 3.29E-02 6.32E-01 1.60E-04 7,709 2.42E+01 4,951 3.49E-02 7.58E-01 1.91E-04 7,186 2.25E+01 4,948 3.71E-02 9.10E-01 2.30E-04 6,685 2.10E+01 4,944 3.95E-02 1.09E+00 2.76E-04 6,210 1.95E+01 4,940 4.21E-02 1.31E+00 3.31E-04 5,759 1.81E+01 4,936 4.50E-02 1.57E+00 3.97E-04 5,332 1.67E+01 4,932 4.81E-02 1.89E+00 4.77E-04 4,927 1.55E+01 4,927 5.15E-02 2.26E+00 5.72E-04 4,548 1.43E+01 4,922 5.52E-02 2.72E+00 6.87E-04 4,194 1.32E+01 4,916 5.93E-02 3.26E+00 8.24E-04 3,866 1.21E+01 4,910 6.38E-02 3.91E+00 9.89E-04 3,561 1.12E+01 4,903 6.86E-02 4.69E+00 1.19E-03 3,278 1.03E+01 4,896 7.39E-02 5.63E+00 1.42E-03 3,015 9.46E+00 4,888 7.97E-02 6.76E+00 1.71E-03 2,770 8.69E+00 4,879 8.60E-02 8.11E+00 2.05E-03 2,545 7.98E+00 4,869 9.29E-02 9.73E+00 2.46E-03 2,337 7.33E+00 4,858 1.00E-01 1.17E+01 2.95E-03 2,147 6.74E+00 4,847 1.09E-01 1.40E+01 3.54E-03 1,972 6.19E+00 4,834 1.17E-01 1.68E+01 4.25E-03 1,813 5.69E+00 4,820 1.27E-01 2.02E+01 5.10E-03 1,666 5.23E+00 4,805 1.38E-01 2.42E+01 6.12E-03 1,531 4.80E+00 4,789 1.49E-01 2.91E+01 7.35E-03 1,407 4.41E+00 4,771 1.62E-01 3.49E+01 8.82E-03 1,294 4.06E+00 4,752 1.76E-01 4.19E+01 1.06E-02 1,191 3.74E+00 4,731 1.90E-01 5.02E+01 1.27E-02 1,097 3.44E+00 4,708 2.07E-01 6.02E+01 1.52E-02 1,013 3.18E+00 4,684 2.24E-01 7.23E+01 1.83E-02 936 2.94E+00 4,657 2.43E-01 8.68E+01 2.19E-02 866 2.72E+00 4,628 2.63E-01 1.04E+02 2.63E-02 802 2.52E+00 4,597 2.85E-01 1.25E+02 3.16E-02 743 2.33E+00 4,564 3.09E-01 1.50E+02 3.79E-02 688 2.16E+00 4,528 3.34E-01 1.80E+02 4.55E-02 638 2.00E+00 4,489 3.62E-01 2.16E+02 5.46E-02 593 1.86E+00 4,448 3.91E-01

154


t [days] tDxf q [stb/day] qD p [psi] pD 2.59E+02 6.55E-02 550 1.73E+00 4,403 4.23E-01 3.11E+02 7.86E-02 512 1.61E+00 4,355 4.57E-01 3.73E+02 9.43E-02 476 1.49E+00 4,303 4.93E-01 4.48E+02 1.13E-01 444 1.39E+00 4,248 5.32E-01 5.38E+02 1.36E-01 414 1.30E+00 4,189 5.74E-01 6.46E+02 1.63E-01 387 1.21E+00 4,126 6.19E-01 7.75E+02 1.96E-01 362 1.14E+00 4,060 6.66E-01 9.30E+02 2.35E-01 339 1.06E+00 3,989 7.16E-01 1.12E+03 2.82E-01 318 9.98E-01 3,914 7.69E-01 1.34E+03 3.38E-01 299 9.38E-01 3,835 8.25E-01 1.61E+03 4.06E-01 282 8.84E-01 3,752 8.84E-01 1.93E+03 4.87E-01 266 8.34E-01 3,665 9.46E-01 2.12E+03 5.36E-01 258 8.09E-01 3,616 9.80E-01 2.31E+03 5.84E-01 251 7.86E-01 3,571 1.01E+00 2.54E+03 6.43E-01 243 7.63E-01 3,521 1.05E+00 2.77E+03 7.01E-01 237 7.43E-01 3,474 1.08E+00 3.05E+03 7.71E-01 230 7.23E-01 3,422 1.12E+00 3.33E+03 8.41E-01 224 7.04E-01 3,374 1.15E+00 3.66E+03 9.25E-01 218 6.86E-01 3,320 1.19E+00 3.99E+03 1.01E+00 213 6.69E-01 3,270 1.23E+00 4.36E+03 1.10E+00 208 6.53E-01 3,219 1.26E+00 4.58E+03 1.16E+00 205 6.44E-01 3,190 1.28E+00 4.79E+03 1.21E+00 203 6.36E-01 3,162 1.30E+00 5.16E+03 1.30E+00 199 6.24E-01 3,119 1.33E+00 5.46E+03 1.38E+00 196 6.15E-01 3,085 1.36E+00 5.75E+03 1.45E+00 193 6.06E-01 3,052 1.38E+00 6.12E+03 1.55E+00 190 5.97E-01 3,014 1.41E+00 6.48E+03 1.64E+00 187 5.88E-01 2,979 1.43E+00 6.69E+03 1.69E+00 186 5.84E-01 2,959 1.45E+00 6.90E+03 1.74E+00 185 5.79E-01 2,939 1.46E+00 7.27E+03 1.84E+00 182 5.72E-01 2,907 1.48E+00 7.63E+03 1.93E+00 180 5.65E-01 2,876 1.50E+00 7.96E+03 2.01E+00 178 5.59E-01 2,849 1.52E+00 8.28E+03 2.09E+00 176 5.54E-01 2,824 1.54E+00 8.65E+03 2.19E+00 175 5.48E-01 9.01E+03 2.28E+00 173 5.43E-01 9.38E+03 2.37E+00 171 5.38E-01 9.66E+03 2.44E+00 170 5.34E-01 9.94E+03 2.51E+00 169 5.30E-01 1.03E+04 2.60E+00 168 5.26E-01 1.07E+04 2.70E+00 166 5.22E-01 1.10E+04 2.79E+00 165 5.18E-01 1.14E+04 2.88E+00 164 5.14E-01

155


t [days] tDxf q [stb/day] qD 1.17E+04 2.95E+00 163 5.11E-01 1.19E+04 3.02E+00 162 5.09E-01 1.23E+04 3.11E+00 161 5.05E-01 1.27E+04 3.20E+00 160 5.02E-01 1.30E+04 3.29E+00 159 4.99E-01 1.34E+04 3.38E+00 158 4.96E-01 1.38E+04 3.48E+00 157 4.93E-01 1.40E+04 3.55E+00 156 4.91E-01 1.43E+04 3.62E+00 156 4.88E-01 1.47E+04 3.71E+00 155 4.86E-01 1.51E+04 3.80E+00 154 4.83E-01 1.54E+04 3.90E+00 153 4.81E-01 1.58E+04 3.99E+00 152 4.78E-01 1.61E+04 4.08E+00 152 4.76E-01 1.65E+04 4.17E+00 151 4.74E-01 1.68E+04 4.26E+00 150 4.72E-01 1.72E+04 4.34E+00 150 4.70E-01 1.75E+04 4.43E+00 149 4.68E-01 1.79E+04 4.53E+00 148 4.66E-01 1.83E+04 4.62E+00 148 4.64E-01 1.86E+04 4.71E+00 147 4.62E-01 1.90E+04 4.80E+00 147 4.60E-01 1.94E+04 4.90E+00 146 4.58E-01 1.97E+04 4.99E+00 145 4.56E-01 2.01E+04 5.08E+00 145 4.55E-01 2.04E+04 5.14E+00 145 4.54E-01 2.06E+04 5.21E+00 144 4.52E-01 2.10E+04 5.30E+00 144 4.51E-01 2.13E+04 5.39E+00 143 4.49E-01 2.17E+04 5.49E+00 143 4.48E-01 2.21E+04 5.58E+00 142 4.46E-01 2.24E+04 5.67E+00 142 4.45E-01 2.28E+04 5.76E+00 141 4.43E-01 2.32E+04 5.85E+00 141 4.42E-01 2.35E+04 5.95E+00 140 4.40E-01 2.39E+04 6.04E+00 140 4.39E-01 2.43E+04 6.13E+00 140 4.38E-01 2.45E+04 6.19E+00 139 4.37E-01 2.47E+04 6.25E+00 139 4.36E-01 2.51E+04 6.34E+00 139 4.35E-01 2.55E+04 6.43E+00 138 4.34E-01 2.58E+04 6.53E+00 138 4.32E-01 2.62E+04 6.62E+00 137 4.31E-01

156



157



158


t [days] tDxf q [stb/day] qD 5.71E+04 1.44E+01 119 3.74E-01 5.75E+04 1.45E+01 119 3.73E-01 5.79E+04 1.46E+01 119 3.73E-01 5.82E+04 1.47E+01 119 3.72E-01 5.86E+04 1.48E+01 119 3.72E-01 5.89E+04 1.49E+01 118 3.72E-01 5.93E+04 1.50E+01 118 3.71E-01 5.97E+04 1.51E+01 118 3.71E-01

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] p [psi] tDxf pD t [days] p [psi] tDxf pD 0 5,000.00 n/a n/a 1.09E+00 4,909.81 2.76E-04 6.39E-02

1.03E-04 4,992.24 2.60E-08 5.49E-032.27E-04 4,990.58 5.74E-08 6.67E-03 1.31E+00 4,901.91 3.31E-04 6.95E-023.76E-04 4,990.04 9.50E-08 7.05E-03 1.57E+00 4,893.26 3.97E-04 7.56E-025.54E-04 4,989.81 1.40E-07 7.22E-03 1.89E+00 4,883.73 4.77E-04 8.23E-027.68E-04 4,989.65 1.94E-07 7.33E-03 2.26E+00 4,873.37 5.72E-04 8.97E-021.03E-03 4,989.48 2.59E-07 7.45E-03 2.72E+00 4,862.00 6.86E-04 9.77E-021.33E-03 4,989.29 3.37E-07 7.58E-03 3.26E+00 4,849.61 8.24E-04 1.07E-011.70E-03 4,989.08 4.30E-07 7.74E-03 3.91E+00 4,836.05 9.88E-04 1.16E-012.15E-03 4,988.82 5.43E-07 7.92E-03 4.69E+00 4,821.26 1.19E-03 1.27E-012.68E-03 4,988.52 6.77E-07 8.13E-03 5.63E+00 4,805.09 1.42E-03 1.38E-013.32E-03 4,988.17 8.39E-07 8.38E-03 6.76E+00 4,787.37 1.71E-03 1.51E-014.09E-03 4,987.76 1.03E-06 8.67E-03 8.11E+00 4,768.10 2.05E-03 1.64E-015.01E-03 4,987.30 1.26E-06 9.00E-03 9.73E+00 4,747.04 2.46E-03 1.79E-016.11E-03 4,986.76 1.54E-06 9.38E-03 1.17E+01 4,724.00 2.95E-03 1.95E-017.43E-03 4,986.15 1.88E-06 9.81E-03 1.40E+01 4,698.87 3.54E-03 2.13E-019.02E-03 4,985.47 2.28E-06 1.03E-02 1.68E+01 4,671.57 4.25E-03 2.33E-011.09E-02 4,984.69 2.76E-06 1.08E-02 2.02E+01 4,641.74 5.10E-03 2.54E-011.32E-02 4,983.83 3.34E-06 1.15E-02 2.42E+01 4,609.29 6.12E-03 2.77E-011.60E-02 4,982.88 4.03E-06 1.21E-02 2.91E+01 4,573.92 7.35E-03 3.02E-011.93E-02 4,981.83 4.87E-06 1.29E-02 3.49E+01 4,535.50 8.82E-03 3.29E-012.32E-02 4,980.69 5.87E-06 1.37E-02 4.19E+01 4,493.69 1.06E-02 3.59E-012.80E-02 4,979.46 7.07E-06 1.45E-02 5.02E+01 4,448.38 1.27E-02 3.91E-013.37E-02 4,978.12 8.51E-06 1.55E-02 6.02E+01 4,399.52 1.52E-02 4.25E-014.05E-02 4,976.67 1.02E-05 1.65E-02 7.23E+01 4,346.28 1.83E-02 4.63E-014.87E-02 4,975.10 1.23E-05 1.76E-02 8.68E+01 4,288.90 2.19E-02 5.04E-015.86E-02 4,973.41 1.48E-05 1.88E-02 1.04E+02 4,227.00 2.63E-02 5.47E-017.04E-02 4,971.56 1.78E-05 2.01E-02 1.25E+02 4,160.51 3.16E-02 5.95E-018.46E-02 4,969.55 2.14E-05 2.16E-02 1.50E+02 4,088.75 3.79E-02 6.45E-011.02E-01 4,967.35 2.57E-05 2.31E-02 1.80E+02 4,011.49 4.55E-02 7.00E-011.22E-01 4,964.94 3.08E-05 2.48E-02 2.16E+02 3,928.38 5.46E-02 7.59E-011.46E-01 4,962.30 3.70E-05 2.67E-02 2.59E+02 3,839.10 6.55E-02 8.22E-011.76E-01 4,959.40 4.44E-05 2.88E-02 3.11E+02 3,743.36 7.86E-02 8.90E-012.11E-01 4,956.23 5.34E-05 3.10E-02 3.73E+02 3,640.69 9.43E-02 9.63E-012.53E-01 4,952.77 6.41E-05 3.34E-02 4.48E+02 3,530.76 1.13E-01 1.04E+003.04E-01 4,948.98 7.69E-05 3.61E-02 5.38E+02 3,413.38 1.36E-01 1.12E+003.65E-01 4,944.80 9.23E-05 3.91E-02 6.46E+02 3,287.72 1.63E-01 1.21E+004.38E-01 4,940.22 1.11E-04 4.23E-02 7.75E+02 3,154.61 1.96E-01 1.31E+005.26E-01 4,935.19 1.33E-04 4.59E-02 9.30E+023,013.14 2.35E-01 1.41E+006.31E-01 4,929.68 1.60E-04 4.98E-02 1.12E+03 2,863.37 2.82E-01 1.51E+007.57E-01 4,923.68 1.91E-04 5.41E-02 1.34E+03 2,705.39 3.38E-01 1.63E+009.09E-01 4,917.04 2.30E-04 5.88E-02

1.61E+03 2,539.42 4.06E-01 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] p [psi] tDxf pD 1.93E+03 2,364.91 4.87E-01 1.87E+002.12E+03 2,267.88 5.36E-01 1.93E+002.31E+03 2,177.39 5.84E-01 2.00E+002.54E+03 2,076.76 6.43E-01 2.07E+002.77E+03 1,982.99 7.01E-01 2.14E+003.05E+03 1,878.72 7.71E-01 2.21E+003.33E+03 1,781.67 8.41E-01 2.28E+003.66E+03 1,674.00 9.25E-01 2.36E+003.99E+03 1,573.90 1.01E+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] tDxf qD t [days] q [stb/day] tDxf qD 1.03E-04 57,731.64 2.60E-08 181.149 1.31E+00 3,381.58 3.31E-04 10.611 2.27E-04 45,780.68 5.74E-08 143.650 1.57E+00 3,060.39 3.97E-04 9.603 3.76E-04 44,300.47 9.50E-08 139.005 1.89E+00 2,774.68 4.77E-04 8.706 5.54E-04 43,639.16 1.40E-07 136.930 2.26E+00 2,519.97 5.72E-04 7.907 7.68E-04 43,039.15 1.94E-07 135.047 2.72E+00 2,292.29 6.86E-04 7.193 1.03E-03 42,374.53 2.59E-07 132.962 3.26E+00 2,088.78 8.24E-04 6.554 1.33E-03 41,616.37 3.37E-07 130.583 3.91E+00 1,905.76 9.88E-04 5.980 1.70E-03 40,751.01 4.30E-07 127.868 4.69E+00 1,740.14 1.19E-03 5.460 2.15E-03 39,772.57 5.43E-07 124.797 5.63E+00 1,589.66 1.42E-03 4.988 2.68E-03 38,679.54 6.77E-07 121.368 6.76E+00 1,452.11 1.71E-03 4.556 3.32E-03 37,470.74 8.39E-07 117.575 8.11E+00 1,327.37 2.05E-03 4.165 4.09E-03 36,153.84 1.03E-06 113.443 9.73E+00 1,214.18 2.46E-03 3.810 5.01E-03 34,743.22 1.26E-06 109.016 1.17E+01 1,111.43 2.95E-03 3.487 6.11E-03 33,263.45 1.54E-06 104.373 1.40E+01 1,018.41 3.54E-03 3.196 7.43E-03 31,736.06 1.88E-06 99.581 1.68E+01 934.05 4.25E-03 2.931 9.02E-03 30,187.47 2.28E-06 94.722 2.02E+01 856.77 5.10E-03 2.688 1.09E-02 28,655.41 2.76E-06 89.914 2.42E+01 786.07 6.12E-03 2.467 1.32E-02 27,173.31 3.34E-06 85.264 2.91E+01 721.38 7.35E-03 2.264 1.60E-02 25,762.59 4.03E-06 80.837 3.49E+01 662.60 8.82E-03 2.079 1.93E-02 24,437.02 4.87E-06 76.678 4.19E+01 609.34 1.06E-02 1.912 2.32E-02 23,200.53 5.87E-06 72.798 5.02E+01 561.33 1.27E-02 1.761 2.80E-02 22,034.77 7.07E-06 69.140 6.02E+01 518.11 1.52E-02 1.626 3.37E-02 20,919.01 8.51E-06 65.639 7.23E+01 478.61 1.83E-02 1.502 4.05E-02 19,829.81 1.02E-05 62.222 8.68E+01 442.62 2.19E-02 1.389 4.87E-02 18,746.77 1.23E-05 58.823 1.04E+02 409.61 2.63E-02 1.285 5.86E-02 17,653.63 1.48E-05 55.393 1.25E+02 379.35 3.16E-02 1.190 7.04E-02 16,544.00 1.78E-05 51.911 1.50E+02 351.48 3.79E-02 1.103 8.46E-02 15,415.49 2.14E-05 48.370 1.80E+02 325.87 4.55E-02 1.023 1.02E-01 14,281.31 2.57E-05 44.812 2.16E+02 302.37 5.46E-02 0.949 1.22E-01 13,151.09 3.08E-05 41.265 2.59E+02 280.82 6.55E-02 0.881 1.46E-01 12,038.30 3.70E-05 37.774 3.11E+02 261.06 7.86E-02 0.819 1.76E-01 10,958.50 4.44E-05 34.385 3.73E+02 242.92 9.43E-02 0.762 2.11E-01 9,924.63 5.34E-05 31.141 4.48E+02 226.29 1.13E-01 0.710 2.53E-01 8,949.95 6.41E-05 28.083 5.38E+02 211.05 1.36E-01 0.662 3.04E-01 8,041.69 7.69E-05 25.233 6.46E+02 197.05 1.63E-01 0.618 3.65E-01 7,205.55 9.23E-05 22.609 7.75E+02 184.30 1.96E-01 0.578 4.38E-01 6,445.61 1.11E-04 20.225 9.30E+02 172.62 2.35E-01 0.542 5.26E-01 5,762.60 1.33E-04 18.082 1.12E+03 161.96 2.82E-01 0.508 6.31E-01 5,158.12 1.60E-04 16.185 1.34E+03 152.23 3.38E-01 0.478 7.57E-01 4,625.07 1.91E-04 14.512 1.61E+03 143.35 4.06E-01 0.450 9.09E-01 4,155.15 2.30E-04 13.038 1.93E+03 135.22 4.87E-01 0.424 1.09E+00 3,743.63 2.76E-04 11.747

2.12E+03 131.05 5.36E-01 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 qD

2.31E+03 127.42 5.84E-01 0.400

2.54E+03 123.68 6.43E-01 0.388

2.77E+03 120.42 7.01E-01 0.378

3.05E+03 117.04 7.71E-01 0.367

3.33E+03 114.08 8.41E-01 0.358

3.66E+03 111.01 9.25E-01 0.348

3.99E+03 108.32 1.01E+00 0.340

164

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

t

[months] q[Mcf/m] tDxf

qD t

[months] q[Mcf/m] tDxf

qD 1 45,816 2.70E-06 3.82E+01 29 30,127 7.83E-05 2.51E+01 2 95,857 5.40E-06 7.99E+01 30 29,846 8.10E-05 2.49E+01 3 76,889 8.10E-06 6.41E+01 31 28,822 8.37E-05 2.40E+01 4 78,556 1.08E-05 6.55E+01 32 29,362 8.64E-05 2.45E+01 5 69,757 1.35E-05 5.81E+01 33 26,080 8.91E-05 2.17E+01 6 66,602 1.62E-05 5.55E+01 34 29,420 9.18E-05 2.45E+01 7 59,948 1.89E-05 5.00E+01 35 28,784 9.45E-05 2.40E+01 8 51,841 2.16E-05 4.32E+01 36 27,173 9.72E-05 2.26E+01 9 53,079 2.43E-05 4.42E+01 37 29,166 9.99E-05 2.43E+01

10 47,095 2.70E-05 3.92E+01 38 28,066 1.03E-04 2.34E+01 11 49,919 2.97E-05 4.16E+01 39 23,200 1.05E-04 1.93E+01 12 44,319 3.24E-05 3.69E+01 40 26,474 1.08E-04 2.21E+01 13 44,979 3.51E-05 3.75E+01 41 25,212 1.11E-04 2.10E+01 14 43,737 3.78E-05 3.64E+01 42 26,565 1.13E-04 2.21E+01 15 38,344 4.05E-05 3.20E+01 43 24,300 1.16E-04 2.03E+01 16 41,482 4.32E-05 3.46E+01 44 24,601 1.19E-04 2.05E+01 17 39,781 4.59E-05 3.32E+01 45 23,552 1.22E-04 1.96E+01 18 38,887 4.86E-05 3.24E+01 46 23,236 1.24E-04 1.94E+01 19 37,787 5.13E-05 3.15E+01 47 23,874 1.27E-04 1.99E+01 20 39,142 5.40E-05 3.26E+01 48 22,521 1.30E-04 1.88E+01 21 36,193 5.67E-05 3.02E+01 49 21,305 1.32E-04 1.78E+01 22 34,558 5.94E-05 2.88E+01 50 22,362 1.35E-04 1.86E+01 23 34,100 6.21E-05 2.84E+01 51 21,997 1.38E-04 1.83E+01 24 32,084 6.48E-05 2.67E+01 52 22,933 1.40E-04 1.91E+01 25 32,300 6.75E-05 2.69E+01 53 22,379 1.43E-04 1.86E+01 26 32,002 7.02E-05 2.67E+01 54 20,673 1.46E-04 1.72E+01 27 28,602 7.29E-05 2.38E+01 55 26,054 1.49E-04 2.17E+01 28 31,073 7.56E-05 2.59E+01

165

Table 11 – FCD=100 – Results of Numerical Simulation and Dimensionless Time, Pressure and Rate for xf/y=255, Point Source

t [days] tDxf q [stb/day] qD p [psi] pD 0.00E+00 n/a 0 n/a 5,000 n/a1.03E-04 2.60E-08 41,141 1.29E+02 4,989 7.69E-032.27E-04 5.74E-08 35,004 1.10E+02 4,987 8.88E-033.76E-04 9.50E-08 33,913 1.06E+02 4,987 9.26E-035.54E-04 1.40E-07 33,474 1.05E+02 4,987 9.43E-037.68E-04 1.94E-07 33,108 1.04E+02 4,987 9.54E-031.03E-03 2.59E-07 32,708 1.03E+02 4,986 9.66E-031.33E-03 3.37E-07 32,249 1.01E+02 4,986 9.79E-031.70E-03 4.30E-07 31,717 9.95E+01 4,986 9.95E-032.15E-03 5.43E-07 31,107 9.76E+01 4,986 1.01E-022.68E-03 6.77E-07 30,412 9.54E+01 4,985 1.03E-023.32E-03 8.39E-07 29,627 9.30E+01 4,985 1.06E-024.09E-03 1.03E-06 28,749 9.02E+01 4,985 1.09E-025.01E-03 1.26E-06 27,780 8.72E+01 4,984 1.12E-026.11E-03 1.54E-06 26,727 8.39E+01 4,984 1.16E-027.43E-03 1.88E-06 25,597 8.03E+01 4,983 1.20E-029.02E-03 2.28E-06 24,400 7.66E+01 4,982 1.25E-021.09E-02 2.76E-06 23,157 7.27E+01 4,982 1.30E-021.32E-02 3.34E-06 21,892 6.87E+01 4,981 1.37E-021.60E-02 4.03E-06 20,627 6.47E+01 4,980 1.43E-021.93E-02 4.87E-06 19,383 6.08E+01 4,979 1.51E-022.32E-02 5.87E-06 18,181 5.70E+01 4,978 1.59E-022.80E-02 7.07E-06 17,032 5.34E+01 4,976 1.68E-023.37E-02 8.51E-06 15,942 5.00E+01 4,975 1.77E-024.05E-02 1.02E-05 14,911 4.68E+01 4,974 1.87E-024.87E-02 1.23E-05 13,935 4.37E+01 4,972 1.98E-025.86E-02 1.48E-05 13,007 4.08E+01 4,970 2.10E-027.04E-02 1.78E-05 12,121 3.80E+01 4,968 2.24E-028.46E-02 2.14E-05 11,271 3.54E+01 4,966 2.38E-021.02E-01 2.57E-05 10,458 3.28E+01 4,964 2.53E-021.22E-01 3.08E-05 9,684 3.04E+01 4,962 2.71E-021.46E-01 3.70E-05 8,948 2.81E+01 4,959 2.89E-021.76E-01 4.44E-05 8,254 2.59E+01 4,956 3.10E-022.11E-01 5.34E-05 7,602 2.39E+01 4,953 3.32E-022.53E-01 6.41E-05 6,993 2.19E+01 4,950 3.57E-023.04E-01 7.69E-05 6,423 2.02E+01 4,946 3.83E-023.65E-01 9.23E-05 5,892 1.85E+01 4,942 4.13E-024.38E-01 1.11E-04 5,400 1.69E+01 4,937 4.45E-025.26E-01 1.33E-04 4,945 1.55E+01 4,932 4.81E-026.31E-01 1.60E-04 4,528 1.42E+01 4,927 5.20E-027.57E-01 1.91E-04 4,146 1.30E+01 4,921 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] tDxf q [stb/day] qD p [psi] pD 9.09E-01 2.30E-04 3,794 1.19E+01 4,914 6.10E-021.09E+00 2.76E-04 3,474 1.09E+01 4,907 6.61E-021.31E+00 3.31E-04 3,179 9.98E+00 4,899 7.17E-021.57E+00 3.97E-04 2,908 9.13E+00 4,890 7.78E-021.89E+00 4.77E-04 2,659 8.34E+00 4,881 8.46E-022.26E+00 5.72E-04 2,432 7.63E+00 4,870 9.19E-022.72E+00 6.86E-04 2,225 6.98E+00 4,859 9.99E-023.26E+00 8.24E-04 2,036 6.39E+00 4,846 1.09E-013.91E+00 9.88E-04 1,865 5.85E+00 4,833 1.18E-014.69E+00 1.19E-03 1,708 5.36E+00 4,818 1.29E-015.63E+00 1.42E-03 1,564 4.91E+00 4,802 1.40E-016.76E+00 1.71E-03 1,431 4.49E+00 4,784 1.53E-018.11E+00 2.05E-03 1,310 4.11E+00 4,765 1.66E-019.73E+00 2.46E-03 1,200 3.77E+00 4,744 1.81E-011.17E+01 2.95E-03 1,100 3.45E+00 4,721 1.98E-011.40E+01 3.54E-03 1,008 3.16E+00 4,696 2.15E-011.68E+01 4.25E-03 925 2.90E+00 4,668 2.35E-012.02E+01 5.10E-03 849 2.67E+00 4,639 2.56E-012.42E+01 6.12E-03 780 2.45E+00 4,606 2.79E-012.91E+01 7.35E-03 716 2.25E+00 4,571 3.04E-013.49E+01 8.82E-03 658 2.06E+00 4,532 3.31E-014.19E+01 1.06E-02 605 1.90E+00 4,491 3.61E-015.02E+01 1.27E-02 557 1.75E+00 4,445 3.93E-016.02E+01 1.52E-02 514 1.61E+00 4,396 4.27E-017.23E+01 1.83E-02 475 1.49E+00 4,343 4.65E-018.68E+01 2.19E-02 440 1.38E+00 4,286 5.06E-011.04E+02 2.63E-02 407 1.28E+00 4,224 5.50E-011.25E+02 3.16E-02 377 1.18E+00 4,157 5.97E-011.50E+02 3.79E-02 349 1.10E+00 4,086 6.48E-011.80E+02 4.55E-02 324 1.02E+00 4,008 7.02E-012.16E+02 5.46E-02 300 9.43E-01 3,925 7.61E-012.59E+02 6.55E-02 279 8.75E-01 3,836 8.24E-013.11E+02 7.86E-02 259 8.14E-01 3,740 8.92E-013.73E+02 9.43E-02 241 7.57E-01 3,638 9.65E-014.48E+02 1.13E-01 225 7.06E-01 3,528 1.04E+005.38E+02 1.36E-01 210 6.58E-01 3,410 1.13E+006.46E+02 1.63E-01 196 6.14E-01 3,284 1.21E+007.75E+02 1.96E-01 183 5.75E-01 3,151 1.31E+009.30E+02 2.35E-01 172 5.38E-01 3,010 1.41E+001.12E+03 2.82E-01 161 5.05E-01 2,860 1.52E+001.34E+03 3.38E-01 151 4.75E-01 2,702 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] tDxf q [stb/day] qD p [psi] pD 1.61E+03 4.06E-01 143 4.47E-01 2,536 1.74E+001.93E+03 4.87E-01 134 4.22E-01 2,362 1.87E+002.12E+03 5.36E-01 130 4.09E-01 2,265 1.94E+002.31E+03 5.84E-01 127 3.98E-01 2,174 2.00E+002.54E+03 6.43E-01 123 3.86E-01 2,074 2.07E+002.77E+03 7.01E-01 120 3.76E-01 1,980 2.14E+003.05E+03 7.71E-01 116 3.65E-01 1,876 2.21E+003.33E+03 8.41E-01 113 3.56E-01 1,778 2.28E+003.66E+03 9.25E-01 110 3.46E-01 1,671 2.36E+003.99E+03 1.01E+00 108 3.38E-01 1,571 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] p [psi] tDxf pD t [days] p [psi] tDxf pD 0 5,000.00 n/a n/a 2.90E+01 3,476.51 1.19E-01 1.08E+00

1.18E-02 4,957.43 4.86E-05 3.02E-02 3.48E+01 3,352.52 1.43E-01 1.17E+002.60E-02 4,941.38 1.07E-04 4.15E-02 4.18E+01 3,220.49 1.72E-01 1.26E+004.30E-02 4,926.60 1.77E-04 5.20E-02 5.01E+01 3,080.39 2.06E-01 1.36E+006.34E-02 4,912.28 2.61E-04 6.21E-02 6.01E+01 2,932.67 2.48E-01 1.46E+008.79E-02 4,897.89 3.62E-04 7.23E-02 7.21E+01 2,776.96 2.97E-01 1.57E+001.17E-01 4,883.12 4.83E-04 8.28E-02 8.66E+01 2,612.36 3.57E-01 1.69E+001.53E-01 4,867.75 6.28E-04 9.37E-02 1.04E+02 2,440.07 4.29E-01 1.81E+001.95E-01 4,851.55 8.03E-04 1.05E-01 1.25E+02 2,260.05 5.15E-01 1.94E+002.46E-01 4,834.40 1.01E-03 1.17E-01 1.50E+02 2,072.51 6.18E-01 2.07E+003.06E-01 4,815.98 1.26E-03 1.30E-01 1.80E+02 1,878.02 7.42E-01 2.21E+003.80E-01 4,796.29 1.56E-03 1.44E-01 2.16E+02 1,676.82 8.91E-01 2.35E+004.67E-01 4,775.11 1.92E-03 1.59E-01 2.60E+02 1,469.32 1.07E+00 2.50E+005.72E-01 4,752.31 2.36E-03 1.75E-01 3.12E+02 1,255.68 1.28E+00 2.65E+006.98E-01 4,727.67 2.88E-03 1.93E-01 3.74E+02 1,036.80 1.54E+00 2.81E+008.50E-01 4,700.91 3.50E-03 2.12E-01 4.49E+02 812.67 1.85E+00 2.97E+001.03E+00 4,672.03 4.25E-03 2.32E-01 5.38E+02 583.76 2.22E+00 3.13E+001.25E+00 4,640.79 5.15E-03 2.54E-01 6.46E+02 350.13 2.66E+00 3.29E+001.51E+00 4,606.86 6.23E-03 2.78E-01 7.75E+02 113.66 3.19E+00 3.46E+001.83E+00 4,570.07 7.52E-03 3.04E-01 7.91E+02 85.843.26E+00 3.48E+002.20E+00 4,530.11 9.08E-03 3.33E-01 8.10E+02 51.92 3.34E+00 3.50E+002.66E+00 4,486.84 1.09E-02 3.63E-01 8.13E+02 46.64 3.35E+00 3.51E+003.20E+00 4,439.87 1.32E-02 3.97E-01 8.17E+02 40.07 3.37E+00 3.51E+003.85E+00 4,389.05 1.59E-02 4.33E-01 8.23E+02 29.88 3.39E+00 3.52E+004.63E+00 4,333.97 1.91E-02 4.72E-01 8.24E+02 28.29 3.39E+00 3.52E+005.57E+00 4,274.38 2.29E-02 5.14E-01 8.25E+02 26.31 3.40E+00 3.52E+006.70E+00 4,209.65 2.76E-02 5.60E-01 8.27E+02 23.22 3.41E+003.52E+008.05E+00 4,139.93 3.32E-02 6.09E-01 8.30E+02 18.41 3.42E+00 3.53E+009.67E+00 4,064.61 3.98E-02 6.62E-01 8.30E+02 17.65 3.42E+00 3.53E+001.16E+01 3,983.13 4.79E-02 7.20E-01 8.31E+02 16.72 3.42E+00 3.53E+001.39E+01 3,895.77 5.75E-02 7.82E-01 8.32E+02 15.25 3.43E+00 3.53E+001.67E+01 3,801.71 6.90E-02 8.49E-01 8.32E+02 15.02 3.43E+00 3.53E+002.01E+01 3,700.74 8.28E-02 9.20E-01 8.32E+02 14.74 3.43E+00 3.53E+002.41E+01 3,592.52 9.95E-02 9.97E-01

169

A P P E N D I X C

170

Figure 1 – Finite conductivity type curve with deviations for fracture face interference for constant flow rate for and FCD=1 – Line source – Vertical well

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

0.001

0.01

0.1

1

10

0.000001 0.00001 0.0001 0.001 0.01 0.1 1


Dim

ensi

on

less

Pre

ssu

re P

D

0.028

4.0

16.0

63.8

255




xf

y5

25

10

FCD

1

100

500

FCD=1

0.001

0.01

0.1

1

10

0.000001 0.00001 0.0001 0.001 0.01 0.1 1


Dim

ensi

on

less

Pre

ssu

re P

D

0.028

4.0

16.0

63.8

255




xf

y5

25

10

FCD

1

100

500

FCD=5

171

Figure 3 – Finite conductivity type curve with deviations for fracture face interference for constant flow rate and FCD=10 – Line source – Vertical well Figure 4 – Finite conductivity type curve with deviations for fracture face interference for constant flow rate and FCD=25 – Line source – Vertical well

0.001

0.01

0.1

1

10

0.000001 0.00001 0.0001 0.001 0.01 0.1 1


Dim

ensi

on

less

Pre

ssu

re P

D

0.028

4.0

16.0

63.8

255




xf

y5

25

10

FCD

1

100

500

FCD=10

0.001

0.01

0.1

1

10

0.000001 0.00001 0.0001 0.001 0.01 0.1 1


Dim

ensi

on

less

Pre

ssu

re P

D

0.028

4.0

16.0

63.8

255




xf

y5

25

10

FCD

1

100

500

FCD=25

172

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

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

0.001

0.01

0.1

1

10

0.000001 0.00001 0.0001 0.001 0.01 0.1 1


Dim

ensi

on

less

Pre

ssu

re P

D

0.028

4.0

16.0

63.8

255




xf

y5

25

10

FCD

1

100

500

FCD=500

0.001

0.01

0.1

1

10

0.000001 0.00001 0.0001 0.001 0.01 0.1 1


Dim

ensi

on

less

Pre

ssu

re P

D

0.028

4.0

16.0

63.8

255




xf

y5

25

10

FCD

1

100

500

FCD=100

173

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

0.01

0.1

1

10

100

1.E-06 1.E-05 1.E-04 1.E-03 1.E-02 1.E-01 1.E+00


Dim

ensi

on

less

Flo

w R

ate

qD

0.028

4.0

16

63.8

255


y

FCD=1

0.001

0.01

0.1

1

10

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


Rec

ipro

cal D

imen

sio

nle

ss F

low

Rat

e 1/

qD

xf

y




5

FCD

1

500

100

1025

FCD=1

Rec

ipro

cal D

imen

sio

nle

ss F

low

Rat

e F

or

Tw

o F

ract

ure

Sys

tem

2/q

Dtf

s

0.028

4.0

16

63.8

128

255

174


0.1

1

10

100

1.E-06 1.E-05 1.E-04 1.E-03 1.E-02 1.E-01 1.E+00


Dim

ensi

on

less

Flo

w R

ate

qD

0.028

4.0

16.0

63.8

255


y

FCD=5

0.001

0.01

0.1

1

10

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


Rec

ipro

cal D

imen

sion

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Flo

w R

ate

1/q

D

xf

y




5

FCD

1

500

100

10

25

FCD=5

0.028

4.0

16.0

63.8

128

255

Rec

ipro

cal D

imen

sion

less

Flo

w R

ate

For

Two

Frac

ture

Sys

tem

2/q

Dtf

s

175

Figure 9 – Finite conductivity type curve with deviations for fracture face interference for constant pressure case and FCD=10 – Line source – Vertical well

Figure 9A – Dimensionless rate versus dimensionless time with deviations for fracture face interference for constant pressure case and FCD=10 – Line source – Vertical well

0.1

1

10

100

1.E-06 1.E-05 1.E-04 1.E-03 1.E-02 1.E-01 1.E+00


Dim

ensi

on

less

Flo

w R

ate

qD

0.028

4.0

16.0

63.8

255


y

FCD=10

0.001

0.01

0.1

1

10

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


Rec

ipro

cal D

imen

sio

nle

ss F

low

Rat

e 1/

qD

xf

y



xf - fracture half-length

y - distance between two fractures

5

FCD

1

500

100

10

25

FCD=10

0.028

4.0

16.0

63.8

128

255

Rec

ipro

cal D

imen

sio

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ss F

low

Rat

eF

or

Tw

o F

ract

ure

s S

yste

m 2

/qD

tfs

176

Figure 10 – Finite conductivity type curve with deviations for fracture face interference for constant pressure case and FCD=25 – Line source – Vertical well

Figure 10A – Dimensionless rate versus dimensionless time with deviations for fracture face interference for constant pressure case and FCD=25 – Line source – Vertical well

0.1

1

10

100

1.E-06 1.E-05 1.E-04 1.E-03 1.E-02 1.E-01 1.E+00


Dim

ensi

on

less

Flo

w R

ate

qD

0.028

4.0

16.0

63.8

255


y

FCD=25

0.001

0.01

0.1

1

10

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


Rec

ipro

cal D

imen

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nle

ss F

low

Rat

e 1/

qD

xf

y




5

FCD

1

500

100

10

25

FCD=25

0.028

4.0

16.0

63.8

128

255

Rec

ipro

cal D

imen

sio

nle

ss F

low

Rat

eF

or

Tw

o F

ract

ure

s S

yste

m 2

/qD

tfs

177


0.1

1

10

100

1000

0.000001 0.00001 0.0001 0.001 0.01 0.1 1


Dim

ensi

on

less

Flo

w R

ate

qD

0.0284.016.063.8255128


y

FCD=100

0.001

0.01

0.1

1

10

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


Rec

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cal D

imen

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ss F

low

Rat

e 1/

qD

xf

y




5

FCD

1

500

100

10

25

FCD=100

0.028

4.0

16.0

63.8

128

255

Rec

ipro

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imen

sio

nle

ss F

low

Rat

e fo

r T

wo

Fra

ctu

re S

yste

m 2

/qD

tfs

178


0.1

1

10

100

1000

1.E-06 1.E-05 1.E-04 1.E-03 1.E-02 1.E-01 1.E+00


Dim

ensi

on

less

Flo

w R

ate

qD

0.028

4.0

16.0

63.8

255


y

FCD=500

0.001

0.01

0.1

1

10

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


Rec

ipro

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imen

sio

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ss F

low

Rat

e 1/

qD

xf

y




5

FCD

1

500

100

10

25

FCD=500

0.028

4.0

16.0

63.8

255

128

Rec

ipro

cal D

imen

sio

nle

ss F

low

Rat

eF

or

Tw

o F

ract

ure

s S

yste

m 2

/qD

tfs

179

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

0.001

0.01

0.1

1

10

0.000001 0.00001 0.0001 0.001 0.01 0.1 1


Dim

ensi

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less

Pre

ssu

re P

D

4.0

16.0

63.8

255




xf

y

FCD=1

5

25

10

FCD

1

100

500

0.001

0.01

0.1

1

10

0.000001 0.00001 0.0001 0.001 0.01 0.1 1


Dim

ensi

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Pre

ssu

re P

D

4.0

16.0

63.8

255




xf

y

FCD=5

5

25

10

FCD

1

100

500

180


0.001

0.01

0.1

1

10

0.000001 0.00001 0.0001 0.001 0.01 0.1 1


Dim

ensi

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less

Pre

ssu

re P

D

4.0

16.0

63.8

255




xf

y

FCD=10

5

25

10

FCD

1

100

500

0.001

0.01

0.1

1

10

0.000001 0.00001 0.0001 0.001 0.01 0.1 1


Dim

ensi

on

less

Pre

ssu

re P

D

4.0

16.0

63.8

255




xf

y

FCD=25

5

25

10

FCD

1

100

500

181


0.001

0.01

0.1

1

10

0.000001 0.00001 0.0001 0.001 0.01 0.1 1


Dim

ensi

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less

Pre

ssu

re P

D

4.0

16.0

63.8

255




xf

y5

25

10

FCD

1

100

500

FCD=100

0.001

0.01

0.1

1

10

0.000001 0.00001 0.0001 0.001 0.01 0.1 1


Dim

ensi

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less

Pre

ssu

re P

D

4.0

16.0

63.8

255




xf

y

FCD=500

5

25

10

FCD

1

100

500

182

Figure 19 – Finite conductivity type curve with deviations for fracture face interference for constant pressure case and FCD=1 – Point source – Horizontal well Figure 19A – Dimensionless rate versus dimensionless time with deviations for fracture face interference for constant pressure case and FCD=1 – Point source – Horizontal well

0.1

1

10

1.E-06 1.E-05 1.E-04 1.E-03 1.E-02 1.E-01 1.E+00


Dim

en

sio

nle

ss

Flo

w R

ate

qD

4.0

16

63.8

255

FCD=1


y

0.001

0.01

0.1

1

10

0.000001 0.00001 0.0001 0.001 0.01 0.1 1

4.0

16.0

63.8

128

255

Rec

ipro

cal

Dim

ensi

on

less

Flo

w R

ate

1/q

D


5

FCD

1

500

100

10

25


xf

y



FCD=1

Rec

ipro

cal

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Flo

w R

ate

Fo

r T

wo

Fra

ctu

res

Sys

tem

2/q

Dtf

s

183


0.1

1

10

100

1.E-06 1.E-05 1.E-04 1.E-03 1.E-02 1.E-01 1.E+00


Dim

ensi

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less

Flo

w R

ate

qD

4.0

16.0

63.8

255

FCD=5


y

Rec

ipro

cal

Dim

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less

Flo

w R

ate

Fo

r T

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Fra

ctu

res

Sys

tem

2/q

Dtf

s

0.001

0.01

0.1

1

10

0.000001 0.00001 0.0001 0.001 0.01 0.1 1

4.0

16.0

63.8

128

255

Rec

ipro

cal

Dim

ensi

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less

Flo

w R

ate

1/q

D


FCD=5

5

FCD

1

500

100

10

25


xf

y



184


0.1

1

10

100

1.E-06 1.E-05 1.E-04 1.E-03 1.E-02 1.E-01 1.E+00


Dim

ensi

on

less

Flo

w R

ate

qD

4.0

16.0

63.8

255

FCD=10


y

Rec

ipro

cal

Dim

ensi

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less

Flo

w R

ate

Fo

r T

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Fra

ctu

res

Sys

tem

2/q

Dtf

s

0.001

0.01

0.1

1

10

0.000001 0.00001 0.0001 0.001 0.01 0.1 1

4.0

16.0

63.8

128

255

Rec

ipro

cal

Dim

ensi

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less

Flo

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ate

1/q

D


FCD=10

5

FCD

1

500

100

10

25


xf

y



185


0.1

1

10

100

1.E-06 1.E-05 1.E-04 1.E-03 1.E-02 1.E-01 1.E+00


Dim

ensi

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less

Flo

w R

ate

qD

4.0

16.0

63.8

255

FCD=25


y

Rec

ipro

cal

Dim

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less

Flo

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ate

Fo

r T

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Fra

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Sys

tem

2/q

Dtf

s

0.001

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0.1

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10

0.000001 0.00001 0.0001 0.001 0.01 0.1 1

4.0

16.0

63.8

128

255

Rec

ipro

cal

Dim

ensi

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less

Flo

w R

ate

1/q

D


FCD=25

5

FCD

1

500

100

10

25


xf

y



186


0.1

1

10

100

1.E-06 1.E-05 1.E-04 1.E-03 1.E-02 1.E-01 1.E+00


Dim

ensi

on

less

Flo

w R

ate

qD

4.0

16.0

63.8

255

FCD=100


y

0.001

0.01

0.1

1

10

0.000001 0.00001 0.0001 0.001 0.01 0.1 1

4.0

16.0

63.8

128

255

Rec

ipro

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ensi

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ate

1/q

D


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FCD

1

500

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10

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xf

y



FCD=100

Rec

ipro

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less

Flo

w R

ate

Fo

r T

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Fra

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res

Sys

tem

2/q

Dtf

s

187


0.1

1

10

100

1000

1.E-06 1.E-05 1.E-04 1.E-03 1.E-02 1.E-01 1.E+00


Dim

ensi

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less

Flo

w R

ate

qD

4.0

16.0

63.8

255

FCD=500


y

Rec

ipro

cal

Dim

ensi

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Flo

w R

ate

Fo

r T

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Fra

ctu

res

Sys

tem

2/q

Dtf

s

0.001

0.01

0.1

1

10

0.000001 0.00001 0.0001 0.001 0.01 0.1 1

4.0

16.0

63.8

128

255

Rec

ipro

cal

Dim

ensi

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ate

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D


FCD=500

5

FCD

1

500

100

10

25


xf

y



188

A P P E N D I X 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]

Simulation of Hydraulic Fracturing

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Transcript of Simulation of Hydraulic Fracturing