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UNIVERSITY OF OKLAHOMA GRADUATE COLLEGE
FRACTURE FACE INTERFERENCE OF FINITE CONDUCTIVITY FRACTURED
WELLS USING NUMERICAL SIMULATION
A THESIS
SUBMITTED TO THE GRADUATE FACULTY
in partial fulfillment of the requirements for the
Degree of
MASTER OF SCIENCE
IN NATURAL GAS ENGINEERING AND MANAGEMENT
By SVJETLANA LALE Norman, Oklahoma
2008
FRACTURE FACE INTERFERENCE OF FINITE CONDUCTIVITY FRACTURED WELLS USING NUMERICAL SIMULATION
A THESIS APROVED FOR THE MEWBOURNE SCHOOL OF PETROLEUM AND GEOLOGICAL ENGINEERING
BY _______________________________ Dr. Jeffrey G. Callard – Chair
_______________________________
Dr. Djebbar Tiab
_______________________________ Dr. Samuel Osisanya
_______________________________
Dr. Dean S. Oliver
©Copyright by SVJETLANA LALE 2008 All Rights Reserved.
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.
vi
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
vii
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
viii
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
Table 3 – FCD=10 – Results of Numerical Simulation and
Dimensionless Time, Pressure and Rate ……………………….… 128
Table 4 - FCD=25– Results of Numerical Simulation and
Dimensionless Time, Pressure and Rate ………………………..… 134
Table 5 – FCD=100 – Results of Numerical Simulation and
Dimensionless Time, Pressure and Rate ………………….……… 140
x
Table 6 – FCD=500 – Results of Numerical Simulation and
Dimensionless Time, Pressure and Rate ………………………… 147
Table 7 – FCD=100 – Results of Numerical Simulation and Dimensionless Time,
Pressure and Rate for Point Source …………………………….... 152
Table 8 – FCD=100 – Results of Numerical Simulation and Dimensionless Time,
and Pressure for xf/y=255, Constant Rate Case, Vertical Well ….. 160
Table 9 – FCD=100 – Results of Numerical Simulation and Dimensionless Time,
and Rate for xf/y=128, Constant Pressure Case, Vertical Well ….. 162
Table 10 – Production Data and Dimensionless Time
and Flow Rate for McAlister O.H. 16 ………………………….. 164
Table 11 – FCD=100 – Results of Numerical Simulation and Dimensionless Time,
Pressure and Rate for xf/y=255, Point Source ……………..……... 165
Table 12 – FCD=100 – Results of Numerical Simulation and Dimensionless
Time, and Pressure for Constant Rate Case,
xf2 =xf1/4 and xf/y= 255 ………………………………………… 168
Appendix C…………………………………………………………………… 169
Figure 1 – Finite conductivity type curve with deviations for fracture face
interference for constant flow rate for and FCD=1 – Line source –
Vertical well ……………………………………………………. 170
Figure 2 – Finite conductivity type curve with deviations for fracture face
interference for constant flow rate and FCD=5 – Line source –
Vertical well ……………………………………………………… 170
Figure 3 – Finite conductivity type curve with deviations for fracture face
xi
interference for constant flow rate and FCD=10 – Line source –
Vertical well …………………………………………………….. 171
Figure 4 – Finite conductivity type curve with deviations for fracture face
interference for constant flow rate and FCD=25 – Line source –
Vertical well ……………………………………………………... 171
Figure 5 – Finite conductivity type curve with deviations for fracture face
interference for constant flow rate for and FCD=100 – Line source –
Vertical well …………………………………………………….. 172
Figure 6 – Finite conductivity type curve with deviations for fracture face
interference for constant flow rate and FCD=500 – Line source –
Vertical well ……………………………………………………… 172
Figure 7 – Finite conductivity type curve with deviations for fracture face
interference for constant pressure case and FCD=1 – Line source –
Vertical well ……………………………………………………… 173
Figure 7A – Dimensionless rate versus dimensionless time with deviations for
fracture face interference for constant pressure case and FCD=1 –
Line source – Vertical well …………………………………….. 173
Figure 8 – Finite conductivity type curve with deviations for fracture face
interference for constant pressure case and FCD=5 – Line source –
Vertical well ………………………………………………………. 174
Figure 8A – Dimensionless rate versus dimensionless time with deviations for
fracture face interference for constant pressure case and FCD=5 –
Line source – Vertical well ………………………………………. 174
xii
Figure 9 – Finite conductivity type curve with deviations for fracture face
interference for constant pressure case and FCD=10 – Line source –
Vertical well ……………………………………………………… 175
Figure 9A – Dimensionless rate versus dimensionless time with deviations for
fracture face interference for constant pressure case and FCD=10 –
Line source – Vertical well ……………………………………….. 175
Figure 10 – Finite conductivity type curve with deviations for fracture face
interference for constant pressure case and FCD=25 – Line source –
Vertical well ……………………………………………………… 176
Figure 10A – Dimensionless rate versus dimensionless time with deviations for
fracture face interference for constant pressure case and FCD=25 –
Line source – Vertical well ………………………………………. 176
Figure 11 – Finite conductivity type curve with deviations for fracture face
interference for constant pressure case and FCD=100 – Line source –
Vertical well …………………………………………………….. 177
Figure 11A – Dimensionless rate versus dimensionless time with deviations for
fracture face interference for constant pressure case and FCD=100 –
Line source – Vertical well ………………………………….….. 177
Figure 12 – Finite conductivity type curve with deviations for fracture face
interference for constant pressure case and FCD=500 – Line source –
Vertical well …………………………………………………….. 178
Figure 12A – Dimensionless rate versus dimensionless time with deviations for
fracture face interference for constant pressure case and FCD=500 –
xiii
Line source – Vertical well ………………………………………. 178
Figure 13 – Finite conductivity type curve with deviations for fracture face
interference for constant rate case and FCD=1 – Point source –
Horizontal well ………………………………………………….. 179
Figure 14 – Finite conductivity type curve with deviations for fracture face
interference for constant rate case and FCD=5 – Point source –
Horizontal well ………………………………………………….. 179
Figure 15 – Finite conductivity type curve with deviations for
fracture face interference for constant rate case and FCD=10 –
Point source – Horizontal well …………………………………… 180
Figure 16 – Finite conductivity type curve with deviations for fracture
face interference for constant rate case and FCD=25 –
Point source – Horizontal well ………………………………….. 180
Figure 17 – Finite conductivity type curve with deviations for fracture
face interference for constant rate case and FCD=100 –
Point source – Horizontal well ………………………………….. 181
Figure 18 – Finite conductivity type curve with deviations for fracture
face interference for constant rate case and FCD=500 –
Point source – Horizontal well ………………………………….. 181
Figure 19 – Finite conductivity type curve with deviations for fracture
face interference for constant pressure case and FCD=1 –
Point source – Horizontal well ………………………………….. 182
Figure 19A – Dimensionless rate versus dimensionless time with deviations
xiv
for fracture face interference for constant pressure case and FCD=1 –
Point source – Horizontal well ………………………………….. 182
Figure 20 – Finite conductivity type curve with deviations for fracture
face interference for constant pressure case and FCD=5 –
Point source – Horizontal well ………………………………….. 183
Figure 20A – Dimensionless rate versus dimensionless time with deviations
for fracture face interference for constant pressure case and FCD=5 –
Point source – Horizontal well ………………………………….. 183
Figure 21 – Finite conductivity type curve with deviations for fracture
face interference for constant pressure case and FCD=10 –
Point source – Horizontal well ………………………………….. 184
Figure 21A – Dimensionless rate versus dimensionless time with deviations
for fracture face interference for constant pressure case and
FCD=10 – Point source – Horizontal well ……………………….. 184
Figure 22 – Finite conductivity type curve with deviations for fracture
face interference for constant pressure case and FCD=25 –
Point source – Horizontal well ………………………………….. 185
Figure 22A – Dimensionless rate versus dimensionless time with deviations
for fracture face interference for constant pressure case and
FCD=25 – Point source – Horizontal well ……………………….. 185
Figure 23 – Finite conductivity type curve with deviations for fracture
face interference for constant pressure case and FCD=100 –
Point source – Horizontal well ………………………………….. 186
xv
Figure 23A – Dimensionless rate versus dimensionless time with deviations
for fracture face interference for constant pressure case and
FCD=100 – Point source – Horizontal well …………………………186
Figure 24 – Finite conductivity type curve with deviations for fracture
face interference for constant pressure case and FCD=500 –
Point source – Horizontal well ………………………………….. 187
Figure 24A – Dimensionless rate versus dimensionless time with deviations
for fracture face interference for constant pressure case and
FCD=500 – Point source – Horizontal well ……………………. 187
Appendix D …………………………………………………………………... 188
Nomenclature …………………………………………………………………. 189
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
for fracture face interference for constant pressure case and
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
xx
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
k [md] – formation permeability
xf [ft] – fracture half-length
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:
k [md] – formation permeability
t [days] – production time
φ [-] – reservoir porosity, fraction
ct [psi-1] – total system compressibility
µ [cp] – fluid viscosity
xf [ft] – fracture half-length
Correlation for dimensionless pressure:
( )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
k [md] – formation permeability
h [ft] – total reservoir thickness
pi [psi] – initial pressure
pwf [psi] – well bore flowing pressure
Rate multiplier calculated using equation (10) and based on numerical model data set:
qM=3.14E-03 [day/STB]
42
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
Dimensionless Time tDxf
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
Dimensionless Time tDxf
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 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
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
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 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
Dimensionless Time tDxf
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
Dimensionless Time tDxf
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 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]
335 100,646 618 201,956
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
Added Part of Reservoir
Enlarged Fracture
Enlarged Fracture
201,
000
100,
500
Well 1
Well 2
Initial Reservoir
Added Part of 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
Dimensionless Time tDxf
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
Dimensionless Time tDxf
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
Dimensionless Time tDxf
Rec
ipro
cal D
imen
sio
nle
ss F
low
Rat
e 1/
qD
xf
y
length to distance ratio
FCD - dimensionless fracture conductivity
xf - fracture half-lengthy - distance between two fractures
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 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
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
Dimensionless Time tDxf
Dim
ensi
on
less
Pre
ssu
re P
D
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
68
This type curve is developed for the ratio fracture half-length versus reservoir thickness
xf/h =2043/100
The same analysis was done for the FCD=1, 5, 10, 25, and 500 and graphical results are
presented in the Appendix C – Figures 13 to 18 .
6.3. Constant Pressure Case
The research methodology is almost identical to the one described in the case of the
constant flow rate. The only difference is the control mode and in this case it is
BHP=500 psi. Reservoir geometry has already been described and reservoir and fracture
physical characteristics and fluid PVT properties were constant. The only variable was
the distance between two fractures, y which was the maximum at the start of the
research. Simulation methodology was the same, the distance between fractures has
been decreased by removing grid blocks between fractures.
Eclipse simulator data input file for length to distance ratio, xf/y=255 and dimensionless
fracture conductivity, FCD=100 is given in the Appendix A8 for point source. Results of
numerical simulation and dimensionless time and flow rate are given in Table 11 in
Appendix B. Plot of well production rate versus time with length to distance ratio as
parameter is presented in Figure 6.3.
69
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
Dimensionless Time tDxf
5
FCD
1
500
100
10
25
length to distance ratio
xf
y
FCD - dimensionless fracture conductivity
xf - fracture half-lengthy - distance between two fractures
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
Dimensionless Time tDxf
Dim
ensi
on
less
Flo
w R
ate
qD
4.0
16.0
63.8
255
FCD=100
length to distance ratioxf
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
Dimensionless Time tDxf
5
FCD
1
500
100
10
25
FCD - dimensionless fracture conductivity
xf - fracture half-lengthy - distance between two fractures
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
Dimensionless Time tDxf
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
Dimensionless Time tDxf
Dim
ensi
on
less
Pre
ssu
re p
D
pi=1,000
pi=2,000
pi=5,000
76
7.2. Constant Pressure Case
Additional verification of the numerical model was performed by changing well
productivity indexes by adding keyword WELPI in Schedule section of Data File for
FCD=5 and FCD=100. The three different well productivity indexes were set up: initial
one, twice higher and twice lower than initial ones. Results are given in the Figures 7.3
and 7.4.
Results of the analysis are plotted on Bennett finite conductivity type curve for FCD=5
and FCD=100 and they match the numerical simulation solution and provide numerical
model verification.
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
Dimensionless Time tDxf
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
Dimensionless Time tDxf
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
Dimensionless Time tDxf
Dim
ensi
on
less
Pre
ssu
re p
D
2 2
8 8
36 36
255 255
length to distance ratio
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
Dimensionless Time tDxf
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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Bennett, C.O., Camacho, R.G., Reynolds, A.C., Raghavan, R., – Approximate
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Bennett, C.O., Reynolds, A.C., Raghavan,R., Jacques, L.E., – Performance of Finite
Conductivity, Vertically Fractured Wells in Single-Layer Reservoirs – SPE
Formation Evaluation – August 1986, 399-412
Bennion, D.B., Thomas, F.B., Bietz, R.F. – Low Permeability Gas Reservoirs:
Problems, Opportunities and Solutions for Drilling, Completion, Stimulation and
Production - SPE 35577 presented at Gas Technology Conference, Calgary, Canada
– May 1996
84
Bennion, D.B., Thomas, F.B. and Ma, T. – Formation Damage Processes Reducing
Productivity of Low Permeability Gas Reservoirs – SPE 60325 presented at the SPE
Rocky Mountain Regional/Low Permeability Reservoir Symposium and Exhibition,
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Cinco-Ley, H., Samaniego, V.F., Dominguez, N.,– Transient Pressure Behavior for
a Well With Finite-Conductivity Fracture – SPE Journal, August 1978, 253-264,
Trans. AIME 265
Cinco-Ley, H., Samaniego, V.F., – Transient Pressure Analysis for Fractured Wells
– SPE paper 7490, Journal of Petroleum Technology, September 1981, 1749-1766
Daniels, J., Waters, G., LeCalvez, J., Lassek, J., Bentley, D. – Contacting More of
Barnett Shale Through an Integration of Real-Time Microseismic Monitoring,
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Advanced Stimulation Technologies – PhD Dissertation, TUniversity Bergakademie
Freiberg, July 2004
Naik, G.C. - Tight Gas Reservoirs – An Unconventional Natural Energy Source for
the Future –2006
85
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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 /
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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
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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
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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
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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 /
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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 /
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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 ==========================================================================
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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 /
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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 ==========================================================================
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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
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 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
Table 1 – FCD=1 – Results of Numerical Simulation and Dimensionless Time, Pressure and Rate - continued
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
Table 1 – FCD=1 – Results of Numerical Simulation and Dimensionless Time, Pressure and Rate - continued
t [days] tDxf q [stb/day] qD 2.97E+04 7.50E+00 98 3.08E-01 3.00E+04 7.59E+00 98 3.07E-01 3.04E+04 7.69E+00 98 3.07E-01 3.08E+04 7.78E+00 98 3.06E-01 3.11E+04 7.87E+00 97 3.06E-01 3.15E+04 7.96E+00 97 3.05E-01 3.19E+04 8.05E+00 97 3.05E-01 3.22E+04 8.15E+00 97 3.04E-01 3.26E+04 8.24E+00 97 3.04E-01 3.30E+04 8.33E+00 97 3.03E-01 3.33E+04 8.42E+00 96 3.03E-01 3.37E+04 8.52E+00 96 3.02E-01 3.41E+04 8.61E+00 96 3.02E-01 3.44E+04 8.70E+00 96 3.01E-01 3.48E+04 8.79E+00 96 3.01E-01 3.52E+04 8.88E+00 96 3.00E-01 3.54E+04 8.94E+00 96 3.00E-01 3.56E+04 9.00E+00 96 3.00E-01 3.60E+04 9.09E+00 95 2.99E-01 3.63E+04 9.18E+00 95 2.99E-01 3.67E+04 9.28E+00 95 2.99E-01 3.71E+04 9.37E+00 95 2.98E-01 3.74E+04 9.46E+00 95 2.98E-01 3.78E+04 9.55E+00 95 2.97E-01 3.82E+04 9.65E+00 95 2.97E-01 3.85E+04 9.74E+00 94 2.96E-01 3.89E+04 9.83E+00 94 2.96E-01 3.93E+04 9.92E+00 94 2.96E-01 3.96E+04 1.00E+01 94 2.95E-01 4.00E+04 1.01E+01 94 2.95E-01 4.04E+04 1.02E+01 94 2.94E-01 4.07E+04 1.03E+01 94 2.94E-01 4.11E+04 1.04E+01 94 2.94E-01 4.15E+04 1.05E+01 94 2.93E-01 4.18E+04 1.06E+01 93 2.93E-01 4.22E+04 1.07E+01 93 2.93E-01 4.25E+04 1.07E+01 93 2.92E-01 4.27E+04 1.08E+01 93 2.92E-01 4.31E+04 1.09E+01 93 2.92E-01 4.35E+04 1.10E+01 93 2.91E-01 4.38E+04 1.11E+01 93 2.91E-01 4.42E+04 1.12E+01 93 2.91E-01 4.46E+04 1.13E+01 93 2.90E-01 4.49E+04 1.14E+01 92 2.90E-01 4.53E+04 1.14E+01 92 2.90E-01
121
Table 1 – FCD=1 – Results of Numerical Simulation and Dimensionless Time, Pressure and Rate - continued
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
Table 2 – FCD=5 – Results of Numerical Simulation and Dimensionless Time, Pressure and Rate
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
Table 2 – FCD=5 – 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 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
Table 2 – FCD=5 – Results of Numerical Simulation and Dimensionless Time, Pressure and Rate - continued
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
Table 2 – FCD=5 – Results of Numerical Simulation and Dimensionless Time, Pressure and Rate - continued
t [days] tDxf q [stb/day] qD 1.40E+04 3.55E+00 142 4.44E-01 1.43E+04 3.62E+00 141 4.42E-01 1.47E+04 3.71E+00 140 4.40E-01 1.51E+04 3.80E+00 140 4.38E-01 1.54E+04 3.90E+00 139 4.36E-01 1.58E+04 3.99E+00 138 4.34E-01 1.61E+04 4.08E+00 138 4.32E-01 1.65E+04 4.17E+00 137 4.30E-01 1.68E+04 4.26E+00 137 4.28E-01 1.72E+04 4.34E+00 136 4.27E-01 1.75E+04 4.43E+00 135 4.25E-01 1.79E+04 4.53E+00 135 4.23E-01 1.83E+04 4.62E+00 134 4.22E-01 1.86E+04 4.71E+00 134 4.20E-01 1.90E+04 4.80E+00 133 4.18E-01 1.94E+04 4.90E+00 133 4.17E-01 1.97E+04 4.99E+00 132 4.15E-01 2.01E+04 5.08E+00 132 4.14E-01 2.04E+04 5.14E+00 132 4.13E-01 2.06E+04 5.21E+00 131 4.12E-01 2.10E+04 5.30E+00 131 4.11E-01 2.13E+04 5.39E+00 130 4.09E-01 2.17E+04 5.49E+00 130 4.08E-01 2.21E+04 5.58E+00 130 4.07E-01 2.24E+04 5.67E+00 129 4.05E-01 2.28E+04 5.76E+00 129 4.04E-01 2.32E+04 5.85E+00 128 4.03E-01 2.35E+04 5.95E+00 128 4.02E-01 2.39E+04 6.04E+00 128 4.01E-01 2.43E+04 6.13E+00 127 4.00E-01 2.45E+04 6.19E+00 127 3.99E-01 2.47E+04 6.25E+00 127 3.98E-01 2.51E+04 6.34E+00 127 3.97E-01 2.55E+04 6.43E+00 126 3.96E-01 2.58E+04 6.53E+00 126 3.95E-01 2.62E+04 6.62E+00 126 3.94E-01 2.66E+04 6.71E+00 125 3.93E-01 2.69E+04 6.80E+00 125 3.92E-01 2.73E+04 6.90E+00 125 3.91E-01 2.77E+04 6.99E+00 124 3.90E-01 2.80E+04 7.08E+00 124 3.89E-01 2.84E+04 7.17E+00 124 3.88E-01 2.87E+04 7.26E+00 123 3.87E-01 2.91E+04 7.36E+00 123 3.87E-01 2.94E+04 7.43E+00 123 3.86E-01
126
Table 2 – FCD=5 – Results of Numerical Simulation and Dimensionless Time, Pressure and Rate - continued
t [days] tDxf q [stb/day] qD 2.97E+04 7.50E+00 123 3.85E-01 3.00E+04 7.59E+00 122 3.84E-01 3.04E+04 7.69E+00 122 3.83E-01 3.08E+04 7.78E+00 122 3.83E-01 3.11E+04 7.87E+00 122 3.82E-01 3.15E+04 7.96E+00 121 3.81E-01 3.19E+04 8.05E+00 121 3.80E-01 3.22E+04 8.15E+00 121 3.80E-01 3.26E+04 8.24E+00 121 3.79E-01 3.30E+04 8.33E+00 120 3.78E-01 3.33E+04 8.42E+00 120 3.77E-01 3.37E+04 8.52E+00 120 3.77E-01 3.41E+04 8.61E+00 120 3.76E-01 3.44E+04 8.70E+00 120 3.75E-01 3.48E+04 8.79E+00 119 3.74E-01 3.52E+04 8.88E+00 119 3.74E-01 3.54E+04 8.94E+00 119 3.73E-01 3.56E+04 9.00E+00 119 3.73E-01 3.60E+04 9.09E+00 119 3.72E-01 3.63E+04 9.18E+00 118 3.72E-01 3.67E+04 9.28E+00 118 3.71E-01 3.71E+04 9.37E+00 118 3.70E-01 3.74E+04 9.46E+00 118 3.70E-01 3.78E+04 9.55E+00 118 3.69E-01 3.82E+04 9.65E+00 117 3.69E-01 3.85E+04 9.74E+00 117 3.68E-01 3.89E+04 9.83E+00 117 3.67E-01 3.93E+04 9.92E+00 117 3.67E-01 3.96E+04 1.00E+01 117 3.66E-01 4.00E+04 1.01E+01 117 3.66E-01 4.04E+04 1.02E+01 116 3.65E-01 4.07E+04 1.03E+01 116 3.64E-01 4.11E+04 1.04E+01 116 3.64E-01 4.15E+04 1.05E+01 116 3.63E-01 4.18E+04 1.06E+01 116 3.63E-01 4.22E+04 1.07E+01 115 3.62E-01 4.25E+04 1.07E+01 115 3.62E-01 4.27E+04 1.08E+01 115 3.61E-01 4.31E+04 1.09E+01 115 3.61E-01 4.35E+04 1.10E+01 115 3.60E-01 4.38E+04 1.11E+01 115 3.60E-01 4.42E+04 1.12E+01 115 3.59E-01 4.46E+04 1.13E+01 114 3.59E-01 4.49E+04 1.14E+01 114 3.58E-01 4.53E+04 1.14E+01 114 3.58E-01
127
Table 2 – FCD=5 – Results of Numerical Simulation and Dimensionless Time, Pressure and Rate - continued
t [days] tDxf q [stb/day] qD 4.57E+04 1.15E+01 114 3.57E-01 4.60E+04 1.16E+01 114 3.57E-01 4.64E+04 1.17E+01 114 3.57E-01 4.67E+04 1.18E+01 113 3.56E-01 4.71E+04 1.19E+01 113 3.56E-01 4.75E+04 1.20E+01 113 3.55E-01 4.78E+04 1.21E+01 113 3.55E-01 4.82E+04 1.22E+01 113 3.54E-01 4.86E+04 1.23E+01 113 3.54E-01 4.89E+04 1.24E+01 113 3.53E-01 4.93E+04 1.25E+01 112 3.53E-01 4.97E+04 1.26E+01 112 3.53E-01 5.00E+04 1.26E+01 112 3.52E-01 5.04E+04 1.27E+01 112 3.52E-01 5.08E+04 1.28E+01 112 3.51E-01 5.10E+04 1.29E+01 112 3.51E-01 5.13E+04 1.30E+01 112 3.51E-01 5.16E+04 1.31E+01 112 3.50E-01 5.20E+04 1.31E+01 111 3.50E-01 5.24E+04 1.32E+01 111 3.49E-01 5.27E+04 1.33E+01 111 3.49E-01 5.31E+04 1.34E+01 111 3.49E-01 5.35E+04 1.35E+01 111 3.48E-01 5.38E+04 1.36E+01 111 3.48E-01 5.42E+04 1.37E+01 111 3.47E-01 5.46E+04 1.38E+01 111 3.47E-01 5.49E+04 1.39E+01 110 3.47E-01 5.53E+04 1.40E+01 110 3.46E-01 5.57E+04 1.41E+01 110 3.46E-01 5.60E+04 1.42E+01 110 3.46E-01 5.64E+04 1.43E+01 110 3.45E-01 5.68E+04 1.43E+01 110 3.45E-01 5.71E+04 1.44E+01 110 3.45E-01 5.75E+04 1.45E+01 110 3.44E-01 5.79E+04 1.46E+01 110 3.44E-01 5.82E+04 1.47E+01 109 3.43E-01 5.86E+04 1.48E+01 109 3.43E-01 5.89E+04 1.49E+01 109 3.43E-01 5.93E+04 1.50E+01 109 3.42E-01 5.97E+04 1.51E+01 109 3.42E-01 6.00E+04 1.52E+01 109 3.42E-01 6.04E+04 1.53E+01 109 3.41E-01 6.08E+04 1.54E+01 109 3.41E-01 6.11E+04 1.55E+01 109 3.41E-01 6.14E+04 1.55E+01 109 3.41E-01 6.16E+04 1.56E+01 108 3.40E-01
128
Table 3 - FCD=10 – Results of Numerical Simulation and Dimensionless Time, Pressure and Rate
t [days] tDxf q [stb/day] qD p [psi] pD 0 0 0 0 5,000 n/a
1.16E-05 2.92E-09 36,748 1.15E+02 4,988 8.59E-032.55E-05 6.44E-09 20,490 6.43E+01 4,982 1.27E-024.22E-05 1.07E-08 16,945 5.32E+01 4,978 1.56E-026.22E-05 1.57E-08 15,472 4.85E+01 4,975 1.77E-028.62E-05 2.18E-08 14,731 4.62E+01 4,973 1.93E-021.15E-04 2.91E-08 14,326 4.50E+01 4,971 2.05E-021.50E-04 3.78E-08 14,092 4.42E+01 4,970 2.14E-021.91E-04 4.83E-08 13,949 4.38E+01 4,969 2.20E-022.41E-04 6.09E-08 13,851 4.35E+01 4,968 2.24E-023.01E-04 7.60E-08 13,773 4.32E+01 4,968 2.27E-023.72E-04 9.41E-08 13,698 4.30E+01 4,968 2.29E-024.58E-04 1.16E-07 13,619 4.27E+01 4,967 2.31E-025.61E-04 1.42E-07 13,529 4.25E+01 4,967 2.33E-026.85E-04 1.73E-07 13,424 4.21E+01 4,967 2.34E-028.34E-04 2.11E-07 13,301 4.17E+01 4,967 2.37E-021.01E-03 2.56E-07 13,158 4.13E+01 4,966 2.39E-021.23E-03 3.10E-07 12,991 4.08E+01 4,966 2.42E-021.48E-03 3.75E-07 12,797 4.02E+01 4,965 2.45E-021.79E-03 4.53E-07 12,575 3.95E+01 4,965 2.49E-022.16E-03 5.46E-07 12,321 3.87E+01 4,964 2.54E-022.61E-03 6.58E-07 12,034 3.78E+01 4,963 2.59E-023.14E-03 7.93E-07 11,713 3.68E+01 4,962 2.66E-023.78E-03 9.54E-07 11,358 3.56E+01 4,961 2.73E-024.54E-03 1.15E-06 10,972 3.44E+01 4,960 2.81E-025.46E-03 1.38E-06 10,559 3.31E+01 4,959 2.91E-026.56E-03 1.66E-06 10,126 3.18E+01 4,957 3.01E-027.88E-03 1.99E-06 9,680 3.04E+01 4,956 3.13E-029.47E-03 2.39E-06 9,229 2.90E+01 4,954 3.25E-021.14E-02 2.88E-06 8,784 2.76E+01 4,952 3.39E-021.37E-02 3.46E-06 8,355 2.62E+01 4,950 3.54E-021.64E-02 4.15E-06 7,949 2.49E+01 4,948 3.70E-021.97E-02 4.98E-06 7,572 2.38E+01 4,945 3.87E-022.37E-02 5.99E-06 7,223 2.27E+01 4,943 4.04E-022.84E-02 7.19E-06 6,902 2.17E+01 4,940 4.22E-023.41E-02 8.63E-06 6,602 2.07E+01 4,938 4.41E-024.10E-02 1.04E-05 6,319 1.98E+01 4,935 4.60E-024.92E-02 1.24E-05 6,049 1.90E+01 4,932 4.80E-025.90E-02 1.49E-05 5,790 1.82E+01 4,929 5.02E-027.08E-02 1.79E-05 5,538 1.74E+01 4,926 5.24E-028.50E-02 2.15E-05 5,295 1.66E+01 4,923 5.48E-021.02E-01 2.58E-05 5,061 1.59E+01 4,919 5.73E-021.22E-01 3.09E-05 4,839 1.52E+01 4,915 5.99E-021.47E-01 3.71E-05 4,628 1.45E+01 4,912 6.26E-02
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
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 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
Table 3 - FCD=10 – Results of Numerical Simulation and Dimensionless Time, Pressure and Rate - continued
t [days] tDxf q [stb/day] qD 1.40E+04 3.55E+00 149 4.67E-01 1.43E+04 3.62E+00 148 4.65E-01 1.47E+04 3.71E+00 147 4.62E-01 1.51E+04 3.80E+00 147 4.60E-01 1.54E+04 3.90E+00 146 4.58E-01 1.58E+04 3.99E+00 145 4.56E-01 1.61E+04 4.08E+00 144 4.53E-01 1.65E+04 4.17E+00 144 4.51E-01 1.68E+04 4.26E+00 143 4.49E-01 1.72E+04 4.34E+00 143 4.48E-01 1.75E+04 4.43E+00 142 4.46E-01 1.79E+04 4.53E+00 141 4.44E-01 1.83E+04 4.62E+00 141 4.42E-01 1.86E+04 4.71E+00 140 4.40E-01 1.90E+04 4.80E+00 140 4.39E-01 1.94E+04 4.90E+00 139 4.37E-01 1.97E+04 4.99E+00 139 4.35E-01 2.01E+04 5.08E+00 138 4.34E-01 2.04E+04 5.14E+00 138 4.33E-01 2.06E+04 5.21E+00 138 4.32E-01 2.10E+04 5.30E+00 137 4.30E-01 2.13E+04 5.39E+00 137 4.29E-01 2.17E+04 5.49E+00 136 4.27E-01 2.21E+04 5.58E+00 136 4.26E-01 2.24E+04 5.67E+00 135 4.25E-01 2.28E+04 5.76E+00 135 4.23E-01 2.32E+04 5.85E+00 134 4.22E-01 2.35E+04 5.95E+00 134 4.21E-01 2.39E+04 6.04E+00 134 4.19E-01 2.43E+04 6.13E+00 133 4.18E-01 2.45E+04 6.19E+00 133 4.17E-01 2.47E+04 6.25E+00 133 4.17E-01 2.51E+04 6.34E+00 132 4.16E-01 2.55E+04 6.43E+00 132 4.14E-01 2.58E+04 6.53E+00 132 4.13E-01 2.62E+04 6.62E+00 131 4.12E-01 2.66E+04 6.71E+00 131 4.11E-01 2.69E+04 6.80E+00 131 4.10E-01 2.73E+04 6.90E+00 130 4.09E-01 2.77E+04 6.99E+00 130 4.08E-01 2.80E+04 7.08E+00 130 4.07E-01 2.84E+04 7.17E+00 129 4.06E-01 2.87E+04 7.26E+00 129 4.05E-01 2.91E+04 7.36E+00 129 4.04E-01 2.94E+04 7.43E+00 129 4.03E-01
132
Table 3 - FCD=10 – Results of Numerical Simulation and Dimensionless Time, Pressure and Rate - continued
t [days] tDxf q [stb/day] qD 2.97E+04 7.50E+00 128 4.03E-01 3.00E+04 7.59E+00 128 4.02E-01 3.04E+04 7.69E+00 128 4.01E-01 3.08E+04 7.78E+00 127 4.00E-01 3.11E+04 7.87E+00 127 3.99E-01 3.15E+04 7.96E+00 127 3.98E-01 3.19E+04 8.05E+00 127 3.97E-01 3.22E+04 8.15E+00 126 3.97E-01 3.26E+04 8.24E+00 126 3.96E-01 3.30E+04 8.33E+00 126 3.95E-01 3.33E+04 8.42E+00 126 3.94E-01 3.37E+04 8.52E+00 125 3.93E-01 3.41E+04 8.61E+00 125 3.93E-01 3.44E+04 8.70E+00 125 3.92E-01 3.48E+04 8.79E+00 125 3.91E-01 3.52E+04 8.88E+00 124 3.90E-01 3.54E+04 8.94E+00 124 3.90E-01 3.56E+04 9.00E+00 124 3.89E-01 3.60E+04 9.09E+00 124 3.89E-01 3.63E+04 9.18E+00 124 3.88E-01 3.67E+04 9.28E+00 123 3.87E-01 3.71E+04 9.37E+00 123 3.87E-01 3.74E+04 9.46E+00 123 3.86E-01 3.78E+04 9.55E+00 123 3.85E-01 3.82E+04 9.65E+00 123 3.85E-01 3.85E+04 9.74E+00 122 3.84E-01 3.89E+04 9.83E+00 122 3.83E-01 3.93E+04 9.92E+00 122 3.83E-01 3.96E+04 1.00E+01 122 3.82E-01 4.00E+04 1.01E+01 122 3.81E-01 4.04E+04 1.02E+01 121 3.81E-01 4.07E+04 1.03E+01 121 3.80E-01 4.11E+04 1.04E+01 121 3.80E-01 4.15E+04 1.05E+01 121 3.79E-01 4.18E+04 1.06E+01 121 3.78E-01 4.22E+04 1.07E+01 120 3.78E-01 4.25E+04 1.07E+01 120 3.77E-01 4.27E+04 1.08E+01 120 3.77E-01 4.31E+04 1.09E+01 120 3.76E-01 4.35E+04 1.10E+01 120 3.76E-01 4.38E+04 1.11E+01 120 3.75E-01 4.42E+04 1.12E+01 119 3.75E-01 4.46E+04 1.13E+01 119 3.74E-01 4.49E+04 1.14E+01 119 3.74E-01 4.53E+04 1.14E+01 119 3.73E-01
133
Table 3 - FCD=10 – Results of Numerical Simulation and Dimensionless Time, Pressure and Rate - continued
t [days] tDxf q [stb/day] qD 4.57E+04 1.15E+01 119 3.73E-01 4.60E+04 1.16E+01 119 3.72E-01 4.64E+04 1.17E+01 118 3.72E-01 4.67E+04 1.18E+01 118 3.71E-01 4.71E+04 1.19E+01 118 3.71E-01 4.75E+04 1.20E+01 118 3.70E-01 4.78E+04 1.21E+01 118 3.70E-01 4.82E+04 1.22E+01 118 3.69E-01 4.86E+04 1.23E+01 118 3.69E-01 4.89E+04 1.24E+01 117 3.68E-01 4.93E+04 1.25E+01 117 3.68E-01 4.97E+04 1.26E+01 117 3.67E-01 5.00E+04 1.26E+01 117 3.67E-01 5.04E+04 1.27E+01 117 3.66E-01 5.08E+04 1.28E+01 117 3.66E-01 5.10E+04 1.29E+01 117 3.66E-01 5.13E+04 1.30E+01 116 3.65E-01 5.16E+04 1.31E+01 116 3.65E-01 5.20E+04 1.31E+01 116 3.64E-01 5.24E+04 1.32E+01 116 3.64E-01 5.27E+04 1.33E+01 116 3.64E-01 5.31E+04 1.34E+01 116 3.63E-01 5.35E+04 1.35E+01 116 3.63E-01 5.38E+04 1.36E+01 115 3.62E-01 5.42E+04 1.37E+01 115 3.62E-01 5.46E+04 1.38E+01 115 3.62E-01 5.49E+04 1.39E+01 115 3.61E-01 5.53E+04 1.40E+01 115 3.61E-01 5.57E+04 1.41E+01 115 3.60E-01 5.60E+04 1.42E+01 115 3.60E-01 5.64E+04 1.43E+01 115 3.60E-01 5.68E+04 1.43E+01 114 3.59E-01 5.71E+04 1.44E+01 114 3.59E-01 5.75E+04 1.45E+01 114 3.58E-01 5.79E+04 1.46E+01 114 3.58E-01 5.82E+04 1.47E+01 114 3.58E-01 5.86E+04 1.48E+01 114 3.57E-01 5.89E+04 1.49E+01 114 3.57E-01 5.93E+04 1.50E+01 114 3.56E-01 5.97E+04 1.51E+01 113 3.56E-01 6.00E+04 1.52E+01 113 3.56E-01 6.04E+04 1.53E+01 113 3.55E-01 6.08E+04 1.54E+01 113 3.55E-01 6.11E+04 1.55E+01 113 3.55E-01 6.14E+04 1.55E+01 113 3.54E-01 6.16E+04 1.56E+01 113 3.54E-01
134
Table 4 - FCD=25– Results of Numerical Simulation and Dimensionless Time, Pressure and Rate
t [days] tDxf q [stb/day] qD p [psi] pD 0 0 0 0 5,000 n/a
1.16E-05 2.92E-09 58,234 1.83E+02 4,992 5.42E-032.55E-05 6.44E-09 32,424 1.02E+02 4,989 8.02E-034.22E-05 1.07E-08 26,827 8.42E+01 4,986 9.84E-036.22E-05 1.57E-08 24,508 7.69E+01 4,984 1.12E-028.62E-05 2.18E-08 23,350 7.33E+01 4,983 1.22E-021.15E-04 2.91E-08 22,724 7.13E+01 4,982 1.30E-021.50E-04 3.78E-08 22,364 7.02E+01 4,981 1.35E-021.91E-04 4.83E-08 22,143 6.95E+01 4,980 1.39E-022.41E-04 6.09E-08 21,989 6.90E+01 4,980 1.41E-023.01E-04 7.60E-08 21,865 6.86E+01 4,980 1.43E-023.72E-04 9.41E-08 21,747 6.82E+01 4,980 1.44E-024.58E-04 1.16E-07 21,621 6.78E+01 4,979 1.45E-025.61E-04 1.42E-07 21,479 6.74E+01 4,979 1.47E-026.85E-04 1.73E-07 21,314 6.69E+01 4,979 1.48E-028.34E-04 2.11E-07 21,120 6.63E+01 4,979 1.49E-021.01E-03 2.56E-07 20,894 6.56E+01 4,979 1.51E-021.23E-03 3.10E-07 20,631 6.47E+01 4,978 1.52E-021.48E-03 3.75E-07 20,326 6.38E+01 4,978 1.54E-021.79E-03 4.53E-07 19,976 6.27E+01 4,978 1.57E-022.16E-03 5.46E-07 19,576 6.14E+01 4,977 1.60E-022.61E-03 6.58E-07 19,124 6.00E+01 4,977 1.63E-023.14E-03 7.93E-07 18,618 5.84E+01 4,976 1.67E-023.78E-03 9.54E-07 18,060 5.67E+01 4,976 1.72E-024.54E-03 1.15E-06 17,452 5.48E+01 4,975 1.77E-025.46E-03 1.38E-06 16,801 5.27E+01 4,974 1.83E-026.56E-03 1.66E-06 16,118 5.06E+01 4,973 1.89E-027.88E-03 1.99E-06 15,414 4.84E+01 4,972 1.97E-029.47E-03 2.39E-06 14,702 4.61E+01 4,971 2.05E-021.14E-02 2.88E-06 13,999 4.39E+01 4,970 2.13E-021.37E-02 3.46E-06 13,321 4.18E+01 4,969 2.22E-021.64E-02 4.15E-06 12,679 3.98E+01 4,967 2.32E-021.97E-02 4.98E-06 12,080 3.79E+01 4,966 2.43E-022.37E-02 5.99E-06 11,526 3.62E+01 4,964 2.53E-022.84E-02 7.19E-06 11,014 3.46E+01 4,963 2.65E-023.41E-02 8.63E-06 10,535 3.31E+01 4,961 2.76E-024.10E-02 1.04E-05 10,083 3.16E+01 4,959 2.89E-024.92E-02 1.24E-05 9,653 3.03E+01 4,957 3.01E-025.90E-02 1.49E-05 9,236 2.90E+01 4,956 3.15E-027.08E-02 1.79E-05 8,831 2.77E+01 4,954 3.29E-028.50E-02 2.15E-05 8,439 2.65E+01 4,951 3.44E-021.02E-01 2.58E-05 8,062 2.53E+01 4,949 3.59E-021.22E-01 3.09E-05 7,703 2.42E+01 4,947 3.76E-021.47E-01 3.71E-05 7,360 2.31E+01 4,945 3.93E-02
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
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 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
Table 4 - FCD=25– Results of Numerical Simulation and Dimensionless Time, Pressure and Rate - continued
t [days] tDxf q [stb/day] qD 1.40E+04 3.55E+00 154 4.82E-01 1.43E+04 3.62E+00 153 4.80E-01 1.47E+04 3.71E+00 152 4.78E-01 1.51E+04 3.80E+00 151 4.75E-01 1.54E+04 3.90E+00 151 4.73E-01 1.58E+04 3.99E+00 150 4.71E-01 1.61E+04 4.08E+00 149 4.68E-01 1.65E+04 4.17E+00 149 4.66E-01 1.68E+04 4.26E+00 148 4.64E-01 1.72E+04 4.34E+00 147 4.62E-01 1.75E+04 4.43E+00 147 4.60E-01 1.79E+04 4.53E+00 146 4.58E-01 1.83E+04 4.62E+00 145 4.56E-01 1.86E+04 4.71E+00 145 4.55E-01 1.90E+04 4.80E+00 144 4.53E-01 1.94E+04 4.90E+00 144 4.51E-01 1.97E+04 4.99E+00 143 4.49E-01 2.01E+04 5.08E+00 143 4.48E-01 2.04E+04 5.14E+00 142 4.46E-01 2.06E+04 5.21E+00 142 4.45E-01 2.10E+04 5.30E+00 141 4.44E-01 2.13E+04 5.39E+00 141 4.42E-01 2.17E+04 5.49E+00 140 4.41E-01 2.21E+04 5.58E+00 140 4.39E-01 2.24E+04 5.67E+00 140 4.38E-01 2.28E+04 5.76E+00 139 4.36E-01 2.32E+04 5.85E+00 139 4.35E-01 2.35E+04 5.95E+00 138 4.34E-01 2.39E+04 6.04E+00 138 4.32E-01 2.43E+04 6.13E+00 137 4.31E-01 2.45E+04 6.19E+00 137 4.30E-01 2.47E+04 6.25E+00 137 4.30E-01 2.51E+04 6.34E+00 136 4.28E-01 2.55E+04 6.43E+00 136 4.27E-01 2.58E+04 6.53E+00 136 4.26E-01 2.62E+04 6.62E+00 135 4.25E-01 2.66E+04 6.71E+00 135 4.24E-01 2.69E+04 6.80E+00 135 4.23E-01 2.73E+04 6.90E+00 134 4.21E-01 2.77E+04 6.99E+00 134 4.20E-01 2.80E+04 7.08E+00 134 4.19E-01 2.84E+04 7.17E+00 133 4.18E-01 2.87E+04 7.26E+00 133 4.17E-01 2.91E+04 7.36E+00 133 4.16E-01 2.94E+04 7.43E+00 132 4.15E-01
138
Table 4 - FCD=25– Results of Numerical Simulation and Dimensionless Time, Pressure and Rate - continued
t [days] tDxf q [stb/day] qD 2.97E+04 7.50E+00 132 4.15E-01 3.00E+04 7.59E+00 132 4.14E-01 3.04E+04 7.69E+00 132 4.13E-01 3.08E+04 7.78E+00 131 4.12E-01 3.11E+04 7.87E+00 131 4.11E-01 3.15E+04 7.96E+00 131 4.10E-01 3.19E+04 8.05E+00 130 4.09E-01 3.22E+04 8.15E+00 130 4.08E-01 3.26E+04 8.24E+00 130 4.07E-01 3.30E+04 8.33E+00 130 4.07E-01 3.33E+04 8.42E+00 129 4.06E-01 3.37E+04 8.52E+00 129 4.05E-01 3.41E+04 8.61E+00 129 4.04E-01 3.44E+04 8.70E+00 129 4.03E-01 3.48E+04 8.79E+00 128 4.03E-01 3.52E+04 8.88E+00 128 4.02E-01 3.54E+04 8.94E+00 128 4.01E-01 3.56E+04 9.00E+00 128 4.01E-01 3.60E+04 9.09E+00 128 4.00E-01 3.63E+04 9.18E+00 127 3.99E-01 3.67E+04 9.28E+00 127 3.99E-01 3.71E+04 9.37E+00 127 3.98E-01 3.74E+04 9.46E+00 127 3.97E-01 3.78E+04 9.55E+00 126 3.96E-01 3.82E+04 9.65E+00 126 3.96E-01 3.85E+04 9.74E+00 126 3.95E-01 3.89E+04 9.83E+00 126 3.94E-01 3.93E+04 9.92E+00 125 3.94E-01 3.96E+04 1.00E+01 125 3.93E-01 4.00E+04 1.01E+01 125 3.92E-01 4.04E+04 1.02E+01 125 3.92E-01 4.07E+04 1.03E+01 125 3.91E-01 4.11E+04 1.04E+01 124 3.91E-01 4.15E+04 1.05E+01 124 3.90E-01 4.18E+04 1.06E+01 124 3.89E-01 4.22E+04 1.07E+01 124 3.89E-01 4.25E+04 1.07E+01 124 3.88E-01 4.27E+04 1.08E+01 124 3.88E-01 4.31E+04 1.09E+01 123 3.87E-01 4.35E+04 1.10E+01 123 3.87E-01 4.38E+04 1.11E+01 123 3.86E-01 4.42E+04 1.12E+01 123 3.85E-01 4.46E+04 1.13E+01 123 3.85E-01 4.49E+04 1.14E+01 122 3.84E-01 4.53E+04 1.14E+01 122 3.84E-01
139
Table 4 - FCD=25– Results of Numerical Simulation and Dimensionless Time, Pressure and Rate - continued
t [days] tDxf q [stb/day] qD 4.57E+04 1.15E+01 122 3.83E-01 4.60E+04 1.16E+01 122 3.83E-01 4.64E+04 1.17E+01 122 3.82E-01 4.67E+04 1.18E+01 122 3.82E-01 4.71E+04 1.19E+01 121 3.81E-01 4.75E+04 1.20E+01 121 3.81E-01 4.78E+04 1.21E+01 121 3.80E-01 4.82E+04 1.22E+01 121 3.80E-01 4.86E+04 1.23E+01 121 3.79E-01 4.89E+04 1.24E+01 121 3.79E-01 4.93E+04 1.25E+01 120 3.78E-01 4.97E+04 1.26E+01 120 3.78E-01 5.00E+04 1.26E+01 120 3.77E-01 5.04E+04 1.27E+01 120 3.77E-01 5.08E+04 1.28E+01 120 3.76E-01 5.10E+04 1.29E+01 120 3.76E-01 5.13E+04 1.30E+01 120 3.75E-01 5.16E+04 1.31E+01 120 3.75E-01 5.20E+04 1.31E+01 119 3.75E-01 5.24E+04 1.32E+01 119 3.74E-01 5.27E+04 1.33E+01 119 3.74E-01 5.31E+04 1.34E+01 119 3.73E-01 5.35E+04 1.35E+01 119 3.73E-01 5.38E+04 1.36E+01 119 3.72E-01 5.42E+04 1.37E+01 119 3.72E-01 5.46E+04 1.38E+01 118 3.71E-01 5.49E+04 1.39E+01 118 3.71E-01 5.53E+04 1.40E+01 118 3.71E-01 5.57E+04 1.41E+01 118 3.70E-01 5.60E+04 1.42E+01 118 3.70E-01 5.64E+04 1.43E+01 118 3.69E-01 5.68E+04 1.43E+01 118 3.69E-01 5.71E+04 1.44E+01 117 3.69E-01 5.75E+04 1.45E+01 117 3.68E-01 5.79E+04 1.46E+01 117 3.68E-01 5.82E+04 1.47E+01 117 3.67E-01 5.86E+04 1.48E+01 117 3.67E-01 5.89E+04 1.49E+01 117 3.67E-01 5.93E+04 1.50E+01 117 3.66E-01 5.97E+04 1.51E+01 117 3.66E-01 6.00E+04 1.52E+01 116 3.65E-01 6.04E+04 1.53E+01 116 3.65E-01 6.08E+04 1.54E+01 116 3.65E-01 6.11E+04 1.55E+01 116 3.64E-01 6.14E+04 1.55E+01 116 3.64E-01 6.16E+04 1.56E+01 116 3.64E-01
140
Table 5 – FCD=100 – Results of Numerical Simulation and Dimensionless Time, Pressure and Rate
t [days] tDxf q [stb/day] qD p [psi] pD 0 0 0 0 5,000 n/a
1.16E-05 2.92E-09 116,839 3.67E+02 4,996 2.70E-032.55E-05 6.44E-09 65,262 2.05E+02 4,994 3.99E-034.22E-05 1.07E-08 54,250 1.70E+02 4,993 4.89E-036.22E-05 1.57E-08 49,678 1.56E+02 4,992 5.55E-038.62E-05 2.18E-08 47,341 1.49E+02 4,991 6.04E-031.15E-04 2.91E-08 46,040 1.44E+02 4,991 6.41E-031.50E-04 3.78E-08 45,279 1.42E+02 4,991 6.67E-031.91E-04 4.83E-08 44,809 1.41E+02 4,990 6.86E-032.41E-04 6.09E-08 44,488 1.40E+02 4,990 6.98E-033.01E-04 7.60E-08 44,231 1.39E+02 4,990 7.07E-033.72E-04 9.41E-08 43,991 1.38E+02 4,990 7.14E-034.58E-04 1.16E-07 43,735 1.37E+02 4,990 7.19E-035.61E-04 1.42E-07 43,448 1.36E+02 4,990 7.24E-036.85E-04 1.73E-07 43,113 1.35E+02 4,990 7.30E-038.34E-04 2.11E-07 42,722 1.34E+02 4,990 7.37E-031.01E-03 2.56E-07 42,266 1.33E+02 4,989 7.44E-031.23E-03 3.10E-07 41,734 1.31E+02 4,989 7.53E-031.48E-03 3.75E-07 41,118 1.29E+02 4,989 7.64E-031.79E-03 4.53E-07 40,410 1.27E+02 4,989 7.76E-032.16E-03 5.46E-07 39,600 1.24E+02 4,989 7.91E-032.61E-03 6.58E-07 38,684 1.21E+02 4,989 8.07E-033.14E-03 7.93E-07 37,659 1.18E+02 4,988 8.27E-033.78E-03 9.54E-07 36,526 1.15E+02 4,988 8.49E-034.54E-03 1.15E-06 35,290 1.11E+02 4,988 8.75E-035.46E-03 1.38E-06 33,965 1.07E+02 4,987 9.04E-036.56E-03 1.66E-06 32,572 1.02E+02 4,987 9.36E-037.88E-03 1.99E-06 31,133 9.77E+01 4,986 9.72E-039.47E-03 2.39E-06 29,672 9.31E+01 4,986 1.01E-021.14E-02 2.88E-06 28,223 8.86E+01 4,985 1.06E-021.37E-02 3.46E-06 26,819 8.42E+01 4,984 1.10E-021.64E-02 4.15E-06 25,480 8.00E+01 4,984 1.15E-021.97E-02 4.98E-06 24,223 7.60E+01 4,983 1.21E-022.37E-02 5.99E-06 23,052 7.23E+01 4,982 1.26E-022.84E-02 7.19E-06 21,955 6.89E+01 4,981 1.32E-023.41E-02 8.63E-06 20,917 6.56E+01 4,980 1.38E-024.10E-02 1.04E-05 19,923 6.25E+01 4,980 1.45E-024.92E-02 1.24E-05 18,959 5.95E+01 4,979 1.52E-025.90E-02 1.49E-05 18,015 5.65E+01 4,978 1.59E-027.08E-02 1.79E-05 17,088 5.36E+01 4,976 1.67E-028.50E-02 2.15E-05 16,179 5.08E+01 4,975 1.75E-021.02E-01 2.58E-05 15,296 4.80E+01 4,974 1.84E-021.22E-01 3.09E-05 14,441 4.53E+01 4,973 1.94E-021.47E-01 3.71E-05 13,617 4.27E+01 4,971 2.05E-02
141
Table 5 – FCD=100 – 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 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
Table 5 – FCD=100 – Results of Numerical Simulation and Dimensionless Time, Pressure and Rate - continued
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
Table 5 – FCD=100 – Results of Numerical Simulation and Dimensionless Time, Pressure and Rate - continued
t [days] tDxf q [stb/day] qD 1.40E+04 3.55E+00 156 4.91E-01 1.43E+04 3.62E+00 156 4.89E-01 1.47E+04 3.71E+00 155 4.86E-01 1.51E+04 3.80E+00 154 4.84E-01 1.54E+04 3.90E+00 153 4.81E-01 1.58E+04 3.99E+00 153 4.79E-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+00 149 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 4.59E-01 1.97E+04 4.99E+00 146 4.57E-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 4.53E-01 2.10E+04 5.30E+00 144 4.51E-01 2.13E+04 5.39E+00 143 4.50E-01 2.17E+04 5.49E+00 143 4.48E-01 2.21E+04 5.58E+00 142 4.47E-01 2.24E+04 5.67E+00 142 4.45E-01 2.28E+04 5.76E+00 141 4.44E-01 2.32E+04 5.85E+00 141 4.42E-01 2.35E+04 5.95E+00 141 4.41E-01 2.39E+04 6.04E+00 140 4.40E-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 4.37E-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 4.33E-01 2.62E+04 6.62E+00 138 4.32E-01 2.66E+04 6.71E+00 137 4.30E-01 2.69E+04 6.80E+00 137 4.29E-01 2.73E+04 6.90E+00 136 4.28E-01 2.77E+04 6.99E+00 136 4.27E-01 2.80E+04 7.08E+00 136 4.26E-01 2.84E+04 7.17E+00 135 4.25E-01 2.87E+04 7.26E+00 135 4.24E-01 2.91E+04 7.36E+00 135 4.23E-01 2.94E+04 7.43E+00 135 4.22E-01
144
Table 5 – FCD=100 – Results of Numerical Simulation and Dimensionless Time, Pressure and Rate - continued
t [days] tDxf q [stb/day] qD 2.97E+04 7.50E+00 134 4.21E-01 3.00E+04 7.59E+00 134 4.20E-01 3.04E+04 7.69E+00 134 4.19E-01 3.08E+04 7.78E+00 133 4.18E-01 3.11E+04 7.87E+00 133 4.17E-01 3.15E+04 7.96E+00 133 4.17E-01 3.19E+04 8.05E+00 132 4.16E-01 3.22E+04 8.15E+00 132 4.15E-01 3.26E+04 8.24E+00 132 4.14E-01 3.30E+04 8.33E+00 132 4.13E-01 3.33E+04 8.42E+00 131 4.12E-01 3.37E+04 8.52E+00 131 4.11E-01 3.41E+04 8.61E+00 131 4.10E-01 3.44E+04 8.70E+00 131 4.10E-01 3.48E+04 8.79E+00 130 4.09E-01 3.52E+04 8.88E+00 130 4.08E-01 3.54E+04 8.94E+00 130 4.08E-01 3.56E+04 9.00E+00 130 4.07E-01 3.60E+04 9.09E+00 129 4.06E-01 3.63E+04 9.18E+00 129 4.05E-01 3.67E+04 9.28E+00 129 4.05E-01 3.71E+04 9.37E+00 129 4.04E-01 3.74E+04 9.46E+00 129 4.03E-01 3.78E+04 9.55E+00 128 4.03E-01 3.82E+04 9.65E+00 128 4.02E-01 3.85E+04 9.74E+00 128 4.01E-01 3.89E+04 9.83E+00 128 4.00E-01 3.93E+04 9.92E+00 127 4.00E-01 3.96E+04 1.00E+01 127 3.99E-01 4.00E+04 1.01E+01 127 3.98E-01 4.04E+04 1.02E+01 127 3.98E-01 4.07E+04 1.03E+01 127 3.97E-01 4.11E+04 1.04E+01 126 3.96E-01 4.15E+04 1.05E+01 126 3.96E-01 4.18E+04 1.06E+01 126 3.95E-01 4.22E+04 1.07E+01 126 3.95E-01 4.25E+04 1.07E+01 126 3.94E-01 4.27E+04 1.08E+01 125 3.94E-01 4.31E+04 1.09E+01 125 3.93E-01 4.35E+04 1.10E+01 125 3.92E-01 4.38E+04 1.11E+01 125 3.92E-01 4.42E+04 1.12E+01 125 3.91E-01 4.46E+04 1.13E+01 125 3.91E-01 4.49E+04 1.14E+01 124 3.90E-01 4.53E+04 1.14E+01 124 3.90E-01
145
Table 5 – FCD=100 – Results of Numerical Simulation and Dimensionless Time, Pressure and Rate - continued
t [days] tDxf q [stb/day] qD 4.57E+04 1.15E+01 124 3.89E-01 4.60E+04 1.16E+01 124 3.88E-01 4.64E+04 1.17E+01 124 3.88E-01 4.67E+04 1.18E+01 123 3.87E-01 4.71E+04 1.19E+01 123 3.87E-01 4.75E+04 1.20E+01 123 3.86E-01 4.78E+04 1.21E+01 123 3.86E-01 4.82E+04 1.22E+01 123 3.85E-01 4.86E+04 1.23E+01 123 3.85E-01 4.89E+04 1.24E+01 122 3.84E-01 4.93E+04 1.25E+01 122 3.84E-01 4.97E+04 1.26E+01 122 3.83E-01 5.00E+04 1.26E+01 122 3.83E-01 5.04E+04 1.27E+01 122 3.82E-01 5.08E+04 1.28E+01 122 3.82E-01 5.10E+04 1.29E+01 122 3.81E-01 5.13E+04 1.30E+01 121 3.81E-01 5.16E+04 1.31E+01 121 3.81E-01 5.20E+04 1.31E+01 121 3.80E-01 5.24E+04 1.32E+01 121 3.80E-01 5.27E+04 1.33E+01 121 3.79E-01 5.31E+04 1.34E+01 121 3.79E-01 5.35E+04 1.35E+01 121 3.78E-01 5.38E+04 1.36E+01 120 3.78E-01 5.42E+04 1.37E+01 120 3.77E-01 5.46E+04 1.38E+01 120 3.77E-01 5.49E+04 1.39E+01 120 3.76E-01 5.53E+04 1.40E+01 120 3.76E-01 5.57E+04 1.41E+01 120 3.76E-01 5.60E+04 1.42E+01 120 3.75E-01 5.64E+04 1.43E+01 119 3.75E-01 5.68E+04 1.43E+01 119 3.74E-01 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 3.73E-01 5.86E+04 1.48E+01 119 3.72E-01 5.89E+04 1.49E+01 119 3.72E-01 5.93E+04 1.50E+01 118 3.71E-01 5.97E+04 1.51E+01 118 3.71E-01 6.00E+04 1.52E+01 118 3.71E-01 6.04E+04 1.53E+01 118 3.70E-01 6.08E+04 1.54E+01 118 3.70E-01 6.11E+04 1.55E+01 118 3.70E-01 6.14E+04 1.55E+01 118 3.69E-01 6.16E+04 1.56E+01 118 3.69E-01
146
Table 6 – FCD=500 – Results of Numerical Simulation and Dimensionless Time, Pressure and Rate
t [days] tDxf q [stb/day] qD p [psi] pD 0 0 0 0 5,000 n/a
1.16E-05 2.92E-09 264,200 8.29E+02 4,998 1.20E-032.55E-05 6.44E-09 148,579 4.66E+02 4,998 1.76E-034.22E-05 1.07E-08 122,893 3.86E+02 4,997 2.16E-036.22E-05 1.57E-08 111,672 3.50E+02 4,997 2.45E-038.62E-05 2.18E-08 105,603 3.31E+02 4,996 2.68E-031.15E-04 2.91E-08 101,959 3.20E+02 4,996 2.86E-031.50E-04 3.78E-08 99,657 3.13E+02 4,996 3.00E-031.91E-04 4.83E-08 98,169 3.08E+02 4,996 3.10E-032.41E-04 6.09E-08 97,166 3.05E+02 4,996 3.17E-033.01E-04 7.60E-08 96,421 3.03E+02 4,995 3.23E-033.72E-04 9.41E-08 95,772 3.01E+02 4,995 3.27E-034.58E-04 1.16E-07 95,116 2.98E+02 4,995 3.30E-035.61E-04 1.42E-07 94,389 2.96E+02 4,995 3.33E-036.85E-04 1.73E-07 93,545 2.94E+02 4,995 3.36E-038.34E-04 2.11E-07 92,557 2.90E+02 4,995 3.39E-031.01E-03 2.56E-07 91,405 2.87E+02 4,995 3.43E-031.23E-03 3.10E-07 90,058 2.83E+02 4,995 3.48E-031.48E-03 3.75E-07 88,495 2.78E+02 4,995 3.54E-031.79E-03 4.53E-07 86,694 2.72E+02 4,995 3.61E-032.16E-03 5.46E-07 84,630 2.66E+02 4,995 3.69E-032.61E-03 6.58E-07 82,289 2.58E+02 4,995 3.78E-033.14E-03 7.93E-07 79,662 2.50E+02 4,995 3.89E-033.78E-03 9.54E-07 76,742 2.41E+02 4,994 4.01E-034.54E-03 1.15E-06 73,543 2.31E+02 4,994 4.16E-035.46E-03 1.38E-06 70,093 2.20E+02 4,994 4.32E-036.56E-03 1.66E-06 66,446 2.08E+02 4,994 4.51E-037.88E-03 1.99E-06 62,649 1.97E+02 4,993 4.73E-039.47E-03 2.39E-06 58,762 1.84E+02 4,993 4.97E-031.14E-02 2.88E-06 54,879 1.72E+02 4,993 5.24E-031.37E-02 3.46E-06 51,086 1.60E+02 4,992 5.54E-031.64E-02 4.15E-06 47,450 1.49E+02 4,992 5.87E-031.97E-02 4.98E-06 44,024 1.38E+02 4,991 6.23E-032.37E-02 5.99E-06 40,832 1.28E+02 4,991 6.62E-032.84E-02 7.19E-06 37,874 1.19E+02 4,990 7.05E-033.41E-02 8.63E-06 35,127 1.10E+02 4,989 7.52E-034.10E-02 1.04E-05 32,554 1.02E+02 4,989 8.02E-034.92E-02 1.24E-05 30,132 9.45E+01 4,988 8.57E-035.90E-02 1.49E-05 27,835 8.73E+01 4,987 9.18E-037.08E-02 1.79E-05 25,657 8.05E+01 4,986 9.84E-038.50E-02 2.15E-05 23,595 7.40E+01 4,985 1.06E-021.02E-01 2.58E-05 21,672 6.80E+01 4,984 1.14E-021.22E-01 3.09E-05 19,891 6.24E+01 4,983 1.22E-021.47E-01 3.71E-05 18,254 5.73E+01 4,981 1.32E-02
147
Table 6 – FCD=500 – 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 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
Table 6 – FCD=500 – Results of Numerical Simulation and Dimensionless Time, Pressure and Rate - continued
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
Table 6 – FCD=500 – Results of Numerical Simulation and Dimensionless Time, Pressure and Rate - continued
t [days] tDxf q [stb/day] qD 1.40E+04 3.55E+00 157 4.93E-01 1.43E+04 3.62E+00 157 4.91E-01 1.47E+04 3.71E+00 156 4.89E-01 1.51E+04 3.80E+00 155 4.86E-01 1.54E+04 3.90E+00 154 4.83E-01 1.58E+04 3.99E+00 153 4.81E-01 1.61E+04 4.08E+00 153 4.79E-01 1.65E+04 4.17E+00 152 4.76E-01 1.68E+04 4.26E+00 151 4.74E-01 1.72E+04 4.34E+00 151 4.72E-01 1.75E+04 4.43E+00 150 4.70E-01 1.79E+04 4.53E+00 149 4.68E-01 1.83E+04 4.62E+00 149 4.66E-01 1.86E+04 4.71E+00 148 4.64E-01 1.90E+04 4.80E+00 147 4.63E-01 1.94E+04 4.90E+00 147 4.61E-01 1.97E+04 4.99E+00 146 4.59E-01 2.01E+04 5.08E+00 146 4.57E-01 2.04E+04 5.14E+00 145 4.56E-01 2.06E+04 5.21E+00 145 4.55E-01 2.10E+04 5.30E+00 144 4.53E-01 2.13E+04 5.39E+00 144 4.52E-01 2.17E+04 5.49E+00 143 4.50E-01 2.21E+04 5.58E+00 143 4.49E-01 2.24E+04 5.67E+00 142 4.47E-01 2.28E+04 5.76E+00 142 4.46E-01 2.32E+04 5.85E+00 142 4.44E-01 2.35E+04 5.95E+00 141 4.43E-01 2.39E+04 6.04E+00 141 4.41E-01 2.43E+04 6.13E+00 140 4.40E-01 2.45E+04 6.19E+00 140 4.39E-01 2.47E+04 6.25E+00 140 4.38E-01 2.51E+04 6.34E+00 139 4.37E-01 2.55E+04 6.43E+00 139 4.36E-01 2.58E+04 6.53E+00 139 4.35E-01 2.62E+04 6.62E+00 138 4.34E-01 2.66E+04 6.71E+00 138 4.32E-01 2.69E+04 6.80E+00 137 4.31E-01 2.73E+04 6.90E+00 137 4.30E-01 2.77E+04 6.99E+00 137 4.29E-01 2.80E+04 7.08E+00 136 4.28E-01 2.84E+04 7.17E+00 136 4.27E-01 2.87E+04 7.26E+00 136 4.26E-01 2.91E+04 7.36E+00 135 4.25E-01 2.94E+04 7.43E+00 135 4.24E-01
150
Table 6 – FCD=500 – Results of Numerical Simulation and Dimensionless Time, Pressure and Rate - continued
t [days] tDxf q [stb/day] qD 2.97E+04 7.50E+00 135 4.23E-01 3.00E+04 7.59E+00 135 4.22E-01 3.04E+04 7.69E+00 134 4.21E-01 3.08E+04 7.78E+00 134 4.20E-01 3.11E+04 7.87E+00 134 4.19E-01 3.15E+04 7.96E+00 133 4.18E-01 3.19E+04 8.05E+00 133 4.17E-01 3.22E+04 8.15E+00 133 4.16E-01 3.26E+04 8.24E+00 132 4.16E-01 3.30E+04 8.33E+00 132 4.15E-01 3.33E+04 8.42E+00 132 4.14E-01 3.37E+04 8.52E+00 132 4.13E-01 3.41E+04 8.61E+00 131 4.12E-01 3.44E+04 8.70E+00 131 4.11E-01 3.48E+04 8.79E+00 131 4.11E-01 3.52E+04 8.88E+00 131 4.10E-01 3.54E+04 8.94E+00 130 4.09E-01 3.56E+04 9.00E+00 130 4.09E-01 3.60E+04 9.09E+00 130 4.08E-01 3.63E+04 9.18E+00 130 4.07E-01 3.67E+04 9.28E+00 130 4.06E-01 3.71E+04 9.37E+00 129 4.06E-01 3.74E+04 9.46E+00 129 4.05E-01 3.78E+04 9.55E+00 129 4.04E-01 3.82E+04 9.65E+00 129 4.04E-01 3.85E+04 9.74E+00 128 4.03E-01 3.89E+04 9.83E+00 128 4.02E-01 3.93E+04 9.92E+00 128 4.01E-01 3.96E+04 1.00E+01 128 4.01E-01 4.00E+04 1.01E+01 127 4.00E-01 4.04E+04 1.02E+01 127 3.99E-01 4.07E+04 1.03E+01 127 3.99E-01 4.11E+04 1.04E+01 127 3.98E-01 4.15E+04 1.05E+01 127 3.97E-01 4.18E+04 1.06E+01 126 3.97E-01 4.22E+04 1.07E+01 126 3.96E-01 4.25E+04 1.07E+01 126 3.96E-01 4.27E+04 1.08E+01 126 3.95E-01 4.31E+04 1.09E+01 126 3.95E-01 4.35E+04 1.10E+01 126 3.94E-01 4.38E+04 1.11E+01 125 3.93E-01 4.42E+04 1.12E+01 125 3.93E-01 4.46E+04 1.13E+01 125 3.92E-01 4.49E+04 1.14E+01 125 3.92E-01 4.53E+04 1.14E+01 125 3.91E-01
151
Table 6 – FCD=500 – Results of Numerical Simulation and Dimensionless Time, Pressure and Rate - continued
t [days] tDxf q [stb/day] qD 4.57E+04 1.15E+01 124 3.91E-01 4.60E+04 1.16E+01 124 3.90E-01 4.64E+04 1.17E+01 124 3.89E-01 4.67E+04 1.18E+01 124 3.89E-01 4.71E+04 1.19E+01 124 3.88E-01 4.75E+04 1.20E+01 124 3.88E-01 4.78E+04 1.21E+01 123 3.87E-01 4.82E+04 1.22E+01 123 3.87E-01 4.86E+04 1.23E+01 123 3.86E-01 4.89E+04 1.24E+01 123 3.86E-01 4.93E+04 1.25E+01 123 3.85E-01 4.97E+04 1.26E+01 123 3.85E-01 5.00E+04 1.26E+01 122 3.84E-01 5.04E+04 1.27E+01 122 3.84E-01 5.08E+04 1.28E+01 122 3.83E-01 5.10E+04 1.29E+01 122 3.83E-01 5.13E+04 1.30E+01 122 3.83E-01 5.16E+04 1.31E+01 122 3.82E-01 5.20E+04 1.31E+01 122 3.82E-01 5.24E+04 1.32E+01 121 3.81E-01 5.27E+04 1.33E+01 121 3.81E-01 5.31E+04 1.34E+01 121 3.80E-01 5.35E+04 1.35E+01 121 3.80E-01 5.38E+04 1.36E+01 121 3.79E-01 5.42E+04 1.37E+01 121 3.79E-01 5.46E+04 1.38E+01 121 3.78E-01 5.49E+04 1.39E+01 120 3.78E-01 5.53E+04 1.40E+01 120 3.77E-01 5.57E+04 1.41E+01 120 3.77E-01 5.60E+04 1.42E+01 120 3.77E-01 5.64E+04 1.43E+01 120 3.76E-01 5.68E+04 1.43E+01 120 3.76E-01 5.71E+04 1.44E+01 120 3.75E-01 5.75E+04 1.45E+01 119 3.75E-01 5.79E+04 1.46E+01 119 3.75E-01 5.82E+04 1.47E+01 119 3.74E-01 5.86E+04 1.48E+01 119 3.74E-01 5.89E+04 1.49E+01 119 3.73E-01 5.93E+04 1.50E+01 119 3.73E-01 5.97E+04 1.51E+01 119 3.73E-01 6.00E+04 1.52E+01 119 3.72E-01 6.04E+04 1.53E+01 118 3.72E-01 6.08E+04 1.54E+01 118 3.71E-01 6.11E+04 1.55E+01 118 3.71E-01 6.14E+04 1.55E+01 118 3.71E-01 6.16E+04 1.56E+01 118 3.71E-01
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
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 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
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 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
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 2.66E+04 6.71E+00 137 4.30E-01 2.69E+04 6.80E+00 137 4.29E-01 2.73E+04 6.90E+00 136 4.28E-01 2.77E+04 6.99E+00 136 4.27E-01 2.80E+04 7.08E+00 136 4.26E-01 2.84E+04 7.17E+00 135 4.25E-01 2.87E+04 7.26E+00 135 4.24E-01 2.91E+04 7.36E+00 135 4.23E-01 2.94E+04 7.43E+00 134 4.22E-01 2.97E+04 7.50E+00 134 4.21E-01 3.00E+04 7.59E+00 134 4.20E-01 3.04E+04 7.69E+00 134 4.19E-01 3.08E+04 7.78E+00 133 4.18E-01 3.11E+04 7.87E+00 133 4.17E-01 3.15E+04 7.96E+00 133 4.16E-01 3.19E+04 8.05E+00 132 4.15E-01 3.22E+04 8.15E+00 132 4.14E-01 3.26E+04 8.24E+00 132 4.13E-01 3.30E+04 8.33E+00 132 4.13E-01 3.33E+04 8.42E+00 131 4.12E-01 3.37E+04 8.52E+00 131 4.11E-01 3.41E+04 8.61E+00 131 4.10E-01 3.44E+04 8.70E+00 130 4.09E-01 3.48E+04 8.79E+00 130 4.08E-01 3.52E+04 8.88E+00 130 4.08E-01 3.54E+04 8.94E+00 130 4.07E-01 3.56E+04 9.00E+00 130 4.07E-01 3.60E+04 9.09E+00 129 4.06E-01 3.63E+04 9.18E+00 129 4.05E-01 3.67E+04 9.28E+00 129 4.04E-01 3.71E+04 9.37E+00 129 4.04E-01 3.74E+04 9.46E+00 128 4.03E-01 3.78E+04 9.55E+00 128 4.02E-01 3.82E+04 9.65E+00 128 4.02E-01 3.85E+04 9.74E+00 128 4.01E-01 3.89E+04 9.83E+00 128 4.00E-01 3.93E+04 9.92E+00 127 3.99E-01 3.96E+04 1.00E+01 127 3.99E-01 4.00E+04 1.01E+01 127 3.98E-01 4.04E+04 1.02E+01 127 3.97E-01 4.07E+04 1.03E+01 126 3.97E-01 4.11E+04 1.04E+01 126 3.96E-01 4.15E+04 1.05E+01 126 3.95E-01
157
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 4.18E+04 1.06E+01 126 3.95E-01 4.22E+04 1.07E+01 126 3.94E-01 4.25E+04 1.07E+01 125 3.94E-01 4.27E+04 1.08E+01 125 3.93E-01 4.31E+04 1.09E+01 125 3.93E-01 4.35E+04 1.10E+01 125 3.92E-01 4.38E+04 1.11E+01 125 3.92E-01 4.42E+04 1.12E+01 125 3.91E-01 4.46E+04 1.13E+01 124 3.90E-01 4.49E+04 1.14E+01 124 3.90E-01 4.53E+04 1.14E+01 124 3.89E-01 4.57E+04 1.15E+01 124 3.89E-01 4.60E+04 1.16E+01 124 3.88E-01 4.64E+04 1.17E+01 124 3.88E-01 4.67E+04 1.18E+01 123 3.87E-01 4.71E+04 1.19E+01 123 3.86E-01 4.75E+04 1.20E+01 123 3.86E-01 4.78E+04 1.21E+01 123 3.85E-01 4.82E+04 1.22E+01 123 3.85E-01 4.86E+04 1.23E+01 123 3.84E-01 4.89E+04 1.24E+01 122 3.84E-01 4.93E+04 1.25E+01 122 3.83E-01 4.97E+04 1.26E+01 122 3.83E-01 5.00E+04 1.26E+01 122 3.82E-01 5.04E+04 1.27E+01 122 3.82E-01 5.08E+04 1.28E+01 122 3.81E-01 5.10E+04 1.29E+01 121 3.81E-01 5.13E+04 1.30E+01 121 3.81E-01 5.16E+04 1.31E+01 121 3.80E-01 5.20E+04 1.31E+01 121 3.80E-01 5.24E+04 1.32E+01 121 3.79E-01 5.27E+04 1.33E+01 121 3.79E-01 5.31E+04 1.34E+01 121 3.78E-01 5.35E+04 1.35E+01 120 3.78E-01 5.38E+04 1.36E+01 120 3.77E-01 5.42E+04 1.37E+01 120 3.77E-01 5.46E+04 1.38E+01 120 3.77E-01 5.49E+04 1.39E+01 120 3.76E-01 5.53E+04 1.40E+01 120 3.76E-01 5.57E+04 1.41E+01 120 3.75E-01 5.60E+04 1.42E+01 119 3.75E-01 5.64E+04 1.43E+01 119 3.74E-01 5.68E+04 1.43E+01 119 3.74E-01
158
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 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
159
160
Table 8 – FCD=100 – Results of Numerical Simulation and Dimensionless Time, and Pressure for xf/y=255, Constant Rate Case, Vertical Well
t [days] 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
Dimensionless Time tDxf
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=1
0.001
0.01
0.1
1
10
0.000001 0.00001 0.0001 0.001 0.01 0.1 1
Dimensionless Time tDxf
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=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
Dimensionless Time tDxf
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=10
0.001
0.01
0.1
1
10
0.000001 0.00001 0.0001 0.001 0.01 0.1 1
Dimensionless Time tDxf
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=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
Dimensionless Time tDxf
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=500
0.001
0.01
0.1
1
10
0.000001 0.00001 0.0001 0.001 0.01 0.1 1
Dimensionless Time tDxf
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
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
Dimensionless Time tDxf
Dim
ensi
on
less
Flo
w R
ate
qD
0.028
4.0
16
63.8
255
length to distance ratioxf
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
Dimensionless Time tDxf
Rec
ipro
cal D
imen
sio
nle
ss F
low
Rat
e 1/
qD
xf
y
length to distance ratio
FCD - dimensionless fracture conductivity
xf - fracture half-lengthy - distance between two fractures
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
Figure 8 – Finite conductivity type curve with deviations for fracture face interference for constant pressure case and FCD=5 – Line source – Vertical well Figure 8A – Dimensionless rate versus dimensionless time with deviations for fracture face interference for constant pressure case and FCD=5 – 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
Dimensionless Time tDxf
Dim
ensi
on
less
Flo
w R
ate
qD
0.028
4.0
16.0
63.8
255
length to distance ratioxf
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
Dimensionless Time tDxf
Rec
ipro
cal D
imen
sion
less
Flo
w R
ate
1/q
D
xf
y
length to distance ratio
FCD - dimensionless fracture conductivity
xf - fracture half-lengthy - distance between two fractures
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
Dimensionless Time tDxf
Dim
ensi
on
less
Flo
w R
ate
qD
0.028
4.0
16.0
63.8
255
length to distance ratioxf
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
Dimensionless Time tDxf
Rec
ipro
cal D
imen
sio
nle
ss F
low
Rat
e 1/
qD
xf
y
length to distance ratio
FCD - dimensionless fracture conductivity
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
nle
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
Dimensionless Time tDxf
Dim
ensi
on
less
Flo
w R
ate
qD
0.028
4.0
16.0
63.8
255
length to distance ratioxf
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
Dimensionless Time tDxf
Rec
ipro
cal D
imen
sio
nle
ss F
low
Rat
e 1/
qD
xf
y
length to distance ratio
FCD - dimensionless fracture conductivity
xf - fracture half-lengthy - distance between two fractures
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
Figure 11 – Finite conductivity type curve with deviations for fracture face interference for constant pressure case and FCD=100 – Line source – Vertical well Figure 11A – Dimensionless rate versus dimensionless time with deviations for fracture face interference for constant pressure case and FCD=100 – Line source – Vertical well
0.1
1
10
100
1000
0.000001 0.00001 0.0001 0.001 0.01 0.1 1
Dimensionless Time tDxf
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
Dimensionless Time tDxf
Rec
ipro
cal D
imen
sio
nle
ss F
low
Rat
e 1/
qD
xf
y
length to distance ratio
FCD - dimensionless fracture conductivity
xf - fracture half-lengthy - distance between two fractures
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
178
Figure 12 – Finite conductivity type curve with deviations for fracture face interference for constant pressure case and FCD=500 – Line source – Vertical well Figure 12A – Dimensionless rate versus dimensionless time with deviations for fracture face interference for constant pressure case and FCD=500 – Line source – Vertical well
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
Dimensionless Time tDxf
Dim
ensi
on
less
Flo
w R
ate
qD
0.028
4.0
16.0
63.8
255
length to distance ratioxf
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
Dimensionless Time tDxf
Rec
ipro
cal D
imen
sio
nle
ss F
low
Rat
e 1/
qD
xf
y
length to distance ratio
FCD - dimensionless fracture conductivity
xf - fracture half-lengthy - distance between two fractures
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
Dimensionless Time tDxf
Dim
ensi
on
less
Pre
ssu
re P
D
4.0
16.0
63.8
255
FCD - dimensionless fracture conductivity
xf - fracture half-lengthy - distance between two fractures
length to distance ratio
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
Dimensionless Time tDxf
Dim
ensi
on
less
Pre
ssu
re P
D
4.0
16.0
63.8
255
FCD - dimensionless fracture conductivity
xf - fracture half-lengthy - distance between two fractures
length to distance ratio
xf
y
FCD=5
5
25
10
FCD
1
100
500
180
Figure 15 – Finite conductivity type curve with deviations for fracture face interference for constant rate case and FCD=10 – Point source – Horizontal well Figure 16 – Finite conductivity type curve with deviations for fracture face interference for constant rate case and FCD=25 – 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
Dimensionless Time tDxf
Dim
ensi
on
less
Pre
ssu
re P
D
4.0
16.0
63.8
255
FCD - dimensionless fracture conductivity
xf - fracture half-lengthy - distance between two fractures
length to distance ratio
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
Dimensionless Time tDxf
Dim
ensi
on
less
Pre
ssu
re P
D
4.0
16.0
63.8
255
FCD - dimensionless fracture conductivity
xf - fracture half-lengthy - distance between two fractures
length to distance ratio
xf
y
FCD=25
5
25
10
FCD
1
100
500
181
Figure 17 – Finite conductivity type curve with deviations for fracture face interference for constant rate case and FCD=100 – Point source – Horizontal well Figure 18 – Finite conductivity type curve with deviations for fracture face interference for constant rate case and FCD=500 – 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
Dimensionless Time tDxf
Dim
ensi
on
less
Pre
ssu
re P
D
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
0.001
0.01
0.1
1
10
0.000001 0.00001 0.0001 0.001 0.01 0.1 1
Dimensionless Time tDxf
Dim
ensi
on
less
Pre
ssu
re P
D
4.0
16.0
63.8
255
FCD - dimensionless fracture conductivity
xf - fracture half-lengthy - distance between two fractures
length to distance ratio
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
Dimensionless Time tDxf
Dim
en
sio
nle
ss
Flo
w R
ate
qD
4.0
16
63.8
255
FCD=1
length to distance ratioxf
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
Dimensionless Time tDxf
5
FCD
1
500
100
10
25
length to distance ratio
xf
y
FCD - dimensionless fracture conductivity
xf - fracture half-lengthy - distance between two fractures
FCD=1
Rec
ipro
cal
Dim
ensi
on
less
Flo
w R
ate
Fo
r T
wo
Fra
ctu
res
Sys
tem
2/q
Dtf
s
183
Figure 20 – Finite conductivity type curve with deviations for fracture face interference for constant pressure case and FCD=5 – Point source – Horizontal well Figure 20A – Dimensionless rate versus dimensionless time with deviations for fracture face interference for constant pressure case and FCD=5 – Point source – Horizontal 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
Dimensionless Time tDxf
Dim
ensi
on
less
Flo
w R
ate
qD
4.0
16.0
63.8
255
FCD=5
length to distance ratioxf
y
Rec
ipro
cal
Dim
ensi
on
less
Flo
w R
ate
Fo
r T
wo
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
on
less
Flo
w R
ate
1/q
D
Dimensionless Time tDxf
FCD=5
5
FCD
1
500
100
10
25
length to distance ratio
xf
y
FCD - dimensionless fracture conductivity
xf - fracture half-lengthy - distance between two fractures
184
Figure 21 – Finite conductivity type curve with deviations for fracture face interference for constant pressure case and FCD=10 – Point source – Horizontal well Figure 21A – Dimensionless rate versus dimensionless time with deviations for fracture face interference for constant pressure case and FCD=10 – Point source – Horizontal 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
Dimensionless Time tDxf
Dim
ensi
on
less
Flo
w R
ate
qD
4.0
16.0
63.8
255
FCD=10
length to distance ratioxf
y
Rec
ipro
cal
Dim
ensi
on
less
Flo
w R
ate
Fo
r T
wo
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
on
less
Flo
w R
ate
1/q
D
Dimensionless Time tDxf
FCD=10
5
FCD
1
500
100
10
25
length to distance ratio
xf
y
FCD - dimensionless fracture conductivity
xf - fracture half-lengthy - distance between two fractures
185
Figure 22 – Finite conductivity type curve with deviations for fracture face interference for constant pressure case and FCD=25 – Point source – Horizontal well Figure 22A – Dimensionless rate versus dimensionless time with deviations for fracture face interference for constant pressure case and FCD=25 – Point source – Horizontal 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
Dimensionless Time tDxf
Dim
ensi
on
less
Flo
w R
ate
qD
4.0
16.0
63.8
255
FCD=25
length to distance ratioxf
y
Rec
ipro
cal
Dim
ensi
on
less
Flo
w R
ate
Fo
r T
wo
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
on
less
Flo
w R
ate
1/q
D
Dimensionless Time tDxf
FCD=25
5
FCD
1
500
100
10
25
length to distance ratio
xf
y
FCD - dimensionless fracture conductivity
xf - fracture half-lengthy - distance between two fractures
186
Figure 23 – Finite conductivity type curve with deviations for fracture face interference for constant pressure case and FCD=100 – Point source – Horizontal well Figure 23A – Dimensionless rate versus dimensionless time with deviations for fracture face interference for constant pressure case and FCD=100 – Point source – Horizontal 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
Dimensionless Time tDxf
Dim
ensi
on
less
Flo
w R
ate
qD
4.0
16.0
63.8
255
FCD=100
length to distance ratioxf
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
Dimensionless Time tDxf
5
FCD
1
500
100
10
25
length to distance ratio
xf
y
FCD - dimensionless fracture conductivity
xf - fracture half-lengthy - distance between two fractures
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
187
Figure 24 – Finite conductivity type curve with deviations for fracture face interference for constant pressure case and FCD=500 – Point source – Horizontal well Figure 24A – Dimensionless rate versus dimensionless time with deviations for fracture face interference for constant pressure case and FCD=500 – Point source – Horizontal well
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
Dimensionless Time tDxf
Dim
ensi
on
less
Flo
w R
ate
qD
4.0
16.0
63.8
255
FCD=500
length to distance ratioxf
y
Rec
ipro
cal
Dim
ensi
on
less
Flo
w R
ate
Fo
r T
wo
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
on
less
Flo
w R
ate
1/q
D
Dimensionless Time tDxf
FCD=500
5
FCD
1
500
100
10
25
length to distance ratio
xf
y
FCD - dimensionless fracture conductivity
xf - fracture half-lengthy - distance between two fractures
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]