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Catalog No. L00400e
New Concepts In Underground Storage Of Natural Gas
PO-50
Prepared for the
Pipeline Research Technical Committee
Of
Pipeline Research Council International, Inc.
Prepared by the following Research Agencies:
UNIVERSITY OF MICHIGAN
Authors: Various
Publication Date: March 1966
“This report is furnished to Pipeline Research Council International, Inc. (PRCI) under the terms of PRCI PO-50, between PRCI and UNIVERSITY OF MICHIGAN. The contents of this report are published as received from UNIVERSITY OF MICHIGAN. The opinions, findings, and conclusions expressed in the report are those of the authors and not necessarily those of PRCI, its member companies, or their representatives. Publication and dissemination of this report by PRCI should not be considered an endorsement by PRCI or UNIVERSITY OF MICHIGAN, or the accuracy or validity of any opinions, findings, or conclusions expressed herein. In publishing this report, PRCI makes no warranty or representation, expressed or implied, with respect to the accuracy, completeness, usefulness, or fitness for purpose of the information contained herein, or that the use of any information, method, process, or apparatus disclosed in this report may not infringe on privately owned rights. PRCI assumes no liability with respect to the use of, or for damages resulting from the use of, any information, method, process, or apparatus disclosed in this report. The text of this publication, or any part thereof, may not be reproduced or transmitted in any form by any means, electronic or mechanical, including photocopying, recording, storage in an information retrieval system, or otherwise, without the prior, written approval of PRCI.”
Pipeline Research Council International Catalog No. L00400e
Copyright, 1966 All Rights Reserved by Pipeline Research Council International, Inc.
PRCI Reports are Published by Technical Toolboxes, Inc.
3801 Kirby Drive, Suite 340 Houston, Texas 77098 Tel: 713-630-0505 Fax: 713-630-0560 Email: info@ttoolboxes.com
ACKNOWLEDGMENTS
The Project PO-50 "New Concepts in Underground Storage of Natural
Gas" was initiated during February 1963. It was terminated on March 31,
1966. The initiation of the project was largely due to the efforts of
Messrs. Thomas Walsh and William Morse of the Pipeline Research Council International.
During the three year tenure of this project, many individuals from the
Pipeline Research Council International, Natural Gas Storage Industry and The Univer-
sity of Michigan have actively participated in several phases of the
program.
The authors would like first to acknowledge assistance support
and direction provided by Dr. Donald L. Katz who served as principal
consultant. In many ways the entire research project, from inception
to conclusion, has been influenced and enhanced by his active participa-
tion in the form of suggestions, constructive criticism, guidance and
encouragement,
The work of the Supervising Committee, Messrs. Brooks (Chairman),
El l is,Grow,Jr. , Hedges, Stout and Walsh in maintaining continuity of
liaison with the industry and general direction of the program is
gratefully acknowledged.
Some of the cores, data and grouting materials were provided by
Consumers Power, Michigan Consolidated Gas, Natural Gas Storage Company
of Illinois, Southeastern Michigan Gas, Northern Illinois Gas, Diamond
A l k a l i , Halliburton and American Cyanamid Companies. Some of the in-
formation presented in Chapter 6 on Subsurface Nuclear Explosions was
provided by the U.S. Atomic Energy Commission, San Francisco Operations
Of f ice .
The work of Messrs. Wayne Dupree, Shiv Arora and Robert Reid in
the early phases of the program is also acknowledged.
i
This Page Intentionally Left Blank
SUPERVISING COMMITTEE FOR PROJECT PO-50
"New Concepts of Underground Gas Storage"
C. E. BROOKS, (CHAIRMAN)
Con-Gas Service Corporation
Four Gateway Center, Pittsburgh, Pennsylvania
J. H. N. ELLIS, Superintendent of Operations
Pacific Lighting Gas Supply Company
720 West Eighth Street, Los Angeles, California
G. C. GROW, JR., Chief Geologist - Eastern Area
Transcontinental Gas Pipe Line Corporation
744 Broad Street, Newark, New Jersey 07102
15222
90017
E. B. HEDGES, Manager - Gas Production & Transmission
Consumers Power Company
212 West Michigan Avenue, Jackson, Michigan 49201
C. E. STOUT, Vice President - Geology & Gas Production
Columbia Gas System Service Corporation
120 East 41st Street, New York, New York 10017
T. E. WALSH, (SECRETARY), Manager of Pipeline Research
Pipeline Research Council International
605 Third Avenue, New York, New York 10016
i i
This Page Intentionally Left Blank
PIPELINE RESEARCH COMMITTEE
W. B. HAAS, (CHAIRMAN), Vice President
Corporate Engineering Division, Northern Natural Gas Company
2223 Dodge Street, Omaha, Nebraska 68102
A. J. SHOUP, (VICE CHAIRMAN), Senior Vice President
Texas Eastern Transmission Corporation
2521, Houston, Texas 77001
KEITH BENTZ, Vice President - Transmission & Engineering
Natural Gas Pipeline Company of America
120 South Michigan Avenue, Chicago, Illinois 60603
S. A. BRADFIELD, Manager of Engineering
Southern California Gas Company
Box 3249 Terminal Annex, Los Angeles, California 90054
O. W. CLARK, Senior Vice President
Southern Natural Gas Company
Box 2563, Birmingham, Alabama 35202
R. H. CROWE, Chief Engineer
Transcontinental Gas Pipe Line Corporation
Box 1396, Houston, Texas 77001
J. F. EICHELMANN, Vice President & Executive Engineer
El Paso Natural Gas Company
Box 1492, El Paso, Texas 79999
J. L. GERE, Vice President - Research & Engineering
Cities Service Gas Company
Box 1995, Oklahoma City, Oklahoma
i i i
L. E. HANNA, Chief Engineer
Panhandle Eastern Pipe Line Company
Box 1348, Kansas City, Missouri 64141
G. E. MCKINLEY, Vice President - Engineering & Facility Planning
Con-Gas Service Corporation
Four Gateway Center, Pittsburgh, Pennsylvania 15222
R. D. MOREL, General Superintendent
Algonquin Gas Transmission Company
1284 Soldiers Field Road, Boston, Massachusetts 02135
L. D. MYERS, Chief Engineer
United Gas Pipeline Company
Box 1407, Shreveport, Louisiana 71102
S. ORLOFSKY, Vice President - Engineering & Research
Columbia Gas System Service Corporation
120 East 41 Street, New York, New York 10017
T. L. PELICAN, Senior Vice President
Colorado Interstate Gas Company
Box 1087, Colorado Springs, Colorado 80901
A. W. STANZEL, Chief Design Engineer
Michigan Wisconsin Pipe Line Company
One Woodward Avenue, Detroit, Michigan 48226
H. L. STOWERS, Vice President of Engineering
Texas Gas Transmission Corporation
Box 1160, Owensboro, Kentucky 43201
T. E. WALSH, (SECRETARY), Manager of Pipeline Research
Pipeline Research Council International
605 Third Avenue, New York, New York 10016
i v
TABLE OF CONTENTS
Page
CHAPTER 1. INTRODUCTION. . . . . . . . . . . . . . . . . . . . . 1
References. . . . . . . . . . . . . . . . . . . . . . . . . . 7
CHAPTER 2. PROBLEMS ASSOCIATED WITH LEAKS FROM OVERPRESSUREDSTORAGE RESERVOIRS. . . . . . . . . . . . . . . . . .
2.1 Leakage or Spill From Overpressured Reservoirs. . . .
2.2 Concept of Threshold Pressure . . . . . . . . . . . .
2.3 Prediction of Threshold Pressure From Core Properties
Example Calculations . . . . . . . . . . . . . . .
Preparation of Consolidated Natural Cores. . . . .
Preparation of Super-Permeable Laboratory Cores. .
Measurement of Porosity. . . . . . . . . , . . . .
Measure of Permeability. . . . . . . . . . . . . .
Measurement of Threshold Pressure. . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . .
CHAPTER 3. MECHANISM OF GAS LEAKAGE ACROSS A CAP ROCK. . . . . . 33
3.1 Introduction. . . . . . . . . . . . . . . . . . . . .
Equations Governing Gas-Water Flow in a Porous Medium
33
3.2 34
Darcy's Law. . . . . . . . . . . . . . . . . . . . 35
Mass Balance . . . . . . . . . . . . . . . . . . . 36
Relation between Saturations . . . . . . . . . . . 37
Capillary Pressure . . . . . . . . . . . . . . . . 37
Definit ion of Potentials . . . . . . . . . . . . . 38
Rearranged Equations . . . . . . . . . . . . . . . 38
v
9
10
14
15
17
18
19
21
21
24
32
Table of Contents (contd)
Page
3.3 Application of Equations to Leak across Cap Rock. . . 39
Initial Conditions (t = 0, . . . . . . . . 40
Boundary Conditions. . . . . . . . . . . . . . . . 40
3.4 Equations in Terms of Dimensionless Variables . . . . 41
Potential Equations. . . . . . . . . . . . . . . . 42
Initial Conditions . . . . . . . . . . . . . . . . 42
Boundary Conditions (for T>0). . . . . . . . . . . 42
Injection Rates. . . . . . . . . . . . . . . . . . . 42
3.5 Rearrangement of Equations in Terms of P and RVariables . . . . . . . . . . . . . . . . . . . . . . 44
3.6 Finite Difference Equations . . . . . . . . . . . . . 45
3.7 Finite Difference Approximation to Parabolic orR-Equation. . . . . . . . . . . . . . . . . . . . . . 46
3.8 Finite Difference Approximation to El l ipt ic orP-Equation. . . . . . . . . . . . . . . . . . . . . . 48
3.9 Solut ion of the Finite Dif ference Equations . . . . . 50
3.10 Computation of the Injection Terms. . . . . . . . . . 52
3.11 Iterative Technique for Improving Estimate ofInjection Terms . . . . . . . . . . . . . . . . . . . 54
3.12 Relative Permeability and Capillary PressureRelations , . . . . . . . . . . . . . . . . . . . . . 57
3.13 Results . . . . . . . . . . . . . . . . . . . . . . . 59
CHAPTER 4. PERFORMANCE OF STORAGE RESERVOIRS SUBJECT TO LEAKAGE. 7 5
4.1 Introduction and Model. . . . . . . . . . . . . . . . 75
Notation . . . . . . . . . . . . . . . . . . . . . 77
Governing Equations . . . . . . . . . . . . . . . . . 77
Dimensionless Form . . . . . . . . . . . . . . . . 78
Change of Variable . . . . . . . . . . . . . . . . 79
4.2
4.3 Formulation of Leak . . . . . . . . . . . . . . . . . 79
v i
Table of Contents (contd)
4.4
4.5
4.6
4.7
4.8
4.9
Special Case of Cylindrical Symmetry. . . . . . . .
Boundary and Initial Conditions. . . . . . . . .
Computed Results . . . . . . . . . . . . . . . .
Two-Dimensional Leakage Problem . . . . . . . . . .
Finite Difference Approximations. . . . . . . . . .
First Half Time Step (Implicit in Angular Direction)
Second Half Time Step (Implicit in Radial Direction)
Results . . . . . . . . . . . . . . . . . . . . . .
Run 6 (No Leak). . . . . . . . . . . . . . . . .
Run 5 (Single Leak). . . . . . . . . . . . . . .
Run 9 (Single Leak, with Threshold PressureEf fec t ) . . . . . . . . . . . . . . . . . . . . .
Run 10 (Row Leak). . . . . . . . . . . . . . . .
Page
80
81
82
86
87
88
91
93
94
94
95
95
CHAPTER 5. IMPERMEATION OF UNDERGROUND FORMATIONS. . . . . . . 107
5.1 The Grouting Materials. . . . . . . . . . . . . . . 109
Si l icate Grouts. . . . . . . . . . . . . . . . . 109
Chrome - Lignin Grouts . . . . . . . . . . . . . 110
Furfural Grouts. . . . . . . . . . . . . . . . . 110
AM- 9 Chemical Grouts . . . . . . . . . . . . . 112
SIROC Chemical Grouts. . . . . . . . . . . . . . 112
Polymer Grouts . . . . . . . . . . . . . . . . . 113
Herculox Grout . . . . . . . . . . . . . . . . . 113
5.2 Properties and Performance of Grouting Materials. . 113
Polymerization Mechanism . . . . . . . . . . . . 115
AM - 9 Gel Properties. . . . . . . . . . . . . . 115
AM - 9 Solution Properties . . . . . . . . . . . 116
SIROC Gel Properties . . . . . . . . . . . . . . 117
SIROC Solution Properties. . . . . . . . . . . . 119
v i i
Table of Contents (contd)
Page
5.3 Adhesion Between Grouts and Porous Formations. . . . 121
Mechanism of Bonding and Adhesion . . . . . . . . 121
Proposed Models for Polymer Grout Adhesion. . . . 127
5.4 Evaluation of Grouts by Laboratory Experiments . . . 128
Grout Inject ion - Curing and Testing of Cores . . 130
Grouting Core Samples . . . . . . . . . . . . . . 131
Results of Experimental Work on Evaluation ofG r o u t s . . . . . . . . . . . . . . . . . . . . . . 134
Compressive Adhesion Tests on Grouted Cores . . . 137
Microscopic Observations on Grouted CoreSpecimens . . . . . . . . . . . . . . . . . . . . 147
Photomicrographic Observations on Grouted CoreSpecimens . . . . . . . . . . . . . . . . . . . . 150
5.5 Practical Reservoir Engineering Calculations onInjection of Grouts. . . . . . . . . . . . . . . . . 152
Economic Evaluation in Aquifer Grouting . . . . . 160
5.6 Well Fracturing as Related to Grouting . . . . . . . 161
Literature Survey and Fracture DesignCalculations. . . . . . . . . . . . . . . . . . . 163
Fracture Extent . . . . . . . . . . . . . . . 163
Fracture Width. . . . . . . . . . . . . . . . 168
Pressure and Horsepower Requirements. . . . . . . 178
Fracturing Fluids . . . . . . . . . . . . . . . . 179
Propping Agent. . . . . . . . . . . . . . . . . . 182
Design Procedure. . . . . . . . . . . . . . . . . 186
Example Problem . . . . . . . . . . . . . . . . . 187
Data . . . . . . . . . . . . . . . . . . . . . 187
Nomenclature. . . . . . . . . . . . . . . . . 191
Laboratory Fracture - Grout Experiments . . . . . 195
Axial Fracture Data . . . . . . . . . . . . . . . 200
Radial Fracture Data. . . . . . . . . . . . . . . 201
References. . . . . . . . . . . . . . . . . . . . . . . . 203
v i i i
Table of Contents (contd)
Page
CHAPTER 6. UNDERGROUND STORAGE IN NON-POROUS SPACE. . . . . . . 207
6.1 Storage of Natural Gas in Salt Caverns . . . . . . . 210
Locations for Dissolved Salt Caverns. . . . . . . 210
Creation of Salt Cavern Reservoirs. . . . . . . . 212
Determination of the Size of Dissolved SaltCaverns . . . . . . . . . . . . . . . . . . . . . 213
Deliverability of Natural Gas from SaltCavern Storage. . . . . . . . . . . . . . . . . . 216
Stress Considerations . . . . . . . . . . . . . . 218
Stresses Induced in Formations Surrounding aSpherical Cavity. . . . . . . . . . . . . . . . . 220
Simplified Stress Calculations for Non-SphericalCavit ies in Salt . . . . . . . . . . . . . . . . . 222
Strength Data for Salt. . . . . . . . . . . . . . 224
Safety Considerations . . . . . . . . . . . . . . 225
Recovery of LP Gas from Caverns . . . . . . . . . 225
Applicat ion - A Case Study and Observationson Marysville Salt Cavern Gas Storage . . . . . . 228
Relationship between Cavern Pressure-GasProduction-Brine Injection. . . . . . . . . . . . 228
Review of Storage Data from Marysville Cavern . . 229
Effect of Gas Solubility in Brine . . . . . . . . 231
Stress Calculations . . . . . . . . . . . . . . . 232
6.2 Storage of Natural Gas in Cavities Induced byNuclear Explosions . . . . . . . . . . . . . . . . . 237
Storage Capacity of Nuclear Explosion Cavities. . 240
Size and Shape of Cavities Caused by NuclearExplosions. . . . . . . . . . . . . . . . . . . . 241
Damage from Seismic Effect of UndergroundNuclear Explosions. . . . . . . . . . . . . . . . 249
Radio Activi ty Distr ibution . . . . . . . . . . . 253
Economic Considerations . . . . . . . . . . . . . 255
6.3 Storage in Natural or Mined Cavities . . . . . . . . 255
Coal Mine Storage . . . . . . . . . . . . . . . . 257
Hard Rock Mined Storage . . . . . . . . . . . . . 257
i x
Table of Contents (contd)
Page
6.4 Underwater Storage of Natural Gas. . . . . . . . . . . . 258
Engineering Calculations for Underwater Storage . . . 260
Anchoring Force Requirements. . . . . . . . . . . . . 265
Storage Vessel Configuration. . . . . . . . . . . . . 266
Pressure Loading on Storage Vessel. . . . . . . . . . 269
General Design Concept and Considerations . . . . . . 271
References. . . . . . . . . . . . . . . . . . . . . . . . . . 274
APPENDIX A. THEORETICAL RESERVOIR ENGINEERING CALCULATIONSRELATED TO GROUT INJECTION. . . . . . . . . . . . . . . 279
Introduction. . . . . . . . . . . . . . . . . . . . . . . . . 280
Mathematical Models . . . . . . . . . . . . . . . . . . . . . 281
Equations Describing Grout Injection in OneDimension - (Cartesian Coordinates) . . . . . . . . . . . . . 285
Solutions for a Constant Viscosity Grout Solution-Plane Front . . . . . . . . . . . . . . . . . . . . . . . . 286
Solutions for a Variable Viscosity Grout Solution-Plane Front . . . . . . . . . . . . . . . . . . . . . . . . . 292
Solutions for Constant and Variable Viscosity GroutSolutions - Cylindrical Front . . . . . . . . . . . . . . . . 298
Conclusions and Recommendations for Future Work . . . . . . . 303
Sample Calculations for Analytical and TabulatedSolutions . . . . . . . . . . . . . . . . . . . . . . . . . . 306
APPENDIX A-1. NUMERICAL TECHNIQUES FOR THE GROUT INJECTIONPROBLEM WITH A VARIABLE VISCOSITY GROUTSOLUTION-PLANE FRONT. . . . . . . . . . . . . . . . . 309
APPENDIX A-2. NUMERICAL TECHNIQUES FOR THE GROUT INJECTIONPROBLEM WITH A CONSTANT OR VARIABLE VISCOSITYGROUT SOLUTION - CYLINDRICAL FRONT. . . . . . . . . . 313
APPENDIX B SOLUTION OF A SYSTEM OF LINEAR EQUATIONSHAVING A TRIDIAGONAL COEFFICIENT MATRIX . . 321
APPENDIX C COMPUTER PROGRAMS FOR CHAPTER 3 . . . . . . 323
APPENDIX D COMPUTER PROGRAMS FOR CHAPTER 4 . . . . . . 337
x
NEW CONCEPTS IN UNDERGROUND STORAGE OF NATURAL GAS
CHAPTER 1
INTRODUCTION
During the last two decades some very significant developments
have taken place in domestic and international scene in underground
storage of natural gas. Where there was practically no gas produced
from underground storage operations prior to 1949, 143 billion cubic
feet of gas were produced from storage during 1950, an amount which
was more than doubled during 1954. The gas produced from storage in-
creased to 492 billion cubic feet during 1956 and to much higher
figures during the early sixties owing particularly to development of
new means and concepts, such as “overpressuring” and “aquifer storage.”
The last decade has seen further significant developments in gas stor-
age, discovery and production in North Africa, Europe, Soviet Union,
and the United States. The proved reserves of natural gas for 1963*1.1were enough to last for 19 years. The rate of consumption of natural
gas was 12 trill ion cubic feet as of 1963. This figure is expected to
double by 1980. The current rate of consumption in natural gas indus-
try is 15 tr i l l ion cubic feet per year. The amount of gas produced
from storage during 1965 was about 963 billion SCF.
Original underground storage projects were located in depleted
gas or oil reservoirs; media with well proved ability to retain hydro-
carbons under pressure. A significant breakthrough was made in under-
ground storage when the concept of “overpressuring” became an engineer-
ing possib i l i ty . Through operation of depleted storage reservoirs at
*The numbers in upper script refer to literature citations given
at the end.
- 1 -
New Concepts in Underground Storage of Natural Gas
-2-
pressure levels reasonably higher than “discovery pressures,” large in-
creases in the capacity of storage fields were realized. Early exper-
iments with the practice of overpressure and simultaneous advances in
our ability to understand and analyze the movement of water in contact1.2with natural gas led to the development of “aquifer storage,” where
the pore volume for storing natural gas was obtained through the expul-
sion of water from its native formation by injection of gas at pressures
above the discovery pressure. Applications of digital computing tech-
niques to study of gas storage reservoirs permitted significant new1.3,1.4,1.5
contr ibutions in our understanding of the behavior of gas
reservoirs subject to water drive.
Along with all the new data, solutions and experience, a large
number of new problems have been uncovered. The actual mechanics of
the development of the storage bubble during early stages of gas injec-
t ion into aquifers, the microscopic physics of gas-water displacement
process, properties of cap rocks related to leakage or breakage, nature
of threshold pressure phenomena, instability and “fingering” in gas-
water displacement are but a few of such typical problems indicated by
recent advances in storage technology.
Ever increasing developments in processing and consumption of
natural gas resulted in large expansion of marketing areas near highly
populated industrial centers. Many such areas located in the Northeast1.6,1.7
and North Central United States and Eastern Canada have little
proved, on location, natural gas reserves. These areas are normally
supplied by long distance pipelines, some as long as 2,000 miles from
gas reserves in Southwestern United States and Western Canada. Recent
spectacular discoveries of natural gas in Northern Africa and North Sea
vis-a-vis the highly populated industrial consuming market in Europe
suggest similar problems with respect of the logistics of gas movement.
Economic considerations in the operation of long distance pipelines
require that maximum feasible usage must be maintained in order to re-
duce unit transportation cost. The trend toward high “load factors”
indicates desirability of underground storage of natural gas in areas
where depleted gas or oil reservoirs are not generally available.
Introduction
-3-
While the expansion of the gas market does and will continue to foster
the search for new gas reserves in areas of acute need, development of
new ideas, new techniques and new concepts for gas storage must be ex-
plored if the industry is to meet the long range requirements of the
expanding gas market. One must also realize that there are areas in
the United States, Canada and Europe where sedimentary rocks do not
exist, making it impossible for any oil or gas reserves to be found.
At present, aquifer storage, if and when operated successfully,
appears to be the most economical method for areas devoid of depleted
o i l o r gas f ie lds . It is well known, on the other hand, that the
success of aquifer storage depends critically on the presence of suit-
able geological conditions. Suf f ic ient poros i ty , adequate permeability
and good cap rock are among the prime requirements for such storage.
In many areas, the above factors do not simultaneously coexist. Suf-
ficient porosity and permeability but lack of adequate structural clo-
sure, adequate closure but leaky cap rock, semi-open structure, com-
municating faults or no anticl ine at al l are typical of such condit ions.
The storage of gas in such strata must require new techniques and new
concepts not yet explored to date. There has been some work, reported
in t he l i t e ra tu re , in storage of gas in aquifers with no structural clo-
s u r e . 1 . 8 , 1 . 9 , 1 . 1 0 To date these methods have not yet been explored suf-
f ic ient ly for a s ign i f icant eva luat ion o f the i r potent ia l .
Storage of natural gas in subsurface but nonporous media has re-
cently been suggested. Storage in dissolved salt caverns, in mined or
natural caverns, in underground cavities induced by nuclear explosions
and underwater storage have all been subject to studies, some on paper,
some at laboratory, and some on field, in order to evaluate their
engineering possibi l i ty and economic feasibi l i ty. More recently, use
of chemical grouts to impermeate subsurface strata has offered inter-
est ing possibi l i t ies to provide art i f ical cap rocks or perhaps to re-
pair existing cap rock subject to leakage. Analysis and evaluation of
all these new concepts for potential use in gas storage have been the
objective of a 3-year research program supported by the American Gas
Association at the University of Michigan.
New Concepts in Underground Storage of Natural Gas
-4-
The Project PO-50 “New Concepts on Underground Storage” was ini-
tiated at The University of Michigan during 1963 to "...formulate and
explore new concepts” applied to underground storage of natural gas.
The research effort carried out during the tenure of the project was
devoted to three major areas designated in the original proposal. These
were:
I . Detection and Remedy of Leaks from Conventional StorageReservoirs
I I . Soil Impermeation by Grouting
I I I . Underground Storage in Non-Porous Media
While there were practically no underground storage reservoirs in
the for t ies , depleted gas or oil field storage gained prominence during
t h e f i f t i e s . One of the most significant “breakthroughs” in storage
technology was obtained when the practice of “overpressuring” was re-
cently tried and successfully developed. The injection of gas into
formations to pressure levels above the discovery resulted not only in
large increases of the storage capacity of these fields but also paved
the way to then unconventional but now conventional concept of “aquifer
storage.” A second breakthrough in gas storage was made when the move-
ment of water in contact with natural gas was quantitatively related to
the performance of the gas storage bubble. By successfully applying
high speed digital computing, significant new contributions were made
to our understanding of the behavior of gas storage reservoirs subject
to water drive. On the practical side the control of growth of gas
bubble beyond areas of minimum structural closure, location, completion
and treatment of observation wells, maintenance of storage reservoir
boundaries within the limits of protective acreage by control of “cush-
ion gas,” optimum well density in heterogeneous storage reservoirs,
interpretation of pressure survey data from non-homogeneous reservoirs
are among problems of current, day-to-day interest. In the case of
apparent leaks from storage reservoirs due to one reason or another the
location, pressure testing and evaluation of possible collector zones
Introduction
-5-
is another problem of interest where field data must be analyzed with
reservoir engineering calculations of special type.
In this project the problem of leakage has been approached from
two viewpoints :
a. The mechanism of gas leakage across a cap rockstudied on purely theoretical grounds by for-mulating a mathematical model describing theunsteady state two-phase flow through the por-ous matrix constituting the cap rock.
b. By specifically determining the effect of gasleakage on the performance of gas storage bubbleas related to the movement of water in and outof the gas sand.
In the area of soil impermeation by grouting the experimental pro-
gram carried was directed to evaluate various grouts as to their effec-
tiveness in sealing formations along controlled geometries. I t i s we l l
known that the success of aquifer storage depends critically on the1.5
presence of suitable subsurface geology. Suff icient porosity, ade-
quate permeability, satisfactory cap rock and complete structural clo-
sure are among the necessary requirements for aquifer storage. Even in
areas where sedimentary rocks abound, the above factors do not always
simultaneously co-exist. The storage of gas in such strata requires
artifical creation of boundaries impervious to the flow of natural gas.
After reviewing during the first year on the project the existing
methods of gas storage related to geographic, geologic and geophysical
conditions to which they are best suited, the potential advantages of
grouts applied through the porous media and from the surface of a cavity
surrounded by porous media became apparent. 1.5Study of various grouts
available, determination of their desirable or undesirable physical
propert ies, their effect on threshold pressures and their suitabi l i ty
to a particular type of formation have been the primary objectives of
experimental work carried out during the progress of research work.
New Concepts in Underground Storage of Natural Gas
-6-
In the area of unconventional underground storage, significant
exploratory studies have been devoted to assess and evaluate the merits
of several new concepts such as storage in dissolved salt caverns, in
natural or mined caves, in cavities induced by nuclear explosions and
finally underwater storage in the bottom of deep lakes and oceans.
In reporting the final results to date, the presentation of var-
ious topics mentioned above have been organized as independent sections
in a sequence approximately ranging from current and conventional in-
terest to future, completely novel and unconventional aspects.
The second chapter consists of a comprehensive review of problems
associated with leaks from conventional storage reservoirs subject to
“overpressure.” In this chapter the concept of overpressuring, its
advantages as well as limitations are discussed in reasonable detail.
The relation between capillary imbibition and drainage phenomena and
leakage across cap rock, the concept of threshold pressure and problems
related are also discussed.
The Chapters 3 and 4 relate to mechanism of gas leakage across
cap rocks and to the performance of gas storage reservoirs subject to
leakage. The complete mathematical formulation of leak problems as
two-phase two-dimensional unsteady state flow across cap rocks is enter-
tained in detail in Chapter 3. Equations which reconcile the material
balances in gas inventory in the storage bubble with the movement of
water in and out of the gas sand with leakage occurring across the cap
rocks are given in detail in Chapter 4.
The Chapter 5 presents the work currently underway on evaluation
of grouts and reservoir engineering calculations related to grout injec-
t ion.
The storage of natural gas in subsurface, non-porous storage
cav i t ies , unconventional methods and new concepts such as storage in
cavities resulting from underground nuclear explosions and deep under-
water storage near the bottom of oceans are featured in Chapter 6.
Introduction
-7-
REFERENCES
1.1 Jacobs, J. C., The Future of the Natural Gas Industry, 1964,Transmission Conference, Vancouver, B. C., Canada.
1.2 Katz, D. L., Coats, K. H., and M. R. Tek, Effect of Unsteady-State Aquifer Motion on the Size of an Adjacent Gas StorageReservoir, Pet. Trans. AIME, 216, 18, 1959.
-
1.3 Katz, D. L., Tek, M. R., Coats, K. H., Jones, S. C., and M.Miller, Movement of Underground Water in Contact with NaturalGas, Pipeline Research Council International Monograph (February 1963).
1.4 Rzepczynski, W., Katz, D. L., Tek, M. R., and K. H. Coats, TheMount Simon Gas Storage Reservoir in the Herscher Field, Oiland Gas Journal, 86-91, June 1961.
1.5 Tek, M. R., and D. L. Katz, Development Recents Dans Le StockageSouterrain du Gaz Naturel, Revue de l' Institut Francais duPetrole et Annales des Combustibles Liquides, Vol. XVIII, No.11, November 1963.
1.6 6th Annual Report on Statistics, PRCI Committee on UndergroundStorage, December 31, 1956.
1.7
1.8
1.9
1.10
Katz, D. L., et al . , Handbook of Natural Gas Engineering,McGraw Hill, (1959).
Private Communication, D. L. Katz.
Private Communication, D. L. Katz.
Nielsen, R. L., On the Flow of Two Immiscible IncompressibleFluids in Porous Media, Ph.D. Dissertation, University ofMichigan.
This Page Intentionally Left Blank
CHAPTER 2
PROBLEMS ASSOCIATED WITH LEAKS FROM OVERPRESSURED
STORAGE RESERVOIRS
The practice of operating storage reservoirs at pressures above
the level corresponding to discovery of the part icular f ield is cal led
“overpressuring.” During the last decade the storage capacity of and
gas deliverability from a large number of depleted oil or gas producing
reservoirs have been substantially increased through the practice of
overpressuring. Early experiments with the practice of overpressure and
simultaneous advances in our ability to understand and analyze the move-
ment of water in contact with natural gas2.1
led to the development of
“aquifer storage” where the pore volume for storing natural gas was
obtained through the expulsion of water from its native formation by
injection of gas at pressures above the discovery pressure. Through
proper applications of digital computation on high speed electronic com-
puters, much has been added to our understanding of the performance of
gas reservoirs subject to water drive. 2.2,2.3,2.4 These computational
procedures which proved quite informative and practical revolved around
the use of superposition principle to handle time-varying boundary con-
d i t ions, the material balances to reconcile the inventory gas in storage
any time and unsteady state fluid flow equations describing the rate of
movement of water in and out of the gas sand. These equations show that a
storage reservoir maintained at “overpressure” over a long period of time
will continue to grow. The rate of growth is found to be a function of
physical, geometric, and geologic parameters of reservoir-aquifer system
as well as the extent and duration of the “overpressure.” While such
extended overpressure in usually practiced at early stages of develop-
ment of storage reservoirs and is practiced with caution regarding the
movement of gas water interface, it often causes concern in regard to
-9-
New Concepts in Underground Storage of Natural Gas
-10-
spill of gas across areas of minimum structural closure. Judicious con-
trol of extent and duration of overpressure to delimit the areal expan-
sion of the gas bubble and confine it to areas adequately covered by
protective acreage is almost foremost in future planning of any storage
operation. Field data gathered during the last decade on overpressured
reservoirs also indicate on the other hand that just as effectively as
the extended overpressure will continue to grow the gas storage bubble,
operation at pressures below discovery or “underpressure” will shrink
the gas bubble back until safely within structural closure or protective
acreage.
“Overpressure” viewed as a means to increase gas deliverability
and storage capacity and “underpressure” as a means to control and con-
tain the bubble within prescribed boundaries may be regarded as key
factors entering into the overall logistics of a company’s supply and
transportation of natural gas. While one must recognize that increased
deliverability on an existing system may also be obtained by means
other than overpressuring such as development of new fields and peak-
shaving storage capability, increased well density, increased well
stimulation or more compression horsepower, the extent and nature of
overpressure and underpressure operations must be analyzed on the basis
of overall economic merits as well as current technical and practical
engineering considerations.
2.1 Leakage or Spill From Overpressured Reservoirs
In depleted oi l or gas reservoir storage or in “aquifer storage”
the presence of a suitable cap rock is of paramount importance for the
retention of natural gas within the structural boundaries of the reser-
vo i r . The cap rock that constitutes the overburden to a natural petro-
leum reservoir obviously does possess proved integrity to retain the gas
at least up to discovery pressure. If overpressure conditions are sus-
tained in a field, depending upon the extent of overpressure, possibility
exists of gas leaking across the cap rock or moving in uncontrolled man-
ner to formations beyond areas of minimum structural closure.
Problems Associated with Leaks fromOver-pressured Storage Reservoirs
-11-
The leakage or spill of gas from a storage reservoir may be due
to:
1. Exceeding the threshold pressure of the cap rock,
2. Mechanically fracturing the cap rock because of exces-sive overpressure
3. By having “overpressure” of excessive extent or dura-tion to cause water to be pushed beyond the seal ofstructural closures
4. By fractures extending through and across the cap,induced during drilling or formation stimulation
5. By poor bonding of the cement between casing and hole
6. By existing permeable faults or incipient fracturesin the native formation.
An excessive overpressure when sustained a long time, may cause
the cap rock to leak or it may lift off the overburden causing nearly
horizontal fracturing along bedding planes. Sometimes overpressure
applied to deep wells may induce vertical fractures or oblique fractures
as well. In aquifer storage the gas bubble is always at pressures above
the discovery pressure. The discovery pressure of a blanket sand con-
taining water or petroleum reservoir containing hydrocarbon is usually
related to the hydraulic gradient corresponding to the depth of the
reservoir,. Figure 2.1 shows the relationship between depth and discov-
ery pressure for various petroleum reservoirs. It can be seen that the
discovery pressure for most of the reservoirs lie between two limiting
lines (A) and (B). The line (A) corresponds to the approximate upper
limit corresponding to the weight of the overburden which is about 1.0
p s i / f t . The line (B), the lower limit corresponds to the hydraulic
gradient which is about 0.433 psi/ft for pure water.
The fact that most petroleum reservoirs are discovered at equilib-
rium values between the above 2 curves is considered in itself as a
supporting evidence to theories of underwater sedimentation and compaction
New Concepts in Underground Storage of Natural Gas
-12-
Fig. 2.1. Discovery and hydraulic pressure gradientsfor some fields.5.7
processes advanced by most geologists and petroleum engineers to explain
the origin and occurrence of petroleum deposits. The more or less minor
variations frequently observed between the discovery and hydrostatic
gradients are usual ly attr ibuted to variat ions in the density of salt
water and geothermal gradients existing on the crust of the earth. It
must be recognized that it is not uncommon to observe discovery pressures
outside the range delineated by curves (A) and (B). The abnormally high
pressures are usually attributed to compaction of shales surrounding the
strata bearing the hydrocarbons. The abnormally low pressures may be
due to hydrology of underground water movement in as much as it may or
may not communicate pressure wise with outcropping strata.
It is generally accepted that a pressure gradient of 1 psi/ft is
high enough to lift the overburden and will open a fracture along the
bedding planes.
The difference of the limiting value of 1 psi/ft and the discovery
gradient is basically the amount that brackets the extent to which
Problems Associated with Leaks fromOver-pressured Storage Reservoirs
-13-
“overpressure” may be available on a given field. For an aquifer 4000
feet deep, as an example, this difference could be as high as 4000 x
(1 - 0.433) = 2270 psi.
For the example above somewhere between 1730 psia discovery and
4000 psia upper limit lies a safe and reasonable maximum pressure where
over-pressure can be safely and economically practiced.
In deciding the (psi/ft) at which the reservoir may be operated
one must consider both the problems of leakage across the cap rock and
breakage of the cap rock. It must also be pointed out that if storage
is in semi-open structures, the movement of water in the adjacent aquifer
and the relation of water level to the spill point becomes equally im-
portant.
The problem of detection of leaks, and sealing of leaks in reser-
voirs through special techniques by soil impermeation are currently under
study. The progress to date and current thinking on this area indicate
that the leakage from cap rocks is related to water saturation in the cap
as well as i ts imbibit ion and drainage characterist ics related to capi l-
lary behavior.
The possibility of fracture of cap rock by mechanical failure due
to pressure load on the gas bubble side may not be ignored for it may
happen long before water is pushed out of the interstices of rock strata.
It appears that the relationship between “overpressure” and the
leak phenomena, whether due to drying out of the cap or structural stress
failure or excessive water movement, is the next logical area where some
basic engineering research effort must be deployed.
The first significant advance in our understanding of gas storage
reservoirs was accomplished when the reservoir pressure was quantitatively
and correct ly related to inventory gas quanti t ies including the effect of
water movement.
New Concepts in Underground Storage of Natural Gas
-14-
The second milestone was reached when it was discovered empirically
that gas can be stored in aquifers and depleted reservoirs at pressures
above the discovery.
The third breakthrough, hopefully, would be containment of gas in
large storage quantities through new concepts such as soil impermeation
by art i f ic ial means.
The next logical effort toward adding more storage capacity to
existing fields may be through the development of safe, reliable and
rational methods permitting the quantitative evaluation of “overpres-
sure” for each storage reservoir.
2.2 Concept of Threshold Pressure
Figure 2.2 shows a typical imbibition-drainage capillary pressure
curve representing water distribution found in a reservoir standing
over geologic times and consequently at capillary equilibrium. The
threshold pressure noted on Fig. 2.2 is the pressure required for gas
to start moving water out of the cap rock previously fully saturated
with water. In the drainage type of capillary pressure curve AB the
two end points A and B are quite significant for our understanding of
the performance of cap rocks. If the capi l lary pressure, i .e., pressure
difference between gas and water exceeds the threshold pressure the cap
rock would lose its 100 percent water saturation and begin to leak along
channels of high permeability. The other end point B where the satura-
tion asymptotically approaches “connate water” saturat ion S i is signif-
icant for the establishment of storage volume through the drying of
sandstone in the aquifer. The value S i is roughly correlated in the
literature using the permeability as a parameter. 2.2,2.3 At present
there exists no general correlation or method permitting the prediction
of the threshold pressure from the first principles. An approximate
correlation such as the one for connate water saturation has been devel-
oped by L. K. Thomas. 2.4A recent attempt to predict the threshold
Problems Associated with Leaks fromOver-pressured Storage Reservoirs
-15-
Fig. 2.2. Imbibition and drainage capillary pressure curves.
pressure from the equation of general unsteady state two phase flow
through porous media has not been conclusive.
Extensive laboratory data on samples of cap rock materials of
high shale content and extremely low permeability is at present prac-
t ica l ly non-ex is tent . Data on capillary pressures and the concept of
relat ive permeabil i ty for t ight cap rock materials are further subject
to questionable accuracy and validity because of chemical effects
associated with hydration of shales.
2.3 Prediction of Threshold Pressure From Core Properties
Because it was found impossible to predict the threshold pressures
of cap rock materials purely on theoretical grounds and from the first
principles, data exist ing in the L i tera ture 2.1and supplied by the In-
dustry have been collected and analyzed along with measurements on
samples of cap rocks and reservoir materials given the project by var-
ious storage companies.
Ne
w
Co
nce
pts
in
U
nd
erg
rou
nd
S
tora
ge
o
f N
atu
ral
Ga
s
-16
-
Problems Associated with Leaks fromOver-pressured Storage Reservoirs
-17-
The Table 2.1 is a complete summary of core properties such as
porosity, permeability and threshold pressure, analyzed, collected,
measured and compiled for purposes of correlation during this research
pro jec t . This table includes core properties and threshold pressure
values provided by Northern Illinois Gas Company as well as our own
laboratory measurements from reservoir sand and cap rock samples given
to the project by various gas storage companies. Some of the cores
whose properties are listed in Table 2.1 are synthetically prepared in
our Laboratories as will be discussed later.
Various correlations giving threshold pressure (p t) as a function
of porosity and permeability (k) were attempted.
It was found that the plot of P t v/s approximately followed
a linear pattern as shown in the Fig. 2.3. The core samples selected
were from very permeable to almost impermeable ones with varying
from 10-3 to 107 md-1 and threshold pressure from a few tenths of a
psia to about a thousand psia. Standard deviation and bias for the
plot are 147.3% and 9.778%. A minimum correlation curve giving the
lower limit of threshold pressure values as a function of core proper-
ties is also shown in Fig. 2.3.
The graph should serve as a first-order approximation for the
threshold pressure values from the porosity and permeability of the
core samples.
Example Calculations
1) Part iculars of the Core:
Source - Northern I l l inois Gas Co., Iroquois, I l l inois
Mater ia l - St. Peters Sand
Depth - 1281-9 feet
= 6.97%
k = 4.9 md
New Concepts in Underground Storage of Natural Gas
-18-
= 1.42 md-1
Pt (graph) = 4.8 psia
Pt (experiment) = 7.5 psia
2) Particulars of the Core:
Source - Northern I l l inois Gas Co., LaSalle, I l l inois
Material - Mount Simon
Depth - 1450-6 feet
= 8.9%
k = 15.6 md
= .57 md-1
Pt (graph) = 3.6 psia
Pt (experiment) = 6.0 psia
3) Data supplied by Northern Illinois Gas Co.
= 3.9%
k = 7.1 x 10-4 md
= 5.5 x 103 md-1
Pt (graph) = 130 psia
Pt (experiment) = 140 psia
Preparation of Consolidated Natural Cores
Field samples of core from underground formations are usually in
the forms of solid cylinders 4-6 inches in diameter and about 6 inches
to 2 feet in length. The cores are first cut by a diamond saw which is
a steel disc with diamond bits embedded on the edge. An oil emulsion
is used to cool the saw. The pieces thus reduced in length to 4-6 inches
Problems Associated with Leaks fromOver-pressured Storage Reservoirs
-19-
are then drilled by a diamond drill to obtain cylindrical core samples
of diameter 1.5 inches and length 2.5 inches which are further given
a smooth finish. Water is used to cool the dri l l during the dri l l ing
operations. In order to avoid subsequent difficulties during testing,
care should be taken to get ends of the core perpendicular to its axis.
The core samples are finally washed with an organic solvent such as
ether, to remove the residual oil remaining in pores.
Preparation of Super-Permeable Laboratory Cores
Natural cores normally have a low permeability and porosity and
as such do not provide the best media for evaluating grouting experi-
ments. Therefore, a method has been developed to cast super-permeable
cores in the laboratory.
These laboratory cores have the additional advantage of their
properties threshold pressures, porosity and permeability falling within
a very narrow and duplicable range. These cores are prepared in the
following manner:
White river sand 80 parts by weight with small enough particles
to pass through ASTM sieve No. 100 are dry-mixed with portland cement
20 parts by weight. Water 8% by weight on dry basis is then added and
a uniformly wet mixture is obtained by proper mixing and kneading. A
previously greased metallic mould of 1.5 inches diameter and 4.5 in
length is filled with the wet sand-cement mixture and compressed under
6000 pound-force using thick end-pieces to obtain a sample of 2.5 inches
in length. After compression the mould with sand-cement core is im-
mersed in water. After two days the core is extruded and further cured
under water for 4 days. The cores thus made are found to possess a
high compressive strength, porosity and permeability and very low thres-
hold pressures.
Ne
w
Co
nce
pts
in
Un
de
rgro
un
d
Sto
rag
e
of
Na
tura
l G
as
-20
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Problems Associated with Leaks fromOver-pressured Storage Reservoirs
-21-
Measurement of Porosity
Porosity is the ratio between void volume and bulk volume. I t
is measured by determining the volume of water required to saturate
the core and bulk volume. The core is first dried by keeping it over-
night in an oven at 230°F. After the core is adequately dried a vacuum
of 75.5 cm of mercury is applied for about 30 minutes. Then while
maintaining this vacuum the core is submerged in water. The saturated
core is taken out and weighed from time to time until a constant weight
is obtained. The apparatus consists of a container connected to a suc-
tion pump and a funnel containing water as shown in the Fig. 2.4. Bulk
volume of the core is determined by immersing the core under water in a
graduated cylinder and noting the increase in the volume.
where Wcs = weight of the saturated core
Wcd = weight of the dry core
Yw = specific weight of water at the temperatureof the experiment
V = bulk volume of the core
M e a s u r e m e n t o f P e r m e a b i l i t y
Permeability is determined by means of the Darcy’s Equation
where K = permeabi l i ty , mi l l idarcy
q = f l ow ra te , cc/hr. at room temperature
(2.1)
(2.2)
New Concepts in Underground Storage of Natural Gas
-22-
F i g . 2 . 5 The Rubber S leeved Core Holder
Pro
ble
ms
Asso
ciate
d
with
L
ea
ks fro
mO
verp
ressu
red
S
tora
ge
R
ese
rvoirs
-23
-
New Concepts in Underground Storage of Natural Gas
-24-
A = cross sectional area of the core, cm2
= v iscos i ty o f the f lu id , cent i -po ise
L = length of core, cm
P- Pouti n , = pressure at the inlet and outlet, psia
The fluid used in the experiment is nitrogen and its flow rate
is measured by a Fisher Porter flow meter. As variations in the room
temperature and pressure are small, correction of the flow rate to STP
was not significant.
The method consists of drying the core by keeping it overnight
in an oven at 230°F. The dried core is then placed in the Rubber
Sleeved Core Holder shown in Fig. 2.5 and the pressure exceeding the
inlet pressure by 100 psia is supplied on the rubber sleeve. Nitrogen
gas is allowed to pass through the core and rate q, inlet pressure
(P i n ) , outlet pressure (Pout) are measured. Viscosity of nitrogen gas
is also determined at the corresponding room temperature. Permeability
is finally computed by eq. 2.2.
Measurement of Threshold Pressure
The pressure at which a gas just starts pushing water out from
the pores of a saturated core is called threshold pressure.
Method for its determination in the laboratory consists of sat-
urating the core using the vacuum technique as in the measurement of
porosity and placing it in the rubber sleeved core holder which has a
graduated capil lary tube part ial ly f i l led with water at top exit end.
As in the test on permeability, pressure is applied on to the rubber
sleeve as shown in Fig. 2.6. Then a very slight pressure of nitrogen
gas is applied on the inlet end and sufficient time is allowed for
equilibrium to be reached. Inlet pressure is then gradually increased
Problems Associated with Leaks fromOver-pressured Storage Reservoirs
-25-
till the water level in the capillary tube moves up and continues to
move. The minimum pressure sufficient to cause the water to be con-
tinually expulsed from the cores is the “Threshold Pressure.”
New Concepts in Underground Storage of Natural Gas
-26-
TABLE 2.1
LABORATORY DATA FOR USE IN CORRELATION OF THRESHOLD PRESSURES
Problems Associated with Leaks fromOverpressured Storage Reservoirs
-27-
New Concepts in Underground Storage of Natural Gas
-28-
Problems Associated with Leaks fromOverpressured Storage Reservoirs
-29-
New Concepts in Underground Storage of Natural Gas
-30-
Problems Associated with Leaks fromOverpressured Storage Reservoirs
-31-
New Concepts in Underground Storage of Natural Gas
-32-
REFERENCE
2.1 Nouveaux Aspects du Stockage Souterrain du Gaz, M. R. Tek, Revue
de l'Institut Francais du Petrole, pp. 1623-1640, Novembre, 1965.
3 . 1
CHAPTER 3
MECHANISM OF GAS LEAKAGE ACROSS A CAP ROCK
I n t r o d u c t i o n
T h e s u c c e s s o f a n y p r o j e c t f o r s t o r i n g g a s i n u n d e r g r o u n d f o r -
m a t i o n s d e p e n d s l a r g e l y o n l o c a t i n g a c o n t i n u o u s n a t u r a l b a r r i e r
w h e r e b y t h e n a t u r a l u p w a r d s m o t i o n o f t h e g a s m a y b e r e s t r a i n e d .
T y p i c a l l y , s u c h a b a r r i e r o r " c a p r o c k " w i l l c o n s i s t o f a s t r a t u m o f
a l m o s t i m p e r v i o u s s h a l e ; o c c a s i o n a l l y , s a n d s t o n e s a n d d o l o m i t e s o f
v e r y l o w p e r m e a b i l i t y m a y a l s o s e r v e t o p r e v e n t l e a k a g e . From the
v i e w p o i n t o f s t o r i n g a s m u c h g a s a s p o s s i b l e i n a g i v e n f o r m a t i o n ,
i t i s d e s i r a b l e t o o v e r p r e s s u r e t h e g a s b u b b l e s o t h a t i t s p r e s s u r e
c o n s i d e r a b l y e x c e e d s t h e f o r m a t i o n d i s c o v e r y p r e s s u r e . C l e a r l y , t h e
c a p r o c k m u s t p o s s e s s b o t h m e c h a n i c a l s t r e n g t h a n d a n a b i l i t y t o w i t h -
s t a n d g a s l e a k a g e . O n l y t h e l a t t e r p r o p e r t y w i l l b e d i s c u s s e d i n
t h i s c h a p t e r .
U p o n d i s c o v e r y o f a r e s e r v o i r , t h e c a p r o c k i s u s u a l l y f u l l y
s a t u r a t e d w i t h w a t e r , t o t h e e x t e n t t h a t i t s m i n u t e p o r o s i t y w i l l
a l l o w . T h e p r e s e n c e o f t h i s w a t e r g r e a t l y e n h a n c e s t h e a b i l i t y o f
t h e c a p r o c k t o a c t a s a s e a l a g a i n s t e s c a p i n g g a s . The p ressu re
d i f f e r e n t i a l a c r o s s t h e l a y e r i s f a i r l y s m a l l , u s u a l l y c o r r e s p o n d i n g
t o t h a t d u e t o t h e o r d i n a r y h y d r o s t a t i c g r a d i e n t . A s m a l l i n c r e a s e
i n g a s p r e s s u r e o n t h e u n d e r s u r f a c e o f t h e c a p r o c k w i l l n o t c a u s e
a n i m m e d i a t e l e a k . R a t h e r , a c e r t a i n t h r e s h o l d p r e s s u r e , d i c t a t e d
b y c a p i l l a r y f o r c e s w i t h i n t h e c a p r o c k , m u s t b e a t t a i n e d i n t h e g a s
b u b b l e b e f o r e t h e g a s s t a r t s t o d i s p l a c e w a t e r . A t a s u f f i c i e n t l y
h i g h o v e r p r e s s u r e , t h e g a s w i l l w o r k i t s w a y t o d i s p l a c e w a t e r f r o m
-33-
New Concepts in Underground Storage of Natural Gas
-34-
p e r m e a b l e c h a n n e l s w i t h i n t h e c a p r o c k a n d t h u s e s t a b l i s h c o m m u n i c a t i o n
w i t h t he more po rous and pe rmeab le f o rma t i ons above .
I n p r a c t i c e , t h e g a s b u b b l e p r e s s u r e w i l l n o t a l w a y s e x c e e d
t h e t h r e s h o l d p r e s s u r e , b u t w i l l v a r y w i t h t i m e ( p o s s i b l y i n a n
a p p r o x i m a t e l y p e r i o d i c m a n n e r ) a c c o r d i n g t o t h e p a r t i c u l a r i n j e c t i o n /
p r o d u c t i o n s c h e d u l e . U n d e r t h e s e c i r c u m s t a n c e s , t h e f o l l o w i n g
ques t i ons may be asked :
( a ) W i l l t h e g a s d i s p l a c e w a t e r f r o m t h e c a p r o c k a t a l l ?
( b ) I f s u c h a d i s p l a c e m e n t o c c u r s , - w i l l t h e g a s p e n e t r a t e t h e c a p
r o c k c o m p l e t e l y , a n d t h e r e b y c r e a t e a l e a k ?
( c ) O n c e a g a s b r e a k t h r o u g h h a s o c c u r r e d , c a n w a t e r b e r e a b s o r b e d
i n t o t h e c a p r o c k b y r e d u c i n g t h e g a s p r e s s u r e ?
( d ) I s i t p o s s i b l e f o r t h e g a s / w a t e r i n t e r f a c e t o o s c i l l a t e , s o
t h a t i t a l w a y s r e m a i n s w i t h i n t h e c a p r o c k ?
( e ) I s t h e i n t e r f a c e s h a r p l y d e f i n e d o r d i f f u s e i n n a t u r e ?
( f ) W i l l t h e i n t e r f a c e a d v a n c e u n i f o r m l y , o r w i l l t h e r e b e a
t e n d e n c y f o r " f i n g e r i n g " t o o c c u r ?
T h e d i s c u s s i o n w h i c h f o l l o w s w i l l a t t e m p t t o a n s w e r s o m e o f t h e s e
q u e s t i o n s . A m o r e p r e c i s e s t a t e m e n t o f t h e p r o b l e m i s p r e s e n t e d
i n S e c t i o n 3 . 3 . F i r s t , however , we mus t cons ide r t he ma thema t i ca l
e q u a t i o n s w h i c h d e s c r i b e i m m i s c i b l e t w o - p h a s e f l o w i n a p o r o u s
medium.
3 .2 Equa t i ons Gove rn ing Gas -Wate r F low i n a Po rous Med ium
T h e f o l l o w i n g n o t a t i o n w i l l b e u s e d i n t h i s s e c t i o n . The
i n t r o d u c t i o n o f s p e c i f i c u n i t s w i l l b e d e l a y e d u n t i l a c t u a l n u m e r i c a l
c a l c u l a t i o n s a r e m a d e . F o r t h e p r e s e n t , i t i s u n d e r s t o o d t h a t a n y
c o n s i s t e n t s e t o f u n i t s m a y b e e m p l o y e d .
Mechanism of Gas Leakage Across a Cap Rock
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Symbol D e f i n i t i o n
G r a v i t a t i o n a l a c c e l e r a t i o n .
H e i g h t .
P e r m e a b i l i t y .
R e l a t i v e p e r m e a b i l i t y f o r g a s , w a t e r .
C a p i l l a r y p r e s s u r e , p c = p g - p w .
P r e s s u r e s i n g a s , w a t e r , p h a s e s .
L o c a l r a t e o f i n j e c t i o n o f g a s o r w a t e r , c o n s i d e r e dp o s i t i v e i n t o c a p r o c k , v o l u m e o f f l u i d p e r u n i tt i m e p e r u n i t v o l u m e o f c a p r o c k .
W a t e r s a t u r a t i o n , S = S W .
G a s a n d w a t e r s a t u r a t i o n s .
T ime.
S u p e r f i c i a l v e l o c i t y v e c t o r s f o r g a s , w a t e r , e q u a li n m a g n i t u d e t o t h e v o l u m e t r i c f l o w r a t e p e r u n i ta r e a n o r m a l t o t h e f l o w .
P o r o s i t y ( f r a c t i o n o f t o t a l v o l u m e w h i c h i s v o i d ) .
G a s a n d w a t e r v i s c o s i t i e s .
G a s a n d w a t e r d e n s i t i e s .
G a s a n d w a t e r p o t e n t i a l s , = p + pgh.
T h e s i m p l i f y i n g a s s u m p t i o n i s m a d e t h a t t h e g a s a n d w a t e r b e h a v e
a s i n c o m p r e s s i b l e f l u i d s . T h e f o l l o w i n g e q u a t i o n s g o v e r n t h e t w o -
p h a s e f l o w .
Darcy ' s Law
For each phase , t h e s u p e r f i c i a l v e l o c i t y v e c t o r i s p r o p o r t i o n a l
t o t h e p o t e n t i a l g r a d i e n t o f t h a t p h a s e :
( 3 . 1 )
( 3 . 2 )
New Concepts in Underground Storage of Natural Gas
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T y p i c a l l y , t h e r e l a t i v e p e r m e a b i l i t i e s k g a n d k w w i l l d e p e n d
o n t h e w a t e r s a t u r a t i o n i n t h e m a n n e r s h o w n i n F i g . 3 . 1 .
Fig. 3.1 Relative permeabil i ty.
Mass Balance
For each phase , t h e r a t e o f a c c u m u l a t i o n p e r u n i t v o l u m e o f
m e d i u m e q u a l s t h e n e t r a t e o f i n f l u x i n t o t h a t v o l u m e :
( 3 . 3 )
( 3 . 4 )
A s w i l l b e s e e n l a t e r , t h e i n j e c t i o n t e r m s q w a n d q g a r e i n c l u d e d
t o a l l o w f o r t h e e f f e c t s o f g a s o r w a t e r l e a k a g e a c r o s s t h e c a p
r o c k b o u n d a r i e s .
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R e l a t i o n b e t w e e n S a t u r a t i o n s
T h e v o i d s p a c e s m u s t b e c o m p l e t e l y f i l l e d b y w a t e r a n d / o r g a s ,
s o t h a t
( 3 . 5 )
H e n c e f o r t h , w e s h a l l w o r k m a i n l y i n t e r m s o f t h e w a t e r s a t u r a t i o n ,
s = SW.
C a p i l l a r y P r e s s u r e
T h e p r e s s u r e i n t h e g a s p h a s e e x c e e d s t h a t i n t h e w a t e r b y t h e
c a p i l l a r y p r e s s u r e :
( 3 . 6 )
A s i n d i c a t e d i n F i g . 3 . 2 , t h e c a p i l l a r y p r e s s u r e d e p e n d s o n t h e w a t e r
s a t u r a t i o n a n d a l s o o n t h e d i r e c t i o n o f t h e d i s p l a c e m e n t .
F i g . 3 . 2 . C a p i l l a r y P r e s s u r e
New Concepts in Underground Storage of Natural Gas
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D e f i n i t i o n o f P o t e n t i a l s
w h e r e h i s t h e h e i g h t a b o v e a n a r b i t r a y d a t u m .
l o w i n g r e l a t i o n s :
( 3 . 7 )
( 3 . 8 )
N o t e a l s o t h e f o l -
( 3 . 9 )
( 3 . 1 0 )
( 3 . 1 1 )
Rear ranged Equa t i ons
F r o m e q u a t i o n s ( 3 . 3 ) , ( 3 . 4 ) , ( 3 . 5 ) , ( 3 . 9 ) , a n d ( 3 . 1 1 ) , w e o b t a i n
t h e f o l l o w i n g s i m u l t a n e o u s n o n - l i n e a r p a r t i a l d i f f e r e n t i a l e q u a t i o n s
i n t h e w a t e r a n d g a s p o t e n t i a l s a s t h e d e p e n d e n t v a r i a b l e s :
( 3 . 1 2 )
( 3 . 1 3 )
T h e s a t u r a t i o n i s a c t u a l l y a f u n c t i o n o f t h e t w o p o t e n t i a l s s i n c e , i f
a n d a r e k n o w n , p c i s d e t e r m i n e d f r o m ( 3 . 1 0 ) a n d S c a n t h e n b e
f o u n d f r o m t h e c a p i l l a r y p r e s s u r e c u r v e . N o t e t h a t a l t h o u g h e q u a t i o n s
( 3 . 1 2 ) a n d ( 3 . 1 3 ) w i l l s h o r t l y b e s i m p l i f i e d , t h e y h o l d g e n e r a l l y f o r
t h r e e - d i m e n s i o n a l s p a c e .
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3 . 3 A p p l i c a t i o n o f E q u a t i o n s t o L e a k a c r o s s C a p R o c k
F i g . 3 . 3 Mode l f o r Leak ac ross Cap Rock
C o n s i d e r a h o r i z o n t a l c a p r o c k o f u n i f o r m t h i c k n e s s Assume
t h a t t h e l a y e r i s b o u n d e d t o p a n d b o t t o m b y s t r a t a o f m u c h h i g h e r
p e r m e a b i l i t i e s w h i c h c o n t a i n w a t e r a n d g a s r e s p e c t i v e l y . The po ten -
t i a l o f t h e w a t e r a n d g a s a b o v e a r e m a i n t a i n e d c o n s t a n t a t v a l u e s
a n d r e s p e c t i v e l y . T h e p o t e n t i a l o f t h e g a s b e l o w m a y , h o w -
e v e r , f l u c t u a t e w i t h t i m e , p o s s i b l y i n a p e r i o d i c m a n n e r , a c c o r d i n g
t o s o m e p r e s c r i b e d f u n c t i o n A t t i m e . t = 0 , t h e c a p r o c k i s
i n i t i a l l y a l m o s t c o m p l e t e l y s a t u r a t e d w i t h w a t e r .
Assume t ha t no wa te r f l ows down ac ross t he bo t t om face o f t he
c a p r o c k , b u t t h a t g a s m a y l e a k a c r o s s t h e u p p e r b o u n d a r y . The mot ion
w i l l b e t r e a t e d a s o n e - d i m e n s i o n a l , i n t h e v e r t i c a l o r x - d i r e c t i o n .
T h e p r o b l e m i s t o d e t e r m i n e t h e s u b s e q u e n t m o t i o n o f g a s a n d w a t e r i n
t h e c a p r o c k .
Fo r one space d imens ion , e q u a t i o n s ( 3 . 1 2 ) a n d ( 3 . 1 3 ) b e c o m e
( 3 . 1 4 )
New Concepts in Underground Storage of Natural Gas
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( 3 . 1 5 )
T h e a s s o c i a t e d i n i t i a l a n d b o u n d a r y c o n d i t i o n s n o w f o l l o w .
I n i t i a l C o n d i t i o n s ( t = 0 ,
T h e w a t e r p o t e n t i a l i s c o n s t a n t t h r o u g h o u t t h e c a p r o c k a n d
i s e q u a l t o i t s v a l u e i n t h e w a t e r j u s t a b o v e t h e c a p r o c k :
( 3 . 1 6 )
T h e g a s p o t e n t i a l i s a l s o s p e c i f i e d t o b e u n i f o r m t h r o u g h o u t t h e c a p
r o c k a t a l e v e l a p p r o p r i a t e t o t h e s u b s e q u e n t p o t e n t i a l v a r i a t i o n s
i n s i d e t h e g a s b u b b l e , a n d a l s o e q u a l t o t h e g a s p o t e n t i a l o n t h e
u p p e r s u r f a c e o f t h e c a p r o c k :
( 3 . 1 7 )
T h e c o r r e s p o n d i n g i n i t i a l s a t u r a t i o n d i s t r i b u t i o n m a y b e o b t a i n e d b y
u s i n g t h e k n o w n c a p i l l a r y p r e s s u r e r e l a t i o n i n c o n j u n c t i o n w i t h
e q u a t i o n ( 3 . 1 0 ) .
B o u n d a r y C o n d i t i o n s
T h e g a s p o t e n t i a l o n t h e l o w e r s u r f a c e a n d b o t h g a s a n d w a t e r
p o t e n t i a l s o n t h e u p p e r s u r f a c e a r e s p e c i f i e d f o r a l l v a l u e s o f t i m e :
( 3 . 1 8 )
T h e r e i s n o f l o w o f w a t e r a c r o s s t h e l o w e r b o u n d a r y :
( 3 . 1 9 )
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A n o v e r a l l m a t e r i a l b a l a n c e m u s t b e s a t i s f i e d . T h a t i s , t h e v o l u m e t -
r i c r a t e o f i n j e c t i o n o f g a s t h r o u g h t h e l o w e r b o u n d a r y e q u a l s t h e
r a t e o f d i s p l a c e m e n t o f w a t e r a n d g a s t h r o u g h t h e t o p b o u n d a r y , i . e .
( 3 . 2 0 )
Here , t h e q ' s a r e c o n s i d e r e d p o s i t i v e f o r f l o w i n t o t h e c a p r o c k , w i t h
3 . 4 E q u a t i o n s i n T e r m s o f D i m e n s i o n l e s s V a r i a b l e s
( 3 . 2 1 )
N o t e f i r s t t h a t t h e r e l a t i v e p e r m e a b i l i t i e s k g a n d k w , a n d t h e
w a t e r s a t u r a t i o n S a r e a l r e a d y d i m e n s i o n l e s s q u a n t i t i e s . We next
i n t r o d u c e d i m e n s i o n l e s s t i m e , d i s t a n c e , i n j e c t i o n r a t e s , p o t e n t i a l s ,
a n d c a p i l l a r y p r e s s u r e d e f i n e d b y
I n t e r m s o f t h e s e n e w v a r i a b l e s , t he gove rn ing equa t i ons may be
r e - e x p r e s s e d i n t h e f o l l o w i n g d i m e n s i o n l e s s f o r m s .
( 3 . 2 2 )
New Concepts in Underground Storage of Natural Gas
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P o t e n t i a l E q u a t i o n s
( 3 . 2 3 )
( 3 . 2 4 )
I n i t i a l C o n d i t i o n s
T h e i n i t i a l s a t u r a t i o n d i s t r i b u t i o n w i l l b e t h a t c o r r e s p o n d i n g t o t h e
s t a r t i n g d i m e n s i o n l e s s c a p i l l a r y p r e s s u r e d i s t r i b u t i o n c o m p u t e d f r o m
B o u n d a r y C o n d i t i o n s ( f o r T > 0 )
I n j e c t i o n R a t e s
( 3 . 2 5 )
( 3 . 2 6 )
( 3 . 2 7 )
Mechanism of Gas Leakage Across a Cap Rock
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E q u a t i o n s ( 3 . 2 3 ) t h r o u g h ( 3 . 2 7 ) , t o g e t h e r w i t h r e l a t i o n s
e x p r e s s i n g r e l a t i v e p e r m e a b i l i t i e s a n d c a p i l l a r y p r e s s u r e a s f u n c t i o n s
o f s a t u r a t i o n , c o n s t i t u t e t h e p r o b l e m s t a t e m e n t .
T h e f o l l o w i n g i s a s u m m a r y o f t h e a d d i t i o n a l n o t a t i o n w h i c h h a s
b e e n i n t r o d u c e d i n t h i s s e c t i o n :
Symbol D e f i n i t i o n
T h i c k n e s s o f c a p r o c k .
D i m e n s i o n l e s s c a p i l l a r y p r e s s u r e .
V o l u m e t r i c f l o w r a t e s , p e r u n i t a r e a , o f g a s a n dw a t e r i n t o t h e c a p r o c k , f r o m b e l o w a n d a b o v e ,r e s p e c t i v e l y .
D i m e n s i o n l e s s g a s a n d w a t e r i n j e c t i o n r a t e s .
D i m e n s i o n l e s s t i m e .
V e r t i c a l d i s t a n c e a b o v e b o t t o m o f c a p r o c k .
D i m e n s i o n l e s s v e r t i c a l d i s t a n c e .
G a s p o t e n t i a l a t l o w e r s u r f a c e .
W a t e r p o t e n t i a l a t u p p e r s u r f a c e ( c o n s t a n t ) .
G a s p o t e n t i a l a t u p p e r s u r f a c e ( c o n s t a n t ) .
D i m e n s i o n l e s s g a s a n d w a t e r p o t e n t i a l s .
New Concepts in Underground Storage of Natural Gas
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3 . 5 R e a r r a n g e m e n t o f E q u a t i o n s i n T e r m s o f P a n d R V a r i a b l e s
T o f a c i l i t a t e t h e s o l u t i o n o f t h e p r o b l e m , w e n o w i n t r o -
d u c e t w o n e w v a r i a b l e s , P a n d R , d e f i n e d a s f o l l o w s .
( 3 . 2 8 )
( 3 . 2 9 )
I f w e a l s o l e t
( 3 . 3 0 )
( 3 . 3 1 )
t h e n s u b t r a c t i o n a n d a d d i t i o n , r e s p e c t i v e l y , o f e q u a t i o n s ( 3 . 2 3 )
a n d ( 3 . 2 4 ) y i e l d s
Here , t h e d e r i v a t i v e o f w a t e r s a t u r a t i o n w i t h r e s p e c t t o t h e
d i m e n s i o n l e s s c a p i l l a r y p r e s s u r e h a s b e e n a b b r e v i a t e d a s
N o t e t h a t t h e R e q u a t i o n ( 3 . 3 2 ) i s p a r a b o l i c , w h e r e a s t h e P
( 3 . 3 2 )
( 3 . 3 3 )
( 3 . 3 4 )
e q u a t i o n ( 3 . 3 3 ) , w h i c h c o n t a i n s n o t i m e d e r i v a t i v e , i s e l l i p t i c .
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3 . 6 F i n i t e D i f f e r e n c e E q u a t i o n s
A n a p p r o x i m a t i o n t o t h e s o l u t i o n o f t h e P a n d R e q u a t i o n s ,
( 3 . 3 2 ) a n d ( 3 . 3 3 ) r e s p e c t i v e l y , t o g e t h e r w i t h t h e a s s o c i a t e d i n i t i a l
a n d b o u n d a r y c o n d i t i o n s , c a n b e o b t a i n e d b y a f i n i t e d i f f e r e n c e
t e c h n i q u e . W e i n t r o d u c e a s e r i e s o f g r i d p o i n t s , s p a c e d b y
i n t o t h e c a p r o c k , a s s h o w n i n F i g . 3 . 4 . T h e s u b s c r i p t s 0 a n d n
d e n o t e p o i n t s w i t h i n t h e g a s b u b b l e a n d s u p e r n a t a n t
t i v e l y .
w a t e r , r e s p e c -
F i g . 3 . 4 S y s t e m o f G r i d P o i n t s
I t i s m a t h e m a t i c a l l y e x p e d i e n t t o c o n s i d e r t h e b o u n d a r i e s X = 0 a n d
X = 1 o f t h e c a p r o c k a s b e i n g i m p e r v i o u s t o f l u i d f l o w . Any
a c t u a l l e a k a g e o f g a s o r w a t e r i n t o o r f r o m t h e c a p r o c k i s t h e n
h a n d l e d v i a t h e i n j e c t i o n t e r m s Q g , 1 , Q g , n - 1 , and Qw,n-1 at grid
p o i n t s 1 a n d n - 1 . A l l o t h e r i n j e c t i o n s a r e z e r o ; i n p a r t i c u l a r ,
Qw , 1 i s z e r o b e c a u s e t h e r e i s n o w a t e r f l o w a c r o s s t h e l o w e r b o u n d a r y .
W e a l s o c o n s i d e r t h e v a l u e s o f P a n d R a t s u c c e s s i v e t i m e l e v e l s
e tc . T h u s b y u s e o f d o u b l e s u b -
s c r i p t s s u c h a s P i , m w e c a n d e n o t e t h e v a l u e o f P a t t h e i t h s p a c e
p o i n t a t t h e t i m e l e v e l
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-46-
3 . 7 F i n i t e D i f f e r e n c e A p p r o x i m a t i o n t o P a r a b o l i c o r R - E q u a t i o n
A t a n y i n t e r i o r g r i d p o i n t i n o t a d j a c e n t t o t h e b o u n d a r i e s
( i . e . , i = 2 , 3 , . . . , n - 3 , n - 2 ) , a s u i t a b l e f i n i t e d i f f e r e n c e
a p p r o x i m a t i o n o f t h e R e q u a t i o n ( 3 . 3 2 ) i s
(3 .35)
i n w h i c h
E q u a t i o n ( 3 . 3 5 ) m a y b e r e w r i t t e n a s
w h e r e , f o r i = 2 , 3 , . . . , n - 3 , n - 2 ,
( 3 . 3 6 )
( 3 . 3 7 )
( 3 . 3 8 )
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-47-
(eqn . ( 3 . 3 8 ) , c o n t d . )
S i n c e , f r o m t h e i m p e r v i o u s o r r e f l e c t i n g - t y p e b o u n d a r y c o n d i t i o n s ,
RO = R 1 a n d R n - 1 = R n , t h e f i n i t e d i f f e r e n c e a p p r o x i m a t i o n o f
e q u a t i o n ( 3 . 3 3 ) a t p o i n t s i = 1 a n d i = n - l h a s t h e f o l l o w i n g
s p e c i a l c o e f f i c i e n t s f o r u s e i n ( 3 . 3 7 ) :
At Point i = 1
A t P o i n t i = n - 1
( 3 . 3 9 )
( 3 . 4 0 )
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W r i t t e n o u t f o r i = 1 , 2 , . . . , n - 1 w i t h a p p r o p r i a t e c o e f f i c i e n t s ,
e q u a t i o n ( 3 . 3 7 ) g e n e r a t e s a s e r i e s o f n - 1 s i m u l t a n e o u s a l g e b r a i c
equa t i ons i n t he unknowns R 1 , m + 1 ,
( m + 1 ) t h t i m e l e v e l .
R 2 , m + 1 ) , . . . , R n - 1 , m + 1 a t t h e
T h i s p a r t i c u l a r s y s t e m o f e q u a t i o n s h a s
a t r i d i a g o n a l c o e f f i c i e n t m a t r i x a n d c a n b e s o l v e d b y t h e s t a n d a r d
a l g o r i t h m g i v e n i n A p p e n d i x B .
3 . 8 F i n i t e D i f f e r e n c e A p p r o x i m a t i o n t o E l l i p t i c o r P - E q u a t i o n
I t m a y l i k e w i s e b e s h o w n t h a t t h e f i n i t e d i f f e r e n c e a p p r o x -
i m a t i o n t o t h e P e q u a t i o n a t p o i n t i i s
( 3 . 4 1 )
w h e r e t h e c o e f f i c i e n t s h a v e t h e f o l l o w i n g v a l u e s :
A t I n t e r i o r P o i n t s i = 2 , 3 , . . . , n - 3 , n - 2
(3.42)
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-49-
At Point i = 1
A t P o i n t i = n - l
( 3 . 4 3 )
( 3 . 4 4 )
N o t e t h a t t h e s y s t e m o f s i m u l t a n e o u s e q u a t i o n s g e n e r a t e d
f r o m e q u a t i o n ( 3 . 4 1 ) b y c o n s i d e r i n g i = 1 , 2 , . . . , n - 1 i n t u r n
i s a g a i n t r i d i a g o n a l i n f o r m , and has t he s tanda rd me thod o f
s o l u t i o n g i v e n i n A p p e n d i x B .
New Concepts in Underground Storage of Natural Gas
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3 . 9 S o l u t i o n o f t h e F i n i t e D i f f e r e n c e E q u a t i o n s
S u p p o s e t h a t a t a t i m e l e v e l m ( i . e . , t = all potentials,
i n j e c t i o n t e r m s , a n d p h y s i c a l p r o p e r t i e s a r e k n o w n t h r o u g h o u t t h e
c a p r o c k . I n p a r t i c u l a r , t h e v a l u e s a t t = 0 w i l l c o r r e s p o n d t o
t h e s p e c i f i e d i n i t i a l c o n d i t i o n s . S u p p o s e f u r t h e r t h a t t h e i n -
j e c t i o n t e r m s a n d p h y s i c a l p r o p e r t i e s r e m a i n a p p r o x i m a t e l y c o n s t a n t
a c r o s s a s i n g l e s t e p o f d u r a t i o n A t . T h e c o e f f i c i e n t s A i , B i ,
C i , a n d D i ( i = 1 , 2 , . . . , n - 1 ) a p p e a r i n g i n t h e s y s t e m g e n e r a t e d
b y e q u a t i o n ( 3 . 3 7 ) a r e t h e n k n o w n q u a n t i t i e s ; s o l u t i o n o f t h i s
t r i d i a g o n a l s y s t e m t h e n y i e l d s t h e n e w v a l u e s o f R , v i z . R 1 , m + 1 ,
R 2 , m + 2 , . . . , R n - 1 , m + 1 , a t a l l i n t e r i o r g r i d p o i n t s a t t h e n e w t i m e
l e v e l t = T h e c o r r e s p o n d i n g c o e f f i c i e n t s c a n n o w b e
c o m p u t e d f o r u s e i n t h e s y s t e m o f e q u a t i o n s g e n e r a t e d b y ( 3 . 4 1 ) .
S o l u t i o n o f t h i s t r i d i a g o n a l s y s t e m y i e l d s t h e n e w v a l u e s o f P ,
v i z . P1 , m + 1 , P 2 , m + 1 , . . . , P n - 1 , m + 1 . The new po ten t i a l s ( and hence
t h e n e w s a t u r a t i o n d i s t r i b u t i o n a n d a l s o t h e d i s t r i b u t i o n o f
p h y s i c a l p r o p e r t i e s ) c a n r e a d i l y b e o b t a i n e d b y n o t i n g f r o m e q u a t i o n s
( 3 . 2 8 ) a n d ( 3 . 2 9 ) t h a t
( 3 . 4 5 )
( 3 . 4 6 )
The above p rocedu re may be repea ted f o r success i ve t ime s teps , and
t h u s a p r e d i c t i o n m a d e o f w h a t h a p p e n s i n s i d e t h e c a p r o c k .
I n p r a c t i c e , t h e c o m p u t a t i o n a l p r o c e d u r e i s r a t h e r m o r e
c o m p l i c a t e d t h a n t h a t j u s t d e s c r i b e d , p a r t l y b e c a u s e t h e r e l a t i v e
p e r m e a b i l i t e s a p p e a r i n g i n t h e c o e f f i c i e n t s E a n d F d o v a r y a c r o s s
a t i m e s t e p . T h e m a i n t r o u b l e , h o w e v e r , i s i n s u p p o s i n g t h a t t h e
i n j e c t i o n t e r m s r e m a i n c o n s t a n t ( a t t h e i r i n i t i a l v a l u e ) a c r o s s a
t i m e s t e p . A l l c a l c u l a t i o n s i n t h i s r e s e a r c h b a s e d o n s u c h a n
assump t i on soon became uns tab le and gene ra ted mean ing less va lues .
Mechanism of Gas Leakage Across a Cap Rock
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S e c t i o n 3 . 1 1 s h o w s h o w t h i s d i f f i c u l t y m a y b e o v e r c o m e b y r e p e a t e d
i t e r a t i o n o v e r a t i m e s t e p , e a c h t i m e r e v i s i n g t h e l a s t a v a i l a b l e
e s t i m a t e o f t h e i n j e c t i o n t e r m s . F i r s t , however , i n S e c t i o n 3 . 1 0 ,
we show how the i n j ec t i on t e rms t hemse l ves a re compu ted .
New Concepts in Underground Storage of Natural Gas
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3 . 1 0 C o m p u t a t i o n o f t h e I n j e c t i o n T e r m s
T h e v o l u m e t r i c f l o w o f , f o r e x a m p l e , g a s i n t o t h e c a p r o c k
f r o m b e l o w , p e r u n i t i n t e r r a c i a l a r e a p e r u n i t t i m e , i s a p p r o x i m a t e l y
( 3 . 4 7 )
Here, a r e t h e g a s p o t e n t i a l s i n t h e g a s b u b b l e a n d
a t t h e f i r s t g r i d p o i n t i n s i d e t h e c a p r o c k , r e s p e c t i v e l y . The
co r respond ing i n j ec t i on t e rm q g , 1 (volume o f g a s p e r u n i t t i m e
p e r u n i t v o l u m e o f c a p r o c k ) a t g r i d p o i n t 1 i s o b t a i n e d b y s u p -
p o s i n g t h a t t h e f l o w g i v e n b y ( 3 . 4 7 ) e n t e r s a r e c t a n g u l a r b l o c k
o f u n i t c r o s s - s e c t i o n a n d l e n g t h A x w i t h p o i n t 1 a t i t s c e n t e r .
We have
I n d i m e n s i o n l e s s f o r m ,
where
L i k e w i s e , t h e o t h e r d i m e n s i o n l e s s i n j e c t i o n r a t e s a r e
where
A s m e n t i o n e d p r e v i o u s l y , Q w , 1 = 0 .
( 3 . 4 8 )
( 3 . 4 9 )
( 3 . 5 0 )
( 3 . 5 1 )
Mechanism of Gas Leakage Across a Cap Rock
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N o w t h e c o m p u t a t i o n o f t h e i n j e c t i o n t e r m s i s i n t i m a t e l y
i n v o l v e d w i t h t h e s o l u t i o n o f t h e P e q u a t i o n s . I t may be shown
t h a t t h e t r i d i a g o n a l s y s t e m a r i s i n g f r o m ( 3 . 4 1 ) i s a c t u a l l y s i n g u l a r
( i . e . t h e d e t e r m i n a n t o f t h e c o e f f i c i e n t m a t r i x i s z e r o ) , a n d s o
t h e r e i s a n i n f i n i t e n u m b e r o f s o l u t i o n s . S ince P =
t h i s a m o u n t s t o s a y i n g t h a t s o l u t i o n o f t h e P e q u a t i o n s i s i n s u f -
f i c i e n t t o d e t e r m i n e t h e o v e r a l l p o t e n t i a l l e v e l . Thus i f we add
a c o n s t a n t t o a n y s o l u t i o n o f t h e P e q u a t i o n s , w e h a v e a n o t h e r
s o l u t i o n o f t h e s a m e P e q u a t i o n s . R e a l i z i n g t h a t a f i n a l a d j u s t m e n t
must be made, w e a r b i t r a r i l y s e t P n - 1 t o z e r o , w h i c h t h e n a l l o w s
a u n i q u e s o l u t i o n ( s o t o s p e a k ) t o b e o b t a i n e d f o r P a t t h e r e m a i n -
i n g i n t e r i o r p o i n t s i = 1 , 2 , . . . , n - 2 . S i n c e t h e R v a l u e s a t t h e
n e w t i m e l e v e l h a v e a l r e a d y b e e n o b t a i n e d , t h e c o r r e s p o n d i n g p o t e n -
t i a l s a n d a t a l l i n t e r i o r g r i d p o i n t s c a n b e d e r i v e d b y
a p p l i c a t i o n o f e q u a t i o n s ( 3 . 4 5 ) a n d ( 3 . 4 6 ) .
B u t f r o m w h a t h a s j u s t b e e n s a i d , t h e s e p o t e n t i a l s a r e a s y e t
u n k n o w n t o t h e e x t e n t o f a c o n s t a n t t o b e a d d e d t o e a c h o f t h e m .
T h i s c o n s t a n t a d j u s t m e n t ( c , s a y ) c a n b e f o u n d b y p e r f o r m i n g a n
o v e r a l l m a t e r i a l b a l a n c e o n t h e c a p r o c k a n d r e q u i r i n g t h a t t h e
s u m o f t h e i n j e c t i o n t e r m s b e z e r o , i . e . t h a t
( 3 . 5 2 )
S p e c i f i c a l l y , i f t h e c o m p u t e d p o t e n t i a l s a t t h e e n d p o i n t s a r e
d e n o t e d b y ( w i t h a p p r o p r i a t e s u b s c r i p t s ) , t h e n t h e a d j u s t m e n t
c i s g i v e n b y
i . e . ,
( 3 . 5 3 )
( 3 . 5 4 )
New Concepts of Underground Storage of Natural Gas
-54-
3 . 1 1 I t e r a t i v e T e c h n i q u e f o r I m p r o v i n g E s t i m a t e o f I n j e c t i o n T e r m s
B r i e f l y , t h e p r o c e d u r e d e s c r i b e d s o f a r f o r a d v a n c i n g t h e
p o t e n t i a l s a c r o s s a s i n g l e t i m e s t e p i n v o l v e s t h e f o l l o w i n g c o m -
p u t a t i o n s :
( a ) A s s u m e t h e l a t e s t a v a i l a b l e e s t i m a t e s f o r t h e p h y s i c a l
p r o p e r t i e s a n d t h e i n j e c t i o n t e r m s . T h e s e w i l l b e a t t h e t i m e
l e v e l t =
( b ) D e t e r m i n e t h e c o e f f i c i e n t s f o r u s e i n t h e R e q u a t i o n s ,
a n d s o l v e f o r v a l u e s o f R 1 , m + 1 a t t h e e n d o f t h e t i m e s t e p , i . e .
at t =
( c ) D e t e r m i n e t h e c o e f f i c i e n t s f o r u s e i n t h e P e q u a t i o n s ,
a n d s o l v e f o r v a l u e s o f P i , m + 1 a t t h e e n d o f t h e t i m e s t e p .
( d ) C o m p u t e t h e n e w p o t e n t i a l s f r o m t h e v a l u e s o f P a n d R
' v i a e q u a t i o n s ( 3 . 4 5 ) a n d ( 3 . 4 6 ) .
( e ) D e t e r m i n e t h e p o t e n t i a l a d j u s t m e n t s c f r o m e q u a t i o n ( 3 . 5 4 3 ,
a n d a d d t h i s t o a l l t h e n e w l y c o m p u t e d i n t e r i o r p o t e n t i a l s .
( f ) C o m p u t e t h e n e w s a t u r a t i o n d i s t r i b u t i o n a n d h e n c e t h e
n e w p h y s i c a l p r o p e r t i e s ( i n p a r t i c u l a r , v a l u e s o f k g , k w and
d S / d P c a t e a c h g r i d p o i n t ) .
A l t h o u g h i t a p p e a r s t h a t t h i s p r o c e s s c a n b e r e p e a t e d i n d e f -
i n a t e l y o v e r s u c c e s s i v e t i m e s t e p s , i n s t a b i l i t i e s w o u l d s o o n a r i s e
i n t h e c a l c u l a t i o n s . I n o r d e r t o a v o i d t h i s d i f f i c u l t y , i t i s
n e c e s s a r y t o i t e r a t e o r r e p e a t t h e c o m p u t a t i o n s s e v e r a l t i m e s o v e r
t h e e x i s t i n g t i m e s t e p , e a c h t i m e u s i n g w h a t a r e t h o u g h t t o b e t h e
m o s t a p p r o p r i a t e v a l u e s o f t h e i n j e c t i o n t e r m s f o r u s e o v e r t h e s t e p .
E x p e r i e n c e i n d i c a t e s t h a t i f t h e v a l u e s o f t h e i n j e c t i o n t e r m s
u s e d f o r a d v a n c i n g t h e p o t e n t i a l s o v e r t h e t i m e s t e p c a n b e m a d e t o
a g r e e w i t h t h o s e c o m p u t e d a t t h e e n d o f t h e s t e p , t h e n t h e r e s u l t s
a r e l i k e l y t o b e s t a b l e . U n f o r t u n a t e l y , i t d o e s n o t s u f f i c e
m e r e l y t o r e p e a t t h e c a l c u l a t i o n s u s i n g i n j e c t i o n t e r m s c o m p u t e d a t
t h e e n d o f t h e p r e v i o u s i t e r a t i o n . I t i s s o o n f o u n d t h a t t h e u s e
Mechanism of Gas Leakage Across a Cap Rock
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o f i n j e c t i o n t e r m s w h i c h a r e , f o r e x a m p l e , t o o l o w , w i l l g e n e r a t e ,
o v e r a s i n g l e i t e r a t i o n , i n j e c t i o n t e r m s w h i c h a r e t o o h i g h - a n d
w h i c h a l s o d e v i a t e f u r t h e r f r o m t h e t r u e c o n v e r g e d v a l u e s . T h e s i t u -
a t i o n i s s h o w n i n F i g . 3 . 5 .
F i g . 3 . 5 D e t e r m i n a t i o n o f I n j e c t i o n T e r m s b y I t e r a t i o n
New Concepts in Underground Storage of Natural Gas
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A s s u m e t h a t t h e g a s i n j e c t i o n a t t h e l o w e r f a c e i s u n d e r
d i s c u s s i o n . R e f e r r i n g t o F i g . 3 . 5 , i f a v a l u e s u c h a s Q A 1 w e r e
a s s u m e d t o h o l d o v e r t h e f i r s t i t e r a t i o n a c r o s s a t i m e s t e p , t h e n
t h e p o t e n t i a l s c o m p u t e d a t t h e e n d o f t h a t s t e p w o u l d p r o d u c e a
n e w i n j e c t i o n t e r m s u c h a s Q B l . I f a s e c o n d i t e r a t i o n w e r e
emp loyed ove r t he same s tep , a second assumed va lue QA2 (= QB 1 )
m i g h t g e n e r a t e Q B 2 a t t h e e n d o f t h e s t e p . I f t h e p r o c e d u r e i s
r e p e a t e d w i t h Q A 3 = Q B 2 , e t c . , t h e n t h e c o m p u t e d v a l u e s r a p i d l y
b e c o m e m e a n i n g l e s s ( e . g . , Q B 3 m i g h t b e n e g a t i v e ) .
C l e a r l y , t h e p o i n t P o f i n t e r s e c t i o n o f t h e 4 5 ° l i n e Q A = Q B
w i t h t h e c u r v e i s r e q u i r e d . A l t h o u g h t h e p o s i t i o n o f t h e e n t i r e
cu rve i s unknown , o n c e t h e p o i n t s ( Q A l , Q B l ) a n d ( Q A 2 , Q B 2 ) h a v e b e e n
e s t a b l i s h e d , t h e n b y s i m i l a r t r i a n g l e s a n a p p r o x i m a t i o n P ' c a n b e
l o c a t e d a t w h i c h t h e i n j e c t i o n t e r m i s
( 3 . 5 5 )
A t h i r d i t e r a t i o n c a n n o w b e b a s e d o n Q A 3 = Q P ' , a n d a n e w v a l u e
Q B 3 o b t a i n e d a t t h e e n d o f t h e s t e p . A f u r t h e r a p p r o x i m a t i o n c a n
b e f o u n d f r o m e q u a t i o n ( 3 . 5 5 ) b y r e p l a c i n g Q A 2 a n d Q B 2 w i t h Q A 3
and Q B 3 , r e s p e c t i v e l y . I n t h e c o m p u t e r p r o g r a m , t h i s i t e r a t i v e
p r o c e d u r e i s r e p e a t e d s e v e r a l t i m e s ( u s u a l l y , s i x t i m e s ) b e f o r e
s a t i s f a c t o r y c o n v e r g e n c e i s o b t a i n e d . O n l y t h e n i s i t p o s s i b l e t o
p r o c e e d w i t h t h e c o m p u t a t i o n s o v e r t h e s u c c e e d i n g t i m e s t e p .
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3 . 1 2 R e l a t i v e P e r m e a b i l i t y a n d C a p i l l a r y P r e s s u r e R e l a t i o n s
T o f a c i l i t a t e t h e c o m p u t a t i o n , t h e f o l l o w i n g s p e c i f i c m o d e l s
a r e u s e d t o r e p r e s e n t t h e r e l a t i v e p e r m e a b i l i t i e s t o w a t e r a n d g a s :
( 3 . 5 6 )
( 3 . 5 7 )
E a c h t e n d s t o ( a ) a v e r y l o w v a l u e w h e n t h e o t h e r p h a s e p r e d o m i n a t e s ,
a n d ( b ) a p p r o x i m a t e l y u n i t y w h e n t h e p h a s e i t s e l f p r e d o m i n a t e s .
T h e f o l l o w i n g t w o - p a r a m e t e r m o d e l i s u s e d f o r r e p r e s e n t i n g
c a p i l l a r y p r e s s u r e :
O r , i n d i m e n s i o n l e s s f o r m :
( 3 . 5 8 )
( 3 . 5 9 )
i n w h i c h
E q u a t i o n ( 3 . 5 8 ) p r e d i c t s a n e v e r - i n c r e a s i n g c a p i l l a r y p r e s s u r e a s
t h e w a t e r s a t u r a t i o n f a l l s , a n d i t a l s o g i v e s a t h r e s h o l d p r e s s u r e
o f ( c l - c2) at S = 1. T h e c o n s t a n t s c l a n d c 2 m a y b e a d j u s t e d i n
o r d e r t o i n v e s t i g a t e t h e e f f e c t o f v a r i o u s t y p e s o f c a p i l l a r y
p r e s s u r e c u r v e s . E q u a t i o n ( 3 . 5 9 ) y i e l d s t h e f o l l o w i n g e x p r e s s i o n
f o r t h e d e r i v a t i v e o f s a t u r a t i o n w i t h r e s p e c t t o d i m e n s i o n l e s s
c a p i l l a r y p r e s s u r e :
New Concepts in Underground Storage of Natural Gas
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( 3 . 6 0 )
A m o r e d e s c r i p t i v e t h r e e - p a r a m e t e r m o d e l , w h i c h i s n o t u s e d h e r e ,
wou ld be :
( 3 . 6 1 )
T h i s w o u l d m o r e r e a l i s t i c a l l y y i e l d a s t e e p e r c u r v e i n t h e v i c i n i t y
o f 1 0 0 % w a t e r s a t u r a t i o n .
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3 . 1 3 R e s u l t s
T h e c o m p u t e r p r o g r a m w h i c h i s w r i t t e n t o s o l v e t h e p r o b l e m
d e s c r i b e d i n t h e p r e c e d i n g s e c t i o n s i s p r e s e n t e d i n A p p e n d i x C .
The re , d e f i n i t i o n s a r e a l s o g i v e n o f a l l t h e m a i n v a r i a b l e s i n v o l v e d ,
t o g e t h e r w i t h a c o m p l e t e l i s t i n g o f s e v e n s e t s o f i n p u t d a t a w h i c h
a r e i n v e s t i g a t e d , a n d e x a m p l e s o f t h e p r i n t e d c o m p u t e r o u t p u t . The
p a r t i c u l a r r u n s w h i c h m e r i t d i s c u s s i o n h e r e a r e t h o s e n u m b e r e d 2 , 4 ,
5, 6, and 7. The f o l l ow ing va lues a re a lways assumed :
( a ) G a s v i s c o s i t y : u g = 0 . 0 2 c p .
( b ) G a s d e n s i t y : p g = 4 . 0 l b . / c u / f t .
( c ) W a t e r v i s c o s i t y : u w = 1 . 1 2 c p .
( d ) W a t e r d e n s i t y : p w = 6 2 . 4 l b . / c u . f t .
( e ) C a p r o c k t h i c k n e s s : = 1 0 f t .
( f ) W a t e r p o t e n t i a l a b o v e c a p r o c k : = 1000 psi.
( g ) N u m b e r o f g r i d s p a c i n g s w i t h i n c a p r o c k : 1 0 ( i . e . , n = 1 1 ) .
T h e p r i n c i p a l r e m a i n i n g v a r i a b l e s c o n s t i t u t i n g t h e i n p u t d a t a a r e :
( a ) P o r o s i t y ,
( b ) P e r m e a b i l i t y , k m d .
( c ) C o n s t a n t s c 1 , c 2 ( p s i ) i n c a p i l l a r y p r e s s u r e e q u a t i o n ( 3 . 5 8 ) .
( d ) G a s p o t e n t i a l a b o v e c a p r o c k , ps i .
( e ) M a x i m u m t i m e , t m a x d a y s , f o r w h i c h c a l c u l a t i o n s a r e t o b e p e r f o r m e d .
( f ) I n f o r m a t i o n a s t o h o w g a s p o t e n t i a l varies with time in the
gas bubb le .
At time t = 0, t h e g a s a n d w a t e r p o t e n t i a l s t h r o u g h o u t t h e c a p
rock a re assumed cons tan t and equa l t o a n d r e s p e c t i v e l y .
T h e c o r r e s p o n d i n g s a t u r a t i o n d i s t r i b u t i o n i s c o m p u t e d f r o m e q u a t i o n s
( 3 . 2 5 ) a n d ( 3 . 5 9 ) .
New Concepts in Underground Storage of Natural Gas
-60-
T h e p r o g r a m i s c u r r e n t l y s e t u p s o t h a t t h e g a s p o t e n t i a l
i n t h e g a s b u b b l e v a r i e s s i n u s o i d a l l y w i t h t i m e a c c o r d i n g t o
( 3 . 6 2 )
w h e r e t h e a m p l i t u d e a a n d p e r i o d T a r e r e a d a s d a t a . E q u a t i o n ( 3 . 6 2 )
i s i n t e n d e d a s a n a p p r o x i m a t i o n t o c y c l i c p r e s s u r e f l u c t u a t i o n s
w i t h i n t h e g a s b u b b l e , a l t h o u g h t o d a t e t h e i n v e s t i g a t i o n h a s o n l y
u s e d ( 3 . 6 2 ) o v e r a f r a c t i o n o f a p e r i o d t o g e n e r a t e a n e v e r - i n c r e a s i n g
g a s p o t e n t i a l o n t h e u n d e r - s u r f a c e o f t h e c a p r o c k . I n t h e r u n s
u n d e r d i s c u s s i o n , a and T a re g i ven t he va lues 0 .25 and 800 days ,
r e s p e c t i v e l y , a n d t h e g a s b u b b l e p o t e n t i a l i s p l o t t e d i n F i g . 3 . 1 6 .
T h e r e s u l t s f o r r u n s 2 , 4 , 5 , 6 , a n d 7 a r e s u m m a r i z e d i n F i g s .
3 . 6 t h r o u g h 3 . 1 5 . T w o f i g u r e s a r e g i v e n f o r e a c h r u n ; t h e f i r s t
s h o w s h o w t h e c o m p u t e d s a t u r a t i o n s t h r o u g h o u t t h e c a p r o c k v a r y w i t h
t i m e , and t he second shows how the d imens ion less gas l eakage ra te
Q g , 1 i n t o t h e c a p r o c k v a r i e s w i t h t i m e . T h e m a i n c o n d i t i o n s f o r
e a c h r u n a r e t a b u l a t e d b e l o w .
The c l ose r approaches (1000 psi.), the more nearly will
t h e i n i t i a l w a t e r s a t u r a t i o n a p p r o a c h u n i t y t h r o u g h o u t t h e c a p r o c k .
T h e p a r t i c u l a r c h o i c e c l = c 2 g i v e s z e r o t h r e s h o l d c a p i l l a r y p r e s s u r e
at S = 1. T h a t i s , t h e c o m p u t e d r e s u l t s c a n b e i m a g i n e d t o o c c u r
e f f e c t i v e l y a f t e r a t h r e s h o l d p r e s s u r e h a s b e e n r e a c h e d .
Mechanism of Gas Leakage Across a Cap Rock
-61-
A s t i m e p r o g r e s s e s , a l l r u n s a r e c h a r a c t e r i z e d b y ( a ) a d e c -
r e a s i n g w a t e r s a t u r a t i o n t h r o u g h o u t t h e c a p r o c k ( w i t h t h e m o s t
p r o n o u n c e d e f f e c t n e a r t h e g a s b u b b l e ) , a n d ( b ) a g a s l e a k a g e w h i c h
i n c r e a s e s r a t h e r s l o w l y a t f i r s t , b u t w h i c h l a t e r a c c e l e r a t e s .
F r o m t h e s t a n d p o i n t o f r e t a i n i n g w a t e r i n t h e c a p r o c k , t h e r e s u l t s
a r e m u c h a s e x p e c t e d , v i z . , s m a l l p o r o s i t i e s a n d p e r m e a b i l i t i e s a n d
h i g h c a p i l l a r y p r e s s u r e s a r e b e n e f i c i a l . T h e c a p r o c k o f r u n 4
b e h a v e s p a r t i c u l a r l y b a d l y , w h e r e a s t h a t o f r u n 7 h a s d r i e d o u t
c o m p a r a t i v e l y l i t t l e a f t e r 6 0 d a y s o p e r a t i o n . I n t h e o t h e r c a s e s ,
w a t e r s a t u r a t i o n s n e a r t h e u n d e r - s u r f a c e o f t h e c a p r o c k f a l l t o
a b o u t 6 0 % a f t e r 1 0 0 d a y s o p e r a t i o n .
I n s t u d y i n g t h e g a s l e a k a g e r a t e s , n o t e t h a t t h e a c t u a l l e a k a g e
r a t e q g 0 t h r o u g h t h e u n d e r - s u r f a c e i s r e l a t e d t o t h e d i m e n s i o n l e s s
l e a k a g e r a t e Q g , 1 b y
( 3 . 6 3 )
A g a i n , l o w p o r o s i t i e s a n d p e r m e a b i l i t i e s , a n d h i g h c a p i l l a r y p r e s s u r e s
a r e m o s t f a v o r a b l e f o r p r e v e n t i n g l e a k a g e . B y s u b s t i t u t i n g n u m b e r s
i n t o e q u a t i o n ( 3 . 6 3 ) a n d r e f e r r i n g t o F i g . 3 . 1 5 , t h e
rate at t = 6 0 d a y s f o r r u n 7 i s
a c t u a l l e a k a g e
New Concepts in Underground Storage of Natural Gas
-62-
Now, s i n c e t h i s w a s d e r i v e d f r o m t h e i n j e c t i o n r a t e i n t o t h e f i r s t
" b l o c k " o f h e i g h t A x = 1 . 0 f t . , t h i s i s a l s o e q u i v a l e n t t o a v o l u m e t r i c
l e a k a g e r a t e o f
q = 1 . 6 9 x 1 0 6 c u . f t . o f g a s ( a t r e s e r v o i r p r e s s u r e ) p e r s q . f t .
o f u n d e r - s u r f a c e o f c a p r o c k p e r d a y .
A f t e r 6 0 d a y s , f r o m F i g . 3 . 9 , t h e c o r r e s p o n d i n g f i g u r e f o r r u n 4
i s a p p r o x i m a t e l y
q = 7 . 8 8 x 1 0 - 2 c u . f t . / s q . f t . / d a y ,
i . e . , r o u g h l y 5 0 , 0 0 0 t i m e s a s l a r g e a s t h e l e a k i n r u n 7 .
W e f i n a l l y e m p h a s i z e t h a t t h e a b o v e r e s u l t s a r e b a s e d o n a o n e -
d imens iona l mode l . D u e t o l o c a l h o m o g e n e i t i e s , i t i s p r o b a b l e t h a t
t h e a c t u a l d i s p l a c e m e n t o f w a t e r b y g a s o c c u r s b y a p r o c e s s i n w h i c h
" f i n g e r s " o f g a s a r e f o r m e d . Such wou ld need a t h ree -d imens iona l
t r e a t m e n t f o r i t s c o m p l e t e e x p l a n a t i o n . However, i t i s p r o b a b l e t h a t
i f t h e m o s t a d v e r s e r o c k p r o p e r t i e s l i k e l y t o b e e n c o u n t e r e d a r e u s e d
i n t h e p r e s e n t t r e a t m e n t , t h e n a n u p p e r l i m i t f o r t h e c a p r o c k l e a k a g e
r a t e c a n s t i l l b e o b t a i n e d . The one -d imens iona l me thod wou ld a l so
b e e x p e c t e d t o h o l d f a i r l y w e l l f o r a d o w n w a r d s d i s p l a c e m e n t o f w a t e r
by gas , i n w h i c h " f i n g e r s " o f g a s a r e n o t s o l i k e l y t o o c c u r .
Mechanism of Gas Leakage Across a Cap Rock
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CHAPTER 4
PERFORMANCE OF STORAGE RESERVOIRS SUBJECT TO LEAKAGE
4 . 1 I n t r o d u c t i o n a n d M o d e l
T h e p u r p o s e o f t h i s c h a p t e r i s t o i n v e s t i g a t e t h e e f f e c t o f
v a r i o u s t y p e s o f c a p r o c k l e a k a g e o n t h e p e r f o r m a n c e o f s t o r a g e
r e s e r v o i r s . C o n v e r s e l y , o b s e r v a t i o n s o n r e s e r v o i r p e r f o r m a n c e m a y
a s s i s t i n p i n - p o i n t i n g a p a r t i c u l a r l e a k . C l e a r l y , i n s u c h a n i n v e s -
t i g a t i o n , a l l owance mus t be made fo r va r i ous t ypes o f gas seepage
t h r o u g h t h e c a p r o c k . Fo r examp le , t he l eak may be ve ry much
l o c a l i z e d , o r i t m a y b e d i s t r i b u t e d o v e r a w i d e a r e a . T h e a c t u a l
r a t e o f l e a k a g e m a y b e s m a l l o r l a r g e , a n d m a y p o s s i b l y n o t o c c u r a t
a l l u n l e s s a c e r t a i n t h r e s h o l d p r e s s u r e i s e x c e e d e d . Thus, s e v e r a l
f a c t o r s m u s t b e t a k e n i n t o a c c o u n t i n f o r m u l a t i n g t h e l e a k a g e e f f e c t .
W e n o w s p e c i a l i z e t o t h e i d e a l i z e d s i t u a t i o n i l l u s t r a t e d i n F i g .
4 . 1 , w h i c h s h o w s a h o r i z o n t a l g a s b u b b l e o f u n i f o r m t h i c k n e s s a n d
c o n s t a n t r a d i u s s u r m o u n t e d b y a c a p r o c k . T h e o u t e r p e r i p h e r y o f
t h e g a s b u b b l e i s t a k e n t o b e i m p e r v i o u s t o g a s f l o w , a s i s t h e r o c k
f o r m a t i o n u n d e r l y i n g t h e r e s e r v o i r . T h e s i n g l e w e l l - b o r e i s s u b j e c t
t o a v a r i a b l e p r e s s u r e w h i c h i s a k n o w n f u n c t i o n o f t i m e . Since gas
d e n s i t y i n c r e a s e s w i t h p r e s s u r e , a n i n c r e a s e i n w e l l p r e s s u r e w i l l
c a u s e a d d i t i o n a l g a s t o e n t e r s t o r a g e . T h e f l o w i n t h e f o r m a t i o n
c o n t a i n i n g t h e b u b b l e i t s e l f i s t a k e n t o b e i n t h e r a d i a l a n d a n g u l a r
d i r e c t i o n s o n l y . The l eakage ra te t h rough t he cap rock i s assumed
t o d e p e n d o n l y o n t h e p a r t i c u l a r p o s i t i o n a n d o n t h e p r e v a i l i n g l o c a l
p r e s s u r e i n t h e g a s b u b b l e . T h e p r o b l e m i s t o d e t e r m i n e t h e s u b -
s e q u e n t p r e s s u r e v a r i a t i o n s i n t h e g a s b u b b l e .
-75-
New Concepts in Underground Storage of Natural Gas
-76-
F i g . 4 . 1 M o d e l f o r A r e a - D i s t r i b u t e d G a s L e a k .
Performance of Storage ReservoirsSubject to Leakage
-77-
N o t a t i o n
P e r m e a b i l i t y o f r o c k c o n t a i n i n g b u b b l e .
P o r o s i t y o f r o c k c o n t a i n i n g b u b b l e .
G a s v i s c o s i t y .
G a s d e n s i t y .
P r e s s u r e o f g a s i n b u b b l e .
Time
S u p e r f i c i a l r a d i a l v e l o c i t y c o m p o n e n t .
S u p e r f i c i a l a n g u l a r v e l o c i t y c o m p o n e n t .
R a d i a l d i s t a n c e f r o m c e n t e r o f w e l l - b o r e .
R a d i i o f w e l l a n d g a s b u b b l e .
L e a k a g e m a s s f l o w r a t e p e r u n i t s u r f a c e a r e a o f c a p r o c kp e r u n i t d e p t h o f b u b b l e .
F o r t h e p r e s e n t , a s s u m e t h a t a n y c o n s i s t e n t s e t o f u n i t s i s
employed.
4 . 2 G o v e r n i n g E q u a t i o n s
D a r c y ' s l a w i s o b e y e d i n t h e r a d i a l a n d a n g u l a r d i r e c t i o n s :
( 4 . 1 )
( 4 . 2 )
T h e e q u a t i o n o f c o n t i n u i t y , t a k i n g i n t o a c c o u n t r a d i a l a n d a n g u l a r
f l o w s , a c c u m u l a t i o n d u e t o d e n s i t y c h a n g e a n d l e a k a g e t h r o u g h t h e
c a p r o c k , g i v e s
( 4 . 3 )
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-78-
The gas may be non - i dea l , b u t i s t a k e n t o o b e y t h e f o l l o w i n g s i m p l e
e q u a t i o n o f s t a t e :
i n w h i c h c i s a c o n s t a n t .
( 4 . 4 )
E l i m i n a t i n g v r , a n d p b e t w e e n e q u a t i o n s ( 4 . 1 ) , ( 4 . 2 ) , ( 4 . 3 )
a n d ( 4 . 4 ) , w e o b t a i n
( 4 . 5 )
D imens ion less Fo rm
I t i s c o n v e n i e n t t o i n t r o d u c e t h e f o l l o w i n g d i m e n s i o n l e s s
v a r i a b l e s :
P ressu re :
T ime:
Rad ius :
L e a k a g e r a t e :
I n t h e a b o v e , p 0 r e f e r s t o t h e i n i t i a l u n i f o r m p r e s s u r e t h r o u g h o u t t h e
r e s e r v o i r . E q u a t i o n ( 4 . 5 ) t h e n b e c o m e s
( 4 . 6 )
Performance of Storage ReservoirsSubject to Leakage
-79-
E q u a t i o n ( 4 . 6 ) i s a n o n - l i n e a r s e c o n d - o r d e r p a r t i a l d i f f e r e n t i a l
e q u a t i o n w h i c h g o v e r n s t r a n s i e n t v a r i a t i o n s o f p r e s s u r e w i t h r a d i a l
a n d a n g u l a r p o s i t i o n s i n s i d e t h e g a s b u b b l e .
Change o f Va r i ab le
F o r t h e p u r p o s e o f a l a t e r n u m e r i c a l f i n i t e d i f f e r e n c e s o l u t i o n ,
w e a l s o c h a n g e t h e r a d i a l s p a c e v a r i a b l e t o
R =
E q u a t i o n ( 4 . 6 ) i s t h e n t r a n s f o r m e d t o g i v e
( 4 . 7 )
T h e f i r s t p a r t o f t h i s i n v e s t i g a t i o n , s e c t i o n 4 . 4 , c o n s i d e r s
p r e s s u r e t o b e a f u n c t i o n o f r a d i a l l o c a t i o n a n d t i m e o n l y . I n
t h i s c a s e , t h e s i m p l i f i e d e q u a t i o n t o b e s o l v e d i s t h u s
i n w h i c h
P = dimensionless pressure,
R = d imens ion less rad ius ,
T = d imens ion less t ime ,
M = d i m e n s i o n l e s s l e a k a g e r a t e ,
4 . 3 F o r m u l a t i o n o f L e a k
( 4 . 8 )
T h e t y p e o f l e a k s t i l l r e m a i n s t o b e s p e c i f i e d . A s i m p l e , y e t
New Concepts in Underground Storage of Natural Gas
-80-
f a i r l y d e s c r i p t i v e m o d e l i s t h e f o l l o w i n g :
where
( 4 . 9 )
m = l e a k a g e m a s s f l o w r a t e p e r u n i t c a p r o c k s u r f a c e a r e ap e r u n i t d e p t h o f b u b b l e ,
p = l o c a l p r e s s u r e i n t h e g a s b u b b l e ,
pL = t h r e s h o l d p r e s s u r e b e l o w w h i c h n o l e a k o c c u r s ,
k* = c o e f f i c i e n t d e p e n d i n g o n l e a k a g e r a t e .
N o t e t h a t p L a n d k * m a y b e f u n c t i o n s o f r a d i a l a n d a n g u l a r l o c a t i o n ,
s o t h a t t h e e n t i r e c h a r a c t e r o f t h e l e a k m a y v a r y f r o m o n e p o s i t i o n
t o a n o t h e r . I n d i m e n s i o n l e s s f o r m , w e h a v e
( 4 . 1 0 )
4 . 4 S p e c i a l C a s e o f C y l i n d r i c a l S y m m e t r y
W e s t a r t t h e i n v e s t i g a t i o n b y s u p p o s i n g t h a t t h e r e a r e n o
p r e s s u r e v a r i a t i o n s i n t h e a n g u l a r d i r e c t i o n , s o t h a t t h e o n l y i n d e -
p e n d e n t v a r i a b l e s t o b e c o n s i d e r e d a r e r a d i a l d i s t a n c e a n d t i m e .
T h i s c o r r e s p o n d s t o t h e r a t h e r u n l i k e l y c a s e o f a l e a k b e i n g i n t h e
f o r m o f a r i n g o r s e r i e s o f c o n c e n t r i c r i n g s , w i t h t h e w e l l - b o r e a t
t h e c e n t e r . However, i t i s h o p e d t h a t t h i s s i m p l e c a s e w i l l a t l e a s t
g i ve some c l ues as t o wha t may be expec ted i n t he much more comp l i -
c a t e d c a s e o f a t r u l y t w o - d i m e n s i o n a l l e a k , w h i c h i s s t u d i e d i n
s e c t i o n s 4 . 5 e t s e q .
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-81-
S e p a r a t e c o m p u t e r p r o g r a m s a r e w r i t t e n t o s o l v e t h e e q u a t i o n s
g o v e r n i n g t h e o n e - a n d t w o - d i m e n s i o n a l l e a k a g e e f f e c t s , b u t s i n c e
t h e f o r m e r i s b u t a s p e c i a l c a s e o f t h e l a t t e r , t h e o n e - d i m e n s i o n a l
p r o g r a m i t s e l f w i l l n o t b e d i s c u s s e d i n d e t a i l . H o w e v e r , t y p i c a l
r e s u l t s f o r t h e o n e - d i m e n s i o n a l c a s e a r e p r e s e n t e d b e l o w .
B o u n d a r y a n d I n i t i a l C o n d i t i o n s
T h e c o m p u t e r p r o g r a m w r i t t e n f o r t h e f i n i t e d i f f e r e n c e a p p r o x i -
m a t i o n t o t h e s o l u t i o n o f e q u a t i o n ( 4 . 8 ) c a n c o p e w i t h a n y p r e s c r i b e d
w e l l p r e s s u r e a s a f u n c t i o n o f t i m e a n d a n y r a d i a l d i s t r i b u t i o n o f
l e a k s . F o r t h e p u r p o s e s o f c o m p a r i n g r e s u l t s i n t h i s i n v e s t i g a t i o n ,
i t i s e x p e d i e n t t o c o n s i d e r t h e f o l l o w i n g c o n d i t i o n s .
I n i t i a l l y , t h e p r e s s u r e i s t a k e n t o b e u n i f o r m a n d e q u a l t o p o ,
t h r o u g h o u t t h e r e s e r v o i r :
F o r s u b s e q u e n t t i m e s , t h e w e l l p r e s s u r e p l i s a p r e s c r i b e d f u n c t i o n
o f t i m e :
O n e s p e c i a l c a s e i n p a r t i c u l a r w i l l b e c o n s i d e r e d , w i t h w e l l p r e s s u r e
m a i n t a i n e d a t a c o n s t a n t v a l u e p w . T h e o u t e r b o u n d a r y o f t h e b u b b l e
i s i m p e r v i o u s t o g a s f l o w :
I n d i m e n s i o n l e s s f o r m , t h e s e i n i t i a l a n d b o u n d a r y c o n d i t i o n s b e c o m e
( 4 . 1 1 )
New Concepts in Underground Storage of Natural Gas
-82-
Computed Resul ts
T h e m e t h o d w i l l b e i l l u s t r a t e d u s i n g t h e f o l l o w i n g v a l u e s :
0 . 2 5 ,
5 0 m i l l i d a r c i e s ,
0 . 0 5 c e n t i p o i s e ,
0 . 0 0 3 l b m / ( c u . f t . p s i ) ,
0 . 2 5 f t ,
1 0 0 0 f t ,
= 8 .294 ,
1 0 0 0 p s i ,
1 2 0 0 p s i ,
0 . 1 l b m / ( c u . f t . d a y . p s i ) .
I n t h e f i n i t e d i f f e r e n c e c o m p u t a t i o n , t h e r a n g e R = 0 t o R m a x
= 8 . 2 9 4 i s s u b d i v i d e d i n t o t e n e q u a l i n c r e m e n t s b y t h e i n t r o d u c t i o n
o f e l e v e n g r i d p o i n t s . A l o c a l i z e d l e a k i s a s s u m e d t o o c c u r a t t h e
f i f t h g r i d p o i n t , c o r r e s p o n d i n g t o a l e a k s p r e a d o v e r t h e f a i r l y
n a r r o w r a n g e R / R m a x f r o m 0 . 4 5 t o 0 . 5 5 . E v e r y w h e r e e l s e , t h e c a p
r o c k i s t a k e n t o b e c o m p l e t e l y i m p e r v i o u s t o g a s l e a k a g e .
A s i n d i c a t e d a b o v e , t h e i n i t i a l f o r m a t i o n p r e s s u r e i s 1 0 0 0 p s i .
T h e w e l l p r e s s u r e i s t h e n s u d d e n l y m a i n t a i n e d a t 1 2 0 0 p s i , c o r r e s p o n -
d i n g t o a d i m e n s i o n l e s s p r e s s u r e P = 1 . 2 . T w o t y p e s o f l e a k a r e
s t u d i e d , w i t h t h r e s h o l d p r e s s u r e s o f p L = 1 0 0 0 a n d 1 1 0 0 p s i , s h o w n i n
F i g s . 4 . 2 a n d 4 . 3 r e s p e c t i v e l y . These f i gu res show how the p ressu re
t h r o u g h o u t t h e g a s b u b b l e v a r i e s w i t h t i m e . The co r respond ing cu rves
f o r a c o m p l e t e l y i m p e r v i o u s c a p r o c k a r e a l s o s h o w n f o r p u r p o s e s o f
compar ison. T h e s e i n d i c a t e a c o n s t a n t l y i n c r e a s i n g p r e s s u r e w h i c h
a p p r o a c h e s P = 1 . 2 u n i f o r m l y f o r l a r g e v a l u e s o f t i m e .
Performance of Storage ReservoirsSubject to Leakage
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F i g . 4 . 2 E f fec t o f Leak on Gas Bubb le P ressu re .
New Concepts in Underground Storage of Natural Gas
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F i g . 4 . 3 E f fec t o f Leak on Gas Bubb le P ressu re .
Performance of Storage ReservoirsSubject to Leakage
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I n t h e c a s e o f p L = 1 0 0 0 ( F i g . 4 . 2 ) , t h e e f f e c t o f t h e l e a k i s
soon appa ren t . F o r r a d i i b e y o n d t h e l e a k , t h e p r e s s u r e t e n d s t o
become un i f o rm as t ime i nc reases . N o t e a l s o t h a t w i t h t h e p r e s e n t
c o o r d i n a t e s i t s o h a p p e n s t h a t t h e p r e s s u r e d i s t r i b u t i o n i s e s s e n t i a l l y
l i n e a r o n b o t h s i d e s o f t h e l e a k , w i t h a s h a r p c h a n g e i n s l o p e a t t h e
l e a k i t s e l f . T h i s s u g g e s t s a m e t h o d f o r d e t e c t i n g a n d p i n - p o i n t i n g
a l e a k b y m a k i n g p r e s s u r e r e c o r d i n g s a t j u s t f o u r o b s e r v a t i o n w e l l s .
F o r , i s t h e p r e s s u r e s a r e k n o w n a t t h e p o i n t s A , B , C , a n d D o f F i g .
4 . 2 , t h e n t h e i n t e r s e c t i o n o f A B a n d C D g i v e s t h e l o c a t i o n o f t h e l e a k .
The dev ia t i on be tween the s l opes o f AB and CD a l so depends on t he
m a g n i t u d e o f t h e l e a k . T h e f i n a l s t e a d y s t a t e o f P = 1 . 2 i s n e v e r
reached , s i n c e e v e n t u a l l y a l l t h e g a s b e i n g p u m p e d i n t o t h e w e l l
e s c a p e s t h r o u g h t h e c a p r o c k . F o r t h e h i g h e r t h r e s h o l d p r e s s u r e o f
p L = 1 1 0 0 p s i ( F i g . 4 . 3 ) t h e e f f e c t i s a g a i n s i m i l a r , b u t l e s s p r o -
nounced.
T h e o n e - d i m e n s i o n a l m o d e l i s o b v i o u s l y r a t h e r r e s t r i c t i v e , a n d
t h e m e t h o d w i l l n e x t b e e x t e n d e d t o t w o s p a c e d i m e n s i o n s .
New Concepts in Underground Storage of Natural Gas
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4 .5 Two-D imens iona l Leakage P rob lem
We now remove t he p rev i ous res t r i c t i on and no l onge r assume
c y l i n d r i c a l s y m m e t r y . T h a t i s , t h e l e a k a g e e f f e c t a n d p r e s s u r e
m a y n o w b e f u n c t i o n s o f t i m e , r a d i a l d i s t a n c e , a n d a n g u l a r p o s i t i o n .
P r e s s u r e f l u c t u a t i o n s a r e g o v e r n e d b y e q u a t i o n ( 4 . 7 ) , w h i c h c a n b e
r e w r i t t e n a s
( 4 . 1 2 )
where ( 4 . 1 3 )
( 4 . 1 4 )
To s imp l i f y t he ma thema t i cs somewha t , b u t w i t h o u t r e m o v i n g m u c h i n
t h e w a y o f u s e f u l n e s s f r o m t h e m o d e l , w e s h a l l s o l v e e q u a t i o n ( 4 . 1 2 )
f o r o n e v e r t i c a l h a l f o f t h e g a s b u b b l e , i . e . i n t h e s e m i - c i r c u l a r
r e g i o n d e p i c t e d i n F i g . 4 . 4 . Bo th t he cu rved bounda ry a t R =
l n ( r 2 / r l ) , i . e . t h e g a s / w a t e r i n t e r f a c e , a n d t h e m e r i d i a n p l a n e
represented by w i l l b e e f f e c t i v e l y i m p e r v i o u s t o
g a s f l o w .
A n a p p r o x i m a t i o n t o t h e s o l u t i o n o f ( 4 . 1 2 ) a n d i t s a s s o c i a t e d
i n i t i a l a n d b o u n d a r y c o n d i t i o n s c a n b e o b t a i n e d b y a f i n i t e d i f -
f e r e n c e t e c h n i q u e . W e i n t r o d u c e a s e r i e s o f g r i d p o i n t s s p a c e d
u n i f o r m l y b y a d i m e n s i o n l e s s r a d i a l i n c r e m e n t A R a n d a n a n g u l a r
increment S u b s c r i p t s i a n d j m a y t h e n b e u s e d t o d e n o t e t h a t
g r i d p o i n t h a v i n g c o o r d i n a t e s T h e r e i s a
t o t a l o f m a n d n r a d i a l a n d a n g u l a r i n c r e m e n t s , r e s p e c t i v e l y .
T h u s a p o i n t s u c h a s ( m , j ) w i l l l i e o n t h e c u r v e d b o u n d a r y , ( 0 , j )
w i l l b e a t t h e w e l l , a n d ( i , 0 ) a n d ( i , n ) w i l l b e o n t h e " s p o k e s "
r e s p e c t i v e l y .
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W e a l s o c o n s i d e r t h e v a l u e o f t h e d i m e n s i o n l e s s p r e s s u r e P
a t d i s c r e t e t i m e s Then, by use
o f t r i p l e s u b s c r i p t s s u c h a s P i , j , p , we can denote the computed
v a l u e o f P a t
F i g . 4 . 4 P l a n o f H a l f o f t h e G a s B u b b l e
4 . 6 F i n i t e D i f f e r e n c e A p p r o x i m a t i o n s
E q u a t i o n ( 4 . 1 2 ) w i l l b e s o l v e d h e r e b y t h e i m p l i c i t a l t e r -
n a t i n g d i r e c t i o n m e t h o d , i n w h i c h e a c h t i m e s t e p o f d u r a t i o n A t
i s s p l i t i n t o t w o h a l f t i m e s t e p s e a c h o f d u r a t i o n Ove r t he
f i r s t h a l f t i m e s t e p , e q u a t i o n ( 4 . 1 2 ) w i l l b e r e p r e s e n t e d i m p l i c i t l y
i n t h e e - d i r e c t i o n a n d e x p l i c i t l y i n t h e R - d i r e c t i o n . S t a r t i n g
f rom va lues such as P i , j , p a t t h i s w i l l e n a b l e i n t e r -
mediate values P* i , j t o b e o b t a i n e d a t t h e e n d o f t h e f i r s t h a l f
t i m e s t e p . O v e r t h e s e c o n d h a l f t i m e s t e p , e q u a t i o n ( 4 . 1 2 ) w i l l
New Concepts in Underground Storage of Natural Gas
-88-
b e r e p r e s e n t e d i m p l i c i t l y i n t h e R - d i r e c t i o n a n d e x p l i c i t l y i n t h e
B - d i r e c t i o n , e n a b l i n g v a l u e s P i , j , p + 1 t o b e c o m p u t e d a t t i m e l e v e l
T h e u s e o f t w o h a l f t i m e s t e p s i n s e r i e s a l l o w s
t h e p r o c e d u r e d e s c r i b e d i n A p p e n d i x . B t o b e u s e d f o r t h e s o l u t i o n
o f t h e r e s u l t i n g t r i d i a g o n a l s y s t e m s o f e q u a t i o n s . We now presentt h e r e l e v a n t f i n i t e d i f f e r e n c e a p p r o x i m a t i o n s f o r u s e o v e r e a c h
h a l f t i m e s t e p .
4 . 7 F i r s t H a l f T i m e S t e p ( I m p l i c i t i n A n g u l a r D i r e c t i o n )
A t a g e n e r a l i n t e r i o r g r i d p o i n t ( i , j ) , t h e f i n i t e d i f f e r e n c e
a p p r o x i m a t i o n t o ( 4 . 1 2 ) i s
( 4 . 1 5 )
M u l t i p l y i n g t h r o u g h b y and letting
w e o b t a i n
( 4 . 1 6 )
( 4 . 1 7 )
( 4 . 1 8 )
Performance of Storage ReservoirsSubject to Leakage
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i . e . ,
where
I n a d d i t i o n ,
( 4 . 1 9 )
t h e c o e f f i c i e n t s h a v e t h e f o l l o w i n g s p e c i a l v a l u e s
( 4 . 2 0 )
a t t h e v a r i o u s b o u n d a r y p o i n t s .
i = m:
( 4 . 2 1 )
A , B , a n d C h a v e t h e v a l u e s g i v e n b y ( 4 . 2 0 ) , e x c e p t a t j = 0 o r
j = n ( s e e b e l o w ) .
j = 0:
T h e c o e f f i c i e n t A 0 i s n o t r e q u i r e d .
( 4 . 2 2 )
New Concepts in Underground Storage of Natural Gas
-90-
i = n:
T h e c o e f f i c i e n t C n i s n o t r e q u i r e d .
i = 1:
( 4 . 2 3 )
A , B , a n d C h a v e t h e v a l u e s g i v e n b y ( 4 . 2 0 ) , e x c e p t t h a t
P* 0 , j s h o u l d b e s e t e q u a l t o t h e k n o w n w e l l p r e s s u r e P 0 , j , p + 1 .
N o w c o n s i d e r e q u a t i o n ( 4 . 1 9 ) a p p l i e d t o e a c h p o i n t j = 0 , 1 ,
. . . , n a l ong a sem ic i r c l e o f r ad ius T h e r e s u l t i n g t r i -
d i agona l sys tem may be so l ved by t he me thod desc r i bed i n Append i x
B , giving the intermediate pressures P*0,i, P*1,i,..., P*n,i at that
p a r t i c u l a r r a d i u s . T h i s p r o c e d u r e i s r e p e a t e d f o r s u c c s s i v e
r a d i i i = 1 , 2 , . . . , m , t h u s e n a b l i n g a l l t h e i n t e r m e d i a t e p r e s s u r e s
to be computed t h r o u g h o u t t h e s e m i c i r c u l a r g a s s b u b b l e .
Performance of Storage ReservoirsSubject to Leakage
-91-
4 . 8 S e c o n d H a l f T i m e S t e p ( I m p l i c i t i n R a d i a l D i r e c t i o n )
A t a n i n t e r i o r g r i d p o i n t , t h e f i n i t e d i f f e r e n c e a p p r o x i m a t i o n
i s
( 4 . 2 4 )
i . e . ,
i n w h i c h
( 4 . 2 5 )
( 4 . 2 6 )
New Concepts in Underground Storage of Natural Gas
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I n a d d i t i o n , t h e c o e f f i c i e n t s h a v e t h e f o l l o w i n g s p e c i a l v a l u e s
a t t h e v a r i o u s b o u n d a r y p o i n t s .
i = m:
T e x t
T h e c o e f f i c i e n t C m i s n o t r e q u i r e d .
j = 0:
( 4 . 2 8 )
A , B , a n d C h a v e t h e v a l u e s g i v e n b y ( 4 . 2 6 ) .
j = n:
A , B , a n d C h a v e t h e v a l u e s g i v e n b y ( 4 . 2 6 ) .
( 4 . 2 9 )
i = 1:
A l i s n o t r e q u i r e d , a n d C lA l i s n o t r e q u i r e d , a n d C l h a s t h e v a l u e g i v e n b y ( 4 . 2 6 ) .h a s t h e v a l u e g i v e n b y ( 4 . 2 6 ) .
D l h a s t h e v a l u e g i v e n b y ( 4 . 2 6 ) , m i n u s A l P 0 , j p + 1 , w h e r e A l i s t h e
v a l u e t h a t w o u l d b e p r e d i c t e d b y ( 4 . 2 6 ) , a n d P 0 , j , p + 1i s t h e k n o w ni s t h e k n o w n
we l l p ressu re a t t ime l eve l
Performance of Storage ReservoirsSubject to Leakage
-93-
N o w c o n s i d e r e q u a t i o n ( 4 . 2 5 ) a p p l i e d t o e a c h p o i n t i = 1 , 2 ,
. . . , m a l o n g a r a d i a l s p o k e a t a n a n g l e T h e r e s u l t i n g t r i -
d i agona l sys tem may be so l ved by t he me thod desc r i bed i n Append i x B ,
g i v i n g t h e f i n a l p r e s s u r e s P 1 , j , p + 1 , P 2 , j , p + 1 , . . . . , P m , j , p + 1 a t t h a t
p a r t i c u l a r a n g l e . T h i s p r o c e d u r e i s r e p e a t e d f o r s u c c e s s i v e a n g u l a r
subscripts j = 0, 1, . . . . n , t h u s e n a b l i n g a l l t h e p r e s s u r e s a t t h e
e n d o f t h e w h o l e t i m e s t e p t o b e c o m p u t e d t h r o u g h o u t t h e s e m i c i r c u l a r
gas bubb le .
4 . 9 R e s u l t s
T h e c o m p u t e r p r o g r a m f o r i m p l e m e n t i n g t h e p r o c e d u r e o u t l i n e d
i n s e c t i o n s 4 . 5 t h r o u g h 4 . 8 i s f u l l y d e s c r i b e d i n A p p e n d i x D . W e
p r e s e n t h e r e t h e r e s u l t s c o m p u t e d f o r f o u r d i f f e r e n t s e t s o f c o n -
d i t i o n s , i d e n t i f i e d b y t h e " r u n " n u m b e r s 6 , 5 , 9 , a n d 1 0 ( i n t h a t
o r d e r ) . A l t h o u g h t h e c o m p u t e r p r o g r a m c a n a c c e p t a w i d e v a r i e t y o f
c o n d i t i o n s , t h e f o l l o w i n g p a r a m e t e r s a r e h e l d c o n s t a n t f o r t h e r e s u l t s
g i v e n h e r e :
0 . 2 5 f t ,
1 0 0 f t ,
1 0 0 0 p s i ,
5 0 0 m i l l i d a r c i e s ,
0 . 0 0 3 l b m / ( c u . f t . p s i ) ,
0 . 0 2 c e n t i p o i s e ,
0 .25
0 .0125 days .
A t t i m e t = 0 , t h e w e l l p r e s s u r e i s s u d d e n l y r a i s e d t o 1 2 0 0 p s i , c o r -
r e s p o n d i n g t o a d i m e n s i o n l e s s p r e s s u r e P = 1 . 2 . T e n g r i d i n c r e m e n t s
a r e u s e d i n b o t h t h e r a d i a l a n d a n g u l a r d i r e c t i o n s .
T h e r e s u l t s f o r t h e f o u r r u n s a r e r e p r o d u c e d d i r e c t l y f r o m t h e
p r i n t e d c o m p u t e r o u t p u t , a n d a r e d i s p l a y e d i n F i g s . 4 . 5 t h r o u g h 4 . 8 .
I n e a c h c a s e , t h e o u t p u t c o n s i s t s o f ( a ) a c h e c k o f d a t a i t e m s r e a d
New Concepts in Underground Storage of Natural Gas
-94-
b y t h e p r o g r a m , ( b ) a l i s t o f c e r t a i n a u x i l i a r y p a r a m e t e r s c o m p u t e d
by the p rog ram, a n d ( c ) t a b l e s o f t h e c o m p u t e d d i m e n s i o n l e s s p r e s s u r e s
a t s e l e c t e d v a l u e s o f t i m e . M o s t o f t h e s y m b o l s a r e s e l f - e x p l a n a t o r y ,
b u t a l l a r e c o m p l e t e l y d e f i n e d i n A p p e n d i x D . F i g . 4 . 5 i l l u s t r a t e s
h o w t h e c o m p u t e d p r e s s u r e s a r e t o b e i n t e r p r e t e d . B e a r i n g i n m i n d
t h a t i a n d j a r e t h e r a d i a l a n d a n g u l a r s u b s c r i p t s r e s p e c t i v e l y , a n y
c o l u m n o f f i g u r e s c o r r e s p o n d s t o t h e c o m p u t e d p r e s s u r e s o n a " s p o k e "
g o i n g r a d i a l l y o u t w a r d s f r o m t h e w e l l . A s c a n h o r i z o n t a l l y a c r o s s a
r o w c o r r e s p o n d s t o o b s e r v i n g t h e p r e s s u r e s a l o n g a s e m i - c i r c u l a r a r c
o f c o n s t a n t r a d i u s . I n p a r t i c u l a r , t h e f i r s t r o w ( i = 0 ) g i v e s t h e
w e l l p r e s s u r e , a n d t h e l a s t r o w g i v e s t h e p r e s s u r e a t t h e o u t e r
p e r i p h e r y o f t h e g a s b u b b l e . T h e p a r t i c u l a r c h o i c e o f t i m e s t e p
= 0 . 0 1 2 5 d a y s ) i s m a i n l y d i c t a t e d b y t h e n e e d t o e n s u r e c o m p u -
t a t i o n a l s t a b i l i t y . F o r a p p r e c i a b l y l a r g e r t i m e s t e p s , m e a n i n g l e s s
v a l u e s f o r p r e s s u r e a r e q u i c k l y g e n e r a t e d . T h i s d i f f i c u l t y i s l a r g e l y
d u e t o t h e p r e s e n c e o f t h e e x p o n e n t i a l f a c t o r e 2 Ri n h e r e n t i n e q u a t i o n
( 4 . 1 2 ) .
Run 6 (No Leak)
The case o f no l eak se rves as a conven ien t check on t he compu-
t a t i o n a l p r o c e d u r e , a n d t h e r e s u l t s a r e s h o w n i n F i g . 4 . 5 . The
s u d d e n i n c r e a s e i n w e l l p r e s s u r e i s e v e n t u a l l y t r a n s m i t t e d t h r o u g h o u t
the who le gas bubb le . W i t h i n a s m a l l d e g r e e o f c o m p u t a t i o n a l r o u n d -
o f f e r r o r , t h e d i m e n s i o n l e s s p r e s s u r e s a s y m p t o t i c a l l y a p p r o a c h 1 . 2 .
T h i s v a l u e i s e s s e n t i a l l y r e a c h e d e v e r y w h e r e a f t e r 0 . 2 d a y s . Note
t h a t , a s e x p e c t e d , t h e p r e s s u r e i s c o n s t a n t a t a l l p o i n t s l y i n g a t t h e
s a m e r a d i a l d i s t a n c e f r o m t h e w e l l .
R u n 5 ( S i n g l e L e a k )
W e n o w s u p p o s e t h a t a l e a k p r e v a i l s o v e r a s m a l l s e g m e n t a l a r e a
w i t h t h e g r i d p o i n t ( i = 8 , j = 5 ) a t i t s c e n t e r , a s s h o w n i n F i g . 4 . 9 .
O v e r t h i s a r e a , a l e a k o c c u r s a c c o r d i n g t o e q u a t i o n ( 4 . 9 ) , w h e r e i n
k * = 2 . 0 l b m / ( c u . f t . d a y . p s i ) a n d t h e t h r e s h o l d p r e s s u r e i s p L = 1 0 0 0
p s i . T h e r e s u l t s a r e s h o w n i n F i g . 4 . 6 . F o r s m a l l t i m e s . t h e
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c o m p u t e d p r e s s u r e s a p p r o x i m a t e t h o s e f o r n o l e a k , b u t t h e e f f e c t o f
t h e l e a k i s s o o n a p p a r e n t , a n d t h e s u b s e q u e n t r i s e s i n p r e s s u r e
become s lower and s lower . T h e v a l u e s a t t = 0 .25 days have a lmos t
r e a c h e d a s t e a d y s t a t e . I t i s o b v i o u s t h a t t h e p o s t u l a t e d l e a k i s
enormous, f a r g r e a t e r t h a n a n y t h i n g t h a t c o u l d b e t o l e r a t e d i n p r a c -
t i c e , s i nce mos t o f t he gas bubb le neve r r eaches a p ressu re app roach -
i n g t h a t a t t h e w e l l . F o r m o r e m o d e r a t e l e a k s , i t a p p e a r s t h a t i f
t h e l o c a t i o n o f t h e l e a k w e r e u n k n o w n , i t w o u l d p r o b a b l y b e d i f f i c u l t
t o p i n p o i n t i t s l o c a t i o n f r o m p r e s s u r e o b s e r v a t i o n s m a d e a t a f e w
a u x i l i a r y b o r e h o l e s . T h e p r e s s u r e f a l l s s i g n i f i c a n t l y , o f c o u r s e ,
i n t h e i m m e d i a t e n e i g h b o r h o o d o f t h e l e a k . A l s o , t h e r e i s a c t u a l l y
a n u l t i m a t e i n c r e a s e i n p r e s s u r e g o i n g r a d i a l l y b e y o n d t h e l e a k , d u e
t o l a t e r a l c o m m u n i c a t i o n w i t h i n t h e g a s b u b b l e .
R u n 9 ( S i n g l e L e a k , w i t h T h r e s h o l d P r e s s u r e E f f e c t )
Here , t h e l o c a t i o n a n d k * v a l u e f o r t h e l e a k a r e t h e s a m e a s
t h o s e i n r u n 6 . However, a t h r e s h o l d p r e s s u r e o f p L = 1 1 0 0 p s i i s
used, w h i c h m e a n s t h a t t h e l e a k a g e i s c o n s i d e r a b l y r e d u c e d a n d t h a t t h e
p r e s s u r e s r i s e t o g e n e r a l l y h i g h e r v a l u e s a t s t e a d y s t a t e . F o r s m a l l
t i m e s , t h e r e s u l t s a r e i d e n t i c a l w i t h t h o s e f o r t h e n o - l e a k c a s e .
Run 10 (Row Leak)
Here , we assume a row of leaks a long the wedge-shaped area
shown i n F ig . 4 . 1 0 , w i t h k * = 2 . 0 l b m / ( c u . f t . d a y . p s i ) a n d p L = 1 0 0 0
p s i t h r o u g h o u t . T h e a c t u a l l e a k a g e r a t e w o u l d t e n d t o b e c o n c e n t -
r a t e d t o w a r d s t h e l a r g e r v a l u e s o f r a d i u s , o w i n g t o t h e l a r g e r a r e a
o f t h e " w e d g e " t h e r e . Hence, i t i s n o t s u r p r i s i n g t h a t t h e r e s u l t s
a r e s o m e w h a t s i m i l a r t o t h o s e f o r t h e s i n g l e l e a k o f r u n 5 . There
i s p a r t i c u l a r l y l i t t l e v a r i a t i o n i n p r e s s u r e a l o n g c i r c u l a r a r c s
c l o s e t o t h e w e l l - b o r e .
A l t h o u g h n o t i n v e s t i g a t e d h e r e , n o t e t h a t l o c a l v a r i a t i o n s i n
t h e p r o p e r t i e s o f t h e r o c k f o r m a t i o n w o u l d i n t r o d u c e a d d i t i o n a l
New Concepts in Underground Storage of Natural Gas
-96-
d e v i a t i o n s . A l s o , f o r m o r e m o d e r a t e l e a k s , t h e p r e s s u r e v a r i a t i o n s
wou ld be much sma l l e r t han t hose p red i c ted above . T h e a b o v e r e s u l t s
t h e r e f o r e s u p p o r t t h e c o n c l u s i o n t h a t i t w o u l d p r o b a b l y b e e x t r e m e l y
d i f f i c u l t t o l o c a t e a l e a k o f m o d e r a t e s i z e b y o b s e r v i n g p r e s s u r e s a t
( s a y ) a h a l f - d o z e n o r s o a u x i l i a r y b o r e - h o l e s o n l y . Howeve r , a suc -
c e s s f u l c o m p u t a t i o n a l t e c h n i q u e h a s b e e n d e m o n s t r a t e d f o r e s t i m a t i n g
t r a n s i e n t p r e s s u r e v a r i a t i o n s w i t h i n a c o m p r e s s i b l e b u b b l e o f g a s .
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F i g . 4 . 9 L o c a t i o n o f L e a k i n R u n s 5 a n d 9 .
F i g . 4 . 1 0 L o c a t i o n o f L e a k i n R u n 1 0 .
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CHAPTER 5
IMPERMEATION OF UNDERGROUND FORMATIONS
A l a r g e n u m b e r o f p r a c t i c a l e n g i n e e r i n g p r o b l e m s e n c o u n t e r e d
i n s u b s u r f a c e c o n s t r u c t i o n , d r i l l i n g , s t o r a g e a n d m i n i n g r e q u i r e
s t a b i l i z a t i o n o f s o i l s u s i n g a r t i f i c i a l t e c h n i q u e s t o i m p e r m e a t e
t h e p o r o u s f o r m a t i o n a l o n g s p e c i f i c a l l y c o n t r o l l e d g e o m e t r i e s .
U n d e r g r o u n d s t o r a g e o f n a t u r a l g a s w h e t h e r i n d e p l e t e d o i l
o r g a s r e s e r v o i r s o r i n a q u i f e r s d e p e n d s c r i t i c a l l y u p o n t h e e x i s -
t e n c e o f a n i m p e r v i o u s c a p t o p r e v e n t e s c a p e o f n a t u r a l g a s t o
s h a l l o w e r o r a d j a c e n t f o r m a t i o n s u n d e r t h e i n f l u e n c e o f b u o y a n c y .
T h e p o s s i b i l i t y o f g a s l e a k a g e a c r o s s t h e c a p r o c k o r t h e p r o s p e c t
o f u s i n g s u b s u r f a c e s a n d s w i t h h i g h p o r o s i t y a n d p e r m e a b i l i t y b u t
n o s u i t a b l e c a p f o r u n d e r g r o u n d s t o r a g e p o i n t t o t h e p o s s i b i l i t y
a n d d e s i r a b i l i t y o f a p p l i c a t i o n o f s p e c i a l c h e m i c a l s t o p e r m i t s o i l
s t a b i l i z a t i o n a l o n g d e s i r e d g e o m e t r i c c o n f i g u r a t i o n s .
T h e p r o c e s s b y w h i c h a s p e c i a l c h e m i c a l s o l u t i o n i s i n j e c t e d
i n t o t h e p o r e s o f a p o r o u s m a t e r i a l t o r e n d e r i t i m p e r v i o u s t o f l u i d
f l o w a c r o s s i t i s c a l l e d " G r o u t i n g . "
P r e v i o u s r e s e a r c h w o r k o n t h e c o m p o s i t i o n , p r o p e r t i e s , a n d
a p p l i c a t i o n s o f g r o u t s h a v e b e e n q u i t e l i m i t e d t o s h a l l o w s u b s u r -
f a c e c o n s t r u c t i o n w o r k a n d s o m e l a b o r a t o r y s t u d i e s b y m a n u f a c t u r e r s
o f g r o u t s . T h e i n f o r m a t i o n a v a i l a b l e i n t h e l i t e r a t u r e o n t h e a p p l i -
c a t i o n a n d s u c c e s s o f g r o u t i n g p r o c e s s e s h a v e b e e n l i m i t e d t o a l a r g e
ex ten t on impe rmea t i on on an exposed su r face . The re has been ve ry
l i t t l e , i n d e e d i f a n y i n f o r m a t i o n o n t h e a p p l i c a t i o n o f g r o u t s t o
p r o b l e m s e n c o u n t e r e d i n u n d e r g r o u n d s t o r a g e o f n a t u r a l g a s .
-107-
New Concepts in Underground Storage of Natural Gas
-108-
I n o r d e r t o e x p l o r e a n d e v a l u a t e p h y s i c a l p o s s i b i l i t y a n d
e c o n o m i c f e a s i b i l i t y o f s u b s u r f a c e g r o u t i n g t o r e m e d y l e a k s f r o m
u n d e r g r o u n d s t o r a g e r e s e r v o i r s , t o p r o v i d e s t o r a g e p o r e v o l u m e s o f
s u f f i c i e n t s i z e , s h a p e , a n d c h a r a c t e r i s t i c s , p a r t o f t h e r e s e a r c h
e f f o r t on t he "New Concep ts on Unde rg round S to rage " P ro jec t has been
d i r e c t e d t o w a r d t h e s t u d y o f p r o b l e m s a s s o c i a t e d w i t h g r o u t i n g .
T h e r e a r e t w o k i n d s o f g r o u t i n g m a t e r i a l s g e n e r a l l y a v a i l a b l e
i n t h e i n d u s t r y t o d a y . T h e s e m a y b e c l a s s i f i e d a s s u s p e n s i o n g r o u t s
a n d t r u e - s o l u t i o n c h e m i c a l g r o u t s . The suspens ion g rou t s such as
c e m e n t a n d b e n t o n i t e h a v e r a t h e r w i d e s p r e a d a p p l i c a t i o n a s s u r f a c e
i m p e r m e a t i n g m a t e r i a l s . These g rou t s , o n t h e o t h e r h a n d , c a n n o t
b e u s e d i n a r e a s w h e r e i t i s d e s i r e d t o i n j e c t t h e g r o u t b e y o n d
t h e s u r f a c e p o r e s o f t h e f o r m a t i o n s . T h e l i t h o l o g y o f t y p i c a l p o r -
o u s f o r m a t i o n s w h e r e g r o u t s m u s t b e u s e d i s s u c h t h a t t h e s i z e o f
t h e p o r e s a r e s m a l l e r t h a n t h o s e o f t h e p a r t i c l e s i n s u s p e n s i o n i n
t h e g r o u t s . I n o r d e r t o i m p e r m e a t e s a n d s t o n e , l i m e s t o n e , d o l o m i t e
o r s h a l e t y p e o f f o r m a t i o n s , o n e m u s t g o t o a p p l i c a t i o n s o f t r u e
s o l u t i o n g r o u t s .
A l a b o r a t o r y s t u d y o f p h y s i c a l p r o p e r t i e s o f v a r i o u s g r o u t i n g
agen t s , a s y s t e m a t i c c o m p i l a t i o n o f t h e i r s i g n i f i c a n t c h a r a c t e r i s -
t i c s a n d e v a l u a t i o n o f t h e i r i n j e c t a b i l i t y i n t o p o r o u s p l u g s a n d
t y p i c a l f i e l d f o r m a t i o n s , t h e i r f l o w p r o p e r t i e s , i n j e c t i o n , s e t t i n g ,
i m p e r m e a t i o n p r o p e r t i e s h a v e b e e n t h e p r i m a r y o b j e c t i v e s o f t h e r e -
s e a r c h w o r k r e p o r t e d i n t h e f o l l o w i n g c h a p t e r .
T h e c o n t a i n m e n t o f a l a r g e s t o r a g e b u b b l e i n u n d e r g r o u n d s t r a t a
h a v i n g g o o d p o r o s i t y , g o o d p e r m e a b i l i t y b u t l i t t l e o r n o s t r u c t u r a l
c l o s u r e o r c a p r o c k m a y i n d e e d b e p o s s i b l e i f f o r m a t i o n s c a n b e i m -
p e r m e a t e d a l o n g c o n t r o l l e d g e o m e t r i e s t h r o u g h i n j e c t i o n o f s o m e c h e m -
i c a l g r o u t s . F i g u r e 5 . 1 s h o w s t w o a r e a s w h e r e g r o u t i n g m a t e r i a l s
m a y b e a p p l i e d i n c o n v e n t i o n a l u n d e r g r o u n d s t o r a g e . The c ross sec -
t i o n s h o w s a s e m i - o p e n s t r u c t u r e w h e r e i n j e c t i o n o f t h e g r o u t i n t o
t h e s a d d l e a r e a w o u l d d e f i n i t e l y i n c r e a s e t h e s t o r a g e c a p a c i t y o f
t h e f i e l d b e y o n d t h e " s p i l l p o i n t " l e v e l .
Impermeation of Underground Formations
-109-
F i g . 5 . 1 C o n t r o l o f G a s S p i l l A c r o s s S a d d l e b yG r o u t I n j e c t i o n
5 . 1 T h e G r o u t i n g M a t e r i a l s
V a r i o u s m a t e r i a l s a v a i l a b l e i n t h e m a r k e t f o r g r o u t i n g p o r o u s
f o r m a t i o n s w i l l b e l i s t e d a n d d i s c u s s e d i n t h e f o l l o w i n g . W h i l e
s o m e o f t h e s e m a t e r i a l s a r e w e l l k n o w n a s t o t h e i r c h e m i c a l c o m p o s i -
t i o n o t h e r s a r e n o t r e l e a s e d t o t h e p u b l i c f o r p r o p r i e t a r y a n d p a -
t e n t p r o t e c t i o n r e a s o n s .
S i l i c a t e G r o u t s
T h e f i r s t c h e m i c a l g r o u t s w e r e b a s e d o n s o d i u m s i l i c a t e , b u t
a l t h o u g h s o d i u m s i l i c a t e i s t h e b a s i s o f t h e s i l i c a t e g r o u t s , m a n y
f o r m u l a v a r i a t i o n s h a v e b e e n p a t e n t e d . A s u r v e y 5 ' 1 * i n 1 9 5 7 r e v e a l e d
p a t e n t s o n 2 7 n o n - s o l u b l e a n d 1 7 s o l u b l e s i l i c a t e f o r m u l a s . Examples
* T h e n u m b e r s i n u p p e r s c r i p t p a r e n t h e s e s r e f e r t o l i t e r a t u r e
c i t a t i o n s g i v e n a t t h e e n d o f e a c h c h a p t e r .
New Concepts in Underground Storage of Natural Gas
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a n d p r o p e r t i e s o f t h e o l d e r s i l i c a t e g e l s a r e l i s t e d i n T a b l e 1 . 1 .
A s c a n b e s e e n f r o m t h e t a b u l a t i o n , t h e c o m p o s i t i o n s g i v i n g h i g h
s t r e n g t h g e l s h a d t o b e i n j e c t e d b y t h e i m p r a c t i c a l a n d e x p e n s i v e
two -sho t me thod . H o w e v e r , D i a m o n d A l k a l i a n d H a l l i b u r t o n h a v e r e -
c e n t l y i n t r o d u c e d a o n e - s h o r t s i l i c a t e g e l w h i c h h a s t h e s t r e n g t h
o f t h e t w o - s h o t g e l s . A l i m i t a t i o n , h o w e v e r , o f t h e s i l i c a t e s i s
t h e v i s c o s i t y o f t h e s i l i c a t e s o l u t i o n , a b o u t 5 - 1 2 c e n t i p o i s e s .
A l t h o u g h s i l i c a t e g e l s a r e s t i l l b e i n g i n v e s t i g a t e d , t h e y d o n o t
s e e m t o h a v e t h e l o w v i s c o s i t y r e q u i r e d f o r i n j e c t i o n i n t o t i g h t
p o r o u s m a t r i c e s e n c o u n t e r e d i n c a p r o c k m a t e r i a l s .
Chrome - L i g n i n G r o u t s
T h e l e a s t e x p e n s i v e c h e m i c a l g r o u t i s b a s e d o n c a l c i u m l i g n o -
s u l f o n a t e , a b y - p r o d u c t o f t h e p a p e r p u l p i n d u s t r y . When ca ta l yzed
by sod ium d i ch roma te , a g e l c a l l e d c h r o m e - l i g n i n f o r m s . A n a c c e l -
e r a t o r s u c h a s f e r r i c c h l o r i d e i s s o m e t i m e s u s e d . T h e g e l t i m e c a n
b e c o n t r o l l e d b y v a r y i n g t h e a m o u n t o f c a t a l y s t a n d a c c e l e r a t o r a s
w e l l a s t h e a m o u n t o f w a t e r . S i n c e w a t e r i n f l u e n c e s t h e s e t t i n g
p r o p e r t i e s , t h e m a t e r i a l i s h i g h l y s e n s i t i v e t o d i l u t i o n w i t h g r o u n d
w a t e r . T h e v i s c o s i t y o f t h e c h r o m e - l i g n i n s o l u t i o n i s 3 - 1 2 c e n t i p o i s e s
a t r oom tempe ra tu re ; howeve r , t h i s v i s c o s i t y i n c r e a s e s f r o m t h e m o -
m e n t o f m i x i n g ' u n t i l t h e g e l f o r m s . I f d i l u t i o n d o e s n o t o c c u r ,
t h e s e g e l s s e t w i t h r e a s o n a b l e s t r e n g t h , A l t h o u g h t h i s m a t e r i a l
h a s m a n y d e s i r a b l e p r o p e r t i e s , i t s u s e m a y b e l i m i t e d i n t h e g a s
s t o r a g e f o r m a t i o n s b e c a u s e o f i t s s e n s i t i v i t y t o d i l u t i o n .
F u r f u r a l G r o u t s
T h e f u r f u r a l g e l w a s o r i g i n a l l y d e v e l o p e d b y P h i l l i p s P e t r o -
l e u m f o r s e a l i n g p o r o u s w a l l s i n o i l w e l l s . L a b o r a t o r y t e s t s i n d i -
c a t e t h a t t h i s m a t e r i a l m a y b e u s e f u l i n i m p e r m e a t i n g s a n d s t o n e o r
o t h e r t y p i c a l f o r m a t i o n s i n u n d e r g r o u n d s t o r a g e a p p l i c a t i o n s , The
g e l i s p r e p a r e d w i t h f u r f u r a l a n d t h i o u r e a w i t h h y d r o c h l o r i c a c i d
Impermeation of Underground Formations
-111-
TABLE 5.1
P r o p e r t i e s o f C o n v e n t i o n a l S i l i c a t e G e l s
New Concepts in Underground Storage of Natural Gas
-112-
a s a c a t a l y s t . G e l t i m e s o f 4 t o 6 h o u r s h a v e b e e n o b t a i n e d i n t h e
l a b o r a t o r y w i t h a b o u t 3 h o u r s r e q u i r e d f o r c o m p l e t e f o r m a t i o n o f
t h e g e l . T h e v i s c o s i t y h a s n o t y e t b e e n m e a s u r e d , b u t t h e s o l u t i o n
a p p e a r s v e r y f l u i d a l t h o u g h t h i c k e n i n g o c c u r s o v e r a p e r i o d o f t i m e .
S i n c e f u r f u r a l i s r e l a t i v e l y i n e x p e n s i v e ( o b t a i n e d f r o m c o r n c o b s ) ,
t h i s g e l c o u l d b e a n i n e x p e n s i v e s o l u t i o n i n p r o b l e m s w h e r e l a r g e
i n j e c t i o n v o l u m e s a r e r e q u i r e d .
AM - 9 Chemica l G rou ts
A n o t h e r g r o u t i n g m a t e r i a l o n t h e m a r k e t a t p r e s e n t i s A m e r i c a n
Cyanamid 's AM-9 Chemica l Grout . T h e A M - 9 i s s u p p l i e d a s a f i n e
w h i t e p o w d e r a n d i s a m i x t u r e o f N , N ' - m e t h y l e n e - b i s a c r y l a m i d e a n d
a c r y l a m i d e . T h e c a t a l y s t f o r t h e s y s t e m i s B - d i m e t h y l a m i n o p r o p i o n i -
t r i l e ( D M A P N ) , t h e i n i t i a t o r i s a m m o n i u m p e r s u l f a t e ( A P ) , a n d t h e
i n h i b i t o r i s p o t a s s i u m f e r r i c y a n i d e ( K F e ) . The AM-9, DMAPN, and KFe
a r e d i s s o l v e d i n a w a t e r s o l u t i o n w h i c h i s s t a b l e a n d c a n b e s t o r e d
f o r 2 4 h o u r s i f k e p t o u t o f s u n l i g h t . T h e A P i s d i s s o l v e d i n a s e p a r -
a t e s o l u t i o n a n d a d d e d t o t h e A M - 9 s o l u t i o n i m m e d i a t e l y b e f o r e i n j e c -
t i o n . 5 . 2
SIROC Chemical Grouts
A r e l a t i v e l y n e w p o l y m e r i c g r o u t m a t e r i a l o n t h e m a r k e t i s
Diamond Alka l i 's SIROC. S I R O C 1 i s t h e p r i m a r y o r g e l - f o r m i n g m a t e r -
i a l . I t s s t r u c t u r e h a s n o t a s y e t b e e n d i s c l o s e d . S IROC 2 - r eac ts
s l o w l y w i t h S I R O C 1 t o c o n t r o l t h e s e t t i n g t i m e , p l a s t i c i t y a n d f i n a l
c u r e d s t r e n g t h . S I R O C 3 - r e a c t s r a p i d l y w i t h S I R O C 1 t o s p e e d s o l i d i -
f i c a t i o n a t l o w t e m p e r a t u r e s a n d t o m a k e t h e f i n a l s e t i n s o l u b l e a n d
permanent . SIROC 1 and SIROC 2 are suppl ied as l iqu ids and are ready
t o u s e . S IROC 3 i s a f i ne wh i t e powder and mus t be m ixed be fo re use .
P r i o r t o i n j e c t i o n t h e t h r e e c o n s t i t u e n t s a r e m i x e d i n a p p r o p r i a t e
r a t i o s . 5 . 3
Impermeation of Underground Formations
-113-
Po l ymer Grou ts
M a n y p o l y m e r m a t e r i a l s h a v e a l s o b e e n u s e d a s c h e m i c a l g r o u t s .
A t l e a s t t w e n t y d i f f e r e n t m a t e r i a l s h a v e b e e n u s e d r a n g i n g f r o m a
f o r m u l a c o n s i s t i n g o f f u r f u r a l + u r e t h a n e t o o n e c o n s i s t i n g o f ' u n -
s a t u r a t e d f i s h o i l + p e t r o l e u m d i s t i l l a t e + c a r b o n t e t r a c h l o r i d e +
s u l f u r m o n o c h l o r i d e . I n g e n e r a l , t h e s e m a t e r i a l s f a l l i n t o t h r e e
c l a s s e s : ( 1 ) c o m p l e t e l y p o l y m e r i z e d m o l t e n m a t e r i a l s , ( 2 ) p a r t i a l l y
p o l y m e r i z e d m a t e r i a l s t h a t c o m p l e t e t h e i r p o l y m e r i z a t i o n i n t h e
g r o u n d f o r m a t i o n , a n d ( 3 ) u n p o l y m e r i z e d m a t e r i a l s t h a t p o l y m e r i z e
i n t h e g r o u n d f o r m a t i o n . I n r e g a r d t o t h e p r e s e n t p r o b l e m , c l a s s
( 1 ) a n d c l a s s ( 2 ) p o l y m e r s p r o b a b l y a r e t o o v i s c o u s t o f l o w t h r o u g h
t h e p o r o u s m e d i a a t r e a s o n a b l e r a t e s . However, c l a s s ( 3 ) p o l y m e r s
s e e m t o o f f e r t h e m o s t p r o m i s i n g p e r f o r m a n c e i f a n d w h e n i n j e c t e d
in to a po rous med ium.
H e r c u l o x G r o u t
H e r c u l o x i s a h a r d - s e t t i n g r e s i n g r o u t a v a i l a b l e f r o m H a l l i -
b u r t o n w h i c h c a n d e v e l o p h i g h c o m p r e s s i v e s t r e n g t h s . A n e x t r e m e l y
h i g h s t r e n g t h i s d e v e l o p e d e v e n t h o u g h t h e i n i t i a l v i s c o s i t y o f t h e
s o l u t i o n i s r e l a t i v e l y l o w . T h i s v i s c o s i t y , h o w e v e r , m a y b e t o o
h i g h f o r i n j e c t i o n i n t o p o r o u s m e d i a . I n a d d i t i o n , t h e h i g h s t r e n g t h
o f t h i s g e l i s p r o b a b l y n o t n e e d e d f o r i m p e r m e a t i o n o f s a n d s t o n e b u t
t h i s g r o u t m a y b e e x c e l l e n t f o r t h e p l u g g i n g o f c r a c k s a n d f r a c t u r e s .
T h e T a b l e 5 . 2 s u m m a r i z e s t h e t y p e s o f c h e m i c a l g r o u t s a v a i l a b l e
f o r v a r i o u s a p p l i c a t i o n s .
5 . 2 P r o p e r t i e s a n d P e r f o r m a n c e o f G r o u t i n g M a t e r i a l s
A s i n d i c a t e d a b o v e , p e r h a p s t h e m o s t p r o m i s i n g g r o u t i n g a g e n t s
f o r t h e i m p e r m e a t i o n o f u n d e r g r o u n d r o c k f o r m a t i o n s a r e t h e o n e - s h o t
p o l y m e r g r o u t s , w h i c h g e l i n p o s i t i o n . T h e p r i m a r y r e a s o n s a r e ( 1 )
P e r m a n e n c e o f t h e f i n a l s e t - - P o l y m e r s f o r m e d a r e i n s o l u b l e u n d e r a l l
Ne
w
Co
nce
pts
in
Un
de
rgro
un
d
Sto
rag
e
of
Na
tura
l G
as
-11
4-
Impermeation of Underground Formations
-115-
c o n d i t i o n s a n t i c i p a t e d ( 2 ) C o n t r o l l a b l e g e l t i m e - - S i n c e t h e p o l y -
m e r s r e s u l t f r o m a c a t a l y z e d r e a c t i o n , t h e a m o u n t o f c a t a l y s t a d d e d
i s a d i r e c t l i m i t i n g f a c t o r ' o n t h e r a t e o f t h e r e a c t i o n ( 3 ) C o m p l e t e -
n e s s o f g e l a t i o n - -T h e g e l l i n g f l u i d r e m a i n s l i q u i d u n t i l g e l a t i o n
w h e r e u p o n a r i g i d g e l i s f o r m e d a n d c u r e d w i t h i n s e v e r a l h o u r s ( 4 )
V i s c o s i t y - - C o m p o s i t i o n m a y b e v a r i e d t o g i v e a f l u i d w h i c h m a y b e
i n j e c t e d i n t o a n y p o r e s p a c e i n t o w h i c h t h e s o l u t i o n w o u l d e a s i l y
p e n e t r a t e .
Po l ymer i za t i on Mechan i sm
M o s t o f t h e p o l y m e r g r o u t s w h i c h g e l i n p o s i t i o n f o r m b y a
f r e e r a d i c a l m e c h a n i s m . A n i n i t i a t o r , a c a t a l y s t , a n d a n i n h i b i -
t o r a r e a d d e d t o t h e m o n o m e r s o l u t i o n t o i n d u c e a n d c o n t r o l t h e
f r e e r a d i c a l p o l y m e r i z a t i o n . T h e i n i t i a t o r c a u s e s t h e c a t a l y s t t o
d e c o m p o s e i n t o f r e e r a d i c a l s w h e r e u p o n t h e i n d u c t i o n p e r i o d f o l l o w s
b e f o r e p o l y m e r i z a t i o n b e g i n s . T h e i n h i b i t o r p r o l o n g s t h e i n d u c t i o n
p e r i o d . T h e l e n g t h o f i n d u c t i o n i s c o n t r o l l e d b y r e l a t i v e a m o u n t s
o f i n i t i a t o r , c a t a l y s t , a n d i n h i b i t o r p r e s e n t . W h e n p o l y m e r i z a t i o n
f i n a l l y s t a r t s , t h e c o m p l e t e p o l y m e r i s f o r m e d i n a r e l a t i v e l y s h o r t
t i m e .
A M - 9 G e l P r o p e r t i e s
One advantage of AM- i s t h a t i t i s i m p e r m e a b l e t o w a t e r ,
g a s e s , a n d h y d r o c a r b o n s i n t h e g e l l i n g s t a t e . I n a d d i t i o n , i t w i l l d i s -
p l a c e w a t e r w h e n i t i s i n t h e l i q u i d s t a t e . However, i f t h e g r o u n d
w a t e r i s f l o w i n g r a p i d l y t h e s o l u t i o n w i l l b e d i l u t e d o n t h e p e r i -
p h e r y a n d w i l l b e d i s p l a c e d s o m e w h a t , I f t u r b u l e n t f l o w c o n d i t i o n s
a re encoun te red , t h e d i l u t i o n e f f e c t c a n b e m i n i m i z e d b y i n c r e a s i n g
t h e A M - 9 c o n c e n t r a t i o n t o 1 5 - 2 0 % a n d b y u s i n g s p e c i a l f o r m u l a t i o n s
t o o b t a i n s h o r t g e l t i m e s . On the o the r hand , i f t h e g e l i s i n j e c t e d
i n t o v e r y d r y m a t e r i a l s , g r a v i t a t i o n a l a n d c a p i l l a r y f o r c e s w i l l
a c t t o g r e a t l y d i s p e r s e t h e s o l u t i o n , r e n d e r i n g i t i n e f f e c t i v e .
New Concepts in Underground Storage of Natural Gas
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W h i l e t h e A M - 9 g e l h a s g o o d w a t e r r e t e n t i o n p r o p e r t i e s , u n d e r
d r y c o n d i t i o n s a s m a l l a m o u n t o f w a t e r w i l l e s c a p e c a u s i n g s h r i n k -
a g e o f t h e g e l . S i n c e s o i l d o e s n o t s h r i n k , t h e i n d u c e d s t r e s s e s
m a y c a u s e r u p t u r e o f t h e g e l - s o i l b o n d s . Th i s may appea r as a v i s -
i b l e s h r i n k a g e c r a c k . W h e n e x p o s e d t o h u m i d c o n d i t i o n s , t h e g e l
w i l l e x p a n d a n d f i l l t h e v o i d s b u t t h e r u p t u r e w i l l n o t b e h e a l e d .
S t r e n g t h a n d g o o d i m p e r m e a b i l i t y w i l l t h e n b e l o s t . Th i s howeve r
w o u l d n o t b e a p r o b l e m i n d e e p f o r m a t i o n s a s t h e w a t e r w i l l a l w a y s
b e p r e s e n t d u r i n g a n d a f t e r g r o u t a p p l i c a t i o n .
T a b l e 5 . 3 b e l o w g i v e s a t y p i c a l c o m p o s i t i o n f o r A M - 9 m a t e r i a l
T a b l e 5 . 3
A T y p i c a l A M - 9 F o r m u l a t i o n
A M - 9 S o l u t i o n P r o p e r t i e s
T h e o u t s t a n d i n g c h a r a c t e r i s t i c s o f A M - 9 a r e i t s l o w
i n d u c t i o n p e r i o d v i s c o s i t y ( c o n s t a n t o v e r t h e i n d u c t i o n p e r i o d a t
1 . 6 c p ) a n d i t s a c c u r a t e l y c o n t r o l l a b l e g e l t i m e s . H o w e v e r , i t
does no t appea r t ha t t he ge l t imes can be made l onge r t han two hou rs
Impermeation of Underground Formations
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i f c o m p l e t e p o l y m e r i z a t i o n i s t o b e i n s u r e d . A n o t h e r d i s a d v a n t a g e
i s t h e g e l t i m e s e n s i t i v i t y t o p H a n d t e m p e r a t u r e . Be low a pH o f
6 . 5 , t h e g e l t i m e s b e c o m e l o n g a n d i n d e f i n i t e . F u r t h e r m o r e , t e m -
p e r a t u r e i n c r e a s e o f t e n d e g r e e s c a n c u t t h e g e l t i m e i n h a l f .
T h e f a c t o r s a f f e c t i n g t h e g e l t i m e a r e s u m m a r i z e d i n T a b l e
5 . 4 .
B e c a u s e o f i t s l o w v i s c o s i t y , A M - 9 i s v e r y e f f e c t i v e i n i m -
p e r m e a t i n g f i n e m a t e r i a l s . However, c o n s i d e r i n g t h e r e l a t i o n s h i p
b e t w e e n p e r m e a b i l i t y a n d p u m p i n g p r e s s u r e , i t i s u n e c o n o m i c a l t o
i m p e r m e a t e p o r o u s f o r m a t i o n s n e a r t h e s u r f a c e b e c a u s e o f t h e d i f -
f i c u l t y o f a t t a i n i n g h i g h p r e s s u r e s . On the other hand, AM-9 was
v e r y e f f e c t i v e i n s e a l i n g f i n e s a n d s t o n e i n a d e e p s h a f t w h e r e
2 0 0 0 p s i p u m p i n g p r e s s u r e w a s f e a s i b l e . M o r e c o m p l e t e i n f o r m a t i o n
o n t h e e f f e c t s o f t e m p e r a t u r e , p H , a n d c o m p o s i t i o n w i l l b e f o u n d
i n C y a n a m i d ' s p a m p h l e t " C h e m i c a l P r o p e r t i e s o f A M - 9 " 5 . 4 .
S IROC Ge l P rope r t i es
I n m a n y w a y s S I R O C i s s u p e r i o r t o o t h e r p o l y m e r g r o u t s t e s t e d .
F i r s t , a n d o f p r i m a r y i m p o r t a n c e , i s t h e f a c t t h a t l o n g g e l t i m e
m a y b e a c h i e v e d w i t h o u t s a c r i f i c i n g f i n a l s e t s t r e n g t h . Second,
i n i t i a l v i s c o s i t i e s o f 3 . 2 t o 5 . 2 c p p e r m i t i t t o b e i n j e c t e d i n t o
a n y f o r m a t i o n i n t o w h i c h w a t e r w i l l p e n e t r a t e . T h i r d , t h e t h r e s -
ho ld p ressu re o f co re spec imens have been i nc reased by as much as
300 t imes , w h i c h i s m u c h b e t t e r t h a n t h e p e r f o r m a n c e o f o t h e r g r o u t s
t e s t e d . F o u r t h , t h e g e l i s r i g i d a n d p e r m a n e n t . O t h e r p r o p e r t i e s
o f S I R O C s u c h a s d e h y d r a t i o n ' u p o n d r y i n g a n d d i l u t i o n i n r e g i o n s
o f r a p i d l y f l o w i n g g r o u n d w a t e r a r e e s s e n t i a l l y t h e s a m e a s t h o s e
o f A M - 9 5 . 3 .
New Concepts in Underground Storage of Natural Gas
T a b l e
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5 . 4
F a c t o r s A f f e c t i n g AM-9 Gel T ime2
F a c t o r
1.
2.
3 .
4.
R e d u c t i o n o f A M - 9 c o n c e n t r a -t i o n
DMAPN, AP, and KF e concent ra-t i o n
Temperature
pH
5. A i r
6 . M e t a l s
7 . M ix Wa te r
8 . S u n l i g h t
9 . I n h i b i t o r s
10. H y d r o g e n s u l f i d e
11. S a l t s
E f f e c t o f G e l T i m e
S l i g h t i n c r e a s e
T o o m u c h o r t o o l i t t l e w i l l p r o -d u c e w e a k g e l s o r n o n e a t a l l .Lower l im i t i s 0 .4% fo r DMAPN,0.25% for AP, a n d u p p e r l i m i tf o r KFe i s 0 .035%.
1 0 % r i s e c u t s t h e t i m e i n h a l f .
B e s t r a n g e i s 7 - 1 1 . DMAPN main-t a i n s p H a t 8 - 9 e x c e p t a t h i g ha c i d c o n c e n t r a t i o n . Below pH =6 . 5 , g e l t i m e s a r e l o n g a n di n d e f i n i t e .
I f s o l u t i o n i s s a t u r a t e d w i t ha i r , t h e g e l t i m e i s l o n g e r .
I r o n , coppe r , a n d c o p p e r a l l o y sd e c r e a s e g e l t i m e . Most usealuminum, s t a i n l e s s s t e e l , p l a s -t i c o r r u b b e r e q u i p m e n t .
I m p u r i t y i n m i x w a t e r m a y a f f e c tt i m e . T e s t s h o u l d b e c a r r i e do u t w i t h w a t e r t h a t w i l l b e u s e di n t h e f i e l d .
S u n l i g h t w i l l g e l A M - 9 s o l u t i o n sl e f t u n c o v e r e d .
A l t h o u g h m o s t p o l y m e r i z a t i o ni n h i b i t o r s c a n b e ' u s e d , t h e s ew i l l r e s u l t i n w e a k g e l s . K F ed o e s n o t h u r t t h e s t r e n g t h .
S h o r t e n s g e l t i m e .
S o l u b l e s a l t s ( N a C l , C a C l 2 , e t c ) ,p r e s e n t i n t h e f o r m a t i o n decreaseg e l t i m e a l t h o u g h i n c r e a s i n gs t r e n g t h .
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T a b l e 5 . 4 ( c o n t i n u e d )
F a c t o r s A f f e c t i n g A M - 9 G e l T i m e 2
12. F r e e z i n g Preven t by us ing any commerc ia la n t i f r e e z e .
13. I n s o l u b l e m a t e r i a l F i n e i n s o l u b l e p a r t i c l e s s u c h a sc l a y o r b e n t o n i t e s l o w d o w n t h eg e l a t i o n t o s o m e e x t e n t .
S I R O C S o l u t i o n P r o p e r t i e s
S o m e p r e c a u t i o n s s h o u l d b e n o t e d a s t o t h e l i m i t a t i o n s o f t h e
pH range over which SIROC grouts may be 'used. A SIROC grout was
p r e p a r e d i n a . l N H C L s o l u t i o n w i t h l o c a l g e l l i n g . H o w e v e r , l i t t l e
e f f e c t w a s n o t e d i n t h e u s e o f a . l N N a O H s o l u t i o n . The presence
o f N a 2 C O 3 a p p e a r s t o h a v e l i t t l e e f f e c t o n e i t h e r g e l a t i o n o r s e t
s t r e n g t h . C u + + i o n , however , causes immed ia te ge la t i on and m igh t
s e r i o u s l y l i m i t t h e u s e o f S I R O C g r o u t s i n f o r m a t i o n s b e a r i n g t h i s
i o n .
O n l y o n e p r e c a u t i o n s h o u l d b e n o t e d i n t h e p r e p a r a t i o n o f
S IROC so lu t i ons . The t h ree componen ts mus t be m ixed i n t he co r rec t
o r d e r a n d w i t h v i g o r o u s s t i r r i n g t o p r e v e n t t h e f o r m a t i o n o f l o c a l
g e l m a s s e s w h i c h c o u l d p l u g i n j e c t i o n e q u i p m e n t o r p r e v e n t p r o p e r
d i s p e r s i o n o f t h e g r o u t b y p l u g g i n g p o r e s i m m e d i a t e l y s u r r o u n d i n g
t h e i n j e c t i o n t u b e . T h e T a b l e 5 . 5 i n t h e f o l l o w i n g s h o w s a t y p i c a l
S IROC fo rmu la t i on .
T a b l e 5 . 5
A Typ i ca l S IROC Formu la t i on
Component Volume %
SIROC 1 60SIROC 2 5SIROC 3 5WATER 30
G E L T I M E = 1 2 h r s . a t 7 0 ° F
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T h e t y p i c a l g e l a t i o n t i m e p r o f i l e i s i l l u s t r a t e d i n F i g . 5 . 2 ,
F i g s . 5 . 3 t h r o u g h 5 . 6 i l l u s t r a t e t y p i c a l v a r i a t i o n s o f g e l t i m e
w i t h c o m p o s i t i o n a t a c o n s t a n t t e m p e r a t u r e .
T a b l e 5 . 6 b e l o w s u m m a r i z e s p h y s i c a l p r o p e r t i e s , c o s t a n d c h a r -
a c t e r i s t i c s f o r v a r i o u s g e l s .
T a b l e 5 . 6
* S o d i u m s i l i c a t e g e l s m a d e f r o m l i g n i n l i q u o r t r e a t e d w i t h a m m o n i u m h y d r o x i d eg i v e ' u n s t a b l e g e l s , w h i l e t h o s e t r e a t e d w i t h l i m e y i e l d s t a b l e g e l s .
* * N o n e o f t h e a b o v e g r o u t s p r e s e n t s e r i o u s c o r r o s i o n p r o b l e m s .
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5 .3 Adhes ion Be tween Grou ts and Po rous Fo rma t i ons
T h e s o l u t i o n g r o u t s w h i c h p o l y m e r i z e i n t h e p o r e s p a c e a f t e r
h a v i n g b e e n i n j e c t e d w h e n p r o p e r l y d i s t r i b u t e d a n d c u r e d , i m p e r m e a t e
t h e r o c k w i t h r e s p e c t t o f l o w o f w a t e r o r g a s a n d s u b s t a n t i a l l y i n -
c r e a s e t h e t h r e s h o l d p r e s s u r e f o r t h e c a p i l l a r y d r a i n a g e o f w a t e r .
Th i s phenomena mus t obv ious l y be re l a ted t o t he bond ing and adhe -
s i o n b e t w e e n t h e p o l y m e r a n d t h e s a n d g r a i n .
Mechanism of Bonding and Adhes ion
I n v e s t i g a t i o n s r e l a t i n g t o t h e c o n s t r u c t i o n o f c l a y - b a s e a n d
b i t u m e n ( a s p h a l t ) r o a d s h a v e d i v u l g e d m u c h r e l e v a n t i n f o r m a t i o n
w h i c h b e a r s o u t t h e t h e o r y o f v a n d e r W a a l ' s f o r c e s a n d t h e i r i m -
p o r t a n c e i n a d h e s i o n i n p e r m e a b l e g r o u n d f o r m a t i o n s . I n s t u d i e s
m a d e o n c l a y s a m p l e s i t w a s f o u n d t h a t , w h e n s h r i n k a g e i s p e r m i t t e d
i n t h e s a m p l e a n d t h e f i l m a n d c a p i l l a r y s i z e t h u s a l l o w e d t o d e -
c r e a s e , bond fo r ces may reach t he equ i va len t o f 1000 a tmosphe res
a t a i r d r y n e s s . One reason fo r t h i s may be t ha t common c lays and
roads tone possess a weak nega t i ve cha rge . T h i s c a u s e s i m m e d i a t e
o r i e n t a t i o n o f w a t e r m o l e c u l e s a n d s t r o n g h y d r o g e n b o n d i n g . 5 . 5
New Concepts in Underground Storage of Natural Gas
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F i g . 5 . 2 . V i s c o s i t y P r o f i l e f o r S I R O C G r o u t
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S i m h a , F r i s h , a n d E i r i c h h a v e m a d e a s t u d y o f t h e p r o b a b i l i t y
o f a t t a c h m e n t o f h i g h p o l y m e r s t o s o l i d s5 . 6
. The ma thema t i c s i n -
v o l v e d a r e v e r y c o m p l e x a n d w i l l n o t b e g i v e n h e r e . T h e y p r e d i c t
t h a t t h e n u m b e r o f a d h e s i v e a t t a c h m e n t s w i l l b e p r o p o r t i o n a l t o
t h e s q u a r e r o o t o f t h e m o l e c u l a r w e i g h t o f t h e p o l y m e r . The assump-
t i o n s m a d e w e r e ( 1 ) t h e p o l y m e r w a s d i l u t e r e l a t i v e t o t h e s o l v e n t .
( 2 ) t h e s u r f a c e c o n s i s t s o f a c t i v e s p o t s f o r w h i c h t h e n u m b e r o f
s p o t s i s p r o p o r t i o n a l t o t h e s u r f a c e a r e a . T h e s e a c t i v e s p o t s i n
o u r c a s e w o u l d p r o b a b l y r e f e r t o s o m e o f t h e i n d i v i d u a l g r a i n s ,
s i n c e o n e w o u l d e x p e c t t h e c h a r g e d e n s i t y t o b e g r e a t e s t t h e r e .5 . 6
T h e m a t h e m a t i c a l m o d e l p r e d i c t s t h e f o r m a t i o n o f a m o n o l a y e r w h i c h
o n e m a y t h i n k o f a s a f i e l d c o v e r e d w i t h " i n c h - w o r m s " s i d e b y s i d e .
The es t ima ted b r i dge l eng th be tween a t t achmen ts i s t en monomer i c
u n i t s .
P roposed Mode ls f o r Po l ymer Grou t Adhes ion
A s w a s p o i n t e d o u t i n t h e p r e c e d i n g , s i l i c a - c o n t a i n i n g
a g g r e g a t e s m a y r e t a i n a n e g a t i v e r e s i d u a l c h a r g e . T h i s c h a r g e
c a n i n t u r n e x p l a i n t h e h i g h v a n d e r W a a l ' s f o r c e o f a t t r a c t i o n
f o r w a t e r i n t h e c a p i l l a r i e s o f c l a y a n d c e r t a i n t y p e s o f r o c k .
I t m a y a l s o b e o f ' u s e i n e x p l a i n i n g t h e a d h e s i o n o f p o l y m e r i c
g r o u t i n g m a t e r i a l s . The manufacturers o f both SIROC and AM-9
r e p o r t t h a t t h e i r m a t e r i a l s c a p t u r e a n d r e t a i n w a t e r i n i t i a l l y
p r e s e n t i n t h e p a r t i c u l a r g r o u n d f o r m a t i o n .5 . 7
S i n c e , t h e v a n d e r
W a a l ' s f o r c e s h a v e a m u c h g r e a t e r a t t r a c t i o n f o r p o l a r w a t e r m o l e -
c u l e s t h a n f o r n o n - p o l a r p o l y m e r s , one poss ib l e mechan i sm o f t he
a d h e s i o n m a y b e t h a t t h e s e e n t r a i n e d w a t e r m o l e c u l e s a r e r e s p o n -
s i b l e f o r t h e l a r g e f o r c e s h o l d i n g t h e p o l y m e r i n t h e c a p i l l a r i e s
a n d p r o h i b i t i n g f r a c t u r e o r d i s p l a c e m e n t u n d e r p r e s s u r e . T h i smechan i sm i s deno ted as t he " c l ay mechan i sm. " I f t h i s i s i n d e e d
the mechanisms, o n e m i g h t e x p e c t t h a t t h e p r e s e n c e o f H + o r o t h e rp o s i t i v e i o n s i n t h e g r o u t s o l u t i o n s h o u l d i n c r e a s e t h e a t t r a c t i o n .
New Concepts in Underground Storage of Natural Gas
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t h i s
draw
B e c a u s e t h e p r e s e n c e o f c a t i o n s d i s t u r b s t h e p o l y m e r i z a t i o n p r o c e s s ,
e f f e c t m a y b e d i f f i c u l t t o t e s t i n p r a c t i c e . S e e t h e a t t a c h e d
i n g f o r a n i l l u s t r a t i o n o f t h i s t y p e o f m e c h a n i s m . ( F i g . 5 . 7 )
A s e c o n d p l a u s i b l e m e c h a n i s m c o u l d b e d u e t o t h e a c t u a l a d -
h e s i o n a n d i n t e r t w i n i n g o f t h e p o l y m e r i c s u b s t a n c e . E n t r a i n m e n t
o f w a t e r c o u l d b e a n i m p o r t a n t c o n t r i b u t i n g f a c t o r , b u t t h e m o d e l
i t s e l f w o u l d r e s t ' u p o n t h e S i m h a , F r i s h , E i r i c h p i c t u r e o f t h e d i -
r e c t a d h e s i o n o f t h e p o l y m e r . Th i s second mode l cou ld be used t o
e x p l a i n t h e a p p a r e n t f r a c t u r e o f g r o u t b o n d s w h i c h h a v e b e e n a i r -
d r i e d a n d r e - h y d r a t e d . T h i s m o d e l i s v e r y a n a l o g o u s t o t h e b i t u m e n
a d h e s i o n p r o b l e m i n w h i c h t h e p o l y m e r i s d i s p l a c e d b y w a t e r . T h i s
m o d e l i s i l l u s t r a t e d b e l o w a s " m e c h a n i s m 2 . " ( F i g . 5 . 8 ) .
5 . 4 E v a l u a t i o n o f G r o u t s b y L a b o r a t o r y E x p e r i m e n t s
I n o r d e r t o e v a l u a t e t h e e f f e c t i v e n e s s o f g r o u t s i n c h a n g i n g
t h e c h a r a c t e r i s t i c s o f n a t u r a l a n d s y n t h e t i c a l l y p r e p a r e d c o r e s a m -
p l e s , a s e r i e s o f l a b o r a t o r y e x p e r i m e n t s h a v e b e e n p e r f o r m e d a n d a n -
a l y z e d .
T h e p r e p a r a t i o n o f s u p e r - p e r m e a b l e , l a b o r a t o r y c a s t c o r e s ,
a n d c u t t i n g o f t h e " p e r m - p l u g s " f r o m f i e l d c o r e s f o r m o u n t i n g a n d
t e s t i n g o n t h e r u b b e r - s l e e v e d c o r e b a r r e l h a v e b e e n d i s c u s s e d i n
Chap te r 2 . T h e f i g u r e 5 . 9 r e p r e s e n t s t h e s e t - u p u s e d f o r c u t t i n g
t h e c o r e s b y a d i a m o n d b i t m o u n t e d o n a d r i l l c o l l a r t h r o u g h w h i c h
w a t e r i s c i r c u l a t e d t o c o o l t h e c o r e a n d e n t r a i n t h e c u t t i n g s .
M e a s u r e m e n t s o f p o r o s i t y , p e r m e a b i l i t y a n d t h r e s h o l d p r e s s u r e
p e r f o r m e d o n n a t u r a l c o n s o l i d a t e d a n d l a b o r a t o r y c a s t s u p e r p e r m e -
a b l e c o r e s h a v e b e e n d e s c r i b e d i n C h a p t e r 2 .
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Fig. 5.7 Clay M e c h a n i s m
F i g . 5 . 8 M e c h a n i s m “ 2 ” P o l y m e r B o n d i n g
New Concepts in Underground Storage of Natural Gas
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G r o u t I n j e c t i o n - C u r i n g a n d T e s t i n g o f C o r e s
I n o r d e r t o t e s t a n d e v a l u a t e t h e e f f e c t i v e n e s s o f g r o u t s o -
l u t i o n s i n i m p e r m e a t i n g t h e p o r o u s m e d i a , a s e r i e s o f c a r e f u l l y
p l a n n e d a n d c o n t r o l l e d e x p e r i m e n t s o n c o r e s a m p l e s w e r e d e v i s e d .
T h e s e e x p e r i m e n t s c e n t e r e d a r o u n d p r o v i d i n g c o m p a r a t i v e b a s e s f o r
t h e v a l u e s o f " T h r e s h o l d p r e s s u r e " b e f o r e a n d a f t e r g r o u t i n g .
A d d i t i o n a l l y s p e c i a l e x p e r i m e n t s t o m e a s u r e p e r t i n e n t p h y s i c a l p r o -
p e r t i e s o f g r o u t s a n d p h o t o m i c r o g r a p h i c s t u d i e s o f g r o u t d i s b r i b u -
t i o n i n p o r e s a n d o n t h e c o m p r e s s i v e s t r e n g t h o f g r o u t s w e r e p e r -
f o rmed .
F i g . 5 . 9 C o r e C u t t i n g A p p a r a t u s S e t - u p
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Grout ing Core Samples
The g rou t componen ts we re m ixed and pou red i n to a g rou t con -
t a i n e r . T h e w a t e r s a t u r a t e d c o r e s a m p l e s w e r e p u t i n t o t h e c o r e
h o l d e r w h e r e g r o u t s o l u t i o n w a s i n j e c t e d i n f r o m t h e g r o u t c o n t a i n e r
t o t h e c o r e s b y r e g u l a t i n g t h e i n j e c t i n g p r e s s u r e . U s u a l l y g r o u t -
i n g w a s c o n t i n u e d u n t i l a b o u t 3 0 c c d i s p l a c e d w a t e r w a s c o l l e c t e d
a t t h e o u t l e t . I n s o m e c a s e s t h e p e r m e a b i l i t y a n d g e l t i m e o f
g r o u t w a s s u c h t h a t t h e co re had t o be removed f r om the ho lde r
be fo re t h i s much had accumu la ted . The appa ra tus and expe r imen ta l
l a y o u t o f e q u i p m e n t u s e d f o r g r o u t i n g c o r e s a m p l e s a r e s h o w n i n
F i g . 5 . 1 0 a n d F i g . 5 . 1 1 .
A f t e r g r o u t i n g , co re samp les we re cu red by keep ing i n a
1 0 0 % h u m i d i f i e d c o n t a i n e r t o p r e v e n t d r y i n g o f t h e g e l . I nduced
s t r e s s e s , wh i ch wou ld have caused rup tu re o f t he ge l and co re bonds ,
w e r e t h e r e b y p r e v e n t e d . U s u a l l y , 1 4 d a y s f o r c u r i n g w a s f o u n d
s u f f i c i e n t f o r t h e g r o u t t o r e a c t a n d f o r m a s t a b l e g e l .
V i s c o s i t y p r o f i l e s o f t h e g r o u t s w e r e d e t e r m i n e d w i t h t h e
s tanda rd Os twa ld v i s come te r s . T h e t e m p e r a t u r e o f t h e g r o u t w a s
k e p t a t 2 5 ° C . T h e g e l t i m e w a s r e c o r d e d a s t h e l e n g t h o f t i m e a f t e r
m i x i n g t h a t t h e f i r s t g e l f o r m e d i n t h e l i q u i d .
T h e g r o u t e d c o r e s a m p l e s a f t e r c u r i n g i n t h e w e t ( 1 0 0 % h u m i d -
i t y ) d e s i c c a t o r f o r r e q u i r e d l e n g t h o f t i m e , w e r e t h e n s a t u r a t e d
w i t h w a t e r i n t h e b e l l j a r u n d e r v a c u u m . T h e s a t u r a t e d g r o u t e d c o r e
s a m p l e s w e r e t e s t e d i n t h e c o r e h o l d e r f o r t h r e s h o l d p r e s s u r e .
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R e s u l t s o f E x p e r i m e n t a l W o r k o n E v a l u a t i o n o f G r o u t s
T h e r e s u l t s o f e x p e r i m e n t s d e s i g n e d t o c o m p a r e t h e t h r e s h o l d
p r e s s u r e s b e f o r e a n d a f t e r g r o u t i n g a r e s u m m a r i z e d i n T a b l e s 5 . 6 ,
5 . 7 , 5 . 8 .
T h e T a b l e 5 . 6 s h o w s t h e r e s u l t s o f g r o u t i n g e x p e r i m e n t s a n d
t h r e s h o l d p r e s s u r e m e a s u r e m e n t s o n f i e l d c o r e s a m p l e s f r o m n a t u r -
a l l y o c c u r r i n g f o r m a t i o n s . T h e s o u r c e o f t h e c o r e a l o n g w i t h l a -
b o r a t o r y d e s i g n a t i o n , t h e m a t e r i a l , dep th f r om wh i ch t he co re was
c u t , p o r o s i t y , p e r m e a b i l i t y , t h r e s h o l d p r e s s u r e b e f o r e g r o u t i n g ,
a n d a f t e r g r o u t i n g a n d t h e g r o u t u s e d a r e s h o w n . The Tab le 5 .9
s h o w s t h e v a r i o u s g r o u t m i x e s a n d f o r m u l a t i o n s r e f e r r e d t o i n
T a b l e s 5 . 6 , 5 . 7 , a n d 5 . 8 .
T h e T a b l e 5 . 7 s u m m a r i z e s t h e r e s u l t s o f s i m i l a r t e s t s c o n -
d u c t e d o n l a b o r a t o r y - m a d e s y n t h e t i c c o r e s . The Tab le 5 .8 shows
t h e e f f e c t o f g r o u t i n g o n t h r e s h o l d p r e s s u r e o f f r a c t u r e d c o r e s
s u b s e q u e n t l y h e a l e d b y p r o p e r a p p l i c a t i o n o f g r o u t s . S p e c i a l
f r a c t u r e g r o u t i n g e x p e r i m e n t s w i l l b e d i s c u s s e d l a t e r i n t h i s c h a p -
t e r .
T h e r e s u l t s o f m o s t e x p e r i m e n t s t e n d e d t o i n d i c a t e t h a t A M - 9
w a s m o r e s u i t a b l e f o r g r o u t i n g t i g h t c o r e s b e c a u s e o f i t s l o w v i s -
c o s i t y w h i l e S I R O C g r o u t w a s m o r e e f f e c t i v e o n s u p e r p e r m e a b l e l a -
b o r a t o r y c o r e s . T h e v i s c o s i t y o f A M - 9 g r o u t w a s i n m o s t c a s e s v e r y
c l o s e t o t h a t o f w a t e r a t t h e s a m e t e m p e r a t u r e . The d i sadvan tage
o f AM-9 compared to S IROC was re la t i ve l y sho r t ge l t ime and weak -
n e s s o f g e l d u e t o p a r t i a l g e l a t i o n . The F igu re 5 .12 shows the
v i s c o s i t y p r o f i l e o f a t y p i c a l S I R C C g r o u t w i t h a p a r t i c u l a r S I R O C
f o r m u l a t i o n .
-135-
Impermeation of Underground Formations
F ig . 5 .12 . Typ i ca l V i scos i t y P ro f i l e f o r S i roc Grou t
New Concepts in Underground Storage of Natural Gas
-136-
F ig . 5 .13 Typ i ca l V i scos i t y P ro f i l e f o r AM -9 Grou t .
Impermeation of Underground Formations
-137-
T h e F i g u r e 5 . 1 3 s h o w s a t y p i c a l v i s c o s i t y p r o f i l e f o r A M - 9
g r o u t . I t m a y b e n o t e d t h a t f o r n e a r l y 5 0 m i n u t e s t h e A M - 9 g r o u t
r e m a i n s a t n e a r l y c o n s t a n t w a t e r v i s c o s i t y w h i l e t h e S I R O C g r o u t
s o l u t i o n s h o w n i n F i g u r e 5 . 1 2 s t a r t s a t 5 . 5 c e n t i p o i s e a n d e x h i b i t s
a c o n t i n u o u s l y i n c r e a s i n g v i s c o s i t y p r o f i l e . T h e s e t w o d i s t i n c t l y
d i f f e r e n t p r o f i l e s b e c o m e q u i t e i m p o r t a n t i n c a l c u l a t i o n s o f g r o u t
i n j e c t i o n i n t o p o r o u s m e d i a b y s u i t a b l e u n s t e a d y s t a t e f l o w e q u a -
t i o n s . T h e m a t h e m a t i c a l d e r i v a t i o n s a n d p h y s i c a l i n t e r p r e t a t i o n
o f t h e s e e q u a t i o n s a r e g i v e n i n A p p e n d i x A .
T h e T a b l e s 5 . 7 , 5 . 8 a n d 5 . 9 c l e a r l y s h o w t h e i n c r e a s e i n t h r e s -
h o l d p r e s s u r e s f o r e a c h p a r t i c u l a r g r o u t i n g e x p e r i m e n t . A g e n e r a l
c o r r e l a t i o n b e t w e e n t h e v a l u e s o f t h r e s h o l d p r e s s u r e s a f t e r a n d
b e f o r e g r o u t i n g w a s n o t p o s s i b l e . On a q u a l i t a t i v e b a s i s , h o w -
e v e r , i t m a y b e s a i d t h a t g r o u t i n g b e c o m e s m o r e e f f e c t i v e ( i . e .
r a t i o o f t h r e s h o l d p r e s s u r e s a f t e r a n d b e f o r e g r o u t i n g g e t s l a r g e r )
f o r t i g h t e r f o r m a t i o n s .
A l o n g w i t h t h i s o b s e r v a t i o n o n e m u s t a l s o r e c o g n i z e t h a t i t
w o u l d b e r e l a t i v e l y h a r d e r t o g r o u t a t i g h t f o r m a t i o n t h a t a p e r -
meable one. P e r h a p s e f f e c t i v e g r o u t i n g o f h o r i z o n t a l a n d v e r t i c a l
f r a c t u r e s i n d u c e d i n t i g h t f o r m a t i o n s w i l l u l t i m a t e l y p r o v i d e a
f e a s i b l e a p p l i c a t i o n .
Compress ive Adhes ion Tests on Grouted Cores
T o o b t a i n a q u a l i t a t i v e c o m p a r i s o n o f t h e a d h e s i v e a n d c o h e -
s i v e f o r c e s i n v o l v e d i n a c t u a l t e s t s p e c i m e n s , p o r o u s s e c t i o n s
w e r e p r e p a r e d f r o m a b e n t o n i t e - s a n d m i x t u r e . The samples were
molded, compacted, a n d f i r e d f o r 2 4 h r s . a t 1 8 0 0 ° F . G r o u t i n g
w a s a c c o m p l i s h e d b y a l l o w i n g t h e s p e c i m e n s t o s t a n d i n t h e v a r i o u s
g r o u t s o l u t i o n s f o r 8 h r s . P r i o r t o t e s t i n g , t h e g r o u t e d c o r e s
Ne
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-14
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-14
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as
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-14
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4-
Impermeation of Underground Formations
-145-
T a b l e 5 . 9
New Concepts in Underground Storage of Natural Gas
-146-
Compress i ve Adhes ion Tes t s on Grou ted Co res
T o o b t a i n a q u a l i t a t i v e c o m p a r i s o n o f t h e a d h e s i v e a n d c o h e -
s i v e f o r c e s i n v o l v e d i n a c t u a l t e s t s p e c i m e n s , p o r o u s s e c t i o n s w e r e
p r e p a r e d f r o m a b e n t o n i t e - s a n d m i x t u r e . The samples were molded,
compacted, a n d f i r e d f o r 2 4 h r s . a t 1 8 0 0 ° F . Grou t i ng was accomp l i shed
b y a l l o w i n g t h e s p e c i m e n s t o s t a n d i n t h e v a r i o u s g r o u t s o l u t i o n s
f o r 8 h r . P r i o r t o t e s t i n g , t h e g r o u t e d c o r e s w e r e c u r e d u n d e r
r o o m c o n d i t i o n s f o r 4 8 h r s . T h e s e c t i o n s w e r e t h e n s u b j e c t e d t o
c o m p r e s s i v e s t r e n g t h t e s t s .
I n e a c h o f t h e f o l l o w i n g i n w h i c h a d d i t i v e s a r e i n d i c a t e d ,
t h e g r o u t s o l u t i o n s w e r e . 1 6 7 N i n t h e a d d i t i v e a g e n t w h e n i n j e c t e d .
25 ml . o f t h e . 5 N r e a g e n t i n d i c a t e d w e r e a d d e d t o 5 0 m l . o f g r o u t
s o l u t i o n .
C o r e a n d g r o u t c o m p o s i t i o n s ' u s e d i n t h e s e t e s t s a r e s h o w n
i n t h e f o l l o w i n g :
CORE COMPOSITION:
S a n d ( s i l i c a ) - - - - - 6 2 . 5 %B e n t o n i t e - - - - - - - - - 2 6 . 8 %W a t e r - - - - - - - - - - - - - 1 0 . 7 %
GROUT COMPOSITION:
SIROC 1 - - - - - - - - - - - 6 0 . 0 %SIROC 2 - - - - - - - - - - - 5 . 0 0 %SIROC 3 - - - - - - - - - - - 5 . 0 0 %WATER - - - - - - - - - - - - - 3 0 . 0 %
For t he pu rpose o f compar i son 7 samp les we re used . These were:
SAMPLE 1 - - - - S t a n d a r d , u n g r o u t e d , a i r - d r y s p e c i m e n
SAMPLE 2 - - - - S p e c i m e n g r o u t e d a i r - d r y
SAMPLE 3 - - - - S p e c i m e n g r o u t e d s a t u r a t e d w i t h w a t e r
SAMPLE 4 - - - - S p e c i m e n g r o u t e d w i t h H 2 S 0 4 - a d d i t i v e g r o u t
Impermeation of Underground Formations
-147-
SAMPLE 5 - - - - S p e c i m e n g r o u t e d w i t h H 2 S O 4 - a d d i t i v e g r o u t
SAMPLE 6 - - - - S p e c i m e n g r o u t e d w i t h N a O H - a d d i t i v e g r o u t
SAMPLE 7 - - - - S p e c i m e n g r o u t e d w i t h N a 2 C O 3 - a d d i t i v e g r o u t
SAMPLE NO. YIELD FORCE YIELD STRESS
1 2 8 4 . 4 l b 1 5 4 . 0 p s i2 1705.0 9 5 6 . 03 5 5 2 . 5 3 0 8 . 6
2 8 4 . 04 5 0 7 . 55 5 4 5 . 5 3 0 6 . 16 9 6 5 . 0 5 4 2 . 07 9 4 0 . 0 5 2 7 . 0
F r o m t h e a b o v e d a t a o n e c a n c o n c l u d e t h a t t h e i n c l u s i o n o f
g r o u t i n p o r o u s c o r e s h a s a g r e a t e f f e c t o n s t r e n g t h . T h i s i n c r e a s e
i n s t r e n g t h i s d u e t o b o t h c o h e s i v e a n d a d h e s i v e f o r c e s , c o n s e q u e n t -
l y , t h e t e s t s c a n n o t b e i n t e r p r e t e d a s a d i r e c t m e a s u r e o f t h e a d -
h e s i v e f o r c e s . Even ' unde r t he mos t adve rse cond i t i ons i n wh i ch
t h e c o r e w a s i n i t i a l l y s a t u r a t e d w i t h w a t e r , t h e g r o u t d o u b l e d t h e
y i e l d s t r e n g t h . As m igh t be expec ted , t h e g r o u t e d d r y c o r e g a v e
t h e h i g h e s t y i e l d s t r e s s . T h e a c i d g r o u t s , i n p r a c t i c e , w o u l d c r e -
a t e p r o b l e m s b e c a u s e o f l o c a l g e l a t i o n . T h e b a s i c g e l s w e r e s t r o n g e r
t h a n t h e a c i d g r o u t s , a n d t h e r e w a s l i t t l e d i f f e r e n c e b e t w e e n t h e
weak base (Na 2 C0 3 ) and t he s t rong base (NaOH) .
M ic roscop i c Obse rva t i ons on Grou ted Co re Spec imens
T h e f o l l o w i n g s e c t i o n s w a s p r e p a r e d a n d s t u d i e d u n d e r l o w
m a g n i f i c a t i o n ( 1 0 x a n d 3 0 x ) . T h e s a n d s t o n e s e c t i o n s r e f e r r e d t o
w e r e g r o u t e d b y a b s o r p t i o n . The g rou t s used and t he p rocedu re em-
p l o y e d w e r e e s s e n t i a l l y t h e s a m e a s t h o s e u s e d i n t h e c o m p r e s s i v e
t e s t s . T h e s a n d s t o n e u s e d w a s h i g h l y p o r o u s ; t h e i n d i v i d u a l s e c -
t i o n s w e r e a p p r o x i m a t e l y 2 " x 2 " x 1 / 2 " . T h e s e c t i o n s w e r e a l l o w e d
t o c u r e f o r 7 2 h r s . u n d e r c o n d i t i o n s o f 1 0 0 % h u m i d i t y p r i o r t o t e s t -
i n g . A d i a m o n d s a w w a s u s e d t o c u t v e r t i c a l s e c t i o n s f r o m t h e i n -
d i v i d u a l s l a b s a s i l l u s t r a t e d i n F i g . 5 . 1 4 . A n a g g r e g a t e o f s i l i c a
New Concepts in Underground Storage of Natural Gas
-148-
sand w i t h g rou t po l ymer as t he cemen t i ng agen t was p repa red us ing
t h e g r o u t c o m p o s i t i o n c i t e d i n t h e s e c t i o n o n c o m p r e s s i v e t e s t s .
T h i s a g g r e g a t e w a s t h e s o u r c e o f a l l t h e p a r t i c l e s s t u d i e d .
O b s e r v a t i o n s o n t h e m i c r o s c o p e o n v a r i o u s s p e c i m e n s a r e l i s t e d
be low :
SPECIMEN 1 - - - - S i l i c a p a r t i c l e a g g r e g a t e .
I n t h e t o t a l l y s a t u r a t e d s a m p l e , t h e g r o u t p o l y m e r a d h e r e s o n l y
w e a k l y t o t h e p a r t i c l e s . I n d i v i d u a l g r a i n s a r e e a s i l y r e m o v e d w i t h
a n y s h a r p i n s t r u m e n t r e v e a l i n g o n t h e n e w o p e n g r o u t f a c e s t h e p o w -
d e r y t e x t u r e o f g r o u t w h i c h i s p r o b a b l y r e s p o n s i b l e f o r i t s w a t e r -
r e t a i n i n g p r o p e r t i e s .
F ig . 5 .14. Sketch o f Sample for MicroscopicObse rva t i on
SPECIMEN 2 - - - - P a r t i c l e s r e m o v e d f r o m t h i s a g g r e g a t e ,
U n d e r h i g h m a g n i f i c a t i o n i t w a s o b s e r v e d t h a t t h e g r o u t w h i c h s t i l l
a d h e r e d w a s f o u n d p r i m a r i l y o n t h e e l e v a t e d " h i l l s " o f t h e p a r t i c l e s .
T h e a p p e a r a n c e w a s m a r k e d l y d i f f e r e n t f r o m u n g r o u t e d p a r t i c l e s .
Impermeation of Underground Formations
-149-
SPECIMEN 3 - - - - S a n d s t o n e i n i t i a l l y s a t u r a t e d b e f o r e g r o u t i n g .
U n d e r 3 0 x m a g n i f i c a t i o n o n e c o u l d e a s i l y s e e , u p o n s c r a p i n g , t h e
f r i a b l e n a t u r e o f t h e g r o u t p o l y m e r . When t he f ace , wh i ch had
b e e n m a i n t a i n e d t o t a l l y s a t u r a t e d s i n c e g r o u t i n g , w a s a l l o w e d t o
d r y , o n e c o u l d o b s e r v e p l a n a r f r a c t u r e s i n t h e g r o u t i n t e r s t i c e s
a n d a t t h e i n t e r f a c e s b e t w e e n t h e g r o u t a n d t h e p a r t i c l e s . These
f r a c t u r e s w e r e n o t f u l l y h e a l e d b y r e h y d r a t i o n . I n t e r n a l p e n e t r a -
t i o n o f g r o u t w a s m u c h g r e a t e r t h a n e x p e c t e d .
SPECIMEN 4 - - - - S a n d s t o n e a i r - d r y b e f o r e g r o u t i n g ,
The genera l appearance was much the same as in the above spec imen.
F r a c t u r e s a s i n t h e a b o v e w e r e n o t e d .
SPECIMEN 5 - - - - S a n d s t o n e g r o u t e d d r y w i t h a n a c i d - a d d i t i v e g r o u t .
T h e g e n e r a l a p p e a r a n c e o f t h e g r o u t w a s d i f f e r e n t f r o m t h e a b o v e .
I n s t e a d o f a w h i t e , p o w d e r y s u b s t a n c e , t h e a c i d g r o u t w a s n e a r l y
t r a n s p a r e n t i n p o s i t i o n . D r y i n g f r a c t u r e s w e r e a g a i n o b s e r v e d .
SPECIMEN 6 - - - - S a n d s t o n e g r o u t e d d r y w i t h N a 2 C C 3 - a d d i t i v e g r o u t .
N o l o c a l g e l a t i o n o c c u r e d . T h e p h y s i c a l a p p e a r a n c e o f t h e g r o u t
i n p o s i t i o n i s e s s e n t i a l l y t h e s a m e a s i n t h e a b o v e s a m p l e s . Upon
d r y i n g , t h e a d h e s i o n a t t h e i n t e r f a c e a p p e a r e d b e t t e r .
SPECIMEN 7 - - - - S a n d s t o n e g r o u t e d d r y w i t h C u S O 4 - a d d i t i v e g r o u t .
L o c a l g e l a t i o n w a s e x c e s s i v e a n d w o u l d d e f i n i t e l y p r o h i b i t t h e ' u s e
O f C u S O 4 i n a p p l i c a t i o n s i n t h e f i e l d . T h e i d e a o f s t u d y i n g i t
w a s t o o b s e r v e t h e e f f e c t s o f a d e l i q u e s c e n t a d d i t i v e i n r e d u c i n g
t h e f r a c t u r e s o b s e r v e d a b o v e d u r i n g d r y i n g . Low power obse rva t i on
i n d i c a t e d t h a t t h e g r o u t w h i c h h a s p e n e t r a t e d t h e s a n d s t o n e w a s
h a r d e r a n d s u f f e r e d f e w e r f r a c t u r e s .
I n a d d i t i o n t o C u S O 4 , C a C l 2 w a s t r i e d a s a n o t h e r p o s s i b l e
d e l i q u e s c e n t a d d i t i v e . T h e r e s u l t s w e r e s i m i l a r ; l o c a l g e l a t i o n
New Concepts in Underground Storage of Natural Gas
-150-
w o u l d b e p r o h i b i t i v e t o f i e l d a p p l i c a t i o n s . The sea rch shou ld be
c o n t i n u e d t o f i n d a s a t i s f a c t o r y d e l i q u e s c e n t w h i c h w o u l d n o t p r o -
d u c e l o c a l g e l a t i o n . T h i s a d d i t i v e c o u l d p r o v e i m p o r t a n t i n a p p l i -
c a t i o n s w h e r e d r y i n g i s p o s s i b l e .
Pho tom ic rog raph i c Obse rva t i ons on Grou ted co re Spec imens
V e r t i c a l c r o s s - s e c t i o n s w e r e m a d e f r o m t h e g r o u t e d s a n d s t o n e
s e c t i o n s ' u s e d f o r m i c r o s c o p i c f a c e s t u d i e s , a n d g r o u n d t o s t a n d a r d
p e t r o g r a p h i c t h i n s e c t i o n s ( . 0 4 m m ) . W i t h t h i s t h i n a s l i c e i t
w a s e a s y t o d i s t i n g u i s h t h e s i l i c a g r a i n s f r o m t h e g r o u t i n t e r s t i c e s .
A l l p h o t o m i c r o g r a p h s w e r e m a d e u s i n g a i r - d r y s l i d e s .
CAMERA AND MICROSCOPE: Bausch and Lomb Opt ica ls
MAGNIFICATION: 60x
FILM: K o d a k T r i - X ( f i l m p a c k - 1 2 e x p o s u r e s )
EXPOSURE TIME: N o . 1 - 5 a t 1 5 s e c . No. 6 at 7 sec.
PHOTOGRAPHY: R e f l e c t e d l i g h t
T y p i c a l p h o t o m i c r o g r a p h s o n 6 f r a m e s i n c l u d e d i n t h i s r e p o r t
( F i g . 5 . 1 5 ) a r e l i s t e d w i t h o b s e r v a t i o n s i n t h e f o l l o w i n g :
FRAME 1 - - - - S a n d s t o n e g r o u t e d w i t h a c i d - a d d i t i v e g r o u t .
T h e l o c a l g e l a t i o n i n t h e b u l k o f t h e g r o u t s o l u t i o n a p p e a r s n o t
t o h a v e d i s t u r b e d t h e g e l a t i o n i n t h e s a m p l e . One may see that
t h e g r o u t h a s f i l l e d e v e n t h e l a r g e r v o i d s ( ' u p p e r r i g h t c o r n e r ) .
FRAME 2 - - - - S a n d s t o n e g r o u t e d i n i t i a l l y d r y . One may observe
i n b o t h f r a m e s N o . 1 a n d N o . 2 t h e g l a s s - l i k e r e f l e c t i o n s o f g r o u t
i m m e d i a t e l y a d j a c e n t t o t h e i n d i v i d u a l g r a i n s . Th i s may rep resen t
e i t h e r a d h e s i o n t o t h e g r a i n s o r r e f l e c t i o n f r o m t h e g r a i n s .
FRAME 3 - - - - S a n d s t o n e g r o u t e d w i t h C u S O 4 - a d d i t i v e g r o u t .
L o c a l g e l a t i o n o c c u r s i n t h e b u l k g r o u t . T h i s a p p e a r s t o a f f e c t
t h e i n t e r i o r o f t h e s a m p l e t o o . O n e m a y n o t e t h e i n c o m p l e t e f i l l -
i n g o f e v e n t h e s m a l l e r v o i d s .
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F i g . 5 . 1 5 Photo Micrographs on Grouted Core Specimens
New Concepts in Underground Storage of Natural Gas
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FRAME 4 - - - - S a n d s t o n e g r o u t e d i n i t i a l l y s a t u r a t e d .
T h e s p e c i a l i n t e r e s t o f t h i s s l i d e i s t h a t o n e c a n s e e t h e a c t i o n
o f g r o u t i n f i l l i n g e v e n t h e l a r g e v o i d i n t h e c e n t e r o f t h e s l i d e .
FRAME 5 - - - - Sandstone grouted with an Na2CO3-additive grout.
I t a p p e a r s t h a t t h i s a d d i t i v e g r o u t i s a b o u t a s e f f i c i e n t a s t h e
g r o u t w i t h o u t t h e a d d i t i v e i n f i l l i n g t h e v o i d s .
FRAME 6 - - - - G r o u t e d a n d u n g r o u t e d p a r t i c l e s .
T h e s i l i c a p a r t i c l e o n t h e r i g h t w a s r e m o v e d f r o m a d r y - g r o u t e d
s i l i c a a g g r e g a t e . T h e o n e i n t h e u p p e r l e f t c o r n e r w a s u n t r e a t e d .
O n e m a y o b s e r v e h e r e t h e n a t u r e o f t h e a d h e r e n t o n t h e p a r t i c l e
i t s e l f . W h e n a i r - d r y , t he po l ymer seems to adhe re more un i f o rm ly
to the part icle than when the specimen is saturated. This may be caused
b y a r e d u c t i o n o f c a p i l l a r y s i z e s i m i l a r t o t h a t f o u n d i n c l a y .
O n e c a n o b s e r v e f r o m t h i s s l i d e t h e a l m o s t - p o w d e r y t e x t u r e o f t h e
g r o u t .
5 . 5 P r a c t i c a l R e s e r v o i r E n g i n e e r i n g C a l c u l a t i o n s o n I n j e c t i o n o f
G rou t s
B e f o r e t h e g r o u t i n g m a t e r i a l c a n b e i n j e c t e d i n t o a p o r o u s
f o r m a t i o n , p r e s s u r e r e q u i r e m e n t s a n d w e l l s p a c i n g m u s t b e d e t e r m i n e d
t o p r o v i d e a d e q u a t e g r o u t p e n e t r a t i o n . T h e f o l l o w i n g e x a m p l e p r o -
b l e m i l l u s t r a t e s a m e t h o d o f d e t e r m i n i n g t h e r a d i u s o f p e n e t r a t i o n
a s a f u n c t i o n o f t h e w e l l b o r e p r e s s u r e . T h e o r e t i c a l c a l c u l a t i o n s
o n t h e m a t h e m a t i c s o f g r o u t p e n e t r a t i o n , i n c l u d i n g t h e e f f e c t o f
v i s c o s i t y v a r i a t i o n s a r e g i v e n i n d e t a i l i n t h e a p p e n d i x .
I t i s d e s i r e d t o i m p e r m e a t e a n a q u i f e r 5 0 f e e t i n t h i c k n e s s
l o c a t e d a t a d e p t h o f 2 0 0 0 f e e t . T h e s a n d s t o n e h a s a p e r m e a b i l i t y
o f 2 5 0 m i l l i d a r c y s a n d a p o r o s i t y o f 0 . 1 5 . T h e g r o u t i n g m a t e r i a l
Impermeation of Underground Formations
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i s t o b e i n j e c t e d f r o m a w e l l 1 / 4 f e e t i n r a d i u s t o t h e p o r o u s m a -
t r i x . U s i n g a g r o u t s o l u t i o n o f 1 . 6 c p v i s c o s i t y , a g e l t i m e o f
10 hours is specif ied. F i g u r e 5 . 1 6 i l l u s t r a t e s t h e f i e l d p r o b l e m w h e r e
such an impe rmea t i on m igh t be des i red .
F o r m a t i o n p a r t i n g p r e s s u r e i s ' u s u a l l y t a k e n a s 1 p s i / f t . o f
d e p t h s o t h i s p r e s s u r e i s t h e m a x i m u m f o r g r o u t i n j e c t i o n w i t h o u t
f r a c t u r i n g t h e f o r m a t i o n . A 10% sa fe t y f ac to r may be used mak ing
t h e w e l l b o r e p r e s s u r e 0 . 9 p s i / f t . o f d e p t h . A d i m e n s i o n l e s s f l o w
r a t e i s d e f i n e d a s :
where:
= total cumulative influx
= po ros i t y = 0 .15
= c o m p r e s s i b i l i t y o f g r o u t s o l u t i o n ,
as H20) = 7 x 10-6 p s i - 1 .
h = f o r m a t i o n t h i c k n e s s f t .
= r a d i u s o f p e n e t r a t i o n ( f t . )
( 5 . 1 )
(assumed same
= p r e s s u r e a t t h e w e l l p o r e ( p s i ) = 1 8 0 0 p s i g .
= f o r m a t i o n p r e s s u r e ( p s i ) = p g / g c V = 8 6 6 p s i g
= w e l l b o r e r a d i u s ( f t ) = 1 / 4
A d i m e n s i o n l e s s t i m e i s d e f i n e d a s
(5.2)
( 5 . 3 )
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F i g . 5 . 1 6 T y p i c a l A p p l i c a t i o n o f G r o u t i n g t o A q u i f e r
S t o r a g e ( I m p e r m e a t i o n o f S p i l l P o i n t )
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F i g . 5 . 1 7 R a d i a l G r o u t P e n e t r a t i o n a s a F u n c t i o n o f S e t t i n g T i m ea n d A q u i f e r D e p t h f o r a P r e s s u r e o f 0 . 9 p s i / f t .
New Concepts in Underground Storage of Natural Gas
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F i g . 5 . 1 8 A p p l i c a t i o n o f F r a c t u r i n g a n d G r o u t i n g t o A q u i f e r
S to rage (No c a p r o c k o r i g i n a l l y )
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where:
t = i n j e c t i o n t i m e ( h o u r s ) = 1 0 h o u r s
k = p e r m e a b i l i t y ( m d ) = 2 5 0 m d .
u = g r o u t s o l u t i o n v i s c o s i t y ( c p ) = 1 . 6 c p .
Q t a n d t oa r e r e l a t e d a n d t h e c o r r e s p o n d i n g v a l u e s a r e t a b u l a t e d
o n p a g e s 4 2 4 - 4 2 6 o f r e f e r e n c e 5 . 7 ( f o r r a d i a l f l o w , i n f i n i t e a q u i -
f e r ,
1 0 5 .
c o n s t a n t t e r m i n a l p r e s s u r e ) . F o r t o = 6 . 2 8 x 1 0 6 , Q t = 8 . 1 9 x
S o l v i n g e q u a t i o n 5 . 2 f o r r w e o b t a i n :
( 5 . 4 )
F i g u r e 5 . 1 7 p r e s e n t s a p l o t o f g r o u t p e n e t r a t i o n r a d i u s a s a f u n c -
t i o n o f t o t a l p u m p i n g t i m e f o r v a r i o u s d e p t h s .
A n o t h e r t y p i c a l a p p l i c a t i o n o f g r o u t i n g t o a q u i f e r s t o r a g e
i s i l l u s t r a t e d i n f i g u r e 5 . 1 8 . T h i s a p p l i c a t i o n f o l l o w s a f r a c t u r -
i n g t r e a t m e n t i n a s a n d s t o n e f o r m a t i o n . S i n c e t h e p r o p p e d f r a c -
t u r e w i l l h a v e a v e r y l a r g e p e r m e a b i l i t y , t h e p r e s s u r e t h r o u g h o u t
t h e f r a c t u r e c a n b e a s s u m e d e q u a l t o t h e w e l l b o r e p r e s s u r e . T h e r e -
f o r e , l i n e a r f l o w f r o m t h e f r a c t u r e b e c o m e s t h e p r i m a r y c o n s i d e r a -
t i o n . O n c e t h e f r a c t u r e h a s b e e n c r e a t e d , w e l l b o r e p r e s s u r e m u s t
New Concepts in Underground Storage of Natural Gas
-158-
b e m a i n t a i n e d b e l o w t h e f r a c t u r i n g p r e s s u r e t o p r e v e n t p o s s i b l e
u n d e s i r a b l e f r a c t u r e e x t e n s i o n . T h e r e f o r e , i n t h e f o l l o w i n g e x -
ample, 0 . 9 p s i / f o o t w a s u s e d f o r t h e w e l l b o r e p r e s s u r e .
I n t h i s c a s e u s i n g e q u a t i o n f o r l i n e a r f l o w5 . 8
where
t o t a l c u m u l a t i v e i n f l u x ( f t 3 ) =
p o r o s i t y = 0 . 1 5c o m p r e s s i b i l i t y o f g r o u t s o l u t i o n ,
a r e a
w e l l b o r e p r e s s u r e = ( 0 . 9 ) ( 2 0 0 0 f t ) = 1 8 0 0 p s i
p e r m e a b i l i t y = 2 5 0 m d
i n j e c t i o n t i m e ( d a y s ) = 1 0 / 2 4
v i s c o s i t y = 1 . 6 c p
r a d i u s o f f r a c t u r e = 6 0 0 f t .
d e p t h o f p e n e t r a t i o n o f g r o u t , f t .
= formation pressure ( p s i g ) = p g / g c = 866 psig
( 5 . 5 )
(5.6)
F i g u r e 5 . 1 9 i s a p l o t o f g r o u t p e n e t r a t i o n a s a f u n c t i o n o f t o t a l
p u m p i n g t i m e f o r v a r i o u s a q u i f e r d e p t h s .
O n c e t h e r e q u i r e d w e l l - b o r e p r e s s u r e h a s b e e n d e t e r m i n e d t h e
p o w e r r e q u i r e m e n t f o r t h e p u m p s m u s t b e c a l c u l a t e d . A l t h o u g h t h e s e
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F i g . 5 . 1 9 L i n e a r G r o u t P e n e t r a t i o n a s a F u n c t i o n o f S e t t i n g T i m e
a n d A q u i f e r D e p t h f o r a P r e s s u r e o f 0 . 9 p s i / f t .
New Concepts in Underground Storage of Natural Gas
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power requ i remen ts have been de te rm ined , t h e y a r e n o t p r e s e n t e d h e r e .
S u c h p o w e r r e q u i r e m e n t s w o u l d o n l y b e o f i n t e r e s t i n t h e f i n a l m e c h a n -
i c a l d e s i g n f o r a f i e l d t e s t a n d t h e r e f o r e a r e n o t c o n s i d e r e d w i t h i n
t h e s c o p e o f c u r r e n t i n v e s t i g a t i o n .
A m a t t e r o f e c o n o m i c i m p o r t a n c e i n a q u i f e r g r o u t i n g i s t h e
o p t i m u m w e l l s p a c i n g f o r l o w e s t c o s t .
C o s t = g r o u t c o s t + w e l l c o s t .
E c o n o m i c E v a l u a t i o n i n A q u i f e r G r o u t i n g
T h e t o t a l c o s t o f g r o u t i n g m a y b e f o r m u l a t e d a s f o l l o w s .
C o s t = g r o u t c o s t + w e l l c o s t .
( 5 . 7 )
where:
d = d e p t h o f f o r m a t i o n , f t .
h = h e i g h t o f a q u i f e r , f t .
= p o r o s i t y o f a q u i f e r
r = r a d i u s o f p e n e t r a t i o n , f t .
L = l e n g t h o f g r o u t w a l l , f t .
A = c o s t p e r g a l o f g e l s o l u t i o n
B = c o s t p e r f o o t f o r w e l l s ( i n c l u d e s p u m p i n g )
F i n d i n g t h e m i n i m u m o f e q u a t i o n ( 5 . 7 ) , w e o b t a i n :
( 5 . 8 )
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Example:
d = 1000 ft.
h = 30 ft.
= 0 . 2 0
A = $ 1 . 0 0 / g a l .
B = $ 1 0 . 0 0 / f t .
F i g u r e 5 . 2 0 p r e s e n t s a p l o t o f o p t i m u m w e l l s p a c i n g a s a f u n c t i o n
o f a q u i f e r t h i c k n e s s f o r v a r i o u s d e p t h s .
5 . 6 W e l l F r a c t u r i n g a s R e l a t e d t o G r o u t i n g
W h e n s o i l i m p e r m e a t i o n t h r o u g h a p p l i c a t i o n o f g r o u t i n g i n t o
p o r o u s m a t e r i a l s s u c h a s s a n d s t o n e i s c o n s i d e r e d f o r c r e a t i o n o f
s t o r a g e r e s e r v o i r s t h e n e e d t o i n d u c e f r a c t u r e s t o p r o v i d e c o n t i n u u m
f o r g r o u t i n g m a t e r i a l s b e c o m e s i m p o r t a n t . T h i s i s w h y i t w a s p o i n t e d
o u t e a r l i e r t h a t h y d r a u l i c f r a c t u r i n g c a n b e o f d e f i n i t e a s s i s t a n c e
i n d e s i g n a n d d e v e l o p m e n t o f s t o r a g e r e s e r v o i r s . Once an underground
f o r m a t i o n h a s b e e n f r a c t u r e d , g r o u t c a n b e i n j e c t e d i n t o t h e f r a c -
t u r e t o p r o v i d e a n i m p e r v i o u s l a y e r . G r o u t e d h o r i z o n t a l f r a c t u r e s
m a y s e r v e a s c a p r o c k w h i l e g r o u t e d v e r t i c a l f r a c t u r e s w i l l a c t a s
w a l l s t o p r o v i d e i m p e r v i o u s b o u n d a r i e s a r o u n d o r a m i d s t t h e p e r m e -
ab le sands tone .
H y d r a u l i c f r a c t u r i n g w a s f i r s t i n t r o d u c e d t o t h e p e t r o l e u m
i n d u s t r y i n 1 9 4 9 a s a m e t h o d o f i n c r e a s i n g o i l w e l l p r o d u c t i v i t y .
S i n c e 1 9 4 9 , m a n y m e t h o d s a n d t h e o r i e s c o n c e r n i n g f r a c t u r i n g h a v e
been proposed. I n f o r m a t i o n d e a l i n g w i t h f r a c t u r i n g o c c u r s f r e q u e n t l y
i n t h e l i t e r a t u r e , b u t i t a p p e a r s t h a t n o o n e h a s s u m m a r i z e d t h e
m a t e r i a l t o t h e e x t e n t t h a t p r a c t i c a l a n d r e a l i s t i c f r a c t u r e
New Concepts in Underground Storage of Natural Gas
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F i g . 5 . 2 0 O p t i m u m W e l l S p a c i n g a s a F u n c t i o n o f A q u i f e r H e i g h t
f o r V a r i o u s D e p t h s .
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ca l cu la t i ons can be made . I n o r d e r t o p r o v i d e a b a s i s f o r t h e d e -
s i g n o f f r a c t u r e s i n t h e c r e a t i o n o f g a s s t o r a g e r e s e r v o i r s , t h i s
p a p e r p r e s e n t s m e t h o d s o f c r e a t i n g f r a c t u r e s , e q u a t i o n s t o c a l c u -
l a t e f r a c t u r e e x t e n t , a d i s c u s s i o n o f f l u i d s a n d p r o p p i n g a g e n t s
to be used , a l o n g w i t h a c o m p l e t e p r o c e d u r e f o r d e s i g n .
L i t e r a t u r e S u r v e y a n d F r a c t u r e D e s i g n C a l c u l a t i o n s
F r a c t u r e E x t e n t
T h e b a s i c e q u a t i o n f o r f r a c t u r e d e s i g n w a s p r e s e n t e d b y
H o w a r d a n d F a s t i n 1 9 5 7 . 5 . 9 , 5 . 1 0
( 5 . 9 )
where
x = d i m e n s i o n l e s s t i m e
A = t o t a l a r e a o f o n e f a c e o f t h e f r a c t u r e , f t 2
Q i = c o n s t a n t i n j e c t i o n r a t e d u r i n g t r e a t m e n t , f t 3 / m i n ,o r ( b b l / m i n ) ( 5 . 6 1 4 )
t = t o t a l p u m p i n g t i m e , m i n .
W = c o n s t a n t f r a c t u r e c l e a r a n c e o r w i d t h , f t .
C = a c o n s t a n t w h i c h i s a m e a s u r e o f f l o w r e s i s t a n c e o ft h e f l u i d l e a k i n g o f f i n t o t h e f o r m a t i o n d u r i n g f r a c -t u r i n g t r e a t m e n t , f t / ( m i n . ) 0 . 5
e r f c ( x ) = c o m p l e m e n t a r y e r r o r f u n c t i o n o f x .
T h e f r a c t u r i n g f l u i d c o e f f i c i e n t , C , i n e q u a t i o n ( 5 . 9 ) i s
t h e r a t e o f f l u i d l o s s f r o m t h e f r a c t u r e t o t h e f o r m a t i o n . There
a r e t h r e e m e c h a n i s m s w h i c h c o n t r o l r a t e o f f l u i d f l o w i n t o a f o r -
m a t i o n . These are:5 . 9 , 5 . 1 0
New Concepts in Underground Storage of Natural Gas
-164-
1. F r a c t u r i n g f l u i d v i s c o s i t y a n d r e s e r v o i r p e r m e a b i l i t y .
2. V i s c o s i t y a n d c o m p r e s s i b i l i t y o f r e s e r v o i r - f l u i d .
3 . W a l l b u i l d i n g e f f e c t s .
C o e f f i c i e n t s f o r m e c h a n i s m s 1 , 2 , a n d 3 a r e c a l c u l a t e d a n d
t h e l o w e s t v a l u e i s t a k e n a s c o n t r o l l i n g . T h e c o e f f i c i e n t f o r
m e c h a n i s m 3 i s c o n s i d e r e d o n l y w h e n w a t e r l o s s a d d i t i v e s a r e a d d e d
i n o r d e r t o c o n t r o l f l u i d l o s s i n t o t h e f o r m a t i o n .
F r a c t u r i n g f l u i d v i s c o s i t y a n d f o r m a t i o n p e r m e a b i l i t y c o e f f i c i -
e n t , C i s c a l c u l a t e d a s f o l l o w s :5 . 9 , 5 . 1 0
I '
( 5 . 1 0 )
where
K = p e r m e a b i l i t y o f f o r m a t i o n t o f r a c t u r i n g f l u i d , d a r c y s
= p o r o s i t y o f f o r m a t i o n , f r a c t i o n a l q u a n t i t y
A P = d i f f e r e n c e i n p r e s s u r e b e t w e e n t h e f l u i d a t t h e f o r m a -t i o n , p s i
u F = v i s c o s i t y o f f r a c t u r i n g f l u i d , C P .
E q u a t i o n ( 5 . 1 0 ) c a n b e s o l v e d b y t h e n o m o g r a m i n F i g u r e 5 . 2 1 .
R e s e r v o i r f l u i d v i s c o s i t y a n d c o m p r e s s i b i l i t y e f f e c t , C I I ,
can be exp ressed by :5 . 9 , 5 . 1 0
where
CF = c o m p r e s s i b i l i t y o f r e s e r v o i r f l u i d , l / p s i
uR =v i s c o s i t y o f r e s e r v o i r f l u i d , C P .
( 5 . 1 1 )
E q u a t i o n ( 5 . 1 1 ) c a n b e so lved by the nomogram in F igure 5.22.
Note : F i g u r e 5 . 2 2 h a s i n c o r p o r a t e d i n i t a c o m p r e s s i b i l i t y o f
1 x 1 0 - 5 p s i - 1 , a n a p p r o x i m a t e v a l u e f o r c r u d e o i l s . I f t h e a c t u a l
c o m p r e s s i b i l i t y o f t h e r e s e r v o i r f l u i d i s s i g n i f i c a n t l y d i f f e r e n t
C I I v a l u e o b t a i n e d f r o m t h e n o m o g r a m s h o u l d b e m u l -f r o m t h i s , t h e
t i p l i e d b y
Impermeation of Underground Formations
-165-
W a l l b u il d i n g e f f e c t , C I I I , i s c a l c u l a t e d a s f o l l o w s :5 . 9 , 5 . 1 0
( 5 . 1 2 )
where
m = s l o p e o f e x p e r i m e n t a l f l u i d l o s s l i n e w h e n c m 3 f l u i d
l o s s i s p l o t t e d v e r s u s
a = a r e a o f f i l t e r m e d i u m , c m 2 .
T h e f l u i d l o s s t e s t i s c a r r i e d o u t i n a f i l t e r p r e s s w h i c h c o n s i s t s
o f a p r e s s u r e c e l l w i t h o n e e n d c o n t a i n i n g f i l t e r p a p e r o r a t h i n
c o r e w a f e r . A s t a n d a r d p r e s s u r e d i f f e r e n c e , e . g . , 1 0 0 0 p s i , i s
a p p l i e d a c r o s s t h e c e l l a n d t h e v o l u m e o f f i l t r a t e c o l l e c t e d i s
r e c o r d e d a s a f u n c t i o n o f t i m e . A s a m p l e f l u i d l o s s t e s t i s s h o w n
i n F i g u r e 5 . 2 3 . T h e t e s t m u s t b e c o r r e c t e d t o t h e a c t u a l r e s e r v o i r
p r e s s u r e d i f f e r e n c e , A P . F i g u r e 5 . 2 4 i s u s e d t o m a k e t h i s c o r r e c -
t i o n o f C I I I , b a s e d u p o n t h e r e l a t i o n s h i p t h a t f l u i d l o s s t h r o u g h
a f i l t e r i s p r o p o r t i o n a l t o t h e p r e s s u r e d i f f e r e n c e . T h e s e f l u i d
l o s s t e s t s a l m o s t a l w a y s g i v e a p o s i t i v e i n t e r c e p t a t t i m e e q u a l s
z e r o , a s i n d i c a t e d i n F i g u r e 5 . 2 3 . A m e t h o d f o r i n c l u d i n g t h i s
s p u r t l o s s i n c a l c u l a t i n g f r a c t u r e a r e a w i l l b e p r e s e n t e d b e l o w .
T h e s o l u t i o n t o e q u a t i o n ( 5 . 9 ) c a n b e s i m p l i f i e d a n d p u t i n
n o m o g r a p h i c f o r m i n t h e f o l l o w i n g m a n n e r . B y r e a r r a n g e m e n t , i t i s
p o s s i b l e t o e x p r e s s e q u a t i o n ( 5 . 9 ) i n t h e f o r m :5 : 1 1
New Concepts in Underground Storage of Natural Gas
-166-
F i g . 5 . 2 1 Nomogram f o r D e t e r m i n a t i o n o f C I 5 . 9 , 5 . 1 0
F i g . 5 . 2 2 Nomogram f o r D e t e r m i n a t i o n of CII 5.9, 5.10
Impermeation of Underground Formations
-167-
F i g . 5 . 2 5 F r a c t u r i n g E f f i c i e n c y v s . X5 . 9 , 5 . 1 0
New Concepts in Underground Storage of Natural Gas
-168-
( 5 . 1 3 )
where
A = t o t a l a r e a o f o n e f a c e o f t h e f r a c t u r e , f t 2
V = v o l u m e o f f r a c t u r i n g f l u i d p u m p e d , f t 3
W = w i d t h o f f r a c t u r e , f t
E = e f f i c i e n c y , t h e v o l u m e o f f r a c t u r e c r e a t e d e x p r e s s e da s a f r a c t i o n o f t h e v o l u m e o f f l u i d p u m p e d . E is af u n c t i o n o n l y o f x .
A p l o t o f e f f i c i e n c y v e r s u s x i s p r e s e n t e d i n F i g u r e 5 . 2 5 . E f f i c i -
e n c y a s a f r a c t i o n i s c a l c u l a t e d f r o m t h e e q u a t i o n :
(5 .14 )
F i g u r e 5 . 2 6 p r e s e n t s a n o m o g r a p h i c s o l u t i o n f o r e f f i c i e n c y a n d t h e n
a l l o w s s o l u t i o n o f e q u a t i o n 5 . 2 5 f o r f r a c t u r e a r e a .
F r a c t u r e W i d t h
T h e p r e c e d i n g e q u a t i o n s f o r f r a c t u r e a r e a d e p e n d ' u p o n t h e
k n o w l e d g e o f t h e f r a c t u r e w i d t h . P e r k i n s a n d K e r n 5 . 1 2 h a v e d e v e l o p e d
a m e t h o d f o r t h e d e t e r m i n a t i o n o f f r a c t u r e w i d t h s . One case con-
s i d e r e d b y P e r k i n s a n d K e r n w a s t h a t o f a v e r t i c a l f r a c t u r e c r e a t e d
f r o m N e w t o n i a n f l u i d s i n l a m i n a r f l o w . T h e c o n d i t i o n f o r l a m i n a r
f l o w e x i s t s w h e n t h e R e y n o l d s n u m b e r i s l e s s t h a n ( 7 . 8 1 x 1 0 3 ) ( 0 . 3 2 ) =
2500, where
( 5 . 1 5 )
I t m u s t b e n o t e d t h a t i n E q . 5 . 1 5 t h e n u m e r i c a l c o n s t a n t 8 . 7 0 x
1 0 3 h a s t h e d i m e n s i o n s o f d e n s i t y ( i . e . m a s s / c u b i c l e n g t h ) .
Imp
erm
ea
tion
o
f U
nd
erg
rou
nd
F
orm
atio
ns
-16
9-
New Concepts in Underground Storage of Natural Gas
-170-
N R E = Reynolds number
Q = t o t a l i n j e c t i o n r a t e , b b l / m i n .
S p G r = s p e c i f i c g r a v i t y o f f r a c t u r i n g f l u i d
H = h e i g h t o f f r a c t u r e , f t .
u = v i s c o s i t y o f f r a c t u r i n g f l u i d , c p .
W h e n t h e c o n d i t i o n f o r l a m i n a r f l o w i s s a t i s f i e d , e q u a t i o n 5 . 1 6
( 5 . 1 6 )
a p p l i e s .
where
W = m a x i m u m c r a c k w i d t h a t t h e w e l l b o r e , i n .
Q = t o t a l p u m p r a t e , b b l / m i n .
u = e f f e c t i v e f r a c t u r i n g f l u i d v i s c o s i t y , c p .
L = l e n g t h o f a v e r t i c a l f r a c t u r e m e a s u r e d f r o m t h e w e l lb o r e , f t .
E Y = Y o u n g ' s m o d u l u s o f f o r m a t i o n r o c k , p s i .
( v a l u e s p r e s e n t e d i n T a b l e 5 . 1 0 )
T h i s e q u a t i o n i s p r e s e n t e d g r a p h i c a l l y i n F i g u r e 5 . 2 7 . I n t h e c a s e
o f a homogeneous f o rma t i on , t h e c r a c k w o u l d t a k e t h e s h a p e o f a
d i sc mak ing L = 1 /2 H . T h i s i s i l l u s t r a t e d i n F i g u r e 5 . 2 9 . How-
e v e r , i f a v e r y t i g h t l a y e r i s w i t h i n t h e p o t e n t i a l f r a c t u r e r a d i u s ,
T h e f r a c t u r e w i l l b e r e s t r i c t e d y i e l d i n g d i f f e r e n t v a l u e s f o r H a n d
L .
I f ( Q ) ( S p G r ) / ( H ) ( u ) i s g r e a t e r t h a n 0 . 3 2 , t h e n t h e f l u i d w i l l
b e i n t u r b u l e n t f l o w w i t h i n t h e f r a c t u r e . F o r t h i s c a s e , t h e w i d t h
i s g i v e n b y e q u a t i o n 5 . 1 7 . 5 ' 1 2
( 5 . 1 7 )
T h i s e q u a t i o n i s p r e s e n t e d g r a p h i c a l l y i n F i g u r e 5 . 2 8 .
Impermeation of Underground Formations-171-
FIG 5.27 CRACK WlDTHS FOR RESTRlCTED VERTICAL FRACTURES RESULTING FROM NEWTONIAN FLUIDSIN LAMINAR FLOW. 5.12
FIG. 5.28 CRACK WIDTHS FOR RESTRICTED VERTICAL FRACTURES RESULTING FROM NEWTONIAN FLUIDSTURBULENT FLOW. 5.12
New Concepts in Underground Storage of Natural Gas
-172-
FIG. 5.29 RESTRICTED AND UNRESTRICTED VERTICAL FRACTURES.5.12
Impermeation of Underground Formations
-173-
FIG. 5.30 CRACK WIDTHS FOR RESTRICTED VERTICAL FRACTURE RESULTING
FROM NON-NEWTONIAN FLUIDS IN LAMINAR FLOW.5.12
New Concepts in Underground Storage of Natural Gas
-174-
I f n o n - N e w t o n i a n f l u i d s s u c h a s g e l l e d o i l s o r e m u l s i o n s a r e
u s e d , t h e n i t i s n e c e s s a r y t o d e t e r m i n e t h e f l u i d ' s f l o w p r o p e r t i e s
b e f o r e e s t i m a t i n g c r a c k w i d t h . From Fann meter (measures shear
s t r e s s a s a f u n c t i o n o f s h e a r r a t e ) d a t a , t w o c o n s t a n t s , k a n d
n a re de te rm ined and t hese cons tan t s used i n p l ace o f v i scos i t y . 5 . 1 2
O n c e k ' a n d n ' h a v e b e e n d e t e r m i n e d , c r a c k w i d t h s c a n b e e s t i m a t e d
f r o m F i g u r e 5 . 3 0 .
I f a f r a c t u r e i s o r i e n t e d h o r i z o n t a l l y , c r a c k w i d t h m a y r e -
s u l t f r o m t w o t y p e s o f r o c k m o v e m e n t . I f t h e f r a c t u r e i s d e e p
w i t h i n t h e e a r t h , c r a c k s r e s u l t p r i n c i p a l l y f r o m c o m p r e s s i o n o f
r o c k i n t h e v i c i n i t y o f t h e f r a c t u r e . However, i f t h e f r a c t u r e
i s v e r y s h a l l o w , c r a c k w i d t h m a y a l s o r e s u l t f r o m f l e x i n g a n d l i f t -
i n g o f t h e o v e r b u r d e n . I t h a s b e e n s h o w n 5 . 1 2 t h a t c o m p r e s s i o n o f
s u r r o u n d i n g r o c k i s t h e p r i n c i p a l m e c h a n i s m g o v e r n i n g t h e c r a c k w i d t h
i f t h e d e p t h i s g r e a t e r t h a n a b o u t t h r e e - f o u r t h s o f t h e f r a c t u r e
r a d i u s . T h e r e f o r e , t h i s i s t h e m e c h a n i s m t h a t c o n t r o l s d u r i n g
m o s t f r a c t u r e t r e a t m e n t s . F o r t h i s c o n d i t i o n t h e w i d t h i s g i v e n
a p p r o x i m a t e l y b y e q u a t i o n 5 . 1 8 .5 . 1 2
( 5 . 1 8 )
where
C Y = r a d i u s o f f r a c t u r e , f t .
F i g u r e 5 . 3 1 p r e s e n t s t h i s e q u a t i o n g r a p h i c a l l y . L a m i n a r f l o w
o f t h e f l u i d a t e v e r y p o i n t i n a h o r i z o n t a l f r a c t u r e i s p r o b a b l y
e n c o u n t e r e d o n l y r a r e l y i n f i e l d o p e r a t i o n s . H e n c e , t u r b u l e n t f l o w
m u s t a l s o b e c o n s i d e r e d b e f o r e a g e n e r a l l y a p p l i c a b l e e q u a t i o n c a n
b e d e r i v e d . However, t h e t u r b u l e n t z o n e u s u a l l y w i l l n o t e x t e n d
f a r f r o m t h e w e l l b o r e ; t h e r e f o r e , F i g u r e 5 . 3 1 i s a p p r o x i m a t e l y
c o r r e c t i n m o s t c a s e s .5 . 1 2
Imp
erm
ea
tion
o
f U
nd
erg
rou
nd
F
orm
atio
ns
-17
5-
New Concepts in Underground Storage of Natural Gas
-176-
FIG. 5.32 VISCOSITY OF A SLURRY CONTAlNING SUSPENDED SOLID
MATERIAL COMPARED TO THE VISCOSITY OF THE BASIC FLUID.
Impermeation of Underground Formations
-177-
T h e c r a c k w i d t h s e s t i m a t e d f r o m F i g u r e s 5 . 2 7 , 5 . 2 8 , 5 . 3 0 a n d
5 . 3 1 a p p l y w h e n p u r e f l u i d s a r e b e i n g p u m p e d a l o n g a f r a c t u r e .
T h e s e e s t i m a t e d w i d t h s a r e a l s o v a l i d w h e n t h e r e i s a s p a r s e d i s -
t r i b u t i o n o f p r o p p i n g a g e n t s u s p e n d e d i n t h e f l u i d . H o w e v e r , i f
a l a r g e a m o u n t o f s a n d i s i n j e c t e d a s a p r o p p i n g a g e n t , t h e n i t s
p r e s e n c e i n t h e f r a c t u r e w i l l i n f l u e n c e p r e s s u r e d r o p a n d t h e r e b y
c r a c k w i d t h . 5 . 1 2 T h e f o l l o w i n g e q u a t i o n w i l l g i v e t h e a v e r a g e
s l u r r y c o n c e n t r a t i o n t a k i n g i n t o a c c o u n t f l u i d l e a k - o f f .
( 5 . 1 9
where
cs = s l u r r y c o n c e n t r a t i o n v o l / v o l
vs = v o l u m e o f s a n d i n j e c t e d i n t o f r a c t u r e , f t3
A = f r a c t u r e a r e a , f t 2
W = f r a c t u r e w i d t h , f t .
T h e a v e r a g e s l u r r y v i s c o s i t y c a n t h e n b e e s t i m a t e d f r o m F i g u r e 5 . 3 2 .
T h e c r a c k w i d t h i s t h e n e s t i m a t e d f r o m F i g u r e s 5 . 2 7 , 5 . 2 8 , 5 . 3 0 o r
5 . 3 1 u s i n g t h e a v e r a g e s l u r r y v i s c o s i t y a n d d e n s i t y r a t h e r t h a n
t h e v i s c o s i t y a n d d e n s i t y o f t h e p u r e f r a c t u r i n g f l u i d . I n t h e
a c t u a l c a s e , s l u r r y p r o p e r t i e s v a r y f r o m p o i n t t o p o i n t i n t h e
f r a c t u r e . Hence, t h e w i d t h c a l c u l a t e d a s j u s t s h o w n m u s t b e i n t e r -
p r e t e d o n l y a s a n a p p r o x i m a t e w i d t h .
I f i t i s d e s i r e d t o i n c l u d e t h e s p u r t l o s s , V S p , i n t h e c a l -
c u l a t i o n o f f r a c t u r e a r e a , i t m a y b e d o n e b y i n c l u d i n g t h i s v a l u e
a s a n i n c r e a s e d f r a c t u r e c l e a r a n c e 5 . 9 , 5 . W '
( 5 . 2 0
New Concepts in Underground Storage of Natural Gas
-178-
where
W = f r a c t u r e w i d t h , f t .
V S p = spu r t l o ss i n cm 3
a = f i l t e r a r e a , c m 2
A g roup o f Russ ian eng inee rs5 . 1 3h a s a d v a n c e d t h e o r i e s c o n -
c e r i n g t h e i n i t i a t i o n a n d d e v e l o p m e n t o f f r a c t u r e s . H o w e v e r , t h e r e
a p p e a r s t o b e a c o n t r o v e r s y a s t o t h e v a l i d i t y o f t h e i r a p p r o a c h ,
a n d f o r t h i s r e a s o n t h e s e e q u a t i o n s h a v e b e e n o m i t t e d f r o m t h e d i s -
c u s s i o n .
Pressure and Horsepower Requi rements
O t h e r i m p o r t a n t f a c t o r s i n t h e d e s i g n o f a f r a c t u r i n g j o b
a re t he p ressu re and ho rsepower requ i remen ts . S u r f a c e p r e s s u r e
r e q u i r e m e n t s t o c r e a t e a n d e x t e n d a f r a c t u r e m a y b e d i v i d e d i n t o
t w o b a s i c c a t e g o r i e s ;5 . 1 4
b r e a k d o w n a n d t r e a t i n g p r e s s u r e . The
b r e a k d o w n p r e s s u r e i s d e p e n d e n t o n s e v e r a l v a r i a b l e s s u c h a s f o r -
m a t i o n f a c e c o n t a m i n a t i o n ( f i l t e r c a k e , c e m e n t , e t c . ) , e x i s t e n c e
o r a b s e n c e o f f o r m a t i o n f r a c t u r e s , b e d d i n g p l a n e s , r o c k s t r e n g t h
a n d t y p e , e t c . a n d g e n e r a l l y c a n n o t b e p r e d i c t e d a c c u r a t e l y . How-
e v e r , s i n c e i n j e c t i o n r a t e s a r e n o t s i g n i f i c a n t l y i m p o r t a n t d u r i n g
b r e a k d o w n o t h e r t h a n i n s u r i n g t h a t t h e f o r m a t i o n h a s r u p t u r e d ,
h o r s e p o w e r r e q u i r e m e n t s a r e n o r m a l l y c o m p u t e d u s i n g p r e d i c t e d
t r e a t i n g p r e s s u r e s a n d i n j e c t i o n r a t e s .
T r e a t i n g p r e s s u r e o r p u m p p r e s s u r e ( P s ) r e q u i r e m e n t i s e q u a l
t o t h e s u m o f t h e h y d r a u l i c p r e s s u r e ( P r ) r e q u i r e d t o m a i n t a i n f r a c -
t u r e p a r t i n g p l u s f l u i d f r i c t i o n l o s s e s ( P f ) m i n u s t h e h y d r o s t a t i c
f l u i d h e a d ( P h ) , o r
Impermeation of Underground Formations
-179-
( 5 . 2 1 )
T h e p r e s s u r e r e q u i r e d t o m a i n t a i n f r a c t u r e p a r t i n g a n d e x t e n s i o n
( P r ) m a y b e e s t i m a t e d a s 1 p s i / f o o t o f d e p t h t o a p p r o x i m a t e l y 5 , 0 0 0
f e e t a n d 0 . 7 p s i / f o o t o f d e p t h f o r d e e p e r f o r m a t i o n s . A minimum
p r e s s u r e r e q u i r e m e n t o f 0 . 6 p s i / f o o t o f d e p t h m a y b e e s t i m a t e d f o r
a l l d e p t h s . P r e s s u r e l o s s e s d u e t o f r i c t i o n ( P f ) i n t h e c o n d u c -
t o r p i p e a n d t h r o u g h t h e p e r f o r a t i o n s ( i f a n y ) a r e d e p e n d e n t o n
f l u i d v i s c o s i t y , i n j e c t i o n r a t e , s a n d c o n c e n t r a t i o n a n d s i z e o f
c o n d u c t o r p i p e a n d m a y b e e s t i m a t e d f r o m c h a r t s a v a i l a b l e f r o m
f l u i d s u p p l i e s s u c h a s t h o s e a p p e a r i n g i n F i g u r e 4 . 1 2 . M a n y f l u i d
f r i c t i o n c h a r t s d o n o t a c c o u n t f o r t h e s a n d c o n t e n t o f t h e f l u i d
a n d a n a p p r o x i m a t e c o r r e c t i o n f o r s a n d c a n b e m a d e b y i n c r e a s i n g
f r i c t i o n l o s s e s o f s a n d - l a d e n f l u i d s b y 8 . 0 % / l b o f s a n d p e r g a l l o n . 5 . 1 4
S t a t i c f l u i d h e a d m a y t h e n b e o b t a i n e d f r o m F i g u r e 5 . 3 4 . P s can
t h e n b e c a l c u l a t e d f r o m e q u a t i o n ( 5 . 2 1 ) .
T h e c h o i c e o f a g o o d i n j e c t i o n r a t e i s a l s o i m p o r t a n t f o r
a s u c c e s s f u l f r a c t u r e t r e a t m e n t . R e c o m m e n d e d i n j e c t i o n r a t e s f o r
v a r i o u s s i z e d t u b i n g a r e g i v e n i n T a b l e 5 . 1 1 .
A f t e r t h e i n j e c t i o n p r e s s u r e a n d t h e i n j e c t i o n r a t e h a v e b e e n
de te rm ined , e q u a t i o n ( 5 . 2 2 ) c a n b e u s e d t o g i v e t h e h y d r a u l i c h o r s e -
power requ i remen t .
( 5 . 2 2 )
where
Ps = i n j e c t i o n s u r f a c e p r e s s u r e i n p s i
vi = i n j e c t i o n r a t e i n b b l / m i n .
F r a c t u r i n g F l u i d s
A n o t h e r i m p o r t a n t f a c t o r i n t h e d e s i g n o f f r a c t u r e t r e a t m e n t s
i s t h e s e l e c t i o n o f a g o o d f r a c t u r i n g f l u i d . S o m e b a s i c r e q u i r e -
m e n t s f o r a f l u i d a r e a s f o l l o w s : 5 . 1 4
New Concepts in Underground Storage of Natural Gas
-180-
1. M u s t b e a b l e t o p h y s i c a l l y o p e n a n d e x t e n d a f r a c t u r e .
2. M u s t b e c a p a b l e o f c a r r y i n g a " p r o p p i n g " a g e n t w h i c h c a n b e
l e f t i n t h e f o r m a t i o n t o p r e v e n t " h e a l i n g " o f t h e f r a c t u r e .
3 . S h o u l d b e e a s i l y b a c k - f l u s h e d f r o m t h e f o r m a t i o n .
4 . S h o u l d b e c o m p a t i b l e w i t h n a t i v e f o r m a t i o n f l u i d s .
5. S h o u l d c r e a t e a m i n i m u m o f p e r m e a b i l i t y d a m a g e w i t h i n t h e f o r -
m a t i o n .
6 . S h o u l d h a v e l o w f r i c t i o n - l o s s p r o p e r t i e s a n d b e e a s i l y p u m p e d .
F r a c t u r i n g f l u i d s m a y b e d i v i d e d i n t o t h r e e b a s i c c a t e g o r i e s ,
w a t e r , o i l , a n d a c i d .
A l t h o u g h w a t e r i s t h e c h e a p e s t o f t h e f r a c t u r i n g f l u i d s , i t s
‘ u s e u s u a l l y i s r e s t r i c t e d t o t r e a t m e n t s w h e r e e m u l s i o n s o r s w e l l -
i n g c l a y m i n e r a l s a r e n o t c o n s i d e r e d t o b e a p r o b l e m . E v e n i f
s w e l l i n g i s n o t a p r o b l e m , ' u s e o f u n m o d i f i e d w a t e r s h o u l d b e r e -
s t r i c t e d t o t r e a t m e n t s i n w h i c h h i g h i n j e c t i o n r a t e s a r e o b t a i n -
a b l e 5 . 1 4
O n e m e t h o d t o m o d i f y w a t e r i s t o a d d f l u i d l o s s c o n t r o l a d d i -
t i v e s t o i n h i b i t f l u i d l o s s f r o m t h e f r a c t u r e s f o r m e d a n d t h e r e b y
i n c r e a s e t h e e f f i c i e n c y o f t h e h y d r a u l i c f l u i d a s a " p r y i n g ' a g e n t
i n o p e n i n g a n d e x t e n d i n g a f r a c t u r e . W a t e r m a y a l s o b e g e l l e d t o
i n c r e a s e s a n d s u s p e n s i o n a b i l i t y w i t h o u t i n c r e a s i n g e f f e c t i v e v i s -
c o s i t y .5 . 1 4 G e l s u s u a l l y e x h i b i t h i g h a p p a r e n t v i s c o s i t i e s w h e n
m e a s u r e d i n a l a b o r a t o r y ; h o w e v e r , t h e i r e f f e c t i v e v i s c o s i t y d e -
c r e a s e s r a p i d l y w i t h i n c r e a s e d f l o w r a t e s . S p e c i a l c h e m i c a l s s u c h
a s s u r f a c t a n t s , b e n t o n i t e a n d c l a y s w e l l i n g i n h i b i t o r s , e m u l s i o n
b r e a k e r s , p r e c i p i t a t e s o l v e n t s , e t c . , a re some t imes added t o ge l l ed
a n d u n g e l l e d w a t e r a n d a r e u s e d i f l a b o r a t o r y t e s t s j u s t i f y t h e
a d d i t i o n a l e x p e n s e .
Imp
erm
ea
tion
o
f U
nd
erg
rou
nd
F
orm
atio
ns
-18
1-
New Concepts in Underground Storage of Natural Gas
-182-
O i l i s t h e m o s t c o m m o n f l u i d u s e d i n f r a c t u r i n g o p e r a t i o n s
b e c a u s e o f i t s c o m p a t i b i l i t y w i t h m o s t r e s e r v o i r f l u i d s a n d g o o d
s a n d - c a r r y i n g a b i l i t y . B o t h r e f i n e d o i l a n d l e a s e c r u d e s a r e u s e d
and a re mod i f i ed i n t he same manne r as wa te r . F i g u r e 5 . 3 5 i l l u s -
t r a t e s t h e e f f e c t o f w a t e r l o s s a d d i t i v e s o n l e a s e c r u d e f l u i d l o s s .
G e l l e d o r u n g e l l e d a c i d i s a c o m m o n f r a c t u r i n g f l u i d i n s o m e
a r e a s f o r h y d r a u l i c a l l y f r a c t u r i n g c a r b o n a t e r e s e r v o i r s o r c a r b o n -
a t e b e a r i n g s a n d s t o n e s . One techn ique 5 . 1 5 i n v o l v e s t h e a d d i t i o n
o f d r y c r y s t a l s o f s u l f o n i c a c i d t o t h e f r a c t u r i n g f l u i d . Because
o f t h e a d d i t i o n a l e x p e n s e i n v o l v e d , a c i d t r e a t m e n t i s n o t n o r m a l l y
' u s e d i f o t h e r f l u i d s c a n b e s a t i s f a c t o r i l y a p p l i e d .
I n f r a c t u r i n g t o c r e a t e a g a s s t o r a g e r e s e r v o i r , i t m a y b e
m o r e e x p e d i e n t t o u s e c h e m i c a l g r o u t a s t h e f r a c t u r i n g f l u i d . Then,
a f t e r t h e f r a c t u r e h a s b e e n o p e n e d a n d p r o p p e d , t h e g r o u t w o u l d
s e t t o p r o v i d e a n i m p e r v i o u s b o u n d a r y . T h e a d d i t i o n o f w a t e r l o s s
a d d i t i v e s t o a v i s c o u s c h e m i c a l g r o u t w o u l d p r o b a b l y b e s u f f i c i e n t
t o g i v e t h e a d d e d p r o p e r t i e s n e c e s s a r y t o m a k e t h e s o l u t i o n a f r a c -
t u r i n g f l u i d a s w e l l a s a g r o u t i n g m a t e r i a l .
P ropp ing Agen t
T h e e v a l u a t i o n a n d s e l e c t i o n o f t h e f r a c t u r e p r o p p i n g a g e n t
i s a l s o a n i m p o r t a n t p a r t o f f r a c t u r e t r e a t m e n t d e s i g n . The main
p r o p p i n g a g e n t i s s a n d a l t h o u g h a l u m i n u m p e l l e t s5 . 1 8 , 5 . 1 9 a n d
c r u s h e d w a l n u t s h e l l s 5 . 1 6 , 5 . 1 9 a re ' used occas iona l l y and have
p r o v e d s a t i s f a c t o r y . M o s t o f t h e p a p e r s p r e s e n t e d o n p r o p p i n g
a g e n t s e l e c t i o n , h o w e v e r , h a v e b e e n c o n c e r n e d w i t h t h e p r o p p i n g
a g e n t c o n c e n t r a t i o n f o r m a x i m u m f r a c t u r e f l o w c a p a c i t y i n o i l w e l l
s t i m u l a t i o n .5 . 2 1
However, i n t h e c r e a t i o n o f a n ' u n d e r g r o u n d s t o r -
a g e r e s e r v o i r , t h i s e v a l u a t i o n p r o b a b l y w o u l d b e l e s s i m p o r t a n t .
Impermeation of Underground Formations
-183-
F i g . 5 . 3 4 E f f e c t o f S a n d C o n c e n t r a t i o n
o n F l u i d H e a d i n p s i / f t . 5 . 1 4
New Concepts in Underground Storage of Natural Gas
-184-
F o r t h e p u r p o s e o f t h i s i n v e s t i g a t i o n , t h e a m o u n t o f s a n d n e e d e d
c a n b e t a k e n a s 4 . 4 l b s a n d / g a l . o f f r a c t u r e v o l u m e c r e a t e d .5 . 1 1
I n d e e p w e l l s w i t h h i g h o v e r b u r d e n p r e s s u r e a n d / o r s o f t f o r m a t i o n s ,
i t may be des i rab le t o ' use more sand . The maximum amount that can
b e a d d e d i s t h a t w h i c h w i l l f i l l t h e e n t i r e v o l u m e o f f r a c t u r e
c r e a t e d . T h e a m o u n t o f s a n d n e c e s s a r y t o " s a n d p a c k " t h e f r a c t u r e
i s o b t a i n e d b y m u l t i p l y i n g t h e r e c o m m e n d e d v a l u e b y 2 . 7 .5 . 1 1
When
a n d i f a f r a c t u r e f i e l d t e s t i s m a d e , a m e t h o d e x i s t s f o r d e t e r m i n -
i n g t h e i n j e c t i o n s c h e d u l e f o r a f r a c t u r e t r e a t m e n t5 . 2 7
(i.e., the
t i m e a n d a m o u n t o f f l u i d a n d p r o p p i n g a g e n t i n j e c t i o n ) .
T h e c o n v e n t i o n a l w e l l b o r e p r e p a r a t i o n f o r f r a c t u r i n g h a s
b e e n t o ' u s e a p e r f o r a t e d p i p e , t h u s c r e a t i n g m a n y s m a l l f r a c t u r e s
o n a p p l i c a t i o n o f p r e s s u r e . The me thod , howeve r , p robab l y wou ld
n o t p r o v i d e s a t i s f a c t o r y c o n t r o l o f t h e l o c a t i o n o f t h e f r a c t u r e
i n c r e a t i n g f r a c t u r e s f o r s o i l i m p e r m e a t i o n t h r o u g h g r o u t i n g .
Seve ra l new techn iques , howeve r , a l l o w s e l e c t i v e f r a c t u r i n g . One
m e t h o d i n v o l v e s t h e ' u s e o f h i g h v e l o c i t y p r o j e c t i l e s5 . 2 2
t o s t a r t
t h e f r a c t u r e , t h u s p r o v i d i n g a w e a k p o i n t f o r t h e h y d r a u l i c p r e s -
s u r e t o i n i t i a t e t h e f r a c t u r e . Ano the r me thod i nvo l ves t he ' use
o f a n e v a c u a t e d c y l i n d e r t h a t i m p l o d e s ' u n d e r t h e h y d r a u l i c p r e s -5 . 2 3
s u r e , a n d t h u s c a u s e s a s u d d e n b u r s t o f v e r y h i g h p r e s s u r e a s
t h e f l u i d i n t h e p i p e a c c e l e r a t i n g t o f i l l t h e v o l u m e p r e v i o u s l y
o c c u p i e d b y t h e c y l i n d e r . S t i l l a n o t h e r m e t h o d c o n s i s t s o f n o t c h -
i n g t h e w e l l b o r e t o a l l o w s i n g l e - p o i n t e n t r y .5 . 2 4
Another method
i n v o l v e s t h e i n j e c t i o n o f s m a l l r u b b e r b a l l s i n t o t h e w e l l .5 . 2 5
A s p r e s s u r e i s a p p l i e d , t h e b a l l s a r e p u s h e d i n t o t h e p e r f o r a t i o n s
t h u s p r e v e n t i n g a f r a c t u r e i n t h a t s e c t i o n o f t h e w e l l .
Some da ta have been pub l i shed i n t he l i t e r a t u r e5 . 2 6
t o a i d
i n t h e c a l c u l a t i o n o f t h e c o s t f o r a f r a c t u r i n g j o b . T a b l e 5 . 1 2
g i v e s t h e c o s t o f o i l f r a c t u r i n g f l u i d w i t h w a t e r l o s s a d d i t i v e s
( C I I I v a l u e s a r e a l s o l i s t e d f o r t h e o i l s ) . O n e d o l l a r p e r h y d r a u -
l i c ho rsepower can be assumed f o r pump ing cos t .5 . 2 6
Impermeation of Underground Formations
-185-
F i g . 5 .35 E f f e c t o f F l u i d L o s s C o n t r o l A d d i t i v e
C o n c e n t r a t i o n o n F l u i d L o s s f o r S p e c i f i c C r u d e 5 . 1 4
New Concepts in Underground Storage of Natural Gas
-186-
Des ign P rocedu re
size, pumping1. S e l e c t p r o p p i n g a g e n t , f r a c t u r i n g f l u i d , t u b i n g
t i m e .
2. R u n a f l u i d l o s s t e s t a n d ‘ u s e t h e r e s u l t s t o p l o t
ume versus
f i l t r a t e v o l -
3 .
4 .
5.
6 .
7 .
8 .
9 .
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
U s e t h e r e c o m m e n d e d i n j e c t i o n r a t e , V i , f r o m T a b l e 5 . 1 1 .
K n o w i n g t h e d e p t h , c a l c u l a t e P b ( 1 p s i / f t a t d e p t h s l e s s t h a n
5 , 0 0 0 f t ) .
O b t a i n f r i c t i o n l o s s , P f , f r o m a g r a p h s u c h a s F i g u r e 5 . 3 3
( a l s o c a n b e c a l c u l a t e d v i a R e y n o l d s N u m b e r a n d f r i c t i o n f a c -
t o r ) .
O b t a i n s t a t i c f l u i d h e a d , P h , f r o m F i g u r e 5 . 3 4 .
C a l c u l a t e P s f r o m e q u a t i o n 5 . 2 1 .
C a l c u l a t e h o r s e p o w e r r e q u i r e m e n t b y e q u a t i o n 5 . 2 2 .
C a l c u l a t e C I b y F i g u r e 5 . 2 1 .
C a l c u l a t e C I I b y F i g u r e 5 . 2 2 .
C a l c u l a t e C I I I b y e q u a t i o n 5 . 1 2 a n d F i g u r e 5 . 2 4 .
S e l e c t t h e l o w e s t o f t h e s e ( C I , C I I , o r C I I I ) a s t h e c o n t r o l -
l i ng mechan i sm.
Compu te t he t o ta l vo lume pumped (V = V i t ) .
G u e s s a f r a c t u r e w i d t h ( p r o b a b l y i n t h e r a n g e 0 . 1 - 0 . 4 i n . ) .
C o r r e c t t h e w i d t h f o r s p u r t l o s s b y e q u a t i o n 5 . 2 0 .
U s e F i g u r e 5 . 2 6 t o d e t e r m i n e t h e f r a c t u r e a r e a .
C a l c u l a t e t h e f r a c t u r e r a d i u s f r o m t h e a r e a .
C a l c u l a t e N R E ( R e y n o l d s N u m b e r ) f r o m e q u a t i o n 5 . 1 5 i f t h e
f r a c t u r e i s t o b e v e r t i c a l .
C a l c u l a t e W f r o m F i g u r e s 5 . 2 7 , 5 . 2 8 , 5 . 3 0 , o r 5 . 3 1 u s i n g t h e
s l u r r y p r o p e r t i e s d e t e r m i n e d u s i n g e q u a t i o n 5 . 1 9 a n d F i g u r e
5 . 3 2 .
Impermeation of Underground Formations
-187-
21. I f t h e c a l c u l a t e d w i d t h v a r i e s s i g n i f i c a n t l y f r o m t h e a s s u m e d
w id th , guess a new w id th and repea t s t eps 13 -17 .
22. Compu te t he cos t us ing Tab le 5 .12 and ' us i ng one do l l a r pe r
h y d r a u l i c h o r s e p o w e r .
Example Problem
D a t a
D e p t h o f z o n e t o b e t r e a t e d 2 0 0 0 f e e t
F o r m a t i o n p e r m e a b i l i t y .050 da rcys
F o r m a t i o n p o r o s i t y 0 . 1 5
F o r m a t i o n F l u i d V i s c o s i t y 1 cp
F r a c t u r e T y p e D e s i r e d h o r i z o n t a l
1. F r a c . F l u i d - L e a s e O i l
G r a v i t y 35° AP I
V i s c o s i t y 500 cp
Propp ing agen t - sand
C o n c e n t r a t i o n 1 . 5 l b m / g a l
Pumping Time 60 min.
P a c k c a s i n g t o a l l o w s i n g l e p o i n t h o r i z o n t a l e n t r y
Cas ing S i ze 5 1 / 2 i n .
2 . A s s u m e t h a t a f l u i d l o s s t e s t g i v e s t h e c u r v e o f F i g u r e 5 . 2 3
( a r e a o f f i l t e r = 2 0 c m 2 ) .
3 . F r o m T a b l e 4 . 2 , V i = 2 5 b b l / m i n .
4 . P r = ( 2 0 0 0 f t ) ( 1 p s i / f t ) = 2 0 0 0 p s i g .
5. S i n c e a f r i c t i o n l o s s g r a p h s u c h a s F i g u r e 5 . 3 3 w a s n o t
a v a i l a b l e f o r t h i s f l u i d , P k w a s a s s u m e d e q u a l t o 2 5 0 p s i g .
6 . F r o m F i g u r e 5 . 3 4 , f l u i d h e a t = 0 . 4 2 0 p s i / f t .
P h = ( 0 . 4 2 0 ) ( 2 0 0 0 f t ) = 8 4 0 p s i g .
New Concepts in Underground Storage of Natural Gas
7 . Ps = Pb + Pf - Ph ( 5 . 2 1 )
8 .
P s = 2000 + 250 - 840
PS = 1410 ps ig
HHP = 0.0245 PsVi
HHP = (0.0245)(1410)(25)
HHP = 863 horsepower.
P G =
( 5 . 2 2 )
P G = 865 ps ig
10.
11.
12.
From F i g u r e 5 . 2 1 , C I = 6 x 10 - 3
F r o m F i g u r e 5 . 2 2 , C I I = ( 1 1 x 1 0 - 3 f t
( 5 . 1 2 )
F r o m F i g u r e 5 . 2 4 , C I I I c o r r e c t e d = 1 . 7 x 1 0 - 3
13.
14.
c = CIII = 1.7 x
V = V i t
1 0 - 3 f t /
V = (25 bb l /m in . )
V = 6 3 , 0 0 0 g a l .
( 6 0 m i n ) ( 4 2 g a l / b b l )
15. Guess W = 0 .15 in .
16. ( 5 . 2 0 )
-188-
Impermeation of Underground Formations
-189-
17. F r o m F i g u r e 5 . 2 6 , A = 1 5 0 , 0 0 0 f t 2.
18.
19. N o t n e e d e d
20. (5 .19 )
F r o m F i g u r e 5 . 3 2 , s l u r r y v i s c . / p u r e f r a c . f l u i d v i s c . = 2
S I u r r y v i s c . = ( 2 ) ( 5 0 0 ) = 1 0 0 0 c p .
Q u C r = ( 2 5 ) ( 1 0 0 0 ) ( 2 1 8 ) = 5 , 4 5 0 , 0 0 0
F r o m F i g u r e 5 . 3 1 , W = 0 . 2 8 5 i n .
Second T r i a l
15. Guess W = 0.20
16.
17. From F i g u r e 5 . 2 6 , A = 1 6 5 , 0 0 0 f t 2 .
18.
19. Not needed.
20. V s = 4 4 1 f t 3 .
C s = ( 4 4 1 ) ( 1 2 ) / ( 0 . 2 8 1 ) ( 1 6 5 , 0 0 0 )
C s = 0 .114
F r o m F i g u r e 5 . 3 2 s l u r r y v i s c / p u r e f r a c . f l u i d v i s c . = 1 . 7 .
New Concepts in Underground Storage of Natural Gas
-190-
S l u r r y v i s c . = ( 1 . 7 ) ( 5 0 0 ) = 8 5 0 c p .
Q u C r = ( 2 5 ) ( 8 5 0 ) ( 2 2 9 ) = 4 . 8 7 x 1 0 6
F r o m F i g u r e 5 . 3 1 , W ' = 0 . 2 8 0 i n . ( a s s u m e d s a t i s f a c t o r y w i t h i n
t h e a c c u r a c y o f g r a p h s ) . ( assumed co r rec ted w id th was 0 .281 )
21. F l u i d c o s t f r o m T a b l e 5 . 1 2 ; a s s u m e f l u i d i s o f a v e r a g e c o s t
= 0 . 0 4 c e n t s / g a l .
C o s t = ( 0 . 0 4 ) ( 6 3 , 0 0 0 g a l . ) + ( 8 6 3 6 ) ( $ 1 . 0 0 / h p )
Cost = 2520 + 863
Cos t = $3383 .00
Impermeation of Underground Formations
-191-
Nomencla ture
A =
a =
C =
CI =
CII =
C I I I =
CF =
CR =
E =
EY =
H =
h =
HHP =
K =
L =
m =
NRE =
=
Pf =
Pg =
Ph =
t o t a l a r e a o f o n e f a c e o f f r a c t u r e , f t 2 .
a r e a o f f i l t e r m e d i u m , c m 2 .
a c o n s t a n t w h i c h i s a m e a s u r e o f t h e f l o w r e s i s t a n c e o f
t h e f l u i d l e a k i n g o f f i n t o t h e f o r m a t i o n d u r i n g f r a c t u r e
t r e a t m e n t ,
c o n s t a n t C f o r f r a c t u r i n g f l u i d v i s c o s i t y a n d r e l a t i v e
p e r m e a b i l i t y e f f e c t ,
c o n s t a n t C f o r r e s e r v o i r f l u i d v i s c o s i t y a n d c o m p r e s s i -
b i l i t y e f f e c t ,
c o n s t a n t C f o r f l u i d l o s s a d d i t i v e s e f f e c t ,
c o m p r e s s i b i l i t y o f r e s e r v o i r f l u i d , 1 / p s i .
r a d i u s o f a h o r i z o n t a l f r a c t u r e , f t .
e f f i c i e n c y , t h e v o l u m e o f f r a c t u r e c r e a t e d e x p r e s s e d a s
a f u n c t i o n o f t h e v o l u m e o f f l u i d p u m p e d .
Y o u n g ' s m o d u l u s o f f o r m a t i o n r o c k , p s i ,
h e i g h t o f a v e r t i c a l f r a c t u r e , f t .
d e p t h o f f o r m a t i o n , f t .
h y d r a u l i c h o r s e p o w e r r e q u i r e m e n t , h o r s e p o w e r .
p e r m e a b i l i t y o f f o r m a t i o n t o f r a c t u r i n g f l u i d , d a r c y s .
l e n g t h o f a v e r t i c a l f r a c t u r e m e a s u r e d f r o m t h e w e l l b o r e ,
f - t .
s l o p e o f e x p e r i m e n t a l f l u i d l o s s l i n e w h e n c m 3f l u i d l o s s
i s p l o t t e d v e r s u sReyno ld ' s number .
d i f f e r e n c e i n p r e s s u r e b e t w e e n t h e f l u i d a t t h e f o r m a t i o n
f a c e a n d t h e f l u i d i n t h e f o r m a t i o n , p s i .
f l u i d f r i c t i o n l o s s i n p i p e , p s i .
f o r m a t i o n p r e s s u r e , p s i g .
h y d r o s t a t i c f l u i d h e a d o f f r a c t u r i n g f l u i d i n p i p e , p s i g .
New Concepts in Underground Storage of Natural Gas
-192-
N o m e n c l a t u r e , c o n t i n u e d
pr =
ps =
Q , Q i , V i =
S p G r =
t =
v =
vs =
vSp =
w =
W’ =
X =
u , u F =
uR =
=
p =
h y d r a u l i c p r e s s u r e r e q u i r e d t o m a i n t a i n f r a c t u r e p a r t -
i n g , p s i g .
s u r f a c e p u m p p r e s s u r e , p s i g .
c o n s t a n t i n j e c t i o n r a t e d u r i n g t r e a t m e n t , b b l / m i n , f t 3 / m i n
b b l / m i n .
s p e c i f i c g r a v i t y o f f r a c t u r i n g f l u i d , d i m e n s i o n l e s s .
t o t a l p u m p i n g t i m e , m i n .
v o l u m e o f f r a c t u r i n g f l u i d p u m p e d , f t 3 .
v o l u m e o f s a n d i n j e c t e d i n t o f r a c t u r e , f t3 .
s p u r t l o s s i n f l u i d l o s s t e s t , c m 3 .
f r a c t u r e w i d t h , f t . o r i n .
c o r r e c t e d f r a c t u r e w i d t h , f t . o r i n .
v i s c o s i t y o f f r a c t u r i n g f l u i d , c p .
v i s c o s i t y o f r e s e r v o i r f l u i d , c p .
p o r o s i t y o f f o r m a t i o n .
d e n s i t y o f w a t e r , l b m . / f t 3 .
Impermeation of Underground Formations
-193-
T a b l e 5 . 1 0
Estimates of Young's M o d u l i o f F o r m a t i o n R o c k s5 . 1 2 *
T a b l e 5 . 1 1
I n j e c t i o n R a t e s ( R e c o m m e n d e d ) f o r V a r i o u s T u b i n g S i z e s5 . 1 4
* S u p e r s c r i p t s r e f e r t o r e f e r e n c e f r o m w h i c h g r a p h o r t a b l e w a st a k e n .
New Concepts in Underground Storage of Natural Gas
-194-
T a b l e 5 . 1 2
F r a c t u r i n g - t r e a t m e n t C o s t C o m p a r i s o n s f o r F l u i d s o f V a r i o u s
F r a c t u r i n g - f l u i d C o e f f i c i e n t s P u m p e d a t D i f f e r e n t I n j e c t i o n R a t e s5 . 2 4
* A s s u m e $ 0 . 0 1 fo r f l u i d w i t h C = 10 x 10-3
$ 0 . 0 3 C = 5 x 1 0 - 3
$ 0 . 0 5 C = 10 x 1 0 - 3
+ 3 , 0 0 0 p s i s u r f a c e p r e s s u r e a n d $ 1 . 0 0 p e r h y d r a u l i c h o r s e p o w e r .
+ + F r a c t u r i n g - f l u i d c o s t .
+ + + A v e r a g e w i d t h o f 0 . 1 i n .
Impermeation of Underground Formations
-195-
L a b o r a t o r y F r a c t u r e - G r o u t E x p e r i m e n t s
I n o r d e r t o e x p l o r e t h e p r a c t i c a l f e a s i b i l i t y a n d e f f e c t i v e -
ness o f g rou t s wh i ch may be used t o impe rmea te f r ac tu res , a number
o f e x p e r i m e n t s h a v e b e e n c o n d u c t e d i n t h e l a b o r a t o r y . T h e s e e x p e r i -
m e n t s l a r g e l y c o n s i s t e d o f p r o d u c i n g h o r i z o n t a l a n d v e r t i c a l f r a c -
t u r e s o n c o r e s a m p l e s i n t h e l a b o r a t o r y a n d o f s u b s e q u e n t l y g r o u t i n g
t h e f r a c t u r e d a r e a s . T h e e v a l u a t i o n o f t h e e f f e c t i v e n e s s o f g r o u t -
i n g w a s m a d e i n t e r m s o f t h e t h r e s h o l d p r e s s u r e s m e a s u r e d a f t e r t h e
g r o u t h a d g e l l e d a n d c u r e d .
A x i a l f r a c t u r e s w e r e o b t a i n e d s u c c e s s f u l l y b y a p p l i c a t i o n
o f l o c a l i z e d v e r t i c a l c o m p r e s s i v e l o a d s o n c o r e s u s i n g a w e d g e a n d
a b e n c h v i s e a s a f o r c e m u l t i p l i e r . T h e l a y o u t o f t h e m e c h a n i c a l
s y s t e m ' u s e d t o f r a c t u r e t h e c o r e i s s h o w n i n F i g u r e 5 . 3 6 . The
p e r m p l u g u s e d w a s 2 . 5 i n c h e s l o n g a n d h a d a d i a m e t e r o f 1 . 5 . i n .
A 1 / 8 ” d e e p c u t i s i n t r o d u c e d i n t h e a x i a l d i r e c t i o n o n t h e
s i d e s o f t h e c y l i n d r i c a l s p e c i m e n s t o d i r e c t t h e f r a c t u r e . The
w e d g e w a s p l a c e d i n t h e c u t a n d t h e s e c o n d p l a t e ( b e a r i n g ) w a s
p l a c e d o p p o s i t e . F o r c e w a s a p p l i e d a n d t h e a x i a l f r a c t u r e w a s
p r o d u c e d i n a n a p p r o x i m a t e l y v e r t i c a l p l a n e . T h e m e t h o d o f j o i n i n g
a n d s e a l i n g t h e t w o s e c t i o n s t o g e t h e r i s s h o w n i n F i g . 5 . 3 7 . C u r -
i n g t i m e f o r t h e e p o x y w a s 7 2 h o u r s . The epoxy was then m i l l ed
t o f i t t h e c o r e h o l d e r . T h e t e s t s e c t i o n w a s s a t u r a t e d w i t h w a t e r
a n d g r o u t i n j e c t e d t h r o u g h F a c e N o . 1 . Th resho ld p ressu re was f ound
a f t e r 1 4 4 h r . T h e a n t i c i p a t e d d i s p e r s i o n p a t t e r n i s s h o w n a t t h e r i g h t .
I t w a s a n t i c i p a t e d t h a t d i s p e r s i o n i n r e g i o n B s h o u l d b e g r e a t e r
t h a n t h a t i n r e g i o n A b e c a u s e r e g i o n B r e p r e s e n t s t h e " p a t h o f l e a s t
r e s i s t a n c e " ' c o t h e f r e e f a c e ( F a c e N o . 2 ) . T h i s m a y b e c o n t r o l l e d
t o s o m e e x t e n t b y i n c r e a s i n g t h e w i d t h o f t h e s e a l o n t h a t f a c e .
New Concepts in Underground Storage of Natural Gas
-196-
F i g . 5 . 3 6 T e c h n i q u e U s e d f o r O b t a i n i n g V e r t i c a l F r a c t u r e s
W h e n t h e g r o u t w a s i n j e c t e d f r o m t h e t o p w i t h t h e c o r e h a v -
i n g i t s s u r f a c e s e a l e d b y e p o x y i n t h e f r a c t u r e d r e g i o n t h e s o l u -
t i o n f i r s t f i l l e d t h e f r a c t u r e t h e n d i s p e r s e d s o m e w h a t u n e v e n l y
t h r o u g h t h e p o r o u s m a t r i x i n a p l a n e p e r p e n d i c u l a r t o t h e f r a c t u r e .
A n u m b e r o f s y n t h e t i c c o r e s w e r e c a s t w i t h a h o l e a t t h e c e n -
t e r s i m u l a t i n g t h e w e l l b o r e i n a p o r o u s p e r m e a b l e f o r m a t i o n . The
t e c h n i q u e u s e d i n t h e p r e p a r a t i o n o f t h e s e c o r e s i s i l l u s t r a t e d i n
F i g u r e 5 . 3 8 . S o m e o f t h e s e c o r e s w e r e f r a c t u r e d h o r i z o n t a l l y a r o u n d
t h e w e l l b o r e . T h e F i g u r e 5 . 3 9 s h o w s a n e x p e r i m e n t s i m u l a t i n g t h e
g r o u t i n g o f a n h o r i z o n t a l f r a c t u r e a r o u n d t h e w e l l b o r e . A s m a l l
n i p p l e o f p r e s s u r e t u b i n g w a s i n s e r t e d t h r o u g h t h e h o l e t o t h e f r a c -
t u r e p l a n e a n d c e m e n t e d i n p l a c e w i t h e p o x y . W i t h t h e t w o h o r i z o n -
t a l l y f r a c t u r e d s e c t i o n s h e l d t o g e t h e r w i t h a n e p o x y c o l l a r a t t h e
o u t e r f a c e t h e g r o u t w a s i n j e c t e d t h r o u g h t h e t u b i n g a n d f o r c e d t o
d i s p e r s e t h r o u g h t h e p o r o u s m a t r i x a s s h o w n i n F i g u r e 5 . 3 9 . Samples
o f d a t a a n d o b s e r v a t i o n s f r o m e x p e r i m e n t s w i t h a x i a l a n d r a d i a l
f r a c t u r e s a r e i n c l u d e d i n t h e f o l l o w i n g . G r o u t e v a l u a t i o n d a t a f r o m
l a b o r a t o r y f r a c t u r e d c o r e s a m p l e s w e r e s u m m a r i z e d e a r l i e r i n T a b l e
5 . 8 .
Imp
erm
ea
tion
o
f U
nd
erg
rou
nd
F
orm
atio
ns
-19
7-
Ne
w
Co
nce
pts
in
Un
de
rgro
un
d
Sto
rag
e
of
Na
tura
l G
as
-19
8-
Imp
erm
ea
tion
o
f U
nd
erg
rou
nd
F
orm
atio
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-19
9-
New Concepts in Underground Storage of Natural Gas
-200-
A x i a l F r a c t u r e D a t a
GROUT USED: SIROC (No. 1-45%, No. 2 -5$ , No . 3 -5$ , H 2 O-45%)
CURING TIME: 5 days a t 100% humid i t y
COMMENTS: F a c e 1 w a s g r o u n d d o w n t o r e m o v e a l l e p o x y p r i o r t o t e s t -
i n g . G a s w a s f o r c e d t o e n t e r t h r o u g h f a c e N o . 2 t o p r e v e n t
a n y d i r e c t b l e e d i n g t h r o u g h t h e f r a c t u r e . The obse rved i nc rease
o f t h r e s h o l d p r e s s u r e i s t h e r e s u l t o f a c t u a l p e n e t r a t i o n o f g r o u t
i n t o t h e s p e c i m e n .
CORE X l 3 3
Vo id Vo lume = 19 .148 cm 3 , Bu l k Vo lume = 74 cm 3 , Po ros i t y = 25 .7%
Permeab i l i t y = 3500 md .
T h r e s h o l d P r e s s u r e = . 8 6 9 p s i g ( b e f o r e g r o u t i n g )
T h r e s h o l d P r e s s u r e = 3 . 6 0 p s i g ( a f t e r g r o u t i n g )
G r o u t I n j e c t i o n R a t e = 3 . 0 8 c m 3 / m i n .
W a t e r D i s p l a c e d = 4 2 . 5 m l .
CORE X134
Void Volume = 17.904 cm 3 , B u l k V o l u m e = 6 9 . 7 c m 3 , P o r o s i t y =
25.68%
Permeab i l i t y = 1650 md .
T h r e s h o l d P r e s s u r e = . 9 6 4 p s i g . ( b e f o r e g r o u t i n g )
T h r e s h o l d P r e s s u r e = 4 . 5 0 0 p s i g . ( a f t e r g r o u t i n g )
G r o u t I n j e c t i o n R a t e = . 9 0 9 c m 3 / m i n .
W a t e r D i s p l a c e d = 3 2 . 3 m l .
CONCLUSIONS:
G r o u t i n g e f f i c i e n c y i s a p p r o x i m a t e l y t h e s a m e a s f o r i n j e c t i o n
th rough one end o f a comparab le samp le . T h e s l i g h t d e c r e a s e s i n
i n c r e a s e i n t h r e s h o l d p r e s s u r e m a y b e a t t r i b u t a b l e t o f i n g e r i n g i n
r e g i o n B ( F i g . 5 . 3 7 ) , a s a n t i c i p a t e d . F r o m t h e r e s u l t s , h o w e v e r ,
o n e m a y c o n c l u d e t h a t t h e c r o s s - s e c t i o n ( a t l e a s t i n r e g i o n B ) h a s
b e e n t o t a l l y g r o u t e d .
Impermeation of Underground Formations
-201-
R a t e o f i n j e c t i o n a n d t h e t o t a l v o l u m e o f f l u i d s e e m t o b e
o f l i t t l e i m p o r t a n c e . A p p r o x i m a t e l y t w o p o r e v o l u m e s o f f l u i d w e r e
c o l l e c t e d i n e a c h c a s e .
R a d i a l F r a c t u r e D a t a
NOTE: B e c a u s e o f t h e p h y s i c a l c o n f i g u r a t i o n s o f t h e s a m p l e s , d e t e r -
m i n a t i o n o f t h e b a s i c p r o p e r t i e s i s m a d e d i f f i c u l t u s i n g t h e a x i a l
methods. R e p r e s e n t a t i v e v a l u e s f r o m o t h e r t e s t s e c t i o n s p r e p a r e d
u n d e r s i m i l a r c o n d i t i o n s m a y b e u s e d f o r a r o u g h e s t i m a t e o f t h e
i n c r e a s e i n t h r e s h o l d p r e s s u r e w h e n g r o u t i s a p p l i e d r a d i a l l y .
Spec imen Z3
T h i s s p e c i m e n w a s m a d e t o r e p r e s e n t a c a v i t y o r l a r g e d i s c o n -
t i n u o u s f r a c t u r e b y c h o p p i n g m a t e r i a l f r o m t h e f r a c t u r e f a c e s t o
f o r m a s m a l l p o c k e t . T h e s e c t i o n s w e r e j o i n e d b y a n e p o x y c o l l a r
and t r ea ted as above . B e f o r e t e s t i n g P t h t h e t u b e w a s c u t o f f l e v e l
w i t h t h e t o p s u r f a c e a n d s e a l e d .
P t h = 1 5 0 p s i g
Spec imen Z4
Th i s spec imen w a s t r e a t e d a s a b o v e b u t w i t h o u t i n t r o d u c i n g
a c a v i t y .
P th = 55 psig
CONCLUSIONS
T h e s e r e s u l t s i n d i c a t e t h a t S I R O C d o e s s e a l c a v i t y f r a c t u r e s
v e r y w e l l a n d t h a t i t c o u l d b e o f g r e a t i m p o r t a n c e i n s c a l i n g h o r -
i z o n t a l f r a c t u r e s i n s t r a t a c o v e r i n g g a s s t o r a g e b e d s .
New Concepts in Underground Storage of Natural Gas
-202-
The compos i t i on o f S IROC ' used i n t hese expe r imen ts was as
f o l l o w s :
Impermeation of Underground Formations
-203-
References
5 . 1
5 . 2
5 . 3
5 . 4
5 . 5
5 . 6
5 . 7
5 . 8
5 . 9
5 .10
5 .11
5 .12
5 .13
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H o w a r d , G . C . , a n d F a s t , C . R . , Op t imum F lu i d Cha rac te r i s t i c sf o r F r a c t u r e E x t e n s i o n , A P I D r i l l i n g a n d P r o d u c t i o n P r a c t i c e ,1957.
H o w a r d , G . C . , a n d F a s t , C . R . , F a c t o r s C o n t r o l l i n g F r a c t u r eE x t e n s i o n , C o n . M i n . a n d M e t . B u l . , V o l . 5 4 , N o . 5 8 6 ,F e b r u a r y , 1 9 6 1 .
H u n t , D . D . , a n d C r a w f o r d , H . R . , H y d r a u l i c F r a c t u r e D e s i g n ,ASME Paper No. 5 9 - P E T - 4 5 , f o r m e e t i n g S e p t . 2 0 - 2 3 , 1 9 5 9 .
P e r k i n s , J . K . , a n d K e r n , L . R . , W i d t h s o f H y d r a u l i c F r a c t u r e s ,J o u r n a l o f P e t r o l e u m T e c h n o l o g y , V o l . 1 3 , N o . 9 , S e p t e m b e r1961.
C h r i s t i a n o v i t c h , S . A . , Z h e t o v , Y . P . , B a r e n b l a t t , G . I . , a n dM a x i m o v i t c h , G . K . , T h e o r e t i c a l P r i n c i p l e s o f H y d r a u l i c F r a c -t u r i n g o f O i l S t r a t a , P r o c e e d i n g s o f t h e F i f t h W o r l d P e t -r o l e u m C o n g r e s s , S e c t i o n I I , N e w Y o r k , 1 9 5 9 .
New Concepts in Underground Storage of Natural Gas
5 .14
5 .15
5 .16
5 .17
5 .18
5 .19
5 .20
5 .21
5 .22
5 .23
5 .24
5 .25
5 .26
-204-
E s s a r y , R . L . , F rac tu re T rea tmen ts i n S .E . New Mex i co , Wor l dO i l , V o l . 1 5 4 , N o . 4 , M a r c h 1 9 6 2 .
Rev iew A r t i c l e : New F rac tu re -Ac id Me thod Looks Good , O i l andG a s J o u r n a l , V o l . 5 7 , N o . 2 3 , J u n e 1 , 1 9 5 9 .
A lexande r , B .G . , and Wagne r , J .M . , Wa lnu t She l l s G i ve GoodA s s i s t i n M e r a m e c F r a c t u r e s , O i l a n d G a s J o u r n a l , V o l . 6 0 ,No . 43 , Oc tobe r 22 , 1962 .
M c G l o t h l i n , H u i t t , J . L . , a n d J e n n i n g s , J . W . , F l u i d a n dP r o p p i n g - A g e n t I n j e c t i o n S c h e d u l e f o r H i g h C a p a c i t y F r a c t u r e s ,O i l a n d G a s J o u r n a l , V o l . 5 9 , N o . 3 8 , S e p t e m b e r 1 8 , 1 9 6 1 .
K a s t r o p , J . E . , New Fracture Propping Process Uses AluminumP e l l e t s , P e t r o l e u m E n g i n e e r , V o l . 3 2 , N o . 1 2 , N o v e m b e r 1 9 6 0 .
H u i t t , J . L . , M c G l o t h l i n , B . B . , a n d M c D o n a l d , J . F . , E v a l u a t i o na n d S e l e c t i o n o f F r a c t u r e P r o p p i n g A g e n t s , P e t r o l e u m E n g i n e e r ,V o l . 3 1 , J u n e 1 9 5 9 .
K e r n , L . R . , W y a n t , R . E . , a n d P e r k i n s , T . K . , P r o p p i n g F r a c t u r e sw i th A lum inum Pa r t i c l es , A IME Pape r No . 1573G, p repa red f o rm e e t i n g O c t o b e r 2 - 5 , 1 9 6 0 .
Romero -Jua rez , A . , Sand Concen t ra t i on f o r Max imum F rac tu reF low Capac i t y , A IME Paper No . 1574G, p repa red f o r mee t i ngOc tobe r 2 - 5 , 1 9 6 0 .
R i n e h a r t , J . S . , a n d M a u r e r , W . C . , F r a c t u r e s a n d C r a t e r s P r o -d u c e d i n S a n d s t o n e b y H i g h - V e l o c i t y P r o j e c t i l e s , A I M E P a p e rNo. 1534G, p repa red f o r mee t i ng Oc tobe r 2 - 5 , 1960 .
S e t s e r , D . D . , I m p l o s i o n T e c h n i q u e I m p r o v e s F r a c t u r i n g P e r f o r -mance, W o r l d O i l , V o l . 1 5 0 , N o . 4 , M a r c h 1 9 6 0 .
S w i f t , V . N . , B a u m a n , W . E . , J e n n i n g s , J . W . , a n d H u i t t , J . L . ,S o m e R e s u l t s o f F r a c t u r i n g w i t h t h e S i n g l e - P o i n t E n t r y T e c h -n ique, AIME Paper No. 1570G, p repa red f o r mee t i ng Oc tobe r2 - 5 , 1 9 6 0 .
B rown , R .W. , and Lope r , R .G . , S t i m u l a t i o n T r e a t m e n t S e l e c t -i v i t y t h r o u g h P e r f o r a t i o n B a l l S e a l e r T e c h n o l o g y , P e t r o l e u mE n g i n e e r , V o l . 3 1 , N o . 6 , J u n e 1 9 5 9 .
S h e l l , F . J . , a n d B o d i n e , O . K . , E c o n o m i c s o f H y d r a u l i c F r a c t -u r i n g U s i n g W a l l - b u i l d i n g A d d i t i v e s , A P I D r i l l i n g a n d P r o d u c -t i o n P r a c t i c e , 1 9 6 0 .
Impermeation of Underground Formations
-205-
5 .27
5 . 2 8 *
5 .29
5 .30
5 .31
B l e a k l e y , W . B . , F l o w C h a r t s P i n p o i n t P r e s s u r e L o s s e s o fF r a c t u r i n g F l u i d s , O i l a n d G a s J o u r n a l , V o l . 6 0 , N o . 4 2 ,Oc tobe r 15 , 1962 .
C a r s l a w , H . S . , a n d J a e g e r , J . C . , C o n d u c t i o n o f H e a t i n S o l i d s ,C la rendon P ress , London , 1960 .
R i c h t m y e r , R . D . , D i f f e r e n c e M e t h o d s f o r I n i t i a l V a l u e P r o b l e m s ,I n t e r s c i e n c e , N e w Y o r k , 1 9 5 7 .
J a e g e r , J . C . , N u m e r i c a l V a l u e s f o r t h e T e m p e r a t u r e i n R a d i a lH e a t F l o w , J o u r n a l o f M a t h e m a t i c s a n d P h y s i c s , V o l . 3 4 ,p . 3 1 6 , 1 9 5 6 .
A r d e n , B . , G a l l e r , B . , a n d G r a h a m , R . , T h e M i c h i g a n A l g o r i t h mDecode r , C o m p u t i n g C e n t e r , U n i v . o f M i c h i g a n , A n n A r b o r , 1 9 6 3 .
* Re fe rences 5 .28 - 5 . 3 1 r e f e r t o c i t a t i o n s i n A p p e n d i x A .
This Page Intentionally Left Blank
CHAPTER 6
UNDERGROUND STORAGE IN NON-POROUS SPACE
The problems of storage for manufactured or produced natural gas
remained closely related through the years in scope and magnitude to
the development and growth of gas industry. This pattern has followed
the sequence from storage in surface tanks, to underground storage in
depleted gas or oil reservoirs and, more recently, to storage in aqui-
fe rs . For reasons of capacity, economy and safety, storage in surface
tanks is now completely out of the picture. On the other hand, wherever
possible, depleted natural gas producing reservoirs offer the most simple
and reliable media for underground storage of natural gas. The depleted
gas reservoir not only has the distinct advantage of having been proved
by the nature throughout the geologic ages to retain the gas in-place
but has, on the average, fairly high capacity. Such subsurface forma-
tions also have the added feature of being well known, geologically and
structurally, well studied and documented during their producing life
from the viewpoints of geology, geophysics, reservoir engineering, rock
mechanics. For instance, the cap rock becomes well known as to its
nature, porosity, permeabil i ty, threshold pressure, water saturat ion,
thickness, susceptibi l i ty to fracturing during dri l l ing and coring opera-
t ions. The storage sand, on the other hand, becomes known as to its
porosity, permeabil i ty, connate water saturation, water drive, homogene-
i t y , inc ip ient f rac tures , fau l ts , e tc . The response to particular well
completion and that of the formation with respect to particular forma-
tion stimulation becomes ascertained. The optimum method of well bore
cleaning with respect to salt or other mineral deposits becomes deter-
mined. The cost of drill ing and well completion, the life expectancy
of well with respect to corrosion, the capacity of wells become known.
As all this experience and knowledge are constantly added, reviewed and
-207-
New Concepts in Underground Storage of Natural Gas
-208-
replenished the prospective storage reservoir becomes a well known
engineering entity subject and amenable to fairly precise economic
evaluation. When the field is reverted to storage service, the prac-
tice of “overpressuring ” within safe limits usually adds appreciably
more storage capacity to the depleted gas producing field.
The depleted oil reservoir affords approximately the same advan-
tages when viewed as a prospect for underground storage. Existence of
rel iable cap, su i tab le s t ruc tura l t rap, reservoir properties determined
from data collected throughout the producing life of the reservoir,
estimation of original in-place and actually recovered reserves, thus
of the available pore space for the storage of natural gas, all result
from studies stemming from production and engineering of such reser-
vo i rs . Along with these conveniences, however, storage in depleted oil
reservoirs pose some rather difficult and unique engineering problems
stemming from the presence of unrecovered residual crude oil in the pore
space. The miscibi l i ty of residual oi l with injected gas, low relat ive
permeability due to two phase flow, handling of the production of oil
along with gas, dif f icult ies in reconci l ing the inventory gas with pro-
duction pressure behavior of total reserves are typical of problems
which must be dealt with in storing the natural gas in depleted oil
reservoirs.
During the last decade the phenomenal growth of domestic consump-
tion of natural gas precipitated the acute need for storage of natural
gas in areas where there are no depleted oil or gas reservoirs present.
The advent of aquifer storage where the pore volume necessary to store
the gas is created by pressurizing aquifers above their discovery pres-
sure through gas injection provided during the last decade very sub-
stantial storage reservoirs in such areas of need.6 . 4 0
The storage of natural gas in aquifers depends upon existence of
suitable structure, closure, and cap rock. These factors, though per-
haps more readily or more frequently available along with suitable
porosity and permeability close to areas of high domestic gas consump-
tion than producing or depleted gas or oil reservoirs, are not always
Underground Storage in Non-Porous Space
-209-
altogether present. Quite often an aquifer sand of high porosity and
permeability will exist at some reasonable depth but will not have
adequate structural closure. Sometimes, on the other hand, such a
structure will have adequate closure, but unfortunately no suitable
cap rock. Other times, impervious cap rock, good porosity, high perme-
ability will be present but only a semi-open structure will delimit the
subsurface geology.
There has been a substantial amount of field data gathered re-
cently indicating that cap rocks overlying many storage reservoirs are
subject to leakage of gas to shallower formations when the gas pressure
exceeds a certain critical value. Such leaks are sometimes area dis-
tributed and related to drainage capillary pressure characteristics of
the shale, At other times, it appears that the leak may be along a
fracture line and primarily due to having the gas overpressure in ex-
cess of the threshold pressure of the shale.
The possibility and desirability of storage in aquifer sands
where one or more of the prime requirements is nonexistent, or storage
in depleted petroleum reservoirs above the, original content of hydro-
carbons, storage in aquifers below levels and beyond limiting bound-
aries indicated by areas of minimum structural closure, all point to
the urgent need for new concepts in underground storage where either
soil impermeation or storage in new media through novel engineering
methods should play a significant role.
There are many areas in the United States6 . 3 9
where sedimentary
rocks are known to be practically nonexistent. As shown in Fig. 6.1
some of these areas are located near industrial complexes in the den-
sely populated areas where underground storage must depend on relatively
or entirely new concept ideas.
Subsurface storage of natural gas in non-porous, void continua
has been suggested and tried to a limited extent in the past. Near
the surface, porous storage in shallow sand beds, gravel pits, sand
dunes and surface sands have been suggested at one time or another in
New Concepts in Underground Storage of Natural Gas
-210-
the past as possible new media for storage of natural gas. Of these
various ideas and possibilities the storage in dissolved salt caverns,
in mined caverns, storage in cavities induced by underground nuclear
explosions and storage in containers held at the bottom of oceans will
be discussed. It is hoped that through documentation of data, theory
and pertinent engineering considerations on hand a starting point will
be provided for those who may consider a more precise economic and
engineering evaluation of these media for peak shaving or large scale
storage service.
6.1 Storage of Natural Gas in Salt Caverns
The use of salt caverns for underground storage of liquified
petroleum gases such as propane, butane, etc., has been in practice
for over ten years. As early as 1952 the Natural Gasoline Association
of America prepared standards for operating and testing such under-
ground storage wells.6 . 2 8
Considerat ions related to location, creation, operation, size,
capacity and deliverability of salt cavern storage will be discussed
in the fol lowing. Considerations of stress and safety and a case study
of the only existing commercial salt cavern storage reservoir in
Michigan are included.
Locations for Dissolved Salt Caverns
Underground salt strata occur in the form of salt beds of salt
domes. Salt beds are geologically known to occur in the western part
of Texas, Oklahoma, Kansas, eastern New Mexico and in the Great Lakes
region of Michigan, Ohio and Canada. Salt beds also occur in
Pennsylvania, New York and in Uintah Basis in Utah and finally in
Colorado, These salt beds outcrop at the surface in some areas while
they may be found at depths up to 7000 feet in others. In the Texas-
Oklahoma-Kansas-New Mexico area the salt beds range in depth from 1000
Underground Storage in Non-Porous Space
-211-
Fig. 6.1 Favorable and Unfavorable Areas For Productionand Storage of Natural Gas
New Concepts in Underground Storage of Natural Gas
-212-
to 2000 feet and in thickness from 50 to 100 feet. In the Great Lakes
region, the beds range in depth from 1500 to 7000 feet and in thickness
from 1 to 400 feet. Almost all of the beds contain thin layers of
shale, anhydrite, etc. This causes some problems in the formation of
cav i t ies . Salt domes are for the most part located along the Gulf Coast
of Texas, Louisiana, Mississippi, and Alabama. These domes vary in
depth from near surface to depths that lie beyond the reach of present
day drilling equipment. The dome usually consists of nearly pure rock
s a l t . Salt domes are usually the best formations in which to provide
underground storage.6 . 3 3
Creation of Salt Cavern Reservoirs
The process of washing out a cavern in rock salt is inexpensive
(from 19 cents to $1.80 per bbl) and quite simple. A shaf t is dr i l led
into a subterranean salt stratum and the salt is dissolved and brought
to the surface by pumping in fresh water and pumping out the brine,
leaving an opening of the desired size within the stratum.6 . 7
In theory,
every 6.03 bbl of fresh water will dissolve one bbl of salt. In prac-
tice, about 11% capacity is experienced with each unit volume of wash-
water. It has been observed that in about seven wash volume turnovers,
the cavern size doubles, in 11 it triples.6 . 5
Salt cavities should be formed in an area free from shale ledges
i f poss ib le . I f present, these ledges may break off and kink or snap
any brine or product tubes that might be located in the cavity. In
some operations, the amount of insolubles may be excessive and tubing
may have to be raised from the bottom of the cavity to prevent plugging.
There are three methods of developing the salt cavity (Fig. 6.2)
a) bottom injection; b) reverse circulation, and c) progression tech-
nique.6 . 2 1
When creating caverns in salt layers, fracturing may be employed
to faci l i tate the cavern construct ion. Two or more wells may be sunk
Underground Storage in Non-Porous Space
-213-
and connected by fracturing. The bed may then be washed out to provide
a large storage area. Fracturing probably cannot be used in salt domes
because the general homogeneity of physical properties of salt may not
lend i tself to control led horizontal fracturing. Figure 6.2 shows
graphically three distinct methods of developing salt cavities of con-
trolled shape.
Determination of the Size of Dissolved Salt Caverns
After a cavern has been washed, accurate dimensions of the caverns
are usually desired. If the cavern shape is assumed to be without
stringers of insoluble material and that it approximates a cylinder,
then volume height data will fix the diameter.6 . 1 5
This volume height
relationship can be obtained by measuring the difference in shut-in
annulus pressures of the liquefied hydrocarbons at the surface after
successive additions of known volumes to the cavern. The rise in pres-
sure together with the calculated base pressure required on the annulus
to balance the brine column from the bottom of the casing permits the
calculation of an equivalent diameter at various test points. Referring
to Figure 6.3,6 . 1 6
i t wi l l be recognized that
(6.1)
(6.2)
where h = depth to the LP gas brine contact, feet
P = casing pressure, psig
dB = specif ic gravity of brine,
Ne
w C
on
cep
ts inU
nderg
round
Sto
rage
of
Natu
ral
Gas
-21
4-
Underground Storage in Non-Porous Space
-215-
FIG. 6.3 A SIMPLIFIED VIEWOF A SALT STORAGE WELL.
FIG. 6.4 AVERAGE CAVERNDIAMETERS AS DETERMINED FOREQUILIBRIUM PRESSURE CALCULATIONS.
New Concepts in Underground Storage of Natural Gas
-216-
Deliverability of Natural Gas from Salt Cavern Storage
When natural gas is stored in a salt cavern, the stress condition
induced in formations surrounding the dissolved cavity depends upon
mechanical characteristics of these formations, weight of the overbur-
den, shape of the cavity and the pressure of the gas inside the cavity.
In the early phases of development of salt cavern storage, because
of uncertainty of stress calculat ions, it was felt that one would prob-
ably have to maintain full hydrostatic pressure at the salt cavern in
order to prevent collapse due to the weight of the overburden. To make
this possible, saturated brine would have to be pumped in and out during
each production-injection phase of the storage. In other words, a dis-
solved salt cavern 2000 feet deep would be maintained say at 860 psig
al l the t ime. This cavern pressure can be maintained if every cu. ft.
of gas pumped out is replaced by a cubic ft of brine pumped in and vice
versa. Accordingly, it becomes interesting from a storage engineering
viewpoint to determine the brine pumping requirement necessary to main-
tain cavern pressure at varying rates of gas deliverability. I f t he
water pumped into the cavern only partially replaces the gas being pro-
duced, then the cavern pressure drops as a function of time accordingly.
In the following example, a relat ionship wil l be derived for pre-
dicting the decline of cavern pressure versus time for various fractions
of brine replacing the gas withdrawn.
In order to fix the ideas, let us assume that: 1) cavern has a
volume of 109 ft3 and is located 2000 feet below the surface; 2) with-
drawal rate is 50 x 106 SCF/day; 3) cavity temperature is constant at
100oF.
QS= withdrawal rate = 50 x 106 SCF/day
Vc= cavity volume = 109 ft3
T = cavity temperature = 100oF
Underground Storage in Non-Porous Space
-217-
Ts, Zs, Ps= standard temperature, compressibility factor,
and pressure, respectively
To, Zo, Po= original temperature, compressibi l i ty factor,
and pressure, respectively
t = time in days after withdrawal has started
T i , Z i , P i = temperature, compressibi l i ty factor, and
pressure, respectively, at time t
VG, nG= volume and moles of gas, respectively, at time t
no = initial number of moles of gas
f * = fraction of gas withdrawn that is replaced by
water (corrected to actual reservoir conditions)
QL= water inject ion rate, f t3 /day
QS= gas production rate STD cu. ft/day
Material balance:
but
where
(6.3)
(6.4)
(6.5)
(6.6)
New Concepts in Underground Storage of Natural Gas
-218-
Substi tut ing (6.3) in (6.7) and solving for P i yields
This equation must be solved by trial and error since Zi is
(6.7)
(6.8)
a
function of P i and T i. Z values are functions of pseudo-reduced tem-
perature and pressure and are available in the literature.6 . 3 9
Equation (6.8) is presented graphically in Figure 6.5 for various
values of f* .
Stress Considerations
When the creation or enlargement of a salt cavern is considered,
it is important to know the cavern volume that can be washed out with-
out danger of cavern collapse. Induced stresses must also be considered
when the cavern pressure is reduced below formation pressure. Approx-
imate but rel iable correlat ions exist in the l i terature giving the in-
duced stresses as functions of geometry, rock properties and reservoir
variables. More specif ical ly, these correlations give induced stresses
from cavern dimensions, shape, pressure, depth, thickness of overlying
consolidated formations, and physical properties of the surrounding
formations. With such a correlation, one could create maximum cavern
volume without danger of collapse. In addit ion, one could calculate
the minimum storage pressure which would be permissible for a given dis-
solved salt cavern. Once the operating maximum and allowable minimum
Underground Storage in Non-Porous Space
-219-
FIG. 6.5 SALT CAVERN STORAGE VARIATION OF RESERVOIR PRESSURE WITHTIME FOR VARIOUS DEGREES OF WATER REPLACEMENT.
(Vc = 109 FT3, Qs = 50x106 SCF/DAY )
New Concepts in Underground Storage of Natural Gas
-220-
cavern pressures are determined then the deliverability of gas from the
cavern and the necessary pumping requirement for maintenance of cavern
pressure may be readily calculated.
In considering stress problems related to formation and operation
of salt caverns, it is important to know what radius, height and shape
a salt cavern should be leached to provide an adequate storage volume
while insuring that collapse will not occur. Two geometries, specifi-
ca l ly d i f ferent , one of spherical shape and another one of non-spherical
shape will be considered next.
Stresses Induced in Formations Surrounding a Spherical Cavity
Creep deformation in a salt cavity becomes significant when the
cavity depth exceeds 1000 feet. Whenever the maximum shearing stress
exceeds a certain value, the salt starts to creep rather than fracture.
Whenever present, addition of thermal stresses in general produce a
further extension of the plastic zone. The following equations have
been developed for calculation of radial and tangential stress compo-
nents when deformation is in the elastic zone:
where S t and S t are radial and tangential elastic stresses,
psig
p o= the overburden pressure on the cavity, psig
p i = internal pressure on the cavity
= equivalent yield strength of the salt , psig
r = radial distance, f t
(6.9)
(6.10)
Underground Storage in Non-Porous Space
-221-
a = cavity radius, f t
= plast ic front radius, f t .
At the boundary between plastic and elastic deformation zones
(6.11)
Substituting the values of Sr and St from equations (6.9) and (6.10)
into equation (6.11) gives
or
(6.12)
(6.13)
Now substituting the value of p from (6.13) into (6.9) and (6.10):
(6.14)
and
(6.15)
From equations 6.14 and 6.15, i t can be seen t ha t t he t angen t i a l s t r esses .
would be controlling.
It is interesting to note that the point of maximum stress does
not occur at the cavern wall but at a point inside the surrounding
formation. This is due to the transition which occurs between elastic
and plastic deformations. The equations giving stress in the plastic
zone are:
(6.16)
New Concepts in Underground Storage of Natural Gas
-222-
(6.17)
where denote respectively radial and tangential stresses,
p s i , p i the cavity pressure, psi, and the yield stress for the
material .
Simplified Stress Calculations for Non-Spherical Cavities in Salt
While the size and shape of cavities washed out in salt formations
vary a great deal, two typical shapes will be considered for stress cal-
culat ions in non-spherical cavit ies.
These are shown in Figure 6.6, a and b. Both cavities are over-
laid by anhydrite t feet thick located D feet from the surface. The
cavern shown in “a” somewhat conically tapers down from a radius r1 a t
the top immediately adjacent to anhydrite cap. The cavern shown in “b”
is in the form of an inverted cone with the largest radius r1 at the
bottom.
The stress calculations for non-spherical cavities are based on
shear loading of the anhydrite cap due to the weight of the overburden.6 . 6 4
The overburden weight usually results in a vertical downward pressure
force of 1.0 psi per foot of depth. The support in case 6.6a is provided
by the anhydrite while in case 6.6b anhydrite plus the salt above the
maximum diameter occurring at the bottom jointly contribute to the sup-
po r t .
In case a: shear stress SS = load/area in shear
(6.18)
(6.19)
Underground Storage in Non-Porous Space-223-
Fig. 6.6 Typical Shapes of Non-Spherical Cavities
New Concepts in Underground Storage of Natural Gas
-224-
In case b, the entire height (h) of the cavern above the maximum
diameter at the bottom contributes to the support of the cavern roof.
In this case Ss = load/area in shear
(6.20)
(6.20a)
Because the shear force is distributed over a larger area in case b,
it can be seen that the shape “b” is structural ly more stable than
shape “a.”
Using equations 6.19 and 6.20 and ultimate stress data for salt
and anhydrite and a suitable safety factor, size limitation for cavities
of each type can be computed.
Strength Data for Salt
A specific safe limit to cavern size is, at the present time,
virtual ly impossible to calculate. Inhomogeneity of rock compacted
through complex geological processes confronts the designer with such
variables as fracturing, slams, and faults. Some data for anhydrites
and salt formations are available from core samples. A typical sample
taken from salt formation at 2000 feet in West Texas, for instance, is
reported to yield the following data on unconfined compressive
strength.6 . 2 1
Sal t - 2600-4000 psi
Anhydr i t e - 4500-23,000 psi
Underground Storage in Non-Porous Space
-225-
Ultimate compressive strength when confined, was found to be:6 . 2 1
Sal t - 17,000 psi @ 2000 atm and 150°C.
Anhydr i t e - 82,000 psi @ 2000 atm and 150°C.
Triaxial tests on anhydrite under formation pressure conditions indicate
ultimate shear strength in the range 12,000 to 14,200 psi while measured
compressive strengths reached 28,000 psi.6 . 2 1
The following funda-
mental properties of the aggregate salt from the Grand Saline salt mine
were determined:6 . 3 6
The maximum compressive stress 2300 psi
with the standard deviation 200 psi
The 0.5% yielding stress 2000 psi
Young's Modulus 0.14x106 psi
with the standard deviation .03x106 psi
Poisson's rat io
with the compressive stress up to 300 psi: 0.25-0.5
with the compressive stress over 300 psi: 0.5
Safety Considerations
The acid industry, as well as others, have brine wells exceeding
1,000,000 bbl. in cavern volume. A cave-in in a cavity containing
brine usually does not involve serious damage to surface equipment.
Collapse, fracture, or leakage of an LPG or natural gas storage cavity,
however, may result in serious leaks, explosions or fires.6 . 2 1
I t i s ,
therefore, important to design the cavern within the limits of various
safety considerations.
Recovery of LP Gas from Caverns
While this report is primarily concerned with storage of natural
gas in caverns there are many aspects of L.P.G. storage which may
New Concepts in Underground Storage of Natural Gas
-226-
also be applicable to natural gas storage. It is for this reason that
literature in formation on recovery of L.P. gas is included in the
fol lowing. When the reservoir is filled, some hydrocarbons will be
lost in permanent storage. This loss is relat ively small6 . 1 5
and most
of the gas can be recovered. Gas recovery can be accomplished by:
(1) Brine displacement, (2) Pumping, (3) Vaporization, (4) Gas displace-
ment. Brine displacement has one advantage and a serious disadvantage.
If the brine used is saturated, the cavern size will not change. Surface
storage or disposal of the brine usually constitutes a problem. In some
instances simultaneous operation of brine wells near underground storage
cavities provides the gas reservoir with ample brine for displacement.
A second alternative is to have a large asphalt lined surface basin for
temporary brine storage. Prefabricated asphalt lining can be installed
for about 30 to 35 cents per square foot.6 . 5
The static pressure dif-
ference between the brine and hydrocarbons is approximately 600#. It is
necessary, therefore, to have a pump with about a 1000# discharge pres-
sure to handle the pipeline flow rates.6 . 1 5
In LPG storage, product removal by pumping has several disadvan-
tages although brine is eliminated in the storage process. The amount
of product that can be handled by a centrifugal pump decreases rapidly
with depth. In caverns up to 1000 feet deep, discharge rates are as
high as 1500 GPM. If the depth is increased to 1500 feet, the rate
drops off to about 20 GPM; and beyond 3000 feet the centrifugal pump
loses a l l pract ica l i ty .6 . 5
In addition, the products being pumped are
poor lubricants, creating high maintenance costs. Furthermore, pumping
will allow the cavern pressure to drop to a point where cavern collapse
may occur.
Product removal can also be accomplished by vaporization lift.
In this method, hydrocarbon vapors are withdrawn from a relatively small
diameter tube that extends to the bottom of the storage chamber. These
vapors are recompressed and injected into the vapor space above the
liquid in the cavern. Bubbles caused by the vaporization of liquid
hydrocarbons under reduced pressure within the tube create, in effect,
Underground Storage in Non-Porous Space
-227-
a gas lift which will carry products out of the hole. Although this
method is relatively expensive and is limited in depth of operation to
1400 feet, it eliminates the need for downhole pumping equipment and
at least partially preserves the cavern pressure.
Product removal by gas displacement forces the liquid out of the
cavity by virtue of high injection pressures. It does not involve ex-
tra pumping equipment and is adaptable to any depth and rate. However,
it requires a gas source capable of both giving and taking large quanti-
t ies of gas at i rregular rates.
Table 6.1 summarizes cost data on LPG storage.
TABLE 6.1. COSTS OF LPG STORAGE6.15
New Concepts in Underground Storage of Natural Gas
-228-
Application - A Case Study and Observations
on Marysville Salt Cavern Gas Storage
In 1960, Southeastern Michigan Gas Company began development of
an underground gas storage project in an abandoned solution cavern in
a salt bed at Marysville near Port Huron, Michigan. The depth of the
salt bed is 2100 feet. The cavern had been previously washed out by
Morton Salt Company. The Marysvi l le Project, the f irst of i ts kind,
provided much direct field data on the operation and stability of salt
caverns in storage of natural gas. It answered many questions and
posed others. For instance, how much gas is lost by leakage and by
dissolution in the brine? How far could the cavern pressure be lowered
without danger of overburden collapse? Would additional leaching be
safe from the standpoint of structural stabi l i ty?
An analysis of the data available from Marysville will be presented
in the following with special emphasis on gas loss and stress considera-
t ions .
Relationship between Cavern Pressure-Gas Production-Brine Injection
Since the cavern is ini t ial ly f i l led with brine, the volume of
the gas in the cavern at any time must equal the volume of brine with-
drawn:
where
(6.21)
QL= brine injected (+) or withdrawn (-), cu ft/day
vG = volume of gas in cavern at any time, cu ft
t = time, days
From the gas laws.,
Underground Storage in Non-Porous Space
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(6.22)
where
nG= the number of moles of gas in cavern at any time
(6.23)
where
QS= gas withdrawal (+) or injection (-), std cu ft/day.
In the above and the following, subscript i wi l l refer to condit ions in
the reservoir and subscript s will refer to standard conditions at base
pressure Ps and base temperature Ts.
Combining Eqs. 6.21, 6.22 and 6.23
which gives
(6.24)
(6.25)
Equation 6.25 above gives the cavern pressure resulting from injection
of Qs std cu ft of gas per day while producing QL cu ft of brine per
day.
Review of Storage Data from Marysville Cavern
After 166 days of storage at Marysville, the data on hand indi-
cates 133,738,000 SCF of gas had been injected and 10,727,090 gallons
of brine had been removed. These data have been summarized in Table 1.
New Concepts in Underground Storage of Natural Gas
-230-
Assuming a ground temperature of 60°F and taking a geothermal
gradient of 1°F per 100 ft of depth, the temperature of the cavern may
be estimated at 82°F.
If there were no leakage and if the gas were completely insoluble
in brine, the pressure in the cavern would be:
(6.26)
(6.27)
A simple tr ial and error calculat ion using the compressibi l i ty factor
charts gives:
These calculations were repeated for 442 days of storage and 557 days
of storage. The results along with actual reservoir pressures are given
in Table 6.2. It may be observed that the measured pressures are
slightly lower than the calculated pressures.
TABLE 6.2. INVENTORY PRESSURE DATA ON MARYSVILLE STORAGE
Underground Storage in Non-Porous Space
-231-
Effect of Gas Solubility in Brine
The discrepancy between estimated and observed cavern pressures
may be due to gas migration through salt permeability or fractures,
through unmetered venting or loss through dissolution in brine. An
estimate of the percentage of gas loss through solution in brine at
cavern pressures and evolution with brine pumped to surface will be
discussed next,
The Marysville cavern was initially full of brine and contained
4,122,700 ft3 or 30,890,000 gallons of brine. The brine withdrawn after
166 days of storage was 10,727,090 gallons leaving a balance of
20,122,910 gal lons or 479,500 bbls st i l l in the cavern. The solubi l i ty
of natural gas in brine at 80°F and 1200 psia is 7.5 std cu ft of gas
per bbl of brine.6 . 3 9
Accordingly the amount of gas in brine would be:
7.5 x 479,500 = 3,595,000 std cubic feet
During this interval 133,738,000 SCF were injected. The volume of
brine withdrawn equals the volume of gas in the cavern or 10,727,090
gal lons (1,438,000 ft3) . This corresponds to
The above figures then indicate:
or 2.69 percent in solution
Total percent gas unaccounted for was
or 3.92%.
This means that only 1.23% of gas was unaccounted for once the
amount gone into solution is determined. Similar calculat ions for
storage periods of 442 days and 557 days are given in Table 6.3.
New Concepts in Underground Storage of Natural Gas
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TABLE 6.3. ANALYSIS OF GAS LOSS FROM MARYSVILLE STORAGE
It can be seen from Table 6.3 above that once the amount of gas
gone in solution is accounted for, then the calculated pressures are
significantly closer to observed pressures.
Stress Calculations
Using the equations given for a spherical cavern and the depth and
pressures at Marysville, the following numerical example shows the mag-
nitude of estimated stresses:
= 1500 psi
= 1 psi/ft x 2100 ft = 2100 psi
= 344 psi
Using equation 6.13
Underground Storage in Non-Porous Space
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In the plastic zone,
In the elastic zone
The stress distr ibut ion along the radial direct ion for the cavity
assumed spherical is shown in Fig. 6.7. Shear stresses estimated on
the basis of non-spherical geometry will be presented in the following.
Underground Storage in Non-Porous Space
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Using the model shown in Fig. 6.6a and pi = 921 psig cavern
pressure and 2100 ft cavern depth, t = 100 ft and r1 = 125 ft.
If the model given in Fig. 6.4b is used, then Eq. 6.21 with
t = 140 ft
In the calculations above it can be seen that the extent of the
thickness of consolidated anhydrite cap would significantly affect the
magnitude of induced shear stress. Because i t is di f f icult to deter-
mine accurately from logs the precise magnitude of t, the effect of (t)
on the induced shear stress is computed and given in Table 6.4.
TABLE 6.4. EFFECT OF CAP THICKNESS ON INDUCED STRESSES
New Concepts in Underground Storage of Natural Gas
-236-
The caliper logs indicate incidentally that the Marysville cavern
is more closely represented by case “b” than “a” of non-spherical geo-
metry. Further calculations using salt strength data on Marysville
cavern with assumed zero pressure in the cavern indicate that for struc-
tural stability the thickness of the cap need only be about 43.75 ft.
Conversely if the effective cap thickness is assumed to be 100 ft (the
effective thickness of cap supporting the Marysville cavern is unknown
at present), then the diameter of the cavern can be increased to 571
feet. Similarly with 921 psia prevailing in the cavern the minimum
thickness for 250 feet diameter and maximum allowable cavern diameter
per 100 feet of cap are 24.5 and 1020 feet respectively. Concerning
these calculations it must be emphatically noted, however, that these
values are merely results of crude estimates based on oversimplified
theoret ical considerat ions. Accurate theory on stresses induced in
porous media, full of fluids in pore space, usually heterogeneous and
mostly anisotropic, has not yet been fully developed for practical de-
sign calculations and was beyond the original intent of this research
pro jec t . Consequently the values quoted in this report are only intended
as indications of the orders of magnitude and should not be construed in
any way as specific recommendations for engineering operations at
Marysville Storage. Furthermore, the equations given for spherical
cavities do not take into account the weight of the overburden. Large
cavities of spherical shape would probably be critically subject to
cave-ins due to overburden collapse. In the equations concerning the
shear stresses in non-spherical cavities, “the thickness of consolidated
cap” is frequently referred to. Whether this refers to anhydrite cap
including the consolidated formations above it or not is not clear. In
considering caverns with arching geometry (as in case 6.4b), it is prob-
ably erroneous to add the entire cavern height to the cap thickness once
more indicating the danger of relying on generalizations of oversimplified
geometry.
Underground Storage in Non-Porous Space
-237-
6.2 Storage of Natural Gas in Cavities Induced by Nuclear Explosions
As a result of recent developments in engineering by subsurface
nuclear explosives the possibility and potential of using such cavities
for underground storage of natural gas recently came into attention.
Possible use of such nuclear explosions to enhance the productive capa-
city and deliverability of oil and gas producing formations through
creation of large cavities or extensive fractures has been under study
by the oil industry in general and by governmental agencies such as the
Atomic Energy Commission and U. S. Bureau of Mines for some time.
It is interesting to note that in a shallow cavity the primary
limitation for storage is the maximum pressure while in a deep cavity
the controlling limit is probably the minimum pressure. Throughout the
recent years considerable engineering and research effort has been de-
ployed toward the technology and analysis of subsurface nuclear explo-
sions.6 . 4 1 , 6 . 4 2
Some of the data collected in various experiments
have been analyzed with a view toward possible use of caverns created
in storage of natural gas.
In 1957, a major program called “Project Plowshare” was established
by the U.S. Atomic Energy Commission to develop and demonstrate peaceful
uses for nuclear explosives. Under this program, the f i rst contained
nuclear explosion was the “Event Rainier” which took place on September
19, 1957, at the Nevada test-site. Since the “Event Rainier,” numerous
other underground tests have been conducted either for weapons testing,
seismic detection or for peaceful uses of nuclear explosions. Prominent
among these were events “Logan,” “Blanca,” “Gnome” and “Hardhat.” Many
of these tests have involved explosions at shallow depths attended by
cratering phenomena for possible applications in earth moving. A large
number of the shots, on the other hand, have been contained in under-
ground strata. Data from 34 such contained shots in tuff, salt, and
granite (granodiorite), yielded much data and substantial know-how on
the effect of these explosions, on various formations. When a nuclear
explosion occurs underground, it may be a cratering or contained event
New Concepts in Underground Storage of Natural Gas
-238-
depending upon many factors. A cratering explosion is characterized
by the eruption of blast into the atmosphere. A contained event shows
very l i t t le surface effect and no detectable radioactivi ty is vented
to the surface.
Although a nuclear explosion device produces enormous explosive
force, yet its size is very small making it quite handy for subsurface
emplacement. For example, one kiloton nuclear device will produce ex-
plosive force equal to that of 1,000 tons of TNT which if pelletized
(density = 1.0 gm/c.c) would have a volume of more than 50,000 cu ft
and would fill a room 50 by 100 ft. with a 10 ft. ceiling or a spherical
chamber with a diameter of about 40 ft. On the other hand, a device
with a yield up to 10 kiltons can be fabricated with an outside diameter
of 12 inches.6 . 4 3
Figure 6.8 shows comparative emplacement techniques
for nuclear and chemical explosives. Beside the advantage in size, the
energy of a nuclear explosive is generated in a few tenths of a micro-
second or a thousand times faster than the fastest chemical explosive.
Nuclear energy is released in the form of blast, shock wave, heat radia-
t ion, l ight-rays and residual nuclear radiat ion. Only a very small
percentage appears as residual nuclear radiation over a period.6 . 4 4
Five of the specific subsurface experiments have been of par-
t i c u l a r in teres t . These are event “Rainier” the f irst major
nuclear explosion completely contained underground, “Event Logan,”
specially conducted to get seismic data and events “Blanca,” “Gnome”
and “Hardhat”. The major program to which the above five experiments
belonged was called “Project Plowshare.” A study of the data reported
on the five events indicate that the “Event Gnome” was the first con-
tained nuclear explosion which created a deep cavity which did not
collapse. In that part icular experiment the fact that an aquifer si t-
uated 650 feet above the device which exploded and a tunnel shaft
located 1300 feet from the explosion were not damaged is of particular
in teres t .
When a nuclear explosion occurs underground it may be a cratering
or contained event depending upon many factors. A cratering explosion
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New Concepts in Underground Storage of Natural Gas
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is characterized by the eruption of blast into the atmosphere. A con-
tained event shows very little surface effects. Empirical data col-
lected from Project Plowshare indicates that the depth of a contained
explosion is a direct function of the yield of explosion:
(6.28)
where H = minimum depth of exploding device for contain-
ment, feet
W = yield of the device, ki lotons.
In a contained nuclear explosion the shape and size of the cavity
obtained depend on properties of subsurface area near the shot, the
depth of exploding device, yield of explosion. The yield of nuclear
energy causing an adiabatic shock wave attended by temperature and
pressures in the order of a m i l l i o n d e g r e s s F a n d a m i l l i o n a t m o s p h e r e s .
Propagation of the shock wave to the surface and reflection down to the
cavity, first vaporization then condensation and flow of molten phase,
al l result in a stable, approximately spherical cavity i f the internal
pressure can hold-up both the overburden weight and the shock wave
returning from the surface. Otherwise the phenomenon called “chimney
formation” will occur where the roof collapses leaving the cavity full
of rubble. The chimney may or may not extend to the surface.
Storage Capacity of Nuclear Explosion Cavities
The “Gnome Event” resulted in a stable cavity of nearly spher-
ical shape with an average radius of 87 ft at a depth of 1200 ft.
If gas pressures up to 1 psi/ft could be maintained in such a cav-
ity, the approximate amount of natural gas in storage would be about
290 MMCF. For a cavity twice the size at twice the depth, the storage
capacity would be about 5 billion standard cubic feet. The nature of
the interior surface of the cavity, i ts impermeabil i ty, threshold pres-
sure, susceptibi l i ty to fractures during dri l l ing or due to overpressure,
Underground Storage in Non-Porous Space
-241-
the decay of radiation and many other problems must be carefully and
exhaustively considered before the economics of gas storage in nuclear
cavit ies can begin to crystal l ize.
Size and Shape of Cavities Caused by Nuclear Explosions
Exploratory mining and dri l l ing indicate that the cavit ies pro-
duced by underground nuclear detonations are roughly spherical, although
departures from the spherical symmetry do occur primarily due to differ-
ential movements of the overburden towards the free surface. For com-
parison purposes, radii are usually measured from the region below the
shot point.
Nuckrolls (1959) showed that the radius of the Event Rainier could
be explained by having the cavity expand until the pressure of the gas
in it is balanced by the weight of the overburden. The formula for
scaling radii on this assumption is
(6.29)
R = cav i ty rad ius in f t
c = a constant
W = y ie ld i n k i l o tons
= overburden density in gm/cc
h = dep th o f bu r i a l i n f t .
Study of 34 underground detonations has shown that with the ex-
ception of one very low yield shot, the constant C and consequently
cavity radii is predictable within ± 20% regardless of whether the
surrounding rock is tuff , salt , al luvium or granodiori te. I f the data
New Concepts in Underground Storage of Natural Gas
-242-
are limited to a single rock type, the cavity radii become predictable
within ± 8%. Experimental values for the constant C were found as
follows :
TABLE 6.5. CONSTANT FOR PREDICTION OF CAVITY RADIUS
Table 6.5 shows the effect of burial on the cavity radius for
a given yield and rock type.
Fig. 6.9 shows the effect of yield and overburden pressure on
the cavity-radius,
It appears that the density, porosity and elastic properties of
the containing rock medium have very little effect, if any, on the size
of the cavity.
Fig. 6.10 shows the Gnome Cavity profile.
When the natural pressure of the cavity cannot hold either or
both the overburden pressure and the shock wave returning from the sur-
face, a phenomenon called “Chimney Formation” occurs where the roof
co l lapses, par t ia l ly or complete ly f i l l ing the cav i ty . The chimney
may or may not extend to surface.
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New Concepts in Underground Storage of Natural Gas
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Fig. 6.10. Gnome cavity profile.6 . 4 2
Underground Storage in Non-Porous Space
-245-
TABLE 6.6. CAVITY RADII CALCULATED AS A FUNCTION OF ROCK TYPE,
YIELD AND DEPTH OF BURIAL
New Concepts in Underground Storage of Natural Gas
-246-
Fig. 6.11. Hardhat schematic cross section.6 . 4 2
Underground Storage in Non-Porous Space
-247-
Fig. 6.12. Rainier schematic cross section.6 . 4 2
New Concepts in Underground Storage of Natural Gas
-248-
Fig. 6.13. Blanca schematic cross section.6 . 4 2
Underground Storage in Non-Porous Space
-249-
When the collapse does not extend to the surface, the top of the
chimney appears to be cone shaped as shown in Figures 6.11 and 6.12.
Surface subsidence craters indicate that the top flares outward how-
ever, when the chimney does communicate with the surface. This is
i l lus t ra ted in F ig . 6 .13.
The Hardhat (granodiorite) cavity stood for 11 hours after detona-
tion and it has been suggested that changing of thermal stresses as the
cavity Slowly cooled influenced collapse. Properties of the rock as
well as its preshot structure are important factors in cavity collapse.
The gnome event, a detonation in a f lat- lying strat i f ied evaporite
deposite in New Mexico, resulted in a standing open chimney (Fig. 6.10)
which extended only 89 feet above the original shot point.6 . 4 6
The
collapse of the rock salt beds was limited to several tens of feet from
the roof of the cavity and was influenced by separations at clay seams.
When collapse occurs, the cavity void is translated into the chimney of
broken rocks in the form of interst i t ial rubble porosity. I t is pos-
sible to predict the height of the chimney as follows:
H = 4 / 3 k R (6.30)
R = cav i t y r ad ius , f t
H = ch imney he ight , f t
k = a constant, determined by properties and geologicalstructure of the rock. It is equal to chimneyvolume divided by cavity volume.
Damage From Seismic Effect of Underground Nuclear Explosions
The degree of seismic damage to underground mine shafts and tun-
nels for a given yield is a function of the elastic properties and
strength of the rock as well as its weakness. The data in the Table
6.7 serve as a guide in estimating major damage to mining.
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New Concepts in Underground Storage of Natural Gas
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TABLE 6.7. NUCLEAR EXPLOSION TUNNEL DAMAGE DATA
However, the damage is not very severe and the shafts can be re-
habilitated to preshot conditions in two or three weeks' time on a
f ive-day, two-shif t basis.
On the basis of data collected, it has been now possible to esti-
mate surface motion in a variety of geological environments. The area
affected seems to depend upon the shot depth, yield and rocktype.
Figure 6.14 shows permanent displacement measured by comparing pre and
post-shot elevations and Figure 6.15 contains a map showing the extent
of surface effects. It is interesting to note that in the "Event Gnome,"
window panes in a Butler prefab steel building located 900 ft. from
surface zero or a radial distance from the explosion of' 1690 ft. were
unbroken and there was no discernible damage.6 . 4 9
A bolted steel tank,
30 ft. in height and diameter located 145 feet from this surface zero
was also undamaged.
Underground Storage in Non-Porous Space
-253-
In an area directly over the shot-point, surface spall occurs and
beyond the spall area, the surface motion is less violent. Studies
have shown that at a particle velocity below which little, if any,
damage occurs6.49 is 11cm/sec. The approximate maximum radial dis-
tance at which this particle velocity was attained on the surface
shown in the Table No. 6.7 for four events.
TABLE 6.8
Distance at which measured particle velocity drops below 11cm/sec,
the damage threshold for residential-type building (extrapolated
from U.S. Coast and Geodetic survey record)
R a d i o a c t i v i t y D i s t r i b u t i o n
Exploration has shown that the bulk of the radio activity produced
is concentrated in a puddle below the shot point and that an optimum
mining level within the chimney as far as external gamma radiation dos-
age is concerned is one cavity radius above the shot point. Post-shot
mining at Rainier and Hardhat has shown that, in all cases, radiation
dosages were below the minimum tolerance set by the Atomic Energy Com-
mission and Public Health Service. Groundwater contamination does not
appear to be a problem in geohydrological environments where nuclear
explosives might be employed. Development of methods of trapping the
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Underground Storage in Non-Porous Space
-255-
radioactive melt as well as development of devices with very low fis-
sion yield will greatly reduce the hazard associated with the fission
product. In th is d i rect ion, development of thermonuclear explosives
with very low f ission yields is signif icant.
Economic Considerations
Cost of underground nuclear explosion is still high, being
$350,000 and $600,000 for nuclear explosivesof 10 kiloton and 2-megaton
yields respectively. However, with the improvement in the design of
nuclear explosives, and in their emplacement as well as in the technol-
ogy of explosion and its effects, and production on a commercial scale
the price is very l ikely to reduce6 . 5 0
appreciably. A plot showing
projected charges for thermonuclear explosives is contained6 . 5 0 i n
Fig. 6.16.
6.3 Storage in Natural or Mined Cavities
There are 11,791 known caves in the United States.6 . 3
Some of
these are known to be over 400 feet deep, to cover acres of subsurface
area and b e j o i n e d through miles and miles of tortuous passage
ways. Most of the caverns are formed in limestone beds through dis-
solving action of underground water combined with stream erosion and
subsequent drying as the level of underground water table drops. The
prospect of storing natural gas in caverns located near areas of great
need for storage depends upon many factors. The maximum pressure that
can be safely carried in the caverns, the condition of cavern walls
with respect to permeability, incipient fractures and groutabi l i ty, the
nature and condition of overburden above the cavern must all be con-
sidered before their economic feasibility can be assessed. The problems
in mined caverns are quite similar to those of natural caverns. There
has been some use in the United States of abandoned coal mines for the
storage of natural gas.
New Concepts in Underground Storage of Natural Gas
-256-
FIG, 6.17 GEOLOGICAL FORMATIONS SUITABLE FOR CONSTRUCTION OF THESE UNDERGROUNDSTORAGE CAVERNS ARE FOUND IN MANY PARTS OF THE U.S. AND BROADAREAS HAVE BEEN TENTATIVELY CLASSIFIED AS FAVORABLE OR UNFAVORABLE.
Underground Storage in Non-Porous Space
-257-
The proximity of mined or natural caverns near the surface creates
a serious limitation for their storage capacity due to the maximum pres-
sure that can be used as related to the formation parting pressure at
the ceiling of the cavern. This limitation is not present in the pros-
pect of using a cavern induced by nuclear explosion at some satisfactory
depth for the storage of natural gas. Geographic formations suitable
for construction of underground caverns are indicated in Figure 6.17.
Coal Mine Storage
A well-constructed coal mine may constitute a ready-made storage
reservoir. First, however, test drillings should be made to determine
the permeability of the formations around the mine. Tests on a Denver
mine 6 . 1 3 showed that the strata above the coal seam were of sufficient
strength and of low enough permeability to serve as a “cap rock” for
the reservoir. The vertical distance between the surface outcrop and
the mine operating level fixes the pressure at which the reservoir is
to be operated. At the Denver mine, an upper limit of 300 psig for
storage was selected while the operating pressure used for storage was
200 psig.
Hard Rock Mined Storage
Hard rock mining is often the only solution to a storage problem
due to the conditions of the subsurface formation. This type of stor-
age is exemplified by Sinclair Oil and Gas Co.‘s 5,000,000 gallon hard
rock underground storage of LPG near Seminole, Oklahoma.6 . 1 2
Conven-
tional mining methods and equipment were used to construct the mine.
The mine was tested for leakage with dehydrated air at 150 psig and 90
to 100 psig was used for storage.
A hard rock mine was also constructed at the Bayway refinery of
Esso at New Jersey6 . 7
at a depth of 330 ft. Total capacity is about
New Concepts in Underground Storage of Natural Gas
-258-
675,000 bbl. Large pillars of shale were left in place during the ex-
cavation to provide support, The support pi l lars are actual ly larger
than the excavated area, making the reservoir a series of interconnected
tunnels.
The techniques presented for LPG storage possibly can be used in
the storage of natural gas. The Gulf Coast, Mid West, and Great Lakes
regions with salt formations offer the salt dome and salt bed storage,
while the coal and ore mining areas might be considered for storage in
abandoned mines. In the eastern states, many of the large centers of
population are over solid rock formations. Hard rock mining may pro-
vide some possibilities for gas storage of the type first discussed.
6.4 Underwater Storage of Natural Gas
One of the most recent and promising ideas advanced at The Univer-
sity of Michigan research project on new concepts is the storage of
natural gas at the bottom of lakes or oceans. The sea offers interest-
ing advantages over methods of conventional underground storage because
of the following reasons:
1. More storage space per bulk storage volume (100 percentporos i ty) .
2. Avai labi l i ty of pressure for storage.
3. Minimum safety, leakage, collection, contamination problems.
4. Low and constant temperature of ocean bottom permitting moregas to be stored for same depth and pressure.
5. Presence of salt water which would prevent hydrate formation.
6. Possibi l i ty to pract ice overpressuring in special containersto a much larger extent than in underground storage.
Figure 6.18 represents our current concept for a peak shaving
storage installation located at the bottom of the ocean at some suit-
able depth. It merely and simply consists of an inverted container
Underground Storage in Non-Porous Space
-259-
F I G . 6 . 1 8 P E A K S H A V I N G S T O R A G E
A T O C E A N B O T T O M
New Concepts in Underground Storage of Natural Gas
-260-
open at the bottom and supplied from the top by a pipeline. As the
pressure in the pipeline is increased, the natural gas flows in the
container, pushing the sea water out. During periods of peak demand as
the gas is pulled through the distribution network, the water moves in
and displaces the gas out. One of the critical requirements of such a
storage facility is to provide suitable and sufficient anchor design
to hold the container and the pipeline at the bottom.
Figure 6.19 shows the reservoir volume necessary to store one
billion standard cubic feet of natural gas under water as a function
of depth. It can be seen that for 0.6 gravity gas at 3,000 feet, one
would need a container having a volume of 7.8 x 106 cub i c f ee t . A
vessel of spherical shape open at the bottom of this volume would have
a radius of about 125 feet. The total anchoring force to hold such a
vessel at the bottom would be equal to about 4.8 x 108 pounds-force as
can be seen from Figure 6.20. While these anchoring requirements seem
to be enormous at first with the rapidly advancing underwater construc-
tion technology, it is believed that mechanical and design problems
associated with underwater storage are definitely within the reach of
technically possible and economically feasible solutions.
Engineering Calculations for Underwater Storage
A brief literature search was made in an attempt to establish
some information about the sea environment, such as, the temperature,
and the pressure in the Atlantic and Pacific ocean. Figure 6.21 shows
the effect of depth on the pressure, and represents the following
equation:
(6.31)
where Pv = deep sea pressure (psia)
Pa = atmospheric pressure (14.7 psia)
PW = density of sea water (64 lb mass/ft3)
d = depth (feet)
Underground Storage in Non-Porous Space
-261-
FIG. 6.19 R e s e r v o i r V o l u m e V S D e p t h N e c e s s a r y
t o S t o r e 1 0 9 S C F o f N a t u r a l G a s i n
S e a W a t e r
New Concepts in Underground Storage of Natural Gas
-262-
F I G , 6 . 2 0 A n c h o r i n g F o r c e R e q u i r e m e n t s f o r 1 0 9
S C F V e s s e l s i n S e a W a t e r .
Underground Storage in Non-Porous Space
-263-
Fig. 6.21 Effect of Depth on the Pressure
New Concepts in Underground Storage of Natural Gas
-264-
Therefore, Pv = 14.7 + 0.445 d (6.32)
The temperature of the ocean of the Atlantic coast of the U.S.
at depths below 300 feet does not vary more than one degree during the
year. The temperature gradient with depth goes from 50°F at 650 feet
to 40°F at 3,000 feet and below. The Pacific Ocean (Monterey, California
at 300 feet) varies in temperature annually between 43 and 48°F.
Because these represent the major geographic areas of interest
and because of the relatively small variation of temperature with time
and depth (below 600 feet), it was decided that an average value of
45°F (505°R) would be assumed in all calculations.
Figure 6.19 summarizes the calculated reservoir volume to hold
109 S. C. F. (standard cubic feet) natural gas by the depth and the spe-
cif ic gravity of gas. It was assumed that the reservoir pressure is
the sea water hydrostatic pressure at the lowest point of the reser-
vo i r .
If Vc represents vessel volume required at the bottom, Tc ocean
bottom temperature, Zc, compressibility factor of gas at bottom condi-
t ions , R the gas constant, nc no. of moles in storage, then by the
gas law:
(6.33)
and
(6.34)
where ZS = 1.0 and subscript s refers to standard conditions.
(6.35)
Underground Storage in Non-Porous Space
-265-
and for Vs = 109 std. cu. ft ., Ts = 520°R abs., Tc = 505°R abs. and
Pc = 14.7 + 0.445 d (d = depth, feet), by substitution:
(6.36)
The values of Zc are given in Table 6.9 for various depths and gas
grav i t ies .
Anchoring Force Requirements
In order to determine the anchoring force necessary to hold the
storage vessel down at the bottom against the buoyancy of the gas bubble
it is necessary to determine the net buoyant force acting on the storage
vessel.
Consequently, the relationship between the anchoring force and the
vessel volume assumed full of natural gas is given by:
where pw = density of sea water, = 64 lbm / f t3
pG= density of natural gas at bottom pressure
and temperature, l b m / f t 3
Vc = volume of vessel, cu. ft.
S inceC
(6.37)
(6.38)
(6.39)
New Concepts in Underground Storage of Natural Gas
-266-
and
(6.40)
which is the basis for Figure (6.18).
Storage Vessel Configuration
Using an inverted saucer shape open at the bottom and consequently
subject to no pressure differential inside-out at the bottom across the
vessel wall, optimum shape can be determined by observing surface to
volume ratio on vessels of various geometric shapes.
Optimum volume-to-surface configurations were calculated using
d i f fe rent ia l ca lcu lus . Expressions for the surface area and volume were
written and the configuration parameters (i.e. radius and height of
right cylinder) were optimized. The ca lcu la t ions were car r ied out
for various geometric shapes. The resu l ts ind icate that a spher ica l
system (hemi- or full sphere) results in the lowest surface area per
unit volume. Figure 6.22 shows the plots of surface area requirements
versus depth for the different gravity gases for the hemisphere.
Figure 6.23 shows the radius of the hemi-sphere and Figure 6.24 that of
the full sphere versus depth. Figure 6.22 is a plot of the equation:
(6.41)
where
Vc is calculated from equation 6.36 and C = 3.84.
Underground Storage in Non-Porous Space
-267-
F i g . 6.22 Area of a Spherical Vessel Required to Contain
1 BSCF Gas Underwater
Underground Storage in Non-Porous Space
-269-
Figure 6.23 is a plot of the equations
w i t h
(6.42)
(6.43)
where
Pressure Loading on Storage Vessel
When a vessel of spherical shape and of 2 rs height open at the
bottom is considered, the pressure across the vessel wall at the bottom
is same inside-out. However, with gas pressure inside the vessel the
pressure differential across the vessel wall at the top may be readily
computed as
where = pressure dif ferential across vessel wal l attop, psi
r =S radius of sphere, ft.
Pw = density of sea water = 64 lbm/ft3
PG = density of gas in storage lbm/ft3
(6.44)
The Figure 6.25 shows the pressure differential for a spherical vessel
containing one billion cubic feet of gas at various depths for differ-
ent gas gravities.
New Concepts in Underground Storage of Natural Gas
-270-
Fig. 6.25 Pressure Differential for a Spherical Vessel
Containing 1 BSCF Gas Underwater
Underground Storage in Non-Porous Space
-271-
General Design Concept and Considerations
Figure 6.25 shows what might be considered to be a typical example
of an overall design diagram for a deep sea storage reservoir.
Specific attention is called to the polymeric lower half of the
reservoir because only a small pressure differential exists along the
boundary of the reservoir, only a strong cap is needed to withstand the
tremendous anchoring force requirement. The use of a polymeric material
for one-half of the reservoir would greatly reduce the cost of the
reservoir. As the gas is produced, the bag will collapse, eliminating
any possibility of implosion by the hydrostatic pressure on the empty
reservoir.
It may be perhaps at least equally feasible to use a submerged
pipel ine, say about 200 feet below to surface all the way to a platform
at some point out at sea at the surface and then convey the gas ver-
tically down to the bottom anchored storage vessel.
The Figure 6.26 shows the general concept of an underwater peak-
shaving gas storage facility. The container may be constructed either
open at the bottom or equipped at the bottom with an expandable/col-
lapsable plastic bag. It appears that the design of such a container
of optimum shape and dimensions may be critically affected by its
structural integrity to sustain stresses due to anchor loading versus
the buoyancy of gas in storage. The anchoring of pipeline at the bottom
and the on-shore compression-distribution dispatch facility are also
schematically shown. As in the technology of off-shore oil and gas dril-
l ing-produc ing fac i l i t ies , deep well dr i l l ing platform construct ion
techniques are advancing almost daily. The system proposed above should
sometime in the future become a technically possible and economically
feas ib le real i ty .
Problems concerning construct ion, corrosion, durabi l i ty, resis-
tance to earthquakes, tidal waves, sabotage, other risks and optimiza-
tion of materials, however still remain to be solved in the future.
-273-
Un
de
rgro
un
d
Sto
rag
e
in
No
n-P
oro
us
Sp
ace
This Page Intentionally Left Blank
New Concepts in Underground Storage of Natural Gas
-274-
REFERENCES
6.1 Jennings, G. P., Underground Storage Ideal for LPG, The Oil andGas Journal, Vol. 59, No. 18, May 1, 1961.
6.2 Newman, B. F., Underground LPG Storage, The Petroleum Engineer,Vol. 24, No. 13, December 1952.
6.3 Scisson, S. E., Planning for Mined Underground LPG Storage, TheOil and Gas Journal, Vol. 58, No. 18, May 2, 1960.
6.4 Kinney, Gene T., Underground Gas Storage Still Rising, The Oiland Gas Journal, April 23, 1956.
6.5 Brandt, C. T., 5 Ways to Recover Stored LPG, The Oil and GasJournal, Vol. 59, No. 15, April 10, 1961.
6.6 Famous Texas Salt Dome to Become LPG Cavern, The Oil and GasJournal, Vol. 57, No. 15, April 6, 1959.
6.7 Carving Out a Cavern Through a "Needles' Eye," Engineering NewsRecord, Vol. 160, No. 4, January 23, 1958.
6.8 Vance, Thaddeus B., Salt Cavern Gas Storage Boosts Profits, TheOil and Gas Journal, Vol. 60, No. 42, October 15, 1962.
6.9 Bil lue, G. H., and Roberts, T. E., How N.C.R.A. Operates ItsUnderground LPG Storage, The Oil and Gas Journal, Vol. 53, No.19, September 13, 1954.
6.10 Can Gas Be Stored Under Flat Caprock?, Petroleum Week, December23, 1960.
6.11 Wheeler, Henry P., Jr., and Eckard, William E., UndergroundStorage of Natural Gas in Coal-Mining Areas, Information Circular7654, United States Department of the Interior, December 1952.
6.12 Counts, E. H., and Childress, C. L., Underground LPG Storage, TheOil and Gas Journal, Vol. 53, No. 17, August 30, 1954.
6.13 Bleanley, W. B., Old Coal Mine Converted to Gas Storage, The Oiland Gas Journal, December 13, 1961.
6.14 Daugherty, Patr ick F., How Sun Created Underground Storage, TheOil and Gas Journal, Vol. 53, No. 43, February 28, 1955.
Underground Storage in Non-Porous Space
-275-
6.15
6.16
6.17
6.18
6.19
6.20
6.21
6.22
6.23
6.24
6.25
6.26
6.27
6.28
6.29
6.30
Henderson, G. R., and Dougherty, P. F., Underground StorageCreated in Salt Beds at Sun's Sarnia Refinery, The CanadianJournal of Chemical Engineering, Vol. 35, No. 2, August 1957.
Branyan, Stuart G., How Anchor Recovers 97% of LPG Stored Under-ground, World Oil, Vol. 142, No. 2, February 1, 1956.
How LPG Was Stored in a Producing Brine Well, World Oil, Vol.139, No. 4, September 1954.
Wilson, W. M., Lion Oil Co.'s Experience with Underground Stor-age, The Oil and Gas Journal, Vol. 52, No. 43, March 1, 1954.
Wiederker, A. M., Barker Dome Gas Storage Project, The PetroleumEngineer, Vol. 24, No. 13, December 1952.
Richards, A. W., San Diego Goes Underground to Increase StorageFacilities, Gas, Vol. 32, No. 6, May 1956.
Johns, D. F., Formation Strength in Salt Cavern Storage, ThePetroleum Engineer, Vol. 29, No. 9, August 1957.
Landes, Kenneth K., LPG, Fuel Oil for Natural Gas Storage Pos-sibilities in Western New York, 1005 Berkshire Rd., Ann Arbor,Michigan, April 19, 1955.
Erickson, A. R., and Svoboda, R. F., Redfield Gas Storage Struc-ture, AAPG Annual Meeting Program, 1956.
Chisholm, J. P., and Patterson, G. D., Sonar Caliper SimplifiesLPG Storage Surveys, The Petroleum Engineer, Vol. 30, No. 1,January 1958.
Landes, Kenneth K., International Salt Company-Ludlowville BrineField, 1005 Berkshire Rd., Ann Arbor, Michigan, December 3, 1962.
Todd, Raymond W.,May 1962.
Progress in Gas Storage, Gas, Vol. 38, No. 5,
Galpin, Sidney S., and Montgomery, Palmer H., Unique Tools andMethods Used in Gas Well Workovers, The Petroleum Engineer,Vol. 28, No. 10, September 1956.
N.G.A.A. Prepares Standards for Underground LP-Gas Storage, ThePetroleum Engineer, Vol. 24, No. 13, December 1952.
Huff, Rable L., Here's How Texas Gas Recovers More LPG, ThePetroleum Refiner, Vol. 35, No. 5, May 1956.
Dougherty, Pat, and Fenix, Gilbert J., How Sun Oil Co. Mines andOperates LPG Storage Caverns,No. 21, May 22, 1961.
The Oil and Gas Journal, Vol. 59,
New Concepts in Underground Storage of Natural Gas
6.31
6.32
6.33
6.34
6.35
6.36
6.37
6.38
6.39
6.40
6.41
6.42
6.43
6.44
6.45
-276-
Gentry, H. L., Natural Gas Successfully Stored in Salt Cavern,The Oil and Gas Journal, Vol. 60, No. 42, October 15, 1962.
Givens, Homer C., Depleted Sands Make Dual Reservoir for LPGProduct, The Oil and Gas Journal, Vol. 56, No. 73, September 24,1956.
Doughty, K. V., and Cole, Charles M., Jr., Status and Progress ofUnderground Storage, The Petroleum Engineer, September 1954.
Chapin, Earl V., LPG in Volume Can Be Stored Underground, ThePetroleum Engineer, Vol. 26, No. 4, April 1954.
Goebel, E. D., and Jewett, J. M., Possibilities for UndergroundStorage of Natural Gas Near the Kansas River Valley, Universityof Kansas Publications, State Geological Survey of Kansas, Oiland Gas Investigations No. 27, 1962.
Serata, Shosei, and Gloyna, Earnest, Design Principles for Under-ground Salt Cavities, Trans. ASCE, Part III, Paper No. 3146,1961.
Sippel, Robert F., and Hodges, H. Darwin, LPG Storage Well Log-ging, The Petroleum Engineer, April 1958.
Grow, George C. Jr., and Zack, Julia, Bibliography on UndergroundStorage, Pipeline Research Council International, July 1, 1958.
Katz, D. L., et al . , Handbook of Natural Gas Engineering, McGrawHill Book Co., Inc., New York, 1959.
Katz, D. L., et al . , Movement of Water in Contact with NaturalGas, Monograph, Pipeline Research Council International, 1963.
Sanders, Ralph, Project Plowshare, Public Affairs Press, WashingtonD. C. (1962).
AEC Reports, UCRL - 5675, 5676, 6588, 6240, 7515.
Watkins J. Wade and Charles H. Atkinson, Can Nuclear Explosionsbe used to Stimulate Gas Production?, Paper presented at AnnualMeeting, New Mexico Oil and Gas Association, Santa Fe, NewMexico; October 6, 1964.
The Effects of Nuclear Weapons, U.S. Department of Defense andU.S. Atomic Energy Commission, April 1962, 730 pp.
Boardman, Charles R., David D. Raab, and Richard D. McArthur,Characteristic Effects of Contained Nuclear Explosions forEvaluation of Mining Applications, University of Cali forniaErnest O. Lawrence Radiation Lab., UCRL-7350, Rev. 1, September1963, 43 pp.
Underground Storage in Non-Porous Space
6.46
6.47
6.48
6.49
6.50
6.51
6.52
6.53
6.54
6.55
6.56
6.57
6.58
6.59
6.60
-277-
Rawson, Donald E., Charles R. Boardman, and Nanette Jaffe, TheEnvironment Created by a Nuclear Detonation in Salt, P.N.E.-107 F, to be Published.
Foose, R. M., and R. B. Hoy, Air and Ground Inspection Techniquesfor the Detection of Underground Explosive Test, Operation Hard-tack II, Project 2614, ITR-1715, January 5, 1959.
McArthur, Richard D., Geologic and Engineering Effects, theHardhat Event (preliminary) GN 1-63, February 1963.
Cauthen, Lewis J., Jr., Damage to Residential Type StructuresResulting from Industrial Blasting (to be published).
Policy Statement on Projected Charges for Peaceful Nuclear Explo-sions, A AEC Press Release, May 6, 1964.
New Concepts on Underground Storage, Progress Report, 05625-2-P,University of Michigan, 1965.
Tek, M. R., New Concepts in Underground Storage of Natural Gas,Paper presented in the Second International Colloquium of ARTFPRueil-Malmaison, June 1965.
Atkinson, Charles H., Robert T. Johnson, A Study of the Feasi-bility of Using Nuclear Explosions to increase Petroleum recovery.United State Department of the Interior, Bureau of Mines, 1966.
Hoy, R. B., and R. M. Foose, Earth Deformation From a NuclearDetonation in Salt, PNE - 109P, January 22, 1962.
Johnson, Gerald W., and Charles E. Violet, Phenomenology of Con-tained Nuclear Explosions, Lawrence Radiation Laboratory(Rivermore) Dept. UCRL- 5124, Rev. 1, December 1958.
Nuckolls, John H., A Computer Calculation of Rainier. LawrenceRadiation Lab. (Rivermore) Dept. UCRL-5675, May 1959.
Schroeder, Melvin C., Interior Report on Research on RadioactiveContamination of Groundwater Aquifers, Lawrence Radiation Lab.(Berkeley) Dept. UCRL - 130 48, July 1962.
Swi f t , L . M. , e t a l . , Measurements of Close-in Earth Motion,Hardhat Event, VUP-2101, March, 1962.
Thomson, Thomas J., and John O. Misz., Geologic Studies of Under-ground Nuclear Explosions Rainier and Neptune, Lawrence RadiationLab. (Rivermore) Dept. UCRL-5757, October 28, 1959.
Bennett, W. P., B. L. Smith, D. W. Roberts, University ofCalifornia (L.R.L.) Gnome Postshot Temperature and RadiationStudies. AEC Dept. PNE-106P, June 1962.
New Concepts in Underground Storage of Natural Gas
-278-
6.61
6.62
6.63
6.64
6.65
6.66
Bennett, W. P., B. L. Smith, A. L. Anderson, University ofCalifornia (L.R.L.) Cavity-Deformation, Radiation and-Tempera-ture Distribution and resulting from Logan Event. AEC ReportUCRL-6240, December 1960.
Radiological Health Handbook, U. P. Dept. of Health, Educationand Welfare, September 1960.
Charles E. Mohr, "Exploring America Underground," NationalGeographical Magazine, 125, 6, 803 June (1964).
Johns, D. F., Formation Strength in Salt Cavern Storage, ThePetroleum Engineer, August 1952.
Henry Stammel, "The Gulf Stream," University of CaliforniaPress, 1960.
Sverdrux, H. V., Johnson, M. W. and R. H. Flemming, The Oceans,Prentice Hall, Inc., New York 1942.
This Page Intentionally Left Blank
APPENDIX A
THEORETICAL RESERVOIR ENGINEERING CALCULATIONS
RELATED TO GROUT INJECTION
This section describes the techniques developed for calculation
of the rate of injection of a fluid with a time dependent viscosity
(grout) into a porous medium saturated with another fluid (water).
Three methods were developed. They are for
(a) constant viscosity solutions in one dimensional,
Cartesian coordinate (analytical method),
(b) variable viscosity solutions in one dimensional,
Cartesian coordinates (finite difference method),
(c) variable viscosity solutions in one dimensional,
cyl indrical coordinates (f ini te dif ference method).
The finite difference methods were checked with some analytical and
tabulated solutions and found to be accurate.
Several computer runs were made using the finite difference
methods to give qualitative indications of the effects of grout vis-
cosi ty . For grout/water viscosity ratios below 10 the effect of
viscosity is small .
The methods can be applied in comparing the rate of injection
of di f ferent grout solut ions.
-279-
New Concepts in Underground Storage of Natural Gas
-280-
Introduction
The grouting fluid undergoes a chemical change during and after
the inject ion period, The resistance of the porous medium to further
fluid flow may be measured in terms of a “threshold pressure.” The
threshold pressure being the maximum pressure difference which can be
withstood without allowing fluid flow. The “strength” or threshold
pressure of the grouted region depends in part on the thickness of this
region. Hence the selection of a grouting material requires informa-
tion about the rate of injection given the pressures, viscosities and
other physical properties of the grout and the porous medium.
The objective of the study reported here is to predict the thick-
ness of the impermeated region as it increases with the time elapsed
since the start of the injection process, given the fol lowing informa-
t i on :
(a) viscosity-t ime relat ion for the grouting solut ion,
(b) viscosity of the native reservoir f luid (brine),
(c) permeability and porosity of the formation,
(d) compressibility of the porous material plus inter-
s t i t i a l f l u i d ,
(e) injection and formation pressures,
( f ) shape of the grout front.
The problem is then one of calculating the displacement of one fluid
(usually brine) by another f luid (a grout solut ion). The major com-
plication is due to the difference between the viscosities of the two
fluids and the fact that the viscosity of the grouting fluid increases
with time. A typical grouting fluid might have the viscosity-time
behavior shown on Figure A-l.
Two types of fronts are being considered, planar and cylindrical.
The planar front would apply to grouting from a plane fracture to
Theoretical Reservoir Engineering CalculationsRelated to Grout Injection
-281-
produce vertical or horizontal boundaries while a cylindrical front
would be encountered in grouting from a well to produce a vertical
and cylindrical boundary.
F i g . A - l . T y p i c a l G r o u t V i s c o s i t y - t i m e R e l a t i o n s h i p
Mathematical Models
The mathematical models used to describe the injection and dis-
placement processes are based on the following assumptions:
(a)
(b)
(c)
(d)
superf icial f luid velocity and pressure are related
according to Darcy's Law,
effects of capi l lary and gravitat ional forces can
be neglected,
injected and displaced fluids have the same density
and compressibility,
fluid-porous medium combination can be treated as
sl ightly compressible,
New Concepts in Underground Storage of Natural Gas
-282-
(e) porous medium is homogeneous and isotropic,
( f ) interface between the injected and displaced
fluids is a surface.
One shortcoming of models based on these assumptions is that the
effects of grout dilution due to molecular diffusion, bypassing of the
fluid in place and “fingering' are not accounted for. Hence the pre-
dicted grout thickness is probably greater than the “effective” thick-
ness. However, since grout solutions normally contain a large amount
of water and the fluid being displaced is water, these effects are not
expected to be large enough to invalidate the results.
The following notation will be employed in this section with the
understanding that any consistent set of units may be employed.
A, B
a
C
k
p
P
P1
Po
q
t
x , r ,R
XF , r F ,RF
dimensionless constants
radius of well
compressibility of the fluid plus porous medium
permeabil i ty
dimensionless density, (p-po) /(p1-po)
local pressure
pressure at the injection surface
pressure far from the surf ace
volumetr ic inject ion rate per unit area
time
supe r f i c i a l ve l oc i t y , ve loc i t y vec to r , i n t e r s t i t i a l
ve l oc i t y
coordinate normal to injection face
pos i t ion o f f ront
space step sizes
Theoretical Reservoir Engineering CalculationsRelated to Grout Injection
-283-
time step sizes
poros i ty
parameter
viscosity of f luid beyond the front
viscosity of fluid between the front and the injec-
t ion face (grout)
3.14159
density
density far from the injection face
density at inject ion face
Ko t / a2 , dimensionless time
The continuity equation, Darcy's Law and the equation of state
are:
(A.1)
(A.2)
(A.3)
New Concepts in Underground Storage of Natural Gas
-284-
Then since
(A. 4)
these equations may be combined to give a partial differential equa-
tion involving density, position and time as
(A.5)
In the “grouted” region the viscosity is denoted by u1(t) (a function
of the time elapsed since the start of the injection process) and in
the region beyond the interface by uo (a constant). At the interface,
x F ( t ) , continuity of density and velocity result in
(A.6)
Equation A.5 applied to the two regions of interest (between the injec-
tion point and the interface and beyond the interface) and the contin-
uity condit ions, equations A.5 and A.7 are assumed to describe the flow
processes. These equations, along with certain initial and boundary
conditions, will predict the front movement (grout thickness) given the
data specified in the previous section.
Theoretical Reservoir Engineering CalculationsRelated to Grout Injection
-285-
Equations Describing Grout Injection in One Dimension -
(Cartesian Coordinates)
F i g . A - 2 . C o o r d i n a t e S y s t e m f o r “ L i n e a r G r o u t i n g .
The geometry is indicated on Figure A.2. The grout front is
assumed to move as a plane, i ts posit ion noted as xF( t) . I n i t i a l l y
the semi-infinite region is at some uniform pressure (po) and contains
a f luid with viscosity uo. The pressure at the plane of the origin is
raised to a level p1 (p1 > po) and kept there. The grouting solution,
with viscosity u1(t), begins to invade the porous medium. The viscosity
of the grouting solution depends only on the elapsed time measured from
the start of the injection process. The local density may be found from
the solut ion of the fol lowing problem in part ial di f ferential equations.
(A.8)
(A.9)
New Concepts in Underground Storage of Natural Gas
-286-
where
(A.10)
w i th
(A.11)
(A.12)
(A.13)
(A.14)
(A.15)
This problem is di f f icult to solve for u l(t) an arbitrary function of
time and numerical methods will be employed, however, for constant
u l(t) an analyt ical solut ion is possible. Furthermore, the solution
has an interesting physical interpretat ion.
Solutions for a Constant Viscosity Grout Solution-Plane Front
The grout solution is assumed to have a constant viscosity ul
(u1 > uo) for a period of time after which the viscosity increases
rapidly. (Fig. A-3)
Theoretical Reservoir Engineering CalculationsRelated to Grout Injection
-287-
F i g . A - 3 . V i s c o s i t y - t i m e C u r v e
The part ial dif ferential equations and al l boundary and init ial
conditions except at the front are satisfied by
where "erf" and "erfc" denote the error function and complimentary
error function respectively. The constants A and B are found from
the conditions at the front namely,
(A.16)
(A.17)
(A.18)
-288-
(A.19)
If both of these conditions are to hold for all values of time then the
equation for the movement of the front must be
(A.20)
where is a constant to be determined. It may be noted in passing
that this problem is quite similar in structure to the "freezing front"
problem discussed by Carslaw and Jaeger.5 . 7
From A.18 and A.19
(A.21)
Since the movement of the front is given from Darcy's Law as
(A.22)
we have
(A.23)
(A.24)
where p(xF, t) is found either A.16 or A.17. Since
Theoretical Reservoir Engineering CalculationsRelated to Grout Injection
-289-
it can be seen that A depends on two dimensionless parameters,
c(p1 - po) and u1/uo. For small values of c(p l - po) and u1/uo~1
the result is
Hence the position of the front is approximately
Th i s i s , of course, the same result as would have been obtained i f t he
injected fluid were considered to have the same viscosity as the fluid
in place. Since c(p l - po) is usually small due to the compressibility
o f t h e f l u i d , i t is of interest to f ind the error between this re-
sult and the exact result for u1/uo > 1. The implication being that
the rate of front movement may be controlled by c(p l - po) rather than
by the viscosity rat io. The roots (A) of Equation A.24 are shown in the
following table for several values of u1/uo.
(A.25)
(A.26)
(A.27)
New Concepts in Underground Storage of Natural Gas
-290-
TABLE A-1
The error in the approximate result (equation A.26) is seen to be less
than 10% for ul/uo < 20.
The previous result may be interpreted by considering the injec-
tion rate in the following manner. The rate, q, in units of volume
injected per unit area per unit time is given by:
(A.28)
Theoretical Reservoir Engineering CalculationsRelated to Grout Injection
-291-
Terms (1) and (2) represent the rate of change of mass in the regions
on either side of the interface. From the exact solution the result
i s
For small
(A.29)
(A.30)
Hence the rate of front movement is (since
(A.31)
For A small this reduces to the rate given by Equation A.27. The con-
clusion is that the rate of inject ion in this instance (A small) is
controlled entirely by the rate of "compression" in the region beyond
the front and that viscosity plays a minor role in the region between
the injection face and the front.
This interpretation is of importance in that it may provide an
approximation method for situations where no analytical solutions are
avai lable (cyl indrical coordinates or viscosity varying with t ime).
New Concepts in Underground Storage of Natural Gas
-292-
Solutions for a Variable Viscosity Grout Solution-Plane Front
When the viscosity of the grout solution increases with time in
an arbitrary fashion i t is not possible to obtain analyt ic solut ions
in closed form, hence, it is necessary to employ approximate methods
to predict the movement of the front. The problem may be stated as
(A.31)
(A.32)
(A.33)
(A.34)
(A.35)
(A.36)
(A.37)
where
Theoretical Reservoir Engineering CalculationsRelated to Grout Injection
-293-
In order to apply finite difference techniques the problem stated
in equations A.31 through equation A.37 is reformulated in terms of a
rectangular grid, Then instead of seeking the values of the dependent
variable at every "point" in space and time, approximate values are
found at the grid nodes. The finite difference technique employed is
a variant of the DuFort-Frankel5 . 8
procedure. The details of the
finite difference technique and the computer program are explained
later in the Appendix.
Several runs were made to establish the accuracy of the program
and to evaluate the effects of varying viscosity.
TABLE A-2
SUMMARY OF COMPUTER RUNS
Ne
w
Co
nce
pts
in
Un
de
rgro
un
d
Sto
rag
e
of
Na
tura
l G
as
-29
4-
Theoretical Reservoir Engineering CalculationsRelated to Grout Injection
-295-
Table A-2 is a summary of the runs made. Figure A-5 indicates the
ef fec t o f on front location. It appears that there is an
optimum size for Figure A-6 shows the comparison between the
analyt ical and numerical solut ion for the front location. The best
values of seem to be The agree-
ment is seen to be good.
TABLE A-3
COMPARISON OF DIMENSIONLESS DENSITIES
Table A-3 compares the dimensionless density at several points near
the grout front as predicted by the analytical solution and by the
numerical solution, the agreement is seen to be good. These compari-
sons establish the capability of the numerical method to predict the
front location for the problem under study.
The study of the effects of having a grout fluid with a time
varying viscosity were made with a model grout having the viscosity-
time behavior shown on Figure A-4. Two studies were made, the dif-
ference being the size of the time step used. The results for the
Ne
w
Co
nce
pts
in
Un
de
rgro
un
d
Sto
rag
e
of
Na
tura
l G
as
-29
6-
Th
eo
retic
al
Re
se
rvo
ir E
ng
ine
erin
g
Ca
lcu
latio
ns
Re
late
d
to
Gro
ut
Inje
ctio
n-2
97
-
New Concepts in Underground Storage of Natural Gas
-298-
front location are shown in Figure A-7. For the purposes of comparison
the front locations for two constant viscosity grout solutions are also
shown. These latter viscosity ratios correspond to that at the begin-
ning and end of the curve shown on Figure A-4. As expected these re-
sults bound the results for the case of a time dependent viscosity. A
result which might have been anticipated from the study conducted with
the analytical solutions is that the almost 10 fold increase
grout solution viscosity has not affected the front location
tial ly, at least over the range of viscosity studied.
in the
substan-
Figure A-8 shows the dimensionless densities at several points
beyond the grout front as functions of time. Here the results are sur-
prising in that the densit ies f irst increase and then decrease sl ight ly.
This effect was not observed in the constant viscosity studies, which
were made with the same parameters. The effect does not appear to be
due to the numerical method, although a more exhaustive study should
be made to establish this conclusion beyond any doubt.
Solutions for Constant and Variable Viscosity Grout Solutions -
Cylindrical Front
Analyt ical solut ions are apparently not possible for the cyl in-
drical geometry and numerical solutions were employed. The physical
s i tuat ion is s imi lar to that descr ibed ear l ie r , except that
cylindrical coordinates are used and the solution is injected from a
well and the front is cyl indrical. The problem to be solved is obtained
by expression equations A.1 through A.7 in cylindrical coordinates as
(A. 38)
(A. 39)
Th
eo
retic
al
Re
se
rvo
ir E
ng
ine
erin
g
Ca
lcu
latio
ns
Re
late
d
to
Gro
ut
Inje
ctio
n
-29
9-
Ne
w
Co
nce
pts
in
Un
de
rgro
un
d
Sto
rag
e
of
Na
tura
l G
as
-30
0-
Theoretical Reservoir Engineering CalculationsRelated to Grout Injection
-301-
(A.40)
(A.41)
The finite difference forms of equations A.38 through A.41 are
given later in the Appendix. The DuFort-Frankel technique was employed
in much the same fashion as for the plane front problem. This problem
has, however, a characteristic which makes a complete solution rather
time consuming. For typical values of the parameters describing the
problem the dimensionless time, T, reaches values of 106 to 108. Since
the accuracy of the location of the front depends on the smallness of
A-r, a large quantity, of computer time may be required for each run.
Table A-4 summarizes the runs made for grouting in cylindrical
coordinates.
TABLE A-4
Summary of Computer Runs in Cylindrical Coordinates
New Concepts in Underground Storage of Natural Gas
-302-
The accuracy of the method was checked for uo/ul = 1 using the
tabulated exact solution of Jaeger5 . 9
for a related heat transfer prob-
lem which is described by the same differential equations and boundary
conditions. The front location for the exact solution is calculated
from the solution tabulated by Katz, Tek, et al.5 . 1 0
The comparison
is shown in Table A-S.
TABLE A-S
Comparison of Exact and Numerical Solutions
The agreement is seen to be good thus
numerical method.
establ ishing the val idi ty of the
The effect of the viscosity ratio, u l/uo, was checked with runs
fo r u l / u o= 1, 10, 100. Some evidence of instability was found in the
run with ul/uo = 100, = 10 - 3 , = .1 and the run was repeated with
= 1. No direct evidence of instabi l i ty was noted in this latter
case.
Theoretical Reservoir Engineering CalculationsRelated to Grout Injection
-303-
The relation between front location and time for parametric
values of ul/uo is shown in Figure A-8. Calculations were not carried
past a dimensionless time of 1, which is small compared with the values
to be expected in practice (106 - 108). The conclusions to be drawn
here must be qualified due to small time interval studied but are in
qualitative agreement with the results shown on Figures A-5 and A-6.
A 10 to 1 difference in viscosity between the grout solution and the
fluid in place has not caused a substantial change in the rate of front
movement while a 100 to 1 change has reduced the rate substantially.
No runs were made with a grout solution whose viscosity varied
with time due to the necessity for extensive computation.
Conclusions and Recommendations for Future Work
Grouting in a one-dimensional Cartesian coordinate system (a
plane grout-water interface) was studied in some detail with analytical
and numerical solutions of the differential equations governing the
grouting process.
(a) For constant viscosity grout solut ions or for those
whose viscosity is relatively constant for a period
of t ime after which i t increases rapidly, the analyt-
ical solution embodied in equations A.20 through
A.25 can be used to predict front movement.
(b) For variable viscosity solut ions where the viscosity
varies slowly with time but u l(t)/uo < 10 an average
viscosity may be used as in (a).
(c) For variable viscosity solut ions wherein u l( t ) /uo > 10
the numerical scheme developed and outlined in Appendix
should be used.
New Concepts in Underground Storage of Natural Gas
-304-
F i g . A - 9 . F r o n t L o c a t i o n - T i m e R e l a t i o n s h i p
Theoretical Reservoir Engineering CalculationsRelated to Grout Injection
-305-
Grouting in a one-dimensional cylindrical coordinate system
(cylindrical grout-water interface) was studied with a numerical cal-
culation scheme developed specifically for that purpose. The MAD
program is called RGROUT. For constant viscosity grout solutions
where ul/uo < 10 the tabulated solution of Katz, Tek, et al.5 . 1 0
may
be used as a first approximation.
The process of injection of a “high” viscosity grout solut ion
into a water saturated formation appears to be limited by the proper-
ties of the water saturated porous medium beyond the grout-water inter-
face for viscosity ratios (u l/uo) less than 10. This conclusion was
verified for the case of plane front grouting but is subject to qual-
i f icat ion for the case of cyl indrical front grouting due to the t ime
intervals studied. For higher viscosity grout solutions the viscosity
plays a greater role which is, however, smaller than might be anticipated
solely on the basis of Darcy’s law where the rate of movement is inver-
sely proport ional to the viscosity. This small viscosity effect seems
to be due to the fact that compression of the region beyond the front
is responsible for the flow. This is to be contrasted with flow through
a porous region where fluid is injected at one location and withdrawn
at another. In this latter si tuat ion viscosity must st i l l be a dominant
fac tor .
The results of this study, in particular the numerical methods
developed, provide a means for the systematic evaluation of a grout
insofar as the “ in jectab i l i ty ” of one grout is to be compared with
another. The effects of grout thickness on the strength of the grout
are still matters for experimental study,
The following recommendations are made in connection with the
results of this study.
(a) The computer program for grouting in cylindrical
coordinates should be used to study the effect of
space and time step size on the accuracy of front
New Concepts in Underground Storage of Natural Gas
-306-
location to extend the usefulness of the program
to large times (106 ~ 108).
(b) The effect of the viscosity rat io on front loca-
tion in cylindrical coordinates should be studied
at large times,
Sample Calculations for Analytical and Tabulated Solutions
(a) Prediction of the movement of a plane grout front.
G i ven : p l -p o = 1000 ps i
R = 250 mi l l idarc ies
E = .15
u o = 1 centipoise
u1= 10 centipoise
C = 7 x 10 - 6 ( ps i a ) - 1
t = time in hours
From equation A.26, the approximate value of is (as a first
approximation)
Hence B(X) (equation A.22) is .96 and equations A.24 and A.25 yield
The d i f fus iv i ty Ko i s
Theoretical Reservoir Engineering CalculationsRelated to Grout Injection
-307-
Hence
(b) Prediction of the movement of a cylindrical front
for low viscosity grout solut ions, (u l( t ) /uo < 10).
From Katz et al.5 . 1 0
(pg. 54)
where Qt is tabulated in the reference cited above,
This Page Intentionally Left Blank
APPENDIX A-1
Numerical Techniques for the Grout Injection Problem
with a Variable Viscosity Grout Solution-Plane Front.
Fig, A-10 Shows the Sequence of Numbers Used in Numerical
I terat ions
The region x > 0 is divided into segments of length Ax. A gen-
eral point is denoted by the subscript i, so the distance from the
inject ion face ( i = 0) is The point immediately behind the front
is denoted by the subscript M, and the point immediately ahead by M + 1.
The distance from the point M to the front is Both M and may
change with time. The time domain is divided into increments of At,
hence t = the subscript n being the “time level.” A DuFort-Frankel
type5 . 8
of approximation procedure was chosen to represent the system
of equations describing the injection process (equations A.31 to A.37).
This particular procedure was chosen since it is an explicit method and
possess the property of uncondit ional stabi l i ty for the parabol ic dif-
fusion equation. The explicit nature is of particular importance in a
“moving boundary” problem of this type since the value of the dependent
variable at the front cannot be advanced in time until the new front
location is known, This is, however, no guarantee that the method will
be a stable for the problem at hand due to the introduction of a time
-309-
New Concepts in Underground Storage of Natural Gas
-310-
dependent diffusivity and a "moving boundary" (the front). A few tests
were made to examine the accuracy and stability and for good accuracy
must be less than 1/4. No evidence of instability was noted.
The finite difference approximations (FDA) used are at a general
po in t : 0 < i < M , n > l
at a general point, M+1 < i, n > l
at point M, n > 1 ("*" subscript denotes the front)
at point M+1, n > 1-
(A.42)
(A.43)
(A.44)
(A.45)
Numerical Techniques for the Grout Injection Problemwith a Variable Viscosity Grout Solution-Plane Front
-311-
The movement of the front is calculated from
The dimensionless density, P*n+1, , is obtained from
The problem is solved by taking the starting values as
Then new values of etc. , are calculated using equations A.42
through A.47. Special provisions are included in the program for the
situation when the front is close to a grid point. The program was
written in the MAD language.5 . 1 1
The runs were made using the Univer-
sity of Michigan IBM 7090.
The data required for implementations of the program GROUT are
Variable
DTAU
DX
K
Mode Uni t
f loat ing po in t hoursII II feet11 II ( f t ) 2 / h r .
Problem Variable
time step
space step
diffusivity,
New Concepts in Underground Storage of Natural Gas
-312-
Variable
CP
DELTAP
Mode
f loat ing po in t
II 11
Unit Problem Variable
( l b F / ( i n ) 2 ) - 1 compressibi l i ty
l b F / ( i n ) 2 c h a r a c t e r i s t i cpressure
TAUMAX
LIM
11 II hours maximum time
in teger init ial range onsubscript
DLIM II change in range onsubscript i
PMIN f loat ing po in t whenever the valueof P at point LIMis above PMIN, LIM =LIM + DLIM
FPRNT in teger frequency of print-ing
The main program calls on an external function called MU, which
has as an argument the variable TAU and returns the value of uo/ul(t).
The output is in the form of a table giving the time in hours, the
viscosity rat io uo/u l( t) , the subscript of the point nearest the front
in the grouted region, the location of the front in feet, the dimension-
less density at the seven points nearest the grout interface, including
the interface. For DTAU = .01, DX = 50, K = 1 x 105, CP = 7 x 10-6,
DELTAP = 1000, TAUMAX = 7., LIM = 10, DLIM = 10, PMIN = 1 x 10-5
,
FPRNT = 10 the execution time was 14.7 sec. or .021 sec/time step.
APPENDIX A-2
Numerical Techniques for the Grout Injection Problem with a Constant or
Variable Viscosity Grout Solution - Cylindrical Front
The grid and notation are similar to that used for the plane front
problem and only the finite difference approximations will be noted here.
At a general point 0 < i < M, n > 1
(A-48)
At a general point M+1 < i, n > 1, the equation is identical to
A.48 with uo/u l(r) = 1.
At point M, n > 1
(A.49)
-313-
New Concepts in Underground Storage of Natural Gas
-314-
At point M+1, n > 1
(A.50)
The movement of the front is found from
(A.51)
The dimensionless density, P*n+1
(A.52)
The starting values are
The starting value for the front location is not taken as
being at R = 1 but at the location calculated from the numerical solu-
tion tabulated in Katz, Tek, et al.5 . 1 0
for the case uo/u1(T) = 1.
This feature is an attempt to allow the front to be located accurately
when large time steps are used.
Numerical Techniques for the Grout InjectionProblem with a Constant or Variable ViscosityGrout Solution - Cylindrical Front
-315-
'The data required for implementation of the program RGROUT. are
Variable
DTAU
DR
CP
DELTAP
Mode Uni t
FP none
II none
II (lbF/(in2)) -1
II l b F / ( i n ) 2
TAUMAX II none
RFnone
Problem Variable
time step
space step
compressibi l i ty
characterist ic pressured i f ference (p l -po)
maximum time
location of front atTAU = DTAU
LIM, DLIM} as explained in A-1.
PMIN, FPRNT
The main program calls MU. and the output is in the same form as
for the linear case except that the variables are all dimensionless.
For DTAU = .001, DR = .1, CP = 7 x 106, DELTAP = 1000., TAUMAX = 1.,
LIM = 10, DLIM = 10, PMIN = 1 x 10-5, FPRNT = 10, RF = 1.0002. 17.7
sec of execution time was required or .017 sec/time step.
This Page Intentionally Left Blank
New Concepts in Underground Storage of Natural Gas
-316-
COMPUTER PROGRAMS FOR EVALUATION OF GROUTS
A.1 Program for Calculation of Eq. A.24
This Page Intentionally Left Blank
Computer Programs for Evaluation of Grouts
-317-
A.2 Program for Grouting with a Plane Front
New Concepts in Underground Storage of Natural Gas
-318-
Computer Programs for Evaluation of Grouts
-319-
A.3 Program for Grouting with a Cylindrical Grout Front
New Concepts in Underground Storage of Natural Gas
-320-
A p p e n d i x B
SOLUTION OF A SYSTEM OF LINEAR EQUATIONS
HAVING A TRIDIAGONAL COEFFICIENT MATRIX
T h e f i n i t e d i f f e r e n c e m e t h o d s d e s c r i b e d i n C h a p t e r s 3 a n d 4
r e s u l t i n s y s t e m s o f s i m u l t a n e o u s e q u a t i o n s o f t h e f o r m :
(B-1)
Here , t h e v a r i a b l e s v i w i l l b e u n k n o w n v a l u e s o f P o r R ( C h a p t e r 3 ) ,
o r u n k n o w n v a l u e s o f d i m e n s i o n l e s s p r e s s u r e ( C h a p t e r 4 ) , a l o n g a
s i n g l e r o w o f g r i d p o i n t s . T h e v a l u e s o f t h e c o e f f i c i e n t s a i , b i ,
c i , a n d d i a r e g i v e n b y e x p r e s s i o n s s u c h a s ( 3 . 3 8 ) , ( 3 . 4 2 ) , ( 4 . 2 0 ) ,
a n d ( 4 . 2 6 ) . I t m a y b e s h o w n t h a t t h e s o l u t i o n t o e q u a t i o n s ( B - 1 )
i s g i v e n b y t h e a l g o r i t h m
( B - 2 )
w h e r e t h e ß ' s a n d y ' s a r e d e t e r m i n e d b y t h e r e c u r s i o n f o r m u l a e
-321-
New Concepts in Underground Storage of Natural Gas
-322-
( B . 3 )
Append ix C
COMPUTER PROGRAMS FOR CHAPTER 3
T h e p r o g r a m f o r c o m p u t i n g t h e l e a k a g e e f f e c t s i n t h e c a p
r o c k c o n s i s t s o f a m a i n p r o g r a m a n d e i g h t s u b r o u t i n e s . These
p r o g r a m s a r e w r i t t e n i n t h e M A D ( M i c h i g a n A l g o r i t h m D e c o d e r )
l a n g u a g e a n d a r e l i s t e d b e l o w , t o g e t h e r w i t h d e f i n i t i o n s o f t h e
m a i n v a r i a b l e s i n v o l v e d . A s h o r t s t a t e m e n t i s g i v e n a t t h e
b e g i n n i n g o f e a c h p r o g r a m i n d i c a t i n g t h e p u r p o s e f o r w h i c h i t i s
used. I n a d d i t i o n , t h e s t a t e m e n t s o f t h e m a i n p r o g r a m a r e i n t e r -
s p e r s e d w i t h " r e m a r k " c a r d s w h i c h d e s c r i b e t h e m a j o r c o m p u t a t i o n a l
s t e p s .
-323-
This Page Intentionally Left Blank
A, B, C, D
AMP
BASE
C1, C2
CARD
CHECK
DATANO
DT
DTDIM
DX
DXSQ
E, F
EPS
F1
F2
F3
F4
F5
F6
F7
F8
New Concepts in Underground Storage of Natural Gas
-324-
L i s t o f P r i n c i p a l V a r i a b l e s
MAD Symbol D e f i n i t i o n
C o e f f i c i e n t a r r a y s i n t r i d i a g o n a l s y s t e m s
r e s u l t i n g f r o m e q u a t i o n s ( 3 . 3 7 ) a n d ( 3 . 4 1 ) .
F r a c t i o n a l a m p l i t u d e f o r s i n e w a v e r e p r e s e n t i n g
g a s p o t e n t i a l a t l o w e r i n t e r f a c e .
D i m e n s i o n l e s s g a s p o t e n t i a l a t l o w e r i n t e r f a c e ,
upon wh i ch a s i ne wave i s supe r imposed .
C o n s t a n t s i n c a p i l l a r y p r e s s u r e e q u a t i o n , ( 3 . 5 8 )
B o o l e a n v a r i a b l e ; if CARD = 1 B , t h e n p r e v i o u s l y
c o m p u t e d p o t e n t i a l d i s t r i b u t i o n s a r e r e a d a s d a t a .
O t h e r w i s e , t h e p r o g r a m w i l l g e n e r a t e i t s o w n
s t a r t i n g p o t e n t i a l s .
B o o l e a n v a r i a b l e ; if CHECK = 1 B , t h e n t h e r e s u l t s
o f m a n y i n t e r m e d i a t e c o m p u t a t i o n s w i l l b e p r i n t e d
f o r p r o g r a m c h e c k i n g p u r p o s e s .
I n d i c a t e s a p a r t i c u l a r s e t o f p r o p e r t i e s f o r t h e
c a p r o c k .
T i m e i n c r e m e n t ( d a y s ) .
D i m e n s i o n l e s s t i m e i n c r e m e n t ,
D i m e n s i o n l e s s s p a c i n g b e t w e e n g r i d p o i n t s ,
C o e f f i c i e n t s d e f i n e d i n e q u a t i o n s ( 3 . 3 0 ) a n d ( 3 . 3 1 ) .
P o r o s i t y o f c a p r o c k ,
C o n v e r s i o n f a c t o r , ( c e n t i p o i s e . s q . f t . ) / ( p . s . i .
m i l l i d a r c y . d a y ) .
u w / u g .
(n - 1)2, ß o f e q u a t i o n ( 3 . 5 1 ) .
( n - 1 ) 2 u w / u g , o f e q u a t i o n ( 3 . 4 9 ) .
u g / u w .
Computer Programs for Chapter 3
-325-
L i s t o f P r i n c i p a l V a r i a b l e s ( c o n t d . )
MAD Symbol D e f i n i t i o n
FRPNCH Number o f t ime i nc remen ts e l aps ing be tween
s u c c e s s i v e p u n c h i n g o f p o t e n t i a l d i s t r i b u t i o n s
o n t o c a r d s .
FRPRNT
G
I
ITER
ITMAX
K
KG
KPRINT
KPUNCH
KW
MUG
MUW
N
P
PERIOD
PHIG
Number o f t ime i nc remen ts e l aps ing be tween
s u c c e s s i v e p r i n t i n g s o f p o t e n t i a l d i s t r i b u t i o n s .
I n t e r m e d i a t e a r r a y , d e f i n e d b y e q u a t i o n s ( 3 . 3 8 ) ,
( 3 . 3 9 ) , a n d ( 3 . 4 0 ) , i n v o l v e d i n s o l u t i o n o f t h e
p a r a b o l i c P e q u a t i o n s .
I n d e x d e n o t i n g t h e i t h g r i d p o i n t .
I t e r a t i o n c o u n t e r f o r u s e i n c o m p u t i n g i n j e c t i o n
t e r m s .
Max imum a l l owab le va lue o f ITER.
C a p r o c k p e r m e a b i l i t y , m i l l i d a r c i e s .
R e l a t i v e p e r m e a b i l i t y o f c a p r o c k t o g a s .
I ndex used when coun t i ng number o f t ime s teps
e l a p s i n g b e t w e e n s u c c e s s i v e p r i n t i n g s o f p o t e n -
t i a l d i s t r i b u t i o n s .
I ndex used when coun t i ng number o f t ime s teps
e l a p s i n g b e t w e e n s u c c e s s i v e p u n c h i n g s o f p o t e n -
t i a l d i s t r i b u t i o n s .
R e l a t i v e p e r m e a b i l i t y o f c a p r o c k t o w a t e r .
T h i c k n e s s o f c a p r o c k , f e e t .
G a s v i s c o s i t y , u c e n t i p o i s e .
W a t e r v i s c o s i t y , u c e n t i p o i s e .
S u b s c r i p t d e n o t i n g t h a t g r i d p o i n t w h i c h i s i n
t h e w a t e r a b o v e t h e c a p r o c k .
H a l f t h e s u m o f t h e d i m e n s i o n l e s s p o t e n t i a l s ,
P e r i o d i n d a y s o f s i n e w a v e r e p r e s e n t i n g t h e g a s
p o t e n t i a l a t t h e l o w e r i n t e r f a c e .
D i m e n s i o n l e s s g a s p o t e n t i a l ,
New Concepts in Underground Storage of Natural Gas
-326-
L i s t o f P r i n c i p a l V a r i a b l e s ( c o n t d . )
MAD Symbol D e f i n i t i o n
PHIGN
PHIGX
PHIW
PHIWN
PHIWX
POLD
QG
QGX
QW
QWX
R
RHOG
RHOW
ROLD
RUN
S
SP
T
TMAX
TZERO
D i m e n s i o n l e s s g a s p o t e n t i a l a t u p p e r b o u n d a r y ,
A r r a y u s e d f o r t e m p o r a r y s t o r a g e o f n e w l y c o m -
p u t e d g a s p o t e n t i a l s .
D i m e n s i o n l e s s w a t e r p o t e n t i a l ,
D i m e n s i o n l e s s w a t e r p o t e n t i a l a t u p p e r b o u n -
d a r y ,
A r r a y u s e d f o r t e m p o r a r y s t o r a g e o f n e w l y c o m -
p u t e d w a t e r p o t e n t i a l s .
A r r a y u s e d f o r s t o r a g e o f P v a l u e s a t t h e b e g i n -
n i n g o f a t i m e s t e p .
A r r a y c o n t a i n i n g d i m e n s i o n l e s s g a s i n j e c t i o n r a t e s .
A r r a y u s e d f o r t e m p o r a r y s t o r a g e o f Q G v a l u e s .
A r r a y c o n t a i n i n g d i m e n s i o n l e s s w a t e r i n j e c t i o n
r a t e s .
A r ray used f o r t empora ry s to rage o f QW va lues .
H a l f t h e d i f f e r e n c e b e t w e e n t h e d i m e n s i o n l e s s
p o t e n t i a l s ,
G a s d e n s i t y , p g l b m . / c u . f t .
W a t e r d e n s i t y , p w l b m . / c u . f t .
A r r a y u s e d f o r s t o r a g e o f R v a l u e s a t t h e b e g i n -
n i n g o f a t i m e s t e p .
R u n i d e n t i f i c a t i o n n u m b e r .
W a t e r s a t u r a t i o n , S .
D e r i v a t i v e o f w a t e r s a t u r a t i o n w i t h r e s p e c t t o
d i m e n s i o n l e s s c a p i l l a r y p r e s s u r e , d S / d P c .
T ime, t d a y s .
U p p e r l i m i t o n T f o r w h i c h r e s u l t s a r e r e q u i r e d .
I n i t i a l v a l u e o f T ( u s u a l l y z e r o ) .
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Compu te r P rog rams fo r Chap te r 3
- 3 2 7 -
New Concepts in Underground Storage of Natural Gas
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Compu te r P rog rams fo r Chap te r 3
- 3 2 9 -
New Concepts in Underground Storage of Natural Gas
-330-
Compu te r P rog rams fo r Chap te r 3
- 3 3 1 -
New Concepts in Underground Storage of Natural Gas
-332-
Compu te r P rog rams fo r Chap te r 3
- 3 3 3 -
New Concepts in Underground Storage of Natural Gas
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Compu te r P rog rams fo r Chap te r 3
- 3 3 5 -
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Append ix D
COMPUTER PROGRAMS FOR CHAPTER 4
T h e c o m p l e t e p r o g r a m f o r c o m p u t i n g t h e e f f e c t o f t h e a r e a -
d i s t r i b u t e d l e a k c o n s i s t s o f a m a i n p r o g r a m a n d t h r e e s u b r o u t i n e s .
These p rog rams a re w r i t t en i n t he MAD (M ich igan A lgo r i t hm Decode r )
l a n g u a g e a n d a r e l i s t e d b e l o w , t o g e t h e r w i t h d e f i n i t i o n s o f t h e
m a i n v a r i a b l e s i n v o l v e d . A s h o r t s t a t e m e n t i s g i v e n a t t h e
b e g i n n i n g o f e a c h p r o g r a m i n d i c a t i n g t h e p u r p o s e f o r w h i c h i t i s
used. I n a d d i t i o n , t h e s t a t e m e n t s o f t h e m a i n p r o g r a m a r e i n t e r -
s p e r s e d w i t h " r e m a r k " c a r d s w h i c h d e s c r i b e t h e m a j o r c o m p u t a t i o n a l
s t e p s .
-337-
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New Concepts in Underground Storage of Natural Gas
-338-
L i s t o f P r i n c i p a l V a r i a b l e s
MAD Symbol D e f i n i t i o n
A, B, C, D
CHECK
C o e f f i c i e n t a r r a y s i n t r i d i a g o n a l s y s t e m s
r e s u l t i n g f r o m e q u a t i o n s ( 4 . 1 9 ) a n d ( 4 . 2 5 ) .
T h e c o n s t a n t C i t s e l f i s a l s o u s e d f o r i n i t i a l
s t o r a g e o f t h e c o n s t a n t c i n t h e e q u a t i o n p = c p .
B o o l e a n v a r i a b l e ; if CHECK = 1 B , t h e n a l l
c o m p u t e d p r e s s u r e s ( b o t h a t t h e e n d o f t h e f i r s t
a n d s e c o n d h a l f t i m e s t e p s ) w i l l b e p r i n t e d f o r
check ing pu rposes . Normal ly CHECK wi l l have
t h e v a l u e 0 B .
COUNT
DR
DRSQ4
DT
DTDAYS
DTHETA
EPS
FACTOR
FREQ
GAM4, GAM8
GAMMA
I, J
IMAX, JMAX
K
LAMBDA
MU
MZERO
P
Coun te r on t he number o f who le t ime s teps .
D i m e n s i o n l e s s r a d i a l i n c r e m e n t ,
D i m e n s i o n l e s s t i m e i n c r e m e n t ,
T i m e i n c r e m e n t i n d a y s .
A n g u l a r i n c r e m e n t ,
P o r o s i t y .
C o n v e r s i o n f a c t o r , ( m i l l i d a r c i e s . p s i . d a y /
c e n t i p o i s e . s q . f t . ) .
Number o f t ime i nc remen ts e l aps ing be tween suc -
c e s s i v e p r i n t i n g s o f p r e s s u r e f i e l d .
4y and 8y , r e s p e c t i v e l y .
G r i d p o i n t s u b s c r i p t s f o r r a d i a l a n d a n g u l a r
d i r e c t i o n s , r e s p e c t i v e l y .
U p p e r l i m i t s f o r I a n d J , b e i n g t h e m a n d n
r e f e r r e d t o i n S e c t i o n 4 . 5 .
P e r m e a b i l i t y , k m i l l i d a r c i e s .
V i s c o s i t y , u c e n t i p o i s e .
D i m e n s i o n l e s s p r e s s u r e , P .
Computer Programs for Chapter 4
-339-
L i s t o f P r i n c i p a l V a r i a b l e s ( c o n t d . )
MAD Symbol D e f i n i t i o n
PLEAK
PSTAR
PZERO
Q
R1, R2
RMAX
RUN
SAVEP
SMALLK
T
TMAX
TYPE
TZERO
U
A r r a y c o n t a i n i n g v a l u e s o f p L , t h e t h r e s h o l d
p r e s s u r e b e l o w w h i c h n o l e a k o c c u r s , a t e a c h
g r i d p o i n t , f o r u s e i n e q u a t i o n ( 4 . 9 ) .
A r r a y c o n t a i n i n g t h e d i m e n s i o n l e s s p r e s s u r e s
P * a t t h e e n d o f a h a l f t i m e s t e p .
T h e i n i t i a l f o r m a t i o n p r e s s u r e , p o .
Leak term, Q = Me2R = ( m / m o ) e 2 R , d i m e n s i o n l e s s .
I n n e r a n d o u t e r r a d i i , r l , r 2 , f e e t .
M a x i m u m v a l u e o f d i m e n s i o n l e s s r a d i u s , l n ( r 2 / r l ) .
N u m b e r o f p a r t i c u l a r i n v e s t i g a t i o n .
U s e d f o r t e m p o r a r y s t o r a g e o f w e l l p r e s s u r e .
A r r a y c o n t a i n i n g c o n s t a n t k * i n e q u a t i o n ( 4 . 9 )
f o r t h e l e a k t e r m a t e a c h g r i d p o i n t .
T ime, t d a y s .
M a x i m u m v a l u e o f t i m e i n d a y s f o r w h i c h c a l -
c u l a t i o n s a r e t o b e p e r f o r m e d .
A r r a y c o n t a i n i n g e i t h e r t h e v a l u e 1 o r 2 a t e a c h
g r i d p o i n t ; t h e v a l u e 1 m e a n s t h a t t h e r e i s n o
l e a k a t t h a t p o i n t ; t h e v a l u e 2 m e a n s t h a t t h e r e
i s a l e a k , a c c o r d i n g t o e q u a t i o n ( 4 . 9 ) .
A r ray con ta in ing t he va lue e 2 R a t s u c c e s s i v e
p o i n t s i n t h e r a d i a l d i r e c t i o n .
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- 3 4 1 -
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