New Concepts in Underground Storage of Natural Gas

Catalog No. L00400e

New Concepts In Underground Storage Of Natural Gas

Prepared for the

Pipeline Research Technical Committee

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

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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.

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

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

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

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

TABLE OF CONTENTS

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

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

Table of Contents (contd)

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

4.3 Formulation of Leak . . . . . . . . . . . . . . . . . 79

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). . . . . . . . . . . . . . . .

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

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

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

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

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.

New Concepts in Underground Storage of Natural Gas

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

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.

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

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

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.

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

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.

Katz, D. L., et al . , Handbook of Natural Gas Engineering,McGraw Hill, (1959).

Private Communication, D. L. Katz.

Nielsen, R. L., On the Flow of Two Immiscible IncompressibleFluids in Porous Media, Ph.D. Dissertation, University ofMichigan.

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

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

The leakage or spill of gas from a storage reservoir may be due

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

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

“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.

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

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.

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

= 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

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.

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

F i g . 2 . 5 The Rubber S leeved Core Holder

ks fro

rvoirs

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

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.”

TABLE 2.1

LABORATORY DATA FOR USE IN CORRELATION OF THRESHOLD PRESSURES

Problems Associated with Leaks fromOverpressured Storage Reservoirs

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.

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

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

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 )

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 .

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

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 .

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 )

( 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 )

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 )

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 )

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 .

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 .

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

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 )

(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 )

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)

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 .

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 .

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 .

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 ,

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

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 )

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

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

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

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 .

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

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

( 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 .

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 ) .

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 .

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

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 .

ts in U

e of N

atural G

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 .

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

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 .

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 )

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 )

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

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

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 :

( 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 .

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 )

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 .

F i g . 4 . 2 E f fec t o f Leak on Gas Bubb le P ressu re .

F i g . 4 . 3 E f fec t o f Leak on Gas Bubb le P ressu re .

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 .

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 .

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

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 )

i . e . ,

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 )

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 .

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

( 4 . 2 4 )

i . e . ,

i n w h i c h

( 4 . 2 5 )

( 4 . 2 6 )

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

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 .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

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

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

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 .

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 .

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 .

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

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 .

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

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

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

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

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 .

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

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 .

T a b l e

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

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

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 .

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

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 .

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

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

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 .

t h i s

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 .

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

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

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 .

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 .

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

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 .

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

T a b l e 5 . 9

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

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

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 .

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

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 .

F i g . 5 . 1 5 Photo Micrographs on Grouted Core Specimens

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

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 . 3 )

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 )

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 .

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 )

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

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

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 )

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

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 .

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 )

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

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 .

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 )

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

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

( 5 . 1 0 )

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

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

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 )

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

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

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

( 5 . 1 3 )

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 ) .

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 .

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

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

FIG. 5.29 RESTRICTED AND UNRESTRICTED VERTICAL FRACTURES.5.12

FIG. 5.30 CRACK WIDTHS FOR RESTRICTED VERTICAL FRACTURE RESULTING

FROM NON-NEWTONIAN FLUIDS IN LAMINAR FLOW.5.12

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 )

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

FIG. 5.32 VISCOSITY OF A SLURRY CONTAlNING SUSPENDED SOLID

MATERIAL COMPARED TO THE VISCOSITY OF THE BASIC FLUID.

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

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

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

( 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 )

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

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 .

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 .

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

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

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

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

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 -

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 .

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 .

7 . Ps = Pb + Pf - Ph ( 5 . 2 1 )

P s = 2000 + 250 - 840

PS = 1410 ps ig

HHP = 0.0245 PsVi

HHP = (0.0245)(1410)(25)

HHP = 863 horsepower.

( 5 . 2 2 )

P G = 865 ps ig

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

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 )

17. F r o m F i g u r e 5 . 2 6 , A = 1 5 0 , 0 0 0 f t 2.

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

17. From F i g u r e 5 . 2 6 , A = 1 6 5 , 0 0 0 f t 2 .

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 .

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

Nomencla ture

C I I I =

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 .

N o m e n c l a t u r e , c o n t i n u e d

Q , Q i , V i =

S p G r =

W’ =

u , u F =

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 .

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 .

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 .

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 .

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 .

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 .

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 .

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 :

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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 .

5 . 2 8 *

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 .

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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 .

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

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

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

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

Fig. 6.1 Favorable and Unfavorable Areas For Productionand Storage of Natural Gas

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

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

where h = depth to the LP gas brine contact, feet

P = casing pressure, psig

dB = specif ic gravity of brine,

ts inU

FIG. 6.3 A SIMPLIFIED VIEWOF A SALT STORAGE WELL.

FIG. 6.4 AVERAGE CAVERNDIAMETERS AS DETERMINED FOREQUILIBRIUM PRESSURE CALCULATIONS.

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

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:

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

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

FIG. 6.5 SALT CAVERN STORAGE VARIATION OF RESERVOIR PRESSURE WITHTIME FOR VARIOUS DEGREES OF WATER REPLACEMENT.

(Vc = 109 FT3, Qs = 50x106 SCF/DAY )

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,

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

(6.10)

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

(6.12)

(6.13)

Now substituting the value of p from (6.13) into (6.9) and (6.10):

(6.14)

(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)

(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

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

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

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,

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

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:

(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.,

(6.22)

nG= the number of moles of gas in cavern at any time

(6.23)

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

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.

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

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.

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

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.

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

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.

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

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

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,

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

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.

Fig. 6.10. Gnome cavity profile.6 . 4 2

TABLE 6.6. CAVITY RADII CALCULATED AS A FUNCTION OF ROCK TYPE,

YIELD AND DEPTH OF BURIAL

Fig. 6.11. Hardhat schematic cross section.6 . 4 2

Fig. 6.12. Rainier schematic cross section.6 . 4 2

Fig. 6.13. Blanca schematic cross section.6 . 4 2

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

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.

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.

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

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.

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.

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

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

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

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)

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

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 .

Fig. 6.21 Effect of Depth on the Pressure

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)

(6.34)

where ZS = 1.0 and subscript s refers to standard conditions.

(6.35)

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)

(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)

Vc is calculated from equation 6.36 and C = 3.84.

F i g . 6.22 Area of a Spherical Vessel Required to Contain

1 BSCF Gas Underwater

Figure 6.23 is a plot of the equations

w i t h

(6.42)

(6.43)

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.

Fig. 6.25 Pressure Differential for a Spherical Vessel

Containing 1 BSCF Gas Underwater

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.

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.

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,

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.

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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.

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

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:

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,

(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.

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

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

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

Then since

(A. 4)

these equations may be combined to give a partial differential equa-

tion involving density, position and time as

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

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.

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.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)

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)

(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

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)

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)

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

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).

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)

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

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

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)

(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

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

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.

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

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

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

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

(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,

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

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

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

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,

Variable

DELTAP

f loat ing po in t

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

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)

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

'The data required for implementation of the program RGROUT. are

Variable

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.

COMPUTER PROGRAMS FOR EVALUATION OF GROUTS

A.1 Program for Calculation of Eq. A.24

Computer Programs for Evaluation of Grouts

A.2 Program for Grouting with a Plane Front

Computer Programs for Evaluation of Grouts

A.3 Program for Grouting with a Cylindrical Grout Front

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 :

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

( 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 .

A, B, C, D

C1, C2

DATANO

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

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 . )

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

KPRINT

KPUNCH

PERIOD

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 ,

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 ) .

Compu te r P rog rams fo r Chap te r 3

- 3 2 7 -

- 3 2 9 -

- 3 3 1 -

- 3 3 3 -

- 3 3 5 -

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 .

L i s t o f P r i n c i p a l V a r i a b l e s

A, B, C, D

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 .

DTDAYS

DTHETA

FACTOR

GAM4, GAM8

IMAX, JMAX

LAMBDA

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

R1, R2

SMALLK

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 .

- 3 4 1 -

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