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SINGLE- AND CROSS-HOLE PNEUMATIC INJECTION TESTS IN
UNSATURATED FRACTURED TUFFS AT THE APACHE LEAP RESEARCH SITE
NEAR SUPERIOR, ARIZONA
by
Walter Arthur Illman
(Sho Yamamoto)
Copyright © Walter Arthur Illman 1999
A Dissertation Submitted to the Faculty of the
DEPARTMENT OF HYDROLOGY AND WATER RESOURCES
In Partial Fulfillment of the Requirements
For the Degree of
DOCTOR OF PHILOSOPHY
WITH A MAJOR IN HYDROLOGY
In the Graduate College
THE UNIVERSITY OF ARIZONA
1999
DMX Niunber: 9927499
Copyright 1999 by
Illmsm, Walter Arthur
All rights resorved.
UMI Microform 9927499
Copyright 1999, by UMI Company. All rights reserved.
This microform edition is protected against unauthorized
copying under Title 17, United States Code.
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2
THE UNIVERSITY OF ARIZONA «
GRADUATE COLLEGE
As members of the Final Examination Committee, we certify that we have
read the dissertation prepared by
entitled
Walter Arthur Illman
Single- and Cross-hole Pneumatic Injection Tests in Unsaturated
Fractured Tuffs at the Apache Leap Research Site near Superior, AZ
and recommend that it be accepted as fulfilling the dissertation
requirement for the Degree of
Shlomo
Doctor of Philosophy
Da^
Neuman
'
D a t e ^ ^
Peter J. Wierenga
2
Date
G. Zreda
Dona.m E. Mye
Date
Arthur W. Warrick
Date
Final approved, and acceptance of this dissertation is contingent upon
the candidate's submission of the final copy of the dissertation to the
Graduate College.
I hereby certify that I have read this dissertation prepared under my
direction and recommend that it be accepted as fulfilling the dissertation
rei
Dissertation Director
Shlomo P. Neuman
'ate
3
STATEMENT BY AUTHOR
This dissertation has been submitted in partial fulfillment of requirements for an
advanced degree at the University of Arizona and is deposited in the University Library
to be made available to borrowers under rules of the Library.
3rief quotations from this dissertation are allowable without special permission,
provided that accurate acknowledgement of source is made. Requests for permission for
extended quotation from or reproduction of this manuscript in whole or in part may be
granted by the copyright holder.
SIGNED:
[
4
ACKNOWLEDGEMENTS
I am indebted to Regents' Professor Shiomo Peter Neuman, my dissertation
director and finend for iiis guidance, support, patience and time taken from his very busy
schedule. I will never forget his lectures (HWR 503, 504, and 603), conference
presentations, numerous discussions on various topics pertaining to hydrogeology, and
his commitment to teaching and research. He always took time to listen to my problems,
whether it pertained to academics, research, or personal matters. His skills in challenging
students to improve their understanding of hydrogeology/mathematics are urunatched and
commendable.
I would like to thank Professors Peter J. Wierenga, Donald E. Myers, Arthur W.
Warrick, and Marek G. Zreda for serving on my dissertation committee. I would also
like to thank Professor James E. Smith for his personal advice in teaching and research.
I wish to acknowledge with gratitude the unwavering and effective support,
advice and encouragement provided to me throughout this study by our NRC Project
Manager, Mr. Thomas J. Nicholson, without whom this work would never have
materialized.
I would also like to thank Mr. Dick L. Thompson, for his commitment and
excellence as a field technician during my field project. Our success in conducting crosshole pneumatic tests and its substantial cost-savings were accomplished through his
ingenious inventions and persistence in the field.
I must acknowledge and thank Dr. Daniel M. Tartakovsky for being my good
friend, improving my understanding of mathematics, and teaching me the simplicity and
beauty of analytical solutions.
My parents (William and Hatsue Illman) must be acknowledged for their support
throughout my life and studies. Difficult times encountered throughout my research and
studies at the University of Arizona were always resolved with telephone calls to Japan.
Finally, I would like to thank Ms. Carla Marie Frascari for providing me
emotional support during my last year at the University of Arizona. It has been very
difficult this last year, but her love, positive attitude, kindness, forgiveness and patience
is leading me to become a better person.
The majority of this research was supported by the United States Nuclear
Regulatory Commission under contracts NRC-04-95-038 and NRC-04-97-056. It was
also supported in part by a National Science Foundation Graduate Traineeship during
1994-1995, a University of Arizona Graduate College Fellowship during 1997-1998, the
Horton Doctoral Research Grant from the American Geophysical Union during 19971998, and the John and Margaret Harshbarger Doctoral Fellowship from the Department
of Hydrology and Water Resources at The University of Arizona during 1998-1999.
5
Those who understand classical music have a good grasp of mathematics.
(translated from Japanese)
Hatsue Yamamoto Illman
Dreaming of success leads to success. One must also remember that success is a result of
diligence, patience, honesty, and persistence.
William Richard Illman
6
To My Carla
7
TABLE OF CONTENTS
STATEMENT BY AUTHOR
3
ACKNOWLEDGEMENTS
4
TABLE OF CONTENTS
7
LIST OF ILLUSTRATIONS
10
LIST OF TABLES
19
LIST OF PLATES
21
ABSTRACT
22
1. INTRODUCTION
24
I. I ROLE OF PNEUMATIC TESTS IN SITE CHARACTERIZATION
1.2 THE ALRS AND PREVIOUS SITE INVESTIGATIONS
1.2.1
Site Description
1.2.2
Previous Work in Study Area of ALRS
1.2.3
Single-Hole Pneumatic Injection Tests
1.2.4
Kriged Estimates of Air Permeability at the ALRS
1.2.5
Geophysical Surveys Conducted at Site
1.3 TYPE-CURVE ANALYSIS OF SINGLE-HOLE PNEUMATIC
INJECTION TESTS
1.4 CROSS-HOLE PNEUMATIC INJECTION TESTS
1.5 SCOPE OF THE DISSERTATION
2. GAS FLOW IN POROUS AND FRACTURED MEDIA
2.1 THEORY OF GAS FLOW IN POROUS MEDIA
2.1.1 Literature Review
2.1.2 Two-Phase and Single-Phase Representations
2.1.3 Linearization of the Gas Flow Equation
3
TYPE-CURVE MODELS FOR THE INTERPRETATION OF SINGLEHOLE TEST DATA
3.0 CONCEPTUALIZATION OF FLOW THROUGH FRACTURED
ROCKS
3.1 SPHERICAL GAS FLOW MODEL
3.2 RADIAL GAS FLOW MODEL
3.3 UNIFORM FLUX HORIZONTAL AND VERTICAL FRACTURES
4
TYPE-CURVE INTERPRETATION OF SINGLE-HOLE PNEUMATIC
INJECTION TEST DATA
24
26
26
28
44
49
53
61
62
64
65
65
67
71
78
80
80
85
88
94
101
8
TABLE OF CONTENTS - Continued
4.1 SINGLE-HOLE TEST METHODOLOGY
4.2 PHENOMENOLOGY OF SINGLE-HOLE TESTS
4.3 TYPE-CURVE INTERPRETATION OF SINGLE-HOLE TESTS
5
TYPE-CURVE MODELS FOR THE INTERPRETATION OF CROSSHOLE TEST DATA
5.1
5.2
5.3
5.4
5.5
5.6
6
129
INTRODUCTION
129
POINT-INJECTION/POINT-OBSERVATION
133
LINE-INJECTION/POINT-OBSERVATION
137
POINT-INJECTION/LINE-OBSERVATION
142
LINE-INJECTION/LINE-OBSERVATION
145
MODIFIED FORM OF HSIEH AND NEUMAN (1985a) SOLUTION TO
ACCOUNT FOR OBSERVATION WELLBORE STORAGE AND SKIN
EFFECTS
148
TYPE-CURVE INTERPRETATION OF CROSS-HOLE PNEUMATIC
TEST DATA
6.1 CROSS-HOLE TEST METHODOLOGY
6.1.1 Instrumentation Used In Cross-Hole Tests
6.2 CROSS-HOLE TESTING PROCEDURE WITH EMPHASIS ON
TEST PP4
6.3 TYPE-CURVE INTERPRETATION OF CROSS-HOLE TEST PP4
7
101
107
115
FINDINGS AND CONCLUSIONS
APPENDIX A;
ASSESSMENT OF ELECTRICAL RESISTIVITY
TOMOGRAPHY (ERT) IN OBTAINING THE
SUBSURFACE DISTRIBUTION OF WATER CONTENT IN
THE VADOSE ZONE
157
157
158
171
189
211
218
A.1 INTRODUCTION-APPLICATIONS TO ENVIRONMENTAL
MONITORING
218
A.2 THEORY AND CONCEPTS
220
A.3 INVERSION OF TRANSFER RESISTANCES - VARIOUS
RECONSTRUCTION ALGORITHMS
221
A.4 APPLICATION OF ERT TO MONITOR VADOSE ZONE WATER
MOVEMENT
225
A.5 DISCUSSION OF RESULTS: RELATIVE MERITS OF APPLYING ERT
TO MONITOR VADOSE ZONE WATER MOVEMENT
229
A.6 REFERENCES PERTAINING TO APPENDIX A
234
APPENDIX B:
NEUTRON PROBE METHODOLOGY
B.l INTRODUCTION
B.2 THEORY BEHIND THE NEUTRON ATTENUATION METHOD
237
237
238
9
TABLE OF CONTENTS - Continued
B.3 THE STANDARD CALIBRATION METHOD OF NEUTRON PROBE IN
GRANULAR MATERIALS
241
B.4 FACTORS THAT AFFECT NEUTRON COUNTS
243
B.5 REFERENCES PERTAINING TO APPENDIX B
248
APPENDIX C;
SLIP FLOW
250
APPENDIX D:
AIR COMPRESSIBILITY AND VISCOSITY IN RELATION
TO PRESSURE AND TEMPERATURE AT THE ALRS
256
DERIVATION OF SINGLE-PHASE GAS FLOW
EQUATION
259
APPENDIX F:
TYPE-CURVE SOLUTION OF SPHERICAL GAS FLOW
262
APPENDIX G:
CODE FOR NUMERICAL INVERSION OF LAPLACE
TRANSFORM
269
CODE FOR NUMERICAL DIFFERENTIATION OF
PNEUMATIC TEST DATA
278
MODIFICATION OF HSEEH AND NEUMAN (1985a)
SOLUTION TO ACCOUNT FOR STORAGE AND SKIN IN
MONITORING INTERVALS
282
RECORD OF BOREHOLE TELEVISION (BHTV) SURVEY
CONDUCTED AT THE ALRS
285
APPENDIX E:
APPENDIX H:
APPENDIX I:
APPENDIX J:
REFERENCES
298
10
LIST OF ILLUSTRATIONS
Figure 1.1;
Location map of the covered site (indicated as red star)
30
Figure 1.2:
Plan view of boreholes, plastic cover, and field laboratory at
ALRS
33
Figure 1.3;
Three-dimensional perspective of the boreholes at the site
34
Figure 1.4:
Lower hemisphere Schmidt equal-area projection of fractures
identified by Rasmussen et al. (1990). Contours indicate
number of fi^actures per unit area of projection circle
39
Geometric parameters associated with a single-hole hydraulic
injection test (adapted from Tidtwell, 1988)
42
Air permeability versus fi:acture density (data fi^om Rasmussen
elal. 1990)
43
Perspective toward the Northeast showing center locations of 1m single-hole pneumatic test intervals; overlapping circles
indicate re-tested locations (adapted fi-om Guzman et al. 1996)
48
Kriged log k estimates obtained using \-m scale data from
boreholes X2, Y2, Y3, Z2, V2, and W2A (left) and same
together with 3-m scale d a t a firom boreholes X I , X3, YI, Z I ,
and Z3 (right) (adapted from Illman et al. 1998)
51
Three-dimensional representation of kriged log k. (adapted from
Illman et al. 1998)
52
Image of cross-hole, electrical resistivity tomography (ERT)
obtained between boreholes VI and V3
56
Variation of volumetric water content with depth in W-series
boreholes
57
Variation of volumetric water content with depth in X-series
boreholes
57
Variation of volumetric water content with depth in Y-series
boreholes
58
Variation of water content with depth in Z-series boreholes
58
Figure 1.5:
Figure 1.6:
Figure 1.7:
Figure 1.8:
Figure 1.9:
Figure 1.10:
Figure 1.11:
Figure 1.12:
Figure 1.13:
Figure 1.14:
11
LIST OF ILLUSTRATIONS - Continued
Figure 1.15:
Histogram of volumetric water content values from the W-, X-,
Y-, and Z-series boreholes
59
Figure 1.16;
EM survey conducted along borehole Y2
60
Figure 2.1;
Variation of compressibility factor Z with pressure and
temperature
74
Figure 2.2:
Variation of air viscosity fj. *vith pressure and temperature
75
Figure 2.3:
Variation of fiZ with pressure and temperature
75
Figure 3.1:
Type-curves of dimensionless pseudopressure in injection
interval versus normalized dimensionless time for various Co
and 5 = 0 under spherical flow
91
Type-curves of dimensionless pseudopressure and its derivative
in injection interval versus dimensionless time for various Co
and s under spherical flow
92
Type-curves of dimensionless pseudopressure and its derivative
in injection interval versus dimensionless time for various Co
under radial flow
93
Geometry of idealized horizontal fracture in an infinite flow
domain
98
Type-curves of dimensionless pseudopressure in injection
interval, normalized by dimensionless height, versus
dimensionless time for various ho in uniform flux horizontal
fracture model
99
Figure 3.2:
Figure 3.3:
Figure 3.4:
Figure 3.5:
Figure 3.6:
Figure 4.1:
Figure 4.2:
Figure 4.3:
Type-curves of dimensionless pseudopressure and its derivative
in injection interval versus dimensionless time in uniform flux
vertical fracture model at center of fracture (x^ =
=0)
100
Schematic diagram of the air injection system (adapted from
Guzman et al. 1996)
105
Example of multi-rate single-hole test (VCClOOl in borehole
V2 at 10.37 m from LL marker)
111
Arithmetic plot of pressure data from CGAl 120
112
12
LIST OF ILLUSTRATIONS - Continued
Figure 4.4:
Logarithmic plot of pressure data CGAI120
112
Figure 4.5:
Logarithmic plot of pressure data from test JGA0616
113
Figure 4.6:
Logarithmic plot of pressure data from test JHB0612
113
Figure 4.7:
BHTV image taken in borehole Y2 (high permeability zone)
114
Figure 4.8:
Logarithmic plot of pressure data from test ZDC0826
114
Figure 4.9a:
Type-curve match of data from single-hole pneumatic test
CAC0813 with liquid, spherical flow model
120
Type-curve match of data from single-hole pneumatic test
CAC0813 with gas, spherical flow model
120
Figure 4. lOa: Type-curve match of data from single-hole pneumatic test
CHB0617 with liquid, spherical flow model
121
Figure 4. lOb: Type-curve match of data from single-hole pneumatic test
CHB0617 with gas, spherical flow model
121
Figure 4.9b:
Figure 4.11:
Figure 4.12:
Figure 4.13:
Scatter diagram of results of permeability obtained from steadystate and /7-based spherical models
122
Scatter dia^am of results of permeability obtained from steadystate and -based spherical models
122
Scatter diagram of results of permeability obtained from steadystate and p-znd p'-based spherical models
123
Figure 4.14a: Type-curve match of data from single-hole pneumatic test
JGA0605 with liquid, radial flow model
124
Figure 4.14b: Type-curve match of data from single-hole pneumatic test
JGA0605 with gas, radial flow model
124
Figure 4.15a: Type-curve match of data from single-hole pneumatic test
JJA0616 with gas, spherical flow model
125
Figure 4.15b: Type-curve match of data from single-hole pneumatic test
JJA0616 with liquid, radial flow model
125
13
LIST OF ILLUSTRATIONS - Continued
Figure 4.16:
Figure 4.17;
Figure 4.18:
Figure 4.19:
Figure 4.20:
Figure 4.21:
Figure 5.1:
Figure 5.2:
Figure 5.3:
Figure 5.4:
Figure 5.5:
Type-curve match of data from single-hole pneumatic test
J T O 0 6 I 2 t o the horizontal fracture model
126
Type-curve match of data from single-hole pneumatic test
JHB0612 to the vertical fracture model
126
Type-curve match of data from single-hole pneumatic test
CDB1007 to the gas, radial flow model
127
Type-curve match of data from single-hole pneumatic test
JNA0713 to the gas, radial flow model
127
Type-curve match of data from single-hole pneumatic test
JKC0625 to the gas, spherical flow model
128
Type-curve match of data from single-hole pneumatic test
YFB0621 to the gas, spherical flow model
128
The point source solution (solid curve) and its time derivative
(dashed curve)
136
Logarithmic plot of
(solid curves) and d
(dashed curves) versus
for a range of or, values. The Theis
(1935) solution (open circles) and its derivative (open triangles)
are included for purposes of comparison
140
Plot of normalized pressure head for the line-injection/pointobservation case (solid curves) and its derivatives (dashed
curves) plotted against . The TT/c/j (1935) solution (open
circles) and its derivative (open triangles) are included for
purposes of comparison
141
Logarithmic plot of(solid curves) and
(dashed curves) versus
where the point injection/pointmonitoring solution (open circles) and its derivative (open
triangles) have also been plotted
144
Logarithmic plot for a, = yS, = 0 for various values of or, = ySJ.
The Theis (1935) solution (open circles) and its derivative (open
triangles) are included for purposes of comparison
147
14
LIST OF ILLUSTRATIONS - Continued
Figure 5.6:
Type-curves of dimensionless pressure in observation interval
(solid curves) and its derivative (dashed curves) versus
dimensionless time for various Q and fit = 5.0, ySo = 0.01. Open
circles represent dimensionless pressure in the solution of Hsieh
and Neuman (1985a) and open triangles represent their
derivatives
152
Type-curves of dimensionless pressure in observation interval
(solid curves) and its derivative (dashed curves) versus
dimensionless time for various Q and Pi = OA,
0.01. Open
circles represent dimensionless pressure in the solution of Hsieh
and Neuman (1985a) and open triangles represent their
derivatives
153
Type-curves of dimensionless pressure in observation interval
(solid curves) and its derivative (dashed curves) versus
dimensionless time for various
and Pi = 0.01, ^2= 0.01.
Open circles represent dimensionless pressure in the solution of
Hsieh and Neuman (1985a) and open triangles represent their
derivatives
154
Type-curves of dimensionless pressure in observation interval
(solid curves) and its derivative (dashed curves) versus
dimensionless time for various Q and or/ = 0.01, orj = 0.01. The
red solid line represents dimensionless pressure in the solution
of Hsieh and Neuman (1985a) and red dashed line their
derivatives. The open circles represent the Theis (1935)
solution while the open triangles represent their derivatives
155
Type-curves of dimensionless pressure in observation interval
(solid) and its derivative (dashed) versus dimensionless time for
various H and ai = 0.01, a2= 0.01. The blue solid line
represents dimensionless pressure in the solution of Theis
(1935) and the blue dashed line their derivatives. The open
circles represent the Black andKipp (1977) solution while the
open triangles represent their derivatives
156
Figure 6.1:
Injection and monitoring systems used for cross-hole tests
163
Figure 6.2:
Air injection system installed in the field laboratory
169
Figure 5.7:
Figure 5.8:
Figure 5.9:
Figure 5.10:
15
LIST OF ILLUSTRATIONS - Continued
Figure 6.3:
Locations of centers of injection and monitoring intervals.
Large solid circle represents injection interval, small solid
circles represent short monitoring intervals, and open circles
represent long monitoring intervals
174
Figure 6.4:
Barometric pressure during cross-hole test PP4
178
Figure 6.5:
Flow rate during cross-hole test PP4
178
Figure 6.6:
Packer pressure during cross-hole test PP4
179
Figure 6.7:
Battery voltage during cross-hole test PP4
179
Figure 6.8:
Fluctuations in relative humidity during cross-hole test PP4
180
Figure 6.9:
Surface injection pressure and pressure in injection (Y2M) and
monitoring (Y2U, Y2B) intervals within borehole Y2 during
cross-hole test PP4
180
Figure 6.10:
Pressure in monitoring interval VI during cross-hole test PP4
181
Figure 6.11:
Pressure in monitoring intervals V2U, V2M, and V2B during
cross-hole test PP4
181
Pressure in monitoring intervals V3U, V3M, and V3B during
cross-hole test PP4
182
Figure 6.13:
Pressure in monitoring interval W1 during cross-hole test PP4
182
Figure 6.14:
Pressure in monitoring interval W2 during cross-hole test PP4
183
Figure 6.15:
Pressure in monitoring intervals W2AU, W2AM, W2AL, and
W2AB during cross-hole test PP4
183
Pressure in monitoring intervals W3U, W3M, and W3B during
cross-hole test PP4
184
Figure 6.17:
Pressure in monitoring interval XI during cross-hole test PP4
184
Figure 6.18:
Pressure in monitoring intervals X2U, X2M, and X2B during
cross-hole test PP4
185
Pressure in monitoring interval X3 during cross-hole test PP4
185
Figure 6.12:
Figure 6.16:
Figure 6.19:
16
LIST OF ILLUSTRATIONS - Continued
Figure 6.20;
Pressure in monitoring intervals YIU, YIM, and YIB during
cross-hole test PP4
186
Pressure in monitoring intervals Y3U, Y3M, and Y3B during
cross-hole test PP4
186
Figure 6.22:
Pressure in monitoring interval Zl during cross-hole test PP4
187
Figure 6.23:
Pressure in monitoring intervals Z2U, Z2M, Z2L, and Z2B
during cross-hole test PP4
187
Pressure in monitoring intervals Z3L, Z3M, and Z3B during
cross-hole test PP4
188
Figure 6.25:
Type-curve match of pressure data from monitoring interval VI
195
Figure 6.26:
Type-curve match of pressure data from monitoring interval
V2M
195
Type-curve match of pressure data from monitoring interval
V3U
196
Type-curve match of pressure data from monitoring interval
V3M
196
Type-curve match of pressure data from monitoring interval
V3B
197
Figure 6.30:
Type-curve match of pressure data from monitoring interval Wl
197
Figure 6.31:
Type-curve match of pressure data from monitoring interval
W2AU
198
Type-curve match of pressure data from monitoring interval
W2AM
198
Type-curve match of pressure data from monitoring interval
W2AL
199
Type-curve match of pressure data from monitoring interval
W2AB
199
Type-curve match of pressure data from monitoring interval
W3U
200
Figure 6.21:
Figure 6.24:
Figure 6.27:
Figure 6.28:
Figure 6.29:
Figure 6.32:
Figure 6.33:
Figure 6.34:
Figure 6.35:
17
LIST OF ILLUSTRATIONS - Continued
Figure 6.36:
Type-curve match of pressure data from monitoring interval
W3M
200
Figure 6.37;
Type-curve match of pressure data from monitoring interval XI
201
Figure 6.38:
Type-curve match of pressure data from monitoring interval
X2U
201
Type-curve match of pressure data from monitoring interval
X2M
202
Type-curve match of pressure data from monitoring interval
X2B
202
Type-curve match of pressure data from monitoring interval
YIU
203
Type-curve match of pressure data from monitoring interval
YIM
203
Type-curve match of pressure data from monitoring interval
Y2U
204
Type-curve match of pressure data from monitoring interval
Y2B
204
Type-curve match of pressure data from monitoring interval
Y3U
205
Type-curve match of pressure data from monitoring interval
Y3M
205
Type-curve match of pressure data from monitoring interval
Y3B
206
Figure 6.48:
Type-curve match of pressure data from monitoring interval Z1
206
Figure 6.49:
Type-curve match of pressure data from monitoring interval
Z2U
207
Type-curve match of pressure data from monitoring interval
Z2M
207
Figure 6.39:
Figure 6.40:
Figure 6.41:
Figure 6.42:
Figure 6.43:
Figure 6.44:
Figure 6.45:
Figure 6.46:
Figure 6.47:
Figure 6.50:
18
LIST OF ILLUSTRATIONS - Continued
Figure 6.51;
Type-curve match of pressure data from monitoring interval
Z2L
208
Type-curve match of pressure data from monitoring interval
Z2B
208
Type-curve match of pressure data from monitoring interval
Z3U
209
Type-curve match of pressure data from monitoring interval
Z3M
209
Type-curve match of pressure data from monitoring interval
Z3B
210
Relative frequency of equivalent mean pore diameter plotted on
semi-logarithmic scale, showing bi modal distribution of pores
in tuff matrix at the ALRS (data from Rasmussen et al. 1990)
255
Figure F.1:
Mass conservation in injection system
267
Figure F.2:
Impact of positive skin on pseudopressure around the injection
interval
268
Figure 6.52:
Figure 6.53;
Figure 6.54:
Figure 6.55:
Figure C.1:
19
LIST OF TABLES
Table 1.1:
Statistics of effective porosity measurements on ALRS core.
Medium size sample measurements were obtained using
mercury intrusion method which underestimates effective
porosity (after Rasmussen et al. 1990)
36
Statistics of hydraulic conductivity values were obtained from
105 large core samples at the ALRS (after Rasmussen et al.
1990)
36
Table 1.3:
Statistics for van Genuchten's a (after Rasmussen et al. 1990)
36
Table 1.4;
Statistics of laboratory determined air permeabilities for various
values of suction and at oven-dried (OD) conditions (after
Rasmussen et al. 1990)
37
Statistics of laboratory determined Klinkenberg slip flow
coefficient (after Rasmussen et al. 1990)
37
Statistics of fracture density and orientation obtained from
ALRS cores (after Rasmussen et al. 1990)
37
Statistics of field determined outflow rates and of saturated
hydraulic conductivity values at the ALRS (after Rasmussen et
al., 1990)
38
Statistics of 3-m scale air permeability data. One interval
exceeded the capacity of the measurement device, while a
second interval was less than the measurement threshold (after
Rasmussen et al. 1990)
38
Field water content on various days at the ALRS (after
Rasmussen et al. 1990)
38
Table 1.2;
Table 1.5;
Table 1.6;
Table 1.7;
Table 1.8;
Table 1.9;
Table 4.1;
Table 4.2;
Table 4.3;
Table 6.1;
Coordinates of boreholes subjected to air permeability testing
(adapted from Guzman et al. 1996)
103
Nominal Scale and Number of Single-Hole Pneumatic Injection
Tests at the ALRS.(adapted from Guzman et al. 1996)
103
Air permeabilities obtained usingp- and p"-based spherical flow
models
119
Cross-hole tests completed at the ALRS
160
20
LIST OF TABLES - Continued
Table 6.2;
Table 6.3:
Table 6.4:
Table 6.5:
Table C.1:
Table D. 1:
Table D.2:
Table D.3:
Table D.4:
Information on injection and monitoring intervals with pressure
transducer types during phase 3 cross-hole tests
164
Coordinates of centers of monitoring intervals relative to origin
at center of injection interval, interval lengths, radial distances
between centers of injection and monitoring intervals, geometric
parameters Pi and ^2 and maximum recorded pressure change
177
Pneumatic parameters obtained from type-curve analysis of
pressure buildup data collected during cross-hole test PP4
193
Sample statistics of directional air permeabilities and air-filled
porosities obtained from type-curve interpretation of cross-hole
test PP4, and of air permeabilities from steady state
interpretations of l-m scale single-hole tests. Numbers in
parentheses represent corresponding actual values
194
Cumulative mercury intrusion volume as a function of
equivalent pore diameter (adapted from Rasmussen et al. 1990)
254
Gaseous composition of U. S. standard atmosphere {CRC
Handbook, 1992-1993)
257
Data concerning stable pressures encountered during \-m scale
single-hole pneumatic injection tests at the ALRS (summarized
from Guzman et al. 1996)
257
Data concerning stable temperatures encountered during \-m
scale single-hole pneumatic injection tests at the ALRS
(summarized from Guzman et al. 1996)
257
Variation of air dynamic viscosity with temperature and
pressure (data from Vasserman et al. 1966)
258
21
LIST OF PLATES
Plate 1.1;
Photograph of the Apache Leap
31
Plate 1.2:
Photograph of the covered borehole site (CBS) at ALRS where
pneumatic testing program took place
32
Photograph of the single-hole packer system employed during
the study of Guzman et al. (1996)
106
Plate 6.1;
Photograph of the single packer monitoring system
161
Plate 6.2:
Photograph of the modular packer system
162
Plate 6.3:
Photograph of the air manifold installed at the site
167
Plate 6.4:
Photograph of the injection interval with neutron probe
installed
168
Photograph of the air injection system and electronic equipment
installed in the field laboratory
170
Plate 4.1:
Plate 6.5:
22
ABSTRACT
This dissertation documents research results from a series of field experiments
and analyses used to test interpretive models for investigating the role of fractures in fluid
flow through unsaturated, fractured tuffs.
It summarizes the experimental design of
single- and cross-hole pneumatic injection tests, including borehole configuration and
testing schedules, data collection system, interpretive models developed and tested, data,
and conclusions. Single-hole tests were interpreted by Guzman et al. (1996) by means of
steady-state analysis to obtain permeability values based solely on late pressure data.
This dissertation and Illman et al. (1998) employ pressure and pressure-derivative typecurves to analyze transient data. Air permeabilities determined from transient analyses
agree well with those derived from steady-state analyses. Cross-hole pneumatic tests
were analyzed by means of a graphical matching procedure using newly-developed
pressure and pressure-derivative type-curves. Analyses of pressure data from individual
monitoring intervals using these new type-curves, under the assumption that the rock acts
as a uniform and isotropic fractured porous continuum, yield results that are comparable
with parameters obtained from a numerical inverse procedure described in Illman et al.
(1998).
The results include information about pneumatic connections between the
injection and monitoring intervals, corresponding directional air permeabilities, and airfilled porosities. Together with the results of earlier site investigations, single- and crosshole test analyses reveal that at the Apache Leap Research Site in central Arizona: (I) the
pneumatic pressure behavior of fractured tuff is amenable to analysis by methods that
treat the rock as a continuum on scales ranging from meters to tens of meters; (2) this
23
continuum is representative primarily, but not exclusively, of interconnected fractures;
(3) its pneumatic properties vary strongly with location, direction and scale, in particular,
the mean of pneumatic permeabilities increases, and their variance decreases with scale;
(4) this scale effect is most probably due to the presence in the rock of various size
fractures that are interconnected on a variety of scales; and (5) given a sufficiently large
sample of spatially varying pneumatic rock properties on a given scale of measurement,
these properties are amenable to analysis by geostatistical methods, which treat them as
correlated random fields defined over a continuum.
24
1. INTRODUCTION
1.1 ROLE OF PNEUMATIC TESTS IN SITE CHARACTERIZATION
Issues associated with the site characterization of fractured
rock terrains, the
analysis of fluid flow and contaminant transport in such terrains and the efficient
handling of contaminated sites are typically very difficult to resolve. A major source of
this difficulty is the complex nature of the subsurface "plumbing systems" of pores and
fractures through which flow and transport in rocks take place. There is at present no
well-established field methodology to characterize the fluid flow and contaminant
transport properties of unsaturated fractured rocks. In order to characterize the ability of
such rocks to conduct water, and to transport dissolved or suspended contaminants, one
would ideally want to observe these phenomena directly by conducting controlled field
hydraulic injection and tracer experiments within the rock. In order to characterize the
ability of unsaturated fractured
rocks to conduct non-aqueous phase liquids such as
chlorinated solvents, one would ideally want to observe the movement of such liquids
under controlled conditions in the field. In practice, there are severe logistical obstacles
to the injection of water into unsaturated geologic media, and logistical as well as
regulatory obstacles to the injection of non-aqueous phase liquids.
There also are
important technical reasons why the injection of liquids, and dissolved or suspended
tracers, into fractured rocks may not be the most feasible approach to site characterization
when the rock is partially saturated with water.
Injecting liquids and dissolved or
suspended tracers into an unsaturated rock would cause them to move predominantly
25
downward under the influence of gravity, and would therefore yield at best limited
information about the ability of the rock to conduct liquids and chemical constituents in
directions other than the vertical. It would further make it difficult to conduct more than
a single test at any location because the injection of liquid modifies the ambient
saturation of the rock, and the time required to recover ambient conditions may be
exceedingly long.
Many of these limitations can be overcome by conducting field tests with gases
rather than with liquids, and with gaseous tracers instead of chemicals dissolved in water.
Experience with pneumatic injection and gaseous tracer experiments in fractured rocks is
limited; much of this experience has been accumulated in recent years by The University
of Arizona at the Apache Leap Research Site (ALRS) near Superior, Arizona and by the
U.S. Geological Survey (USGS) near the ALRS {LeCain, 1995), and at Yucca Mountain
in Nevada {LeCain, 1996a,b; LeCain, 1998). Two research groups fi-om the Lawrence
Berkeley National Laboratory (LBNL) also conducted numerous air injection tests at
Yucca Mountain {Wang et aL 1998; Cook and Wang 1998) and in unsaturated fractured
basalts at Box Canyon in Idaho {Benito et al. 1998). To our knowledge, the earliest
pneumatic injection tests were conducted by Boardman and Skrove (1966) to determine
fracture permeability following a contained nuclear explosion. Their analysis was based
on the assumption of steady-state, isothermal, radial flow of ideal gases. Other early
testing of fractured
rocks with air is described by Montazer (1982), Trauz (1984), Mishra
et aL (1987), and Guzman et aL (1996).
26
This dissertation focuses on single-hole and cross-hole pneumatic injection tests
conducted by our group at the ALRS under the auspices of the U.S. Nuclear Regulatory
Commission (NRC). These tests were part of confirmatory research in support of NRC's
role as the licensing agency for a potential high-level nuclear waste repository in
unsaturated fractured tuffs at Yucca Mountain. However, unsaturated fi-actured porous
rocks similar to tuffs are found at many locations, including some low-level radioactive
waste disposal sites, nuclear decommissioning facilities and sites contaminated with
radioactive as well as other hazardous materials.
The test methodologies we have
developed, and the understanding we have gained concerning the pneumatic properties of
tuffs at the ALRS, are directly relevant to such facilities and sites. It should be stated that
test methodologies developed by us for unsaturated fi-actured rocks are equally applicable
to unsaturated porous media (sand, gravel, fractured clays and alluvium).
1.2 THE ALRS AND PREVIOUS SITE INVESTIGATIONS
1.2.1 Site Description
The Apache Leap Research Site is situated near Superior in central Arizona,
approximately 160 km north of Tucson at an elevation of 1,200 m above sea level (Figure
1.1-covered site indicated with a red star). The site is similar in many respects to Yucca
Mountain in southern Nevada where the candidate high-level nuclear waste (HLW)
repository site is being characterized. ALRS is located near the extreme western edge of
the Pinal Mountains. Lying immediately east of Superior, Arizona, is the Apache Leap,
which forms a 600-/n west-facing escarpment that exposes a volcanic zoned ash-flow tuff
27
sheet and an underlying carbonate stratum (see Plate 1.1). The dacite ash-flow sheet
{Peterson, 1961) covers an area of 1,000 km^ and varies considerably in thickness with an
average of about 300 m. The tuff is a consolidated deposit of volcanic ash with particle
diameters of less than 0.4 mm, resulting from a turbulent mixture of gas and pyroclastic
materials at high temperature about 19 million years ago. The climate is temperate and
dry, with a mean annual precipitation of less than 50 cm. Most of the precipitation occurs
during two periods, from mid-July to late-September and from mid-November to lateMarch.
During periods of high temperature and evapotranspiration demand in the
summer, rain is characterized by high intensity, short duration thunderstorms. During
cooler periods with much lower evapotranspiration demand in the winter, storms are of
longer duration and lower intensity. The regional water table lies at a variable depth of
more than 600 m. Except for a relatively thin perched zone of saturation at a depth of
approximately 150 m, the rock above the water table is unsaturated.
The covered site (Plate 1.2) under investigation consists of a cluster of 22 vertical
and inclined (at 45") boreholes that have been completed to a maximum vertical depth of
30 m within a layer of slightly welded unsaturated tuff. A plan view of the boreholes is
shown in Figure 1.2 and a three-dimensional perspective in Figure 1.3.
Recently
surveyed wellhead locations in Figure 1.2, and borehole geometries in Figure 1.3, are
given with references to Cartesian coordinates (x, y, z) with origin at the lower lip of the
casing in borehole Z3 and a vertical r-axis pointing downward. Boreholes having the
designations X, Y and Z were drilled during the initial stages of the project, prior to those
designated V, W and G. Shortly after the completion of drilling, a surface area of 1500
28
m' that includes all boreholes was covered with a thick plastic sheet to minimize
infiltration and evaporation. The V, W, X, Y and Z boreholes span a surface area of
approximately 55 OT by 35 m and a volume of rock on the order of 60,000 m^. The
vertical G boreholes were drilled with conventional rotary equipment using water as
cooling fluid; are about 20 m deep; and lie to the west of the plastic cover. A total of 270
m of oriented core was retrieved fi'om the boreholes and stored at the University of
Arizona Core Storage Facility. The upper 1.8 m of each borehole was cased. Borehole
television images are available for boreholes VI, V2, V3, Wl, W2, W2A, W3, XI, X2,
X3, Yl, Y2, Y3, ZI, Z2 and Z3.
1.2.2 Previous Work in Study Area of ALRS
Earlier work related to our area of study at the ALRS is described by Evans
(1983), Schrauf and Evans (1984), Huang and Evans (1984), Green and Evans (1987),
Rasmussen and Evans (1987, 1989, 1992), Yeh et al. (1988), Weber and Evans (1988),
Chuang et al. (1990), Rasmussen et al. (1990, 1993, 1996), Evans and Rasmussen {\99\),
and Bassett et al. (1994). The earlier work included drilling 16 boreholes (VI, V2, V3,
Wl, W2, W2A, W3, XI, X2, X3, Yl, Y2, Y3, Zl, Z2, and Z3) and conducting numerous
field and laboratory investigations. Laboratory measurements of matrix properties were
conducted on core segments of various sizes taken from 3-m borehole intervals at 105
locations (indicated in Figure 5 of Evans and Rasmussen, 1991) within nine of the
boreholes (XI, X2, X3, Yl, Y2, Y3, ZI, Z2, Z3). The measurements include interstitial
properties such as bulk density (Table 1 in Rasmussen et al. 1990), effective porosity
29
(Table 1.1), skeletal density (Table 3 in Rasmussen et al. 1990), pore surface area (Table
4 in Rasmussen et al. 1990), and pore size distribution (Table 5 in Rasmussen et al.
1990); hydraulic properties such as saturated and unsaturated hydraulic conductivity
(Table 1.2) and moisture retention characteristics (Table 8 in Rasmussen et al. 1990); a value of the van Genuchten moisture characteristic function (Table 1.3); pneumatic
properties such as oven-dry and unsaturated air-phase permeability (Table 1.4); and
Klinkenberg slip-flow coefficient (Table 1.5).
Moisture retention properties of the matrix were characterized by the van
Genuchten (1980) parzuneter a while holding other parameters of the van Genuchten
retention model constant: the dimensionless parameter n at 1.6, and the residual water
content Or at zero {Rasmussen et al. 1990).
30
Figure l.l: Locatioii map of the covered sfte (indicated as red star)
31
Pbte 1.1: Photogci^hofthe Apache Leap
32
Plate 1.2: Pfaotogci^ of the coveted boidioie sfte (CBS) at ALRS where (meumatic
testing program took place.
33
J
W3
1\«
Field
N
Labontoty
W2
W2A
Black PlHtk Cover
»G4
• G3
06 • G2
• *05
«
Wl
• • «
VI V2 V3
Gl
«
-20
-10
•
0
•
•
X2
X3
•
•
Y1
Y2
•
•
10
20
Z2
Z3
•30
•
XI
Y3
Z1
30
X f/ml
F^;iire 1.2: Plan view of botdules, plastic cover, and field laboratory at ALRS.
40
34
* Iml
Figure 1.3; Three-dimensional perspective of the boreholes at the site.
35
Information about the location and geometry of fractures in the study area has
been obtained from surface observations, the examination of oriented cores, and borehole
televiewer records. A summary of data concerning the orientation, dip and density of
fractures in boreholes can be found in Rasmussen et al. (1990). According to Rasmussen
et al. (1990), a total of 79 fractures have been identified in boreholes at the site. The
fractures appear to be exponentially distributed in a manner consistent with a Poisson
process of fracture locations. Fracture density, defined by Rasmussen et al. (1990) as
number of fractures per meter in a 3-/n borehole interval, ranges from zero to a maximum
of 4.3 per meter (Table 1.6). A Schmidt equal-area projection of fracture orientations,
with contours indicating number of fractures per unit area of the projection circle, is
shown in Figure 1.4. Though the fractures exhibit a wide range of inclinations and
trends, most of them are near vertical, strike north-south and dip steeply to the east. The
stereonet in Figure 1.4 is based on the data of Rasmussen et al. (1990); an earlier
stereonet presented by Yeh et al. (1988) has the strikes of all fractures rotated by 180°
degrees.
36
Table l.l: Statistics of effective porosity measurements on ALRS core. Medium size
sample measurements were obtained using mercury intrusion method which
underestimates effective porosity jdiAex Rasmussen et al. 1990).
Effective Porosity [%1
Large
Small
Medium
17.54
Mean
17.15
14.62
Coef. Var.
13%
16%
26%
14.30
Minimum
11.02
9.18
Median
16.52
17.21
14.31
Maximum
24.73
47.58
27.51
Table 1.2: Statistics of hydraulic conductivity values were obtained from 105 large core
Mean
Coef. Var.
Minimum
Median
Maximum
Hydraulic Conductivity [*10*^ /n/s]
Suction [kPd\
0
25
10
3.346
1.475
21.31
301%
105%
156%
0.69
0.126
0.110
2.610
0.556
4.24
25.750
14.588
438.28
50
0.908
115%
0.002
0.498
5.041
100
0.364
112%
0.005
0.235
2.541
Table 1.3: Statistics for van Genuchten's a (after Rasmussen et al. 1990).
Van Genuchten's Moisture Characteristic Function
a-value [kPa']
0.0224
Mean
31.T/0
Coef Var.
0.0102
Minimum
0.0203
Median
0.0643
Maximum
37
Table 1.4; Statistics of laboratory determined air permeabilities for various values of
suction and at oven-dried (QD) conditions (after Rasmussen et al. 1990).
Air Permeability^rni^*io-"'i
Suction kPd\
300
10
25
50
100
500
OD
1.54
26.67
35.11
Mean
11.20
16.88
38.23
57.12
309%
Coef. Var.
434%
436%
344%
326%
295%
272%
<0.01
<0.01
Minimum
0.02
0.25
1.29
1.91
3.81
0.05
5.09
Median
0.10
0.39
2.10
6.04
12.08
41.90
780.50
333.10
678.70
780.50 1012.60
Maximum
389.80
Table 1.5: Statistics of laboratory determined Klinkenberg slip flow coefficient
Rasmussen et al. 1990).
Klinkenberg Coefficient
{kPa\
Mean
322.1
Coef Var.
82%
35
Minimum
217
Median
1277
Maximum
Table 1.6: Statistics of fracture density and orientation obtained from ALRS cores
{Z&&V Rasmussen et al. 1990).
Fracture
Fracture Orientation
Strike [deg]
Density [m"']
Dip \deg\
Mean
0.77
214.4
64.5
108%
56%
Coef Var.
37%
0.00
Minimum
3
I
Median
0.67
109
55
Maximum
359
4.33
89
38
o
Table 1.7: Statistics of field determined outflow rates and of saturated hydraulic
conductivity values at the ALRS (after/towmssew et al. 1990).
Outflow
Hydraulic Conductivity [•
r*10-^#nVjl
Philip
Glover
Dachler
Mean
26.349
59.42
29.10
30.20
Coef Var.
612%
729%
729%
662%
Minimum
0.016
0.27
0.48
0.41
Median
0.633
11.63
5.64
10.33
Maximum
1232
39224
19126
17900
Table 1.8; Statistics of 3-/n scale air permeability data. One interval exceeded the
capacity of the measurement device, while a second interval was less than the
measurement threshold (after Rasmussen et al. 1990).
Field Air Permeability
[•10*"'m^l
178.1
Mean
667%
Coef Var.
<0.420
Minimum
4.02
Median
>13366
Maximum
Table 1.9: Field water content on various days at the ALRS (after Rasmussen et al. 1990).
Borehole Water Content (m /m )
Julian Day
279
372
41
236
255
406
448
505
After 1/1/87
Mean
13.75
14.59
14.50
14.36
14.26
14.12
14.32
14.32
Coef Var.
11%
12%
11%
12%
12%
12%
12%
12%
Minimum
10.77
9.99
9.85
10.19
10.85
10.89
9.53
10.14
Median
13.71
14.04
14.46
14.48
14.33
14.35
14.18
14.23
Maximum
17.63
17.29
18.75
18.42
17.74
17.77
17.57
17.51
39
Figure 1.4; Lower hemisphere Schmidt equal-area projection of fractures identified by
Rasmussen et al. (1990). Contours indicate number of fractures per unit area of
projection circle.
40
Surface fracture traces include an additional, steeply dipping east-west set. Laboratory
experiments have also been conducted on large blocks of fractured tuff, including a study
of aperture distribution in a large natural fracture (Vickers et al. 1992).
Single-hole hydraulic injection tests were performed by Tidwell (1988) in 87 out
of 105 3-m intervals in boreholes XI, X2, X3, Yl, Y2, Y3, Zl, Z2, and Z3 from which
core samples had been taken. The hydraulic tests were conducted by maintaining a
constant water level near the top of a borehole until a constant injection rate was
established. The injection rate was converted into equivalent hydraulic conductivity
using three different formulae for steady state flow. One formula, due to Glover (1953),
was modified by Tidwell (1988, p. 65, eq. 4.26) to account for borehole inclination
according to
O sinh"'(Z,/r)- Ljh
(1.1)
where K is hydraulic conductivity {LT'\ Q is flow rate [L^T'], L is length of injection
interval [Z,], r is borehole radius [Z,], h is borehole length [£.], 2, is distance from
bottom of borehole to lower edge of test interval [Z,], and fi
is angle of borehole
inclination relative to ground surface (Figure 1.5), and the dimensions of each variable
are specified in terms of mass \M\, length [Z,] and time [7] throughout this dissertation.
Another formula, due to Philip (1985), was modified in a likewise manner by Tidwell
(1988, p. 79, eq. 4.61) to read
41
where Q is a geometric factor related to the eccentricity of an assumed prolate spheroid
representing the borehole, and z, is distance along the borehole from its bottom to the
upper edge of the test interval [Z,]. The third formula, due to Dachler (1936), is written
by Rasmussen et al. (1990; p.27, eq. 19) as
(1.3)
where
is pressure head in the injection interval [Z,]. Summary statistics are listed in
Table 1.7 and show that calculated hydraulic conductivities range over five orders of
magnitude.
According to Rasmussen et al. (1990), the corresponding hydraulic
conductivities are log-normally distributed and strongly skewed toward high values.
Single-hole pneumatic injection tests were conducted in 87 intervals of length 3 m
in 9 boreholes (XI, X2, X3, YI, Y2, Y3, Zl, Z2 and Z3) by Rasmussen et al. (1990,
1991, 1993). According to Rasmussen et al. (1993), the tests were conducted by injecting
air at a constant mass rate between two inflated packers while monitoring pressure within
the injection interval. Pressure was said to have reached stable values within minutes in
most test intervals. Air permeability was calculated using Dachler's (1936) steady state
formula, adapted by Rasmussen et al. (1990, p.35, eq. 24 and 1993) to isothermal airflow,
.
where
Q.M.P.
rlrf\ p 2 - P a2\)
,,4\ • /
is intrinsic permeability to air [L^], Qa is volumetric airflow rate at standard
temperature and pressure [L^T^
water \ML'^T\
is air viscosity
y is the specific weight of
is atmospheric pressure \ML''T\ and p is absolute air pressure
42
[ML''T^'\ in the test interval. Summary statistics are listed in Table 1.8. Figure 5b of
Rasmussen et al. (1993) suggests a good correlation (r = 0.876) between pneumatic and
hydraulic permeabilities at the ALRS.
Ground Surfoce
H sin B
Figure 1.5; Geometric parameters associated with a single-hole hydraulic injection test
(adapted from Tidwell, 1988).
43
Figure 1.6 shows a scatter plot of pneumatic permeability versus fracture density
for 3-m borehole intervals based on the data of Rasnnissen et al. (1990). It suggests a
lack of correlation between fracture density and air permeability.
1E-11
1E-12 -=
1E-13 ^
£
§
1E-14
• • :
g.
^
1E-15 -4
:! •
•
I.
1E-16 ^
: i
5
•
•
1E-17
0.0
2.0
4.0
Fracture density [counts/m]
6.0
Figure 1.6: Air permeability versus fracture density (data from Rasmussen et al. 1990).
Rasmussen et al. (1990) conducted in situ determinations of volumetric water
content by means of neutron probes at 105 locations within the nine boreholes listed
earlier. They took measurements on eight separate occasions 41, 236, 255, 279, 372,
406, 448, and 505 Julian days following 1/1/87 (Table 1.9). Temporal variations in
44
neutron readings were slow during this period. As matrix porosities are much larger than
the porosity of fractures, and pore sizes within the matrix are generally much smaller than
fracture openings, most of the water resides in the matrix and its water content is much
higher than that of the fractures. We therefore attribute the neutron probe measurements,
and corresponding water contents, primarily to the matrix.
A gaseous tracer experiment was conducted at the ALRS (Yeh et al. 1988) by
injecting helium into one borehole and monitoring its arrival in neighboring boreholes by
means of a thermal conductivity meter (utilizing the low thermal conductivity of helium
relative to that of standard atmosphere).
A test performed by injecting helium into
borehole X2, below a packer set at a distance of about 20 m from the surface, showed
breakthrough of helium into borehole XI at a distance of 9.5 - 10 m from the surface.
The breakthrough was attributed by the authors to a fracture that had been encountered in
the injection and the detection intervals. No helium was detected in borehole X3 during
the test, but the data collected in borehole XI are summarized in Figure 10 of Yeh et al.
(1988).
1.2.3 Single-Hole Pneumatic Injection Tests
Single-hole pneumatic injection tests conducted at the ALRS by Rasmussen et al.
(1990, 1991, 1993) were of relatively short duration and involved relatively long test
intervals. Guzman et al. (1994, 1996) and Guzman and Neuman (1996) conducted a
much larger number of single-hole pneumatic injection tests of considerably longer
duration over shorter borehole intervals.
Their tests were conducted under highly
45
controlled conditions, subject to strict quality assurance, within six boreholes (V2, W2A,
X2, Y2, Y3, and Z2) that extend over a horizontal area of 32 m by 20 m. Five of the
boreholes are 30-m long (V2, W2A, X2, Y2, Z2) and one has a length of 45 m (Y3); five
are inclined at 45° (W2A, X2, Y2, Y3, Z2) and one is vertical (V2). A total of 184
borehole segments were tested by setting the packers 1 m apart as shown in Figure 1.7.
Additional tests were conducted in segments of lengths 0.5, 2.0 and 3.0 m in borehole Y2,
and 2.0 m in borehole X2, bringing the total number of tests to over 270. The tests were
conducted by maintaining a constant injection rate until air pressure became relatively
stable and remained so for some time. The injection rate was then incremented by a
constant value and the procedure repeated. Three or more such incremental steps were
conducted in each borehole segment while recording the air injection rate, pressure,
temperature and relative humidity. For each relatively stable period of injection rate and
pressure, air permeability was estimated by treating the rock around each test interval as a
uniform, isotropic continuum within which air flows as a single phase under steady state,
in a pressure field exhibiting prolate spheroidal symmetry.
The results of these steady state interpretations of single-hole air injection tests
are listed in Guzman el al. (1996). The authors found, and noted, that {Guzman et al.
1994, \996-, Guzman andNeuman, 1996)
I. Air permeabilities determined in situ fi-om steady state single-hole test data are much
higher than those determined on core samples of rock matrix in the laboratory,
suggesting that the in situ permeabilities represent the properties of fractures at the
site.
46
2. It is generally not possible to distinguish between the permeabilities of individual
fractures, and the bulk permeability of the fractured rock in the immediate vicinity of
a test interval, by means of steady state single-hole test data.
3. The time required for pressure to stabilize in the injection interval typically ranges
from 30 to 60 min, increases with flow
rate, and may at times exceed 24 hrs,
suggesting that steady state permeability values published in the literature for this and
other sites, based on much shorter air injection tests, may not be entirely valid.
4. Steady state interpretation of single-hole injection tests, based on the assumption of
radial flow, is acceptable for intervals of length equal to or greater than 0.5 m in
boreholes having a radius of 5 cm, as is the case at the ALRS.
5. Pressure in the injection interval typically rises to a peak prior to stabilizing at a
constant value, due to a two-phase flow effect whereby water in the rock is displaced
by air during injection.
6. In most test intervals, pneumatic permeabilities show a systematic increase with
applied pressure as air displaces water under two-phase flow.
7. In a few test intervals, intersected by widely open fractures,
air permeabilities
decrease with applied pressure due to inertia! effects.
8. Air permeabilities exhibit a hysteretic variation with applied pressure.
9. The pressure-dependence of air permeability suggests that it is advisable to conduct
single-hole air injection tests at several applied flow rates and/or pressures.
10. Enhanced permeability due to slip flow (the Klinkenberg effect) appears to be of little
relevance to the interpretation of single-hole air injection tests at the ALRS.
47
11. Local-scaie air permeabilities vary by orders of magnitude between test intervals
across the site.
12. Spatial variability in permeability is much greater than that due to applied pressure
and lends itself to meaningful statistical and geostatistical analysis.
13. Air permeabilities are poorly correlated with fracture densities, as is known to be the
case for hydraulic conductivities at many water-saturated fractured
rock sites
worldwide {Neuman, 1987), providing further support for Neuman's conclusion that
the permeability of fractured rocks cannot be reliably predicted from information
about fracture geometry (density, trace lengths, orientations, apertures and their
roughness) but must be determined directly by means of hydraulic and/or pneumatic
tests.
14. Air permeabilities vary systematically with the scale of measurement as represented
nominally by the distance between packers in an injection interval.
The work of Guzman el al. (1994, 1996) and Guzman and Neuman (1996)
strongly suggests that air injection tests yield properties of the fracture system, which are
relevant to both unsaturated and saturated conditions.
In particular, numerical
simulations by these authors show that, whereas the intrinsic permeability one determines
from such tests is generally lower than the intrinsic permeability to water of fractures
which surround the test interval, it nevertheless approaches the latter as the applied
pressure goes up. This is so because capillary forces tend to draw water from fractures
into the porous (matrix) blocks of rock between the fractures, thereby leaving the latter
saturated primarily with air. Water saturation in the matrix blocks is therefore typically
48
much higher than that within the fractures, making it relatively difHcuIt for air to flow
through such blocks. It follows that, during a pneumatic injection test, the air moves
primarily through fractures (most of which contain relatively little water) and the test
therefore yields flow and transport parameters which reflect the intrinsic properties of
these largely air-fllled fractures.
Locatians of 1.0 m Air
MmmmwiimiU
(raria* do net indicf «phf» of influ»ne*)
Figure 1.7; Perspective toward the Northeast showing center locations of l-m single-hole
pneumatic test intervals; overlapping circles indicate re-tested locations (adapted from
Guzman etal. 1996).
49
1.2.4 Kriged Estimates of Air Permeability at the ALRS
Core and single-hole measurements, conducted over short segments of a borehole,
provide information only about a small volume of rock in the immediate vicinity of each
measurement interval.
Available data from the ALRS indicate that rock properties,
measured on such small scales, vary erratically in space in a manner, which renders the
rock randomly heterogeneous and pneumatically anisotropic. A major question is how to
describe this spatial and directional dependence of medium properties in untested
portions of the rock.
Analyses to date suggest (Bassett et al. 1994, 1997; Guzman et al. 1996, Illman et
al. 1998) that it is possible to interpolate some of the core and single-hole measurements
at the ALRS between boreholes by means of geostatistical methods, which view the
corresponding variables as correlated random fields. This is especially true about air
permeability, porosity, fracture density, water content, and the van Genuchten water
retention parameter a, for each of which we possess enough measurements to constitute a
workable geostatistical sample. Standard geostatistical analysis provides best (minimum
variance) linear unbiased estimates of how each such quantity varies in three-dimensional
space, together with information about the quality of these estimates.
A geostatistical analysis of the above site variables has been conducted by Chen
et al. (1997) and by Illman et al. (1998) in which the details are provided. Here, kriged
maps of single-hole air permeability values are provided for the reader to give a
subsurface picture of air permeability distribution at the ALRS.
50
Figure 1.8 compares kriged images of log air permeability generated by lllman et
al. (1998) along four vertical sections at
= 0, 5, 7, and 10 m using \-m data (left
column) and the combined set o f \ - m and 3-m data, the latter from boreholes X I , X3, YI,
Z1 and Z3 (right column). The figure shows boreholes intersected by, or located very
close to, each cross-section. The two sets of kriged images are considerably different
from each other. This is most pronounced at
= 0 m, which passes through the Z-series
of boreholes; here the inclusion of data from boreholes Z1 and Z3 has caused estimated
permeability in the upper right comer of the section to be much higher than it is v/ithout
these data.
The effect extends to all four cross-sections, which exhibit elevated
permeabilities near the upper-right comer. Along sections ^X.y = S m and 7 m, which pass
close to the Y and V series, respectively, the addition of data from YI affects the shape
and size of a prominent high-permeability zone which extends through Y2. The addition
of data from X3 to the set reveals corresponding high- and low-permeability zones in
section= 10 /n which correlate well with similar zones, intersected by Y3, in section j/ =
5 m.
A three-dimensional representation of kriged log permeability based on the
combined set of data is shown in Figure 1.9.
Illmcm et al. (1998) show that the above data are amenable to continuum
geostatistical analysis and exhibit distinct spatial correlation stmctures. This suggests
that the data can be viewed as samples from a random field, or stochastic continuum, as
proposed over a decade ago by Meuman (1987) and affirmed more recently by Tsang et
al. (1996). This is so despite the fact that the rock is fractured and therefore mechanically
discontinuous. Our finding may be contrary to geologic intuition, but it is supported
51
strongfy
similar findings in many other ftactured tock terrains including crystalline
rocks at Oracle, Arizona; Strqia, Sweden; and Fanay-Ai^res, France.
It strongly
siqiports the triplication ofcontinuum flow and tramqwrt theories and models to fiactured
porous tuffi on scales of a metn or more.
Figure 1.8: Kriged log k estimates obtained using l-m scale data from boreholes X2, Y2,
Y3, Z2, V2, and W2A (left) and same together with 3-m scale data fiom boreholes XI,
X3, Yl, Zl, and Z3 (pg^) {adapted Smmlllnumetal. 1998).
Log ifc where it [m'l
Figure 1.9: Three-dimeiisioiial rqnesentation of kriged tog k (ad^ed finm Ulnum et al.
1998).
53
1.2.5 Geophysical Surveys Conducted at Site
Modeling of gas/liquid transport through the vadose zone by analytical and
numerical techniques requires inter alia, the quantification of spatial and temporal
variations in water content. Water content can be obtained in situ by means of neutron
probes and possibly through the employment of cross-hole electrical resistivity
tomography (ERT).
Neutron probe measurements were taken using Boart-Longyear CPN Model 503
at 1-m intervals in boreholes Wl, W2A, W3, XI, X2, X3, Yl, Y2, Y3. Zl, Z2, and Z3
during July 1996. A cross-hole ERT survey was also conducted by student researchers
from the University of Arizona, during January 1996. Here, results from both techniques
are briefly discussed and summarized. Appendix A describes the use of ERT to estimate
in situ values of water content. A review of the neutron probe methodology is given in
Appendix B.
Cross-hole electrical resistivity tomography (ERT) was employed (measurements
tak^n by G. Morelli from Mining Engineering and his coworker) during January 1996 to
obtain estimates of electrical resistivity (which can then be possibly related to water
content) at the covered site.
Figures 1.10 is an image of cross-hole ERT obtained through a numerical
inversion technique described in Morelli and LaBrecque (1996). Data were obtained
along the V-series boreholes (VI, V2, and V3). Contours of electrical resistivity are
given in [Qni\. A relatively large area of high resistivity measurements seen in figure
54
1.10 suggests a possible (dry) angled fracture plane. Spatial variability in resistivity can
result from a number of causes including textural variability of subsurface materials (e.g.,
sand versus clay; large voids such as fractures), distribution of capillary water, variability
in the ionic content of water, etc.
Neutron probe measurements were taken in boreholes W, X, Y, and Z to quantify
the amount of moisture present in the unsaturated fractured
tuffs, prior to the
instrumentation of boreholes for cross-hole pneumatic testing. Due to time and financial
constraints, a detailed calibration of neutron probes in unsaturated fractured
rocks
proposed by Illman (1996) could not be completed. For purposes of obtaining crude
estimates of water content from thermalized neutron counts, a field calibration curve
obtained by Rasmussen el al. (1990; p.3l, eq. 21) has been employed to convert the
neutron counts to estimates of volumetric v.'ater content. Figures 1.11 through 1.14 show
the spatial distribution of volumetric water content down the boreholes. The figures
clearly show that downhole variability of water content in 13 boreholes is quite small
(between 0.08 - 0.12), suggesting that capillary water is quite uniformly distributed
throughout the fractured rock mass. The uniformity in the spatial distribution of capillary
water may perhaps be due to the black, plastic cover shown on Plate 1.2. A study by
McTigue et al. (1993) also showed relatively uniform moisture in unsaturated alluvium
under a concrete slab at a field site near Albuquerque, NM.
Figure 1.15 is a histogram of the 336 values of water content. One unusually high
value obtained from the bottom of borehole Y3 (probably filled with water) was excluded
from the statistical sample.
The histogram clearly shows a bimodal distribution of
55
volumetric water content which is related to the pore size distribution of fractured tuffs
shown in Figure C.l, supporting the notion that neutron probes measure primarily the
properties of the matrix. This may further imply qualitatively that capillary water resides
in the porous matrix and not in fractures, which are thus relatively dry.
An EM survey was conducted along borehole Y2 by researchers from
the
Geosciences Department at the University of Arizona (personal conmiunication, A.
Guzman, 1999). Figure 16 shows the results from the survey in which both small and
large-scale oscillations of resistivity are visible. Approximately 54 feet from the lower
lip of the casing an anomaly in resistivity is evident which corresponds to a large fracture
seen in BHTV (or cavity - see Figure 4.7).
^
56
Figure 1.10: Image of cross-hole, electrical resistivity tomogrtq>hy (ERT) obtained
between bordioles VI andVB.
57
a 14
at2
ai
ao8
ao6
ao4
• wt I
' W2A|
W3 i
ao2
0
S
I
O
U
2
0
2
S
X
3
S
4
O
4
S
S
O
Figure 1.11: Variation of vohunetric water content with depth in W-series boreholes.
0.14
ai2
ai
aos
ao6
aoi
• XI
• X2
X3
0.02
10
IS
20
25
10
35
40
4S
SO
F^:ure 1.12: Variation of vohunetric water content widi depth in X-series boreholes.
58
0.14
0.12
I K >•
0.1
0.0S
0.06
0.04
• Yl
0.02
• Y2
Y3
10
iS
25
20
30
35
40
45
SO
Dtmyrni
Figure 1.13: Variatioii of vohimetric water conteoEt wfth depth in Y-series boreholes.
0.14
0.12
•
•
0.1
I •
a •
0.08
0.06
0.04
• Zl
0.02
m
Z2
Z3
10
15
20
25
30
35
40
45
50
Otptkyml
Figure 1.14: Variation of vohmietric water content wfth dqidi in Z-series boreholes.
59
8
f
20
'
0,
Figine 1.15: Histogcam of vohimetric water oontCTt vahies from the W-, X-, Y-, and Zseries boreholes.
Y2 high k zone
»V CCMSfJCWTL ?« ttpis
tv
«« «K?ir?
Figure 1,16: EM survey conducted along borehole Y2.
61
1.3 TYPE-CURVE ANALYSIS OF SINGLE-HOLE PNEUMATIC INJECTION
TESTS
In this dissertation, interpretations of transient pressure data from the single-hole
air injection tests previously conducted at the site by Guzman el al. (1994, 1996) and
Guzman and Neuman (1996) are presented. These interpretations are based on typecurves derived analytically for single-phase gas flow
by linearizing the otherwise
nonlinear partial differential equations, which govern such flow in uniform, isotropic
porous continua. The type-curves correspond to three different flow geometries: threedimensional flow with spherical symmetry, two-dimensional flow with radial symmetry,
and flow in a continuum with an embedded high-permeability planar feature (a major
fracture). Included in the type-curves are effects of gas storage in the injection interval
(known as borehole storage effect) and reduced or enhanced permeability in the
immediate vicinity of this interval (known as positive or negative skin effects). Our test
interpretations rely not only on standard type-curves but also on the type-curves of
pressure derivative versus the logarithm of time. Such pressure derivative curves have
become popular in recent years because they accentuate phenomena which might
otherwise be missed, help diagnose the prevailing flow regime, and aid in constraining
the calculation of corresponding flow parameters.
Transient analyses of single-hole pneumatic tests yield information about air
permeability, air-filled porosity, skin factor and dimensionality of the flow regime on a
nominal scale of I m in the immediate vicinity of each test interval. We show that
transient air permeabilities agree well with previously determined steady state values.
62
1.4 CROSS-HOLE PNEUMATIC INJECTION TESTS
Cross-hole pneumatic injection tests were conducted at the ALRS by the author
with the help of a technician, Mr. Dick L. Thompson. These tests span the entire volume
of fractured rock previously subjected to single-hole testing, and beyond. Their purpose
was to a) directly characterize the pneumatic properties of the rock on a site-wide scale;
b) determine the spatial extent and connectivity of fractures and/or high-permeability
flow chaimels across the site; and c) compare the results with corresponding information
that one might deduce from smaller-scale (laboratory and single-hole) tests. A total of
forty-four cross-hole tests of diverse types (constant injection rate, step injection rates,
instantaneous injection rate) have been conducted between various boreholes and
borehole intervals at the site as part of this dissertation work.
Several tests were
conducted without inflating the packers in monitoring holes in order to ascertain the
effect that such open holes may have on pressure response in the injection test interval,
i.e., on single-hole pneumatic injection tests. Additional single- and cross-hole tests were
conducted to investigate the effects of: 1) drierite (calcium sulfate which acts as a drying
agent) on humidity in the injection air stream and two-phase flow effects seen in Guzman
and Neuman (1994 and 1996); 2) air injection into a zero-permeability PVC pipe to
quantify air leakage from the equipment; 3) barometric pressure fluctuations
on air
pressure in packed-off monitoring intervals under ambient conditions; 4) air injection
while measuring changes in neutron counts in the rock surrounding the injection interval;
and 5) variations in injection rate on pressure in the injection interval. To design our
63
cross-hole tests, we relied heavily on information about fiacture locations and medium
properties derived from core, borehole televiewer, and single-hole air injection test data.
A complete list of all cross-hole tests conducted by us at the ALRS is included in
Table 6.1. In most of these tests, air was injected at a constant mass flow rate into a
relatively short borehole interval of length I - 2 m while monitoring a) air pressure and
temperature in the injection interval; b) barometric pressure, air temperature and relative
humidity at the surface; and c) air pressure and temperature in 13 short (0.5-2 m) and 24
longer (4 - 20 m) intervals within the injection and surrounding boreholes. Only one of
these tests, labeled PP4, was fiilly analyzed to date. During this test, which we describe
in detail, pressure responses were detected in 12 of the 13 short monitoring intervals and
20 of the 24 longer intervals.
We use newly developed, analytically-derived type-curves to interpret the PP4
cross-hole test data.
An automatic parameter estimation method based on a three-
dimensional finite volume code (FEHM), coupled with an inverse code (PEST) has been
employed to analyze the same data set in Illman et al. (1998). The type-curve approach
treats short and longer borehole intervals as either points or lines, depending on distance
between injection and monitoring intervals, while accounting indirectly for storage
effects in monitoring intervals due to the compressibility of air. The finite volume code
allows representing borehole geometry and storage more realistically, and directly, by
treating each borehole as a high-permeability cylinder having finite length and radius.
Analyses of pressure data from individual monitoring intervals by the two methods,
under the assumption that the rock acts as a uniform and isotropic fractured
porous
64
continuum, yield comparable results. These results include information about pneumatic
connections between the injection and monitoring intervals, corresponding directional air
permeabilities, and air-filled porosities, all of which are found to vary considerably from
one monitoring interval to another on a scale of 10 - 20 /n. Together with the results of
earlier site investigations, our single- and cross-hole test analyses provide important new
insight into the phenomenology of air flow through unsaturated fractured rocks at the
ALRS, and about the spatial variability and scale-dependence of corresponding
pneumatic parameters such as permeability, connectivity and air-filled porosity.
1.5 SCOPE OF THE DISSERTATION
Chapter 2 of this dissertation presents the theory of gas flow in porous media with
an emphasis on the case where water is relatively immobile. This theory forms the basis
for all the analytical and numerical methods employed here to interpret single-hole and
cross-hole pneumatic injection tests at the ALRS. Analytical formulae and corresponding
type-curves for the interpretation of single-hole tests are presented in Chapter 3. Chapter
4 describes the type-curve interpretation of transient single-hole test results and their
comparison with corresponding steady state test interpretations. Analytical formulae and
corresponding type-curves for the interpretation of cross-hole tests are presented in
Chapter 5. In Chapter 6, the design and conduct of cross-hole pneumatic interference
tests at the site and interpretation of one of them by means of newly developed typecurves are described. The dissertation ends with a comprehensive list of flndings and
conclusions in Chapter 7.
65
2. GAS FLOW IN POROUS AND FRACTURED MEDIA
2.1 THEORY OF GAS FLOW IN POROUS MEDIA
The theory of gas (air) flow in porous and fi^ctured media has gained great
importance in recent years. There are several advantages to using gases instead of liquids
to conduct laboratory and field determinations of flow and transport parameters in
unsaturated media. Injecting liquids and dissolved or suspended tracers into unsaturated
media would cause them to move predominantly downward under the influence of
gravity, and would therefore yield at best limited information about the ability of
geologic media to conduct liquids and chemical constituents in directions other than the
vertical. It would further make it difficult to conduct more than a single test at any
location because the injection of liquid modifies the ambient saturation of the geologic
medium, and the time required to recover ambient conditions may be exceedingly long.
In the laboratory, the flow of liquids can cause plugging of a porous medium upon
continuous flow of water {Mttskat, 1946).
Many of these limitations can be overcome by conducting field and laboratory
tests with gases rather than with liquids, and with gaseous tracers instead of chemicals
dissolved in water. Gases also have much lower densities in comparison to liquids which
makes it possible for one to employ lighter (or less durable yet more flexible) materials
(PVC as opposed to steel) to conduct experiments both in the field and in the laboratory.
In unsaturated flow systems, gases fill all void space not occupied by liquid. This
may be one reason why air injection experiments at the ALRS (single- and cross-hole
66
tests - to be discussed later) almost always reach steady-state or three-dimensional flow.
If one were able to disregard nonlinear effects such as slip flow, inertia, and multi-phase
flow, one would expect to see similar pressure responses in saturated fi-actured media if
water was employed as the extraction/injection fluid.
This is because intrinsic
permeability (not relative permeability in multi-phase flow) is independent of the
physico-mechanical properties of homogenous fluids passing through a stable, nonreactive material. Intrinsic permeability is by definition a constant fully determined by
the structure of the medium. The apparent permeability, on the other hand is a pressure
dependent quantity and is usually associated with gas flow at low velocities in geologic
formations of low permeability or when other fluids coexist within the medium.
The equations that describe gas flow in partially saturated porous media are
nonlinear due to the compressible nature of gases, its capillary interaction with water, and
non-Darcian behavior at high Reynolds numbers. A complete description of gas-water
(in our experiments air-water) interaction requires two systems of coupled partial
differential equations, one for each phase. The resulting two-phase flow equations can
then be solved by numerical methods of the kind developed by Mendoza and Frind
(1990), Pruess (1991) or Zyvoloski et al. (1988, 1996, 1997). In special cases it may be
possible to solve simplified versions of these equations analytically. We are particularly
interested in the use of analytically derived type-curve methods for the interpretation of
single-hole and cross-hole pneumatic test results. The development of corresponding
analytical formulae requires that two-phase flow is approximated as single-phase gas
flow and that water is treated as immobile. The resulting single-phase gas flow equations
67
must additionally be linearized to allow solving them either in terms of pressure, as is
customary for liquids, or in terms of pressure-squared or pseudopressure, as is more
common for gases.
We provide a literature survey and describe this process of
simplification and linearization in the subsequent sections.
2.1.1 Literature Review
Early attempts to solve/study gas-flow problems were limited to solutions
involving a single gas phase. Research concerning gas flow through porous media was
conducted
primarily in the petroleum/natural gas industry (also in chemical
engineering/industry; ceramics and materials) that involved approximate analytical,
graphical, analogue and numerical solutions. Muskat (1934) suggested a method, which
employs a succession of steady-state solutions.
Others {Heatherington et al. 1942;
MacRoberts 1949, Cornell and Katz 1953, and others) have obtained approximate
solutions by assuming constant gas properties throughout the duration of a well test and
small pressure gradients. Variations of gas properties such as dynamic viscosity {fx) and
compressibility factor (Z) with pressure have been neglected due to analytical difficulties
in solving the nonlinear partial differential equation. Katz et al. (1959) list linearized
versions of the gas flow equation for an ideal gas.
Aronofsky ( \ 9 S A r ) considered the effect of gas slip on transient flow of gas through
one and two-dimensional porous media.
The gas slip permeability (or apparent
permeability) was assumed to vary as a linear function with the reciprocal pressure. As
discussed by Aronofsky (1954), Guzman and Neuman (1996), and in Appendix C of this
68
dissertation, gas slip most probably does not impact pressures during well tests; it is more
important in the conduct and analysis of laboratory gas flow experiments.
Other
nonlinear effects such as two-phase flow and inertia have been found by Guzman et al.
(1994, 1996) and Guzman and Neuman (1996), as well as lllman et al. (1998), to
manifest themselves primarily in the immediate neighborhood of injection intervals.
During the early period (petroleum industry in the 1930's through the 1950's) in
which analytical solutions were sought for the gas flow equation, some authors such as
Aronofsky and Jenkins (1952) Jenkins and Aronofsky (1953), Bruce et al. (1953),
Aronofsky and Jenkins (1954), Aronofsky and Ferris (1954), and Aronofsky and Porter
(1956) attempted to solve the nonlinear equation using numerical techniques.
In
particular, Douglas, Peaceman, andRachford, Jr. (1955) numerically solved the transient
gas flow equation through porous media assuming a square reservoir with a single well in
the center.
The alternating direction implicit (ADI) method was employed for the
calculation in which the numerical solutions compared favorably with published solutions
for radial reservoirs. A numerical study by Aronofsky and Jenkins (1954) yielded one of
the most important conclusions in gas flow through porous media during this period. The
important conclusion was that the numerical solution of radial flow of ideal gas in porous
media for various rates can be approximated quite well by the analytical solution for
transient liquid flow in a radial system of van Everdingen and //wrs/ (1949).
A major advance in theory was made when Al-Hussainy et al. (1966) published
their application of the real gas pseudopressure concept to rigorously linearize the
nonlinear flow equation of real gases through porous media. Pseudopressure is a term
69
employed in the petroleum industry, but is more widely known as the Kirchhoff
transformation (Kirchhoff, 1894).
According to Raghavan (1993), the concept of
pseudopressure in the context of gas flow has been used by Carter (1966) prior to Al-
Hussainy et al. (1966) in a slightly different form. It is Al-Hussainy et al. (1966) who
first applied this concept to understand pressure behavior and well responses. We show
later in this chapter how the Kirchhoff transformation is applied to our nonlinear gas
diffusion equation to linearize it in a more tractable form. Before the introduction of
KirchhofF transformation, data collected fi-om gas well tests were plotted in terms of
pressure-squared (^*) in the petroleum industry.
Recently, Massmcom (1989) and KicWhorter (1990) derived analytical solutions
to unsteady radial flow of gas in the vadose zone. Massmann (1989), not concerned with
the compressible nature of gases, instead addressed the issue of the validity of Darcy's
law in describing the flow of gases through porous media and gave some conditions
under which Darcy's law is applicable to gas flow. One of the conclusions reached by
this author was that Darcy's law is valid and groundwater equations can be used to model
gas flow in the vadose zone as long as slip flow can be neglected.
In the literature, one finds data from gas injection or withdrawal tests plotted and
analyzed in terms of pressure (p\ pressure-squared
or pseudopressure (vf). Tracey
(1956) and others (e.g., Matthews, 1961) suggested that when pressure in the system is
sufficiently low, such data can be analyzed in terms of p as if the gas behaved like a
slightly compressible liquid.
This approach is quite common in the field
of
environmental engineering and hydrology. For example, Massmann (1989), Johnson et
70
al. (1990), Edwards (1994), Massmann and Madden, (1994), Beckett and Huntley (1994)
and Edwards (1996) analyzed transient radial flow of gas in the vadose zone by means of
standard groundwater flow equations written in terms of pressure without a gravity term.
Massmann (1989) concluded that this is appropriate as long as gas pressure between two
points is less than 0.5 atm (1 a/m=101,325 Pa). Above 0.5 atm, he recommends to write
groundwater equations in terms of pressure-squared as suggested by the analytical
investigation by Kidder (1957). In the same study, Massmann (1989) also gave condition
under which the Klinkenberg effect may become important.
A more rigorous analysis under steady-state conditions has been carried out by
Baehr and Hult (1991) who had developed two analytical solutions for steady-state, twodimensional, axisymmetric airflow to a single well partially screened in the unsaturated
zone. Other steady-state solutions have been developed by Shan et al. (1992), Baehr and
Joss (1995), Kaluarachchi (1995), DePaoli et al. (1996), Ge and Liao (1996), and Ge
(1998). Warrick and Rojano (1999) examined the effects of source cavity shape on
steady, three-dimensional flow of soil gases. Transient solutions have been developed by
McWhorter (1990) for a two-dimensional flow system and Falta (1995) for injection and
extraction wells from horizontal wells. Guo (1997) obtained a quasi-analytical solution
to the transient, nonlinear radial, gas flow equation using the Boltzman transformation in
which the pressure is calculated using the Newton-Raphson method.
71
2.1.2 Two-Phase and Single-Phase Representations
Consider the flow of two homogenous immiscible fluids, a liquid and air, in a
rigid porous medium under isothermal conditions.
In the absence of mass transfer
between the fluid phases, the liquid and air mass conservation equations take the forms
=
(2.1-1)
where the subscripts / and a refer to liquid and air, respectively; p is mass density
q is flux density [LT'\, ^ is porosity [Z,®]; S is fluid saturation [L\ t is time
[7]; V- is the divergence operator
where the dimensions of each variable are
specified in terms of mass [M\, length [Z,] and time [7].
In practice, it is common to assume that Darcy's law applies to both liquids and
air. In the case of air, Darcy's law may not apply when Knudsen diffusion or slip flow
(the so-called Klinkenberg effect) are important. Both of these phenomena are discussed
briefly in Appendix C. Laboratory experiments by Alzaydi et al. (1978) have shown that
air flux through a column of Ottawa sand is linear in the applied pressure gradient {i.e.,
Darcian) except at early time following the start of each experiment. They found the
same to be true for kaolinite clay, except that here the time required to establish a Darcy
regime was longer. As mentioned in Section 1.2.3 of the Introduction, the steady state
interpretation of multistep single-hole pneumatic injection tests at the ALRS have
72
revealed inertial (nonlinear, non-Darcian) behavior of air flow in a few test intervals
intersected by highly permeable fractures. In most other test intervals, the flow appeared
to be multiphase Darcian. We therefore adopt the following standard forms of Darcy's
law for liquid and air, respectively,
,,=-A£i,Vh,
Ml
q. = -—vp,
f.
where g is the gravitational constant [LT'\,
is dynamic viscosity of air
(2.1-2)
k
is intrinsic permeability [Z,*] for both air and water; V is the gradient operator [Z,"']; hi is
hydraulic head [L\ defined as
h,=-^ + z
P,g
p is pressure
Z
(2.1-3)
is elevation about an arbitrary datum [Z,]; and the effect of
gravity on air flow is neglected. Substituting (2.1-2) into (2.1-1) yields the liquid flow
and air flow equations,
V •[p, ^
(2.1-4)
respectively. The two equations are coupled via the relationships
5,+^„ = l
(2.1-5)
Pc = Pa-Pi
(2.1-6)
73
where
is capillary pressure. To solve them requires the additionzd specification of
appropriate equations of state (functional relationships between fluid properties and
pressure, and between permeability, capillary pressure and saturation, for each phase) and
forcing functions (source terms, initial and boundary conditions). As the equations of
state are generally nonlinear, so are the above flow equations.
For purposes of analyzing airflow around wells, it is common to treat the liquid
phase as being immobile. Then Sa = I and the airflow equation simplifies to
where the subscript a has been omitted as all equations now refer to a single gas phase.
The corresponding equation of state is commonly written as {Burcik, 1957)
pV = Z{pj)nRT
(2.1-8)
where V is volume [Z,^]; Z(/7, t) is a dimensionless compressibility factor; n is mass in
moles [mo/]; and R is the universal gas constant [Jottles "FC'mor']. Upon rearranging
eq. (2.1-8) to read
where
= V j n is specific volume
the dependence of Z o n p and T is made
explicit. One can thus readily calculate Z for any pressure, temperature and specific
volume.
A sample calculation of Z for conditions typically found at the ALRS
(p = 200 kPa, 7 = 200° k) is included in Appendix D. Figure 2.1 shows corresponding
variations of Z with pressure and temperature. The value of Z is seen to be 1 for a large
74
range of pressures ^10^ - \Qi*kPc^ and temperatures ^270-400 'K^
Dik^ sin^e-hole
tests at the ALRS, pressures widim the iiyection mterval ranged between 80 kPa and 360
itPa (Table D.2) and tenpecatuies between 288.10" ATand 302.74' K (Table D.3). Thk
allows us to treat ak, for purposes of test analysis at the ALRS, as an ideal gas with Z= 1.
15
ZM fldtai Ih"a fcbljr
hign«y6ffiiwn
L5
N
111 • •••
05
UBHB
Figure 2.1: Variation of con^nessibility fiictor Z widi pressure and temperature.
P^29(K
T=30(K
LIB45
Pim]
F^;ure 2J2: Variotioa of air viscosity // with pressure and tenqierature.
-O— T-270K
-
MSOK
—T-290K
—T-300K
2MM
1M«M
P [*^«l
F^;ure 2.3: Variation of
with i»essure and tempoature.
76
Figure 2.2 shows the variation of air viscosity // with pressure and temperature.
Air viscosity is seen to increase slowly with pressure, and to increase with temperature at
low pressures but decrease at higher pressures. Figure 2.3 has been included to show
how the corresponding product /zZ varies with pressure for a range of temperatures. The
product is seen to remain constant (equal to 1.81 x lO'^fa .v) for the range of pressures
and temperatures encountered under field-testing conditions at the ALRS. Since Z is
virtually constant under these conditions, so is jj, (equal to 1.81 x 10"'/'a 5).
Upon considering isothermal conditions, and treating Z and n as constants, one
finds from eq. (2.1-8) that mass density is proportional to pressure. For purposes of
solving the gas flow equation analytically, it is additionally common to treat ^ as a
constant. This allows rewriting (2.1-7) as
(2.1-10)
which can be rewritten as
^
k dt
(2.1-11)
or, alternatively, as
^2 2 ^^^£2
^
k dt
(2.1-12)
where c is gas compressibility, defmed as
c = -L^ = l
~ pdp~ p
(2.1-13)
77
In the more general case where /iZ is not a constant, the equation in terms of
includes an additional logarithmic term {Al-Hussairy et al. 1966; Raghavan^ 1993; see
derivation in Appendix E),
V2 2 _
dp
^ <l>fjc dp
k dt
(2.1-14)
Yet another form of the gas flow equation can be obtained upon introducing the
pseudopressure
defined as {Al-Hussainy et al., 1966; Raghavan, 1993)
wi
where vv(p) has dimensions [ML''T%
This is akin to the well-known Kirchhoff
transformation {Kirchhqff, 1894) and has, according to Raghavan, been used earlier by
Carter (1966) in a slightly different form. For our purposes, the lower limit of zero on p
represents barometric pressure. In the case where /iZ is constant,
w{p) =
fjZ^Po
(2.1-16)
iXL
where Pj, is the barometric pressure. Figure 2.3 shows that, at high pressures, /xZ varies
more-or-less linearly with p. It then follows that {Gradshteyn and Ryzhik, 1994, p. 79,
eq. 2.152),
M p )« '
f ———dp = —
h^a-^bp
b
In general.
+ bpJ
(2.1-17)
78
dt
_ dw{p) d p _ I p d p
dp dt
piZ dt
Vw(p) = ^ ^ V p = ^ V p
(2.1-18)
and so the gas flow equation in terms of pseudopressure takes on a quasilinear form,
k
dt
2.1.3 Linearization of the Gas flow Equation
Under steady state, equations (2.1-10) and (2.1-11) are linear in p', and equation
(2.1-19) is linear in w. Linearization of the transient p-based equation (2.1-10) requires
that the leftmost p be set equal to a constant, p (typically some average pressure). The
equation can then be rewritten as
k
dt
(2.-20)
where, by virtue of (2.1-13), c = \l p. This equation is similar to that typically used for
liquids in the absence of gravity (linear dif^usion equation). Setting c equal to a constant
i n (2.1-10) o r (2.1-19), s a y c =\ l p , renders these equations linear i n p ' o r w ,
respectively,
a at
a dt
where a = kjpS^. is pneumatic diffusivity in terms of a gas storage factor
(2.1-21)
(2.1-22)
79
(2-1-23)
P
The latter differs from the specific storage commonly used in hydrology which, in our
case, would be defined asSs = pgSaIn Chapter 3 we modify a number of analytical type-curve expressions, derived by
various authors for liquid flow under conditions similar in principle to those we
encounter during single-hole and cross-hole pneumatic injection tests at the ALRS, in a
manner which renders them applicable to air-flow.
Among others, we recast these
expressions in terms of pseudopressure, which by virtue of (2.1-16) is analogous to
rewriting them in terms ofp' when /iZ is constant, as is the case at the ALRS. Our typecurve analyses of single-hole pneumatic air injection tests at the ALRS, described in
Chapter 4, are conducted using both the /7-based equation (2.1-20), and the
'-based
equation (2.1-22) with pseudopressure defined according to (2.1-16). As the two sets of
results are consistent, we conduct our type-curve analysis of cross-hole tests, described in
Chapter 5, using only the simpler p-based form.
80
3. TYPE-CURVE MODELS FOR THE INTERPRETATION OF
SINGLE-HOLE TEST DATA
3.0 CONCEPTUALIZATION OF FLOW THROUGH FRACTURED ROCKS
For purposes of analyzing fluid flow and solute transport through fractured rocks
on the fleld scale, it has become common to think of matrix blocks as forming a
continuum, and of fractures as forming another, overlapping continuum {Barenblatt et al.
1960; Warren and Root, 1963).
If one of these overlapping entities dominates all
relevant aspects of flow and transport, one treats the rock as a single matrix- or fracturedominated continuum.
Otherwise, one must allow for the possibility that fluids and
solutes could migrate from one entity to the other under a pressure and/or concentration
differential between the two. If such fluid and mass transfer between matrix and fractures
is fast in comparison to flow and transport through the rock, one considers the two
entities to be at equilibrium and treats the rock as an equivalent or effective (single)
matrix-fracture continuum {Dykhuizen, 1990; Peters and Klavetter, 1988; Pruess et al.
1990). In the absence of such equilibrium, it is common to adopt the dual porosity model
(Bibby, 1981; Moench, 1984; Zimmerman et al. 1993) in which the matrix acts as a
nonconducting storage reservoir; the fractures form a conducting medium with negligible
storage capacity; and the transfer of fluids (or solutes) between these two overlapping
continua is proportional to the pressure (or concentration) differential between them at
each point in space-time. A more general version of the latter is the dual permeability
model {Duguid and Lee, 1977; Gerke and van Germchten, 1993a,b) in which both the
matrix and fracture continua conduct fluids and solutes; still another version is one that
81
accounts for internal gradients of pressure (or concentration) within matrix blocks,
referred to as interacting multiple continua by Pruess and Narasimhan (1985; see also
Kazemi, 1969; Berkowitz et al. 1988; and Birkholzer et al. 1993). Carrera et al. (1990)
were able to successfully reproduce pumping tests in a fractured block of monzonitic
gneiss by treating discrete fracture zones, and the rest of the fractured rock mass, as
juxtaposed (nonoverlapping) fracture-dominated (single) continua.
The extent to which continuum (single, equivalent, dual, multiple, juxtaposed,
deterministic or stochastic) concepts may or may not apply to fractured rocks has been
the subject of intense research and debate for over two decades. In such rocks, flow and
transport often take place preferentially through discrete fractures and channels. Some of
these discontinuities can usually be identified and mapped in surface outcrops, boreholes,
and subsurface openings. This has led to the belief on the part of some that it should be
possible to delineate the geometry of the subsurface "plumbing system" through which
most flow and advective transport take place. Several hydrologists consider it especially
feasible to construct realistic models of fracture
networks deterministically or
stochastically. Typically, such networks consist of discrete polygonal or oval-shaped
planes of finite size, embedded in an impermeable, or at times permeable, rock matrix.
Each plane is assigned effective flow and transport properties, usually at random; in some
single fracture studies, these properties are further treated as random fields defined at
each point in the fracture plane. Fracture network models containing thousands of planes
have been used to simulate flow and tracer migration at several experimental sites, most
notably in crystalline rocks of the Site Characterization and Validation (SCV) complex at
82
the Stripa mine in Sweden {Dershowitz et al. 1991; Dverstorp et al. 1992; National
Research Council, 1996) and the Fanay-Augeres mine in France {Cacas et al. 1990a,b).
The conceptual-theoretical framework
behind the discrete fracture
modeling
approach was seen by Neuman (1987, 1988) as lacking firm experimental support.
Neuman also questioned the practicality of the approach on the grounds that existing field
techniques make it extremely difficult, if not impossible, to reconstruct with any
reasonable degree of fidelity either the geometry of the subsurface plumbing system
(which consists of porous blocks, fractures, and channels that are known to evolve
dynamically along fracture planes and fracture intersections) or the flow and transport
properties of its individual components. Indeed, Tsang and Neuman (1995) recently
reached a similar conclusion based on extensive experience gained during the six-year
international ENTRAVAL project. The authors pointed out that several ENTRAVAL field
hydraulic and tracer experiments have proven equally amenable to analysis by discrete
and continuum models, rendering the validation of either approach difficult. The best
models appeared to be those that were neither too simplistic nor too complex. A recent
summary of the international Stripa project {SKB, 1993) has concluded that while it has
been possible to construct working fracture network models with thousands of discrete
planes for the SCV site by calibrating them against observed hydraulic and tracer data,
these models have not necessarily performed better than much simpler and more
parsimonious continuum models (only very elementary continuum models were
considered in this comparison; more sophisticated stochzistic continuum models were not
considered and could potentially perform much better).
The idea of representing
83
fi^ctured rocks as stochastic continua, originally proposed by Neuman (1987), has been
adopted by Tsang et al. (1996) in their recent analysis of flow and transport at the Aspo
Island SKB Hard Rock laboratory in Sweden.
Most single-hole and cross-hole pneumatic test data at the ALRS have proven
amenable to analysis by means of a single fracture-dominated continuum representation
of the fractured-porous tuff at the site. This is in line with the more general conclusion
by Neuman (1987, 1988) that flow and transport in many fractured rock environments is
amenable to analysis by continuum models which account in sufficient detail for medium
heterogeneity and anisotropy. Only when one can distinguish clearly between distinct
hydrogeologic units or features such as layers, faults, fracture zones, or major individual
fractures of low or high permeability, on scales not much smaller than the domain of
interest, should one in our view attempt to model them discretely (delineate their
geometry deterministically); one should then still consider treating the internal properties
of each such discrete unit as random fields.
Recent evidence that the latter idea often
works can be found in the University of Arizona theses of Kostner (1993) and Ando
(1995), and in the work of Guzman and Neuman (1996). The two theses demonstrate that
hydraulic and tracer tests in saturated fractured granites at the Fanay-Augeres mine in
France can be reproduced by means of continuum indicator geostatistics {Joumel, 1983,
1989), and fracture-dominated single-continuum stochastic flow and transport models,
with better accuracy than has been achieved previously with discrete fracture network
models.
Illman et al. (1998) demonstrate that small-scale, single-hole, steady state
pneumatic permeability data from unsaturated fractured tuffs at the ALRS are likewise
84
amenable to continuum geostatistical analysis, exhibiting both anisotropic and random
fractal behaviors.
Most type-curve models currently available for the interpretation of single-hole
and cross-hole fluid injection (or withdrawal) tests in fractured rocks fall into three broad
categories; I) those that treat the rock as a single porous continuum representing the
fracture network; 2) those that treat the rock as two overlapping continua of the dual
porosity type; and 3) those that allow an additional major fracture
to intersect the
injection (or withdrawal) test interval at various angles. The prevailing interpretation of
dual continua is that one represents the fracture network and the other embedded blocks
of rock matrix. We take the broader view that multiple (including dual) continua may
represent fractures on a multiplicity of scales, not necessarily fractures and matrix. When
a dominant fracture is present in a type-curve model, it is usually pictured as a highpermeability slab of finite or infinitesimal thickness. To allow developing analytical
solutions in support of type-curve models, the continua are taken to be uniform and either
isotropic or anisotropic. The test interval is taken to intersect a dominant fracture at its
center. Either flow across the walls of such a fracture, or incremental pressure within the
fracture, are taken to be uniform in most models. Flow is usually taken to be transient
with radial or spherical symmetry, which may transition into near-uniform flow as one
approaches a major fracture that intersects the test interval. Some models account for
borehole storage and skin effects in the injection (or withdrawal) interval.
In this dissertation, interpretation of transient data from single-hole pneumatic
injection tests at the ALRS are conducted by means of modified single-continuum type-
85
curve models developed for spherical flow by Joseph and Koederitz (1985); for radial
flow by Agarwal et al. (1970); for a single horizontal fracture by Gringarten and Ramey
(1974); and for a single vertical fracture by Gringarten et al. (1974). We interpret crosshole injection tests by means of a type-curve model developed for spherical flow in an
anisotropic continuum by Hsieh artd Neuman (1985a). Our modifications consist of
recasting the first three models in terms of pseudopressure; developing corresponding
expressions and type-curves in terms of (pseudo)pressure-derivatives; and in Chapter 5
adding compressible storage in monitoring wells to the model of Hsieh and Neuman
(1985a). A brief description of these modified models is given in the following sections
and in Chapter 5.
3.1 SPHERICAL GAS FLOW MODEL
Single-hole pneumatic pressure data tend to stabilize with time in most injection
intervals at the ALRS. As recharge boundaries are not likely to be a major cause of such
stabilization at the site, we consider instead a model in which air is injected at a constant
volumetric flow rate 0 [L^T'] fi-om a spherical source of (equivalent or pseudo) radius
[Z,] into a uniform, isotropic continuum of infinite extent.
This results in a
spherically symmetric flow regime governed by (2.1-22) and subject to the following
initial and boundary conditions (Appendix F)
vi/(r,0) = 0;
r>r^
(3.1-1)
lim M/(r,/) = 0;
/>0
(3.1-2)
86
2Q„rp^
dt
"
QJTp^\
dr
2;zArsw I T '
^
(3.1-3)
(3.1-4)
JC
where most variables have been defined in Chapter 2; p,„ is pressure in the injection line;
Cw is storage coefficient
of the injection interval w;
is pseudopressure in
the rock just outside the skin; ^ is a dimensionless skin factor; the subscript sc denotes
standard conditions; and (Guzman and Neuman, 1994, eq. (C.7), p. C-2; see also Energy
Resources Conservation Board, p.2-34)
(3.1-5)
~
T^P
Eq. (3.1-3) represents mass balance in the injection interval under the assumption that gas
density within it is the same as in the injection system and the rock. Eq. (3.1-4) relates
pressure in the injection interval to that in the rock by means of an additive skin factor.
The latter represents an infinitesimal skin or membrane that resists flow but does not
store fluid so that flow across it takes place at a steady rate. Reduced permeability
around the injection interval is represented by a positive skin factor, enhanced
permeability by a negative skin. The above formulation is analogous to that of Joseph
andKoederitz (1985) who additionally write
87
where b is the length of the actual cylindrical injection interval; this follows from
equating the spherical source volume to that of a prolate-spheroid that fits snugly into the
cylindrical interval.
Upon defining dimensionless pseudopressure, time, radius and
borehole storage coefficient as
\yv
,
__
'd ~
r
,
<^^ir
•
4 »
=1 -—
r
_ C^p
C.=
A7r<f>rl
(3.1-7)
r>r_
(3.1-8)
r >r„
(3.1-9)
\
3^
(3.1-10)
respectively, it is possible to recast the problem in dimensionless form. Here, we have
4
_
defined the storage coefficient of the injection interval to be Q = -^LP
The
corresponding solution has the same form as that given by Joseph andKoederitz
m'.d =
- !
(3.1-11)
/1^2|VXCd [(i -
+ jVI(i +
+ (l +
where W ^D is dimensionless pseudopressure in the well, L' is inverse Laplace transform
and X is the corresponding transform parameter. Inversion of the Laplace transform
solution was accomplished numerically using an algorithm due to DeHoog et al. (1982).
A computer code for numerical inversion of Laplace transformation based on this
algorithm is provided in Appendix G.
88
Figure 3.1 is a logarithmic plot of
versus
for various values of C^,
when s = 0. The early unit slope of the type-curve is a diagnostic feature of borehole
storage. The stabilization of
case spherical) flow.
at late time is diagnostic of three-dimensional (in our
Type-curves corresponding to Co> 2.5x10^ are difficult to
distinguish from each other. Under test conditions at the ALRS, this means that one
cannot use type-curves based on the spherical model to determine porosity from singlehole test data.
The dimensionless time derivative of pseudo-pressure is obtained directly from
(3.1-11) in the form
/1^|VICd[(I Figure 3.2 shows type-curves of
+ (l +
(solid) and
(dashed) versus
^
for
various values of Q and s. The derivative is seen to climb to a peak and then to
diminish asymptotically toward zero at a rapid rate.
3.2 RADIAL GAS FLOW MODEL
Some single-hole pneumatic test data from the ALRS appear to fit a radial flow
model during part or the entire test. The model we use has been modified after Agarwal
et al. (1970; see also Raghavan, 1993, p. 68, eq. 4.105) in the manner mentioned earlier.
Their model is in turn a modification of an earlier solution due to Papadopulos and
89
Cooper (1967) by adding to it the effect of a skin. Airflow is governed by eq. (2.1-22)
subject to
\
P
C
'2-QJ'Ps. <Py)f J
w(r,0) = 0
(3.2-1)
Hm w{r,t) = 0
(3.2-2)
f
dt
-Tdcb
7- — f r dw\ =1
Q^Tp^V dr)^
a IeAS
Tdcbr^ r. J
(3.2-3)
(3.2-4)
The dimensionless pseudo-pressure and its derivative are given by
w
"wD
K„{A.) + sy[lK,[-jA)
= / '^
^
(3.2-5)
A|VI/:.(>/I) + /iCo[^o(VI)+iVI/:,(VI)]|^
d w wD
d \x\ t r
_ T-i
K^{X) + SyflK,[4J)
(3.2-6)
+ ACo[^o(VI)+iVI^,(VI)]}^
which, in this case, can be inverted analytically to read
Vf WD
4 f«
r^ ^
1 - exp(- v-/o)
^ ° ^'{[^^D"^o(v')-(i-Co^')y,(v)]"+[vCoK„(v)-(i-Cojrv-)?;(v)]"|
(3.2-7)
4 f
=—
^In/r
w- Jc
°
,(l-exp(-v'fo))
dv
+[vC„r„(v)-(i-Co^=)?;{v)]'}
(3.2-8)
90
where now
nkb(
w
Q^yTpJ
(3.2-9)
(3.2-10)
r
r>r
r.
C ^ 27d)r^<f>
where
^
2<l>
= jArlp . The solution is plotted on logarithmic paper versus
in Figure 3.3. At early time,
Asymptotically,
(3.2-11)
(3.2-12)
-^ = 0
(solid) exhibits a unit slope due to borehole storage.
becomes proportional to In/^, and its derivative (dashed) with
respect to In/^ becomes a constant. This makes radial flow easy to distinguish from
spherical flow.
sensitive to
Both the pseudo-pressure and derivative type-curves are sufficiently
to allow extracting porosity values from time-pressure data that match the
radial type-curves.
91
l.C&Ol
l.CEKX)
I.G&Q2
I.C&04
l.QE+OO
l.O&Ol
1.0&02
l.CGKB
Figure 3.1; Type-curves of dimensionless pseudopressure in injection interval versus
normalized dimensionless time for various Co and ^ = 0 under spherical flow.
92
I.0EH)5
1.0&-10
Figure 3.2: Type-curves of dimensionless pseudopressure and its derivative in injection
interval versus dimensionless time for various Co and s under spherical flow.
approxiinatB b^inning of
pseudo-steacty state
unit slope
behavior
l.O&Ol
1.0EK)I
1.0EK)3
tc/Oo
Figure 3.3: Type-curves of dimensionless pseudopressure and its derivative in injection
interval versus dimensionless time for various Co and j = 0 under radial flow.
94
3.3 UNIFORM FLUX HORIZONTAL AND VERTICAL FRACTURES
Some of the early singie-hole pneumatic time-pressure data from the ALRS
delineate a half-slope on logarithmic paper.
Such behavior is typically observed in
boreholes that are intersected by a high-permeability planar feature like a major fracture.
Gringarten and Ramey (1974) developed a uniform flux model for a horizontal
fracture in an infinite, uniform confined aquifer. This model was developed in the
petroleum industry to model pressure responses in wells that were completed with large
fractures induced by means of hydraulic fracturing
to enhance oil recovery.
This
particular model can be classified as a hybrid model, which differs from the continuum
models in a sense that a discrete feature is built explicitly into a porous continuum.
In this model, flow rate across the fracture is uniform and equal to the rate of
injection or withdrawal. The fracture has finite thickness
is disk-shaped with radius
/y, centered about the well, and lies at an elevation Zj. above the bottom of the aquifer
(Figure 3.4). The presence of the fracture renders the aquifer anisotropic with horizontal
permeability
greater than the vertical permeability k,. Our modified version of this
solution is
r
I
4 A
• '^
fD
/
>I
Jexpl
n
>
^
( n - T T T n ^
—
^
I'd
/o
4rJ I 2^-0 J
(
^
if P
J •H — J •H — J •' ' " I J
where the dimensionless variables are now defined as
95
(3.3-2)
r
(3.3-3)
(3.3-4)
=
_
(3.3-5)
The derivative with respect to In/^ is (Gradshteyn and Ryzhik, 1965, p. 23, eq. 0.410)
Figure 3.5 shows type-curves of
(solid) versus
for various values of
.
Another model by Gringarten et al. (1974) considers a pumping well intersected
by a vertical fracture of zero thickness that completely penetrates a confined, uniform and
isotropic aquifer.
Under the assumption that flux is distributed uniformly across the
fracture, whose edges are at equal distances from
the well, their solution at the center
of the fracture where it intersects the active well reads, in modified form.
(3.3-7)
where
96
Tikbf T„
w
Qsc^Tp^
(3.3-8)
kfp
erf
is
error
function,
Ei
is
exponential
(3.3-9)
integral,
and
is
volumetric
withdrawal/injection rate written in terms of standard conditions, from/into the fracture
wliich, in turn, is equal to that from/into the well. The time derivative is given by
Mathematica™ as
(3.3-10)
Figure 3.6 shows type-curves of
(solid) and d-w^^ldXnt^ ^ (dashed) versus
a pumping well at the center of the fracture.
for
Half-slope behavior is evident during early
time when uniform flow into the fracture dominates. At late time, the flow pattern is
predominantly radial. The derivative of pseudopressure (dashed line) reaches a constant
value of V-i as in the radial flow case without a fracture.
Solutions have also been developed for partially penetrating (Raghavan et al.,
1978) and finite conductivity (Cinco-Ley et al., 1978) vertical as well as inclined
(Cinco-Ley etal., 1975) fractures. These models are more complex than the ones already
presented and include a larger number of parameters. They do not necessarily provide a
more realistic representation of pneumatic test conditions at the ALRS than do the
simpler models we just described, and many of their parameters are difficult to either
define or measure at the site. We are therefore guided by the principle of parsimony in
97
working with the simplest models that nevertheless help us interpret the available
pneumatic test data in a satisfactory manner.
Figure 3.4; Geometry of idealized horizontal fracture in an infinite flow domain.
99
l.OEKE
ho^^.OS
hD=0.2
I.OEH)I
hD^-3
hD=0.4
hD=0.5
hD=K).7
Q
Q
hD=l.O
hD=2-I0
I.OEKX)
l.OE-01
1.0&Q2
l.OE-OI
I.OEKX)
l.OE+OI
1.0EH12
to
Figure 3.5; Type-curves of dimensionless pseudopressure in injection interval,
normalized by dimensionless height, versus dimensionless time for various /loin uniform
flux horizontal fracture model.
100
i.(e«i2
I.(eK)l :
I.CKHX) :
•s
l.OEOl •
I (^02
I.(®<12
—
^
l.OBOl
l.OBKX)
1.0B401
l.OEKG
l.OEKB
1.0&04
iorf
Figure 3.6: Type-curves of dimensionless pseudopressure and its derivative in injection
interval versus dimensionless time in uniform flux vertical fracture model at center of
fracture (Xo = y ^ = 0 ) .
lOl
4. TYPE-CURVE INTERPRETATION OF SINGLE-HOLE
PNEUMATIC INJECTION TEST DATA
4.1 SINGLE-HOLE TEST METHODOLOGY
Single-hole pneumatic tests were conducted to help characterize unsaturated
fractured rocks at the ALRS on nominal scales (borehole interval lengths) of 0.5, 1.0, 2.0
and 3.0 m.
The tests and their analyses by means of steady state formulae, were
conducted by Guzman et al. (1994, 1996) and Guzman and Neuman (1996). Each test
involved the injection of air into a section of a borehole isolated by means of inflatable
packers. The corresponding air injection system is depicted schematically in Figure 4.1.
The system included a straddle packer assembly consisting of two inflatable rubber
bladders (each 3 m in length-see Plate 4.1), a set of flow meters and flow controllers,
pressure valves and regulators, and an electronic monitoring system to automatically
record field data.
The distance between bladders was adjustable.
Air pressure,
temperature and relative humidity were recorded in the test interval. Mass flow meters
and controllers were used to cover two ranges o f equivalent volumetric rate, 0 - 0 . 1
SLPM (standard liters per minute) and 0-20 SLPM. Flow rotameters served as visual
backup to help maintain a constant volume rate between 0.01 SLPM and 20 SLPM.
Measurements were recorded using a data logger connected to an optically isolated
interface, which allowed periodic downloading onto an on-site personal computer.
Recording was done at small intervals during early times when changes in pressures were
expected to be largest and at successively longer intervals thereafter.
102
Table 4.1 lists the coordinates of six boreholes subjected to single-hole air
permeability testing at the ALRS. The coordinate system is shown in Figure 1.2 and its
origin coincides with the lower lip of the near-surface casing in borehole Z3. The table
also lists the approximate length and dip of each borehole. A three-dimensional
perspective of all l-m scale air permeability test locations, viewed from the Southwest
toward the Northeast, is shown in Figure 1.7. Each point in the perspective represents the
center of a borehole test interval between packers. Each test interval is identified by
borehole and the distance along the borehole from its center to the L.L. point listed in
Table 4.1. There were neither gaps nor significant overlaps between test intervals in a
borehole.
Prior to each air injection test, the packers were inflated to isolate the test interval,
and the resulting pressure was allowed to dissipate. The test commenced by injecting air
into the packed off interval at a constant mass flow rate.
This continued until the
pressure stabilized so as to vary by not more than the equivalent of 1 mm of mercury in
30 min. The test continued by incrementing the mass flow rate and monitoring pressure
until it attained a new stable value. Most tests included three or more such incremental
steps of mass flow rate. Injection was then discontinued and the pressure allowed to
recover back to atmospheric.
103
Table 4.1: Coordinates of boreholes subjected to air permeability testing (adapted from
Guzman etal. 1996)
Bore L.L. X
Borehole B B . r
BB.y
L.L. Z
General
L.L.y
B.B.Z
-hole
[m]
[m]
length
[m]
dip
[m]
[m]
[m]
[m]
direction
V2
30
6.8
4.24
6.84
0.01
4.2
-30.0
Vertical
W2a
5.42
5.4
-0.03
30
0.2
21.46
SSE
-21.2
X2
20.44
30
-0.8
10.0
WSW
10.03
-0.02
-21.2
Y2
20.04
30
5.2
WSW
5.20
-0.31
-1.2
-21.5
Y3
30.07
5.3
WSW
45
-1.8
5.35
-0.27
-32.0
9.80
Z2
31.0
0.0
ENE
0.03
30
-0.20
-21.4
Coordinates are shown in Figure 1.2 and their origin is at lower lip of Z3 casing.
Borehole length is approximate. L.L. marks lower lip of near-surface casing in each
borehole; B.B. marks borehole bottom (approximate).
Table 4.2; Nominal Scale and Number of Single-Hole Pneumatic Injection Tests at the
ALRS adapted from Guzman et al. 1996).
Nominal Scale
Number of Intervals
j
Borehole
(Length of Test
Tested
Interval)
1.0 m
21
V2
37
1.0 m
W2A
30
1.0 m
X2
2.0 m
10
X2
54
0.5 m
Y2
28
l.O m
Y2
2.0 m
14
Y2
9
3.0 m
Y2
39
1.0 m
Y3
1.0 m
28
Z2
270
Total
The packers were deflated, the instrument was repositioned in the borehole, and
testing resumed until the entire uncased length of a borehole has been tested. The method
has proved reliable in that repeated tests of selected intervals, over several years, have
104
given highly reproducible permeability estimates. This was due in part to a strict quality
assurance and quality control protocol at each stage of testing.
Tests were conducted along borehole intervals of lengths 0.5, I.O, 2.0 and 3.0 m in
borehole Y2; 1.0 and 2.0 m in borehole X2; and 1.0 m in boreholes V2, W2A, Y3 and Z2.
A total of more than 270 single-hole injection tests have been completed, of which 180
were conducted along l-m sections in six boreholes (Figure 1.7 and Table 4.2) that span
20,000
of unsaturated porous fractured tuff. Guzman et al. (1996) relied on a steady
state analysis of late data during each step to obtain a corresponding pneumatic
permeability value. They used a modified formula originally developed by Hvorslev
(1951; see also Hsieh et al. 1983) which assumes that, during each relatively stable
period of injection rate and pressure, air is the only mobile phase within the rock near the
test interval and its steady-state pressure field has prolate-spheroidal symmetry. This
implies that the rock forms a uniform, isotropic porous continuum. The formula reads
(after substituting eq. 3.1-5 into eq. 1, p. 10, of Guzman et al. (1996))
^ _ Qpft\n{blrJ)
(4-1)
where k is permeability [Z,"], Q is volumetric flow rate at standard conditions [L^T'\
H
is dynamic viscosity of air at standard conditions (l.81 x 10 ^ Ns/m'^, b is length of the
test interval [Z,],
interval,
is borehole radius (0.05 m ) , p \ s steady-state pressure in the injection
o is ambient air pressure.
105
During single-hole tests, air injection was conducted at a number of incremental
volumetric flow rates. The type-curve analyses of transient single-hole test data that we
present below concern only the first of any such multistep sequence.
Date Logger
CR-IO
Flow CoatroUer
Lme Dner, Futer,
Pieanre Tmuducer.
Thermictor
Pressure
Reservoir
Pieaiure
Tnmaducer
(S)
Valve
Thennitrnr ^
Relative
Hbunidity CeU
KMolator
Figure 4.1: Schematic diagram of the air injection system (adapted from Guzman et al.
1996).
106
Plate 4.1: Photogn^of single-hole packer system en^toyed during the study of Guzman
«/a/.(1996).
107
4.2 PHENOMENOLOGY OF SINGLE-HOLE TESTS
Figure 4.2 shows how air pressure and temperature varied during a typical
multistep single-hole pneumatic injection test, labeled VCClOOl, conducted within a \-m
interval whose center was located 10.37 m below the LL marker (Table 4.1) in borehole
V2 at the ALRS (our designation of single-hole tests follows that established by Guzman
et al. 1994, 1996).
Whereas temperature was nearly constant throughout the test,
pressure first rose to a peak and then declined toward a stable value. The same pressure
phenomenon is seen more clearly in a plot of pressure versus time during the first step of
test CGA1120 (Figure 4.3), conducted within a X-m interval whose center was located
20.15-/W below the LL marker (Table 4.1) in borehole X2. Simulations of two-phase
flow in a porous medium by Guzman et al. (1996) have confirmed that this phenomenon
is due to displacement of water by air in the immediate vicinity of the injection interval.
When air is injected into a rock that contains water at partial saturation, the latter acts to
block its movement. Hence the permeability one computes for air is lower than what one
would compute in the absence of a water phase.
It follows that the computed air
permeability is less than the intrinsic permeability of the rock. Indeed, Guzman et aL
(1996) were able to demonstrate computationally that the higher is the applied pressure
during a test, the closer the computed air permeability is to its intrinsic value. They also
found that, in most test intervals, pneumatic permeabilities show a systematic increase
with applied pressure as air displaces water under two-phase flow. Only in a few test
intervals, which were intersected by high-permeability fractures, did air permeability
108
decrease with applied pressure due to inertial efTects. In many cases, air permeability
exhibited a hysteretic variation with applied pressure.
The observed stabilization of pressure at late time during each step recurs in more
than 90 percent of single-hole tests at the ALRS. This could, in principle, be due either to
the establishment of three-dimensional flow around the test interval or the presence of an
atmospheric pressure boundary in its vicinity. As pressure and pressure derivative data
from most test intervals fit type-curves based on a spherical flow model in an infinite
domain, we attribute the observed pressure stabilization to three-dimensional flow
(Jllman et al. 1997; Illman et al. 1998).
Figure 4.4 shows the same data as those in Figure 4.3 re-plotted on logarithmic
paper. It reveals that, at early time, the pressure increases linearly with a unit slope that is
diagnostic of borehole storage due to gas compressibility (van Everdingen and Hurst,
1949; Papadopulos and Cooper, 1967, Agarwal et al. 1970, Joseph and Koederitz, 1985).
Only when this straight line starts curving does air start penetrating into the rock in a
measurable way. The ensuing period of transitional flow, prior to pressure stabilization,
is seen to be extremely short and dominated by two-phase flow effects.
As such, it
cannot be analyzed to yield reliable values of fracture porosity or borehole skin factor. A
similar difficulty is encountered in the majority of single-hole tests at the ALRS in which
the pressure stabilizes in a manner typical to three-dimensional flow.
The two-phase
flow efifect is more pronounced on an arithmetic plot than on a logarithmic plot of
pneumatic pressure versus time.
109
Figure 4.5 shows a relatively rare example of a single-hole test (JJA06I6 in a \-m
interval with center located 17.77 m from the LL mark along borehole) at the ALRS in
which the pressure does not stabilize but continues to increase in a manner that is
characteristic of radial flow (on semi-logarithmic paper, the late pressure data would
delineate a straight line). Here the transient period, following that dominated by borehole
storage, is sufficiently long to allow extracting from the corresponding data information
about air-filled porosity and skin factor. Our attempts to do so, for this and other data
that appear to fit radial type-curves, led to air-filled porosity values that are much smaller
than those we obtain later by a numerical inverse analysis of similar data and from crosshole tests (Illman et ai 1998); we therefore suspect the reliability of porosity values
obtained from
radial type-curve fits at the ALRS and do not quote them in this
dissertation. In a still smaller number of tests, such as that illustrated in Figure 4.6
(JHB0612 in a \-m interval with center located 15.81 m from the LL mark along
borehole), pneumatic pressure data delineate a straight line with a half-slope at early time
on logarithmic paper. Such behavior is diagnostic of a highly conductive planar feature
such as a wide fracture.
Indeed, a televiewer image of the test interval in Figure 4.7
reveals the presence of a wide-open fracture or cavity within it. At late time, the pressure
in Figure 4.6 continues to climb as is typical of radial flow. We analyze this test in this
chapter with the aid of various analytical continuum (radial, spherical) and fracture
(horizontal, vertical) flow type-curve models. In contrast. Figure 4.8 shows an example
of a test (ZDC0826 in a \-m interval with center located 13.56-m from the LL mark along
borehole Z2 in which an early half-slope indicates the presence of a conductive planar
110
feature, and the stable late pressure data are indicative of a three-dimensional flow regime
or possibly a recharge boundary.
Some single-hole tests, during which the pressure eventually stabilizes, contain a
sufficiently long transition period that is amenable to analysis by type-curves based on a
radial flow model. This is true about numerous \-m test data from boreholes W2A and
Y2. It suggests to us that flow around such test intervals evolves from radial to threedimensional with time. Though it seems to allow extracting from these tests information
about air-filled porosity and skin factor, we consider the results of such analyses
unreliable, for reasons mentioned earlier, and do not quote the corresponding parameter
values in this dissertation.
The fact that transient pressure behavior is not entirely consistent across the site,
but varies from one test interval to another, provides a qualitative indication that the site
is pneumatically nonuniform and the local rock is heterogeneous.
soo
§ 40U) oc
Ui
01
-aoap 2
o
UJ
o
m
Ui
QC
oA
-4oaj) "3^
Q.
Ui
o
200 3
-— ffitchgwH
— Protur*
-OO
TIME (MIN)
Figure 4.2; Example of multi-rate single-hole test (VCCIOOI in borehole V2 at 10.37 /n
from LL marker) (adapted from Guzman et al, 1996).
112
70000
60000
\
SOOOO
40000
/
pcMan TMdiM Maaify vriiw
£
a. 30000
20000
10000
5000
15000
10000
20000
25000
«^1
Figure 4.3: Aritfamedc plot ofpfessuie data fituii test CGAl 120
IBHJS
Hernfy^itme
1EH)4
2piiaMflaw
lEHO
i.
lE+02
lE+01
lE+00
1&02
lE-Ol
lE+00
1EH)1
IEHJ2
/fuel
lEHB
Hgure4.4: LogahtfamicplotofptessuredatafiantestCGAl 120
lEW
1EH)5
16+06
113
1EHJ5
lE+Ot
ibhb
g.
Q.
lE+02
lE+OI
1E«»
IBOl
IfiHX)
1EH}1
1BH)2
1E«<)3
IE+04
1EH>S
1EH)6
Figure 4.5: Logarithmic plot ofpiessure data fiom test JGA0616
lE+OS
lEHM
paendo-iteKfy
1E«B
1EH)2
IB«)1
lE+00
lE-01
lEHW
lEtOl
lE+02
lB+03
/(MCI
Figure 4.6: Logaridmnc plot of pressure data fiom test XHB0612
lEt04
lEHJS
lEt06
Figure 4.7: BETTV image taken in b<KdioleY2(hi^penneabiIily zone)
1.<E«)5
T
l.CEHM
l.OE+iB
I
1.0Et02
l.(eH)I
l.(BH)0
l.Q&Ol
' •
•
l.flE+00
• • i
• • i ....i
l.(E+02
1.QEH13
. • • ...ii
l.QBHM
t\suc\
Figute4.8; Logaiithmic plot of pressure data from sinde-hcrie test ZDC0826
-•
l.(E+05
l.QE+06
115
4.3 TYPE-CURVE INTERPRETATION OF SINGLE-HOLE TESTS
Over 40 sets of l-m scale single-hole pneumatic injection test data were
interpreted by means of the spherical, radial, vertical and horizontal fracture flow models
described in Chapter 3. Some of the data were interpreted using both p-based and p'based formulae in order to check the extent to which the method of linearizing the gas
flow equations affects test result. As will be illustrated below, the single-hole test data
examined showed little sensitivity to the method of linearization.
As was already
mentioned, the majority of our data conform to the spherical flow model regardless of
number or orientation of fractures in a test interval. We interpret this to mean that flow
around most test intervals is controlled by a single continuum, representative of a threedimensional network of interconnected fractures, rather than by discrete planar features.
Only in a small number of test intervals, known to be intersected by widely open
fractures, have the latter dominated flow as evidenced by the development of an early
half-slope on logarithmic plots of pressure versus time; unfortunately, the corresponding
data do not fully conform to available type-curve models of fracture flow. Some pressure
records conform to the radial flow model during early and intermediate times but none do
so fully at late time.
Figure 4.9a shows visual fits between pressure (circles) and pressure derivative
(triangles) data from test CAC08I3 (in a l-m interval with center located 4.47 m from the
LL mark along borehole X2), and /;-based type-curves corresponding to the spherical
flow model. Pressure derivatives have been calculated using an algorithm described in
Appendix H.
Figure 4.9b shows similar fits between incremental squared pressure
116
(circles) and derivative (triangles) data, and type-curves corresponding to a spherical flow
model
expressed
in
derivative ^^w^/^ln/o »
terms
of
dimensionless
pseudopressure
and
its
recall from (2.1-16) that pseudopressure, w, is proportional to
incremental squared pressure, Ap', when /zZ is constant as we take it to be at the ALRS.
Both the />-based and /7*-based sets of data exhibit a good match with type-curves that
correspond to zero skin (s = 0); indeed, most test data from the ALRS show little
evidence of a skin effect. The two sets of data yield similar values of air permeability,
1.29 X 10"''
based on p and 1.56 x 10"^'
based on p^. Each set of early-time data
falls on a straight line with unit slope, indicative of compressible air storage within the
test interval. A pressure peak due to two-phase flow is not discernible on the logarithmic
scale of Figures 4.9a,b.
Figures 4.10a,b show similar type-curve matches for test CHB0617 (in a l-m
interval with center located 24. l-m from
the LL mark along borehole X2). Both the p-
and p^-based sets of data exhibit a fair match with type-curves that correspond to zero
skin for early to intermediate data. The two matches yield comparable permeabilities,
6.11
X
10"'^ m' based on p and 6.01 x 10"'^ m' based on p'. Late data do not match the
type-curves due to apparent displacement of water by air, which manifests itself as a
gradually decreasing skin effect; this two-phase flow effect is most clearly discernible
when derivative data are plotted in terms of pressure squared. Each set of early-time data
is strongly affected by compressible air storage within the test interval. A pressure peak
due to two-phase flow is now discernible on the logarithmic scale of Figures 4. IOa,b.
117
A complete list of air permeability values and skin obtained by means of /7-based
and />^-based spherical type-curve analyses is given in Table 4.3. Skin factors that appear
to evolve with time due to two-phase flow are not included in the table. Figures 4.11
compares permeabilities obtained by steady-state and p-based transient analyses; Figure
4.12 compares those obtained by steady-state and p'-based transient analyses; and Figure
4.13 does so for permeabilities obtained by p'-based and /?-based analyses. Overall, all
three data sets agree reasonably well with each other. The good agreement between p^based and ^-based results suggests that the method of linearization has little influence on
our interpretation of \-m single-hole pneumatic air injection tests at the ALRS. This
suggests that nonlinear effects due to gas compressibility have little effect on pneumatic
pressure propagation through the rock, though our analysis shows that they have a major
impact on gas storage within borehole injection and monitoring intervals.
Figures 4.14 a,b show p- and p'-based type-curve matches for single-hole test
JGA0605 in a \-m interval with center located 11.89 /n fi*om the LL mark along borehole
Y2. In this case, the early and intermediate data appear to fit the radial flow model but
the late pressure data stabilize, and the late pressure derivative data drop, in manners
characteristic of three-dimensional flow. The same is seen to happen when we consider
in Figures 4.15a and b data from single-hole test JJA0616 in a \-m interval with center
located 17.77-m from the LL mark along borehole Y2. We take this to indicate that the
flow regime evolves from radial to spherical with time.
The \-m test interval JHB0612, with center located 15.81-/n from the LL mark
along borehole Y2, intersects a fracture which on the televiewer (Figure 4.7) appears to
118
be widely open. Figure 4.16 depicts an attempt on our part to match the corresponding
incremental squared pressure data to type-curves of pseudopressure based on the
horizontal fracture flow model described in Chapter 3. Only the early time data appear to
match one of these curves. We suspect that deviation of the late data from the typecurves is due to the fact that whereas in reality the flow evolves with time to become
three-dimensional, in the model it evolves to become radial. Upon ignoring the late data
and considering only the early match, we obtain an air permeability of 1.32 x 10"'^ m^.
This is about four times the value of 4.8 x 10"''* m' obtained by Guzman et ai (1996) on
the basis of a steady state analysis of the late data.
An unsuccessful attempt to match the same data with a type-curve corresponding to
the vertical fracture flow model, described in Chapter 3, is depicted in Figure 4.17.
Pressure and, to a much greater extent, pressure derivative data from several
single-hole pneumatic injection tests, some of which are illustrated in Figures 4.18
through 4.21, exhibit inflections that are suggestive of dual or multiple continuum
behaviors. If so, we ascribe such behavior not to fractures and rock matrix as is common
in the literature {Warren and Root, 1963; Odeh, 1965; Gringarten, 1979, 1982), but to
fractures associated with two or more distinct length scales. However, these inflections
show some correlation with barometric pressure fluctuations, implying that they might
have been caused by the latter rather than by dual or multiple continuum phenomena.
119
Table 4.3; Air permeabilities obtained using p- and ^'-based spherical flow models.
Test
CAC0813
CBA0902
CDB1007
CGA1204
CHA0121
CHB0617
CIA0621
JFC0604
JGA0605
JGB0608
JGC0609
JHA0611
JHB0612
JJA0616
JJB0618
JKB0623
JKC062S
JNA0713
2JG092
VCB0924
VCClOOl
VFB0318
VHA0422
VHB0429
WFClOl
YAA0301
YDA0426
YFB0621
YGB070S
YGC0709
YIC1002
YLB1108
ZCB0819
ZDB0825
ZDC0826
ZFC0918
ZHA1006
ZIB1109
borehole location (m
4.5
X2
5.5
X2
12.3
X2
20.2
X2
X2
23.1
X2
24.1
26.0
X2
10.9
Y2
11.9
Y2
12.9
Y2
13.9
Y2
14.8
Y2
15.8
Y2
17.8
Y2
18.8
Y2
21.7
Y2
22.7
Y2
Y2
26.6
16.1
Y2
V2
9.4
V2
10.4
V2
18.4
V2
23.4
V2
24.4
W2A
19.6
Y3
2.6
Y3
11.6
Y3
18.6
21.6
Y3
Y3
22.6
Y3
28.6
Y3
36.6
9.6
Z2
12.6
Z2
13.6
Z2
19.5
Z2
23.5
Z2
27.5
Z2
p-based
p^-based
k [m'\
k [m'\
1.3E-15
7.7E-16
l.lE-14
3.1E-16
1.4E-15
6.1E-17
5.8E-17
1.8E-15
4.3E-15
9.2E-15
2.2E-15
3.6E-15
5.1E-14
6.5E-15
3.9E-15
1.6E-15
8.2E-16
1.3E-14
3.4E-16
1.5E-15
6.0E-17
6.4E-17
1.9E-15
4.6E-15
1.2E-14
2.9E-15
4.0E-15
6.5E-14
8.0E-15
4.3E-15
9.1E-16
4.3E-15
1.7E-16
2.6E-14
3.2E-16
1.4E-15
8.3E-17
1.6E-17
2.9E-15
1.2E-16
3.1E-16
1.3E-15
1.5E-16
3.1E-17
2.5E-17
9.0E-16
4.9E-14
5.0E-16
1.2E-17
1.3E-17
4.4E-17
2.1E-17
l.lE-14
1.2E-14
4.6E-14
7.2E-17
7.4E-17
l.lE-15
9.1E-16
5.0E-16
l.lE-17
1.3E-17
3.5E-17
1.2E-14
5.9E-14
1.6E-16
1.2E-15
skin
1.4E-15
l.lE-15
1.8E-15
3.6E-16
1.6E-15
7.8E-17
5.9E-17
1.6E-15
3.0E-15
2.1E-15
2.0E-15
2.7E-15
4.8E-14
5.6E-15
3.4E-15
8.7E-16
1.2E-15
9.1E-17
2.8E-14
4.5E-16
1.3E-15
l.lE-16
2.7E-17
2.0E-17
8.0E-16
5.2E-14
5.6E-16
l.OE-17
l.OE-17
7.5E-18
3.9E-17
1.6E-17
1.3E-14
1.3E-14
5.3E-14
1.4E-16
8.2E-17
1.3E-15
0
negative
3 and incieasine
negative
0
negative
0
0
0 and increasing
3 and increasing
0 and increasing
0 and increasing
0 and increasing
0 and increasing
0 and increasing
0
1
2 and increasing
0 and increasing
0
neagtive
0 and decreasing
0 and increasing
?
7
0
0 and decreasing
?
0
?
7
0
0
0
negative
0
120
\\
l.QE-02
TGAC 0
Figure 4.9a: T^ipe-curve matdi of data fiom siii{^e-hde pneumatic test CAC0813 to the
1.0&02
lo/c/t
Figure 4.9b: Type-curve match. oCdata ftom-siii^e-faQle pneumatic test CAC0813 to the
gas, qihencal flow modd.
121
10
IE-01
lE-02
lE-02
lE-01
Figiire4.10a: Type<urve match ofdatafiomsmg^e-hdepneamatic test CHB0617
Kqoid, sphencal flow modd.
o-s, 10
%•
i
L€&02
l.(E-01
l.(e400
LOE«01
l.(£^
l.<e^
1.(^404
ta/cjt
Figure 4.1Ob: Type<urve matdi of data fiom sin^e-hcle pneumatic test CHB0617
>with gas, qihencal flow modd.
L<eH>S
122
l.OE-13
1.0E-I4
"jl l.OE-15
sv
2 l.OE-16
«5
1.0E-I7
1.0E-I8
I.OE-18
I.OE-17
l.OE-16
l.OE-15
l.OE-14
l.OE-13
l.OE-12
p-bastdk [jr 'I
Figure 4.11;Scatter diagram of results of permeability obtained from steady-state and p -based
spherical models.
l.OE-13
l.OE-14
l.OE-15
s
3 l.OE-16
m
l.OE-17
l.OE-18
l.OE-18
l.OE-17
l.OE-16
l.OE-15
l.OE-14
l.OE-13
l.OE-12
p'-bascdik [m'l
Figure 4.12:Scatter diagram of results of permeability obtained from steady-state and p ^ -based
spherical models
t.OE-14
I.OE-15
9 l.OE-16
I.0E-I7
l.OE-18
1.0E-I8
I.OE-17
l.OE-16
I.0E-1S
1.0E-I4
l.OE-13
p-based ft [m']
Figure 4.13;Scatter diagram of results of permeability obtained firom p - andp -based
spherical models.
I.OE-
124
««
a
a.
lE+04
TO/CO
FigDie4.14a: Type^nrve match of data from singte-hdeiaieuiiialic test JGA060S
with Ucpud. racfi^ flow model
l.(EM)2
4
««
1.QB02
IX&Ol
IXeHX)
lUlEH)!
IJOBKa
lXSE*<a
t/f/co
Figure 4.14b: Tjpe-carve match of data frmn single-hole pneumatic test JGA0605
with gas, radial flow modeL
IXBHM
125
l.OEtOl
l.OB+Ol
l.OE+02
1.0E«)4
Figiue 4. ISa: Type-curve match of data fixnn single-hcde pneumatic test JJA0616 to the
gas, spbeiical flow model
i^ppnBOBMte begimiog cf
lE+01 Qy-10'
lE+04
1&02
o
Figure 4.1Sb; Type-curve match of data from single-hole pneumatic test JJA0616 to the
gas, radial flow model.
126
VMcii fiadnra aohnioa
>.05
LOE-tOl
LOE-HU
'o
Figiire4.16; Type-curve matdi of data fiom singje-hole pneumatic test JHB0612 to the horizontal
fiacdue model.
LOE-KU
LOE-HM
LdE-fOl
1.0E-HM
Figuie4.17; Type-curve match of data from single-hole pneumatic test JiiB0612 to the vertical
fracture mode!.
127
^ LOE-HW
to/co
Figure 4.18: Type<urve matdi of data fiom sii^e-hde pneumatic test CDB1007 to the
gas, radiai flow modd.
LOE-HW
LOE-HU
LOE-HU
Figute4.19: Type-curvematchofdatafiomsin^e-holepneumatictestJNA0713tottte
gas, radial flow modd.
LOE-HM
128
O-S.10
tf/c^
Fi0ire4.2O: lype-cmve match ofcfala from single-hole pnemnatK test JKC0625 to die
9B, spherical flow model
O-S. 10
jkoeli^
tyc,
Fi9iie4.21: TVpe-corvsmatchofpiessiiiedatafiamtestYFB0621 todie
g^s, spheiical flowmodeL
129
CHAPTER 5: TYPE-CURVE MODELS FOR THE
INTERPRETATION OF CROSS-HOLE TESTS
5.1: INTRODUCnON
Subsurface materials including fractured media often exhibit anisotropy with
respect to permeability. In the field, such anisotropy is due to fracturing or stratification
in sedimentary materials. The permeability of an anisotropic material is described by a
second-rank, symmetric, positive-definite tensor. Because anisotropy has an important
effect on fluid flow and contaminant transport, the determination of this tensor is
important for subsurface hydrology.
Existing methods to determine the hydraulic conductivity or permeability tensor
by means of pumping/injection tests are limited to horizontal aquifers in which one of the
principal directions is vertical. Papadopulos (1965), Hantush (I966a,b), Hantush and
Thomas (1966), Neuman et aL (1984), and Loo et al. (1984) developed methods to
determine the two-dimensional anisotropic transmissivity tensor of an aquifer under
conditions of horizontal flow.
Weeks (1969) and Wc^ and McKee (1982) proposed
methods to determine the vertical and horizontal hydraulic conductivities of confined
aquifers.
Hantush (1961), using partially penetrating wells presented a method to
determine vertical anisotropy. A method that enables one to do the same for unconfined
aquifers exhibiting delayed gravity response (or delayed yield) was developed by
Neuman (1975).
Interests in situations where the principal directions of the hydraulic conductivity
tensor are not necessarily horizontal and vertical have risen primarily among geotechnical
130
engineers concerned with flow through fractured rocks. Virtually all the methods that
have been proposed to date require that the principal directions be known prior to the test.
In the absence of more reliable alternatives, these directions are usually predicted by the
basis of fracture geometry information obtained from boreholes and surface or subsurface
rock exposures. In the past, individual boreholes were drilled parallel to the conjectured
principal directions in order to determine the corresponding principal hydraulic
conductivities by conventional packer tests {Snow, 1966).
Because the principal
directions are orthogonal, the boreholes must be perpendicular to each other, which is a
difiHcult drilling requirement. Furthermore, single-hole packer tests only give results that
are valid in the close vicinity of the borehole. For example, Butler and Healey (1998)
discuss the reliability of parameters obtained from single-hole tests. According to Butler
and Healey (1998), skin effects may adversely affect single-hole tests such that parameter
estimates from single-hole tests may be biased.
They state that interference/cross-
hole/pumping tests give more accurate estimates of medium properties. Illman et al.
(1998) clearly show through their type-curve analysis that their single-hole data from the
ALRS are not largely affected by skin effects (see section 4.3).
To enable the measurement of principal hydraulic conductivities on a larger scale
(between boreholes), Louis (1974) proposed a method that requires drilling one injection
hole and two monitoring holes parallel to any one of the three (known) principal
directions. The two monitoring holes are drilled so that the shortest lines between them
and the injection hole are parallel to the remaining two principal directions. A "hydraulic
triple probe", consisting of four packers, is inserted into the injection hole to create three
131
isolated intervals (see Figure 2 in Hsieh and Neumcm, 1985a). Fluid is injected into all
three intervals at a known rate and pressure. The role of the two "guard intervals" is to
minimize deviations from radial flow around the central interval. The measurement of
pressure inside the short intervals between long packers in the monitoring holes makes it
possible to compute all the three unknown principal hydraulic conductivities.
In this chapter, a field method originally proposed by Hsieh and Neuman (1985a)
is applied to the determination of the three-dimensional permeability tensor of an
unsaturated anisotropic medium. Their method differs from previous approaches in two
important ways: (1) it does not require that the principal directions be known prior to the
tests and (2) the boreholes may be drilled in any directions that are technically feasible
(e.g., they may be vertical, slanted or horizontal).
Here, we present an extension of
solutions given in Hsieh and Neuman (1985a), which account for gas flow conditions,
observation wellbore storage and skin effects. In the absence of observation wellbore
storage and skin effects, the solution approaches that given by Hsieh and Neuman
(1985a).
While the method is applicable to both porous and fi-actured media, we
concentrate on the latter because it is fi-acturing that most often prevents one from
determining a priori the principal directions. Because the permeability of fractured rocks
is often quite low, we consider fluid injection rather than fluid withdrawal. However, for
more permeable media in which pumping is possible, the same test could be performed
by reversing the direction of flow without any changes in the method of interpretation.
The proposed method is referred to as cross-hole testing {Hsieh and Neuman,
1985a), to distinguish it from the more conventional single-hole packer. It consists of
132
injecting fluids into packed-off intervals in a number of boreholes and monitoring the
transient pressure response in monitoring intervals in neighboring boreholes.
An
important aspect of the method is that it provides direct field information on whether or
not the rock behaves as a uniform, anisotropic continuum on the scale of the test. We
first describe the mathematical model, then systematically present theoretical expressions
for transient type-curves given in Hsieh omd Neuman (1985a) modified to account for gas
flow conditions. Expressions of pressure derivatives not given in Hsieh and Neuman
(1985a) are also given. We then describe a method to modify the Hsieh and Neuman
(1985a) solutions to account for observation wellbore storage and present its pressure
derivatives. The method to account for observation wellbore storage is based on the
theory by Black and Kipp (1977). After we discuss the characteristics of type-curves, we
employ those new curves to analyze a set of pressure data obtained fi"om a cross-hole
pneumatic interference test conducted in unsaturated fractured rocks at the ALRS.
For purposes of cross-hole test analysis, we represent the fractured rock by an
infinite three-dimensional uniform, but anisotropic continuum, as was done by Hsieh and
Neuman (1985a).
In terms of pressure, single-phase airflow is then governed
approximately (due to linearization)
Ot
where the permeability is written in terms of a tensor and the variables were defined
previously. We choose a convenient set of "working" cartesian coordinate system, x,,
/ = 1,2,3, so that a point in the flow domain is located by the vector x = (x,}, and the
133
permeability tensor is represented by the matrix k =
definite and symmetric.
j. We assume that k is positive-
The initial pressure distribution in the medium (and hence
pressure) is assumed to be uniform:
p{x,t) = 0-
t=0
(5-2)
We also begin by assuming that the flow domain is infinite, so that
p ( x j )= 0 ;
1
*
1
(
5
-
3
)
The injection and observation intervals are idealized, depending on their lengths,
as points or lines. In the following analysis, we deal only with fluid injection into the test
interval. The analysis is identical for fluid withdrawal if the increase in pressure is
replaced by decrease in pressure. The following four cases are considered; (1) injection
at a point, observation at a point; (2) injection along a line, observation at a point; (3)
injection at point, observation along a line; and (4) injection along a line, observation
along a line.
5.2: POINT INJECTION/POINT OBSERVATION
Consider a point source injecting at a constant volumetric rate, Q, at the origin of
the working coordinate system. The increase in pressure at a point x and at time / is
given in dimensionless form by
=e^c{l/(4/o)"'}
(5-4)
where etfc[ ) is the complementary error function and
is the dimensionless pressure
increase.
134
(5-5)
The subscript denotes the case number and
is the dimensionless time
ta=Dt/(S^^J
while
D
(5-6)
is the determinant of k, i.e.,
D — ^11^22^33
^11^23
^33^12
(5-7)
and G„ is the quadratic form, defined as
(5-8)
= x ^ \ x = x , x ^ A,J
where the superscript
T
indicates transpose. Unless stated otherwise, summation of
repeated indices is implied. The matrix A in (5-8) is known as the adjoint of k, its
components are given by (no summation implied below)
(5-9)
^•j = ^j' =
(5-10)
where /,y,and k are cyclic arrangements of 1,2, and 3 (i.e., /,7,^ = 1,2,3, or 2,3,1, or
3,1,2). It can be shown that A, like k, is symmetric and positive-definite. The time
derivative of the point injection/point observation solution is
c/ln/o
= - M e x p ( - u)
2V;r
where u = 1/4/^ . For isotropic media, ^12 =
the (scalar) permeability.
Consequently,
Thus d = k ^ , k =
= R^k', where
R
'
= ^13 = 0.
(5-11)
i = ^22 = ^33 = ^ ^ where k is
, where I is the identity matrix.
is the radial distance from the point source, and the
135
dimensionless expressions (5-5) and (5-6) take the simple form
= AjipkR/Q/j and
= k f p j . The solution for the isotropic case is seen to possess spherical symmetry
about the point source.
A logarithmic plot of
(solid curve) and dpa^fdXvit^ (dashed curve) versus
is shown in Figure 5.1. It is interesting to note that a comparison of the point source
and spherical flow solutions with wellbore storage and skin (Figure 3.2) shows that the
derivative decays much sooner for the latter model.
This is because when borehole
storage in the injection borehole is dominant (in comparison to borehole storage in
observation intervals - shown later), the transient fluid behavior becomes very short in
duration and approaches steady-state at a much faster rate (which is clearly shown
through the derivatives).
136
l.OE+01
l.OE+00 •
I.0E^2
1.0Er02
I.OE-Ol
l.OE+00
l.OE+Ol
l.OE+02
I.OE+03
l.OE+04
to
Figure 5.1; The point source solution (solid curve) and its time derivative (dashed curve).
137
5.3: LINE-INJECnON/POINT-OBSERVATION
The solution for a finite line source injecting at a constant volumetric rate, Q, can
be obtained from the solution for a point source by integration. Let a line source of
length L be centered at the origin of the x, coordinate system and be represented by the
line BOA (see Figure 4 in Hsieh and Neuman, 1985a). Let OA be represented by the
vector
(5-12)
so that OB is -1. In addition, let e and e, be unit vectors in the direction x and I
respectively, so that
x = Re
and I = (Ll2)e,. If we assume that flux along the line source
is uniform, then the solution at point x is given in the following dimensionless form
Pd,=]: J -expf- (l - a2')y\-+ l/a.)]" *:rf[y"'{a2 - l/a,)]}«^
(5-13)
where erf{^ ) is the error function, and the two geometric parameters a, and a,, are
defined as
(5-14)
(5-15)
and
Gj = x'"AI = x,/^4,
(5-16)
138
G„=rAl = /./,4
(5-17)
The dimensionless pressure increase is now
= inpQ^'-lQix
the dimensionless time,
(5-18)
is the same as in (5-6).
parameters a, and a, become evident when
The meanings of geometric
, and G„ are expressed in terms of
R^L, e, and e,:
=/?-(e'"Ae)
(5-19)
G^={RL/2)le''\e,)
(5-20)
G^=(^V4Xe/Ae,)
(5-21)
Then
a, = (2 ^//:)[e'"Ae/e/ Ae,
(5-22)
a, = (e'^Ae,)^e'^Ae/e/Ae,j'
(5-23)
For an isotropic medium, (5-22) reduces to a, =(2/?//,), and (5-23) becomes
OTj = e^e, = cos 0, where 0 is the angle between x and I. Thus a, is related to the ratio
between
R
and L, and a, is related to the angle between x and I. For anisotropic cases,
the effect of anisotropy is incorporated into the definitions. Since A is positive-definite,
we have or, > 0 and 0 <
^ 1.
139
The partial derivative of eq. (5-13) can be calculated by means of Leibnitz's rule
of differentiation of a definite integral with respect to a parameter {Gradshteyn and
Ryzhik, p.23, eq. 0.410)
0~
Figure 5.2 shows a logarithmic plot of
+ I/a,)]- erf[u"-{a,_ - I/a,)]}
and ^Po^l^XntQ versus
(5-24)
(^solid and dashed
curves) for a range of or, values and the Theis (1935) solution and its derivative (open
circles and triangles) are included for purposes of comparison. The line injection/pointobservation case approaches the solution of Theis (1935) when the observation point is
close to the injection line. This can be seen from the plot of d/^In /j, for a, = 0.01.
At late times, the derivative approaches a constant value of unity when flow reaches
pseudo-steady state in which the pressure gradient becomes constant with time. As the
distance between the injection line and the observation point increases, we observe that
the derivative decreases with time indicating that flow becomes three-dimensional at late
time. It is shown in Hsieh and Neuman (1985a - see figure 9 in their paper) that as
ex, -^00, the quantityck,h' /2 approaches the point-injection/point-observation solution.
o
Figure 5.3 is a plot of normalized pressure head for the line-injection/point-observation
case and its derivatives plotted against
. We have also plotted the point-injection/point
observation solution and its derivative (solid circle and triangles), the Theis (1935) model
and its derivatives (open circles and triangles) to show the variability of the lineinjection/point observation solution with a,. Examination of the derivative curves shows
140
that this decrease becomes constant
becomes constant) at late times
owing to the three-dimensional nature of flow.
a,-0.0 J
& W(u)
l.OE+00
a,=2.0
a,=5.0
3
Q.
a, =2.0
Figure 5.2: Logarithmic plot of
(solid curves) and d(dashed curves)
versus
for a range of a, values. The Theis (1935) solution (open circles) and its
derivatives (open triangles) are included for purposes of comparison.
141
a,=0.2
l.OE-01
a,=0.2
o'
l.OE-02
•''A
a,=0.01. W{u)
a,~O.OK dW(uydlnt|
l.OE^OI
Figure 5.3; Plot of normalized pressure head for the line-injection/point-observation case
(solid curves) and its derivatives (dashed curves) plotted against
The Theis (1935)
solution (open circles) and its derivatives (open triangles) are included for purposes of
comparison.
142
5.4: POINT-INJECnON/LINE-OBSERVATION
The solution for the point injection/line observation case can be obtained by
averaging the solution for the point injection/point observation case along the observation
line. Consider an observation line of length B, such as B'CA' (see Figure 7 in Hsieh and
Neuman, 1985a), and let x be the radius vertor of point C (midpoint of the observation
line). Let CA' be represented by the vector
b" =(A„A,>3)
so that CB' is - b. In addition, let
(5-25)
be a unit vector in the direction of b so that
b = (5/2)e,
(5-26)
The average pressure increase, Ap, over B'CA' can be obtained by integrating the
dimensional form of the point-injection/point-observation solution along this line and
dividing the result by B. However, due to our assumption of uniform flux along the line
source, there is a symmetric relationship between the point-injection/line-observation
case and the line-injection/point-observation case. In other words, the solution for the
former case can be obtained directly from (5-13) by merely redefining some of the
variables;
pd, = ^ f - e x p f - ( i - A ' j j ' ] •
ii=i/4fo y
+ y P x ) ]-
)^dy
(5-27)
where
143
(5-28)
(5-29)
Pz=Gj{GJG^f-
(5-30)
= x^'Ab = (/25/2)(e'"Ae,)
(5-31)
and
G,, = b'" Ab = (5V4)e/Ae,
(5-32)
The only difference between (5-13) and (5-27) is the replacement of I by b. Note,
however, that the dimensionless pressure increases,
and
, are defined in different
manners, which accounts for the difference in the appearances of the respective
dimensionless solutions. The meanings of
and
are similar to those of a, and or,
A=(2^S)[e''Ae/e/Ae,]'"
(5-33)
A=(e''Ae.)[e''Ae/e/Ae.]'"
(5-34)
In other words, of >9, is related to the ratio between the magnitudes of x and b, while
is related to the angle between the two vectors.
For an isotropic medium the
dimensionless expressions (5-5) and (5-6) take the simple form
= k^l[<(>fjR^^.
= A7q}kR[Q jj. and
The solution for the isotropic case is seen to possess spherical
symmetry about the point source. The time derivative of (5-27) is
^^ = ^exp[-(l-A')"] M"'"(A
-l/A)]} (5-35)
144
Figure 5.4 is a logarithmic plot of
and dversus
for various values of
pi and p2 was icept at 0.01 (solid and dashed lines). The point injection/point-monitoring
solution and its derivative has also been plotted (open circles and triangles).
P5=0.01
p,=0.01.0.1.0.Z OJ. 0.4. 0.5. 1.0. 5.0
Pi-S.O and point source
^ —''"'^Pi=5.0 and derivative
source solution
P. =0.0
1.0E-Q2
P.=0.0
1.0E-Q2
l.OE+00
to
Figure 5.4: Logarithmic plot of p,^ (solid curves) and
(dashed curves)
versus
where the point injection/point-monitoring solution (open circles) and its
derivative (open triangles) have also been plotted.
145
5.5: LINE-INJECnON/LINE-OBSERVATION
The solution for the line injection/line observation case is obtained by averaging
the solution for the line-injection/point-observation case along the observation line. The
averaging procedure is given in Hsieh and Neuman (1985a).
The final result in
dimensionless form is
pd , = \ f -exp[-(l-a,/);/1- |exp{-[(l-c-)/lY^,'+2(A-«2'^)A/>0;M
^ ii=i/4/o y
i=-i
(5-36)
where
^
(i'-it)
ai, a2. Pi and P2 are defined in (5-14), (5-15), (5-29), and (5-30), respectively, and
c = GJ(G,IGJ'-
(5-38)
where
= I'Ab = (iB/4)(e,''Ae.)
(5-39)
Note that c has a similar meaning as a? and P2, it is related to the angle between 1 and
b. The time derivative for the line-injection/line-monitoring case is given by
146
^ ^ = ^exp[-(l-a/)tt]- Jexp|-[(l-c-)ylY)ff,' +2(^2 -a2c)A/^,]wJ
•
+ V«i + '^/A)](5-40)
For the general case, the inner integral in (5-36) and (5-40) cannot be evaluated
analytically. However, when b is parallel to I (the line source and observation line are
in parallel boreholes), simplification is possible. For this case, a, =
and c = 1. The
exponent in the inner integral of (5-36) and (5-40) vanishes and the remaining error
function can now be integrated analytically to yield (see Gradshteyn and Ryzhik, 1965,
p.633, eq. 5.41)
^ ii=i/4fp y
[exp(- yC^')- exp(- >C,') -exp(- yCi) + exp(- y C { n y ^ ' ^ d y
(5-41)
Similarly the pressure derivative simplifies as
- C^erf[u''-C^) + C,erf[u"^'C,)+
Jexp(- mC,^ )- exp(- «C^) -exp^- uCj*) + exp(- uC,^
|du
(5-42)
where
147
C, = a, + 1/a, + \/fi^
C=a,+l/a,-I/^
C3=a,-l/a,+iM
^
^
Q =«2-V«l -1/A
Due to the large number of parameters in (5-41) and (5-42), numerous solution plots may
be constructed. Here we consider only the case where 1 = b, so that a, = >55
Figure 5.5
shows a logarithmic plot (5-41) and (5-42) (solid and dashed lines) Oj = ^ =0 and
various values of or, =^.
Again, the Theis (1935) solution has been included for
purposes of comparison.
a, =P ,= 0.01, 02, 0.5, 1.0. Za 5.0
Oti= P2=0.01
1.0E-Q2
Figure 5.5: Logarithmic plot for
l.OE+00
l.OE+02
for various values of
l.OE+04
cc^=P\-
148
5.6 MODIFIED FORM OF HSIEH AND NEUMAN (1985A) SOLUTION TO
ACCOUNT FOR OBSERVATION STORAGE AND SKIN EFFECTS
The solutions for various injection and monitoring interval configurations derived
by Hsieh and Neuman (198Sa) do not account for observation well storage and skin
effects. We have seen from the analysis of single-hole data that borehole storage effects
in the injection interval can be important and in some cases can cause significant delays
for pressure transients to enter the formation. Here, we expect similar effects in the
observation intervals arising from compounding effects of large interval volume and gas
compressibility. Tongpenyai and Raghavan (1981) present a solution for radial flow,
which considers storage and skin effects in both the pumping and the observation wells.
We follow a much simpler approach due to Black and Kipp (1977) that is based on a
concept introduced by Hvorslev (1951). According to their approach (see also Neuman
and Gardner, 1989), the pressure p in the rock is related to observed pressure p^^
(5-44)
dt
where tg = C^^fFk, is the characteristic response time of the instrument, known as
basic time lag;
of volume F;
- Vjp^ is storage coefficient
associated with an interval
is permeability (which we attribute to skin); and F is a geometric shape
factor [Z,]. The basic time lag can be determined by means of a pressurized slug test so
that there is no need to know either
or F. Once tg has been determined, one can
correct p^ for the effects of storage and skin by means of (5-44).
The general solution of (5-44), in dimensionless form, is (Appendix I)
149
where 0 = 4^^/^/^^///?* is a dimensionless well response time, cr is a constant related to
the surface area of the observation well intake (for example, o" = l/2;zr^6 for radial flow
in which b is the length of the injection interval), and Sa is the gas storage factor [M''LT^'\
defined in Chapter 2 (eq. 2.1-23). Substituting (5-27) into (5-45) yields for dimensionless
pressure, and its derivative,
^ »=l/4fo ^
• {l - exp[- I/a(l/w - l/^)]}i/^
(5-46)
D TF=|/4FP ^
• {exp[- I/fi(l/« (5-47)
Figures 5.6 through 5.8 depict corresponding type-curves for various values of Q,
>^2 =0.01, and Pi = 5.0, 0.1 and 0.01, respectively where
is represented by solid
curves and^Pj^f^Xnti^ by dashed curves. There clearly is a delay in response as Q
increases. The original solution (open circles, eq. 5-27) of Hsieh and Neuman (1985a)
and its derivative (open triangles, eq. 5-35) are included for comparison.
150
In the case of Figure 5.6 where Pt = 5.0, radial distance between the centroids of
the injection and observation intervals is large compared to the length of the observation
interval.
Here the pressure derivative decays towards zero as is typical of three-
dimensional flow, which develops around the point injection interval. As Pi diminishes,
so does the response, which is additionally delayed in time. In Figure 5.8 where y?/=O.OI,
so that the monitoring interval is long relative to its distance from the injection interval,
the pressure derivative corresponding to n= 1.0 and 10.0 is constant during intermediate
time as is typical of radial flow (see Figure 3.3). We use these type-curves to interpret
cross-hole test data from the ALRS in Chapter 6.
The solution for the line-injection/point-observation with observation wellbore
storage can be likewise obtained by substituting eqn. (5-13) into (5-45).
P d 2 = \ J 7 ® * P [ ~ +1 / a , ) ] - - I / a , ) ] }
^ u=\/4to S
- {1 - exp[- l/Q(l/tt -1/^)]}«/^
(5-48)
D
U=L/4F0 ^
• {exp[- l/n (l/M (5-49)
151
Figure 5.9 shows a family of type-curves for the line-injection/point-observation
case with at = 0.01, aj = 0.01. The red solid line represents the dimensionless pressure
in the solution of Hsieh and Neuman (1985a) while the red dashed line represents their
derivatives. We notice that the type-curve with Q = 0.1 and the Hsieh and Neuman
(1985a) solution coincide at late time but is slightly delayed at early time. This delay in
pressure increases with the magnitude of Q as it did with the point-injection/lineobservation solution. The Theis (1935) solution and its derivatives (as open circles and
triangles) are also included for purposes of comparison in this figure. We notice from
this plot that the Hsieh andNettman
solution matches exactly with the solution of
Theis (1935) from the early to late time (except at to greater than 2000). The derivative
of the Theis (1935) solution remains constant at late time as the pressure response reaches
a pseudosteady state. The derivatives of the Hsieh and Neitman (1935) and our modified
solution (5-51) begins to decrease as the pressure response "seen" in the point
observation interval in response to line injection evolves from a radial to a spherical flow
regime. This is true for all values of Q in our modified solution (5-50). The Black and
Kipp (1977) solution (integral independently evaluated numerically) is included in Figure
5.10 to show the correspondence with the Hsieh and Neuman (1985a) at ai = 0.01, a2 =
0.01.
152
l.OEH)!
0=1000
0=10
I.OEtOO
o=i
Q
2 l.OBOI
£3
O.
1.0E02
l.OBOI
l.OE«)l
I.OEKC
l.OEKM
'o
Figure 5.6: Type-curves of dimensionless pressure in observation interval (solid curves)
and its derivative (dashed curves) versus dimensionless time for various Q and Pi = 5.0,
>?2= 0.01. Open circles represent dimensionless pressure in the solution of Hsieh and
Neuman (1985a) and open triangles their derivatives.
153
Q=IO
Q=1
l.OE+OO
l.OEtOI
1.0EK)2
I.OEKW
to
Figure 5.7: Type-curves of dimensionless pressure in observation interval (solid curves)
and its derivative (dashed curves) versus dimensionless time for various O and ^/ = 0.1,
^2= 0.01. Open circles represent dimensionless pressure in the solution of Hsieh and
Neuman (1985a) and open triangles their derivatives.
154
l.CEKX) :
0=100
1.0EK)l
1.0E-+03
to
Figure 5.8; Type-curves of dimensionless pressure in observation interval (solid curves)
and its derivative (dashed curves) versus dimensionless time for various £2 and >3/ = 0.01,
>?2= 0.01. Open circles represent dimensionless pressure in the solution of Hsieh and
Neuman (1985a) and open triangles their derivatives.
155
UEKU
« L(E01
L(EC2
L(EOI
L(EOI
L(E(B
L(EHB
Figure 5.9: Type-cucves of dimmaonless ptessure in obsovstiim intovai (solid cuives)
and its doivadve (dashed carves) versus dimensionless time for various and ai = 0.01,
0.01. The red solid line represents dimensionless pressure in die solution of Hsieh
oite/JVisuman (1985a) and die red dadied line dieir derivatives. The opw circles represoit
die Theis (1935) sohtion while the open triangles their daivatives.
156
i a I = 0.01
a2=qsi\
•
£2=10
Q = 0.01;0.1
••
l.GB-04
1.0E-Q3
l.OE-Ol
l.OE-HW
to
F^nue 5.10: Type-caives of dimoiafHiless piessme in obsovation interval (solid curves)
and its dqivative (dashed corves) vctsus dimqisionless time for various
and or/= 0.01,
02=0.01. The bine solid line r^resents dimaisionless pressure in the sohition of Theis
(1935) and die blue dashed line their derivatives. The open circles represent die
and Kipp (1977) sohition^lile die open tiiai^es dieir derivatives.
157
6. TYPE-CURVE INTERPRETATION OF CROSS-HOLE
PNEUMATIC INJECTION TEST DATA
6.1 CROSS-HOLE TEST METHODOLOGY
Single-hole air injection tests provide information only about a small volume of
rock in the close vicinity of the injection interval. Our data indicate (see sections 1.2.4
and 4.2) that rock properties, measured on such small scales, vary rapidly and erratically
in space so as to render the rock strongly and randomly heterogeneous. To determine the
properties of the rock on a larger scale, we conducted cross-hole interference tests by
injecting air into an isolated interval within one borehole, while monitoring pressure
responses in isolated intervals within this and other boreholes. Of the 16 boreholes we
used for cross-hole testing, 6 were previously subjected to single-hole testing.
The
results of the single-hole tests (primarily spatial distribution of air permeabilities and
local flow geometry) together with other site information (primarily borehole televiewer
images-see Appendix J) served as a guide in our design of the cross-hole tests.
As there is little prior experience with pneumatic cross-hole tests in unsaturated
fractured rocks, we conducted our tests at the ALRS in three phases. Phase 1 included
line-injection/line-monitoring (LL) tests in which injection and monitoring took place
along the entire length of a borehole that had been isolated from the atmosphere by
means of shallow packers. Phase 2 consisted of point-injection/line-monitoring (PL) tests
in which air was injected into a 2-m section in one borehole while pressure was recorded
along the entire length of each monitoring borehole.
During Phase 3, we conducted
158
point-injection/point-monitoring (PP) tests in which both the injection and the monitoring
intervals were short enough to be regarded, for purposes of type-curve analysis, as points.
A total of 44 cross-hole pneumatic interference tests of various types (constant injection
rate, multiple step injection rates, instantaneous injection) have been conducted during
the years 1995 - 1997 using various configurations of injection and monitoring intervals
(LL, PL and PP). The type of cross-hole test; injection borehole, interval and rate;
monitoring borehole and intervals; as well as brief comments on each test are listed in
Table 6.1. A test was considered to be successful when all equipment functioned reliably
throughout the entire period; injection rate was adequately controlled; and all data were
recorded properly.
6.1.1 Instrumentation Used in Cross-Hole Tests
Cross-hole tests were conducted using modular straddle packer systems that were easily
adapted to various test configurations and allowed rapid replacement of failed
components, modification of the number of packers, and adjustment of distances between
them in both the injection and monitoring boreholes. For their construction we relied on
relatively inexpensive PVC pipes, which can be worked on in the field using hand tools
and are sufficiently flexible to slide with relative ease up and down an uneven borehole.
Figure 6.1 is a schematic diagram of the main cross-hole injection string of packers. The
air-filled volume of the injection interval was made smaller than it had been during
single-hole testing so as to minimize borehole storage effects. The main injection string
installed in borehole Y2 consisted of three packers, one near the soil surface to isolate the
borehole from the atmosphere, and two to enclose the injection interval.
Pressure
159
transducers were used to monitor absolute pressure and temperature in each of the three
borehole sections that had been isolated from each other in this manner. In general, the
most sensitive transducers (GEOKON-4500H-00I0) were placed close to the injection
interval where we expected to see relatively pronounced pressure responses. Sensitive
transducers were also placed furthest away fi-om the injection interval, and in monitoring
intervals YIM, Z2M, Z2L, Z3M and Z3B, where we expected to see pronounced
barometric pressure effects.
Two types of borehole monitoring systems were employed, one with a single
packer near the soil surface to monitor pressure along the entire length of a borehole (6
units - see Plate 6.1) and another, modular system with three or four packers to monitor
pressure in several isolated segments of a borehole (9 units-see Plate 6.2). Monitoring
intervals with a single packer near the soil surface are identified by borehole designation;
for example VI, XI and Wl. Where a modular system separates a borehole into three
isolated intervals, we append to the borehole designation a suffix U, M or B to identify
the upper, middle or bottom interval, respectively; for example V3U, V3M and V3B.
Where a modular system separates a borehole into four isolated intervals, we append to
the borehole designation a suffix U, M, L or B to identify the upper, middle, lower or
bottom interval, respectively; for example Z2U, Z2M, Z2L, and Z2B. Of the 9 modular
monitoring systems, 7 were equipped with two tracer-sampling ports each, which we had
use for air injection.
160
Table 6.1: Cross-hole tests completed at ALlRS
Q Uipm\
Y2
Y2
Int. Int. [ml
100-30 0
10.0-30.0
I5.0-I70
I5.0-I70
I5.0-I70
21.0-230
IS.O-200
IS.O-200
IS.0-200
IS.O-200
26.0-28.0
23.0-25.0
21 0-23.0
CR
Barometric
CR
CR
CR
Slug
SlUR
CR
CR
SiuK
Bamnetnc
Barometric
CR
CR
Y2
Y2
Y2
Y2
Y2
Y2
Y2
Y2
Y2
Y2
Y2
Y2
Y2
Y2
21.0-23.0
21.0-23.0
21.0-23.0
21.0-23.0
21.0-23.0
21.0-23.0
21 0-23.0
21 0-23 0
21.0-23 0
21 0-23 0
21 0-23 0
21 0-23 0
21 0-23 0
21 0-23 0
10
00
10
10
15
10
2.0
10
10
00
0.0
00
PL26
PL27
PL28
PPI
PP2
Step
SH-neulron
SH-aeutron
CR
CR
Y2
W3
W3
Y2
Y2
21 0-23 0
14.7-169
16 9-19 I
15 0-170
15 0-170
05.10.15.10
PP3
PP4
PPS
PP6
PP7
PPS
PP9
PPIO
CR
CR
Step
Step
Step
CR
Step
CR
Y2
Y2
X2
Z3
W3
Y2
Y2
Y2
15 0-170
150-170
18.5-207
15 9-179
19.2-204
15 0-170
15 0-170
15 0-170
150
500
5.0 and 10 0
5 0 and 10 0
5 0 and 10.0
500
75.0 and 50.0
250
PPI I
Slug
Y2
15 0-170
1000
PPI 2
PPI3
PPM
CR
CR
SH-neutron; Step
Y2
Y2
W3
15.0-17.0
15.0-17.0
25.7-269
250
05
0.5, 10. 13. 19
Test
LL2
PLI
PL2
PL3
PL4
PL5
PL6
PL7
PL8
PL9
PLIO
PLII
Flow
CR
CR
CR
CR
CR
CR
CR
CR
CR
CR
CR
CR
CR
Ini. Hole
Y2
Y2
Y2
PLI2
PLI3
PLI4
PLIS
PL 16
PL17
PLIS
PLI9
PL20
PL2I
PL22
PL23
PL24
PL25
LLI
1
]
1
rz
Y2
Y2
Y2
rz
T2
Y2
n
50-0
100.0
8.S.I00
lao
2ao
1.0
1.0
1.0
l.o
10
10
1.0
10
?
10
OS
OS
IS
18.0
Comments
Pressure increase appears to be
proportional to Q
Could not maintain constant O
RupciinnR of packer lines
First successful test
Responses in W2A and V2
Generator problems
Generator problems
Generator problems
Response m VI
[nadvertent lack of proper data
recordmg
No air iniection
First point*to>point connection
Same as PL14 but without driente
Same with larRer O
First cross>hole slufc test
Repeat of PLI 7 with more data points
Packer tnnation SIUK test
Data lost?
Added 3-packer monitoring system m
V2
Isolated V2-W2 intersection
No change in neutron counts
No change m neutron counts
Unknown flow rale
Y3 and X2 surface guard packers kept
uninflated
All packers inflated
Ail packers inflated
All packers inflated
All packers inflated
All packers inflated
Flow rate changes towards end of test
Flow rale changes towards end of test
All packers deflated except Y2 mjection
stnng
Monitored Y2M. YIM. W2AM. V3M
and X2M at fastest possible rate
All packers inflated
Slight decrease ui neutron counts
CR = constant rate
SH-neutron = single-hole test with neutron probe installed in the injection interval
Slug = slug injection of air
Barometric = barometric test in which all packers were inflated, pressure in the intervals
were monitored while there was no air injection at the injection interval
Plate 6.1: Photogtiqrii of a single packer monitorii^ system.
162
Plate 6.2: Pfaotogcqrfi of tfie modular packer ss^em.
163
Upper Guard Packer
Upper Monitoring Interval
(with pressure transducer)
A
Middle Guard Packer^
Injection / Monitoring Interval
(with pressure transducer and
CPN neutron probe for W3)
Lower Guard Packer
t
Lower Monitoring Interval
Down-hole direction
(with pressure transducer)
Note; [>awtng Is not to scale
Figure 6.1: Injection and monitoring systems used for cross-hole tests.
164
Table 6.2; Information on injection and monitoring intervals with pressure
transducer types during phase 3 cross-hole tests
Pressure transducer installed in injection and monitoring intervals
Interval
Interval
type
FT manufactioer
Model
PT
Range
PT Resolution
VI
V2U
V2M
V2B
V3U
V3M
V3B
W1
W2
W2AU
W2AM
W2AL
W2AB
W3U
W3M
W3B
XI
X2U
X2M
X2B
X3
YIU
YIM
YIB
Y2U
Y2M
Y2B
Y3U
Y3M
Y3B
Z1
Z2U
Z2M
Z2L
Z2B
Line
Line
Point
Line
Line
Point
Line
Line
Line
Line
Point
Point
Line
Line
Point
Line
Line
Line
Point
Line
Line
Point
Point
Line
Line
Point
Line
Line
Point
Line
Line
Line
Point
Point
Line
Line
Point
Line
Geokon
Honeywell
Geokon
Honeywell
Druck
Geokon
Druck
Geokon
Honeywell
Honeywell
Geokon
Geokon
Honeywell
Honeywell
Geokon
Honeywell
Geokon
Geokon
Geokon
Geokon
Geokon
Druck
Geokon
Geokon
Honeywell
Geokon
Honeywell
Geokon
Geokon
Geokon
Geokon
Geokon
Geokon
Geokon
Geokon
Geokon
Geokon
Geokon
4500AL-25
Microswitch
4500AL-25
Microswitch
PDCR
4500AL-25
PDCR
4500AL-25
Microswitch
Microswitch
4500AL-25
4500AL-25
Microswitch
Microswitch
4500AL-25
Microswitch
4500AL-25
4500AL-25
4500AL-25
4500AL-25
4500AL-25
PDCR
4500H-0010
4500AL-25
Microswitch
4500AL-25
Microswitch
4500AL-25
4500H-(K)10
4500H-0010
4500AL-25
4500AL-25
4500H-0010
4500H-OOIO
4500AL-25
4500AL-25
4500H-0010
4500H-0010
0-25
0-15
0-25
0-15
0-15
0-25
0-15
0-25
0-15
0-15
0-25
0-25
0-15
0-15
0-25
0-15
0-25
0-25
0-25
0-25
0-25
0-15
O-IO
0-25
0-15
0-25
0-15
0-25
0-10
0-10
0-25
0-25
0-10
O-IO
0-25
0-25
0-10
0-10
HijOi
Medium
HiRh
Medium
Medium
High
Medium
HiRh
Medium
Medium
High
High
Medium
Medium
High
Medium
High
High
High
High
High
Medium
Very High
High
Medium
High
Medium
High
Very High
Very High
High
High
Very High
Very High
High
High
Very High
Very High
Z3U
Z3M
Z3B
Pressure transducers installed within injection system in field laboratory
Injection
system
Packer line
Barometer
Geokon
4500AL-25
0-25
High
Honeywell
Geokon
MS
4580-1-2.5
0-60
0-2.5
Medium
Very High
Tracer sampling
ports (injection
port)
No
No
Yes
No
No
No
No
No
No
No
No
No
No
No
No
No
No
No
Yes
No
No
No
Yes
No
No
Yes
No
No
Yes
No
No
No
No
Yes
No
No
Yes
No
165
In total, thirty-eight packers were used in each cross-hole test. The packers were
approximately \-m long in all PP tests performed during phase 3 of the project except in
monitoring system V3 where shorter packers, 50-cm in length, had been employed. An
air manifold was constructed to distribute air pressure evenly and efficiently through the
system (see Plate 6.3 - pressure gauge was added later). A pressure gauge was placed in
the manifold to allow visual check of packer inflation. The system allowed inflating
packers individually and, in this way, monitoring and recording their pressure as well as
varying the lengths and configurations of monitoring intervals without necessarily
moving the down-hole equipment. Packer inflation pressure was maintained at 60 Psi
throughout each cross-hole test. Temperature readings were taken by downhole and
surface pressure transducers that are temperature compensated as noted in Table 6.2.
The equipment included twenty-seven Geokon 4500 pressure transducers which
compensate pressure for fluctuations in temperature, one Geokon 4580 barometric
transducer which provides temperature compensation, three Druck PDCR™ pressure
transducers without such compensation, and ten Honeywell-MICROSWITCH™ pressure
transducers without temperature compensation. The type of pressure transducers, their
locations, range and relative resolutions are listed in Table 6.2.
To monitor relative changes in volumetric water content around the injection
interval, we installed a BOART LONGYEAR CPN* model 503 neutron probe in the
injection interval toward the end of our project. As the probe was sensitive to PVC, the
latter was replaced by galvanized steel (see Plate 6.4).
166
The air injection system installed in the field laboratory is illustrated
schematically in Figure 6.2 (see Plate 6.5 also). It included a mass flow meter, several
pressure valves and regulators, an oil and a water filter, a 1-ijm particle filter, a pressure
transducer, a relative humidity sensor, drierite (anhydrous calcium sulfate) and electronic
equipment to automatically collect field data. Pressure and temperature in the injection
air stream were measured using a Geokon 4500 pressure transducer compensated for
temperature.
Relative humidity (RH) in the air stream was measured using a
VAISALA™ SOY probe. Various Sierra Instruments Sidetrack™ 830/840 mass flow
controllers and meters operated over ranges of 0-0.1 SLPM (standard liters per minute),
0-20 SLPM, and 0-100 SLPM, supplying a constant injection (volumetric) flow rate that
could be adjusted to between O.I to 100 SLPM. For simplicity of test interpretation, we
tried to maintain a constant injection (volumetric) flow rate during each test. As the mass
flow controller was sensitive to variations in moisture content in the air stream, we used
dry air for injection to help maintain the rate constant. This, we believe, had little if any
effect on pressure variations in the monitoring intervals. Not shown in Figure 5.2 are a 02 Psi barometric transducer, two \0-kW generators, four compressors, and solar panels
that made up the rest of the system.
Data were recorded at 1- to 10-5ec intervals throughout the duration of each crosshole test using three Campbell Scientific™ CRIO dataloggers connected to a Campbell
Scientific™ SC32a optically isolated interface, which allowed periodic downloading of
data onto an on-site personal computer with a removable, large capacity disk drive.
Three Campbell Scientific
AM416 multiplexers were concatenated to allow
167
simuftaiieous recofdmg of downhole pressure and temperature data, packer pressure,
battery voltage, mass flow rate, air temperature, air relative humidity and barometric
pressure throu^mut the duration of each test.
The electronic system was tested
thorou^ily for defects. The computer program, which controls each datalogger was
optimized for maximum efSciency.
Plate 63: Photogrq)h of tiie air manifold installed at the site (does not show pressure
gauge).
Plate 6.4: Photogn^of the mjectioii interval with neutron probe installed.
169
1. pressure gauge
3. pressure gauge
1 r
4. oil & water filter
2. pressure regulator
5. Drierite:
CaS04 _ 11H2O
from air source
10. valve
7. mass flowmeter
8. pressure transducer
flow to the injection
interval
n
Va
R
o
6. 7 (un particle filter
9. relative humidity sensor
Figure 6.2; Air injection system installed in the field laboratory.
170
Musflow
contfdkr
Plate 6.5: Photogr^di of the air injection system and electronic equ^ment installed in the
&ld hbocatory.
171
6.2 CROSS-HOLE TESTING PROCEDURE WITH EMPHASIS ON TEST PP4
A typical cross-hole test consisted of packer inflation, a period of pressure
recovery (due to packer inflation), air injection and another period of pressure recovery.
Our system allowed rapid release of packer inflation pressure when the corresponding
recovery was slow, but this feature was never activated even though recovery had
sometimes taken several hours. Once packer inflation pressure had dissipated in all
(monitoring and injection) intervals, air injection at a constant mass flow rate began. It
generally continued for several days, until pressure in most monitoring intervals appeared
to have stabilized.
In some tests, injection pressure was allowed to dissipate until
ambient conditions have been recovered.
In other tests, air injection continued at
incremental flow rates, each lasting until the corresponding pressure had stabilized,
before the system was allowed to recover.
In this dissertation we focus on the analysis of test PP4 conducted during the third
phase of our program. This test was selected for type-curve analysis because it involved
1) injection into a high-permeability zone in borehole Y2 (see Figures 1.8, 1.9, and 1.16),
which helped pressure to propagate rapidly across much of the site; 2) injection at a
relatively high flow rate which led to unambiguous pressure responses in a relatively
large number of monitoring intervals; 3) the largest number of pressure and temperature
monitoring intervals among all tests; 4) a complete record of relative humidity, battery
voltage, atmospheric pressure, packer pressure, and injection pressure; 5) the least
number of equipment failures among all tests; 6) flow conditions (such as injection rate,
fluctuations in barometric pressure, battery voltage, and relative humidity) that were
172
better controlled, and more stable, than in all other tests; 7) minimum boundary effects
due to injection into the central part of the tested rock mass; 8) a relatively long injection
period; 9) rapid recovery; and 10) a test configuration that allowed direct comparison of
test results with those obtained from two line-injection/line-monitoring tests (LLI and
LL2), and a point-injection/line-monitoring test (PL3), at the same location (see Table 6.1
for a list of cross-hole tests). Stable flow rate and barometric pressure made type-curve
analysis of test PP4 results relatively straightforward.
Test PP4 was conducted by injecting air at a rate of 50 SLPM into a 1-m interval
located 15 - Mm below the LL marker (Table 4.1) in borehole Y2 as indicated by a large
solid circle in Figure 6.3. The figure also shows a system of Cartesian coordinates x, y, z
with origin at the center of the injection interval which we use to identify the placement
of monitoring intervals relative to this center. Responses were monitored in 13 relatively
short intervals (0.5 - 2 m) whose centers are indicated in the figure by small white circles,
and 24 relatively long intervals (4 - 42.6 iri) whose centers are indicated by small solid
circles, located in 16 boreholes. Several of the short monitoring intervals (V2M, V3M,
W2AM, W2AL, W3M, X2M, Z2M, Z2L and Z3M) were designed to intersect a high
permeability region (Figures 1.8, 1.9, and 1.16) that extends across much of the site at a
depth comparable to that of the injection interval. Table 6.3 lists coordinates of the
centers of all monitoring intervals, their lengths 5, radial distances R from the center of
the injection interval, geometric parameters /?/ and P2 defined in (5-33) and (5-34), and
maximum pressure during the test.
173
Data recorded at the surface during the test included barometric pressure (Figure
6.4), mass flow rate (Figure 6.5), packer pressure (Figure 6.6), battery voltage (Figure
6.7) and relative humidity in the injection air stream (Figure 6.8).
Fluctuations in
barometric pressure are seen to have been quite regular with an amplitude of about 0.25
kPa during the first 250,000 sec of the test. It later dropped by about I kPa and stayed
nearly constant until the end of the test. Mass flow rate remained constant except for a
slight drop of I to 2 SLPM zi about 175,000 sec. Packer pressure remained constant at 60
Psi throughout the first part of the test but dropped approximately 275,000 sec into the
test, apparently in response to a concurrent drop in barometric pressure. Battery voltage,
supplied by solar panels, increased during the day and decreased at night to form a square
wave. Relative humidity in the injection stream varied diumally.
Arithmetic plots of pressure responses in all monitoring intervals as well as in the
injection interval are presented in Figures 6.9 - 6.24. Pressure responses in the 2.0-m
injection interval Y2M, 7.1-m monitoring interval Y2U above it, and \2.9-m monitoring
interval Y2B below it are depicted in Figure 6.9. The figure also shows back-pressure at
the surface, recorded behind the mass flow controller (Figure 6.2), which is seen to be
about twice as high as pressure in the injection interval. Fluctuations in back pressure
could be due to variations in relative humidity and air temperature within the injection
line at the surface. This is one primary reason why we placed our pressure transducers
within the injection and monitoring intervals and not at the soil surface. As injection
mass flow rate is constant, these fluctuations have no adverse effect on pressure in the
injection interval.
174
Figure 6.3; Locations of centers of injection and monitoring intervals. Large solid circle
represents injection interval, small solid circles represent short monitoring intervals, and
open circles represent long monitoring intervals.
Pressure in the injection interval (Figure 6.9) is seen to reach a stable value almost
immediately after the start of injection and then to decline slowly with time. A similar
pressure behavior was observed in many single-hole tests by Guzman et al. (1994, 1996)
and attributed by them to a two-phase flow effect, as discussed in section 4.2. No such
effect is seen in any of the monitoring intervals, not even in those situated inrunediately
above and below the injection interval in borehole Y2. On the other hand, the Y2M
175
injection interval shows an inflection (175,000-200,000 sec) that we consider
characteristic of dual continuum behavior.
Barometric pressure fluctuations are included when they are deemed to have had a
potential impact on downhole pressure. There was no measurable pressure response in
monitoring interval YIB, and pressure transducers in monitoring intervals V2U, V2B,
W3B and YIU appeared to have malfunctioned during cross-hole test PP4. In general,
pressure responses tended to be largest in monitoring intervals with lengths ranging fi-om
0.5-m to 2-m and to diminish with distance fi-om the injection interval.
Pressure in monitoring interval VI (Figure 6.10) shows an inflection at about
150,000 sec which may be an indication of dual continuum behavior; a similar inflection
occurs in monitoring intervals V2M (Figure 6.11), V3M (Figure 6.12), W1 (Figure 6.13),
W2AM (Figure 6.15), W3M (Figure 6.16), XI (Figure 6.17), X2U (Figure 6.18), X2M
(Figure 6.18), X2B (Figure 6.18), Y2M (Figure 6.9), Y3B (Figure 6.21), and Z3M
(Figure 6.24).
At late time, the pressure in VI declines in apparent response to a
concurrent decline in barometric pressure; this too is seen in several intervals including
V2M (Figure 6.11), W1 (Figure 6.13), W2AM (Figure 6.15), W2AL (Figure 6.15), W3M
(Figure 6.16), XI (Figure 6.17), X2U (Figure 6.18), X2M (Figure 6.18), X2B (Figure
6.18), X3 (Figure 6.19), YIM (Figure 6.20), YIB (Figure 6.20), Y3U (Figure 6.21), Y3M
(Figure 6.21), Y3B (Figure 6.21), Z1 (Figure 6.22), Z2U (Figure 6.23), Z2M (Figure
6.23), Z2L (Figure 6.23), Z2B (Figure 6.23), Z3U (Figure 6.24), Z3M (Figure 6.24) and
Z3B (Figure 6.24). Otherwise, barometric pressure fluctuations seem to have only a
small effect on pressure in VI as well as in V2M (Figure 6.11), V3U (Figure 6.12), V3M
176
(Figure 6.12), V3B (Figure 6.12), W1 (Figure 6.13), W2AM (Figure 6.15), W2AL
(Figure 6.15), X2U (Figure 6.18), X2M (Figure 6.18), X2B (Figure 6.18), Y2U (Figure
6.9), Y2M (Figure 6.9), Y2B (Figure 6.9), Z3U (Figure 6.24) and Z3M (Figure 6.24). On
the other hand, such fluctuations seem to have a measurable impact on pressure in W3M
(Figure 6.15), YIM (Figure 6.19), Y3B (Figure 6.21), Z3U (Figure 6.24) and Z3M
(Figure 6.24), and to dominate pressure variations in X3 (Figure 6.19), Y3U (Figure
6.21), Y3M (Figure 6.21), Z1 (Figure 6.22), Z2U (Figure 6.23), Z2M (Figure 6.23), Z2L
(Figure 6.23), Z2B (Figure 6.23) and Z3B (Figure 6.24). These intervals must have
excellent pneumatic communication with the atmosphere through high-permeability
fractures.
We do not have a complete record of pressure recovery for cross-hole test PP4.
We did observe, however, that the rate of recovery was much faster in intervals V2M,
V3M, Wl, W2AM, W2AL, W3M, XI, Y2M, Y3M, Y3B and Z3M than in VI, V2B,
V3U, V3B, W2AU, W2AB, W3U, YIU, Y2U, Y2B, Y3U, Z2U, Z2M and Z3U; was
nearly identical in intervals X2U, X2M and X2B; and was imperceptible in interval YIU.
177
Table 6.3: Coordinates of centers of monitoring intervals relative to origin at center of
injection interval, interval lengths, radial distances between centers of injection and
monitoring intervals, geometric parameters Pi and P2 and maximum recorded pressure
Interval
VI
V2U
V2M
V2B
V3U
V3M
V3B
W1
W2AU
W2AM
W2AL
W2AB
W3U
W3M
W3B
XI
X2U
X2M
X2B
X3
YIU
YIM
YIB
Y2U
Y2M
Y2B
Y3U
Y3M
Y3B
Z1
Z2U
Z2M
Z2L
Z2B
Z3U
23M
Z3B
r [OT|
-7.2
-4.2
-4.2
-4.2
-1.2
-1.2
-1.2
-3.6
-3.2
-3.2
-3.2
-3.2
-3.4
-3.4
-3.4
-6.0
2.6
-1.8
-7.9
4.2
-0.7
-2.8
-7.3
4.0
0.0
-6.0
14.8
10.7
-0.5
18.9
10.5
14.8
16.9
21.1
-0.3
3.5
14.4
y[m\
1.6
1.6
1.6
1.6
1.6
1.6
1.6
0.3
4.8
1.2
-0.8
-3.7
16.5
12.9
1.8
4.8
4.8
4.8
4.8
4.8
-0.1
-0.1
-0.1
0.0
0.0
0.0
0.1
0.1
0.1
-5.2
-5.2
-5.2
-5.2
-5.2
-5.2
-5.2
-5.2
PT = Pressure Transducer
z{m\
-6.8
8.4
3.9
-7.2
6.1
-0.3
-9.5
4.9
0.4
-3.2
-5.2
-8.0
1.5
-2.1
-13.3
4.0
2.4
-1.9
-8.0
-5.9
9.4
7.3
2.8
4.0
0.0
-6.0
4.8
0.7
-10.5
3.8
2.6
-1.7
-3.8
-8.0
3.8
0.0
-10.9
B[m\
29.5
5.0
2.0
18.2
9.6
0.5
18.1
11.9
6.7
1.5
2.1
4.0
6.8
1.2
28.6
14.4
8.2
2.2
12.9
42.6
2.1
1.8
8.8
7.1
2.0
12.9
7.5
2.0
27.8
13.0
8.1
2.0
2.0
7.7
6.8
2.0
26.0
10.0
9.6
6.0
8.5
6.4
2.0
9.7
6.1
5.7
4.7
6.1
9.4
16.9
13.5
13.8
8.7
6.0
5.5
12.2
8.7
9.4
7.8
7.8
5.6
0.0
8.5
15.6
10.8
10.5
19.9
12.0
15.8
18.1
23.2
6.4
6.3
18.8
P,
07
0.7
3.9
5.9
0.9
1.3
8.2
l.l
1.0
1.7
6.1
5.9
4.7
5.0
22.0
1.0
1.2
1.5
5.0
1.9
0.4
8.9
8.5
1.8
1.6
0.7
0.9
0.7
0.8
0.9
0.2
1.0
0.6
0.6
0.3
0.7
0.9
0.8
0.6
0.6
0.2
0.6
0.5
0.9
O.l
0.7
0.4
0.4
0.0
1.3
4.2
10.6
0.8
3.1
3.0
15.8
18.0
6.0
1.9
6.3
1.4
0.0
0.9
0.8
0.7
0.5
0.5
0.7
0.8
0.9
0.4
0.4
1.0
max Ap \kPa\
8.3
Broken PT
9.8
Broken PT
6.3
49.2
23.5
9.2
2.9
13.8
7.5
6.9
4.5
4.8
Broken PT
7.0
9.2
17.9
7.3
0.2
broken PT
7.2
0.0
16.9
116.6
21.5
0.6
0.7
3.0
0.2
0.7
0.3
O.l
O.l
6.5
6.9
0.2
i
100000
u
3
I
200000
150000
230000
300000
330000
-05
time [sec]
F^nie 6.4: Barametxic piessore dnxii^cross-hole test PP4.
60
so
45
40
30
20
10
0
50000
100000
150000
200000
dme [MC]
Figure 6.5: Flow rate during cross-hole test PP4.
230000
300000
350000
400000
179
70
60
SO
I :
40 -
30
20
10 -
0
—
0
SOOOO
100000
ISOOOO
200000
250000
300000
3SOOOO
400000
230000
300000
330000
400000
tfane {ser I
Figuce 6.6: Padcer pressure dnring ctoss^iQle test PP4.
0
30000
ISOOOO
200000
tteetMc]
Figure 6.7: Battery vottage during oossp^Kiie test PP4.
\
A
ji
V
50000
100000
IJOOOO
^
!
<f \
200000
/
'
230000
%
^
300000
330000
400000
>(s*c|
Figure 6.8; Fluctuations in relative humidity dnting cross-hole test PP4.
24&0
/
19&0
14&0
98L0
Y2B
4&0
Y2U
-ZO
0
50000
100000
150000
200000
250000
300000
350000
400000
/ |Me|
Figure 6.9: Surface injection pressure and pressure in injection (Y2M) and monitoring (Y2U, Y2B)
intervals within borehole Y2 during cross-hcde test FP4.
181
10.0
8.0
6.0
VI
£ 4.0
2Xt
0.0
SOOOO
130000
30000Q
^30000
400300
330000
400000
Figure 6.10: Pressure in monitonQg interval VI during cross-hole test PP4.
V2M
V2B
OJ
in bnometnc presswe
0
SOOOO
100000
ISOOOO
200000
2S0000
300000
ffmr]
Figure 6.11:Pressure inmooitonag intervals V2U, V2M, and V2B duni^ cross^e test PP4.
182
58^
483
383
V3M
V3B
283
A
183
V3U
\
•
•
-13
0
30000
100000
150000
200000
250000
300000
350000
400000
'6wcl
Figure 6.12; Pressure in mocitorii^ intervals V3U, V3M, and V3B during cross4ude test PP4.
10.0
/
CMW
3OOOO0
Figure 6.13; Pressure in moiiitorii^interval W1 durii^ cross-hole test PP4.
^50000
4mMW
W2
OS
ao
aaqj^V
i
300000
*^QOOOO
4OC|DOO
320000
400000
iabioomriciiiiMiiii
-1.0
Figoie 6.14: Pressure in monitanng interval W2 <biring cross-hole test H>4.
16
14
12
W2AB
10
W2AL
W3AM
8
6
W2AU
4
2
0
-2
-4
0
soooo
100000
150000
200000
230000
300000
Figure 6. IS; Pressure in numitning intervals W2AU, V^AM, W2AL, and W2AB during cross-hole
testl?4.
184
9
7
I
W3M
'
3
W3U
W3B
1
-2
0
50000
100000
130000
200000
250000
300000
350000
400000
Figure 6.16; Pressure in monitoring intervals W3U, W3M, and W3B during cross^ide test PP4.
XI
£
%
-1
soooo
ISOOOO
300000
Figure 6.17: Pressure in monitoring interval XI during cros&4ide test PP4.
350000
40«D00
185
23J
l&S
13.5
-IJ
SOOOO
100000
150000
200000
250000
300000
350000
400000
Figure 6.18; Piessuie in monitonng intervals X2U, X2M, and X2B during cross4icde test PP4.
300000
•03
-1.0
-IJ
t^]
Figure 6.19; Pressure in mooitonng interval X3 dunog cros&lurie test PP4.
350000
400DOO
186
30000
100000
130000
200000
230000
300000
330000
400000
<Ehc|
Figure 6.20 Pressure in nuwiiftwing intervals YIU, YIM, and YIB during cross-hole test FP4.
30000
100000
130000
200000
230000
300000
330000
400000
Figure 6.21 Pressure in monitonng intervals Y3U, Y3M, and Y3B daring cross-hole test PP4.
03
Zl
330000
I
-IJ
Figme 6.22; Pressure in manitariiig interval Zl during cioss-hole test PP4.
1.0
Z2U
0.0
i
-1.0
Z2B
0
50000
100000
130000
200000
vc
230000
300000
330000
400000
/^I
Figure 6.23; Pressure in monitoring intervals Z2U, Z2M, 22L, and Z2B during cross-iKrie test PP4.
.2
0
30000
100000
130000
200000
230000
300000
330000
400000
tpecl
Figure 6.24;Pressure in monitoring intervals Z3U, Z3M, andZ3B durii^ cross-bcde test PP4.
189
6.3 TYPE-CURVE INTERPRETATION OF CROSS-HOLE TEST PP4
Type-curve interpretation of pressure data from cross-hole test PP4 included
intervals VI, V2M, V3U, V3M, V3B, WI, W2AU, W2AM, W2AL, W2AB, W3U,
W3M, XI, X2U, X2M, X2B, YIU, YIM, Y2U, Y2B, Y3U, Y3M, Y3B, Zl, Z2U, Z2M,
Z2L, Z2B, Z3U, Z3M and Z3B. Pressure data from intervals V2U, V2B, W2, W3B and
YIB were not amenable for type-curve interpretation and have therefore been excluded.
A special set of type-curves was developed for each pressure monitoring interval based
on the point-injection/line-monitoring solution, modified for storage and skin in the
monitoring interval, given by equations (5-46) and (5-47) in Chapter 5. Evaluation of the
corresponding integrals was done by Rhomberg integration. The geometric parameters
and yS, were calculated according to
P^=1RIB
(6.6-1)
P = cos^
(6.6-2)
where R is radial distance between the centroids of the injection and monitoring
intervals, B is the length of the monitoring interval, and 6 is the angle between the
corresponding radius vectors. Equations (6.6-1) and (6.6-2) are obtained from (5-33) and
(5-34), respectively, upon treating the medium as if it was pneumatically isotropic.
Indeed, our type-curve analysis additionally treats the rock as if it was pneumatically
uniform. However, since the analysis of pressure data from different monitoring intervals
yield different values of pneumatic parameters, our analysis ultimately yields information
about the spatial and directional dependence of these parameters.
190
Figures 6.25 - 6.55 show how we matched each record of pressure buildup from
cross-hole injection test PP4 to corresponding type-curves on logarithmic paper. Though
the figures include type-curves of pressure derivatives, many of the pressure derivative
data are noisy compared to the single-hole counterparts (in Chapter 4) and some that are
not excessively so. Pressure derivatives have been computed using the algorithm by
Bourdet et al. (1989), described in Appendix H. Those derivatives were calculated using
variable intervals of pressure data (10, 20, 50, 100, and 150 data points) depending on the
quality of data. Pressure derivatives were calculated using fewer pressure data points for
good quality data from monitoring intervals with the strongest pressure responses. As the
pressure response became noisier, a larger number of data points were employed for the
calculation. Pressure derivatives allowed us to guide our type-curve match and to obtain
better fits.
The match between pressure data and its derivatives with corresponding typecurves in Figures 6.25, 6.27, 6.28, 6.32, 6.36, 6.38, 6.40, 6.42, 6.43, 6.44, 6.46, 6.47, 6.49
and 6.53 are excellent over the entire length of the buildup record; those in Figures 6.30,
6.31, 6.37, 6.43, and 6.44 are good except for intermediate time where the data exhibit an
inflection, which suggests dual continuum behavior; in Figures 6.26, 6.33, 6.45, 6.50, and
6.54 intermediate time and late pressure data match the type-curves well, but early data
lie above the type-curves; in Figures 6.39, intermediate time and late pressure data match
the type-curves well but early data lie below the type-curves; in Figures 6.36, 6.40, and
6.47, early and intermediate time data fit the type-curves well, but late data fall below the
type-curves; and in Figures 6.29 and 6.41 the matches are poor.
191
Fluctuations in barometric pressure (Figure 6.4) affect almost all of the late
pressure buildup data. It is accentuated in the pressure derivatives for late data. The
timing and magnitude of this barometric effect varies between intervals; it tends to be
most pronounced in monitoring intervals that show a weak response to air injection into
Y2M, in intervals close to the soil surface, and within the Z holes (close to the cliff face).
The rapid and haphazard fluctuation in pressure derivatives also may be due to noise in
pressure data.
Our finding that most pressure buildup data match the type-curves well is a clear
indication that the majority of cross-hole test PP4 results are amenable to interpretation
by means of a continuum model, which treats the rock as being pneumatically uniform
and isotropic while describing airflow by means of linearized, pressure-based equations.
The fact that some of our data do not fit this model shows that the latter does not provide
a complete description of pneumatic pressure behavior at the site. That the site is not
pneumatically uniform or isotropic on the scale of cross-hole test PP4 is made evident by
pneumatic parameters derived fi-om our type-curve matches. Table 6.4 lists values of the
dimensionless well response time
defined in section 5.6, pneumatic permeability, and
air-filled porosity derived from these matches. The latter two parameters represent bulk
properties of the rock between the corresponding monitoring interval and the injection
interval. The permeabilities additionally represent directional values along lines that
connect the centers of these intervals. Corresponding statistics are listed in Table 6.5,
which compares them with similar statistics of 1-m scale permeabilities, obtained fi^om
steady sate interpretations of single-hole test data. The directional permeabilities range
192
from l.lxlO""^m" to 4.6 X
with a mean of-13.5 for logio-based k while a
corresponding (anti-log) value is 2.9 x
.
The corresponding variance and
coefHcient of variation (CV) are 53 x 10'' and - 5.4 x 10 ", respectively. Corresponding
air-filled porosities range from 1.7 x 10'^ to 2.2 x 10"' with a geometric mean of 3.5 x 10"'
while the variance and coefficient of variation are 8.0 x 10"' and - 3.6 x 10"',
respectively. Permeabilities derived from cross-hole tests are seen to have a much higher
mean, and lower variance, than those from the smaller-scale single-hole tests.
193
Table 6.4: Pneumatic parameters obtained from type-curve analysis of pressure buildup
data collected during cross-hole test PP4
k[m']
Interval
n
1.1E-I4 4.5E-03
VI
1.0
2.0E-14 5.7E-03
V2M
1.0
3.4E-14 8.2E-03
V3U
1.0
1.2E-14 5.4E-03
V3M
0.0
6.2E-15 5.7E-04
V3B
1.0
2. IE-14 1.4E-03
WI
10.0
1.2E-02
7.6E-14
W2AU
1.0
1.5E-14 2.0E-02
W2AM
1.0
2.4E-14 2.1E-02
W2AL
l.O
1.9E-14 5.4E-03
10.0
W2AB
W3U
1.6E-14 4.6E-03
1.0
1.5E-14 4.1E-03
W3M
1.0
l.lE-03
1.8E-14
10.0
XI
3.4E-03
2.0E-14
lO.O
X2U
I.lE-16 2.4E-05
1.0
X2M
9.8E-15 2.6E-03
1.0
X2B
3.2E-14 2.2E-01
YIU
1.0
7.7E-15 2.5E-03
lO.O
YIM
l.lE-14 7.4E-03
Y2U
1.0
1.5E-03
5.4E-I5
Y2B
1.0
7.6E-04
100.0
1.3E-13
Y3U
4.2E-02
1.4E-13
Y3M
1.0
1.5E-02
2.6E-14
Y3B
1.0
1.8E-02
4.2E-13
10.0
Z1
1.7E-05
1000.0
1.4E-13
Z2U
2.3E-13 2.9E-04
10000.0
Z2M
1.3E-03
4.6E-I3
1000.0
Z2L
3.7E-03
3.6E-I3
100.0
Z2B
2.4E-14 3.7E-02
Z3U
1.0
2.5E-14 2.3E-02
Z3M
1.0
1.3E-02
10.0
4.0E-13
Z3B
194
Table 6.S; Sample statistics of directional air permeabilities and air-filled porosities
obtained from type-curve interpretation of cross-hole test PP4, and of air permeabilities
from steady state interpretations of \-m scale single-hole tests. Numbers in parentheses
Statistic
Minimum
Maximum
Mean (Logio)
Mean
Variance
CV
Cross-hole values
Logio k [m-]
Logio 4>
-16.0 (5.4 X 10-'^)
-3.1 (7.6 X lO"')
-12.3(4.6 X 10*'^)
-13.5
2.9 X
lO-'"
-0.7 (2.2 X 10 ')
-2.5
Single-hole values
Logio k
-17.1
-13.1
-15.3
lO-'"
3.5 X 10*^
5.6 X
5.3 X 10-'
1.5 X 10-^
7.6 X 10"'
-5.4 X 10-^
-3.6 X 10*'
-5.8 X 10"^
195
i.(eH>i
Q = lo
QflOO
Q^IOOO
l.(EH)Q
Q
I
a i.a&(n
•S*
a
«.
i.a&a2
i.a&«3
xxem
I.(£-OI
l.(»00
i.ce+oi
I.(£H12
I.(£+<B
l.(£-H>4
Figiiie6.25: Type-curve match afpiessure and its derivative fipominonitoniig interval VI.
Q =10
1.0&03
1.0EH12
Figure6.26: Type-curve match afpressure and its derivative from monitoring interval V2M.
196
l.(£K)l
/r, = IJ4
/»,=OiW
1.0EH)0
a
I
ft 1.C&01
ijobm
i.a&03
1.0&<I3
l-OErOZ
I.(£-OI
UCCHX)
l.OEHB
l.OEHM
Figure 6.27;Type-curve matdiafpiessuie and its derivativefianmamtoni^ interval V3U.
/?;=S.l9
^, = 0.16
0=10
0=100
0=1000
i.(eHX)
h
I
ft
I.(Kr01
'5
l.(£pQ2
1.0E-a3
l.OErOI
I.O&OI
l.(SH)0
\SS&<a.
l.0E*C3
l.OEHM
F^;uie 6.28;Type-curve match ofpressure and its derivative fircnnmanitonng interval V3M.
197
/I, = 1.07
^,=058
Q=100
a
1
'5
I.0EH>4
'o
Figure 6.29:Type-curve matcfaofpressnre and its derivative ftewnmonitofiiig interval V3B.
Q=i
1
Figure 6.30: Type-curve matdi of pressure and its denvative fiom monitorii^ intenral W1.
198
0=100
O-l
l-OBOI
a«
l.OEHW
'o
Figare6Jl;Type-ciirve match ofpressuieazxlits derivative firaninaaitoringinterval W2AU.
£>?10
Q=100
1.0&03
1.0EH)1
Figuie 6.32:Type-curve match ofpiessuie and its derivative fiommamtonng interval W2AM.
199
1.0EK>1
^,=592
p2=0j»
Q =10
l-OEHX)
Q=IOO
0=^1000
Q
I
^ l.(£«l
*
i.a&a2
i.a&a3
l.(£-Q3
O
Oq
I.(£<C
i.(£«l
l.(£H)l
I.^HX2
l.(£H)3
1.0EHM
Figure 6.33;Type-curve matdi of pressure audits derivative fixim mouitorii^ interval W2AL.
1.0EH)1
ft I =4.75
fij =0m
Q = 100
Q = lOOO
Q»10
1.(£4<W
I
^ 1.0&01
'5
i.a&az
l.(£^
i.(£-a2
I.(£ 01
p
l.OEHM
l.ae-kxz
F^uie 6.34:Type-curve matdi of pressure andits derivative 6om monitoring interval W2AB.
200
p , =4J»7
p,=o.n
Q=10
Q=IOO
QjlOOO
1.CEHI0
I
a i.«KOi
•
1.0B<X2
l.a&413
i-oeroz
i.(ep(»
uoepOI
l.CSHC
t.0EH)3
l.QEHM
'o
Figure 6.35;Type-curve matdi of pressure audits derivative from monitOTiiig interval W3U.
i.aEHn
P,
=21.99
P,=0SJ
Q=i
Q=io
Q^lOO
Q"1000
I.aEH)0
I
•fr
'5
i.a&02
••
l.(£03
I.OE-02
'll '
: /
//
l.(EtOO
I.0EH>3
l.OEHM
Figure 6.36;Type-curve matdi of pressure and its derivative from monitotiiig interval W3M.
201
I.0EH)1
Bj =0.16
l.OE-01
i.a&c2
I.0E<I3
l.OBrm
l.(£-01
I.OEtOO
i.cbkj3
1.0Et04
Figure 6.37;Type<uive match ofpressure andits denvative from nusiitoniig interval XI.
/?, = 1.47
Q=1
Q=10
Q = 100
Q=1000
I-OEHX)
l.(Er01
l-OErCa
l.(£-02
l.OE-01
l.oeha
l.OEHB
l.(£t04
Figure 6.38;Type-curve match of pressure and its derivative frcm manitoring interval X2U.
202
1.0EH)1
/»,=0.48
Q=tO
Q-IOO
Q = 1000
i.aEH»
I
^ I-OEPOI
•
1.0642
I.OEr^B
1.0&a2
l.(E-01
l.ffi+OO
1.(^1
l.OEHC
1.0EHB
l.(SH)4
Figure 6.39;Type-curve match of pressure andits derivative from monitoiiug interval X2M.
i.(e«<)i
fij'usa
Q°10
Q=100
Q=1000
LOEHX)
1.0Er01
L-OEROZ
1.0E43
I.(£P02
1.(£P01
l.OEHX)
I.OE+01
1.^+02
l.(£K>3
l.(£HM
Figure 6.40; Type-curve match of pressure and its derivative from monitonng intOTval X2B.
203
1.0EH)1
/?,=8.90
fij
=0.65
Q=l
l.aEH)0
QclO
Q=100
Q=1000
^ 1.0&01
Q
«i
ijo&m
I.(£^
I.(£pa2
I.aE41
l.(£H)0
l.OEH)!
I.(£+<B
l.OEHB
Figure 6.41;Type-curve matdi cf pressure audits derivative fixjmmanitonnginterval
l.OEHM
YIU.
I.0EH)1
^2=0.40
Q=10
Q=100
Q=1000
I-C&OZ
I.0EM12
l.CeHX)
1.(^*03
l.OEHM
Figure 6.42;Type-curve matdi of pressure andits derivative firom monitonng interval YIM.
204
p, =\st
pj=om
Q=1
Q=100
Q=10
Q=1000
Q l.OErOl
ix&m.
l.OE-02
I.OE-01
1.0EH)0
l.OEM)I
LOEHG
l.OEHB
l.OEHM
Figure 6.43;Type-curve matdi of pressure andits derivative firom monitoring interval Y2U.
I.OE+OI
P, = 131
pi =0.01
Q=l
1.0EH)0
Qfio
,= 1000
QtIOO
«> l.OE-01
t
"5
1.0Er02
1.0Ep03
l.OE-02
l.OE-OI
I.OE+00
l.OEHG
l.OEHB
l.OEHM
Figure 6.44:Type-curve xnatdiafptessure and its derivative fixminuxiitoiing interval Y2B.
205
/r,=4.i8
Q=l
l-OEHW
Q=10
Q^lOO
Q=IOOO
Q
I
6 I.(]&01
A
l-OErOS
1.0E^
l.(lBr(a
l.OErOl
1.0EHX)
1.0EH)1
l.OE+02
l.OEtOS
l.OEHM
Figure 6.45;Type-curve match of pressure audits derivative firom monitoringinterval Y3U.
fi, = 11163
Q=i
a=io
Q=100
Q=1000
l-CEHX)
A
I
"% i.a&oi
•
*5
I-OEr^B
ijo&m
I-OEFOZ
i.(e4>i
l.(eH)0
uoeka
Figure 6.46;Type<urve matdi of pressure audits derivative finon monitoring interval Y3M.
206
p, =0.76
/I, =0.74
= 10
en
'b
Figure 6.47;Type-curve match of pressure and its derivative fi:om monitonag interval Y3B.
Q=i
1.0&03
1.0EHX)
Figure 6.48;Type-curve match ofpressure and its draivativefixnnmcnitoring interval Zl.
207
Q=10
Q=1
//
l.OEHB
Figure 6.49:Type-curve matdiof pressure andits derivative fixim manitonnginterval Z2U.
tf,=0.74
Q=°10
Q=1000
Q=100
Q= 10000
I.CEHX)
I.OE-02
1.0EP03
l.(K«K)
l.(E+01
LOEHB
I.QEHQ
l.«®+04
l.OE+05
*0
F^pre 6.50: Type-curve matdiafpressure andits derivative fixwnmoaitoung interval Z2M.
208
l.<£t01
p, =isjn
Q = .o
Q=»«>
0=^1000
KKHX)
1.0E01
l.oe-oz
xsmrol
l.OEHX)
1.0EH)I
1.0ehz2
I.OEHB
l.OEHVt
r i^e 6.51 Type-curve match ofpiessuie and its derivative from monitonng interval Z2L.
Q=1
Q=10
piooo
I
ft 1.0&01
•»
//
1.C&01
Figure 6.S2; Type-curve matchof pressure and its derivative from monitoring interval Z2B.
209
Q=1
Q=10
Q=100
Figure 6.53:Type-curve match of pressure and its derivative firom monitom^ interval Z3U.
P,=6.29
Q=10
Q=100
1.QE-Q3
Figure 6.54;Type-curve match of pressure and its derivative fiom manttoiing interval Z3M.
210
P, = 1.14
B,=03S
Q = 10
l.OE-02
l.OE-01
l.OEHX)
Q=100
I.OE+OI
Q = 1000
LOEHB
I.OEMB
I.OEHM
Figure 6.55; Type-curve match of pressure and its derivative fiom mamtoni^ interval 23 B.
211
7. FINDINGS AND CONCLUSIONS
Our work, together with earlier studies concerning the pneumatic behavior of unsaturated
fractured
tuffs at the ALRS, most notably those by Guzman et al. (1994, 1996), Guzman
and Neuman (1996), and Illman et al. (1998) lead to the following major findings and
conclusions;
1. Issues associated with the site characterization of fractured rock terrains, the analysis
of fluid flow and contaminant transport in such terrains, and the efficient handling of
contaminated sites are typically very difficult to resolve.
A major source of this
difficulty is the complex nature of the subsurface "plumbing systems" of pores and
fractures through which flow and transport in rocks take place. There is at present no
well-established field
methodology to characterize the fluid flow and contaminant
transport properties of unsaturated fractured rocks.
2. In order to characterize the ability of unsaturated fractured
rocks to conduct water,
and to transport dissolved or suspended contaminants, one would ideally want to
observe these phenomena directly by conducting controlled field hydraulic injection
and tracer experiments within the rock.
In order to characterize the ability of
unsaturated fractured rocks to conduct non-aqueous phase liquids such as chlorinated
solvents, one would ideally want to observe the movement of such liquids under
controlled conditions in the field.
In practice, there are severe logistical obstacles to
the injection of water into unsaturated geologic media, and logistical as well as
regulatory obstacles to the injection of non-aqueous liquids. There also are important
technical reasons why the injection of liquids, and dissolved or suspended tracers.
212
into fractured rocks may not be the most feasible approach to site characterization
when the rock is partially saturated with water. Many of these limitations can be
overcome by conducting field tests with gases rather than with liquids, and with
gaseous tracers instead of chemicals dissolved in water.
3. The University of Arizona has conducted successfully numerous single-hole and
cross-hole pneumatic injection tests in unsaturated fi'actured tuffs at the Apache Leap
Research Site (ALRS) near Superior, Arizona, under the auspices of the U.S. Nuclear
Regulatory Commission (NRC).
These tests were part of confirmatory research in
support of NRC s role as the licensing agency for a potential high-level nuclear waste
repository in unsaturated fractured tuffs at Yucca Mountain. However, unsaturated
fractured porous rocks similar to tuffs (and alluvium) are found at many locations,
including some low-level radioactive waste disposal sites, nuclear decommissioning
facilities and sites contaminated with radioactive as well as other hazardous materials.
The test methodologies we have developed, and the understanding we have gained
concerning the pneumatic behavior and properties of tuffs at the ALRS, are directly
relevant to such facilities and sites.
4. We found it possible to interpret both single-hole and cross-hole pneumatic injection
tests at the ALRS by means of analytically derived type-curves, which account only
for single-phase airflow through the rock while treating water as if it was immobile.
Our type-curves are additionally based on linearized versions of the nonlinear partial
differential equations that govern single-phase airflow in uniform, isotropic porous
continua under three regimes; three-dimensional flow with spherical symmetry, two-
213
dimensional flow with radial symmetry, and flow in a continuum with an embedded
high-permeability planar feature (a major fracture).
The particular method of
linearization appears to have only a minor impact on the results of our type-curve
analyses. Included in our type-curves are effects of compressible air storage and skin
in the injection interval during single-hole tests, and in monitoring intervals during
cross-hole tests. Our analytical tools include type-curves of pressure derivative versus
the logarithm of time, which accentuate phenomena that might otherwise be missed,
help diagnose the prevailing flow regime, and aid in constraining the calculation of
corresponding flow parameters.
5. Steady state type-curve interpretations of single-hole pneumatic tests yield air
permeability values for the rock in the immediate vicinity of the test interval.
Transient type-curve analyses of such tests provide additional information about the
dimensionality of the corresponding flow regime, skin factors and compressible air
storage effects. Under radial flow, or in the absence of a significant borehole storage
effect, transient type-curve analyses may also yield values of air-filled porosity. At
the ALRS, air permeabilities obtained from
steady state and transient type-curve
interpretations of single-hole pneumatic injection tests, conducted in borehole
intervals of 1-m, agree closely with each other but correlate poorly with fracture
density data. Airflow around the vast majority of these relatively short test intervals
appears to be three-dimensional; borehole storage due to air compressibility is
pronounced; and skin effects are minimal.
The combined effects of three-
214
dimensional flow and borehole storage make it difficult to obtain reliable air-fiiled
porosity values from these tests by means of type-curves.
6. During a pneumatic injection test, air moves primarily through fractures
most of
which contain relatively little water, and the test therefore yields permeabilities and
porosities which reflect closely the intrinsic properties of the surrounding fractures.
This is so because capillary forces tend to draw water from fractures
into porous
(matrix) blocks of rock, leaving the fractures saturated primarily with air, and making
it difficult for air to flow through matrix blocks.
Since the fractures
contain some
residual water, the corresponding pneumatic permeabilities and air-filled porosities
tend to be somewhat lower than their intrinsic counterparts. The former nevertheless
approach the latter as the rate of injection goes up. This is due to displacement of
water by air, which under a constant rate of injection manifests itself in a rapid
increase in pressure within the injection interval, followed by a gradual decrease.
Two-phase flow of water and air additionally causes air permeabilities from
single-
hole pneumatic injection tests to exhibit a hysteretic variation with applied pressure.
7. In most single-hole pneumatic injection tests at the ALRS, pneumatic permeabilities
increase systematically with applied pressure as air displaces water under two-phase
flow.
In a few single-hole tests, where the injection intervals are intersected by
widely open fractures, air permeabilities decrease with applied pressure due to inertial
effects. This pressure-dependence of air permeability suggests that it is advisable to
conduct single-hole air injection tests at several applied flow rates and/or pressures.
Pneumatic parameters derived from
pressure data recorded in monitoring intervals
215
during cross-hole tests appear to be much less sensitive to the rate of injection,
suggesting that two-phase flow and inertial phenomena decay rapidly with distance
from the injection interval. Enhanced permeability due to slip flow (the Klinkenberg
effect) appears to be of little relevance to the interpretation of single-hole or crosshole air injection tests at the ALRS.
8. Flow in the vicinity of most \-m single-hole pneumatic test intervals at the ALRS
appears to be three-dimensional regardless of the number or orientation of fractures in
the surrounding rock.
We interpret this to mean that such flow is controlled by a
single continuum, representative of a three-dimensional network of interconnected
fractures, rather than by discrete planar features. Indeed, most single-hole and crosshole pneumatic test data at the ALRS have proven amenable to analysis by means of
a single fracture-dominated
continuum representation of the fractured-porous
tuff at
the site. Only in a small number of single-hole test intervals, known to be intersected
by widely open fractures,
have the latter dominated flow as evidenced by the
development of an early half-slope on logarithmic plots of pressure versus time;
unfortunately, the corresponding data do not fiilly conform to available type-curve
models of fracture flow.
Some pressure records conform to the radial flow model
during early and intermediate times, but none do so fiilly at late time.
9. It is generally not possible to distinguish between the permeabilities of individual
fractures, and the bulk permeability of the fractured rock in the immediate vicinity of
a test interval, by means of pneumatic injection tests.
Hence there is little
justification for attempting to model flow through individual fractures at the site. The
216
explicit modeling of discrete features appears to be justified only when one can
distinguish clearly between layers, faults, fracture zones, or major individual fractures
on scales not much smaller than the domain of interest.
10. Air permeabilities obtained from single-hole tests are poorly correlated with fracture
densities, as is known to be the case for hydraulic conductivities at many watersaturated fractured
rock sites worldwide {Neuman, 1987).
This provides further
support for Neuman's conclusion that the permeability of fractured rocks cannot be
reliably predicted from information about fracture geometry (density, trace lengths,
orientations, apertures and their roughness) but must be determined directly by means
of hydraulic and/or pneumatic tests.
1 i. Cross-hole pneumatic injection test data from individual monitoring intervals at the
ALRS have proven amenable to analysis by type-curves, which treat the rock as a
uniform and isotropic fractured porous continuum. Analyses of pressure data yielded
information concerning pneumatic connections between injection and monitoring
intervals, corresponding directional air permeabilities, and air-filled porosities. All of
these quantities were found to vary considerably from one monitoring interval to
another in a given cross-hole test on scales ranging from
a few to over 20 meters.
Thus, even though our type-curve analysis treats the rock as if it was pneumatically
uniform and isotropic, it ultimately yields information about the spatial and
directional dependence of pneumatic permeability and connectivity across the site.
12. Some single- and cross-hole pressure records reveal an inflection that is characteristic
of dual continuum behavior. The prevailing interpretation of dual continua is that one
217
represents the fracture network and the other embedded blocks of rock matrix. We
take the broader view that multiple (including dual) continua may represent fractures
on a multiplicity of scales, not necessarily fractures
and matrix.
However, these
inflections show some correlation with barometric pressure fluctuations, implying
that they might have been caused by the latter rather than by dual or multiple
continuum behavior.
13. The pneumatic permeabilities of unsaturated fractured
strongly with location, direction and scale.
tuffs at the ALRS vary
In particular, the mean of pneumatic
permeabilities increases, and their variance decreases, with distance between packers
in a single-hole injection test, and with distance between injection and monitoring
intervals in cross-hole injection tests. This scale effect is most probably due to the
presence in the rock of various size fractures that are interconnected on a variety of
scales.
218
APPENDIX A: ASSESSMENT OF ELECTRICAL RESISTIVITY
TOMOGRAPHY (ERT) IN OBTAINING THE SUBSURFACE
DISTRIBUTION OF WATER CONTENT IN THE VADOSE ZONE
A.1: INTRODUCTION-APPLICATIONS TO ENVIRONMENTAL
MONITORING
Electrical resistivity tomography (ERT) has been recently applied to a number of
field studies in subsurface hydrology. Interests in employing ERT in conjunction with
hydraulic tests and remediation activities have increased due to its potential in providing
high-resolution, two- and three-dimensional images of the subsurface that cannot be
usually obtained by means of hydraulic and/or tracer tests.
Needless to say,
hydraulic/tracer tests can provide high-resolution images of the subsurface on the
condition that data samples are large enough for interpolation and stochastic/geostatistical
analysis. In practice, intensive field data collection is expensive and time consuming;
hence, geophysical techniques such as ERT is becoming an attractive alternative.
The underlying principle behind ERT is straightforward. Electrical currents are
passed through electrodes (transmitters) placed on the soil surface and/or in the
subsurface and voltage differences are measured in another set of electrodes (receivers).
The known voltage difference and input current is related to the resistivity of subsurface
materials. Voltage differences obtained from a large number of cross-hole measurements
are subsequently inverted to reproduce the subsurface distribution of electrical resistivity.
This variability in electrical resistivity is then related to flow and transport parameters
through empirical laws.
219
Spatial variability in resistivity can result from textural variability of subsurface
materials (e.g., sand versus clay; large voids such as fractures at the ALRS - see 1.2.5),
distribution of capillary water, variability in the ionic content of water, etc. Emphasis is
placed on monitoring changes in resistivity over time (it requires a succession of
measurements to see changes in subsurface conditions).
In the saturated zone, Rcanirez et al. (1993) employed ERT to map the subsurface
distribution of a steam flood as a function time. Injected steam induced changes in soil
resistivity because the application of steam displaces some of the capillary water,
increasing the pore water and soil temperature.
According to Ramirez et al. (1993),
changes in resistivity are also due to alterations in the ionic content of capillary water. In
another field
study, Lundegard and LaBrecque (1995) employed ERT to monitor air
sparging. This study showed that the radius of influence of air sparging can be delineated
from reconstructed images of resistivity. In a more recent study, Stubben et al. (1998)
applied ERT to monitor changes in groundwater conductivity in relation to in situ
leaching of copper.
Applications of ERT in the unsaturated zone include monitoring of infiltration
experiments by Daily et al. (1992) and Stubben et al. (1998).
Daily et al. (1992)
conducted two types of infiltration experiments (point and line infiltration) in which they
employed ERT to monitor the movement of water. More recently, Stubben et al. (1998)
applied the technique to monitor an infiltration experiment conducted on a large scale in
unsaturated alluvium at a site in Southern Arizona. In this appendix, a journal article by
Daily et al. (1992) will be reviewed for purposes of evaluating ERT to obtain adequate
220
estimates of vadose zone water movement through monitoring infiltration experiments.
Prior to the evaluation, an introduction to the theory and concepts behind ERT is
provided, followed by a description of the reconstruction algorthim (numerical inverse
procedure).
The conduct of the infiltration experiments will not be discussed but a
discussion on the relative merits of ERT will be provided.
A.2: THEORY AND CONCEPTS
ERT is a method of determining the distribution of electrical resistivity within
some volume of subsurface materials fi^om discrete measurements of current and voltage.
Electrodes are placed in contact with the formation on the soil surface and/or in
boreholes. Two adjacent electrodes are driven by a known electrical current in which the
resulting voltage differential is measured between all other adjacent pairs of electrodes
(in both boreholes). Once the voltage differential is measured between two points in the
transmitting and receiving boreholes, two other electrodes are picked sequentially and the
process is repeated until electrical current is applied to all pairs of adjacent electrodes.
For each pair of measurement, a transfer resistance is computed. Transfer resistance is
the ratio of voltage at one pair of electrodes to the induced current {Daily and Owen,
1991). For n electrodes, there are n(n-3)/2 independent transfer resistances. The current
imposed on electrodes in ERT follows the path of least resistance unlike X-rays
employed primarily in medicine.
This implies that current can flow in curved paths
rendering the inversion process a non-unique problem.
221
Following Daily and Owen (1991), we define the mathematical problem, which
describes the steady distribution of electrical potential in three-dimensional space. The
distribution of electrical potential in a conductive material is governed by the Possion
equation
(A-1)
subject to the following Neumann boundary condition
(A-2)
p \s)r] V
where V - is a divergence operator, V is the gradient operator, p is the resistivity of the
region s, y/ is the electric potential,
tj is the outward normal unit vector, F is the current
source, and g is the current density. There are slight differences in the statement of the
mathematical problem when the works of various researchers (Yorkey, 1987, Daily and
Owen, 1991, Daily et al. 1992, LaBrecque et al. 1996) are compared but the underlying
principle is the same. The inversion problem is to estimate the spatial variability of
resistivity p, given multiple sets of measurements of n/
and g.
A.3: INVERSION OF TRANSFER RESISTANCES - VARIOUS
RECONSTRUCTION ALGORITHMS
As described briefly in the previous section, data collected through resistivity
surveys must be inverted (analytically or numerically) to obtain a spatial distribution of
resistivities.
Here, we outline different inversion techniques and discuss some key
characteristics.
Early analyses included tabulation of measured apparent resistivity,
which is difficult to interpret for all but the simplest subsurface structures and the trial
222
and error method {Daily and Owen, 1991). In the trial and error method, a conceptual
model is heuristically proposed and the resistivity distribution resulting from this model
is computed.
The calculated values of resistivity are then compared to data and the
model is calibrated by adjusting the parameters so that the model fits the data in a trial
and error fashion.
According to Daily and Owen (1991), a more sophisticated method
was proposed by Tripp et al. (1988). They formulated a nonlinear inversion technique to
adjust resistivity for a set of blocks, each of fixed size, shape, and position to fit a surface
sampling of voltage and current.
One key disadvantage to this technique is that the
geometry of the model must be known a priori and cannot be determined from
data
collected in the field.
LaBrecque and Ward (1988) studied borehole to borehole inversion data but their
work was also limited to a specific geometry. Daily and Owen (1991) employed a finite
element Newton-Raphson algorithm proposed by Yorkey (1986) in which there was no
need for a priori knowledge of subsurface resistivities.
The main advantage of their
method was the flexibility of the inversion process, which can be applied to a variety of
measurement geometries (cross-borehole, surface, and borehole to surface).
Other
researchers such as Beasley (1989), LaBrecque (1990), Sasaki (1990), Tang and Yang
(1990) applied different inversion techniques (Marquardt's method, smoothest inversion,
and other related methods) with varying degrees of success.
More recently, LaBrecque et al. (1996) employed an inversion algorithm in which
the geological formation is discretized into parameter blocks, each containing one or
more elements.
A finite
element technique solves the forward problem and Occam's
223
inversion is employed to search for the smoothest two-dimensional model for which the
Chi-squared statistic equals an a priori value. This data-weighting scheme effectively
removes outliers (perhaps not a good thing). The inverse problem solved is an ill-posed
problem yielding non-unique solutions. According to LaBrecque et al. (1996), solutions
of least-squares inversions are often highly erratic and geologically unreasonable.
Occam's inversion, on the other hand, controls the form of the final solution by finding
an optimal, yet conservative solution that fits field
data. Further details to the procedure
are given in LaBrecque et al. (1996) and Morelli and LaBrecque (1996).
LaBrecque et al. (1996, 1998) discuss the effects of data and procedural noise on
the inverse modeling using Occam's method. Here, they refer to noise as variance or
distribution of random and systematic errors in data collected in the field.
Random
errors, inter alia, arise fi'om fluctuations in applied current, and inherent variability in
voltage differential in the subsurface.
Systematic or field
procedural errors are more
severe and include interference between electrical cables within the borehole and
misplacement of electrodes. Another systematic error commonly neglected includes the
application of a crude two-dimensional inversion model to data collected in a threedimensional field environment.
One intriguing aspect of the inversion procedure described is how the process
handles outliers in transfer resistances. As stated earlier, Occam's inversion efficiently
removes outliers in data noise in an effort to obtain smoother images.
The removal of
outliers fi'om data may achieve greater efficiency in the numerical inversion procedure.
However, there is no guarantee that those outliers represent real data points collected in
224
highly heterogeneous formations. In many environmental applications, those outliers are
data that most interest hydrogeologists (esp. in heterogeneous formations), thus the
removal of outliers may contribute to over-smoothing of inverted images.
Determination of noise in data appears to be the focus in increasing accuracy and
of inverted images.
According to LaBrecque et al. (1996), noise estimates can be
obtained by comparing reciprocal data points (those with the receiver and transmitter
interchanged). However, this may not be the optimal method to determine the noise level
in the data due to the nonlinear nature of current path. Another (possible) way to test the
validity of collected data is to vary current systematically to see if the transfer resistance
changes.
If the data collection system is flinctioning correctly and assuming a linear
relationship of resistance with current and voltage differential, resistance should remain
constant with changes in applied current.
Increasing the current should also decrease
random noise, as clearer signals should be obtained at the receiving end. In any case, the
estimation of noise level is crucial in achieving greater accuracy. LaBrecque et al. (1996)
state that overestimates of the noise will decrease the resolution of the image, particularly
away from
the boreholes. The method appears to be highly dependent on the correct
estimation of noise. Even modest overestimates of the noise can cause dramatic increases
in the resolution of the images. Morelli and LaBrecque (1996) discuss the effect of noise
on reconstructed images through synthetic data with both random and systematic errors
while applying their robust inversion technique to a field site in California.
The inversion algorithms discussed above are deterministic in a sense that
statistical information about the correlation among model parameters and observed data
225
are not incorporated. One promising alternative to previous approaches is the stochastic
inversion of three-dimensional ERT data proposed by Xianjin and LaBrecque (1998).
According to those authors, one significant advantage of this new algorithm is that any
prior knowledge about the model parameters can be incorporated into the inversion
process in the form of covariance model type and correlation length. Formulation of the
inversion process in the context of an iterative geostatistical inversion procedure (KcA et
al. 1996) should allow one to conduct joint hydrological-geophysical inversion with
larger confidence.
A.4: APPLICATION OF ERT TO MONITOR VADOSE ZONE WATER
movemem-daily et al. (1992) STUDY
Infiltration experiments were conducted near Lawrence Livermore National
Laboratory (LBNL) by Daily et al. (1992) to demonstrate the capabilities and limitations
of cross-hole ERT in monitoring the movement of vadose zone water.
The plume of
wetted soil can be monitored using ERT because coarser, well-drained soils (sands and
gravels) appear as more resistive zones, whereas finer grained soils (silts and clays) that
hold more water are imaged as less resistive zones. The secondary goal of this study was
to leam about specifics of unsaturated flow in alluvial materials.
Two types of infiltration experiments were conducted at the site. One was a point
infiltration experiment in which a constant head of tap water was maintained in a shallow
borehole for 2.5 hrs. The point infiltration event was monitored to a depth of about 17.3
m during a 22-hr period.
Two boreholes adjacent to the injection borehole were
instrumented with electrodes to conduct a cross-borehole ERT survey to monitor the
226
movement of water in the unsaturated zone. Another experiment, conducted at a nearby
site, consisted of a 25-m long trench filled with tap water again maintained at constant
head for 74 dc^s.
This second infiltration experiment was monitored in two image
planes; one perpendicular to the trench and the other image plane parallel to the trench.
On both image planes, ERT data were collected to about 27-m below the soil surface.
The authors refer to this second experiment as a line or one-dimensional infiltration
experiment. In both experiments, images of resistivity distribution were obtained prior to
and during infiltration.
Cross-hole ERT data taken prior to infiltration in both
experiments were considered as the initial distribution of resistivity.
This background
image was then employed to compare/separate the static resistivity distribution from the
changes caused by water infiltration.
Daily et al. (1992) evaluated consistency in their results by comparing inverted
images to well log data, considered repeatability of their experiments, showed agreement
of adjacent image planes along their common edges, considered the similarity of field
scale images and the reasonableness of the time sequence of images.
In addition, the
authors claimed that site-wide average hydrogeologic properties were similar and that
previous results {Toney, 1990) can be used to evaluate results from cross-hole ERT
surveys.
The authors stated that induction logs were employed to validate the
reconstructed images.
Daily et al. (1992) provided an adequate introduction to the concept and theory
behind ERT including the mathematical procedure employed to conduct the numerical
inversion of field data. The experimental procedure was explained in sufficient detail.
227
Information
on
site-geology,
data
acquisition
and
experimental
results
were
systematically illustrated.
The mathematical procedure employed in Daily et al. (1992)'s study is outlined
below. A forward solution to eqn. (A-1) is obtained in Fourier transform space in which
the potentials are calculated for a discrete number of transform variables at the nodes of a
mesh consisting of quadrilateral elements.
Potentials calculated in the transformed
domain are then inverted to Cartesian domain using a method described by LaBrecque
(1989). A numerical inversion procedure similar to that described in LaBrecque et al.
(1996) was employed to obtain images of changes in resistivity.
Experimental results from
the point infiltration experiment show that cross-
borehole ERT appears to capture the resistive and less resistive regions quite effectively.
It is interesting to see in the background image (Plate la - see Daily et al. 1990), both
resistive zones (one near the soil surface at about 3 m, while the other one is at about 13
m in depth) thinning out equidistant from the transmitting and receiving boreholes.
The
authors do not comment on the thinning out of the resistive zone but it is suspected that
the sensitivity of ERT appears to decrease with distance between the transmitting and
receiving points. Subsequent images (Plate lb through li) show snapshots of changes in
resistivity, which the authors relate heuristically to changes in water content due to the
infiltration event. Images of interest are Plates lb through Id, which show the effect of a
capillary barrier, temporarily arresting the downward migration of water and Plates le
through li which show preferential flow of water to the right side of the image caused by
subsurface heterogeneity.
The authors acknowledge that cross-hole ERT data was
228
collected on only this one two-dimensional plane.
Anomalous "islands" of decreasing
resistivity seen in Plates le through li may be due to: I) ERT not capturing the narrow
zone of infiltrating water; and/or 2) water may be flowing out of the image plane
(channeled flow).
This effectively illustrates one limitation of cross-hole ERT as it was
conducted by Daily et al. (1992).
In heterogeneous materials (such as alluvium and
fiactured rocks), this technique will most likely not be able to capture preferential flow in
sufficient detail due to the lack of resolution and over-smoothing of inverted images.
Cross-hole ERT, however, may be useful as a monitoring tool in obtaining images (to
capture sufficient detail) in relatively homogeneous materials.
One interesting finding
from this study is the much faster wetting front speed inferred from the ERT images as
compared to a rate reported by Taney {^1990).
The line source infiltration experiment was conducted at another site using 3
image planes. The background data (Plate 4a) shows several regions of resistive and less
resistive regions.
The authors claim that there is "^good continuity at the image plane
edges along BHl.
This correlation is one important consistency check for the
tomographs, and this is evidence that the reconstructions represent the in situ resistivity".
However, there appears to be a discontinuity in resistivity where tomograph I was taken
suggesting some problems with the inversion of data to obtain those images. Subsequent
images (Plates 4b through 4d) show irregularities and discontinuities in the images
rendering them not too reliable. In addition, there is an apparent increase in the lower
part of plane 1-4 in Plate 4d in which the authors claim that there is undoubtedly a
reconstruction artifact. They explain that '^''artifacts occur -when the algorithm generates
229
laarge contrasts in the image; a change in one part of the image is accompanied by a
change of the opposite sense, in an adjacent region".
The fact that there are
discontinuities and artifacts in the images renders (on my part) further analysis and
review of the line infiltration experiment fiitile.
The authors also compare resistivity
obtained from cross-hole ERT and induction logs showing a lack of consistency in the
results. To obtain values of water content, Archie's law modified for the vadose zone
was employed to relate resistivity and water content. The authors found a weak negative
correlation between imaged resistivity and water content. This is another indication inter
alia (such as discontinuities, reconstruction artifacts) that water content values cannot be
accurately inferred from cross-hole ERT surveys.
A.5: DISCUSSION OF RESULTS: RELATIVE MERITS OF APPLYING ERT TO
MONITOR VADOSE ZONE WATER MOVEMENT
The paper by Daily et al. (1992) discussed one application of cross-hole ERT to
monitor point and line infiltration experiments conducted in the vadose zone.
Results
from this study showed that the technique was able capture the downward migration of
water in a general sense. Major conclusions reached by the author include;
1. Images of resistivity before infiltration were consistent with what is known of the
geology at the site.
2. From images of resistivity changes during infiltration, they inferred the size and
shape of the plume in the subsurface over a 2A-hr period.
230
3. The development of the plume was consistent with what they knew of lithology from
the driller's log and what could be concluded from these about capilary suction and
permeability distribution.
4. The time sequence of images was internally consistent, showing a reasonable
development of plume with time.
5. Numerical simulations further confirmed the accuracy of tomographs and revealed
that tomography and numerical models can be complementary tools for understanding
infiltration in heterogeneous formations.
6. The results showed that water moved through the unsaturated sediments more rapidly
than previous investigations had estimated.
7. Results from the line infiltration experiment were reasonably consistent; tomographs
of background conditions as well as infiltration yielded similar results for both
experiments. For the most part, adjacent image planes matched along their common
edges. There was, however, some disagreement in the line source experiment once
the wetted regions produced fairly complex resistivity distributions and also in the 1-4
plane where the resistivity constrast was very large.
8. Tomographs matched qualitatively with the borehole induction logs.
9. On the basis of the internal consistency and agreement with other data, they believe
that ERT has been shown to have a useful capability to delineate an infiltration plume
in a hydrologically complex environment. This subsurface imaging technique will
not only be a useful tool for studying contaminant transport but may find use in other
fields of geophysics.
231
Based on Daily et al. (1992)'s conclusions, cross-hole ERT surveys appears to be useful
in delineating the downward migration of the wetted front.
However, there are several
inconsistencies and questions arising from their results. The approach of validating the
utility of ERT in monitoring vadose zone water movement using site-wide average
hydrogeologic parameters appears to be acceptable if one were to employ ERT to obtain
a general picture of the subsurface and changes in subsurface conditions induced by
water infiltration. However, the authors appear to contradict themselves by trying to
obtain a realistic image of the vadose zone using site-wide (or homogeneous)
hydrogeologic parameters. In addition, when testing a new field technique in a complex
geologic setting, it appears that independent validation of calculated parameters by means
of neutron probes and tensiometers may be advisable for added confidence in results.
The authors of the journal article criticize traditional approaches (one based on
Darcy's law and the other based on tracers) to determine flow velocity and associated
moisture distribution on the following grounds: 1) an approach based on Darcy's law
requires information of hydraulic conductivity, hydraulic gradient, and porosity which is
difficult to obtain; and 2) an approach based on tracer tests is simple and accurate but
requires correction for dispersion of solutes, requires multiple measurement points, and
inadequate sampling can lead to misleading flow fields.
Their criticisms on traditional
approaches are valid, yet troublesome, because ERT does not provide us with values of
water content, capillary pressure and other important flow and transport parameters.
Only those traditional approaches criticized by Daily et al. (1992) give us those
estimates.
232
A more recent study by Stubben et al, (1998) conducted at the Maricopa Site in
Southern Arizona appears to overcome this shortcoming. In this study, a variety of other
hydrogeological and geophysical methods were employed in conjuction with cross-hole
ERT which effectively evaluated the performance of ERT to monitor vadose zone water
movement. Daily et al, (1992) should have instrumented the tomographed region with
tensiometers, neutron probe access tubes and perhaps obtain estimates of air permeability
and air-filled porosity through air injection tests to independently validate results from
cross-hole ERT.
The two-dimensional inversion of data can be troublesome because the collected
data reflect water flowing in three-dimensional space. An alternative sampling plan and
inversion technique covering three-dimensional space is probably needed to adequately
characterize vadose zone water movement and to comment about capabilities and
limitations of ERT.
Regardless of Daily et al. (1992)'s questionable results, there are several
advantages in employing ERT to monitor vadose zone water movement. Some of those
include; 1) the technique is not an invasive procedure requiring less boreholes; 2) high
resolution can be achieved with large number of electrodes; 3) rapid estimates of plume
location in real-time can be obtained through automation {LaBrecque et al. 1998), and 4)
if background data is available, one can effectively monitor changes in subsurface water
movement.
There are, however, numerous problems that must be addressed and corrected.
Disadvantages in employing ERT to monitor vadose zone water movement include; 1)
233
complexity of instrumentation can have a high failure rate that can cause systematic
errors; 2) collection of reliable data in the field
requires good electrical contact of
electrodes with the formation; 3) many subsurface variables such as porosity, water
content, water chemistry, temperature fluctuations, etc., affect resistance; 4) unknown
relationship(s) among variables such as spacing of electrodes, data noise and spatial
variability of subsurface materials can cause artifacts in reconstructed images; S)
interference of cable signals can cause excessive noise in data; 6) inversion of data can be
difficult and faces the problem of non-uniqueness; 7) arbitrariness in weighting scheme
for Occam's inverse procedure-particular concern is in the removal of outliers because
those outliers may be of most interest to hydrogeologists; and 8) sensitivity of ERT
decreases with distance which may cause artifacts in the images (needs to be put in the
context of geostatistics so that one can quantify the variance in data points allowing the
optimal placement of sensors in boreholes); and 9) mapping resistivity does not
necessarily give the subsurface distribution of vadose zone water-one has to still rely on
traditional approaches to obtain reliable estimates of water content.
The largest problem in employing cross-hole ERT to monitor vadose zone water
movement is in its data inversion.
Inversion of transfer resistances yield stable, and
unique solutions in medical applications because of our knowledge of anatomy of
humans, geometry and vast experience in conditions that can cause anomalies in
resistivity measurements. In environmental applications, the knowledge of subsurface
distribution of flow and transport parameters is limited or unknown, hence rendering
every inversion ill-posed, which will then yield solutions that are non-unique.
234
Despite (albeit) the complexity of equipment, logistics, and problems in inverting
data to obtain reliable images, ERT appears to be a promising environmental monitoring
tool once the problems discussed above are addressed and better quantified. According
to LaBrecque et al. (1998), near-real time monitoring environmental processes is being
undertaken including the monitoring of water movement in the vadose zone. Large data
sets will be collected in the field, processed and imaged in less than 24 hours and still
produce reliable, high-quality images. This particular paper describes the automation of
data collection and inversion of collected data. We must remember that even if one were
able to obtain accurate images of the subsurface, ERT does not give us direct measures of
flow and transport parameters.
Only traditional hydraulic/pneumatic and tracer tests
provide us with estimates of flow and transport parameters that will allow us to model
fluid and contaminant transport in the subsurface.
A.6: REFERENCES PERTAINING TO APPENDIX A
Beasley, C.W., Cross-borehole resistivity inversion: Theory and applications to
monitoring enhanced oil recovery, Ph.D. Dissertation, Univ. of Utah, 1989.
Daily, W. and E. Owen, Cross-borehole resistivity tomography. Geophysics, 56(8), 12281235, 1991.
Daily, W., A. Ramirez, D. J. LaBrecque, and J. Nitao, Electrical Resistivity Tomography
ofVadose Water Movement, Water Resour. Res., 28(5), 1429-1442, 1992.
LaBrecque, D., and Ward, S., Two-dimensional inversion of cross-borehole resistivity
data using multiple boundaries. Presented at the 58"' Ann. Intemat. Mtg., Soc. Expl.
Geophys., Expanded Abstracts, 194-197, 1988.
LaBrecque, D. J., Cross-borehole resistivity modeling and model fitting: Ph.D Thesis,
Univ. of Utah, 1989.
235
LaBrecque, D. J., M. Miletto, W. Daily, A. Ramirez, and E. Owen, The effects of noise
on Occam's inversion of resistivity tomography data. Geophysics, <5/(2), 538-548, 1996.
LaBrecque, D., J. Bennett, G. Heath, S. Schima, and H. Sowers, Electrical Resistivity
Tomography For Processs Control In Environmental Remediation, SAGEEP, Meeting of
the Environmental and Engineering Geophysical Society - March 22-26, 1998, Palmer
House EBlton Hotel, Chicago, Illinois, Conference Proceedings, 613-622, 1998.
Morelli G. and D. J. LaBrecque, Advances in ERT Inverse modeling, European Journal
of Environmental and Engineering Geophysics, 1, 171-186, 1996.
Ramirez, A. and W. Daily, Monitoring an Underground Steam Injection Process Using
Electrical Resistance Tomography, Water Resoitr. Res., 29(1), 73-87, 1993.
Sasaki, Y., Model studies of resistivity tomography using boreholes: Presented at the
Internal. Symp. On borehole geophysics: Petroleum, hydrogeology, mining, and
engineering applications, Soc. Expl. Geophys., 1990.
Stubben, M.A. and D. J. LaBrecque, 3-D ERT Inversion Used To Monitor An Infiltration
Experiment, SAGEEP, Meeting of the Environmental and Engineering Geophysical
Society - March 22-26, 1998, Palmer House Hilton Hotel, Chicago, Illinois, Conference
Proceedings, 593-601, 1998.
Stubben, M. A. and D. J. LaBrecque, 3-D ERT Inversion Used To Monitor An Injection
Experiment, SAGEEP, Meeting of the Environmental and Engineering Geophysical
Society - March 22-26, 1998, Palmer House Hilton Hotel, Chicago, Illinois, Conference
Proceedings, 603-612 1998.
Sullivan, E. and D. J. LaBrecque, Optimization of ERT Surveys, SAGEEP, Meeting of
the Environmental and Engineering Geophysical Society - March 22-26, 1998, Palmer
House Hilton Hotel, Chicago, Illinois, Conference Proceedings, 571-581, 1998.
Toney, K.C., A hydrogeological investigation of groundwater recharge near the drainage
retention basin, Lawrence Livermore National Laboratory, Livermore, California, thesis.
Dept. ofGeol., San Jose State University, San Hose, Calif, 1990.
Tong, L., and Yang, C., Incorporation of topography in two-dimensional resistivity
inversion: Geophysics, 55, 354-361, 1990.
Tripp, A.C., Hohmann, G.W., and Swift, C.M. Jr., Two-dimensional resistivity inversion:
Geophysics, A9, 1708-1717, 1984.
236
Xianjin, Y. and DJ. LaBrecque, Stochastic Inversion of 3D ERT Data, SAGEEP,
Meeting of the Environmental and Engineering Geophysical Society - March 22-26,
1998, Palmer House Hilton Hotel, Chicago, Dlinois, Conference Proceedings, 221-228,
1998.
Yeh, T-C. J., M. Jin, and S. Hanna, An iterative stochastic inverse method: conditional
effective transmissivity and hydraulic head fields. Water Resour., Res., 32, I, 85-92,
1996.
Yorkey, T. J., J. G Webster, and W. J. Tompkins, Comparing Reconstruction Algorithms
for Electrical Impedance Tomography, IEEE Transactions on Biomedical Engineering,
BME-34, II, 843-852, 1987.
237
APPENDIX B: NEUTRON PROBE METHODOLOGY
B.1: INTRODUCTION:
One of the recommended indirect approaches to determine water content, both in
the laboratory and field is the neutron attenuation method.
The neutron attenuation
method is a standardized method, recommended by agencies such as the American
Standard of Testing Materials
(ASTM D30I7-88) and the U.S. Dept. of Interior (1977).
Previous studies and a comprehensive review of the methodology can be found in
Visvalingam and Tandy
(1972) and {Gardner, in Klute 1986). Employment of neutron
probe to measure water content yields a value of volumetric water content {9J) instead of
the gravimetric water content
obtained by weighing the sample. Field determination
of water content is necessary because 1) the collection of sample for laboratory analysis
is a destructive process, 2) handling numerous samples for laboratory analysis may be
cumbersome and inefficient and 3) in sitit measurements of water content may be more
meaningful. A neutron probe may be operated with relative ease with a number of point
measurements taken over a large volume of the geologic medium. It gives the analyst the
capability to obtain a large data set of water content in a nondestructive manner.
One of the traditional applications of a neutron probe is to lower inside an access
tube at successive intervals to quantify the vertical distribution of moisture down the
hole. Measurements taken over long periods (months) can be used to delineate the spatial
and temporal variations of water content at a field site. For example, a neutron probe was
238
employed at the Apache Leap Research Site (ALRS) to measure water content in
fractured tuffs (Rasmussen et al. 1990; see also 1.2.5 of this dissertation). Collected data
was helpful in simulating the proposed infiltration experiment using a computer model
{Chen et al.
1995 in Bassett et al. 1995). In another example, a neutron probe has been
used extensively to determine the spatial and temporal variations of soil moisture under
an impermeable cap at a field site in New Mexico for purposes of waste isolation
{McTigue et al.
1994).
Measurements of water content and temperature in seven
boreholes over three years have led the workers to conclude that the spatial variability in
water content decreases as depth increases and moves downward by diffusive vapor
distillation. In each example, a calibration curve was fiimished to relate neutron counts
to water content. Calibration of neutron probes for use in alluvial materials is well
documented, practiced, and has become a standardized procedure. Calibration of neutron
probes in consolidated rocks, however, has not been attempted previously, since the
device is used primarily by soil physicists and agronomists, whom are usually not
interested in vadose zone investigations in consolidated rocks.
We begin with a
discussion of the theory of neutron attenuation and a brief description of the standard
calibration procedure using alluvial materials. Factors that affect neutron counts are then
discussed in attempt to understand the capabilities and limitations of the neutron probe.
B.2 TEDEORY BEHIND THE NEUTRON ATTENUATION METHOD
The method for measuring volumetric water content in partially saturated soils
and rocks is commonly referred to as the neutron attenuation method.
The neutron
239
attenuation method was first proposed by Belcher et al. (1950). Gardner et al. (1952),
however, first stated the theory and described the physical and operational aspects of the
methodology.
The device used for the investigation is a cylindrical probe, which consists of a
small radiation source and a detector. The radiation source, which is usually a Ra-Be or
Am-Be, both emit high energy neutron in the range of few million electron volts (MeV)Emitted neutrons collide with the nuclei of atoms that surround the neutron source. The
collision of fast or "high-energy" neutrons with heavier nuclei such as
C'*', and
Mg^^ is virtually inelastic and the reduction of momentum of traveling neutrons is
minimal. The neutrons become "thermalized" or slowed down, when the neutrons collide
with smaller atoms such as hydrogen (H").
The number of thermalized neutrons Is
proportional to the number of hydrogen atoms found in the surrounding material.
A
detecting device, which consists of boron trifluoride (BF3), registers neutrons moving at
thermal velocities. Hydrogen atoms in soils and rocks are mostly in the form of water
and neutron thermalization is dominated by neutron-hydrogen collisions, hence a relation
between neutron counts and volumetric water content can be established for the
surrounding material whether the water is in solid, liquid or gaseous phases. We must
note, however, that some hydrogen is in the form of organic material, plastics (casing
material and current injection interval for cross-hole tests) and hygroscopic water that is
attached to the mineral grains. Consequently, measurements of thermalized neutrons may
not be a true reflection of the amount of moisture surrounding the detector. Neutron
counts are amenable to statistical analysis due to the inherent variance associated with
240
neutron-hydrogen collisions and the number of thermalized neutrons that are counted by
the detector.
For a homogenous soil with a uniform moisture distribution, the cloud of thermal
neutrons surrounding the source will resemble a sphere.
The phrase "sphere of
influence" (Van Bavel et al, 1956; Glasstone and Edlund, 1957) or the "sphere of
importance" (Olgaard, 1965) denotes the zone of neutron flux that is 95% of the total
flux obtained in an infinite medium. An empirical expression suggested by [Olgaard,
1965) relates the surrounding water content to the radius of the "sphere of importance".
1.4 + 10^.
where 9^ is the volumetric water content
and R is the radius of the sphere [L] of
importance in centimeters. The radius calculated in this manner will provide the analyst
a rough estimate on the size of the container to be used for calibration purposes. For wet
soils, the sphere of influence is smaller compared to dry soils. For example, a completely
dry soil will have a R of 70cm, while a wet soil with a volumetric moisture content of
= 0.35) will decrease the R to 22cm. The shape of the thermal neutron cloud will not
be spherical for a heterogeneous soil. In a soil profile consisting of strata with large
differences in lithology, the moisture content profile is discontinuous across those layers.
The vertical resolution of the neutron probe is the ability of the neutron probe to
distinguish changes in moisture content with depth and has been examined by McHenry
(1963). The study was conducted by placing a soil type with higher moisture content
between two dry soils. One of the most important conclusions reached by this author was
241
that a neutron probe can distinguish up to 1 inch of wet soil that intervenes two very dry
soil layers.
B.3: THE STANDARD CALIBRATION METHOD OF NEUTRON PROBE IN
GRANULAR MATERIALS
The neutron-scattering method as stated previously, is an indirect method of
obtaining water content. Thermal neutron counts are related to volumetric water content
through a site-specific calibration curve. The accuracy of the calculated water content
values hence depends on the accuracy of the calibration curve. Calibration, although
important, is often neglected in some applications when neutron probes are employed to
measure moisture content. A factory calibration curve is usually shipped with the device
and in some applications, this calibration curve is sufficient.
Calibration of the
instrument, if possible, should be taken seriously if one is to obtain credible results fi'om
field measurements. Relative changes in water content from one sampled region to the
other may be obtained, but an absolute water content cannot be obtained unless a detailed
calibration study is conducted.
The standard calibration procedure suggested by the American Society for Testing
and Materials will be described below.
Calibration should be performed following
ASTM-D3017, so that results can be compared and replicated. The first step involved in
the calibration of the instrument is to standardize the equipment to a homogeneous
sample with a known value of water content. In order to standardize the equipment, the
ratio of the measurement count rate to the count rate made on a reference standard is
recorded. Standardization is required each time the instrument is employed to ensure that
242
the equipment is functioning correctly. Radioactive decay, leakage, failure of electrical
components and other factors that may affect the performance of the instrument should
be monitored continuously by standardizing the probe before the investigation
commences.
A calibration curve/line can be obtained from at least two samples of different
known water content values. A large container of well-mixed, compacted material with a
known water content and density should be prepared for the calibration study.
Dimensions of the container that exceeds 610 mm * 460 mm. * 360 mm. has been
suggested to be satisfactory.
The use of a 55 Gal drum is widely accepted and an
aluminum access tube is installed at the center of the drum. Measurements are taken by
carefully lowering the probe down the access tube. The probe is lowered to the middle of
the access tube, where neutron counts are taken. It is suggested that the measurements
are taken at the central portion of the container to avoid boundary effects. Boundary
effects at the top and bottom of the container can cause deviance from the true value of
water content. Repetitive measurements should be taken to minimize instrumental errors.
The water content should be determined by the evaporative method (ASTM D2216) and
the wet density calculated fi-om the mass of the wet soil and the volume of the container.
The water content of the material employed for calibration purposes should be varied
through a range of water content values encountered in the field and the density of the
material tested should be in the range of 1,600 to 22^Qkg/m^. A calibration line can be
established with merely two points. However, multiple measurements of widely varying
243
water content values should be taken for a more reliable calibration curve. A best-fit line
is obtained through linear regression analysis.
The precision of the instrument should also be calculated periodically. Precision
can be determined from the ratio of the standard deviation of thermal neutron counts to
the slope of the calibration curve.
B.4: FACTORS THAT AFFECT NEUTRON COUNTS
The neutron attenuation method is valid based on the premise that I) the hydrogen
atoms surrounding the probe is the only source of moderation of fast neutrons and that 2)
the majority (95% or more) of thermalized neutrons is registered by the detector.
In
natural soils, however, thermal neutrons can be produced by other means and/or can be
deflected or absorbed by other constituents of the surrounding material. Organic material
in shallow soils, is one source of hydrogen and has been shown to affect neutron counts.
Elements such as lithium (Li^), cadmium (Cd^*), and chlorine (CI") tend to absorb
neutrons that are emitted fi-om the source. All of the aforementioned factors affect the
reliability of the measurement. Therefore, an absolute water content, in some cases, may
not be obtained with confidence.
Early researchers such as Gardner et al. (1952) and Van Bavel et al. (1955) have
claimed that a universal calibration curve can be employed in determining soil water
content from thermal neutron counts. In contrast, research conducted by Holmes (1966),
Lai
(1974), Greacen and Hign&XX (1979) and others have shown that the bulk density
(pj) of the tested material influences the thermal neutron counts. The bulk density of the
244
soil depends on factors such as the mineral constituent, texture, presence of fractures and
clay content.
Holmes
(1966) compared calibration curves obtained from four different soil
types and concluded that the slope of the calibration curve flattened with increasing clay
content. The slope becomes flatter because more thermalized neutrons are absorbed for
soils with higher clay content which have a larger "macroscopic cross section for
absorption of slow neutrons." This author reports that the accuracy of the calibration can
be improved by taking into account the absorption of thermalized neutrons.
Lai
(1974) studied the effect of bulk density and soil texture on the calibration
curve. The study was conducted using five different ferrallitic soils obtained in a tropical
region. This author reported that both bulk density and texture have a very large effect
on the calibration curve and questioned the validity of a universal calibration curve for all
soil types. The author clearly showed that coarse textured soils have greater slopes
compared to fme-textured soils. The intercept of the calibration curve was found to be
higher for higher clay and iron content.
Greacen and Hignett
(1979) studied the heterogeneous nature of bulk density in
fractured clays. They reported that the intercepts of calibration curve can vary from site
to site and that the bias resulting fi'om the heterogeneous nature of fractured clays can
lead to misleading values of soil water content. The bias can be reduced significantly, if
the calibration curve is corrected for variations in bulk density.
Organic material in soil has also been found to affect neutron counts. It is known
that the average hydrogen content of humus is approximately 5% of its weight, while the
245
hydrogen content in pure water is 11% of its weight. Calibration curves of soils with
high organic content tend to have a best-fit line with a larger intercept.
Therefore,
hydrogen atoms found in humus can play a large role in the production of thermal
neutrons and must be taken into account.
Elements such as boron (B^^, cadmium (Cd^^, chlorine (CI ), lithium (Li"^),
magnesium (Mg^^) and potassium (K^ are known to absorb neutrons, thus a lower
thermal neutron count is observed in soils with high concentrations of those elements.
Access tube characteristics on neutron counts are often neglected or taken very
lightly in most hydrologic and agronomic applications.
Research has shown that the
characteristics of access tubes such as the diameter, construction material, compaction of
surrounding soil and backfill (skin) may have a significant contribution to either an
increase or decrease in the soil moisture content. Access tubes for shallow vadose zone
investigations are made of aluminum and are usually two inches in diameter. In some
cases, neutron probes are sometimes lowered into monitoring boreholes that have a much
larger diameter (4 to 6 inches). Air space between the probe and the borehole wall seems
to have a quite significant effect - a 35% reduction in neutron counts (personal
communication, technician from Boart-Longyear CPN corp.). Tyler (1988) studied the
effects of large air gaps between the detector and the surrounding material and came to a
conclusion that a calibration of the instrument is possible with a nonlinear calibration
curve.
An in-house study performed by Boart-Longyear CPN corp. (1988), the
manufacturer of a neutron (and density) probe, demonstrated that the calibration curve for
246
an over-sized tubing had a flatter slope compared to the factory calibration standard.
Researchers compared the factory calibration to calibration curves obtained for the case
when 1) the probe was placed against the wall and 2) when the probe was centered in the
borehole. The calibration curves for the two cases are nearly identical with the slope
slightly steeper for the case when the probe is placed against the wall.
One of the
conclusions reached by the workers was with proper calibration, the probe can be
employed in access tubing larger than the suggested nominal 2 inch diameter.
The material of access tubes and compaction of surrounding soil during the
installation of access tubes have also shown to affect moisture content data. Aluminum is
the obvious choice for the material because it is virtually invisible to the fast and
thermalized neutrons. In some cases, PVC pipes have been used at field sites and found
to be satisfactory, when the calibration is performed correctly.
Compaction of
surrounding soil and presence of water on access tube wall (from condensation) may
increase the apparent moisture content.
The sphere of influence or importance discussed previously is spherical for a
homogenous soil with uniform moisture distribution. A value of water content (for this
idealized case) obtained from neutron investigation in homogenous soils represents a
unique value due to the uniform distribution of moisture.
The sphere of influence,
however, loses its symmetry when neutron probes are employed in layered or laterally
heterogeneous soils/rocks. Water content determined in heterogeneous soils, is therefore,
an averaged value of some volume depending on the sphere of influence.
247
Textural differences among layers sometimes cause discontinuities in water
content across strata. McHenry (1963) studied the effect of layering of soils on the
vertical moisture distribution using cadmium tape. He reported that simulating vertical
heterogeneity by placing intervening wet layers between very dry layers was technically
difficult. Therefore, sharp moisture boundaries were simulated by using cadmium tape in
the access tube. Cadmium has the property of absorbing most neutrons emitted fi^om the
source. He reasoned that portions of access tube taped with cadmium can be made to
appear as a dry layer, while the untaped portion can be used to simulate a wet layer. He
admitted that the loss of thermal neutrons by absorption through cadmium tape was not
identical to the situation in which thermal neutrons were lost in very dry soils nevertheless, it was a good approximation. He considered three cases: I) the influence of
a dry soil layer between two wet layers, 2) the influence of a wet layer between two dry
layers and 3) multiple wet layers placed between dry layers. The thickness of intervening
layers were varied from one to twelve inches. From his study, he found that a minimum
thickness of one inch wet or dry layer can be distinguished, but an absolute water content
cannot be determined because of the dominant influence of the over and underlying
strata. He also determined that a minimum thickness (12.5 inches) of a soil layer to yield
a measurement of neutron density unaffected by adjacent layers.
248
B.5: REFERENCES PERTAINING TO APPENDIX B
ASTM D2216-7I Standard Method of Laboratory Determination of Moisture Content of
Soil, 290-291
ASTM D3017-88 Standard Test Method for Water Content of Soil and Rock in Place by
Nuclear Methods (Shallow Depth), 411-415
Belcher, D. J., T. R. Cuykendall, and H. S. Sack, The measurements of soil moisture and
density by neutron and gamma-ray scattering. Civil Aero. Adm. Tech. Dev. Rept. v. 127,
Washington D.C., 1950.
Cotecchia V., A. Inzaghi, E. Pirastru, and R. Ricchena, Influence of the physical and
chemical properties of soil on measurements of water content using neutron probes.
Water Resour. Research, A, 1025-1028, 1968.
CPN, An in-house study on the effect of air gaps on the 501 DR probe, 1988.
Elder, A. N., 1988, Neutron Gauge Calibration Model for Water Content of Geologic
Media, Unpublished M.S. Thesis, University of Arizona, Tucson, AZ, 181 pp., 1988.
Gardner, Wilford, and D. Kirkham, Determination of soil moisture by neutron scattering.
73;391-401, 1952.
Soil Sci.
Glasstone, E., and M. C. Edlund, 1957, The elements of nuclear theory. Van Nostrand
Co. Inc., New Jersey.
Greacen, E. L., and C. T. Hignett, Sources and bias in the filed calibration of a neutron
meter. Anst. J. Soil Res. 17:405-415, 1979.
Greacen E. L. and G. Schrale, The Effect of Bulk Density on Neutron Meter Calibration,
Vol. 14, pp. 159-169, 1976.
Aust. J. Soil Res.,
Holmes, J. W., Influence of bulk density of the soil on neutron moisture calibration. Soil
Sci. 102:355-360, 1966.
Klute, A., (editor). Methods of Soil Analysis Part I-Physical and Mineralogical Methods,
SSSABook Series: 5, 1986.
Lai, R. The effect of soil texture and density on the neutron and density probe calibration
for some tropical soils. Soil Sci. 117:183-190, 1974.
249
McHenry, J. R. 1963, Theory and application of neutron scattering in the measurement of
soil moisture. Soil Sci. 95:294-307, 1963.
McTigue D. F., C. L. Stein, W. A. Illman, M. J. Martinez, and T. D. Burford, Heat and
Vapor Transport Beneath and Impermeable Barrier, Abstract presented at the 1993 Fall
Meeting, San Francisco, CA, 1993.
Olgaard, P. L., On the theory of the neutronic method for measuring the water content of
soil. Danish Atomic Energy Comm., Riso Rep. 97, 1965.
Rawitz, E., Installation and Field Calibration of Neutron-Scatter Equipment for
Hydrological Research in Heterogeneous and Stony Soils, Water Resour. Res., 5(2), 519523, 1969.
Sinclair, D. F., and J. Williams, Components of variance involved in estimating soil
water content and water content change using a neutron moisture meter. Aust. J. Soil Res.
17:237-47, 1979.
Tyler, S. W., Neutron Moisture Meter Calibration in Large Diameter Boreholes, Soil Sci.
Soc. Am. J., V.52, pp.890-893, 1988.
U.S. Dept. of Interior, National Handbook of Recommended Methods for Water Data
- A United States Contribution to the International Hydrological Program,
Chapter 6 - Soil Water, 6-1 to 6-3, 1977.
Acquisition
Van Bavel, C. H. M., N. Underwood, and R. W. Swanson. 1955, Soil moisture
measurement by neutron moderation. Soil Sci., 82:29-41, 1955.
Visvalingam, M., and J. D. Tandy, The neutron method for measuring soil moisture
content-a review. Soil Sci., 13 A99-S\\, 1972.
250
APPENDIX C: SLIP FLOW
Consider single-phase flow of a Newtonian fluid in a porous or fractured medium.
Knudsen
(1934) deflned a dimensionless number K„ = X/d (called Knudsen number)
where A is the mean free path of fluid molecules (average distance between molecules)
and ^ is a characteristic length. In porous or fractured media, d is typically a measure of
pore diameter or firacture aperture {Present, 1958; Bear, 1972). When K„ <0.01, the fluid
behaves (on the pore or fi-acture aperture scale) as a viscous continuum and satisfies the
Navier-Stokes equation. At sufficiently small Reynolds numbers, a linear Stokes regime
develops within pores or fractures in which viscous forces are large in comparison to
nonlinear inertia so that the latter can be disregarded. The fluid continuum satisfies a noslip, or zero velocity, condition at its contact with the solid walls of pores or fi-actures.
On a macroscopic scale that includes many pores or fracture apertures the Stokes regime,
under the no-slip condition, manifests itself in the form of Darcy's law. As the Knudsen
number increases toward 1, viscosity and drag along solid walls diminish sufficiently to
allow slippage of fluid past solid walls to occur. This allows flow rate to be higher than
predicted by Darcy's law, a phenomenon known as slip flow or the Klinkenberg (1941)
effect. As K„ increases further, the fluid behaves less and less as a viscous continuum
a n d m o r e a n d m o r e a s a diffusing collection o f fi-ee-flowing molecules. W h e n K„ > 1 ,
Knudsen diffusion becomes dominant. As the mean free path of gas molecules is much
larger than that of liquid molecules, the former is associated with much larger Knudsen
numbers and one must therefore consider the possibility that gas may exhibit the
Klinkenberg effect or, less commonly, be subject to Knudsen diffusion.
251
A decrease in gas pressure brings about an increase in the mean free path of its
molecules and a concomitant increase in the Knudsen number. If one uses Darcy's law to
describe the associated Klinkenberg effect, one finds that permeability appears to
increase as pressure decreases. Indeed, based on an analogy to gas flow in a capillary
tube, Klinkenberg {\9AV) proposed that the apparent permeability of gas varies according
to
(C-1)
where k is intrinsic permeability at high pressures (small Knudsen numbers), c ^ 1 is a
proportionality factor, r is the radius of the capillary tube, and m is an empirical
coefficient valid for porous media that depends on k and pore size. As pore size and
permeability are related, m can be considered a function of k.
Though such inverse
relationship between apparent permeability and pressure is often observed in the
laboratory, it was pointed out by Aronofsky (1954) that the Klinkenberg effect seldom
affects the interpretation of gas well tests in the field.
It was mentioned in the
Introduction (section 1.2.3) that, according to Guzman et al. (1994), Knudsen diffusion
and the Klinkenberg effect play virtually no role in the observed pressure dependence of
air permeabilities obtained from single-hole pneumatic injection tests at the ALRS; the
observed decrease in apparent permeability with pressure is due to inertia effects, and the
observed increase is due to two-phase flow.
Most pneumatic injection tests at the ALRS "see" fractures, which have mean
apertures that are considerably larger than the mean pore diameter of intact matrix at the
252
site (see Table C. I for pore size distribution and Figure C. I for mean pore sizes of intact
matrix cores; ydgt, 1988; Rasmussen et al. 1990). We suspect that this is a major reason
why these tests appear to be unaffected by the Klinkenberg effect. To understand how
this effect is related to pore size, we consider the work of Scott and Dullien (1962a) who
proposed the following quasi-Darcian relationship for gasflow.
F=-
^ rp
_2
ArRT
8//
' Vp
(C-2)
RT
where F is molar flux of gas [molL'^T'], r is mean pore radius [Z-], M is molecular
weight
and
is mean molecular velocity [LT'^. Volume flux is related to
molar flux via {Massmann, 1989)
q=—
(c-3)
P
This, coupled with the Boyle-Mariotte law for real gases {Burcik, 1957)
P=—
(C-4)
ZRT
transform (C-2) into
q=
(C-5)
8//
In the absence of slip flow, (C-5) reduces to Darcy's law in the form
8/i
(C-6)
The first term on the right hand sides of (C-2) and (C-5) accounts for Darcian flow and
the second for slip flow.
These equations imply that (a) as r becomes large, the
253
importance of slip flow relative to Darcian flow diminishes; (b) the opposite happens as r
becomes small; and (c) gas-flow involving light molecules that travel at low velocities
through large pores can exhibit a large slip component.
According to Harleman et al. (1963), intrinsic permeability is proportional to the
square of mean pore diameter d,
k
= cd^
(C-8)
where c is a constant. Therefore, eq. (C-6) can be rewritten as
q = --Vp
(C-9)
which is Darcy's law expressed in terms of pressure when the flowing gas is assumed to
b e ideal ( Z = l ) .
254
Table C.1: Cumulative mercury intrusion volume as a function of equivalent pore
diameter (adapted from Rasmussen et al. 1990)
Pore Size
[fm\
Cumulative Volumetric
Porosity
Volumetric
Porosity [m^/m^]
Relative
Frequency
13.8
9.19
7.64
6.3
5.08
4.06
3.27
2.61
2.07
1.65
1.3
1.03
0.802
0.627
0.494
0.385
0.302
0.237
0.187
0.146
0.115
0.0896
0.07
0.0548
0.0429
0.0336
0.0262
0.0205
0.016
0.0125
0.0098
0.0077
0.0003
0.0005
0.0011
0.0024
0.006
0.0125
0.0213
0.0308
0.041
0.0487
0.0544
0.059
0.0631
0.0666
0.0696
0.0726
0.0756
0.0787
0.0819
0.0859
0.0907
0.0969
0.1042
0.1116
0.1182
0.1237
0.1284
0.1327
0.1369
0.1397
0.1417
0.1428
Sum
0.0003
0.0002
0.0006
0.0013
0.0036
0.0065
0.0088
0.0095
0.0102
0.0077
0.0057
0.0046
0.0041
0.0035
0.003
0.003
0.003
0.0031
0.0032
0.004
0.0048
0.0062
0.0073
0.0074
0.0066
0.0055
0.0047
0.0043
0.0042
0.0028
0.002
0.0011
0.1428
0.002101
0.001401
0.004202
0.009104
0.02521
0.045518
0.061625
0.066527
0.071429
0.053922
0.039916
0.032213
0.028711
0.02451
0.021008
0.021008
0.021008
0.021709
0.022409
0.028011
0.033613
0.043417
0.05112
0.051821
0.046218
0.038515
0.032913
0.030112
0.029412
0.019608
0.014006
0.007703
1
255
0.2
0.18
0.16
0.14
i 0.12
er
£
u, 0.1
u
I 0.08
OS
*• slip flow is negligible
slip flow needs to be
accounted for
0.06
•
0.04
•
•
•
••
0.02
0
0.001
A
•
0.01
0.1
1
•
10
Equivalent Mean Pore Diameter (^m)
Figure C.1: Relative frequency of equivalent mean pore diameter plotted on semilogarithmic scale, showing bimodal distribution of pores in tuff matrix at the ALRS (data
Rasmussen et al. 1990).
100
256
APPENDIX D: AIR COMPRESSIBILITY AND VISCOSITY IN
RELATION TO PRESSURE AND TEMPERATURE AT THE ALRS
In this appendix we include a sample calculation of the compressibility factor Z for
air under standard conditions, tables of pressures and temperatures encountered during
single-hole pneumatic injection tests over \-m intervals at the ALRS, and a table of air
viscosity values under a range of temperatures.
The compressibility factor for gas is defined in (2.1-9) as
where V, = V/n. To assess Z for air, we note from Table D. 1 (CRC Handbook, 19921993) that the molecular weight of standard atmospheric air in the United States is
28.96443 g/mol.
= 1.246629 •
Its specific volume is therefore {Vasserman et
Imol.
As
al.
1966)
the universal gas constant is /? = 8.31441 Nm /" AT mol
we find that, under standard conditions of /? = 200 kPa and T = 300°A!',
(200,000/'a)(l246629 x lO'"'"'
(8314510 A/m/°A:/no/)(300°^)
Otherwise Z varies with p and T as shown in Figure 2.1. Table D.2 lists the minimum,
maximum, average and range of stable injection pressures attained in each borehole within
which single-hole pneumatic injection tests took place over \-m intervals at the ALRS.
Table D.3 shows the same for temperatures.
257
Table D.4 lists values of dynamic viscosity for air over a range of pressures and
temperatures comparable to those encountered at the ALRS. The same data are shown
graphically in Figure 2.2.
Table D.l: Gaseous composition of U.S. standard atmosphere
Molecular Weight
\kg/kmo[\
28.0134
31.9988
39.948
44.00995
20.183
4.0026
83.8
131.3
16.04303
2.01594
Gas Species
N2
O2
AT
CO2
Ne
He
Kr
Xe
CH4
H,
Total
Fractional Volume
Idimensionlessl
0.78084
0.209476
0.00934
0.000314
0.00001818
0.00000524
0.00000114
0.000000087
0.000002
0.0000005
1"
Fractional Molecular
Weight \g/mo[\
21.87398
6.702981
0.373114
0.013819
0.000367
2.1e-5
9.55e-5
1.14e-5
3.21e-5
I.Ole-6
28.96443
•1
Table D.2: Data concerning stable pressures encountered during \ - m scale single-hole
Borehole
V2
W2A
X2
Y2
Y3
Z2
Minimum Pressure
\Pa\
88886.04
88419.42
89805.97
89632.65
88379.42
90072.61
Maximum Pressure
\Pa\
248592.9
240073.6
265058.3
233380.9
361610.3
240273.6
Mean Pressure [Pa\
159460.9
147661.9
144005.1
146031.9
162972.3
152740.5
Range ofPressxues [
\Pa\
159706.9
151654.2
175252.3
143748.2
273230.9
150201.0
Table D.3; Data concerning stable temperatures encountered during l-nt scale single-hole
pneumatic in ection tests at the ALRS (summarizec from Guzman et al. 1996)
Borehole
V2
V/2A
X2
Y2
Y3
Z2
Minimum
Temp r'AH
293.09
292.57
293.04
288.50
288.10
291.32
Maximum Temp ["K]
300.90
299.07
302.60
293.93
298.93
302.74
Mean
Temp m
294.40
293.40
294.82
293.14
293.02
294.22
Range of
Temps fATI
7.81
6.50
9.56
5.43
10.83
11.42
258
Table D.4: Variation of air dynamic viscosity with temperature and pressure (data from
?{kPa\
l.OE+02
l.OE+03
2.0E-H)3
3.0E-H)3
4.0E-K)3
5.0E+03
6.0E-K)3
7.0E-K)3
8.0E-H)3
9.0E-H)3
1.0E-H)4
TfAl
T = 270TC
1.6960E-0S
1.7110E-05
1.7280E-05
1.7480E-05
1.7690E-05
1.7930E-05
1.8190E-05
1.8460E-05
1.8750E-05
1.9070E-05
1.9410E-05
T[''iq
T = 280°K
I.7460E-05
1.7600E-05
1.7770E-05
1.7950E-05
1.8160E-05
1.8380E-05
1.8620E-05
1.8880E-05
1.9140E-05
1.9440E-05
1.9760E-05
TC/n
T = 290'TC
I.7960E-05
1.8100E-05
1.8260E-05
1.8430E-05
1.8630E-05
1.8830E-05
1.9060E-05
I.9300E-05
1.9550E-05
1.9830E-05
2.0120E-05
TL'An
T = 300°K
I.8460E-05
I.8590E-05
1.8740E-05
1.8910E-05
I.9100E-05
1.9290E-05
1.9510E-05
1.9730E-05
1.9970E-05
2.0230E-05
2.0500E-05
TI'Al
T = 310'X
1.8960E-05
1.9100E-05
1.9230E-05
1.9390E-05
I.9570E-05
1.9760E-05
1.9960E-05
2.0I70E-05
2.0400E-05
2.0640E-05
2.0890E-05
TCAH
T = 320TC
1.9450E-05
1.9580E-05
1.9710E-05
1.9860E-05
2.0030E-05
2.0210E-05
2.0400E-05
2.0610E-05
2.0820E-05
2.1050E-05
2.1290E-05
259
APPENDIX E: DERIVATION OF SINGLE-PHASE GAS FLOW
EQUATION
The continuity equation for single-phase gas flow in a rigid porous medium is
-V.{pq) = ,S^
(E-1)
Substituting the Boyle-Mariotte relationship {Burcik, 1957)
pM
P=
(E-2)
TRT
and Darcy's law (2.1-2) for a gas into (E-I) yields
M p
M p k^ ^
V-
vp
As M and
R are
(E-3)
dt
RTZ n
constant for a homogeneous gas, and
T
is constant under isothermal
conditions, this reduces to
V-
(E-4)
Z //
The latter form of the single-phase gas flow equation is found, for example, in
Hussainy et al.
(1966) and
Bear
Al-
(1972, p. 200, eq. 6.2.26). In a medium with uniform
permeability.
= £Af £
kaivz
(E-5)
which can be written as
(E-6)
V.
2pZ
J
k dt\ZJ
260
Expanding the left-hand-side of (E-6) yields
\.2fjZ) ^
^
=-—f-1
(E-7)
kdt\z)
Multiplying both sides by 2/zZ gives
•vp' ^ ^ 2 = ^ m £ a ( p ]
^il
^
k
(E-8)
dAz)
Remembering that /j Z is pressure dependent, we note that
d
\
<?xL/izJ
d ( \ ^dp- _
d
^ 1 ^ d{^iZ) dp-
dp'Ki^j dx
1
dp-
d{^)dp-
(pZ)'
dp
dx
(E-9)
<ip'
(E-IO)
dp'
and therefore
d? '' 1 ^ _
^xv/izj
1 d\n{jij^dp'
(/iZ) dp'
dx
(E-11)
It follows from (E-8) and (E-11) that
dp'
k
dt
(I)
(E-12)
Upon taking p « constant and Z « 1,
vv-
2it>p dp
k
dt
(E-I3)
261
which is the same as (2.1-11) and a form found in
Scheidegger
(1974). Defining the
compressibility of air as
(E-I4)
P dp
we obtain from (E-2) and (E-14), for isothermal conditions.
_Z{p)RT d ^ pM ^ _Z(p) d ^ P
'W =
pM dp Z{p)RT,
P dp zip)}
(E-IS)
This simplifies to
/ \ I
c\p)=
p
I dZ{p)
\ V
Z{p) dp
(E-16)
If Z is a weak function of pressure, air behaves almost ideally and compressibility can be
approximated by
c{p)~P
(E-I7)
1_( P \- P^{p)^P
Z{p) dt
(E-18)
Otherwise, it follows from (E-15) that
Substitution of (E-18) into (E-12) yields (Raghavari, 1993, p. 27, eq. 3.21)
dp^
vp-
k
dt
(E-19)
262
APPENDIX F: TYPE-CURVE SOLUTION OF SPHERICAL GAS
FLOW
In a manner analogous to Joseph andKoederitz (1985) who dealt with a liquid, we
consider the injection of gas at a constant mass rate
Q
from a spherical source of radius
into a uniform, isotropic continuum of infinite extent.
The flow is spherically
symmetric so that surfaces of equal pressure (isobars) form concentric spheres about the
source. Pseudopressure is then governed by
— + 2 Sw = I dw
dr'
r 3r
_ ,.
(F-I)
a dt
where r is radial distance from the center of the source.
In our case the source is closer to a cylinder of length
L
and radius r„. Under
steady state, the isobars around such a cylinder can be closely approximated by prolatespheroids or confocal ellipsoids
{Moron and Finklea,
1962,
Culham,
1974, and
Joseph
and Koederitz (1985).
It is convenient to view the source as the innermost ellipsoid with
semi axes a and b
The radius
= c.
Joseph andKoedertiz
r^^
of a sphere having similar volume is then given by
(1985) as
r
L
(F-2)
Equation (F-1) is solved subject to the initial and outer boundary conditions
w(r,0) = 0;
r>r^
(F-3)
lim w(ir,f) = 0;
/>0
(F-4)
263
Mass balance within the injection interval (Figure F.1) is written in terms of pressure as
0mA„ - QwPw =
^
(F-5)
where the subscripts in and w refer to the injection line and injection interval, respectively,
and
is a borehole storage coefficient
Mass flow rate from a spherical source into the surrounding rock is given in terms
of pressure by Darcy's law, in combination with (E-2), as
=
(F.6)
'n.
For a low-pressure system such as ours, eq. (2.1-16) implies that
dr
2p dp
/iZ dr
dM/
dt
2p dp
fjiZ. 3t
so that (F-6) can be written in terms of pseudopressure as
= -2^^^
dw
dr.
(F-7)
By virtue of (E-2), the rate at which mass within the interval increases with time can be re­
written in terms of pseudopressure as
^ r
^
" dt
ZRT
"2/7^ dt
^
pM dw^
2RT dt
Substituting (F-7) and (F-8) into (F-5) yields the inner boundary condition in terms of
pseudopressure.
264
^2RT dt
RT\
dr
(F-9)
QmPm
If air density in the injection system and in the formation are the same
{pi„
pj),
(F-9)
simplifies to
(
2 dw
=a
(F-IO)
2/?^ dt
Equation (3.1-5),
^
(F-11)
t„p
allows rewriting (F-10) as
'^qj'p^ p^ di
2 dw
qjpk \ ~d?
=1
(F-12)
'•n.
An infinitesimal skin may cause pressure and pseudopressure differences
(F-13)
(F-14)
to develop across it, as shown schematically in Figure F.2, where the subscripts s and
wf
represent skin and the rock just outside the skin, respectively. As the skin is infinitesimally
thin, it cannot store gas and therefore the pressure drop across it is constant, as required
by the Laplace equation. This allows defining a constant dimensionless skin factor
through
s
265
r t- ^
Aw, = —qsc te^
27dcr_ V t y
(F-15)
where 5 is a dimensionless skin factor. Substituting (F-15) into (F-14) yields the auxiliary
inner boundary condition
= "'w + a. (tp.\
(F-16)
27dcr„ KT„)
The complete mathematical problem for spherical flow with storage and skin effects
consists of the governing diffusion equation (F-1), initial condition (F-3), outer boundary
condition (F-4), inner boundary condition (F-12) and auxiliary inner boundary condition
(F-16).
The following dimensionless variables reduce the dimensional boundary value
problem into a dimensionless form:
D~
r
'd
,
= 1 -—
'
r>r_
4
•
r
r >r
(F-17)
The dimensionless form of the complete mathematical problem is
dlr
=o) = 0; 0<rD < 1
(F-18)
(F-I9)
266
lim{H'o(ro,/o)} = 0; t ^ > 0
r
°
du
=1
drr.
»^«o('D)=»^D('b=0>'D)-
where
(F-20)
(F-21)
ro=0
(^)
(F-22)
=C^/'/4;r^^ is a dimensionless wellbore storage coefficient. Note that the
dimensionless form of the boundary value problem is virtually identical to that of liquid
flow given in Joseph
and Koederitz
(1985). The definition of the radial flow boundary
value problem follows in a similar manner to the above development.
267
Q,rP„
i
QwPw ^
•
pC ^
* dt
Figure F.1: Mass conservation in injection system.
qmpm
268
w
wwf
/N
^w(pj =
V
w
W.
Infinitesimal
skin zone
•> r
r^+5r
Figure F.2; Impact of positive skin on pseudopressure around the injection interval.
269
APPENDIX G: CODE FOR NUMERICAL INVERSION OF
LAPLACE TRANSFORM
To invert analytical expressions from Laplace to real time domain we employ the
algorithm of DeHoog et
al.
(1982). This algorithm relies on accelerated convergence of
Fourier series obtained from the Laplace inversion integral by means of the trapezoidal
rule. It is similar in principle to that of Crump (1976) but is more accurate, converges
faster, and is thus computationally more efficient.
The Fortran code included below was modified slightly fi-om
Press et al.
(1992)
for double complex arithmetic to allow accurate computation of Bessel functions. The
program computes one value of
for each value of tp listed as a data statement in the
main program. After the end of the computation, a data file is written in the same
directory in which the executable file resides.
Numerically inverted results are then
imported into a standard spreadsheet program for the purposes of generating type curves.
The sample program given below is the one that inverts the Laplace transform for
the radial flow problem. The program consists of three main parts: 1) a main program
which assigns a value of dimensionless time, evaluates the Laplace transform in a
subroutine, and writes the calculated value into an output file, 2) subroutines used to
evaluate Bessel functions, and 3) a function statement which does the actual inversion of
the Laplace transform to real-time space.
A few comments are added before key
operations in the main programs for clarity. This computer program can be applied to
essentially any problem, which requires the inversion of the Laplace transform into real­
time space by rewriting the Laplace transform listed in the subroutine (RADIAL).
270
PROGRAM WELLBORE
IMPLICIT REAL*8 (A-H, 0-Z)
REAL niax_time, time
DOUBLE PRECISION inversion
DOUBLE COMPLEX conc_lap( 15), p i
DIMENSION TDW(8)
DATATDW/ LODOO. 1.25DOO, 2.50D00, 3.750D00, 5.0D00, 6.250D00,
"
7.50D00,8.250D00/
WRITE (*,*) 'max_time=?'
READ (*,*) max time
WRITC (*,*) Enter a rD value;'
READ (*,*) rD
WRITE (*,*) "Enter the dimensionless storage value (CD);'
READ (•,•) CD
C
C
Opening output data file
OPEN (6,FILE= '2ider-l.dat', STATUS='UNXNOWN')
C
C
C
Writing file headings into file
WRITE (6,*) 'rD=', rD, •CD=', CD
WRITE (6,*) 'tDw,
tDCD,
C
C
C
C
C
C
C
mD'
Operation which takes the value of data statement given at the beginning of the main
program and assigns a value on the logarithmic scale. Eight values of dimensionless
time are input for each log cycle. This number can be changed in the data
statement and in the DO statement given below for better resolution of the type
curves.
DO 3000 j= 1,30
DO 3000 k = 1, 8
tD= tDw(K) * (10.0D00**G-5))
max_time = sngl(tD)
p0=8.634694/max_time
271
DOi=l, 15
p i = dcniplx( pO, (i-l)*3.141592654/(0.8*max_time) )
CALL RADIAL (p_i, CD, RD, conc_lap(i))
END DO
time = maxtime
wow = inversion(conc_lap, time, max time)
mD = wow
tDCD = tD / CD
C
C
C
Writing calculated results into the output data file
WRITE (6,5000) tD, tDCD, mD
WRITE (*,5000) tD, tDCD, mD
3000 CONTINUE
5000 FORMAT (6(E15.7, IX))
STOP
END
C
C
C
C
C
Definition of the Laplace transform for the radial flow problem
in Laplace space. Note that all input variables are converted to double
-precision, complex arithmetic including numbers for improved accuracy.
SUBROUTINE RADIAL (p_i, CD, RD, concjap)
IMPLICIT DOUBLE COMPLEX (A-H, 0-Z)
REAL*8 CD, RD
DCCD=DCMPLX(CD)
DIMR=DCMPLX(RD)
RMD_i= CDSQRT(pJ)
CALL IBESSKO (RMDJ*DIMR, SOLI)
CALL IBESSKO (RMDJ, S0L2)
CALL IBESSKl (RMDJ, SOLS)
S0L4 = (p_i*SOL2*DCCD+RMD_i»SOL3»DCMPLX(0.5))
conclap = SOL1/SOL4
RETURN
END
272
C
C
C
C
C
A series of subroutines borrowed from Numerical Recipes (1986) that evaluate
various Bessel function for a particular argument. Note that all
functions and numbers are rewritten in double-precision, complex arithmetic.
SUBROUTINE IBESSKO(X,ZZ)
IMPLICIT DOUBLE COMPLEX (a-h,o-z)
REAL*8 RP,RQ
DIMENSION RP(7),RQ(7),P(7),Q(7)
DATARP/-O.57721566DO,0.4227842ODO,O.23O69756DO,
* 0.3488590D-l,0.262698D-2,0.10750D-3,0.74D-5/
DATA RQ/1.25331414D0,-0.7832358D-1,0.2189568D-1,
* -0.1062446D-l,0.587872D-2,-0.251540D-2,0.53208D-3/
C
DO 10 1=1,7
P(I) = DCMPLX(RP(I))
10
QG) = DCMPLX(RQ(I))
IF (CDABS(X) LE. 2.0D00) THEN
Y=X*X/DCMPLX(4.0)
CALL IBESSIO(X,BESSIO)
X2 = X/DCMPLX(2.0)
X3 = -CDLOG(X2)*BESSIO
ZZ=X3+(P( 1)+Y*(P(2)+Y*(P(3)+
*
Y*(P(4)+Y*(P(5)+Y*(P(6)+Y*P(7)))))))
ELSE
Y=(DCMPLX(2.0)/X)
ZZ=(CDEXP(-X)/CDSQRT(X))»(Q( 1 )+Y*(Q(2)+Y»(Q(3)+
Y*(Q(4)+Y*(Q(5)+Y*(Q(6)+Y*Q(7)))))))
ENDIF
RETURN
END
SUBROUTINE IBESSK1(X,ZZ)
IMPLICIT DOUBLE COMPLEX (A-H,0-Z)
REAL»8 RP,RQ
DIMENSION RP(7),RQ(7),P(7),Q(7)
DATA RP/1.0D0,0.15443144D0,-0.67278579D0,
• -0.18156897D0,-0.1919402D-1,-0.1 l0404D-2,-0.4686D-4/
DATA RQ/1.25331414D0,0.23498619D0,-0.3655620D-1,
» 0.1504268D-l,-0.780353D-2,0.325614D-2,-0.68245D-3/
C
DO 10 1=1,7
273
10
P(I) = DCMPLX(RPa))
QO) = DCMPLX(RQ(I))
IF ( CABSCX) LE. 2.0 ) THEN
Y=X»X/DCMPLX(4.0)
CALL IBESSI1(X,BESSI1)
ZZ=(CLOG(X/DCMPLX(2.0))*BESSI 1)+(DCMPLX( 1.0)/X)*(P( 1)+Y*(P(2)+
•
Y*(P(3)+Y*(P(4)+Y*(P(5)+Y*(P(6)+Y*P(7)))))))
ELSE
Y=DCMPLX(2.0)/X
ZZ=(CEXP(-X)/CSQRT(X))*(Q( I )+Y*(Q(2)+Y*(Q(3)+
*
Y*(Q(4)+Y*(Q(5)+Y*(Q(6)+Y*Q(7)))))))
ENDIF
RETURN
END
SUBROUTINE IBESSIO(X,ZZ)
IMPLICIT DOUBLE COMPLEX (A-H,0-Z)
REAL»8 RP,RQ,AX
DIMENSION RP(7),RQ(9),P(7),Q(9)
DATA RP/1.0D0,3.5156229D0,3.0899424D0,1.2067492D0,
• 0.2659732D0,0.360768D-l,0.45813D-2/
DATA RQ/0.39894228D0,0.1328592D-1,
• 0.225319D-2,-0.157565D-2,0.916281D-2,-0.2057706D-1,
» 0.26355370-1,-0.1647633D-l,0.392377D-2/
C
DO 10 1=1,7
10
P(I) = DCMPLX(RP(I))
DO 20 1=1,9
20
Q(I) = DCMPLX(RQ(I))
C
IF (CABS(X).LT.3.75) THEN
Y=(X/DCMPLX(3.75))**2
ZZ=P(1)+Y*(P(2)+Y*(P(3)+Y*(P(4)+Y*(P(5)+Y*CP(6)+Y*P(7))))))
ELSE
AX=CABS(X)
Y=DCMPLX( 3.75/AX )
ZZ=DCMPLX( EXP(AX)/SQRT(AX) )*(Q(I)+Y*(Q(2)+Y*(Q(3)+Y*(Q(4)
•
+Y*(Q(5)+Y*(Q(6)+YnQ(7)+Y»(Q(8)+Y*Q(9)))))))))
ENDIF
RETURN
END
274
SUBROUTINE IBESSIl(X,ZZ)
IMPLICIT DOUBLE COMPLEX (A-H,0-Z)
REAL*8 RP,RQ,AX
DIMENSION RP(7),RQ(9),P(7),Q(9)
DATA RP/0.5D0,0.87890594D0,0.51498869D0,
• 0.I5084934D0,0.2658733D-l,0.30I532D-2,0.3241 ID-3/
DATA RQ/
• 0.39894228D0, -0.3988024D-1,
» -0.362018D-2,0.163801D-2,-0.1031555D-1,0.2282967D-1,
• -0.2895312D-l,0.1787654D-1,-0.4200590-2/
C
DO 10 1=1,7
10
P(I) = DCMPLX(RP(I))
DO 20 1=1,9
20
QO) = DCMPLX(RQ(I))
C
IF (CABS(X).LT.3.75) THEN
Y=( X / DCMPLX(3.75) )**2
ZZ=X*(P( 1 )+Y*(P(2)+Y*(P(3)+Y*(P(4)+Y*(P(5)+Y*CP(6)+Y*P(7)))))))
ELSE
AX=CABS(X)
Y=DCMPLX( 3.75/AX )
ZZ=DCMPLX(DEXP(AX)/DSQRT(AX))*(Q( 1)+Y*(Q(2)+Y»(Q(3)+Y*(Q(4)+
•
Y*(Q(5)+Y*(Q(6)+Y*(Q(7)+Y*(Q(8)+Y*Q(9)))))))))
ENDIF
RETURN
END
C
C Beginning of the DeHoog et al. (1982) inversion function
C
double precision function inversion(conc_lap,time,max_time)
implicit none
integer max_lap,i
parameter (maxjap=15)
real pi, max_time, pO
parameter (pi=3.141592654)
real time
double complex record(max_lap), conc(max_lap),conc_lap(max_lap)
common /transform/conc
double complex temp_time, AM, BM
lis
p0=8.634694/max_time
do i=l, max_lap
conc(i)=conc_lap(i)
end do
call coeff_QD(record)
temp_time=dcmplx(0.0, pi*time/0.8/max_time)
teinp_time=exp(temp_time)
call ab(am,bm,record,temp_time)
if ( (am.eq.O).and.(bm.eq.O) ) then
inversion=0
else if (bm.eq.O) then
write(*,«)'ERRORS EXIST IN THE INVERSION CALCULATIONS'
stop INVERSION ERROR*
else
inversion= exp(p0*time)/0.8/max_time*Dreal(AM/BM)
end if
return
end
subroutine coefF_QD(record)
implicit none
integer maxlap
parameter (maxJap=15)
double complex record(max_lap)
integer count
integer i, j, trafic
common /trafic/trafic
double complex epsilon(max_lap, 2), Term
count=l
doj=l,2
do i=l,maxjap
epsilon(ij)=(0.0,0.0)
end do
end do
record(count)=Term(O)
do while (count.lt.max_lap)
call add(count, epsilon)
count=count+l
record(count)=-epsilon(1,1)
end do
return
end
subroutine ab(am,bm,record,z)
implicit none
integer max_lap, i
parameter (maxjap=15)
double complex record(max_lap)
double complex z, am, bm
double complex temp_l(maxjap), tenip_2(max_lap), h
temp_l (1)=record(1)
temp_1(2)=record(1)
temp_2(l HI-0,0.0)
temp_2(2)= I +record(2)*z
do i=3, max_lap
if (i.lt.maxjap) then
temp_l (i)=temp_ 1(i-1)+record(i)*temp_ 1 (i-2)*z
temp_2(i)=temp_2(i-1 )+record(i)*temp_2(i-2)*z
else if (i.eq.max lap) then
h=0.5-K).5*z*( record(i-l) - record(i) )
h=-h*( 1 - sqrt(l+z*record(i)/h**2))
temp_1(i)=temp_1(i-1)+h*temp_ 1(i-2)*z
temp_2(i)=temp_2(i-1)+h*temp_2(i-2)*z
end if
end do
am=temp_1(maxjap)
bm=temp_2(maxjap)
return
end
subroutine add(count, epsilon)
implicit none
integer maxlap
parameter (max_lap=l 5)
double complex epsilon(maxJap, 2)
integer count
integer trafic, i, switch
common /trafic/trafic
double complex Term
if (trafic.eq.1) then
epsilon(count,2)=Term(count)/Term(count-1)
switch=l
do i=count-1,1,-1
if (switch.gt.O) then
epsilon(i,2)=epsilon(i+1,1)+epsilon(i+1,2)-epsilon(i, I)
Ill
else if (switch.It.O) then
epsiIon(i,2)=epsilon(i+1,1)*epsiIon(i+1,2)/epsilon(i, 1)
end if
switch=-switch
end do
do i=l, count
epsilon(i, l)=epsilon(i,2)
end do
c
end if
return
end
double complex function Term(l)
implicit none
integer maxlap
parameter (maxjap= 15)
double complex conc(max_lap)
common /transform/conc
integer 1, trafic
conunon /trafic/trafic
if (l.eq.O) then
Term=
0.5*Conc(l)
else
Term=Conc(l+l)
end if
if ( abs(Term).lt.exp(-50.0)) then
trafic=0
else
trafic=l
end if
return
end
C
C
C
End of the inversion program
278
APPENDIX H: CODE FOR NUMERICAL DIFFERENTIATION OF
PNEUMATIC TEST DATA
Numerical differentiation of data can be accomplished either directly or indirectly.
Direct numerical differentiation is applied to data relatively free of noise that do not
require extensive pre-processing. Numerical differentiation, when performed indirectly,
involves pre-processing of data b'm(3Uits application. Data that are very noisy may be
first smoothed using spline fitting.
A number of researchers in the petroleum industry
have suggested various differentiation algorithms. Smoothing of data, however, needs to
be conducted with extreme caution, because the procedure tends to mask the slight
changes in pressure data that may be of interest to the analyst.
Researchers such as Blasingame et
al.
(1989) and Onur
et al.
(1989) have recently
proposed integrating pressure data prior to the approximation of derivatives. The main
advantage of employing this technique is that the integration procedure tends to smooth
data and makes type-curve matching easier. This technique, as in the case of spline fitting
schemes, may cause the loss of character of the original data. The loss of the character in
original data will make the subsequently performed pressure derivative analysis futile.
Bourdet et al.
(1989) suggested a method based on a weighted central difference
approximation to calculate the derivative at a reference point i. This algorithm, over the
last decade, has become the standard algorithm among petroleum engineers because of its
simplicity to implement and has been used to calculate numerical derivatives of pneumatic
pressure data colleaed at the ALRS. The algorithm takes the form:
279
dp _
axj
dx,
where AP,
= P, - P,_^ •,
APj =
P,,i- P. \
-
v^xj
(H-1)
(A^,+A^2)
AA',
= X, - X,_, ;
AAf, = A",,, - A",. Data before
(i-1) and after (i+1) the reference data point (i) is employed to calculate the numerical
derivative. The weighted mean of the two calculated derivatives (one between the point
before and the reference point; the other between the point after and the reference point)
are calculated.
One disadvantage in using the algorithm suggested by
Bourdet et al.
(1989) is that when consecutive points are put to use for the calculation, the derivative can
be very noisy and may be unsuitable for type-curve analysis. This dilemma can be usually
alleviated by choosing a larger interval length (L = AX1+AX2) between successive data
points.
The interval length cannot be increased blindly, when applied to selected
pneumatic data. Manual selection of interval length becomes a very important component
of the calculation and it is when the judgment of the analyst counts the most when
employing this algorithm. Noisy data will require a larger interval length if this algorithm
is employed.
Increasing the interval length blindly will increase truncation errors as
pointed out by Lane
et al.
(1991). The selection of the interval length is perhaps one of
the few uncertain aspects of this algorithm.
280
CSdebug
PROGRAM BOURDET
C
C
C
C
C
C
A program which reads values from a given file with two columns (time and
pressure). The program calculates the difference of every N pressure readings.
If the difference is greater than zero, then the difference is written on
an output file. The natural logarithm of the time is also calculated
simultaneously and written in an output file.
by Walter Illman
IMPLICIT REAL*8 (A-H, O-Z)
CHARACTER*12 INFILE, OUTFILE
DIMENSION T(1500), deIP(1500)
C
write( *,*) "Enter input filename(.txt):'
READ(*,'(A12y) INFILE
write( *,*) "Enter output filename(.out):"
READ(*,'(A12)") OUTFILE
write( *,*) "Enter interval length;"
READ •, NLEN
write( *,*) "Enter the number of rows;"
READ •, NROW
C
C
C
C
C
Number of rows must be specified accurately. If the number of
rows is not specified accurately""subscript out of range"" run-time
error will be displayed.
NUM = NROW
OPEN(UNIT= I ,FILE=INFILE,STATUS="OLD")
0PEN(UNIT=2,FILE=0UTFILE,STATUS="UNKNO\VN")
C creates input and output files
WRITE (2,2000)
2000 FORMAT (lOX, "t15X, "Int", 15X, "delP", I5X, "dP/dlnt")
numm=num
DO 10I0i=l, NUMm
READ (l.*,end=l 111) T(i), delP(i)
write(*,*) i, T(i), delP(i)
numi=i
1010 continue
1111 continue
281
num=numi
write(*,*)'— num = num
if(NUM-NLEN .LT. NLEN+1) stop
DO 1000 i=NLEN+l, NUM-NLEN
il=i-NLEN
i2=i
i3=i+NLEN
C
C
C
C
Do calculations in this loop; use an algorithm suggested by
Bourdet et al (1989) "Use of Pressure Derivative in Well-Test Interpretation"
til =0.0
ti2 = 0.0
ti3 = 0.0
Dint I =0.0
Dlnt2 = 0.0
Dlnt3 = 0.0
delPl =0.0
delP2 = 0.0
DERTV = 0.0
til =DLOG(T(il))
ti2 = DL0G(T(i2))
ti3 = DL0G(T(i3))
Dlntl = ti2 - til
Dlnt2 = ti3 - ti2
DInt3 = ti3 - ti 1
delPl = delP(i2) - delP(il)
delP2 = delP(i3) - delP(i2)
DERIVl =(deIPl/DlntI)»Dlnt2
DERIV2 = (deIP2/Dlnt2)*DlntI
DERIV = (DERTV1+DERIV2)/Dlnt3
IF (DERIV .GT. 0.0) then
WRITE (2,500) T(i2), ti2, DELP(i2), DERIV
WRITE (*,500) T(i2), ti2, DELP(i2), DERTV
ENDIF
1000 CONTINUE
500 FORMAT (IX, E15.6, 3X, E15.7, 3X, E15.7, 3X, E15.7)
STOP
END
282
APPENDIX I: MODIFICATION OF HSIEH AND NEUMAN (1985a)
SOLUTION TO ACCOUNT FOR STORAGE AND SKIN IN
MONITORING INTERVALS
In this appendix we modify the cross-hole solution of Hsieh and Neuman (1985a)
to account for storage and skin effects in monitoring intervals by following an approach
originally proposed by Black and Kipp (1977).
Mass balance in the monitoring interval can be expressed as
q^ = c„^
(i-i)
dt
where
is volumetric flow rate into the interval,
= Vjp^^ is storage coefficient
associated with an interval of volume V, and p^^ is pressure in the interval
Hvorslev
(1951) expresses volumetric flow rate into the observation interval
as
(1-2)
where F is a shape factor [L],
is permeability [L'] which we take to represent a skin,
and Ap = /; - pow where p is pressure in the rock outside the skin. Substituting (1-2) into
(I-l) and rearranging yields
^p =
^
(1-3)
Fk.
dt
This can be rewritten as
(1-4)
283
where tg = C^^lFk/\s basic time lag [7] as defined by Hvorslev.
The latter is a
characteristic constant of the monitoring interval, which reflects its response time (lag) to
changes in pressure within the rock. If tg is known, one can use (1-4) to calculate
pressure in the rock outside the skin, based on measurements of pressure within the
monitoring interval. Note that when the latter is stable, the two pressures are equal unless
(s is infinite due to an impermeable skin. The basic time lag can be determined by means
of a pressurized slug test in the monitoring interval according to a graphical method
proposed by Hvorslev (1951).
For known p, (1-4) is an ordinary differential equation in terms ofpow. In the case
where pow = /? at / = 0, its solution is
(1-5)
Now suppose that p is given by the solution of Hsieh and Neuman (1995a) described in
Chapter 5.
Dimensionless pressure and time are defined in Chapter 5 as
p^ = A7q}kRlOn and
^kfpj^fpP^ where R is distance between the centroids of the
injection and monitoring intervals. Introducing a new variable ^
l^kp-q then
transforms (1-5) into
(1-6)
where
u=
1/4/^ , Q = Ak^tg/Sf~pR^ is a dimensionless well response time,
= (f>lp is a
gas storage factor defined in equation 2.1-23, and Po(^) ^^ts as the kernel of a Fredholm
284
integral equation of the 2"** kind. For the case of point-injection/line-observation, the
solution oiHsieh and Neuman (1985a, p. 1658, eq. 27) reads
Pd = y J -exp[- (l - P:;)y\ • [erf[y""-(p, +1/;3,)] - erf[y"^{p, - l/A)]}'^'
^ »=i/4Jo y
(1-7)
Substituting this into (1-6) gives (5-46) and (5-47).
285
APPENDIX J: RECORD OF BOREHOLE TELEVISION (BHTV)
SURVEY FROM ALRS
A record of borehole television survey (BHTV) from boreholes at the ALRS is
provided to document discernible downhole features such as fractures, drill rings, cavities,
and mineralizations. A. Caster from COLOG (Golden, Colorado) surveyed the boreholes
under the supervision of E. Hardin from the University of Arizona. Depths are provided
in feet and inches with respect to the lower lip of each borehole casing (LL marker-see
Chapter 4) for consistency with other data available at the site. No attempt is made to
quantify geometrical properties (aperture width, orientation, etc.) of each fracture due to
the moderate to poor quality of borehole videos. In some descriptions, the orientation of
each fracture is given by the depth at which the 3 prongs (0, 120, and 240 wands) of the
survey camera intersect the fracture.
Borehole VI
Borehole length;
Date of investigation by COLOG;
Date of investigation by author;
Single-hole pneumatic test data; none
Video corrupted
Borehole V2
Borehole length; 92.7'
Date of investigation by COLOG; 9/12/89
Date of investigation by author; 1/24/96
Smgle-hole pneumatic test data; \-m nominal scale (Guzman et al., 1996)
Note; found steel tape at the bottom of the borehole
Downhole features
high angle fracture 7'7.5"
fracture 9'2.0"
286
smooth surface from 10' to 25'
high angle fracture top limb 25'2.5", bottom limb 26' 1.0"
high angle fracture top limb 26'4.5", bottom limb 27'5.0"
near horizontal fracture 28'5.25"
high angle fracture top limb 29'7.0", bottom limb 30'9.0"
high angle fracture top limb 34'3.0", bottom limb 35'9.25"
fracture intersecting third of the borehole at 37'10.25" (top limb?), 38'4.75" (bottom
limb?)
borehole intersect W2 at 44'8.0"
couple of hairline fractures at 62.5 to 63.0 ft.
hairline fracture at 78'
82' start of zone with numerous hairline fractures
relatively smooth surface from 85' - 92.7' bottom of borehole
Borehole V3
Borehole length; 100'
Date of investigation by COLOG; 9/12/89
Date of investigation by author: 1/24/96
Single-hole pneumatic test data; none
Downhole features
high angle fracture 1T
fracture 19'0.25"
fracture 34'
fracture 37'
fracture 39'7.5"
major horizontal fracture 40'2.75" through 40'1.25"
fracture 42.4.75"
fracture 43'1.5"
fracture 45'8"
fractures 46.5'
hairline fractures 59'7.5"
fracture 66'7.75"
fairly smooth surface 70' - 87'
Borehole W1
Borehole depth; 52'
Date of investigation by COLOG; 9/12/89
Date of investigation by author; 1/24/96
Single-hole pneumatic test data; none
Downhole features
fracture 10'7.25"
287
fracture 13'3.6"
fracture I5'3.25"
series of fractures: 1st 17'5.7", 2nd \T1.V\ 3rd ir9.35"
high angle hairline fracture 18'7.3"
smooth surface (no visible fractures, drill rings) 19' to 39'
fracture 0 deg. wand at 39'4.4" - fracture intersecting perpendicular to borehole
fracture 40'8.4"
fracture 120 wand 4r3.25"
fracture 5r2.0"
Borehole W2
Borehole length; 102'
Date of investigation by COLOG: 9/14/89
Date of investigation by author: 1/26/96
Single-hole pneumatic test data: none
Downhole Features
end of survey 6.0'
relatively smooth surface 6.0' - 16'
fracture at 16.39'
smooth surface below fracture to right above next fracture
fracture at 37.74'
fracture at 42.53'
fracture at 43.81'
loss of video from 49' to 58'
fracture at 59'5.2"
drill hole intersection at 68'7.3"
fracture at 72'5.25"
fracture at 80'2.25"
fracture at 93'0.4"
possible fracture at 95'4.6"
96' level a lot of mud on wall
at the bottom of the well - covered with mud with mudcracks - COLOG analyst thought
initially that bottom of borehole was being intersected with numerous large fractures
Borehole W2A
Borehole length: 98.9'
Date of investigation by author: 1/26/96
Date of investigation by COLOG: 7/15/93 (Black and white)
Single-hole pneumatic test data: \-m nominal scale (Guzman et al., 1996)
Downhole Features
7.1' casing
288
relatively smooth surface 7.1' through 12'
series of drilling rings/fractures - camiot tell from video
drill ring 12'
break in wall 28.3' - 28.8'
high angle fracture 52'
inclusions 64', 64.9'
fracture 79.6'
fracture 84'
bottom of hole 98.9'
Borehole W3
Borehole length; 149 to 150'
Date of investigation by COLOG; 9/14/89
Date of investigation by author; 1/26/96
Single-hole pneumatic test data; see Table 6.1 in this dissertation
Downhole Features
14ft end of tape
fracture at 19.9'
fracture at 22.36'
fracture with 0 wand 29.65', 120 wand 29.74', 240 wand 29.85'
fracture at 36.46'
fracture at 36.69'
fracture at 37'
fracture at 41.25'
note: measuring scale changes
smooth from 41' to 61'
fracture at 6r4.5"
large fracture at 63'4.3"
large open fracture at 63'8.5"
fracture at 64'0.6"
fracture at 64'2.3"
65' start of breakout zone - end of breakout zone 64'6.25"
fracture 65'6.1"
break in wall 66'1"
opening - break 78'7.8"
possible fracture 84'9.0"
fracture with 0 wand 93'2.64", 120 wand 93'5.2", 240 wand 93'1.5"
drilling ring at 108'
fracture at 112'6.35"
fracture at 0 wand 123'10.5", 120 wand 124.0', 240 wand 124'1.1"
drilling ring 139'0.7"
289
standing water I49'8.5" camera submerged in water!!
Borehole XI
Borehole length: 63' 6.0"
Date of investigation by COLOG; 9/12/89
Date of investigation by author: 1/24/96
Single-hole pneumatic test data: 3-/n nominal scale (Rasmussen et ai, 1990)
Downhole Features
7.0 ft. fracture
120 wand 8'6.77" 240 wand at same depth, 0 wands at 8'5.65"
240 wand at 8' 11.25", 0 and 120 wands at 8' 10.25"
240 wand at 10'6.25" 120 wand at 10'7.8", 0 deg wand at 10' 5.4"
all wands on same line 11' 7.1'
fracture at 14.0'
all wands on same line 15' 6.4"
120 wand 16' 2.9", 240 wand at 15'11.1", 0 deg wand??? 15' 7 1"
large fracture 120 degree wand 17'0.9", 0 deg wand 16' 10"
fracture 240 wand 17'6.2", 0 wand 17'7.3"
unknown feature (possibly a fracture) 20' 10.65"
fracture 2r7.0 ",2r4.5"
fracture 26'3.5"
drilling ring at 26.5'
drilling rings at 27' - 28'
fracture 32'6.5'to 32'5.7"
high angle fracture 35'2.5" - 34'11.4"
high angle fracture 32'3.7" - 31 '9.75"
fracture 36'5.1" to 36'8.2"
drilling rings at 38'
fracture 39'8.25"
drilling rings (smeared) 43.5'
another series of drilling rings 48'
hairline fracture 50'3.25"
drilling ring? 53'
drilling ring 58'
fracture 120 wand 60', 0 and 240 wands 59' 11"
Borehole X2
Borehole length: 100'
Date of investigation by COLOG: 9/13/89
Date of investigation by author: 1/25/96
Single-hole pneumatic test data: 3-/n nominal scale {Rasmussen et al., 1990); \-m and 2-m
nominal scale (Guzman et al., 1996)
290
Downhole Features
large fracture at 7.0'
fracture at 7'5.0"
fractures at 8'7.3"
fractures at 9'0.8"
several fractures at 12'4.9"
break out zone - large fractures 13' - - 120 wand 12' 10", 240 wand 18'5.0"
hairline fractures 14'
possible hariline fracture 17'1.9"
fracture with 0 wand 18'1.9", 120 wand 18'8.6", 240 wand 18'5.0"
mineralized zone at 22'
fracture at 33'
hairline fracture at 37'6.8"
series of drilling rings 41'
unknown feature 46'2.0"
unknown feature 46'7.75"
fracture? series of drilling rings 47'
fracture 50.0'
unknown feature 52'10.5"
small fracture 55'8.0"
opening (cavity) 55' 11.0"
hairline fracture 57'
hairline fracture 58'3.0"
hairline fracture 58'8.8"
two possible hairline fractures near 61' level
fracture with 0 wand 63'6.6", 120 wand 62'7.75", 240 wand 63' 1.8"
drilling ring at 64' level
possible fracture at 66'2.25"
small fracture at 66'9.1"
medium size fracture with 240 wand 75' 11.25"
two small fractures at 76'4.5"
fracture at 77'2.0"
relatively rough surface 77' - 92'
series of drilling rings between 91' and 92'
Borehole X3
Borehole length: 150'
Date of investigation by COLOG; 9/14/89
Date of investigation by author: 1/26/96
Single-hole pneumatic test data: 3-/w nominal scale {Rasmussen et al., 1990)
291
Downhole Features
fractures at 5.78'
fracture at 6.87'
fi:acture at 7.71'
hairline fractures at 8'
high angle fracture at 15.46'
drilling rings at 16'
fracture at 16.55'
fracture at 22.2'
possible fracture at 23.7'
possible hairline fracture at 24'
mineralized zone at 25' bands of different mineralogy
possible hairline fracture at 26'
hairline fracture at 28' several in this area
fracture at 29.2' Owand 29.37', 120 wand 28.84'
possible fracture at 35.37'
drilling ring at 36' series of 6 to 7
fracture at 37.38'
fracture at 38.12'
drilling ring at 42'
fracture at 43.2'
fracture with 0 wand 44', 120 wand 44.54', 240 wand 44.07'
fracture with 240 wand at 52'0.25"
drilling ring at 55'
fracture Avith 0 wand 60'8.75", 120 wand 61' 1.6", 240 wand 61'!"
hairline fractures just passed below 62' level
possible drilling rings? above 68'
fracture with 0 wand 69'4.7", 120 wand 69'6.0", 240 wand 69'
fracture at 84'5.75"
drilling ring at 87'
large fracture 105'5.8" part of last fracture set - multiple and intersect large opening at
105'0.2"
possible fracture at 111 '3.0"
fracture at 114'0.2"
fracture at 114'3.0"
fracture at 115'7.0" (could be part of fracture at 117'0.5")
fracture at 117'7.5"
hairline fracture at 125'9.5"
drilling ring at 130'
fracture at 138'1.8"
small fractures at 142' (intersecting each other on borehole face)
fracture at 143' 11.0"
fracture at 147'5.6"
292
bottom 150'
Borehole Y1
Borehole length: 56.5'
Date of investigation by COLOG: 9/13/89
Date of investigation by author: 1/25/96
Single-hole pneumatic test data: 3-/ii nominal scale {Rasmussen et al., 1990)
Downhole Features
casing 4.5'
fracture at 7'3.75"
two small hairline fractures at 10'3.2"
fracture at higher angle 0 wand 10'11.4" 120 wand 10'8.3" 240 wand 10'9.5"
fracture all wands at same level H'1.5"
fracture will all wands at the same level 14'3.5"
hairline fracture 15'2.5"
filled fracture' 18'6.7"
fracture 0 wand 20'7.75", 120 wand 20'10.75", 240 wand 20'8.4"
unknown feature at 24'2.0" (possibly a fracture?)
hairline fracture 28'7.3"
hairline fracture 29' 1.0"
drilling ring 30.0'
drilling ring 35.0'
drilling ring 45.0'
fractureOwand 48'5.6", 120 wand 48'10.1", 240 wand 48'9.1"
relatively smooth borehole surface from 49' to the bottom of the hole 56'6"
Borehole Y2
Borehole length: 100.5'
Date of investigation by COLOG: 7/15/93
Date of investigation by author: 1/24/96
Single-hole pneumatic test data: 3-m nominal scale {Rasmussen et ai, 1990); 0.5, 1, 2,
and 3-m nominal scale; 20-m nominal scale {Illman et al., 1998)
casing depth: 4.7'
Downhole Features
rough surface with some fractures 6-10', 10' some fractures filled with some minerals
(silica?)
11-15' cavities with minerals, fracture at 12.2' (horizontal - high angle in reality because
borehole is slanted 45 deg.)
18.2' filled fracture
21-30' no major feature: relatively smooth surface
293
30.8' series of small near horizontal fractures
31-35' relatively smooth
42.7' single, near horizontal fracture
46-50' relatively smooth borehole surface
52.4' massive cavitation (high k zone - undetermined number of fractures) 52.4 - 52.9'
smooth surface 53-55'
seris of fractures 58.5'
high angle fracture 58.8'
high angle fracture 64.1'
near horizontal fracture 71'
relatively smooth surface with some hairline fractures 76 - 80'
relatively smooth surface 81-85'
sealed high angle fracture 92.5 - 93'
horizontal and near vertical fractures 96 - 100' zone
moderate angle fracture 100'
Borehole Y3
Borehole length; 150'
Date of investigation by COLOG: 9/13/89
Date of investigation by author: 1/25/96
Single-hole pneumatic test data: 3-m nominal scale {Rasmussen et al., 1990)
Downhole Features
fracture at 8.59'
fracture with 120 wand 9.13', 240 wand 9.44'
fracture at 10.8'
zone of several fractures at 14.58'
several fractures at 16.03'
fracture at 16.47'
fracture at 18.4'
hairline fracture at 21.43'
fracture at 23.41'
minral bands at 24' layer of tuff with different lithology
unknown feature at 28' (drilling ring or fracture)
possible fracture at 28.45'
fracture at 31.44'
fracture at 33.4'
fracture at 34.49'
fracture at 38.45'
cavity at 39.20' - break in wall: lithologic weathering? vug? fracture 40.5'
(measurements above this level are in feet only - no inches)
fracture at 43'7.5"
break in wall (cavitation) 47'4.25"
294
break in fracture wall?? 50'1.2"
fracture at 52'5.0"
fracture with 0 wand 53'3.8", 120 wand? 53'2.25", 240 wand 52'9.50"
fracture at 52'5.0"
fracture with 0 wand 57'9.3", 120 wand 57'7.5", 240 wand 57'4.0"
fracture at 60'
fracture at 0 wand 65'6.8". 120 wand 65'9.75". 240 wand 65'3.5"
fracture with 120 wand 67'5.2", 240 wand 66'7.0" - 2nd fracture with 0 wand 67'3.5",
120 wand 67'
fracture with 240 wand at 77' with lower limit of fracture at 76'8.5"
fracture with 0 wand 78'8.8", 120 wand 78'6.5", 240 wand 78' 1.0"
fracture with 0 wand 81'!.0", 120 wand 81240 wand 80' 1.0"
possible fracture at 86' 10.0"
possible fracture at 104'8.25"
drilling ring at 109'
drilling ring at 119'2.6"
possible fracture at 129'
due to twisted tape all depths should be 7.0ft shallower - depths adjusted already for this
document
Borehole Z1
Borehole length; 55.0'
Date of investigation by COLOG: 9/13/89
Date of investigation by author; 1/26/96
Single-hole pneumatic test data; 3-m nominal scale (Rasmussen et al., 1990)
Downhole Features
fracture with 240 wand 7'6.5"
fracture with 240 wand 8'8.25"
cavitation at 11 '3.5" (break out zone)
several fractures near surface of borehole 5' - 10'
hairline fractures 12'
high angle fracture 12'8.7"
hairline fractures 14' - 15'
fracture with 0 wand 16'9.5", 120 wand 17'0.0", 240 wand 17'0.25"
several hairline fractures 17'3.3"
possible fracture 18'3.25"
cavity? at 21'
fracture with 240 wand 25' 1.9"
fracture at 25'5.9"
possible fracture at 28* 1.5"
possible fracture at 28'5.3"
295
fracture with 240 wand 30'2.0"
fiacture with 0 wand 31*1.5", 120 and 240 wands at 31 '4.2"
fracture with 120 wand 31'6.5", 240 wand 31 '8.2" - note another fracture - total of 3?
borehole breakoff at 32'
fracture with 0 wand 32'2.0"
small fracture 34'
fracture with 0 wand 34'6.75", 120 wand 34*10.25", 240 wand 34*8.6"
fracture with 0 wand 36*2.0*', 120 wand 36*6.0*', 240 wand 36'3.25"
hairline fracture at 37*5.25**
hairline fracture at 39*0.5"
fracture with 0 wand? 40*4.9**, 120 and 240 wands at 40*1.0"
possible fracture at 40*5.5**
fracture with 240 wand 41*
high angle fracture at 41*7.5*'
fracture with 120 wand 42*0.5", 240 wand 41*10.6"
small fracture or a small cavity? at 42*7.3*'
small fracture at 45*0.5**
fracture with 0 wand 48*7.35", 120 wand 48*2.75", 240 wand48'5.1"
fracture with 120 wand 50*3.7**
fracture at 51*10.6"
fracture with 0 wand 54*1.5", 120 wand 54'3.2", 240 wand 54'5.6"
Borehole Z2
Borehole length; 100.0*
Date of investigation by COLOG: 9/13/96; also 7/15/93 (black and white)
Date of investigation by author; 1/25/96
Single-hole pneumatic test data; 3-/n nominal scale {Rasmussen et al., 1990); \-m nominal
scale (Guzman et al.., 1996)
Downhole Features
fracture at 20*3.75**
hairline fractures above 26* level
possible fracture at 27* 1.0**
fracture at 30*6.8**
possible hairline fractures at 32*
fracture with 0 wand 39*, 120 wand 39*2.5**, 240 wand 39*4.3**
hairline fracture at 40*
possible two drilling rings slightly above the 41* level
fracture at 41*6.2**
fracture zone at roughly 42.5* to 45' (approximately 20 fractures - counted)
uneven (ragged) fracture zone 44*5.4" several intersecting fractures ~ end of zone at 44*
- note high angle fracture
296
intersecting fractures wand pos. 0 wand 47'5.7", 120 wand 47'4.25", 240 wand
47'7.25"
large fracture 0 wand 47'9.8", 120 wand 47'10.25", 240 wand 47'7.25"
fracture 48'9.5" (horizontal - with respect to borehole)
fracture with 0 wand?? 48'2.3", 120 wand 48'5.7", 240 wand 48'3.25"
fracture at 50*6.35"
fracture at 51'
fracture at 5r4.5"
fracture with 0 wand 51*8.5", 120 wand 5 I'll.2", 240 wand 51*9.5" - same level with
120 wand of second fracture
hairline fracture 52*8.25**
hairline fracture 54*3.0**
hairline fracture 55*6.5**
hairline fracture 56*0.85**
hairline fracture 59*2.79**
at 60*8.8** level several fractures comprise a broken (cavity/cavernous) zone
fracture with 0 and 240 wands 62*6.25*' and 120 wand 62*7.75"
hairline fracture 63*7.25**
hairline fracture 65*7.7**
drilling rings at 71*
top of intersecting fracture at 73*5.75** - large fracture with an estimated aperture of
quarter of an inch intersecting another fracture
large fracture 0 wand 74*1.5**, 120 wand 74*4.75**, 240 wand 74*0.6**
at 75' level a possible third fracture
fracture with 120 wand 75'5.0", 240 wand 75'4.25"
drilling ring at 81'
hairline fracture at 86*
hairline fracture at 91 * 1.7* *
large fracture with 0 wand 91 * 10.25**, 120 wand 91* 11.25**, 240 wand 92*2.0**
fracture with 0 wand 95*1.5**, 120 wand 95*0.0**, 240 wand ???
fracture with 0 wand ?? 97'1.1", 120 wand 97'4.0", 240 wand 97*5.5**
Borehole Z3
Borehole length: 149.0*
Date of investigation by COLOG: 9/14/89
Date of investigation by author: 1/26/96
Single-hole pneumatic test data: 3-/w nominal scale (Rasmussen et al., 1990)
Downhole Features
drilling ring at 6*
fracture at 6.8*
drilling ring at 8*
fracture with 0 wand 31.9*, 120 wand 31.71', 240 wand 31.52'
297
fi^cture 0 wand 36.75', 120 wand 36.8', 240 wand 36.38'
fracture at 38.73'
drilling ring at 39'
drilling ring right below 44'
fracture at 51'5.0"
fracture and breakout zone (from Y2?) 55'6.0"
drilling ring 56'
drilling ring and fracture 64'3"
fracture at 72'8"
fracture (lower) with 120 wand 77'7.0" upper fracture 120 wand 77'3.75"
drilling ring at 79.5'
fracture at 0 wand and 240 wand approx. same spot 81*9 I" 120 wand 81 '5.75"
fracture with 0 wand 85'2.6", 120 wand 84'H", 240 wand 84'7.25" fracture intersecting
drilling ring (smearing?)
hairline fracture 86'3.75'
hairline fracture 86'10.1"
large fracture 0 wand 9r2.5", 120 wand 9r4", 240 wand 90'8.9"
possible fracture at 93' ( possible mineral band or a filled fracture)
drilling ring at 95'
fracture with 0 wand 99'1.25", 120 wand 99', 240 wand 98'7.5"
drilling ring at 100'
drilling ring at 105'
fracture intersects drilling ring at 110'3.0" with drilling ring at 110'
drilling ring at 115'
drilling ring at 120'
drilling ring at 125'
fracture with 0 wand I26'4.5", 120 wand 126'3.r', 240 wand 126'0.0"
drilling ring at 130'
fracture with 0 wand and 240 wands at I33'5" 120 wand I33'3.25"
drilling ring at 135'
drilling ring at 140'
drilling ring at 145'
fracture at 147'6.7"
298
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