NUMERICAL INVERSE INTERPRETATION OF PNEUMATIC TESTS IN UNSATURATED FRACTURED

NUMERICAL INVERSE INTERPRETATION OF PNEUMATIC TESTS IN UNSATURATED FRACTURED
NUMERICAL INVERSE INTERPRETATION OF
PNEUMATIC TESTS IN UNSATURATED FRACTURED
TUFFS AT THE APACHE LEAP RESEARCH SITE
by
Velimir Valentinov Vesselinov
Copyright
C) Velimir Valentinov Vesselinov
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
200 0
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
Velimir Valentinov Vesselinov
Numerical inverse interpretation of pneumatic
tests
in
unsaturated fractured tuffs at the Apache Leap Research Site
and recommend that it be accepted as fulfilling the dissertation
requirement for the Degree of
Doctor of Philosophy
A
Teum
d
Peter J Wierenp.
Arthur W. Warrick
Donald E.
Date
Myers
Date
Final approval 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
requireme t
1
Dissertation Director Shlomo P. Neuman
Date
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.
Brief quotations from this dissertation are allowable without special permission,
provided that accurate acknowledgment 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
This dissertation would have been impossible without the support of many. First
and foremost, my advisor Dr. Shlomo P. Neuman. I am indebted for his guidance,
patience, and support. I acknowledge his invaluable contributions to the accomplished
research. He was always available to answer my questions and discuss everything from
science to politics. He was the best mentor I have ever had and helped me a lot to
develop myself as a researcher and person.
I am thankful to everybody who contributed to the research of Apache Leap
Research Site through the years, especially to Amado Guzman, Walter A. Illman,
Dick Thompson and Guoliang Chen who conducted most of pneumatic tests I have
analyzed. I am also thankful to the support and guidance of our Project Manager,
Mr. Thomas J. Nicholson, United States Nuclear Regulatory Commission.
I would also like to thank Drs. Thomas Maddock III, Donald E. Myers, Arthur
W. Warrick, and Peter J. Wierenga for serving on my dissertation committee, and
Dr. Jesus Carrera for the beneficial comments and discussions on my research.
A substantial part of my research would have been impossible without the support of High-Speed Computing Committee and Computer Operations Group at the
Center for Computing and Information Technology, The University of Arizona. I run
my inverse model in parallel using the full capabilities of our supercomputer, which
extended significantly my analysis.
Part of the simulation and inverse modeling work was conducted during a summer
internship with the Geoanalysis Group at Los Alamos National Laboratory. I am
grateful to Dr. George A. Zyvoloski for his help in the implementation of FEHM,
and to Dr. Carl W. Gable for his assistance in the use of X3D.
Finally, I am thankful for the continued support of my family and friends.
This research was financially supported by the United States Nuclear Regulatory
Commission under contracts NRC-04-95-038 and NRC-04-97-056.
5
To Rumi and Neda.
6
TABLE OF CONTENTS
LIST OF FIGURES
8
LIST OF TABLES 14
ABSTRACT 16
CHAPTER
1. INTRODUCTION CHAPTER 2. APACHE LEAP RESEARCH SITE 2.1. Site description
2.2. Previous investigations
2.2.1. Laboratory and field investigations
2.2.2. Single-hole pneumatic tests
2.2.3. Cross-hole pneumatic tests
2.3. Geostatistical analysis of spatial variability
2.3.1. Air permeability
2.3.2. Fracture density, matrix porosity, water content and
van Genuchten a 2.3.3. Conditional simulations
2.4. Conceptualization of hydrogeologic conditions during
pneumatic tests
18
CHAPTER 3. NUMERICAL INVERSE METHODOLOGY 3.1. Background
3.1.1. Governing equations
3.1.2. Inverse problem
3.2. Stochastic numerical inverse model
3.2.1. Pilot point formulation
3.2.2. Maximum likelihood approach 3.2.3. Optimization
3.2.4. Linearized analysis of estimation errors
3.2.5. Calculation of sensitivity matrix
3.2.6. Model identification criteria
3.3. Numerical inverse model for ALRS 3.3.1. Computational domain
3.3.2. Boundary and initial conditions
3.3.3. Borehole effects
3.3.4. Computational grid
26
26
31
31
33
39
41
41
54
54
68
71
71
71
74
78
79
81
83
88
91
92
93
93
93
95
96
7
TABLE OF CONTENTS—Continued
3.3.5. Program FEHM 3.3.6. Parameterization 3.3.7. Program PEST 3.3.8. Parallel processing 3.3.9. Code interaction 3.4. Linearization of governing equations 105
108
109
112
113
113
INTERPRETATION OF SINGLE-HOLE TESTS . . . . 115
4.1. Single-hole test JG0921 118
4.2. Single-hole test JGC0609 125
4.3. Single-hole test JHB0612 132
4.4. Single-hole test JJA0616 138
4.5. Discussion 145
CHAPTER 4.
CHAPTER 5. INTERPRETATION OF CROSS-HOLE TESTS 146
5.1. Qualitative analysis 147
5.2. Inverse analysis of cross-hole tests 158
5.2.1. Uniform medium 158
5.2.2. Non-uniform medium 176
5.3. Discussion 210
5.3.1. Comparison between numerical and analytical analyses .
210
5.3.2. Comparison between present stochastic inverse model and previous applications of pilot-point method 211
5.3.3. Comparison between different models 5.3.4. Scale effects 5.3.5. Comparison between spatial patterns of permeability obtained
by various methods 5.3.6. Comparison between permeability and porosity estimates .
5.3.7. Borehole effects CHAPTER 6.
REFERENCES CONCLUSIONS 212
212
218
222
222
229
236
8
LIST OF FIGURES
FIGURE
2.1.
Location map of Apache Leap Research Site [after Guzman
et al., 1996])
27
Plan view of tested boreholes and plastic cover
FIGURE 2.2.
29
FIGURE 2.3.
Three-dimensional perspective of boreholes at the site
30
FIGURE 2.4.
Three-dimensional perspective showing center locations of 3-m
test intervals employed by Rasmussen et al. [1990]
32
FIGURE 2.5. 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.
34
FIGURE 2.6.
Three-dimensional perspective showing center locations of neutron probe measurements performed by Thompson et al. [1995] 35
Scatter plot of air permeability versus fracture density [data
FIGURE 2.7.
from Rasmussen et al., 1990]
37
FIGURE 2.8.
Three-dimensional perspective showing center locations of 1-m
single-hole test intervals employed by Guzman et al. [1996]
38
FIGURE 2.9.
Omni-directional sample and model variograms for various parameters at a minimum separation distance of 3 m 43
FIGURE 2.10. Kriged estimates and kriging variances of log io k at y = 7 m
using a power model (left), and an exponential model with second order
drift (right)
44
FIGURE 2.11. Variation of air permeability with depth and support scale
(data from Guzman et al. [1996] and Rasmussen et al. [1990])
46
FIGURE 2.12. Histograms of log io k 1-m and 3-m scale data (data from Guzman et al. [1996] and Rasmussen et al. [1990])
47
FIGURE 2.13. Histograms of log io k at various scales along borehole Y2 (data
from Guzman et al. [1996])
49
FIGURE 2.14. Omni-directional sample variograms for log lo k with various
supports and a power variogram model fitted to 1-m data 50
FIGURE 2.15. Three-dimensional perspective showing center locations of singlehole test intervals for combined set of 1-m (circles) and 3-m (squares)
51
scale measurements FIGURE 2.16. Kriged log lo k estimates obtained using 1-m scale data from
boreholes X2, Y2, Y3, Z2, V2 and W2A (left) and same together with 3-m scale data from boreholes Xl, X3, Yl, Z1 and Z3 (right)
53
FIGURE 2.17. Kriged estimates of various parameters at y = 7 m 57
FIGURE 2.18. Three-dimensional representation of kriged log lo k [m 2 ].
58
FIGURE 2.19. Kriged log io k [m 2 ] along various y-z planes
59
60
FIGURE 2.20. Kriged log io k [m 2 ] along various x-z planes
9
LIST OF FIGURES—Continued
FIGURE 2.21. Kriged log io k [m 2 ] along various x-y planes
FIGURE 2.22. Three-dimensional representation of kriged fracture
61
density
{coun ts /m].
62
2.23. Three-dimensional representation of kriged matrix porosity
63
log io 0,, [m 3 /m 3 ]
FIGURE 2.24. Three-dimensional representation of kriged water content 8
[m 3 /m 3 ] 64
FIGURE 2.25. Three-dimensional representation of kriged van Genuchten's
log io a [k P al 65
FIGURE 2.26. Three-dimensional representation of estimated air-filled matrix
porosity log io çb,,,, [m 3 /m 3 ]
66
FIGURE 2.27. Random conditional realizations of various parameters at y = 7
m
67
FIGURE
Computational domain and explicitly simulated boreholes. . .
Side views of computational grid for injection borehole Y2. . .
Vertical cross-sections through computational grid for injection
borehole Y2 FIGURE 3.4.
Three-dimensional representation of computational along injection borehole Example of log io k distribution throughout computational doFIGURE 3.5.
main. Example of log lo 0 distribution throughout computational doFIGURE 3.6.
main. FIGURE 3.1.
FIGURE 3.2.
FIGURE 3.3.
94
97
98
100
102
103
4.1.
Pressure data from first injection step of test JG0921 interpreted by various inverse models 121
FIGURE 4.2. Relative change of borehole storage effects due to variation of
air compressibility pressure versus relative time for all injection steps
122
and recovery of test JG0921
FIGURE 4.3.
Relative pressure versus relative time for all injection steps and
recovery of test JG0921 123
FIGURE 4.4.
Pressure data from for all injection steps and recovery of test
124
JG0921 interpreted by inverse model. Pressure data from first injection step of test JGC0609 interFIGURE 4.5.
127
preted by various inverse models FIGURE 4.6. Relative change of borehole storage effects due to variation of
air compressibility pressure versus relative time for all injection steps
128
and recovery of test JGC0609.
Relative pressure versus relative time for all injection steps and
FIGURE 4.7.
129
recovery of test JGC0609 FIGURE
10
LIST OF FIGURES—Continued
Pressure data from for all injection steps and recovery of test
4.8.
JGC0609 interpreted by inverse model. 130
Pressure data from first injection step of test JHB0612 interFIGURE 4.9.
preted by various inverse models
134
FIGURE 4.10. Relative change of borehole storage effects due to variation of
air compressibility pressure versus relative time for all injection steps
and recovery of test J11B0612.
135
FIGURE 4.11.
Relative pressure versus relative time for all injection steps and
recovery of test JHB0612 136
FIGURE 4.12. Pressure data from for all injection steps and recovery of test
JHB0612 interpreted by inverse model. 137
FIGURE 4.13. Pressure data from first injection step of test JJA0616 inter
preted by various inverse models
141
FIGURE 4.14. Relative change of borehole storage effects due to variation of
air compressibility pressure versus relative time for all injection steps
142
and recovery of test JJA0616
Relative pressure versus relative time for all injection steps and
FIGURE 4.15.
143
recovery of test JJA0616. FIGURE 4.16. Pressure data from for all injection steps and recovery of test
144
JJA0616 interpreted by inverse model FIGURE
FIGURE 5.1.
Location of monitoring intervals and packers along tested bore-
148
holes. Log-log plots of observed pressure buildups (dots; kP a) versus
5.2.
150
time (days) and match points (circles) during test PP4. kP
Log-log
plots
of
observed
pressure
buildups
(dots;
a)
versus
FIGURE 5.3.
151
time (days) and match points (circles) during test PP5. Log-log plots of observed pressure buildups (dots; kP a) versus
FIGURE 5.4.
152
time (days) and match points (circles) during test PP6. Log-log plots of observed pressure buildups (dots; kP a) versus
FIGURE 5.5.
153
time (days) and match points (circles) during test PP7. Log-log plots of observed pressure buildups (dots; kP a) versus
FIGURE 5.6.
154
time (days) and match points (circles) during test PP8. Measured variations in barometric pressure and pressure reFIGURE 5.7.
155
sponses in some of intervals during test PP4. test
PP4.
156
temperature
measurements
collected
during
Some air
FIGURE 5.8.
Separate matches between observed (small dots) and simulated
FIGURE 5.9.
(thick curves) responses for test PP4 assuming uniform medium. . . . 160
FIGURE 5.10. Separate matches between observed (small dots) and simulated
(thick curves) responses for test PP5 assuming uniform medium. . . . 161
FIGURE
11
LIST OF FIGURES—Continued
FIGURE 5.11. Separate matches between observed (small dots) and simulated
(thick curves) responses for test PP6 assuming uniform medium. . . . 162
FIGURE 5.12. Separate matches between observed (small dots) and simulated
(thick curves) responses for test PP7 assuming uniform medium. . . 163
FIGURE 5.13. Separate matches between observed (small dots) and simulated
(thick curves) responses for test PP8 assuming uniform medium. . . . 164
FIGURE 5.14. Separate matches between observed (small dots) and simulated
(thick curves) responses for test PP4 assuming uniform medium and
adjusting borehole storage parameters along injection and observation
intervals. 171
FIGURE 5.15. Estimates of pneumatic properties for test PP4 obtained with
and without borehole effects. 172
FIGURE 5.16. Estimates of pneumatic properties for test PP4 obtained for
different initial barometric pressure. 174
FIGURE 5.17. Analytically versus numerically derived pneumatic properties
for test PP4. 175
FIGURE 5.18. Three-dimensional representation of 32 pilot points. 178
FIGURE 5.19. Matches between observed (small dots) and simulated (thick
curves) responses obtained by simultaneous inversion of PP4 data with
32 pilot points; log io q is assumed to be uniform 180
FIGURE 5.20. Three-dimensional representation of kriged log io k estimated
by simultaneous inversion of PP4 data with 32 pilot points; log io is
assumed to be uniform 181
FIGURE 5.21. Matches between observed (small dots) and simulated (thick
curves) responses obtained by simultaneous inversion of PP4 data with
32 pilot points 182
FIGURE 5.22. Histogram of residuals between observed and simulated responses 184
FIGURE 5.23. Two-dimensional representation of covariance matrix of estimation errors at the pilot points 187
FIGURE 5.24. Two-dimensional representation of correlation matrix of estimation errors at the pilot points 188
FIGURE 5.25. Absolute components of the estimation covariance matrix versus distance between the pilot points 189
FIGURE 5.26. Absolute components of the estimation correlation matrix ver189
sus distance between the pilot points FIGURE 5.27. Two-dimensional representation of eigenvectors of covariance
191
matrix of estimation errors at the pilot points. 12
LIST OF FIGURES—Continued
FIGURE
5.28. Maximum absolute eigenvector components associated with pa-
rameter estimates FIGURE 5.29. Three-dimensional representation of kriged log o k estimated
by simultaneous inversion of PP4 data with 32 pilot points. FIGURE 5.30. Three-dimensional representation of kriged log o q5 estimated
by simultaneous inversion of PP4 data with 32 pilot points. FIGURE 5.31. Sample variograms of kriged log o k and log o q5 estimates obtained by simultaneous inversion of PP4 data with 32 pilot points. . .
FIGURE 5.32. Forward simulation of test PP5 applying parameter fields estimated by simultaneous inversion of PP4 data with 32 pilot points. . .
FIGURE 5.33. Forward simulation of test PP6 applying parameter fields estimated by simultaneous inversion of PP4 data with 32 pilot points. . .
FIGURE 5.34. Forward simulation of test PP7 applying parameter fields estimated by simultaneous inversion of PP4 data with 32 pilot points. . .
FIGURE 5.35. Matches between observed (small dots) and simulated (thick
curves) responses obtained by simultaneous inversion of PP4 data with
64 pilot points FIGURE 5.36. Three-dimensional representation of 72 pilot points. FIGURE 5.37. Matches between observed (small dots) and simulated (thick
curves) responses for test PP4 obtained by simultaneous inversion of
PP4, PP5 and PP6 data.
FIGURE 5.38. Matches between observed (small dots) and simulated (thick
curves) responses for test PP5 obtained by simultaneous inversion of
PP4, PP5 and PP6 data.
FIGURE 5.39. Matches between observed (small dots) and simulated (thick
curves) responses for test PP6 obtained by simultaneous inversion of
PP4, PP5 and PP6 data.
FIGURE 5.40. Three-dimensional representation of kriged log o k estimated
by simultaneous inversion of PP4, PP5 and PP6 data. FIGURE 5.41. Three-dimensional representation of kriged log o 0 estimated
by simultaneous inversion of PP4, PP5 and PP6 data. FIGURE 5.42. Histograms of air permeability data obtained using various
approaches FIGURE 5.43. Histograms of air-filled porosity data obtained using various
approaches FIGURE 5.44. Air permeability along boreholes estimated from single-hole
tests and by simultaneous inversion of PP4, PP5 and PP6 data FIGURE 5.45. Sample variograms of log o k estimated from single-hole data
and through stochastic inversion of cross-hole data 192
193
194
195
197
198
199
200
202
205
206
207
208
209
216
217
220
221
13
LIST OF FIGURES—Continued
FIGURE 5.46.
Air permeability versus air-filled porosity estimated by stochastic inversion of PP4 data at pilot points (open big circles) and within
the computational domain 223
FIGURE 5.47. Simulated responses with (solid curve) and without (dashed
curve) borehole effects for cross-hole test PP4 in log-log plots 225
FIGURE 5.48. Simulated responses with (solid curve) and without (dashed
curve) borehole effects for cross-hole test PP4 in semi-log plots 226
FIGURE 5.49. Simulated air pressure [MPa] in computational region at the
end of cross-hole test PP4 with borehole effect 227
FIGURE 5.50. Simulated air pressure [MPa] in computational region at the
end of cross-hole test PP4 with borehole effect
228
14
LIST OF TABLES
2.1.
TABLE 2.2.
TABLE 2.3.
Borehole coordinates and length.
28
Summary statistics of the geostatistically analyzed data. . . . 46
Summary statistics for log io k [772, 2 ] from single-hole tests at various scales along borehole Y2
48
TABLE 2.4.
Results from Kolmogorov-Smirnov test of Gaussianity
56
TABLE
Single-hole pneumatic tests analyzed by inverse modeling [after
TABLE 4.1.
Guzman et al., 1996] 116
Single-hole test data analyzed by inverse modeling [after GuzTABLE 4.2.
man
et al., 1996]
116
119
TABLE 4.3.
Parameter estimates for test JG0921. Eigenanalysis of covariance matrix of estimation errors obtained
TABLE 4.4.
from interpretation of the first step of test JG0921 120
Eigenanalysis of covariance matrix of estimation errors obtained
TABLE 4.5.
from interpretation of all the injection steps and recovery of test JG0921. 120
126
Parameter estimates for test JGC0609. TABLE 4.6.
Eigenanalysis of covariance matrix of estimation errors obtained
TABLE 4.7.
131
from interpretation of the first step of test JGC0609 Eigenanalysis of covariance matrix of estimation errors obtained
TABLE 4.8.
from interpretation of all the injection steps and recovery of test JGC0609. 131
133
TABLE 4.9. Parameter estimates for test JHB0612.
TABLE 4.10. Eigenanalysis of covariance matrix of estimation errors obtained
from interpretation of all the injection steps and recovery of test JHB0612. 133
TABLE 4.11. Parameter estimates for test JJA0616
140
TABLE 4.12. Eigenanalysis of covariance matrix of estimation errors obtained
140
from interpretation of the first step of test JJA0616. TABLE 4.13. Eigenanalysis of covariance matrix of estimation errors obtained
from interpretation of all the injection steps and recovery of test JJA0616. 140
Point-to-point cross-hole tests interpreted by numerical inverse
TABLE 5.1.
model Parameters identified for cross-hole test PP4 treating the medium
TABLE 5.2.
as spatially uniform Parameters identified for cross-hole test PP5 treating the medium
TABLE 5.3.
as spatially uniform Parameters identified for cross-hole test PP6 treating the medium
TABLE 5.4.
as spatially uniform Parameters identified for cross-hole test PP7 treating the medium
TABLE 5.5.
as spatially uniform 147
165
166
167
168
15
LIST OF TABLES—Continued
Parameters identified for cross-hole test PP8 treating the medium
as spatially uniform 169
Summary statistics for log io k [n1, 2 ] identified for cross-hole tests
TABLE 5.7.
170
treating the medium as spatially uniform Summary statistics for log io 0 [m 3 /m 3 ] identified for cross-hole
TABLE 5.8.
170
tests treating the medium as spatially uniform TABLE 5.9. Parameters identified for cross-hole test PP4 treating the medium
as spatially uniform and adjusting borehole storage at injection and ob171
servation intervals
TABLE 5.10. Parameters identified at the pilot points by numerical inversion
of data from cross-hole test PP4 using 32 pilot points. 185
TABLE 5.11. Parameters identified at the pilot points by numerical inversion
of data from cross-hole tests PP4, PP5 and PP6 using 72 pilot points. 204
TABLE 5.12. Comparison of different numerical inverse models using model
identification criteria (numbers in brackets show model ranking). . . . 213
Summary statistics for log io k [772. 2 ] identified using different data
TABLE 5.13.
214
and methods TABLE 5.14. Summary statistics for log io q5 [m 3 /m 3 ] identified using different
214
data and methods TABLE
5.6.
16
ABSTRACT
A three-dimensional stochastic numerical inverse model has been developed for characterizing the properties of unsaturated fractured medium through analysis of singleand cross-hole pneumatic tests. Over 270 single-hole [Guzman et al., 1996] and 44
cross-hole pneumatic tests [Illman et al., 1998; Inman, 1999] were conducted in 16
shallow vertical and slanted boreholes in unsaturated fractured tuffs at the Apache
Leap Research Site (ALRS), Arizona. The single-hole tests were interpreted through
steady-state [Guzman et al., 1996] and transient [Illman and Neuman, 2000b] analytical methods. The cross-hole tests were interpreted by analytical type-curves
[Illman and Neuman, 2000a]. I describe a geostatistical analysis of the steady-state
single-hole data, and numerical inversion of transient single-hole and cross-hole data.
The geostatistical analysis of single-hole steady-state data yields information about
the spatial structure of air permeabilities on a nominal scale of 1 m. The numerical inverse analysis of transient pneumatic test data is based on the assumption of
isothermal single-phase airflow through a locally isotropic, uniform or non-uniform
continuum. The stochastic inverse model is based on the geostatistical pilot point
method of parameterization [de Marsily, 1978], coupled with a maximum likelihood
definition of the inverse problem [Carrera and Neuman, 1986a]. The model combines
a finite-volume flow simulator, FEHM [Zyvoloski et al., 1997], an automatic mesh
generator, X3D [Trease et al., 1996], a parallelized version of an automatic parameter
estimator, PEST [Doherty et al., 1994], and a geostatistical code, GSTAT [Pebesma
and Wesseling, 1998]. The model accounts directly for the ability of all borehole
intervals to store and conduct air through the system; solves the airflow equations in
their original nonlinear form accounting for the dependence of air compressibility on
absolute air pressure; can, in principle, account for atmospheric pressure fluctuations
at the soil surface; provides kriged estimates of spatial variations in air permeability
17
and air-filled porosity throughout the tested fractured rock volume; and is applied
simultaneously to pressure data from multiple borehole intervals as well as to multiple cross-hole tests. The latter amounts to three-dimensional stochastic imaging,
or pneumatic tomography, of the rock as proposed by Neuman [1987] in connection
with cross-hole hydraulic tests in fractured crystalline rocks near Oracle, Arizona.
The model is run in parallel on a supercomputer using 32 processors. Numerical in-
version of single-hole pneumatic tests allows interpreting multiple injection-step and
recovery data simultaneously, and yields information about air permeability, air-filled
porosity, and dimensionless borehole storage coefficient. Some of this cannot be accomplished with type-curves [Inman and Neuman, 2000b]. Air permeability values
obtained by my inverse method agree well with those obtained by steady-state and
type-curve analyses. Both stochastic inverse analysis of cross-hole data and geostatistical analysis of single-hole data, yield similar geometric mean and similar spatial
pattern of air permeability. However, I observe a scale effect in both air permeabil-
ity and air-filled porosity when I analyze cross-hole pressure records from individual
monitoring intervals one by one, while treating the medium as being uniform; both
pneumatic parameters have a geometric mean that is larger, and a variance that is
smaller, than those obtained by simultaneous stochastic analysis of multiple pressure records. Overall, my analysis suggests that (a) pneumatic pressure behavior of
unsaturated fractured tuffs at the ALRS can be interpreted by treating the rock as
a continuum on scales ranging from meters to tens of meters; (b) this continuum is
representative primarily of interconnected fractures; (c) its pneumatic properties nevertheless correlate poorly with fracture density; and (d) air permeability and air-filled
porosity exhibit multiscale random variations in space.
18
Chapter 1
INTRODUCTION
An important part of current environmental problems is related to flow and transport in the vadose zone. Still, there is no well-established methodology for the characterization of unsaturated, especially fractured medium properties. Among major
difficulties are those associated with the complexity of the governing processes and
the pronounced non-uniformity and scale effects of medium properties. A potentially
advantageous approach is to conduct gaseous pressure and tracer field tests which
overcome many of the limitations associated with hydraulic flow and tracer tests
[Guzman and Neuman, 1996]. Experience with gaseous field tests in fractured rocks
is limited and much of it has been accumulated at the Apache Leap Research Site
(ALRS) near Superior, Arizona [Rasmussen et al., 1990; Guzman et al., 1994, 1996;
Guzman and Neuman, 1996; Inman et al., 1998]. Similar studies have also been
performed at Los Alamos, New Mexico [Cronk et al., 1990], the Yucca Mountain
site, Nevada [LeCain, 1996, 1998; Wang et al., 1998; Huang et al., 1999] and at Box
Canyon, Idaho [Benito et al., 1998, 1999]. The objective of my dissertation is to characterize unsaturated fractured rocks at the ALRS through analysis and interpretation
of previously conducted pneumatic (air-injection) tests. To accomplish this objective,
I have developed a novel geostatistical three-dimensional stochastic numerical inverse
model.
The major questions that I address are: Is it plausible to represent the fractured
medium at the site as an equivalent continuum, either deterministic or stochastic? Is it
possible to simulate the pneumatic tests, and identify equivalent pneumatic properties
of the medium, by means of a numerical inverse model? Is it possible to resolve the
spatial variability of these properties in detail across a site? Is it feasible to obtain
high-resolution images of subsurface medium heterogeneity by interpreting several
19
pneumatic tests simultaneously (which would amount to "pneumatic tomography"
or "high-resolution stochastic imaging" of the subsurface [Neuman, 1987])? How
accurately can the non-uniformity of the medium be estimated? Is there a scale
effect when rock properties are identified on a fine and a coarse scale? What is the
influence of open borehole intervals on pressure propagation and storage in the rock?
Can a numerical inverse model complement, or be an alternative, to the standard
analytical type-curve method of field test analysis?
Flow in fractured rocks occurs predominantly through open fractures which tend
to be more permeable than the surrounding rock matrix. There are two basic approaches to simulating flow in fractured rocks: discrete and equivalent continua. In
the discrete approach, flow takes place through a network of intersected planes whose
properties are prescribed either deterministically [e.g. Long et al., 1982; Dershowitz,
1984; Wang and Narasimhan, 1985] or stochastically [e.g. Chiles and de Marsily,
1993]. In practice, it is difficult to define in detail the geometry of a fracture network:
size, shape, orientation, density, apertures and roughness of individual fractures. Even
when some such data are available, it is still difficult to correctly represent flow within
each fracture, along its intersections, and between the fractures and surrounding matrix blocks. There is a limited number of actual applications of this approach [Dyerstorp and Andersson, 1989; Cacas et al., 1990; Dershowitz et al., 1991]. As discussed
in Chapter 2, at the ALRS, it has not been possible to determine the pneumatic prop-
erties of individual fractures. The bulk pneumatic properties do not correlate with
fracture occurrence in boreholes at the site. This supports an earlier conclusion by
Neuman [1987, 1988b] that the hydraulic properties of fracture rocks must be deter-
mined directly by means of hydraulic field tests, rather than surmised indirectly from
fracture geometry. Similar results have been also obtained at other fractured rock sites
worldwide such as Oracle, Arizona [Jones et al., 1985]; Chalk River, Canada [Raven
et al., 1985]; and Stripa, Sweden [Gale et al., 1987]. To address spatial variability, he
20
proposed to consider the fractured rock as an equivalent stochastic continuum with
properties that form spatially correlated (regionalized) random fields, which is valid
if flow on the scale of interest is not dominated be a small number of individual fractures. If flow is dominated by a few major fractures or fracture zones, one can adopt
a mixed discrete-continuum approach [cf. Andrews et al., 1986; Carrera et al., 1990a;
Carrera and Martinez-Landa, 2000] which treats them as thin "aquifers" embedded
with in a less permeable rock continuum. Equivalent approaches can be categorized by
the number of continua they consider. Fractures and porous blocks can be simulated
individually (as I do) or simultaneously as dual or multiple continua. Most common
are the double-porosity model of Barenblatt et al. [1960] and Warren and Root [1963],
and double-porosity/double-permeability model of Duguid and Lee [1977].
A major task of our field of study is to understand and predict the fluid flow and
contaminant transport through natural hydrogeological systems. This requires the
definition of a conceptual model and corresponding model parameters, which represent adequately hydrogeological processes of interest. For a given model, the identification of model parameters and associated uncertainties is called an inverse problem.
Solution of the inverse problem is preferable to be obtained through analysis of data
representing system behavior under controlled conditions; quite often, however, the
analyzed data represent arbitrary, uncontrolled fluctuations of state variables. The
inverse problem is often not well posed, i.e. the solution is non-unique or unsta-
ble with respect to the model parameters, due to both model errors and insufficient
quality and quantity of observational data.
Numerical inverse methods have been used widely in hydrogeological research
and, more recently, application. Milestone papers include those of Neuman [1973],
Neuman and Yakowitz [1979], Neuman et al. [1980], Neuman [1980], Carrera [1984]
and Carrera and Neuman [1986a, b, c]. Reviews and summaries can be found in
the publications of Yeh [1986], Kool et al. [1987], Carrera [1988], Ginn and Cushman
[1990], Carrera et al. [1993a], Sun [1994], McLaughlin and Townley [1996] and Carrera
21
et al. [1997]. In order to address the spatial non-uniformity of medium properties,
inverse techniques have been coupled with geostatistics so that medium properties can
be treated as spatially correlated (regionalized) random fields. This allows to capture
in one parsimonious way heterogeneity at a relatively high resolution with only a
small number of unknown parameters. de Marsily [1978] proposed the pilot-point
method (discussed in Chapter 3) and Clifton and Neuman [1982] developed method
which penalizes departures of inverse estimates from prior kriging estimates. Later
developments of geostatistical inverse methodology were based on both analytical
[Rubin and Dagan, 1987] and numerical [Kitanidis and Vomvoris, 1983; Hoeksema and
Kitanidis, 1984; Sun and Yeh, 1992; Yeh et al., 1996; Yeh and Zhang, 1996; Zhang and
Yeh, 1997] co-kriging techniques. G6mez-Herncindez et al. [1997] and Capilla et al.
[1997] developed a sequential self-calibrated method based on "master locations",
which is somewhat similar to an extension of the pilot-point method proposed by
LaVenue and Pickens [1992]. Important comparisons between various geostatistical
inverse methodologies have been performed by Rubin and Dagan [1987], Keidser and
Rosbjerg [1991] and Zimmerman et al. [1998]. However, applications have been few
in number and virtually all were two-dimensional [e.g. Clifton and Neuman, 1982;
de Marsily et al., 1984; Certes and de Marsily, 1991; LaVenue et al., 1995; Carrera
et al., 1997; Zimmerman et al., 1998]. Some of the applications were related to
Performance Assessment of the Waste Isolation Pilot Project (WIPP) near Carlsbad,
New Mexico, conducted by the U.S. Department of Energy.
There have been very few three-dimensional numerical inverse simulations of transient flow, and these have not considered the heterogeneity of medium porosity; uniform values of porosity were either estimated by an inverse model or defined a priori.
Carrera et al. [1990b] analyzed a cross-hole test conducted at the Chalk River site,
Canada. Data from multiple observation points were inverted simultaneously to es-
timate uniform transmissivities of four major fracture zones and the permeability of
the surrounding fractured medium. Carrera et al. [1993b] and Sanchez-Vila et al.
22
[1993] developed a mixed discrete-continuum model for the El Cabril site (for storage
of low and intermediate level radioactive waste in Spain). The model was calibrated
against a 5-year record of natural head fluctuations, and it was successfully tested
against a 1-month pumping test. Huang et al. [1999] interpreted a series of pneumatic
cross-hole tests at Yucca Mountain by means of an inverse model that considers porosity as known and uniform, and used manual trial-and-error to estimate permeability
within a series of zones. Martinez-Landa et al. [2000] designed an inverse model for
the Grimsel test site for the Full Engineered Barrier Experiment (FEBEX). Uniform
permeabilities of major fractures and the surrounding matrix were estimated through
simultaneous analysis of five cross-hole tests. Sauer et al. [2000] interpreted a series
of cross-hole tests in a system of aquifers and aquitards having uniform anisotropic
properties. In all these studies, medium permeability was not regarded as a stochastic random field. Only G6mez-Herncindez et al. [2000] used a geostatistical inverse
model [Gómez-Hernandez et al., 1997] to interpret three-dimensional saturated flow
in a synthetic fractured medium.
Interpretation of welltest data has been traditionally performed by means of typecurves derived from analytical solutions [e.g. Gringarten, 1982; Neuman, 1988a, 1998;
Batu, 1998] which allow simple, fast and reliable means of assessing pneumatic prop-
erties of the medium. Such analytical models, however, are restricted to simplified
representations of the hydrogeological environment. An alternative is to apply numerical inverse methods, which, in general, are complex, difficult and time consuming.
Though computationally intensive and prone to discretization and round-off errors,
numerical methods are much more flexible in terms of their ability to simulate realistic
hydrogeological conditions. There is a limited number of interpretations of welltests
by numerical inverse models. Lebbe et al. [1992], Lebbe and de Breuck [1995, 1997],
Hvilshoj et al. [1999] and Lebbe and van Meir [2000] applied numerical inverse mod-
els to interpret well tests conducted in various hydrogeological conditions assuming
axi-symmetric radial flow in isotropic or anisotropic, uniform or locally uniform multi-
23
layer (aquifer/aquitard) medium (see also Sauer et al. [2000] discussed in the previous
paragraph). Lebbe and van Meir [2000] also demonstrated the importance of simul-
taneous interpretation of multiple tests for the accurate characterization of medium
properties. Cardenas et al. [1999] analyzed by a numerical inverse model "seal performance" air-injection tests in packed-off intervals at the WIPP site near Carlsbad,
New Mexico. Numerical inverse models have been also successfully applied to ana-
lyze various types of laboratory tests [e.g. Finsterle and Pruess, 1995; Finsterle and
Persoff,, 1997; Finsterle and Faybishenko, 1999].
Simultaneous interpretation of a series of welltest by means of a stochastic inverse
model which accounts for medium heterogeneity with a high resolution can be defined
as "tomography" or "stochastic imaging." It has been recognized in the literature [e.g.
Neuman, 1973; Carrera and Neuman, 1986a] that as much information as possible
should be collected and simultaneously interpreted to obtain accurate characterization
of medium properties. The idea of "high-resolution stochastic imaging," however,
was originally proposed over a decade ago by Neuman [1987] in connection with
interpretation of hydraulic cross-hole tests in saturated fractured crystalline rocks at
the Oracle site near Tucson, Arizona [Hsieh et al., 1985]. Later, the concept has
been explored by Tosaka et al. [1993], Bohling [1993], Gottlieb and Dietrich [1995],
Masumoto et al. [1995, 1996, 1998], Butler et al. [1999] and Liu and Yeh [1999] (see
also Gottlieb [1992]). Tosaka et al. [1993] analyzed a series of transient cross-hole
tests in two- and three-dimensional synthetic media. While porosity was assumed
a known constant, the inverse model estimated uniform permeability at the grid
blocks of a computational region through simultaneous interpretation of the test data.
This procedure identified properly the spatial pattern but not the magnitude of the
permeability field. Gottlieb and Dietrich [1995] did a similar two-dimensional analysis
considering transient flow between a source and a sink, and obtained similar results.
Masumoto et al. [1998] extended the analysis of Tosaka et al. [1993] to synthetic
multi-rate injection and recovery tests. For two-dimensional synthetic cases, Liu and
24
Yeh [1999] studied the influence of injection rate, distance between test boreholes
and lengths of injection and observation intervals on inverse estimates depending on
stochastic properties of the medium. They used a numerical inverse model based on
iterative co-kriging of pressure and permeability data, while requiring that the spatial
correlation of permeability field known a priori [Yeh et al., 1996]. There were also
studies on tomographic inversion of tracer test data [e.g. Kunstmann et al., 1997;
Vasco arid Datta-Gupta, 1999].
Tomographic imaging has been initially developed and is widely applied in physics,
geophysics and medicine [e.g. Jackson, 1996; Ramm, 1997], where in contrast to hy-
drogeology, the tomographic data represent steady-state measurements. Steady-state
data, however, do not allow identifying the storage properties of a medium. Timevariation in the measurements complicates the inversion, but provides additional information and, therefore, in principle should improve the quality of inverse estimates.
There have been a few applications of geophysical tomographic methods in hy-
drogeological research. At the Oracle site, Ramirez [1986] demonstrated that tomographic images of electromagnetic resistivity correlate with kriging images of hydraulic
conductivity obtained from single-hole tests [Jones et al., 1985; Neuman et al., 1985].
This is the first study to compare geophysical tomographic images with measured
and kriged permeabilities.
In Chapter 2, I provide a description of the Apache Leap Research Site and field
and laboratory studies conducted there previously, with special emphasis on singlehole and cross-hole pneumatic tests. I discuss the statistical and geostatistical analyses of various ALRS rock properties and define the hydrogeologic conditions that
prevail during pneumatic tests at the site. In Chapter 3, I develop a stochastic numerical inverse model for the simulation and interpretation of both single-hole and
cross-hole transient pneumatic tests at the ALRS. Chapter 4 presents and discusses
corresponding results from single-hole tests, and Chapter 5 does so for the cross-hole
25
tests. The dissertation ends with a comprehensive list of findings and conclusions in
Chapter 6.
26
Chapter 2
APACHE LEAP RESEARCH SITE
2.1 Site description
The Apache Leap Research Site (ALRS) is located in central Arizona near Superior
approximately 160 km north of Tucson (Figure 2.1). Regional geology is characterized
by a dacite zoned ash-flow tuff sheet overlying carbonate rocks. The volcanic sheet
covers an area of 1,000 km 2 and varies considerably in thickness about an average
of 300 m [Peterson, 1961]. This dissertation concern field studies conducted in the
upper part of the tuff sheet, within a layer of slightly welded fractured tuffs.
The site is located within the Pinal Mountains at an elevation of 1,200 m above sea
level. The average barometric pressure at the site is approximately 87 kPa. Climate
is temperate and dry, with a mean annual air temperature of about 30°C and a mean
annual precipitation of less than 500 mm. Most of the precipitation occurs during
the summer (from mid-July to late-September) and the winter (from mid-November
to late-March). Infiltration conditions are not favorable in the summer, when not
only air temperatures and evapotranspiration demand are high but also precipitation
events have high intensity and short duration. However, in the winter due to lower
air temperatures and evapotranspiration demand as well as longer duration and lower
intensity precipitation events, the conditions for infiltration and groundwater recharge
are much more favorable. 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 site is similar in many respects to Yucca Mountain in southern Nevada where
a candidate high-level radioactive waste repository site is still being characterized by
the U.S. Department of Energy.
27
FIGURE
1996]).
2.1. Location map of Apache Leap Research Site [after Guzman et al.,
28
Borehole
2.1. Borehole coordinates and length.
Top
Bottom
Length
x [m]
y [m]
z [rr ]
x [m]
y [m]
z [m]
[m]
X1
X2
X3
Y1
Y2
Y3
Z1
Z2
Z3
V1
V2
V3
W1
W2
W2A
W3
10.324
20.44
30.427
10.204
20.036
30.068
19.52
9.8
0.
1.248
4.238
7.238
5.24
4.85
5.24
5.24
TABLE
10.038
10.031
10.037
5.084
5.2
5.347
0.
0.032
0.
6.844
6.844
6.844
11.46
22.43
21.46
31.46
0.02
-0.02
0.04
-0.03
-0.31
-0.27
-0.33
-0.2
0.
0.
0.
0.
0.
0.
-0.03
0.
-2.715
-2.619
-2.552
-1.951
-1.842
-1.681
31.568
32.074
31.975
1.248
4.238
7.238
5.24
3.815
5.24
5.24
10.038
10.031
10.037
5.084
5.2
5.347
0.
0.032
0.
6.844
6.844
6.844
0.252
-3.931
0.148
-3.846
-13.02
-23.07
-32.94
-12.18
-22.18
-32.02
-12.38
-22.47
-31.98
-33.53
-28.25
-30.48
-11.24
-23.08
-21.34
-35.34
18.44
32.61
46.64
17.19
30.94
44.90
17.04
31.50
45.22
33.53
28.25
30.48
15.85
35.05
30.14
49.93
The site includes 22 vertical and inclined (at about 45 0 ) boreholes. In my study,
I discuss data from 16 of these boreholes-X1, X2, X3, Yl, Y2, Y3, Z1, Z2, Z3, V1,
V2, V3, Wl, W2, W2A and W3. The remaining boreholes are located outside the
pneumatic test area. Borehole locations (Table 2.1) are defined in a local Cartesian
coordinate system (x, y, z) with origin at the lower lip of the casing in borehole
Z3, and a vertical z-axis pointing downward. A plan view and a three-dimensional
perspective of the boreholes are shown in Figures 2.2 and 2.3, respectively. Due to
a drilling error, boreholes V2 and W2 intersect at depth. All boreholes are open,
except for the upper 1.8 m of each borehole which is cased. The boreholes span a
rock volume of approximately 35 x 36 x 35 (44,100) m 3 . A total of 270-m oriented
core was retrieved from the boreholes and stored at the University of Arizona Core
Storage Facility.
Shortly after the completion of drilling, a surface area of 47 x 27.5 (1,293) m 2 that
includes all boreholes was covered with a thick plastic sheet (Figure 2.2) to minimize
infiltration and evaporation.
29
-
• W3
-
30
-
-
-
W2
1111W2A
-
-
20
-
-
-
-
W1
0
-
-
V1 V2 V3
41 411 .
-
--
X2
• X1
0
• X3
eY1•y2•Y3
-
-
0
_
• Z3•Z2
• Z1
-
-
Plastic cover
-
-
-10
_
-
i
1
1
0
10
x[m]
FIGURE 2.2. Plan view of tested
20
30
boreholes and plastic cover.
40
30
0
-10
N
-20
B
-30
-40
FIGURE 2.3. Three-dimensional perspective of boreholes at the site.
31
2.2 Previous investigations
Early work related to my area of study at the ALRS included the drilling of 16
boreholes and the conduct of various laboratory and field experiments. Detailed
summary of these studies can be found in the Ph.D. dissertation of Illman [1999]. Here
I concentrate mainly on laboratory and field investigations conducted by Rasmussen
et al. [1990] and Thompson et al. [1995], single-hole pneumatic tests conducted by
Rasmussen et al. [1990, 1993], Guzman et al. [1994, 1996] and Guzman and Neuman
[1996], and cross-hole pneumatic tests conducted by Illman et al. [1998] and Inman
[1999].
2.2.1 Laboratory and field investigations
Rasmussen et al. [1990] conducted a series of laboratory and field determinations of
rock properties over 105 3-m borehole intervals in 9 of the boreholes—X1, X2, X3,
Yl, Y2, Y3, Z1, Z2 and Z3 (Figure 2.4).
Laboratory measurements on core segments from the 3-m borehole intervals included various matrix properties. For the purposes of my study the most relevant
are the "oven-dry" matrix air permeability k [m 2 ], the effective matrix porosity O m
[m 3 /m 3 ] and the parameter a [kPct -1 ] of the van Genuchten [1980] retention model
(other parameters of this retention model are considered to be constant: the dimensionless parameter n = 1.6 and the residual water content 0, = 0). The measured
values of k, On, and a lie in the following respective ranges: 3.8 x 10' 6 - 1.0 x 10'
m 2 , 0.14 - 0.28 m 3 /m 3 and 0.0102 - 0.0642 kPa -1 .
Rasmussen et al. [1990] defined the location and geometry of fractures intersected
by the above nine boreholes through examination of oriented cores. They observed a
total of 224 fractures. Using these data, I plotted fracture orientations on a Schmidt
equal-area plot as shown in Figure 2.5 (another version of a similar plot by Yeh et al.
[1988] has the strikes of all fractures rotated by 180 ° degrees). Though the fractures
32
o
-10
N
-20
2
-30
-40
2.4. Three-dimensional perspective showing center locations of 3-m test
intervals employed by Rasmussen et al. [1990].
FIGURE
33
exhibit a wide range of inclinations and trends, most are near vertical, strike northsouth and dip steeply to the east. Fracture density for the 3-m intervals ranges from
0 to 4.3 counts I m.
Thompson et al. [1995] determined the water content 19 [m 3 /m 3 ] at 159 locations
along all boreholes, except W2A and W3, using neutron probes (Figure 2.6). The
measurements were repeated eight times between May 1990 and April 1991. The
results show that water content is nearly constant over time. Measured values vary
from 0.071 to 0.124 m 3 /m 3 and are consistently lower than the matrix porosity ck n, (see
above). The support scale of neutron probe measurements is inversely proportional
to the local water content [cf. Jury et al., 1991].
2.2.2 Single-hole pneumatic tests
Over the 3-m borehole intervals defined in the previous subsection (Figure 2.4), Rasmussen et al. [1990, 1993] conducted a series of single-hole pneumatic tests by injecting
air at a constant mass rate between two inflated packers while monitoring pressure
within the injection interval. The injection flow rate varied from 6 x 10 -8 to 2 x 10 -5
kg/s (from 0.003 to 0.98 l I min under standard conditions) and the steady-state pres-
sure buildup in the injection interval ranged from 86.1 to 151 kPa. Pressure is said
to have reached stable values within minutes in most test intervals. Air permeability
was calculated using steady state formula derived by Dachler [1936] and adapted
to isothermal airflow by Rasmussen et al. [1990, 1993]. The air permeability values
defined from 87 successful tests vary from 4.2 x 10 -17 to 2.4 x 10- 12 m2. The sample
locations span a rock volume of approximately 32 x 10 x 28 (8,960) m 3 .
Figure 2.7 shows a scatter plot of air permeability versus fracture density for the
3-m borehole intervals. There clearly is no correlation between fracture density and
air permeability. Similar results have been obtained at other fractured rock sites
worldwide such as Oracle, Arizona [Jones et al., 1985]; Chalk River, Canada [Raven
34
\
E Xi
A >OE
+ X3
o 11
x 12
-0- 13
-0- Z1
+72
11 73
FIGURE 2.5. 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.
35
o
-10
N
-20
3
-30
-40
FIGURE 2.6. Three-dimensional perspective showing center locations of neutron
probe measurements performed by Thompson et al. [1995].
36
et al., 1985]; and Stripa, Sweden [Gale et al., 1987].
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 borehole intervals with different lengths. Their tests were performed under highly
controlled conditions, subject to strict quality assurance, within 6 of the boreholesX2, Y2, Y3, Z2, V2 and W2A. A total of 184 borehole segments were tested by setting
the packers 1 m apart (Figure 2.8); additional tests were performed 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 more than 270. The sample locations span a rock volume of
approximately 31 x 20 x 28 (17,360) m 3 . 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. Two or more such incremental steps were performed in each
borehole segment while recording the air injection rate, pressure, temperature and
relative humidity during injection and, in most cases, recovery. The injection flow
rate varied 9.8 x 10 -8 to 4.4 x 10 -4 kgls (from 0.0049 to 22.0 //min under standard
conditions) and the steady-state pressure buildup in the injection interval ranged
from 0.49 to 273 kPa. The time required for pressure in the injection interval to
stabilize typically ranged from 30 to 60 min, increased with flow rate, and might have
at times exceeded 24 h, suggesting that steady-state permeability values published
in the literature for this [Rasmussen et al., 1990] and other sites, based on much
shorter air injection tests, may not be entirely valid. Guzman and Neuman [1996]
have shown that the measured pressure responses are affected by two-phase flow
and inertia effects. For each relatively stable period of injection rate and pressure,
air permeability was estimated by treating the rock around each test interval as
an infinite three-dimensional uniform, isotropic continuum within which air flows as
a single phase under steady state, in a pressure field exhibiting prolate spheroidal
symmetry. For each test interval, I have defined the midrange value for the logio-
37
CS"
• •
•
•
•
•
•
• •
•
• •
•
•
•
•
• •
1-C
\
I
I—
I
LIJ
•
•
CO
'71-
1—
Lli
,
•
•
• • •
•
111111
1—
SD •
••
• •
•
1111111 1
•
•
•
•
11111111 1
•
ai,_
[w] Ameewied
ID OMB
•
•• • •
IMO
se • • •
1111111 1'
1.0
iii
i_
71'1
C.0
1—
LI,
,_
,-
N-
w
r-
38
0
-10
N
-20
2
-30
-40
FIGURE 2.8. Three-dimensional perspective showing center locations of 1-m singlehole test intervals employed by Guzman et al. [1996].
transformed air permeability estimates. In terms of these midrange values, local-scale
air permeabilities vary by orders of magnitude, from 7.5 x 10' to 8.7 x 10- 14 m2,
across the site.
Illman et al. [1998] and Illman and Neuman [20001)] have interpreted data from
the first step of 40 single-hole pneumatic tests by means of analytically derived typecurves, based on the assumption of single-phase airflow through a uniform, isotropic
porous continuum. They considered the case where injection takes place at a point,
and accounted for the effects of storage and skin in the injection interval. Their anal-
39
ysis was performed by means of type-curves based on previously published analytical
solutions for liquid flow under a spherical and radial flow regime, including cases
where a single horizontal or vertical fracture intersects the injection interval. To render them applicable to air, the authors recast the original analytical solutions in terms
of pseudo-pressure [Al-Hussaing et al., 1996; Raghavan, 1993] and developed expressions and type-curves in terms of pseudo-pressure derivatives. Although the governing
airflow equation was linearized in terms of either pressure, p, or pressure-squared, p 2 ,
(see Section 3.4) their results demonstrated that the two approaches produce similar
results. They also concluded that during single-hole tests, skin effects are negligible
but storage effects are significant. Using the same methodology, Illman and Neuman
[2000b] also analyzed independently the data from all injection steps and recovery of
a single test. This analysis confirmed the earlier findings of Guzman and Neuman
[1996] and Vesselinov and Neuman [2000] that, although two-phase and inertia effects
are pronounced for some of the tests, they do not impact significantly the estimation
of air permeability. The type-curve analysis did not allow reliable identification of
air-filled porosity or dimensionless borehole storage. Nevertheless, air permeability
values obtained by type-curve and steady-state analyses agreed very well. Some of
these results will be discussed in Chapter 4.
2.2.3 Cross hole pneumatic tests
-
Illman et al. [1998] and Illman [1999] designed and conducted a total of 44 cross-hole
pneumatic tests, incorporating 16 of the boreholes at the site. The tests span a volume
of fractured rock larger than that previously subjected to single-hole testing. In each
cross-hole test, air was injected at a constant mass flow rate into a relatively short
borehole interval while monitoring air pressure and temperature in the injection and
observation intervals and barometric pressure, air temperature and relative humidity
at the surface. Pressure measurements are relative, representing changes in absolute
40
air pressure with time. The injection flow rate varied from 1 x 10 -5 to 2 x 10 -3 kgls
(from 0.5 to 100 //min under standard conditions) and the pressure buildup in the
injection interval reached a maximum of 140 kPa. A more detailed description of
some of the tests and results is given in Chapter 5.
Assuming single-phase airflow through a uniform, isotropic porous continuum,
Illman et al. [1998] analyzed the cross-hole tests by means of analytically derived
type-curves. These type-curves were a modified version of those developed by Hsieh
and Neuman [1985] for the interpretation of cross-hole hydraulic tests in saturated
fractured crystalline rocks at their Oracle site near Tucson [Hsieh et al., 1985]. The
type-curves allow the injection and observation intervals to be represented by lines
having arbitrary lengths and spatial orientations. Illman et al. [1998] modified them
for single-phase airflow in the presence of storage and skin in observation intervals.
Since both pressure, p, and pressure-squared, p 2 , based methods for linearization
of the governing airflow equation (see Section 3.4) gave similar results for single-
hole tests, Illman et al. [1998] adopted the simpler p-based type-curves for their
interpretation of cross-hole tests. They considered the case where injection takes
place at a point, and observation along a line. To date, only one of these tests
(labeled PP4) was fully analyzed in this manner [Inman et al., 1998; Illman, 1999;
Inman and Neuman, 2000a]. The most up-to-date results are those given by Illman
and Neuman [2000a]. They are discussed in Chapter 5.
41
2.3 Geostatistical analysis of spatial variability
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. A major question is how to describe this
spatial and directional dependence of medium properties in untested portions of the
rock.
The analyses of Guzman et al. [1994, 1996], Chen et al. [1997] and Illman et al.
[1998] suggest that it is possible to interpolate some of the core and single-hole test
measurements at the ALRS between boreholes by geostatistical methods, which view
the corresponding variables as spatially correlated random fields. This is especially
true for air permeability k, matrix porosity O m , fracture density, water content 0, and
the van Genuchten water retention parameter a, for each of which there are 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
uncertainty of these estimates.
A geostatistical analysis of the above site variables was originally conducted by
Chen et al. [1997]. I have repeated, slightly modified, and extended their interpreta-
tions.
2.3.1 Air permeability
An omni-directional sample variogram y(h) of log io air permeability k [m 2 ] data as
-
a function of separation distance h [in], obtained from steady state interpretation of
1-m scale single-hole pneumatic injection tests, is shown in Figure 2.9. The shape of
the sample variogram implies statistical non-homogeneous random log io k field with
42
homogeneous spatial increments (also called in the geostatistical literature "intrinsic"
random field of "second order"). It was analyzed by Chen et al. [1997] using three
structural models:
• power variogram -y(h) = 0.2720 4 5 ;
• exponential variogram -y(h) = 0.495 [1 — exp(-1.26h)] of residuals about a firstorder polynomial trend given by —16.45 0.0489x + 0.0561y — 0.182z; and
• exponential variogram ry(h) = 0.581 [1 — exp(-1.67h)] of residuals about a
second-order polynomial trend given by —16.76 H- 0.0566x H- 0.0463y — 0.233z
0.00126x 2 0.00154y 2 — 0.00718z 2 0.0000501xy 0.00530xz 0.00720yz.
To select the best among these three models, Chen et al. [1997] employed the
Maximum Likelihood Cross Validation approach of Samper and Neuman [1989a, b],
coupled with the generalized least squares drift removal approach of Neuman and Ja-
cobson [1984]. The analysis of Chen et al. [1997] showed that whereas the exponential
variogram model with a second-order drift fits the data most closely, all the model
discriminating criteria (AIC, BIC, O m and dm ; these criteria are also discussed in
Chapter 3, page 92) consistently rank the power model as best. The reason is that the
power model is most parsimonious with only two parameters, while the exponential
variogram model with second-order drift is least parsimonious with twelve parameters. As is shown in Figure 2.10, both models yield very similar kriged estimates of
log lo k, but rather different measures of the associated estimation uncertainty.
The geostatistical analysis of Chen et al. [1997] was based on 1-m scale air permeability data [Guzman and Neuman, 1996] in six boreholes—X2, Y2, Y3, Z2, V2
and W2. I have augmented the 1-m data with air permeabilities obtained from 3-m
test intervals [Rasmussen et al., 1990] in four additional boreholes Xl, X3, Yl, Z1
and Z3.
43
k) gio k [rn 2 ]
2.0
1-m support data
0.003
Power model
1.5
o
Second-order residuals
Spherical model
• 0.002
e
as
0.001
0.0
Fracture density [counts/m]
,
0 [M 3/M 3]
•
1.5
•
0.8
I.
•
- -
0
1.0
E
El
sr
0,
0
.
0.5
- - -o- - - Data
•
-O.
-••
,
,o
1
.
/c) ......
.. ..
œ
>
0.2
,o
'S.
e
..
-e- - - Original data
o Second-order residuals
Spherical model
- -
Exponential model
0.0
0.02
FIGURE 2.9. Omni-directional sample and model variograms for various parameters
at a minimum separation distance of 3 m.
44
-14A
-14.9
-15 4
-15.9
-16A
Estimation variance
0
I
0.48
0.38
028
0.18
-15
20
25
FIGURE 2.10. Kriged estimates and kriging variances of log io k at y = 7 m using a
power model (left), and an exponential model with second order drift (right).
45
To check whether this augmentation is justified, I compared 1-m and 3-m scale
single-hole air permeability data from boreholes X2, Y2, Y3 and Z2. Figure 2.11
shows how these vary with depth in each borehole. The figure also shows 0.5-m, 2-m
and 3-m scale data obtained by Guzman et al. [1996] in borehole Y2. The figure
demonstrates that as the support scale increases, the amplitude and frequency of
spatial variations in air permeability decrease. Nevertheless, the 3-m data captures
the overall trend of non-uniformity observed by 1-m data. One also notes that 3-m
scale permeabilities obtained by Rasmussen et al. [1990] are consistently lower than
those obtained by Guzman et al. [1996]; I attribute this systematic difference to the
relatively short duration of tests conducted by Rasmussen et al. [1990].
I have included in Figure 2.11 information about the location of fractures intersected by the boreholes. One notes that correlation between measured local permeabilities and fracture locations/densities is poor.
Histograms obtained from all the 1-m and 3-m scale data are presented in Figure
2.12. They suggest that population distributions associated with the two data sets
are not significantly different. Summary statistics of the data samples are listed in
Table 2.2. Recall that not all data come from the same boreholes. A qualitative
comparison of the underlying population distributions can be performed by means of
the non-parametric Kolmogorov-Smirnov test [cf. Hollander and Wolfe, 1973]. The
Kolmogorov-Smirnov Z value is defined as the largest absolute difference between
the sample distributions. The computed Z is compared to a critical value depending
on sample sizes. If Z is greater than the critical value, the null hypothesis that the
population distributions are similar is rejected. In practice, computer programs such
as SPSS [1997] directly compute the two-tailed asymptotic significance level of the
probability that the population distributions are not substantially different from each
other. The significance level in our case is 0.2 and therefore the null hypothesis cannot
be rejected at the typically accepted significance level of 0.05. A similar conclusion
is reached using the parametric two-sample T test.
46
o
Y3
10 —
20
30—
1-m scale
ao
- - - 3-m scale (1990)
8-
I
1E-16
1E-14
1E-15
1E-13
1E-18
I
I I 11111
I
1E-17
I 1 I 11111J
1E-16
Air permeability [mq
I 1 1 11111
Fracture
I
1 1 1 11111
1E-15
I
1 I I 1111
1E-14
1E-13
k (m2]
"k2
kt
10
20
30
1
1 1 1 1111
1
1
I I I I 111
1
1
1 1 1 I 11
permeability with depth and support scale (data from
Guzman et al. [1996] and Rasmussen et al. [1990]).
FIGURE 2.11. Variation of air
TABLE 2.2. Summary statistics of the geostatistically analyzed data.
Sample size
Parameter
k
[m
2
log io
184
- 1-m data
87
data
- 3-m
227
and
3-m
data
- 1-m
105
Fracture density
[count s/m]
105
log io O m [m 3 /m, 3 ]
Minimum
Maximum
Mean
Variance
CV
-17.03
-16.38
-17.13
0.0
-13.06
-11.62
-11.62
4.33
-15.25
-15.11
-15.22
0.767
0.763
1.037
0.870
0.695
-0.05
-0.0687
-0.0572
1.09
-0.845
0.071
-1.99
-0.561
0.124
-1.19
-0.760
0.0986
-1.68
2.94 x 10 -3
1.21 x 10 -4
2.09 x 10 -2
-0.0713
0.112
-0.0861
]
0 [m 3 /m, 3 ]
log io a [kPa -1 ]
159
105
47
O
Cn1
C•I
"or
ar dir
.dor Air .or
.0" .0"
099999999119999
ii imi 00119911999999-99
07
Air, Jr , .ordrir .de Fir Z.", .0"
MI1111111111111111111111111111111111111111111
ArZ .44;.a.Z.de
Ar.e.e.or
4B
1 RITIIJIB IBUTE IMPRIBRINIB IJIJI
11101119 111
,
1111
o
O
CN1
6O
6o
Aouenbeal a/0q 9
6
o
48
TABLE 2.3. Summary statistics for log io k [m 2 ] from single-hole tests at various scales
along borehole Y2.
Minimum
Maximum
Mean
Variance
Support scale Sample size
CV
0.5 m
1m
2m
3m
54
28
14
9
—16.31
—16.19
—16.15
—15.80
-13.12
—13.32
—13.55
—13.61
—15.10
14.99
—14.95
—14.78
—
0.424
0.584
0.498
0.378
—0.0431
—0.0501
—0.0472
—0.0416
Histograms of the log io -transformed air permeability data at various (0.5, 1, 2 and
3 Tri) scales along borehole Y2 [Guzman et al., 1996] are presented in Figure 2.13 (these
data have already been presented in different format in Figure 2.11). Summary statistics of the data are listed in Table 2.3. The histograms suggest that the underlying
population distributions are not significantly different from each other; nevertheless,
as support scale increases, the range of the data decreases, and the sample histograms
becomes less skewed and closer to log-normal. The four distributions are compared
qualitatively using the Kruskal-Wallis H test [cf. Hollander and Wolfe, 1973], which
is a non-parametric alternative to the one-way analysis of variance (ANOVA) test.
The H statistic is calculated on the basis of sums of ranks for combined samples. It is
approximately chi-square (x 2 ) distributed with degrees of freedom equal to the total
number of samples minus one. The null hypothesis that the population distributions
do not differ in mean rank is rejected if the critical value of X 2 for the desired significance level is less than the computed H value. SPSS [1997] directly computes the
corresponding significance value, which in our case is equal to 0.62. Therefore, the
underlying population distributions do not differ statistically and do not indicate a
statistically significant scale effect.
Finally, I compare in Figure 2.14 the omni-directional sample variograms of 1-m,
3-m and combined 1-m and 3-m log io k data. Though the sample variograms differ
somewhat from each other at large separation distances, they are otherwise quite
close. Attempts on my part to represent the 3-m data by a variogram model that
views them as a sample from a statistically homogeneous random field with a linear
49
MOM
dor
1 111111111111111111
111111111111111111111
IIIIIIIIIIIIIIIIIlIIII11 1111111111111i1111 111111111111111111
/ZZYYZZZZ/ZZZZZ/ZZZ/ZZ ZZ FZZZZZLIZZ
‘„..\\.\:\ N.'"\ N.N.\.\\.\\X\N"\\X\NNN.N.
111111111111111 11111111 11111111111111111111111111111111111111
bWFJW
1=11111111111111111111111
"ZZZ/ZZZZZZZZZZZZ
erZZZZZZ Z
LA
0
LO
0
in
CICIINNr
6
6
6
6 6 6
Ao u en bail angel au
0
r
6
00
6
6
50
);.*..;°'
2.0
1.5
- - -e- - - 1-m support data
- --o----- 3-m support data
.....
fr
/...
o
1-m and 3-m support data/: ,/
/.:
Power model
.1
0.5
0.0
FIGURE
5
10
15
Distance [m]
20
2.14. Omni-directional sample variograms for log
io
25
k with various supports
and a power variogram model fitted to 1-m data.
or quadratic spatial drift were not successful. This supports the earlier variogram
analysis of 1-m log io k data by Chen et al. [1997].
My analysis justifies the addition of the 3-m scale air permeability data of Ras-
mussen et al. [1990] to the 1-m data of Guzman et al. [1996]. Locations of the
combined set of 227 air permeability measurements are depicted in Figure 2.15; log io
permeability [m 2 ] values range from —17.13 to —11.62 with mean, variance and coefficient of variation equal to —15.2, 8.7 x 10 -1 and —6.1 x 10 -2 , respectively (Table
2.2). The sample locations span a rock volume of approximately 32 x 20 x 32 (20,480)
m 3 . I kriged these data using the power variogram model obtained by Chen et al.
[1997] for the 1-m scale data.
51
0
-10
-20
N
2
-30
-40
2.15. Three-dimensional perspective showing center locations of single-hole
test intervals for combined set of 1-m (circles) and 3-m (squares) scale measurements.
FIGURE
52
Figure 2.16 compares kriged images of log io k I have generated along four vertical
sections at y = 0, 5, 7 and 10 m using 1-m data (left column) and the combined set
of 1-m and 3-m data (right column). Boreholes providing information to produce the
respective kriged maps are included in the figure. The two sets of kriged images differ
substantially from each other. This is most pronounced at y = 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 corner of the section to be
much higher than it is without these data. The effect extends to all four cross-sections,
which exhibit elevated permeabilities near the upper-right corner. Along sections at
y = 5 m and 7 m, which pass close to the Y and V series, respectively, the addition of
data from Yl affects the shape and size of a prominent high-permeability zone which
extends through Y2 (see corresponding peak in Figure 2.15). The addition of data
from X3 to the set reveals corresponding high- and low-permeability zones in section
y = 10 m, which correlate well with similar zones intersected by Y3 in section y = 5
M.
53
0
ly= O ml
Z2
0
-5
-5
-10
-10
î_ 15
-15
Z3
Z2
Z1
-20
-25
-25
-30
-30
x [m]
Y2
0
Y3
-5
-10
I -15
-20
-25
-30
x [m]
X2
0
-5
-5
-10
-10
7 -15
-20
-20
-25
-25
-30
-30
x [m]
x [m]
FIGURE 2.16. Kriged log lo k estimates obtained using 1-m scale data from boreholes
X2, Y2, Y3, Z2, V2 and W2A (left) and same together with 3-m scale data from
boreholes Xl, X3, Yl, Z1 and Z3 (right).
54
2.3.2 Fracture density, matrix porosity, water content and
van Genuchten a
Chen et al. [1997] performed geostatistical analyses of fracture density [counts I m],
log lo matrix porosity Om, [m 3 /m 3 ], water content 8 [m 3 /m 3 ] and log io van Genuchten
a [kPal data. Summary statistics of the data are listed in Table 2.2 on page 46.
Figure 2.9 (page 43) shows corresponding omni-directional sample variograms and
models. The latter for each rock parameter are
• fracture density [counts/m], exponential model with variance 0.69 and an inte-
gral scale 2.5 m;
• 1og 10 (q5,, [m 3 /m 3 ]), exponential model with variance 0.00315 and an integral
scale 3 m;
• 0 [m 3 /m 3 ], spherical model (sill 0.561 and correlation scale 6.29 m) of residuals about a second-order polynomial trend defined by 8.3145 + 0.2234x —
0.05905y+0.010397z-0.0049492x 2 +0.00077045y 2 +0.0028264z 2 -0.001986xy+
0.0039864xz — 0.0080923yz; and
• log io (a [k P a 1), exponential model with variance 0.0223 and integral scale 2.25
-
rn.
2.3.3 Conditional simulations
Figure 2.17 shows kriged images of log io k (utilizing the combined set of 1-m and 3-m
data), fracture density, log io O m , 0 and log io a in a vertical plane corresponding to
y = 7.0 m around which many of the available data are clustered. There clearly is
no correlation between log lo air permeability and fracture density along this plane
(nor anywhere else in the domain of investigation). Matrix porosity is consistently
55
higher than water content throughout the tested rock mass, reflecting the fact that
the medium is not fully saturated.
A three-dimensional representation of kriged log io k based on the combined set of
data is shown in Figure 2.18. Figures 2.19 through 2.21 show corresponding sections
in the y-z, z-x and x-y planes, respectively.
Similar three-dimensional representations of kriged fracture density, matrix porosity log io Om , water content 0 and van Genuchten's log io a are presented in Figures
2.22 — 2.25, respectively.
To estimate the pore space available for airflow, I subtracted the kriged estimates
for water content (Figure 2.24) from those of matrix porosity (Figure 2.23) to obtain
a three-dimensional representation of log io air-filled matrix porosity, q5,„, as shown
in Figure 2.26.
The three-dimensional images of kriging estimates in Figures 2.18 —2.26 represent
a rock volume of approximately 45 x 33 x 33 (49,005) m 3 , encompassing the locations
of all the analyzed data sets.
Kriged estimates of hydrogeologic variables are smooth relative to the corresponding random fields. To generate less smooth and more realistic images that honor the
available data, I used GCOSIM3D, a sequential Gaussian conditional simulation code
developed for three-dimensional data by Gómez-Herndndez and Cassiraga [1994]. The
code is applied separately to log io k (conditioned on the combined set of 1-m and 3-
m data), fracture density, log io çbm , 19 and log io a data on the assumption that each
of these variables is Gaussian. The normality test is performed using a one-sample
Kolmogorov-Smirnov goodness-of-fit test [cf. Hollander and Wolfe, 1973], where the
Kolmogorov-Smirnov D value, similarly to Z value defined above, is determined by
the largest absolute difference between the sample and hypothesized theoretical distributions. Again, the computed D is compared to a critical value depending on the
sample size. If D is greater than the critical value, the null hypothesis that the population distribution is Gaussian should be rejected. Computer programs such as SPSS
56
TABLE
2.4. Results from Kolmogorov-Smirnov test of Gaussianity.
Parameter
log io k [m 2 ]
1-m data
— 3-m data
— 1-m and 3-m data
—
Fracture density [counts I m]
log lo O m [m 3 /m 3 ]
0 [m 3 /m 3 ] residuals
log io a [kPar-1 ]
Most extreme
absolute difference
Kolmogorov-
Smirnov Z
Asymptotic
significance
0.046
0.135
0.065
0.194
0.114
0.075
0.076
0.652
1.263
0.978
1.984
1.173
0.947
0.778
0.789
0.082
0.294
0.001
0.127
0.331
0.581
[1997] directly computes the two-tailed significance level of the probability that the
population distribution is not substantially different from the theoretical distribution.
Calculated significance levels for all data samples are given in Table 2.4. They show
that all data except fracture density pass the test of Gaussianity at a significance
level of 0.05. Figure 2.27 shows conditionally simulated images of parameters in a
vertical plane corresponding to y = 7 m. These images are clearly much less smooth
than are their kriged counterparts in Figure 2.17.
The results demonstrate that all the above data are amenable to continuum geo-
statistical analysis and exhibit distinct spatial correlation structures. This suggests
that each data set can be viewed as a sample from a random field, or stochastic
continuum. This is so despite the fact that the rock is fractured and therefore me-
chanically discontinuous. The conclusion is supported strongly by similar findings in
many other fractured rock sites including crystalline rocks at Oracle, Arizona [Jones
et al., 1985; Neuman and Depner, 1988]; Stripa, Sweden [SKB, 1993; Neuman, 1998];
and Fanay-Augeres, France [Cacas et al., 1990; Neuman, 1998]. It justifies the ap-
plication of continuum flow and transport theories and models to fractured porous
tuffs on scales of a meter or more, as proposed over a decade ago in the context of
crystalline rocks by Neuman [1987] and affirmed more recently by Tsang et al. [1996].
57
log io k [m 2 ]
0-0.695
-0.72
-0.745
-0.77
-0.795
-0.82
o
10
x [m]
X
10
[m]
115 20
e [m 3 /m 3 }
Fracture density [counts/m
0.3
N
-20
FIGURE
2.17. Kriged estimates of various parameters at y = 7 m.
25
58
30
-10
FIGURE
2.18. Three-dimensional representation of kriged log io k [m 2 ].
-12.5
-13
-13.5
-14
-14.5
-15
-15.5
-16
-16.5
-17
59
x = 12 m
x = -3 m
x = 27 m
-10
-20
-30
-30
(.1
X
=
X = 30m
x= 15m
M
-10
-10
-20
'4 -20
-30
-30
1
y Cm]
x=3m
x= 18m
-10
-20
-30
n
10
0
y
x = 6 m
-10
x = 21 m
'
"§:
-20
N -20
-30
10
2
Y Irn1
x=9m
x = 24 m
-10
X
-20
FIGURE 2.19. Kriged log lo k [m 2 ]
along various y-z planes.
60
a
-10
" -20
-30
-30
-20
0
0
-10
-10
-10
" -20
" -20
-20
.30
-30
-30
T.
[m]
0
.10
-10
" .20
" -20
-30
-30
T,
FIGURE 2.20. Kriged log lo k [m 2 ]
along various x-z planes.
61
z = -30 m
[011
z = -18m
20
20
10
-E- 10
7.
0
20
r
10
0
20
[ mj
FIGURE 2.21. Kriged log io k [m 2 ]
along various x-y planes.
62
30
1.85
1.6
1.35
1.1
0.85
0.6
0.35
0.1
-
FIGURE
2.22.
[counts lm].
10
Three-dimensional representation of kriged fracture density
63
-10
FIGURE 2.23. Three-dimensional representation of kriged matrix porosity log io O m
[m 3 1 n13].
64
-10
FIGURE 2.24. Three-dimensional representation of kriged water content 0 [m3/m3].
65
-1.4
-1.5
-1.6
-1.7
-1.8
-10
FIGURE 2.25. Three-dimensional representation of
[kPa].
kriged van Genuchten's log io a
66
-10
FIGURE 2.26. Three-dimensional representation of estimated air-filled matrix poros-
ity log lo O ma [m3/m3].
67
FIGURE 2.27. Random conditional realizations of various parameters at y
= 7 m.
68
2.4 Conceptualization of hydrogeologic conditions during
pneumatic tests
The conceptual hydrogeological model of airflow at the site during pneumatic tests
was originally defined by Guzman et al. [1996] and Guzman and Neuman [1996].
Here, I follow their findings and conclusions. The conceptualization is consistent
with pneumatic-test interpretations by Rasmussen et al. [1990], Illman et al. [1998],
Inman [1999], Illman and Neuman [2000b] and Inman and Neuman [2000a].
Pneumatic tests at the ALRS were conducted in a fractured unsaturated rock.
To prevent evaporation and infiltration, a plastic cover was placed on the ground
surface. Due to the high capillary retention properties of the porous (matrix) blocks
of rock (see Section 2.2.1), water is drawn from fractures into the matrix, leaving the
fractures primarily air-filled, and making it difficult for air to flow through matrix
blocks. As a result, airflow during pneumatic tests is considered to occur primarily
through fractures, most of which appear to contain little water. Moreover, since
matrix blocks appear to be virtually saturated with water, air storage effects in the
matrix blocks are not expected to play a role during pneumatic tests. While twophase flow of air and water may be taking place, it appears valid to disregard the
movement of water and treat air as a single gas phase. This is so despite findings by
Guzman and Neuman [1996] that, during single-hole tests, (1) pressure in the injection
interval typically rises to a peak prior to stabilizing at a constant value, and (2) air
permeability increases systematically with applied pressure. These observations are
most probably due to a two-phase flow effect whereby water in the rock is displaced by
air during injection. The air entry pressure of matrix can be estimated approximately
[van Genuchten, 1980] as the inverse of van Genuchten's a. This yields air entry
values that range from 15.6 to 98 kPa. These compare with air pressures in injection
intervals during pneumatic tests, which however die rapidly with distance from these
intervals. Therefore, one can expect two-phase flow effects to be important at most
69
in the close vicinity of the injection interval.
Guzman and Neuman [1996] conducted numerical simulations demonstrating that
the permeability determined from pneumatic tests under unsaturated conditions is
generally lower than intrinsic permeability. Nevertheless pneumatic permeability approaches intrinsic permeability as the applied pneumatic pressure increases. It follows
that air permeabilities and porosities derived from pneumatic test data should reflect
closely the intrinsic properties of the surrounding fractures, which are relevant to
both unsaturated and saturated conditions.
It has also been demonstrated that:
• Air permeabilities determined in situ from steady-state single-hole test data
are much higher than those determined on core samples of rock matrix in the
laboratory [Rasmussen et al., 1990], suggesting that the in situ permeabilities
represent the properties of fractures at the site [Guzman and Neuman, 1996].
• The permeabilities of individual fractures cannot be distinguished from the bulk
permeability of fractured rock in the immediate vicinity of a single-hole test
interval [Guzman and Neuman, 1996; Rlman et al., 1998; Illman and Neuman,
2000b].
• Air permeabilities are poorly correlated with fracture densities (Figure 2.7), as
is known to be the case for hydraulic conductivities at many water-saturated
fractured rock sites worldwide [Neuman, 1987].
• Along the boreholes, there is a poor correlation between the location and density
of fractures and local permeabilities from single-hole tests (Figure 2.11).
• The geostatistical interpretation of air permeability and fracture density data
[Chen et al., 1997] demonstrates that both parameters are characterized by very
different spatial structure functions (variograms) and (Figure 2.9) and kriged
estimates are quite distinct (Figure 2.17).
70
All these observations provide further support for the conclusion of Neuman [1987]
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. In addition, the fractures appear to form a dense interconnected threedimensional network (Table 2.2, Figure 2.5). As a result, it is reasonable to represent
fractured rock at the site as a single equivalent porous continuum.
Inertial effects were observed by Guzman and Neuman [1996] in only a few singlehole test intervals intersected by widely open fractures.
Enhanced permeability due to slip flow (the Klinkenberg effect) appears to be of
little relevance to the interpretation of pneumatic tests at the ALRS [Guzman and
Neuman, 1996; Illman et al., 1998].
I further assume that airflow at the ALRS takes place under isothermal conditions, ignoring adiabatic effects and differences in temperatures between injected and
ambient air in the rock.
Variation of atmospheric pressure at the ground surface clearly affect pressure in
the rock under both ambient and test conditions. They should be taken into account
to fully describe pressure responses during pneumatic tests.
71
Chapter 3
NUMERICAL INVERSE METHODOLOGY
The theoretical discussion on inverse methodology follows the principles and defini-
tions established by Neuman [1973], Neuman and Yakowitz [1979], Neuman et al.
[1980], Neuman [1980], Carrera [1984] and Carrera and Neuman [1986a, b, c].
3.1 Background
3.1.1 Governing equations
Gas flow in porous medium is governed by the following partial differential equation
over a three-dimensional domain St with boundary F [cf. Bear, 1972, page 200],
v . (kp vp +
/
J
a
(kgp 2q.= o op
OZ\/J
at
(3.1)
subject to initial and generalized boundary conditions
P = po on Si at t = 0,
(3.2)
kPVp) • n = V (p f — p) qb along F
(---
(3.3)
where p is absolute air pressure [M LT 2 ], p is air density [M/L 3 ], ,a is air dynamic
viscosity [M I LT], k is air permeability [L 2 ], 0 is air-filled porosity [—], qu, is a source
term [M/L 3 T], n is unit vector normal to r, p f is prescribed boundary air pressure
[M LT 2 ] on the boundary r, qb is prescribed air mass flux [M/L 2 T] normal to the
boundary F, y [T I L] is a parameter controlling the type of boundary conditions (first
or second type if y 0 or y oo, respectively; third type otherwise) and g is
72
acceleration due to gravity [L/T 2 ; 9.8 m/s 2 ]. The absolute air pressure p [N1m 2 ] and
air density p [kg/m 3 ] are related through the equation of state
PM
P = ZRT
(3.4)
where Z is a dimensionless compressibility factor depending on air pressure and tem-
°
perature, M is molecular mass [kg], T is absolute temperature [ K] and R is the
universal gas constant [8.314 J/( ° K • mol)]. In the case of isothermal gas flow (T is
taken to be constant), the problem is nonlinear due to dependence of p and on
pressure p. For the ranges of pressure and temperature monitored during pneumatic
test, the dimensionless compressibility factor, Z, is approximately constant and equal
to 1.
To characterize the borehole storage effect of the injection interval, the mass flow
rate Q from the interval into the medium (proportional to qa, in Equation 3.1) satisfies
the following mass-balance equation [e.g Neuman, 1988a]
dM
dl
(3.5)
where Q in3 is the rate of air mass injected in the packed-off interval and M is the
air mass confined in the injection interval. The right side of the last equation can be
rewritten as
dM dM dp„ , dp dp w
= vw
(3.6)
=
dpw dt
dpw dl
dt
where pw is air pressure at the injection interval and V,„ [L 3 ] is nominal volume of the
injection interval. Since by definition, air compressibility C a [LT 2 /M] is
(3.7)
one obtains
73
Qi 713 — Q =17",„pC„
dp,
(3.8)
dt
If the dimensionless compressibility factor, Z, in (3.4) is approximately constant, Ca
is approximately equal to the inverse of absolute air pressure. Assuming that the
pressures of injected air and air within the borehole interval are equal, the respective
air densities are also equal (T taken to be constant). Therefore, one can write a
volumetric version of (3.8)
dp,,
at
—
Q' = V„,ClaV„
1 dp„,
dt
(3.9)
where Q' = Q I p, [L 3 171 ]. The term on the right side in (3.9) represents volumetric
rate of change in storage within the injection interval. The borehole storage depends
on air compressibility, and decreases with the increase of absolute air pressure in
the injection interval. During air injection, the storage effect also decreases at late
times when air pressure in the injection interval stabilizes. During pressure recovery,
qi n3 = 0 and theoretically there is no borehole storage effect.
The borehole storage effect of the observation interval can be defined in a similar
way
Q , t vwcia
dt
dp„,
v
1 di:6
p,, dt
(3.10)
where Q/2„,,, and Q' t are the rate volumetric airflow into and out of the observation
interval.
If there is no flow from the packed-off interval into the medium (Q' = 0 in 3.9 or
Q'o„t = 0 in 3.10), the following dimensionless expression can be derived from (3.9) or
(3.10)
42-5 exp
— 1
(3.11)
74
where A 75 = P po
P° is dimensionless pressure buildup, 7= t 9 '—n' is dimensionless time.
-
-
When pneumatic test data are analyzed on a log-log plot of pressure versus time, the
borehole storage effect is characterized by an exponential curve. Only when pressure
buildups are small (Ap 1), T is also close to 1 and (3.11) can be approximated up
--
to first-order as
T
(3.12)
and the characteristic borehole storage curve on a log-log plot is a straight line with
unit gradient, which is similar to the case of a constant fluid compressibility [e.g
Neuman, 19884 Therefore, pressure records having significant pressure buildups at
early times might demonstrate deviations from the linear behavior due to the impact
of air compressibility on borehole storage.
3.1.2 Inverse problem
As defined in previous section, the governing air pressure p (x, t) depends, among other
factors, on the spatial distribution of air permeability, k (x), and air-filled porosity,
(x), throughout Q. One can express this relationship through
p (x,t) =
where a = [a', al
T
(a)
(3.13)
is a vector of discrete k and 0 values, a', as well as other parameters
describing their spatial distribution, a", and the forward operator ,F is a functional
that maps a into p (x, t) through Equations 3.1 - 3.4. When the forward problem
is solved numerically, ,F is a matrix and (3.13) is a set of algebraic equations. The
inverse problem can be defined formally as solving (3.13) for a based on a knowledge
of the state variable p (x,t)
a =
p (x, t)]
(3.14)
75
where is an inverse operator.
Neuman [1973] categorized methods of solving inverse problem into direct and
indirect. Direct methods employ a direct inversion of the forward operator „F in line
with Equation 3.14. This typically leads to a system of inverse equations in which
the parameters a are the dependent variables. Quite often the system is overdetermined, in which case it is solved approximately by optimization. When the system
of direct inverse equations is underdetermined, one typically augments it with constraints and/or includes a penalty criterion in the objective function to be minimized.
Neuman [1973] proposed to base this penalty function on prior measurements of the
parameters and/or prior ideas concerning their mode of spatial distribution (various
degrees/modes of smoothness). Indirect methods adjust the parameters a in the forward problem (Equation 3.13) so as to minimize the difference between computed
and measured
p(x,t) values at discrete points in space-time. This is usually done
iteratively by non-linear optimization. Adding a penalty criterion based on prior information [Neuman, 1980; Carrera and Neuman, 1986a, b, c] helps insure that the
resulting parameters are well-defined.
According to Hadamard [1932], a well-posed mathematical problem is one for
which a solution exists, is unique and stable. Although the forward problem defined
by Equations 3.1 — 3.4 is generally well-posed, the corresponding inverse problem
tends to be ill-posed. This is due to lack of sufficient information about the state of
the system (pressure, flux), measurement and interpolation errors, as well as computational errors associated with solving the forward problem. This can lead to
non-unique and unstable inverse solutions [Carrera and Neuman, 1986b].
Stability of the inverse solution requires that small variations in the state variable
p(x,t) produce only small variations in the estimated parameters a [Carrera and
Neuman, 1986b]. Mathematically, this means that for every positive E, there exists a
positive
S such that
76
11/33. (x, t) —P2 (x, t) II <€
> Hal — a2II <S
(3.15)
or the more rigorous Lipschitz condition
Hai — a2 11 < clIn. (x,t) —p2 (x, t) II
(3.16)
where p i (x, t) /3 2 (x, t), a la 2 , c is a constant and 11' 11 is a norm of appropriate
space for p and a. Typically, a Hilbert space LP (R) is applied,
L) .7=- {f
(x) E R 1 Ilf (RA' =
[f If
(x)1 2 dx]
2
}
(3.17)
R
where 11f (R)Il < oc and R is a real-valued domain.
The notion of uniqueness implies that if there are two solutions to the inverse
problem Equation 3.14, a l and a 2 , then
Hai — adl
0
11Pi (x,t) —P2 (x, t) H 0
(3.18)
If the solution of the inverse problem is unique, different parameter sets will produce
different pressures.
Identifiability differs from uniqueness, implying that different spatial distributions
of the parameters within a given subdomain, a l a 2 , produce different spatial con-
figuration of the state variable, p i (x, t) p 2 (x, t), or
11 .13 1 (x, t) — P2 (x, t)11 0
Hai — az (x)
(.1
(3.19)
Chicone and Gerlach [1987] proved theoretically that, for linear elliptic partial differ-
ential equations, the unknown spatially non-uniform conductance parameter (in my
case k) is identifiable within a given subdomain of the model region if every char-
acteristic that crosses its boundary stays in the subdomain. Dietrich and Newsom
77
[1990] extended the idea to the time-averaged transient case. In the case of most hy-
draulic and pneumatic tests, the parameters are usually not identifiable at locations
far from a pumping or injection wells. Identifiability is a necessary but not a sufficient
condition for well-posedness of the inverse problems. The existence of a unique and
stable solution requires that when the problem is solved by optimization, the objec-
tive function has a well-defined global minimum, which depends continuously on the
parameters [Carrera and Neuman, 1986b].
I already mentioned that the inverse problem can be properly posed by augmenting
its statement with supplementary information about the system. This may include
prior information about the parameters or constrains on their sign and magnitudes.
Neuman [1973] and Carrera and Neuman [1986a, b, c] demonstrate theoretically, and
by examples, that ill-posed inverse problems can be made well-posed through such
means.
A functional representation of parameters across the model domain is termed
parameterization. The number of discrete coefficients required to parameterize a
given medium can vary from 1 when the medium is assumed to be uniform to as many
as (or more than) the number of nodes or elements in the computational grid. For
best results, the parameterization must be complex and flexible enough to represent
adequately the spatial non-uniformity of the medium, and simple enough to yield a
well-posed inverse problem. The number of unknown parameters must never exceed
the number of observations (of both system state variables and parameters). The
difference between the former and the latter is referred to as degrees of freedom. In
general, as the number of degrees of freedom goes up, one can reproduce observed
behavior more accurately, but at the expense of loosing confidence in the parameter
estimates.
Parameterization can be either deterministic or stochastic (geostatistical). The
simplest deterministic approach is to assume the medium is uniform, which greatly
enhances well-posedness but causes all details about medium heterogeneity to be lost.
78
Some such details can be recovered by adopting piecewise constant parameterization,
known as zonation in which the model region is subdivided into a set of zones assigned uniform properties [Carrera and Neuman, 1986c]. Another compromise is to
define model parameters as low-order (e.g. linear or quadratic) polynomial functions
of space. Other popular options include interpolation based on inverse-distance or
piecewise weighting, splines, etc. The number of coefficients (discrete parameters)
utilized in the characterization of medium non-uniformity defines the degree of parameterization.
Stochastic (or geostatistical) parameterization provides a parsimonious way to
capture medium heterogeneity at a relatively high resolution with only a small number
of unknown parameters. The underlying idea is to represent spatially varying medium
properties as random fields, conditioned on discrete measurements of these properties.
Both analytical [Rubin and Dagan, 1987] and numerical [de Marsily, 1978; Clifton and
Neuman, 1982; Kit anidis and Vomvoris, 1983; Hoeksema and Kit anidis, 1984; Carrera
and Glorioso, 1991; Sun and Yeh, 1992; Yeh et al., 1996; Yeh and Zhang, 1996; Zhang
and Yeh, 1997; Gómez -Herncindez et al., 1997; Capilla et al., 1997] geostatistical
methods have been described in the literature. Zimmerman et al. [1998] provide an
excellent description of many of these methods, as well as a thorough comparison
among them based on a series of specially designed synthetic test cases. I adopt one
of these methods, based on the idea of pilot points, as proposed by de Marsily [1978].
3.2 Stochastic numerical inverse model
My stochastic inverse model is based on the geostatistical pilot-point method of parameterization [de Marsily, 1978], coupled with a maximum likelihood definition of
the inverse problem [Carrera and Neuman, 1986a]. Whereas the latter authors include prior information about parameter values in their statement of the problem, I
find it possible to pose the problem well, and solve it reliably, without the use of such
79
information (though its inclusion could, in principle, improve my estimates).
3.2.1 Pilot
point formulation
In the pilot-point method of de Marsily [1978], each medium property is viewed as
a spatially correlated random field. A smooth estimate of this field is obtained by
kriging of real and fictitious measurements of medium properties at discrete spatial
points. De Marsily refers to the locations of fictitious measurements as "pilot points."
As I use the method to analyze cross-hole pneumatic test data that do not represent
point measurements, I do not include any real but only fictitious (pilot) points in my
inverse model. Though I could, in principle, include the single-hole pneumatic test
data as real measurements to better condition my inverse solution, I have purposefully
not done so in order to explore the ability of my stochastic inverse solver to analyze
cross-hole pneumatic test data without the help of single-hole data. A joint analysis
of both sets of data would be an important future extension of my work. The location
of pilot points is arbitrary and can be assessed either on the basis of site knowledge
[de Marsily et al., 1984; Certes and de Marsily, 1991] or by optimization [La Venue
and Pickens, 1992]. The unknown values of medium properties at the pilot points
are estimated by the inverse algorithm.
Let the unknown random property of the medium, z, be a non-homogeneous
random field with homogeneous spatial increments (also called in the geostatistical
literature "intrinsic" random field of "second order") such that
E [z (xp + h)
—
z (xp)]
= 0
var [z (xp + h) — z (xP)1 -= 2-yz (PO
(3.20)
(3.21)
where z is defined at a discrete set of locations xp, called pilot points. The theoretical
variogram -y z (111111) depends only on separation distance 11h11 between the pilot points,
80
xp . Existence of mean and variance of increments does not imply their in the case of
z.
Consistent with the geostatistical analysis of single-hole permeability data, I utilize
an omni-directional power variogram
-y.(11h11)= allh11 13
(3.22)
where a (a > 0) and 0 (0 < < 2) are constants. In some of inverse analyses, I
treat a and [3 as unknown parameters to be estimated jointly with log permeability
and porosity at the pilot points by the inverse model.
A linear kriging estimator at a point x o is defined as
(xolz) = > (3.23)
(xo, xP) zi; Zi == Z ( Xpi )
i=1
where P is the number of pilot points and A i (x o ) are so-called kriging weights. The
latter are computed so as to insure that the estimator is exact (reproduces exactly
values at the pilot pints), unbiased and "best" (minimizes the estimation variance).
This leads to the following "ordinary kriging equations" for A z [Deutsch and Journel,
1992],
'yz
(xpi—xp)
i(xo)+/(xo) =
-
Yz (11xPi
—
P
(3.24)
j= 1
>2, A i (x o ) = 1
i=1
(3.25)
where i (x 0 ) is a Lagrange multiplier (determined jointly with A i ). Kriging produces
a maximum likelihood estimate of z provided that the field is multivariate normal
and Y. (IN ) is fully defined. Since I treat the parameters of -y, as unknowns
-
to be determined jointly with values at the pilot points, my solution is not strictly
81
maximum likelihood even if the estimated variables are multivariate Gaussian. The
kriging variance at point x o is given by
6-.2
(x0) = > (xPi xo) (x0)
1 (x0)
— 1
(3.26)
i=1
I apply these equations independently to log io k and log io ck, ignoring any crosscorrelation between these two variables. The vector of estimated unknown parameters
a include discrete log io k and log io q5 values, a', as well as the parameters a and 3
of their power variogram models, a". The kriging estimates in Equation 3.23 are
influenced by /3 but not by a. Therefore, a cannot be estimated directly by the
inverse model, but it can be evaluated a posteriori from the sample variogram of
kriging estimates within the computational domain.
3.2.2 Maximum likelihood approach
Define a vector of residuals
r (x, t; a) = p* (x, t) — (a) = pt (x, t) — p (x, t; a)
(3.27)
where p* and p are vectors of measured and simulated system responses (in my case
pressures) at points (x, t) in space-time, respectively. In general, both p and r depend
in a nonlinear fashion on a. The vectors pt, p, and r have dimensions N equal to the
number of pressure observations included in the model (I shall refer to these as match
points in Chapters 4 and 5). The vector a has dimension M equal to the number of un-
known parameters. The residuals r are affected by observation and simulation errors.
According to Carrera and Neuman [19864 observation errors represent differences
between field measurements and their unknown "true" counterparts. Simulation errors are differences between simulated and unknown "true" system responses which
are due to conceptual and computational errors. Conceptual errors arise from incor-
rect description of physicochemical processes and their mathematical representation
82
(mass-balance equations, initial and boundary conditions, equations of state) as well
as improper parameterization of medium properties. Computational errors arise from
numerical approximations and numerical round-off errors. Both types of error have
random and systematic components. A detailed analysis of various error types and
their effect on inverse solutions is given by Carrera [1984].
The likelihood function L(alp*) of unknown parameters a given measurements
p* and a forward model ‘F is proportional to the conditional probability P (p*Ia) of
observing p* given a. The maximum likelihood estimate of a, a, is defined by the
maximum of its likelihood function L(alp*), L(alp*). The likelihood concept implies
that parameters a are unknown deterministic quantities, and their uncertainty is due
to insufficient measurement data p* as well as observation and simulations errors
[Carrera and Neuman, 1986a]. If r is characterized by only random, non-systematic
components, and if a and r are mutually independent, then
L(alp*) = P (P * 1a) = P (r) =
P(1
3*
— P)
(3.28)
If P (r) is Gaussian with mean E[r] = 0 and covariance matrix C r ,. = E[rr T ], one
obtains
2
1 r T C;r1 r
L(alp*) = (27r 1C„1) - 21 exp --
(3.29)
Since the logarithm is a monotonically increasing function of its argument, it is convenient to search for the minimum of the function S defined as
S = —2 log [L(alp*)] =
= r T C;r1 r + log (C„) + log (27r)
(3.30)
Decomposing the covariance matrix C„ into unknown scaling factor, 3, and known
"weight" matrix, V„, such that C r ,. = s r2 V,,, the last equation can be rewritten as
83
rTv-ir
= 2" + log (1V„I) N log + log (27r)
S
The minimizing S is equivalent to minimizing the objective function 0 given by
r
=
(3.31)
Sr
T
v r-rl r
(3.32)
Since the errors r are assumed to be uncorrelated, the matrix V„ (respectively C„)
has a diagonal form and the objective function simplifies to a weighted least-square
problem
rTwr
(3.33)
where W is a diagonal weight matrix. The objective function can also be rewritten
as
0
=
—
wij (/):.; —P(x0i,ti; a)) 2(3.34)
where Np is the number of observation points with coordinates x o and NT is the
number of measurements in time t. The total number of observations N is the product
of Np and NT. The diagonal terms of W represent the relative weight of each
measurement in the optimization process. If measurement errors (diagonal terms
of C„) were known, the diagonal terms of W could be defined as w = 1/4 .
3.2.3 Optimization
Overview of optimization algorithms
Various automated optimization algorithms can be employed for the minimization of
the objective function 0 (a). All optimization algorithms require an initial estimate
of the unknown parameter vector a o . They then perturb this estimate by some La
so as to bring about a reduction in 0, and continue doing so iteratively until a given
84
convergence criterion is satisfied. Methods differ from each other by the way this
process is accomplished. Taylor expansion of the objective function to second order
about a current estimate a3 of the unknown parameter vector a at iteration j yields
1
0 (a3+1 ) -= 0 (a3 )+V a 0 (a )•(a3+1 — a 3 )+— (a +1 — a 3 )-V a2 0 (
2 3
) (a3+1 — a 3 ) (3.35)
where a3+1 is the vector of parameters at iteration j 1. Here Aa = a 3+1 — a 3 ,
Va 0 (a ) is the gradient vector and V E2, 0 (a 3 ) the Hessian matrix of I. Depending
on how many terms are retained in Equation 3.35, the search is either zero-, first-, or
second-order.
A zero-order search seeks the minimum of 0 without making any assumptions
about its topography, which has advantages as well as disadvantages. The downhill
simplex and simulated-annealing techniques are two examples [Press et al., 1992].
The former is very easy to program and implement, while the second is very efficient
in avoiding local minima.
First-order (gradient) search methods require computing of the gradient vector Va 0 (a3 ). The simplest but least efficient is the steepest descent method [e.g.
Chavent, 1971]. An algorithm proposed by Powell [1964] searches along conjugate
directions without calculating explicitly the derivatives of 0. There are several other
conjugate-gradient algorithms, but the one proposed by Fletcher and Reeves [1964]
was demonstrated by Carrera [1984] and Carrera and Neuman [1986b] to be the most
efficient.
Second-order (inverse-Hessian) search methods require evaluating the Hessian matrix 'ÇqL 0 (ai ). Direct computation of the Hessian matrix is not generally practical.
Most second-order methods therefore approximate it on the basis of the Jacobian
(sensitivity) matrix of derivatives, which consists of simulated pressure responses with
respect of the unknown model parameters. Consequently, the computational effort
required for each iteration is much larger than that for first-order search methods. It
85
is demonstrated in the literature [Cooley, 1985; Zou et al., 1993; Carrera et al., 1997;
Finsterle, 1999] that the overall rate of convergence and computational efficiency of
second-order methods are generally higher than those of first-order methods.
More efficient and robust optimization algorithms can be developed through a
combination of different-order search methods. For example, the Levenberg-Marquardt method, described below, combines first- and second-order search methods.
Alcolea et al. [2000] propose an algorithm that jointly uses the Levenberg-Marquardt
and simulated-annealing techniques.
Levenberg-Marquardt algorithm
The Levenberg-Marquardt algorithm is a robust method of nonlinear optimization
that combines the inverse-Hessian and steepest-descent methods. Inverse-Hessian is
a second-order optimization method which is very efficient close to the minimum.
Steepest-descent is a first-order (gradient-based) method which is more robust, albeit inefficient. The algorithm proposed by Levenberg [1944] and later improved by
Marquardt [1963] allows smooth transition between the two methods in the process
of optimization. It has become almost a standard of nonlinear optimization and
has been implemented in most groundwater parameter estimation programs such as
MODFLOWP [Hill, 1992], PEST [Doherty et al., 1994], UCODE [Poeter and Hill,
1997] and iTOUGH2 [Finsterle, 1999].
Starting from an initial parameter estimate a o , the algorithm minimizes the objective function iteratively until convergence is achieved. In my case, the objective
function 1) is (see Equation 3.33)
Np NT
Np
NT
= (Pi*j
(1. = r T Wr = tv,jri2
J
i=1 j=1 i=1 j=1
P(X0i7 tj;
a)) 2 (3.36)
Sufficiently close to the minimum, 43, and its gradient can be approximated roughly
to second order by
86
, 1
= 0 (a i )d-V a 0 (ai )•(ai+i — ai )-k(
—
2
Va 0 (a
i
— ai )•V a2 0 (ai ) (ai+i — ai ) (3.37)
) = V.0 (ai) • (a 3+1 — ai) V a2 0 (ai) (ai + 1 — ai)
(3.38)
Close to the minimum, this quadratic approximation of 0 should be satisfactory, and
the inverse-Hessian method allows calculating the minimum within a single step by
setting
Aa =- [Va l. (a)] 1 Va 0 (a)
(3.39)
where Aa = a 3+1 — a i . However, if the approximation is not satisfactory, an alternative is to use steepest descent which is more robust far from the minimum,
La = —cV a 0 (a)
(3.40)
where c is a vector of unknown positive constants. From Equation 3.36, the gradient
of 0 is given by
D
(a)
(a)
Np NT
ap(x0i, ti; a)
= 2 Yjw iip(x0i, t i ; a))
Da jacid
j=1
, / = 1,
N
(3.41)
Taking second derivative,
32 0 (a)
Np NT
wii[ aP(X0i,t jj a) ap(x 0i , t j ; a)
act /&tn.,
i=1 j=1
,\ a 2 p((xoi, ti; a)1
03:j — p(xoi, tj; a))
p ai a am
1
N,m
N
(3.42)
The second term in Equation 3.42, which consists of second derivatives of p with
respect to a, is generally ignored. Close to the minimum, this is justified because
87
(p7
—
p(t , a)) is close to zero and second derivatives of p are negligible compared to
i
first derivatives. Considering second-derivatives would have a destabilizing effect if
there are significant measurement errors (outliers) or if the model fails to simulate the
system response with a high enough accuracy [Press et al., 1992]. To calculate the
second derivatives would also be computationally intensive. It is therefore appropriate
to replace Equation 3.42 by
a20 ( a )
[ap(x0i,
actiaa ni
[
act/
t 3 ;
ap(xo„ t 3 ;
Dam
i, ... ,
,i
,m = 1,
N (3.43)
Consider the Jacobian matrix as J = V a r (x, t; a) = —V a p (x, t;a) (i.e., Ji+i , / =
ar(c oi ,ti; a)
ap(x0i,tj;a)
a.,
a.,
)
and rewrite Equations 3.41 and 3.42 in matrix notation as
Va 0 = —2JWr
(3.44)
V a2 0 = 2J T WJ
(3.45)
where J T WJ is a so-called "normal" or curvature matrix. Substituting the last two
expressions into Equations 3.39 and 3.40, yields
Aa = [J T WJ] J T Wr
La = 2cJ T Wr
(3.46)
(3.47)
Marquardt proposed that the vector of unknown constants c in the steepest descent equation (3.40) be related to the diagonal elements of [J T WJ]1 , multiplied
by some positive fudge factor a, called the Marquardt parameter. Defining a square
diagonal matrix D as
3.48)
D= [J T WJ 1J(
ii
88
thus yields
1
c = aDTD
2
—
(3.49)
Marquardt also suggested combining Equations 3.46 and 3.47 into one expression
Aa = [J T WJ + aD T D] 1 J T Wr (3.50)
Depending on the Marquardt parameter a, the algorithm ranges from inverse-Hessian
(a — 0) to steepest descent (a >> I). At each iteration, the algorithm searches for
an a that minimizes the objective function. The largest element of aD T D is typically
called the Marquardt parameter A.
In many problems, the observations and optimized parameters differ significantly
in magnitude. The same may happen to the elements of J. To minimize round-
off errors, it is advisable to scale the Jacobian matrix J by the diagonal matrix D
according to
Aa. = [(JD) T W (JD) + aD T D1 1 (JD) T WDr (3.51)
When a is large and the algorithm is close to steepest descent, convergence may
be accelerated by defining an upgrade vector (Aa, where ( is an adjustment factor
given by [Doherty et al., 1994]
C=
rTDTWJDAa
AaTDTJTWJDAa
(3.52)
3.2.4 Linearized analysis of estimation errors
In the inverse methodology, the analysis of estimation errors is of critical importance.
Here I follow the linearized analysis of estimation errors proposed by Carrera and
89
Neuman [1986a]. It assumes that the forward model .F (a) is linear close to the max-
imum likelihood parameter estimate, 'a, and parameter estimation errors are multiGaussian. Therefore, the parameter estimation errors are fully characterized by their
mean (equal to zero) and covariance matrix —aa
E (E
,—aa — E [(a — a) (a — a.)1). The
generalization of the Cramér-Rao inequality for the multivariate case demonstrates
that E aa is such that E aa — F -1 is semi-positive definite, where F is the Fisher
information matrix [SiIvey, 1975]. Therefore, F -1 defines 'a lower bound' for the
covariance matrix of estimation errors. In terms of the maximum likelihood function,
F is estimated as [SiIvey, 1975]
F=
2
1
E [ 82
0a 2
(3.53)
where S is defined at the optimum -6... The Fisher information matrix F represents a
measure of information about the parameters that is contained in the inverted data,
and E aa is a measure of estimation uncertainty. The more information is contained
in the data, the less uncertain are the parameter estimates. Ignoring second order
terms, the partial derivatives of S with respect to pneumatic parameters are
325 c) (Ory ,
(Or ) _
aa 2 — , , --e.k., 1
-
2j T c7.7,1 j
(3.54)
and as a result
F = J T C,7,1 J
(3.55)
The Fisher information matrix F may become singular when some of the parameter
estimates are highly correlated. The correlation between estimates will render the
corresponding matrix components in J, and respectively in F, to be also correlated.
If F is not singular, the covariance matrix of estimation errors E aa can be defined as
E aa = F -1 = [J T C 77,1 J]
1
= s, [J T V 777,1 J] -1(3.56)
90
When the scaling factor of covariance matrix .9 72. is unknown, it can be assessed a
posteriori, after optimizations have been completed as
s2 =
r T V„r
rTWr
(N — M) (N — M)
(3.57)
The 100(1 —10% confidence region of a is defined as
S (a) — S (a) <
where
N — M)
(3.58)
(M, N — M) is a quantile of the F-distribution. The shape of the con-
fidence region in general is arbitrary, defined by the contour of S on level S (a) +
s r2 MF1 _,(M,N — M). Due to our assumption that the model ,F (a) is linear in the
vicinity of a (.F (a)
(a) (aaS:)
(a — a)), the confidence region can be approxi-
mated by an ellipsoid defined as
(a — a) T E;c,1 (a —)< MFi _„ (M, N — M)
(3.59)
The linearization produces an overprediction of the size of the confidence region along
the longer ellipsoidal axes [Carrera, 1984; Carrera and Neuman, 1986b]. The diago-
nal terms of E aa define the variances of estimation errors for respective parameters.
The parameter uncertainties can be represented by separate 100(1 — n)% confidence
intervals defined as
—
(N — M) , = 1, M (3.60)
where cr i is a square root of the i'th diagonal terms of E aa and t i „ /2 (N — M) is a
quantile of Student's t-distribution. When the estimation errors are highly correlated,
the variances of estimation errors significantly underestimate parameter uncertainties,
and a computation of simultaneous confidence intervals is required [Carrera, 1984;
Carrera and Neuman, 1986b; Vecchia and Cooley, 1987].
91
3.2.5 Calculation of sensitivity matrix
Three different methods can be employed to compute the sensitivity, or Jacobian,
matrix J required for implementation of the Levenberg-Marquardt algorithm.
Partial derivatives in the sensitivity matrix can be replaced by their finite dif-
ference approximation. This is easy to implement, but is computationally intensive
because it requires a series of forward model runs (M + 1 or 2M + 1 for forward and
central difference schemes, respectively, M being the number of unknown parame-
ters). However, the model runs are independent and can be conducted in parallel,
which allows efficient simulation on multi-processor computers. Most importantly,
this method does not require any change in the forward simulator.
It is also possible to compute sensitivity coefficients directly by differentiating the
original partial differential equation with respect to the unknown parameters. The
resultant sensitivity equation is similar to the original equation and therefore a model
similar to the forward model can be used to compute the sensitivity coefficients. Additional computational and programming efforts are required to calculate the source
term of the sensitivity equation. The method requires M +1 model runs, which again
can be easily parallelized. Compared to the finite-difference method, this approach is
more computationally intensive, but more accurate. Carrera et al. [1990c] proposed
an efficient approach which computes the sensitivity matrix simultaneously with the
solution of a nonlinear forward problem. However, this approach requires further
programming modifications in the code of the forward simulator.
A third option is to use the adjoint-state method proposed by Chavent [1971]
and extended by Neuman [1980] to finite-elements. This method is widely used in
combination with first-order algorithms because it requires only 2 model runs to compute the gradient of the objective function with respect to the unknown parameters.
To compute the sensitivity matrix, the method needs Np + 1 model runs (Np being the number of measurement points), which renders it computationally intensive
92
when Np > M, as is typically the case. The implementation of the adjoint-state
method requires additional computational and programming efforts to calculate the
parameters of the adjoint-state equation. In the case of our governing equation (3.1),
the adjoint-state partial differential equation has the form of a convective-diffusion
equation [Sun, 1994], which renders its numerical solution more difficult.
3.2.6 Model identification criteria
Carrera and Neuman [1986b] used model identification criteria to differentiate be-
tween models having different structures (parameterizations). These criteria support
the principle of parsimony in that among alternative models producing similar re-
sults, that with the smallest number of unknown parameters is generally ranked the
highest. The four criteria, due to Akaike [1974], Akaike [1977], Hannan [1980] and
Kashyap [1982], are (in the same order)
AIC (â) = S (â) + 2M
BIC
* = S (a)
(3.61)
M ln N
(3.62)
Om (â) = S (a) + 2M ln (ln N)
dm (a) = S (a)
(3.63)
M ln (27r ) + 1n11 1
1
(3.64)
where â is a maximum likelihood estimate of the parameter vector a, and S is from
Equation 3.31.
93
3.3 Numerical inverse model for ALRS
A three-dimensional numerical inverse model was created to interpret single-hole and
cross-hole pneumatic tests at the ALRS and to identify the spatial distribution of
fractured rock properties. The inverse model is based on an indirect nonlinear approach and incorporates a forward flow simulator coupled to a parameter-estimating
algorithm, which are used in conjunction with various parameterization schemes.
3.3.1 Computational domain
The three-dimensional computational domain measures 63 m in the x direction, 54
m in the y direction and 45 m in the z direction, encompassing a rock volume of
153,090 m 3 (Figure 3.1). The medium is treated as a uniform or non-uniform continuum that is locally isotropic. Flow through the medium is taken to be single-phase
isothermal. Medium properties to be evaluated include air permeability k [m 2 ] and
air-filled porosity 0 [m 3 /m 3 ].
3.3.2 Boundary and initial conditions
As I consider only single-phase airflow, the saturation of air and associated pneumatic
properties of the rock, remain constant during each simulation. The side and bottom boundaries of the flow model are impermeable to airflow. These boundaries are
placed sufficiently far from the injection and observation borehole intervals to have
virtually no effect on simulated pneumatic tests. The top boundary coincides with
the ground surface and is maintained at a constant and uniform barometric pressure.
During the cross-hole pneumatic tests, the absolute barometric pressure at the ALRS
was not measured; records exist only for relative variations of barometric pressure. In
my inverse simulations, I assumed that the barometric pressure is equal to 100 kPa,
which is the average barometric pressure at mean sea level. In fact, the ALRS is at
an elevation of 1,200 m above mean sea level, and the average barometric pressure
94
o
-10
N
-20
3
-30
-40
FIGURE 3.1.
Computational domain and explicitly simulated boreholes.
95
should be approximately 87 kPa, but can vary with meteorological conditions. I
performed inverse analyses on the influence of this barometric pressure discrepancy
on the parameter estimates. Barometric pressure fluctuated during each pneumatic
test; in some of these tests (e.g. cross-hole test PP4) the fluctuations were significant
enough to influence substantially air pressure responses in some of the monitoring
intervals. For the moment, barometric pressure fluctuations have not been simulated.
It could be that had I incorporated these fluctuations in the inverse model, my estimates would have been somewhat different (especially in the region close to ground
surface). Initial air pressure was set equal to the barometric pressure throughout the
flow domain.
3.3.3 Borehole effects
Due to the high compressibility of air, and the non-uniform nature of the rock, I
expected both borehole storage and conductance to have an important impact on
airflow and storage in the modeled region. The total volume of open boreholes is 4.1
m 3 , which is small in comparison to the rock volume between the boreholes, approximately 10 x 10 x 10 (1,000) m 3 , within which most of airflow takes place. Nevertheless,
its effect is important. Borehole effects, especially conductance, are often ignored in
the interpretation of field tests. There is little information in the literature about the
effect that open borehole intervals may have on pressure propagation and response
during such tests. Paillet [1993] noted that the drilling of an additional observation
borehole had an effect on drawdowns during an aquifer test [see also Pickens et al.,
1987; Carrera et al., 1996]. In Illman et al. [1998] and Illman and Vesselinov [1998],
I demonstrated through numerical simulations that the presence of open borehole intervals has a considerable impact not only on pressure propagation through the site,
but also on measured pressure responses within monitoring borehole intervals during cross-hole pneumatic tests. Similar results were later presented by Huang et al.
96
[1999]. My numerical model accounts directly for conductive and storage effects of
all open borehole intervals by treating them as high-permeability and high-porosity
cylinders of finite length and radius. How exactly the boreholes are incorporated in
the model is described in the next section.
3.3.4 Computational grid
The grid generator X3D [Cherry et al., 1996; Trease et al., 1996] was adopted to
automatically subdivide the computational domain into tetrahedral elements. Grids
were created in a manner that enhances the computational efficiency of the FEHM
simulator, described below. One of the computational grids employed in my study
is illustrated, by means of two-dimensional images, in Figures 3.2 and 3.3. Figure
3.2 shows three views of the grid perpendicular to the x-y, x-z and y-z planes. As
the grid in the vicinity of boreholes is relatively fine, the corresponding areas appear
dark in the figures. Figure 3.3 shows four cross-sectional views of the grid along
vertical planes that contain selected boreholes. Since the grid is three-dimensional,
its intersections with these planes do not necessarily occur along nodal points (i.e.,
what may appear as nodes in the figure need not be such).
The grid shown in Figures 3.2 and 3.3 is employed for inverse analyses of both
single-hole and cross-hole pneumatic tests performed by injecting air into borehole
Y2. Other grids used in my study are very similar. All of them can be divided into
three zones (Figures 3.2 and 3.3): (1) a regular grid at the center of the modeled
area, which measures 30 x 20 x 25 (15,000) m 3 and has a node spacing of 1 x 1 x 1
m 3 ; (2) a surrounding regular grid haying a node spacing of 3 x 3 x 3 m 3 ; and (3)
a much finer and more complex unstructured grid surrounding each borehole. In
the last mentioned zone, different discretizations are applied around injection and
monitoring boreholes. As a result, different grids are adopted for the simulation of
air injection into different boreholes. I have interpreted pneumatic tests performed in
97
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4 different boreholes (Y2, X2, Z3 and W3) and, for each case, I have used a different
computational grid. In each grid, the number of nodes and tetrahedral elements are
respectively: Y2 — 39,264 and 228,035; X2 — 39,288 and 227,961; Z3 — 39,797 and
231,366; W3 — 43,812 and 254,310. The number of nodes and tetrahedral elements
is largest in the case of injection borehole W3 because it is longer than the other
injection holes. Node spacing at the center of the computational domain, where
most of airflow takes place, is equivalent to the 1-m support scale of the majority of
single-hole tests (Chapter 2).
A 5-m long segment of the three-dimensional computational grid along an injection
borehole is shown in Figure 3.4. This grid is wider and finer than those associated
with monitoring boreholes so as to allow accurate resolution of the relatively highpressure gradients that develop around the former. However, the grids around both
injection and monitoring boreholes have similar structures. Along the central axis
of each borehole, nodes are spaced 0.5 m apart. The surrounding grid is designed
so that the sum of computational volumes associated with these nodes is close to
the actual volume of the borehole. Additional nodes are located along radii that are
perpendicular to this axis within each borehole grid. The intervals between these
nodes grow sequentially with distance from the borehole axis by a factor of 1.6,
thus forming a geometric series. The number of rays and nodes associated with
the injection borehole are larger than those associated with monitoring boreholes.
Each borehole grid is additionally refined near the ground surface so as to obtain an
accurate resolution of conditions near this atmospheric boundary. Where boreholes
are located close to each other (as in the cases of V2 and the W-series of boreholes;
W2A and W2; W1 and Yl; W3 and Y3), the grid between them is made finer in
order to resolve correctly processes that take place within this grid volume. The most
complicated of the grid structures is that representing the region between boreholes
V2 and W2, which intersect each other (Figure 3.3).
To simulate the effect of open borehole intervals on pressure propagation, these
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intervals are treated as porous media having much higher permeability and porosity than the surrounding rock. The permeability and porosity of nodes along the
open borehole intervals are set to 3.23 x 10 m 2 and 1.0 m 3 /m 3 , respectively; these
correspond to an ideal tube with radius equal to that of a typical borehole. The
permeability and porosity of instrumented borehole intervals are set to 3.23 x 10 -5
m 2 and 0.5 m 3 /m 3 , respectively. At the intersection of boreholes V2 and W2, a lower
permeability of 10 -1 0 m 2 is assigned to avoid numerical difficulties; this value is still
orders of magnitude higher than that of the surrounding rock. Packers are assigned
zero permeability and a porosity of 10 -5 m 3 /m 3 . Figures 3.5 and 3.6 show examples
of permeability and air-filled porosity distributions within the computational domain.
In the figures, open borehole intervals are shown to be of high permeability/porosity,
and packers of low permeability/porosity. The spatial distributions of log lo k and
log lo 0 in the figures are those shown earlier in Figures 2.18 (page 58) and 2.26 (page
66), respectively.
In most inverse runs, the values of permeability and porosity along boreholes were
kept constant as specified above. In a few instances, I allowed the inverse model to
optimize the porosity of either the injection interval or both the injection and the
observation intervals. I also ran test cases in which the borehole effects were excluded
by specifying values of permeability and porosity along boreholes equal to those of
the surrounding rock.
The dimensionless borehole storage coefficient CD is directly related to the porosity
0„, along the borehole intervals. In the case of three-dimensional airflow from a point
source, CD is defined [Inman et al., 1998] as
CD =
C, p
ll
(3.65)
where C„, = Vs Ca [m 5 /N] is storage coefficient of the injection interval, Vs [m 3 ]
is effective storage volume, Ca [m 2 /N] is air compressibility (as discussed in Section
cv
102
In
(-V
[ w] z
in
co ri
in
•zi
Lc,
Lc)
Lri (z) cc> N-
103
Ca)
0,1
mom
N
.._
E
u
x
CD CO
cr
N
(0 CO
104
3.1.1, equal approximately to 1/p where p is absolute air pressure), Vu, [m 3 ] is nominal
volume of the injection interval and q,. is effective air-filled porosity of the surrounding
rock medium (for uniform medium,
o
r
is simply equal to the air-filled porosity
Since I define the effective storage volume Vs as
0).
Vu,0„„ the above equation takes the
form
CD
Vs Ow
=3 O
r
30,
(3.66)
The effective porosity of the borehole interval, çb w , is allowed to take on values in
excess of 1.0, as a way of accommodating effective interval volumes that are larger
than those built into the model. In this way, Ow can account for air storage not only
in the injection interval, but also in the surrounding rock.
The net result is a complex three-dimensional grid which represents quite accurately the geometry, flow properties, and storage capabilities of vertical and inclined
boreholes at the ALRS; is capable of resolving medium heterogeneity on a support
scale of 1 m across the site; is able to represent, with a high degree of resolution, steep
gradients around the injection test interval, as well as pressure interference between
boreholes, no matter how closely spaced; and assures smooth transition between fine
borehole grids having radial structures and surrounding coarser grids having regular structures. Such complex three-dimensional grids incorporating boreholes with
various spatial orientations have not been previously applied in the hydrogeological
research [see also Trease et al., 1996]. For example, Carrera et al. [1996] simulated
boreholes on a much coarser computational grid utilizing one-dimensional elements
interconnecting nodes located along the boreholes. A fine grid around boreholes is
required to accurately represent flow from and into both injection and observation
intervals.
105
3.3.5 Program FEHM
I simulated the pneumatic test data by means of a three-dimensional finite-element /
finite-volume heat- and mass-transfer code, FEHM version 96-05-07, developed at Los
Alamos National Laboratory, New Mexico [Zyvoloski et al., 1988, 1996, 1997]. My
decision to use FEHM was based in part on the ability of this code to simulate nonisothermal two-phase flow of air and water in dual porosity and/or dual permeability
continua, and to account for discrete fractures, should the need to do so arise (it never
did).
The code is employed to simulate three-dimensional isothermal gas flow defined
by Equations 3.1 - 3.4. The problem is nonlinear due to dependence of air compressibility, Ca , and air viscosity, ,a, on absolute air pressure, p. Variations in air
viscosity are ignored in FEHM, which is justified for the ranges of pressure and temperature recorded during the pneumatic tests. Air dynamic viscosity ti, is set equal
to 1.82 x 10 -8 Pa • s [Zyvoloski et al., 1997].
Spatial discretization of the governing equations is done on a three-dimensional
computational grid of tetrahedral elements. Since the medium is assumed to be
isotropic, the code is applied using its finite-volume (integrated finite-difference) option which is computationally more efficient than the finite-elements. The volume
associated with each node is defined using three-dimensional Voronoï diagrams based
on Delaunay tessellation [Watson, 1981]. Model parameters, here air permeability k
and air-filled porosity 0, are defined at the nodes of the grid, not over the elements.
Parameters are viewed as averages over control volumes associated with respective
nodes. Modifying Equation 3.1 by replacing spatial and time derivatives with their
finite-difference approximations and multiplying by the finite volume V3 , one obtains
an equation representing local mass balance for node j at time step s,
106
wk
i=1
ILL
(Pis) 2 Szi qiVi =
(PI p;s ) P:Ali ps3 Si; + kjg
1LL
p s.-1
3
At,
V
(3.67)
Here superscripts and subscripts define time step and node, respectively (e.g. ps3 -1
represents air pressure calculated at node j at time step s — 1), W is the number of
nodes in the neighborhood of node j, ki 3 is the upgradient air permeability between
nodes i and j, Al i 3 and Si j are respectively the distance and effective flow area between
nodes i and j, Sz3 is the effective area for gravity flow associated with node j, and
At s is the size of the s'th time step. Parameters W, V3 , A/ ii , Si3 and Sz , are obtained
from the Voronoi: grid [Zyvoloski et al., 1997]. The time derivative is discretized
using a fully implicit backward difference scheme, which is unconditionally stable and
first-order accurate in time; for nonlinear problems, it is generally preferred to the
higher-order central-difference scheme.
Rearranging the mass balance equation (3.67) and taking into account initial
conditions, boundary conditions and equation of state 3.4, a system of nonlinear
algebraic equations is obtained for time step s [Zyvoloski et al., 1997]
A(Ps)Ps = b(p5)
(3.68)
where A is a so-called "conductance" matrix and b is a vector representing boundary
conditions and internal sources. Both A and b depend in a nonlinear fashion on air
pressure p at the nodes. In the most general case, A is a square matrix of size U x U,
and p and b are vectors of size U, where U is the number of nodes used to discretize
the computational domain. The system of nonlinear algebraic equations is solved by
means of the Newton-Raphson iterative method. Let
d(Ps) = A(Ps)Ps — b(Ps
(3.69)
be a vector of residuals. The residuals are to be minimized iteratively. For the m'th
iteration, the last equation can be rewritten as
107
dm(P s ) = Am(P s )Pis-ri — bni(P s )
(3.70)
The residual for the next, m + l'st iteration is estimated to first order with respect
to d m (ps) through
clin+i( ps) = dm(Ps) + (vpd(P 5 )) A PL+ 1 (3.71)
where Ap m
s +1 = p m
' +1 — p m
s and (V p d(ps)) m is a Jacobian square matrix of size
Ux
U, representing the partial derivatives of residuals d with respect to the current
estimate of unknown pressures p,
(Vpd(P s )) —
adm(Ps)
Op
(3.72)
Assuming that d m+i = 0, Equation 3.71 is solved for Ap m+1 by inverting (V p d(ps)),
= [(Vpd(138 ))] 1 €1 .(P s ) (3.73)
In general, the Jacobian matrix (V p d) m is neither symmetric nor sparse. In FEHM,
successive iterations are performed until the L2 norm of residuals d(ps) becomes less
than a prescribed tolerance. The code uses the algorithm GZSOLVE [Zyvoloski and
Robinson, 1995] to provide a robust implementation of the Newton-Raphson sparse
systems of nonlinear equations. Based on the convergence of the Newton-Raphson
method, FEHM employs a semi-automatic time-step control. If convergence is not
achieved within a given number of iterations, the time step is reduced by a certain
factor and the procedure repeated. If convergence is attained with a specified number
of iterations, the next time step is incremented by a certain factor. The user also
specifies limits for the minimum required and maximum allowed number of iterations
in each time step.
108
3.3.6 Parameterization
The interpretation of pneumatic tests is performed assuming that the medium is either
uniform or non-uniform. In the first case, the unknown parameters are uniform values
of air permeability k and air-filled porosity 0 throughout the computational domain.
It provides information about medium properties on the scale of the cross-hole test,
and allows comparing the numerical model with analytical type-curve methods for
the interpretation of both single-hole and cross-hole tests.
For the simultaneous interpretation of cross-hole tests, I treat the medium as a
non-uniform stochastic continuum. In some cases, I do so for air permeability while
keeping air-filled porosity uniform, in other cases, for both air permeability and airfilled porosity. I have also explored some other approaches to characterize medium
non-uniformity.
The geostatistical analysis in Chapter 2 shows that fractured rock properties at
the ALRS can be viewed as spatially correlated (regionalized) random variables, and
the rock can be represented as a stochastic continuum. Based on this finding, I define
spatial variations in the unknown medium properties using a stochastic inverse model
based on the pilot-point method.
Pilot points are located within the tested region where the monitoring intervals
are concentrated. Most are located along boreholes; some are between borehole in-
tervals. The number of pilot points varies between the inverse runs from 32 to 72. At
each pilot point, there is an unknown log-transformed air permeability and air-filled
porosity. Admissible ranges for the estimates are set between —20 and —10 for log lo k
and between —5 and —0.5 for log io 0. The inverse model provides covariances and
confidence intervals for the corresponding estimation errors.
I treat the variogram parameters as unknowns, estimating the exponent
0 either
separately or simultaneously with unknown medium properties at the pilot points.
In the simultaneous case, I assign an upper limit of 1 on the values of 0, in order to
109
avoid convergence problems in the forward problem. The constant a of the power
variogram model does not influence the kriging estimates, and therefore cannot be
evaluated directly by the inverse model. However, a can be evaluated a posteriori
from the sample variogram of the kriging estimates obtained by the inverse model.
In my inverse model, kriging is performed by means of the geostatistical code
GSTAT Pebesma and Wesseling [1998]. GSTAT is preferred to other kriging programs
such as GSLIB [Deutsch and Journel, 1992] due to its simple input and output,
easy batch-mode processing, option to simulate over unstructured grids, and (most
importantly) ability to perform global kriging which in my case, is computationally
much more efficient than employing a search neighborhood. For example, over a given
structured grid with given input data, GSLIB requires 6 hours and GSTAT 2 minutes
to compute the kriging field.
3.3.7 Program PEST
A detailed description of the PEST code is given in its manual [Doherty et al., 1994].
Here I discuss only major features of the code, which are required to run the numerical
inverse model.
The code uses a variant of the Levenberg-Marquardt algorithm to estimate model
parameters a by minimizing .1) (a). For each unknown parameter, one must specify
an initial value and a weight. PEST allows the parameters to be internally (1) scaled
and offset, (2) log transformed, (3) tied to another parameter, or (4) fixed. PEST
also requires information about an initial value of the Marquardt parameter a and a
series of parameters defining the optimization of (1. (a) for each iteration.
There are two ways to limit the variation of optimized parameters in PEST. The
first one is to define upper and lower bounds for each parameter. The second is to
use prior information with weights equal to the inverse variances of prior estimation
errors. In my case, prior kriging errors are mutually correlated, a situation that PEST
110
cannot handle. This is one reason why I do not rely on prior information.
PEST allows limiting the amount by which a parameter is accepted to change
during any iteration. This helps the optimization of highly nonlinear models. Two
types of limits can be employed: factor and relative. Let a i be an unknown parameter
and a io its current "best" estimate. In the factor-limited case, the plausible range of
a i for the next iteration is given by
ai E [
fi
aiofil if ai o > 0
, azoi .,
a, E [ azoji,
—
fi
if aio < 0
(3.74)
where fi is the maximum allowed factor change (fi > 1). Factor-limited parameters
can not change sign. In the case of relative limits, the plausible range of a i is
(3.75)
where r i is the maximum allowed factor change (r i > 0). It is not recommended to
use r i > 1, which allows parameters to change substantially, including a change in
sign or becoming zero. The magnitude of parameter upgrade vector is defined such
as neither of plausible parameter ranges is violated.
PEST uses a technique proposed by Cooley [1983] to reduce parameter over-
adjustment and to damper potential oscillations in parameter changes. Let Aa i be
the maximum change for the i'th parameter within the current iteration. In the
previous iteration, the actual parameter upgrade was associated with a maximum
change Aa 03 for the j'th parameter. The new parameter upgrade vector is defined as
•Aa, where 6 is such that
6 _
4i : I ii ff
s:—
{ 33,
- -FF
S
where
11
(3.76)
111
s =
z= j
Aao,
if i j
0
(3.77)
Accurate computation of derivatives of observations with respect to parameters is
critical for success of the Levenberg-Marquardt algorithm. PEST calculated derivatives directly using a finite-difference method, which requires a series of simulation
runs. For each optimized parameter, it employs either a forward or a central difference
scheme. For given group of parameters, PEST can switch from forward to central
differences during the optimization when the relative reduction in the objective function between two successive iterations is less than a given value. In the case of the
central scheme, PEST can calculate the derivatives using three different approaches.
The most accurate approach is to fit a parabola to three parameter-observation pairs
and then compute its derivative. Another approach is to fit a straight line instead
of a parabola, which is recommended for cases when simulated observations can not
be calculated with good precision. The third possibility is to use standard central
differences, which ignores the middle parameter-observation pair. For each group of
parameters, increments for derivative calculations can be defined (1) absolutely, (2)
in relation to current parameter values, (3) in relation to the current maximum values
for the parameters.
To terminate the optimization process, PEST utilizes 7 criteria:
1. The objective function 0 becomes zero.
2. A given number of iterations has elapsed since the lowest objective functions 0
was achieved.
3. A given number of local minima for 0 are within a specified distance in param-
eter space of each other relative to the lowest 0.
4. The gradient of the objective functions 0 with respect to all parameters is zero.
112
5. The upgrade vector components for each parameter is zero.
6. The maximum relative change of parameters within a given number of iterations
is less than a specified tolerance.
7. The number of iterations exceeds a specified limit.
After termination of the optimization process, if the normal matrix is non-singular,
PEST computes the covariance matrix of optimized parameters. It defines the sepa-
rate 95% confidence intervals of optimized parameters as discussed in Section 3.2.4.
It also calculates the correlation matrix and performs eigenvector analysis of the covariance matrix. Since the covariance matrix is positive definite, eigenvalues are real
and eigenvectors are mutually orthogonal, representing the axes of an M-dimensional
probability ellipsoid (M being the number of optimized parameters).
As explained earlier, the error analysis assumes that (a) the measurements are
mutually uncorrelated, (b) the estimation errors are Gaussian, and (c) the model
is linear. In my case, none of these assumptions are expected to be fulfilled and,
therefore, I consider the corresponding statistics merely as crude approximations.
I have modified PEST to calculate the four model identification criteria discussed
in Section 3.2.6.
3.3.8 Parallel processing
Enhanced computational efficiency can be achieved by parallelizing the evaluation of
the Jacobian matrix J. Doherty [1997] created a parallel UNIX version of PEST. I
have modified this parallel version so as to utilize more optimally the computational
resources of a standard UNIX multi-processor environment. I have further altered
PEST to allow efficient restarting of the optimization process, if and when it ter-
minates prematurely, so as to virtually eliminate loss of computational time. The
113
parallelized version of the inverse model has been run on the University of Arizona
SGI Origin 2000 32-processor cluster.
3.3.9 Code interaction
My numerical inverse model incorporates FEHM, PEST and GSTAT. In addition,
I use X3D to generate the computational grids, GMV [Ortega, 1995] to visually
analyze them, and TECP LOT [Amtec, 1997] to present results graphically. I have
supplemented these codes with a series of data-processing programs, which facilitate
the generating, reformatting, handling, analysis and visualization of massive input
and output data files.
PEST is designed to interact with any kind of external simulator. Its features
allow not only easy generation of one or a number of input files (which include the
current values of optimized parameters) but also elaborate reading of one or a number
of output files (which contain the simulation results for defined parameters). Since
my inverse model is rather large and the corresponding FEHM input and output
files are huge, these very nice features of PEST could not be fully implemented.
Hence, additional pre- and post-processing codes for FEHM inputs and outputs were
developed to allow direct and e ffi cient (in terms of disk-space and computer-time)
interfacing of PEST, GSTAT and FEHM.
3.4 Linearization of governing equations
The numerical inverse model solves the equations governing gas flow in their original
nonlinear form (Equations 3.1 — 3.4, page 71). However, the analytical type-curve
results of [Illman et al., 1998], Inman and Neuman [2000b] and Illman and Neuman
[2000a] with which I compare some of my numerical inverse estimates are obtained
using a linearized version of the governing equations. Ignoring the gravity and source
114
terms, the so called p and p 2 based linear representations of (3.1) for spatially uniform
medium are defined as [Inman et al., 1998]
• p based:
V 2p
1.10Ca Op
kp at
(3.78)
POCa 3p 2
kp at
(3.79)
• p 2 based:
v2,n2
where air compressibility C a is equal to
h 5 (see Section 3.1.1) and 19 is some constant
- -
absolute air pressure; Illman et al. [1998] took p to be the barometric pressure. During
-
single-hole and cross-hole pneumatic tests, pressure buildups not only were significant
relative to the barometric pressure but also varied substantially in time and space,
and therefore, so did air compressibility. The assumption that air compressibility
is constant should impact the analytical identification of medium flow and storage
properties as well as the storage properties of the packed-off intervals.
115
Chapter 4
INTERPRETATION OF SINGLE-HOLE TESTS
As discussed in Chapter 2, Guzman et al. [1994, 1996] and Guzman and Neuman
[1996] conducted a series of multiple-step single-hole pneumatic tests at the ALRS.
Guzman et al. [1996] applied steady-state formulae for single-phase airflow in a uni-
form, isotropic porous continuum to interpret late data from each step of an injection
test. Illman et al. [1998] and Illman and Neuman [2000b] employed transient type-
curves for single-phase airflow in a similar continuum to analyze all data from the
first step of several injection tests as well as some later steps. Their analysis did not
-
allow reliable identification of air-filled porosity and borehole storage coefficient.
Here I describe inverse analyses of four single-hole tests conducted by Guzman
and Neuman [1996] (Tables 4.1 and 4.2). In each case, I present a separate analysis
of pressure data from the first step of the test (labeled A) and of data from the entire
test, including multiple injection steps (labeled A, B, C, D) and recovery (labeled
R). My analyses of the first step consider three cases: (1) no open borehole intervals,
(2) an open injection interval, and (3) open intervals in all boreholes. In the first
case, I ignore open borehole effects; permeability and porosity are uniform over the
entire computational grid (including nodes along boreholes). In the second case, I
consider the effect of an open injection interval by assigning to it high permeability
and high porosity values. In the last case, the effects of all open borehole intervals are
considered by treating them as high-permeability and high-porosity porous cylinders.
To analyze the single-hole tests numerically, a set of match points is defined for
each injection step and recovery. The match points are distributed more or less evenly
along a log-transformed time axis. On the average, there are 10 match points per
injection step. These are assigned unit weights w i when analyzing pressure data from
116
4.1. Single-hole pneumatic tests analyzed by inverse modeling [after Guzman
et al.,
1996].
TABLE
JG0921
JGC0609
JHB0612
JJA0616
TABLE
[m]
Distance from top of Y2
to center of injection interval [m]
2.0
1.0
1.0
1.0
16.10
13.85
15.81
17.77
Length of injection interval
Test
4.2. Single-hole test data analyzed by inverse modeling [after Guzman et al.,
1996].
Test
JG0921
Injection
step
A
B
R
JGC0609
A
B
C
D
R
JHB0612
A
B
C
R
JJA0616
A
B
C
R
Duration
[min]
66.0
63.05
11.12
144.9
161.1
220.9
195.1
286.0
109.65
100.05
95.25
62.05
91.9
132.1
100.0
726.0
Injection rate
Injection rate
[cm3Imin]
400.7
1983.6
0.0
499.2
999.8
1501.0
1801.2
0.0
502.3
1201.1
1951.7
0.0
300.3
800.8
1301.0
0.0
[kg I s]
8.014 x 10 -6
3.967 x 10 -5
0.0
9.984 x 10 -6
2.000 x 10 -5
3.002 x 10 -5
3.602 x 10 -5
0.0
1.005 x 10 -5
2.402 x 10 -5
3.903 x 10 -5
0.0
6.001 x 10 -6
1.602 x 10 -5
2.602 x 10 -5
0.0
117
the first step of a test. When analyzing pressure data from all steps simultaneously,
the weights w i are made inversely proportional to the observed pressure value p:.
Applying weights inversely proportional to the observed pressures is similar to defining
the objective function as the sum of squared differences between the logarithms of
observed measurements 137 and simulated pressures pi [Doherty et al., 1994]. This
type of objective function allows better representation of weak pressure responses
that characterize early-time pressure buildup and recovery.
In my inverse analysis of single-hole tests, the medium is assumed to be uniform.
The vector a consists of either two unknown parameters, air permeability k and air-
filled porosity 0, or three parameters k, 0 and the effective borehole porosity Ow of
the injection interval, as defined in Chapter 3, Equations 3.65 and 3.66, page 101.
118
4.1 Single-hole test JG0921
Parameters estimated for test JG0921 by both analytical and numerical approaches
are summarized in Table 4.3. In the table, and through the rest of the text, the
+ range represents separate 95% confidence intervals identified by my numerical inverse model. Steady-state interpretation of single-hole test JG0921A by means of
an analytical formula gives a pneumatic permeability of 2.8 x 10- 14 m2 [Guzman
et al., 1996], and transient type-curve analysis based on the spherical flow model
gives 2.6 x 10 -14 M 2 [Inman et al., 1998]; neither of these two analyses have yielded
air-filled porosity estimates. The type-curve interpretation by
Illman et al. [1998] did
not produce satisfactory match of the measured pressure responses.
The matches between pressure values computed by the numerical inverse model,
and measured values from the first injection step of the test, are depicted in Figure 4.1.
When open borehole intervals (including that used for injection) are not considered
in the simulation, my numerical inverse model yields a match that is not entirely
satisfactory (Figure 4.1), with air permeability k = 2.3 x 10 -14 + 2.6 x 10 -16 m 2 and
air-filled porosity 0 = 4.5 x 10 -1 + 1.9 x 10 -3 . The porosity estimate appears to
be too high for fractures. When only steady-state pressure data are included in the
inverse analysis, the match at early time is poor, yielding k = 2.8 x 10- 14 m2 and
0 = 4.6 x 10 -3 , respectively. Here the estimate of porosity is based entirely on the time
at which steady state commences, which I have specified to be 0.008 days, as indicated
by open circles in Figure 4.1. The model is therefore quite insensitive to
0 and fails to
yield a finite confidence interval for either parameter since the normal matrix becomes
singular. When the effect of all open borehole intervals is included in the analysis, and
the effective porosity of the injection interval is allowed to vary simultaneously with
k and 0, the match improves significantly, yielding k = 2.2 x 10 -14 + 4.4 x 10 -16 m 2 ,
0 = 6.7 x 10 -3 1 4.7 x 10 -3 and 0„ = 7.0 x 10 -1 + 6.7 x 10 -2 . The large confidence
interval associated with
0 (of the same order as its estimate) shows low sensitivity to
119
TABLE
Type of analysis
Analytical steady-state (A)
Analytical transient
- Spherical flow (A)
Inverse modeling
- No open intervals (A)
4.3. Parameter estimates for test JG0921.
k [m 2 ]
2.8 x 10 -14
95 [m3/m3]
Ow [77/3 /m3 ]
CD
4.5 x 10 -1 +
1.9 x 10 -3
4.6 x 10 -3
6.7 x 10 -3 ±
4.7 x 10 -3
1.4 x 10 -2 ±
1.7 x 10 -3
7.0 x 10 -1 ±
6.7 x 10 -3
8.0 x 10 -1 +
4.6 x 10 -2
3.5 x 10 1
[-]
2.6 x 10 -14
2.3x 10 -14 ±
2.6 x 10 -16
- No open intervals (A; steady-state) 2.8 x 10 -14
- All intervals open (A)
2.2 x 10 -14 ±
4.4 x 10 -16
- All intervals open (A,B,R)2.4x 10 -14 ±
7.1 x 10 -16
1.9 x 10 1
this parameter and indicates that its estimate is highly uncertain. It reflects the fact
that transient data is influenced by borehole storage effects.
Eigenvalues and eigenvectors of the covariance matrix of the parameter estimates
are listed in Table 4.4. The first eigenvector is dominated entirely by permeability,
indicating that this parameter is very well defined. The second and third eigenvectors
show correlation between 0 and Ow , rendering both estimates less certain.
As discussed in Section 3.4, the analytical type-curves of [Illman et al., 1998] considered air compressibility to be a constant. However, the increase of absolute air
pressure at the injection interval decreases the interval storage as shown by Equation
3.9 (page 73). Figure 4.2 depicts relative changes in storage during injection steps and
recovery due to changes in air compressibility, plotted versus time measured relative
to the end of the preceding step. Since the relative pressure buildups during injection steps were not significant compared to initial air pressure, changes in borehole
storage were not substantial. However, the effect of air compressibility on borehole
storage appears to be important for obtaining good matches between observed and
numerically simulated pressures during the first injection step.
Figure 4.3 depicts pressure during each step and recovery, relative to that established during the preceding step, plotted versus time measured relative to the end of
the preceding step. We see a very distinct unit slope during the first step (labeled A)
120
4.4. Eigenanalysis of covariance matrix of estimation errors obtained from
interpretation of the first step of test JG0921.
TABLE
Parameters
k [m 2 ]
[m3 /m3 ]
[m3 /m 3 ]
Eigenvalues
Eigenvectors
1.000
3.0424 x 10 -14
5.7346 x 10 -14
1.5993 x 10 -32
5.4264 x 10 -14
—0.8766
—0.4812
2.2001 x 10 -6
—3.5632 x 10 -14
—0.4812
0.8766
1.0564 x 10 -5
TABLE 4.5. Eigenanalysis of covariance matrix of estimation errors obtained from
interpretation of all the injection steps and recovery of test J 00921.
Parameters
k [m 2 ]
q5 [770 Im 3 ]
[m 3 /m3 ]
Eigenvalues
Eigenvectors
1.000
—2.3431 x 10 -13
—2.7675 x 10 -15
8.2641 x 10 -32
—2.3433 x 10 -13
—1.000
—6.1345 x 10 -3
6.6215 x 10 -7
1.3300 x 10 -15
—6.1345 x 10 -3
1.000
5.1651 x 10 -4
of the test. To the extent that storage affects the other two pressure records in the
figure, this effect is not clearly discernible. Air storage in the injection interval has
theoretically no effect on recovery data (see Section 3.1.1), which should therefore
be ideal for the estimation of air-filled porosity and borehole storage coeffi cient. I
therefore expect a simultaneous analysis of pressure data from the entire test to yield
a more reliable estimate of parameters than is possible based only on data from the
first step.
A reasonably good fit of my model (which now accounts for all open borehole
intervals) to the entire two-step pressure buildup and recovery record is shown in
Figure 4.4. The corresponding parameter estimates are k = 2.4 x 10 -14 + 7.1 x
10 -16 m2 ,
o „:„ 1.4 x 10
-2
+ 1.7 x 10 -3 and q5„, = 8.0 x 10 -1 + 4.6 x 10 -2 . Table 4.5
shows that each eigenvector of the corresponding estimation covariance matrix is now
dominating almost entirely by one parameter, showing lack of correlation between
the estimates and suggesting that they are of high quality.
121
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125
4.2 Single-hole test JGC0609
Results of analyzing pressure data from test JGC0609 by various methods are listed in
Tables 4.3. Steady-state analysis gives a pneumatic permeability of k = 2.0 x 10 -15 m 2
[Guzman et al., 1996], and transient type-curve analysis with a spherical flow model
yields k = 2.9x 10 -15 m 2 [Illman et al., 1998]. As for the previous test, the type-curves
could not match properly the entire pressure record [Illman et al., 1998].
Numerical inverse results are compared with measured values in Figure 4.5. In
the absence of open borehole intervals, the inverse model yields a poor fit with k =
1.8 x 10 -15 +3.9 x 10 -15 m 2 and 0 = 5.0>< 10 -1 +4.2 x 10 -2 ; upon including the injection
interval, the fit improves dramatically to yield k = 1.6 x 10 -15 + 2.6 x 10' 7 m 2 ,
q5 = 4.8 x 10 -3 +9.4 x 10 -3 and 0„, = 1.3+2.6 x 10 -2 ; incorporating the storage effects
of all open boreholes results in an equally good fit with k = 1.6 x10 -15 +1.4 x 10 -17 m 2 ,
0 = 5.3 x 10 -3 + 5.1 x 10 -3 and Ow = 1.3 + 1.9 x 10 -2 . As I have already discussed in
Chapter 3 (Equations 3.65 and 3.66 on page 101), Ow is allowed to take on values in
excess of 1.0, as a way of accounting for an effective borehole volume that is larger than
the one originally built into the computational grid. Therefore, a borehole porosity in
excess of 1 is plausible, implying that the effective storage volume Vs of the injection
interval exceeds its nominal volume Vto . The excess is most probably due to openings
in the surrounding rock. However, the available single-hole test data do not allow me
to distinguish unambiguously between the roles that the borehole and the surrounding
rock (fractures and matrix) may play in controlling the observed air storage effect.
Despite the good fit, the air-filled porosity estimates do not seem reliable because of
the relatively large confidence intervals. Indeed, for the last match, the eigenanalysis
results displayed in Table 4.7 indicate that there is a correlation between 0 and 0.„„
most probably due to the influence of storage effects during the transient period of
the test.
Figure 4.6 presents changes in borehole storage due to air compressibility versus
126
TABLE 4.6. Parameter
k [m2]
Type of analysis
2.0 x 10_ 15
Analytical steady-state (A)
Analytical transient
- Spherical flow (A)
2.9 x 10 -15
Inverse modeling
1.8 x 10 -15 ±
- No open intervals (A)
3.9 x 10 -15
- Open injection interval (A)
1.6 x 10 -15 ±
2.6 x 10 -17
- All intervals open (A)
1.6 x 10 -15 +
1.3 x 10 -17
- All intervals open (A,B,C,R) 1.7 x 10 -15 +
4.1 x 10 -17
estimates for test JGC0609.
Ow [M3 I M3 ]
0 [m 3 / 77-43 ]
5.0 x 10 -1 ±
4.2 x 10 -2
4.8 x 10 -3 ±
9.4 x 10 -3
5.5 x 10 -3 ±
4.7 x 10 -3
3.6 x 10 -3 ±
2.8 x 10 -4
CD [-]
1.3 ± 2.6 x 10 -2
9.0 x 10 1
1.3 ± 1.7 x 10 -2
7.9 x 10 1
1.5 ± 6.3 x 10 -2
1.4 x 10 2
relative time. Pressure buildups were much higher than the initial air pressure, and as
a result, changes in borehole storage were substantial even during the first injection
steps. In a similar way, the variation of air compressibility modifies the medium
storage in the vicinity of injection interval. This explains the poor matches obtained
by type-curves, and therefore, the analytical estimates are biased. The effect of
air compressibility on storage is important for obtaining good matches and reliable
parameter estimates by means of the numerical inverse model.
Figure 4.7 depicts relative pressure versus relative time for injection steps 1 - 4
(labeled A - D, respectively) and recovery (labeled R). Here again we see that only
data corresponding to step A exhibit an unambiguous one-to-one slope at early time,
while all other data appear to be less influenced by storage. I therefore expect a
simultaneous analysis of pressure data from the entire test to yield a more reliable
estimate of parameters, especially air-filled porosity, than is possible based only on
data from the first step.
A fit of my model to the entire four-step pressure record, including recovery data,
yields a good match (Figure 4.8) with k = 1.7 x 10 - ' 5 + 4.1 x 10 - ' 7 m 2 , 0 = 3.6 x
10 -3 + 2.8 x 10 -4 and çbv, = 1.5 + 6.3 x 10 -2 . The model appears sensitive to all three
parameters (Table 4.8) whose estimates seem reasonable.
127
o
o
o
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6
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129
0
0
0
ujr
NNW
0
1-
1+
-zr
uJ
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< co
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TABLE 4.7. Eigenanalysis of covariance matrix of estimation errors obtained from
interpretation of the first step of test JGC0609.
Eigenvectors
Parameters
4.9941 x 10 -16
2.9243 x 10 -15
1.000
k [m 2 ]
—0.9758
—0.2188
2.9627
x
10
-15
95 [m3/m3]
1.5247
x
10
-16
—0.2188
0.9758
[m
3
/m
3
]
Ow
1.7338 x 10 -6
6.7666 x 10 -5
Eigenvalues
3.0691 x 10 -36
TABLE 4.8. Eigenanalysis of covariance matrix of estimation errors obtained from
interpretation of all the injection steps and recovery of test JGC0609.
Eigenvectors
Parameters
k [m 2]
—1.1142 x 10 -13
1.3489 x 10 -16
1.000
—4.8913 x 10 -4
—1.000
—1.1142 x 10 -13
qS [m3irn3]
1.000
—4.8913
x
10
-4
-16
—1.8939
x
10
]
[rn3 /rn 3
9.7135 x 10-4
1.8990 x 10 -8
Eigenvalues
1.6206 x 10 -34
132
4.3 Single-hole test JHB0612
Table 4.9 lists parameters obtained by various methods of analysis from pressure
data recorded during test JHB0612. Steady-state analysis gives k = 4.8 x 10-14 m2
[Guzman et al., 1996], and transient type-curve analysis using a spherical flow model
yields k -= 6.5 x 10 -14 m 2 and using a radial flow model yields k = 1.3 x 10 -13 m 2 ,
0 = 4.0 x 10 -5 and CD -= 1.0 x 10 4 [Illman et al., 1998]. Both type-curve models did
not match the observed pressures record [Inman et al., 1998]: the spherical model
failed at the late times, the radial model at the early times. Illman et al. [1998] also
applied type-curves considering a single horizontal or vertical fracture intersects the
injection interval; these models also failed to represent the pressure record.
Figure 4.9 indicates that inverse analysis without open boreholes gives a poor fit
with k = 5.2 x 10 -14 + 1.1 x 10 -16 m 2 and 0 = 5.0 x 10 -1 1 1.2 x 10 -3 ; incorporating
the injection interval yields a much improved fit with k = 4.0 x 10 -14 +4.4 x 10' m 2 ,
0 = 8.1 x 10 - 2
+ 1.6 x 10 6 and 0„ = 1.2 + 3.4 x 10 6 (the huge confidence intervals
reflecting a virtual lack of sensitivity to 0 and q5 , which are negatively correlated
with each other); and including all open borehole intervals gives an equally good fit
with k = 4.1 x 10- 14 m2,
0
8.8 x 10 -2 and 0„, = 1.2 (here the normal matrix as
defined in Equation 3.56 on page 89 is singular and cannot be inverted to compute
the covariance matrix, due to correlation between q5 and 0,,). The porosity estimates
are clearly unreliable.
Similarly to the case of test J 00921, pressure buildups were small compared to
the initial air pressure, and therefore, changes in borehole storage due to air com-
pressibility were minor (Figure 4.10). Plots of relative pressure versus relative time
for injection steps 1 - 3 (labeled A - C, respectively) and recovery (labeled R) are
shown in Figure 4.11. As before, only data corresponding to step A exhibit an unambiguous borehole storage effect at early time, while all other data appear to be less
influenced by this effect. A simultaneous analysis of pressure data from all stages of
133
TABLE
Type of analysis
Analytical steady-state (A)
Analytical transient
- Spherical flow (A)
- Radial flow (A)
Inverse modeling
- No open intervals (A)
4.9. Parameter estimates for test J11B0612.
k [m 2 ]
g5 [m 3 /m,3]
4.8 x 10 -14
6.5 x 10 -14
1.3 X 10 -13
5.2 x 10 -14 +
1.1 X 10 -16
- Open injection interval (A) 4.0 x 10 -14 ±
4.4 x 10 -16
4.1 x 10 -14
- All intervals open (A)
- All intervals open (A,B,C,R) 3.9 x 10 -14 ±
1.6 x 10 -15
Ow [m 3 i m 3 ]
4.0 x 10 -5
5.0 x 10 -1 ±
1.2 x 10 -3
8.1 x 10 -2 ±
1.6 X 10 6
8.8 x 10 -2
9.6 x 10 -2 ±
7.0 x 10 -3
CD
[-]
1.0 X 10 4
1.2 ± 1.6 x 10 6
4.9
1.2
1.3 ± 2.2 x 10 -2
4.5
4.5
TABLE 4.10. Eigenanalysis of covariance matrix of estimation errors obtained from
interpretation of all the injection steps and recovery of test JHB0612.
Eigenvectors
Parameters
9.6439 x 10 -15
-7.5978 x 10 -16
k [m 2 ]
1.000
-2.2232 x 10 -2
-0.9998
0 [m 3 /m3 ]
9.6247 x 10 -15
-2.2232
x
10
0.9998
-16
-2
9.7400
x
10
Ow [rn3irn3]
6.1682 x 10 -6
1.1690 x 10 -2
Eigenvalues
6.3155 x 10 -31
the test should therefore yield a reliable estimate of all parameters, including air-filled
porosity.
A joint analysis of pressure data from all three steps of the test, including recovery,
gives a good match (Figure 4.12) with k = 3.9 x 10 -14 + 1.6 x 10 -15 m 2 , 0 = 9.6 x
10 -2 + 7.0 x 10 -3 and 0„ = 1.3 + 2.2 x 10 -2 . As implied by the eigenvectors in
Table 4.10, the model now is sensitive to all three parameters, whose estimates seem
reasonable.
134
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T
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138
4.4 Single-hole test JJA0616
Results for test JJA0616 are listed in Tables 4.11. The steady-state analysis of Guzman et al. [1996] gives k = 5.6 x 10 -15 m 2 ; a transient type-curve analysis with a
spherical flow model results in a relatively poor fit with k = 8.0 x 10 -15 m 2 [Illman
et al., 1998]; and a similar analysis with a radial flow model yields an equally poor
fit with k = 7.1 x 10 -15 m 2 , q5 = 3.1 x 10 -2 and CD = 1.0 X10 1 [Illman et al., 1998].
Numerical inverse results are compared with measured values in Figure 4.13. Upon
ignoring open borehole intervals in my numerical inverse model, I obtain a poor fit
with k = 5.4 x 10 -15 +7.9 x 10 -16 m 2 and q5 = 5.0 x 10 -1 +3.9 x 10 -2 . The fit improves
greatly when I include the injection interval, leading to k = 4.0 x 10 -15 +7.8 x 10' m 2 ,
= 1.3 x 10 -1 + 5.8 x 10 -3 and 0 = 1.1 + 5.2 x 10 -2 ; the fit is equally good when
I incorporate all open borehole intervals, yielding k = 4.1 x 10 -15 + 7.1 x 10 -17 m 2 ,
= 1.3x 10'+6.4x 10 and q5,,, = 1.1+3.9x 10 -2 . Here the numerical inverse model
is capable of producing estimates with low errors for all three parameters based solely
on data from the first injection step. This conclusion is confirmed by an eigenanalysis
of the corresponding covariance matrix (Table 4.12). Figure 4.14 presents relative
change of storage due to variation of air compressibility versus relative time. Similarly
to the case of test JGC0609, obtained pressures buildups were much higher than the
initial air pressure, and therefore, changes in borehole storage due to changes in air
compressibility were substantial. This may explain the poor matches obtained by
analytical type-curves. The effect of air compressibility on storage is important to
obtain good matches by means of the numerical inverse model. Still, borehole storage
dominates pressure transients and makes it difficult to obtain reliable estimates of
air-filled porosity; identified value of q5 is much too high for fractures.
Plots of relative pressure versus relative time for various injection steps in Figure
4.15 demonstrate that data corresponding to the first injection step (A) exhibit very
strong borehole storage effect. Therefore, though the model calculates low estimation
139
errors, the porosity estimates are not reliable. Fitting the inverse model to data from
all three steps of the test, including its recovery stage, leads to a good overall fit
(Figure 4.16) with k = 4.3 x 10 - ' 5 + 1.9 x 10 -16 m 2 , 0 = 1.8 x 10 -2 + 3.9 x 10 -4 and
0,,, = 1.6 + 7.2 x 10 -2 . The eigenanalysis of covariance matrix demonstrates reliable
estimation for all parameters (Table 4.13). The estimated 0 value is now much lower,
and can be considered as representative of fracture properties.
140
TABLE 4.11. Parameter estimates for test JJA0616.
k [m 2 ]
Ow [m3/m3]
0 [m3/m3]
5.6 x 10 -15
Type of analysis
Analytical steady-state (A)
Analytical transient
- Spherical flow (A)
- Radial flow (A)
Inverse modeling
- No open intervals (A)
8.0 x 10 -15
7.1 x 10 -15
5.4 x 10 -15 ±
7.9 x 10 -16
4.0 x 10 -15 ±
- Open injection interval (A)
7.8 x 10 -17
4.1 x 10 -15 +
- All intervals open (A)
7.1 x 10 -17
- All intervals open (A,B,C,D,R) 4.3 x 10 -15 ±
1.9 x 10 -15
CD
3.1 x 10 -2
5.0 x 10 -1 ±
3.9 x 10 -2
1.3 x 10 -1 ±
5.8 x 10 -3
1.3 x 10 -1 +
6.4 x 10 -3
1.8 x 10 -2 ±
3.9 x 10 -4
[-]
1.0 x 10 1
1.1 ± 5.2 x 10 -2
2.8
1.1 + 3.9 x 10 -2
2.8
1.6+ 7.2 x 10 -2
3.0 x 10 1
TABLE 4.12. Eigenanalysis of covariance matrix of estimation errors obtained from
interpretation of the first step of test JJA0616.
Parameters
k [m 2 ]
0 [m3 /m3 ]
Ow [rn 3 /m3 1
Eigenvalues
Eigenvectors
1.000
3.5461 x 10 -15
3.8852 x 10 -16
8.9518 x 10 -34
3.5628 x 10 -15
-0.9983
-5.8609 x 10 -2
7.0241 x 10 -8
-1.8002 x 10 -16
-5.8609 x 10 -2
0.9983
2.9162 x 10 -4
TABLE 4.13. Eigenanalysis of covariance matrix of estimation errors obtained from
interpretation of all the injection steps and recovery of test JJA0616.
Eigenvectors
Parameters
-4.4528 x 10 -16
3.5963 x 10 -13
1.000
k [m 2 ]
6.9652 x 10 -4
-1.000
-13
3.5963
x
10
]
0 [m3 /rn 3
-2
1.000
6.9652
x
10
1.9479
x
10
-16
]
Ow [m 3 /rn3
1.2518 x 10-3
-8
3.6030
x
10
-33
3.5004
x
10
Eigenvalues
141
Cn1
EedIll d
142
0
e's'l
I=MM
I=MMM•111 MIMI=
:u
==0111=
MINIM == NM
IMMIM MIM =MI
MN= IMMMMM MMIn IMMM
MM/ MI/1
/W.
1.1111....111.1111W MUM 71.1111 MIME
11=111111111MW L•N•• MO., BEM Ili
11111 IErA11111ow 7 31111I
111111"1111111
1111111111111111 II 11111111
,.. 0
Q)
s.
0
•
T
I
•
I.0
e—
<-.
0
11
or)
1.0
co
o
aBeJols alogaJoq a6uego anReim
o
co
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7.)
143
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0
0
W
1.0
0
9
LLI
LLJ
[e cpi] ainssaid anneieN
144
o
0
T
[edm] d
145
4.5 Discussion
\
Transient multi-step single-hole pneumatic tests in unsaturated fractured tuffs at the
ALRS are amenable to analysis by means of a three-dimensional numerical inverse
model which considers only isothermal single-phase airflow and treats the rock as a
uniform, isotropic porous continuum representative of interconnected fractures. Bore-
hole storage depends on compressibility of air in the injection interval, and decreases
with the increase of absolute air pressure. Borehole storage effect influences pressure
transients during the first step of each test and makes it difficult to obtain reliable
estimates of air-filled porosity and borehole storage coefficient from such data. Pressure transients during subsequent injection steps are less affected by borehole storage
than are those during the first step. Recovery data are theoretically free of borehole storage effects. A consideration of the effect of air compressibility on storage
and a joint analysis of pressure data from all stages of a single-hole pneumatic test
are therefore essential for the reliable estimation of air-filled porosity and borehole
storage coefficient. The values of 0 estimated using complete pressure records for
the four tests range from 9.6 x 10 -2 to 3.6 x 10 -3 ; the respective estimates of CD
are between 4.5 and 140. The storage effect observed during single-hole tests seems
related not only to the injection interval, but also to openings in the surrounding
rock. Open borehole intervals surrounding the injection interval have little impact
on pressures within the latter. Air permeabilities obtained by our inverse method
are comparable to those obtained by means of steady-state formulae [Guzman et al.,
1996] and transient type-curve analyses [Illman et al., 1998]. Due to linearization
of airflow equations, analytical transient type-curves [Illman et al., 1998; Illman and
Neuman, 2000b] did not match observed pressure records and did not allow reliable
identification of air-filled porosity and borehole storage coefficient.
146
Chapter 5
INTERPRETATION OF CROSS-HOLE TESTS
Illman et al. [1998] and Illman [1999] conducted 44 cross-hole pneumatic tests at the
ALRS. To date, only one of the cross-hole tests, labeled PP4, has been fully analyzed
by means of analytically derived type-curves [Illman et al., 1998; Illm,an, 1999; Inman
and Neuman, 2000a]. Due to the pronounced non-uniformity of the fractured rock
at the site, this approach allows only separate interpretation of pressure records from
individual observation intervals, under the assumption that air-filled fractures form a
uniform continuum.
Here, I perform comprehensive numerical inverse analyses of five of the cross-hole
tests labeled PP4, PP5, PP6, PP7 and PP8 (Table 5.1). Tests PP4, PP5, PP6 and
PP7 were conducted with a packer configuration shown in Figure 5.1. In test PP8,
the packers in X3 and Y3 were placed lower along the inclined boreholes by 5.5 and
5.0 m, respectively. In the 16 tested boreholes (Xl, X2, X3, Yl, Y2, Y3, Z1, Z2,
Z3, Vi, V2, V3, Wl, W2, W2A and W3), there were 37 monitored intervals with
lengths between 0.5 and 42.6 m. During tests PP4 and PP8, air was injected at a
constant mass rate of 1 x 10 -3 kg/s (50 //min under standard conditions) into the
middle interval of borehole Y2. During tests PP5, PP6 and PP7, air was injected
into the middle intervals of boreholes X2, Z3 and W3, respectively. In the latter three
tests, the injection consisted of two constant rate steps; however, I process only data
from the first step, at a mass-rate of 1 x 10 -4 kg/s (5 l I min). The injection intervals
were 1 m (PP7) and 2 m (PP4, PP5, PP6 and PP8) in length (Table 5.1). Distances
between injection and observation intervals varied from 1 to 30 m (Figure 5.1).
Parts of this analysis have been presented earlier by Illman et al. [1998], Illman
and Vesselinov [1998], Vesselinov et al. [1999], Neuman et al. [2000], Vesselinov et al.
147
TABLE 5.1. Point-to-point cross-hole tests interpreted by numerical inverse model
Cross-hole
test
Injection
interval
PP4
PP5
PP6
PP7
PP8
Y2-2
X2-2
Z3-2
W3-2
Y2-2
Length
Duration
Injection rate
Injection rate
[m]
[days]
2.0
2.0
2.0
1.0
2.0
3.95
1.95
1.70
1.90
0.84
[1Imin]
50.0
5.0
5.0
5.0
50.0
1.0x10-3
0.1x10-3
0.1x10-3
0.1x10-3
1.0x10-3
[kgls]
[2000a], Vesselinov et al. [2000b] and Chen et al. [2000].
The labeling of test intervals in my dissertation differs from that of Illman et al.
[1998]. In both studies, when a single packer is used to isolate a borehole segment, the
corresponding interval is named after the borehole (e.g. X1). In the case of multiple
intervals, these are labeled by Illman et al. [1998] from top to bottom as UMB, UML
and UMLB (e.g. X2U, X2M, X2B), respectively. Instead, I numbered test intervals
sequentially from top to bottom (e.g. X2-1, X2-2, X2-3).
5.1 Qualitative analysis
During the cross-hole tests, pressure and temperature data were collected by means
of 38 measuring devices located below the respective confining packers [Illman et al.,
1998]. As boreholes V2 and W2 are interconnected, the intervals V2-3 and W2 are
linked, forming one relatively large and complex observation interval (Figure 5.1).
Therefore, the actual number of test intervals is 37. Four of the intervals—Y1-1, V21, V2-3/W2 and W3-3—gave unreliable readings in all the cross-hole tests [Illman
et al., 1998] (initially, data from interval Y1-1 have been interpreted analytically by
Illman et al. [1998] and Illman [1999]; later, Illman and Neuman [2000a] excluded
Y1-1 from their analysis). Most probably, due to a measurement problem, interval
Y1-3 produced responses that are perfectly correlated with variations in atmospheric
pressure. Pressure data from these five monitoring intervals are ignored in my study.
This leaves 32 test intervals-1 injection and 31 monitoring. Discrete pressure data
148
0
-10
N
-20
3
-30
-40
FIGURE 5.1.
Location of monitoring intervals and packers along tested boreholes.
149
recorded in these intervals are shown by small solid dots on log-log plots in Figures
5.2 - 5.6. Pressure measurements are relative, representing changes in absolute air
pressure with time. Pressure data were collected every 20 s during each test [Illman
et al., 1998]. I filtered the recorded data so that the minimum time separation between
two consequent measurements in these figures is (log t 2 - log t 1 ) > 5 x 10 -3 , where t i
and t 2 are in days. In Figures 5.2 - 5.6 and all similar figures presented later, vertical
axes show relative pressure buildup in kPa, and horizontal axes show elapsed time
from the start of the test in days, respectively.
In the case of test PP4, pressure responses due to air-injection into Y2-2 (Figure
5.2) are well-defined in all 32 intervals. Barometric pressure varied significantly and
dropped by more than 1 kPa during this test (Figure 5.7). Late-time pressure data in
intervals X3, Y3-1, Y3-2, Z1, Z2-1, Z2-2, Z2-3, Z2-4 and Z3-3 are visibly affected by
this barometric pressure decline. Measured pressure responses in some of the intervals
and the atmospheric pressure are plotted versus time in Figure 5.7. Injection into
interval X2-2 during test PP5 produces responses in most intervals (Figure 5.3).
However, 11 of these pressure records (those corresponding to intervals X3, Y1-2,
Y3-2, Z1, Z2-1, Z2-2, Z2-3, Z2-4, Z3-3, W2A-4 and W3-1) are difficult to interpret.
Pressure data from nine intervals during test PP6-X3, Y1-2, Y3-2, Z1, Z2-2, Z2-
3, Z2-4, Z3-3 and W3-2 (Figure 5.4)-are weak and difficult to interpret; most of
these monitoring intervals are close to the injection interval, Z3-2. During test PP7
(injection into W3-2), pressure responses in all but three of the monitoring intervals
Y1-2, W2A-1 and W2A-4-are amenable to interpretation (Figure 5.5). Test PP8 is
similar to test PP4 and all corresponding pressure responses, except that in X3, are
well defined (Figure 5.6). In contrast with PP4, there are no significant variations
in barometric pressure during test PP8. As a result, there are no late time pressure
drops in intervals Y3-1, Y3-2, Z1, Z2-1, Z2-2, Z2-3, Z2-4 and Z3-3 (Figure 5.2) which
were affected by barometric pressure during test PP4 (Figure 5.2). In summary, the
number of intervals that are amenable to analysis are as follows: PP4 - 32; PP5 -
150
X1
X2 - 3
10°
10°
o
10'
10 0
10.0
10'10°
Y2-2
X3
1 00
10.
10 "
• •.: :47
.
10'
-
0
110
./fe-e.NNikt
1.
•
• •
100
1OE°
10'
10'
10°
10'
10-2
lo-
1U
10
zi
1U
0
10 Z2-2
8:0
-
eAstr.e*4.9
10 •
10
Z2-1
0.4
0.2
10 10
10'
10 •
10'
10 4
10 -°-
10°
1U
100Z2-3
10'•
10 010-1
Z2-4
P
10°
10 0
10'
o
10°10'
10"
10'
10°
Z3-1
10
1 ,0
10 °
10 •
10'
10'
10 0
10 2
10°
Z3-3
10'
o.
10 2 .
10°
V1
10-210-'
10 210'10°
10°
V3-2
V3-1
10°
40
•
10'
10'
o
30
o
.
•
•
10'
10
10'
10'
FIGURE 5.2. Log-log plots of observed pressure buildups (dots; kPa) versus time
(days) and match points (circles) during test PP4.
151
X1
100X2-1
1 0'
1 0'
1 0 -0
10"
10'
1.5
10'
10 -0
10 -2
0.5
10°
10'
0.08
0.06
0.06
0.04
1 0'
10 -2
1o 2
10 -0
10'
1 0'
1OE'
10 •
10 -2
10 -0
.
.
16'
-
'
16 0
(':1
W2A-1
02
0 00
0.15
10°
OA
10'
10 - '
0.05
•
10 -310
•• • ••
10'
1 0°
10-0
10'
10 0
W3-1
W2A-4
0.3
0.2
10'
0.1
10 0
10 -2
1 0 01 0 010'
10°
10 -0
1o 0
10'
1 0°
10 0
io'
lo'
lo°
FIGURE 5.3. Log-log plots of observed pressure buildups (dots; kPa) versus time
(days) and match points (circles) during test PP5.
1 52
X1
X2-2
X2-3
10'
1 0 -0
10 -2
1 00
10
10 -210"
10 0o'
Y2-1
Y2-2
0.6
0.4
0.6
0.2
__sf•
0.5
10'
10"
Y3-2
0.6
0.4
10 -210°
1 0 -2
10
Y3-3
16-0
15'100
10 -2
0.2
1 0 -4
10.0
1 0 -0
16-3
Z2 - 2
16-2
16'
-Co.
0.2
io'
Z3-2
0.2
52
0.15
51
0.1
50
100
49
48 c;
0.05
180
V3-1
0.3
0.2
0.3
0.2
0.1
0.1
18-0 . 18:r;
V3-3
Iv
16.
0.2 0.1
10'
10'
10"
10 °
10'
0.15
0.1
0,1
.•
•
•
lo°
10'
0.6
0.4
101'
10 0
._
10"
o
0.05
.• -
10" 10 -210°
W2A-4
10; 10' 1 0 01
00
10"
W2A- 1
0.3
0.2
W2A- 3
100
41°4°4
W1
8:1
10-2
• 11..
10 -0
.:•••
10°
0.3
0.2
10-0
W3-1
10°
W3-2
10
0.1
0.2
•
' • '1'1::
'
o'
10 -0
10"
10
10010
10°
10'
18-0
18'
16°
10 -2
o
10°
Log-log plots of observed pressure buildups (dots; k P a) versus time
(days) and match points (circles) during test PP6.
FIGURE 5.4.
153
X2-2
X2-3
10'
. • :
10'
Y2-1
10.
10 - '
10°
L
10 - '
10-0
10'
‘7r.
10 3
10'
10'
.
' ,. 2.:n, , ,
10'
V3-1
10'
10'
,
10'
V3-2
10 0
10'
•
10 . °
102
:"1-A
10'
10°
10-'
__ • •••••-.0.;..grf)
10
100
-..--:
•
.
10"
f i:
•
.. " ...r:.:,'.• :
•
10'10010'10 °
V3-3
eta* to
10'
,
-
10'
10°
to-0
W2A-3
10°
10 - '
10'
10 - '
10 -2
10
o
10
1 0 10
10
16
10'
1 0 '
10 °
1 0 310'
10'
10°
5.5. Log-log plots of observed pressure buildups (dots; k P a) versus time
(days) and match points (circles) during test PP7.
FIGURE
154
X2-1
X1
10
°
10°
10"
10'
10
10
010 -°
o'
X3
. .
• •: •
10 1
100
10 4• • •
,
•
0-3
.
°
•
10°
80
•
• • •
10 -' 10 - s
to'
10 -
Y2-3
10'
10°
10'
Z2-2
Z1
10 4
10 -0
10 .310"
10 - '
10' Z2-4
10'
•
10"
10 -0
•
-1&"
Z3-1
Z3-2
10°
•
16'
Z3-3
10'
10°
10 -°
10'
1 0 -0
10 -3
10 0
V2-2
10'
10°
10°
10'
10'
V3-2
V3-1
10'
10'
10°
.erw
-
10°
e°
10'
W2A- 4
10
10°
W3-1
10°
°
10'
10 -1
10 - °
10'
10
10-
10'
16' 16'
16 '
-
10 - °
10"
16 1 0 '
-0
-
Log-log plots of observed pressure buildups (dots; kPa) versus time
(days) and match points (circles) during test PP8.
FIGURE 5.6.
155
1
,646p,66,64NA66,66646A666,AA64,666.66446.6.61664,
114466,6,
6•A6:6A&
A
A
A
A
A
▪
A
O
•
O
-1
Y1-3
Z2-1
Z2-2
Z2-3
Z2-4
barometric
0
1
2
3
4
Time [days]
FIGURE 5.7. Measured variations in barometric pressure and pressure responses in
some of intervals during test PP4.
21; PP6 - 23; PP7 - 29; and PP8 - 31. The total number of such intervals is 136.
Illman et al. [1998] measured variations in air temperature during each pneumatic
test at ground level and within most monitoring intervals. Air temperatures in the
intervals varied insignificantly during the tests, except in the injection interval (Y2-2)
during tests PP4 and PP8. Some of the temperature data from PP4 are presented
in Figure 5.8; temperature data from PP8 look similar. Though the injected air
was much warmer than ambient air in the rock, air injection at a high rate causes
temperature in the injection interval to decrease. This could be either an adiabatic
effect (rapid expansion of injected compressed air) or an evaporation effect (two-phase
vapor flow) or both. The effect is localized and not observed in the surrounding
156
el
- atmosphere I I
injected
11
Y1-3
0
Y2-2
i
\I
12i
-1
i
i
i
0
I
I
j
i
i
I
\ I\
I
\
\
I
\
il1
k l
n
i
Ii
2
1
I
3
1
, \ l,
I
4
Time [days]
FIGURE 5.8.
Some air temperature measurements collected during test PP4.
intervals (Figure 5.8).
Overall, most monitoring intervals respond consistently to air injection during
each pneumatic test. Since injection took place from several intervals during these
tests, one can conclude that air-filled fractures at the ALRS are pneumatically well
connected. Most air pressure responses show behavior typical of a three-dimensional
continuum. Pressure in most injection intervals (Y2-2 in Figure 5.2, X2-2 in Figure
5.3, Z3-2 in Figure 5.4, Y2-2 in Figure 5.6) exhibits a decline at late time. This
can be attributed to two-phase flow (as suggested by Guzman and Neuman [1996]
and supported by the above temperature data), barometric pressure variations or
instrumentation problems. The exception is injection interval W3-2 during test PP7
157
(Figure 5.5). Slopes of the early-time response curves are close to 1:1, demonstrating
significant storage effect in the injection intervals and in some monitoring intervals
located close to the injection interval (e.g. X2-2, X2-3, Y2-1, Y2-3, Y3-3 and V3-3
during PP4; Figure 5.2). The pressure record of the injection interval during PP6
(Z3-2 on Figure 5.4) shows an exponential pressure increase at early times which
reflects the impact of the variable air compressibility (Ca ;---' 1h3, ) on borehole stor-
11
age as described in Section 3.1.1 (Equation 3.11, page 73). Some of the responses
show early slopes close to 1:2—Y3-1, Z1, Z2-1, Z2-2, Z2-3, Z2-4 and Z3-3 during
tests PP4, PP7 and PP8—which suggests that they are intersected by highly perme-
able zones. In these intervals and in those intervals located close to them, measured
records are clearly influenced by barometric pressure variations, suggesting that the
highly permeable zones are connected to the ground surface. The data indicate highly
permeable zones between some of the intervals producing very similar pressure measurements in all tests: X1 and W3-2; Y3-1, Z2-3, Z2-4 and Z3-3. The X3 records are
very weak, implying locally reduced permeability. It thus appears that the medium
is highly non-uniform. High-permeable zones around Y3-1 and the Z boreholes seem
connected to the ground surface. The existence of these highly permeable zones is
confirmed by geostatistical analysis of air permeability (Figure page 2.18, page 58)
and fracture density (Figure 2.22, page 62) data in Chapter 2.
158
5.2 Inverse analysis of cross-hole tests
To invert numerically cross-hole test data, match points are defined subjectively for
portions of the measured pressure records that clearly demonstrate response due to
air injection. Variations in pressure that result from either a change in barometric
pressure or measurement problems are ignored. The match points are distributed
more or less evenly along the log-transformed time axis so as to capture both early
and late time responses correctly. The number of match points defined for each test
are: PP4 - 252; PP5 - 132; PP6 - 105; PP7 - 155; PP8 - 210. They are indicated
by open circles in Figures 5.2 - 5.6. Each point is assigned a weight of 1. The same
set of match points and weights are applied in all inverse analyses.
In the inverse model, all open borehole intervals at the site, regardless of their
response, are simulated as high-permeability and high-porosity cylinders.
I performed inverse analyses of the pneumatic tests by treating the medium as if it
was either spatially uniform or non-uniform. Inverse runs were conducted in parallel
on the University of Arizona 32-processor SCI Origin 2000 supercomputer.
5.2.1 Uniform medium
I started by analyzing separately pressure data from each monitoring interval while
treating the medium as if it was uniform across the site. Two uniform parameters
were estimated on the basis of each individual pressure record in a test interval:
log io transformed air permeability k and air-filled porosity q5. This interpretation
is similar to the analytical type-curve analysis performed on some of the same data
by Inman et al. [1998], Illman [1999] and Illman and Neuman [2000a]. On average,
each numerical inversion required about 50 forward simulations, and the effective
computational time was about 4 hours.
The resulting matches between simulated and observed pressures for the five crosshole pneumatic tests are shown in Figures 5.9 - 5.13. Some of these matches are
159
very poor due to instrumentation problems and barometric pressure effects; some
are of intermediate quality; and some are good to excellent. Tables 5.2 — 5.6 list
log io k and log io 0 estimates and their respective separate 95% confidence limits.
All permeability estimates appear to be reasonable. Unrealistically high porosity
estimates (log io ck = —0.3, equal to the prescribed upper bound) were obtained for
some intervals that appear to be pneumatically connected to the atmosphere. The
corresponding estimation uncertainties are large, and fits between calculated and
observed responses are poor. These data are ignored in the rest of my analysis. The
remaining estimates are characterized by low uncertainties. Summary statistics for
log lo k and log io q5 estimates are presented in Tables 5.7 and 5.8.
160
X1
1 o°
X2-1
10°
X2-2
X2-3
10°
10"
10°
10 0
10"
10°
10'
10'
10 -0
10°
1 0'
5.9. Separate matches between observed (small dots) and simulated (thick
curves) responses for test PP4 assuming uniform medium.
FIGURE
161
X2 - 2
X1
X2-3
10 '
10°
2.5
2
1.5
10.0
1 0 •°
lo
Z2-2
lw
10
lo
0.08
0.04
1/1
10
1 0. 0
W3-1
W2A-4
0.3
0.2
1 0 .0
10 2
r
10'
0.1
1 0°
5.10. Separate matches between observed (small dots) and simulated (thick
curves) responses for test PP5 assuming uniform medium.
FIGURE
162
X2-2
X1
X2-3
1 0'
10'
0.8
1 0 -2
0.7
0.6
10 -3
0.2
0.5
1 0 -s
0.4
0.6
0.4
0.2
10
0
1'0°
16'
Z2-2
0.3
0.25
0.1
0.2
10'
0.2
10°
10'
10°
los
1o0
10'
o°
10'
10 -210'
10°
5.11. Separate matches between observed (small dots) and simulated (thick
curves) responses for test PP6 assuming uniform medium.
FIGURE
163
X1
X2-1
X2 - 2
X2-3
10'
10 0
0.8
0.6
•••
•
10 0
0.4
100
0.2
10 0
10 4
1 0'
10 -0
10'
10 -0
10'
10 .0
10 -2
10'
10'
10'
5.12. Separate matches between observed (small dots) and simulated (thick
curves) responses for test PP7 assuming uniform medium.
FIGURE
164
X2-2
X2-3
1 00
10'
1 0 -2
10 0
10 -3
10'
10 -a
o'
5.13. Separate matches between observed (small dots) and simulated (thick
curves) responses for test PP8 assuming uniform medium.
FIGURE
165
TABLE 5.2. Parameters identified for cross-hole test PP4 treating the medium as
spatially uniform.
log io k [m 2 ]
Interval
k [rn 2 ]
loglo 0 [m 3 /m 3 ]
0 [m 3 /m 3 ]
X1
-14.08+ 0.007
8.3 x 10 -15
-2.12 1 0.021
7.7 x 10 -3
X2-1
1.1 x 10 -14
-13.94 ± 0.017
2.6 x 10 -2
-1.59 ± 0.054
X2-2
-14.26+ 0.050
5.5 x 10 -15
1.2 x 10 -2
-1.93 ± 0.170
X2-3
-2.04 ± 0.131
8.7 x 10 -15
-14.06 ± 0.054
9.2 x 10 -3
-12.18 + 0.032
6.6 x 10 -13
X3
-0.30 ± 0.111
5.0 x 10 -1
Y1-2
9.4 x 10 -15
-1.21 + 0.047
-14.02 ± 0.017
6.2 x 10 -2
Y2-1
9.3 x 10 -15
-1.20+ 0.140
-14.03 ± 0.039
6.3 x 10 -2
Y2-2
2.1 x 10 -14
-1.22 ± 0.049
-13.68 ± 0.001
6.0 x 10 -2
Y2-3
5.0 x 10 -15
-1.70 + 0.266
2.0 x 10 -2
-14.30+ 0.073
-1.64 ± 0.018
Y3-1
-13.24 + 0.015
5.8 x 10 -14
2.3 x 10 -2
Y3-2
6.6 x 10 -14
-1.01 ± 0.047
-13.18 ± 0.012
9.8 x 10 -2
-1.19 + 0.127
2.1 x 10 -14
Y3-3
6.4 x 10 -2
-13.68 ± 0.032
Z1
-12.42 ± 0.029
-1.22 ± 0.028
3.8 x 10 -13
6.0 x 10 -2
-1.28 + 0.151
Z2-1
-13.19 + 0.047
6.5 x 10 -14
5.2 x 10 -2
2.4 x 10 -13
-1.47 ± 0.027
Z2-2
-12.62 ± 0.018
3.4 x 10 -2
Z2-3
-11.52 ± 0.090
3.0 x 10 -12
-0.30+ 0.080
5.0 x 10 -1
1.2 x 10 -12
Z2-4
-11.92 1 1.528
-0.30 + 0.670
5.0 x 10 -1
1.4 x 10 -14
-0.99 + 0.035
Z3-1
-13.87+ 0.017
1.0 x 10 -1
1.6
x
10
Z3-2
-13.80+ 0.015
-14
-1.39 ± 0.037
4.1 x 10 -2
-12.58+ 0.099
2.6 x 10 -13
-0.52 + 0.101
Z3-3
3.0 x 10 -1
-14.11 ± 0.027
-1.90 ± 0.150
V1
7.8 x 10 -15
1.3 x 10 -2
-1.94 ± 0.075
-13.95+ 0.021
1.1 x 10 -14
V2-2
1.1 x 10 -2
-1.45 + 0.058
-13.65+ 0.014
2.3 x 10 -14
V3-1
3.5 x 10 -2
-2.29 + 0.066
-14.21 + 0.007
V3-2
6.1 x 10 -15
5.1 x 10 -3
-14.42
±
0.041
-1.62 ± 0.109
-15
2.4 x 10 -2
V3-3
3.8 x 10
-1.82
-13.94
±
0.025
1.1
x
10
+
0.077
W1
-14
1.5 x 10 -2
W2A-1
-13.36 1 0.011
4.3 x 10 -14
-1.55 + 0.042
2.9 x 10 -2
-14.11 ± 0.041
-1.48 ± 0.119
W2A-2
7.7 x 10 -15
3.3 x 10 -2
-1.35 ± 0.043
W2A-3
-13.82 ± 0.018
1.5 x 10 -14
4.5 x 10 -2
-1.22 ± 0.050
1.2 x 10 -14
W2A-4
-13.93 ± 0.024
6.0 x 10 -2
-15
-2.08
±
0.038
-14.21
±
0.025
6.2
x
10
8.3 x 10 -3
W3-1
±
0.114
10
-15
-2.08
-14.08
1
0.048
8.4
8.3 x 10-3
x
W3-2
166
TABLE
5.3. Parameters identified for cross-hole test PP5 treating the medium as
spatially uniform.
Interval
logio k [m2]
X1
-13.72 ± 0.010
X2-1
-13.83 ± 0.033
X2-2
-14.90 ± 0.112
X2-3
-14.31 + 0.045
X3
Y1-2
-14.18 + 0.124
Y2-1
Y2-2
-14.16 ± 0.077
Y2-3
-14.81 1 0.034
Y3-1
-13.48 ± 0.136
Y3-2
Y3-3
-13.35 ± 0.101
Z1
Z2-1
Z2-2
Z2-3
Z2-4
Z3-1
-13.42 + 0.132
Z3-2
-13.43 ± 0.094
Z3-3
V1
-14.01 ± 0.039
V2-2
-13.73 ± 0.031
-13.90 + 0.097
V3-1
-13.99 ± 0.023
V3-2
-15.15 ± 0.117
V3-3
W1
-13.91 + 0.045
-12.22 ± 0.024
W2A-1
W2A-2
-14.55 ± 0.040
W2A-3
-13.80 ± 0.090
W2A-4
W3-1
-13.94 ± 0.038
W3-2
k [m2]
1.9 x 10 -14
1.5 x 10 -14
1.3 x 10 -15
4.9 x 10 -15
logio 0 [m 3 /m 3 ]
-1.34 ± 0.030
-1.10 ± 0.129
-2.92 ± 0.741
-1.87 ± 0.097
0 [m 3 /m 3 ]
4.6 x 10 -2
7.9 x 10 -2
1.2 x 10 -3
1.3 x 10 -2
6.7 x 10 -15
6.9 x 10 -15
1.6 x 10 -15
3.3 x 10 -14
-2.07 ± 0.154
-1.98 ± 0.263
-2.78 ± 0.045
-1.33 ± 0.138
8.5 x 10 -3
1.1 x 10 -2
1.7 x 10 -3
4.7 x 10 -2
4.5 x 10 -14
-1.68 1 0.203
2.1 x 10 -2
3.8 x 10 -14
3.8 x 10 -14
-2.20 ± 0.127
-2.06 ± 0.108
6.4 x 10 -3
8.6 x 10 -3
9.7 x 10 -15
1.9 x 10 -14
1.3 x 10 -14
1.0 X 10 -14
7.1 x 10 -16
1.2 x 10 -14
6.1 x 10 -13
2.8 x 10 -15
1.6 x 10 -14
-1.75 ± 0.169
-1.42 ± 0.079
-0.98 ± 0.399
-1.46 ± 0.163
-2.79 ± 0.146
-1.67+ 0.093
-1.68 ± 0.170
-1.84 ± 0.068
-1.46 ± 0.215
1.8 x 10 -2
3.8 x 10 -2
1.0 x 10 -1
3.4 x 10 -2
1.6 x 10 -3
2.1 x 10 -2
2.1 x 10 -2
1.5 x 10 -2
3.5 x 10 -2
1.2 x 10 -14
-1.61 ± 0.114
2.5 x 10-2
167
TABLE 5.4. Parameters identified for cross-hole test PP6 treating the medium as
spatially uniform.
k [ m 2]
loglo 0 [m 3 /m 3 ]
logio k [m 2 ]
Interval
0 [m/m]
-14
-2.28
X1
-13.51 ± 0.038
3.1 x 10
+ 0.040
5.2 x 10 -3
-14
-1.47
±
X2-1
3.2
x
10
0.016
3.4 x 10 -2
-13.50 ± 0.015
1.6
x
10
2.6 x 10 -2
X2-2
-13.80 ± 0.030
-14
-1.59 ± 0.009
-14
-1.47
X2-3
-13.31 + 0.076
5.0 x 10
± 0.033
3.4 x 10 -2
X3
Y1-2
Y2-1
Y2-2
Y2-3
Y3-1
Y3-2
Y3-3
Z1
Z2-1
Z2-2
Z2-3
Z2-4
Z3-1
Z3-2
Z3-3
V1
V2-2
V3-1
V3-2
V3-3
WT1
W2A-1
W2A-2
W2A-3
W2A-4
W3-1
W3-2
-13.78 + 2.309
-13.58 + 0.755
-13.98 ± 1.499
-13.27± 0.033
1.6 x 10 -14
2.6 x 10 -14
1.0 X 10 -14
5.4 x 10 -14
-1.55 ± 1.454
-1.50 ± 0.729
-2.18 + 1.164
-1.20 + 0.024
2.8 x 10 -2
3.1 x 10 -2
6.5 x 10 -3
6.3 x 10 -2
-13.27 + 0.032
5.4 x 10 -14
-0.58 ± 0.013
2.6 x 10 -1
-12.00 + 0.043
1.0 X 10 -12
-1.15 ± 0.060
7.0 x 10 -2
-14.40
-14.22 + 0.006
4.0 x 10 -15
6.0 x 10 -15
-1.52
-5.00 ± 0.100
3.0 x 10 -2
1.0 X 10 -5
-13.43 ± 0.028
-13.71 ± 0.033
-13.97 ± 0.418
-13.66 ± 0.015
-14.13 ± 0.057
-13.89 1 0.015
-13.49 ± 1.966
-13.61 ± 0.062
-13.89 ± 0.097
-13.84 1 0.098
-13.89 ± 0.033
3.7 x 10 -14
2.0 x 10 -14
1.1 X 10 -14
2.2 x 10 -14
7.5 x 10 -15
1.3 x 10 -14
3.2 x 10 -14
2.4 x 10 -14
1.3 x 10 -14
1.4 x 10 -14
1.3 x 10 -14
-2.14± 0.063
-2.41 ± 0.058
-2.02 + 0.472
-2.31 + 0.049
7.2 x i0
3.8 x 10 -3
9.4 x 10 -3
4.9 x 10 -3
2.4 x 10 -2
2.0 x 10 -2
1.0 X 10 -2
5.9 x 10 -3
8.9 x 10 -3
1.2 x 10 -2
2.3 x 10-3
-1.63 1 0.014
-1.70 ± 0.026
-2.00 1 0.642
-2.23 ± 0.037
-2.05 ± 0.052
-1.92 ± 0.029
-2.64 ± 0.008
-
168
TABLE 5.5. Parameters identified for cross-hole test PP7 treating the medium as
spatially uni form.
k [m 2]
logio 0 [m 3 /m 3 ]
log io k [m 2 ]
Interval
0 [m 3 im 3 ]
2.5 X 1 0 -16
-5.00
1.0 x 10 -5
X1
-15.61
-14.03 ± 0.016
-1.89 ± 0.021
1.3 x 10 -2
X2-1
9.3 X 1 0 - ' 5
-14.00 + 0.012
1.0 X i o -14
-1.62 ± 0.024
2.4 x 10 -2
X2-2
-1.91 ± 0.098
1.2 x 10 -2
X2-3
-14.04 + 0.055
9.2 X i o -15
X
0
-12
-12.00
±
0.076
1.0
1
-0.63
±
0.091
2.4
x 10 -1
X3
Y1-2
Y2-1
-14.63 + 0.001
2.3 X i o -15
-2.29 ± 0.000
5.1 x io -3
1.2 x i o -14
-2.21 + 0.100
6.1 x 1 0 -3
Y2-2
-13.92 + 0.075
-2.02 ± 0.036
Y2-3
-14.31 + 0.138
4.9 X 1 0 - ' 5
9.6 x i o -3
1.0 x 1 0 -2
1.5 X
-1.99 ± 0.030
Y3-1
-13.82 + 0.063
-1.24 ± 0.018
1.1 X 1 0 -13
5.7 x 1 0 -2
Y3-2
-12.96 + 0.030
2.7 x 1 0 -2
5.3 X b o - ' 4
-1.58 1 0.060
Y3-3
-13.28 + 0.128
1.9 X i o - ' 4
-2.06 ± 0.349
8.7 x i o -3
Z1
-13.72 + 0.481
-1.95 + 0.146
1.1 x 10 -2
Z2-1
-13.36 ± 0.224
4.3 X i o - i 4
7.8 X i o -14
-1.19 ± 0.926
Z2-2
-13.11 ± 1.380
6.5 x 10 -2
7.1 X i o -1- 3
-0.30 + 0.334
5.0 x 10 -1
Z2-3
-12.15 + 0.516
2.3 X i o - ' 3
2.3 x 1 0 -1
-0.64 ± 0.495
Z2-4
-12.64 + 0.367
2.7 x 1 0 -2
Z3-1
-13.88 + 0.004
1.3 X i o - ' 4
-1.56 ± 0.003
-1.28 + 0.047
-12.74 + 0.612
1.8 X
Z3-2
5.3 x 10 -2
2.2 X
-0.49 ± 0.281
3.3 x 10 -1
Z3-3
-12.66 ± 0.374
1.1 X
-1.83 ± 0.071
1.5 x 10 -2
V1
-13.96 ± 0.023
-2.13 ± 0.083
7.5 x 10 -3
V2-2
-14.08 ± 0.065
8.3 X 10 -15
1.6 x 10 -2
9.0 X io -- ' 5
-1.79 ± 0.090
-14.05 ± 0.074
V3-1
1.2 X 1 0 -14
-2.03 ± 0.130
9.3 x 10 -3
V3-2
-13.93 + 0.090
9.9 X i o -15
-1.71 ± 0.023
1.9 x 10 -2
-14.00 ± 0.044
V3-3
-2.26 ± 0.071
6.4 X i o -15
5.5 x 10 -3
-14.20 + 0.062
W1
W2A-1
2.6 x 10 -14
-1.97 + 0.071
1.1 x 10 -2
W2A-2
-13.59 ± 0.060
-1.29 ± 0.059
5.1 x 10 -2
-13.52 ± 0.112
3.0 x 10 -14
W2A-3
W2A-4
1.6 x 10 -2
2.5 x 10 -15
-1.80 ± 0.377
-14.61 + 0.094
W3-1
3.2 x 10 -14
-0.30 ± 0.846
5.0 x 10-1
W3-2
-13.50 ± 0.021
169
TABLE 5.6. Parameters identified for cross-hole test PP8 treating the medium as
spatially uni form.
k [m2]
logio 0 [m3 /m3 ]
log lo k [m 2 ]
Interval
45 [m 3 /m3 ]
-2.12 + 0.018
7.5 x 10 -3
8.1 x 10 -15
-14.09 ± 0.008
X1
2.6 x 10 -2
1.1 x 10 -14
-1.59 ± 0.032
-13.96 ± 0.013
X2-1
1.3 x 10 -2
-14.31 ± 0.047
4.8 x 10 -15
-1.90 ± 0.111
X2-2
-2.02 + 0.082
-14.11 + 0.045
7.7 x 10 -15
9.5 x 10 -3
X2-3
X3
-1.19 ± 0.020
-14.12 ± 0.013
6.4 x 10 -2
Y1-2
7.5 x 10 -15
-14.07± 0.058
-1.16 ± 0.142
6.9 x 10 -2
8.5 x 10 -15
Y2-1
-1.12 ± 0.100
2.1 x 10 - ' 4
7.6 x 10 -2
-13.68 + 0.005
Y2-2
-1.62 ± 0.264
2.4 x 10 -2
4.3 x 10 -15
Y2-3
-14.37 ± 0.089
8.3 x 10 -2
-1.08 ± 0.005
-13.20 ± 0.004
6.4 x 10 -14
Y3-1
1.7 x 10 -1
-0.76 ± 0.148
-13.41 + 0.027
3.9 x 10 -14
Y3-2
-1.21 + 0.075
1.5 x 10 -14
6.1 x 10 -2
-13.83 ± 0.026
Y3-3
-1.18 ± 0.047
2.8 x 10 -13
6.7 x 10 -2
Z1
-12.55 ± 0.052
-1.18 ± 0.069
6.7 x 10 -2
-13.29 ± 0.025
5.2 x 10 -14
Z2-1
-1.40 ± 0.019
4.0 x 10 -2
2.1 x 10 -13
-12.68 ± 0.014
Z2-2
3.1 x 10 -12
-0.30 ± 0.028
5.0 x 10 -1
-11.50 ± 0.039
Z2-3
4.7 x 10 -1
1.0 x 10 -12
-0.33 ± 0.450
Z2-4
-11.98 + 0.843
1.3 x 10 -14
8.5 x 10 -2
-1.07 ± 0.057
Z3-1
-13.89 + 0.042
4.1 x 10 -2
1.6 x 10 -14
-1.39 + 0.034
Z3-2
-13.81 ± 0.018
3.0 x 10 -1
2.7 x 10 - ' 3
-0.52 ± 0.054
Z3-3
-12.56 ± 0.058
1.4 x 10 -2
-1.87 + 0.059
6.5 x 10 -15
-14.18 + 0.015
V1
1.1 X 10 -2
1.1 x 10 -14
-1.96 + 0.070
V2-2
-13.95 ± 0.029
2.5 x 10 -2
1.1 X 10 -14
-1.60 ± 0.159
-13.98 + 0.040
V3-1
-2.31 ± 0.099
4.9 x 10 -3
6.1 x 10 -15
-14.21 + 0.019
V3-2
2.7 x 10 -15
3.3 x 10 -2
-1.48 ± 0.082
-14.57 ± 0.048
V3-3
1.2 x 10 -2
1.1 x 10 -14
-1.91 ± 0.041
-13.94 + 0.017
W1
-1.62 ± 0.056
2.4 x 10 -2
4.4 x 10 -14
W2A-1
-13.36 1 0.022
-1.46 ± 0.088
3.5 x 10 -2
7.0 x 10 -15
W2A-2
-14.15 ± 0.040
4.5 x 10 -2
1.5 x 10 -14
-1.35 ± 0.046
-13.83 ± 0.027
W2A-3
-1.24 ± 0.024
1.1 x 10 -14
5.8 x 10 -2
W2A-4
-13.94 ± 0.026
1.0 X 10 -2
-1.98 + 0.106
5.2 x 10 -15
-14.28 ± 0.109
W3-1
8.8 x 10-3
-2.06 + 0.038
7.0 x 10 -15
-14.15 + 0.023
W3-2
170
TABLE 5.7. Summary statistics for log lo k [m 2 ] identified for cross-hole tests treating
the medium as spatially uniform.
Minimum
Test
Sample size
Maximum
Mean
Variance
CV
PP4
-14.42
-11.52
-13.57
0.568
32
-0.0555
-12.22
PP5
21
-15.15
-13.94
0.397
-0.0452
-14.4
-12.0
PP6
-13.66
0.220
-0.0343
23
PP7
-15.61
-12.0
29
-13.67
0.560
-0.0562
-14.57
-11.5
-13.68
PP8
31
0.530
-0.0533
-11.5
All
-15.61
-13.69
0.479
136
-0.0505
TABLE 5.8. Summary statistics for log io 0 [m 3 /m 3 ] identified for cross-hole tests
treating the medium as spatially uniform.
Test
Sample size
Minimum
Maximum
Mean
Variance
CV
PP4
-2.29
29
-0.52
-1.53
0.168
-0.267
PP5
21
-2.92
-0.98
-1.81
0.279
-0.292
PP6
22
-2.64
-0.58
-1.80
0.237
-0.271
PP7
26
-2.29
-0.49
-1.67
0.254
-0.302
-1.46
-2.31
PP8
30
-0.33
0.225
-0.326
128
-2.92
0.241
-0.301
All
-0.33
-1.63
It is of interest to note that the injection intervals in cross-hole test PP4 (Y22) and single-hole test JG0921 (discussed in Chapter 4) virtually coincide. Though
the injection rate during PP4 (1 x 10 -3 kg/a) had significantly exceeded that during
JG0921 (8.014 x 10 -6 and 3.967 x 10 -5 kg/a), both tests yielded similar permeabilities
and porosities for the injection interval (Tables 5.2 and 4.11). The estimates are
k = 2.1 x 10' and 0 = 6.0 x 10 - 2 for the cross-hole test, and k = 2.4 x 10' and
0 = 1.4 x 10 -2 for the single-hole test.
Some of the poor matches obtained for test PP4 (Figure 5.9) can be significantly
improved by adding an adjustable borehole storage parameter to both the injection
and observation intervals. In my analysis of PP4, borehole porosities along intervals
are fixed as defined in Chapter 3 (Equations 3.65 and 3.66, page 101). In my analysis
of single-hole tests (Chapter 4), there was only one adjustable borehole storage pa-
rameter at the injection interval, Ow . Here I consider two such parameters, log io 0„/
and log lo 0„°. The new matches for eight of the intervals are depicted in Figure 5.14,
and the corresponding parameters are listed in Table 5.9. Variations in the two stor-
171
X2-3
Y2-1
io '
10°
10 -1
10 -2
10 -.3
10 -2 16-1
W2A-2
'-i"o°'
10 1
lo°
10°
10 1
-
1 0°
10 -1
10 -1
10 -2
10 -2
10 210 110°
FIGURE 5.14. Separate matches between observed (small dots) and simulated (thick
curves) responses for test PP4 assuming uniform medium and adjusting borehole
storage parameters along injection and observation intervals.
TABLE 5.9. Parameters identified for cross-hole test PP4 treating the medium as
spatially uniform and adjusting borehole storage at injection and observation intervals.
loglo 0 {m 3 /m3
Interval
logio k [m 2
logio OL, [770/m 3 ]
loglo O w° [m 3 /m 3 ]
X2-2
-14.18 ± 0.010
-3.03 ± 1.484
1.83 ± 0.210
1.37 ± 0.544
X2-3
-2.69 ± 1.327
-13.95 ± 0.038
1.97 ± 0.531
1.42 ± 0.754
Y2-1
-14.00
-1.69
1.75
1.30
Y2-3
-14.21 ± 0.013
1.71 + 0.343
-2.57 ± 1.176
0.97 + 0.474
-2.99 ± 1.224
2.59 1 0.118
2.67 ± 0.073
Y3-3
-13.47 ± 0.013
-14.26
±
-3.83 ± 1.353
1.46 ± 0.065
V3-3
0.016
1.97 ± 0.028
1.82
W2A-2
-14.00
-2.38
1.86
-2.59 ± 0.018
W3-2
-13.96 ± 0.002
1.96 ± 0.009
1.85 ± 0.012
]
]
age parameters are an indication of variations in effective geometry of test intervals
and medium heterogeneity.
To check the influence of borehole conductance and storage effects, the analysis
of test PP4 was repeated without including open borehole intervals in the model.
Pressure in each test interval is calculated as the average of pressures at all nodes
along the interval. Estimates of log io k and log io q obtained with and without borehole
effects are compared in Figure 5.15. Neglecting the influence of boreholes causes a
systematic increase in the estimates of both parameters by a factor of about 1.4. This
means that open test intervals enhance the conductive and storage properties of the
172
i]
t
1Mth borehole effects
FIGURE 5.15. Estimates of pneumatic properties for test PP4 obtained with and
without borehole effects.
medium and must not be ignored.
As I have already discussed in Chapter 3, a constant uniform pressure was specified
at the top boundary of the computational region. Absolute barometric pressure was
not measured at the ALRS during the cross-hole pneumatic tests. I therefore assumed
that barometric pressure was equal to 100 kPa, which is the average barometric
pressure at sea level. In fact, ALRS is at an elevation of 1,200 m above sea level,
and the average barometric pressure should be lower, approximately 87 kPa, but can
vary with meteorological conditions. To test the effect of this barometric pressure
discrepancy on my estimates, I repeated the inverse analysis of test PP4
with a
barometric pressure equal to 87 kPa. The two sets of estimates are compared in Figure
5.16. The smaller barometric pressure produced larger estimates of k by a factor of
about 1.1, but did not affect the log io 0 estimates (except for a few intervals indicated
in Figure 5.16 which are characterized by poor matches and uncertain estimates). The
factor 1.1 is close to the ratio between the adopted barometric pressures (101/87 =
1.16). Thus, uncertainty in barometric pressure may cause a small systematic error
in log io k estimates, which can be corrected a posteriori. Using analytical type-curves
173
based on linearized airflow equations, Illman and Neuman [2000a] however obtained
opposite results. They concluded that a decrease in barometric pressure would not
alter k estimates but would decrease 0 estimates by a factor of 1.1. To investigate this
discrepancy, I examine the non-linearized governing airflow equations (3.1 — 3.4, page
71) neglecting the gravity term for consistency with analysis of Illman and Neuman
[2000a]. An increase or a decrease in barometric pressure produces a similar change
in air pressures within the rock during a pneumatic test. Absolute air pressure and
air density are proportional to each other, as indicated by Equation 3.4. The pressure
change would not modify significantly the gradient Vp and time derivatives 3p/at of
air density p; it would influence predominantly the product of k and p in Equations
3.1 and 3.3. Therefore, a decrease in barometric pressure would produce an equivalent
increase in permeability estimates. This is consistent with the results yielded by my
numerical inverse model. On the other hand, the linearized p-based equation (3.78
on page 114) applied by Illman and Neuman [2000a] produces different estimates for
different initial barometric pressures only due to change in the air compressibility C..
Illman and Neuman [2000a] took Ca to be a constant in space and time, and equal
to the inverse of barometric pressure. Therefore, the analytical results are biased by
linearization of the governing airflow equations.
To date, analytical type-curves were used to analyze test PP4 [Illman et al., 1998;
Illman, 1999; Illman and Neuman, 2000a]. The most up-to-date results are those
given in [Illman and Neuman, 2000a]. The analytical and numerical estimates of
log lo k and log io 0 are compared in Figure 5.17. Though type-curve results for k
consistently exceed numerical results by a factor of about 1.7, the two data sets correlate quite well if one ignores intervals Z1, Z2-2, Z2-3, Z2-4 and Z3-3 (which, as
discussed earlier, are located in a highly permeable zone and connected to the atmosphere). The difference can be due in part to round-off errors in both the numerical
and analytical methods (the analytical solution relies on numerical inversion of the
Laplace transform). When open monitoring boreholes are excluded from the numer-
174
1E+0
1E-11
1E-1
1E-2
1E-15
1E 15
1E-14
1E-13
1E-12
Barometric pressure 100 kPa
1E-11
1E-3
1E-3
1E-2
1E-1
1E+0
Barometric pressure 100 kPa
FIGURE 5.16. Estimates of pneumatic properties for test PP4 obtained for different
initial barometric pressure.
ical model, the systematic difference between numerical and analytical k estimates
is reduced to a factor of 1.4. This suggests that type-curve analysis assigns higher
permeabilities to the rock so as to compensate for its inability to fully account for
the influence of open borehole intervals on pressure. Further, the analytical solution
solves a linearized version of the governing equation (3.78, page 114) as discussed in
Section 3.4, in which air compressibility is taken to be a constant. During cross-hole
tests, pressure buildups were significant in magnitude compared to the barometric
pressure, and varied substantially in time and space. Air compressibility is approxi-
mately equal to the inverse of absolute air pressure (see Section 3.1.1), and therefore,
also varied considerably. This can be another reason for the discrepancy between
numerical and analytical estimates. Other differences between the two methods of
solution include the effect of constant-pressure boundary on the ground surface, and
the need to prescribe arbitrary boundary conditions in the numerical model.
Overall, there is a good agreement between log io q5 estimates obtained by numerical and analytical methods, except for intervals Y2-3, Y3-1, Z2-1, Z2-2, Z2-3, Z2-4,
Z3-3 and V3-3. This is due to the ambiguous pressure records in these intervals: stor-
175
1E-11
-
1E+°
1E-1
Air-filled porosity [m3/ mil
-
1E-12
-
ir
Z
1E-13
ZF5-4
-
Z -n '3
1E-14
1E-4
Zp
1E-15
1E15
1E-14
1E-13
1E-12
1E-11
1E-5
1E-5
Numerical solution
FIGURE
1E-4
1E-3
1E-2
1E-1
1E+0
Numerical solution
5.17. Analytically versus numerically derived pneumatic properties for test
PP4.
age effects heavily influences Y2-3 and V3-3; the Z intervals are located in a highly
permeable zone connected to the atmosphere.
I attempted to interpret pressure records from all monitoring intervals simultaneously while treating the rock as uniform [Illman et al., 1998]. This led to very poor
matches, which is a sign of medium non-uniformity.
176
5.2.2 Non-uniform medium
I used the stochastic inverse model described in Chapter 3 to interpret simultaneously
pressure records from each of cross-hole tests PP4, PP5 and PP6. Here I describe in
detail the inversion of test PP4. I also performed a simultaneous inversion of pressure
records from all three tests. These analyses amount to high-resolution stochastic
imaging, or pneumatic tomography, of the fractured rock mass, an idea proposed
over a decade ago by Neuman [1987] in connection with hydraulic cross-hole tests
in saturated fractured crystalline rocks at the Oracle site near Tucson [Hsieh et al.,
1985].
The stochastic inverse model regards medium properties as spatially correlated
random fields, characterized by variogram models with adjustable parameters, and
conditioned on adjustable pneumatic parameters at a set of pilot points. In my study,
I utilize between 32 and 72 pilot points located subjectively in the computational
domain. Most points are placed along test intervals. To test the influence of pilotpoint locations, I inverted the same test data (PP4) using two different sets of pilot
points with equal size. In the first case, all pilot points are located at middle of test
intervals, while in the second case, the points are in middle between the injection
and corresponding observation intervals (plus there is again a pilot point at center
of injection interval). The first case produced much more satisfactory results than
the second one in terms of objective function and estimation errors. This suggests
that pressure responses in monitoring intervals are influenced substantially by the
properties of the medium in the vicinity of corresponding intervals. Locating the pilot
points predominantly along boreholes allows also better comparison between inverse
estimates and the available ALRS data (single-hole air permeability, core matrix airfilled porosity, fracture density) which are also defined along the boreholes.
The inverse model requires specifying initial values for the unknown parameters.
In all numerical inversion presented below, initial values for log io k and log lo
0 were
177
set equal to the geometric average of each, as obtained from the earlier uniform inverse
interpretations of PP4 pressure data. The numerical inversion started with spatially
uniform values that were then successively perturbed and adjusted throughout the
optimization process. The exponents of power variograms of the parameter fields
were estimated both simultaneously with and independently of the other unknowns.
When the exponents were assessed simultaneously with pneumatic parameters, their
initial values for both log io k and log io ci) were set equal to 0.45, the value obtained
through geostatistical analysis of single-hole air permeability data (Chapter 2). To
check the influence of initial values on final parameter estimates, the initial values were
either decreased by an order of magnitude, or set equal to random values uniformly
distributed from —16 to —13 for log io k and from —3 to —1 for log io q5. This analysis
proved that initial values have a minor effect on the final estimates.
All numerical inversions converged in less than 50 iterations. Depending on the
number of tests and unknown parameters considered simultaneously, the inversions
required from 1,000 to 5,000 forward model runs which took from 10 to 50 hours of
computational time when using a supercomputer with 32 processors.
Cross-hole test PP4
Before starting the inverse analysis, I simulated test PP4 using the kriging estimates
of air permeability (Figure 2.18, page 58) and matrix air-filled porosity (Figure 2.26,
page 66) discussed in Chapter 2. The calculated pressure responses differed markedly
from those observed in the field [Illman et al., 1998]. An attempt to improve the
matches using trial and error proved to be difficult and unsuccessful.
I also tried to parameterize the unknown spatial medium properties through firstand second-order polynomials. These attempts failed.
The stochastic inverse model was applied to a set of 32 pilot points located within
each of the 32 intervals that had responded during test PP4. Most of the pilot points
were located at the centers of the corresponding intervals. In the guard intervals,
178
0
-10
N
-20
3
-30
-40
FIGURE 5.18. Three-dimensional representation of 32 pilot points.
Y2-1 and Y2-3, the points are offset from the center toward injection interval Y2-2,
to define better medium properties between these intervals. In the long interval V1,
the point is offset toward the ground surface where most airflow takes place during
pneumatic tests. The location of pilot points is depicted in Figure 5.18.
First, I treated log io k as non-uniform and log io 0 as uniform. The log io k random
field was described through unknown values at pilot points and an unknown exponent of the corresponding power variogram. Also unknown was the uniform value of
log lo 0, leading to 34 unknowns. Figure 5.19 shows simultaneous matches between
computed and monitored pressures in all 32 intervals obtained with the corresponding
179
parameter estimates. The sum of squared pressure residuals 0 was 679 kPa 2 . Most of
the computed responses capture in a satisfactory manner the late-time observed pressures. However, there are substantial deviations at early time, which suggests that
the uniform log io 0 assumption is not realistic. The corresponding kriged estimate of
log io k is shown by a three-dimensional fence diagram in Figure 5.20; the exponent of
the power variogram was estimated to be equal to 1, its specified upper bound. The
uniform log io
0 estimate was equal to —2.0. The region depicted in Figure 5.20 and
in all similar figures presented later has dimensions 45 x 33 x 33 (49,005) m 3 , and
includes all the test boreholes.
To take into account spatial non-uniformity of log io 0, the number of unknowns
was increased to 66 by including log-transformed air-filled porosities at the 32 pilot
points, and the exponent of their power variogram. Figure 5.21 shows the resulting
simultaneous matches between computed and recorded pressures. Compared to the
previous case with uniform log io
0 (Figure 5.19), the fits are significantly improved,
especially at early time.
I compared the new matches with those obtained through separate analysis of
individual responses (Figure 5.9). The sum of squared differences, 0, between computed and recorded pressures for all intervals was equal to 131.2 kPcg in the first case
and 104.4 kPa 2 in the second case. Both the number of unknowns and the number of
match points were the same in the two cases. Upon comparing the matches in Figures 5.9 and 5.21, one notes that for some of the intervals—X2-2, Y2-3, Y3-2, Z3-2,
V3-3 and Wl—the simultaneous matches are better. The simultaneous matches for
intervals Y3-1, Z1, Z2-1, Z2-2, Z2-3, Z2-4 and Z3-3 under- or over-estimate the observed pressure responses; however, the simulated curves represent more adequately
the slope of observed responses compared to the separate analysis.
A histogram of residuals between computed and recorded pressures obtained by
stochastic inverse model is shown in Figure 5.22. The figure suggests that the population distribution of residuals is not normal, which is confirmed by a Kolmogorov-
180
X2-1
X1
X2 - 3
10°
X3
10°
10 - '
10 °
110
1 0-2
10'
100
102
10 -2
10'
10
0
10 •
Y2-3
O
1 0°
Y3-1
io° Y3-2
10'
10°
10'
10
10°
'
10 -0
10. 0
-2
104
10 -2
10-'
10 0
10'
Z1
10-'
10°
8:0
io°
1 0 '
10 0
Z2-3
10
10 4 r
10
100
10'
10 -0
10
Z2-2
10'
0.2
10 4
10 210 -110 01 0 '
10 0
Z2-1
10-'
102
Z3-1
10°
o Z2-4
r
10 _1
10'
10 0
10°
1 0'
10'
10
10°
1
•
'
5.19. Matches between observed (small dots) and simulated (thick curves)
responses obtained by simultaneous inversion of PP4 data with 32 pilot points; log io
FIGURE
is assumed to be uniform.
181
30
-10
-10
-11
-12
-13
-14
-15
-16
-17
-18
-19
-20
FIGURE 5.20. Three-dimensional representation of kriged log io k estimated by simultaneous inversion of PP4 data with 32 pilot points; log io 0 is assumed to be uniform.
182
X2-1
X1
10°
X2-3
10°
10°
10 -2
10°
110
100
10"
10'
10°
10°
10 -0
10 -0
10"
10
Z2-4
10° o
10 -1
10 -2
10 -3
10 -2
10'
10°
10°
10°
10 -1
FIGURE 5.21. Matches between observed (small dots) and simulated (thick curves)
responses obtained by simultaneous inversion of PP4 data with 32 pilot points.
183
Smirnov test. Though most of the residuals are close to zero, their sample distribution
exhibits heavy tails due to the matches for intervals Xl, X2-3, V3-1, V3-3 and W2A-1.
To improve these matches, the number of pilot points would have to be increased.
The exponents 0 of power variograms are estimated both simultaneously and
independently with the rest of the unknowns, and in both cases, the best current
estimate was /3 = 1, the specified upper limit. Setting the exponent of the power
variogram to values lower than 1 for both log io k and log io qS causes the matches to
deteriorate. Unfortunately, I could not extend my analysis to values of 0 higher than
1.
The pilot-point estimates of log io k and log io 0, as well as associated separate 95%
confidence limits, are listed in Table 5.10. The points are identified by the intervals in
which they are located. The confidence limits are computed from the diagonal terms
of the covariance matrix of estimation errors E aa . E aa is computed using Equation
3.56 (page 89) without including prior information. Prior information would most
probably have reduced the estimation errors. However, as we are going to see later,
there are other factors that influence estimation uncertainty. Most log io k estimates
seam reasonable and are associated with relatively small errors. Though four of them
reach the specified upper (-10) and lower (-20) limits, their errors are nevertheless
low. Compared to log lo k, the log io 0 estimates have larger errors; more of them are
at the specified upper (-5) and lower (-0.3) limits.
The covariance matrix of estimation errors at pilot points, E„,„ is depicted in Figure 5.23 as a two-dimensional map of square pixels. The parameters are numbered
in the order they are listed in Table 5.10. The first set of 32 parameters describes air
permeabilities, the second set air-filled porosities. Thin solid lines separate between
matrix blocks associated with these two sets of parameters. Figure 5.24 displays the
corresponding correlation matrix. One notes that although the cross-covariances of
porosity estimation errors (upper right block in Figure 5.23) are higher than those
of permeability (lower left block in Figure 5.23), the cross-correlations of porosity
184
80
60
40
20
0
0
0
*000
%
*00 *t50
\
*00 '60
.6‘
0
7
0*00
7
•tS
0 •0
FIGURE 5.22. Histogram of residuals between observed and simulated responses.
185
5.10. Parameters identified at the pilot points by numerical inversion of data
from cross-hole test PP4 using 32 pilot points.
TABLE
Interval
X1
X2-1
X2-2
X2-3
X3
Y1
Y2-1
Y2-2
Y2-3
Y3-1
Y3-2
Y3-3
Z1
Z2-1
Z2-2
Z2-3
Z2-4
Z3-1
Z3-2
Z3-3
V1
V2
V3-1
V3-2
V3-3
W1
W2-1
W2-2
W2-3
W2-4
W3-1
W3-2
logn k [m 2 ]
-11.60 ± 0.23
-15.27 + 0.17
-15.29 ± 0.27
-16.01 ± 0.17
-20.00 + 0.30
-17.76 + 0.32
-16.61 ± 0.17
-13.05 ± 0.03
-14.56 ± 0.07
-12.95 ± 0.17
-20.00 ± 0.25
-20.00 + 0.64
-10.00 ± 0.37
-19.06 1 0.41
-14.37 + 0.33
-14.42 ± 0.38
-14.35 ± 0.44
-12.40 ± 0.41
-11.93 ± 0.20
-13.15 ± 0.36
-18.62 ± 0.23
-15.07 ± 0.37
-11.29 ± 0.18
-15.44 ± 0.07
-14.45 ± 0.66
-12.96 ± 0.74
-18.93 ± 0.46
-16.70 ± 0.15
-14.18 ± 0.18
-13.81 ± 0.14
-16.61 ± 0.03
-14.60 ± 0.19
logio q5 [m 3 /m3 ]
-2.17 ± 0.87
-2.40 ± 0.22
-3.81 ± 0.48
-5.00 ± 0.45
-5.00 1 0.97
-2.72 ± 0.33
-2.53 ± 0.40
-2.38 ± 0.27
-0.30 ± 0.55
-5.00 ± 0.70
-5.00 ± 0.40
-5.00 ± 1.16
-1.25 ± 1.07
-5.00 ± 0.55
-1.63 ± 0.96
-2.76 + 0.79
-1.84 ± 0.93
-0.30 ± 0.46
-0.41 ± 0.25
-0.30 ± 1.15
-5.00 ± 0.78
-4.59 ± 0.94
-1.02 ± 0.39
-4.23 ± 0.16
-1.17 ± 1.15
-1.68 ± 1.43
-5.00 ± 0.76
-4.18 ± 0.51
-2.18 ± 0.37
-1.65 ± 0.44
-2.88 ± 0.58
-5.00 ± 0.51
186
estimation errors (upper right block in Figure 5.24) are lower than those of perme-
ability (lower left block in Figure 5.24). The higher cross-covariances of porosity
errors are consistent with the higher uncertainties of porosity estimates. The high
cross-correlation between permeability errors suggests that the separate confidence intervals, discussed above, underestimate log io k uncertainties. The correct estimation
of log lo k errors require simultaneous, in my case also nonlinear, confidence intervals
which I have not estimated because this would be computationally very demanding. Figures 5.23 and 5.24 suggest that cross-covariances/cross-correlations between
permeability and porosity errors (upper left or lower right blocks) are smaller than
those between permeability errors (lower left block) and those between porosity errors
(upper right block). However, there is cross-correlation (close to 1) between permeability and porosity errors at some of the pilot points (diagonal terms of upper left
matrix block in Figure 5.24). This is as a result of the correlation between estimated
parameters through the governing airflow equations.
Absolute components of the covariance matrix of log io k and log io c/5 estimation
errors are plotted versus spatial distance between the corresponding pilot points in
Figure 5.25. The corresponding absolute correlation matrix components are plotted
in Figure 5.26. The two figures suggest that correlation between estimation errors of
log lo k and log lo 0 does not depend on the spatial configuration of the pilot points.
Figure 5.27 depicts eigenvector components of the covariance matrix E cia . The
eigenvectors are ordered according to the magnitude of their corresponding eigenvalues; that with smallest (2 x 10 -7 ) eigenvalue is numbered 1, and that with highest
eigenvalue (1.3) is numbered 64. A thin solid line separate the first 32 components of
each eigenvector, which correspond to log io k, from the next 32 components, which
correspond to log io q5. Parameters associated with small eigenvalues are less uncer-
tain than those associated with large eigenvalues. The components of each eigenvector
represent the relative contribution to it by various parameter estimates. For example, the first elgenvector (with the smallest eigenvalue) is associated almost entirely
187
0.25
6
0.20
0.15
0.10
5
0.05
-0.00
-0.05
-0.10
-0.15
-0.20
-0.25
2
1
10
20
30
40
50
60
Parameters
FIGURE 5.23. Two-dimensional representation
errors at the pilot points.
of covariance matrix of estimation
188
1.00
0.90
0.80
0.70
0.60
- 0.50
0.40
0.30
0.20
0.10
-0.00
-0.10
-0.20
-0.30
-0.40
-0.50
-0.60
-0.70
-0.80
-0.90
-1.00
10
20
30
40
50
60
Parameters
FIGURE 5.24. Two-dimensional representation of correlation matrix of estimation
errors at the pilot points.
•
189
0.14
logio k [ 1112 ]
0.12
0.1
g
u
c
(a
i 0.08
o
u
g 0.06
Ti
u)
_o
<0.04
*
•
•
s•
• •
• • •
•re s •.• • 8 • ..• • •
• • • ••• .•:•
•:• . :11,• •
•4 eik :.,•* ..: . . :
•
0.02
••
•
f
10
O.
20
30
10
Distance between pilot points [m]
20
30
Distance between pilot points [m]
Absolute components of the estimation covariance matrix versus dis-
FIGURE 5.25.
tance between the pilot points.
1.0
0.9
0
1?; 0.8
o
4=co 0.7
o
0.6
o
.tO 0.5
a)
5 04
.
u
=-' 0.3
'4)
5
A
C 0.2
•ct
0
.1
-
1.0
•
I9g1q
0.9
4
is 0.8
•• •
• •
•
k Ern 2 1
•
•• •
• s
•••• •...•
• • •
• • • • •
•
•: . •
• •
•.
log10 o [m 3 /m 3 ]
-
i3
tcu
cu 0 .i657
oo
•
••
• •
. *. • • ••
• :•••
•
•
• . .. . •
.
• .s. .• • • •
.•••••
• • •.1.
. •
• % . • • %.•••
• • •.; . •••.u: . ••.,
• •
:• •
. •
: : ..4* . 112, • .1„. _..
•%.
. si. ..
• 8... e.b -i
.. ...„
.. • 1, . ..: • . ••• ,:f.• •
.%
.1.1.:
^
,%.: .. • .....•• fti.
...s
..,..:.r.
• .•
•• ....._
. _„,„„%.:-,.....::
. • , I : • •
72
,
,
:
Distance between pilot points
o
to
•
5 0.4
0
23 0.3
'5
m
u) 0.2
<
0.1
[m]
FIGURE 5.26. Absolute components of the estimation correlation matrix versus distance between the pilot points.
190
with the eighth parameter, which is the permeability at the injection interval Y2-2.
Therefore, the corresponding estimate is the least uncertain (as one can also see in
Table 5.10). The last eigenvector (with the largest eigenvalue) is associated with
a number of porosities which are poorly estimated and cross-correlated. Figure 5.27
demonstrates that eigenvectors associated with the smaller eigenvalues are dominated
by permeability estimates. Therefore, the permeabilities are estimated with less uncertainty than porosities. The largest absolute eigenvector components associated
with all parameter estimates are shown in Figure 5.28. The larger is this component
for a given parameter, the smaller is its estimation error. The magnitude of largest
absolute eigenvector components agrees with the size of confidence intervals in Table
5.10.
Kriged images of estimated random log io k and log io 0 fields are depicted by threedimensional fence diagrams in Figures 5.29 and 5.30, respectively. These images were
generated with power variograms having exponents equal to 1. The two pneumatic
parameters exhibit comparable patterns of spatial variability, in that regions of high
and low permeability correspond quite closely. Posterior sample variograms of the
high-resolution estimates in Figures 5.29 and 5.30 are presented in Figure 5.31. Both
increase more or less linearly with separation distance. The coefficients a of log io k
and log io ç variograms are equal to 0.1 and 0.026, respectively.
191
1.00
0.90
0.80
0.70
0.60
0.50
0.40
0.30
0.20
0.10
-0.00
-0.10
-0.20
-0.30
-0.40
-0.50
-0.60
-0.70
-0.80
-0.90
-1.00
10
20
30
40
50
60
Eigenvectors
FIGURE 5.27. Two-dimensional representation of eigenvectors of covariance matrix
of estimation errors at the pilot points.
192
1.0
0.0
Parameters
FIGURE 5.28. Maximum absolute eigenvector components associated with parameter
estimates.
193
30
-10
-10
-11
-12
-13
-14
-15
-16
-17
-18
-19
-20
FIGURE 5.29. Three-dimensional representation of kriged log io k estimated by simul-
taneous inversion of PP4 data with 32 pilot points.
194
-
10
1.5
-2
-2.5
-3
-3.5
-4
-4.5
-5
FIGURE 5.30. Three-dimensional representation of kriged log io estimated by simultaneous inversion of PP4 data with 32 pilot points.
195
2.5
2.0
- - -0-
logio k [m 2 ]
- _ -G
logio 0 [m /m 3 ]
0...
3
/
a
X)
f3"
1.5
0.0
0
5
10
Distance [m]
15
20
FIGURE 5.31. Sample variograms of kriged log io k and log lo 0 estimates obtained by
simultaneous inversion of PP4 data with 32 pilot points.
196
Figures 5.32, 5.33 and 5.34 show what happens when PP5, PP6 and PP7 are sim-
ulated using the above log io k and log io 0 estimates. There are substantial discrepancies between computed and measured pressures, especially in observation intervals
close to injection intervals. However, the calculated curves do not differ substantially
from the measured records for intervals close to the Y2 borehole. This is so because
during test PP4, air was injected into borehole Y2 so that pressure responses close
this borehole were distinct, allowing the local medium non-uniformity to be estimated
with higher accuracy compared to the rest of computational domain (Table 5.10).
To improve the matches between simulated and observed pressures, I analyzed test
PP4 with the aid of 64 pilot points. The additional points were placed between and
along test intervals. Figure 5.35 shows the resultant simultaneous matches between
computed and recorded pressures. The sum of squared pressure residuals, 1), is now
only 17 kPa 2 . Compared to the case of 32 points (Figure 5.21), the new matches are
much better in most intervals, especially in X2-3, Y3-1, Y3-3, Z2-1, Z3-1, V1, V3-1,
V3-3, W2A-1, W2A-2, W2A-3 and W3-2. Some of the latter intervals are within
the previously defined highly permeable zone. I conclude that the numerical inverse
model is capable of representing in a satisfactory manner the highly non-uniform
fracture network at ALRS. Adding pilot points improves the matches, but does not
change the overall spatial distribution of log io k and log io 0, which remain similar to
those obtained using 32 pilot points (Figures 5.29 and 5.30).
197
X1
X2-3
10°
1 0'
10°
10 0
10"
10
to °
10'
0
10°
V3-2
10°
10
•;
10
1 o'
100
10'
10"
10°
10 0
10'
10°
W2A-2
10"
10'
10
102
10 -a
104
10°
10'
10 -°
W3-2
10°
W3-1
10'
10 0
.f-44
100
10
10
16'
10
1 0'
10 0
PP5 applying parameter fields estimated
by simultaneous inversion of PP4 data with 32 pilot points.
FIGURE 5.32. Forward simulation of test
198
X2-1
X2-2
X2-3
10"
10
10'
10
10
10 • .3
Y3-2
10'
10
10
10'
10'
100
10
10'
30
20
10 -0
10
1 00
10"
10 -1
10 -0
10
10'
10 -1
10'
10'
10. 0
1 0 -3
10'
10'
10 -0
10"
5.33. Forward simulation of test PP6 applying parameter fields estimated
by simultaneous inversion of PP4 data with 32 pilot points.
FIGURE
199
X1
1 0'
o • °-
10°
10010'
10"
10°
Y2 - 1
10"
10
10 -0
10'
10'
10 -21
o°
Y2-3
10°-
10
10'
10"
10"
10'
10 -0
10 4
10 -0
16'
•
W2A-1
10°
16-0
10-'10°
16"
4°
1
1 0°
W2A-2
10°
10'
10"
ro'
10°
10 - '
5.34. Forward simulation of test PP7 applying parameter fields estimated
by simultaneous inversion of PP4 data with 32 pilot points.
FIGURE
200
X2-1
X1
X2-2
X2-3
1 00
10°
10'
10 -0
10 -0
110
100
10°
1 0'
10 -0
1 0'
0.4
0.2
1 0 -2
o'
n
10'
v1
FIGURE 5.35. Matches between observed (small dots) and simulated (thick curves)
responses obtained by simultaneous inversion of PP4 data with 64 pilot points.
201
Cross-hole tests PP4, PP5 and PP6
The stochastic inverse model was used to interpret simultaneously pressure records
in all test intervals from three tests labeled PP4, PP5 and PP6, during which air
was injected at different rates into isolated intervals in boreholes Y2, X2 and Z3,
respectively. I employed 72 pilot points located along some of the intervals as well as
between them and the injection intervals. The location of the pilot points is depicted
in Figure 5.36. There was a total of 146 unknown parameters including log lo k and
log io 4) at 72 pilot points and two variogram exponents.
Resulting estimates of log io k and log io 0 and associated separate 95% confidence
limits are listed in Table 5.11. Pilot points located along the boreholes are numbered
from top to bottom. In most of the longer observation intervals, there are more than
one pilot points. Points marked by letters A, B, C, D and E are located as follows: A
— between Y2-2 and X2-2; B — between Y2-2 and Z3-2; C — next to X2-2; D — next to
Z2-2; and E between X1 and W3-2. Most log io k estimates appear to be reasonable
and are associated with relatively small errors; this is especially true for estimates at
pilot points along the boreholes. Only eight estimates reach the specified upper (-10)
and lower (-20) limits, but their confidence limits are narrow. Compared to log io k,
the log io ck estimates have much wider confidence intervals and a larger number of
them (12) have reached the specified upper (-5) and lower (-0.3) limits.
Figures 5.37, 5.38 and 5.39 show simultaneous matches between computed and
recorded pressures during tests PP4, PP5 and PP6, respectively. Of the 96 sets of
observed records in the three figures (32 corresponding to each of the three tests),
only 76 include match points. This not withstanding, only a small number of the 96
matches are poor (due to barometric pressure effects and instrumentation problems),
the rest being of intermediate, good to excellent quality. The sum of squared pressure
residuals, II), for all three tests is only 133.2 kPa 2 . To this PP4 contributes 45.4, PP5
82.7, and PP6 5.1 kPa 2 . We recall that when we matched individual responses under
202
0
-10
N
-20
3
-30
-40
FIGURE 5.36. Three-dimensional representation of 72 pilot points.
203
the assumption that the medium is uniform (Figures 5.9, 5.10, 5.11), the sum of all
squared pressure differences at the same match points was equal to 173.9 kPa 2 of
which 104.4 was contributed by PP4, 68.8 by PP5 and 0.7 by PP6.
Figures 5.40 and 5.41 show three-dimensional perspectives of estimated log io k
and log lo 0 fields, respectively. These estimates are similar to those obtained earlier
through inversion of PP4 test data alone (Figures 5.29 and 5.30).
204
TABLE 5.11. Parameters identified at the pilot points by numerical inversion of data
from cross-hole tests PP4, PP5 and PP6 using 72 pilot points.
Interval
X1
X2-0
X2-1
X2-2
X2-3
X2-9
X3-0
X3-1
Y1
Y2-0
Y2-1
Y2-2
Y2-3
Y2-4
Y3-0
Y3-1
Y3-2
Y3-3
Y3-4
Z1
Z2-0
Z2-1
Z2-2
Z2-3
Z2-4
Z3-0
Z3-1
Z3-2
Z3-3
Z3-4
V1-0
V1-1
V2
V3-1
V3-2
V3-3
WI
W2-1
W2-2
W2-3
W2-4
W3-0
W3-1
W3-2
W3-3
AO
Al
A2
A3
A4
AS
A6
A7
A8
BO
Bl
B2
B3
B4
B5
B6
B7
B8
Cl
C2
C3
C4
D1
D2
D3
D4
El
log, k [m a ]
-12.99 + 0.91
-13.67 + 0.56
-18.08 + 0.29
-13.19 + 0.10
-16.92 + 0.25
-14.24 + 0.36
-12.50 + 0.28
-15.16 + 0.78
-15.42 + 0.37
-16.88 + 0.43
-16.82 + 0.36
-12.70 + 0.10
-14.94 + 0.14
-19.95 + 0.38
-15.01 + 0.48
-14.13 + 0.98
-14.87 + 0.95
-20.00 ± 0.63
-18.11 + 0.87
-15.02 + 0.35
-17.54 + 0.72
-14.01 + 0.62
-14.07 ± 0.66
-14.57 + 0.67
-13.95 + 0.56
-12.63 + 0.63
-14.87 1 0.38
-13.51 + 0.17
-15.79 + 0.53
-16.68 ± 0.46
-20.00 + 0.36
-10.00 + 0.62
-14.33 + 0.80
-10.00 + 0.32
-14.55 + 0.60
-20.00 ± 0.75
-14.34 ± 0.64
-20.00 + 0.96
-13.52 + 0.33
-12.26 + 0.39
-10.00 + 0.49
-19.79 + 0.02
-15.93 + 1.18
-16.01 + 2.95
-20.00 + 1.35
-17.28 + 0.28
-13.42 + 0.39
-15.26 + 0.55
-13.14 + 0.59
-19.02 + 0.63
-14.92 + 0.34
-13.14 + 0.69
-12.11 + 0.68
-14.47 + 0.82
-15.62 + 0.57
-14.24 + 0.55
-19.51 + 1.19
-15.60 + 1.03
-13.92 + 1.15
-15.95 + 0.63
-14.73 + 1.28
-13.29 + 1.79
-18.29 + 1.19
-12.87 + 1.11
-14.81 + 1.06
-14.23 + 1.18
-18.09 + 1.39
-14.60 + 0.98
-17.05 + 1.84
-13.20 + 1.43
-15.73 + 1.79
-11.24 1 0.98
4, [rn 3 /m 3 ]
-2.33 + 1.67
-0.30 + 1.32
-5.00 + 1.50
-3.95 ± 0.41
-5.00 ± 2.11
-5.00 ± 0.87
-2.54 ± 1.78
-2.87 1 2.01
-0.30 ± 0.87
-5.00 + 0.93
-1.11 + 1.27
-5.00 + 0.40
-0.66 ± 1.07
-4.58 1 1.22
-3.04 1 4.40
-2.61 + 3.03
-2.81 + 1.48
-3.04 ± 2.05
-2.67 ± 1.46
-2.61 + 1.27
-2.73 + 2.14
-2.46 + 3.70
-2.47 ± 2.32
-2.50 + 1.31
-1.83 + 1.23
-0.30 + 1.72
-1.36 + 1.13
-5.00 ± 0.61
-4.32 ± 1.85
-1.54 ± 1.50
-3.97 + 2.21
-0.30 ± 2.12
-1.70 + 2.27
-0.39 ± 0.80
-1.67 ± 0.86
-4.90 ± 1.80
-2.67 + 3.18
-4.01 + 2.24
-2.43 ± 1.73
-2.46 ± 1.21
-4.98 ± 0.93
-3.75 + 1.24
-3.36 + 4.95
-3.57 + 4.49
-4.11 + 3.43
-2.52 + 1.59
-1.61 + 1.07
-1.67 ± 2.42
-1.36 + 2.30
-2.90 + 2.33
-0.44 + 2.51
-1.64 ± 2.96
-0.30 1 2.62
-1.96 + 2.97
-2.83 + 1.24
-1.81 ±3.17
-4.34 + 2.00
-1.11 + 2.79
-3.01 + 2.91
-3.52 + 2.76
-1.99 + 2.68
-0.89 1 3.31
-2.74 ± 3.20
-2.63 + 2.02
-2.90 ± 2.42
-3.53 + 2.99
-2.77 + 3.11
-3.33 + 3.80
-5.00 + 3.21
-2.73 + 2.84
-5.00 + 2.91
-3.19 ± 2.73
log"
205
X2 - 1
X1
X2-3
10 .0
1 0'
10'
10.0
1 0°
10
1 0°
10 .0
5.37. Matches between observed (small dots) and simulated (thick curves)
responses for test PP4 obtained by simultaneous inversion of PP4, PP5 and PP6 data.
FIGURE
206
X1
X2-2
0.4
0.3
0.2
0.1
100
10'
50
10'
1 0'
10 0
10'
1 0°
1 0'
lo '
10 -2
10'
10'
FIGURE 5.38. Matches between observed (small dots) and simulated (thick curves)
responses for test PP5 obtained by simultaneous inversion of PP4, PP5 and PP6 data.
207
X2 - 3
X1
X3
10'
• .y. 1
10'
10°
0.2
16r°
10
16'
16°
16'
16 -2
16'
16°
Z1
0.1'5 0.1
0.05 -
Z2-4
0.25
0.2
10°
0.15
0.1
10'
5
10'
10'
10
10
V3-1
0.3
0.2
02
8:4
0.2
0.1
0.1
,...01°*
1 0 ' 16 4
W2A-1
'
1 0°
0.15
0.1
0.05
1
0.76
0.5
W2A-4
10'
0.25
10 -2
10'
10°
10'
5.39. Matches between observed (small dots) and simulated (thick curves)
responses for test PP6 obtained by simultaneous inversion of PP4, PP5 and PP6 data.
FIGURE
208
30
-10
-10
-11
-12
-13
-14
-15
-16
-17
-18
-19
-20
Three-dimensional representation of kriged log io k estimated by simultaneous inversion of PP4, PP5 and PP6 data.
FIGURE 5.40.
209
-10
FIGURE 5.41. Three-dimensional representation of
multaneous inversion of PP4, PP5 and PP6 data.
kriged log lo estimated by si-
210
5.3 Discussion
5.3.1 Comparison between numerical and analytical analyses
Illman and Neuman [2000a] developed analytical type-curves for the interpretation of
cross-hole pneumatic tests. Their results are comparable to those of my inverse model.
Nevertheless, there are major differences between the two approaches. Whereas analytical type-curves represent a linearized solution, the numerical model solves directly
the nonlinear governing airflow equation. Whereas analytical models assume that air
compressibility is constant, the numerical model accounts for its dependence on absolute air pressure. Whereas the particular type-curves used by the above authors
treat the injection interval as a point, my numerical model represents its real shape
and volume. Whereas the type-curves account indirectly for storage of air in pressure
monitoring intervals, my numerical model does so directly and in all open intervals
of each borehole at the site (not only those in which pressure is analyzed). Whereas
the type curves assume that pressure equalizes instantaneously along each monitoring
interval, my numerical model allows for rapid airflow and pressure equalization within
each open borehole interval by modeling it more realistically as a high-permeability
and high-porosity medium. My numerical model additionally accounts for the variation in borehole storage due to variation in air compressibility. My numerical model
considers the effect of an atmospheric boundary at the ground surface, though at the
expense of introducing artificial no-flow boundaries at the sides and the bottom of
the modeled rock volume. The presence of open boreholes in the numerical model
enhances pneumatic connection between the tested rock and the atmosphere. The
most important advantage of the numerical model is that it allows resolving heterogeneities on a small scale, and to consider all test data simultaneously, thus yielding
a high-resolution stochastic (tomographic) image of these heterogeneities.
211
5.3.2 Comparison between present stochastic inverse model and previous
applications of pilot-point method
The pilot-point methodology and its application were first described in the dissertation of Ghislain de Marsily [1978]. Later, the method was modified and used for the
inverse analysis of several synthetic and real data sets [Certes and de Marsily, 1991;
LaVenue and Pickens, 1992; RamaRao et a/., 1995; LaVenue et al., 1995]. There are,
however, some important differences between my approach and those used previously.
In the past, the pilot-point method was applied only to two-dimensional problems. I
work in three dimensions. The method has not been used previously to identify the
spatial distribution of storativities (or, equivalently, porosities). Kriging was commonly performed by utilizing not only pilot points but also actual measurements
of model parameters. I include only pilot points in my inverse algorithm. Kriging
was always performed using given variogram models with predetermined parameters
based on actual data. I treat the variogram model parameters as unknowns to be
determined by the inverse model. Parameters at pilot points were always treated
as exact measurements. I view the pilot points as conditioning points of unknown
stochastic fields, and corresponding values as uncertain estimates. La Venue and Pickens [1992] used sequential optimization to define the location of pilot points within
the computational domain. However, their procedure does not allow identification of
estimation uncertainty at the pilot points. I predefine the location of the pilot points
and estimate the unknowns simultaneously. I also tried to optimize simultaneously
both location and estimates of the pilot points. However, the optimization procedure became computationally very demanding. I believe, however, that this may be
feasible and should be pursued in the future.
212
5.3.3 Comparison between different models
In the previous section, I described various stochastic inverse models utilizing different
sets of pilot points, and analyzing different test data. All produced comparable
results in terms of matches and spatial distributions of parameter estimates. This
demonstrates a certain robustness of my hydrogeologic conceptualization and analysis.
Alternative stochastic inverse models can be compared using the four model identification criteria introduced by Carrera and Neuman [1986a] and discussed in Chapter
3. The values of the four criteria calculated for models of test PP4 using 32 and 64
points as well as the model of PP4, PP5 and PP6 using 72 points are listed in Table
5.12. The table also presents the optimum values of the sum of squared residuals €1.
and natural log determinant of the covariance matrix of estimation errors E act . All
four criteria identify the inverse model interpreting simultaneously tests PP4, PP5
and PP6 as the best by a wide margin. This reflects the fact that using a relatively
small number of pilot points (unknowns), the model provides a satisfactory representation of observed pressures during all three tests. The other 2 models interpret
only PP4 data. Compared to the model of PP4 with 32 points, the PP4 model with
64 points produces a much better representation of observed pressures, but is less
parsimonious. These two models produce close criteria estimates. The first three
criteria—AIC, BIC and Om—rank the 64 pilot-point model higher than the 32 pilotpoint model, and only the criterion dm suggests the opposite. Carrera and Neuman
[1986c] and Samper and Neuman [1989c] suggested that the dm criterion is the most
suitable for model discrimination because it is the most sensitive to model parsimony.
5.3.4 Scale effects
In Tables 5.13 and 5.14, I list statistics of air permeability and air-filled porosity
estimated by different methods. For log io k, single-hole data represent the combined
set of 1-m and 3-m support data as discussed in Chapter 2. The separate analysis of
213
TABLE 5.12. Comparison of different numerical inverse models using model identification criteria (numbers in brackets show model ranking).
Models
Criteria
PP4
PP4
PP4, PP5, PP6
Number of pilot points
32
64
72
Number of unknowns
128
64
144
Number of match points
252
252
489
131.2
cl)
17.03
133.2
-9
-348
-158
In lEaal
AIC
-1131 (3)
-1396 (2)
-2879 (1)
BIC
-905 (3)
-944 (2)
-2275 (1)
-1040 (3)
-1214 (2)
-2642 (1)
Sb m
dm
-1013 (2)
-2382 (1)
-832 (3)
individual pressure records is based on data from cross-hole tests PP4, PP5, PP6, PP7
and PP8 (Tables 5.2 - 5.6); the statistics are derived from 136 and 128 estimates of
log lo k and log io 0, respectively (Tables 5.7 and 5.8). The stochastic numerical inverse
model was used to analyze all pressure records from PP4, PP5 and PP6 separately
from each test and simultaneously from all three tests; the listed log lo k and log lo çb
statistics are calculated from high-resolution stochastic estimates over a portion of
the computational region presented in the three-dimensional fence diagrams above
(for example Figures 5.29 and 5.30). This three-dimensional region (45 x 33 x 33
(49,005) m 3 ) encompasses test boreholes and predominant portion of the flow during
cross-hole tests. It is assumed that this portion of tested rock was characterized by
means of the stochastic inverse analysis of the test data. The same region was used
also for the kriging of the ALRS data in Chapter 2 (Figures 2.18 - 2.26, pages 58 -
66). The region includes 53,176 nodes over a regular structured grid with resolution
1 x 1 x 1 (1) m 3 ; the number of nodes define the sample size.
Geostatistical analysis of single-hole data and stochastic inverse analyses of tests
PP4 and PP5, as well as of tests PP4, PP5 and PP6 simultaneously, produce comparable values of the mean and variance of log ic) k and log io 0. The inversion of test
PP6 only could not define that well overall medium heterogeneity since not all test
intervals produced reliable records. Compared to the rest of stochastic models, corre-
214
TABLE 5.13. Summary statistics for log io k [ml identified using different data and
methods.
Source
Sample size
Mean
Variance
CV
227
—15.22
Single-hole tests (uniform medium)
0.87
—0.061
Cross-hole tests
Uniform medium
-
PP4,5,6,7,8
Non-uniform medium
PP4 (32 pilot points)
PP4 (64 pilot points)
-
-
-
-
-
PP5
PP6
PP4,5,6
TABLE 5.14. Summary statistics for
and methods.
Sample size
Source
Cross-hole tests
136
—13.69
0.48
—0.051
53176
53176
53176
53176
53176
—15.15
—15.07
—15.26
—14.02
—15.69
3.17
1.95
1.64
0.64
2.48
—0.117
—0.093
—0.084
—0.057
—0.100
log io 0 [m 3 /m 3 J identified using different data
Mean
Variance
CV
128
—1.63
0.24
—0.301
53176
53176
53176
53176
53176
—3.00
—2.47
—3.07
—1.82
—2.98
1.45
1.24
2.12
0.66
1.00
—0.402
—0.450
—0.475
—0.469
—0.337
Uniform medium
-
PP4,5,6,7,8
Non-uniform medium
PP4 (32 pilot points)
PP4 (64 pilot points)
-
-
-
-
-
PP5
PP6
PP4,5,6
sponding mean log io k and log io 0 are higher since not only does injection take place
in the highly permeable zone along
Z borehole series, but also pressure responses
are recorded predominantly in observation intervals located in this highly permeable
zone.
Compared to all other estimates, log io k and log io 0 obtained from numerical inverse analyses of individual pressure records, while treating the rock as being uniform,
have a much higher mean value and much lower variance. Compared to the stochas-
tic inverse model of tests PP4, PP5 and PP6 simultaneously, the uniform geometric
mean of k is higher by exactly two orders of magnitude, and the log io k variance is
1.9 times lower; the uniform geometric mean of 0 is higher by more than one order
of magnitude (a factor of
22), and the log io 0 variance is 4.2 times lower. Compared
215
to the single-hole data, the uniform mean and variance of log io k are 34 times higher
and 1.8 times lower, respectively. This scale effect is evident also in the histograms
presented in Figures 5.42 and 5.43. Permeability scale effects were reported by others [Neuman, 1990; Clauser, 1992; Rovey and Cherkauer, 1995; Sanchez-Vila et al.,
1996; Meier et al., 1998; Samper and Garcia, 1998; Schulze-Makuch and Cherkauer,
1998; Martinez-Landa et al., 2000]. The permeability scale effect has been questioned
by Zlotnik [1994] and Butler and Healey [1998] who suggested that it is due to an
inadequate interpretation of small-scale field experiments such as slug and single-hole
tests; they emphasized on the impact of skin effect on the small-scale estimates. However, in my analysis, the scale effect in the medium properties can be defined from
cross-hole data alone, so these problems are not applicable. In addition, the consistency between single-hole estimates and stochastic inverse results suggest the former
were adequately identified. The type-curve analysis of Illman et al. [1998] and Illman
and Neuman [2000b] also indicates that the skin effect is negligible for the single-hole
tests conducted at ALRS. Rovey [1998] and Meier et al. [1998] demonstrated through
two-dimensional synthetic experiments that indeed scale effect in permeability may
result from medium heterogeneity. To date, the analysis of single-hole tests has not
provided air-filled porosity data for comparison with the high-resolution stochastic estimates. As described in Chapter 4, some porosity values were estimated on the basis
of single-hole tests by means of numerical inversion, and they range from 9.6 x 10 -2
to 3.6 x 10 -3 . Therefore, it is not possible to confirm the scale effect in air-filled
porosities with single-hole test estimates.
Treating the rock as being uniform, when it is non-uniform, yields relatively high
estimates of mean and low estimates of variance for both air permeability and airfilled porosity. Scale of inverse analyses is defined by the resolution of the medium
non-uniformity over the computational domain. When the medium is viewed as being
uniform, there is no resolution in medium heterogeneity and the scale of analysis is defined by the size of the computational domain. Clearly, this scale is much larger than
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218
the support scale of singe-hole tests. As a result, there are substantial differences between the permeability and air-filled porosity statistics obtained from uniform analysis
of single-hole and cross-hole test data. On the other hand, my stochastic numerical
model resolves the non-uniformity of the medium at a scale (resolution) defined by
the grid employed to compute kriging estimates. In our case, kriging is performed
at the nodes of a computational grid. In the central portion of the computational
domain, where most airflow takes place, the predominant distance between nodes is
1 m, which corresponds to the nominal support scale of single-hole tests. The grid
along the boreholes has separation distances between nodes less than 1 m, but it
accounts for a relatively small portion of the computational domain. Therefore, there
is consistency between the scale of inverse analysis and the scale of single-hole data,
and there is no scale effect in estimated medium properties. This is consistent with a
earlier conclusion by Neuman [1990, 1994] that permeability (as well as dispersivity)
scale effects diminish as medium non-uniformity is explicitly resolved by parameter
estimation.
5.3.5 Comparison between spatial patterns of permeability obtained by
various methods
The overall pattern of spatial variability of log io k obtained from my stochastic inverse model (Figure 5.29, page 193 and Figure 5.40, page 208) is quite similar to that
obtained from the kriging of single-hole data (Figure 2.18, page 58). There is good
correlation between high-permeability and low-permeability zones. Though the range
of values in Figure 2.18 is narrower than that in Figures 5.29 and 5.40, their mean
values are very close (Table 5.13). The higher variances of high-resolution inverse
estimates, compared to the single-hole data, can be due to several reasons. First, the
domain size of high-resolution images is much larger than the rock region tested by
the single-hole data. Since a power model with an exponent equal to 1 is applied,
219
the variance increases with domain size. Second, there are estimation errors associ-
ated with the values at the pilot points. Therefore, the variance should represent not
only spatial variability of the true unknown parameter field, but also the estimation
errors. Third, the variance is influenced by the parameterization of the spatial distribution of medium properties. In the case of test PP4, the variance decreases as
the number of pilot points increases (Table 5.13; see also Figure 5.45 discussed in the
next section). The larger number of pilot points allows smoother representation of
medium non-uniformity. However, the increase in the number of pilot points enlarges
the uncertainty of pilot-point estimates.
As already discussed in Chapter 2, the kriging estimates of fracture density are
very different than those of air permeability obtained from single-hole data. The
same is true for the high-resolution stochastic images of air permeability obtained
from cross-hole data.
In Figure 5.44, I compare air permeability profiles along four boreholes, obtained
from single-hole tests and by the inverse model of cross-hole tests PP4, PP5 and PP6
simultaneously. Both sets of data represent values on a nominal scale of 1 x 1 x 1 m 3 .
In borehole Y2, the two sets of log io k values correlate quite closely. This is so despite
there being only 5 pilot points along the borehole. The same is true for borehole V2.
In X2 and Y3, the inverse model captures only the general trend of log io k variations
as defined by the single-hole data; both methods define a relatively high permeability
zone close to the top and a relatively low permeability zone at the bottom.
The numerical inverse model yielded power variogram exponents equal to 1, their
specified upper limit. Geostatistical analysis of single-hole air permeability data in
Chapter 2 yielded a lower exponent equal to 0.45. In Figure 5.45, I compare the
sample variograms of 1-m and 3-m single-hole data against the sample variogram of
kriging estimates from stochastic inverse models. The differences in the variograms are
due to higher variability of inverse estimates. In the case of test PP4, the variogram
estimates decrease as the number of pilot points increases.
220
...
X2
OS
•n
Y2
... ..
. .., 1.1.
I
.. -,
V.
0.4
00 ... ...
0
•
- - - - SH data
-5
WE MN MN
Inverse estimate
I=
..
. NM
---
-1
.1•1
4 .
N
1
-
1
.
E
.4
•
N
,- - - - SH data
n
Inverse estimate
-15
.41.,
•
444,
-15
o
40
•
-
il
10, ..
*I s.
-20
i
-20
-17
-18
-16
-15
log io k [m 2 ]
-14
-1 3
-
-18
-17
-16
-15
-14
-13
log lo k [m 2 ]
-
....
Y3
...
-10
n - n n SH data
Inverse estimate
•
..' ...
-
40
••
40
V2
-
-5
-
)
10
t
-10
-
-
E
N-15
N
-
-20
-
-20
-
1
/ „„.
-
-25
/
-
- n ** .44 SH data
40.
,... ... ** "
Inverse estimate
.
1
1
* 1
-17
1
1
-16
1
1
1
1
-15
1
1
-14
1
-13
-12
log 10 k [m 2 ]
FIGURE 5.44. Air permeability along boreholes estimated from single-hole tests and
by simultaneous inversion of PP4, PP5 and PP6 data.
221
3.0
-
-
2.5
1-m SH data
3-m SH data
PP4 32 points
PP4 64 points
PP4, PP5, PP6
--
-
2.0
-
1.5
-
E
cr)
o
as
>
1.0
0.5
-
-
0 00
o
log io k [m 2
]
,
.
FIGURE 5.45. Sample variograms of log lo k estimated from single-hole data and
through stochastic inversion of cross-hole data.
222
In my study, high-resolution estimates of log io k have been obtained independently
by means of two very different test and interpretation methods, using two independent
sets of data with different support scales. Single-hole data yield local estimates of
permeability in the close vicinity of the test intervals. Cross-hole test data represent
three-dimensional flow spanning the rock on a relatively larger scale. Although very
different, the methods of analysis are similar because the medium properties are
viewed as spatially correlated random fields and their heterogeneity is resolved with
equivalent resolution.
5.3.6 Comparison between permeability and porosity estimates
I already mentioned that high-resolution images of log io k and log io 0 obtained by the
numerical inverse model display similar three-dimensional patterns. This is seen on
the three-dimensional fence diagrams in Figures 5.29 - 5.30 and 5.40 - 5.41, where
regions of high and low permeability correspond quite closely to similar regions of
porosity. The inverse estimates in the first two figures are plotted against each other
(small dots) in Figure 5.46. Estimates of log io k and log io 0 at the pilot points are
indicated by open circles. The plot suggests a correlation between log io k and log io 0,
with a slope of approximately 1:2. Using very different methods of analysis, linear
relationships between log io k and log io 0 with similar slopes have been estimated
by laboratory analyses of sedimentary rock samples [cf. Nelson, 1994; Worthington,
1997]. More importantly, Guimera and Carrera [2000] reevaluated published data of
a series of field tracer tests conducted in fractured rocks; their analysis suggested also
a linear relationship between log io k and log io 0 with slope between 1:2 and 1:3.
5.3.7 Borehole effects
I already noted that open borehole intervals impact the interpretation of pneumatic
cross-hole tests due to preferential airflow through, and enhanced storage within,
223
-18
-16
-14
log io k [m 2
-12
-10
]
FIGURE 5.46. Air permeability versus air-filled porosity estimated by stochastic
inversion of PP4 data at pilot points (open big circles) and within the computational
domain.
224
these intervals. To examine this issue in more detail, I simulated test PP4 without
boreholes using log io k and log io ck estimates from the inversion of PP4 with 64 pilot
points. Calculated pressures with and without boreholes are compared in log-log and
semi-log plots in Figures 5.47 and 5.48. The first figure emphasizes differences at
early time, and the second at late time. It is clear that the presence of open borehole
intervals has a considerable effect on pressure propagation through the site, and on
pressure responses within intervals. These responses can be different in shape and
either higher or lower than those calculated without open borehole intervals.
The distribution of pressures across two-dimensional vertical sections through the
numerical model at the end of test PP4 are shown in Figure 5.49 for the case where
open borehole intervals are not accounted for, and in Figure 5.50 for the case where
they are. The open borehole intervals, the packers, and their effects on pressure
distribution are clearly evident in Figure 5.50. The figure also demonstrates that
boreholes X2, X3 and Z3 are venting the system. Both figures show how the atmospheric boundary causes the pressure distributions to exhibit vertical asymmetry.
The effects of lateral and bottom no-flow boundaries on pressure are seen to be slight.
It is clear that borehole effects must be taken into consideration in the interpretation
of pneumatic tests.
225
10'
10
1 01
10°
, X3
10°
10'
1OE°
10
0
10 4
10°
10'
10
100
1
1 0°
5.47. Simulated responses with (solid curve) and without (dashed curve)
borehole effects for cross-hole test PP4 iii log-log plots.
FIGURE
226
X2 - 1
10
X2-2
X3
10
8
8
6
6
120
4
4
110
2
2
10 .2
10
90
10 2
Y3-3
10"
10°
12
10
8
6
4
2
10
10°
Z2 - 3
0.3
0.2
0.1
0.2
0.1
0.2
and without (dashed curve)
borehole effects for cross-hole test PP4 in semi-log plots.
FIGURE 5.48. Simulated responses with (solid curve)
227
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229
Chapter 6
CONCLUSIONS
The following major conclusions arise from my study:
1. It is possible to interpret single- and cross-hole pneumatic tests in unsaturated
fractured tuffs at the ALRS by means of a three-dimensional numerical inverse
model, which treats the rock as a locally isotropic, uniform or non-uniform
continuum and air as a single mobile fluid phase. The rock continuum represents
primarily interconnected air-filled fractures.
2. The inverse model is capable of accounting for storage and conductive effects of
all existing open borehole intervals at the site, both monitored and not monitored. During cross-hole tests, these effects influence airflow significantly, and
neglecting them produces a systematic bias in the estimation of pneumatic parameters. Single-hole tests are affected primarily by conductance and storage
in the injection interval.
3. In the case of single-hole tests, transient records of multiple injection steps and
recovery are amenable to simultaneous analysis by means of my inverse model.
Storage of injection interval depends on air compressibility, and decreases with
the increase of absolute air pressure. Borehole storage dominates pressure tran-
sients during the first step of each test and makes it difficult to obtain reliable
estimates of air-filled porosity and borehole storage coefficient from such data.
Pressure transients during subsequent steps of the test are affected by borehole
storage to a lesser extent than are those during the first step. More importantly,
there is virtually no storage impact on recovery data. A consideration of variable air compressibility and a joint analysis of pressure data from all stages of
230
a single-hole pneumatic test are therefore essential for the reliable estimation
of air-filled porosity and borehole storage coefficient. The storage effect ob-
served during single-hole tests seems related not only to the injection interval,
but also to openings in the surrounding rock. Air permeabilities obtained by
our inverse method are comparable to those obtained by means of steady-state
formulae [Guzman et al., 1996] and transient type-curve analyses [Illman et al.,
1998]. Due to linearization of airflow equations, analytical type-curves [Illman
and Neuman, 2000b] did not match observed pressure records and did not allow
reliable identification of air-filled porosity and borehole storage effects.
4. Currently, the numerical inverse analysis has been performed for only four of the
single-hole tests. The analysis of all the single-hole tests would provide air-filled
porosity data for comparison with stochastic inverse results. The interpretation
of single-hole data could be facilitated through numerical derivation of appro-
priate type-curves, i.e. "numerical type-curves".
5. Qualitative analysis of data from cross-hole tests PP4, PP5, PP6, PP7 and PP8
demonstrates that the medium is well pneumatically interconnected and acts as
a three-dimensional continuum. The medium contains highly permeable zones,
some of which extend to the ground surface.
6. Assuming uniform medium, my three-dimensional inverse model is capable of
analyzing independently individual pressure records from cross-hole tests PP4,
PP5, PP6, PP7 and PP8. For test PP4, which is the only test analyzed to date
by analytical type-curves, the latter and my numerical model produce similar
results, though there is a systematic difference in permeability estimates.
7. Inman and Neuman [2000b, a] developed analytical type-curves for the inter-
pretation of single- and cross-hole pneumatic test data which allow simple, fast
and reliable assessment of pneumatic medium properties. Disadvantages of the
231
type-curve approach include reliance on linearized airflow equations; inability to
consider the dependence of air compressibility on absolute air pressure; ability
to account for borehole storage in only one monitoring interval at a time; inability to consider the effect of open boreholes on pressure distribution through
the system; inability to consider barometric pressure effects; treatment of the
rock as being uniform; and inability to analyze simultaneously pressure data
from multiple monitoring intervals when the rock is non-uniform.
8. In contrast, my numerical inverse model solves the airflow equations in their
original nonlinear form; considers the dependence of air compressibility on abso-
lute air pressure; accounts directly for the ability of all boreholes, and packed-off
borehole intervals, to store and conduct air through the system; can, in principle, account for atmospheric pressure fluctuations at the soil surface; provides
kriged estimates of spatial variations in air permeability and air-filled porosity
throughout the tested fractured rock volume; and can be applied simultaneously to pressure data from multiple borehole intervals as well as to multiple
cross-hole tests. It has the disadvantage of being complex, difficult and time
consuming.
9. Assuming non-uniform medium, my stochastic inverse model is able to produce three-dimensional high-resolution images of air permeability and air-filled
porosity through simultaneous analysis of data from multiple borehole intervals
and multiple cross-hole tests. This interpretation amounts to "pneumatic to-
mography" or "high-resolution stochastic imaging" of the fractured rock mass,
an idea proposed over a decade ago by Neuman [1987] in connection with hy-
draulic cross-hole tests in saturated fractured crystalline rocks at the Oracle
site near Tucson [Hsieh et al., 1985].
10. The stochastic numerical inverse model is capable of representing in a satis-
232
factory manner observed pressure records during cross-hole tests. Stochastic
estimates of air permeability and air-filled porosity obtained using data from
different tests and pilot-point arrangements are similar, which demonstrates the
robustness of our analysis and the proper conceptualization of hydrogeologic
conditions at the site.
11. There is a good correlation between the spatial patterns of air permeability and
air-filled porosity as obtained by the inverse procedure. All of these patterns
differ markedly from the spatial pattern of fracture densities. A correlation
analysis between log-transformed air permeability and log-transformed air-filled
porosity yields a ratio close 1:2.
12. Air permeability data from single-hole pneumatic tests, as well as data con-
cerning fracture density, overall water content, matrix porosity and matrix van
Genuchten's a, are amenable to continuum geostatistical analysis. This means
that each data set can be viewed as a sample from a random field, or stochastic continuum, as proposed for permeability over a decade ago by Neuman
[1987, 1988b] and affirmed more recently by Tsang et al. [1996].
13. The statistical and geostatistical analyses of single-hole air permeabilities at 1-
m and 3-m support scale can be combined, which improves the characterization
of medium non-uniformity.
14. Both geostatistical analysis of single-hole data and geostatistical inverse analysis
of cross-hole data, yield a similar geometric mean and a similar spatial pattern
of air permeability at the ALRS. This consistency demonstrates a certain ro-
bustness of my hydrogeologic conceptualization and stochastic inverse analyses,
and imbues me with some confidence that the obtained estimates constitute a
valid representation of fracture pneumatic properties at the ALRS. However,
the inverse estimates vary over a much broader range than do the single-hole
233
test results. This can explained as a result of (1) larger sample and domain size
of inverse estimates, (2) errors in the inverse estimates and (3) parameterization of the spatial distribution of medium properties. The spatial correlation
of air permeability is characterized by both methods with a power variogram,
which is representative of a random fractal field with multiple scales of spatial
correlation. To date, the analysis of single-hole tests has not provided enough
air-filled porosity data for comparison with stochastic inverse results.
15. Analysis of individual pressure cross-hole records independently, assuming uniform medium, produces higher geometric means and lower variances for pneumatic parameters compared to those estimated from single-hole data and by
stochastic analyses of cross-hole tests PP4, PP5 and PP6. This implies a pronounced scale effect for both air permeability and air-filled porosity.
16. Scale of inverse analyses is defined by the spatial resolution of medium nonuniformity over the computational domain. In the uniform case, there is no
spatial resolution in medium heterogeneity and the scale of cross-hole test analysis is much larger than the support scale of singe-hole test data. This produces a scale effect. The stochastic numerical model resolves non-uniformity
on a scale comparable to the support scale of single-hole test data. This eliminates any scale effect. These findings are consistent with earlier conclusion by
Neuman [1990, 1994] that permeability scale effects diminish as the medium
non-uniformity is explicitly resolved by parameter estimation. Permeability
scale effects were also reported by others [Clauser, 1992; Rovey and Cherkauer,
1995; Sanchez-Vila et al., 1996; Meier et al., 1998; Samper and Garcia, 1998;
Sch,ulze-Makuch and Cherkauer, 1998; Martinez-Landa et al., 2000].
17. There are differences between my pilot-point approach and those used previously. The pilot-point method was applied only to two-dimensional problems. I
234
work in three-dimensions. The method has not been used previously to identify
the spatial distribution of storativities (or, equivalently porosities). Kriging was
commonly performed by utilizing not only pilot points but also actual measurements of model parameters. I include only pilot points in my inverse algorithm.
Kriging was always performed using given variogram models with predetermined
parameters based on actual data. I treat the variogram model parameters as
unknowns to be determined by the inverse model. Parameters at pilot points
were always treated as exact measurements. I view the pilot points as conditioning points of unknown stochastic fields, and the estimates as uncertain
quantities. La Venue and Pickens [1992] used sequential optimization to define
the location of pilot points within the computational domain. However, their
procedure does not allow identification of estimation uncertainty at the pilot
points. I predefine the location of the pilot points and estimate the unknowns
simultaneously. I also tried to optimize simultaneously both location and estimates of the pilot points. However, the optimization became computationally
very demanding. I believe, however, that this may be feasible and should be
pursued in the future.
18. Future developments of the stochastic inverse model could include consideration
of barometric pressure variations and incorporation of prior information for air
permeability and air-filled porosity. The tomographic inverse analysis could be
extended to simultaneous interpretation of both single-hole and cross-hole tests
as wells as to incorporation of additional cross-hole tests. The inverse estimates
could be further tested by simulation of previous and future field experiments
performed at the ALRS.
19. An important and challenging future extension of stochastic inverse methodology could be the analysis of multiphase flow of water and air. An infiltration
experiment is currently conducted at the ALRS, and the processing of collected
235
data by means of such inverse model could further improve the characterization
of medium properties.
236
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