structural geology, propagation mechanics and - Stanford School of ...

STRUCTURAL GEOLOGY, PROPAGATION MECHANICS AND
HYDRAULIC EFFECTS OF COMPACTION BANDS IN SANDSTONE
A DISSERTATION
SUBMITTED TO THE DEPARTMENT OF
GEOLOGICIAL AND ENVIRONMENTAL SCIENCES
AND THE COMMITTEE ON GRADUATE STUDIES
OF STANFORD UNIVERSITY
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS
FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
Kurt Richard Sternlof
March 2006

© Copyright by Kurt Richard Sternlof 2006
All Rights Reserved
ii

Abstract
Low-porosity, low-permeability compaction bands (CBs) form a pervasive planar
fabric throughout the upper 600 meters of the 1,400-meter-thick æolian Aztec sandstone
of southeastern Nevada, an exhumed analog for aquifers and reservoirs of similar
lithology. The existence of CBs, only recently recognized, raises issues of practical
significance. How do they impact local permeability? Can they influence fluid flow and
transport at production scales? When, how and why do they form? Can their presence,
geometry and effects be forecast in the subsurface from sparse data?
To address these questions, we applied a range of techniques, including: outcrop and
thin-section observations and measurements; structural and tectonic analysis; scanning
electron microscopy; detailed field mapping and aerial photograph interpretation;
mechanical analysis and modeling; computational permeability estimation and fluid-flow
simulation. Although derived entirely from the Aztec sandstone, our results offer a more
general phenomenological understanding of CBs and their effects.
Up to centimeters thick and hundreds of meters long, CBs comprise thin discs of
uniaxial porosity-loss compaction that form anastomosing arrays generally symmetric to
the maximum compression. Idealized as anticracks, CB propagation and pattern
development can be modeled using a linear-elastic boundary element method.
Mechanical interaction between CBs is inversely proportional to differential stress, such
that connectivity within a fabric, and its directional impact on fluid flow, can be
estimated from the stress state in which it formed, and vice versa. Computational
estimation corroborates measured permeability reductions of 10 -3 inside CBs. At outcrop
scales, this yields bulk permeability reductions up to 10 -2 and anisotropy up to 10 1 . Flow
simulations performed on a detailed, 40-acre CB map reveal significant practical impacts:
pumping and injection pressures increase three-fold; reservoir production efficiency
depends on proper alignment of the well array to the CB fabric, and contaminant
transport channels along the CBs regardless of the regional pressure gradient.
We interpret CBs in the Aztec as resulting from the tectonic compression of a high-
porosity, low-cohesion sandstone at modest confining pressures (

Acknowledgements
Dissertations can be ephemeral beasts—mythic, elusive and notoriously difficult to
capture on paper. Coaxing a good dissertation into publication has to be one of the more
arduous, solitary journeys a person can make while so entirely swaddled in support. This
volume represents the culmination of one such saga, and there are many people to thank
for seeing me through the often dense fog to that distant, shining shore of “doneness.”
First, I acknowledge the U.S. Department of Energy, Office of Basic Energy Sciences
for its generous funding of my research and tenure at Stanford. Additional support was
provided by the Stanford Rock Fracture Project. The Valley of Fire State Park in Nevada
is a beautiful and convenient place to work—thanks to all the helpful, friendly staff there.
My sincere thanks to the faculty who served on my committees and provided valuable
guidance: Mark Zoback, Steve Graham, Amos Nur and, particularly, Lou Durlofsky and
Atilla Aydin. Lou is great to work with—enthusiastic, incisive and human—and Atilla
imbues every effort with dedication and principle. Thanks also to the administrative staff
for their logistical support, especially Elaine Anderson—font of knowledge, candy and
conversation. My gratitude also goes to Manika Prasad and Bob Jones for their tutelage
in the rock physics lab and on the SEM, respectively.
This truly has been a multidisciplinary effort, and I gratefully acknowledge all my co-
authors and collaborators: John Rudnicki (Northwestern University), Lou Durlofsky,
Mohammad Karimi-Fard, Jeffrey Chapin, Youngseuk Keehm, Tapan Mukerji, Gaurav
Chopra and Ovunc Mutlu. Thanks also to David Lockner (U.S.G.S.), Bill Olsson and
David Holcomb (Sandia National Labs) and Bezalel Haimson (Univ. of Wisconsin) for
opening their minds and their laboratories to my experimental aspirations.
No group deserves more credit than my fellow students and researchers, past and
present. In particular, I acknowledge Eric Flodin, Phil Resor, Nick Davatzes, Jordan
Muller, Juan Mauricio Florez, Frantz Maerten, Peter Eichhubl, Xavier Du Bernard,
Fabrizio Agosta, Ian Mynatt, Ole Kaven, Ashley Griffith, Ghislain de Joussineau and
Ramil Ahmadov. Special thanks to Brita Graham and Tricia Fiore for being “awesome”
office mates, and to everyone for their humor, insights and precious computer skills.
My good friend from Yale daze, John Childs, deserves an enormous thank you for
donating his time and assistance in the field on three occasions over three years—all
v

delivered with fortitude, humor, surprisingly good penmanship, enlightened conversation
and a car. What more could one ask from a history major and wooden objects conservator
who could have spent the extra time on his sailboat headed to Catalina? Thanks John.
This brings me to David Pollard, my principal advisor, collaborator and patron. I
arrived at Stanford 15 years after turning down his initial offer of admission, a story Dave
tells with relish. No matter my reasons for that youthful decision, Dave had made an
impression on me. When I finally decided to hunker down on that Ph.D., I approached
only him. Incredibly, he remembered and welcomed me, offering full funding to plug into
a project in some valley full of “bands.” The details mattered less to me than the pure
educational opportunity of it, the exposure to a mind and philosophy at the cutting edge
of structural geology and scientific inquiry itself. I have not been disappointed, and along
the way managed to carve out my own niche in a fascinating field. Dave enabled this,
with measured patience, open-minded rigor and humanity. No matter how dark the day, I
never left his office without feeling better—more calm, clear and confident—than when I
had entered. Anyone who has experienced graduate school knows what a compliment
that is. And yes Dave, Stanford is an incredible place, golf carts notwithstanding.
Of course family plays a huge, albeit at times less visible, role, and my journey is no
exception. Thanks to my mother and father, Eleanor and Paul, for instilling me with
determination, and to my siblings, Karl, Mark and Erika, for helping with humor and
perspective maintenance. To Sarah, the best mother in law a guy could have, thanks for
all the baked goods, barbeques and babying. And to deep-thinker Rye the dog, who so
often kept my feet warm and pinned to the floor beneath my desk..........he’s a good boy.
Finally, I owe my greatest debt of gratitude and all my love to Beth, who encouraged
and nurtured me throughout the entire Ph.D. saga—the ups and downs from conception
to completion and everything in between. She even exhibited the unmitigated faith and
optimism to marry me smack dab in the middle of it all. Although I can never hope to
compensate her for the years of late nights, lost weekends, fits of distraction and bouts of
insomnia, I promise with all my heart to try—lavishing her with all the extra attention
until so recently squandered on my laptop. Beth is my inspiration, my strength, my love
and my best friend. I dedicate this dissertation to her.
vi
KRS, March 21 st , 2006

Table of Contents
Abstract..............................................................................................................................iv
Acknowledgements............................................................................................................v
Table of Contents.............................................................................................................vii
List of Tables.....................................................................................................................ix
List of Illustrations.............................................................................................................x
Introduction........................................................................................................................1
Chapter 1—Structural geology and tectonic interpretation of compaction bands
in the Aztec sandstone of southeastern Nevada............................................9
1. Abstract......................................................................................................................9
2. Introduction..............................................................................................................10
3. Setting and history of study area..............................................................................11
3.1. Deposition.......................................................................................................11
3.2. Diagenesis.......................................................................................................15
3.3. Deformation....................................................................................................16
4. Structural geology of compaction bands..................................................................19
4.1. Thin-section to outcrop scale..........................................................................19
4.2. Anticrack model..............................................................................................22
4.3. Outcrop to regional scale................................................................................22
5. Compaction band orientations.................................................................................24
6. Tectonic interpretation.............................................................................................28
6.1. Temporal, spatial and material constraints.....................................................28
6.2. Paleostress analysis.........................................................................................31
6.3. Geomechanical implications...........................................................................33
7. Concluding observations..........................................................................................37
9. Acknowledgements..................................................................................................38
Chapter 2—Anticrack-inclusion model for compaction bands in sandstone................39
1. Abstract....................................................................................................................39
2. Introduction..............................................................................................................39
3. Methods....................................................................................................................42
4. Field and petrographic analysis...............................................................................44
4.1. Geological setting and paleostress state..........................................................44
4.2. Outcrop analysis..............................................................................................46
4.3. Petrographic analysis......................................................................................50
5. Anticrack-inclusion conceptual model....................................................................57
6. Elastic properties......................................................................................................60
7. Mechanical analysis.................................................................................................61
7.1. Embedded layer model...................................................................................61
7.2. Eshelby inclusion model.................................................................................64
7.3. Anticrack model..............................................................................................67
8. Discussion................................................................................................................71
9. Acknowledgements..................................................................................................74
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Chapter 3—Energy-release model of compaction band propagation............................75
1. Abstract....................................................................................................................75
2. Introduction..............................................................................................................75
3. Formulation..............................................................................................................78
4. Special cases............................................................................................................81
5. Discussion................................................................................................................82
6. Acknowledgements..................................................................................................84
Chapter 4—Propagation of compaction bands in sandstone as anticracks:
Observational evidence, mechanical theory and numerical simulation.....85
1. Abstract....................................................................................................................85
2. Introduction..............................................................................................................85
3. Field evidence and interpretation.............................................................................88
3.1. Outcrop observations......................................................................................89
3.2. Petrographic observations...............................................................................92
3.3. Anticrack-inclusion interpretation..................................................................96
4. Mechanical theory....................................................................................................98
5. Numerical model....................................................................................................104
6. Propagation simulation..........................................................................................107
6.1. Model calibration and stability testing..........................................................107
6.2. Symmetric propagation.................................................................................112
6.3. Approaching tip interactions.........................................................................114
7. Discussion and conclusions...................................................................................119
8. Acknowledgements................................................................................................123
Chapter 5—Computational estimation of compaction band permeability:
From thin-section estimations to reservoir implications...........................125
1. Abstract..................................................................................................................125
2. Introduction............................................................................................................125
3. Computational method...........................................................................................130
3.1. Image processing..........................................................................................130
3.2. Pore structure realization..............................................................................130
3.3. Permeability estimation................................................................................131
4. Application to the Aztec sandstone........................................................................134
4.1. Simulation results..........................................................................................134
4.2. Comparison to measured values...................................................................137
5. Conclusions............................................................................................................139
6. Acknowledgements................................................................................................140
Chapter 6—Permeability effects of deformation band arrays in sandstone................141
1. Abstract..................................................................................................................141
2. Introduction............................................................................................................141
3. Deformation bands.................................................................................................143
4. Characteristic DB patterns in the Aztec sandstone................................................147
4.1. Parallel..........................................................................................................148
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4.2. Cross-hatch...................................................................................................148
4.3. Anastomosing...............................................................................................150
5. Numerical modeling methods................................................................................150
5.1. Governing equations.....................................................................................152
5.2. Boundary conditions and solution................................................................154
5.3. Finite difference/finite element method........................................................157
6. Effective permeability results................................................................................157
6.1. Parallel..........................................................................................................158
6.2. Cross-hatch...................................................................................................160
6.3. Anastomosing...............................................................................................163
7. Summary................................................................................................................165
8. Discussion and conclusions...................................................................................166
9. Acknowledgements................................................................................................167
Chapter 7—Flow and transport effects of compaction bands in sandstone
at scales relevant to aquifer and reservoir management...........................169
1. Abstract..................................................................................................................169
2. Introduction............................................................................................................169
3. Mapping.................................................................................................................173
4. Flow simulation model..........................................................................................175
5. Simulations............................................................................................................182
5.1. Well-to-well flow..........................................................................................182
5.1.1. Results..................................................................................................182
5.1.2. Discussion............................................................................................185
5.2 Reservoir production......................................................................................187
5.2.1. Results..................................................................................................187
5.2.2. Discussion............................................................................................190
5.3. Contaminant transport...................................................................................190
5.3.1. Results..................................................................................................192
5.3.2. Discussion............................................................................................192
6. General discussion.................................................................................................196
7. Conclusions............................................................................................................198
8. Acknowledgements................................................................................................199
References.......................................................................................................................201
List of Tables
Table 1.1. Summary of compaction band orientation data...............................................29
Table 1.2. Summary of restored compaction band orientation data..................................32
Table 5.1. Summary of permeability estimates...............................................................136
Table 7.1. Relative permeability data for reservoir production simulations...................189
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List of Illustrations
Figure A. Cover photo from AAPG Bulletin (publication of Chapter 6)..........................xii
Figure 1.1. Location and general geology of study area...................................................12
Figure 1.2. Generalized stratigraphic column...................................................................14
Figure 1.3. Schematized fault map for Lake Mead region................................................17
Figure 1.4. Photo collage of compaction bands in the Aztec sandstone...........................20
Figure 1.5. Schematic of anticrack conceptual model......................................................23
Figure 1.6. Collection of orientation data in the field.......................................................25
Figure 1.7. Located orientation data from Valley of Fire.................................................26
Figure 1.8. Orientation data for outcrops with multiple compaction band sets................27
Figure 1.9. Stereographs for combined orientation data...................................................29
Figure 1.10. Stereographs for restored orientation data....................................................32
Figure 1.11. Paleostress orientations and interpretation...................................................36
Figure 2.1. Location of the Valley of Fire State Park, Nevada.........................................41
Figure 2.2. Compaction bands (CBs) in outcrop near Silica Dome..................................43
Figure 2.3. Typical CB fin in outcrop...............................................................................48
Figure 2.4. Compaction band thickness profile data.........................................................49
Figure 2.5. Composite photomicrograph of a CB.............................................................51
Figure 2.6. Porosity profiles across a CB..........................................................................52
Figure 2.7. Backscatter electron images of sandstone and a CB......................................53
Figure 2.8. Distribution of porosity inside a CB...............................................................55
Figure 2.9. Backscatter electron images of grain damage in a CB...................................56
Figure 2.10. Backscatter electron image of incipient CB.................................................55
Figure 2.11. Schematic representations of idealized CB model.......................................58
Figure 2.12. Near-tip stress components for the embedded layer model..........................65
Figure 2.13. Stress distributions for the Eshelby inclusion model....................................66
Figure 2.14. Stress comparison of Eshelby and anticrack models....................................69
Figure 2.15. Near-tip stress comparison of Eshelby and anticrack models......................70
Figure 2.16. Mean normal stress distribution around an anticrack tip..............................70
Figure 3.1. Anatomy of a compaction band in the Aztec sandstone.................................76
Figure 3.2. Model of a semi-infinite CB embedded in an infinite layer...........................79
Figure 3.3. Outcrop basis for the embedded layer CB model...........................................79
Figure 3.4. Hypothetical 1-D distribution of stress near a CB tip....................................84
Figure 4.1. Location of the Valley of Fire State Park, Nevada.........................................87
Figure 4.2. Array of subparallel CBs in the Aztec sandstone...........................................87
Figure 4.3. Typical compaction band patterns exposed in outcrop..................................90
Figure 4.4. Mechanical interaction between CBs and æolian bedding.............................91
Figure 4.5. Tip-to-tip thickness profile of a 25-m-long CB..............................................91
Figure 4.6. Various CB propagation behaviors observed in outcrop................................93
Figure 4.7. Examples of tip-to-tip interactions observed in outcrop................................94
Figure 4.8. Composite photomicrograph of CB anatomy.................................................95
Figure 4.9. Schematic representations of the idealized CB model...................................97
Figure 4.10. Schematic of the global and anticrack tip coordinate systems.....................99
Figure 4.11. Contour plots of near tip stress around an anticrack..................................100
Figure 4.12. Polar near-tip stress as a function of θ for a fixed radius...........................102
Figure 4.13. Effect of differential stress on propagation path stability...........................102
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Figure 4.14. Mechanical interactions between anticrack tips.........................................105
Figure 4.15. Essential schematic of displacement discontinuity BEM...........................105
Figure 4.16. Stiffness dependence of closing-mode displacement discontinuity...........109
Figure 4.17. Near-tip stress comparison of Eshelby and anticrack models....................109
Figure 4.18. Propagation path stability as a function of scale and remote stress............111
Figure 4.19. Scale and remote stress dependence of propagation path stability.............111
Figure 4.20. Element length and stress dependence of propagation path stability.........113
Figure 4.21a. Remote stress and spacing effects on CB tip interactions........................116
Figure 4.21b. Remote stress and spacing effects on CB tip interactions........................117
Figure 4.21c. Remote stress and spacing effects on CB tip interactions........................118
Figure 4.22. Influence of a CB length on approaching tip interactions..........................119
Figure 4.23. Conceptual model of strain and grain damage around a CB......................122
Figure 4.24. Relationship of a CB array to permeability and paleostress.......................124
Figure 5.1. Location of the Valley of Fire State Park, Nevada.......................................126
Figure 5.2. Typical CB in outcrop and its effect on fluid flow.......................................128
Figure 5.3. Representative outcrop array of subparallel, anastomosing CBs.................129
Figure 5.4. Composite photomicrograph of a representative CB...................................132
Figure 5.5. Permeability estimation methodology workflow.........................................133
Figure 5.6. Backscatter electron images of sandstone and CBs......................................135
Figure 5.7. Porosity-permeability estimates versus measured values.............................136
Figure 5.8. Computational estimation of permeability anisotropy.................................138
Figure 6.1. Location of the Valley of Fire State Park, Nevada.......................................144
Figure 6.2. Compactive deformation bands in the Aztec sandstone...............................145
Figure 6.3. Typical outcrop pattern of subparallel compactive DBs..............................149
Figure 6.4. Typical cross-hatch pattern of compactive DBs...........................................149
Figure 6.5. Typical anastomosing pattern of compactive DBs.......................................151
Figure 6.6. Idealized band pattern and model grid representation..................................153
Figure 6.7. Schematic representations of band pattern upscaling...................................153
Figure 6.8. Key parameters for computing effective block permeability.......................156
Figure 6.9. Effective permeability for perfectly parallel band patterns..........................159
Figure 6.10. Effective permeability for orthogonal cross-hatch band patterns...............161
Figure 6.11. Effective permeability for acute cross-hatch band patterns........................162
Figure 6.12. Effective permeability for a real anastomosing band pattern.....................164
Figure 7.1. Location and air photo of the Aztec sandstone, Valley of Fire, NV............170
Figure 7.2. Compaction bands in outcrop and thin section.............................................172
Figure 7.3. Compaction band trace map.........................................................................174
Figure 7.4. Schematic of transmissibility calculation parameters..................................177
Figure 7.5. Triangular discretization of compaction band trace map.............................181
Figure 7.6. Well locations for single-phase flow simulations.........................................183
Figure 7.7. Pressure drop results for single-phase flow simulations..............................184
Figure 7.8. Well configurations for reservoir production simulations............................186
Figure 7.9. Saturation-map snapshots for reservoir production simulations..................188
Figure 7.10. Production efficiency for the two reservoir production scenarios..............189
Figure 7.11. Contaminant leak simulation configurations..............................................191
Figure 7.12. Contaminant plume with bands parallel to regional gradient.....................193
Figure 7.13. Contaminant plume with bands oblique to regional gradient.....................194
Figure 7.14. Contaminant plume with bands normal to regional gradient.....................195
xi

Figure A. Cover photo that accompanied publication of Chapter 6 in AAPG Bulletin
(Sternlof et al., 2004). This view shows steeply east-dipping compaction bands in the
bleached middle Aztec sandstone, just above the red, iron oxide-stained lower Aztec.
Dark Paleozoic rocks of the over-riding Cretaceous Muddy Mountain Thrust loom in the
background. These compaction bands are approximately 1 cm thick and 100 m long.
xii

Introduction
Some brittle structures in rock act to enhance and channel fluid flow, others to
impede and compartmentalize it. Commonly, structures of both types interact with and
overprint each other to form complex fabrics that exert a dominating influence on flow in
subsurface aquifers and reservoirs, even those endowed with high intrinsic permeability
such as porous sandstone. Understanding the distribution, geometry and flow properties
of these structural fabrics is therefore vital to optimal resource production and
management. Unfortunately, data on the attributes of these fabrics—whether from 3-D
seismic surveys, borehole geophysical logs, pump tests and production data, or core
analysis—are generally limited and may be virtually nonexistent for small-scale
structures. One approach to addressing this problem is to study exhumed analog aquifers
and reservoirs, where structures and fabrics can easily be observed (at least in 2-D),
sampled and analyzed. The ultimate goal of such studies is to develop insights and tools
for forecasting the presence and influence of structures in the subsurface based on limited
hard data.
That has been the theme of an ongoing, decade-long research project undertaken by
David Pollard and Atilla Aydin with funding from the U.S. Department of Energy, Office
of Basic Energy Sciences—Structural Heterogeneities and Paleo Fluid Flow in an
Analog Sandstone Reservoir. My thesis work fits into this larger research initiative, just
as my topic—deformation bands and, more specifically, compaction bands—fits into the
framework of structures that did demonstrably impact paleo fluid flow in the Aztec
sandstone and are now breathtakingly exposed in the aptly named Valley of Fire of
southeastern Nevada. My effort follows three earlier Stanford dissertations that also took
shape from the colorful outcrops of the Valley of Fire: Fluid flow and chemical alteration
in fractured sandstone (W.L. Taylor, 1999); Structure and hydraulics of brittle faults in
sandstone (R.D. Myers, 1999); and Structural evolution, petrophysics and large-scale
permeability of faults in sandstone, Valley of Fire, Nevada (E.A. Flodin, 2003).
This dissertation focuses on the oldest structural fabric components present in the
Aztec sandstone—compaction bands—and endeavors to comprehend what they are, why
they formed, and how they influence bulk permeability and fluid flow. In essence, I take
a broad look at a specific structural phenomenon, one which physical logic dictates is
1

likely present in subsurface sandstone equivalents of the Aztec. Coming at this research
as a traditional structural geologist, I have therefore collaborated widely—in reservoir
engineering, petrophysics, and theoretical, experimental and applied mechanics—to bring
the requisite tools to bear. I could not have achieved the breadth of this thesis working in
isolation, and consider these collaborations to be a particular strength of the effort.
The thesis is comprised of seven chapters, each written as a stand-alone manuscript
intended for peer-reviewed publication. As a result, some repetition, particularly with
introductory material, was inevitable. Also, although arranged thematically, the chapters
do not necessarily flow seamlessly one into the next, as they might in a thesis written to
be a coherent book. Indeed, each paper was written with an eye toward the editorial
requirements of a different journal, and they were finished in the order: 6, 2, 3, 7, 5, 4, 1.
A discerning reader might detect evidence for the evolution of my thinking over the past
few years preserved within these pages. For example, Chapter 6 does not differentiate
compaction bands (CBs), which are prevalent in the Aztec sandstone, from the more
general category structural category of deformation bands (DBs). A brief overview of
each chapter follows.
Chapter 1 describes the occurrence and structural geology of CBs and CB arrays, and
considers their significance within the context of the depositional, diagenetic and tectonic
history of the Aztec sandstone through Cretaceous time. As the final paper written, this
draft manuscript represents a preliminary synthesis of my work in the Aztec and its
implications for predicting CB occurrence in sandstone based on material and stress
history or, conversely, interpreting material and stress history based on the occurrence of
CBs. The paper is based on my detailed observations conducted at thin-section, outcrop
and regional scales, as well as on my review and interpretation of the work of others.
David Pollard deserves substantial credit for his tutelage, editing and ability to foment
ideas. Submission of a final manuscript based on Chapter 1 to Tectonics is anticipated.
Chapter 2 presents the evidence and arguments for a mechanical interpretation of CBs
as contractile Eshelby inclusions, and for considering them as anticracks for the purposes
of propagation modeling using linear elastic theory and the displacement discontinuity
boundary element method. All of the data collection, both in the field and the lab, all of
the analytical and numerical modeling, and all of the original writing and figure drafting
2

was performed by me. The ideas expressed were developed in collaboration with my co-
authors, principally David Pollard and secondarily John Rudnicki of Northwestern
University. The embedded-layer model analysis was originated by Rudnicki. The Eshelby
code was generously provided by Pradeep Sharma of the University of Houston, and
adapted for my use by fellow student Ole Kaven. The boundary element code was written
to my specifications by another fellow student, Gaurav Chopra, in close consultation with
Pollard and with substantial input from me. This paper was published in the November
2005 issue of Journal of Geophysical Research (v. 110, n. B11).
Chapter 3 represents an additional fruit of my collaboration with John Rudnicki, who
had conceived of an energy-release analysis for compaction-band propagation, but was
lacking a convincing physical context in which to frame it. As a result of my work with
him on Chapter 2, he came around to the view that his theoretical model could and should
be grounded primarily in field observations rather than in experimental results, which I
contend produce a phenomenology largely distinct from CB formation in nature. This
subtle feat of scientific diplomacy helped to pierce a somewhat insular and
counterproductive feedback loop between theorists and experimentalists working on
compaction localization. I am the second author after Rudnicki on this paper, and
contributed extensive rewrites and editing (both technical and grammatical) to
fundamentally recast the paper as outlined above. I also conducted targeted fieldwork in
the Valley of Fire to produce the second figure of two that appears in the manuscript. As
always, substantial credit is due David Pollard for his guidance and editing. This succinct
contribution, an important step toward establishing a stress/strain/strength activation limit
for compaction propagation in sandstone, was published in the second August 2005 issue
of Geophysical Research Letters (v. 32, n. 16).
Chapter 4 takes the anticrack model for CBs developed in Chapter 2 and applies it
through the boundary element method (BEM) code (also introduced in Chapter 2) to
investigate the mechanics of CB propagation, interaction and pattern development in the
Aztec sandstone. The scientific premise behind this preliminary draft manuscript is that
all the various 2-D patterns of CB interaction revealed in outcrop express the same
fundamental mechanism of propagation. I attempt to recreate some simple diagnostic
patterns—in-plane propagation, “zig-zag” and anastamosing propagation instability and
3

hooking-tip interactions—using the BEM code. The degree of success attained suggests
that the linear elastic anticrack mechanical model provides a valid first approximation for
conceptualizing and simulating CB propagation. Perhaps the most interesting and
potentially useful result of this paper is the model observation that anticrack CBs respond
to high compressive normal stress (σ2 ≈ σ1) oriented parallel to their trend with
propagation-path instability. This is opposite to the instability effect observed for
opening-mode cracks (Olson and Pollard, 1989; Thomas and Pollard, 1993), and could be
used to help forecast the degree of anastomosis in subsurface CB arrays as a function of
stress history. By the same token, the configuration of an exhumed CB array could be
used to constrain the ambient principal paleostress state (orientation and magnitude) in
which it formed. For this paper, I performed all the fieldwork, modeling, writing and
figure drafting, with guidance and editing provided by second-author David Pollard. For
the modeling, I used the BEM code written by third-author Gaurav Chopra (Chapter 2).
Submission of a final manuscript based on Chapter 4 to Journal of Structural Geology is
anticipated.
Chapter 5 represents another outgrowth of my collaborative efforts, in this case with
Tapan Mukerji and Youngseuk Keehm of the Stanford Rock Physics and Borehole
Geophysics (SRB) Group. I had gone to Mukerji for help in using MATLAB® to
perform automated image-analysis measurements of porosity from my thin-sections, the
results of which contributed to every other chapter in this thesis. For the purposes of the
fluid-flow modeling presented in Chapter 7, I had also contemplated collecting
permeability measurements, but was dissuaded by the difficulty of obtaining reliable
results from thin CBs at reasonable expense. Through Mukerji, however, I learned of the
computational permeability estimation algorithm that had been developed in the SRB,
principally by Keehm, and could be used to generate virtual permeability measurements
from my existing thin sections. I recognized the opportunity to perform a practical test of
this new tool within the context of the analog reservoir research concept. Specifically, we
applied the method to my prize thin section and compared the estimation results to
available permeability measurement data for both CBs and host rock from the Aztec and
Navajo sandstones. Correspondence was excellent, suggesting that, for a subsurface
equivalent of the CB-rich Aztec from which only scarce, potentially unconsolidated
4

samples were available, representative flow properties can be gleaned from even a single
thin section. In addition to conceiving of the project, I conducted all the fieldwork and
microscopy involved, and produced all the figures and text (except for the Methods
section, which I edited, and Figure 5). Keehm and Mukerji performed the permeability
estimations. David Pollard provided editing and valuable guidance. This manuscript has
yet to be placed for publication.
Chapter 6 began as a project undertaken in 1999 by Jeffrey Chapin, a visiting
researcher working with David Pollard. Chapin abandoned the work while in draft form
and switched disciplines from geology to mechanical engineering. I inherited the project
and brought it to fruition. Initially I performed a thorough editing and culling job on the
rough draft, including moderate rewriting, and submitted it to American Association of
Petroleum Geologists Bulletin in September 2002 as second author after Chapin, with co-
authors Louis Durlofsky and Pollard. That version of the manuscript was rejected. In
assuming first authorship at Pollard’s behest, I then performed a complete rewrite,
including a strategic reconceptualization, additional culling and the redrafting of all
figures. This reformulated version of the paper, which presents an analytical and
numerical framework for understanding how characteristic patterns of deformation bands
exposed in the Aztec sandstone would transform bulk effective permeability, was
published in the September 2004 issue of American Association of Petroleum Geologists
Bulletin (v. 88, n. 9).
Chapter 7 grew out of reviewer feedback that came with shepherding Chapter 6
through to publication, and my desire to assume more complete ownership of the earlier
work by expanding the effort. Comments by the reviewers, both cogent and off-target,
suggested that, in order convincingly to assess the potential flow effects of compaction
bands, I needed to consider a realistic band pattern at a scale of clear relevance to aquifer
and reservoir modeling, rather than appealing to arguments based on up-scaling
techniques as in Chapter 6. I saw the opportunity to achieve this goal in low-altitude
aerial photographs taken of the Valley of Fire for another purpose. Viewed at full
resolution, the patterns of CBs cropping out in positive relief are clearly visible, which
enabled me to construct a detailed and representative map of cm-thick CBs over an area
of 150,000 m 2 . Using this map in conjunction with realistic flow properties for the bands
5

and sandstone, my co-authors—Mohammad Karimi-Fard, David Pollard and Louis
Durlofsky—and I were able to model the impact of the bands for a variety of practical
scenarios: well-pair pump testing, five-spot reservoir production and contaminant plume
migration. Our results confirm that the anastomosing band pattern as mapped would exert
significant pressure drop and directional fluid transport effects over hundreds of meters.
In addition to conceiving of the project, I conducted all of the fieldwork and photo
mapping (including arranging for additional air photos to be taken), designed the five-
spot and plume scenarios modeled, and produced all the text (except for the Flow Model
section) and final figures. Karimi-Fard ran the flow simulations—an arduous task—using
a model code previously developed, primarily by him. He also collaborated closely in
designing the simulation scenarios and figures, wrote the Flow Model section, and
participated in every sense as an equal research partner. Pollard and Durlofsky provided
guidance and thorough editing. This paper has been accepted for publication in Water
Resources Research and is currently in press.
This introduction would not be complete without touching on my experimental efforts
and collaborations, which failed to bear publishable fruit in time for inclusion in this
dissertation, but did add about a year to my tenure at Stanford. The desire to test the
anticrack CB mechanical interpretation in the lab, and by so doing shed light on the
enigmatic nature of CB tip-zone processes, was irresistible to me—practical timing
concerns voiced by my committee notwithstanding. Besides, as mentioned above in the
introduction to Chapter 3, it seemed that experimental compaction localization research,
which had already become a hot area, was headed in a direction at odds with observable
natural reality. In forging ahead despite my committee’s accurate admonitions, I did
manage to glean valuable “negative result” insights into the balance of material and
loading conditions conducive to CB formation. Specifically, I suggest that natural
compaction bands, such as observed in the Aztec, form in unconsolidated to barely
lithified porous sands subjected to tectonic and burial stress states typical of the shallow
crust. Despite using weakly lithified Aztec specimens in my experiments, which I seeded
with stress-concentrating flaws and loaded uniaxially over a range of geologically
plausible confining pressures, failure invariably occurred by dilational shear. On the other
hand, experiments that force compaction to occur in lithified sandstone at mid-crustal
6

confining pressures produce, in my opinion, a failure phenomenon distinct from
compaction banding as observed in outcrop.
The greater value of my fledgling laboratory efforts probably lies in the collaborative
relationships developed within the experimental community pursuing compaction
localization research. I visited and worked in the labs of Bezalel Haimson (University of
Wisconsin-Madison), William Olsson and David Holcomb (Sandia National Labs) and
David Lockner (USGS-Menlo Park), and corresponded extensively with the laboratory of
Teng-fong Wong (Stony Brook University). Through these interactions and a DOE-
sponsored workshop (October 2004), which led directly to my association with co-author
John Rudnicki (Chapters 2 and 3), David Pollard and I have introduced a much-needed
field-based reality check into compaction localization experimentation. Whether this
influence will become more tangible through my own future experimental efforts remains
to be seen.
Finally, I would be remiss in not observing that, although a strong process-based
mechanics perspective and approach pervade this thesis, it is based primarily on
observations made from outcrops and samples of the Aztec sandstone. While this fact
does not detract from my results and interpretations, the issue of ubiquity is relevant from
a practical applications standpoint. That is, are CB arrays such as observed in the Aztec
common in sandstone aquifers and reservoirs, or are they a relative freak of nature? From
a fundamental physical and mechanical point of view, it is unlikely that CBs in the Aztec
represent an isolated phenomenological fluke. However, shear deformation bands have
been observed in a wide variety of clastic deposits exposed around the world, and similar
observations of CBs are needed to confirm them as a common structure in porous
sandstone.
Beyond this obvious direction for future analog reservoir-based fieldwork, a
concerted effort also is needed to develop borehole geophysical methods for detecting
and characterizing deformation bands of all types in the subsurface. The general
petrophysical attributes of bands should, in theory at least, render them visible to a
variety of imaging technologies, ranging from acoustic to electromagnetic. In conjunction
with a solid mechanical understanding of how CBs and CB arrays form—the foundations
for which are laid here in Chapters 1 through 4—such imaging techniques would provide
7

the hard data from which accurate projections of band configurations within subsurface
sandstones could be made. With reliable configuration projections and flow-property
estimations (Chapter 5) in hand, the fluid flow impacts of any CB array could then
accurately be modeled and generalized for the purposes of aquifer/reservoir-scale
simulation—as demonstrated in Chapters 6 and 7.
8

Chapter 1
Structural geology and tectonic interpretation of compaction bands
in the Aztec sandstone of southeastern Nevada
1. Abstract
The Aztec sandstone in the Valley of Fire of southeastern Nevada comprises an
exhumed analog for active æolian aquifers and reservoirs subjected to tectonic
compression and shortening. A planar fabric of anastomosing compaction bands (CBs)
pervades the upper 600 m of the 1,400-m-thick Aztec. Individual CBs are observed to be
thin, tabular localizations of uniaxial porosity-loss compaction that can be considered
kinematically and mechanically as anticracks. In outcrop, CBs form sometimes complex
patterns that can, nonetheless, be generally understood as resulting from mechanical
interactions between anticracks. Compaction bands are the oldest structures in the Aztec,
and abundant geological evidence indicates that they formed during the earliest period of
regional compression associated with Cretaceous Sevier tectonism in a weakly cemented
material buried less than a few hundred meters.
Viewed at larger scales (> 100 m), a dominant north-south trend and steep east dip are
apparent, as well as localized pockets of a coexisting, coeval CB set trending generally
northeast and dipping steeply northwest. The mean trend of the dominant CB set is
generally orthogonal to the east-vergent direction of Sevier thrusting, while the secondary
set is, on average, orthogonal to the dominant set. Based on this geometry, and our thin-
section to outcrop-based understanding of CBs as anticracks, we interpret the dominant
set as having formed orthogonal to the maximum compressive paleostress (σ1), with the
secondary set orthogonal to σ2. This completely constrains the orientations and relative
magnitudes of the principal paleostresses during CB formation—σ1 → 270°/38°, σ2 →
138°/39°, σ3 → 24°/27°—which, however, depart radically from the classic view of σ3 as
subvertical in thrust-fault environments. We suggest that a diffuse stress perturbation
caused by as yet unrecognized regional structures—e.g. a detachment beneath the Valley
of Fire along which it moved relatively westward—could explain the inferred state of
paleostress.
9

Although the Aztec sandstone experienced less than 1% total shortening as the result
of CB formation, the low-porosity, low-permeability band fabrics developed have been
shown capable of exerting profound fluid flow effects at scales of practical interest. The
analysis presented here of CBs as indicators of paleostress and tectonic structure in this
exhumed analog aquifer/reservoir also applies in reverse. That is, given knowledge of the
stress and material history of a subsurface sandstone, predictions regarding the possible
presence and geometry of CBs can be made.
2. Introduction
Compaction banding as a recognized mode of localized brittle failure in porous
sandstone has a brief history, having first been described and named by Mollema and
Antonellini (1996) based on their work in the Navajo sandstone of south central Utah.
Compaction bands (CBs) represent one kinematic end member of the suite of structures
known collectively as deformation bands, which includes shear and dilation bands (Du
Bernard et al., 2002; Borja and Aydin, 2004). Mollema and Antonellini (1996) described
CBs as long (meters), thin (centimeters) tabular zones of mechanical porosity loss formed
within the compressional quadrants of shear band style faults (Aydin, 1978), interpreting
them as “a structural analog for anti-mode I cracks in æolian sandstone...that
accommodate pure compaction,” following the conceptual anticrack model for pressure
solution surfaces of Fletcher and Pollard (1981).
The recognition of CBs, and their obvious potential to act as low-porosity, low-
permeability impediments to fluid flow in otherwise highly transmissive sandstones, has
excited an ongoing flurry of work both experimental (e.g. Olsson and Holcomb, 2000;
Klein et al., 2001; Vajdova and Wong, 2003; Haimson, 2003; Baud et al., 2004; Tembe et
al., 2006) and theoretical (e.g. Olsson, 1999; Issen and Rudnicki, 2000, 2001; Detournay
et al., 2003; Bésuelle and Rudnicki, 2004; Rudnicki, 2002, 2003, 2004; Katsman et al.,
2005; Rudnicki and Sternlof, 2005; Sternlof et al., 2005; Katsman and Aharonov, 2006)
all of it aimed at comprehending the mechanics of compaction localization in porous,
granular materials. To date, however, the only other field-based study published on CB
occurrence and geology also focused on an æolian Jurassic sandstone of the southwestern
U.S.—the Navajo-equivalent Aztec sandstone of southeastern Nevada (Sternlof et al.,
2005). That paper presents an anticrack-inclusion mechanical model for CBs that is based
10

on detailed outcrop and thin-section observations of the Aztec as extensively exposed in
the Valley of Fire State Park (Figure 1.1).
Stepping back from the wide variety of patterns visible in individual outcrops, the two
most striking aspects of CBs in the Aztec as a whole are their ubiquity and the general
consistency of their planar orientation: north-northwest trending, steeply east dipping.
Considering the Aztec as an exhumed analog for active sandstone aquifers and reservoirs,
the potential for such an anastomosing CB fabric to restrict and channel fluid flow is
substantial (Sternlof et al., 2004; Sternlof et al., 2006). The practical goal behind all CB
investigations is to develop the capability to predict, or at least forecast their occurrence
in the subsurface such that their hydraulic effects can be mitigated. Achieving this goal
will require understanding how, in what kinds of materials and under what loading
conditions CBs form. As a contribution to this endeavor, we present a broad look at the
thin-section to regional scale structural geology of CBs in the Aztec sandstone at the
Valley of Fire, and consider their implications for the tectonic framework and history of
southern Nevada.
3. Setting and history of study area
The following synopsis of the geological setting and history of the Aztec sandstone
was synthesized from data and interpretations published in a wide variety of sources over
the past 85 years as cited below. Given that the methods and perspectives of structural
geology have evolved a great deal since the first research on the area was published
during the Wilson administration (Longwell, 1921), long before any modern notion of
tectonics, we emphasize here more recent work and interpretations.
3.1. Deposition
The Aztec sandstone was deposited during latest Triassic to middle Jurassic time in a
back-arc basin associated with uplift of the ancestral Sierra Nevadan magmatic arc
(Marzolf, 1986; Burchfiel et al., 1992). It comprises the southwestern edge of a much
larger æolian erg system, which includes the Navajo and Nugget sandstones and
blanketed much of the intermountain southwest (Blakey, 1989; Marzolf, 1986).
Prevailing winds during deposition of the Aztec are interpreted to have blown out of the
northeast and over the shallow sea occupying what is now the continental interior
(Marzolf, 1982, 1986).
11

Waterpocket Fault
36
o
26‘
N
0 1
km
114
o
32‘
Willow Tank
Thrust
Park Road
Overton Syncline
Lower Aztec
12
Oregon
Washingto n
California
30
Nevada
Canada
Las Vegas
Idaho
Air photo detail
Valley of Fire
Cretaceous units
25
Upper Aztec
Park Office
25
Montana
Utah
Wyoming
Arizona
Baseline Fault
Route 169

Figure 1.1 (opposite page) Location and general geology of the study area in the Aztec
sandstone at the Valley of Fire State Park, southeastern Nevada. The Aztec crops out as a
1,400-m-thick sequence of gently folded, northeasterly dipping (~25°) æolian deposits
consisting of ~800 m of red-stained sandstone (lower Aztec) and ~600 m of bleached and
variegated sandstone (upper Aztec). The Aztec is immediately, though unconformably
overlain by up to 1,300 m of Cretaceous deposits (principally the quartz-rich Baseline
sandstone in the air photo). The Aztec has been extensively affected by Basin and Range
extension, principally by north-northeasterly trending left-lateral strike slip-faults and
subsidiary northwesterly trending right-lateral strike-slip faults (apparent in the many
offsets of the red/buff alteration front). The major left-lateral bounding faults for the
immediate study area (Waterpocket fault and Baseline fault) are indicated (after
Bohannon 1983), as are the Overton Syncline (after Carpenter and Carpenter, 1994) and
two representative bedding orientations in the lower Cretaceous units (after Bohannon,
1983). Less obvious are remnants of the east-vergent Willow Tank thrust (upper right
corner, teeth denote hanging wall), marking the easternmost extent of compressional
tectonism related to the Cretaceous Sevier ororogeny. The aerial photo base is a mosaic
constructed of images downloaded from Google Earth.
Within the general vicinity of the Valley of Fire (Figure 1.1), the 1,400-m-thick Aztec
unconformably overlies terrigenous clastic, evaporitic and limestone deposits of the
Triassic and Permian, which themselves unconformably overlie miogeoclinal carbonates
of Pennsylvanian age and older. The Aztec in turn is unconformably overlain by up to
1,300 m of upward-coarsening Cretaceous clastics and, following a depositional hiatus
through the Paleogene, another 300 m of Neogene limestones and clastics (Bohannon,
1983; Bohannon et al., 1993; Eichhubl et al., 2004). Figure 1.2 presents a generalized
stratigraphic section of the region from latest Paleozoic time through the present day, as
modified from Bohannon (1977).
The Aztec sandstone itself is a subarkose containing up to 8% orthoclase and minor
amounts of lithic fragments (Eichhubl et al., 2004). It is composed of moderately rounded,
fine to medium-sized detrital grains (0.1 to 0.5 mm, 0.25 mm average) arranged in a
large-scale wedge and tabular-planar cross stratified sedimentary architecture
characteristic of æolian deposition (Marzolf, 1983). The Aztec is generally quite friable,
with porosity averaging about 20-25% (Sternlof et al., 2005) and permeability ranging
from about 100 to 2,500 mD (Flodin et al., 2005; Chapter 5 this thesis). Clay, chiefly
kaolinite, comprises up to 6% of the rock by volume and forms the predominant grain-
bridging, pore-filling cement.
13

Cenozoic
Mesozoic
Paleozoic
Quaternary
Permian Triassic Triassic?
Jurassic Cretaceous Tertiary
Miocene &
Pliocene
Miocene
Thickness (m)
5,400
4,800
4,200
3,600
3,000
2,400
1,800
1,200
600
0
Quaternary Deposits
Muddy Creek Formation
Horse Spring Formation
Baseline Sandstone
Willow Tank Formatio n
Aztec Sandstone
Moenave Formation
Chinle Formation
Moenkopi Formation
Kaibab Limestone
Toroweap Formatio n
Hermit Shale
Lithologic Descriptions
Sandstone
Shale
Siltstone
Calcareous
shale
Limestone
Dolomite
Magnesite
Conglomerate
Breccia
Evaporites
(gypsum)
Calcareous
conglomerate
Figure 1.2. Generalized stratigraphic column for the Valley of Fire area from the
Permian Hermit shale through Quaternary alluvium (adapted from Bohannon, 1977).
14
Chert

Abundant deeply etched and/or kaolin-replaced orthoclase grains indicate that much of
the clay present is internally derived (Eichhubl et al., 2004; Sternlof et al., 2005). Small
amounts of inter-layered smectite and illite are also present (Eichhubl et al., 2004).
Abundant mutual indentation of quartz grains via pressure solution has been reported
(Flodin et al., 2003; Eichhubl et al., 2004), although much of this texture may in fact be
due to mechanical indentation accommodated by pervasive intra-grain microfracturing
(Sternlof et al., 2005).
3.2. Diagenesis
Syndepositional precipitation of hematite grain coats resulted in the Aztec being
stained a uniform red color during initial burial (Eichhubl et al., 2004). The color from
this earliest diagenetic alteration still dominates the lower half of the sandstone. The
upper Aztec, however, has been subjected to at least two stages of bleaching and iron
oxide redistribution apparently associated with upward and eastward expulsion of
reducing basinal brines from beneath Sevier-related thrust sheets advancing from the
west (Eichhubl et al., 2004). These fluid-flow events are responsible for the vividly
colorful patterns of iron mineralization from which the Valley of Fire derives its name
(Figure 1.1).
Despite its long alteration history, however, the presence of only limited quartz
overgrowth cementation and diagenetic illite, both restricted to the very bottom of the
pile (and within the damage zones of some major faults), indicate that the base of the
Aztec in the Valley of Fire has likely never been buried deeper than about 3 km (~ 80° C
for a normal geothermal gradient). Interestingly, this equals the sum of its own thickness
and of all overlying sedimentary deposits through (Eichhubl et al., 2004; Sternlof et al.,
2005). In any case, the Aztec is today at best moderately well cemented toward the
bottom, where it is still red, and poorly cemented toward the middle and top (Flodin et al.,
2003), where it has been extensively bleached and now exhibits an unconfined uniaxial
compressive strength of only 2-3 MPa (Haimson and Lee, 2004). Neither is there
evidence to suggest that the state of lithification anywhere in the Aztec was ever
appreciably greater than it is today. Our own limited triaxial testing, however, does
indicate that the resistance of present-day Aztec to uniform compaction with increasing
pressure approaches that of quartzite.
15

3.3. Deformation
The Aztec also has been subjected to a varied and punctuated history of deformation
driven by tectonic activity, starting with late Mesozoic compression of the Sevier
Orogeny (e.g. Armstrong, 1968; Fleck, 1970; Brock and Engelder, 1977; Bohannon,
1983) and jumping to Basin and Range extension beginning in Miocene time and
continuing to the present (e.g. Bohannon, 1983; Bohannon et al., 1993; Campagna and
Aydin, 1994; Flodin and Aydin, 2004; Myers and Aydin, 2004)). Figure 1.3 presents a
generalized structure and tectonic map for the region, as compiled from multiple sources.
Thin-skinned, east-vergent thrust faulting of the Sevier Orogeny may have initiated
along the flanks of the expanding magmatic arc to the west of the study area during
deposition of the Aztec, and progressively encroached on the Valley of Fire into latest
Cretaceous time (Burchfiel et al., 1992). In general, Paleozoic carbonate rocks were
placed atop Mesozoic clastic rocks along a piggy-backed sequence of low angle thrust
ramps, each one accommodating up to tens of kilometers of shortening (Armstrong,
1968). The easternmost extent of Sevier tectonism, at least insofar as expressed by
exposed thrusts, reached just up to the Valley of Fire (Figure 1.3), first with the thin,
short-traveled (kilometers) Summit-Willow Tank thrust, which placed lower Aztec atop
upper Aztec and as much as 600 m of Cretaceous deposits, then with the kilometers thick,
far-traveled (tens of kilometers) Muddy Mountain thrust, which placed Paleozoic
carbonates over Aztec riding atop the Summit-Willow Tank thrust (Armstrong, 1968;
Bohannon, 1983; Brock and Engelder, 1979; Carpenter and Carpenter, 1994; Longwell,
1949). The preponderance of stratigraphic, structural and diagenetic evidence indicates
that the Aztec sandstone now exposed in the main tourist part of the Valley of Fire State
Park (the primary study area) was never buried by either thrust sheet, only by the upward-
coarsening sequence of Cretaceous clastic deposits shed from them (Bohannon, 1984;
Taylor, 1999; Eichhubl et al., 2004; Sternlof et al., 2005), which consist largely of
reworked Aztec sandstone (Bohannon, 1983).
Structures formed in the Aztec sandstone that are associated with Sevier tectonism
include the north-northwest trending, steeply east-dipping CBs that are the subject of this
paper, and low-angle shear bands that tend to parallel depositional bedding and exhibit a
top-to-the-east sense of shear which mimics the large-scale thrust faults.
16

Clark County Nevada
LVVSZ
40 km
N
Sevier Thrust Front
Las
Vegas
N
20 km
LVVSZ
114 W
36
o
N
o
AHF
MMT
Study Area
WTT
LMFS
CWF
AHF
X
114
o
30’ N
STUDY
AREA
Mead
Lake
LMFS
Figure 1.3. Schematized fault map for the Lake Mead region of southeastern Nevada.
The main map shows major strike-slip and normal faults associated with Basin and
Range extension (LVVSZ = Las Vegas Valley Shear Zone; LMFS = Lake Mead Fault
System; AHF = Arrowhead Fault; CWF = California Wash Fault). Study area box
indicates approximate location of Figure 1.1, while the X (circled in white) marks the
location of measurements made in Aztec sandstone exposed at Buffington Pockets (see
Section 4.3). The inset map indicates the approximate southeastern most extent of
Cretaceous Sevier thrusting, as dissected by the LVVSZ, LMFS and AHF. The study area
just avoided being overridden (MMT = Muddy Mountain Thrust; WTT = Willow Tank
Thrust [also the Summit Thrust]; thrust faults south of the LVVSZ are undifferentiated).
The color base image was downloaded from Google Earth. The structural data was
compiled from Stewart and Carlson (1978); Bohannon (1983b); Campagna and Aydin
(1994) and Flodin (2003).
17
36
o
30’ N

There also are relatively high-angle tabular zones composed of multiple shear bands
(Flodin and Aydin, 2004). Abundant cross-cutting relationships consistently indicate that
shear banding represents a distinct phase of deformation subsequent to CB formation,
leaving CBs as the oldest structural fabric present (Hill, 1989; Taylor and Pollard, 2000;
Myers and Aydin, 2004; Flodin and Aydin, 2004; Eichhubl et al., 2004; Sternlof et al.,
2005). The highest density of low-angle shear banding occurs in close proximity to the
Summit-Willow Tank thrust and extends into the overlying Cretaceous units exposed
beneath it, suggesting a direct genetic relationship with emplacement of the thrust sheet
(Hill, 1989; Eichhubl et al., 2004; Sternlof et al., 2005). Compaction bands, on the other
hand, are restricted to the Aztec sandstone itself (Sternlof et al., 2005).
It appears that the Valley of Fire region was not affected by compression associated
with the early Tertiary Laramide orogeny (Bohannon, 1983), although it is possible that a
distributed tectonic fabric of joints may have formed during this time, when the study
area is interpreted to have been situated on the northeastern flank of a broad, gently
north-plunging arch (Bohannon, 1984). In any case, following some 40 million years of
relative tectonic quiescence, the next major event to impact the area was Basin and Range
extension beginning 15 to 20 Ma. This largely expressed itself as nested systems of both
left and right lateral strike-slip faults (generally with some dip slip) and associated
normal faults (Bohannon 1983, 1984; Burchfiel et al., 1992; Duebendorfer et al., 1998).
The major regional-scale features are the northeast-trending strands of the left-lateral
Lake Mead fault zone and the northwest-trending strands of the right-lateral Las Vegas
Valley shear zone (Figure 1.3). Associated structures developed within the Aztec
sandstone at the Valley of Fire include joints, sheared joints and a hierarchical network of
left-lateral and right-lateral joint-based strike-slip faults (Taylor et al., 1999; Myers and
Aydin, 2004; Flodin and Aydin, 2004). These Basin and Range related structures dissect
and to some degree obscure the CBs of interest, but the effects are generally local and not
difficult to account for.
Finally, the Aztec sandstone and overlying Cretaceous deposits in the central Valley
of Fire study area exhibit a very gentle, northeast-plunging synclinal fold (Figure 1.1)
referred to as the Overton syncline by Carpenter and Carpenter (1994) and attributed to
drag along the east-west trending, left-lateral oblique-slip Arrowhead fault, which bounds
18

the south side of the valley (Figure 1.3). The fold axis currently plunges 20-30° to the
northeast, matching the general dip of the surrounding Cretaceous strata (Bohannon,
1983; Flodin, 2003; Eichhubl et al., 2004). This indicates that the folding largely predates
regional tilting.
4. Structural geology of compaction bands
Extensive, anastomosing arrays of subparallel CBs pervade the upper 600 m of the
Aztec sandstone with a dominant north-northwest trend and steep (~70°) dip to the east.
There also are common, but generally isolated outcrops exhibiting more complicated
patterns of bands, including multiple sets of CBs and abundant shear bands dissecting
them (Sternlof et al., 2004). Phenomenal exposure of the Aztec throughout the main part
of the Valley of Fire State Park permits an integrated view of the structural geology of
CBs from thin-section to regional scale (Figure 1.4).
4.1. Thin-section to outcrop scale
Viewed in thin section (Figure 1.4b), the boundaries of tabular CBs are clearly
delineated as abrupt drops in porosity directly attributable to inelastic mechanical
compaction associated with grain crushing—specifically, micro-fracture accommodated
plasticity of the quartz grains (Sternlof et al., 2005) apparent under backscatter electron
imaging (Figure 1.4c and d). The total volume change realized, as measured by the
relative volume fraction occupied by detrital grains inside versus outside the band, is
about 10% and appears to be consistent along the bands from tip to middle. Judging from
the overall continuity of depositional bedding cutting across the bands (Figure 1.4b), and
the almost complete absence of granular disaggregation within them despite intense grain
fracturing, the volume change appears to have resulted from uniaxial compaction directed
normal to the plane of the band in the absence of appreciable shear accommodated across
it (Sternlof et al., 2005).
In the otherwise weakly lithified Aztec sandstone, CBs tend to weather out in positive
relief as distinctive tabular fins, rendering them readily visible in outcrop and
accentuating the absence of appreciable shear accommodated (Figure 1.4e). The relative
resistance of CBs is due primarily to the preferential precipitation of kaolinite as a pore-
filling, grain-bridging cement (Sternlof et al., 2005). This post-compaction clay
accumulation further exacerbates porosity loss and consequent permeability
19

(e) (f)
(c)
500µm
compaction band trend
depositional bedding
N
0 1
km
(a)
(b)
~ 10 mm
compaction band
20
compaction band trend
500µm
(g)
(d)

Figure 1.4. (opposite page) Photo collage depicting compaction bands in the Aztec
sandstone. (a) Study area air photo from Figure 1.1 with arrows indicating approximate
locations where other photos were taken. (b) Photomicrograph mosaic of a compaction
band at high angle to depositional bedding, which passes through the band with no
apparent shear offset, change in direction or change in thickness (white grains are quartz,
dark grains are mostly stained orthoclase and some hematite, blue is epoxy-filled pore
space. (c) Backscatter electron image (BEI)from just outside the compaction band (light
gray is quartz, off-white is orthoclase, dark gray is kaolinite, stark white is hematite,
black is pore space). (d) BEI from inside the compaction band. Note the abrupt drop in
porosity, pore size and pore connectivity accommodated by intense damage to some
quartz grains. This damage, and attendant grain interpenetration, is predominantly due to
pervasive microfracturing. (e) Close-up view of compaction band fin capturing its
essential tabular-planar aspect, cm thickness and lack of obvious shear displacement. (f)
Typical view of subparallel, anastomosing compaction band array (resistant fins) in
outcrop. View is north-northwest along the dominant trend). (g) Cross-hatch pattern
formed by two distinct band orientations at high angle to each other. Bands belonging to
the dominant trend set (view is northward) pass through both the upper and lower crossbed
boundaries, while the secondary set of bands is constrained within the middle
bedding package.
reduction—from ~25% and ~1,500mD in the sandstone to ~10% and 1.5 mD in the bands
(Sternlof et al., 2004; Sternlof et al., 2006; Chapter 5 this thesis).
In outcrop, individual CBs appear to be grossly penny-shaped and elliptical in profile,
ranging in maximum thickness up to a few cm (averaging ~ 1 cm) and in length to more
than 100 m. They often exhibit well-defined, elliptically tapered tips, and a variety of
meter-scale patterns of near-tip interactions that are diagnostic of in-plane propagation as
anticracks (Sternlof et al., 2005; Chapter 4 this thesis). By far the dominant pattern at the
outcrop-scale, however, is an anastomosing planar fabric trending north-northwest and
dipping steeply eastward (Figure 1.4f). Spacing between adjacent bands within this fabric
can vary from 1 mm to several meters, averaging ~0.7 m (based on 286 m of orthogonal
scan lines intersecting 423 bands at 23 locations—see Section 5 below). Also distinctly
notable, although relatively rare and spatially isolated, are patterns of CBs that cross and
sometimes turn into each other at high angle (Figure 1.4g). These “cross-hatch” patterns
(Sternlof et al., 2004) are comprised of through-going bands belonging to the dominant
orientation set, and a subsidiary local set trending generally southwest, dipping steeply
northwest and spaced ~1.2 m on average (based on 73 m of scan lines intersecting 61
bands at 5 locations). Bands from these two sets generally cross-cut each other at a
21

dihedral angle of 80° or more, and so appear to have formed contemporaneously (Sternlof
et al., 2004).
4.2. Anticrack model
The highly eccentric elliptical tip-to-tip profiles and uniform uniaxial internal
compaction observed for individual CBs corresponds to a pure closing-mode sense of
relative boundary displacement distributed across a very thin band. This geometry and
anti-mode I sense of displacement discontinuity defines CBs kinematically and
mechanically as anticracks (Mollema and Antonellini, 1996; Sternlof et al., 2005). As
such, they would form symmetric with (orthogonal to) the direction of maximum
compressive stress in an otherwise homogeneous, isotropic material (Figure 1.5), except
when mechanically interacting with each other (Sternlof et al., 2005; Chapter 4 this
thesis). The anticrack interpretation suggests that the normal to the dominant planar
orientation of CBs in the Valley of Fire coincides with the direction of maximum
compressive paleostress (σ1) acting regionally when they formed. In locations where a
subsidiary set of cross-hatch CBs also formed, we suggest these reflect the orientation of
the intermediate principal paleostress (σ2), speculating that σ1 ≈ σ2 locally and that the
preferred CB orientation flip-flopped between the two (see Chapter 4, this thesis).
4.3. Outcrop to regional scale
As described in the section on deposition, the Aztec sandstone is neither truly
homogeneous nor isotropic. Rather, it is a granular material deposited in variable bedding
orientations arranged as packages within a complex æolian sedimentary architecture. We
suggest that this heterogeneous reality, coupled with apparently strong mechanical
interactions between adjacent CBs, leads to the highly variable band patterns commonly
observed at outcrop scales. As the scale of observation increases, however, the influence
of these local complications drops away. At a scale of hundreds of meters (Sternlof et al.,
2006) to kilometers, the approximately planar fabric defined by north-northwest trending,
steeply east-dipping bands dominates, suggesting distributed formation in response to,
and generally orthogonal with a paleo direction of regional compression.
Finally, while CBs can be found in upper Aztec sandstone outcrops throughout
southeastern Nevada, it is important to note that large exposures essentially devoid of
22

ands are also fairly common. Generally up to hundreds of meters in lateral extent,
although some may be larger, we speculate that these holes in an otherwise pervasive CB
fabric represent combinations of loading and mechanical properties not conducive to
band formation. Whether they are due primarily to small differences in porosity,
cementation and cohesion, disadvantageous bedding orientations, pore-pressure
variations or location within larger thrust structures remains to be deciphered. However,
abundant evidence that sedimentary architecture can strongly influence CB propagation
(Hill, 1989; Sternlof et al., 2004; Flodin and Aydin, 2004) suggests that mechanical
anisotropy related to depositional bedding, and perhaps also relative movement between
adjacent cross-bed packages that alters the local stress state, might play decisive roles.
(a)
(b)
σ 3
σ 1
x 3
x 1
x 1
Figure 1.5. Schematic of anticrack-inclusion conceptual model. (a) Axisymmetric
geometry of eccentric ellipsoidal band aligned with the principal remote stresses (σ1 > σ2
> σ3, compression positive). (b) Cross-sectional area of the band (solid ellipse) relative to
the pre-compacted area originally occupied by the same detrital grains (dashed ellipse).
Net inward movement of the boundary (u1) during uniaxial compaction corresponds to a
closing-mode (anti-mode I) sense of displacement. Their extreme eccentricity (~ 1)
allows compaction bands to be modeled as anticracks (adapted from Sternlof et al., 2005).
23
u 1
x 2
x 2
σ 2

5. Compaction band orientations
Orientations were collected for 484 CBs intersected along 28 orthogonal scan lines
totaling 359 m in length at 23 locations around the Valley of Fire. One additional set of
data (33 orientations) was collected as a control from outcrops of Aztec sandstone located
~15 km southwest of the Valley of Fire in Buffington Pockets, a window through
Paleozoics of the overriding Muddy Mountain thrust plate (Figure 1.3). All orientation
data were collected as dip azimuth/dip angle, and only deformation bands identifiable as
CBs were measured (Figure 1.6). A band was judged to be a CB if it was relatively thick
(> 5 mm toward the middle), planar and continuous, and did not show discernable shear
offset. These simple criteria were applied to avoid counting low-angle shear bands, as
well as subsidiary bands (e.g. linking structures) judged to be expressions of more
localized stress regimes.
As a point of interest, we note that bands clearly belonging to the dominant CB set do
occasionally show right-lateral offsets on the order of centimeters. These bands are
generally encountered in close proximity to much younger, but similarly north-northwest
trending dextral-slip faults (Myers and Aydin, 2004; Flodin and Aydin, 2004) and display
an unusually high degree of induration. We suggest that these bands represent Cretaceous
CBs reactivated in shear during Miocene-Pliocene faulting, possibly due both to their
favorable orientation and their potential for low shear strength as a result of the grain
damage, comminution and preferential clay accumulation accommodated. Alternatively,
one could argue that decreased porosity within these bands resulted in increased shear
modulus, causing them to concentrate subsequent fault-related stresses and, ultimately, to
fail in synthetic shear themselves. In either case, the greater induration of these bands
today is likely due to frictional heating and healing of the shattered quartz grains,
although this has yet to be confirmed with photomicroscopy. During our collection of
orientation data, these bands were included as CBs.
Figure 1.7 displays all the CB orientation data as planar poles in equal area, lower
hemispheric projection stereograms for each location. These data clearly demonstrate the
pervasive dominance of the steeply east-dipping CB set, although two other distinct
orientation groups are apparent—steeply northwest dipping and very steeply southwest
dipping. Figure 1.8 shows the stereograms for the five locations where multiple CB sets
24

Figure 1.6. Collection of orientation data in the Valley of Fire (view is to the south). The
compaction bands shown are typical of those belonging to the dominant, through-going
set in terms of their cm thickness, sub-meter spacing and planar yet anastomosing pattern.
Acquisition of quality data is made generally easy by the prominent band fins formed in
outcrop. All data were collected as dip azimuth/dip angle, with spot checks of
reproducibility indicating about ± 5° for both.
25

n = 20
M
n = 20
P
n = 22
B
n = 20
R
n = 22
Q
n = 20
S
n = 29
C
n = 20
J
n = 11
N
n = 20
K
n = 30
O
n = 23
L
26
30
n = 20
A
n = 13
U
n = 14
D
N
0 1
km
25
OS
25
n = 20
V
BF
n = 32
X
n = 20
I
n = 20
H
n = 20
W
n = 18
F
n = 20
G
n = 17
T
n = 19
E

Figure 1.7. (opposite page) Compaction band orientation data from the Valley of Fire.
Stereograms are lower hemispheric equal area projections of normals to the CB planes;
lines to the central air photo (same as in Figure 1.1) indicate the approximate location
where each set of data was collected; capital letter designations for each plot reflect the
order of collection. Three distinct orientation groupings are apparent, although only five
locations (C, B, F, L and O) exhibit more than one (see Figure 1.8). Relevant structural
data from Figure 1.1 (Overton syncline [OS], northern tip of Baseline fault [BF] and
Cretaceous bedding orientations) are as indicated. Stereogram X is from the location
marked with a circled X in Figure 1.4, an area of Aztec outcrops called Buffington
Pockets that comprise a window through the Paleozoics of the overriding Muddy
Mountain thrust.
P
n = 22
B
S
dihedral angle
P to S = 80.5
P
n = 29
C
S
T
dihedral angles
o
P to S = 80.0
o
P to T = 32.7
o
S to T = 80.0
P?
n = 18
F
S
T
dihedral angles
o
P to S = 83.1
o
P to T = 7.5
o
S to T = 83.6
P
n = 23
L
S
dihedral angle
o
P to S = 75.9
P
n = 30
O
S
T
dihedral angle
P to S = 87.7
o o
Figure 1.8. Angular relationships between compaction band sets at the five outcrop
locations exhibiting multiple orientations (see Figure 1.7). P = primary set in terms of
park-wide abundance, S = secondary set, T = tertiary set. Dihedral angles were computed
between the mean orientations for each band set. Sets P and S are at high dihedral angle
to each other at all locations, as are sets S and T. Sets P and T, however, are at variably
low dihedral angle.
27

were encountered, giving the dihedral angles between the mean orientations for each
set—labeled P (primary), S (secondary) and T (tertiary), based on relative abundance. P
and S share a dihedral angle of about 80° at all five locations, as do S and T at the two
locations where they coexist. P and T, however share a variably low dihedral angle at
these same two locations, having similar trends, but opposite dip directions.
Figure 1.9 displays the combined data for all locations (n = 484) as poles, density
contours, and Rose diagrams of both strike and dip orientations. Table 1.1 provides a
brief statistical summary of the numbers, including mean and median dihedral angles, and
relative abundance. Two basic conclusions are immediately apparent. Firstly, wherever it
occurs, the secondary CB set forms essentially orthogonal to the primary set. Secondly,
the tertiary CB group is a steeply dipping subset of the primary orientation. Although the
data were not collected to provide a statistically valid measure of relative abundance
between the CB sets, they do also reasonably illustrate the dominance of the primary set
at ~83% of all bands.
6. Tectonic interpretation
Hill (1989) first suggested that high-angle deformation bands in the Aztec sandstone
that do not exhibit macroscopic shear (CBs in current usage) resulted from tectonic
compression related to the Sevier orogeny. His primary arguments were that the bands
trend generally parallel to the encroaching thrust front and orthogonal to the east-vergent
tectonic transport direction, and that they comprise the oldest structures present based on
cross-cutting relationships. Subsequent workers (Taylor et al., 1999; Taylor and Pollard,
2000; Myers and Aydin, 2004; Flodin and Aydin, 2004; Eichhubl et al., 2004; Sternlof et
al., 2004, 2005, 2006) have all come to the same conclusion and we concur, offering a
more in-depth examination of the evidence and implications below.
6.1. Timing, spatial and material constraints
The analysis of Taylor and Pollard (2000) demonstrates that CBs were already
present in the Aztec to influence the initial upward and eastward expulsion of reducing
basinal brines from beneath the advancing Sevier thrust front, which bleached the middle
and upper parts of the sandstone as observed today (Eichhubl et al., 2004) (Figure 1.1).
Thus, CBs formed in the sandstone while it was stained uniformly red with hematite
grain coatings, suggesting that this trace (~1% by volume) cement (Flodin et al., 2003;
28

(a)
(c)
P
P
S
T
S
T
Figure 1.9. Stereograms for combined orientation data (equal area, lower hemispheric
projections). P = primary set, S = secondary set, T = tertiary set, n = 484. (a) Normals to
the band planes. (b) Density contours of planar normals (2.5% contour interval). (c) Rose
diagram of strike azimuth data. (d) Rose diagram of dip azimuth data.
Table 1.1. Summary of compaction band orientation data
Primary Set (P)
(n = 401)
P
Secondary Set (S)
(n = 61)
S
T
S
T
Tertiary Set (T)
(n = 22)
(b)
P
(d)
Control Set
(n = 33)
Azimuth
mean 81.7° 337.8° 241.2° 116.9°
median 82.0° 338.0° 245.0° 116.0°
standard dev. 13.3° 6.7° 11.5° 6.7°
Dip angle
mean 68.7° 56.9° 80.7° 61.6°
median 69.0° 62.0° 80.5° 62.0°
standard dev. 6.7° 13.6° 6.2° 4.9°
Dihedral angles mean data median data
P to S 89.4° 88.2°
P to T 36.6° 34.8°
S to T 89.6° 88.2°
P to Control 32.6° 31.6°
Rel. abundance 82.9% 12.6% 4.5% na
29

Eichhubl et al., 2004) did not lend significant cohesion to the detrital grains (Sternlof et
al., 2005) or, at least that it did not impede CB formation. Why the lower boundary of the
alteration front is located stratigraphically where it is, and why this horizon roughly
coincides with the lower extent of compaction band formation remains a significant open
question. Given that the bands predate alteration and pure coincidence is unlikely, it is
interesting to speculate, however, that the relatively newly formed CB arrays in the
Cretaceous Aztec sandstone played an active role in attracting and concentrating flow of
the expulsing brines.
As described above in Section 3, a preponderance of evidence indicates that the Aztec
sandstone in the immediate study area just avoided being overridden by the Willow Tank
and Muddy Mountain thrust sheets and that, based on diagenetic evidence, has never
been buried by much more than the combined thickness of overlying Cretaceous and
Neogene sediments (~1,600 m). Directly overlying the Aztec is the basal conglomerate of
the Willow Tank formation, interpreted to be a gravel pediment indicative of
erosional/depositional stasis. An upward coarsening, 1,300-m-thick sequence of
Cretaceous deposition followed (Figure 1.2), dominated by more than 1,100 m of the
Baseline sandstone representing aqueous redeposition of the Aztec from western upland
sources into a foreland basin (Longwell, 1949; Bohannon, 1983). Up to 600 m of these
lower Cretaceous sediments are overridden by the Willow Tank thrust, which is
associated with shear banding in the Aztec that post-dates CB formation.
Sternlof et al. (2005) conclude that no more than 600 m of fine-grained foreland basin
sediments buried the Aztec sandstone during CB formation, and likely less. We further
suggest that the evolution of this flexural foreland basin was in its earliest stages during
CB formation and that no more than 200 m of clay-rich swampland deposits of the
Willow Tank formation covered the Aztec. The end-member scenario is that the Aztec
sandstone was essentially unburied, covered only by the gravel pediment, which is
suggestive of mild uplift related to broad regional warping of the crust at the onset of the
Sevier compression (Fleck, 1970; Brock and Engelder, 1977). In any case, it appears
certain that CB formation in the Aztec occurred well before the thrust front reached the
area. This interpretation is further corroborated by the observation that CBs exposed at
Buffington Pockets (Figure 1.3) are substantially similar in geology and orientation to
30

those at the Valley of Fire, despite having been overridden by up to 5 km of the Muddy
Mountain thrust sheet (Brock and Engelder, 1977) and probably rotated in a clockwise
sense during later movement along the Las Vegas Valley shear zone and Lake Mead fault
system (Figure 1.3).
To summarize, we interpret CBs to have formed in the Aztec sandstone when it was
shallowly buried (< 600 m) and well before active east-vergent thrust faulting affected
the area. There is no evidence to suggest that the Aztec at this time was anything more
than a weakly cemented granular aggregate (albeit one with a complex depositional
architecture) topped with relatively thin (

P
(a)
(c)
S
P
S
Figure 1.10. Stereograms for restored orientation data (equal area, lower hemispheric
projections0. P = primary set, S = secondary set, n=484. All orientation data were rotated
25° counterclockwise about an axis with azimuth 315° (NW) and plunge of 0°, in order to
restore them to pre-tilting attitudes. (a) Normals to the band planes. (b) Density contours
of planar normals (2.5% contour interval). (c) Rose diagram of strike azimuth data. (d)
Rose diagram of dip azimuth data.
Table 1.2. Summary of restored compaction band orientation data
Primary Set (P)
(n = 423)
P
S
Secondary Set (S)
(n = 61)
S
(b)
(d)
P
Control Set
(n = 33)
Azimuth
mean 89.9° 317.9° 116.9°
median 91.9° 320.4° 116.0°
standard dev. 16.9° 12.0° 6.7°
Dip angle
mean 51.7° 51.2° 61.6°
median 51.2° 55.2° 62.0°
standard dev. 9.1° 12.1° 4.9°
Dihedral angles mean data median data
P to S 88.8° 86.2°
P to Control 24.5° 22.7°
Rel. abundance 87.4% 12.6% na
32

otated all of the CB orientation data 25° counterclockwise about an axis trending 315°
(northwest) and plunging 0°, in keeping with available bedding orientation data for the
Cretaceous units (Bohannon 1977, 1983). The restored data are presented graphically in
Figure 1.10 and summarized in Table 1.2.
The most obvious result of the restoration is that the 22 measurements originally
attributed to a distinct third set of CBs trending northwest and dipping steeply southwest
(also previously identified by Hill, 1989 and others) can reasonably be categorized as
falling within the periphery of the primary (dominant) orientation group. This leaves just
two sets of CBs with mean orientations that are essentially orthogonal (dihedral angle of
88.8°). On average, the dominant set trends due north and dips 52° east, while the
secondary set trends almost due northeast and dips 51°northwest. The axis of intersection
between these mean CB orientations—azimuth of 24° and plunge of 27°—represents the
approximate line of σ3, which turns out to be more nearly horizontal than vertical. The
inferred azimuth/plunge directions of all three paleostresses are: 270°/38° (σ1), 138°/39°
(σ2) and 24°/27° (σ3) (Figure 1.11a). These inferred paleostress directions represent
approximate estimates, both because σ1 and σ2 (the poles to the dominant and secondary
mean CB orientations) are not quite orthogonal (88.8°), and as a result of the scatter in
the raw orientation data (± 17° in azimuth and 12° in dip) (Table 1.2).
6.3. Geomechanical implications
The radical departure of our inferred paleostress state from the classic Andersonian
expectation that one of the principal stress components is always subvertical, σ3 in thrust-
faulting environments (Anderson, 1951) means one of three things: the reconstruction of
the paleo CB orientations is faulty and the two sets were actually subvertical as well as
orthogonal; the underlying interpretation of CBs as anticracks is incorrect and they do not
reflect the orientations of the principal paleostresses; or the analysis is correct and points
to an as yet unrecognized regional tectonic explanation. We address the two former
possibilities first.
While our first-order approach to the reconstruction is certainly approximate, it is
fundamentally sound and anything more complex is currently unwarranted given the
available data. The best way to refine the reconstruction would be to collect a high
density of bedding orientation data as stratigraphically low in the Cretaceous deposits as
33

possible, and use these to better constrain the azimuth and plunge of the tilting axis, as
well as the amount of tilting. It would also be reasonable to avoid the southeastern limb
of the Overton syncline as adding an unnecessary complication and source of potential
error.
These refinements, however, are unlikely to alter substantially the existing paleostress
assessment. Varying the tilting axis azimuth and plunge by ± 15° around the 315°/0°
orientation used produces a mild range of variation, while the amount of the tilt (~25°) is
reasonably well constrained by field measurements. In any event, no geologically
reasonable restoration can bring both sets of CBs to paleo vertical, as this would also
result in near vertical Cretaceous strata during their deposition.
Could the underlying interpretation of CBs as anticracks be flawed? The specific
mechanics of CB propagation remains an area of active research—observationally,
theoretically and experimentally—and it is possible that the bands will turn out to
propagate obliquely to the principal stresses. This could be the case so long as anti-mode
I deformation dominates such that shear displacements are on the order of the mean grain
size (0.25 mm) and thus nearly impossible to distinguish. Such a result would, of course,
alter our principal paleostress interpretation based on the CB orientations. In this regard,
it is interesting to note that, in individual outcrops containing both CB sets, the mean
dihedral angle is ~80°, and can be less (Figure 1.8). This departure from the more nearly
orthogonal result when looking at the mean band orientations could be a clue that some
degree of resolved shear is involved in propagation at the micromechanical scale. On the
other hand, 80° is closer to 90° than to 60° (the typical dihedral angle between conjugate
shear planes) and the preponderance of available data points strongly to the dominance of
uniaxial compaction (Sternlof et al., 2005; Chapter 4, this thesis). We suggest therefore
that a significantly different interpretation of paleostress direction relative to CB
orientation—such as σ1 bisecting the dihedral angle between the primary and secondary
band sets, and σ2 corresponding to the axis of intersection between them—is highly
unlikely. In any case, no plausible reinterpretation would yield principal paleostress
directions in agreement with Andersonian expectations.
This leaves the third possible interpretation—that the current analysis is substantially
correct and points to an as yet unrecognized tectonic explanation. Any such explanation
34

almost certainly involves movement along regional bounding faults, such that tractions
applied to a tectonic block containing the Aztec sandstone resulted in the orientation of
paleostress inferred from the CBs. Such a configuration is difficult to visualize in 3-D.
A highly schematized 2-D treatment that ignores the secondary CB set, however, serves
to illustrate the essential concept (Figure 1.11b).
For this simple model, assuming homogeneous, isotropic, linear elasticity and using
the finite element method software Abaqus®, we consider a vertical, east-west cross
section through the Mesozoic units of the study area as a block sliced orthogonally to the
mean orientation of the dominant CB set and sitting atop a horizontal detachment at the
top of the Hermit shale. If this 3.6-km-thick block is moving stably westward along the
detachment (i.e. being pushed quasi-statically), the trajectories of σ1 induced a few km in
from its eastern edge are as shown by the tick marks in Figure 1.11b. Within the zone
indicated, 200 to 800 m deep and 5 to 9 km from the eastern edge of the block, which
coincides with a lateral section of the CB-rich upper Aztec, these trajectories roughly
coincide with our inferred orientation of σ1 that produced east dipping CBs. This
rudimentary treatment is not well constrained, but it does demonstrate that the inferred
orientation of σ1 can be produced over a broad, km-scale region in a geologically realistic
way. In fact, any variety of geologically reasonable configurations of mechanical
stratigraphy, and detachment and lateral bounding faults can be expected to produce the
inferred paleostress orientations. Nonetheless, even the simple 2-D model presented here
provides two relatively robust predictions testable by observation. Does the dip of the
dominant band set systematically decrease with increasing depth and increase from west
to east? And is there any independent evidence for westward motion of the study area
along a relatively shallow detachment?
On this last point, we note the interpretation of Carpenter and Carpenter (1994) that
west-vergent reverse faulting along what they called the North Buffington back-thrust
system comprised the earliest expression of Sevier tectonism in southeastern Nevada,
predating the formation of the dominant east-vergent thrusts (Muddy Mountain, Willow
Tank, etc.) now exposed in the study area.
35

1.5(ρgz)
(b)
WEST
Willow Tank
Upper Aztec
(zone of interest)
Lower Aztec
Triassic strata
(a)
σ1
~ 270/38
traction-free ground surface
detachment
0 m
200 m
800 m
3,600 m
36 km 9 km
5 km 0 km
ρgz
σ3
~ 024/27
σ2
~ 138/39
z
EAST
2(ρgz)
Figure 1.11. Paleostress orientations and tectonic interpretation for the Aztec sandstone
during compaction band formation. (a) Equal area, lower hemisphere stereographic
projection of the approximate inferred principal paleostress orientations (σ1 > σ2 > σ3,
compression positive). (b) Simplified 2-D linear elastic, finite element model (using
Abaqus®) of one tectonic scenario that could produce the inferred orientation of the
maximum compressive paleostress (σ1). In this formulation, we consider a homogeneous,
isotropic, linear elastic block representing an east-west cross section through the
Mesozoic strata of the study area, which is orthogonal to the mean orientation of the
dominant CB set. This block, the hanging wall of a horizontal thrust, is subjected to
gravitational forces and unbalanced lateral forces that push it quasi-statically westward
along a rigid detachment, which resists the motion by generating opposing shear tractions.
The resulting trajectories of σ1 induced a few km from the eastern edge of the block and
within the CB-genic zone of the upper Aztec (tick marks) roughly coincide with the
inferred orientation (z is depth, g is gravity, ρ is density, each grid block is 200 m by
1,000 m).
36

7. Concluding observations
The data and analyses presented above indicate that CBs formed as anticracks in the
Aztec while it comprised a shallowly buried, weakly lithified sandstone aquifer subjected
to regional compression associated with the earliest phases of Sevier tectonism in the area.
Interpreted as such, the presence of two orthogonal, coeval and coexisting sets of CBs
can be used to constrain the 3-D paleostress state in which they formed, with the
dominant set orthogonal to the maximum compressive stress (σ1) and the secondary set
orthogonal to intermediate compressive stress (σ2). Furthermore, the mutually cross-
cutting presence of both sets in the same outcrop suggests that σ1 ≈ σ2. That the inferred
paleostress orientations depart radically from classical Andersonian expectations can
reasonably be interpreted as evidence of as yet unrecognized regional structures,
specifically a relatively shallow detachment along which a coherent structural block
containing the Aztec now exposed in the Valley of Fire moved relatively westward.
Based on an average spacing between CBs of the dominant, north-south trending set
of 0.7 m, and an average band thickness of 1 cm representing 10% uniaxial compaction
(~1.1 mm), the Aztec sandstone experienced less than 0.2% total shortening as the result
of CB formation. Even assuming an average band spacing of only 10 cm hardly bumps
the shortening past 1%. If homogeneously distributed, dropping overall porosity from
25% to 24%, such a mild deformation event would be of no consequence to fluid flow.
The highly localized, yet pervasively interconnected nature of the CB fabric formed,
however, constitutes a considerable network of low-permeability baffles to fluid flow
(CB permeability ~0.1% of the inherent sandstone permeability) that has been shown
capable of exerting profound effects at scales of practical interest (Sternlof et al., 2004;
Sternlof et al., 2006).
The analysis presented here of compaction bands as indicators of paleostress in and
tectonic structure around the exhumed analog aquifer/reservoir that is the Aztec
sandstone can also be applied in reverse. That is, given independent knowledge of the
tectonic stress and material history of a sandstone similar to the Aztec, reasonable
forecasts of the possible presence and gross geometry of CBs in the subsurface can be
made. Such insights could prove highly valuable in the efficient development aquifers
and reservoirs.
37

9. Acknowledgements
My sincere thanks go to John Childs for his able and entertaining assistance in the
field, and to David Pollard for many thought-provoking discussions. Thanks also to the
Valley of Fire State Park staff for their indulgence. This work was funded by the U.S.
Department of Energy, Office of Basic Energy Science, Geosciences Research Program
under grant DE-FG03-94ER14462, awarded to Pollard and Atilla Aydin as principal
investigators. Additional support was provided by Stanford Rock Fracture Project.
38

1. Abstract
Chapter 2
Anticrack-inclusion model for compaction bands in sandstone
Detailed observations of compaction bands exposed in the Aztec sandstone of
southeastern Nevada indicate that these thin, tabular, bounded features of localized
porosity loss initiated at pervasive grain-scale flaws, which collapsed in response to
compressive tectonic loading. From many of these Griffith-type flaws, an apparently self-
sustaining progression of collapse propagated outward to form bands of compacted grains
a few centimeters thick and tens of meters in planar extent. These compaction bands can
be idealized as highly eccentric ellipsoidal bodies that have accommodated uniform
uniaxial plastic strain parallel to their short dimension within a surrounding elastic
material. They thus can be represented mechanically as contractile Eshelby inclusions,
which generate near-tip compressive stress concentrations consistent with self-sustaining,
in-plane propagation. The combination of extreme aspect ratio (>10 -4 ) and significant
uniaxial plastic strain (~10%) also justifies an approximation of the bands as anticracks—
sharp boundaries across which a continuous distribution of closing-mode displacement
discontinuity has been accommodated. This anticrack interpretation of compaction bands
is analogous to that of pressure solution surfaces, except that porosity loss takes the place
of material dissolution. We find that displacement discontinuity boundary element
method modeling of compaction bands as anticracks within a two-dimensional linear
elastic continuum can accurately represent the perturbed external stress fields they
induce.
2. Introduction
Compaction bands represent one kinematic end member of the suite of structures
known collectively as deformation bands, which also includes shear and dilation bands
(Antonellini et al., 1994; Aydin, 1978; Du Bernard, 2002; Mollema and Antonellini,
1996). Due to the internal loss of porosity and permeability they accommodate,
deformation bands can significantly affect bulk sandstone permeability when present as
pervasive arrays, with important implications for the management of both groundwater
and hydrocarbon resources (Sternlof et al., 2004).
39

The term compaction band (CB) was coined by Mollema and Antonellini (1996) to
describe “…tabular zones of localized deformation…that accommodate pure
compaction” in sandstone, and which constitute “…analogs for anticracks…such as
pressure solution surfaces,” following the analysis of Fletcher and Pollard (1981).
Mollema and Antonellini described two types of compactions bands—straight, thick
bands and wavy, or crooked bands—both exhibiting porosity loss in the absence of shear
accommodated by granular rearrangement, grain cracking, indentation and limited
comminution. They identified these structures as forming primarily within the
compressional quadrants of deformation band-style faults (Aydin, 1978; Aydin and
Johnson, 1978) in the Jurassic æolian Navajo sandstone of the Kaibab Monocline in
Utah, finding them to be on the order of meters in trace length, segmented, and restricted
to a limited outcrop area on the downthrown limb of the monocline in relatively coarse
grained, high porosity layers.
Extensive arrays of CBs pervade the upper half of the Jurassic Aztec sandstone
exposed in and around the Valley of Fire State Park of southeastern Nevada (Figure 2.1)
where, however, they bear no immediate genetic relationship to local faults and are tens
to more than one hundred of meters in trace length (Sternlof et al., 2004). The Aztec is a
1,400-m-thick æolian sandstone composed of greater than 90% detrital quartz and
typified by large-scale tabular and trough cross-bedding, average porosity of about 20%,
and a mean grain diameter of about 0.25 mm within a range of about 0.1 mm to 0.5 mm
(Flodin et al., 2005). Throughout the weakly lithified upper Aztec, two distinct phases of
deformation bands are present. Steeply dipping, cm-thick CBs of the first phase comprise
the oldest structural fabric. The CBs were crosscut and offset by relatively low-angle
shear bands of the second phase. Overprinting by joints and the evolution of joint-based
strike-slip faults associated with Basin and Range extension followed (Eichhubl et al.,
2004; Flodin and Aydin, 2004; Hill, 1989; Myers and Aydin, 2004; Sternlof et al., 2004;
Taylor et al., 1999). Noting their relative age and dominant orientation—north-northwest
trend and steep dip—Hill (1989) first suggested that the CBs formed in response to east-
northeast-directed regional compression associated with the Cretaceous Sevier orogeny.
Building on this earlier work, we present a conceptual and mechanical model of CBs
in porous sandstone as anticrack-inclusions [Sternlof and Pollard, 2001; Sternlof and
40

NV UT
CA
AZ
Park Road
Route 169
0 km 5
Map
Detail
Park Boundary
*
Silica Dome
Park Office
NEVADA
ARIZONA
10 km
Valley
of Fire
State Park
N
LAKE
MEAD
115 o 30'
Figure 2.1. Location of the Valley of Fire State Park, southeastern Nevada (inset), where
more than 20 square kilometers of the 1,400-m-thick æolian Jurassic Aztec Sandtone are
extensively exposed. Photo shows view northward from the location marked * on the
park detail. Throughout the upper half of the Aztec, compaction bands crop out in
positive relief as sub-parallel, centimeter-thick, north-northwest-trending, steeply eastdipping
tabular fins spaced from centimeters to meters apart.
41
36 o 15'

Pollard, 2002). The particular utility of this hybrid interpretation lies in combining the
analytical insights of Eshelby inclusion theory (Eshelby, 1957; Eshelby, 1959) with the
power of boundary element method (BEM) simulations (Crouch and Starfield, 1983) to
understand the mechanics of CB evolution. Starting with targeted field and petrographic
observations of CBs from the Aztec sandstone in the Valley of Fire, we build a
mechanical analysis of these structures as Eshelby-type inclusions to establish the
validity and utility of a two-dimensional anticrack treatment of CBs using a displacement
discontinuity BEM.
The basic hypothesis underlying this research is that a fundamental mechanism of
incremental, quasi-static and (anti)crack-like propagation was at work throughout the
Aztec such that individual CBs tended to form orthogonal to the most compressive
remote stress. The diversity of outcrop patterns observed therefore reflects the
interactions of CBs with local material and stress heterogeneities through this
fundamental propagation mechanism. Beyond contributing to the understanding of an
interesting and generally overlooked mode of structural failure in porous, granular earth
materials, this research is motivated by the practical goal of using mechanical
understanding of the phenomenon to forecast its occurrence and associated fluid-flow
effects in active reservoirs and aquifers. Although the research presented here is built on
detailed CB data acquired from a specific outcrop of Aztec sandstone in the Valley of
Fire, we suggest that the intrinsic attributes identified are common to all CBs in the Aztec
and, by extension, to other porous sandstones with similar depositional and tectonic
histories.
3. Methods
We focused on a “type-locality” outcrop of CBs in Aztec sandstone near Silica Dome
in the Valley of Fire State Park (Figure 2.1, inset) where elementary patterns of
approximately parallel bands are well exposed. Using a steel tape and calipers, we
measured tip-to-tip thickness profiles of the two-dimensional band traces in outcrop
(Figure 2.2), focusing on one example as representing the prototypical compaction band.
Eleven cores were collected along the trace of one CB from tip to middle, as well as
from essentially undisturbed sandstone nearby. These were vacuum-impregnated with
optical-grade blue epoxy and used to make 37 standard and eight polished thin sections.
42

Figure 2.2. Left-hand photo shows an outcrop of widely spaced, relatively planar and
parallel compaction bands along the northeast flank of Silica Dome. Arrows indicate
opposite tips of a single band 62 m long and up to 15 mm thick (illusory gaps in band
continuity are due to outcrop topography and breaks in the telltale fin). A total of 16 tipto-tip
thickness profiles were measured using a steel tape and calipers. Right-hand photo
illustrates that, even when closely spaced, compaction bands in this locale tend to remain
planar (arrow indicates tip).
43

Hue-based image analysis using MATLAB® was performed on more than 3,500 color
pictures taken from the standard sections with a digital camera mounted on a
petrographic microscope. Intensity-based image analysis was performed on over 200
grayscale digital pictures collected with a scanning electron microscope (SEM). The
particular SEM techniques used included backscatter electron imaging (BEI) and
cathodoluminescence.
The data acquired from the field and petrographic methods were used to inform three
mechanical assessments of the idealized anticrack-inclusion conceptual model.
Comparisons of the near-tip states of stress calculated with each of these continuum
approaches constitute the heart of this paper, in which we use the convention of
compression, closing-mode displacement discontinuity and volume-reduction strain as
positive.
The first approach considers the special case of an oblate ellipsoidal heterogeneity in
the limit as its aspect ratio goes to zero, using the embedded layer analysis of Cocco and
Rice (2002) to construct algebraic expressions for the state of stress immediately ahead of
the tip. The second approach employs an exact, closed-form solution to the general three-
dimensional Eshelby problem (Eshelby, 1957; Eshelby, 1959; Mura, 1987) for the oblate
ellipsoidal heterogeneity to calculate how the state of stress varies with distance from a
model CB matching the idealized physical parameters of the field-based conceptual
model. Finally, we turn to a numerical, two-dimensional BEM code in MATLAB® to
investigate the state of stress induced by an anticrack representation of the CB using only
the trace length and effective closing-mode displacement data.
4. Field and petrographic analysis
4.1. Geological setting and paleostress state
The Aztec sandstone, deposited during early to middle Jurassic time in a back-arc
basin setting, comprises the western edge of a continuous æolian system that blanketed
much of the intermountain southwest and includes the Navajo and Nugget sandstones
(Blakey, 1989; Marzolf, 1983). In the Valley of Fire area, the 1,400-m-thick Aztec is
directly and unconformably overlain by some 1,300 m of Cretaceous sediments, which
comprise a generally upward-coarsening sequence of foreland basin deposits derived
44

from Aztec highlands riding atop the two closest Sevier thrust sheets to the west. The
smaller and easternmost of these—the Willow Tank thrust—placed lower Aztec on upper
Aztec and as much as the first 600 m of Cretaceous deposits. The Willow Tank thrust
sheet subsequently was overridden by the regionally extensive, kilometers-thick Muddy
Mountain thrust, which placed Palezoic carbonates over Aztec and provided the source
for the upper Cretaceous units deposited atop the eastern reaches of the lower thrust sheet
(Armstrong, 1968; Bohannon, 1983; Brock and Engleder, 1979; Carpenter and Carpenter,
1994; Longwell, 1949).
Silica Dome is situated approximately 800 m above the bottom of the Aztec, at the
lower end of the CB-rich upper half of the formation and just above a regional, sub-
horizontal alteration front separating uniformly red, hematite-stained sandstone below
from bleached sandstone above (Taylor and Pollard, 2000). The hematite staining is
interpreted as syndepositional, while CBs are observed to be older than the bleaching
front, which represents the lower extent of an alteration event attributed to the upward
and eastward expulsion of reducing basinal brines by the advancing Muddy Mountain
thrust sheet (Eichhubl et al., 2004). This fluid-flow event, along with at least one
subsequent episode, is responsible for the vividly colorful patterns of iron mineralization
for which the Valley of Fire is justly named. Eichhubl et al. (2004) cite a preponderance
of evidence to argue that thrust emplacement stopped just west (Muddy Mountain thrust)
and northwest (Willow Tank thrust) of more than 20 km 2 of Aztec outcrops, which now
comprise the heart of the Valley of Fire and include the study area. They also present
diagenetic evidence from the base of the Aztec indicating that it has never been buried
much deeper than 3 km, a depth which matches the current combined thickness of Aztec
and Cretaceous to lower Miocene deposits preserved in the area.
Vertical loading of the Aztec during CB formation in the area thus appears to have
been due solely to synorogenic deposition. A strong constraint on the upper limit of
burial can be deduced from the relative timing of shear band formation. Everywhere they
occur in the Aztec sandstone, relatively low-angle shear bands of the second phase of
deformation offset and thus postdate CBs (Flodin and Aydin, 2004a; Hill, 1989; Taylor
and Pollard, 2000). The dominant orientation and consistent top-to-the-east sense of shear
displacement on these bands mimics that of the overlying Willow Tank thrust, suggesting
45

a direct genetic relationship (Hill, 1989). In fact, we observe that phase-two shear bands
extend into the Cretaceous deposits exposed directly beneath the thrust, while phase-one
CBs are confined to the Aztec sandstone and all types of deformation bands are absent
higher in the Cretaceous section. From these observations we infer that CB formation
predated deposition of at least the upper 700 m of Cretaceous sediments.
We conclude that the depth of Silica Dome at the time of CB formation ranged from
600 m (thickness of overlying Aztec sandstone) to 1,200 m (Aztec plus up to 600 m of
additional Cretaceous deposits). This corresponds to a range in vertical compressive
stress from about 13 MPa to 27 MPa, given an average overburden density of 2.25 g/cm 3 .
Assuming that well-drained, near-surface water-table conditions prevailed, as suggested
by geologic, diagenetic and paleoclimatic evidence (Eichhubl et al., 2004; Marzolf,
1983), pore pressure would have ranged from less than 6 MPa up to 12 MPa.
Calculations based on Coulomb critical failure criteria (Zoback and Healy, 1984) and
born out by a catalog of insitu stress measurements (Townend and Zoback, 2000) indicate
that the ratio of maximum to minimum regional principal stress does not exceed about
two under hydrostatic pore pressure conditions. Therefore, taking the Andersonian view
(Anderson, 1951) that the minimum principal stress is vertical in thrust-faulting
environments, both the maximum remote horizontal (tectonic) stress and the minimum
horizontal stress could have ranged anywhere from 13 MPa up to 54 MPa.
For the purposes of the mechanical analyses, we posit a “best estimate” paleo stress
state (σ1 > σ2 > σ3) for Silica Dome of 40 MPa, 30 MPa and 20 MPa. The value of σ3
corresponds to an intermediate burial depth of 900 m (600 m of Aztec plus 300 m
overlying Cretaceous deposits). The value of σ1 corresponds to the implied limit of 2σ3
for active thrust-fault environments, and is presumed to have acted parallel to the east-
vergent tectonic transport direction and perpendicular to the dominant northerly CB
trend. The value of σ2 is simply taken as the average of σ1 and σ3. The mean (lithostatic)
stress for this scenario is 30 MPa.
4.2. Outcrop analysis
Viewed individually at Silica Dome and elsewhere, CBs are seen to comprise sharply
delineated tabular, bounded, and grossly penny-shaped structures that tend to weather out
in positive relief as distinctive fins. They are generally between 1 cm and 2 cm thick in
46

the middle and tens of meters in planar extent. Spacing between CBs ranges from
centimeters to more than a meter (Figure 2.1). Truly isolated CBs are rare and close
interactions between adjacent bands are common. Frequently though, as at Silica Dome,
adjacent CB traces remain markedly straight and parallel even in close proximity (Figure
2.2). The appearance of CBs in outcrop, individually and in aggregate, is reminiscent of
distributed opening-mode fractures, and CBs might easily be mistaken for veins or sand
dikes by the casual observer. Closer inspection, however, reveals that they are composed
of the same detrital material as the surrounding sandstone. In fact, depositional bedding is
commonly preserved across the bands with no detectable shear offset (Figure 2.3).
Sedimentary architecture in the Aztec is observed to affect CB distribution (Eichhubl
et al., 2004; Hill, 1989; Sternlof et al., 2004). Bands and patterns of bands frequently
warp and/or terminate at or near cross-bed boundaries, and almost always terminate at
major inter-dune contacts. Also, while many if not most dune packages contain abundant
CBs, some do not.
Thickness profiles of 16 tip-to-tip individual CB traces ranging from one to 62 meters
long in outcrop reveal a characteristic, approximately elliptical shape (Figure 2.4a).
While midpoint thickness for these profiles generally increases with increasing trace
length, it does so along a decreasing trend that suggests a plateau of maximum attainable
thickness independent of trace length (Figure 2.4b). In order to capture the typical
characteristics of a well-developed CB, we focused on one particularly isolated and
symmetric outcrop trace of sufficient length and thickness that it could reasonably be
assumed to represent a sub-horizontal profile through the middle of the structure. This
CB trace, designated CB-A (Figure 2.4c), is 24.75 m in length with a maximum midpoint
thickness of 11.4 mm, yielding an aspect ratio (midpoint thickness/trace length) of 4.6 x
10 -4 and an eccentricity of 0.99. Represented as an elliptical profile fit by the least
squares method (correlation index = 0.87), the midpoint thickness drops to 9.38 mm,
thereby slightly decreasing the aspect ratio and increasing the eccentricity (Figure 2.4c).
While this dimensional data was gathered on relatively low-angle outcrop faces, steep
cliffs in the vicinity of Silica Dome and elsewhere in the Valley of Fire reveal that CBs
can extend at least 10 m vertically. Coupled with the fact that they occur throughout a
700-m-thick section of the Aztec sandstone, this observation supports what intuition
47

Compaction band fin
Depositional
bedding
Cross-bed boundary
Figure 2.3. Close-up of a typical, well-developed compaction band fin in outcrop. Note
that depositional bedding extends relatively undisturbed across the band, and is clearly
visible on the fin.
48

Normalized thickness
1.4
1.2
1.0
0.8
0.6
0.4
0.2
0.0
-1.0 -0.5 0.0 0.5 1.0
(a) Distance from center normalized by trace half-length
Midpoint thickness (mm)
(b)
Thickness (mm)
16
14
12
10
8
6
4
2
0
0 5 10 15 20 25
Trace half-length (m)
30 35
14
12
10
8
6
4
2
0
(c)
-1 -0.5 0 0.5
Distance from center normalized by trace half-length
1
Figure 2.4. (a) Thickness profile data for 16 tip-to-tip compaction band traces (>1,700
measurements). Position is normalized relative to center by the band half length and
thickness is normalized by the midpoint value. The curved line represents an ellipse
similarly normalized. The correlation index of the data to the ellipse is 0.79. (b) Plot of
midpoint thickness versus trace half-length for the 16 compaction band profiles, star
indicates CB shown in (c). Thickness generally increases with increasing length, but
along a decreasing trend. (c) Thickness profile for CB-A, the 24.75-m-long compaction
band trace featured in this paper. The curved line represents the least-squares best fit of
an ellipse to the data. The correlation index for this fit is 0.87.
49

suggests—that to first approximation CBs are grossly penny shaped. Certainly, thickness
is generally several orders of magnitude less than any in-plane, tip-to-tip dimension.
4.3. Petrographic analysis
Viewed in standard thin section, CB-A is every bit as sharply defined by its decreased
porosity as it is in outcrop by its characteristic fin, and thickness measurements made
under the microscope closely match those made with calipers in the field. Particularly
striking is the absence of any well-developed structural fabric within the band, and the
degree to which depositional bedding is preserved within and across it (Figure 2.5). In
fact, nowhere did we find appreciable evidence of bedding offset across the band, despite
considerable natural variability in the relative orientation of bedding to band between
sample locations. Neither did we find any conclusive evidence of systematic bedding
thickness variations within the band. Together these observations indicate that the
inelastic strain accommodated within CB-A as porosity-loss compaction was
predominantly uniaxial and directed parallel to its shortest dimension (i.e. perpendicular
to its trace).
Despite the coarseness inherent to porosity measurements, particularly using two-
dimensional image analysis of granular materials, profiles across thicker parts of CB-A
clearly reveal the abrupt drop in porosity that defines the band (Figure 2.6a). Mean
porosity in the sandstone outside the band was measured to be 24.5%, with a standard
deviation of 2.9%. Mean porosity inside the band was measured to be 11.9%, with a
standard deviation of 3.0%. Toward the tip, the profiles indicate the absence of any
obvious near-tip process zone recognizable as spatial variations in porosity (Figure 2.6b).
Backscatter electron images (BEI) provide much greater definition than the standard
color digital images. In particular, micro-porosity due to grain damage can be
distinguished and clay, which absorbs epoxy and so looks blue in standard sections, can
be differentiated (Figure 2.7). The clay occurs primarily as undeformed grain-bridging
and pore-filling cements, both inside and outside the CB. Clay also frequently in-fills
cracked grains. These observations indicate that clay should be ignored when determining
the distribution of porosity related to mechanical compaction. Image analysis reveals that
the Aztec averages about 4% clay by volume, while thicker parts of the band and the
sandstone immediately adjacent to it average about 7%. This preferential accumulation of
50

5 mm
Figure 2.5. Composite photomicrograph of compaction band sampled 6.0 meters from
the tip, where it is about 9 mm thick (dotted black lines). Blue indicates epoxy-filled pore
space (~ 25% outside the band, ~ 10% inside). The plane of the section is vertical and
orthogonal to the trace of the band in outcrop. This is a mosaic of images taken with
plane-polarized transmitted light at 25X magnification using a digital camera mounted to
a binocular microscope. Mineral constituents identifiable in this image are: detrital quartz
(white to gray), detrital orthoclase (stained dark brown), and sparse hematite (black). As
is typical in foreset dune deposits, bedding generally consists of alternating layers of
coarser and finer grains, here accentuated by the relative concentration of stained
orthoclase in the finer-grained beds. Note that bedding is preserved, perhaps even
visually enhanced within the band. The three black arrows indicate a coarse-fine-coarse
bedding package as it passes through the band at high angle with no apparent change in
thickness. The preservation of bed thickness is observed consistently in all thin sections,
including those taken across the band and orthogonal to the orientation shown above at
four locations. This indicates that compaction strain within the band is predominantly
uniaxial and oriented parallel to the band normal direction. Natural, mm-scale variations
in bed thickness render more exact interpretations of bed distortion within the band
inconclusive.
51

Porosity
Porosity
0.3
0.25
0.2
0.15
0.1
0.05
0
0 5 10 15 20 25 30
(a)
0.3
0.25
0.2
0.15
0.1
0.05
Distance along profile (mm)
0
0 5 10 15 20 25 30
(b)
Distance along profile (mm)
Figure 2.6. Porosity profiles taken orthogonally across the compaction band 8.7 m from
the tip (a) and at the tip (b). Each box plot represents a composite of 10 porosity
measurements from immediately adjacent transects. The horizontal lines comprising the
boxes indicate lower quartile, median and upper quartile values of the group. The
whiskers extend up to 1.5 times the interquartile range to indicate the spread of additional
data points. Outliers are indicated by “+”. A box plot representing the range of
background matrix porosity (minus outliers) appears to the right of each profile. At its
middle (a), the band is clearly defined as a sharp, statistically significant drop in porosity.
At its tip (b), however, while still visually apparent at a fraction of a millimeter thick, the
band can no longer be detected as a measurable reduction in porosity.
52

Middle
500 µm
Band trend Tip
Figure 2.7. Compositional electron backscatter images of a thick part of the compaction
band (top), adjacent sandstone less than 2 mm outside the band (middle), and sandstone
more than 1 m away from the nearest band (bottom). Porosity is black, quartz is medium
gray, feldspar is white, and clay (kaolinite) shows as clusters of dark gray speckles
around the detrital grains. Note the dominant texture of micro-fracture accommodated
plasticity inside the compaction band versus the relative paucity of grain damage
everywhere outside, even immediately adjacent to the band.
53

pore-clogging clay—due presumably to groundwater filtering by the compacted band
coupled with mechanically enhanced feldspar degradation—at least partially explains the
greater resistance to weathering exhibited by CBs, as well as their strong impact on fluid
flow. That clay remains the primary cement in the weakly indurated Aztec today further
suggests that cementation was light to nonexistent during CB formation.
Six sets of backscatter images were collected at locations ranging from the middle to
the tip of CB-A. Each set comprises images taken from inside and immediately outside
the band within the same depositional layer(s). Porosity-reduction volume loss related to
CB formation at these locations ranged from 9.4% to 11.7% (Figure 2.8). These results
suggest that mechanical compaction throughout the band is uniform at about 10%,
regardless of the local variations in sandstone porosity through which it propagated.
Cathodoluminescence imaging indicates that redistribution of quartz via pressure solution
is volumetrically insignificant and has not appreciably contributed to the spatial
distribution of porosity now observed. This is not surprising given the shallow burial
history of the study area (< 2 km), which, for any reasonable estimate of geothermal
gradient, would have made quartz pressure solution an extremely slow to nonexistent
process.
Cracking of quartz grains at point contacts, due both to overburden and tectonic
loading, is pervasive throughout the thin sections. Wholesale micro-fracture-
accommodated plastic deformation of quartz grains, however, comprises the dominant
micro-structural characteristic of the band, and is the obvious mechanism by which
granular rearrangement and porosity loss compaction was accommodated (Figure 2.9).
Roughly half of the quartz grains within the band exhibit plasticity, creating the hint of a
mild shape fabric of deformed grains elongated parallel to the band trace (i.e. orthogonal
to the inferred direction of maximum compression). Far more striking is the relative
absence of granular disaggregation, despite often intense micro-fracturing and distortion.
Taken together, these observations corroborate the field interpretation of the band as
representing nearly pure uniaxial compaction with little or no shear, and suggest a stable,
quasi-static process of plastic collapse.
Isolated examples of grain plasticity and collapse can also be found outside the band
(Figure 2.10). We interpret these as incipient CBs, representing an early stage of
54

0.30
0.25
0.20
0.15
0.10
Porosity 0.35
0.05
0.00
0.1 0.3 0.6 1.8 4.2 8.7
Distance from tip (m)
Figure 2.8. Clustered bar chart showing the distribution of porosity inside (middle bar)
and immediately outside (left bar) the compaction band with distance from the tip (clay
included as porosity. The difference between (right bar) gives the relative volume strain
attributable to mechanical compaction at each location. This compaction strain is
remarkably uniform throughout the band at about 10%.
Figure 2.10. Electron backscatter image of localized quartz-grain plasticity and collapse
(interlocked grains at center of image) that is not part of a well-developed compaction
band. Similar grain-scale features appear scattered throughout the sandstone and can be
interpreted as incipient compaction bands. Black is porosity, medium gray is quartz,
white is feldspar.
55

500 µm
Figure 2.9. Electron backscatter images of microfracture-accommodated quartz grain
plasticity and interpenetration, which comprises the dominant mode of grain-scale
deformation inside the compaction band (white box in upper image indicates area of
detail shown in lower image). Despite intense grain damage and the absence of
appreciable quartz recrystallization, very little disaggregation of detrital grains is
apparent.
56

compaction around an original Griffith-type flaw—weak grain, irregular pore, etc.—
which for any variety of reasons failed to develop further. Similar textural evidence of
incipient grain damage and collapse immediately around the band and at the tip, however,
is notably absent. Opening-mode fractures and faults typically exhibit a process zone of
transitional deformation that grades into the surrounding undamaged host material,
particularly around the tip where induced stress concentrations are highest (Hoagland et
al., 1973; Peck et al., 1985). Despite considerable effort, we have as yet been unable
positively to identify any analogous process zone around the band, expressed either as
grain damage or porosity variations.
5. Anticrack-inclusion conceptual model
Based on the observations, data and constraints presented above, we propose an
idealized conceptual model of isolated compaction bands in porous sandstone as highly
eccentric, roughly axisymmetric ellipsoidal features of sharply defined inelastic porosity-
loss compaction, which form in response to and generally perpendicular to the maximum
remote compressive tectonic stress (Figure 2.11a). We further suggest that CBs initiate
with the collapse of Griffith-type grain-scale flaws—weak grains, irregular pores, etc.—
and then propagate outward to form tabular inclusions of nearly uniform uniaxial plastic
strain and differing elastic moduli embedded within a surrounding elastic medium. This
conceptual model is well represented mechanically as a contractile Eshelby inclusion
(Figure 2.11b), which would generate near-tip compressive stress concentrations in the
surrounding material consistent with self-sustaining in-plane propagation. Inside any
ellipsoidal inclusion embedded within an infinite medium, the state of stress and strain is
uniform (Eshelby, 1957).
The combination of extreme aspect ratio (~10 -4 ) with significant uniaxial plastic
strain (0.1) further suggests a close mechanical approximation of the idealized CB as an
anticrack—that is, a sharp boundary across which a continuous distribution of closing-
mode displacement discontinuity has occurred (Figure 2.11c). This virtual anticrack
interpretation for compaction bands is analogous to that of pressure solution surfaces
(Fletcher and Pollard, 1981; Mollema and Antonellini, 1996), except that porosity loss
takes the place of material dissolution and transport. In neither case does physical
57

(a)
(b)
σ 3
σ 1
x 3
x 1
x 1
(c) u 1
Figure 2.11. Schematic representations of the idealized compaction band model. (a)
Axisymmetric geometry of the eccentric ellipsoidal band aligned with the principal
remote stresses. (b) Cross-sectional area of the band (solid ellipse) relative to the precompacted
area originally occupied by the same detrital grains (dashed ellipse). As
inelastic compaction progresses, the boundary around the grains involved contracts as
indicated by the displacement arrows (u1). This inward displacement of the elliptical
boundary corresponds to the uniform uniaxial plastic strain of an Eshelby inclusion, and
the area between the dashed and solid ellipses corresponds to the volume loss associated
with the compaction. (c) Two-dimensional anticrack representation of the model band as
an elliptical distribution of closing-mode displacement discontinuity equivalent to the
uniform Eshelby compaction strain. In this virtual treatment, two material lines (solid
lines) interpenetrate by an amount equivalent to the volume loss associated with the
compaction (dashed ellipse) as shown by the displacement arrows (u1). Actual
interpenetration does not occur. Because of its extreme eccentricity, however, solutions
for the state of stress induced around the model band using the Eshelby and anticrack
approaches are substantially similar.
58
u 1
x 2
x 2
u 1
σ 2

interpenetration occur, rather the boundary demarcating volume loss migrates outward
and into the surrounding material.
Central to the anticrack-inclusion concept, and the mechanical analyses to follow, is
the prescription of the model CB as an isolated feature of nonlinear inelastic strain that
evolves quasi-statically in an infinite, homogeneous, isotropic, linear elastic continuum
subject to uniform remote loading. All aspects of this model prescription warrant
comment. Firstly, given the regional tectonic nature of the compression and a study area
situated in the midst of a 1,400-m-thick sandstone deposit, the approximation of uniform
remote stresses applied to an infinite material is reasonable. Given a ratio of trace length
to spacing generally less than 0.1, it would be difficult to argue that any CB in the Aztec
sandstone is truly isolated in a mechanical sense. Nonetheless, the specific field data on
which the conceptual model is based comes from planar bands that betray little reaction
to their nearest neighbors.
As with any granular material, the applicability of homogeneous continuity is scale
dependent. In the Aztec sandstone, with an average grain diameter of 0.25 mm, this
becomes reasonable at the cm-scale (Amadei and Stephansson, 1997), which represents a
lower limit of resolution for interpreting the mechanical modeling results. To focus on
grain-scale processes inside and immediately outside a CB, the distinct element method
approach would be appropriate (Antonellini and Pollard, 1995; Morgan, 1999; Morgan
and Boettcher, 1999). Also problematic at a larger scale is the application of
homogeneous isotropy to the complex æolian sedimentary architecture of the Aztec.
Major dune boundaries influence CBs, so this conceptual model is based on data from an
outcrop of bands located within a single dune package. Given that CBs commonly cut
across depositional bedding without apparent effect, we interpret the mechanical
influence of such layering as negligible. Finally, the assumption of quasi-static CB
propagation is based on the coherent nature of plastic quartz grain deformation within the
bands, which suggests stable propagation accommodated by slow, visco-elastic relaxation
(Chester et al., 2004; Karner et al., 2003). By the same token, the paucity of plastic
deformation outside the bands suggests predominantly elastic behavior through time.
59

6. Elastic properties
Despite a long history of diagenesis and deformation, analysis of the Aztec sandstone
reveals a material arguably similar in terms of depositional structure and cementation to
what it was during CB formation—a weakly cemented, cross-bedded, porous quartz
sandstone. That autochthonous clay derived from the degradation of sparse feldspar has
replaced thin hematite grain coatings as the primary cement is unlikely to have
significantly altered the bulk material properties of the Aztec’s close-packed quartz-grain
skeleton. We therefore infer that the elastic properties of the paleo Aztec resemble those
of the present day.
Published values for apparent Young’s modulus in sandstones based on laboratory
testing range from 10 GPa to 46 GPa (mean of 22 GPa), while apparent Poisson’s ratio
ranges from 0.1 to 0.4 (mean of 0.24) (Bieniawski, 1984). We have calculated values of
apparent Young’s modulus for samples collected near Silica Dome from the results of
triaxial compression tests conducted at Sandia National Laboratories. The results were
16.5 GPa and 21.0 GPa at confining pressures of 10 MPa and 50 MPa, respectively.
Given the assumed lithostatic paleostress of 30 MPa, we therefore posit a Young’s
modulus of 20 GPa and a Poisson’s ratio of 0.2 for the Aztec sandstone at Silica Dome
during the period of CB formation.
The CBs themselves, however, are a different matter. Preferential clay cementation of
the bands subsequent to their formation has rendered them significantly more resistant
and, presumably, stiffer than the friable host rock. Indeed, induration of the bands can
occasionally resemble that of quartzite, suggesting that local pressure-solution healing of
damaged grains may have occurred. Even if the current elastic properties of CBs could
reliably be measured in the lab—a difficult challenge given their thin, tabular
dimensions—we suggest that the results would not represent the bands as they formed.
Additionally, the texture of the bands presents a paradox in that reduced porosity argues
for increased shear stiffness, while intense micro-fracturing and effective grain-size
reduction argues for increased shear compliance. It is reasonable to assume, however,
that bulk modulus within CBs, which measures resistance to volume change, would
increase as the result of both porosity loss and effective grain-size reduction. In the
analyses that follow, we therefore consider a range of elastic properties for the model CB
60

y fixing the ratio of band to sandstone bulk moduli at 1.5. The interrelation of the
essential elastic parameters—Young’s modulus (E), shear modulus (G), bulk modulus
(K) and Poisson’s ratio (ν)—is given by E = 2G(1+ν) = 3K(1-2ν).
7. Mechanical analysis
A logical minimum requirement for in-plane propagation of the model CB is that
compressive stress just ahead of the tip—whether the component normal to the band
trace, the mean value, or some other measure—exceeds the remote value. If the
compacted material inside the band becomes elastically stiffer than the surroundings, the
compressive stress at the tip will decrease. If the inside is softer, the compressive stress
will increase. The other critical independent factor is the magnitude of the inelastic strain
represented by the compaction, which would act to increase the magnitude of the near-tip
compressive stress.
In this section, we first use an embedded-layer model to develop these ideas in a
quantitatively intuitive way, examining the relative influence of elastic moduli versus
inelastic strain on the state of stress exactly at the tip of an infinitely thin band. We then
turn to a full analytical solution of the Eshelby problem to examine the state of stress
induced around a band matching the parameters of the field-based conceptual model.
Finally, we compare the results obtained with the Eshelby solution to those derived from
a two-dimensional (plane strain) BEM approximation of the model band as an anticrack,
using a distribution of closing-mode displacement discontinuity elements to represent the
10% homogeneous uniaxial strain of the inclusion.
7.1. Embedded layer model
Expressions for the stress and strain fields in and around an Eshelby inclusion are
given by Rudnicki (1999) in terms of the remote values, the elastic constants of the
inclusion and surrounding material, and an array of factors depending on the geometry of
the inclusion and the Poisson’s ratio of the surroundings. He specifically treats the
limiting case of interest here, where the aspect ratio of the inclusion goes to zero and its
geometry approaches that of an embedded layer. The same result can be obtained more
directly from the conditions given by Cocco and Rice (2002) for a planar fault zone.
61

Comparison of the two approaches establishes the formal equivalence of the very
eccentric ellipsoidal inclusion with the embedded layer.
With the x1-axis normal to the layer—i.e. corresponding to the short axis of the
inclusion (Figure 2.11a)—Cocco and Rice (2002) note that εij b = εij ∞ if neither i nor j is 1
and σij b = σij ∞ if either i or j is 1, where the superscripts “b” and “∞” denote values of
stress (σ) and strain (ε) inside the model band (layer/inclusion) and in the far field,
respectively. The relations of particular relevance here are
σ b
ε b
∞
11 = σ 11
∞
22 = ε 22
(1a)
∞
, ε ε
(1b)
b
33 = 33
Hooke’s law for the linear elastic relation between strain and stress is
1 ⎧ ν ⎫
ε ij = ⎨σ
ij − σ kkδ
ij ⎬
2G
⎩ 1+
ν ⎭
where the repeated index denotes summation, and δij =1 if i = j, and 0 if i ≠ j. The remote
stresses and strains are related by (2). In the band, (2) relates the stresses to the elastic
strains. The total strain inside the band is given by
b total b elastic p
[ ε ] [ ε ] + ε
ij
= (3)
ij
ij
where εij p denotes the plastic strain representing compaction within the band, which
equals zero unless i = j = 1 as stipulated in the field-based conceptual model. Equations
(1), (2) and (3) can be combined to eliminate the elastic strains and yield expressions for
the stresses inside the model band in terms of the far-field stresses and elastic constants:
σ b
σ
∞
11 = σ 11
b
22
1+
ν b =
2
G
G
b
s
( 1−ν
)
b
⎧
⎨
⎩
∞ ∞ ( σ −σ )
22
2
( 1−ν
s )
( 1+
ν )
33
s
G
G
b
s
∞ ∞ ( σ + σ )
22
33
∞ ⎡
+ σ11⎢
⎣
62
2ν
b
G
−
( 1+
ν ) G ( 1+
ν )
b
b
s
2ν
s
s
⎤⎫
⎥⎬
+
⎦⎭
(2)
(4a)
(4b)

σ
b
33
1+
ν b =
2
G
G
b
s
( 1−ν
)
b
⎧
⎨
⎩
∞ ∞ ( σ −σ )
22
2
( 1−ν
s )
( 1+
ν )
33
s
G
G
b
s
∞ ∞ ( σ + σ )
22
33
∞ ⎡
+ σ11⎢
⎣
2ν
b
G
−
( 1+
ν ) G ( 1+
ν )
b
b
s
2ν
s
s
⎤⎫
⎥⎬
−
⎦⎭
where the subscripts “b” and “s” denote the elastic parameters of the band and
surrounding sandstone, respectively.
The state of stress at the point adjacent to and immediately outside the tip of the
model band along the x2-axis can now be obtained from the conditions of displacement
and traction continuity at the interface:
σ = σ
(5a)
t
22
t
11
b
11
b
22
t b
ε = ε , ε = ε
(5b)
33
33
where the superscript “t” refers to the values just outside the tip. In the second equation
of (5b), ε33 t can be replaced by ε33 ∞ from the second equation of (1b). Again, the elasticity
relations can be used to eliminate the elastic strains and yield expressions for the stresses
at the tip of the band in terms of the uniform stresses inside, the elastic constants and the
nonzero plastic strain component ε11 p :
σ
G
ν
⎛ G
⎜ −
⎝
⎞
⎟
⎠
G
ν −ν
t s b s b s s b s b
s p
11 = σ 11 + σ 22 1 σ kk
ε11
Gb
( 1 ν s ) ⎜ G ⎟
+
+
(6a)
−
b Gb
( 1−ν
s )( 1+
ν b ) ( 1−ν
s )
σ = σ
(6b)
σ
t
22
G
b
22
ν
⎛ G
⎜ −
⎝
⎞
⎟
⎠
G
ν −ν
t s b s b s s b s b
s p
33 = σ 33 + σ 22 1 σ kk
ν sε11
Gb
( 1 ν s ) ⎜ G ⎟
+
+
(6c)
−
b Gb
( 1−ν
s )( 1+
ν b ) ( 1−ν
s )
By substituting (4) into (6), expressions for the stresses at the tip in terms of the
remote values also can be derived. It can be seen from (6) that, for given values of the
elastic parameters and remote stresses, σ11 t and σ33 t are linear functions of ε11 p , with
slopes of 2Gs/(1-νs) and [2Gs/(1-νs)] νs, respectively, while σ22 t remains constant. In fact,
for ε11 p > 10 -3 and any realistic range of elastic parameters and remote stresses, the final
63
2G
2G
(4c)

term in (6a) and (6c) begins to dominate the tip stress state, with σ11 t increasing linearly
with ε11 p at a rate 1/ν ∞ times greater than σ33 t .
Figure 2.12 illustrates this relationship using the putative remote stress state and elastic
parameters defined above. Normal stress components at the tip are plotted as functions of
ε11 p from 0 to 0.1 for two cases: Gb = 2.182Gs and νb = 0.5νs (solid lines), and Gb =
0.857Gs and νb = 2νs (dotted lines). Thus Eb = 2Es in the first case and 0.5Es in the
second, while in both cases, Kb/Ks = 1.5. The two sets of lines lie virtually on top of each
other over the entire range of ε11 p , indicating not only that the actual plastic strain of 0.1
dominates the state of stress at the tip, but that the elastic properties inside the model
band have a negligible effect on the state of stress even as ε11 p approaches zero.
The results of this analysis, however, pertain only to a point infinitesimally close to
the tip of an infinitesimally thin ellipsoidal inclusion. The aspect ratio of the idealized
ellipsoidal CB is small (10 -4 ) but not zero, and the approximation of the granular Aztec
sandstone as a homogeneous continuum makes looking very close to the tip problematic.
We therefore turn to a complete analytical solution of the Eshelby problem to match the
exact physical parameters of the field-based conceptual model and extend the analysis
away from the band-sandstone interface.
7.2. Eshelby inclusion model
For this analysis, we use an exact, closed-form solution to the general three-
dimensional Eshelby problem based on the equivalent inclusion method. The solution is
coded in MATLAB® and was made available to us by Pradeep Sharma at the University
of Houston. Within certain limitations, the code accepts any combination of ellipsoidal
inclusion dimensions, internal and external elastic properties and remote loading to
calculate the state of stress for any point outside the inclusion. The embedded layer stress
results in Figure 2.12 can be reproduced to within a fraction of 1% using the Eshelby
code, by inputting an axisymmetric inclusion with a 10 -7 aspect ratio and calculating the
state of stress just outside (within machine precision) of the tip.
To represent the field-based conceptual model, dimensions of 25 m by 25.1 m by 9.38
mm are used, along with a fixed value of ε11 p = 0.1. The orientation of the model band
with respect to the remote stress field, the range of elastic parameters inside and outside
64

MPa
2000
1800
1600
1400
1200
1000
800
600
400
σ 11
200
σ22 0
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1
p
Uniaxial Plastic Strain ( ε11) Figure 2.12. Normal stress components just outside the tip of the model compaction band
as linear functions of the uniaxial plastic strain from the embedded layer solution. Two
scenarios are plotted: one for a band with Young’s modulus twice that of the surrounding
material (solid lines) and one with Young’s modulus half that of the surroundings (dotted
lines). In both cases, the band/surroundings ratio of bulk moduli is 1.5 and the lines plot
virtually on top of each other. The state of stress at the tip is dominated by the plastic
strain and insensitive to differences in elastic properties.
65
σ 33

MPa
MPa
80
70
60
50
40
30
20
10
0
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2
(a)
50
40
30
20
10
(b)
Distance from tip (m)
Distance from flank (m)
σ 11
σ 22
σ 33
σ 11
σ 22
σ 33
0
0 5 10 15 20 25
Figure 2.13. Distributions of the normal stress components with distance from the model
compaction band tip (a) and flank (b) calculated using the Eshelby inclusion solution and
uniform uniaxial strain of 10%. Two scenarios are plotted: one for a band with Young’s
modulus twice that of the surrounding material (solid lines) and one with Young’s
modulus half that of the surroundings (dotted lines). For both scenarios the
band/surroundings ratio of bulk moduli is 1.5 and, at both tip and flank, the lines plot
virtually on top of each other. The state of stress is insensitive to differences in elastic
properties and only significantly perturbed within a few cm of the tip.
66

the band, and the magnitudes of the remote stresses remain the same as in the embedded
layer analysis. Figure 2.13 presents the resulting stress distributions outside the model
band away from the tip along the x2-axis (Figure 2.13a) and away from the midpoint of
the flank along the x1-axis (Figure 2.13b). Once again, there is a negligible difference
between the results obtained over a realistic range of shear stiffness contrast between the
band and surrounding material for Kb/Ks fixed at 1.5. At about one mm from the tip, the
compressive normal stresses are about double their remote values. This stress
concentration dies away rapidly with distance, dropping to about 1.25 times the remote
values at one cm from the tip, and decaying to a residual increase of about 5% within 20
cm. Immediately adjacent to the flank of the band, the compressive normal stresses drop
by less than 3% relative to the remote values, and then smoothly rebound to effective
background levels within about 10 m (σ22 and σ33) and 20 m (σ11).
This Eshelby formulation provides the most accurate possible representation of the
field-based conceptual model, and the results reinforce the conclusion from the embedded
layer analysis that the perturbed state of stress induced is overwhelmingly a function of
the plastic strain accommodated. Even allowing the relative bulk modulus within the
model band to vary outside realistic bounds—from volumetrically very compliant (K =
4.2 GPa) to very stiff (K = 66.7 GPa)—exerts a negligible effect on the resulting near-tip
stress field when ε11 p = 0.1. Departures from the prescribed axisymmetric geometry also
yield minimal effect. Holding the x2 dimension of the model band constant at 25 m and
varying the x3 dimension from 6.25 m (¼x2)to 100 m (4x2) produces a variation of less
than 5% in the magnitude of σ33 realized one cm in front of the tip.
7.3. Anticrack model
The preceding analyses indicate the primary importance of the uniaxial plastic strain
accommodated within an idealized CB in determining the state of stress induced around
it. This suggests that an anticrack treatment of the problem, with a specified distribution
of closing-mode displacement discontinuity taking the place of internal plastic strain,
would also capture the essential mechanical nature of the conceptual model. To test the
validity of this proposition, we turn to a simple, two-dimensional displacement
discontinuity representation of the idealized CB as an anticrack (Sternlof and Pollard,
67

2002) and use a BEM approach (Crouch, 1976; Crouch and Starfield, 1983) coded in
MATLAB® to solve it.
In this analysis, we consider the principal x1-x2 (horizontal) plane through the middle
of the CB, which roughly corresponds to the outcrop face from which the field data were
collected. The 25-m-long CB trace is represented by 2,500 cm-long constant
displacement discontinuity boundary elements laid end-to-end along the x2-axis from -
12.5 m to +12.5 m. The closing-mode displacement discontinuity of each element is
calculated from the elliptical relation
D
2
⎛ Tmax
⎞ ⎛ xi
⎞
i = ⎜ − Tmax
⎟ 1−
⎜ ⎟
(7)
⎝ 0.
9
⎠
⎝ a ⎠
where Di is the closing-mode displacement discontinuity of the ith element, Tmax is the
maximum (midpoint) thickness of the CB trace (9.38 mm), xi is the x2-coordinate of the
midpoint of the ith element, and a is the half length of the trace (12.5 m). This yields a
step-wise distribution of closing-mode displacement discontinuity equivalent to the
homogeneous uniaxial plastic strain of 10% used above. Because displacement is
specified however, the elastic properties of the CB do not enter the formulation. Also, the
assumption of plane strain dictates that, at every point in the x1-x2 plane, σ33 = νb(σ11 +
σ22) and σ13 = σ23 = 0.
Figure 2.14 shows the resulting anticrack distribution of normal stresses away from
the tip along the x2-axis, and away from the flank along the x1-axis as compared to those
computed with the Eshelby model. Within one mm of the tip, where the singularity of the
anticrack solution produces stresses tending toward infinity, correspondence is poor
(Figure 2.14a). At one cm, the two solutions correspond to within 5% for all normal
stress components. Beyond 2cm the mismatch drops to less than 1%, with the anitcrack
values always slightly above the Eshelby values. At the flank, correspondence between
the two solutions is excellent, with less than a 1% mismatch for all normal stress
components at any distance (Figure 2.14b).
Figure 2.15 highlights the correspondence between the near-tip distributions of σ11
for the anticrack and Eshelby models on a log-log plot. Within a distance (r) from the tip
of less than 0.2 mm, the anticrack distribution of σ11 is dominated by the constant
68

MPa
MPa
100
90
80
70
60
50
40
30
20
10
0
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2
(a)
50
40
30
20
10
0
0 5 10 15 20 25
(b)
Distance from tip (m)
Distance from flank (m)
Figure 2.14. Comparison of the normal stress component distributions with distance
from the model compaction band tip (a) and flank (b) calculated using the BEM anticrack
solution (solid lines) and Eshelby inclusion solution (dotted lines). Beyond 2 cm from the
tip, the two solutions agree to within 1%. They increasingly diverge as the anticrack
solution goes toward infinity at the tip, while the Eshelby solution remains finite.
Adjacent to the flank, differences between the two solutions are less than 1% everywhere.
69
σ11 σ22 σ 33
σ 11
σ 22
σ 33

MPa
10 4
10 3
10 2
10 1
10 −5 10 −4 10 −3 10 −2 10 −1
10 0
Distance from tip, r (m)
Figure 2.15. Log-log plot of the distribution of σ11 with distance from the model
compaction band tip: BEM anticrack solution (solid line), Eshelby inclusion solution
(dashed line), crack-like contours of 1/sqrt(r) (dotted lines), dislocation-like contours of
1/r (dash-dot lines). At the distance r = 1 cm from the tip, the two solutions effectively
coincide. As r decreases, the Eshelby solution approaches the 1/sqrt(r) distribution,
although it remains finite. The BEM anticrack solution, which is composed of a step-wise
distribution of constant closing-mode displacement discontinuity elements, goes to
infinity as 1/r.
Distance (m)
0.4
0.3
0.2
0.1
0
−0.1
−0.2
−0.3
−0.4
12.2 12.3 12.4 12.5 12.6 12.7 12.8 12.9 13
Distance (m)
Figure 2.16. Contour plot of mean normal stress at the tip of the model anticrack
compaction band (band represented by white slot). The remote mean normal stress is 30
MPa. Tick marks indicate the local orientation of the maximum principal stress.
70
MPa
38
37
36
35
34
33
32
31
30

displacement (dislocation) tip element, varying roughly as 1/r (Weertman and Weertman,
1964). For distances greater than 0.2 mm and less than about 2 mm, the anticrack
distribution varies roughly as 1/sqrt(r), as expected for a true crack solution (Lawn and
Wilshaw, 1975). For distances greater than 2 mm, the magnitude of σ11 decreases more
gradually toward the remote value of 40 MPa. The Eshelby distribution of σ11, on the
other hand, varies much more gradually with distance from the tip, only beginning to
approach 1/sqrt(r) inside about 0.01 mm. Beyond 1 cm from the tip, the anticrack and
Eshelby distributions of σ11 effectively coincide.
The size, shape and magnitude of the stress perturbation generated at the tip of the
model anticrack CB is entirely consistent with the field-based interpretation of self-
sustaining in-plane propagation oriented generally orthogonal to the maximum remote
principal stress. For example, elevated mean normal (lithostatic) stress, which would tend
to favor compaction, is tightly concentrated immediately in front of the tip, while the
maximum local principal stress acts everywhere nearly perpendicular to the trace of the
existing band (Figure 2.16).
8. Discussion
The results presented above demonstrate the efficacy of the two-dimensional,
numerical BEM anticrack approach in representing the full three-dimensional Eshelby
state of stress generated around an idealized CB. The only significant mismatch occurs
within one cm of the tip, which in any case is the length scale at which the model
prescription of homogeneous material continuity begins to depart from the granular
reality of the Aztec sandstone. We argue therefore that the anticrack formulation is both
sufficient and even preferable for the purposes of modeling CBs from two-dimensional
outcrop data. Other advantages include increased computational efficiency, the ability to
handle asymmetric configurations of multiple CBs with non-uniform distributions of
closing-mode displacement (i.e. plastic strain), and the potential for modeling active
propagation and pattern development at the outcrop scale (Sternlof et al., 2003).
This study also highlights several interrelated and enigmatic issues that warrant
further investigation: the apparent absence of a near-tip endzone expressed either as
variations in porosity or concentrated grain damage; the observation that thickness
profiles of isolated CBs are elliptical for meters to tens of meters from the tip; the marked
71

concentration of quartz plasticity within the bands and the spatial uniformity of the
plastic strain thereby accommodated; and the apparent distribution of perturbed
compressive stress that suggests grain damage could only occur within a few cm of the
tip.
Ultimately, we anticipate that more refined microscopic examinations will reveal the
presence of a near tip CB process zone defined by quartz grain damage and incipient
plasticity, and that the diameter of this softened damage zone may correlate to a
maximum attainable band thickness independent of trace length. Certainly, most of the
uniform inelastic compaction accommodated within a CB occurs well behind the tip-line
in an environment of reduced compressive stress that would seem to preclude thickening
of the band by lateral propagation into undamaged sandstone. We suggest that the
characteristically elliptical profile of an isolated band could be due primarily to relatively
rapid mechanical propagation of the tip-line, compared to relatively slow processes of
plastic relaxation and collapse that lead to progressive thickening within the rind of
damaged grains left in its wake.
In terms of the bulk material and stress-strain conditions conducive to compaction
localization in sandstone, as well as the mechanical characteristics of the phenomenon,
this analysis of natural CBs in the Aztec sandstone presents a scenario seemingly at odds
with results and interpretations reported in the recent experimental and theoretical rock
mechanics literature. Firstly, regarding material and load conditions, we understand the
phenomenon to have occurred in well consolidated, saturated, but essentially uncemented
sandstone at moderate mean compressive stresses consistent with burial of less than 2.5
km in a thrust-faulting tectonic regime. Laboratory efforts to induce compaction
localization in triaxial experiments have tended to focus on moderately to well-cemented
sandstones (e.g. Berea, Bentheim and Castlegate) generally subjected to much greater
confining pressures reaching as high as 300 MPa (e.g. Wong et al., 2001), which is
equivalent to more than 12 km of overburden and enough to induce brittle-ductile
transition behavior. The grain-scale textures of compaction resulting from such
experiments also tend to involve intense grain crushing and comminution far in excess of
that observed in natural CBs.
72

Secondly, the mechanical style of compaction induced in the laboratory differs
fundamentally from that observed in the field, where discrete CBs appear to have
initiated throughout the upper Aztec at widely distributed flaws, and then propagated
outward along their tip lines to form a pervasive fabric grossly symmetric to the
maximum remote compressive stress. The typical experiment, on the other hand, has
produced a compaction front of damage that propagates across the specimen from one
end cap toward the other, traveling in the direction of maximum compression and leaving
fully compacted material in its wake (e.g Baud et al., 2004; Olsson and Holcomb, 2000).
Such results have been convincingly interpreted in theoretical terms as a stress/strain
bifurcation from homogenous to localized compaction (Issen and Rudnicki, 2000; Issen
and Rudnicki, 2001; Olsson, 1999).
We suggest, however, that this snowplow-like progression of compaction is an
experimental phenomenon largely unrelated to natural compaction localization as
observed in the Aztec sandstone at the Valley of Fire. Similarly, the practical
applicability of bifurcation analysis is somewhat limited both by the stipulation of initial
stress/inelastic strain homogeneity, and by the fact that it applies only to the moment of
localization without reference to any geometric constraint other than imposed planarity.
By contrast, the observations and analysis reported here indicate that localization is the
direct result of initial material heterogeneity—the concentration of stress at and failure of
pre-existing grain-scale Griffith flaws—and that the subsequent evolution of realistic CB
arrays can be understood as the simultaneous propagation and interaction of many
discrete bands modeled as anticracks.
Despite the temporal and physical limitations inherent to experimental modeling, we
therefore suggest that a key to progress in compaction localization research lies in
introducing stress-concentrating initial flaws to jumpstart the deformation process in
more representative materials subjected to less extreme loading conditions. The rate and
state effects of pore water (Baud et al., 2000) and heat (Chester et al., 2004) might also
catalyze improved results at laboratory scales and strain rates. A handful of recent
experiments have incorporated the initial flaw idea (Sternlof and Pollard, 2002) as a
circumferential notch designed to concentrate the applied load and localize compaction
away from the end caps (Vajdova et al., 2003; Tembe et al., 2006). Although still
73

conducted on well-cemented sandstones subjected to unrealistically high confining
pressures, these experiments have produced some promising results. Another set of
experiments, although originally designed to investigate borehole breakouts in sandstone,
have produced perhaps the most representative CBs yet achieved in the laboratory
(Haimson, 2003; Haimson and Lee, 2004).
9. Acknowledgements
Our thanks to John Childs (Figure 2.1) and Frantz Maerten for their able assistance in
the field, and to the Valley of Fire State Park staff and Superintendent Jim Hammons for
access, support and permission to collect samples. We also are deeply indebted to Gaurav
Chopra, Pradeep Sharma and Ole Kaven for their computational and coding work using
MATLAB®. Chopra developed the BEM code used in this paper, while Sharma
graciously provided the Eshelby code, which was modified for our use by Kaven. Tapan
Mukerji provided invaluable assistance on issues of image analysis and rock physics,
while Bob Jones enabled the SEM effort. Dave Lockner (U.S. Geological Survey), and
Bill Olsson and David Holcomb (Sandia National Laboratories) graciously allowed us to
conduct preliminary testing of our ideas in the lab, while providing many useful insights.
This work was funded under grants from the U.S. Department of Energy, Office of Basic
Energy Science, Geosciences Research Program to John Rudnicki (DE-FG02-
93ER14344), and to David Pollard and Atilla Aydin (DE-FG03-94ER14462), with
additional support from the Stanford Rock Fracture Project.
74

1. Abstract
Chapter 3
Energy-release model of compaction band propagation
The elastic strain energy released per unit advance of a compaction band in an infinite
layer of thickness h is used to identify and assess quantities relevant to propagation of
isolated compaction bands observed in outcrop. If the elastic moduli of the band and the
surrounding host material are similar and the band is much thinner than the layer, the
p
energy released is simply σ ξhε , where σ + is the compressive stress far ahead of the
band edge, ξ h is the thickness of the band and
+
p
ε is the uniaxial inelastic compactive
strain in the band. Using representative values inferred from field data yields an energy
2
release rate of 40 kJ/m , which is roughly comparable with compaction energies
inferred from axisymmetric compression tests on notched sandstone samples. This
suggests that a critical value of the energy release rate may govern propagation, although
the particular value is likely to depend on the rock type and details of the compaction
process.
2. Introduction
Mollema and Antonellini (1996) reported observations of tabular zones of localized
compactive deformation without evident shear in the Navajo sandstone formation of
southern Utah. They called these structures compaction bands (CBs). Similar structures
have been observed in other field locations and Sternlof et al. (2005) have reported
detailed measurements from CBs cropping out as extensive arrays in the Aztec sandstone
of southeastern Nevada. Localized compaction has also been observed in laboratory
axisymmetric compression tests on sandstones (Olsson, 1999; Olsson and Holcomb,
2000; Baud et al., 2004; Klein et al., 2001) and in simulations of bore hole breakouts
(Hamison, 2001; Haimson and Lee, 2004; Klaetsch and Haimson, 2002). There are a
number of differences among the laboratory observations and between the laboratory and
field observations. Nevertheless, all the observed structures are characterized by localized
porosity reduction in a roughly tabular zone that forms perpendicular to the maximum
compressive stress. In addition to their inherent interest as a mode of localized
deformation in porous rock that has been recognized only recently, CBs have potential
75

~ 62 m
compaction band trend
500µm
Band tip
Band tip
Figure 3.1. Anatomy of a compaction band in the Aztec sandstone. (top) Archetypical fin
exposed in outcrop from tip-to-tip (15 mm max thickness). (bottom) Backscatter electron
images of the uncompacted sandstone (left) and CB (right). Quartz is medium gray,
feldspar is dirty white, clay is dark gray, hematite is bright white and porosity is black.
Note intense microfracture-accommodated plasticity of grains common in the band.
These images were collected from the same thin section collected across the band.
76

importance to applications because of their effect on permeability. Both field (Sternlof et
al., 2004; Taylor and Pollard, 2000) and laboratory measurements (Holcomb and Olsson,
2003; Vajdova et al., 2004) have shown that the permeability for flow across CBs is as
much as several orders of magnitude smaller than that of the surrounding (host) material.
Consequently, the formation of CBs in subsurface porous formations can strongly affect
applications involving the injection or withdrawal of fluids, such as petroleum production,
CO 2 sequestration, contamination clean-up and aquifer management.
Viewed as two-dimensional profiles in outcrops of Aztec sandstone, individual CBs
are generally long (10s of meters), thin (a few centimeters) zones of grains compacted
through inelastic processes of porosity loss, primarily grain fracture and rearrangement
(Figure 3.1). These CBs appear to have propagated outward along planes roughly
orthogonal to the direction of maximum remote compressive stress (Sternlof et al., 2005).
Experimental efforts to induce compaction localization in sandstone cylinders under
axisymmetric compression (Olsson, 1999; Olsson and Holcomb, 2000; Baud et al., 2004;
Klein et al., 2001) have tended to produce zones of compaction that appear to occur
nearly instantaneously across the width of the specimen indicating that propagation, if it
occurs, does so rapidly. On the other hand, in triaxial simulations of bore hole breakouts
in sandstone (Haimson, 2001; Hamison and Lee, 2004; Klaetsch and Haimson, 2002), the
tips of features resembling CBs have been identified extending from elongated, slot-
shaped breakouts. In order to examine band propagation in the laboratory, Vajdova and
Wong (2003) and Tembe et al., (2006) and conducted axisymmetric compression
experiments on cylindrical specimens with an external, circumferential notch. In some
cases, they observed compaction initiating at the notch and extending across the
specimen in increments corresponding to small load drops.
A critical element in reconciling different laboratory observations, differences
between laboratory and field observations, and in making predictions for practical
applications is a better understanding of conditions for propagation. Although theoretical
studies (Olsson, 1999; Issen and Rudnicki, 2000, 2001; Detournay et al., 2003; Bésuelle
and Rudnicki, 2004; Rudnicki, 2002, 2003, 2004) have had some success in identifying
stress conditions and material behavior that are favorable for CB formation, there is less
understanding of the conditions for propagation or arrest. Sternlof and Pollard (2002) and
77

Sternlof et al. (2005) have suggested a linear elastic anticrack model for CB propagation.
An anticrack is the compressive, closing-mode counterpart of the more familiar tensile,
opening mode crack but with the sign of the stresses and crack-surface displacements
reversed. Although the model formally predicts interpenetration of the crack surfaces, for
CBs this interpenetration is interpreted physically as the inelastic compaction
accompanying porosity loss. In this model, a CB propagates because of the compressive
stress concentration at the edge of the band. Currently, little is known about the
magnitude of the stress concentration required to induce propagation in either the field or
laboratory. Vajdova and Wong (2003) and Tembe et al. (2006) also used the anticrack
model to interpret their experiments on notched specimens of Berea and Bentheim
sandstone. From the nominal stress vs. displacement curves, Vajdova and Wong (2003)
2
estimated a lower bound on the “compaction energy” for Bentheim of 16 kJ/m . Tembe
2
et al. (2006) estimated 43 kJ/m for Berea, but because the band propagated at an angle
of 55° from the plane ahead of the notch, it is likely to be a shear band.
The stress at the edge of an anticrack is singular, as it is for a tensile crack.
Consequently, the quantity relevant to the propagation of the band is the coefficient of the
stress singularity, the stress intensity factor K , or, equivalently, the energy release rate E,
related to K by
E =
(1 −ν
)
K
2G
2
, (1)
where G is the shear modulus and ν is Poisson’s ratio. At present the only estimates of
critical values of K or E appear to be those by Vajdova and Wong (2003) and Tembe et al.
(2006) of compaction energies. In addition, it is unclear how these critical values relate to
the parameters of the band, in particular, the inelastic compactive strain accommodated.
This article presents a very simple energy release model of CB propagation as a
framework for identifying the relevant controlling parameters.
3. Formulation
The model is based on an example calculation of Rice (1968) illustrating the use of
the J-integral and is shown in Figure 3.2. Physically, it represents an idealized geometry
based on outcrop observations from the Aztec sandstone (Figure 3.3 and Sternlof et al.,
78

h
strain energy = W -
h
strain energy = W +
Figure 3.2. Model of a semi-infinite CB of maximum thickness ξh embedded in an
infinite layer of thickness h. Boundaries of the layer are displaced toward each other by a
total amount ∆. Propagation of the band a unit distance reduces the energy of a vertical
slice far ahead of the tip (W + ) to that of a slice far behind the band tip (W - ).
CB Tip
Figure 3.3. Typical configuration of three parallel CBs in the Aztec sandstone, forming
the basis for the model in Figure 3.2: left photo corresponds to vertical slice with elastic
strain energy W - , right photo corresponds to vertical slice with energy W + , middle photo
corresponds to transitional zone associated with the elliptical taper of the central band’s
tip. In the photos, the hypothetical boundaries of the model layer surrounding the central
band (Figure 3.2) fall between it and its two bounding neighbors, which are spaced 1. 2m
apart. Thus, h in this case equals 06 . m.
79

2005). It consists of a long (infinite in the x-direction) layer of material of thickness h (in
the y-direction). The layer contains a long (semi-infinite) CB which attains a fixed
thickness ξ h far behind the band edge. Both the layer and the band are uniform in z-
direction (orthogonal to the xy-plane of Figure 3.2), at least over distances much larger
than h. The layer boundaries are then displaced toward each other a total distance ∆. The
geometry constrains all displacements and strains to be uniaxial in the vertical (y)
direction. Because the configuration is self-similar (translationally invariant in the x-
direction), propagation of the CB a unit distance in the x-direction must reduce the strain
energy in a vertical slice (perpendicular to the xy-plane) far ahead of the band edge to that
of a vertical slice far behind the band edge. Because displacement is fixed on the
boundaries of the sample, this difference in strain energy exactly equals the energy
released per unit area swept out by propagation of the CB a unit distance.
Far ahead of the band, the vertical strain is uniform and equal to ∆/ h . If the material
is assumed to be elastic, then the vertical stress is M(∆/h) (horizontal stresses exist to
enforce zero strain in these directions), where M is the modulus governing one-
dimensional strain. For an isotropic material with shear modulus G and Poisson’s ratio
ν , M = 2 G(1 − ν ) / (1− 2 ν ) . The strain energy per unit area of a vertical slice far ahead of
the CB is
2
+ 1 ⎛∆⎞ W = Mh⎜ ⎟ . (2)
2 ⎝ h ⎠
Far behind the band edge, where the CB thickness is ξ h , the strain is uniform both
inside ( ε band ) and outside ( ε out ) the band. The values are different but must be compatible
with the total displacement of the boundary, given by
ξhε + (1 − ξ) hε
=∆. (3)
band out
The stress outside the CB is given by σ out Mε
out
= . The material inside the CB is also
assumed to be elastic, but with a different modulus M b , and to have undergone an
p
inelastic, vertical compactive strain ε . Hence, the stress is given by
σ = M ε −ε
⎛
band b ⎜
⎝ band
p ⎞
⎟
⎠
. (4)
80

Equilibrium requires that σ band = σ out . This equation can be combined with (3) to
determine the strains in terms of ∆/ h , the moduli, the geometry and
strain energy per unit area of a vertical slice far behind the edge of the CB is
p
ε . Therefore, the
1
p
W h band band (1 ) h out out
2 2
1
− ⎛ ⎞
= ξ σ ⎜ε − ε ⎟+
− ξ σ ε . (5)
⎝ ⎠
+ −
Taking the difference W − W yields the energy release per unit area created of CB
with thickness ξ h :
⎧ 2
⎫
⎪ ⎛ ⎞
2 ⎪
⎨ ⎜ ⎟ ⎬
⎪ ⎝ ⎠ ⎪
⎩⎪ ⎭⎪
1 Mh ⎛∆⎞ ⎛ M ⎞ p⎛∆⎞
p
Eband =
⎜ ⎟ξ⎜ − 1⎟+ 2ξε
⎜ ⎟−
ξε
2( MM / b) ξ + (1 −ξ) ⎝ h⎠ ⎝Mb ⎠ ⎝ h⎠
4. Special cases
If the band modulus vanishes (i.e. M b = 0 ), then (6) reduces to
. (6)
Eband = 1
2 M ⎡ ⎛∆⎞⎤ ⎢ ⎜ ⎟ ∆
⎣ ⎝ h ⎠⎦
⎥ , (7)
where σ + = M ( ∆/ h)
is the uniform stress ahead of the band. This is equivalent to the
result for a crack with zero tractions on the surfaces and, notably, does not depend on the
inelastic compactive strain. Equating (7) and (1) yields the following expression for the
stress intensity factor
K
band
=∆
MG
, (8)
(1 −ν
) h
which is independent of the band length.
If the elastic modulus of the CB remains the same as the material outside (i.e.
M b = M ) and we neglect the term (ξε p ) 2 as inconsequentially small, then (6) reduces to
p
Eband = σ + ξε h . (9)
Thus, (9) has the simple interpretation of the stress multiplied by the compactive
displacement accommodated within the CB.
An indication of the effect of the difference in moduli can be obtained from Eband by
simplifying the expression (6) for ξ

⎡ ∆ ⎤ ⎧ p 1 ∆ ⎛ M ⎞⎫
Eband = M ξh ε
1
h 2 h Mb
⎪ ⎛ ⎞ ⎛ ⎞ ⎪
⎢ ⎜ ⎟⎥ ⎨ + ⎜ ⎟⎜
− ⎟⎬.
(10)
⎣ ⎝ ⎠⎦ ⎪⎩ ⎝ ⎠⎝
⎠⎪⎭
Equation (10) suggests that the ratio of moduli is probably not significant unless it
p
exceeds two and ∆/ h is of the same order of magnitude as ε . This result is consistent
with the finding of Sternlof of al. (2005) that the internal stiffness does not appreciably
p
affect the state of stress around a highly eccentric ellipsoidal CB when ε is on the order
of 10 %. If the CB is modeled as a rigid inclusion, then Mb→∞
and the last parenthesis in
(10) becomes −1.
If the inelastic compactive strain is much greater than the nominal
p
strain (i.e. ε >> ∆/ h ), then this reduces to the same expression as (9). Consequently,
unless the stiffness of the CB material is much less than that of the surrounding material,
(9) gives a good approximation to the energy release rate.
5. Discussion
The simple model presented here suggests that CB propagation can reasonably be
considered to occur when the energy released per unit advance of the band Eband is equal
to some critical value Ecrit that reflects the resistance of the material to compaction.
Expressions (6), (7), (9), and (10) for Eband give the energy released in terms of the
compactive strain of the band, the imposed strain, the elastic properties of the band and
the surrounding material, and the thickness of the band and the surrounding material
(spacing between bands). Assuming that band propagation does occur when Eband = Ecrit ,
it is possible to estimate the minimum value of the material parameter Ecrit from
representative values of the parameters derived from the Aztec sandstone (Sternlof et al.,
2005). Taking σ + = 40 MPa, ξ h = 001 . m, corresponding to 1 cm thick CBs spaced 1
p
meter apart, and ε = 01 . , corresponding to a porosity reduction of 10 % , yields Eband
=
2
40 kJ/m . Ecrit
must be at least this large when propagation of the band ceased, otherwise
it would have continued advancing. Using the smallest and largest values for σ +
estimated by Sternlof et al. (2005) and keeping the same values of ξ h and
2
range of Ecrit from 10 to 60 kJ/m . For
ν = 02 . , the value of σ + implies
h
p
ε , yields a
M = 22 GPa, corresponding to G = 8.3 GPa and
18 10
−3
∆/ = . × . However, ∆ is difficult to estimate and
82

h , the CB spacing, is the most variable physical parameter in outcrop, ranging from
millimeters to meters.
Despite many differences between the rock types, circumstances and morphologies of
2
band formation in the field and the laboratory, the value 40 kJ/m is similar to
compaction energies inferred in the experiments on notched samples. Interestingly, the
value inferred by Tembe et al. (2006) for Berea sandstone is about the same, although, as
noted earlier, the localized deformation in this case likely involved significant shear. The
2
value of 16 kJ/m inferred by Vajdova and Wong (2003) for Bentheim sandstone is a
factor of 25 . smaller. They did, however, report it as a lower bound. If Eband
= 16 kJ/m
and the stress is taken as 40 0 MPa, roughly the axial stresses at propagation for the
Bentheim samples, then (9) yields 00 . 4 mm for the compactive displacement. For a
plastic strain of 7% , the porosity change estimated by Klein et al. (2001), the band width
is 057 . mm, about the same as observed by Baud et al. (2004) (see their Figure 9). Thus,
the expression (9) gives results that are consistent with the limited data. To determine
whether there is systematic variation of Ecrit
with different porous sandstones and,
perhaps, differences in the microscale processes of compaction (e.g. fracturing vs. grain
crushing) would require additional observations. Our results, however, indicate that the
energy release rate provides a reasonable framework for comparison.
p
Since the term ξε h in this expression represents the net effective compactive
displacement across the CB far behind the edge, an alternative criterion might be that
propagation occurs at a critical value of this quantity. In its simplicity, the model does not
specify whether this inelastic strain (or compactive displacement) is accumulated in a
small zone near the edge or more gradually over a large zone behind the edge. The model
only assumes that it eventually reaches a constant, asymptotic value far behind the edge.
Far ahead of the edge the stress is σ + = M ( ∆/ h)
. Far behind the edge the stress is
p
∆/ h −ξε
σ − = M b
. (11)
ξ + (1 − ξ)
M / M
b
Presumably, the stress increases from σ + far ahead of the CB edge to a higher value
(theoretically infinite for the anticrack model) near the edge then decreases to σ − , at
83
2

which point the inelastic strain has attained a critical value (
p
ε )crit . This process is
illustrated schematically in Figure 3.4, where the reduction of stress occurs over an
endzone distance R . Further analysis of the spatial and temporal distribution of energy
dissipation suggested by the more than 5 meters of elliptical taper near the ends of the
bands reported by Sternlof et al. (2005) is warranted and should contribute to a better
understanding of the conditions needed for propagation.
σ −
σ
σp
σ −
σ + = M∆/h
R x
σp
σ −
σ
(ε ) crit
p
Figure 3.4. Schematic illustration of the hypothetical spatial distribution of stress near
the edge (tip) of a CB. (a) As the tip is approached from the right (σ+ side), the stress
increases to a maximum value σp , then drops below the remote background value of σ+
to approach the relaxed value of σ— as the total inelastic compactive strain (ε p ) reaches
its critical value (ε p )crit at a distance R behind the tip.
6. Acknowledgements
The authors thank Jim Rice for a suggestion concerning the model and Teng-fong
Wong for discussion of his laboratory results. Principal financial support for this work
was provided by the U. S. Department of Energy, Office of Basic Energy Science,
Geosciences Research Program through grants to Northwestern University (John
Rudnicki) and Stanford University (David Pollard and Atilla Aydin). Partial support
during preparation of the initial manuscript while Rudnicki was a visitor at the Kavli
Institute for Theoretical Physics was provided by the National Science Foundation under
Grant No. PHY99-07949 (Preprint No. NSF-KITP-05-26). Additional support was
provided by the Stanford Rock Fracture Project.
84
ε p

1. Abstract
Chapter 4
Propagation of compaction bands in sandstone as anticracks:
Field evidence, mechanical theory and numerical simulation
Outcrop and petrographic observations of compaction bands exposed in the Aztec
sandstone of southeastern Nevada suggest that the mechanics of their propagation and
interaction can be approximated using anticrack theory. Our numerical simulations of
anticrack band propagation using the boundary element method confirm this, while
suggesting refinements to better account for the compacted-inclusion character of the
bands. Using realistic, field-based ranges for all physical parameters, we find that, as with
opening-mode cracks, the degree to which adjacent anticrack bands interact is inversely
proportional to the magnitude of the remote differential stress acting upon them. This
model observation suggests that the tendency toward anastomosis along the trend of a
subparallel compaction band array—and thus the extent to which it would impede fluid
flow in that direction—can be predicted from knowledge of the remote stress state in
which it formed. Conversely, the orientations and relative magnitudes of all three
principal paleo stresses can be estimated from directional variations in the degree of
anastomosis revealed by a well-exposed (or imaged) compaction band array.
2. Introduction
Compaction bands (CBs) are a phenomenon of localized compressive failure
commonly observed in outcrops of porous sandstone. They represent one kinematic end-
member of a family of structures known collectively as deformation bands (DBs), which
also includes shear and dilation bands, as well as mixed-mode combinations (Antonellini
et al., 1994; Aydin, 1978; DuBernard et al., 2002; Mollema and Antonellini, 1996;
Rudnicki and Sternlof, 2005; Sternlof et al., 2005). As thin, tabular features of porosity-
loss compaction and order-of-magnitude permeability reduction that are millimeters to
centimeters thick and tens of meters or more in planar extent, compactive DBs act as
baffles to subsurface fluid flow under saturated conditions (Pittman, 1981; Freeman,
1990; Antonellini and Aydin, 1994; Crawford, 1998; Gibson, 1998; Taylor and Pollard,
2000; Sigda and Wilson, 2003; Sternlof et al., 2004). Where present as pervasive arrays,
85

they can exert significant effects on fluid flow at scales relevant to the management of
both groundwater and hydrocarbon resources (Matthai et al., 1998; Sternlof et al., 2006).
However, detecting the presence of CBs in the subsurface, let alone gathering the
requisite geometrical data with which to assess their aggregate hydraulic impacts,
presents a challenge. As small-scale structures, CBs are all but invisible to current
seismic and borehole geophysical imaging techniques, while the coherent nature of the
deformation they accommodate makes them difficult to see in both core and borehole
televiewer logs. Even given limited information on CB occurrence, orientation and
spacing derived from sparse borehole data, knowledge of the greater, interconnected
geometry of any subsurface array is essential to accurate modeling of its effects on fluid
flow at production/injection scales (Sternlof et al., 2004; 2006). Geostatistical methods
are commonly used for such spatial extrapolations and interpolations of data, although
the results are only as good as the training models used to calibrate them. For robust
predictions of subsurface CB geometry—or the geometry of any structure—we advocate
projections based on a fundamental mechanical understanding of how individual bands
propagate and interact to form a through-going array.
This paper expands on the anticrack-inclusion model for compaction bands as
proposed by Sternlof et al. (2005) and based on detailed observations of abundant CBs
cropping out in the Aztec sandstone of southeastern Nevada (Figure 4.1). Specifically, we
examine the linear elastic anticrack approximation for CB mechanics using the
displacement discontinuity boundary element method (BEM) to simulate 2-D
propagation and interaction. We then compare the simulation results to diagnostic
patterns observed in Aztec outcrop. This effort is iterative, insofar as observations of CBs
in the Aztec inspired the anticrack-inclusion model and now comprise the standard
against which it can be tested and improved, while insights gleaned from the modeling
effort suggest new perspectives for refined observations.
Ultimately, we conclude that the anticrack model provides a valid first approximation
for conceptualizing CBs, predicting their dominant orientation symmetric to the
maximum remote compressive stress, and simulating their patterns of propagation and
interaction. We also find that, as with opening-mode cracks (e.g. Cruikshank et al., 1991;
Olson and Pollard, 1989; Thomas and Pollard, 1993), the degree to which adjacent CBs
86

NV UT
CA
AZ
Park Road
Map
Detail
Park Boundary
Park Office
Route 169
0 km 5
NEVADA
ARIZONA
10 km
Valley
of Fire
State Park
N
LAKE
MEAD
115 o 30'
Figure 4.1. Location of the Valley of Fire State Park approximately 60 km NE of Las
Vegas, Nevada, where the 1,400-m-thick æolian Aztec Sandtone is extensively exposed.
Figure 4.2. Typical outcrop exposure of a sub-parallel compaction band array in the
Aztec sandstone. Individual bands weather out in positive relief to form distinctive fins
that are generally at high angle to depositional bedding. The bands in this photo are of the
dominant, through-going set, which trends NNW and dips steeply E (photo looking N).
87
36 o 15'

interact is inversely proportional to the magnitude of the remote differential stress acting
upon them. This model observation suggests that the tendency toward anastomosis along
the trend of a subparallel CB array—and thus the extent to which fluid flow in that
direction may be impeded—can be predicted from knowledge of the remote stress state in
which it formed. Conversely, the orientations and relative magnitudes of all three
principal paleo stresses can be estimated from directional variations in the degree of
anastomosis revealed by a well-exposed (or imaged) CB array.
3. Field evidence and interpretation
Extensive arrays of generally sub-parallel, NNW-trending, steeply E-dipping CBs
pervade the upper 600 meters of the 1,400-m-thick Aztec sandstone as exposed in and
around the Valley of Fire State Park of southeastern Nevada (Sternlof et al., 2005)
(Figures 4.1 and 4.2). The Aztec is an æolian, subarkosic, chronostratigraphic equivalent
of the early Jurassic Navajo and Nugget sandstones (Marzolf, 1983). It remains weakly
lithified and is typified by large-scale tabular and trough cross-bedding, average porosity
of 20-25%, and a mean grain diameter of 0.25 mm within a range of 0.1 mm to 0.5 mm
(Flodin et al., 2005). The CBs, which comprise the oldest structural fabric present, are
relatively thick at a centimeter or more, typically extend for tens of meters in outcrop
trace length, and represent highly localized tabular inclusions of compacted detrital grains
and secondary clay accumulation. With no obvious association to pre-existing structures,
the bands appear to represent a pervasive compressional tectonic fabric—in essence the
kinematic and structural opposite of a regional joint set (Sternlof et al., 2005). Noting
their relative age and dominant orientation, Hill (1989) first suggested that the CBs
formed as a result of and perpendicular (symmetric) to ENE-directed regional
compression associated with tectonic shortening during the Cretaceous Sevier orogeny.
The CB fabric was subsequently crosscut and offset by relatively low-angle shear bands
related to final, proximal emplacement of the overlying Sevier thrust sheets, while
overprinting by joints and joint-based strike-slip faults associated with Basin and Range
extension followed in mid Tertiary time (Eichhubl et al., 2004; Flodin and Aydin, 2004;
Hill, 1989; Myers and Aydin, 2004; Sternlof et al., 2005; Taylor et al., 1999).
Sternlof et al. (2005) present a coherent conceptual and mechanical model for CBs as
anticrack inclusions that is grounded in detailed outcrop and petrographic observations
88

made of the Aztec sandstone in the Valley of Fire. A synopsis and expansion of those
observations and interpretations is presented below. Throughout the rest of this paper,
only CBS in the Aztec are considered. All other younger structural fabrics—shear bands,
joints and joint-based faults—are ignored.
3.1. Outcrop observations
Compaction bands in the otherwise weakly lithified Aztec sandstone tend to weather
out in positive relief as sharply delineated tabular fins, rendering them readily visible in
outcrop (Figure 4.2). In areas of outcrop relief, individual CBs are seen to be grossly
penny-shaped (i.e. very thin, oblate discs). Expansive exposures over more than 10 km 2
in the Valley of Fire represent an approximately horizontal observation plane through the
3-D CB array, revealing a detailed 2-D pattern. Viewed in total, the impression is one of
a very consistent, NNW-trending, sub-parallel (anastomosing) pattern of bands exhibiting
meter to centimeter spacing and interspersed with areas up to tens of thousands of m 2 that
are relatively devoid of bands. While the map-view observation plane does not actually
cut exactly perpendicular to the dominant band orientation, which currently dips eastward
at 70-75°, for the purposes of basic pattern analysis, interpretation and the 2-D
mechanical modeling to come, we make the simplifying assumptions that the current map
view represents paleo horizontal at the time of CB formation and that the bands
themselves were vertical.
In outcrop, band patterns range from nearly straight and parallel, to strongly
anastomosing, to essentially dendritic, to markedly uniform checkerboards comprised of
two distinct band sets crosscutting each other at high angle (Figure 4.3). Truly parallel,
evenly spaced band patterns are not observed at any substantial scale, while the dendritic
and checkerboard patterns occur as relatively rare, localized pockets within the dominant,
sub-parallel pattern. In fact, the dominant, through-going band trend—sometimes steeply
W-dipping, but usually steeply E-dipping—is present in every band pattern observed.
The complex æolian sedimentary architecture of the Aztec also clearly influenced the
extent and distribution of individual bands and patterns of bands, which are often
observed to warp, terminate and/or change configuration at or near cross-bed boundaries,
particularly major interdune contacts (Figure 4.4).
89

(a) (b)
(c) (d)
Figure 4.3. Typical compaction band patterns exposed in outcrops of the Aztec
sandstone: (a) essentially parallel; (b) sub-parallel anastomosing (dominant pattern); (c)
meandering “dendritic”; (d) checkerboard.
90

dune boundary
dominant CB set
cross-bed boundary
secondary band set
dominant CB set
Figure 4.4. Example of mechanical interaction between compaction bands and æolian
sedimentary architecture in the Aztec sandstone (view is to the NW). Relatively parallel
compaction bands of the dominant, steeply E-dipping set are plainly visible at both the
top and bottom of the outcrop. At the fine-grained major dune boundary running
diagonally across the bottom half of the photo, bands of the dominant set either terminate
or warp into a drastically different WNW-dipping orientation with a more wavy outcrop
trace. At the upper cross-bed boundary, where the attitude of depostional bedding attitude
reverts to that at the bottom, the dominant band orientation, spacing and relatively
smooth outcrop trace resumes.
Thickness (mm)
14
12
10
8
6
4
2
0
0 6.25 12.5 18.75 25
Distance in meters from left tip
Figure 4.5. Tip-to-tip thickness profile for a 25-m-long, relatively straight (planar)
compaction band (black squares are measurements). The dark gray curve represents the
least-squares best fit of an ellipse to the data. The correlation index for this fit is 0.87, the
midpoint thickness is 9.5 mm (from Sternlof et al., 2005).
91

Viewed individually, CB traces tend to be straight (planar) at the scale of centimeters
to meters and, where visible from tip to tip, reveal extremely eccentric ellipsoidal
thickness profiles with aspect ratios of approximately 10 -4 (Figure 4.5). However, bands
are also commonly observed to curve, kink, zig-zag, meander and veer toward (or away)
from each other, while two or more bands will frequently overlap and run parallel to each
other in close (mm) proximity for meters to tens of meters (Figure 4.6). Separate CB
traces do sometimes merge at oblique intersections into what appears in outcrop to be a
single band, but we rarely observe two CBs crossing each other except at high angle
(>80°), usually as part of a checkerboard pattern. Noteworthy, because of analogous
patterns documented for joint sets, is the ubiquitous presence of hooking-type interactions,
where adjacent bands spaced up to tens of centimeters curve sharply toward each other.
These can involve two curving tips or one curving and one straight tip, as well as
completely closed eye-shaped structures within a through-going band. Straight tips are
also observed in echelon configurations (Figure 4.7).
3.2 Petrographic observations
Viewed in thin section, a representative planar segment of the fins that define CBs in
Aztec outcrops is clearly delineated by an abrupt decrease in porosity, which is directly
attributable to inelastic mechanical compaction associated with micro-fracture
accommodated plasticity of the quartz grains. The total volume change realized, as
measured by the relative volume fraction occupied by detrital grains inside versus outside
the band, is about 10% and appears to be consistent along the bands from tip to middle.
Judging from the overall continuity of depositional bedding cutting across the bands, and
the almost complete absence of granular disaggregation within them (despite intense
grain fracturing), the volume change appears to have resulted from uniaxial compaction
directed normal to the plane of the band in the absence of appreciable shear
accommodated across it (Figure 4.8).
Notably lacking, or at least not yet detected by the petrographic analyses performed to
date, is any halo of incipient grain damage and/or porosity loss around the elliptically
tapered band tip, as might be expected in analogy with veins, dikes, pressure solution
surfaces and faults, which typically exhibit a process zone of transitional deformation
grading into the undamaged host material (e.g. Delaney et al., 1986; Hoagland et al.,
92

(a)
(b)
(c) (d)
(e) (f)
(g) (h)
Figure 4.6. Variety of compaction band propagation behaviors observed in Aztec
sandstone outcrop: (a) curving and intersecting; (b) complex “meandering” interactions;
(c) long, straight, sometimes closely spaced overlaps and straight, acute approaches; (d)
veering and stepping between parallel bands; (e) parallel propagation of multiple bands
spaced mm apart; (f) veering and zig-zag propagation; (g) touch-and-go intersections; (h)
zig-zag propagation of closely spaced bands.
93

(a)
(b)
(c) (d)
(e) (f)
Figure 4.7. Examples of tip-to-tip interactions observed in Aztec sandstone outcrop: (a)
closed eye structure between closely spaced bands; (b) mutual hooking interaction
without closure; (c) one hooking tip, one straight tip; (d) elongated eye; (e) one band
hooking into another for behind its tip; (f) two straight bands with a echelon overlap
(very slight to no apprarent interaction).
94

(a)
(c)
compaction band
Porosity
0.35
0.30
0.25
0.20
0.15
0.10
0.05
0.00
(b)
(d)
0.1 0.3 0.6 1.8 4.2 8.7
Distance from tip (m)
Figure 4.8. Thin section image of a compaction band within the Aztec sandstone taken
with plane polarized light (quartz grains are white, feldspar grains are brownish, hematite
is black, epoxy-impregnated porosity is blue). The band as shown is approximately 9 mm
thick and a coarse-fine-coarse bedding sequence running across it is indicated by the
double-headed arrows. (a) Core from which this thin section was made. (b) Backscatter
electron image of the (relatively) pristine sandstone matrix. (c) Backscatter electron
image of the compaction band. In the BEI images, quartz is gray, feldspar is white,
speckles of dark gray are kaolinite, and black is porosity. Note the micro-fracture
accommodated plasticity of the detrital quartz. (d) Clustered bar chart showing the
distribution of porosity inside (middle bar) and immediately outside (left bar) the
compaction band with distance from the tip (clay included as porosity. The porosity
difference (right bar) gives the relative volume strain attributable to mechanical
compaction at each location. This compaction strain is remarkably uniform throughout
the band at about 10%.
95

1973; Mardon, 1988; Peck et al., 1985). Given that our observations come from the tips
of CBs that did naturally cease to propagate, however, raises the possibility that damage
halos were originally present, but were consumed by subsequent compaction. On the
other hand, small pods of grains involved in what appears to be incipient collapse can be
distinguished outside of established CBs and interpreted as Griffith-type flaws or proto-
bands (Sternlof et al., 2005).
3.3 Anticrack-inclusion interpretation
The elliptical tip-to-tip profiles and uniform uniaxial internal compaction observed
for straight (planar) CBs corresponds to a pure closing-mode sense of relative boundary
displacement, albeit one distributed across a thin, but finite band thickness rather than the
knife-edge surface of displacement discontinuity that defines solution surfaces (Fletcher
and Pollard, 1981; Mardon, 1988). This elliptical distribution of closing-mode (anti-mode
I) displacement (Figure 4.9) defines the bands kinematically and mechanically as
anticracks (Mollema and Antonellini, 1996; Sternlof et al., 2005). In fact, Sternlof et al.
(2005) show that the local state of stress induced in a linear elastic material by the plastic
compaction of a perfectly planar CB oriented symmetric to the maximum remote
compressive stress is, for all intensive purposes, independent of the evolving material
properties inside the band.
Corroborating the anticrack interpretation is the fact that many of the characteristic
CB patterns observed in Aztec outcrop—particularly the hook and eye configurations—
are distinctly reminiscent of those described for opening-mode cracks (joints), many of
which can be explained as resulting from mechanical interactions between propagating
tips within the framework of linear elastic fracture mechanics (e.g. Cotterell and Rice,
1980; Cruikshank et al., 1991; Fleck, 1991; Olson and Pollard, 1989; Pollard and Aydin,
1988; Segall and Pollard, 1983; Sempere and Macdonald, 1986; Sumi et al., 1985; Swain
and Hagan, 1978; Thomas and Pollard, 1993). This suggests by analogy that CB patterns
may also largely express the mechanical interactions of brittle cracks or, in this case,
anticracks.
Complicating this virtual anticrack approximation of CBs, however, is their physical
reality as highly eccentric, oblate ellipsoidal inclusions of compacted grains, whose
internal material properties must exert nonzero tractions along their boundaries with the
96

(a)
(b)
(c)
σ 3
σ 1
x 3
x 1
x 1
Figure 4.9. Schematic representations of the idealized compaction band model (from
Sternlof et al., 2005). (a) Axisymmetric geometry of the eccentric ellipsoidal band
aligned with the principal remote stresses. (b) Cross-sectional area of the band (solid
ellipse) relative to the pre-compacted area originally occupied by the same detrital grains
(dashed ellipse). As inelastic compaction progresses, the boundary around the grains
involved contracts as indicated by the displacement arrows (u1). This inward
displacement of the elliptical boundary corresponds to the uniform uniaxial plastic strain
of an Eshelby inclusion, and the area between the dashed and solid ellipses corresponds
to the volume loss associated with the compaction. (c) Two-dimensional anticrack
representation of the model band as an elliptical distribution of closing-mode
displacement discontinuity equivalent to the uniform Eshelby compaction strain. In this
virtual treatment, two material lines (solid lines) interpenetrate by an amount equivalent
to the volume loss associated with the compaction (dashed ellipse) as shown by the
displacement arrows (u1). Actual interpenetration does not occur. Because of its extreme
eccentricity, however, solutions for the state of stress induced around the model band
using the Eshelby and anticrack approaches are substantially similar.
97
u 1
u 1
x 2
x 2
u 1
σ 2

undamaged host rock. That CBs cannot exactly be represented as traction-free surfaces of
displacement discontinuity suggests that not all their interactions should be crack-like, as
indeed some are not, notably their tendency to approach at low angle to within
millimeters and then continue along parallel trends for meters. One can infer, for example,
that the middle of a cm-thick CB, within which up to 50% of the available porosity has
already been lost, represents a substantial physical barrier to the cross propagation of an
adjacent, porosity-consuming band. Certainly, the complex material evolution occurring
inside a propagating and curving band will affect the magnitude and distribution of
displacement discontinuity (shear and compaction) realized across it. Nonetheless, the
linear elastic anticrack approach to modeling CB mechanical interactions at the outcrop
scale is substantially justified in that the physical inclusions which define them only
accumulate as a direct consequence of the predominant closing-mode sense of
displacement they accommodate.
4. Mechanical theory
Sternlof et al. (2005) provide a thorough development and justification for the 2-D
(plane strain) mechanical model of an isolated, static CB as an anticrack within an
infinite, homogeneous, linear elastic, isotropic material subjected to uniform principal
remote compressive stress loading (σ11 r ≥ σ22 r , σ12 r = 0). So long as the scale of
observation exceeds a few millimeters, this constitutes a reasonable approximation of
CBs in the porous, granular Aztec sandstone as visible in the sub-horizontal plane of
outcrop exposure (Sternlof et al., 2005). In this section, we expand the anticrack model to
consider the theoretical implications for propagation and interaction between multiple
bands, with reference by analogy to classical elastic fracture mechanics (Irwin, 1960;
Lawn and Wilshaw, 1975) as successfully applied to natural opening-mode fractures in
rock and summarized by Olson and Pollard (1989) and Thomas and Pollard (1993).
Throughout the rest of the paper, compression, compaction, and left-lateral senses of
shear stress and displacement are taken as positive, as illustrated in Figure 4.10.
For an isolated, perfectly linear (planar) anticrack oriented symmetric to σ11 r , the
local state of stress induced in the near-tip region and expressed in the global (remote
principal) coordinate system resembles the example distributions shown in Figure 4.11.
High concentrations of compressive normal stress (σ11 and σ22) are readily apparent
98

σ22 r
σ21 x 1
x 3
x 2
r
σ12 D s
r
σ11 D n
θ σ θθ
Figure 4.10. Schematic diagram of the global 3-D cartesian (x1, x2, x3), and the 2-D local
anticrack tip cartesian (x1, x2) and polar (r,θ) coordinate systems. All components of
stress (σij) and displacement discontinuity (Dn, Ds) are shown as positive.
99
D s
D n
x 1 c
r
σ rθ
x 2 c
σ rr

0.5
0.4
0.3
0.2
0.1
0.5
0.4
0.3
0.2
0.1
0
−0.1
−0.2
−0.3
−0.4
−0.5
−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5
0.5
0.4
0.3
0.2
0.1
0
−0.1
−0.2
−0.3
−0.4
0
−0.1
−0.2
−0.3
−0.4
(a)
−0.5
−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5
(b)
(c)
−0.5
−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5
Figure 4.11. Contour plots of the near tip stress fields generated around an anticrack
(white slot) oriented symmetric to the remote principal stresses (in this case an isotropic
state of remote stress). Dimensions are in meters, stress magnitude are normalized by the
remote value: (a) σ11. (b) σ22 (c) σ12.
100
0
2.8
2.6
2.4
2.2
2
1.8
1.6
1.4
1.2
1
2.8
2.6
2.4
2.2
2
1.8
1.6
1.4
1.2
1
0.4
0.3
0.2
0.1
−0.1
−0.2
−0.3
−0.4

immediately ahead of the tip, while shear stress (σ12) is distributed in an anti-symmetric
quadrant pattern around it. Because each new increment of growth extends a small
distance (r) from the existing tip, it is natural to think in terms of a local polar coordinate
system centered at the tip (Figure 4.10), with propagation occurring in the direction for
which the circumferential normal stress (σθθ) is a maximum, provided that this exceeds
the compressive yield strength of the material. Expressed in terms of opening-mode
cracks (tension positive), this is the maximum circumferential stress criterion of Erdogan
and Sih (1963).
For the symmetric anticrack configuration described above, representing pure mode I
loading and displacements, the distribution of the polar-coordinate stresses as a function
of θ resembles those illustrated in Figure 4.12a, where σθθ max occurs at θ = 0. This
indicates that propagation would continue along a straight path. If even a small amount of
positive mode II displacement (left-lateral shear) is resolved across the tip of the
anticrack, however, the next increment of propagation will diverge at a substantial angle
to the negative (clockwise) side of the θ = 0 axis (Figure 4.12b). In effect, the anticrack
tip attempts to reorient itself to remain symmetric to the perturbed local maximum
principal compressive stress (i.e. along a path for which the local shear stress is zero).
Successive small increments of shearing along the propagation path will result in a
smoothly curving CB.
Excluding variations in effective remote loading (e.g. due to tectonic changes,
regional faulting or slip along major dune boundaries), there are two primary ways for
shear to become resolved on the model anticrack tip: either it deviates from its symmetric
path due to some grain-scale heterogeneity, or it propagates into the perturbed stress field
generated by an adjacent anticrack. The first case results in the resolution of shear on the
deviant tip from the remote loading. If σ11 r > σ22 r (i.e. a significant remote differential
stress exists), then the sense of shear resolved rotates σθθ max back toward the original
symmetric propagation path inhibiting further deviation (Figure 4.13a). As σ11 r
approaches σ22 r in compressive magnitude, however (minimal remote differential stress),
little or no path-correcting shear is resolved on the tip (Figure 4.13b). This analysis
corresponds to that of Cotterell and Rice (1980) on the effect of remote crack-parallel
stress on the propagation stability of opening-mode cracks except that, because anticracks
101

Normalized stress magnitude
3
2.5
2
1.5
1
0.5
0
−0.5
−1
(a)
−1.5
−200 −150 −100 −50 0 50 100 150 200
θ in degrees (local tip coordinate system
Normalized stress magnitude
4
3
2
1
0
−1
−2
(b)
−3
−200 −150 −100 −50 0 50 100 150 200
θ in degrees (local tip coordinate system
Figure 4.12. Plots of the near-tip stress magnitudes expressed in polar coordinates as a
function of θ for a fixed r (see Figure 4.10). (a) Distribution of stresses for a pure closingmode
sense of displacement discontinuity. (b) Distribution of stresses for mixed-mode
displacement discontinuity (shear component positive). Blue lines are σθθ, red lines are
σrr and green lines are σrθ.
(a)
r
σ22 (b)
r
σ22 r
σ11 r
σ11 Figure 4.13. Schematic representation of the effect of remote differential stress on
propagation path stability. (a) When the least compressive remote stress acting parallel to
the anticrack band is significantly less than the maximum compressive remote stress
(high differential stress) any deviation from symmetric propagation results in shear
stresses being resolved on the deviant tip such that it turns back toward the original path.
(b) When the remote stress state is nearly isotropic, the tendency for self-correction is
diminished.
102
α
α
β
β

form perpendicular to σ11 r , increasing σ22 r acts to reduce the effective differential stress
and inhibits, rather than enhances path stability.
In the second case, when the stress perturbations generated by adjacent anticracks
impinge, the effect can be understood by holding one stationary and considering how its
near-tip shear stress field (σ12) resolves onto the other tip propagating along a parallel
path. For example, if tip B approaches tip A from the upper right (Figure 4.14a), it first
enters an area where the sense of shear imposed on it by tip A is mildly positive (left
lateral). This rotates the direction of σθθ max in front of B slightly clockwise, causing it to
curve gently away from A. As B continues on its now sub-parallel path to the left, it
enters the area where the sense of shear induced by A becomes distinctly negative (right
lateral), rotating the direction of σθθ max abruptly counterclockwise and causing B to curve
strongly toward A. Conversely, if tip B overtakes tip A from the upper left (Figure 4.14b),
the sense and magnitude of shear imposed at first cause it to curve strongly away from
the plane of A (counterclockwise), and then gently toward it (clockwise). The net result is
that, if B approaches A, it steps closer to it along a curving path, while if B overtakes A,
it steps away. In both cases, the senses of curving and stepping are the same (mirror
images) if B propagates along a parallel path on the other side of A. If both A and B
propagate, then both paths also curve as described.
The path altering effects of the local and remote stress fields superimpose, generally
in opposition: local effects acting to curve propagation paths out of symmetry with the
remote principal stress state; and the remote differential stress acting to maintain
symmetry. Three additional contributors to path stability for a given CB anticrack would
be the geometry of the path it has already traveled, the distribution of inelastic
(unrecoverable) relative boundary displacements realized along that path, and the
physical properties (elastic or otherwise) of the compacted material inside it. All three of
these would tend to enhance propagation path stability, insofar as shear tractions imposed
on a curving tip would be transmitted into the existing band and resisted by its fixed
physical attributes. The nature of these effects can be thought of as the inertia of a CB to
changes in its path. Furthermore, the nonzero shear tractions imposed by the compacted
material inside a CB on its boundaries with the surrounding pristine sandstone preclude
the local principal state of stress from being symmetric to those boundaries. This fact
103

helps to explain why the oblique approach of one band to the flank of another rarely leads
to intersection, as the local direction of σθθ max can never be parallel to the band flank and
thus cannot lead the propagating tip on a direct collision course with the adjacent band
flank.
Finally we note that, unlike opening-mode cracks, which generally are modeled as
shear traction-free boundary surfaces pressed open by an internal fluid pressure that
constitutes an independently variable driving force for propagation (e.g. Cotterell and
Rice, 1980), anticrack CBs are compaction structures whose propagation can only be
driven by near-tip compressive stress concentrations generated as a consequence of
previous compaction, which must ultimately result from the failure of grain-scale flaws
within the ambient remote stress field (Sternlof et al., 2005). This amounts to a stringent
constraint on initiation and propagation that renders the dense variety of band patterns
pervading the Aztec sandstone all the more remarkable.
5. Numerical model
In order to examine the stress and strain fields generated by CBs and model their
propagation and interaction as anticracks, we developed a MATLAB® code based on the
displacement discontinuity boundary element method of Crouch and Starfield (1983). In
our formulation of this 2-D, linear elastic plane strain model, which corresponds well
with the character of the field data and interpretations described above, we consider an
infinite whole space subjected to uniform principal stresses at infinity and represent the
bands as segmented traces of linear boundary elements (Figure 4.15). Across each of
these elements, two independent and uniformly distributed components of displacement
discontinuity, or dislocation, can be accommodated: Dn (normal) and Ds (shear). The
displacements realized on each element affect those on all others and depend on the
remote loading, as well as the traction boundary conditions (normal and shear) which
must be specified for every element. In order to determine the two unknown components
of displacement for all N elements, while matching the specified traction boundary
conditions, requires the formulation and solution of a traction boundary value problem
consisting of a linear system of 2N equations that can be summarized in matrix notation
as:
tn = E·dn + F·ds (1)
104

0.5
0.4
0.3
0.2
0.1
0
−0.1
−0.2
−0.3
−0.4
(a)
−0.5
−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5
0.5 0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0
0
−0.1 −0.1
−0.2
−0.2
−0.3
−0.3
−0.4
−0.4
−0.5
−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5
Figure 4.14. Schematic illustration of how one anticrack band tip influences the
propagation path of an approaching tip through the action of the near-tip shear stress field
it generates. (a) A band entering from the upper right first experiences a slight positive
shear stress which causes its path to diverge. When it enters the zone stronger, negative
shear stress, the approaching band turns to follow a strongly convergent path. (b) A band
entering from the upper left at first strongly diverges and then weakly converges. In both
cases, if the band approached from the bottom side, the resulting propagation paths would
mirror those shown.
r
σ22 x 1
x 2
Figure 4.15. Schematic illustration of the essential aspects of the displacement
discontinuity boundary element method model.
(b)
105
r
σ11 D n
D s
0.4
0.3
0.2
0.1
0
−0.1
−0.2
−0.3
−0.4

and
ts = G·ds + H·dn (2)
where tn and ts are column vectors containing the normal and shear traction boundary
conditions prescribed for the midpoint of each successive element 1 through N; dn and ds
are column vectors representing the unknown components of displacement discontinuity,
Dn and Ds., for each successive element 1 through N; and E, F,G and H are N x N
matrices containing the influence coefficients that relate displacement on one element to
traction on another. For example, the component Eij of E gives the normal traction on the
ith element due to a unit normal displacement occurring on the jth element.
Analytical functions relate the displacement discontinuity on each element to the
stress field it generates in the surrounding elastic medium (Crouch and Starfield, 1983).
The complete state of stress at any point within the medium can be determined by
superimposing the contributions from each anticrack boundary element and the remote
values. Thus, once Dn and Ds have been determined for all N elements representing a
given band geometry, the code uses them to determine the magnitude and location of
σθθ max along a circle of specified radius centered on the model band. If this value exceeds
the prescribed threshold limit for propagation, an additional displacement discontinuity
element is appended in the appropriate direction, creating a new system of 2(N+1)
equations to be solved. By iterating this process, propagation is simulated.
As currently written, the code allows for any combination of traction and
displacement discontinuity boundary conditions to be specified for each element. If only
boundary displacements are specified, then the problem is fully constrained and the
solution is already in hand. If one or both displacement discontinuities are not specified,
compensating traction boundary conditions are determined during the solution step based
on elastic modulii that must be prescribed for each element. Other schemes for applying
boundary conditions on the elements without assuming linear elastic behavior (e.g.
nonlinear elasticity, plasticity and frictional sliding) would allow for the prescription of a
greater variety of potentially more realistic internal CB behaviors. These improvements
are currently being developed and implemented.
Finally we re-emphasize that, while the code treats positive Dn as a material
interpenetration across the elements, actual CBs represent porosity loss across a band of
106

thin, but finite thickness. It is the extreme elliptical eccentricity of the band trace that
justifies the anticrack approximation in a BEM treatment (Sternlof et al., 2005).
Nonetheless, nothing in the theory or implementation of linear elastic fracture mechanics
precludes the virtual interpenetration of anticrack walls, as represented in our treatment
by positive Dn. By the same token, the code works just as well for opening-mode cracks,
with the signs of the stresses and displacements reversed.
6. Propagation simulation
As a first step toward realistic simulation of CB propagation, interaction and pattern
development, we used the model code to investigate the propagation behavior of a few
simple configurations comprised of one or two anticrack bands. In order to generate
results comparable to the field observations, only values for the model parameters
corresponding to estimates for the Aztec sandstone during CB formation are used
(Sternlof et al., 2005). For the remote stress conditions, these are σ11 r = 40 Mpa, 20 Mpa
≤ σ22 r ≤ 40 Mpa, and σ12 r = 0. For the elastic properties of the pristine (CB-free) infinite
medium, these are Young’s modulus (Es) equals 20 GPa and Poisson’s ratio (νs) equals
0.2, such that the shear modulus (Gs) equals 8.3 GPa. Finally, we normalized by 40 MPa
to produce a problem dimensionless in stress, such that σ11 r = 1, 0.5 ≤ σ22 r ≤ 1, σ12 r = 0
and Es = 500.
6.1. Model calibration and stability testing
As described in Section 2, the anticrack model for CBs is based on detailed
observations, and thickness and porosity measurements of relatively isolated, straight
(planar) bands. These data established the physical reality of the anticrack-like elliptical
distribution of pure closing-mode displacement for such CBs (Sternlof et al., 2005).
However, in order to produce unconstrained and potentially predictive simulations of
propagation and interaction that are not perfectly symmetric to the remote stress field, we
must leave all boundary element displacements unspecified and let the model determine
their equilibrium values. Currently, this means specifying linear elastic modulii (Eb and
νb) for every element. Given the inelastic nature of compaction apparent inside CBs, this
approach to applying boundary conditions has limited appeal others are being developed,
as mentioned above. Nonetheless, if the rate of propagation was fast relative to the rate of
107

plastic compaction, as suggested by the elliptical band profiles, then the assumption of at
least quasi-elastic behavior inside the bands on the time scale of propagation might be
justified (Sternlof et al., 2005).
Despite these uncertainties, the distribution of Dn measured for the 25-m-long CB
profile featured by Sternlof et al. (2005) can be used to estimate an approximate effective
elastic stiffness (Eb) for the anticrack elements (Figure 4.16). For Eb = 0 (i.e. a perfect,
traction-free anticrack), the model calculates a distribution of Dn two orders of magnitude
too high (~10 cm versus ~1 mm in the middle and ~1 cm versus ~.1 mm at the tips). The
addition of even a small amount of element stiffness, however, drastically reduces Dn,
and we found that setting Eb = 15 (3% of Es) produced a slight over estimate. Of course,
the measured distribution of Dn could be matched exactly by independently varying Eb
for every element, but the benefits of such specificity do not justify the effort, particularly
as the assumption of elastic behavior is dubious at best. For our immediate purposes, it is
enough that the near-tip magnitudes of Dn calculated by the model generally coincide
with those measured for natural bands, since these displacement discontinuities, in
concert with any Ds accommodated, dictate the stress perturbations produced outside the
band.
The next aspect of essential model calibration involves choosing the characteristic
distance r at which σθθ max will be calculated. As demonstrated by Sternlof et al. (2005),
because the model represents CBs as trains of constant displacement dislocations, stress
magnitudes calculated in the very near-tip field (r < 0.1 mm) vary as 1/r (Figure 4.17).
For r ranging from about 0.5 mm to 5 mm, the calculated stresses vary along a more a
crack-like 1/√r trend. Beyond 5mm, the BEM model-generated stresses basically
coincide with those calculated analytically using the Eshelby inclusion method (Eshelby,
1959; Mura, 1987), which we consider the most accurate representation of the ideal,
isolated CB based on the field data, and which predicts a normalized σθθ max of about 8.75
as r → 0. The stress singularity generated by the model at r = 0 has no significance in
physical reality, both because the nature of deformation at the tip is actually plastic, and
because the assumption of homogeneous, isotropic elasticity cannot be presumed to hold
as r drops below the grain size of the sandstone. In fact, with an average grain size of
~0.2 mm, any r less than about 5 mm becomes suspect (Amadei and Stephansson, 1997).
108

Distribution of Dn in millimeters
2.5
2
1.5
1
0.5
Eb=10
Eb=12.5
Eb=15
Eb=17.5
Eb=20
Field data
0
−15 −10 −5 0 5 10 15
Distance in meters from center of band profile
Figure 4.16. Distributions of closing-mode displacement discontinuity (Dn) predicted by
the BEM model as a function of varying internal element stiffness (Eb) for a 25-m-long
band trace. Best-fit ellipse to Dn data as calculated from field measurements shown by
thick black curve.
MPa
10 4
10 3
10 2
10 1
10 −5 10 −4 10 −3 10 −2 10 −1
10 0
Distance from tip, r (m)
Figure 4.17. Log-log plot of the distribution of σ11 with distance from the model
compaction band tip: BEM anticrack solution (solid line), Eshelby inclusion solution
(dashed line), crack-like contours of 1/sqrt(r) (dotted lines), dislocation-like contours of
1/r (dash-dot lines). At the distance r = 1 cm from the tip, the two solutions effectively
coincide. As r decreases, the Eshelby solution approaches the 1/sqrt(r) distribution,
although it remains finite. The BEM anticrack solution, which is composed of a step-wise
distribution of constant closing-mode displacement discontinuity elements, goes to
infinity as 1/r (from Sternlof et al., 2005).
109

To determine the sensitivity of propagation path to the choice of r, we performed a
series of stability tests using the same model band that produced Figures 4.16 and 17. We
added a 2-cm element to one tip at a variable kink angle of α, and plotted the relationship
between α and the angle at which the next increment of propagation was predicted to
occur (β) for r = 1, 5 and 10 mm, and the bracketing remote differential stress magnitudes
(σ d = σ11 r -σ22 r ) of zero and 0.5 (Figure 4.18). The response in every case is to correct the
deviation such that β is negative for positive α along a nearly linear trend. When the
remote stress state is isotropic, |β| ≤ |α| and symmetry with σ11 r is nearly restored with the
next increment of propagation. When σ d is the maximum of 0.5, |β| ≥ |α| and the next
increment of propagation over-corrects for the original deviation. The effect of varying r
at the two stress states is also opposite: for σ d = 0, |β| varies directly with r; for σ d = 0.5,
|β| varies inversely with r. Discarding r = 1 mm as unrealistically small for the reasons
stated above, we observe that β never varies by more than 4° for a given α between r = 5
mm and r = 10 mm. Due to this limited effect, and to preserve the plausibility of the
linear elastic continuum assumption, we chose r = 5 mm for all simulations.
Finally, as the model uses linear segments to approximate CB paths that generally are
smoothly curving, we examined the stability of the propagation direction as a function of
the dislocation element length used. Naturally, the shorter the elements, the better the
model fit to the ideal (i.e. smooth) path. As a practical matter of computational efficiency,
however, longer elements are desirable. We performed a second suite of kink-angle
stability tests for tip-element lengths of 2, 6 and 10 cm at r = 5 mm (Figure 4.19). Again,
in every case β acts to oppose α along a nearly linear trend, with the response being
strongest for σ d = 0.5. The sense of the effect at the two stress states is also opposite,
with |β| varying directly with element length for σ d = 0.5, and indirectly for σ d = 0. The
spread in the directional variability is about three times greater at σ d = 0.5 and is equal to
about 85% of α. These results are entirely consistent with those reported in previous
studies of propagation path stability for segmented opening-mode cracks using the
kinked-tip approach (Broberg, 1987; Cotterell and Rice, 1980; Melin, 1983, 1987;
Thomas and Pollard, 1993), which indicate that the misorientation of successive
110

Correction angle (β) in degrees
-100
-90
-80
-70
-60
-50
-40
-30
-20
-10
r=1mm, Sx=.5Sy
r=5mm, Sx=.5Sy
r=10mm, Sx=.5Sy
r=1mm, Sx=Sy
r=5mm, Sx=Sy
r=10mm, Sx=Sy
D n
0
0 5 10 15 20 25 30 35 40 45
Kink angle (α) in degrees
Figure 4.18. Relationship between the angle of incremental deviation from symmetric
propagation (kink angle α) to the subsequent angle of incremental path correction (β) as a
function of both r and the remote state of principal stress (Sy = σ1, Sx = σ2). Note that the
sign of β is negative for positive α in all cases.
Correction angle (β) in degrees
-100
-90
-80
-70
-60
-50
-40
-30
-20
-10
2cm, Sx=.5Sy
6cm, Sx=.5Sy
10cm, Sx=.5Sy
2cm, Sx=Sy
6cm, Sx=Sy
10cm, Sx=Sy
0
0 5 10 15 20 25 30 35 40 45
Kink angle (α) in degrees
Figure 4.19. Relationship between the angle of incremental deviation from symmetric
propagation (kink angle α) to the subsequent angle of incremental path correction (β) as a
function of both element length and the remote state of principal stress (Sy = σ1, Sx = σ2).
Note that the sign of β is negative for positive α in all cases, and that geometric
formulation of problem is as shown in Figure 4.18.
111
D n
σ 1
α
Ds
β
+
−
D n
σ 2

segments will self-correct to provide a reasonable, zig-zag approximation to the ideal
path, certainly for |β| less than ~10° so long as σ d ≠ 0.
In all the simulations presented below, we therefore adopted fixed values of Eb = 15
and r = 5 mm, and use 10 cm propagation elements so long as β < 10°. Insofar as
necessary to maintain numerical stability, and capture details of strong curvature and the
close approach of adjacent bands, we reduced the added element length to 2 cm. We also
verified that at no point in any simulation did Dn on any element become positive (i.e.
opening-mode). Finally, given the current lack of data on a critical σθθ threshold for
propagation to occur, we used a model threshold of σθθ max > σ11 r = 1. In practice, the
normalized magnitude of σθθ max never dropped below about 1.25.
6.2. Symmetric propagation
In order to examine the tendency for CBs to form symmetric (orthogonal) to σ11 r , we
simulated the propagation of a short, incipient band (three elements, 6 cm) over a range
of misalignment angles α (relative to the global coordinate system) for each of the remote
stress states σ d = 0.05, 0.25 and 0.5 (Figure 4.20). Not surprisingly, the results indicate
that the tendency of a misaligned incipient CB to propagate into symmetry with the
remote stress field decreases as σ d → 0. For a perfectly isotropic state of remote stress,
all band trends are preferred equally. For the maximum probable remote differential
stress (σ d = 0.5), the band trend orthogonal to σ11 r is strongly preferred. Even at σ d =
0.25 (σ22 r = 0.75σ11 r ), the symmetric orientation is clearly preferred, particularly if one
allows for the fact that 6 cm greatly exaggerates the length a misaligned incipient CB
would attain without starting to curve.
Another factor to consider in assessing the tendency toward symmetric propagation is
how σθθ max at the tips of the incipient model CB varies with α (Figure 4.20d). Even for
σ d = 0.05, a slight but appreciable increase of 5.4% in σθθ max is realized as α goes from
90° to 0°. For σ d = 0.25 and 0.5, the increase is 33.3% and 100%, respectively. These
results demonstrate that, even as σ d → 0, the propagation of incipient bands oriented
more nearly orthogonal to σ11 r is preferred, likely stranding those oriented at higher α.
112

Distance in meters
Distance in meters
0.15
0.1
0.05
0
−0.05
−0.1
0.1
0.05
0
−0.05
−0.1
σ 1
α
σ 2
(a)
1 deg.
5 deg.
15 deg
30 deg.
45 deg.
60 deg.
75 deg.
85 deg.
89 deg.
−0.15
−0.15 −0.1 −0.05 0 0.05
Distance in meters
0.1 0.15
0.15
(c)
1 deg.
5 deg.
15 deg
30 deg.
45 deg.
60 deg.
75 deg.
85 deg.
89 deg.
−0.15
−0.15 −0.1 −0.05 0 0.05 0.1 0.15
Distance in meters
Distance in meters
0.15
0.1
0.05
0
−0.05
−0.1
(b)
1 deg.
5 deg.
15 deg
30 deg.
45 deg.
60 deg.
75 deg.
85 deg.
89 deg.
−0.15
−0.15 −0.1 −0.05 0 0.05
Distance in meters
0.1 0.15
3
2.8
(d)
Maximum tangential stress
2.6
2.4
2.2
2
1.8
1.6
1.4
1.2
Diff. stress=0.5
Diff. stress=0.25
Diff. stress=0.05
1
0 10 20 30 40 50 60 70 80 90
Misalignment angle (α) in degrees
Figure 4.20. Propagation of an incipient anticrack band as a function of misalignment
angle α from the symmetric orientation for normalized remote differential stress
magnitudes of 0.5 (a), 0.25 (b) and 0.05 (c). A schematic of the model geometry is shown
in the upper left of plot (a). Each propagation simulation started from an initially straight
band 6 cm long and centered at the origin. Plot (d) shows the normalized value of the
maximum tangential stress at the tip of the initial band as a function of α and the remote
differential stress magnitude.
113

6.3. Approaching tip interactions
Aside from the effects of heterogeneous stress fields and/or mechanical anisoptropies
related to sedimentary architecture (e.g. shear along fine-grained deflation-plane
boundaries separating major, teardrop-shaped dune sequences), which this paper does not
attempt to address, the primary source of mechanical interaction for a propagating CB is
the close approach of an adjacent tip. Within the framework of our simple model, there
are two primary variables that can affect the mechanical interaction of straight bands
propagating toward each other along parallel trends symmetric to σ11 r : the spacing
between them and the magnitude of the remote differential stress. In order to examine the
interplay of these variables in determining the degree of interaction realized between
bands, we performed a suite of simulations for σ d = 0, 0.25 and 0.5 at spacing intervals
(s) of 2.5 cm, 25 cm and 2.5 m (Figure 4.21), starting with two 25-m-long bands identical
to that used in the calibration section above.
In all cases the model results match our qualitative expectations based on theoretical
considerations—the tips at first curve slightly away from each other and then curve more
strongly toward each other, while the interaction is distinctly more pronounced for σ d = 0
and s = 2.5 cm. The absolute magnitudes of path divergence from the established linear
trend, however, can not be intuited, and the results reveal several interesting relationships.
When the remote stress state is isotropic, interaction is quite pronounced, even at s = 2.5
m for which the final overlapped spacing drops by half to 1.25 m before parallel
propagation is restored. At s = 25 cm and 2.5 cm, the overlapped band tips end up on
intersecting trends, although the approach is distinctly more asymptotic for s = 25 cm.
When σ d = 0.5, interaction is negligible except at s = 2.5 cm, when the final overlapped
spacing drops by about one third to 1.6 cm. Even then, parallel propagation resumes
before intersection occurs. Perhaps most interesting is that the results for σ d = 0.25 are
all but indistinguishable from those of σ d = 0.5. Again, even at s = 2.5 cm, parallel
propagation resumes prior to intersection. In fact, because the initial repulsive effect is
more pronounced than for σ d = 0.5, the final overlapped spacing ends up the same at 1.6
cm. In sum, these results suggest that strongly anastomosing band patterns, particularly
those involving very close encounters and the effective merging of bands, can only occur
114

when the ambient stress state is nearly isotropic and/or bands are spaced less than a meter
apart.
One other factor mentioned in the section on mechanical theory as a potential
influence on the propagation paths of interacting CBs is the effective drag or “inertia”
imposed by their pre-existing length. As a test of the general validity of this notion, we
undertook a series of three simulations identical in every respect except for the initial
lengths of the model bands: two 25-m bands (as above and in Figure 4.21), one 25 m
band and one 5 m band, and two 5 m bands. In order to ensure adequate space and
propagation reactivity to render variations in path distinguishable on a single plot, a
spacing of 2 m and an isotropic remote stress state were chosen (Figure 4.22). When both
bands are 25 m long, their mutual influence is substantially similar to that when spacing
is 2.5 m. When one band is 5 m long, its impact on the continued propagation of the
longer band is perceptibly diminished, both in terms of the initial repulsion and the
subsequent attraction. The effect of the longer band on the shorter one, however, is even
more noticeably enhanced. When both bands are 5 m long, their responses again mirror
each other, but are completely different from either of the first two cases. In particular,
they initially repel and attract each other somewhat less robustly than the pair of 25-m-
long bands, but, at the point when parallel propagation resumes in both earlier examples,
attraction between the shorter bands re-intensifies and they propagate toward intersection
at a relatively high angle of approach (> 30°). The final path has a slight hitch in it that
produces a mild, s-shaped curvature. This simple example demonstrates both that band
length can influence the strength of tip interactions, and that the most general case
involving different length bands interacting is likely to produce noticeably asymmetric
patterns of hooking.
Finally, we note that, although the simulations presented in Figures 4.21 and 4.22
generally predict the resumption of parallel propagation paths following initial overlap,
the normalized magnitude of σθθ max rapidly drops toward one. This suggests that, for any
reasonable critical threshold value, propagation would cease. Outcrop observations,
however, demonstrate that long, sub-parallel overlaps between bands spaced even
millimeters apart are common.
115

0.2
0.15
0.1
0.05
0
−0.05
−0.1
−0.15
−0.2
2
1.5
1
0.5
0
−0.5
−1
−1.5
6
4
2
0
−2
−4
−0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
−3 −2 −1 0 1 2 3 4 5 6
−10 −5 0 5 10
Figure 4.21a. Propagation simulations for two 25-m-long anticrack bands approaching
each other along parallel paths under isotropic remote stress conditions and initial spacing
intervals of 2.5 cm (top), 25 cm (middle) and 2.5 m (bottom). Black lines indicate starting
configurations, colored lines indicate simulated paths, all dimensions in meters.
116
15

0.2
0.15
0.1
0.05
0
−0.05
−0.1
−0.15
−0.2
2
1.5
1
0.5
0
−0.5
−1
−1.5
6
4
2
0
−2
−4
−4
−0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
−3 −2 −1 0 1 2 3 4 5 6
−10 −5 0 5 10
Figure 4.21b. Propagation simulations for two 25-m-long anticrack bands approaching
each other along parallel paths (orthogonal to the maximum remote compressive stress) at
a normalized differential stress of 0.25 and initial spacing intervals of 2.5 cm (top), 25 cm
(middle) and 2.5 m (bottom). Black lines indicate starting configurations, colored lines
indicate simulated paths, all dimensions in meters.
117
15

0.2
0.15
0.1
0.05
0
−0.05
−0.1
−0.15
−0.2
2
1.5
1
0.5
0
−0.5
−1
−1.5
6
4
2
0
−2
−4
−0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
−3 −2 −1 0 1 2 3 4 5 6
−10 −5 0 5 10
Figure 4.21c. Propagation simulations for two 25-m-long anticrack bands approaching
each other along parallel paths (orthogonal to the maximum remote compressive stress) at
a normalized differential stress of 0.5 and initial spacing intervals of 2.5 cm (top), 25 cm
(middle) and 2.5 m (bottom). Black lines indicate starting configurations, colored lines
indicate simulated paths, all dimensions in meters.
118
15

4
3
2
1
0
−1
−2
25 & 5m
5 & 5m
25 & 25m
−4 −2 0 2 4 6 8
Figure 4.22. Propagation simulations for two anticrack bands approaching each other
along parallel paths under isotropic remote stress conditions and initial spacing interval of
2 meters. Black lines indicate starting configurations, colored lines indicate simulated
paths, and all dimensions are in meters. Three simulations are shown: two 25-m-long
bands (blue paths); two 5-m-long bands (red paths); and a 25-m-long band entering from
the upper left and a 5-m-long band entering from the lower right.
7. Discussion and conclusions
The fundamental postulate of this research is that all CBs in the Aztec—and by
analogy sandstone in general—stem from the same fundamental mechanism of Griffith-
flaw induced grain-scale collapse and subsequent anticrack-like propagation within a
consistent remote stress field. This amounts to an assumption of mechanical simplicity.
That being said, CBs clearly are not simple anticracks by any strict definition, rather they
are ellipsoidal inclusions of compacted grains and porosity loss realized across a finite
thickness. As such, they lack the distinct surfaces of displacement discontinuity and
discrete apertures characteristic of opening-mode cracks. However, insofar as CBs do
also represent elliptical distributions of uniaxial volume change, and have very small
thickness-to-length ratios, the stress and strain fields they induce within the surrounding,
undamaged and nominally elastic sandstone can reasonably be modeled as resulting from
virtual anticracks.
The preliminary modeling results of propagation and interaction presented in this
paper largely corroborate our theoretical expectations and are distinctly reminiscent of
119

the hooking patterns commonly observed in outcrop. We conclude that the anticrack
model provides an adequate approximation to the outcrop-scale mechanics of compaction
bands and suggest specifically that:
1. CBs propagate as a consequence of self-generated compressive stress
concentrations along paths that attempt to maintain incremental symmetry
with (orthogonality to) the maximum circumferential compressive stress
immediately ahead of their tips;
2. In the absence of external influences, CBs form symmetric with (orthogonal
to) the maximum remote principal stress and will continue to propagate in
plane;
3. Small-scale directional perturbations of the tip of a well-developed band
caused by material heterogeneities are self-correcting, regardless of the
magnitude of the remote differential stress;
4. Large-scale heterogeneity and anisotropy in the surrounding material aside,
the primary cause of curving (out-of-plane) CB propagation is mechanical
interaction between adjacent bands, particularly between tips;
5. The degree of these interactions is acutely sensitive to the magnitude of the
remote differential stress—nearly isotropic stress conditions allow significant
interaction, while realistically high values of differential stress virtually
eliminate interaction;
6. The degree of interaction is also sensitive to the material properties inside the
propagating CBs, as well as their length (absolute in-plane dimensions)—
lower internal resistance to shear enhances interaction as does shorter band
lengths.
Despite the general success of the current anticrack numerical model in producing
realistic propagation behavior, considerable room for improvement exists. Among the
physical phenomena that cannot as yet adequately be addressed are long overlaps
between bands spaced sometimes mere millimeters apart, the checkerboard patterns
comprised of two distinct CB sets crossing each other at high angle, and the regular zig-
zag behavior exhibited by both individual bands and clusters of bands. As the scope of
observation shrinks to focus on these issues at the grain scale, thereby challenging the
120

assumption of an elastic continuum, distinct element method modeling techniques will
become increasingly important (e.g. Antonellini and Pollard, 1995; Morgan, 1999;
Morgan and Boettcher, 1999).
Nonetheless, we believe that progress can be made toward understanding the outcrop-
scale implications of grain-scale effects within the framework of the current model by
implementing refined petrographic observations as more sophisticated schemes for
applying boundary conditions on the band elements and as more nuanced propagation
criteria. The key to these refined observations lies in detecting and describing the shape
and nature of the damage zone around the uniformly compacted inclusion of grains
currently recognized as a CB (Figure 4.23) or, alternatively, conclusively demonstrating
the absence of any such damage envelope. In either case, the mechanical implications at
both the grain and outcrop scales would be profound.
These various shortcomings aside, the current model simulation results do suggest
and support several interpretations and inferences.
1. The maximum normalized near-tip compressive stress concentration (σθθ at r
= 5 mm) that can be realized for a realistic distribution of Dn is no more than
about 2.5 (equivalent to 100 MPa). Even taking this value as a lower threshold
because the data is derived from field measurements of CBs that stopped
propagating, it appears that large stresses are not required to promote the
quasi-static progression of compaction as accommodated by relatively
coherent quartz grain plasticity in the Aztec sandstone.
2. Judged in comparison to our simulation results, the variety of CB interactions
observed in Aztec outcrop—ranging from gently anastomosing sub-parallel
arrays of bands spaced up to a meter apart to the near ubiquity of hooking tip
and eye structures to strongly meandering patterns of anastomosis—suggest
that the prevailing remote paleo stress state was close to isotropic (σ22 r ≈
0.9σ11 r , σ d ≈ 0.1).
3. Although the remote state of sub-horizontal paleo stress in the Aztec was
apparently nearly isotropic (σ d ≈ 0.1 in the x1-x2 plane), the state of stress in
the x1-x3 plane, where σ33 r was essentially due to overburden, was decidedly
anisotropic (σ d = 0.5 by definition in our model set up). Our simulation
121

(a)
(b)
(c)
σ1
compaction band
maximum thickness
compaction band
damage envelope
propagation of damage zone
propagation of compacted tip
Figure 4.23. Schematic end-member conceptual models for the distribution of
compaction band strain and damage. (a) Band profile as currently recognized--elliptical
inclusion of uniform uniaxial compactive strain = 0.1. (b) Hypothetical band profile as
suggested by mechanical logic—same compacted band as in (a), but surrounded by an
envelope of grain damage and related softening generated at the tips. Uniform
compaction within the damage zone accumulates from the inside out to form the band as
recognized, resulting in a maximum thickness once all the damaged grains have been
accreted, and creating a layered inclusion of material properties that vary widely from
each other and from the surrounding pristine sandstone. (c) Simultaneous propagation of
the compacted band and the surrounding damage halo from the tip.
122

esults therefore suggest that the generally anastomosing pattern of bands
captured in the x1-x2 plane would appear as a more consistently parallel, less
interconnected pattern in the x1-x3 plane (Figure 4.24). This interpretation has
yet to be confirmed by observing outcrop exposures with the requisite
orientation, which are rare at best.
4. Strongly parallel band propagation in the x1-x3 plane would tend also to retard
CB interactions in the x1-x2 plane. Given the degree of anastomosis
nonetheless present in Aztec outcrops, this suggest that the paleo state of
stress in the x1-x2 plane was likely more nearly isotropic than our 2-D
simulations suggest.
5. These interrelationships of magnitude and direction between the components
of principal paleo stress and the resulting patterns of band propagation have
significant implications for predicting 3-D permeability in aquifers and
reservoirs containing CBs. Specifically, because permeability along the trend
of a band pattern decreases with increasing connectivity, and the minimum
principal permeability (k3 ≤ k2 ≤ k1) is directed orthogonal to the dominant
band orientation (Sternlof et al., 2004), k1 due to a subsurface band array will
parallel the presumed direction of paleo σ33 r , k2 will parallel σ22 r and k3 will
parallel σ11 r (Figure 4.24). The relative magnitudes of the principal
permeabilities can likewise be inferred from the presumed relative magnitudes
of the paleo principal stresses.
6. By the same token, the orientations and relative magnitudes of all three
principal paleo stresses can be inferred from directional variations observed in
the degree of anastomosis revealed by a well-exposed (or imaged) CB array.
8. Acknowledgements
We are deeply indebted to John Childs for his ever cheerful and able assistance in the
field, and thank Ovunc Mutlu for help with the numerical code. This work was funded by
the U.S. Department of Energy, Office of Basic Energy Science, Geosciences Research
Program under grant DE-FG03-94ER14462, awarded to David Pollard and Atilla Aydin
at Stanford University. Additional support was provided by the Stanford Rock Fracture
Project.
123

σ 3
σ 2
σ 1
Figure 4.24. Schematic 3-D relationship between the dominant trend of a compaction
band array, and its connectivity along that trend, to the orientations and relative
magnitudes of both the principal remote stresses (σ1 > σ2 > σ3) in which it formed and of
the corresponding principal permeabilities that result (k1 > k2 >k3).
124
k 3
k 1
k 2

1. Abstract
Chapter 5
Computational estimation of compaction band permeability:
From thin-section estimations to reservoir implications
Permeability measurements can be difficult to obtain when sample availability is
restricted, dimensions are limited, or materials poorly consolidated. With subsurface
cores of sandstone containing thin, tabular compaction bands, all three challenges could
arise. Methods for estimating permeability from thin-section provide an alternative. We
evaluate a new physics-based computational technique, in which Lattice-Boltzmann flow
simulations are conducted on stochastic realizations of 3-D pore structure generated from
digital thin-section images. Applied to a representative thin section from the Aztec
sandstone of southeastern Nevada, an exhumed analog for band-rich sandstone aquifers
and reservoirs, the method yields estimates that agree well with available data—a few
millidarcys (band) to a few Darcys (sandstone)—capturing the range of both matrix and
compaction-band permeability from a single thin section. Extracted from a subsurface
equivalent of the Aztec, such data could prove invaluable, as pervasive arrays of
compaction bands in sandstone have been shown capable of exerting substantial fluid-
flow effects at scales relevant to aquifer and reservoir management.
2. Introduction
In porous, granular rocks such as the Aztec sandstone in the Valley of Fire State Park
of southeastern Nevada (Figure 5.1), compaction bands (CBs) crop out as thin, tabular
features of porosity-loss compaction accommodated by grain damage, rearrangement and
preferential clay accumulation (Sternlof et al. 2005). Generally up to a few centimeters in
thickness and tens of meters in planar extent, they represent the kinematic subset of
deformation bands dominated by closing-mode displacement and oriented perpendicular
to the local direction of maximum compression (Mollema and Antonellini 1996; Du
Bernard et al. 2002; Borja and Aydin 2004; Sternlof et al. 2005). Reduced porosity, pore
connectivity and average pore-throat diameter conspire to decrease deformation-band
permeability by one to four orders of magnitude relative to the host rock matrix (Pittman
1981; Freeman 1990; Antonellini and Aydin 1994; Crawford 1998; Gibson 1998; Taylor
125

NV UT
CA
AZ
Park Road
Map
Detail
Park Boundary
*
Route 169
0 5
km
NEVADA
ARIZONA
10 km
Valley
of Fire
State Park
N
LAKE
MEAD
115 o 30'
Figure 5.1. Schematic location map for the Valley of Fire State Park in southeastern
Nevada. Colorful alteration patterns of iron-oxide staining in the Aztec sandstone lend
the park its name (* marks approximate location of photographs shown in Figures 5.2 and
5.3).
126
36 o 15'

and Pollard 2000; Lothe et al. 2002), rendering them impediments to fluid flow under
saturated conditions (Figure 5.2).
Sternlof et al. (2004) have shown that anastomosing arrays of subparallel, subvertical
CBs abundantly exposed in the Aztec sandstone (Figure 5.3) would significantly reduce
bulk permeability and induce permeability anisotropy at outcrop scales. Part of a
widespread Jurassic æolian system that includes the Navajo and Nugget sandstones
(Marzolf 1983; Blakey 1989), the 1,400-m-thick Aztec is a medium-grained sub-arkose
that is typified by large-scale cross stratification (Bohannon 1983) and comprises an
exhumed analog for aquifers and reservoirs in similar lithologies. The pervasive CB
arrays, which cut across depositional bedding as the oldest structural fabric present,
formed in response to compression associated with the Cretaceous Sevier Orogeny (Hill
1989; Eichhubl et al. 2004; Sternlof et al. 2005). If present in an active aquifer or
reservoir, these band arrays would cause substantial fluid-flow effects at scales relevant
to production and management (Sternlof et al. in review).
The specific hydraulic impacts exerted by CBs depend strongly on their porosity and
permeability relative to the surrounding sandstone matrix, as well as on geometrical
factors such as average band thickness, spacing and connectivity (Sternlof et al. 2004).
Their limited thickness, however, renders direct measurement of CB permeability
difficult and potentially inaccurate under even optimal conditions (Antonellini and Aydin
1994). Given the generally limited availability of potentially unconsolidated core samples
from active aquifers and reservoirs, problems proliferate. Methods for estimating
permeability from epoxy-impregnated thin-sections using computational techniques
provide an alternative (e.g. Berryman and Blair 1987; Adler et al. 1990; Bakke and Øren
1997; Blair et al. 1996).
In this paper, we evaluate a new physics-based algorithm for estimating permeability
from thin section images (Keehm et al. 2004). In order to simulate the realistic situation
of limited subsurface sample supply, we apply the method to a single, representative thin
section from the Aztec (Figure 5.4) and compare the resulting estimates of CB and matrix
permeability to available measurements reported for the Navajo and Aztec sandstones
(Antonellini and Aydin 1994; Flodin et al. 2005).
127

compaction band
Figure 5.2. Typical compaction band exposed as a tabular, cm-thick fin at high angle to
depositional bedding in the Aztec sandstone. Sharply decreased internal porosity and
permeability due to grain crushing, rearrangement and preferential clay accumulation
render the band an impediment to fluid flow, as illustrated by its obvious impact on the
distribution of iron-oxide staining.
128

Figure 5.3. Extensive arrays of dominantly north-northwest trending, steeply eastdipping
compaction bands pervade the upper 600 m of the Aztec sandstone, creating a
sub-parallel, anastomosing system of low-permeability baffles capable of affecting fluid
flow over hundreds of meters. View is to the northwest, bands crop out as resistant fins.
129

3. Computational method
The methodology workflow involves creating a 2-D binary map of porosity from a
digital thin-section image, transforming this into multiple, equally probable stochastic
realizations of the full 3-D pore structure, and finally applying a Lattice-Boltzman
method flow simulation to calculate an effective permeability for each realization (Figure
5.5).
3.1. Image processing
The first step in estimating permeability from thin-section is to convert representative
digital images of the material into binary (pure black and white) images (Figures 5.5a, b).
These binary conversions are used for porosity estimation, variogram modeling and as
conditional data for the stochastic 3-D pore-structure realizations. Two types of initial
digital images are common: true-color RGB (red-green-blue) images captured on
standard petrographic microscopes using transmitted light; and grayscale (intensity)
images captured with scanning electron microscopy (SEM). In both cases, the goal is to
produce accurate binary maps representing the spatial distribution of pores and grains.
Impregnation with colored epoxy (generally blue or red) prior to sectioning renders
porosity distinct in RGB images. The original method of Keehm et al. (2004) creates a
suite of index colors from which an operator selects those corresponding to porosity.
With this approach, a degree of uncertainty in setting color thresholds to differentiate
grains from pores is unavoidable. For grayscale images, there is only the intensity plane,
with low intensity (black) corresponding to porosity. This reduces the binary conversion
to a simple choice of threshold value, generally on a scale of 1 (pure black) to 256 (pure
white). SEM images, particularly those collected in backscattered electron composition
(BEC) mode, therefore provide generally sharper resolution than RGB images, but do
require specialized equipment to obtain.
3.2. Pore-structure realization
Various methods for constructing 3-D pore-structure realizations from 2-D binary
images have been developed, including those of Adler et al. (1990), Yeong and Torquato
(1998) and Øren and Bakke (2002). A brief comparison of these is presented by Keehm
et al. (2004), who offer a geostatistical approach based on the sequential indicator
simulation (SIS) method of Deutsch and Journel (1998). The SIS method produces
130

stochastic 3-D realizations based on the two moments of a binary image—its porosity and
a two-point correlation function (variogram). A random path is followed through the
nodes of a model 3-D grid. At each node, a local conditional cumulative distribution
function (ccdf) is estimated by indicator kriging. Each successive ccdf is conditioned to
the binary image and to all previous node selections, and a random selection from the
function gives the value—pore or grain—for the current node. A 3-D (cubic) binary field
with the same pore-grain spatial characteristics as the original 2-D image results (Figure
5.5c). Comparing multiple realizations generated from a single image provides a means
of assessing pore-structure variability.
3.3. Permeability estimation
Permeability is estimated by conducting numerical flow simulations on the 3-D pore-
structure realizations, which tend to be geometrically complex and may contain statistical
noise due to the stochastic nature of their construction (Keehm et al. 2004). The Lattice-
Boltzmann (LB) method is a robust technique that simulates flow according to simple
rules governing local interactions between individual particles (Doolen 1990) and
recovers the Navier-Stokes equations at the macroscopic scale (Ladd 1994). Prime
advantages of the LB method are its ability to handle any discrete geometry without
simplification (Cancelliere et al. 1998), and its accuracy in describing flow through
porous media (Ladd 1994; Bosl et al. 1998; Keehm et al. 2001). Artifacts can occur in the
local velocity fields (Manwart et al. 2002), but applying time-averaged velocities
mitigates these effects (Ladd 1994). The LB simulations are accomplished by assigning a
pressure gradient (∇P) across opposite faces of the cubic pore-structure realization. From
the local flux (Figure 5.5d), a volume-averaged flux 〈q〉 can be calculated. The
macroscopic or bulk permeability (κ) is then calculated according to Darcy’s Law:
κ
〈 q〉 = − ∇P
η
where η is the dynamic viscosity of the fluid. As a statistical measure of κ variability,
LB simulations can be conducted on multiple 3-D pore-structure realizations generated
from the same image. The final permeability estimate is then the average of the values
from all the simulations.
131
(1)

compaction band
A
5 mm
A‘
compaction band
compaction band
depositional bedding
Figure 5.4. Representative thin section of a tabular compaction band in the Aztec
sandstone. White grains are quartz, brown grains are stained feldspar, black grains are
iron oxide, blue is epoxy-filled porosity. Photo inset (upper left) shows the sample core
from which the thin section was made. Transects A-A’ and B-B’ (normal to bedding)
pertain to the calculations shown in Figure 5.8.
132
B
B‘

0.5 mm
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
Figure 5.5. Permeability estimation methodology workflow: A) raw backscattered
electron composition image; B) binary conversion map of porosity (black) versus mineral
constituents (white); C) stochastic 3-D pore-structure realization; D) parallel slices
through the pore-structure realization showing local mass flux predicted by Lattice-
Boltzmann flow simulation (values normalized by maximum).
133

4. Application to the Aztec sandstone
In order to test this computational estimation method under real-world conditions of
limited sample supply, we applied the algorithm to a single, representative thin section of
Aztec sandstone showing a cm-thick CB surrounded by relatively undeformed matrix
(Figure 5.4). As is typical with foreset deposition in æolian sandstones, bedding consists
of alternating, well-sorted coarse-grain (~0.4 mm) and fine-grain (~0.1 mm) layers up to
several millimeters thick. Poorly sorted mixed-grain layers are also common. Within each
of the three distinct bed types, the distribution of grains and pores is relatively
homogeneous, both in the matrix and in the CB.
Digital images were collected in BEC mode on a standard SEM at 15 kV and a
magnification of 80x. Two images were collected from each bed type, with mosaic
composites used as necessary to ensure representative coverage in coarse-grain beds. A
consistent intensity threshold of 72 out of 256 was used for the binary conversions, and
ten 3-D pore-structure realizations were computed for each of the 12 binary images. Flow
simulations were then conducted on each pore-structure realization to yield 120 total
estimates of permeability relative to porosity. Figure 5.6 shows an example BEC image
with its binary conversion for each bed type from both the matrix and the CB.
4.1. Simulation results
The resulting estimates of porosity and permeability are plotted in Figure 5.7; the
averaged permeability results are summarized in Table 5.1.
Porosity and permeability estimates for the matrix range from about 20-27% and 200-
3,500 millidarcys (mD). Not surprisingly, average porosity for the pore-structure
realizations derived from the mixed-grain bed images is lowest (22%), while average
permeability estimated for the coarse-grain pore-structure realizations is highest (1,392
mD). The fine-grain beds, which exhibit the tightest grain-size distribution, produced the
highest average porosity estimate (25.25%) and the lowest average permeability estimate
(406 mD). The combined average estimates for the matrix are 23.6% and 776 mD. This
simple average bulk permeability value, however, does not account for the inherently
anisotropic, layered nature of the sandstone. Figure 5.8 presents a more realistic approach
to estimating directional permeability in the matrix, yielding best-estimate values of
1,053 mD parallel to bedding and 804 mD normal to bedding.
134

Sandstone matrix
Compaction band
Coarse-grain Fine-grain Mixed-grain
Figure 5.6. Backscattered electron composition (BEC) images of the three depositional
bedding types in the Aztec sandstone (coarse-grain, fine-grain and mixed-grain) shown
above their corresponding binary conversions for both the sandstone matrix (top set) and
compaction band (bottom set). In the BEC images black is porosity, dark gray is
kaolinite, medium gray is detrital quartz and light gray is detrital feldspar.
135
BEC Binary BEC Binary

Permeability (mD)
10 4
10 3
10 2
10 1
Coarse-grain
Fine-grain
Mixed-grain
Mean coarse
Mean fine
Mean mixed
Mean CB
(48)
(33)
Porosity (fraction)
Matrix
Compaction band
10
0 0.05 0.1 0.15 0.2 0.25 0.3
0
Figure 5.7. Semi-log scatter plot of permeability versus porosity for the computational
estimates of matrix and compaction band derived from the thin section shown in Figure
5.4. The adjacent box plots represent the corresponding ranges and distributions of
available permeability measurements for the Aztec and Navajo sandstones, with the total
number of data points in parentheses.
Table 5.1. Summary of permeability estimates
Sandstone matrix Compaction band
Coarse Fine Mixed Coarse Fine Mixed
Mean 1,392 mD 406 mD 530 mD 5.22 mD 4.29 mD 5.77 mD
Min 879 mD 253 mD 258 mD 2.04 mD 1.36 mD 1.20 mD
Max 3,459 mD 655 mD 1,574 mD 10.67 mD 9.12 mD 19.73 mD
136

Porosity and permeability estimates for the CB range from about 9-12% and 1-20
mD, with the results from the three bed types substantially overlapping (Figure 5.7). This
suggests that permeability within the band is approximately homogeneous and isotropic.
The combined average estimates for CB porosity and permeability are 10.6% and 5.1
mD. It turned out that exactly half of the 3-D pore-structure realizations for the CB—12
for the coarse-grain beds and nine each for the fine-grain and mixed-grain beds—yielded
no connected flow paths and so returned zero permeability. Given that the minimum non-
zero permeability result was 1.2 mD, we deemed all zero values to be unrepresentative
and discarded them (see Conclusions). Our average estimate for CB permeability of 5.1
mD is therefore a maximum, and we suggest that the true value is probably around 3 mD,
given that the average including zeros is 2.5 mD.
4.2. Comparison to measured values
Our computational approach yields permeability estimates for the Aztec sandstone of
about 1,000 mD along bedding, 800 mD across bedding and 5 mD within CBs, which
generally cut across bedding at high angle. Permeability reduction due to compaction
within the bands is therefore estimated to be at least two orders of magnitude—the
middle of the range reported for deformation bands in general (see Introduction).
Antonellini and Aydin (1994) report minipermeameter measurements for both matrix and
compactive deformation bands from the Navajo sandstone of southeastern Utah,
recognized as a depositional and chronological equivalent of the Aztec (Marzolf 1983;
Blakey 1989). Their results do not distinguish between bedding parallel and bedding
normal permeability. Flodin et al. (2005) provide a handful of additional lab
measurements for the Aztec matrix, also without reference to bedding orientation.
Together, these measurements—ranging from 123 mD to 5,991 mD in the matrix (mean
of 1,768 mD, and from 1.35 mD to 38.33 mD in the bands (mean of 14.14 mD)—
correspond well with our computational estimates, while being somewhat higher (Figure
5.7). We ascribe this discrepancy to spatial permeability variation within the sandstone,
as much as to any systematic inaccuracy in the computational method, noting again that
the estimates derive from a single sample location.
137

B‘
A‘
A
c
f
m
c
m
c
f
c
f
c
f
c
f
4.8 mm
0.9 mm
1.8 mm
3.3 mm
B
4.5 mm
0.9 mm
0.6 mm
1.5 mm
0.9 mm
1.2 mm
1.5 mm
7.2 mm
0.9 mm
K n
L = 3 cm
=
K n
=
K p
k f k m
K p
kc lc + kf lf + km L
l c
k c k f k m L
+ kc km lf +
l m
k c k f
l m
Bulk Permeability Estimates
Kp Kn Mean
Minimum
Maximum
1,053 mD 804 mD
655 mD 482 mD
2,614 mD 1,785 mD
Figure 5.8. Computation of bulk permeability parallel (Kp) and normal (Kn) to
depositional bedding for the composite transect A-A’—B-B’ shown in Figure 5.4. Kp and
Kn are calculated as weighted arithmetic and harmonic averages, respectively (Deutsch,
1989). Coarse-grain permeability and the total thickness of coarse-grain beds along the
transect are connoted by kc and lc, respectively. Similarly, kf, lf, km, and lm represent these
values for fine and mixed-grain beds. The mean, minimum and maximum bulk
permeability estimates are computed using the corresponding mean, minimum and
maximum values of kc, kf and km (Figure 5.7).
138

5. Conclusions
The method applied in this paper comprises a convenient and robust computational
technique for estimating permeability from thin-section images. It is particularly well
suited to small structures and samples that do not lend themselves to direct measurement,
and to situations where sample supplies are limited or unconsolidated materials requiring
impregnation cannot otherwise be analyzed. We have demonstrated that both matrix and
compaction-band permeability estimates derived from a single thin section of Aztec
sandstone reliably reflect the range of measured values reported for comparable samples.
Furthermore, we were able to isolate variations in permeability due to grain-size and
sorting differences between the characteristic depositional layers—well-sorted coarse,
well-sorted fine and poorly sorted mixed—allowing us to assess the inherent permeability
anisotropy due to bedding. Application of the method to low-porosity CBs, however,
does highlight a resolution limitation warranting further comment.
The stochastic 3-D realization algorithm produces pore structures lacking through-
going connectivity with increasing frequency as porosity drops below about 12%. Half of
our CB pore-structure realizations, with a mean porosity of 10.6%, yielded zero
permeability. Sensitivity testing reveals that below about 7% porosity, the probability of
achieving a nonzero permeability estimate becomes negligible. The reality for clastic
materials is that permeability does not begin to disappear until a minimum porosity
threshold of about 2-4% (Mavko and Nur 1997). This limitation of the method could be
mitigated in a variety of ways, such as conducting pore-structure realizations until the
desired number of non-zero effective permeability results is achieved; increasing
resolution of the 3-D grid (i.e. decreasing node spacing), which however would also
greatly increase computation time; or using a different 3-D pore-structure construction
technique, such as multi-point realization.
Nonetheless, we conclude that representative permeability estimations for porous,
granular materials—even from a single bedding layer or compaction band in sandstone—
can be derived from thin-section images using the method demonstrated in this paper. In
fact, the permeability data gleaned in our test case from a single thin section would be of
great value in anticipating the potential impact of CBs on fluid flow in a subsurface
139

equivalent of the Aztec sandstone, with important implications for aquifer and reservoir
management.
6. Acknowledgements
The authors wish to thank David Pollard and Gary Mavko for their comments,
insights and support; John Childs for able assistance in the field; and Robert Jones for
guidance on the SEM. Primary funding for this work was provided by the U.S.
Department of Energy, Office of Basic Energy Sciences, Geosciences Research Program
under grants DE-FG03-94ER14462 (David Pollard and Atilla Aydin) and DE-FG02-
03ER15423 (Amos Nur and Gary Mavko). Additional support came from the Stanford
Rock Fracture Project (RFP) and the Stanford Rock Physics and Borehole Geophysics
Project (SRB).
140

1. Abstract
Chapter 6
Permeability effects of deformation band arrays in sandstone
We apply established numerical modeling techniques to the calculation of effective
permeability for porous sandstone containing systematic arrays of low permeability
deformation bands (DBs). The numerical method, derived from homogenization theory,
can produce effective permeability results for characteristic DB geometries, and provides
a quantitative framework with which to expand these effects to the reservoir-simulation
scale from spatially limited data. The method is demonstrated in two dimensions for each
of three characteristic DB patterns—parallel, cross-hatch and anastomosing—exposed in
the Aztec sandstone at the Valley of Fire, Nevada, which provides an excellent exhumed
analog for active sandstone reservoirs. Our analysis indicates that these extensive,
systematic DB patterns can reduce effective permeability by as much as two orders of
magnitude at scales relevant to reservoir production, while inducing similar magnitudes
of permeability anisotropy. The potential permeability impacts of DB arrays on reservoir
production thus rival those routinely attributed to depositional heterogeneity—e.g.
bedding and shale streaks. We suggest therefore that properly accounting for the
aggregate effects of DBs where they occur could be as important to optimal reservoir
simulation and production management in sandstone as accounting for sedimentary
architecture.
2. Introduction
Sedimentary and structural heterogeneities both large (e.g. shale lenses, seismically
detectable faults) and small (e.g. bedding, fractures) have been shown to profoundly
influence effective permeability in sandstone. Small features, however, cannot explicitly
be modeled in standard flow simulations using coarse (10 to 100 m) grid blocks. For
porous, granular materials like sandstone, the simplifying assumption of isotropic
background permeability is commonly assumed, with perhaps an anisotropy related to
bedding. To more realistically account for the aggregate influence of small-scale features,
an effective permeability must be calculated and assigned to each simulation block. This
141

effective permeability represents an average bulk or “upscaled” permeability for the host
rock and small-scale heterogeneities taken together.
A variety of numerical techniques have been developed to compute effective
permeability for use in coarse scale modeling. Extensive discussion of these can be found
in the reviews of Wen and Gomez-Hernandez (1996) and Renard and de Marsily (1997).
Many of the techniques also have been used to quantify the effects of fine scale features
on large scale permeability. For example, several researchers have computed effective
permeability values for finely interbedded sand-shale sequences, finding that shale streak
volume fraction (Vsh ) is the key controlling parameter. Desbarats (1987) modeled
sand/shale sequences composed of homogeneous, isotropic sandstone (permeability=kss)
and homogeneous, isotropic shale lenses (permeability=ksh). For ksh/kss=10 -4 , he found the
effective vertical permeability to be an order of magnitude less than kss for Vsh ≈30%, and
2 orders of magnitude less for Vsh ≈65%. He found the effective horizontal permeability
to be reduced by an order of magnitude for Vsh ≈70%. Deutsch (1989) determined that for
a relative shale/sandstone permeability of ksh/kss=10 -5 , effective isotropic permeability
drops by an order of magnitude for Vsh ≈50% and 2 orders of magnitude for Vsh ≈65%.
These results suggest that for Vsh greater than 65%, effective horizontal and vertical
permeability drop by about 1 and 2 orders of magnitude, respectively, thus inducing an
order-of-magnitude permeability anisotropy.
Previous studies also have demonstrated that other types of small scale sedimentary
and structural heterogeneities can exert substantial permeability effects in pure sandstone.
Durlofsky (1992) showed that cross bedding in æolian sandstones can reduce effective
permeability by an order of magnitude and induce an anisotropy of maximum to
minimum permeability (kmax/kmin) greater than 5. Similarly, it has been shown that the
presence of open joints can increase effective permeability by 2 or more orders of
magnitude, and that the presence of low porosity deformation bands (DBs) can reduce
effective permeability by 2 or more orders of magnitude (Antonellini and Aydin, 1994;
Lothe et al., 2002; Taylor and Pollard, 2000; Taylor et al., 1999). As a natural
consequence of their dominant tabular/planar geometry, both joints and DBs also can be
expected to induce permeability anisotropy (Antonellini and Aydin, 1995).
142

In this paper, we use numerical methods and computer codes developed by Durlofsky
(1991; 1992) to quantify the influence of realistic DB patterns on bulk sandstone
permeability in two dimensions (2-D). We restrict ourselves to 2-D analyses so as not to
obscure the essence of the approach—or the potentially important impact of DBs on bulk
sandstone permeability—with the additional conceptual and computational complexity
inherent to 3-D treatments. Also, insofar as DBs are themselves grossly tabular/planar
features, with most arrays consisting of just one or two dominant orientations, the DB
pattern visible on any 2-D observation plane (e.g. outcrop face) that is oriented at high
angle to the bands will be substantially similar for all adjacent parallel planes. That is,
pattern and permeability continuity in the third dimension can reasonably be assumed.
Nonetheless, the numerical methods and analyses presented here could naturally be
extended to 3-D.
We apply the 2-D method to three distinct and characteristic DB patterns—parallel,
cross-hatch and anastomosing—as recognized and mapped within the æolian Jurassic
Aztec sandstone of the Valley of Fire State Park, Nevada (Figure 6.1). The Aztec
sandstone constitutes an excellent exhumed analog for active æolian sandstone reservoirs
potentially affected by DBs, such as the Anschutz Ranch East Field in the Nugget
sandstone of Wyoming, Idaho and Utah (Lindquist, 1988). The Nugget is considered a
chronostratigraphic and depositional equivalent of the Aztec (Marzolf, 1986), and
Anschutz Ranch comprises a large anticlinal trap within the Nugget. Production evidence
for Anschutz Ranch indicates that lowered porosities and permeabilities related to
abundant DB fabrics (observed in core and outcrop) have degraded overall reservoir
quality, introduced flow barriers and produced anisotropic behavior (Lewis, 1993).
Production problems attributed to deformation bands have also been reported for the
Nubian Sandstone in Egypt and for sandstone reservoirs located offshore from Nigeria
(Olsson et al., 2003).
3. Deformation bands
Deformation bands are thin, tabular, bounded features that accommodate pore-loss
compaction, often in association with shear displacement, in sandstones via granular
rearrangement, cataclasis and chemical diagenesis (Aydin, 1978; Aydin and Johnson,
1983; Du Bernard, 2002; Engelder, 1974; Jamison and Stearns, 1982) (Figure 6.2).
143

NV UT
CA
AZ
Park Road
Map
Detail
Park Boundary
Route 169
0 5
km
NEVADA
ARIZONA
10 km
Valley
of Fire
State Park
N
LAKE
MEAD
115 o 30'
Figure 6.1. Schematic location map for the Valley of Fire State Park in Nevada, about 60
km northeast of Las Vegas. The colorful Aztec sandstone comprises the dominant
lithology and primary tourist attraction in the park.
144
36 o 15'

DB tip
DB
DBs
Grains
Pores
Figure 6.2. Compactive deformation bands (DBs) as seen in the Aztec sandstone, Valley
Clay
of Fire, NV. These DBs are tabular, bounded features of pore-loss compaction that
typically crop out as systematic and often extensive arrays of resistant, positive-relief
ridges within a relatively undeformed sandstone matrix (top). Individual bands are
approximately planar, displaying distinct tips even when closely spaced—width of
measuring tape is one inch (bottom left). Viewed in thin section, grain cracking,
rearrangement and pore loss are readily apparent, illustrating why DBs act as barriers to
fluid flow. Unlike many DBs, however, those in the Aztec tend to exhibit limited
cataclasis and little or no net shear offset (bottom right).
145

Commonly from ~1 mm to ~1.5 cm in thickness and ~1 to ~100+ m in extent, DBs
generally exhibit displacements, both closing-mode (compaction) and shear, on the order
of millimeters (Antonellini et al., 1994; Aydin and Johnson, 1978; Mollema and
Antonellini, 1996). All types of DBs are characterized by the absence of any distinct
plane of displacement discontinuity. The petrophysical changes accommodated within
DBs—compaction, grain size reduction, chemical diagenesis and clay infiltration—
typically produce internal porosity reductions of about an order of magnitude—often
down to a residual of only 1 or 2% (Ahlgren, 2001; Antonellini and Aydin, 1994). The
porosity loss, in combination with related reductions in mean pore throat diameter and
connectivity, can produce drastic drops in permeability within DBs, and significant
efforts have been made to quantify this phenomenon.
Freeman (1990) used gas permeameter readings on plugs of Aztec sandstone to
determine that the presence of DBs can cause 2-order-of-magnitude reductions in
effective permeability relative to DB-free rock. Antonellini and Aydin (1994) used a gas
injection mini-permeameter to measure individual DB permeability in a variety of
sandstones, finding DBs to be 2 to 4 orders of magnitude less permeable than the host
rock, with an average permeability drop of 3 orders of magnitude. They also found that
the relative permeability drop is greater for DBs in higher porosity sandstones (Entrada
and Navajo at >20%) than in lower porosity sandstones (Morrison and Chinle at

did not report permeability values for individual bands, Taylor and Pollard (2000) used
his data to estimate that average DB permeability is about 2.3 orders of magnitude less
than in the host rock. Gibson (1998) found that the permeability of DBs depends on the
clay content of the parent sandstone and can vary from 1 to 4 orders of magnitude, with
an average value of ~2.2 orders of magnitude.
4. Characteristic DB patterns in the Aztec sandstone
The Valley of Fire State Park lies about 60 km northeast of Las Vegas and about 30
km west of the Utah border (Figure 6.1). The predominant outcrop unit within the park is
the Jurassic Aztec sandstone, commonly considered correlative with the Navajo
sandstone. The Aztec is an æolian, subarkosic sandstone up to 2 km thick typified by
large-scale cross-bed sets up to 10+ meters thick (Marzolf, 1983; Myers, 1999). The
Aztec has a mean grain size of about 0.25 mm, an average porosity of around 20%
(Taylor and Pollard, 2000) and an average matrix permeability ranging from 0.1 to 1.0
darcys (Antonellini and Aydin, 1994; Myers, 1999). The Aztec exhibits a variety of
brittle structural fabrics that include DBs, joints and faults, with DBs consistently
appearing to be the oldest features (Hill, 1989; Myers, 1999; Taylor and Pollard, 2000).
Hill (1989) hypothesized that DB formation within the Aztec was related to widespread
compression during emplacement of the Summit/Willow Tank and Muddy Mountain
thrusts during the Early Cretaceous Sevier orogeny.
Fieldwork in the Valley of Fire has revealed at least three wide spread, systematic and
volumetrically significant patterns of dominantly compactive DBs in the Aztec sandstone
over an area of more than 10 km 2 . All of these characteristic patterns—parallel, cross-
hatch and anastomosing—have previously been described at various locations, though
generally using different nomenclature (Antonellini and Aydin, 1994, 1995; Aydin and
Reches, 1982; Burhannudinnur and Morley, 1997; Davis, 1998; Hill, 1989; Jamison and
Stearns, 1982; Mollema and Antonellini, 1996; Taylor and Pollard, 2000; Underhill and
Woodcock, 1987; Woodcock and Underhill, 1987). Within the Aztec, the 2-D density of
DBs exposed in any given outcrop (area DBs/outcrop area) ranges from ~1% up to ~20%.
Assuming reasonable continuity in the third dimension, as is clearly suggested by field
observations in outcrop areas of high relief, the volume percent density of DBs also
ranges between 1% and 20%.
147

Although systematic arrays of DBs pervade the middle and upper Aztec, DB-poor
exposures tens to hundreds of meters or more on a side are also common. The reasons for
this patchwork quality to the DB fabric at a variety of scales is as yet poorly understood,
but may be related to the æolian sedimentary architecture of the Aztec and differences in
how distinct dune packages responded to tectonic loading during DB formation.
Nonetheless, any given characteristic DB pattern recognized in outcrop can be seen to
persist at length scales ranging from meters (the thickness of small cross-bed packages)
to tens of meters (the thickness of large cross-bed packages) to hundreds of meters (for
DBs that pass through multiple dune boundaries).
4.1. Parallel
Patterns of approximately parallel DBs crop out extensively throughout the Aztec
over areas ranging from tens to hundreds of meters on a side. Individual bands within the
parallel sets commonly range up to ~1.5 cm thick and are spaced anywhere from
centimeters to meters apart (Figure 6.3). It is not uncommon, however, to see groups of
individually distinct DBs running parallel to each other only millimeters apart, sometimes
merging into what appears in outcrop to be a single band 10 cm or more thick. Also,
groups of closely spaced DBs often form distinct sets spaced a meter or more from the
next adjacent, closely spaced set. DBs cropping out in parallel patterns throughout the
Aztec cluster around three mean strike/dip orientations—350°/70°, 210°/60° and
240°/30° (using the convention of dip direction oriented 90° clockwise from the strike
azimuth). The first two of these sets cut across depositional bedding at a high angle,
while the third generally runs sub-parallel to bedding. The north-trending, steeply east-
dipping DB orientation (350°/70°) strongly dominates in terms of abundance, persistence
and distribution, and is the one shown in both Figures 6.2 and 6.3.
4.2. Cross-hatch
Cross-hatch patterns of DBs, which generally crop out over areas from meters to 10s
of meters on a side, consist of two distinct sets of parallel bands that commonly intersect
each other at a high angle (Figure 6.4). In the Aztec, individual bands within the cross-
hatch patterns exhibit the same range of widths as bands in the parallel patterns and the
same tendency to occur as closely spaced pairs. Spacing between bands ranges from 2 cm
148

0 meters
2
Figure 6.3. A typical outcrop pattern of approximately parallel DBs in the Aztec
sandstone. View is to the north, bedding is approximately horizontal.
cm
0 8
1 meter
Figure 6.4. A typical cross-hatch pattern of DBs in the Aztec sandstone. Bands in the two
sets are at high angle to each other and to bedding (approximately coincident with the
outcrop face (main photo). Looking down the axis of intersection of a cross-hatch set
illustrates how planar DBs can be, and how sharp their intersections (inset photo).
149

to 2 m, with both sets in a cross-hatch pattern typically exhibiting similar spacing. The
cross-hatch pattern is both less abundant and less persistent than the parallel and
anastomosing DB patterns, and often at least one DB set within a cross-hatch pattern
terminates at the nearest dune boundary. The two sets of DBs within the cross-hatch
patterns exhibit mean orientations generally similar to those of any two of the three
dominant orientations described above for the parallel pattern case, and commonly
intersect with an acute angle of ~80°.
4.3. Anastomosing
Anastomosing patterns of DBs consist of sub-parallel, intersecting bands generally
exhibiting the same dominant, high-angle-to-bedding orientation described above for the
parallel pattern, but with trends and dips varying by 25 degrees or more over the scale of
centimeters to meters (Figure 6.5). Deformation bands within the anastomosing patterns
exhibit ranges of thickness (up to ~1.5 cm) and spacing (centimeters to meters) similar to
those reported above for the parallel case. Anastomosing band sets have previously been
described in the literature, but always as existing in narrow zones of very closely spaced
bands—typically associated with faults having distinct slip planes (Aydin, 1978; Aydin
and Johnson, 1983). In the Aztec, however, anastomosing DB arrays commonly cover
areas up to hundreds of meters on a side, while passing through multiple dune
boundaries. Indeed, the anastomosing pattern of steeply dipping, generally north-trending
DBs dominates the Aztec, and the arrays of approximately parallel DBs described above
can reasonably be considered as simply more orderly sub zones within the greater,
anastomosing configuration.
5. Numerical modeling methods
Numerical techniques can be used to calculate effective permeability for any
characteristic DB pattern that can be represented as a finely meshed grid of cells, each
cell specifying either DB or sandstone matrix permeability (Figure 6.6). The method
involves solving a single fluid phase, steady state pressure equation subject to spatially
periodic boundary conditions over the entire grid to yield this upscaled or effective
permeability. The fluid flux and pressure boundary conditions are said to be spatially
periodic insofar as they implicitly assume that the domain block in question is surrounded
150

0 meter
Figure 6.5. A typical anastomosing pattern of DBs located along trend, but about 200 m
south of the outcrop shown in Figure 6.3. View is to the north, bedding is very nearly
coincident with the outcrop face in the foreground.
1
151

y identical blocks all subject to the same regional pressure gradient that drives flow.
This modeling approach derives from an application of homogenization theory as
presented in detail by Durlofsky (1991; 1992).
5.1. Governing equations
Single-phase, steady state, incompressible flow through a heterogeneous porous
media in the absence of sources or sinks can be described by Darcy’s law and the
requirements of continuity as:
1
u = − k ⋅∇p
µ
∇⋅u = 0 (2)
where u is the local fluid velocity vector, ∇ p is the local fluid pressure gradient
vector, µ is the fluid viscosity, and k is the tensor representing directional permeability.
The units of k are length squared, as can be demonstrated by dimensional analysis of
Equation (1). In a 2-D x-y coordinate system, k can be represented as:
⎡kxx
kxy⎤
k = ⎢ ⎥ . (3)
⎣kyx
kyy⎦
where kxy is the permeability component relating fluid velocity in the x-direction to
pressure gradient in the y-direction. Permeability is commonly measured in darcys (1
darcy = 9.87 x 10 -9 cm 2 ), and the native permeability of uniform sandstone ranges from
about 10 -2 to 10 2 darcys in any given direction (Freeze and Cherry, 1979).
The homogenization approach considers k to vary over two spatial scales—a fine
scale, f, and a coarse scale, c, with f

1 meter 1 meter
Figure 6.6. An idealized DB pattern displaying characteristics of both the cross-hatch
and anastomosing types at the scale of one square meter (left); and a representation of
this pattern on a 40 x 40 cell finite difference/finite element grid where black corresponds
to DB permeability and white indicates native sandstone permeability (right).
f-scale
f 2
c 1 ~10 to >100m c-scale
f1
100m c-scale
f1

contiguous areas. This type of characteristic pattern replication lends itself naturally to
numerical solution using spatially periodic boundary conditions, as detailed below.
It is important to note that spatial periodicity does not imply fractal scaling. Based on
our observations of DBs in the Aztec, we can state that their characteristic patterns
definitely are not fractal. Also, it is not necessary for actual pattern periodicity in nature
to be nearly as distinct and regular as idealized in Figure 6.7 for the numerical methods
described here to work effectively.
Using the two-scale representation of k and combining Darcy’s law with continuity
yields the pressure equation for steady state, incompressible flow:
[ ( c,f
) ⋅ ∇ = 0
∇ ⋅ k p]
(4)
where k varies on both the fine and coarse scales. Our intent is to calculate an
effective permeability tensor, k * , that varies only on the c-scale but that implicitly
accounts for permeability variations on the f-scale. This allows us to model the system
using an equation of the form of (4) but with k * (c) replacing k (c,f). The solution to this
“upscaled” equation is much less computationally demanding than the solution of
equation (4), because variations on the fine scale f need not be resolved. The essential
step is to transform k (c,f) to k * (c) by calculating the effective permeability tensor of a c-
scale block containing f-scale heterogeneities (DBs). It has been shown (e.g. Bourgeat,
1984)) that homogenized upscaling is exact when two distinct spatial scales of
permeability variation exist. Subsequent work (e.g. Durlofsky, 1991) has demonstrated
that upscaling can also be applied with reasonable accuracy to more realistic situations
such as represented by DBs in the Aztec, where distinct spatial scales of permeability
variation cannot be defined.
5.2. Boundary conditions and solution
In order to drive large-scale flow, pressure variation on the coarse scale c is imposed.
This can be expressed generally as:
p = p 0 + G ⋅ (c − c 0 ) (5)
where c0 is the center of the region under study, p0 is the fluid pressure at c0, and G is
the local fluid pressure gradient ( ∇p ) over the c-scale at c0.
The average flow through a
154

periodic, f-scale block, denoted , is related to the c-scale pressure gradient by the
effective permeability tensor k * :
k G
* 1
u = - ⋅ . (6)
µ
To calculate k * , we solve the pressure equation (4) over the f-scale block and then
compute the average flux through it. The boundary conditions for this solution
assume periodicity of the system up to the c-scale, as well as the imposition of the
pressure gradient G. The boundary specifications require that flow across the domain
boundaries be periodic over the f-scale block, and that the pressure be periodic but with a
jump due to the large-scale gradient. Boundary conditions can be specified explicitly for
an f-scale block (Figure 6.8) by resolving G into its two components: G = G1i1 + G2i2,
where i1 and i2 are unit vectors in the coordinate directions f1 and f2, respectively.
Comparing pressures and velocities on opposing sides with unit normals n1 and n2 yields:
yields:
p ( f , f 0)
= p(
f , f = b)
− G × b
(7a)
1
2 = 1 2
2
( 1 , f 2 = 0)
⋅n
1 = −u(
f1,
f 2 = b)
n2
u f ⋅ . (7b)
Comparing pressures and velocities on opposing sides with unit normals n3 and n4
p ( f 0,
f ) = p(
f = a,
f ) − G × a
(7c)
1 = 2
1 2 1
( 1 = 0, f 2 ) ⋅n
3 = −u(
f1
= a,
f 2 ) n4
u f ⋅ . (7d)
To compute the complete effective permeability tensor k * , two problems must be
solved, one for which G1=0, G2≠0 and one for which G1≠0, G2=0. By specifying a
reference value for pressure at a single point in the domain, equation (4) can be solved
subject to equations (7). The flux through the f-scale block can then be determined by
integrating the velocities, and finally k * computed from equation (6). Durlofsky (1991)
presents this computational approach in full detail.
155

(0,b)
n 3
(0,0)
f 2
n 2
n 1
(a,0)
(a,b)
Figure 6.8. Gridded block representing permeability variation on the fine scale to which
periodic fluid flux and pressure boundary conditions are applied in solving for effective
permeability. The lower left corner is the origin (0,0) and the block has dimensions a in
the f1 coordinate direction and b in the f2 direction, while n1 through n4 are unit normals
to its sides (adapted from Durlofsky, 1991).
156
n 4
f 1

5.3. Finite difference/finite element method
For the purposes of computational efficiency, a coupled, two-step finite
difference/finite element numerical method was used to compute k * . In order to
adequately resolve centimeter-thick DBs within meter-scale patterns, finely discretized
grids a few hundred cells on a side were generally necessary, with each cell being
assigned one of two isotropic permeability values: kmatrix or kband (e.g. Figure 6.6). Again
for the sake of computational efficiency, two coarsening steps were used to compute the
final, upscaled permeability tensor. For example, to compute k * for a pattern meshed at
400 × 400 cells, the first coarsening step might be to a 20 x 20 cell system (yielding 400
intermediate permeability tensors) and then to the final k * in the second step. An efficient
finite difference method was used to make the first coarsening step. The second step to k *
required a finite element procedure that is less efficient, but capable of handling the
matrix of full permeability tensors produced during the first step. Repeat testing revealed
that the final k * computed is relatively insensitive to the size of the coarsening steps used,
varying by less than 10% for the patterns studied.
6. Effective permeability results
The governing equations, boundary conditions and numerical methods detailed above
were used to calculate 2-D effective principal permeabilities for each of the three
characteristic DB patterns described for the Aztec. For the simpler, more regular parallel
and cross-hatch examples, we used idealized patterns to assess how effective
permeability varies as a function of the relevant physical variables—band thickness,
spacing and intersection angle—and present analytical solutions for comparison where
possible. For the more generally irregular anastomosing pattern, which exhibits
substantial spatial variability that defies analytical treatment, we assess effective
permeability for a representative outcrop pattern. In order to emphasize reductions in
permeability attributable to DBs, all permeability values reported below and in the
figures are normalized by that of the undeformed sandstone matrix (~ 0.1 to 1.0 darcys)
to yield dimensionless ratios of relative permeability. Thus in the analyses that follow,
DBs are assigned isotropic internal permeability values of 10 -2 and 10 -3 , representing the
mid range of values for DBs as reported by other workers and summarized above.
157

6.1. Parallel
Effective permeability for any idealized parallel DB pattern can be calculated
analytically, as previously reported (Antonellini and Aydin, 1994; Taylor et al., 1999).
Our numerical modeling results matched these exact analytical solutions for parallel
patterns, so only the latter are described here for the purpose of building intuition. The
analytical methods are based on the general analysis of flow in layered media by Freeze
and Cherry, pages 30-34 (1979).
Band-parallel effective permeability, k , can be calculated as a weighted arithmetic
average of the band and matrix permeability values:
w k + * b b ( W − wb)km k = p
W
*
p
where kb is the DB permeability, km is the matrix permeability, wb is cumulative DB
thickness, and W is the width of the area studied. Note that using wb/W as a measure of
volume fraction assumes DB continuity in the third dimension, as suggested by field
observations in the Aztec. Band-normal effective permeability can be calculated as the
weighted harmonic average:
k
*
n
=
( ( w / k ) + ( W − w ) / k )
b
b
W
b
m
The band-parallel and band-normal effective permeabilities represent the maximum
and minimum (i.e. principal) values, respectively. An effective permeability tensor, k * ,
for the parallel system thus reads:
*
* ⎡k
⎤ p 0
k = ⎢ * ⎥
(10)
⎣ 0 kn
⎦
Band-normal and band-parallel effective permeability values were calculated for kb/km
ratios of both 10 -2 and 10 -3 and DB volume fractions (wb/W) from 0% to 100% (Figure
6.9). Band-normal effective permeability drops much more quickly at low wb/W than
does band-parallel effective permeability. For kb/km=10 -3 , band-normal effective
permeability is already reduced by an order of magnitude at wb/W=1% (1 mm bands
spaced at 10 cm) and 2 orders of magnitude at wb/W=10% (1cm bands spaced at 10 cm).
158
(8)
(9)

Relative Effective Permeability
1.000
0.100
0.010
kn (kb/km= 10-2)
kp (kb/km= 10-2)
kn (kb/km= 10 ) -3
kp (kb/km= 10 ) -3
0.001
0 10 20 30 40 50 60 70 80 90 100
Volume Fraction Deformation Bands (%)
Figure 6.9. Band-normal and band-parallel effective permeability values (equivalent to
the principal permeability values) as a function of DB volume fraction for perfectly
parallel patterns and DB/sandstone matrix permeability ratios of 10 -2 and 10 -3 . The range
of DB volume fraction observed in the Aztec sandstone is highlighted in gray.
159

By contrast, band-parallel effective permeability is not reduced by an order of magnitude
until wb/W=90%, a DB volume density far exceeding the approximate upper limit of 20%
observed in the Aztec. Thus for parallel DB patterns within the realistic wb/W range, a
permeability anisotropy of up to 2 orders of magnitude can be expected, with the
maximum permeability oriented in the band-parallel direction (Figure 6.9). Note also that
band-parallel effective permeability for both kb/km=10 -2 and 10 -3 are virtually
indistinguishable below wb/W=80%.
6.2. Cross-hatch
Cross-hatch DB sets in the Valley of Fire exhibit a consistent acute intersection angle
of ~80°. In other locales, intersection angles ranging from ~20° to ~90° have been
reported (Aydin and Reches, 1982; Hill, 1989; Jamison and Stearns, 1982; Underhill and
Woodcock, 1987). We begin our analysis of the effective permeability of cross-hatch
patterns by studying those of evenly spaced, orthogonal DB sets as a function of kb/km
and wb/W. Figure 6.10 compares our numerical modeling results for kb/km=10 -2 to an
analytical approximation presented by Taylor and Pollard (2000) and band-normal
effective permeability as calculated by harmonic averaging. The numerical and harmonic
averaging results match well over the entire wb/W range, but correspond to the analytical
approximation only for wb/W below about 5%. For kb/km=10 -3 the wb/W range of
correspondence drops to

Relative Effective Permeability
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
Taylor and Pollard (2000)
Numerical Model
Harmonic Average
0 10 20 30 40 50 60 70 80 90 100
Volume Fraction Deformation Bands (%)
Figure 6.10. Isotropic effective permeability for cross-hatch patterns of evenly spaced,
perpendicular DB sets as a function of DB volume fraction for the DB/sandstone matrix
permeability ratio of 10 -2 . Results from the finite difference-finite element method used
in this paper (FD-FE) modeling are plotted against both the analytical approximation of
Taylor and Pollard (2000) and the permeability computed normal to just one of the DB
sets using the harmonic averaging method. The FD-FE and harmonic averaging methods
agree well over the entire volume fraction range, while the approximation holds only for
volume fractions up to about 5%. The range of DB volume fraction observed in the Aztec
sandstone is highlighted in gray.
161

Relative Effective Permeability
1.00
0.10
Volume Fraction Deformation Bands (%)
o
kmax (15 )
kmax (30 o)
kmax (60 o)
k-isotropic (90 o)
kmin (60 o)
kmin (30 o)
o kmin (15 )
0.01
0 10 20 30 40 50 60 70 80 90 100
Figure 6.11. Principal effective permeability values (maximum and minimum) for evenly
spaced cross-hatch DB patterns with a DB/sandstone matrix permeability ratio of 10 -2 .
Relative permeability is plotted as a function of DB volume fraction for acute intersection
angles of 15 o , 30 o , 60 o and 90 o . The range of DB volume fraction observed in the Aztec
sandstone is highlighted in gray. Inset figure (lower left) illustrates the 2-D permeability
ellipse for an evenly spaced cross-hatch set of DBs intersecting at an acute angle θ.
162

where kmax and kmin are the maximum and minimum principal effective permeability
values, kp is the permeability parallel to the bands when θ=0°, kn is the permeability
normal to the bands when θ =0°, and k+ is the isotropic permeability when half of the
bands are set orthogonal to the other half (θ =90°). Since the analytical solution for k+
only holds for low percentages of DBs, we use k+ as calculated by our numerical method
in the equations above to find the maximum and minimum effective permeability values
as functions of wb/W, θ and kb/km.
Figure 6.11 shows the results for kb/km=10 -2 over the entire range of wb/W for
intersection angles of θ =15°, 30°, 60°and 90°. Both principal effective permeability
values decrease as wb/W increases, with the reduction in kmin being most pronounced. The
degree of this permeability anisotropy also decreases as θ increases, with an isotropic
permeability drop attained at θ =90°. For kb/km=10 -3 , the anisotropy introduced is even
more pronounced, approaching 2 orders of magnitude as θ approaches 0° and the pattern
becomes essentially parallel. For the generally high intersection angles (~80) and low
volume percents (~10%) exhibited for cross-hatch DB patterns in the Aztec, relatively
isotropic permeability reductions ranging from about 0.7 to1.5 orders of magnitude
would be expected.
6.3. Anastomosing
No analytical solutions exist for the systematic, but irregular anastomosing pattern, so
the purely numerical method was used to calculate effective permeability tensors for a
finely discretized version of a real, 12 x 15 m DB outcrop pattern mapped in the Aztec on
a low altitude balloon photograph (Figure 6.12). Not surprisingly, the maximum effective
permeability is oriented roughly parallel to the mean trend of the bands, while minimum
permeability is directed roughly perpendicular. We also compared the effective
permeability of this real anastomosing pattern to an idealized parallel pattern using the
same average spacing and wb/W ratio of 15%. As expected, effective band-parallel
*
p
permeability (k ) was somewhat lower for the anastomosing pattern—0.79 versus 0.85 at
kb/km=10
-2 —due presumably to band connectivity and associated compartmentalization
*
along trend. On the other hand, effective band-normal permeability ( k ) was significantly
higher for the anastomosing pattern—0.26 versus 0.06 at kb/km=10
-2 —due presumably to
163
n

(a) (b)
(c)
0 meters 3
= 10-2 kb /km k b /k m = 10 -3
0.26
0.04
0.79
0.77
kmax log = 0.5 ( kmin)
kmax log = 1.3 ( kmin)
Figure 6.12. Modeling results for a characteristic anastomosing DB pattern from the
Aztec sandstone: (a) low altitude balloon photograph of outcrop used as base map; (b)
digitized version of DB pattern used for numerical modeling as a grid of 500 by 375
cells; (c) permeability ellipses and order-of-magnitude permeability anisotropy values for
DB/sandstone matrix permeability ratios of 10 -2 and 10 -3 .
164

the discontinuous nature of the pattern and the availability of preferred flow paths
crossing a minimum number of DBs. Nonetheless, the anastomosing parallel pattern
studied still could induce a significant effective permeability anisotropy between the
maximum (band-parallel) and minimum (band-normal) directions. For kb/km=10 -2 about a
0.5 order-of-magnitude anisotropy results, for kb/km=10 -3 the anisotropy increases to
about 1.3 orders of magnitude.
7. Summary
We have presented a numerical method for calculating the effective permeability
induced by any given pattern of deformation bands (DBs), and then used it to assess the
2-D effective permeability for volumes of porous sandstone containing each of three
characteristic DB patterns present in the Aztec sandstone of the Valley of Fire State Park,
Nevada—parallel, cross-hatch and the dominant anastomosing.
For substantially parallel patterns, we found that band-parallel effective permeability
remains relatively unaffected for realistic DB volume fractions up to 20%. Likewise,
band-parallel effective permeability is only slightly affected by changes in the
permeability contrast between the DBs and the host rock. By contrast, band-normal
effective permeability drops even for low volume fractions of DBs and is extremely
sensitive to varying internal DB permeability. Parallel band patterns such as those
observed in the Aztec can induce permeability anisotropy of up to 2 orders of magnitude,
with the maximum principal permeability aligned in the band-parallel direction. For
idealized parallel patterns, our modeling results exactly match those computed
analytically.
Cross-hatch band patterns can be analyzed numerically, or in a somewhat simplified
way using a combination of numerical and analytical methods. They yield highly
anisotropic effective permeability reductions for acute intersection angles close to 0°, and
isotropic effective permeability for angles close to 90°. For relatively regular, evenly
spaced patterns, the direction of maximum permeability bisects the acute angle of
intersection. For DB volume fractions of ~10% and acute intersection angles of ~80 o ,
such as found in the Aztec, the cross-hatch pattern can produce up to a 1.5-order-of-
magnitude reduction in bulk permeability and more than an order-of-magnitude
permeability anisotropy.
165

Though grossly systematic, anastomosing DB patterns are strongly irregular in detail
and so their permeability impact can only be assessed using the numerical method. For
this study, a single representative outcrop array from the Aztec was studied. The resulting
permeability effects were substantially similar to those of a comparable, idealized parallel
pattern, though somewhat muted. For the anastomosing pattern, DB connectivity along
trend acts to reduce effective band-parallel permeability, while lack of continuity across
the pattern enhances band-normal permeability. Nonetheless, for the representative
pattern studied, permeability anisotropies of up to 1.3 orders of magnitude are possible,
with bulk permeability reductions of more than an order of magnitude.
8. Discussion and conclusions
In numerically computing the effective permeability of deformation band (DB) arrays
in sandstone, we have demonstrated, their potential to induce order-of-magnitude bulk
permeability reductions and anisotropies at scales relevant to reservoir production. But do
such permeability reductions and anisotropies influence fluid flow in actual field
practice? While it is beyond the scope of our current research effort to address this issue
directly—by correlating DB occurrence with pump test or production data, for example—
such an effort clearly is indicated. We encourage petroleum geologists to be vigilant for
evidence of DBs in core, well bore logs and outcrop, and to investigate the connection
between DB occurrence and otherwise unexplained departures from expected production
performance.
This being said, evidence does exist for the substantial impact on permeability and
fluid flow of DBs in sandstone. For example, in the Aztec patterns of diagenetic
alteration fronts indicative of paleo fluid flow clearly demonstrate that DBs and DB
arrays acted as baffles to flow at a variety of scales (Eichhubl et al., 2003; Taylor and
Pollard, 2000). In addition, direct production evidence for the influence of geological
heterogeneities—including DBs—has been noted in the Aztec-equivalent Nugget
sandstone of the Anschutz Ranch East Field in Wyoming, Idaho and Utah (Lewis, 1993).
Production problems attributed to deformation bands have also been reported for the
Nubian Sandstone in Egypt and for sandstone reservoirs located offshore from Nigeria
(Olsson et al., 2003).
166

Nonetheless, DBs are commonly ignored in the practice of sandstone reservoir
management, even though their potential order-of-magnitude permeability effects rival
those routinely attributed to depositional heterogeneities—e.g. bedding and shale
fraction. We suggest therefore that recognizing and characterizing systematic DB patterns
in the subsurface, and then accounting accurately for their aggregate influence on bulk
permeability with an upscaling process such as presented here, should become a regular
component of reservoir simulation and production management in sandstone. Finally, we
observe that, even with only limited DB data gleaned from scattered core studies,
valuable adjustments to the underlying permeability structure of sandstone reservoirs can
be approximated based on the results presented here for the characteristic DB patterns of
the Aztec sandstone.
9. Acknowledgements
The authors would like to thank Eric Flodin and Nick Davatzes for their assistance in
the field, and Herve Jourde for his help in the field and with the numerical modeling. This
work was supported by the U.S. Department of Energy, Office of Basic Energy Science,
Geosciences Research Program under grant DE-FG03-94ER14462, awarded to David
Pollard and Atilla Aydin.
167

168

1. Abstract
Chapter 7
Flow and transport effects of compaction bands in sandstone
at scales relevant to aquifer and reservoir management
Thin, tabular, low-porosity, low-permeability compaction bands form pervasive, sub-
parallel, anastomosing arrays that extend over square kilometers of exposure in the Aztec
sandstone of southeastern Nevada—an exhumed analog for active aquifers and reservoirs.
In order to examine the potential flow and transport effects of these band arrays at scales
relevant to production and management, we performed a suite of simulations using an
innovative discrete-feature modeling technique to capture the exact pattern of compaction
bands mapped over some 150,000 m 2 of contiguous outcrop. Significant impacts related
to the presence of the bands and their dominant trend are apparent: the average pressure
drop required to drive flow between wells exceeds that for band-free sandstone by a
factor of three, and is 10% to 50% higher across the bands versus along them; reservoir
production efficiency varies up to 56% for a typical five-spot well array, depending on its
orientation relative to the dominant band trend; and contaminant transport away from a
point source within an aquifer tends to channel along the bands, regardless of the regional
gradient direction. We conclude that accounting for the flow effects of similar
compaction-band arrays would prove essential for optimal management of those
sandstone aquifers and reservoirs in which they occur.
2. Introduction
The potential for deformation band arrays at the outcrop scale to reduce bulk effective
2-D permeability by more than an order of magnitude, and to induce significant (10x)
permeability anisotropy, has been established (Sternlof et al., 2004). Their potential to
cause asymmetric patterns of well drawdown has also been demonstrated (Matthai et al.,
1998). However, despite growing recognition that deformation band fabrics are common
in both exhumed (e.g. Underhill and Woodcock, 1987; Edwards et al., 1993; Davis, 1998;
Swierczewska and Tokarski, 1998; Cashman and Cashman, 2000; Shipton and Cowie,
2001; Davatzes et al., 2005) and subsurface (e.g. Fossen and Hesthammer, 1998;
Hesthammer et al., 2000; Hesthammer and Henden, 2000) sandstones, their ability to
169

100 meters
5 meters
NV UT
CA
AZ
Park Roa d
Route 16 9
0 km 5
Map
Detail
Air Photo
NEVADA
ARIZON A
10 km
Valley
of Fire
State Park
N
LAKE
MEAD
115 o 30'
Figure 7.1. Aerial photograph showing a largely bare and level outcrop area of Aztec
sandstone in the Valley of Fire State Park of southeastern Nevada. The geographical
location of the study area is indicated in the map (inset, upper right). A high-resolution
digital scan of the negative made it possible to identify (photo inset lower left) and map
(photo inset lower right) the patterns of individual, cm-thick compaction bands cropping
out as distinct fins over an area of more than 150,000 m 2 .
170
36 o 15'

exert appreciable fluid flow and transport effects at scales of practical relevance to
resource management in aquifers and reservoirs has remained an open question.
To begin addressing this issue, we present a series of flow simulation scenarios that
honor the exact details of individual compaction band arrays mapped over an outcrop area
of some 150,000 m 2 (37 acres) in the Aztec sandstone of southeastern Nevada (Figure
7.1). The simulations were conducted using a discrete-feature model in which the bands
are represented as one-dimensional elements assigned realistic petrophysical properties.
This effort represents the application of state-of-the-art flow modeling to a data set of
unprecedented detail and scale, which was derived from a real geological system that
serves as an exhumed analog for active sandstone aquifers and reservoirs. Analog
outcrop-based permeability and flow-modeling studies, whether focused on gross
sedimentological heterogeneities (e.g. White and Barton, 1999; Stephen and Dalrymple,
2002) or on fine-scale structural features (e.g. Taylor et al., 1999; Sternlof et al., 2004),
can provide valuable insights into the natural variability of inter-well flow and the
balance of parameters governing optimal (or at least improved) well placement.
The Aztec sandstone as exposed in the Valley of Fire State Park lies about 60 km
northeast of Las Vegas (Figure 7.1, map inset). The Aztec is a fine to medium-grained,
sub-arkosic æolian sandstone some 1,400 m thick, which was deposited as large
amplitude cross beds in a back arc basin setting from Late Triassic to Middle Jurassic
time (Bohannon, 1983; Marzolf, 1983). It exhibits a mean grain size of about 0.25 mm,
porosity ranging from 15% to 25% and permeability ranging up to several Darcys
(Antonellini and Aydin, 1994; Flodin et al., 2005). Extensive arrays of sub-parallel,
anastomosing compaction bands (CBs) pervade the upper 600 m of the Aztec exposed
over an area of more than 10 km 2 . Apparently associated with tectonism of the
Cretaceous Sevier orogeny, these bands at high angle to bedding comprise the oldest
structural fabric present, which subsequently was overprinted by shear bands, joints,
sheared joints and strike-slip faults (Hill, 1989; Taylor and Pollard, 2000; Eichhubl et al.,
2004; Flodin and Aydin, 2004; Myers and Aydin, 2004; Sternlof et al., 2005).
As observed in the Aztec and similar sandstones, CBs are discrete, tabular, bounded
features of porosity-loss compaction accommodated by granular rearrangement, grain
crushing and chemical diagenesis that exhibit little or no net shear and tend to weather
171

compaction band trend
500µm
Figure 7.2. Typical outcrop pattern of sub-parallel CB fins in the Aztec sandstone (top).
Backscatter electron images from just outside (bottom left) and inside (bottom right) a
CB—white is hematite, light gray is feldspar, medium gray is quartz, dark gray is clay,
black is porosity. Porosity drops from >20% in the sandstone to 1,000 mD (sandstone) to

out in positive relief as distinct fins (Sternlof et al., 2005; Mollema and Antonellini, 1996)
(Figure 7.2). Millimeters to centimeters thick and tens of meters in planar extent, CBs
represent a kinematic end member of the suite of structures known collectively as
deformation bands, which also includes shear and dilation bands (Aydin, 1978;
Antonellini et al., 1994; Mollema and Antonellini, 1996; Du Bernard et al., 2002). The
petrophysical changes accommodating mechanical compaction inside CBs, in conjunction
with subsequent preferential cementation, can reduce porosity within the bands to a few
percent. Decreased porosity, and corresponding reductions in pore throat diameter and
connectivity, can in turn reduce saturated permeability within the bands by 1 to 4 orders
of magnitude relative to that of the surrounding host rock (Pittman, 1981; Freeman, 1990;
Antonellini and Aydin, 1994; Crawford, 1998; Gibson, 1998; Taylor and Pollard, 2000;
Sigda and Wilson, 2003).
Permeability anisotropy due to depositional bedding in the Aztec and Navajo
sandstones is generally less than a factor of two (Freeman, 1990; Antonellini and Aydin,
1994; Flodin et al., 2005), minimal when compared to the reduced permeability
represented by the CBs. There is no data to suggest that significant anisotropy related to
bedding persists inside the bands, and no reason to expect that the consequences of any
such remnant anisotropy would be significant. In the analyses to follow, we therefore
make the simplifying assumption of homogeneous, isotropic permeability for both the
sandstone and the CBs.
3. Mapping
The fins of CBs cropping out in the Aztec sandstone are plainly visible in low-altitude
aerial photographs. A high-resolution (3,000 dpi) digital scan made from one such
standard (23 cm x 23 cm) aerial photo color negative reveals an extensive exposure of
nearly continuous Aztec outcrops at a scale of 1:2,400 (Figure 7.1). This scan captured a
level of detail down to the grain of the negative, producing a digital image with a pixel
dimension relative to outcrop of about 2 cm. Loading this image into Adobe Illustrator®
and enlarging it, we were able to recognize and trace individual cm-thick CBs (or in many
cases, closely space clusters of CBs) for tens to hundreds of meters (Figure 7.1, inset
photos). This photo mapping effort was guided by our familiarity with the appearance of
CBs in outcrop and was checked extensively in the field.
173

Cover
Outcrop
Band
N
100 meters
20 meters
Figure 7.3. Compaction band trace map representing bands unequivocally identifiable as
fins on the air photo (Figure 7.1). Gaps not due to sedimentary cover, fault damage or
shadows represent outcrop areas where bands are scarce, absent or lack identifiable fins.
Much of the band-trend variability apparent on the map is due to topography. The density
of bands captured in the map is approximately one third that visible in outcrop.
174

To produce a consistent and representative CB map for flow modeling, we applied the
simple criteria that only band traces that could unequivocally be identified on the high-
resolution scan were included on the map. In particular, no attempt was made to
interpolate band patterns through areas of poor exposure due to sediment cover or the
damage caused by younger, mid-Tertiary strike-slip faulting. Nor was any attempt made
to extrapolate band patterns in order to increase the area of coverage. In this way, a
consistent and conservative standard of objectivity was maintained while complications
and uncertainties associated with pattern matching and infilling were avoided.
The resulting 150,000 m 2 CB map (Figure 7.3) represents more than 2000 individual
band traces ranging from a few meters to more than 100 m in length, with an average
spacing of about 1.5 to 2 m. Successive intersections between bands occasionally form
through-going trends hundreds of meters long. Average band thickness measured in the
field is about 1 cm while spacing averages about 0.5 to 0.75 m—roughly one third that
captured on the map. Clustering of CBs (mm spacing) is also common, resulting in
composite bands up to 10 cm and more in total thickness. Field checking indicated that
most gaps within the mapped array not attributable to sedimentary cover or fault damage
do represent real interruptions in the continuity of the band fabric, although the absence
of distinct fins and/or poor image resolution was occasionally to blame. All areas of
bands identified and mapped on the scan were found to exist in outcrop.
4. Flow simulation model
In the scenarios considered below, both single-phase and two-phase flow simulations
are performed. The basic governing equation for incompressible single-phase flow in
porous media is obtained by combining Darcy's law with conservation of mass:
⎛ k ⎞
∇ ⋅⎜
∇p⎟
+ Q = 0 , (1)
⎝ µ ⎠
where p is pressure and Q represents sources and/or sinks (e.g., production/injection
wells). The fluid is described by its viscosity µ , while k represents the absolute
(intrinsic) permeability of the rock. Consistent with core measurements for the sandstone
systems considered here, we take k to be locally isotropic for both the compaction bands
175

(CBs) and the matrix rock in all simulations (although these permeabilities are locally
isotropic, large-scale anisotropy will result from the CB orientation).
Equation (1) can be extended to accommodate two-phase flow by considering Darcy's
law and mass conservation for each phase separately:
( φS
)
∂
∂t
( φS
)
∂
∂t
⎛ kk
= ∇⋅
⎜
⎝
∇
⎞
⎟
⎠
n rn
⎜ pn
⎟ + Qn
µ n
⎛ kk
= ∇⋅
⎜
⎝
∇
⎞
⎟
⎠
w rw
⎜ pw
⎟ + Q w
µ w
, (2)
, (3)
where subscripts n and w refer to nonwetting and wetting phases. Saturation (phase
volume fraction) S is defined for each phase. The porosity of the rock is designated byφ
,
and in addition to the absolute permeability , a relative permeability, k , is defined for
k r
each phase. A capillary pressure (pc)
relationship expresses the difference in pressure
between the two phases as a function of S; i.e., pc( Sw)
= pn
− pw
. An additional
constraint requires Sn
+ Sw
= 1 .
For all the flow scenarios, the governing equations are solved using a specialized
finite-volume discretization technique, namely a discrete-feature model, developed by
Karimi-Fard et al. (2004). Although originally designed to model flow in fractured porous
media, where the discrete features provide high-permeability channels, the generality of
the method allows low-permeability features such as CBs to be treated equally well. The
technique is designed for use with unstructured grids (in both two and three dimensions),
which can accurately represent complex natural geometries such as the CB map (Figure
7.3). A particular strength of the method is that it allows small-scale linear features such
as CBs to be represented and modeled using control volumes that are of the same
thickness as the feature, meaning that CBs need not be resolved in the transverse (thin)
direction by the grid. This reduces the overall number of cells required and greatly
simplifies the gridding procedure. In addition, inefficiencies caused by the proliferation of
small control volumes at CB intersections are mitigated using a special connectivity
transformation, which improves numerical stability and allows for larger time steps in
transport problems in some cases (Karimi-Fard et al., 2004).
176

(c)
(a)
(b)
l
d lm
A j
m
d m
A j
A j
m
b
ll bb
d lm
l m
Figure 7.4. Schematic diagrams depicting quantities relevant to the transmissibility
calculations for (a) matrix to matrix flow, (b) matrix to compaction band flow and (c)
compaction band to compaction band flow. The band thickness is exaggerated in (b) and
(c).
177

The finite volume discretization for single-phase flow is derived by integrating
Equation (1) over each control volume (designated Ω) with k assumed constant within the
control volume. The flux q across the control volume boundary ∂Ω is given by:
⎛ k ⎞
q = − ⎜ ∇p⎟
⋅n
dS
= u⋅
n dS
, (4)
∫∂ Ω⎝
µ ⎠ ∫∂Ω
where n is the outward-pointing unit normal on the surface of ∂Ω and u is the Darcy
velocity. For triangular control volumes, as are considered here, the sum of the fluxes
through the three faces (qj, j =1,2,3) is balanced by the source term; i.e., q + q + q = −Qˆ
,
∫ Ω
where Qˆ = Q dV
. In a two-point flux approximation (TPFA), qj
is expressed in terms of
the geometries, permeabilities, and cell-centered pressures of the two triangles that share
edge j.
With reference to Figure 7.4a, for flow between two matrix control volumes l and m,
this gives
Tj
q j = ( pl
− pm)
, (5)
µ
where Tj is the transmissibility and pl and pm designate the pressures at the barycenters of
the two control volumes. Transmissibility is given by:
k
A
j j
T j = , (6)
dlm
where kj is in general the weighted harmonic average of kl and km (though here matrix
permeability is taken as a constant), Aj is the area (length times thickness) of the shared
interface, and dlm = |dlm| is the length of the line connecting the two cell centers. When dlm
is not exactly orthogonal to the shared edge, we take appropriate projections, as described
in Karimi-Fard et al. (2004).
Analogous expressions for Tj can be developed for flow between matrix and CB
control volumes and for flow between two CB control volumes. For the first case, with
reference to Figure 7.4b, Tj is again given by Equation (6), but now kj is computed via the
weighted harmonic average of the matrix and CB permeabilities (km and kb):
178
1
2
3

k
2d
m b
j = , (7)
lb
2d
m
k
b
+
+ l
k
m
where lb is the CB thickness. In addition, dlm in Equation (6) is replaced by m b , the
center to center distance (see Figure 7.4b). The presence of k
d + l / 2
b -1 in the denominator of
Equation (7) results in kj < km. It is in this way that the permeability-barrier effect of the
CBs enters the matrix-CB transmissibility and thus the flow equations. For flow between
two CB control volumes, Equation (6) is again applied, with quantities as defined in
Figure 7.4c. We note that permeability anisotropy in the CBs (i.e., different permeability
along and across the bands) could be readily modeled within the current formulation,
though this is not necessary here because CB permeability is assumed to be isotropic.
Writing the discrete mass balance ( q + q + q = −Qˆ
) for each cell, and expressing the qj
1
2
via Equation (5), gives the finite volume representation for Equation (1). Solution of the
resulting linear system provides the cell-center pressure unknowns.
In the case of two-phase flow, finite volume discretization of Equations (2) and (3)
entails mass conservation statements for each component (phase). The net flux out of the
control volume in this case is balanced by both the source and accumulation terms. The
flux of a particular phase (e.g., water) across edge j is now given by:
w
w w
q = λ T ( p − p )
(8)
j
w
j
l
m
w
w
where λw is the upstream-weighted phase mobility (λw = krw/µw) and p and p are the
water pressures in blocks l and m. The transmissibility Tj
is the same for both phases and
is again given by Equation (6). Expressing the discrete mass balances for each component
using Equation (8) and introducing time discretizations for the accumulation terms
provides the finite volume representation. Pressure and saturation for each cell at each
time step can then be computed.
3
The two-point flux approximation described above is applied for all connections. This
is strictly valid only when the grid is “orthogonal,” which in this context means that dlm
(the line connecting the two cell centers) is orthogonal to the shared interface.
Nonorthogonality (or full-tensor effects, not an issue here since permeability is taken as
179
l
m

isotropic) can lead to numerical discretization errors when TPFA is applied. Such errors
can be eliminated through use of multipoint flux approximations (see Karimi-Fard et al.,
2004, for discussion of this issue). Errors resulting from TPFA are, however, reduced
when the grid is formed via a Delaunay triangulation. Given a set of points (vertices), a
Delaunay triangulation (Shewchuk, 1996) satisfies a Max-Min property, which means
that, for all possible triangulations of the set of points, the Delaunay procedure maximizes
the minimum angle of the triangulation. This tends to avoid very small or very large
angles, which lead to large discretization errors when TPFA is applied, and gives
triangles that are more nearly equilateral. For a grid comprised of equilateral triangles
(and isotropic permeability), a two-point flux approximation is strictly valid.
The overall solution procedure is compatible with general-purpose flow simulators
that apply connection lists for the definition of cell to cell connectivity. The
transmissibilities defined above along with the associated cell volumes are input to the
Stanford General Purpose Research Simulator (GPRS) for flow simulation. See Cao
(2002) for a detailed description of this simulator.
The CB map (Figure 7.3) was gridded using a standard Delaunay triangulation
(Shewchuk, 1996) technique. The resulting 2-D grid (Figure 7.5) contains 146,682
triangular and segment control volumes representing 1,948 discrete band traces and the
surrounding sandstone matrix. To avoid spurious boundary effects, an oval, no-flow
exterior boundary was established well outside the mapped pattern, while all flow effects
are evaluated inside the pattern. We note that these simulations are relatively demanding
computationally, and that it is not currently practical to perform such runs on full-field
models. However, by considering sectors, as is done here, it is possible to quantify the
impact of CBs on large-scale flow and transport.
In all of the simulations presented below, we specify a uniform band thickness (lb) of
3 cm, band porosity of 10%, homogeneous and isotropic internal band permeability of 1.5
mD, sandstone matrix porosity of 25%, and homogeneous, isotropic matrix permeability
of 1.5 Darcys. The stipulation of 3-cm band thickness approximately accounts for the
under-representation of band density in the final map as discussed above. The
band/matrix permeability ratio of 10 -3 falls in the middle of the range reported in the
literature (Sternlof et al., 2004).
180

150 meters
40 meters
Figure 7.5. Unstructured Delaunay triangular discretization of the compaction band map
shown in Figure 7.3, with an oval no-flow boundary added well outside the band pattern
to delimit the model reservoir/aquifer (left). In this discrete-feature representation, the
exact trace of each band is captured as a series of linear segments defined by the shared
sides of adjacent matrix control volumes (bold, segmented lines in the grid enlargement to
the right).
181

5. Simulations
Three sets of 2-D simulations were performed. The first set prescribes single-phase,
incompressible well-to-well flow for a series of well pairs to examine how the band
pattern affects the pressure drop required to maintain a given flow rate. The second set of
simulations prescribes two-phase, incompressible flow for a standard five-spot petroleum
reservoir production scenario with central injector to examine how the orientation of the
well array relative to the band pattern influences production efficiency. For this
production scenario, the effects of relative permeability are considered with water as the
wetting phase and oil as the nonwetting phase. The third set of simulations prescribes
incompressible contaminant transport (mobility ratio = 1) for a point-source release to
examine the relationship of regional gradient direction to band pattern in determining the
shape and distribution of the resulting plume.
5.1. Well-to-well flow
In order to characterize the bulk effects of the CB array as mapped, we first performed
a series of 210 single-phase, incompressible well-to-well flow simulations for a fixed
flow rate using injector-producer well pairs spaced 25, 50, 75, 100, 125, 150 and 200 m
apart. At 15 midpoint locations scattered within the band pattern, two simulations were
performed for each well-spacing—one oriented generally parallel to (along) the dominant
band trend and one oriented generally normal to (across) it (Figure 7.6). For every
simulation, the pressure drop measured between the wells in the presence of the bands
was normalized by the corresponding CB-free (homogeneous, isotropic sandstone matrix)
value. The resulting normalized pressure drop data (∆P) are presented graphically in
Figure 7.7.
5.1.1. Results
The most basic observation is that the pressure drop required to drive flow in the
presence of the bands always exceeds that for the CB-free case (Figure 7.7a). With a
range from 1.7 to 4.2 and a mean of about 2.9, there is a slight trend toward increasing ∆P
with decreasing well spacing. Also, the pressure drop required to drive flow across the
dominant band trend (∆Pn) generally exceeds that required to drive flow along it (∆Pp) by
an average of about 25% (Figure 7.7b). In other words, a modest degree of
182

150 meters
I
I
P
P
Well-pair midpoint
I Injection well
P Production well
Figure 7.6. Fifteen locations within the compaction band pattern around which well-towell
flow simulations were performed—one set roughly normal to the dominant band
trend and one roughly parallel to it—for spacings of 25, 50, 75, 100, 125, 150 and 200 m.
183

Normalized ∆P
∆P ratio (n/p)
4.5
4
3.5
3
2.5
2
1.5
1
0.5
0
(a) Well spacing (meters)
2.5
2
1.5
1
0.5
0
(b)
n
p
n
p
n
p
n
p
25 50 75 100 125 150 175 200
25 50 75 100 125 150 175 200
Well spacing (meters)
Figure 7.7. Pressure drop (∆P) results for the 210 well-to-well flow simulations shown as
boxplots, each representing the 15 well-pair midpoint locations shown in Figure 7.6. The
top, middle and bottom lines of each box give the upper quartile, median and lower
quartile values, respectively, for the corresponding set of data; the whiskers extend to the
maximum and minimum values. (a) Boxplot pairs representing ∆P roughly normal (n)
and parallel (p) to the dominant band trend as a function of well spacing (all values
normalized by the corresponding homogeneous, isotropic, band-free result). (b) Boxplots
representing the ∆P ratio (n/p), or anisotropy, at each well-pair midpoint location as a
function of spacing.
184
n
p
n
p
n p

large-scale anisotropy is associated with the presence of the bands and, except for a drop
to about 10% at 200 m, this ∆P ratio (∆Pn/∆Pp) appears to be relatively insensitive to well
spacing.
5.1.2. Discussion
It is of interest to compare the results illustrated in Figure 7.7 with those of Sternlof et
al. (2004), who found that the anastomosing pattern of CBs characteristic of the Aztec
sandstone at the outcrop scale can reduce bulk effective 2-D permeability by about an
order of magnitude, while inducing a similar degree of anisotropy with minimum
permeability directed normal to the dominant band trend. The greater magnitude of the
effects detected by Sternlof et al. (2004) can largely be attributed to the relatively small
scale of their modeling effort (~10 m), and their application of local boundary conditions
to force flow through the band pattern in prescribed directions. Our simulations represent
a generalization of this earlier effort, insofar as larger scales defined by realistic well
spacings are considered, while remotely applied (global) boundary conditions do not
constrain local flow pathways.
It is no surprise that ∆P decreases with increasing spacing between wells. As spacing
increases, so does the number of minimally obstructed flow pathways through the
intervening band pattern, even as the average length of those pathways relative to the
straight-line distance between the wells decreases. The relatively modest ∆P ratio, or
anisotropy, can be understood in light of the sinuous, anastomosing nature of the CB
pattern, which ensures that any straight-line path between two wells will intersect
multiple bands regardless of its orientation relative to the dominant (mean) band trend.
Of practical importance to the hydrogeologist or petroleum engineer is the possibility
that unexpectedly high pump-test pressures in what otherwise appears to be high-
permeability sandstone might indicate the presence of CBs. In the absence of
corroborating core or analog outcrop data, and given the current lack of down-hole
geophysical techniques for remote CB detection, the relatively modest ∆P ratio
(anisotropy) induced by a band pattern could be used to both confirm its presence and
determine its dominant geometry.
185

P
P
P
P
I
I
P
P
Case 1
P
P
Case 2
I Injection well
P Production well
150 meters
Figure 7.8. Well configurations used for the five-spot production scenarios—square
pattern 150 m on a side with central injector and producers at the corners. In Case 1, one
diagonal of the five-spot is aligned with the dominant band trend (left). In Case 2, both
diagonals are oriented obliquely (~45°) to the bands.
186

5.2. Reservoir production
A standard five-spot production-well configuration—square, with central injector and
producers at the corners—was used to simulate two-phase (oil-water) incompressible
flow within the band pattern. Two case scenarios were modeled using the same 5.5-acre
pattern (150 m on a side)—the only difference being a 45° rotation of the production-
wells about a fixed injector location (Figure 7.8). In Case 1, the straight-line path from
injector to two of the producers runs generally along the dominant band trend, while the
path to the two other producers runs generally across it. In Case 2, the straight-line paths
from injector to all four producers run oblique to the dominant band trend. For both cases,
we specified complete initial saturation of the pore volume with oil, exact balance at all
times between the total volume of water injected and the total volume of fluid produced,
and equal pressure at all four production wells. A total injection/pumping rate of 15.9
m 3 /day was used and both simulations were run for 250 days, yielding a total pore volume
injected (PVI) of 0.15 (i.e. 15% of the total model pore volume). Relative permeability
was accounted for using representative data for light crude oil (viscosity 2 cp) as the
nonwetting phase and formation water (viscosity 1 cp) as the wetting phase in granular
media similar to the Aztec sandstone (Table 1). Capillary pressure is neglected in these
simulations, as attempting to include it produced computational problems. The efficient
handling of strong capillary effects may require the modification of some of our
numerical treatments. The simulation results are presented in Figures 7.9 and 7.10.
5.2.1. Results
Figure 7.9 shows equivalent saturation-map snapshots for the two simulations at
approximately 0.03, 0.09 and 0.15 PVI. In both cases, the elliptical patterns of water
infiltration that develop suggest a strong tendency toward preferred transport along the
dominant band trend. Judging from the aspect ratios of the infiltration patterns, the
transport rate along the band trend exceeds that across it by at least a factor of two. This
effect results in early breakthrough of water to the production wells oriented along the
band trend from the injector in Case 1 at < 0.03 PVI (Figure 7.9a). For Case 2, significant
breakthrough does not occur until 0.09 PVI (Figure 7.9b). These diverging results are
clearly reflected at 0.15 PVI (Figure 7.9c), when the substantially greater area of water
infiltration for Case 2 illustrates that 56% more oil has been produced than in Case 1.
187

Case 1
Case 1
Case 1
(a)
(b)
(c)
Injection well
Production well
200 meters
188
Case 2
Case 2
Case 2

Figure 7.9. (facing page) Saturation-map snapshots for the five-spot scenarios shown in
Figure 7.8: (a) at 0.03 of the total pore volume injected (PVI); (b) at 0.09 PVI; and (c) at
0.15 PVI. Injected water shows as dark gray, with shades of gray denoting a two-phase
mixture with oil. The pattern of water infiltration is initially similar for the two cases (a),
but diverges as early breakthrough of water occurs at the production wells located along
the dominant band trend from the injector for Case 1 (b), culminating in a larger and more
efficient water sweep for Case 2 (c).
Pore volume produced (oil)
0.15
0.1
0.05
0
0 0.05 0.1
Case 1
Case 2
Ideal
0.15
Pore volume injected (water)
Figure 7.10. Oil production for the two scenarios as a function of the pore volume
injected (PVI) of water. The solid line denotes the ideal case where the volume of oil
produced equals the volume of water injected. Production efficiency drops more quickly
for Case 1 than for Case 2. At PVI = 0.15, the volume of oil produced for Case 2 exceeds
that for Case 1 by 56%.
Table 7.1. Relative permeability data for reservoir production simulations
Fraction
Sandstone Matrix Compaction Bands
Water Saturation Rel. Perm. Water Rel. Perm. Oil Rel. Perm. Water Rel. Perm. Oil
0.0 0.000 0.850 0.000 0.850
0.2 0.020 0.544 0.010 0.544
0.4 0.080 0.306 0.040 0.306
0.6 0.180 0.136 0.090 0.136
0.8 0.320 0.034 0.160 0.034
1.0 0.500 0.000 0.250 0.000
189

5.2.2. Discussion
The elliptical patterns of water infiltration shown in Figure 7.9 demonstrate the
dominant impact of the CB pattern on the gross direction of fluid transport. In fact, the
saturation maps for the two cases are at first strikingly similar (Figure 7.9a), indicating
that the production wells exert a minor influence on transport when compared to
channeling of flow along the dominant band trend. These directional transport effects are
more pronounced than the relatively modest ∆P ratios presented in Figure 7.7 would
suggest.
The 56% increase in production efficiency gained in Case 2 for a 45° rotation of the
five-spot well configuration is a significant (but not optimized) result that could vary
substantially within the CB pattern and for different production configurations. In
addition, the scale of our five-spot, as constrained by the available mapped area, is
smaller than would typically be installed in a real reservoir production situation.
Nonetheless, the petroleum engineer can glean two valuable insights from our results—
that the presence of CB arrays can strongly affect production efficiency in reservoirs, but
also that relatively simple adjustments in well placement can be made to mitigate and/or
harness those effects.
5.3. Contaminant transport
In order to examine how the CB array would affect contaminant transport and
influence the evolution of a point-source plume, we established a fixed leak location for
each of three regional pressure gradient directions: parallel to the dominant band trend,
oblique (45°) to the trend, and normal to the trend (Figure 7.11). We considered
incompressible contaminant transport (mobility ratio = 1), and neglected physical
dispersion. A consistent regional gradient of 0.23 Pa/m (enough to yield an average flow
rate of approximately 7.7 m/year in the absence of CBs) was established in each of the
three directions by specifying pressures for an array of eight wells distributed around the
perimeter of the model domain (just inside the oval no-flow boundary). Each simulation
represents 30 years at a constant leak rate of 0.08 m 3 /day. For each scenario, a
corresponding CB-free control simulation was conducted. The results of all the
simulations are presented in Figures 7.12, 7.13 and 7.14.
190

N
Leak
150 meters
P
O
Figure 7.11. Configuration of contaminant leak location with three directions of equal
regional pressure gradient used in examining the impact of the compaction band pattern
on plume migration (P=parallel to the dominant band trend, O=oblique to the trend,
N=normal to the trend).
191
O
P
N

5.3.1. Results
With the regional gradient directed roughly parallel to the dominant band trend, the
resulting contaminant plume is narrower and longer than would be the case in the absence
of the CB array (Figure 7.12a) and extends some 60 m up-gradient from the leak location.
The long axis of the two plumes (with and without CBs) remains coincident, however,
and parallel to the band trend. Looking in detail, the plume has clearly channeled along
the band array and been constrained from spreading laterally (Figure 7.12b).
With the regional gradient directed obliquely to the dominant band trend, the shape
and orientation of the resulting plume departs substantially from that of the CB-free
control case, remaining constrained by and aligned with the band array (Figure 7.13a).
The plume is, however, wider and shorter than in the parallel scenario, extends 80 m
obliquely up-gradient from the leak (along the band trend), and, where gaps in the CB
pattern allow, turns into the regional down-gradient direction (Figure 7.13b).
With the regional gradient directed normal to the dominant band trend, the resulting
plume bears scant resemblance to the CB-free control case, remaining distinctly elongated
along the band array (Figure 7.14a). The plume is shorter and broader again than in the
oblique scenario, and turns strongly into the regional down-gradient direction at a major
gap in the band pattern (Figure 7.14b, upper left). Finally, the trailing edge of the plume is
essentially straight, having pushed uniformly almost 20 m in the regional up-gradient
direction.
5.3.2. Discussion
As with the reservoir-production simulations above, the plume evolution results
reveal a dominating influence on fluid transport by the CB pattern, regardless of the
regional gradient direction. Limited sensitivity testing also indicated that realistic
variations in the magnitudes of the regional gradient and/or contaminant leak rate do not
substantially affect the results. For the hydrogeologist designing a pump-and-treat
remediation system therefore, recognizing and accounting for the effects of the CBs
would prove crucial, particularly with 100% contaminant capture as the goal.
For example, the standard cleanup approach for a leaking underground storage tank is
to install the minimum number of wells that can be spaced immediately down gradient
from the plume front to provide complete capture, assuming radial drawdown at an
192

(a)
(b)
Leak
200 meters
Regional
gradient
No CBs CBs
100 meters
Figure 7.12. Contaminant plumes (dark gray) after 30 years with the regional gradient
directed parallel to the dominant compaction band trend (scenario P from Figure 7.11).
(a) Plume distribution in the absence of bands (left) and with bands present (right). (b)
Detail of the band-impacted plume (shades of gray denote two-phase mixing).
193

(a)
(b)
Leak
200 meters
Regional
gradient
No CBs CBs
100 meters
Figure 7.13. Contaminant plume (dark gray) after 30 years with the regional gradient
directed obliquely to the dominant compaction band trend (scenario O from Figure 7.11).
(a) Plume distribution in the absence of bands (left) and with bands present (right). (b)
Detail of the band-impacted plume (shades of gray denote two-phase mixing).
194

(a)
(b)
Leak
200 meters
No CBs
Regional
gradient
CBs
100 meters
Figure 7.14. Contaminant plume (dark gray) after 30 years with the regional gradient
directed normal to the dominant compaction band trend (scenario N from Figure 7.11).
(a) Plume distribution in the absence of bands (left) and with bands present (right). (b)
Detail of the band-impacted plume (shades of gray denote two-phase mixing).
195

anticipated sustainable pumping rate based on the results of simple slug tests and the
assumption of homogeneous, isotropic permeability. Given any of the CB scenarios
presented in Figures 7.12, 7.13 and 7.14, this approach would lead to misplaced wells,
both in terms of their location relative to the effective contaminant transport direction and
the spacing needed for complete capture given elliptical drawdown patterns (Matthai et
al., 1998). Failure to recognize the transport-distorting effects of CBs would also
complicate efforts to locate the unknown source of a contaminant plume, and/or identify
its extent based on limited well data.
6. General discussion
While the model simulation scenarios presented above in no way constitute a
complete treatment of the flow and transport effects of CBs in the Aztec sandstone, the
results do illustrate a clear potential for substantial impacts. Given the broad practical
implications of these findings, a few key interpretations and assumptions underlying this
work warrant further comment.
Firstly, as acknowledged in the section on mapping, limitations of outcrop exposure,
image resolution and map scale prevent the final CB map (Figure 7.3) from capturing the
full abundance and detail of the band array present in the field. The map does, however,
accurately represent the relative abundance, distribution, orientation and connectivity of
the exposed pattern of oldest CBs that dominates the area and, as such, is adequate to the
purpose of modeling flow and transport effects at relatively coarse scales of practical
interest. In fact, insofar as the true abundance, distribution and connectivity of the bands
are under-represented in the map, the substantial flow and transport effects realized here
can reasonably be considered as conservative, low-end estimates.
Also, our 2-D approach to an undeniably 3-D problem is dictated by the nature of the
outcrop exposure, which presents a roughly horizontal slice through the CB array to
reveal the detailed pattern as mapped. Given accurate 3-D data and sufficient computing
power, the full spatial effects of the entire CB array could be modeled. Nonetheless,
abundant field observations throughout the upper 600 m of the Aztec sandstone,
particularly in areas of topographic relief, attest to the vertical persistence of the overall
CB array. We suggest therefore that, to a reasonable first approximation, the
anastomosing pattern of bands in Figure 7.3 can be assumed to extend hundreds of meters
196

vertically such that both the magnitude and directionality of the flow and transport effects
realized in 2-D would substantially persist into 3-D.
While it might be interesting and useful to assess whether current dispersion theory
could capture the gross plume behavior exhibited in Figures 7.12-7.14, it is unlikely that
this approach would provide quantitatively accurate results. Even for idealized (Gaussian,
log-normal, etc.) permeability descriptions, it has been demonstrated that, as log
permeability variance increases, so does the gap between simulation results and
theoretical predictions (Naff et al., 1998; Dentz et al., 2002). It may nonetheless be
possible to describe the observed plume behavior qualitatively, or even semi-
quantitatively using effective dispersivity methods.
Accounting for capillary pressure effects in our two-phase simulations would, of
course, also be desirable. Although the modeling methods used in this study can in theory
handle these effects, as stated earlier, attempting to include them in the five-spot
production simulations led to significant degradation in computational efficiency. We
believe that this occurs as a result of the simulator attempting to resolve the very short
capillary time scales associated with the bands. Further investigation, testing and possible
modification of the numerical methods will be required to address this issue adequately. It
is important to note, however, that in many practical cases viscous (convective) effects
dominate capillary effects, so the neglect of capillary pressure is physically reasonable in
such systems.
Although not addressed in this paper, coarse-scale descriptions (i.e. upscaled models)
for detailed CB systems such as considered here would comprise the actual input
parameters to real-world aquifer- and reservoir-scale simulations. We note, however, that
coarse-scale flow attributes such as permeability and transmissibility might not be
independent of the flow process under consideration, but rather could depend on specified
parameters such as boundary conditions, well locations and flow rates. This sort of
process-dependent behavior has previously been observed in highly heterogeneous
systems with permeability features that are of a length scale comparable to the system
size or well spacing, as is the case here (Chen et al., 2003). For such systems, accurate
upscaling may require the incorporation of global flow information, which can
significantly complicate the computations.
197

Finally, there is the issue of forecasting where volumetrically extensive compaction
and deformation band arrays are most likely to occur. This is an active area of research
that to date has emphasized discrete deformation band style faults in sandstone (e.g.
Aydin and Johnson, 1983, Shipton and Cowie, 2001; Davatzes et al., 2005) over the
distributed style of tectonic fabric formed by CBs in the Aztec (Sternlof et al., 2005).
Much of the research into both types of bands has focused on their occurrence in
Mesozoic æolian sandstones of the southwestern U.S., although deformation bands are
increasingly being identified in a wide variety of porous, granular geologic materials
ranging from nonwelded ignimbrites (Wilson et al., 2003) to unlithified Holocence beach
sands (Cashman and Cashman, 2000). Compaction bands, however, have yet to be
positively identified outside of the Navajo (Mollema and Antonellini, 1996) and Aztec
(Sternlof et al., 2005) sandstones. As of this writing therefore, the most that can be said is
that CB arrays such as exposed in the Aztec might reasonably be suspected in any high-
porosity sandstone that has been subjected to a differential tectonic stress prior to
substantial lithification.
7. Conclusions
In this study, we performed flow modeling using a state-of-the-art discrete-feature,
finite-volume simulator applied to a CB data set of unprecedented detail and scale, which
was derived from a real geological system that serves as an exhumed analog for active
sandstone aquifers and reservoirs. Our results demonstrate the potential for CB arrays,
such as abundantly exposed in the Aztec sandstone at the Valley of Fire, Nevada, to exert
profound effects on subsurface flow at scales relevant to aquifer and reservoir modeling
and production. Specifically, we found that:
• The CBs as mapped cause an average three-fold increase in the pressure drop
required to maintain a given flow rate between two wells, as compared to CB-free
sandstone;
• There is a mild directional aspect to this pressure effect—an anisotropy—with ∆P
across the dominant band trend exceeding that along the trend by an average of
25%; and
198

• The tendency for transported fluids to channel preferentially along the dominant
band trend can overpower the effects of regional pressure gradients, both natural
(contaminant plume scenarios) and induced (reservoir production scenarios).
Compaction bands, and other types of deformation bands with similar flow
characteristics, almost certainly exist unrecognized in many sandstone aquifers and
reservoirs. Reliable borehole geophysical techniques for detecting these types of
structures in the subsurface and assessing their gross geometry, density, connectivity and
petrophysical characteristics are needed. Nonetheless, our modeling suggests that simple
steps based on even limited data can mitigate the impact of CB arrays. While not an
exhaustive study, the character and magnitude of the flow and transport effects modeled
here demonstrate that accounting for such band fabrics could prove essential to the
optimal management of sandstone aquifers and reservoirs in which they occur.
8. Acknowledgements
Our thanks go to John Childs for his acute and cheerful assistance in the field.
Primary funding for this work was provided by the U.S. Department of Energy, Office of
Basic Energy Sciences under grant DE-FG03-94ER14462 awarded to David Pollard and
Atilla Aydin at Stanford University. Additional support was provided by the Stanford
Reservoir Simulation Consortium (SUPRI-B) and the Stanford Rock Fracture Project
(RFP).
199

200

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