The relationship between damage and localization

The Relationship between Damage and Localization

The Geological Society of London Books Editorial Committee Chief Editor

BOB PANKHURST (UK) Society Books Editors

JOHN GREGORY (UK) JIM GRIFFITHS (UK) JOHN HOWE (UK) PHIL LEAT (UK) NICK ROBINS (UK) JONATHAN TURNER (UK) Society Books Advisors

MIKE BROWN (USA) ERIC BUFFETAUT (France)

JONATHAN CRAIG (Italy) RETO GIERE´ (Germany) TOM MC C ANN (Germany) DOUG STEAD (Canada) RANDELL STEPHENSON (The Netherlands)

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It is recommended that reference to all or part of this book should be made in one of the following ways: LEWIS , H. & COUPLES , G. D. (eds) 2007. The Relationship between Damage and Localization. Geological Society, London, Special Publications, 289. WIBBERLEY , C. A. J., PETIT , J.-P. & RIVES , T. 2007. The mechanics of fault distribution and localization in high-porosity sands, Provence, France. In: LEWIS , H. & COUPLES , G. D. (eds) The Relationship between Damage and Localization. Geological Society, London, Special Publications, 289, 19 –46.

GEOLOGICAL SOCIETY SPECIAL PUBLICATION NO. 289

The Relationship between Damage and Localization

EDITED BY

H. LEWIS and G. D. COUPLES Heriot-Watt University, UK

2007 Published by The Geological Society London

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Contents COUPLES , G. D. & LEWIS , H. Introduction: the relationship between damage and localization GUE´ GUEN , Y. & BE´ SUELLE , P. Damage and localization: two key concepts in rock deformation studies WIBBERLEY , C. A. J., PETIT , J.-P. & RIVES , T. The mechanics of fault distribution and localization in high-porosity sands, Provence, France DESRUES , J., BE´ SUELLE , P. & LEWIS , H. Strain localization in geomaterials LOGAN , J. M. The progression from damage to localization of displacement observed in laboratory testing of porous rocks DI G IOVANNI , A. A., FREDRICH , J. T., HOLCOMB , D. J. & OLSSON , W. A. Microscale damage evolution in compacting sandstone FISHER , Q. J., HARRIS , S. D., CASEY , M. & KNIPE , R. J. Influence of grain size and geothermal gradient on the ductile-to-brittle transition in arenaceous sedimentary rocks: implications for fault structure and fluid flow OLSON , J. E. Fracture aperture, length and pattern geometry development under biaxial loading: a numerical study with applications to natural, cross-jointed systems UEHARA , S. & SHIMAMOTO , T. Permeability of fault rocks from the Median Tectonic Line in Ohshika-mura, Nagano, Japan as studied by pressure-cycling tests COUPLES , G. D., LEWIS , H., OLDEN , P., WORKMAN , G. H. & HIGGS , N. G. Insights into the faulting process from numerical simulations of rock-layer bending HALL , S. A., LEWIS , H. & MACLE , X. Improved seismic identification of inter-fault damage via a linked geomechanics –seismic approach REYNOLDS , M. A., COUPLES , G. D., LEWIS , H. & PICKUP , G. E. Localization processes in a coupled hydrogeomechanically sensitive fractured system HEFFER , K. J. Proximity to a critical point: evidence from, and implications for, hydrocarbon reservoirs Index

1 7 19 47 75 89 105

123 143 161 187 209 227 241

Introduction: the relationship between damage and localization G. D. COUPLES1,2 & H. LEWIS1,2 1

Institute of Petroleum Engineering, Heriot – Watt University, Edinburgh EH14 4AS, UK (e-mail: [email protected])

2

ECOSSE (Edinburgh Collaborative of Subsurface Science and Engineering), a part of the Edinburgh Research Partnership in Engineering and Mathematics

The papers that appear in this Special Publication were assembled to address a topic that was the subject of a conference entitled ‘Damage and Localization’, one of a series of three Euroconferences on rock mechanics and rock physics that were supported by European Commission funding. Some of papers contained herein were derived from the contributions presented at that meeting, but others were solicited subsequently in order to create a coherent volume that illustrates some key facets of the topic as it is now understood. However, the subject is sufficiently broad that a single collection of papers cannot hope to do justice to the whole theme. This Introduction outlines the conceptual threads that underpin the selection of papers that are included in this volume and introduces the cross-scale relationships that are addressed by the individual contributons. We hope that the reader will find these contributions to be stimulating and informative.

The meaning of ‘damage’ and ‘localization’ Two very different uses of the term ‘damage’ are illustrated in this Special Publication. Gue´guen & Be´suelle provide a clear summary of one usage, in which the term ‘damage’ refers to an alteration of the macroscopic material properties of a rock as a consequence of the development of arrays of microfractures. This concept of damage is based on a perspective that is similar to effective-medium theory, where reality is idealized into a system of a few components (perhaps only two: ‘solid’ and cracks that are represented by idealized, flattened ellipsoidal voids) whose interactions can be expressed analytically, thus allowing the calculation of the response of that composite material. The motivation behind the creation of this damage concept came from laboratory studies, which have led to an understanding that many rocks respond to pre-failure, or pre-yield, loading by developing microcracks (grain-scale discontinuities), leading to acoustic emissions that can be externally

recorded. Concurrent with the creation of these microcracks, the experimental observations also reveal reductions in the values of physical properties such as the elastic moduli (and thus the acoustic velocities), and increases in permeability. By adopting the microcrack model, it is possible to develop explanations for the laboratory observations that are dependent on the same fundamental phenomenon: arrays of cracks. In the resulting equations, damage is represented by a parameter, D, that is associated with the amount of work that has been accomplished. The simplest case takes D to be the fraction of voids that are encountered in a representative slice through the material. As more work is accomplished, more cracks initiate, and old cracks can grow, leading to an increase in D. The resulting equations determine the effective properties as a function of the relevant uncracked (original) property and the parameter D. The partnership between experimentalists and theoreticians that has led to the classical damage concept provides a phenomenological explanation for the observations concerning rock-property changes in the run-up to yield. Another community of researchers uses the term ‘damage’ in a very different fashion. In this use, damage refers to a (typically macroscopic) distribution of deformation features that can include small faults, shear bands, compaction bands, fractures, and solution seams. Perhaps the most familiar example of this alternative use of the word damage is in the term ‘fault damage zone’, which refers to a volume of rock near a fault within which there exist deformation features (possibly of many kinds) that distinguish that volume from a less deformed protolith that lies at a greater distance from the fault. This usage of the term damage is also associated with the concept of strain. A volume of rock that has experienced this type of damage has clearly progressed beyond yielding, so the strains that define damage are inelastic and permanent. Wibberley et al., Logan, and DiGiovanni et al. discuss photomicrograph images of the petrofabrics of deformed rocks (from natural and laboratory cases) that reveal microcrack distributions of the

From: LEWIS , H. & COUPLES , G. D. (eds) The Relationship between Damage and Localization. Geological Society, London, Special Publications, 289, 1–6. DOI: 10.1144/SP289.1 0305-8719/07/$15.00 # The Geological Society of London 2007.

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sort that underlie the classical damage concept that is outlined by Gue´guen & Be´suelle. The other version of the damage concept is also illustrated by examples given by Wibberley et al., along with cases described by Uehara & Shimamoto, Couples et al., Hall et al. and Reynolds et al. These latter papers represent examples of how the term damage has come to mean a response mode in which the rocks develop arrays of macroscopic deformation features (of whatever type) distributed throughout a volume. Similar to the situation with the term ‘damage’, the word ‘localization’ has more than one meaning. In one usage, localization refers to a ‘style’ of deformation in which the overall distortion is accomplished by means of movements that occur in spatially restricted, planar or curviplanar zones that occupy only a small part of the bulk material. Within this style, significant fractions of the total rock volume do not experience much strain. In a limiting case, the bulk strain is accomplished by motions that occur only along distinct

discontinuities. In practice, finite-thickness fault and shear zones are considered as being a type of localization if most of the movement occurs along them, and if their fractional volume is small. The volumes of rock that lie between, and are bounded by, localization features are, based on the definition of localisation, not significantly strained. This use of the term localization is essentially as an endmember in a spectrum of possible spatial arrangements of deformation features, where distributed damage lies at the other extreme. The strain-based definitions of damage and localization introduce a set of issues associated with the scale of observation (Fig. 1). Any one deformation feature, or set of features, is part of a larger deformation process, and at the same time is composed of many smaller deformation features. At any particular scale of observation, it may be possible to comfortably assign the visible deformation features into a class that resides along the damage –localization axis (possibly even at an

Fig. 1. Illustration of issues arising in assignment of the descriptive terms damage and localization as a function of the scale of observation. In this figure, the sub-images are intended to highlight the way that individual deformation features, or assemblages of them, are not isolated components, but are, instead, elements in a multi-scale composite. At any single scale of observation, the observed distribution of deformation features can often be classified as being closer to damage or to localization, but a change of observation perspective may suggest that the same feature is now part of a different classification. This image depicts about six orders of magnitude in terms of the scale of observation (i.e. from millimetre-scale grains to kilometre-scale structures). The concept extends beyond either end of the scale shown, ranging down to (at least) nanometre-scale microcracks in mudstones, and up to the dimensions of orogenic belts.

INTRODUCTION

end-member position on that gradation). However, that classification assignment may need to be changed if the point of view, or scale of viewing, is altered. Where the context is very clear, such as for rock cylinders studied in a high-pressure rock mechanics laboratory, there is only a small risk of miscommunication: at the scale of the sample, the observed distribution of deformation can usually be readily described as being of either the localized or the damaged type. In contrast, when we want to describe naturally deformed rocks, or to extrapolate our laboratory understanding to explain what we see in nature, we run the risk of using the terms damage and localization in ways that are appropriate from our own contextual perspective, but that may not succeed in communicating to others in the fashion that we intended. As every geologist knows, geological photographs always require a scale bar because the length scale of the features is frequently not determinable without such an external reference. It is worth noting that the terms brittle and ductile are often used synonymously with this sense of the terms localization and damage, respectively, with similar risks of miscommunication. An additional problem associated with the terms brittle and ductile is that some writers employ these words to distinguish between different operative deformation mechanisms (i.e. cataclasis versus pressure solution–precipitation creep or crystal–plastic mechanisms), with implications of associated knowledge of the conditions of deformation (pressure, temperature, etc.). The processes that lead to localization (see below) do not require the operation of any specific deformation mechanism, so there is an important distinction to be made between the spatial arrangement of strain and the processes that operate to accumulate it. We suggest abandonment of the terms brittle and ductile because they have become ill-defined. If it were possible to pre-define specific scales of observation, then it might be conceivable that the strain-based meanings of damage and localization could be precisely defined. That is unlikely to happen. Deformation features that are otherwise similar are observed to vary in scale in a way that depends on factors such as the lithology, bed thickness and the stacking patterns of different rock layers (or the architectural arrangement of rock materials, in the general case). Once some deformation has occurred, the deformed rock materials add additional architectural complexity. From the present state of knowledge, it is not possible to develop general rules about the length scales that can be expected for deformation features in natural systems, and so it remains as a long-term target to predefine the observation perspectives that would allow the non-subjective classification of a deformation event into damage and localization components.

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Even at a particular scale, rock materials respond to loading such that our definition of damage and localization will need to change during the deformation. Let us consider a cylindrical sample in the rock mechanics laboratory. A typical loading scenario uses conditions that are appropriate to examine how that rock type would respond in an upper crustal setting. As the loading is applied, the rock sample resists the imposed distortions and generates stresses. Presuming an initially intact sample, let us recall the evolution towards macroscopic yielding. At the beginning of deviatoric loading, we can record widely distributed acoustic emissions (which we take as proxies for the generation of, or movement along, microcracks). As loading continues, the distribution of microcracks will tend to become concentrated into zones that eventually become macroscopic surfaces. At the scale of the sample, we could denote the early response as damage, and the eventual response as a localization. An appropriate classification at one point in the deformation history is usually not suitable throughout, so the problem of scale becomes entangled with a consideration of process. There is another definition of localization that has nothing to do with scale. In continuum mechanics, an assumption is made that the material responds to loading in a way that can be represented by differential equations that express certain rules about the constitutive relationship between stress and strain (rate). Plasticity theory (see discussions by Gue´guen & Be´suelle; Desrues et al. and Couples et al.) is fundamentally linked with the notion of yielding, beyond which the material becomes permanently strained, with a significant focus in plasticity on explaining post-yield responses. The equations that represent the theory of plasticity lead to the identification of certain conditions under which there can be a discontinuity in the increments of strain. The term localization is applied when the material evolves to the point where these conditions are met. In this definition, localization is identified as a bifurcation point, after which the strain is no longer continuous; instead, strain components become discontinuous across a physical surface that is created by the process of localization. This theoretical version of localization is one that is centred on materials, and the notion of material behaviour. Of course, real materials often develop localization features (in the sense associated with strain) when they respond to loading, and that has been a major impetus for theoretical advances. Indeed, the combination of theoretical consideration and observational study (particularly in the laboratory) has been very fruitful. However, as observers examine localization features, they notice additional complexity in terms of

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multi-scale, self-organized behaviours (see below) that do not seem to be immediately predictable from the perspective of idealized materials and plastic constitutive relationships.

Evolution of geomaterials A strong thread that weaves through the papers of this Special Publication is the idea that geomaterials evolve as they deform, such that each increment of deformation is occurring in a ‘new’ material that is not the same as the original material. This idea applies both to bulk materials and to localization features. Wibberley et al. document a case study from Provence in which unconsolidated sands initially undergo a compactional deformation response (a distributed damage expressed by arrays of shear bands, which represent localization responses at another scale), and evolve, because of that deformation, such that they are able to respond in a dilational and strongly localized fashion during subsequent deformation. Desrues et al. lead us through the results of laboratory experiments involving a wide range of geomaterials that develop localization, with a consistent theme that high-porosity materials first experience a compactional change in the deformation bands before they next respond in a dilational fashion. In these two papers, the deformations described do not follow the characteristics associated with the laboratory-based version of damage (widely distributed cracking) leading to localization. Instead, there is a general absence of volumetrically distributed microcracks before localization occurs. The microcracks that do occur in the poorly consolidated materials of Wibberley et al. are spatially associated with a small process zone in front of a propagating shear. Fisher et al. pick up on this theme of material evolution and examine a response transition that can be predicted from laboratory-based measurements to occur with increasing depth of burial. The burial produces a nominal increase in mean stress and a reduction in porosity by general compaction, and activates diagenesis. Fisher et al. examine the consequences of this idea by considering the implications for the case of faults that form or move at different depths, and how this understanding can be used to predict the movements of fluid, which ought to be governed by the time or place of a deformation event. DiGiovanni et al. provide details of one type of compaction response that involves the generation of sets of compaction bands whose sites progressively initiate in a ‘front’ that propagates into a laboratory specimen from its ends. Cracking and other processes occur in association with band formation, as revealed by

acoustic emission event location, but there is no sense of a general damage process occurring through the whole sample. It is possible to imagine that the end point of such a process would be a simply compacted material that does not exhibit very clear ‘bands’. Perhaps DiGiovanni et al. have provided us with an example of a process that may operate on every body of sediment as it becomes buried during basin evolution. Logan provides a synthesis of a long-running programme of experimental studies that examine the development of fabrics in synthetic gouge materials deformed in a shear zone. The fabrics that develop in the specimens are interpreted by Logan to be Riedel shears. The presence of clay alters the resulting pattern, but for any given composition of gouge material, the patterns develop in a specific sequence with specific orientations. An idea that emerges from these studies is that, at a particular scale, damage may be an appropriate description for what happens, but that localization processes may take over. There is some similarity between this concept and the specific case of microcracks that occur over a volume but then concentrate into a narrow zone. However, the observational evidence in these papers points to something slightly different. Although the early deformation may be distributed, it may also be distinctly nondilational. In the porous materials studied by Desrues et al., Wibberley et al., DiGiovanni et al. and Fisher et al., there is a sense that the material ‘needs’ to become stronger (i.e. more solid) to be able to move towards a localization response. This perspective is in strong contrast to the classical damage concept where microcrack arrays are observed to cause a weakening of the material prior to localization. A key concept that is strongly supported by these studies is that the available mechanical responses of geomaterials seem to be controlled by what has happened to them previously, and the situation extant at the time of the next increment of deformation. Because initial dilation– weakening and compaction–hardening are such different behaviours, it is interesting to speculate on why both can lead to strong localization. The explanation is likely to be that localization, however it comes about, is an energetically favourable mode that allows rock masses to respond to loads by doing the least total work. The corollary to that idea is that the external work requirement is smaller if the rock system can reach localization by some sort of internal arrangement. It then becomes necessary to consider a geomaterial not as just itself, but as being a composite that has harder and softer (stronger and weaker) parts. This thought leads to the perception of localization features as being internally self-organized systems.

INTRODUCTION

Behaviour of geomechanical systems The theme that geomechanical responses are dependent on history and current situation is further reinforced in the remaining papers contained in this Special Publication. The context of the next set of papers is very definitely that of systems. For example, Olson examines the development of fracture arrays in systems composed of hundreds of growing fractures that evolve as a result of extensional boundary conditions. The simulations reveal that the resulting patterns are sensitive to initial conditions (representing a prior history) and the loading (representing the extant situation). Olson provides a clear illustration of the way that geomechanical systems can exhibit multiple responses to a given loading that are determined by (in this case, subtle) variations in historical events that may not always be considered. Uehara & Shimamoto demonstrate that the idea of historical dependence is also relevant for systems that evolve from deep to shallow conditions. They describe the distribution of fault-rock types created during large-scale strike-slip faulting in Japan. In this example, all of the materials are highly sheared. However, the present permeabilities of the deformed rocks define an extremely heterogeneous distribution that is the result of variations in material type, including materials created by prior deformation, and when that material experienced its last deformation. Couples et al. describe the outcomes of numerical simulations that are based on a poro-plastic material description that captures both hardening or softening, and compaction or dilation responses of geomaterials. One of the deformation cases examined by Couples et al. is a flexure that progressively develops an array of localization features in a crestal graben configuration. The simulation results illustrate that this system, which evolves to have multiple parts, develops from an originally uniform case. The parts of the system interact and control the subsequent increments of deformation. The evolving mechanical state around the fault array allows Couples et al. to question the concept that faults operate because of a stress regime. They find, instead, that the stress state is the dependent variable, and that predictions based on static stress assumptions could be significantly in error. Hall et al. use geomechanical simulations to gain additional understanding of the processes that can lead to spatially complex mechanical states in reservoirs. Seismic anisotropy studies reveal that there are significant variations in direction and intensity of some possible causative factors that are not resolvable from the seismic data themselves. Using geomechanical simulations, Hall et al. show

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that reservoir systems, comprising multiple faultbounded blocks, develop comparable variations in direction and intensity of their mechanical state (expressed in terms of both stresses and strains) to those observed in seismic studies. By assuming the existence of open(able) fractures, they show that it is possible to explain the observed seismic anisotropy as being caused primarily by the evolved mechanical state. Reynolds et al. employ a coupled simulation approach that allows them to consider the way that systems of fracture-bounded blocks respond to a combined loading associated with fluid flow and mechanical boundary conditions. These systems reveal a surprising degree of sensitivity to minor changes in the imposed conditions, with bulk permeabilities showing more than three orders of magnitude variation over small parameter ranges. Heffer completes the collection with his consideration of the large-scale responses of reservoir systems. During production operations (involving fluid withdrawal or injection), subsurface hydrocarbon reservoirs can exhibit long-range effects that occur at length scales that are larger than can be explained as being a direct influence of changes in pore pressure, and over time scales that are too short to be related to that process. Heffer develops an explanation that is linked with ideas concerning self-organized criticality, and that involves load redistributions through volumes of rock that are many times larger than the volume of the reservoir.

Towards a new perspective This collection of papers, written by authors of varying backgrounds (theoreticians, experimentalists, structural geologists, seismologists and reservoir engineers), provides a broad look at the concepts of damage and localization in geomaterials. A major theme is that the deformation of geomaterials is a non-linear process, with the materials evolving as they deform. In effect, geomaterials become systems via self-organized processes. Another key theme is that systems comprising geomaterials, their deformation features and fluids, similarly evolve, with feedback loops providing opportunities for unexpected or extreme responses. Geomaterial systems can develop responses at many different length scales. Damage sensu lato is often a precursor to a manifestation of localization sensu lato. Localization features may have damage within them. It is not obvious that it is possible, or perhaps ever will be, to specify universal length scales within which damage and localization can be precisely defined. Instead, damage and localization are useful concepts that allow us to express

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relative degrees of concentration of the continuing deformation at a suitable observation scale. The concept of self-organization underpins every contribution in this volume. Geomaterials, and systems of geomaterials, all seem to evolve as though they are aiming at a ‘target’. A speculative explanation is that the attractor is a least-work condition. However, changes in conditions, or interactions, can shift that apparent target, so it may not be possible to predict the path of any arbitrary volume of material without knowing about the complete system that surrounds it. Nevertheless, it seems that geomaterials are typically attracted (from one situation) to specific outcomes (another situation) during their deformation, and it ought to be possible to express many of these common pathways in a fashion that has practical utility. As a corollary, perhaps a failure of some geomaterials to reach an expected outcome may be indicating a system constraint that we have not yet appreciated, and so pointing us towards an improved understanding of that particular event.

The papers assembled here represent a spectrum of views concerning the relationships between damage and localization. The characteristics of the damage may differ depending on the situation, and the same holds for localization phenomena. Future work in this topic area will almost certainly include additional studies that examine the evolution of geomechanical systems, with a particular focus aimed at identifying and understanding the way that process interactions lead to some form of localization. Another fruitful area will be the link between theory and the recognition that localization features may actually be miniature systems in their own right. The editors wish to thank C. Wibberley for his thoughtful and comprehensive review of this paper. We would also like to especially thank A. Hills at the Geological Society Publishing House for her patience, her guidance and her continued faith in us through the very long birth of this book. R. Holdsworth also deserves an extra thank you for patience and persistence through this time. Without their support this book would not exist.

Damage and localization: two key concepts in rock deformation studies Y. GUE´GUEN1 & P. BE´SUELLE2 1

Ecole Normale Supe´rieure, Laboratoire de Ge´ologie, 24 rue Lhomond, 75231 Paris cedex 5, France 2

Laboratoire 3S-R, BP 53, F-38041 Grenoble cedex 9, France (e-mail: [email protected])

Abstract: In upper crustal conditions (the brittle field), rocks subjected to a load can experience a deterioration of physical and mechanical properties. This paper treats two conceptual approaches to model the deterioration that results from a crack distribution, whether the distribution is more or less diffuse or is strongly localized. The first part deals with possible ways to characterize diffuse microcracking (damage) to establish how the elastic response reflects the microcracking. This is of direct geophysical interest, as elastic wave velocities carry quantitative information on crack content. The second part summarizes two kinds of theory to predict the failure by crack propagation or strain localization, using either fracture mechanics or bifurcation theory. This is of direct geomechanical interest, as what is looked for is an interpretation of localized structures. The third part presents the complementarities between both concepts, damage and localization, to predict failure in rocks.

The birth of faults (i.e. first localization of deformation) is an event that is rarely observed at the field scale. However, it can easily be observed at the laboratory scale when intact rock samples are submitted to deviatoric stresses. Before such an event occurs, one intuitively expects the rock bulk properties to be modified by damage. This is exactly what is observed from various datasets: acoustic emission records, volumetric strain measurements, or elastic wave velocity variations (Ashby & Sammis 1990; Chelidze 1993; Main et al. 1993; Zang et al. 1996). At low temperature, all of these effects result from crack propagation and opening. We restrict ourselves to the case of low-temperature, upper crustal conditions, where brittle faulting takes place. Obviously this is appropriate close to the Earth’s surface, but becomes less valid at greater depths. Ductile deformation (and the related strain-softening or strain-hardening behaviours) and faulting in the lower crust is beyond the scope of this paper. A similar behaviour is expected at the field scale where rock properties are likely to be modified before an earthquake, although such modifications are likely to be restricted to an area close to the fault (Rudnicki 1988). Again, many observations point to such a behaviour: elastic properties can vary, as indicated by elastic wave velocity records (Scholz 1990; Le Ravalec et al. 1996; Crampin & Zatsepin 1997); transport properties such as permeability and electrical conductivity are also modified (Roeloffs 1988; Park et al. 1993). Fault and earthquake

mechanics is thus a research field where damage and localization concepts are extremely relevant. Other important areas in geology where the same concepts are very useful are those concerned with reservoir rocks and underground storage. Damage and localization are intuitively considered above as referring to a spatial strain organization: homogeneous for damage and inhomogeneous for localization. This intuitive approach attempts to capture the rock structure evolution. From this point of view, damage refers here to a more or less diffuse distribution of cracks and localization refers to a macroscopic strain concentration. There is, however, a clear distinction to be made between these intuitive views and the theoretically based concepts of damage and localization. A considerable amount of work has been conducted on both damage and localization during recent decades. There is more than 30 years between the early work of Kachanov (1958), which introduced the damage parameter D in macroscopic continuum mechanics, for ductile deformation, and the more recent work of Kachanov (1993), which is more appropriate to the brittle field and has established on a clear basis how elastic properties of cracked rocks can be derived using crack distribution at the microscopic scale. There is a shorter time period between the early work of Hill (1962), which gives the theoretical framework for localization, and the fundamental results of Rudnicki & Rice (1975), which provide the conditions for localization in dilatant rocks,

From: LEWIS , H. & COUPLES , G. D. (eds) The Relationship between Damage and Localization. Geological Society, London, Special Publications, 289, 7–17. DOI: 10.1144/SP289.2 0305-8719/07/$15.00 # The Geological Society of London 2007.

Y. GUE´GUEN & P. BE´SUELLE

8

but these results have been recently revisited and extended to compacting porous rocks (Olsson 1999; Issen & Rudnicki 2000; Be´suelle 2001). Although most of the early work on both damage and localization has been developed in fundamental continuum mechanics and not in geology, it has been clear for some time that these concepts are very relevant to understand and quantify rock deformation and rock properties evolution in the upper crust. It is the purpose of this synthesis paper to recall these basic concepts and show why and how they can be of great interest in upper crustal geology and geophysics. The present work focuses first on damage (in upper crustal conditions where microcracking is the main process) at the micro and macro scale, and then on localization. As explained above, we consider the brittle deformation regime so that strains are small ones. The rock is considered to be elastic at low stresses. Increasing stress results in a progressive degradation of elastic moduli as a result of cracking. Beyond a certain stress level, localization takes place. Ductile behaviour and large strains are not considered in this paper.

Fig. 1. Geometric definition of D in the isotropic case: D ¼ dSD/dS.

where dS is the area of intersection of a plane with a representative elementary volume (REV) of the rock and dSD is the area of the intersections of the same plane with all microcracks and microcavities within dS (Fig. 1). D is a continuous variable defined at the scale of the REV, which varies between zero (undamaged material) and unity (fully damaged material). The above definition allows definition of the effective stress seff as the ratio of the load to the surface area that effectively resists the load, i.e. (S 2 SD) ¼ (1 2 D) S: seff ¼ s=ð1  DÞ:

ð2Þ

Damage The choice of a damage parameter is not straightforward. It should as simple as possible and have a clear mechanical sense. The situation will be very different for ductile failure, as originally discussed by Kachanov (1958), or for the brittle failure of rocks that we examine below. From a rock physics and mechanics point of view, the idea of damage is related to the fact that any rock subjected to loading may experience a progressive deterioration of its physical and mechanical properties. Experimentally, such a deterioration can be demonstrated by the measurement of various properties. For instance, if the rock has an elastic behaviour, its elastic moduli decrease progressively with increasing damage. Although it is intuitive that this deterioration results from a more or less diffuse distribution of cracks, the question is: how can damage be quantified? To answer this question, we present first the theoretical background, which has been established at both the macro and micro scales. We then show through a few examples that it provides an appropriate framework for interpreting the experimental results (see also Dresen & Gue´guen 2004).

Damage variables: macroscopic approach How should we define damage? In the simplest isotropic model, the damage parameter D is a scalar quantity defined as D ¼ dSD =dS

ð1Þ

Recalling that, for a uniaxial compression, Young’s modulus E is defined by seff ¼ E1, it results from equations (1) and (2) that in a damaged rock ED ¼ Eð1  DÞ:

ð3Þ

ED can be considered as the effective elasticity modulus of the damaged rock. The introduction of D as explained above is attractive because of its simplicity. It is straightforward to quantify damage through, for instance, elastic wave velocity measurements, which provide directly elastic moduli values and hence D values from equation (3). Such a quantification of damage may be useful for practical reasons but it does not carry any fundamental information on the micromechanics. Moreover, equations (1) and (2) hold only for an isotropic damage, which is not the general case. How can we deal with the 3D situation and anisotropy? An extension of the surface definition of damage can be obtained in two ways. The extension should account for anisotropy and replaces the scalar D by a tensor. The first way to do this is purely geometrical. A second-rank damage tensor Dij is introduced as follows. Let the area dS, oriented by its normal unit vector with components ni, become after damage an area dS0 with normal n0i. If Iij is the unit second-rank tensor, then Dij is given by ðIij  Dij Þn j dS ¼ n0i dS0 :

ð4Þ

DAMAGE AND LOCALIZATION IN ROCKS

Does this second-rank tensor account for the variations of the various physical properties of the rock? There is no fundamental reason why this should be so, as again, the micromechanical processes have not been considered. The second way to generalize the scalar D into a tensor is of greater interest, as it is specific to a given mechanical behaviour and to well-identified micromechanical processes. These processes are, for rocks in upper crustal conditions, those of the micromechanics of cracking. More specifically, let us consider the important case of the elastic properties. Using the above assumption makes it easy to define a fourth-rank damage tensor to quantify the deterioration of the elastic moduli. In the 3D situation, equation (3) is generalized as follows. In the damaged rock, (sij)eff ¼ Cijkl1kl, where Cijkl is the elastic constants tensor of the intact rock. Its inverse is the elastic compliances tensor Sijkl. In the damaged rock, these tensors become (Cijkl)D and (Sijkl)D. Strain can be obtained from 1kl ¼ (Sklmn)Dsmn. Reporting this relation in the expression for (sij)eff we obtain (sij)eff ¼ Cijkl (Sklmn)Dsmn. Generalizing equation (2) to the 3D case, this provides the relation ðIijmn  Dijmn Þ1 ¼ Cijkl ðSklmn ÞD

Fig. 2. Idealized stress –strain plot for a brittle rock where inelastic deformation is assumed to be caused by microcracking. The mechanical behaviour is elastic, both in loading and unloading conditions, up to a limit sc, which is the elastic limit. The elastic limit defines a yield surface in the stress space. The effective elastic modulus at point A corresponds to unloading. The inelastic increment of strain between A and B corresponds to loading and is due entirely to the incremental increase in compliance as a result of crack extension.

ð5Þ

where Dijmn are the components of the fourth-rank damage tensor. It is a natural consequence of anisotropic elasticity that a fourth-rank tensor emerges, as the elastic constants tensor is a fourth-rank one. Inverting equation (5) and introducing DCijkl ¼ Cijkl 2 (Cijkl)D, the components of the damage tensor are obtained in terms of the elastic constants: Dijmn ¼ DCijkl Sklmn :

9

ð6Þ

As for equation (3), this allows determination of the damage from measurements of elastic properties. The above model is valid to describe the mechanical behaviour of a rock that is elastic up to a limit given by a plasticity criterion (yield surface). Above that limit, the mechanical behaviour follows some flow rule during loading and is elastic during unloading, as shown in Figure 2. The flow rule provides the incremental plastic strain and is discussed in the section below.

Damage variables: microscopic theories The above macroscopic approach is phenomenological. It can provide, using equation (5) or (6), the values of the Dijkl components from measurements of elastic constants in both the intact and the damaged rock. However, it does not give any theoretical prediction of the effective elastic constants

of the damaged rock, which have to be experimentally measured. An independent microscopic approach is required to obtain a theoretical prediction. Such an approach is derived from effective media theories (EMTs) of physics. It is impossible to perform exact theoretical calculations because crack interactions should be taken into account. Crack interactions depend on their separation distance. The ratio crack length/crack separation distance is the parameter that controls crack density. Obviously, crack density is smaller than unity and approaches that value only extremely close to fracture. Two main approximations can be used in EMT calculations. One is the approximation of an effective matrix, and the other is the approximation of an effective field (Kachanov 1993). Effective matrix calculations assume that crack interactions can be obtained by considering that each crack is isolated in a medium that is not the matrix but an effective matrix. A possible way to do this is to derive a set of implicit equations by giving a priori to the matrix the (unknown) properties that one wants to calculate (self-consistent method). A simple limiting case is to ignore crack interactions and keep the original matrix as the effective one. This is called the non-interacting crack model and is the simplest model (expected to be valid only at crack densities 1). The effective field approximation averages out crack interactions by assuming that each crack sees an

10

Y. GUE´GUEN & P. BE´SUELLE

average stress field that is the externally applied field modified by the existence of the cracks themselves. Although various choices are possible, a simple one is to consider that the average stress field is the volumetric average. In the case of upper crustal conditions, the damage is called brittle because the ratio of plastic strain to elastic strain is very small and we assume here that the damage at the micro-scale is mainly due to microcracking in rocks so that the idealized stress –strain curve is as represented in Figure 2. This means that if the rock is stressed beyond a critical threshold, its original elastic modulus decreases. Beyond the threshold, the crack microstructure is continuously modified. The inelastic increment of strain corresponding to loading from A to B in Figure 2 is entirely due to the incremental increase in compliance, as a result of microcrack extension. The specific values of the elastic moduli for a given crack microstructure (at any point on the stress –strain plot) can be obtained from an EMT model with the appropriate microcrack distribution. In this calculation, cracks are assumed to be frozen in (i.e. prevented from propagating and nucleating). To keep the theory simple, an important assumption has been introduced above. We assume that at a given point in the material, the stress –strain relation is local; that is, stress s at point M depends on strain 1 at the same point and not on strain at any other point in the rock. This assumption, which was called the strain equivalence principle by Lemaitre (1996), is identical to the well-known classical approximation in EMT (Beran & McCoy 1970). This is satisfactory as long as cracks are frozen in. It is worth pointing out, however, that a non-local damage theory has been developed by Bazant (1994) to address the problem of a non-uniform system of interacting and growing microcracks. This is of particular importance to handle localization from crack interactions. Such non-local theories provide a way to bridge the gap between damage and localization (see below). This is, however, beyond the scope of this introductory paper. A detailed overview of the known results on effective elastic properties of cracked solids has been given by Kachanov (1993). Of special interest is the model of non-interacting cracks, because, as shown by Kachanov (1993), it remains accurate at high crack densities (up to 0.5; Schubnel & Gue´guen 2003), provided the locations of cracks centres are random. This model is an effective matrix model; in fact, it is the simplest one, as the matrix is assumed to be the intact rock as discussed above. We give below the results of this model because of their wide range of validity and their broad field of application. That this model is valid in a broad range of crack densities appears to be a

paradox. There are two independent reasons why this is so. The first one is that at a microscopic level, crack interactions exist but approximately compensate. This means that shielding and amplification effects cancel each other, on average. The second reason is that the simplest effective field method (volumetric average) predicts exactly the same results. The volumetric averaged stress field is known to be unmodified by the presence of cracks so that it predicts exactly the same result as the non-interacting crack model. This reflects again the above-mentioned compensating effects. The calculation of effective elastic moduli in the approximation of non-interacting, randomly oriented cracks in an isotropic matrix was first solved by Bristow (1960). The general method is to calculate the elastic potential energy f (sij) per unit volume for the cracked rock: f ¼ f0 þ Df

ð7Þ

where f0 is the potential energy of the undamaged rock and Df is the additional term related to cracks. By definition of the effective compliances (Sijkl)D ¼ (Cijkl)21 D , f ¼ ð1=2Þsij skl ðSijkl ÞD :

ð8Þ

Equations (7) and (8) directly provide the effective elastic constants (or alternatively compliances) of the damaged rock if Df is known. The key point in this approach is the derivation of Df from micromechanics. As shown by Kachanov (1993), this additional potential energy related to cracks can be expressed as a function of the crack density tensor aij. An approximate expression of Df in the case of non interacting cracks in tensile conditions is Df ¼ ð1=E0 Þ½8ð1  n20 Þ= 3ð1  n0 =2Þsij s jk aik

ð9Þ

where E0 and n0 are Young’s modulus and Poisson’s ratio of the intact isotropic matrix, respectively. The above result is not exact but it is valid with an approximation of less than 10%. The introduction of the crack density tensor aij results from the micromechanical calculation of the elastic potential. For non-random crack orientations in the case of circular cracks, aij ¼ ð1=VÞ

X ðc3 ni n j Þk

ð10Þ

where c is the crack radius and V is the rock volume. The elastic compliances of the cracked rock are found to be modified with respect to the intact

DAMAGE AND LOCALIZATION IN ROCKS

rock by an amount DSijkl. Interestingly, a consequence of the previous approximation is that the effective elastic properties of the cracked rock are orthotropic if the undamaged rock is isotropic in dry conditions. The orthotropy is, in this case, of a simplified type so that only four independent elastic constants exist, corresponding to the four independent components of the ninj distributions (Kachanov 1993). The above approximation introduces an error of less than 10% on the values of the DSijkl components so that it may be considered as acceptable. Within this approximation, the damage is characterized by the crack density tensor aij, which is a second-rank tensor. Although the elastic constants are anisotropic and described by a fourth-rank tensor, it turns out that a second-order tensor is sufficient to quantify damage within the previously specified conditions and approximations. A complete calculation introduces, in addition to the tensor a, a fourth-rank tensor b, which is small for dry cracks (Sayers & Kachanov 1995). If b is small, an additional assumption to be considered is that of an isotropic random orientation distribution of cracks. This is the simplest situation. The commonly used crack density parameter r (Bristow 1960) is sufficient in that case: aij ¼ ðr=3Þdij

ð11Þ

where dij ¼ 0 if i = j and dij ¼ 1 if i ¼ j. In that case, and only in that case, the damage parameter is a scalar quantity r. For an isotropic random orientation distribution of identical circular cracks r ¼ ðN=VÞc3

(a)

ð12Þ

11

where N/V is the number of cracks per unit volume. Then orthotropy does not hold if cracks are allowed to slide. Moreover, if friction is present, deformation is stress path dependent and stress-induced anisotropy develops. Another important case, which may be important in the crust, is that of fluid-saturated cracks. Then the tensor b is not negligible and it provides direct information on fluid nature and saturation (Schubnel & Gue´guen 2003). From a geophysical point of view, the main interest of this approach is that quantitative information on the rock crack and fluid content can be extracted from elastic wave velocities.

Localized deformation The strain distribution in natural materials may generally follow several types of organization. Depending on the conditions of pressure, temperature, strain rate, strain history and other physical and chemical processes, strain may be more or less distributed through a large domain of the material, as described above. However, in the brittle regime, close to a critical loading (in a broad sense), the displacement field becomes concentrated in localized structures, which absorb the main part of the displacement within the material, for a given set of boundary conditions. The different observations of the localization phenomena at a macroscopic scale can be classified into two main groups, as follows. (1) Fractures. These form a discontinuity of the material and correspond to a discontinuity of the displacement field (Fig. 3a). This discontinuity is quantified in term of displacement jump across the fracture surfaces and the notion of strain is locally irrelevant. Fracture mechanics provides a

(b)

Fig. 3. Schematic description of the two types of localization structures. (a) Crack discontinuity; (b) Strain localization band through a solid.

12

Y. GUE´GUEN & P. BE´SUELLE

theoretical approach to analyse the condition of fracture propagation. Fracture mechanics can be applied if initial flaws or cracks exist within the material. This approach is appropriate at several scales, the micro-scale for intragranular cracks or the macro-scale for large fractures. The basic principles of fracture mechanics are outlined below. (2) Localization bands. They have a finite thickness so that the concept of strain within the band is meaningful and can be used. This strain generally has a dominant shear component and the bands are called ‘shear bands’ (Fig. 3b), although compaction bands also exist (e.g. Aydin et al. 2006). At a microscopic scale, these bands are generated by a progressive concentration of microevents: grain sliding and grain rotation for soils; microcracking, grain crushing and cataclastic deformation for rocks (Wong 1982; Mene´ndez et al. 1996; Be´suelle et al. 2000; Mair et al. 2002; Baud et al. 2004; Paterson & Wong 2005). Bulk properties such as permeability or wave speeds are modified in the localization band. Bifurcation theory provides a way to analyse the localization band initiation. This theory considers the localization as an instability, and is presented below. An overview of some experimental results on strain localization in soils and rocks is addressed in more detail by Desrues et al. (2007). Although the two definitions above are clearly different, fractures and shear bands structures are linked in reality. Shear bands, which are initiated by microcrack concentration, can also transform into a macroscopic fracture as a result of the coalescence of cracks (see, e.g. references given by Desrues et al. 2007). In this case, the fracture can be considered as the result of a strong deformation inside the band. Reciprocally, the process zone of propagation at the tip of a fracture has a non-zero width, which implies that the material close to the fracture boundary can be damaged within a zone that has a finite thickness. We may note also that localization structures such as en echelon crack arrays seem to be intermediate between fractures and shear bands.

Fracture mechanics It is well known that the existence of a cavity, in a broad sense including a hole or a fracture, in a bulk solid plays the role of a stress concentrator. Pioneer studies by Kolosov (1909) and Inglis (1913) derived the expression of the stress field around an elliptical cavity. Major contributions that initiated fracture mechanics are the Griffith energy theory (Griffith 1921), which expresses a condition for crack propagation based on the energy required to separate surfaces in a solid, and the notion of a stress intensity factor, proposed by Irwin (1948),

Fig. 4. Schematic illustration of the stresspconcentration ahead of the crack tip, proportional to 1/ r.

to characterize the singularity of the stress field near a crack tip, in the linear elastic solution. Stress concentration in elastic fracture mechanics. To estimate the stress concentration near the crack tip it is helpful to consider a body in which the material has a linear elastic and isotropic behaviour with homogeneous properties. It is convenient to distinguish three types of crack propagation, depending on the loading conditions. Mode I corresponds to a tensile opening of the crack surfaces, mode II to a relative shear sliding of the faces in the direction of the crack propagation (in-plane shear) and mode III to a shear sliding perpendicular to the direction of crack propagation (anti-plane shear). The stress field near the crack tip (Fig. 4) can be approximated by (e.g. Lawn 1993) sij ¼ Ka ð2prÞ1=2 fija ðuÞ:

ð13Þ

Additional terms that depend on the boundary conditions on the body are negligible near the crack tip. r is the distance from the crack tip, u is a polar angle defined from the crack plane, Ka with a ¼ I, II, III are stress intensity factors, and f aij are stress field distribution functions. It is remarkable that, for a given loading mode, the stress field near the crack tip depends on loading applied to the body only through the stress intensity factors. The crack tip is a singular point of the stress field; from equation (13), it results that the maximum principal stress should be infinite at that point. The linear elastic behaviour assumption leads to overestimation of the stress at the tip. In reality, a material has an inelastic, irreversible mechanical behaviour when stress exceeds a certain value. More realistic models can be used for stress intensity analysis by considering energy dissipative processes in a small zone close to the crack tip (e.g. Irwin 1957; Rice 1968). However, within the elastic approximation, the strain energy in any finite volume around the crack tip remains finite. An alternative approach is to consider the existence of cohesive forces between the crack

DAMAGE AND LOCALIZATION IN ROCKS

faces, near the tip. Such forces allow elimination of the singularity in the stress field (e.g. Dugdale 1960; Barenblatt 1962). Energy considerations. A complementary approach for crack propagation is based on the energy balance of the solid including the crack. During an infinitesimal crack propagation, the external loads work exceeds the increase of strain energy in the body. The difference, per unit area of new crack surface, is defined as the energy release G (Griffith 1921). The energy release corresponds physically to the work ‘needed’ to create the new free crack surfaces. Mathematically, this energy corresponds to the displacement of the stress intensity field around the crack tip. For elastic reversible propagation, it is then possible to express G as a linear combination of the square of the stress intensity factors Ka. If irreversible processes are important, the energy release is accounted for using the mathematical device developed by Rice (1968), the so-called path-independent J integral. Finally, the propagation criterion is expressed when energy release reaches a critical value G ¼ Gc

ð14Þ

or when stress intensity factors reach critical values Kac. The Kac values can be considered as material parameters and have been intensively studied for various materials in laboratory experiments. Estimations of critical values have also been proposed at large scale for natural crustal faults (e.g. Rudnicki 1980).

Bifurcation theory In the spirit of the previous approach, a macroscopic fault can be considered as a structure similar to the propagation of a crack or the coalescence of several cracks, which can be nucleated as a result of the existence of initial flaws in the material. Bifurcation theory provides a different way of looking at the localization process. The strain localization area is now considered to be a structure that appears spontaneously within a homogeneous material. At the onset of bifurcation, the initially homogeneous strain field shows a strain concentration in a small area. What are the necessary conditions for such an instability? The theoretical answer goes back to the initial work of Hadamard (1903) on dynamical aspects of wave propagation, and to studies by Thomas (1961), Hill (1962) and Mandel (1964). Rice (1976) reviewed these previous studies for quasi-static conditions and Rudnicki & Rice (1975) identified the strain localization conditions for pressure-sensitive dilatant rocks. The condition

13

of localization for shear banding as developed by Rice and coworkers is reviewed below. A comprehensive review has been given by Be´suelle & Rudnicki (2004). An infinite solid is considered initially to be uniformly deformed. The strain is defined as the symmetrical part of the displacement gradient: 21kl ¼ @uk =@xl þ @ul =@xk where uk is the displacement vector. The bifurcation analysis determines the conditions for which a nonhomogeneous strain field can develop within an infinite planar band. The strain rate inside the shear band 1˙sb is assumed to be equal to the strain rate outside 1˙o plus an additional term, which depends only on the normal direction to the incipient shear band. This additional strain corresponds to the superimposition of a simple shear and a uniaxial deformation and is expressed by ni, the unit normal vector to the band (Fig. 5), and gi, which is, a priori, an arbitrary vector:

1_ sb _ oij þ 1=2ðgi nj þ ni gj Þ: ij ¼ 1

ð15Þ

The equilibrium condition in the solid can be expressed on the boundaries of the band by the equality of the product of the stress tensor and the unit normal vector (this product has the dimension of a force acting on a unit surface of the boundaries) on each side of the boundaries (Rudnicki & Rice 1975): _ oij ni s_ sb ij ni ¼ s

ð16Þ

where s˙sb ˙ oij are the stress tensor’s time derivaij and s tive inside and outside the shear band. The

Fig. 5. Schematic illustration of the shear band. n is the unit normal vector to the band and g is an arbitrary vector that depends on the x-direction.

Y. GUE´GUEN & P. BE´SUELLE

14

bifurcation condition occurs when these conditions can be met for a non-trivial solution (i.e. a nonzero vector gi). The stress time derivative and the strain rate inside and outside the band are assumed to be related by a linear relation using a fourth-rank symmetrical tensor (Lijkl ¼ Ljikl ¼ Lijlk). The rheological response inside and outside the shear zone can be different. Let us consider here, for example, an elastoplastic law for which the material can load with plastic deformation or unload elastically. However, other categories of constitutive laws can be considered as damage laws. With an elastoplastic law, the material inside the shear band can deform plastically while the material outside the shear zone is unloading elastically: sb sb _ kl s_ sb ij ¼ Lijkl 1

and

s_ oij ¼ Loijkl 1_ okl

ð17Þ

Using equations (15)–(17), we obtain o _ okl ¼ Lsb ðLsb ijkl  Lijkl Þni 1 ijkl nl ni gk :

ð18Þ

Rice & Rudnicki (1980) have shown that to study the condition of first localization it was justified to consider an elastoplastic behaviour inside and outside the incipient shear band at the onset of bifurcation (L sb ¼ L o). This condition can be viewed as a condition of neutral loading on the boundaries of the band; that is, the material at the boundaries is in a state between elastic unloading and plastic loading. The equilibrium condition (18) is then expressed by Lijkl nl ni gk ¼ 0:

(a)

(b)

σ1 σ3

ð19Þ

A nonzero solution for gi is then possible when a deformation is reached for which detðni Lijkl nl Þ ¼ 0:

For small load and for any possible orientation ni, the determinant is generally positive and decreases when strain increases. Localization becomes possible when it becomes zero for a particular orientation ni. The vector gi is an eigenvector of the tensor nLn and if g and n are orthogonal, the additional strain inside the incipient band is pure shear. This strain can have also a normal component, dilating or compacting, which corresponds to a lengthening or shortening in the direction of the normal to the band, if g and n are not orthogonal. That situation corresponds to an increase or a decrease of the porosity inside the band (Be´suelle 2001). The band orientations corresponding to a pure extension band, a dilating and compacting shear band, and a pure compaction band are represented in Figure 6 with respect to the principal stress directions in an axisymmetric compression stress state. The bifurcation analysis determines an onset of localization, the angle of the band with respect to the principal stress directions and the volumetric/ shear strain ratio in the band. The theoretical onset is strongly dependent on the constitutive model so that the material behaviour has to be well known. Intensive investigations of various models have been developed recently, for elastoplastic models with several shapes of yield and potential surfaces (e.g. Molenkamp 1985; Vardoulakis & Sulem 1995) or multilinear models (e.g. Chambon et al. 2000). Precise descriptions (c)

σ1 σ3

ð20Þ

σ1 σ3

(d)

σ1 σ3

Fig. 6. Schematic illustration of the shear band orientation for an axisymmetric compression stress state with respect to the major (s1) and minor (s3) principal stress directions, for (a) an extension band, (b) a dilating shear band, (c) a compacting shear band, (d) a compaction band.

DAMAGE AND LOCALIZATION IN ROCKS

15

of localization (onset, orientation) in laboratory experiments can also help to calibrate the constitutive model in the spirit of a compliance analysis (Vermeer 1982; Desrues & Chambon 2002). In such an approach, some parameters of the law are determined using a comparison between experimental results on strain localization and bifurcation predictions. From a geomechanical point of view, the analysis above allows an interpretation of shear band development as well as compaction band development.

Cosserat (1909) with micromechanical considerations for granular material or the second-gradient models developed by Mindlin (1964) and extended to geomaterials (Chambon et al. 2001; Be´suelle et al. 2006) allow predictions of shear band thickness (Vardoulakis & Sulem 1995). Such formalisms can be extrapolated to damage models (e.g. Kotronis et al. 2007). Another approach is the nonlocal damage models, which introduce also an internal length (Bazant 1994) and allow prediction of band thickness (Pijaudier-Cabot & Benallal 1993).

Diffused and localized damage

Conclusion

Damage models, as treated above, link the stress and strain locally in any part of a solid. They are convenient for dealing with rocks that are typically in upper crustal conditions (i.e. in the brittle failure deformation regime). In this case, rock deterioration during loading is induced by a more or less diffuse crack distribution. If we consider a laboratory elementary test, such a law can describe the homogeneous response of the specimen (i.e. before strain localization). In such conditions, rock behaviour is elastic up to a limit where microcracking takes place and degrades the elastic moduli. Strains are small. In counterpart, when strain localization occurs in the specimen, the global response of the specimen becomes a structural response. However, this does not exclude consideration that the damage law is valid in every part of the specimen; that is, inside and outside the incipient deformation band. As shown above, strain localization can be considered as a problem of loss of uniqueness; that is, for the same initial mathematical problem several solutions become possible, without or with a deformation band. The criteria express when a localization band becomes possible. The bifurcation prediction is strongly dependent on the constitutive law and can be applied to different types of model. Above, we considered the case of an elastoplastic law, but, for example, bifurcation prediction can also be applied to damage models (e.g. Dragon et al. 1994). In such a case, strain localization is associated with a damage localization; that is, a concentration of cracks inside a band of finite thickness. As far as the band thickness is concerned, damage models, and more generally all classical continuum models, are unable to predict it. Only constitutive models with internal lengths can do this. This means that if prediction of incipient localization is possible, prediction of the response after localization (i.e. the post-localization regime) is not possible with classical models. Models that take into account higher displacement gradients are useful. The micropolar models proposed by

Damage and localization are two distinct concepts, which are of major importance in geology and geophysics. Considering upper crustal conditions (brittle failure regime), a precise definition of damage parameters is obtained from microcracking processes. Using such parameters allows us to quantify how much a given rock is damaged (i.e. how far it is beyond its elastic limit). Crack and fluid content can be extracted from elastic moduli (or from wave velocities). On the other hand, the completely independent approach of bifurcation theory allows us to know how close a rock is to localization (i.e. how close it is to shear band or compaction band nucleation). To bridge the gap between the concepts of damage and localization, non-local damage theories and second gradient models appear to be very promising. At the field scale, these tools can be used to infer the damage state of a highly stressed area (damaged area) from elastic wave velocity measurements, or to interpret shear band orientations in relation to the in situ stresses and the volumetric strain type, compactive or dilatant (localization).

References A SHBY , M. F. & S AMMIS , C. G. 1990. The damage mechanics of brittle solids in compression. Pure and Applied Geophysics, 133, 489– 521. A YDIN , A., B ORJA , R. I. & E ICHHUBL , P. 2006. Geological and mathematical framework for failure modes in granular rock. Journal of Structural Geology, 28, 83–98. B ARENBLATT , G. I. 1962. Mathematical theory of equilibrium cracks in brittle fracture. Advances in Applied Mechanics, 7, 55– 129. B AUD , P., K LEIN , E. & W ONG , T.-F. 2004. Compaction localization in porous sandstones: spatial evolution of damage and acoustic emission activity. Journal of Structural Geology, 26, 603– 624. B AZANT , Z. P. 1994. Nonlocal damage theory based on micromechanics of cracks interactions. Journal of Engineering Mechanics, 120, 593–617. B ERAN , M. J. & M C C OY , J. J. 1970. Mean field variations in a statistical sample of heterogeneous linearly elastic

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solids. International Journal of Solids and Structures, 6, 1033. B E´ SUELLE , P. 2001. Compacting and dilating shear bands in porous rocks: theoretical and experimental conditions. Journal of Geophysical. Research, 106, 13435– 13442. B E´ SUELLE , P. & R UDNICKI , J. W. 2004. Localization: shear bands and compaction bands. In: G UE´ GUEN , Y. & B OUTE´ CA , M. (eds) Mechanics of FluidSaturated Rocks. Academic Press, New York, 219– 321. B E´ SUELLE , P., D ESRUES , J. & R AYNAUD , S. 2000. Experimental characterisation of the localization phenomenon inside a Vosges sandstone in a triaxial cell. International Journal of Rock Mechanics and Mining Sciences, 37, 1223–1237. B E´ SUELLE , P., C HAMBON , R. & C OLLIN , F. 2006. Switching deformation modes in post-localization solutions with a quasibrittle material. Journal of Mechanics of Materials and Structures, 1, 1115–1134. B RISTOW , J. R. 1960. Microcracks and the static and dynamic elastic constants of annealed and heavily cold-worked metals. British Journal of Applied Physics, 11, 81– 85. C HAMBON , R., C ROCHEPEYRE , S. & D ESRUES , J. 2000. Localization criteria for nonlinear constitutive equations of geomaterials. Mechanics of CohesiveFrictional Materials, 5, 61–82. C HAMBON , R., C AILLERIE , D. & M ATSUSHIMA , T. 2001. Plastic continuum with microstructure, local second gradient theories for geomaterials: localization studies. International Journal of Solids and Stuctures, 38, 8503–8527. C HELIDZE , T. 1993. Fractal damage mechanics of geomaterials. Terra Nova, 5, 421– 437. C OSSERAT , E. 1909. The´orie des corps de´formables. Hermann, Paris. C RAMPIN , S. & Z ATSEPIN , S. 1997. Modelling the compliance of crustal rock—II. Response to temporal changes before earthquakes. Geophysical Journal International, 129, 495–506. D ESRUES , J. & C HAMBON , R. 2002. Shear bands analysis and shear moduli calibration. International Journal of Solids and Structures, 39, 3757– 3776. D ESRUES , J., B E´ SUELLE , P. & L EWIS , H. 2007. Strain localization in geomaterials. In: L EWIS , H. & C OUPLES , G. D. (eds) The Relationship between Damage and Localization. Geological Society, London, Special Publications, 289, 47–73. D RAGON , A., C ORMERY , F., D E´ SOYER , T. & H ALM , D. 1994. Localised failure analysis using damage models. In: C HAMBON , R., D ESRUES , J. & V ARDOULAKIS , I. (eds) Localization and bifurcation theory for soils and rocks. Balkema, Rotterdam, 127 –140. D RESEN , G. & G UE´ GUEN , Y. 2004. Damage and rock physical properties. In: G UE´ GUEN , Y. & B OUTE´ CA , M. (eds) Mechanics of Fluid-Saturated Rocks. Academic Press, New York, 169–218. D UGDALE , D. S. 1960. Yielding of steel sheets containing slits. Journal of Mechanics and Physics of Solids, 8, 100– 104. G RIFFITH , A. A. 1921. The phenomena of rupture and flow in solids. Philosophical Transactions of the Royal Society of London, Series A, 221, 163–197.

H ILL , R. 1962. Acceleration waves in solids. Journal of Mechanics and Physics of Solids, 10, 1– 16. I NGLIS , C. E. 1913. Stresses in a plate due to the presence of cracks and sharp corners. Transactions of the Institute of Naval Architects, 55, 219–241. I RWIN , G. R. 1948. Fracture dynamics. Fracturing of Metals, 147– 166. I RWIN , G. R. 1957. Analysis of stress and strain near the end of a crack traversing a plane. Journal of Applied Mechanics, 24, 361 –364. I SSEN , K. & R UDNICKI , J. 2000. Conditions for compaction bands in porous rocks. Journal of Geophysical Research, 105, 21529– 21536. K ACHANOV , L. 1958. On creep rupture time. Izvestiya Academii Nauk SSSR, Otdelenie Tekhnicheskikh Nauk, 8, 26– 31. K ACHANOV , M. 1993. Elastic solids with many cracks and related problems. Advances in Applied Mechanics, 30, 259–445. K OLOSOV , G. 1909. Sur le proble`me plan dans la the´orie d’e´lasticite´. In: C ASTELNUOVO , G. (ed.) Atti IV Congresso Internationale Matematici, 3. Academia Lincei, Rome, 187– 190. K OTRONIS , P., C OLLIN , F., B E´ SUELLE , P., C HAMBON , R. & M AZARS , J. 2007. Local second gradient models and damage mechanics: 1D post-localisation studies in concrete specimens. In: E XADAKTYLOS , G. & V ARDOULAKIS , I. (eds) 7th International Workshop on Bifurcation, Instabilities and Degradation in Geomechanics, Chania, Greece, June 13– 16, 2005, 127–142. L AWN , B. 1993. Fracture of Brittle Solids. Cambridge University Press, Cambridge. L EMAITRE , J. 1996. A Course on Damage Mechanics. Springer, Berlin. L E R AVALEC , M., G UE´ GUEN , Y. & C HELIDZE , T. 1996. Magnitude of velocities anomalies prior to earthquakes. Journal of Geophysical Research, 101, 11217–11223. M AIN , I., S AMMONDS , P. R. & M EREDITH , P. G. 1993. Application of a modified Griffith criterion to the evolution of fractal damage during compressional rock failure. Geophysical Journal International, 115, 367–380. M AIR , K., E LPHICK , S. & M AIN , I. 2002. Influence of confining pressure on the mechanical and structural evolution of laboratory deformation bands. Geophysical Research Letters, 29, 49. M ANDEL , J. 1964. Conditions de stabilite´ et postulat de Drucker. In: Proceedings of IUTAM Conference on Rheology and Soil Mechanics, Grenoble, Springer, Berlin, 58–68. M ENE´ NDEZ , B., Z HU , W. & W ONG , T.-F. 1996. Micromechanics of brittle faulting and cataclastic flow in Berea sandstone. Journal of Structural Geology, 18, 1–16. M INDLIN , R. D. 1964. Micro-structure in linear elasticity. Archives for Rational Mechanics and Analysis, 4, 50–78. M OLENKAMP , F. 1985. Comparison of frictional material models with respect to shear band initiation. Ge´otechnique, 35, 127– 143. O LSSON , W. A. 1999. Theoretical and experimental investigation of compaction bands. Journal of Geophysical Research, 104, 7219–7228.

DAMAGE AND LOCALIZATION IN ROCKS P ARK , S. K., J OHNSTON , M. J. S., M ADDEN , T. R., M ORGAN , F. D. & M ORRISON , H. F. 1993. Electromagnetic precursors to earthquakes in the ULF band: a review of observations and mechanisms. Review of Geophysics, 31, 117– 132. P ATERSON , M. S. & W ONG , T.-F. 2005. Experimental Rock Deformation—The Brittle Field, Springer, Berlin. P IJAUDIER -C ABOT , G. & B ENALLAL , A. 1993. Strain localization and bifurcation in a non local continuum. International Journal of Solids and Structures, 30, 1761–1775. R ICE , J. R. 1968. A path independent integral and the approximate analysis of strain concentration by notches and cracks. Journal of Applied Mechanics, 35, 379–386. R ICE , J. R. 1976. The localization of plastic deformation. In: K OITER , W. T. (ed.) International Congress of Theoretical and Applied Mechanics. North-Holland Publishing Co., Delft, 207 –220. R ICE , J. R. & R UDNICKI , J. W. 1980. A note on some features of the theory of localization of deformation. International Journal of Solids and Structures, 16, 597– 605. R OELOFFS , E. 1988. Hydrologic precursors to earthquakes: a review. Pure and Applied Geophysics, 126, 177–209. R UDNICKI , J. W. 1980. Fracture mechanics applied to the Earth’s crust. Annual Review of Earth and Planetary Sciences, 8, 489– 525. R UDNICKI , J. W. 1988. Physical models of earthquakes instability and precursory processes. Pure and Applied Geophysics, 126, 465–498.

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R UDNICKI , J. W. & R ICE , J. R. 1975. Conditions for the localization of the deformation in pressure sensitive dilatant materials. Journal of Mechanics and Physics of Solids, 23, 371– 394. S AYERS , C. & K ACHANOV , M. 1995. Microcracks induced elastic waves anisotropy of brittle rocks. Journal of Geophysical Research, 100, 4149– 4156. S CHOLZ , C. H. 1990. The Mechanics of Earthquakes and Faulting. Cambridge University Press, New York. S CHUBNEL , A. & G UE´ GUEN , Y. 2003. Dispersion and anisotropy of elastic waves in cracked rocks. Journal of Geophysical Research, 108, 2101– 2116. T HOMAS , T. Y. 1961. Plastic Flow and Fracture in Solids. Academic Press, New York. V ARDOULAKIS , I. & S ULEM , J. 1995. Bifurcation Analysis in Geomechanics. Blackie, Glasgow. V ERMEER , P. A. 1982. A simple shear-band analysis using compliances. In: VERMEER , P. A. & LOGER , H. J. (eds) Proceedings of the IUTAM Conference on Deformation and Failure of Granular Materials. Balkema, Rotterdam, 493– 498. W ONG , T.-F. 1982. Micromechanics of faulting in Westerly granite. International Journal of Rock Mechanics and Mining Sciences & Geomechanical Abstracts, 19, 49– 64. Z ANG , A., W AGNER , C. F. & D RESEN , G. 1996. Acoustic emission, microstructure, and damage model of dry and wet sandstone stressed to failure. Journal of Geophysical Research, 101, 17507–17521.

The mechanics of fault distribution and localization in high-porosity sands, Provence, France C. A. J. WIBBERLEY1,2, J.-P. PETIT1 & T. RIVES3 1

Laboratoire de Ge´ophysique, Tectonique et Se´dimentologie, c.060, Universite´ Montpellier II, Place E. Bataillon, 34095 Montpellier Cedex 05, France

2

Present address: TOTAL, CSTJF, Avenue Larribau, 64018 Pau Cedex, France (e-mail: [email protected])

3

Department Interpretation Structurale, TotalFinaElf, Avenue Larribau, 64018 Pau, France Abstract: Excellent quarry exposures have been studied to examine the controls on the growth of fault networks in Cretaceous high-porosity sands. An inverse correlation is found at any one locality between the frequency of faults of an earlier tectonic event and the frequency of later faults. The early faults are cataclastic deformation bands with displacements typically up to 300 mm, and have thicknesses approaching their displacements. Later faults are also deformation bands except where present within a high-frequency array of earlier faults, where they are typically clustered high-displacement ultracataclasite zones that are narrower (smaller width/displacement ratios) than for the deformation band faults. A mechanical model using critical state soil mechanics explains the observed distributions and fault zone characteristics in terms of strength changes in the deforming sand unit and the stress path by which the material is subjected to ‘clastic’–plastic yielding. Localized faulting by constant-volume cataclastic flow at the critical state line will result in deviatoric stress reduction as Coulomb plasticity softening occurs within the fault zone. Elastic unloading of the walls will suppress the continued formation of deformation bands. The point at which the stress state reaches the critical state line, governed by the stress state and position of the ‘clastic’–plastic yield envelope, is therefore crucial in controlling the final distribution of deformation bands and larger faults in the system. Within this framework, the field and microstructural data suggest that earlier deformation became distributed by hardening processes such as compaction and grain-size reduction, resulting in a higher bulk yield strength. In a later tectonic event, the unit behaves in a stronger manner and deformation quickly localizes by fault zone softening processes into fewer fault zones that individually grow larger.

The modelling of fluid flow in the crust, and in particular of hydrocarbon migration in reservoir rocks, is becoming increasingly dependent upon the assumed fluid flow behaviour of faults (e.g. Manzocchi et al. 1998; Mattha¨i et al. 1998). The recognition that fault zones can act as conduits and/or as barriers to flow has encouraged research into both the petrophysical properties and the distributions of fault zones in reservoir rocks to understand how to predict the likely control that fault zones exert on fluid flow patterns in a reservoir (e.g. Seeburger et al. 1991; Gibson 1998; Knipe et al. 1998). Fault zone permeability structure is largely dependent upon lithology (Evans et al. 1997; Wibberley & Shimamoto 2003) and in highporosity sandstone fault zones are typically clusters of ‘deformation band’ faults of moderately low porosity and permeability. They are often cut by, or are contemporaneous with, larger cataclastic fault zones and/or discrete slip surfaces of highly reduced porosity and permeability (Pittman 1981; Antonellini & Aydin 1994; Fowles & Burley

1994; Fisher & Knipe 1998; Shipton et al. 2002). Deformation bands are usually considered to act as baffles to dynamic flow or capillary barriers, whereas larger cataclastic fault zones are often considered to provide across-fault sealing, especially where clay is entrained in the fault zone (e.g. Pittman 1981; Leveille et al. 1997; Lehner & Pilaar 1997; Foxford et al. 1998; Gibson 1998). Descriptions of the spatial arrangement of such fault zones within high-porosity sandstone units are varied, ranging from evenly distributed arrays of conjugate or quadrimodal deformation bands (e.g. Aydin 1978; Underhill & Woodcock 1987), to arrays of deformation bands spread throughout a basement-faulted fold system (e.g. Jamison & Stearns 1982), or relatively narrow clusters of deformation bands as ‘damage zones’ around larger cataclastic fault zones (Knott et al. 1996; Shipton & Cowie 2001; van der Zee 2001). Such variability makes it difficult to predict subsurface distributions of faults in high-porosity sandstone reservoirs. Thus an understanding of the mechanical

From: LEWIS , H. & COUPLES , G. D. (eds) The Relationship between Damage and Localization. Geological Society, London, Special Publications, 289, 19– 46. DOI: 10.1144/SP289.3 0305-8719/07/$15.00 # The Geological Society of London 2007.

20

C. A. J. WIBBERLEY ET AL.

controls on these fault distributions is required, including the impact of tectonic history, to provide a more robust framework for predicting fault distributions in reservoir flow simulations. The mechanical evolution of deformation band networks is thought to be dominated by strainhardening (e.g. Aydin 1978; Aydin & Johnson 1978, 1983). Underhill & Woodcock (1987) suggested that strain-hardening is important in deformation band thickness growth, being caused by high friction in the deformation band as a result of the angular nature of the cataclastic fragments. Field observations (e.g. Aydin 1978; Underhill & Woodcock 1987; Davis 1999) have shown the impressive nature of deformation band networks even at low strain. High-porosity sand bodies may easily develop long, extensive, connected deformation band networks, which will locally destroy fluid communication because of the very low porosity and permeability of these structures (Pittman 1981; Antonellini & Aydin 1994; Fowles & Burley 1994). These faults can form impressive networks because they have abnormally high length/ displacement ratios (Fossen & Hesthammer 1997; Wibberley et al. 2000) as a result of their relatively high frictional strength (Wibberley et al. 2000). Higher displacement faults in porous sandstone tend to be very different in character from these lowdisplacement ‘deformation band’ faults, and contain discrete, low-friction slip surfaces (Aydin & Johnson 1978; Krantz 1988; Foxford et al. 1998). Antonellini & Aydin (1995) and Foxford et al. (1998) documented relatively complicated ultracataclastic zones of high shear strain in which strainsoftening resulted in high displacements being accommodated across relatively narrow zones. The change from cataclastic deformation band evolution (with strain-hardening and continued band thickness increase) into an ultracataclastic fault zone with discrete slip surfaces and displacement focusing could therefore reflect a peak in fault strength (i.e. a transition from strain-hardening to strain-softening). However, the causes of this limit and what controls its breaching are still unclear. Previous studies of deformation bands (e.g. Jamison & Stearns 1982; Underhill & Woodcock 1987; Fossen & Hesthammer 1998, 2000) showed that they can accrue significant displacements without such a transition to localized slip. However, for highporosity sandstone in Utah, USA, Shipton & Cowie (2001) found that deformation bands in their study had no visible shear displacements at all, and suggested that discrete slip surfaces formed after sub-millimetre displacements on deformation bands. Hence further studies into the mechanical controls on fault strength evolution are required to understand fault characteristics and the distribution of strain in high-porosity sandstone.

Geological setting Upper Cretaceous sands and sandstones crop out in many areas in Provence (Fig. 1a), deposited in Cretaceous mini-basins within the ‘Bassin du SudEst’ (Debrand-Passard et al. 1984). They are crisscrossed with different generations of deformation bands and larger faults. Orientation and kinematic data (Fig. 1b) suggest that each generation was formed in a different tectonic event. In Provence, north– south Pyrenean compression caused regional foreland thrusting, giving the Cretaceous outcrop an east –west structural trend. This was post-dated by NW– SE Oligocene rifting, which caused normal faulting in parts of the Upper Cretaceous strata. Finally, Miocene left-lateral strike-slip on NE – SW faults affected some areas, notably reactivating the Cevennes and Nıˆmes Faults. Burial of the Upper Cretaceous strata is not well constrained: measured thicknesses of Tertiary and Pliocene strata vary greatly through the region, totalling from 600 m to 6500 m (Dubois & Delfaud 1984). The study areas are too far SW of the Alpine front and north of the Pyrenean front to have experienced significant tectonic burial. The Upper Cretaceous strata in these study areas appear unusual in generally exhibiting very little evidence for the occurrence of diagenetic processes (i.e. many of the units remain as unlithified sand). This suggests that the depth of burial was relatively shallow. The deformation band and fault orientations and kinematics suggest generation in all of the three tectonic events listed above, across the Bassin du Sud-Est. This paper presents studies of two areas in the Bassin du Sud-Est, the Massif d’Urchaux and the area around Be´doin (Fig. 1c), where active and abandoned quarries of Upper Cretaceous sand provide excellent 3D exposure of deformation band and larger fault arrays. The high-porosity sands studied are of medium grain size and very low cohesion. They record variable distributions of strain related to the different tectonic events.

Observational data General characteristics Observations of many deformation bands in highporosity sands in Provence show that their character changes with displacement, similar to deformation bands documented elsewhere (e.g. Aydin & Johnson 1978; Underhill & Woodcock 1987; Antonellini & Aydin 1995). Deformation bands of very low displacement (e.g. ,10 mm) consist of a single cataclastic fault strand typically one to a few millimetres thick with granulation textures (Fig. 2a). SEM studies show that arrays of

FAULT DISTRIBUTIONS IN HIGH-POROSITY SANDS

21

Fig. 1. (a) Location map of the study areas (denoted by circles) in the Bassin du Sud Est, SE France. (b) lower hemisphere equal area stereographic projections (stereograms) of poles to planes for deformation bands and larger faults. , reverse faults; O, normal faults; , strike-slip faults. For the Massif d’Urchaux stereogram the circled symbol denotes the large strike-slip fault shown in Figure 2f. For the Be´doin stereogram the circled symbols denote the large normal faults shown in Figure 2. (c) Map of outcrops around Be´doin.

†

22 C. A. J. WIBBERLEY ET AL. Fig. 2. Field appearance of deformation bands and larger faults in the high-porosity Cretaceous sands of Provence. (a) Conjugate single strand deformation bands at Be´doin (pen for scale is 14 cm long); (b) conjugate single and multiple strand reverse deformation bands at Be´doin (mapping board for scale is 35 cm long, height in the photo is 4 m); (c) arrays of normal deformation band faults in the Massif d’Urchaux; (d) two large normal faults at Be´doin, cross-cutting the earlier network of low-angle reverse deformation bands (person for scale at the base of the outcrop); (e) close-up of the left-hand normal fault shown in (d), showing the clustering of bands into a localized zone where each band has a displacement of 10–30 cm (compass –clinometer for scale is 10 cm long); (f) plan view photomontage and sketch of the fault zone structure of a large strike-slip fault (displacement unknown) in the Massif d’Urchaux (after excavation of the outcrop surface, the photomontage was prepared using a camera attached to a 4 m high pole operated by remote control).

FAULT DISTRIBUTIONS IN HIGH-POROSITY SANDS

23

Fig. 3. Microstructural studies of deformed high-porosity sand or sandstone samples using scanning electron microscopy. (a) A single strand deformation band in coarse gritstone at Orange, taken 2 cm from the visible termination of the deformation band (backscattered electron (BSE) image); (b) a single strand deformation band in high-porosity sand at Be´doin (displacement about 2 mm) located at the base of the pen in Figure 2a (BSE image); (c) a cemented multiple strand deformation band of 30 cm displacement at Be´doin, showing intragranular tensile fractures filled with quartz cement adjacent to minor transgranular slip surfaces that are oblique to the main fault orientation (CL image); (d) a localized large normal fault of 2.5 m displacement at Be´doin, showing the large proportion of very fine-grained fragments surrounding some larger fragments approaching the size of the original sand grains (BSE image).

en echelon fractures formed at and close to the propagating band tip (Fig. 3a). The preferential occurrence of fractures fanning into grains from grain contact points suggests the operation of Hertzian fracturing (Gallagher et al. 1974; Mene´ndez et al. 1996). However, these fractures appear organized into a tabular proto-band zone, where grains appear most likely to be fractured if their contact points with adjacent grains line up parallel to maximum compressive stress trajectories for simple shear (e.g. centre of Fig. 3a). Samples of single strand bands away from their tips show that fragments infilled the porosity space (Fig. 3b), suggesting consequent volume loss by compaction. Within the host sand on either side of the deformation band, intragranular fractures are concentrated in zones approximately one strand’s width away. Field observations suggest that compaction

in the band is sufficient to reduce the porosity so that the strand acts as a barrier (capillary or dynamic) to water, as evidenced by the distribution of oxidized grain coating colorations reflecting the ponding of groundwater adjacent to the deformation bands (Fig. 2a and c). Some of these single strand bands have no visible displacement to the naked eye, although in some cases the lack of offset markers needs to be taken into account. However, they do not match the description of compaction bands given by Mollema & Antonellini (1996), who reported 1–5 mm wide deformation bands with no shear offsets showing morphological resemblance to stylolites. Alternatively, these features could be analogous to Lu¨der’s bands in metals (Nadai 1950) as suggested for similar structures in high-porosity sandstone from Utah, USA, by Friedman & Logan (1973). Larger ‘multiple

24

C. A. J. WIBBERLEY ET AL.

strand’ deformation bands (typically 10 –300 mm in displacement, 10 –100 mm wide) are clusters of single strands that appear to have formed by generation of successive adjacent strands. The adjacent strands in these multiple strand deformation bands are usually spaced 1–10 mm apart, but often merge to a continuous zone so that the distinction between a single strand and a multiple strand deformation band becomes vague. Field evidence shows that the number of individual strands correlates with displacement in an individual deformation band (Wibberley et al. 2000), and in laboratory experiments the number of strands adjacent to one another in a growing deformation band has recently been demonstrated to correlate with sample axial strain (Mair et al. 2000). Sets of multiple strand deformation bands of opposite dip often form impressive intersection zones at outcrop scale (e.g. Fig. 2b and c). At such intersections, an alternate chronology of cross-cutting of individual strands gives an overall mutual cross-cutting relationship between two deformation bands dipping in opposite senses (e.g. far right-hand side of Fig. 2b), as described by Horsfield (1980), Ferril et al. (2000) and others. Some of these deformation bands protrude from the outcrop face as a result of their relatively hard nature (as described by Davis (1999) for deformation bands in the Colorado Plateau), sometimes increased by quartz cementation of the bands. Microstructural study of these multiple strand deformation bands shows that fragmentation continued to play an important part in the evolution of the faults as they became larger deformation bands with cement healing and further intragranular fracturing (e.g. Fig. 3c). Despite the presence of quartz cement in many of the deformation bands studied, no microstructural evidence of pressure solution was observed, nor evidence for diagenetic processes in the host sand. This supports recent work on quartz cementation in faulted sandstones from North Sea cores, which suggests that cataclastic grain-size reduction of quartz encourages authigenic quartz precipitation by greatly increasing the amount of available fracture surface area (Fisher et al. 2000). The largest faults observed, with displacements greater than 1 m, appear to have been generated only in tectonic events later than those forming the array of deformation bands (e.g. Fig. 2d). They often appear different in character to the deformation band faults both in the field and under microscopic examination, generally lacking the granulation textures of the deformation bands and being much finer grained. They often appear homogeneous except for some localized slip surfaces, although sometimes they exist as clusters of deformation bands across 20 cm –2 m wide fault zones (e.g. Fig. 2e), where slip concentration is

evidenced by narrowing of anastomosing ultracataclasite strands into slip surfaces within the fault zone. Microscopic examination of samples from these fault zones (e.g. Fig. 3d) reveals that they have a much higher proportion of finer-grained fragments, with coarser fragments preserving some smooth surfaces that suggest they were large parts of original sand grains. In the area around Be´doin, these largest faults are normal faults, crosscutting the reverse ‘deformation band’ fault network. In the Massif d’Urchaux, a relatively large strike-slip fault (Fig. 2f ) consists of peripheral deformation bands, cross-cutting the earlier normal faults, having a kinematic relationship consistent with their being low-angle R1 Riedel faults around a sinistral main fault. This is supported by the orientation distribution of strike-slip deformation bands generally in the Massif d’Urchaux outcrop around this strike-slip fault (Fig. 1), showing a spread of trends up to 308 anticlockwise of the fault orientation. The main fault itself is nearly parallel to the Cevennes and Nıˆmes Faults (Fig. 1), the principal faults in the region, which underwent sinistral strike-slip reactivation in the Miocene.

Fault distribution data We present data on the distribution of such faults in each set from several outcrops, to assess the likely controls on the density of developing arrays of deformation bands and larger faults. Data were collected from horizontal scanlines along the outcrop faces, with frequency data adjusted for the average fault dip of each set to give a frequency statistic in a direction perpendicular to the faults (i.e. inversely proportional to fault spacing; Table 1). Outcrops typically display two sets of faults. Outcrops that display a high frequency of faults from the first set tend to have a lower frequency of the second set (Fig. 4a). Outcrops with a very low frequency of faults from the first set (,1 m21) generally have a high frequency of the second set, though not always. No outcrop displays high frequencies of both sets (Fig. 4a). For those outcrops that have a high frequency of earlier faults, the later faults tend to have larger displacements than these earlier faults (Fig. 4b). For example, only 24% of early faults have displacements of 100 mm or more whereas more than 50% of later faults have .100 mm displacement. This suggests that deformation is typically accommodated by faulting on a large number of small displacement faults initially, but becomes localized onto a smaller number of larger displacement faults if a significant previous generation of faults already exists. This seems to be the case irrespective of whether the faults are normal or reverse faults.

To examine the spatial distributions of each set in more detail, we use a simple standard deviationstyle statistic called the ‘cluster factor’, which we define here as: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P 2 ffi fi P  fav cf ¼ fi

1.1086 1.19 1.7273 1.5802 1.7586

where fi is the number of faults in each metre of scanline, i representing the ith metre interval along the scanline, and fav being the average fault frequency for the entire scanline. A high cluster factor indicates a higher fault clustering whereas a low cluster factor indicates a more even distribution of faults. This cluster factor should not be a function of the sample size. For those outcrops with a moderate to high frequency of faults from the first set, and a much lower frequency of second set faults, Figure 4c shows that the second set has a higher cluster factor, whereas the first set is more evenly distributed. An outcrop with lower and approximately equal fault frequencies for both sets (Bed 4, Table 1) shows similar cluster factors in each case. For all outcrops, the second set cluster factor is higher than that of the first set. For all outcrops, Figure 4d shows that the ratio of cluster factors of the two sets of faults inversely correlates to the ratio of fault frequencies for the two sets of faults for each area. However, the correlation is different for two study areas (probably with different sand units and structural settings or strain) within the Upper Cretaceous of Provence.

1.75 1 0.99 0.81 0.87 2.66 1.31 0.15 0.17 0.2

1.94 1.19 1.71 1.28 1.53

2.0411 1.46 0.39

2.98

1.4747 0.99 1.22

1.46

1.33 1.15

2.78

2.0902

Low-frequency first set, high-freq distributed second set High-frequency distributed first set, localized second set Low-frequency first set, high-frequency distributed second set High-frequency distributed first set, localized second set High-frequency second set Low sample size of both sets Distributed first set, localized second set Distributed first set, low-frequency second set High-frequency distributed first set, localized second set 1.2788 1.04 3.46

1.33

c.f.1 Freq. 2

25

c.f., cluster factor; s-s, strike-ship; rev, reverse; norm, normal.

0.32 0.63 0.66 0.7 2.85 s-s norm norm norm norm 111 16 202 88 46 Bed3b Bed4 Bed5b Bed5j Bed5h

norm rev rev rev rev

1.22 s-s 78 UrchN2

norm

0.167 norm 78 UrchN1

rev

3.46 s-s 48 UrchS2

norm

0.167 norm 48 UrchS1

rev

Length (m)

Type 1

Type 2

Freq. 1

Fault thickness data

Outcrop

Table 1. Summary of statistical data for the main localities described

c.f.2

c.f.2/c.f.1

Comments

FAULT DISTRIBUTIONS IN HIGH-POROSITY SANDS

Fault zone thickness evolution has been examined in the past by studying fault zone thickness dependence on fault displacement (e.g. Robertson 1983; Scholz 1987; Hull 1988; Evans 1990). Such studies generally showed a linear relationship between fault zone thickness and displacement, although care must be taken in interpreting the data because of problems in defining the edges of fault zones, thickness variability along the fault, and the reduction of scatter when plotted on log–log axes (e.g. Blenkinsop 1989). Fault zone thickness– displacement data are presented in Figure 5 for deformation bands and larger faults from our study, presenting thickness data for ‘multiple strand’ deformation bands and larger faults in two ways. Thicknesses using method 1 are given as the distance between the outermost edges of the outermost strands (Fig. 5ai and 5bi). Thicknesses using method 2 are given as the sum of the thicknesses of the individual strands, to define an ‘effective’ thickness of deformed material (Fig. 5aii and 5bii). In the field it is often difficult to define the edges of strands

26

C. A. J. WIBBERLEY ET AL.

(b)

(a)

100

4 fav 2nd > fav 1st

fav 1st > fav 2nd

3

% of fault population withdisplacement ≥ than d

Average fault frequency of 2nd set (per m)

3.5

2.5

sa

tur

2

ati

1.5

on

in cr str eas ain ing

1

?

1st set 2nd set

10

0.5

Cluster factor

3

1 0

1 2 3 4 5 6 7 Average fault frequency of 1st set (per m)

0.001

2.5

1st set 2nd set

2 1.5 1

0.5 0

1 0.01 0.1 Displacement, d (m)

10

(d)

(c) Cluster factor ratio (2nd set clust. fac./1st set clust. fac.)

0

1 2 3 4 5 6 Average fault frequency (per m)

7

2.2 2

Massif d'Urchaux Bedoin

1.8 1.6 1.4 1.2 1 0.01 0.1 1 10 100 Frequency ratio (2nd set freq./1st set freq.)

Fig. 4. Fault distribution data. (a) Relationship between average fault frequencies of the first and the second sets, for each outcrop studied; (b) cumulative displacement distribution for the first set and the second set, combined from all outcrops where the first set frequency is higher than the second set frequency; (c) relationship between cluster factor and average frequency of the first and second sets, combined from all outcrops where the first set frequency is higher than the second set frequency; (d) ratio of second set:first set cluster factor v. ratio of second set:first set frequency for all outcrops.

in a ‘multiple strand’ deformation band, because they merge into a single continuous zone. They are recorded as ‘single strand’ deformation bands in Figure 5 (because no undeformed sand could be detected within the band), even if they are probably clusters of adjacent single strands in reality. For first set faults (Fig. 5), the single strand deformation bands do not generally follow a linear thickness–displacement trend (represented on a log–log plot by a straight line with a gradient of one), although a lot of scatter is present. When combined with the multiple strand deformation bands, however, the data give the impression of a generally linear thickness –displacement trend for method 1 (Fig. 5ai), and a clearer non-linear trend of decreasing thickness:displacement ratio with increasing displacement for method 2 (Fig. 5aii).

No significant difference is present between normal deformation bands and reverse deformation bands, provided they are first faults to affect each outcrop studied. Second set faults, however, generally have narrower thickness:displacement ratios than first set faults (Fig. 5b) although some overlap exists between the scattered datasets. The trend with increasing displacement is of a poorly defined decrease in thickness:displacement ratio for the single strand deformation bands. However, the larger-displacement multiple strand faults of the second set yield have consistently narrower thickness:displacement ratios, being highly scattered for method 1, but collapsing onto a linear thickness– displacement trend for method 2. Large second set faults from other outcrops of high porosity sand or sandstone in the region also fit this trend,

FAULT DISTRIBUTIONS IN HIGH-POROSITY SANDS

(a)

27

1

1 (ii)

(i)

1:1

1:1

1:10

1:10 Thickness (m)

Thickness (m)

0.1 1:100 0.01

0.001 0.001

(b)

0.01 0.1 Displacement (m)

1

0.1 1:100 0.01

0.001 0.001

10

0.01

+

+

(ii) 1:1

1:1

1:10

1:10 0.1 Thickness (m)

0.1 Thickness (m)

10

1

1

(i)

1:100 0.01

0.001 0.001

0.1 1 Displacement (m)

0.01 0.1 Displacement (m)

1

10

1:100 0.01

0.001 0.001

0.01

0.1 1 Displacement (m)

10

Fig. 5. Thickness– displacement relationships for deformation bands and larger fault zones in the high-porosity Upper Cretaceous sands of Provence, for outcrops with a high frequency of first set faults. Open symbols denote multiple strand bands; filled symbols denote single strand bands;  denotes a clay smear normal fault of variable thickness in the Massif d’Urchaux. (a) First set data for normal and reverse deformation bands from the Massif d’Urchaux (squares) and Be´doin (diamonds), respectively, for two measurements of multiple strand zone thickness: (i) thickness defined as the distance between the outside edges of the outermost strands (method 1), (ii) thickness defined as the sum of individual strand thicknesses (method 2); (b) second set data (circles) of normal faults at Be´doin superimposed on the first set data (triangles), for the two methods for defining multiple strand zone thickness as in (a). Additionally, the datum for a localized ultracataclasite fault zone at Orange is shown by ‘ þ ’. The minimum thickness recorded was 1 mm, as determined by observations with the naked eye.

except for part of a fault that offsets a clay bed, the clay smear being substantially narrower than in the rest of the cataclastic fault zone (Fig. 5bii).

Fault orientation data The general orientation distributions of deformation bands and other faults are presented in Figure 1 for example outcrops from the Massif

d’Urchaux and Be´doin (outcrop Bed 5). They show generally Andersonian orientation patterns (plane strain) of conjugate reverse and normal faults, and a cluster of strike-slip faults trending anticlockwise of the main strike-slip fault trend. Although some scatter in strike is present, for some of the sets, no convincing quadrimodal distribution of faults is present in any of the cases. This contrasts with some of the other cases in the

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C. A. J. WIBBERLEY ET AL.

N

Key reverse deformation bands: single strand bands with no visible offset bands with measurable offset

normal fault deformation bands larger localized normal faults

Fig. 6. Equal area lower hemisphere stereographic projection of poles to fault planes for deformation bands and larger faults in a large quarry near Be´doin (locality Bed 5).

literature on deformation band and larger faulting in high-porosity sandstone, for which orthorhombic fault patterns (suggesting non-plane strain) have been found (Underhill & Woodcock 1987; Krantz 1988; Shipton & Cowie 2001). The orientation data for an outcrop at Be´doin are plotted in Figure 6, where the distinction has been made between negligible offset low-angle single strand deformation bands, deformation bands of

measurable finite offset (low-angle reverse and high-angle normal), and large localized high-angle normal fault zones. The single strand deformation bands with no visible offset match the general description of Lu¨der’s bands generated by experiments on high-porosity sandstone by Friedman & Logan (1973) and could be a natural analogue as they proposed for bands in Entrada sandstone of Utah. For the early low-angle deformation bands, Figure 6 shows that the single strand deformation bands fall within the general distribution of deformation band orientations, but tend to group around the high-dip side of the cluster, for each conjugate set. The apical angle between the average orientation of single strand deformation bands of each conjugate set is 558 (giving an angle, u, between each fault and the obtuse bisector, considered to be the direction of least compressive stress, of 62.58), whereas for the low-angle deformation bands in general it is 498 (u ¼ 65.58). The apical angles between conjugate sets are presented in Table 2 for first- and second-generation deformation bands and faults of two key localities in the Massif d’Urchaux and Be´doin areas. Firstgeneration faults have similar apical angles for the two outcrops, despite the different kinematics. However, the second-generation faults have a noticably lower apical angle between the conjugate sets in the case of faults from Be´doin, second-generation strike-slip faults in the Massif d’Urchaux not forming conjugate sets, but having a distribution similar to Riedel minor faults around the main faults (Figs 1 and 2f). These apical angles between conjugate sets are related in Table 2 to the material internal friction properties using the Coulomb hypothesis (from f ¼ 2u 2 90, where f is the angle of internal friction, and m (¼tan f) is the coefficient of internal friction). Table 2 shows that the material internal friction properties are similar for the first-generation faults in the two areas, but that the second-generation faults, where developed as conjugates, were formed in a material of significantly higher bulk internal friction.

Table 2. Apical angle data for conjugate sets of deformation bands and larger faults for the two localities sampled in the most detail, with implied bulk material properties using the Coulomb hypothesis Locality

Generation

Urchaux Urchaux Be´doin 5 Be´doin 5

First set Second set First set Second set

Fault type norm s-s rev norm

Apical angle

u

Angle of bulk internal friction, f

Coefficient of internal friction, m

538 no conjugates 498 428

63.58 – 65.58 698

378 – 418 488

0.75 – 0.87 1.04

FAULT DISTRIBUTIONS IN HIGH-POROSITY SANDS

Interpretation of data Interpretation of orientation distribution data Single strand deformation bands with no visible offset show conjugate set orientations with an apical angle bisected by the probable direction of maximum compressive stress of 558. This is a much lower angle than found in experimentally formed Lu¨ders’ bands by Friedman & Logan (1973), who found that such apical angles for Lu¨ders’ bands varied systematically from 708 to 1108 with effective confining pressure from 35 to 250 MPa. Nevertheless, Olsson (2000) found a much larger range in apical angle of Lu¨ders’ bands formed in triaxial experiments on cylindrical samples, noting that the apical angle also depends on the state of stress (and by how much s2 differs from s1 or s3) and material properties, and concluded that faults such as the deformation bands described here were the same thing as Lu¨ders’ bands. The bimodality, rather than quadrimodality, of the conjugate sets of deformation bands and larger faults suggests that they formed under plane strain rather than a non-plane strain field generating the orthorhombic type of fault pattern described by Reches (1978). This applies both to reverse faults and normal fault sets. However, strike-slip faults do not form conjugate sets, but smaller deformation bands cluster in R1 Riedel slip orientations around larger localized strike-slip faults that parallel regional reactivated faults. Andersonian fault mechanics is thought to have operated for all generations of deformation bands and larger faults because either the acute bisector of conjugate planes (normal faults), obtuse bisector of conjugate planes (reverse), or the line of intersection of faults (strikeslip) is vertical. The acute bisector angle between conjugate faults is lower for second-generation (normal) faults at Be´doin (outcrop Bed 5) than for first-generation (reverse) faults, implying either that the state of stress was very different, or that the material had higher bulk internal friction during the formation of the second generation (Table 2). Normal fault deformation bands in the Massif d’Urchaux have a much higher apical angle than those at Be´doin (Table 2) despite having been probably generated at a similar time and depth, suggesting that the state of stress may not have been significantly different between the reverse faulting and normal faulting episodes to cause this difference in their characteristics. A higher bulk internal friction during the secondgeneration faulting was probably the cause of this lower apical angle. This could be either because the first generation of deformation bands caused

29

an increase in bulk internal friction, or due to an increase in burial depth affecting the compaction state between the two deformation events. Unfortunately, the depth history is very poorly constrained, making it difficult to distinguish between these two possibilities, but the first of these two possibilities is appealing considering the differences in spatial distribution between first and second sets discussed next.

Interpretation of spatial distribution data Variable strain was accommodated by the Upper Cretaceous high-porosity sands across the study area during each of the three post-Cretaceous tectonic events. Late Cretaceous to Eocene shortening (Pyrenean –Provenc¸ale) caused pervasive deformation banding in some places but not others, possibly related to the reactivation of underlying basement and Mesozoic faults. The second and third events (Oligocene extension and Miocene compression–strike-slip faulting) caused networks of deformation bands to form only in those places that did not suffer deformation in the previous event. Those areas that did undergo an earlier formation of pervasive deformation band arrays suffered localized faulting in a later tectonic event instead of formation of additional deformation band arrays. Although these later localized faults are of lower density, they have higher displacements, are more clustered, and have narrower thickness:displacement ratios, suggesting that they formed during strain localization in the highporosity sand. The microtextures of the larger displacement faults (e.g. Fig. 3d) suggest lower fault strength (sf) than for the deformation bands (Fig. 3a–c), at least relative to the shear strength of the host rock (s0) and possibly also in an absolute sense. We therefore suggest that strain localization is dependent on the hardening of the sand unit (increase in bulk strength s0) by the previous generation of deformation band arrays, so that later faults are operating with a much larger strength contrast (s0 2 sf) than during the formation of the earlier deformation bands.

Deformation band growth We classify the deformation band and larger fault structures observed in the high-porosity sandstones of Provence as being single strand deformation bands of no visible displacement, single strand deformation bands with displacement (although there were some cases where no offset markers were present), multiple strand deformation bands, or larger localized fault zones. The propagation of individual strands is demonstrated from microstructural

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C. A. J. WIBBERLEY ET AL.

evidence to rely on the formation of intragranular microcracks, but their mechanism of propagation in narrow planar zones of such length is unclear. However, larger displacement single strand deformation bands have undergone cataclasis with grain fragments providing an infilling matrix material to the strand, thereby reducing its porosity (Fig. 3b) and hence its permeability. Increase in thickness of deformation bands with displacement suggests that work hardening of the deforming zones occurred (Fig. 5). The linear thickness–displacement data for thicknesses measured across the total width of the zones (method 1) give the impression that band hardening or adjacent sand weakening continues constantly during deformation. However, when thicknesses are defined as the sum of the individual strand widths (method 2) as a guide to the volume of deformed material, the reduction of thickness– displacement ratios with increasing displacement suggests these processes continued at an everslowing rate during deformation band growth. This has important implications for the way the mechanics of fault zone growth are interpreted from thickness– displacement data.

Deformation band and fault network evolution Here we synthesize the above findings to present a model for the evolution of the deformation band arrays. Observations on natural deformation bands suggest that strain-hardening of individual deformation bands occurs from the moment of deformation band generation by fragmentation, with compaction and porosity reduction of the fragmented zone (e.g. Underhill & Woodcock 1987). Given that the early deformation bands are evenly distributed, the onset of tectonic deformation of the sand unit probably resulted in a wide spacing of single-strand deformation bands (Fig. 7a). Continued deformation resulted in the growth of many of these single strands into multiple-strand deformation bands (Fig. 7b), which increased in length significantly with displacement as a result of the low contrast in shear strength between the deformation band and the host sand ahead of the propagating tip (s0 2 sf) (Wibberley et al. 2000). Field observations of the conjugate nature of these deformation bands show that intersection of two bands is extremely common, as a result of their high lengths causing an increased probability of intersection, and in many cases, the intersection zones show superb examples of alternating offsets of individual strands from each of two overall mutually cross-cutting deformation bands. Indeed, such relationships may be very common for faulting in

high-porosity sandstones in general, as this feature of deformation bands has already been documented in central California (Antonellini et al. 1999). Where one deformation band crosscuts another, it becomes significantly narrower in the intersection region, suggesting that the earlier deformation band material was harder than the surrounding undeformed host sand and thus more effectively inhibited thickness growth of the later deformation band. Similarly, where a later deformation band abuts against an earlier one, the later deformation band has a higher displacement gradient than other bands, suggesting that continued propagation of the later band tip was ‘blocked’ by the earlier band (Wibberley et al. 2000), and a similar blocking effect on deformation band tip propagation by stratigraphical layer boundaries has been described by Schultz & Fossen (2002). Later deformation bands therefore propagated through a material whose bulk strength was increasing as the density of compacted deformation bands increases, and such network hardening must have continued throughout the generation of the arrays of deformation bands observed in Provence. The final result is therefore a unit of high-porosity sandstone containing a high density of conjugate deformation bands of relatively high strength, which have effectively bulk-hardened the unit (Fig. 7c). A later tectonic event such as inversion causes additional deformation. However, the bulk properties of the high-porosity sandstone unit have changed so that distributed deformation is not promoted in the same way as in the first tectonic event. Instead, initiation of a small number of faults is followed by continued localized deformation on these faults because the strength of the surrounding material is higher, surpressing distribution of deformation. This gives rise to a smaller number of faults, but each with larger displacement and much finer particle sizes (Fig. 7d). Microtextural evidence for cataclastic flow and the presence of discrete slip surfaces within the fault zones suggest that the fault rock here has lower strength. This is consistent with generally lower thickness–displacement ratios observed in the large-displacement faults (Fig. 5), noted previously in faults from porous sandstone in North Sea reservoirs (Fossen & Hesthammer 2000).

Towards a quantitative mechanical description Pore fluid pressure rises during compaction band formation? Previous researchers have suggested that compaction during deformation band formation can cause

FAULT DISTRIBUTIONS IN HIGH-POROSITY SANDS

31 MICROSTRUCTURE

a Onset of deformation — fault initiation with even spacing

TIP ZONE Displacement = 0

2

Bulk strength increase

Tip softening by initial fracturing

b Deformation band widening by generation of new adjacent strands

SINGLE STRAND Displacement ~ 1 mm

2 Fault hardening by compaction

Multiple strand DEFORMATION BAND Displacement ~ 2 cm

c Grouping of bands into anastomosing multiband structures

W O R K H A R D E N I N G

3

4 Network hardening as faults have to propagate through one another

Fault zone widths not drawn to scale

S O F T E N I N G

d Later deformation localisation within a strain-hardened area of earlier fault arrays

7 MATURE FAULT ZONE Displacement ~ 10 m

8b 6

8b Fault zone softening as deformation localises onto a maturing fault

8a

Fig. 7. Summary model of deformation band fault network evolution and fault zone microstructural changes. The numbers refer to stress states indicated in Figure 9c.

sufficient fluid pressure rise to significantly reduce the effective normal stress across the deformation band (e.g. Davis 1999). Such a reduction in effective normal stress could weaken the deforming zone and is of great importance in understanding the mechanical evolution of deformation band

network formation. This section presents the results of a simple compaction model (detailed in the Appendix) which predicts fluid pressure rise through time as compaction proceeds. Typical porosity and permeability data for high-porosity host sand and deformation bands (e.g. Antonellini &

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C. A. J. WIBBERLEY ET AL.

Aydin 1994; Fowles & Burley 1994) are used to constrain the decrease in pore volume in the band and the rate of consequent fluid flow out of the band (see the Appendix). The model is split into two components. The first component is the pressure rise as a result of only pore volume reduction (the ‘undrained’ component), which is modelled by assuming elastic compressible behaviour of water, and is dependent only on the fractional change in pore volume. The second component is the pressure reduction caused by the fluid outflow, calculated from Darcy’s Law, which is a dynamic component because it depends upon the actual fluid pressure gradient at any one point in time. The model is basically a dynamic interplay between the rate at which water pressure can rise because of pore volume decrease, and the rate at which water escape can occur to dissipate this pressure rise. Thus the essential parameters are the rate of compaction and the permeability of the band. The permeability and storage of the surrounding host sand are assumed high enough to approximate to a permeable reservoir of infinite pore volume, so that the fluid pressure rise at the edges of the band will be negligible. The model calculates

the fluid pressure rise in the centre of the band from the fluid pressure gradient between the centre and the edge of the band. Two models for the porosity–permeability evolution of the band through time are considered. The more realistic Model A allows permeability to vary with porosity as described in the Appendix, with porosity decreasing as an inverse function of time (Fig. 8ai). The less realistic Model B, used for comparison, considers a constant low permeability in the band, with porosity decreasing linearly through time (Fig. 8aii). The results of the modelling are presented in terms of the time, T, for completion of the compaction process (from an initial porosity of 27% to a final porosity of 3.5%), and hence are related to compaction rate. The smaller the time T (i.e. faster compaction rate), the lower the amount of fluid escape and therefore the higher the fluid pressure rise. Incredibly fast compaction rates (e.g. T ¼ nano- to microseconds) would result in approximately ‘undrained’ behaviour and an unfeasibly large fluid pressure rise. Model A predicts very small fluid pressure rises (less than 1 MPa) for a deformation band 0.5 mm wide (such as in Figs 2a and 3b), provided that compaction is not

Fig. 8. Model of fluid pressure rise during deformation band evolution. (a) Input model functions for (i) porosity, and (ii) permeability, within the deformation band as a function of time. (b) Resulting maximum fluid pressure rises predicted by the model as a function of the time (T ) of deformation band formation, for different deformation band thicknesses: (i) Model A, of variable permeability related to a porosity function, with w ¼ z given for different values; (ii) Model B, of fixed permeability, with w ¼ 0.5 mm and z given for different values.

FAULT DISTRIBUTIONS IN HIGH-POROSITY SANDS

faster than around 0.5 ms (Fig. 8bi). The less realistic Model B predicts similarly low fluid pressure rises for such a deformation band provided that compaction is no faster than around 3 ms (Fig. 8bii). For both models, thicker bands are considered, and bands 5–50 cm thick may experience drastic fluid pressure rises (e.g. 10 –200 MPa) if they form in around 1 s. However, field observations show that such thick bands are multiple structures forming in many events that are unlikely to be so fast. The conclusion reached from this modelling is that fluid pressure rises during compaction will probably not seriously affect the mechanical behaviour of the deformation bands except at extremely high compaction rates or unless permeability is much lower than that suggested from published datasets. Compaction and faulting can therefore be assumed to proceed under ‘drained’ conditions.

Mechanical context using critical state soil mechanics theory To put the description of the evolution of the deformation band system (Fig. 7) into a rigorous mechanical framework, deformation of the sand and deformation band network formation are viewed in terms of critical state soil mechanics, for the ‘drained’ case. Two stress systems are considered, as illustrated by the ranges of stress paths in Figure 9a. Horizontal compression starting at a deviatoric stress Q ¼ s1 2 s3 ¼ 0, and a mean effective pressure P0 ¼ (s1 þ s2 þ s3)/3 ¼ Pc, with a steadily increasing s1 at constant s2 ¼ s3 ¼ Pc, results in Q ¼ 3(P0 2 Pc) (Fig. 9ai). A stress state with horizontal minimum compressive stress, from the same starting point, with a steadily decreasing s3 at constant s1 ¼ s2 ¼ Pc results in Q ¼ 3(Pc 2 P0 ) (Fig. 9aii). Deformation tests on soils have demonstrated that plastic yielding of the sample occurs after a period of elastic behaviour, with the plastic yield point being dependent on the previous loading history and/or current stress state (Fig. 9bi). (By plastic deformation we mean deformation that is irrecoverable upon unloading.) Numerous experimental investigations into the yield points of soils of different porosities (n) and mineralogies (e.g. Schofield & Wroth 1968), under different stress states, have led to the description of plastic yielding in terms of yield envelopes in Q, P0 , n space (e.g. Atkinson & Brandsby 1978) (Fig. 9bii). A typical triaxial compression experiment on a high-porosity unlithified sediment shows that plastic yielding occurs with compaction, thus reducing porosity and creating a new yield envelope (compaction hardening). There is therefore a dynamic interplay between changing porosity and changing strength

33

during deformation. Increased compressive stress allows the state of stress to keep track of this changing yield envelope so that plastic deformation continues. However, if the stresses were removed, then the yield envelope would illustrate the stress state that would need to be reapplied to cause further deformation (as in the second yield point in Fig. 9bi). Although these yield envelopes can only be determined experimentally, idealized geometrical fits, such as the Cam clay model (Roscoe & Burland 1968), are often used. Experimental data on pore volume changes show that there are certain porosities and stress states under which pore volume may either decrease or increase during plastic yielding. For example, a loosely packed sand tends to compact and strengthen continuously during deformation until a constant strength and pore volume is achieved. A densely packed sand, after elastic deformation, will dilate and undergo failure accompanied by a stress drop until deformation reaches a steady state at a lower residual shear or deviatoric stress and the rate of volume change decreases to zero (e.g. Casagrande 1936; Wroth 1958; Bolton 1986) as illustrated schematically in Figure 9bi. The yield point at which volume change does not occur is the critical yield point, and the locus of critical yield points for different stress paths defines the critical state line for the material. Yielding at the critical state results in perfectly plastic shear deformation because it can continue with no volume change or strength change (e.g. Wood 1990). An addition to the usual form of critical state soil mechanics allows for the effect of grain fragmentation and the corresponding plastic work required to form the additional fragment surface area as well as that required for plastic strain of an aggregate of rigid particles (‘clastic mechanics’, Bolton & McDowell 1997). McDowell & Bolton (1998) argued from experimental data that the tensile fracturing of grains depends on grain size in an interplay between lower tensile strength of larger grains (because of their greater probability of containing larger flaws), and higher tensile stress concentrations in smaller grains (because of their lower coordination numbers as demonstrated by Jaeger (1967)). The fact that a fractal distribution of grain sizes with fractal parameters of around 2.5 usually results from natural fragmentation (e.g. Turcotte 1986; Sammis et al. 1987) led McDowell & Bolton (1998) to suggest that the coordination of the smaller particles tends to dominate the control of grain fragmentation in granular aggregates. The onset of such tensile grain fracturing marks the initiation of deformation band formation, with the formation of a band representing a particular plastic strain event (called ‘clastic yielding’ by McDowell & Bolton (1998)) at a

34

C. A. J. WIBBERLEY ET AL.

Fig. 9. Illustration of the use of critical state soil mechanics in examining the strength changes during deformation of high-porosity sand. (a) Stress paths on the deviatoric stress (Q)–mean effective pressure (P0 ) diagram for different confining pressure cases: (i) horizontal maximum principal compressive stress; (ii) horizontal minimum principal compressive stress. (b) Concept of the plastic yield point in terms of: (i) stress–strain relationships; (ii) plastic yield envelopes for different porosities, on the Q– P0 diagram; (iii) plastic strain increment vectors on a plastic shear strain pl increment (d1pl q ) v. plastic volumetric strain increment (d1p ) diagram; (iv) superimposition of different plastic potential curves onto a yield envelope on the Q–P0 diagram. (c) Possible stress paths and failure points of high-porosity sand during deformation and tectonic inversion. (d) Illustration of a possible stress path for the case of normal faulting, in which elastic unloading of the walls is followed by a stress state affected by heterogeneous slip distribution on the fault (see text).

FAULT DISTRIBUTIONS IN HIGH-POROSITY SANDS

certain yield point. The resulting strength of the material in the deformation band forms a new yield envelope both because the more efficient packing of the fragments leads to better particle coordination, and because the lower porosity of the new granular material has a higher shear strength as shown by usual soil mechanics principles. Thus cataclastic deformation band generation, initiated by deformation mechanisms such as Hertzian fracturing (Fig. 3a) for the higher P0 parts of the clastic yield envelope, and perhaps grain asperity fragmentation or spalling for the lower P0 parts, can be modelled as a particular type of plastic yield mechanism within the usual framework of critical state soil mechanics. To consider the volumetric behaviour at yield however, the next increment of plastic strain must be considered, and this can be done by plotting plastic shear strain increment against plastic volume strain increment (volume decrease being positive), as in Figure 9biii. Increments of plastic strain are shown as vectors, with lines perpendicular to the vectors defining plastic potential curves. Deformation occurs at constant volume where the plastic potential curve is horizontal (Fig. 9biii). However, such plastic strain increments depend on stress state and porosity, so the direction of the vector will vary with its position on the yield envelope in the Q v. P0 diagram, making it advantageous to plot the plastic strain vectors diagram directly on top of the Q v. P0 diagram (Fig. 9biv). The plastic potential curves may or may not coincide with the plastic failure envelopes (associated v. nonassociated flow), but unless the angle of dilatancy equals the angle of friction, then deformation is non-associative (e.g. Vermeer & de Borst 1984). Although we do not know the form of the clastic yield envelopes for the Cretaceous sands of Provence, we can assume that they will be similar to those determined for other high-porosity sands and sandstones at relevant pressures and stress states. For example, Zhang et al. (1990) and Wong et al. (1997) showed that loading a range of sandstones under hydrostatic and non-hydrostatic conditions caused continuous plastic compaction to occur until a change in the yield characteristics was reached at a stress state C* where a much greater rate of pore volume reduction suddenly started to occur. For the hydrostatic case, this point (where P0 ¼ P*) was shown to be the onset of Hertzian fracturing (corresponding to a special case of the ‘clastic yielding’ of McDowell & Bolton (1998)) and was found to vary significantly with the initial porosity (n) and grain size (R, in mm) of the material used. Zhang et al. (1990) proposed a Hertzian fracture model of P* ¼ A  (n  R)B where best fit to their data gives B  21.5 and A  2 for P* in MPa. For the non-hydrostatic

35

case, this point (C*) occurs typically at P0 , P* (Rutter & Hadizadeh 1991; Wong et al. 1997), and can be mapped out in P0 –Q space in approximately elliptical form for effective mean stresses from approximately 0.45  P* up to P*, with the ellipse having an axial ratio of around 0.6 and a long axis spanning from the origin to P*. In reality, a slight obliquity of this form occurs so that the maximum P0 on the yield envelope is at a deviatoric stress greater than zero. Assuming that this yield surface is analogous to a ‘clastic yield envelope’, then P* may be used to constrain the onset of deformation band generation (Fig. 9c). Figure 9c illustrates the stress state evolution of sand deforming first by compaction related to burial, then by a horizontal, increasing, maximum principal compressive stress, and later by a horizontal, decreasing, minimum principal compressive stress. For the sake of simplicity plane strain conditions are assumed. During burial (points 0 to 1, Fig. 9c), the sands are continuously compacting, meaning that the corresponding stress state is always on the yield surface with progressively lower porosity and higher effective mean stress through time. As the state of stress is not known, a general stress path close to, but not on, the P0 line is drawn to acknowledge that purely hydrostatic stress is highly unlikely. Tectonic horizontal compression at constant vertical stress results in continued plastic yielding by compaction with hardening until the stress state reaches the clastic yield envelope at point 2 (Fig. 9c). Generation of a deformation band (Fig. 7a) results in fragmentation, accelerated compaction and porosity reduction, with consequent strengthening of the band defining a new yield envelope for the band outside the yield envelope of deformation banding in intact sand (Fig. 9c). Continued deformation will occur within the weaker host sand on the clastic yield envelope rather than in the band. Hence a second deformation band will form outside the first strand, possibly adjacent to the first one where a slight stress perturbation may exist, perhaps assisted by peripheral cracking of grains a few grain diameters from the edge of the first strand (Cundell & Strack 1979) as evidenced in Figure 3b, or otherwise in a conjugate orientation (Fig. 7b). Successive strands may group into multi-strand deformation bands in this way with no macroscopic hardening, each strand being a self-arresting plastic bifurcation, until the intersection of conjugate strands (Fig. 7c) leads to network hardening as described in the section on network evolution. The network hardening process requires the stress state to move up to the yield strength of the deformation band (i.e. from point 2 to point 3 in Fig. 9c). Continued generation and propagation of new bands occurs at this higher deviatoric stress level thereafter until deformation

36

C. A. J. WIBBERLEY ET AL.

bands have spread evenly throughout the volume. At some point, a density of bands is reached such that newly forming strands cross-cut so many bands, or form within bands, that significant continued cataclasis of cataclastic deformation band material must occur for more strain to be accommodated. Continued cataclasis of cataclastic deformation band material becomes increasingly difficult because the evolution towards a fractal grain-size distribution will suppress the dominance of coordination number as a cause of tensile stress concentration in smaller clasts so that tensile failure of grain bridges becomes harder (Biegel et al. 1989). A point comes when granular sliding is the preferred mechanism of grain bridge breakdown (Biegel et al. 1989) and compaction ceases; deformation is now occurring at constant volume and the stress state is on the critical state line (point 5, Fig. 9c). Shearing by contant volume cataclastic flow at this point may continue indefinitely provided that tectonic unloading does not occur, as the yield surface will grow no further. If deformation ceased before the critical state line was reached (e.g. at point 4, Fig. 9c), then after elastic unloading, the deformation band network is much harder than the original porous sand. For example, renewed compression would require a much higher deviatoric stress to generate new bands (by returning to point 4, Fig. 9c) than for the initiation of deformation bands at point 2 (Fig. 9c) for the first deformation event (compare the first and second yield points in Fig. 9bi). The amount of additional strain required to reach the critical state line (point 4 to point 5) and possibly localized failure, however, would probably be much smaller than in the original case. The deformation events affecting the Upper Cretaceous porous sands of Provence were not of the same sense, so the stress path in the second deformation event did not follow that of the first. For example, a second deformation event of regional extension with s3 horizontal could cause the stress path to pass through the clastic yield envelope at point 6 (Fig. 9c) on its way to intersecting the new deformation band network yield envelope (second yield envelope) in the dilational field (point 7, Fig. 9c). Strands could start to form in undeformed parts of the sand body between deformation bands, but will not be able to propagate through the earlier harder deformation band network and will not affect the macro-stress evolution. Dilation at point 7 leads to smaller yield envelopes (corresponding to an increase in porosity) until the critical state line is reached (point 8, Fig. 9c), at which point shearing by constant volume cataclastic flow may continue indefinitely.

Analysis of Coulomb plasticity softening Despite previous work describing the changing properties from deformation bands to higher

displacement faults in high-porosity sandstones, few attempts have been made to understand this transition from a mechanical point of view. Aydin & Johnson (1983) used localization theory (e.g. Rudnicki & Rice 1975) to explain the formation of deformation bands in terms of a strain rate contrast between host rock and band. The present study has taken perhaps a more pragmatic approach by considering the evolution of stresses within the framework of critical state soil mechanics until constant volume shearing is achieved. This section suggests that the onset of ‘ideally plastic’ cataclastic flow can lead to fault zone softening, and consequent localization of deformation. The ultracataclastic fault rock consists of poorly sorted generally fine-grained angular fragments suggesting that cataclastic flow operated. Such a macroscopically continuous deformation across the fault zone may be considered perfectly plastic in that it will accommodate strain at constant shear stress. Within the fault zone, co-operative Coulomb slip on conjugate Riedel planes can accommodate a state of stress within the fault zone that is different from the surrounding imposed state of stress (Mandl 1988). Computed tomography of deforming sand experiments demonstrates the 3D evolution of strain heterogeneity and likely stress heterogeneity possibly related to such a process (Desrues et al. 1996). Ring shear experiments on sand packs by Mandl et al. (1977) showed that widening of the shear zone was accompanied by rotation of principal stress axes inside the shear zone until the maximum compressive stress direction was oriented at 458 to the shear zone walls. An abrupt switch in deformation mode to localized strain accommodation in the centre of the shear zone followed, accompanied by no further volume change, as a result of a critical packing arrangement of fine fragments around larger clasts, which allowed efficient through-going slip pathways to propagate during cataclastic flow. The analogy between these experimental findings and the observations presented here suggests that such macroscopically plastic deformation is probably governing the change in fault zone growth mechanism from deformation band widening and network hardening to localized faulting in zones of cataclastic flow. Vermeer (1990) modelled biaxial deformation of sand using classical soil plasticity and found that, when the plastic potential function does not equal the yield function (i.e. non-associated plasticity), softening occurs for cases where the deformation zone is shearing plastically at Coulomb angles. This consequent Coulomb plasticity softening within the shear zone resulted in reduction of deviatoric stress (i.e. elastic unloading) outside the shear zone. This Coulomb plasticity softening concept was used by Byerlee & Savage (1992) to

FAULT DISTRIBUTIONS IN HIGH-POROSITY SANDS

model the strength of lithosphere-scale strike-slip faults such as the San Andreas Fault, using an assumption of constant external normal stress, with systematic build-up and drop of shear stress. The applicability of the Coulomb plasticity model to mesoscale faults such as those observed in Provence is considered best achieved by assuming a constant confining pressure as in the previous section, instead of a constant normal stress. The arrangement of stresses with respect to the fault before (Fig. 10ai), and after (Fig. 10aii) the onset of Coulomb plasticity will be markedly different. Mohr circles illustrate the decrease in shear stress on the edge of the fault zone as the switch in deformation mechanisms to this process occurs (Fig. 10b). For the case of normal faulting at constant burial depth in which the minimum principal stress steadily decreases until failure, constructing the Mohr circles for the stress states inside and outside the shear zone exhibiting Coulomb plasticity requires three simple principles: (1) that the maximum external principal (vertical) effective stress (s1) remains constant; (2) that the angle, u, between external s1 and the normal to the shear zone walls remains constant; (3) that the maximum compressive stress inside the shear zone (sc1) is at 458 to the shear zone walls. The construction in Figure 10b also assumes for convenience that the internal coefficient of friction for brittle failure (superscript ‘b’) in the host sand (mb ¼ tan f) is the same as that in the ultracataclasite within the shear zone, and that there is no stress drop or change from static to dynamic coefficients of friction during the initial brittle failure, although these are not necessary parts of the model. For normal fault development where s1 (assumed to be related to only the overburden pressure) stays constant, s3 decreases (i.e. becomes less compressive) until the deviatoric stress reaches the peak strength of the sand. After Coulomb faulting at a point W (Fig. 10b), the onset of Coulomb plasticity inside the fault zone (with maximum compressive stress at 458 to the fault zone walls), in which failure on conjugate slip planes occurs at X1 and X2 (Fig. 10b), allows the resolved shear stress acting on the fault zone walls to decrease, and s3 to increase. Hence the state of stress outside the fault zone reorganizes so that continued deformation occurs at point Y (Fig. 10b). The external normal stress resolved onto the fault (s pn) will equal the mean stress inside the fault zone (s cm), and the resolved shear stress on the fault zone walls (t p) will be half of the deviatoric stress inside the fault zone. As shown by Byerlee & Savage (1992), the resulting apparent coefficient of friction (f0 ) at the edge of the fault zone will be lower than f and will be given by tan f0 ¼ sin f demonstrated also here in Figure 10c. The effect of Coulomb plasticity operating as a result of

37

cataclastic flow within the fault zone is therefore to reduce both the coefficient of friction and shear stress on the fault (fault weakening), and the external deviatoric stress (system softening). An additional consequence of Coulomb plasticity softening is that the reduction in external deviatoric stress will cause elastic unloading of the material outside the shear zone (e.g. Vermeer 1990). Hence, during cataclastic flow in the large fault zone, the deformation behaviour outside this fault zone will depend upon the magnitude of the unloading (with the minimum principal stress rising from s b3 to s p3 in Fig. 10b), and the position of the failure envelopes.

Implications for deformation band and larger fault distributions The consequence of the stress path meeting the critical state line (in 3D Q –P0 – n space the stress path cannot cross the critical state line, but can only verge into it) is that plastic shear failure occurs, and in the context of high-porosity sand or sandstone deformation this is the generation of relatively large cataclastic fault zones. Coulomb plasticity softening within such fault zones will result in the state of stress within the fault zone moving down the critical state line (from point 8 to 8a in Fig. 9c), whereas elastic unloading of the surrounding material will move it off the critical state line (a point also shown for particulate sediment deformation by Jones & Addis (1986)) to a point well below the clastic yield envelope (point 8b, Fig. 9c). Deformation bands around the fault zone will therefore not continue to be generated once localization and Coulomb plasticity softening in the fault zone have occurred. This mechanical model is therefore consistent with our data showing lower densities of the second-generation faults, but ones tending to have larger displacements and narrower thickness:displacement ratios. Hence, once cataclastic flow becomes the dominant deformation mechanism, Coulomb plasticity softening may result in the suppression of further deformation banding and the localization of deformation into a small number of larger shear zones. This would result in a very different distribution of deformation bands and larger faults to those from the first event (points 1 –4), which did not reach the critical state line. The scenario illustrated in Figures 7 and 9c is only one of many possible patterns of fault system evolution, and other patterns of deformation bands and/or larger cataclastic fault zones may evolve depending on the relationship between the stress path taken and the yield envelope(s) for the material(s) concerned. The temporal relationships between different structures may be affected by

38

C. A. J. WIBBERLEY ET AL.

(a)

(i)

σ1

(ii)

σ1

σ1c

ILU FA LE E ITT BR RFAC SU

θ σnb

45 θ

σnp

X2 c X1 σ3

σ3b

R EA SH NE ZO

RE

τb θ = 45 + φ/2 W

θ = 45 + φ/2 τp = μp = sin φ = tan φ' p σn

τb = μb = tan φ σnb

(b)

θ

σ3p

τp Y

τ

40

W

τb τp τc

20

φ

Y

X1

φ'

σ3c σ3b

σ3p

2θ σmc 40

2θ 2θ σ1c b σm σmp

80

σ1

σn

-τc X2

(c) O-Y-σmc: r = σnp. tan φ' σ1-Y-σmc: r = (σ1-σnp) / tan θ σ σnp = tan θ tan 1φ' + 1

O-X1-σmc: r = σmc. sin φ

τ

O-Y-σmc: r = σmc. tan φ' tan φ' = sin φ

Y

τp

θ

X1

r r

φ' O

φ

2θ σmc, σnp

2θ σmp

σ1

σn

Fig. 10. Analysis of non-associated Coulomb plasticity softening for the case of normal faulting at a constant external maximum compressive (vertical) stress of 100 MPa. (a) Illustration of the fault geometry with respect to the stress regimes: (i) for the case of a Coulomb fault plane generated at a stress state W; (ii) for the onset of shear zone Coulomb plasticity behaviour, in which the stress regime inside the fault zone (Coulomb failure on conjugate slip planes at stress states X1 and X2) is separate from that outside (apparent slip at stress state Y), but the shear zone walls are in the same orientation as the original fault. (b) Mohr circle construction illustrating the effect of shear zone Coulomb plasticity in reducing the resolved shear stress on the fault and hence the effective angle of friction, as well as the deviatoric stress outside the fault zone. The Mohr circle passing through s1 and W represents the original fault plane. The smallest Mohr circle, passing through X1 and X2, represents the stresses inside the shear zone after the onset of Coulomb plasticity, and the circle passing through s1 and Y represents the state of stress outside the shear zone exhibiting Coulomb plastic behaviour. The inclined straight lines represent the Coulomb failure envelopes and the long dashed lines represent construction lines from the points of failure (W, X1, X2 and Y) to the centres of the Mohr circles. (c) Construction of the Mohr circles for the states of stress inside (small circle) and outside (large circle) the shear zone after the onset of Coulomb plasticity, requiring only the three constraints listed in the text.

FAULT DISTRIBUTIONS IN HIGH-POROSITY SANDS

this evolution. For example, inconsistencies seem to exist in the literature as to the timing of throughgoing failure of high-porosity sandstone units by faulting in localized zones of ultracataclasite or on discrete slip surfaces. On the one hand, most models of deformation band and fault development in high-porosity sandstone suggest that deformation band networks develop first in a hardening regime, and if deformation proceeds to a large enough extent, localized deformation on faults or discrete slip surfaces sequentially occurs at a critical stress level when a ‘runaway instability’ develops (e.g. Rudnicki 1977; Aydin & Johnson 1983). The fact that many deformation band networks form even without the presence of larger fault zones (e.g. Jamison & Stearns 1982; Underhill & Woodcock 1987) seems to support this sequential relationship. However, recent work from Utah suggests that deformation bands form contemporaneously with larger fault zones, being generated in clusters around these faults in a damage zone as different parts of the main fault zone undergo slip (Shipton & Cowie 2001). The mechanical evolution of the deformation band– fault system was not explored in the latter case, and appears to conflict with the sequential model suggested previously. One explanation for continued deformation band generation around fault zones undergoing Coulomb plasticity softening can be found from a heterogeneous elastic unloading path of the wall rocks around the fault zone, especially if nonuniform slip on the fault zone exerts additional local stress heterogeneity (e.g. Pollard & Segall 1987). Figure 9d illustrates the possible mechanical evolution of a system during normal faulting associated with horizontal s3. The stress path proceeds to the clastic yield envelope (point 2, Fig. 9d) at which point deformation bands start to form, and network hardening forces the deviatoric stress to continue rising so that the stress path meets the critical state line soon after network hardening commences, resulting in localized faulting (point 3, Fig. 9d). Coulomb plasticity softening allows the stress state inside the fault zone to move down the critical state line (to point 3a, Fig. 9d) while the walls unload elastically down to a stress state below the critical state line (point 3b, Fig. 9d), inhibiting deformation outside the fault zone. However, the local stress heterogeneity associated with non-uniform slip on the fault will move the stress states at points around the fault zone in different directions depending on the position of the point relative to the slip maximum and slip sense. For example, a point close to the fault zone in the compressional quadrant will move up a stress path beyond the previous yield envelope (point 3c, Fig. 9d), and additional deformation bands could form, possibly with an

39

associated slip zone if the stress path of the wall rock meets the critical state line (point 3d, Fig. 9d). Hence ‘damage zones’ of deformation bands and smaller slip zones can form contemporaneously with larger localized fault zones in certain stress history cases.

Conclusions The field and microstructural observations on deformation bands and larger cataclastic fault zones in Cretaceous high-porosity sands from Provence, SE France, suggest that the formation of a network of distributed deformation bands in a first tectonic event work-hardened the sand unit in a bulk sense and promoted the localization of deformation onto a smaller number of faults of higher displacement in a second tectonic event. The key observations are as follows. (1) Outcrops that contain high densities of firstgeneration deformation bands have lower densities of second-generation faults, but these faults are typically of higher displacements, are more clustered, and contain one or two large cataclastic fault zones. (2) Outcrops that contain low densities of firstgeneration deformation band faults often, but not always, have higher densities of second generation deformation band faults but never have large cataclastic fault zones. (3) Second generation structures tend to have narrower thickness:displacement ratios than those of the first generation. A model for the evolution of the network uses these observations to suggest that the bulk strengthening of the sand unit is important in establishing a strength contrast as faulting by cataclastic flow develops. Various aspects of the mechanics of the network formation have been explored to find the controls on strength changes in the localizing fault zone and the host material, as follows. (4) Elastic fluid modelling of the compaction process found that fluid pressures generated during deformation band formation are insignificant in controlling fault mechanics unless the bands formed on microsecond time scales or faster. This finding suggests that drained conditions can be assumed in considering the mechanics of highporosity sand deformation. (5) Coulomb plasticity softening during shearing by cataclastic flow results in a large enough strength contrast between fault zone and bulk host rock that localization of the deformation into the shear zone is promoted. (6) The mechanical evolution of the deformation band–larger cataclastic fault system can be satisfactorily explained using critical state soil

40

C. A. J. WIBBERLEY ET AL.

mechanics, modified to take account of the cataclastic nature of yielding that generated deformation bands. The switch in deformation modes from distributed deformation band formation to localization of faulting in an ultracataclastic zone is thought to occur when the state of stress reaches the critical state line. The amount of strain accommodated between the first ‘clastic’ plastic yielding and the onset of localized failure at the critical state line will dictate the density of deformation bands formed in a network prior to the formation of a through-going ultracataclastic fault zone. The approach using critical state soil mechanics theory to understand fault evolution is not entirely new, as tectonic deformation of particulate sediments had already been modelled in this way (Jones & Addis 1986), and the theory had also been mentioned in the consideration of deformation band microstructural features (Antonellini et al. 1994). However, with the addition of the ‘clastic’ yield concept, the more robust application presented in this paper shows many properties of the evolving strength of a high-porosity sand system and can be used to explain findings presented here and in previous works on faulting in high-porosity sandstone, as follows. (7) Plastic failure and localization of deformation into ultracataclastic fault zones is more likely in a second tectonic event, if the first tectonic event moved the plastic yield envelope closer to the critical state line. (8) This plastic failure in the second tectonic event, resulting in localization, may occur at lower deviatoric stress than final yield strength in the first event, depending on the stress regime and burial depth. (9) When the stress state meets the critical state line, localized failure in a fault zone occurs with constant volume cataclastic flow in the zone resulting in Coulomb plasticity softening. Elastic unloading of the wall material moves the state of stress outside the fault zone off the critical state line, thus suppressing the generation of more deformation bands away from the fault. In some circumstances, however, heterogeneous slip on the main fault zone may locally alter the stress state sufficiently to move it back above the yield envelope, so that further deformation band generation in a ‘damage zone’ around the fault could occur. (10) The point at which the stress state reaches the critical state line, and the behaviour of the wall rock that is unloaded elastically are therefore important in determining the final distribution of deformation bands and larger faults in the system. The work described in this paper was made possible by an Elf EP post-doctorate grant awarded to the first author. M. Baldasseroni from SIFRACO is thanked for granting

access to the active quarries, without which much of the data collection could not have been done. C. Beaudoin is thanked for field assistance. Feedback on plastic deformation theory from T. Shimamoto has greatly helped to improve the mechanical aspects of the paper. G. Couples, W. Olsson and J. Urai are sincerely thanked for providing constructive advice on the manuscript.

Appendix: Modelling of the fluid pressure rise in a deformation band in high-porosity sandstone We model the fluid pressure rise in an active deformation band by considering the elastic behaviour of the pore fluid (taken to be water) as porosity decreases. The model forces porosity (n) to decrease according to a timedependent function (n ¼ f(t)), and considers initially the undrained response (no fluid flow from the band), and then the response if the pore fluid is allowed to escape in a manner controlled by the permeability of the deformation band. Two models are considered: in model A the permeability is time dependent as a function of the time-dependent porosity; in model B the permeability is constant and lower than that of the host porous sand. The porous host sand is considered to act as a fluid reservoir of infinite volume, so that fluid escape from the compacting band does not cause a fluid pressure rise in the adjacent regions. Both of these models consider a representative elementary volume (REV) of deformation band, which spans from the edge to the centre of the actively compacting strand of the deformation band (height w being half the strand thickness), with a surface area A (Fig. A1). The actively compacting volume is then 2  w  A, for a given surface area A of deformation band. So that the generation of successive strands in a multiple strand deformation band may also be considered, the distance z across which a fluid pressure gradient exists may be larger than the half-thickness of the active strand (Fig. 1). Hence, a strand forming within a lower permeability deformation band can also be considered. However, because multiple strand deformation bands typically form by generation of successive strands at the edges of the deformation band, increasing the height z of the fluid pressure gradient so that z . w is probably of minor importance except for thick faults where the actively compacting zone may be localized within the main low-permeability zone. For the case of no fluid escape from the band during porosity decrease (the ‘undrained’ case), the pressure rise in the REV as a function of time, DP(t), as a result of the increasing fluid density, is considered as the sum of incremental pressure rises defined as dPðtÞ

   dVp t   ¼ b Vp t

ðA1Þ

FAULT DISTRIBUTIONS IN HIGH-POROSITY SANDS

z

Zone of low permeability

Area, A

w

central plane of deformation band

Actively compacting zone

Fig. A1. Illustration of model set-up and parameters.

where Vp is the pore volume of the REV (Vp ¼ n Vtotal), and the elastic compressibility of water (b) is considered constant with respect to fluid pressure and time. A decrease in pore volume is considered to be a negative volume change. Because the compressibility of water is considered to be an elastic constant, the finite pressure rise in the undrained case is related linearly to the overall volume change per unit original volume:

DPðtÞ

h    i    DVp t  Vp t  Vp 0   ¼   ¼ b Vp 0 b Vp 0   1 n0  nt ¼ n0 b

dPðtÞ

V_ esc

dVesc kðtÞ nt Vtotal ¼  DPðtÞ : dt mf wz

ðA5Þ

Denoting the fluid pressure rise above its initial value as DP, the rate of fluid pressure rise is: ( " # " #) dDP 1 d ðDVp Þt d ðVesc Þt   þ   ¼ dt b dt Vp t dt Vp t

ðA6Þ

which means that, for a small increment of time, the fluid pressure rise equals the pressure rise caused by decreasing pore volume minus the effect of escaping fluid. By combining equations (A5) and (A6), this rate of pressure rise is seen to depend upon the existing finite pressure rise: "  # kðtÞ dDP 1 d  DVp t   DPðtÞ :  ¼ bmwznðtÞ dt b dt Vp t

dy þ pðxÞ  y ¼ qðxÞ dx

ðA7Þ

ðA8Þ

ðA2Þ where t ¼ x and DP ¼ y, which is solved by  ð ð  ð

 y ¼ exp  pðxÞdx qðxÞ exp pðxÞdx dx þ c ðA9Þ where in this case ðA3Þ pðxÞ ¼

The rate of fluid escape can be estimated from equations deriving from Darcy’s Law: dVesc kðtÞ A  DPðtÞ ¼ ¼ mz dt

for simplicity will therefore overestimate, and therefore provide an upper bound to, fluid pressure increase. The compacting zone in the REV from which water escapes has a total volume of A  w, and hence a pore volume (Vp)t ¼ nt  A  w. Hence

Equation (A7) is of the form

Accounting for fluid escape, equation (A1) needs to be modified by adding the effect of fluid density change in terms of the volume of water escaping from the REV per unit time, V˙esc:    1  dVp t V_ esc   ¼ : bf Vp t

41

ðA4Þ

in which the fluid flux is controlled by the pressure difference across a distance (z) in a medium of uniform permeability at any one point in time (k(t)), with A denoting the surface area perpendicular to flow of a particular region under consideration, and m being the viscosity of water. In reality, fluid should escape from both sides (þz and 2z) of the deformation band, and not all of the fluid in the actively compacting strand will be from the centre, so an average should be calculated by integrating across the width of the strand. The approach taken here

kðtÞ bmwznðtÞ

ðA10Þ

and, from equation (A2), the undrained component q is 1d qðxÞ ¼ b dt

" #   DVp t 1 n0  nt   : ¼ n0 b Vp 0

ðA11Þ

The form of equation (A9) puts particular constraints on p(x) in order to be easily solvable, and weÐ look for an equation of the form p(x) / 1/xÐ so that p(x)dx / Ð ln(x þ c), so that fq(x)  exp p(x)dxgdx can yield a solution. This requires from equation (10) that k(t)/ n(t) / 1/t. However, we also need to use reasonable porosities and permeabilities in this function to make the model results in any way meaningful. (One way of solving this problem is to use a constant permeability in the deformation band, with n(t) / t. Although such a

42

C. A. J. WIBBERLEY ET AL.

situation is not physically realistic, it is useful as a check and we consider it later as our Model B.) After considerable trial and error of different permeability– porosity and porosity– time functions, the porosity function

nðtÞ ¼

1 1 n0

Rt þ

;

1 1  nT n0 R¼ T

B n0 mwz ðn0 Rt þ 1Þ

2 6 4

ðA13Þ

nt ¼ n0  Kt;

Rn0

ðA14Þ

dDP Rn0 Bn0   DPðtÞ ðA15Þ ¼ dt bðn0 Rt þ 1Þ2 bmwzðn0 Rt þ 1Þ which is solved using the method shown by equations (A8) and (A9) to define the fluid pressure rise as DPðtÞ ¼

Rmwz B  bRmwz

h i ðn0 Rt þ 1Þ1 ðn0 Rt þ 1ÞðB=bRmwzÞ : ðA16Þ

K¼

n0  nT T

ðA18Þ

k bmwzðn0  KtÞ

ðA19Þ

K bn0

ðA20Þ

and qðxÞ ¼

where q(x) is the derivative (with respect to t) of the undrained response (Fig. A2ai). Equation (7) can then be rewritten as

ðA17Þ

(Fig. 8ai) which in the format of equation (A7) results in pðxÞ ¼

bðn0 Rt þ 1Þ2

:

n0 R

This maximum fluid pressure rise, DPtmax, can then be found by putting tmax into equation (A16) or by combining equations (A7) and (A11) with t ¼ tmax. Results of this process are shown in Figure 8bi. Equations (A12)–(A17), in which both permeability and porosity decrease as compaction takes place, form Model A. For comparison, a constant permeability model is also constructed, termed model B, where the criterion of k(t)/n(t) / 1/t for allowing equation (A7) to be solvable is satisfied by n(t) / t. Using the same porosity limits as in Model A, the porosity– time function in Model B is then

and equation (A11) is then qðxÞ ¼

3 1= B  1 bRmwz B 7 51 bRmwz



tmax ¼

ðA12Þ

where n0 and nT are the initial and final porosities respectively (Fig. 8ai), and T is the time over which compaction takes place, and a permeability function: k(t) ¼ B(n(t))2 were found to fit these requirements with n0 ¼ 0.27, nT ¼ 0.035 and B ¼ 4.115  10212 m2 giving an initial permeability k0 ¼ 3  10213 m2 and final permeability kT ¼ 5  10215 m2 (Fig. 8aii). These values are chosen as both being within the general limits of published data on porous sand and deformation band porosities and permeabilities (e.g. Antonellini & Aydin 1994; Fowles & Burley 1994), and fitting the criterion for solution of equation (7) that k(t)/n(t) / 1/t. Equation (A10) is then pðxÞ ¼

(A16), setting dDP/dt ¼ 0 and rearranging in terms of t gives

where q(x) is the derivative (with respect to t) of the undrained solution (Fig. A2aii). Hence for Model B, equation (A7) can be rewritten as dDP K k   DPðtÞ ¼ dt bn0 bmwzðn0  KtÞ

ðA21Þ

and gives a fluid pressure rise function of DPðtÞ ¼

" #   Kmwz n0  Kt n0  Kt k=bKmwz  k  bKmwz n0 n0 ðA22Þ

This fluid pressure rise function yields extremely fast fluid pressure rises initially, steadying out and finally dropping as fluid pressure dissipation becomes increasingly important (Fig. A2bi). The different cases of compaction time plotted in Figure A1ci illustrate that fluid pressure rises are lower when compaction times are longer, this being due to the increased role of fluid escape in the slower compaction cases. The peak fluid pressure rise during the compaction can be found by using the fact that the derivative of equation (A16) must equal zero at the time, tmax, of the peak in fluid pressure. Differentiating equation

which is illustrated in Figure A1bii. Figure A1cii shows the pressure rises predicted by Model B for different cases of compaction time, T, and illustrates how shorter compaction times lead to much higher pressure rises, with T ¼ 1026 s approximating the undrained case. It is interesting to note that the time, tmax, of peak fluid pressure rise is later in Model B than in Model A, even though permeability is lower until t ¼ T, and this is because Model A assumes greater compaction rates in the early stages. For Model B, the time of peak fluid pressure, tmax, can

FAULT DISTRIBUTIONS IN HIGH-POROSITY SANDS

(i)

Model A

Model B

(ii) 2 × 109

2 × 109 1.75 × 109

1.75 × 109

1.5 × 109 1.25 × 109

1.5 × 109 1.25 × 109

ΔP(t) (Pa)

ΔP(t) (Pa)

(a)

43

1 × 109 8

7.5 × 10 5 × 108 2.5 × 108 –4

–3

–2

–1

1 × 109 7.5 × 108 5 × 108 2.5 × 108 –4

0

–3

(b)

(i)

ΔP(t) (Pa)

ΔP(t) (Pa)

200 150

2000 1500

100

1000

50

500 –8

–6

–4

–2

0

–7

–6

–5

log[t/T]

–4

–3

log[t/T]

–2

–1

0

(ii)

(i)

10 u –6

8

log[ΔP(t) (Pa)]

log[ΔP(t) (Pa)]

0

2500

250

(c)

–1

(ii)

3000

300

10

–2

log[t/T]

1og[t/T]

–5

6

–4

4

–2

2

0

–3 –1 1

8

u –6 –5 –4

6

–3

4

–1

2

1

–2 0 2

2

–7

–6

–5

–4

–3

–2

–1

0

–7

log[t/T]

–6

–5

–4

–3

–2

–1

0

log[t/T]

Fig. A2. Results of the model: evolution of excess fluid pressure expressed as a function of dimensionless time (t/T ), for Model A (i) and Model B (ii). (a) The ‘undrained’ components only, with w ¼ z. (b) The model solutions (combined ‘undrained’ and fluid pressure dissipation components) for the case of T ¼ 1 s with a 0.5 mm wide deformation band (w ¼ z ¼ 0.25 mm). (c) The model solutions for various cases of T with a 0.5 mm wide deformation band (numbers denote log[T ], with u indicating the undrained case). The peaks of these curves give the maximum excess fluid pressure illustrated in Figure 8b.

be found in a similar way to the case for Model A, giving

tmax

8 2

 39 >  1= k  1 > = bKmwz n0 < 7 6 bKmwz ¼ 14 5 : ðA23Þ > K> k : ;

The maximum fluid pressure rise, DPtmax, predicted by Model B can then be found by putting tmax into equation (A22) or by combining equations (7) and (11) with t ¼ tmax. Results of this process are shown in Figure 8bii.

References A NTONELLINI , M. & A YDIN , A. 1994. Effect of faulting on fluid flow in porous sandstones: petrophysical properties. AAPG Bulletin, 78, 335–377. A NTONELLINI , M. & A YDIN , A. 1995. The effect of faulting on fluid flow in porous sandstones: geometry and spatial distribution. AAPG Bulletin, 79, 642 –671. A NTONELLINI , M. A. A., A YDIN , A. & P OLLARD , D. D. 1994. Microstructure of deformation bands in porous sandstones at Arches National Park, Utah. Journal of Structural Geology, 16, 941– 959.

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A NTONELLINI , M., A YDIN , A. & O RR , L. 1999. Outcrop-aided characterization of a faulted hydrocarbon reservoir: Arroyo Grande oil field, California, U. S. A. In: H ANEBERG , W. C., M OZLEY , P. S., M OORE , J. C. & G OODWIN , L. B. (eds) Faults and Subsurface Fluid Flow in the Shallow Crust. American Geophysical Union, Monograph, 113, 7– 26. A TKINSON , J. H. & B RANSBY , P. L. 1978. The Mechanics of Soils: an Introduction to Critical State Soil Mechanics. McGraw–Hill, London. A YDIN , A. 1978. Small faults formed as deformation bands in sandstone. Pure and Applied Geophysics, 116, 913–930. A YDIN , A. & J OHNSON , D. 1978. Develpment of faults as zones of deformation bands and as slip surfaces in sandstone. Pure and Applied Geophysics, 116, 931– 942. A YDIN , A. & J OHNSON , A. M. 1983. Analysis of faulting in porous sandstones. Journal of Structural Geology, 5, 19–31. B IEGEL , R. L., S AMMIS , C. G. & D IETERICH , J. H. 1989. The frictional properties of a simulated gouge having a fractal particle distribution. Journal of Structural Geology, 11, 827–846. B LENKINSOP , T. G. 1989. Thickness– displacement relationships for deformation zones: discussion. Journal of Structural Geology, 11, 1051–1054. B OLTON , M. D. 1986. The strength and dilatancy of sands. Ge´otechnique, 36, 65–78. B OLTON , M. D. & M C D OWELL , G. R. 1997. Clastic mechanics. In: F LECK , N. A. & C OCKS , A. C. F. (eds) IUTAM Symposium on Mechanics of Granular and Porous Materials. Kluwer, Dordrecht, 35–46. B YERLEE , J. D. & S AVAGE , J. C. 1992. Coulomb plasticity within the fault zone. Geophysical Research Letters, 19, 2341–2344. C ASAGRANDE , A. 1936. Characteristics of cohesionless soils affecting the stability of slopes and earth fills. Journal of the Boston Society of Civil Engineers, 23, 13–32. C UNDALL , P. A. & S TRACK , O. D. L. 1979. A discrete numerical model for granular assemblies. Ge´otechnique, 29, 47– 65. D AVIS , G. H. 1999. Structural Geology of the Colorado Plateau Region of Southern Utah, with Special Emphasis on Deformation Bands. Geological Society of America, Special Papers, 342. D EBRAND -P ASSARD , S., C OURBOULEIX , S. & L IENHARDT , M. J. 1984. Synthe`se Ge´ologique du Sud-Est de la France. Me´moires du Bureau de Recherches Ge´ologiques et Minie`res. (BRGM), 125. D ESRUES , J., C HAMBON , R., M OKNI , M. & M AZEROLLE , F. 1996. Void ratio evolution inside shear bands in triaxial sand specimens studied by computed tomography. Ge´otechnique, 46, 529–546. D UBOIS , P. & D ELFAUD , J. 1984. Le bassin du Sud-Est. In: G UBLER , Y. (ed.) Dynamique des Bassins Se´dimentaires. Livre jubilaire. BRGM, Orleans. E VANS , J. P. 1990. Thickness–displacement relationships for fault zones. Journal of Structural Geology, 12, 1061–1065. E VANS , J. P., F ORSTER , C. B. & G ODDARD , J. V. 1997. Permeability of fault-related rocks, and implications for hydraulic structure of fault zones. Journal of Structural Geology, 19, 1393– 1404.

F ERRIL , D. A., M ORRIS , A. P., S TAMATAKOS , J. A. & S IMS , D. W. 2000. Crossing conjugate normal faults. AAPG Bulletin, 84, 1543–1559. F ISHER , Q. & K NIPE , R. J. 1998. Fault sealing processes in siliclastic sediments. In: J ONES , G., F ISHER , Q. & K NIPE , R. J. (eds) Faulting, Fault Sealing and Fluid Flow in Hydrocarbon Reservoirs. Geological Society, London, Special Publications, 147, 117– 134. F ISHER , Q., K NIPE , R. J. & W ORDEN , R. H. 2000. Microstructures of deformed and non-deformed sandstones from the North Sea: implications for the origins of quartz cement in sandstones. In: W ORDEN , R. H. & M ORAD , S. (eds) Quartz Cementation in Sandstones. International Association of Sedimentologists, Special Publications, 29, 129– 146. F OSSEN , H. & H ESTHAMMER , J. 1997. Geometric analysis and scaling relations of deformation bands in porous sandstone. Journal of Structural Geology, 19, 1479– 1493. F OSSEN , H. & H ESTHAMMER , J. 2000. Possible absence of small faults in the Gullfaks Field, northern North Sea: implications for downscaling of faults in some porous sandstones. Journal of Structural Geology, 22, 851–863. F OWLES , J. & B URLEY , S. 1994. Textural and permeability characteristics of faulted, high porosity sandstones. Marine and Petroleum Geology, 11, 608–623. F OXFORD , K. A., W ALSH , J. J., W ATTERSON , J., G ARDEN , I. R., G USCOTT , S. C. & B URLEY , S. D. 1998. Structure and content of the Moab Fault Zone, Utah, U.S.A., and its implications for fault seal prediction. In: J ONES , G., F ISHER , Q. & K NIPE , R. J. (eds) Faulting, Fault Sealing and Fluid Flow in Hydrocarbon Reservoirs. Geological Society, London, Special Publications, 147, 87– 103. F RIEDMAN , M. & L OGAN , J. M. 1973. Lu¨ders’ Bands in experimentally deformed sandstone and limestone. Geological society of America Bulletin, 85, 1465– 1476. G ALLAGHER , J. J. J R , F RIEDMAN , M., H ANDIN , J. & S OWERS , G. M. 1974. Experimental studies relating to microfracture in sandstone. Tectonophysics, 21, 203–247. G IBSON , R. G. 1998. Physical character and fluid-flow properties of sandstone-derived fault zones. In: C OWARD , M. P., D ALTABAN , T. S. & J OHNSON , H. (eds) Structural Geology in Reservoir Characterization. Geological Society, London, Special Publications, 127, 83–97. H EYNEKAMP , M. R., G OODWIN , L. B., M OZLEY , P. S. & H ANEBERG , W. C. 1999. Controls on fault-zone architecture in poorly lithified sediments, Rio Grande Rift, New Mexico: implications for fault-zone permeability and fluid flow. In: H ANEBERG , W. C., M OZLEY , P. S., M OORE , J. C. & G OODWIN , L. B. (eds) Faults and Subsurface Fluid Flow in the Shallow Crust. American Geophysical Union, Monograph, 113, 27– 49. H ORSFIELD , W. T. 1980. Contemporaneous movement along crossing conjugate normal faults. Journal of Structural Geology, 2, 305–310. H ULL , J. 1988. Thickness–displacement relationships for fault zones. Journal of Structural Geology, 10, 431–435.

FAULT DISTRIBUTIONS IN HIGH-POROSITY SANDS J AEGER , J. C. 1967. Failure of rocks under tensile conditions. International Journal of Rock Mechanics, Mining Sciences and Geomechanics Abstracts, 27, 345–352. J AMISON , W. R. & S TEARNS , D. W. 1982. Tectonic deformation of Wingate Sandstone, Colorado National Monument. AAPG Bulletin, 66, 2584–2608. J ONES , M. E. & A DDIS , M. A. 1986. The application of stress path and critical state analysis to sediment deformation. Journal of Structural Geology, 8, 575–580. K NIPE , R. J., J ONES , G. & F ISHER , Q. 1998. Faulting, fault sealing and fluid flow in hydrocarbon reservoirs: an introduction. In: J ONES , G., F ISHER , Q. & K NIPE , R. J. (eds) Faulting, Fault Sealing and Fluid Flow in Hydrocarbon Reservoirs. Geological Society, London, Special Publications, 147, vii –xxi. K NOTT , S. D., B EACH , A., B ROCKBANK , P. J., B ROWN , J. L., M C C ALLUM , J. E. & W ELBON , A. I. 1996. Spatial and mechanical controls on normal fault distributions. Journal of Structural Geology, 18, 359–372. K RANTZ , R. W. 1988. Multiple fault sets and threedimensional strain: theory and application. Journal of Structural Geology, 10, 225–237. L EHNER , F. K. & P ILAAR , W. F. 1997. The emplacement of clay smears in synsedimentary normal faults: inferences from field observations near Frechen, Germany. In: M ØLLER -P EDERSEN , P. & K OESTLER , A. G. (eds) Hydrocarbon Seals: Importance for Exploration and Production. Norwegian Petroleum Society, Special Publications, 7, 39– 50. L EVEILLE , G. P., K NIPE , R. J., M ORE , C. ET AL . 1997. Compartmentalisation of Rotliegendes gas reservoirs by sealing faults, Jupiter fields area, southern North Sea. In: Z IEGLER , K., T URNER , P. & D AINES , S. R. (eds) Petroleum Geology of the North Sea: Future Potential. Geological Society, London, Special Publications, 123, 87– 104. M AIR , K., M AIN , I. & E LPHICK , S. 2000. Sequential growth of deformation bands in the laboratory. Journal of Structural Geology, 22, 25– 42. M ANDL , G. 1988. Mechanics and Tectonics of Faulting. Models and Basic Concepts. Elsevier, Amsterdam. M ANDL , G., DE J ONG , L. N. J. & M ALTHA , A. 1977. Shear zones in granular material. Rock Mechanics, 9, 95– 144. M ANZOCCHI , T., R INGROSE , P. S. & U NDERHILL , J. R. 1998. Flow through fault systems in high-porosity sandstones. In: C OWARD , M. P., D ALTABAN , T. S. & J OHNSON , H. (eds) Structural Geology in Reservoir Characterization. Geological Society, London, Special Publications, 127, 65– 82. M ATTHA¨ I , S. K., A YDIN , A., P OLLARD , D. D. & R OBERTS , S. G. 1998. Numerical simulation of departures from radial drawdown in faulted sandstone reservoirs with joints and deformation bands. In: J ONES , G., F ISHER , Q. & K NIPE , R. J. (eds) Faulting, Fault Sealing and Fluid Flow in Hydrocarbon Reservoirs. Geological Society, London, Special Publications, 147, 157– 192. M C D OWELL , G. R. & B OLTON , M. D. 1998. On the micromechanics of crushable aggregates. Ge´otechnique, 48, 667–679.

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Strain localization in geomaterials J. DESRUES1, P. BE´SUELLE1 & H. LEWIS2,3 1

Laboratoire 3S, UJF – INPG – CNRS, Grenoble, France (e-mail: [email protected])

2

Heriot – Watt Institute of Petroleum Engineering, Heriot – Watt University, Edinburgh EH14 4AS, UK

3

ECOSSE (Edinburgh Collaborative of Subsurface Science and Engineering) a Joint Research Institute that is part of the Edinburgh Research Partnership in Engineering and Mathematics Abstract: The main purpose of this paper is a broad review of developments in observation and interpretation of localization in geomaterials in the laboratory, with an emphasis on low mean stress situations. Laboratory investigation of strain localization in granular soils and rocks has been pursued extensively and very accurate strain field evolution measurement techniques have been developed, including false relief stereophotogrammetry (FRS) and computed tomography (CT). These permit full characterization of strain localization, from onset to complete shear band formation. This paper reviews studies of sand, clay, sandstone, stiff marl and concrete, and observations of incipient and developed localization in initially ‘homogeneous’ laboratory tests are presented. Development of localization and peak strength, critical stress and strain, shear band orientation and thickness, and complex localization patterns are discussed. Deformation during triaxial compression of sand is shown to develop complex strain localization patterns. Consequently, the critical void ratio concept in granular materials is reconsidered. Void ratio evolution, global and local, monitored by CT, shows a limiting void ratio being rapidly attained in the strain localization zones. In cohesive materials (clays, rocks and concrete), crack development is also commonly observed. Displacement discontinuity measurement techniques are presented and the results for different cohesive geomaterials are discussed.

Geomaterials (soils, rock and concrete) are especially prone to strain localization and many natural examples of localized deformation are seen in outcrop in situations consistent with deformation in the near-surface down to several kilometres or even tens of kilometres depth. Seismic events associated with development of active faults produce observable displacement discontinuities at the Earth’s surface. Large active faults provide evidence of repeated localized deformation processes extending over the kilometre to multi-kilometre scale vertically as well as horizontally, and such faults are demonstrably important in the development of sedimentary basins, the primary source of hydrocarbons. In the civil engineer’s domain, practical situations in which geomaterials are loaded to failure are common: natural or human-made slopes may become unstable and collapse under certain circumstances; excavation works need to disaggregate and remove a part of the in situ material; collapsed foundations and retaining walls usually show clearly localized kinematic mechanisms. In fact, the early work by Coulomb (1773) on the stability of retaining walls already took into account, as an essential aspect of the problem, the localized nature of the collapse mechanism.

Strain in geomaterials can show a number of different aspects: shear bands are usually observed in granular masses and granular specimens at nearsurface conditions and these materials are typically regarded as ductile materials in the soil mechanics community. In near-surface conditions, or their laboratory equivalents, rocks and concrete typically develop cracks at different scales, and these materials are regarded as brittle materials by both the soil mechanics and the geoscience communities. A wide transition zone lies between these ductile and brittle fields, extending from the near-surface down to burial depths of 1 km or more. In addition, rocks that respond in a clearly brittle manner in the near-surface show a more ductile behaviour when burial depths of several kilometres or elevated temperatures have been reached. These deformations, in which deformation bands are produced but the bulk of the material has not deformed plastically, have been the subject of much observational and theoretical study, as discussed below. Unsurprisingly, occurrence of shear bands or deformation bands rather than cracks in a laboratory specimen, or in a natural situation, depends not only on the material considered, but also on the loading process (e.g. mean or deviatoric stress values or

From: LEWIS , H. & COUPLES , G. D. (eds) The Relationship between Damage and Localization. Geological Society, London, Special Publications, 289, 47– 73. DOI: 10.1144/SP289.4 0305-8719/07/$15.00 # The Geological Society of London 2007.

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loading rate). Moreover, observation of the features at different length scales can lead to different diagnostics on the nature of the localized deformation structures involved in the global deformation process. Localization in rocks has been recognized in outcrop and produced in the laboratory for a considerable length of time by a number of workers (see, e.g. Handin et al. 1963; Brace et al. 1966; Brace 1972, 1978) in the more general context of the rock mechanics response of deformed and deforming rocks. In soil mechanics, strain localization has been taken into account for a considerable length of time in practical design, such as in the semi-empirical design methods for slope stability such as Fellenius’s or Bishop’s slip circle methods (Fellenius 1936; Bishop 1955). However, following the pioneering work of Rice and Rudnicki (Rudnicki & Rice 1975; Rice 1976) the emphasis of these studies changed somewhat. The refocused interest in strain localization by shear band development in both granular and cohesive materials has motivated a large amount of scientific work in geomechanics and allied areas, and a range of experimental, theoretical and numerical tools have been developed over the last 35 years (e.g. Chambon et al. 1994b). Most of the experimental work has been focused in a relatively small number of soil and rock mechanics laboratories around the world. A large part of this paper is a review of the main features of this exploration of the strain localization phenomenon in geomaterials. To achieve this goal, broad reference is made to advances in experimental methods and the associated observations, and to the consequent advances in understanding over the last 35 years. In this review paper there is some emphasis on the experimental methods and observations of Laboratoire 3S-IMG in Grenoble. Although the paper focuses on developments in strain localization from laboratory work, and for shallow deformations (to 1 or 2 km) reference is made to strain localization in natural systems in granular materials and rocks, and to deformation under much higher mean stresses, representing considerably deeper deformation sites. The main materials used in the experimental study, the experimental configurations and the main techniques used to study them are described below. Hostun ‘RF’ sand. This sand is a granulometric class (0.5–1.25 mm) natural sand, obtained by sieving the material extracted from a natural sand deposit in Hostun, Droˆme, France. The resulting material is a fine angular siliceous sand, described in detail by Flavigny et al. (1990). The sand has been used extensively in studies on localization

(Desrues 1984; Hammad 1991; Mokni 1992; Roger 2000). Beaucaire Marl. This is the cohesive material used to study localized failure in soft rocks (Tillard 1992; Tillard-Ngan et al. 1993). It is extracted from a quarry near Beaucaire, Gard, France, at a depth c. 70 m below the natural surface and is dated to the Piacenzian Stage of the Tertiary Era. It consists of 36% quartz, 31% calcareous carbonate and 32% clay. The consolidation pressure has been found to be 5 MPa. A detailed description of the material has been given by Tillard (1992) and Tillard-Ngan et al. (1993). Red Vosges Sandstone. This is a porous sandstone consisting of quartz (93%) and few per cent of feldspar and white mica. Is has an average porosity of 22% and the mean grain size is c. 0.3 mm. Grains are slightly angular as well as well sorted, and poorly cemented. The cohesion is due essentially to the interpenetration between grains. A more detailed description has been given by Be´suelle et al. (2000). False relief stereophotogrammetry (FRS). This method is based on analysis of a series of photographs taken during the deformation of a specimen in the laboratory (Desrues & Duthilleul 1984; Desrues 1995a). It takes advantage of the very accurate human sense of depth perception. This sense is the result of an analysis of the slight geometrical differences between the images seen by the left and right eyes. In natural perception this process allows 3D vision, but in FRS the relief is artificial because the two images seen by the left and right eyes are not taken simultaneously from two different locations, but at two successive times. This leads to a perception of relief that is directly linked to the incremental deformation undergone by the specimen. Quantitative measurements can be obtained using stereophotogrammetric apparatus developed for topographic applications, together with a specific data processing developed for the false relief application. When using a high-quality camera together with large film format (typically 10 cm  13 cm glass plates), a resolution of a few microns at the film scale can be achieved. Tomodensitometry or X-ray computed tomography. This is a method of using X-ray attenuation through solid or bulk materials to quantify the compactness. For granular or cohesive geomaterials, as long as the mineralogical composition is homogeneous and does not change during the process, the radiographic density can be calibrated to compactness. Use of the X-ray scanner allows rebuilding of the internal distribution of radiographic density in 3D specimens using a back-analysis process (computed

STRAIN LOCALIZATION IN GEOMATERIALS

tomography; CT) on a set of radiographic profiles taken in different directions. The tests reported here have been performed in collaboration with LMA laboratory, Marseille, France. In the case of these tests, the voxel size was 0.7 mm  0.7 mm  4 mm. An introduction to X-ray CT applied to geomaterials has been given by Desrues et al. (1996). The resolution achievable has improved considerably over recent years. Tomographic applications performed now using a high-energy beam provided by synchrotron radiation facilities allow resolutions within of a few micrometres to be reached (Viggiani et al. 2004). One can readily go below micrometre level, but with the associated restriction of a very limited volume being investigated. Test types. The types of tests reported here are the axisymmetric triaxial test, the plane strain biaxial test and the true triaxial test. The axisymmetric triaxial test is characterized by two equal principal stresses; it is usually performed on a cylindrical specimen, wrapped in a deformable jacket and confined by a fluid pressure in a confining cell. It may be performed on a dry material, but if there is a fluid in the pores (with or without full saturation), the test can be drained, undrained or partially drained, depending on the mobility of the fluid with respect to the solid skeleton. Plane strain biaxial tests are performed on rectangular specimens, subjected to plane strain; that is, zero deformation in one direction. True triaxial tests are performed on cubic specimens, their characteristic being that the three principal stresses (or strains) can be controlled independently, which is a generalization of the other two tests. This paper addresses the development of localization, primarily as observed in the laboratory, introducing the generalities of shear band evolution in both the field and the laboratory, together with the main laboratory apparatus used and the main theoretical developments. It then addresses unconsolidated granular geomaterials followed by consolidated geomaterials, in each case providing observations of the nature of incipient and developed localization.

Shear band evolution in granular and cohesive materials: natural and laboratory observations and their theoretical representation Strain localization is a very common phenomenon in granular materials. It can be observed in natural soil masses, in embankments and in retaining walls. When rupture occurs, the global mechanism usually involves a number of quasi-rigid blocks sliding past each other along so-called rupture

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surfaces or rupture planes. A similar phenomenon can also be observed in other solids such as metals, alloys and rocks, in which the term deformation band is commonly used (see, for example, Aydin 1978). The general term given to the zones of localized deformation and used in both the mechanical engineering and the geomechanics fields is shear bands. It is the term used here. However, as pure compressive (zero shear) bands have been reported (see, e.g. Antonellini & Aydin 1994), named ‘compaction bands’, the general term ‘deformation bands’ for both shear and compaction bands could be preferred in the future. A large number of case histories described in the soil mechanics literature provide a good illustration of the type of kinematics that produces these features (see, e.g. the paper by Scott (1987)). In a number of cases, the failure surfaces develop along preexisting discontinuities such as interfaces between mechanically different layers or zones. In other cases, failure surfaces can be initiated by strong singularities in the geometry, in the loading, or in the mechanical properties. However, localization can appear even when singularities and heterogeneities have been carefully avoided: this is the case when performing rheological tests in the laboratory. Such tests, used to characterize mechanical properties, are typically performed on specimens that are as homogeneous as possible. Over a large range of test conditions (where material tested and test technique both vary), the final state of rupture is reached with a significantly non-uniform kinematic field, involving one or more rupture planes. The axisymmetric triaxial test is the basic laboratory test in soil and rocks mechanics, and much work has been done to improve the homogeneity of this test. However, in spite of this effort, both localized and diffuse modes of heterogeneity can be observed in the vast majority of deformed triaxial specimens; the loss of homogeneity being extremely sensitive to the quality of the test preparation and execution. These questions have been discussed by Colliat-Dangus et al. (1986, 1988). Localization is also observed in other rheological tests, such as the hollow cylinder test and the directional shear cell (Arthur et al. 1977; Arthur & Dunstan 1982), or the true triaxial test. In the true triaxial test normal stresses or displacements can be imposed on a cubic specimen in three orthogonal directions normal to the faces (Lanier 1989). When this type of apparatus is displacement-controlled, loading is applied by six rigid platens. Despite this kinematical restraint, it was shown that localization is active also in true triaxial tests (Desrues et al. 1985). Drainage conditions can significantly affect the response of granular specimens; the effect is also

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present in consolidated geomaterials. In undrained laboratory tests, dilatancy (positive or negative) is constrained by the fluid, but this very constraint also generates changes in pore fluid pressure, inducing effective stress paths that are completely different from those in the drained tests. Liquefaction can also be observed in unconsolidated specimens. As far as localization is concerned, initially most experimental studies addressed drained tests but after 1990 results in undrained condition have become available (Han & Vardoulakis 1991; Mokni 1992; Roger et al. 1998; Mokni & Desrues 1999; Roger 2000). The main conclusions of these studies are given in the following sections. These observations have been brought together through the development of shear band analysis, a theoretical approach, related to bifurcation theory. Starting with the pioneering work of Rudnicki & Rice (1975) and Rice (1976), the concept and practicalities of shear band analysis have developed over the last 30 years, in geomechanics for soil-like materials (see Desrues et al. 1991a; Chambon et al. 1994a; Vardoulakis & Sulem 1995; Be´suelle & Rudnicki 2004) and in mechanical engineering for metals. In this approach, strain localization is considered as a loss of uniqueness of a local boundary value problem. Importantly, the constitutive equations play a central role in the analysis. In that sense, shear banding is considered as a material instability. For soil modellers, shear band analysis has become a test for constitutive equations; it has

the potential to become a guide for identifying, or even for elaborating advanced models in the near future. Examples of this work have been given by Chambon & Desrues (1985), Desrues & Chambon (1989) and Chambon et al. (1994a, 2000).

Localization in granular bodies Although strain localization in geomaterials has been recognized since the late 1970s as an important research field, experimental studies have not been very numerous, most of the results published having been obtained with specially designed biaxial tests (Arthur et al. 1977; Vardoulakis et al. 1978; Drescher & Vardoulakis 1982; Desrues 1984, 1990, 1995b, c; Vardoulakis & Graf 1985; Tatsuoka et al. 1986, 1990; Desrues & Hammad 1989; Han & Vardoulakis 1991; Yoshida et al. 1994; Vardoulakis 1996a, b). All the experimental results discussed in this section are taken from the studies performed in Grenoble IMG–3S laboratory between 1980 and 2000 using Hostun RF-S28 sand. The plane strain biaxial test (Fig. 1) is very useful for studying strain localization, first because it is a true rheological test, and second because it allows a complete determination of the real strain field at any stage of the test. Figure 1 shows the geometry of the specimen, and the boundary conditions imposed on it. Zero strain is imposed normal to the largest face of the specimen

Fig. 1. Biaxial test geometry and loading conditions (adapted from Desrues 1984 with permission). (a) Rectangular sample, with dimensions, showing boundary constraints where 12 is lateral strain in direction 2. In this case 12 ¼ 0. V is displacement in direction 1 and s3 is lateral pressure in direction 3. Directions 1, 2 and 3 are shown on left of figure. (b) Schematic diagram of the plane strain device with the specimen in place. (c) The pressure cell and loading device.

STRAIN LOCALIZATION IN GEOMATERIALS

by a plane strain device using two thick glass plates (12 ¼ 0). Axial, constant rate, vertical displacement (V ) is applied by a mechanical jack, through a platen that allows free lateral translation. The specimen is wrapped in a rubber membrane c. 0.4 mm thick. It is subject to a confining pressure, denoted as s3, with values in the range 100 –1000 kPa. The maximum axial load applicable is 50 kN. The loading rates are chosen in the range 0.04 – 5.00 mm min21. The contacts between the specimen and the plane strain device are lubricated, the same arrangement being used for the contacts with the platens.

A typical test The results of a typical biaxial test (Desrues 1984) illustrate strain localization in granular bodies. Figure 2 shows the axial load v. displacement curve recorded during a test performed on Hostun sand (relative density c. 90%, porosity 40%). The numbers in Figure 2 are the photograph numbers. Figure 3 shows the result obtained by FRS for the set of increments 1–2, 2– 3, 3–4, 4–5 and 5 –6. Photographs are taken when needed to capture the behaviour of the sample and particularly any changes in the response of the sample. Nonhomogeneous deformations are evident at increment 3 –4 and a sliding block mechanism is clearly operating at increment 5–6. This set of

51

pictures give a clearer image of the initialization of the shear band at increment 3–4 and its further evolution. It can be seen that at increment 3–4 the heterogeneous deformation already has the structure of a shear band; in comparison, increment 2 –3 shows a partially homogeneous deformation field, and partial organization of a localized mode. One interpretation of these pictures is that localization is initiated by a prior smooth heterogeneity; in such a case, we could see the process as a kind of ‘condensation’ of the heterogeneity. In other cases, localization can be initiated by local strong heterogeneities, such as inclusions. In these cases the shear band development mode is more like a propagation process. The bottom row of Figure 3 shows that strong dilatancy is associated with the localization process.

Inception and growth of localization In this section an example of localization development is used to illustrate the influence of different parameters on the occurrence of strain localization. The type of test used is a drained, low-pressure, biaxial test (s3  100 kPa), chosen because localization always occurs in biaxial experiments on Hostun RF sand in this pressure range (see below). When a dense sand under 100 kPa confining pressure is used, a rather short period of quasi-homogeneous deformation develops and is

Fig. 2. Plane strain test SHF06. (a). Plot of axial load (N) v. axial displacement (u), and of volumetric strain v. axial displacement (u). Numbers show position of photographs. (b). Detail of upper part of loading curve for test SHF06. Axial loading is in Newtons; axial displacement is in millimetres.

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Fig. 3. Stereophotogrammetric measures of incremental deformation during the plane strain test SHF06. Numbers specify the deformation increments displayed, such that 2 –3 is the increment between photographs 2 and 3. Top row shows incremental displacement. Middle row shows the corresponding incremental shear intensity, defined as d1s ¼ 12 (d1I 2 d1II), with d1I and d1II being the principal strain rates. Bottom row shows volumetric strain, defined as d1v ¼ 12 (d1I þ d1II). Sizes of symbols are proportional to the value of the scalar quantity being plotted.

followed (at about 3% axial strain) by incipient shear bands. No significant bulging is observed before the incipient shear bands appear. The following factors have been addressed. Slenderness ratio. Tests performed on specimens with slenderness (height/length, designated H/L) ratios ranging from 3.3 to 0.5 (Hammad 1991; Mokni 1992) have shown strain localization in every case, even in very short specimens. Figure 4 shows the localization mechanism in test shf68 where H/L ¼ 0.5. The shear band appears to be ‘reflected’ several times at the upper and lower platens, inducing four triangular blocks and two half blocks.

Density. Less dense specimens, usually called loose specimens in soil mechanics, also show localization, but it develops at noticeably higher axial strains. All conditions except density are unchanged, so this result is clearly a density effect. Mean stress level. A set of tests at different lateral pressures was performed. The lateral pressure values used were in the range 100–1000 kPa. In the biaxial test configuration lateral pressure is applied to the specimen’s two vertical edges, playing the same role as confining pressure does in the more typical axisymmetric tests. A higher lateral pressure will produce higher mean effective stress values in the specimen. Localization occurs

STRAIN LOCALIZATION IN GEOMATERIALS

Fig. 4. Stereophotogrammetric measures of incremental deformation during a plane strain test on a short specimen (test shf68). Top row shows incremental strain. Middle row shows the corresponding incremental shear intensity. Bottom row shows volumetric strain. It should be noted that displacement scale is five times the geometric scale and that sizes of symbols are proportional to the local value of the scalar quantity being plotted. Adapted from Mokni (1992) with permission.

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0.10

8 7

σa/σ1

εv 0.08

6 0.06

5 4

0.04

3 0.02 2 0.00

1 0 0.0

εa 0.02

0.04

0.06

0.08

0.1 test

–0.02

0.12 cell press.

shf40 100 kPa shf41 200 kPa shf42 400 kPa shf43 800 kPa Fig. 5. Plot of axial strain against the ratio of axial and lateral stress and against volumetric strain to illustrate the influence of lateral pressure magnitude on onset of localization in dense sand specimens. Each symbol represents a test performed at a different lateral stress; x-axis is overall axial strain (per cent), and y-axes are ratio of axial to longtitudinal stress (left) and volumetric strain (per cent) (right). The vertical lines indicate the onset of localization, as taken from direct and photogrammetric observation. Adapted from Hammad (1991) with permission.

across the whole range of lateral pressures (and hence mean effective stresses) used, but the localization initiation occurs at much larger axial strains for higher lateral pressure values (Fig. 5). Symmetric and antisymmetric diffuse modes of heterogeneity (bulging) can appear at the earlier stages of these tests but localization still remains the ultimate mode. Figure 5 illustrates the progressive delay in localization onset with lateral pressure. The same is true for specimens with low density, prepared with the same Hostun RF sand, as shown on Figure 6: no peak is observed in the stress –strain curves. Only a change in slope and a sudden increase of the fluctuations of the curve indicate the onset of shear banding, although this behaviour is clearly detected using FRS (not shown here). Other results indicate that localization can occur in materials undergoing moderate and even strong contractancy (compaction capability), such as calcareous sands (Desrues & Hammad 1989; Hammad 1991). The situation described above for low lateral pressures in plane strain biaxial tests contrasts with classic observations in axisymmetric tests of

the same materials and under the same range of applied pressures. In the latter tests, bulging and other diffuse modes of bifurcation are almost always observed before localization. Moreover, Mokni (1992) showed that the occurrence of localized strain patterns in dense specimens deformed in an axisymmetric triaxial mode can be delayed significantly beyond the stress peak strain, up to 20% axial strain, if very careful specimen preparation and test procedures are used. This establishes a specific stability of the axisymmetric triaxial test with respect to any other test condition. Theoretical predictions in the framework of bifurcation theory support this experimental observation (Rudnicki & Rice 1975; Vardoulakis 1979). The soil testing community has typically considered that strain localization would be precluded in true triaxial tests because of the rigid platens and the full kinematic control of the apparatus. However, it has been shown (Desrues et al. 1985) that shear bands do develop in cubic specimens subjected to various kind of stress paths in true triaxial tests. The specimen shown in Figure 7 reveals two

STRAIN LOCALIZATION IN GEOMATERIALS

55

5

0.10

4

0.08

3

0.06 0.04 εa

1

0.02

εv

σa/σl

2

0 .0

.02

.04

.06

.08

.10

.12

.14

.16

0.00

–1 –0.02

–2

–0.04

–3 test

cell press.

shf48 100 kPa shf50 200 kPa shf54 400 kPa shf51 800 kPa Fig. 6. Influence of lateral pressure on the onset of localization in loose specimens. The x-axis is overall axial strain (per cent); y-axes are ratio of axial to longitudinal stress (left side) and per cent volumetric strain (right side). Symbols represent individual tests at different lateral pressures. Positive volumetric strain represents dilation and negative volumetric strain represents compaction. Although no marked peak can be seen on the curves, strain localization is observed, and is confirmed by stereophotogrammetry. The vertical lines indicate the onset of localization in the individual tests and they coincide with the start of the irregularities in the curves. The general trend is the same as in the dense specimens: the higher the lateral pressure, the later the localized mode. Adapted from Hammad (1991) with permission.

Fig. 7. Photograph of localization developed in a cubic specimen tested in a rigid-platen true triaxial apparatus. Reprinted from Desrues et al. (1985) with permission from Elsevier.

conjugate features, each of them involving two shear planes intersecting the faces of the cube along shapes resembling an inverted V. Undrained tests have been performed by a range of workers (Han & Vardoulakis 1991; Mokni 1992; Finno et al. 1996; Roger et al. 1998; Mokni & Desrues 1999; Roger 2000). The accumulated results showed that, for strain localization to occur, the stress ratio (deviatoric stress divided by mean effective stress) has to reach the same critical value as that observed in drained tests at the onset of localization. In loose undrained specimens this condition is easily met and strain localizes normally. However, in dense, dilatant specimens, the drop in pore pressure as a result of the dilatation tendency as the specimen undergoes deviatoric strain makes the mean effective stress increase simultaneously with the deviatoric stress, prohibiting the critical stress ratio being reached and, consequently, also prohibiting the onset of localization. This condition continues to hold until eventually cavitation occurs in the pore fluid, triggering shear banding in the

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Fig. 8. Illustration of strain localization development in an undrained biaxial test performed on dense Hostun RF sand. Plot shows axial strain on the x-axis (in per cent) and ratio of axial to longitudinal stress on the y-axis. Numbers on inserts specify locations of photographs that were used for the stereophotogrammetric measurements. Inserts are plots of the incremental displacements during the test, which was performed on a tall specimen (shfnd02). Numbers specify which deformation increments are being displayed. The onset of localization is shown not to take place before increment 9– 10, when a sudden drop in the effective stress ratio occurs. Adapted from Mokni & Desrues (1999) with permission.

specimen (see Fig. 8). These findings have been shown and discussed in more detail by Roger et al. (1998) and Mokni & Desrues (1999).

Patterns of localization The patterns of shear band localization that develop can be viewed in terms of shear band orientation and the spatial relationships of the shear bands. Shear band orientation is measured from photographs taken during the tests and is defined as the angle between the axial loading direction of the biaxial system and the shear band long axis. There is not much change in this orientation after the peak in the force – displacement curve (Fig. 2) has been reached. What is of interest here is the variation in shear band orientation at the onset of localization with sample character and with

experimental conditions. Figure 9a and b shows the orientations observed for dense and loose Hostun RF sand specimens, respectively, v. the mean effective stress. Drained and undrained tests are presented in the same figures, and the relevant data are given in Tables 1 and 2. In both dense and loose specimens, the observations consistently show that the shear band orientation increases with increasing mean effective stress. If coaxiality of the ends of the specimen is imposed by a biaxial apparatus, generally two crossing shear bands will be observed. Both bands will develop simultaneously in loose (and therefore less strong and stiff) specimens (e.g. void ratio 0.90, porosity 47%) (Desrues 1984; Desrues & Chambon 1986). In dense (and therefore stronger and stiffer) materials (e.g. void ratio 0.66, porosity 40%) there will be only one shear band present up to large

STRAIN LOCALIZATION IN GEOMATERIALS

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Fig. 9. Evolution of the deformation band orientation with respect to the effective mean stress, in drained and undrained tests, for (a) dense samples, and (b) loose samples.

value of axial strain, the shear band appearing at the same time as a sharp peak in the force – displacement curve, as shown in Figure 2. If the test is pursued further, eventually a second band will appear, but this is essentially an artefact of the apparatus’ reaction to the lateral displacement induced by the first shear band. This lateral load mobilization was effectively recorded by Desrues (1984). When the relative lateral displacement of the ends of the specimen is not constrained single band mechanisms are likely to occur. More complex shear band patterns, involving a number of parallel shear bands, are sometimes observed in sand specimens. The increment 3–4 in Figure 3 may be considered as a simple example of such patterning, but in that case the final character of deformation was reduced to either a single band or one dominant band. In shorter specimens (slenderness ratio 0.5 illustrated in Fig. 4) tested in a biaxial plane strain device, localization patterns involves multiple reflections of the shear bands. In addition, the incipient localization in increment 4–5 of Figure 3 shows two shear bands both starting to develop at the same time, each from one of the lower corners of the specimen. As is the case for most dense sand samples, one localization feature becomes dominant in the post-localization regime (see increment 5–6 of Fig. 3). Another localization pattern that can be observed, more frequently in clays, consists of parallel and partially overlapping shear bands: the complete shear mechanism eventually looks like a piecewise shear band, with diffuse strain bridges between (Tatsuoka et al. 1990). When similar materials to those discussed in the biaxial tests are deformed in triaxial axisymmetric tests, the triaxial test end restraint plus the

imposed coaxiality can result in very complex patterns. A detailed image analysis of the density maps obtained during a triaxial axisymmetric test, performed on a dense sand specimen under tomographic control, revealed a pattern that seems to be somewhat generic. A complete set of tomograms was generated, covering the whole specimen. Six circular tomograms are shown, partially superimposed, in Figure 10. A schematic diagram of their geometric relationships is shown in Figure 11. In the front-most circular tomogram, entirely visible in Figure 10, we can see, besides an approximately central black circle, a set of pairs of approximately straight and parallel black lines. The long axis of each pair passes through the specimen axis and the line pairs are approximately radial. The spacing between the lines in each pair increases from one tomogram to a lower one, as does the diameter of the central black circle. This relationship can also be seen in Figure 11. On the sixth, hindmost, tomogram of Figure 10 the spacing between each pair of lines is fairly large but each pair still remains approximately parallel. The individual lines do not pass through the specimen axis. A vertical section can be reconstructed from this set of slices. Such a section is displayed at the bottom of Figure 10, showing the trace of an inverted V. The same type of reconstruction was performed on three parallel planes perpendicular to the same pair of lines (only one being shown). It was found that the V shapes could be superimposed, whereas the trace of the inverted V shapes on the specimen boundary could not, as each vertical slice was at a different distance from the central axis. The geometrical interpretation of these observations is as follows. The localization

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Table 1. Shear band orientation with respect to principal stress direction, dense sands s0 ¼ (s01 þ s30 )/2

u

Dense sand, drained tests shf40 0.673 shf44 0.668 shf41 0.67 shf45 0.659 shf42 0.675 shf46 0.669 shf43 0.661 shf47 0.665 hrfd14 0.637

368 380 726 744 1374 1382 2480 2468 859

21.5

Dense sand, undrained tests shfnd01 0.630 shfnd02 0.617 shfnd03 0.626 hrfnd01 0.638 hrfnd02 0.631 hrfnd03 0.632 hrfnd07 0.619 hrfnd08 0.641 hrfnd10 0.645 hrfnd11 0.648 hrfnd13 0.632 hrfnd16 0.648 hrfnd17 0.634 hrfnd18 0.642 hrfnd19 0.643 hrfnd20 0.639 hrfnd21 0.641 hrfnd22 0.635 hrfnd26 0.642 hrfnd27 0.655 hrfnd28 0.640 hrfnd29 0.645 hrfnd31 0.648 hrfnd33 0.646

1498 2215 2728 1361 1793 1596 1782 2192 1544 818 1745 155 1688 2542 1031 2548 2703 1349 1731 1895 1853 1781 1929 1820

30 28.5 30 28 30 28 29 28 26 25 29 26 28 30 28 30 28 23 29 29 29.5 30 30 30

Name

e0

24 24.7 28 25.3 29 31.5 25

Orientation of the shear bands in dense Hostun RF sand specimens, for drained and undrained biaxial tests. Column 1 gives name of test, column 2 the initial void ratio, column 3 the in-plane mean effective stress (s1 þ s3)/2 at localization, and column 4 is shear band orientation expressed as the angle between the band and the principal stress direction.

pattern contains two kinds of features. The first is a cone centred on the specimen axis with its tip located outside the specimen and its trace exactly matching the bottom of the specimen. The second is a set of plane-strain features occurring in pairs, where each pair crosses on a line that is a diameter of the top section. This spatial organization can be seen as a generalization of the cylindrical specimens with the double mechanism (Fig. 7) for cubic specimens tested on axisymmetric stress paths. More details on this study have been given by Desrues et al. (1996). In nature, examples of a geometry like that shown in Figure 4 are rather common, especially in layered rocks shortened in

Table 2. Shear band orientation with respect to principal stress direction, loose sands Name

e0

s0 ¼ (s01 þ s03)/2

u

239 443 443 878 886 1708 1612

27 27.2 27 27 28.5 32 34

148 562

25.5 26

Loose sand, drained tests shf48 0.894 shf50 0.878 shf55 0.883 shf49 0.896 shf54 0.887 shf52 0.899 shf53 0.884 Loose sand, undrained tests shfnd04 0.887 shfnd05 0.87

Orientation of the shear bands in loose Hostun RF sand specimens, for drained and undrained biaxial tests. Column 1 gives name of test, column 2 the initial void ratio, column 3 the in-plane mean effective stress (s1 þ s3)/2 at localization, and column 4 is shear band orientation expressed as the angle between the band and the principal stress direction.

a direction normal to the bedding plane. However, as perfect axial symmetry is highly unlikely in nature, to the authors’ knowledge no observation like the one described above has been reported from the field.

Volumetric strains Volumetric strains inside the shear band itself can be measured experimentally. Strongly dilating volume changes have been reported by several researchers, using various techniques, namely X-ray photography (Arthur et al. 1977; Scarpelli & Muir Wood 1982; Vardoulakis & Graf 1985) gamma-ray local measurements (Desrues et al. 1985), FRS, or X-ray CT (Colliat-Dangus et al. 1986, 1988; Desrues et al. 1991b, 1995). Those tests were performed on different sands; all were either drained or dry, which are equivalent conditions in terms of effective stress, as the development of significant permeability barriers is normally excluded in sand. Figure 3 presents as an example the case of test SHF06 (Desrues 1984). Comparison of the incremental shear intensity maps and incremental volume change in this figure (middle and bottom rows) shows clearly that both large shear and large isotropic deformations are affecting the shear band. These observations have been confirmed using X-ray CT on axisymmetric triaxial tests (Desrues et al. 1996). It has been possible to measure the evolution of local void ratio in the shear bands, and it was proven by quantitative measurements that the void ratio reached a common

STRAIN LOCALIZATION IN GEOMATERIALS

59

Conclusion for granular bodies Strain localization is likely to arise in most cases in granular bodies as the peak load (maximum load supported by the sample during the test) is approached. The localized zone can be a single shear band, but it can also be much more complex, especially when imposed kinematic constraints restrain its natural development. Shear surfaces can be regarded as material structures with a non-vanishing thickness that depends on the grain size. A common ultimate density is reached inside the shear bands in very large strain situations (at a given mean effective stress). Among the factors which have been explored with respect to shear band occurrence in biaxial tests, the influence of initial compactness, grain size, specimen size and shape, and lateral pressure can be summarized as follows: lower compactness, coarser granulosity, smaller specimen size, less slender shape, and higher pressure will induce a delay in the onset of shear banding, together with a steeper orientation of the bands when they develop. Some of these factors, such as grain size or specimen size, have not been discussed here; a more detailed discussion of the results regarding these factors has been given by Desrues & Viggiani (2004).

Localization in cohesive geomaterials Ductile v. brittle behaviour

Fig. 10. Illustration of the variation of density in specimen rfdt8, a dense sand specimen deformed in a triaxial axisymmetric test performed under tomographic control and reconstructed using X-ray CT. Denser areas are brighter in the reconstructed images. The reconstructed specimen’s density is displayed on a plane parallel to the sample long axis and in a series of slices perpendicular to one pair of localization bands. The lowermost image shows an inverted V produced by the two associated planes intersecting on the top platen; the trace of the cone appears as a parabola oriented toward the lower platen. Adapted from Desrues et al. (1996) with permission.

ultimate value at large deformation magnitudes for both loose and dense specimens tested under the same confining pressure. Figure 12 illustrates these results.

The physical properties of cohesive geomaterials vary widely, an extreme range of rheological behaviours being encountered between soft clays at one end of the spectrum and high-strength very lowporosity igneous rock at the other end. A distinction is usually made between the ‘ductile’ and the ‘brittle’ field. Following Paterson (1978), the definition of brittle fracture is taken to be: ‘a discrete event in which the failure of the rock occurs, without significant prior deformation and without warning, at a particular stress’. Conversely, rupture is taken to be ductile when it occurs after significant plastic deformation of the material. What is observed at rupture in axisymmetric triaxial tests? In clay specimens strain localization is commonly observed but it can take different forms: either a single failure plane (or thin zone), or multiple parallel failure planes (or thin zones) are reported by experimentalists, depending on the uniformity of the specimen’s initial density, the tendency to dilate or to contract, and the boundary conditions (Lade & Jamie 1985). Dilatancy and contractancy are directly linked to the overconsolidation ratio of the clay specimen; that is, the ratio of the present stress value to the

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Slice normal to axis

Barrel shape

End of sample

Slice parallel to axis

Fig. 11. Schematic representation of geometry developed in triaxial axisymmetric test as inferred from tomodensitometry study. Diagram show radial paired features and conical feature. These features intersect the top surface of the specimen as paired radial lines and their intersection on vertical surfaces sliced parallel to the axis is shown on the right-hand side. Adapted from Desrues et al. (1996) with permission.

Fig. 12. Comparison of global and local evolution of the void ratio in loose and dense specimens submitted to axisymmetric triaxial testing under 60 kPa effective confining pressure. Global as used here means averaged over a cross-section of the specimen, whereas local means averaged over a zone identified as a localized shear zone. Figure adapted from Mokni (1992) with permission.

STRAIN LOCALIZATION IN GEOMATERIALS

maximum previously applied stress value as recorded in the fabric of the specimen. Therefore the over-consolidation ratio is expected to be a major indicator of the expected failure type. In high-strength, low-porosity rock specimens, failure is usually described as ‘brittle fracture’. We may distinguish two principal types of brittle fracture: shear fracture, in which a single fracture surface inclined to the principal stress directions is observed; and extension fracture characterized by separation normal to the failure surface, which is generally oriented normal to the minor principal stress (Paterson 1978). Axial splitting is a special case of extension fracture and occurs in axisymmetric triaxial compression tests at low confining pressure. In more porous rock specimens, the variation in failure regimes covers a larger spectrum with respect to the mean effective stress value. The stress values at which the transitions between regimes occur is dependent on the rock type, so we here use ‘low’, ‘middle’ and ‘high’ instead of specific values to discuss this variability. At low confining pressures in triaxial axisymmetric compression tests, axial splitting occurs (Paterson 1978). At medium confining pressures shear bands are observed, associated with dilative volumetric strain. As confining pressure increases, the volumetric strain in the bands gradually changes from dilative to zero, and then contractive, although still associated with significant shearing in the band (Be´suelle 2001). At high confining pressures so-called compaction banding becomes the (dominant) failure mode; that is, localization bands with a pure compaction strain and no shear strain (Baud et al. 2004). Diffuse failure by cataclastic flow (grain crushing) has also been reported at very high pressure (Wong et al. 1997). The differences in the physical make-up between the soft clay end of the spectrum and the highly indurated rock end is considerable: grain mineralogy, grain shape and packing, fabric, nature of the bonds between the grains, porosity, and permeability all vary considerably. However, all these materials show a clear dependence of their failure modes on the value of confining pressure, and in all the materials higher confining pressures produce more ductile and less brittle behaviour. Rocks cannot be said to be ‘brittle’ or ‘ductile’ over all ranges of pressure, temperature and strain rate. The same is true for consolidated clays, an essential difference from rocks being their much lower critical confining pressure. In engineering problems the relevance of ductile or brittle behaviour depends on the problem considered: surface rocks will be brittle in most cases, but in deep mines or, in particular, boreholes the same rocks can behave in a ductile way. Time is

61

also a parameter: on geological time scales, ductile behaviour will typically be dominant at depths below the first few kilometres. However, at these same depths very rapid seismic events are generally seen as brittle.

From diffuse to localized cracking and damage Localization results from a concentration of micro-events. In clay, rotation and decohesion of particles are the dominant micro-mechanisms inside shear bands (Hicher & Wahyudi 1994; Raynaud 1995). In rock, coalescence of microcracks, and other local micro-mechanical events finally lead to macrocracking, with development of more well-defined cracks (Wong 1982; Mene´ndez et al. 1996). Paterson (1978) noted that ‘at higher pressure, the microcracks (now mainly inclined ones) tend to form inclined zones which develop into the macroscopic shear fracture, often by linking up en echelon inclined fractures’. At this stage, the localization process is clearly a displacement (velocity) discontinuity rather than a strain (rate) discontinuity. However, in line with Paterson, one can estimate that ‘at least in the shear failure case, the linking-up of the microcracks is probably generally a subsequent development to the local concentration of microcracking in a band of inelastic deformation’.

Shear band observation in laboratory tests The range of length scales at which deformation processes can be considered is very wide, ranging from large continental structures to submicroscopic cracks. The same sample can be described as having shear band-like structures at one scale of observation whereas observations of this sample at a different length scale indicate a dominant cracking process. In laboratory studies, the specimen size is restricted to the centimetre to decimetre range, but because the grain size v. specimen size ratio still can vary widely the length-scale issue still exists. Shear band observations in laboratory tests have been reported by a number of workers and for a wide range of cohesive materials. These materials include Beaucaire Marl, Vosges Sandstone, a siltite and concrete. The localization phenomena in these materials are addressed in more detail in the next section. Other, significantly different, materials described briefly here are granite, marble and several other sandstones. Ord et al. (1991) performed biaxial tests on Gosford Sandstone (porosity 5%, poorly cemented, 30 MPa uniaxial maximum strength) and showed

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that, in the vicinity of the final failure surface, a region exists in which the block material is somewhat disturbed, showing a ‘white, chalky appearance’. Again quoting from Ord et al. (1991), the discrete fracture ‘does not form instantaneously, but is the result of a gradual alteration of the microstructure’. The microstructural observations of Ord et al. showed that, close to peak stress (stress value at the peak of the stress–strain curve), ‘only minor perturbation of the gross structure of the material’ can be detected, and after complete failure, the global shear zone is made of ‘welldefined sections (having) the same orientation, and separated by more diffuse and shorter shear zones’. Similar observations have been reported by Kondo & Henry (1990) on Gre`s de Fontainebleau, a low-strength poorly cemented sandstone. In contrast, for the same rock, El Bied et al. (2002) showed examples of strong degradation of the grain structure by intense grain crushing concentrated in the shear bands. Wawersik & Fairhurst (1970), who worked with Tennessee Marble, reported that at confining pressures higher than 2000 psi (about 14 MPa), the principal mechanisms are local shear failure, intergranular sliding and intercrystalline glide on well-defined, regularly spaced inclined planes. Tullis & Yund (1977) reported that for a very high-strength, very low-porosity rock, Westerly Granite, it is common to observe residual free particles when cracks coalesce, forming a gouge layer inside the failure plane. Similar observations have been reported by Wong (1982) on the same rock.

Localization in Beaucaire Marl Marls are geomaterials that can be considered as transitional between soils and rocks. Studying them in their own right is a valuable exercise, particularly because they are involved in a many civil, environmental and petroleum engineering problems. However, they can also be seen as a model material for the behaviour of deep crustal rocks. The biaxial apparatus has been used to load a parallelepiped-shaped Beaucaire Marl specimen to failure in plane strain conditions under zero lateral pressure (unconfined). As in the case for the sand specimens discussed above, a zero strain condition is imposed by a set of two vertical rigid glass platens bolted together. A speed regulator allows a vertical displacement rate-controlled loading, with a rate ranging between 2 and 7900 mm min21 to be applied to the specimen through a screw jack. The axial displacement is measured by a potentiometer with a resolution of +2 mm. The force, which is measured by a 20 kN

cell, is transmitted to the rock specimen through a suspended knee joint loading head. The dimensions of all the specimens in this suite are 35 mm in thickness, 120 mm in height and c. 60 mm wide. To ensure that the specimen’s deformation history could be retraced from the photographs taken during the test by the stereophotogrammetric process, a thin layer of metallic paint is finely sprayed over the vertical face being photographed. The presence of the metallic granules in the paint allows references points to be assigned on the surface during the stereophotogrammetric process. Furthermore, the vertical faces of the specimen in contact with the glass platens are lubricated with a silicone grease, and anti-friction devices such as greased Teflon or latex bands are used to avoid the generation of shear traction at the specimen ends. As discussed above, the application of FRS to photographs of a specimen undergoing a plane strain process allows the incremental displacement field all over the specimen to be quantified precisely. Then, the incremental strain field between the different (photographed) stages in the loading process can be computed from the displacement data obtained. This, in turn, allows characterization of the onset of strain localization in granular materials, as shown in previous sections. However, in cohesive materials, ‘strong’ discontinuities (i.e. displacement discontinuities variously known as cracks or fractures or joints) can be the essential phenomenon, at least for certain (late) steps of the test. By using a suitable strategy when gathering data along the crack edges and a special data processing method for measured points that are identified as belonging to a crack edge, the relative displacement jump across the observed cracks can be computed (Desrues 1995a, b). The tangential component of the displacement jump across a crack can also be distinguished from the normal component, which gives useful information on crack propagation. Stereophotogrammetric measurements of the photographs taken during biaxial tests on Beaucaire Marl show that macrocracks with origins at the corners of the specimen propagate through the specimen at an angle of c. 258 to the vertical. Figure 13a shows the stress –strain curve recorded during test MB04 and the calculated displacement jump profiles along the length of the left-hand crack in this Beaucaire Marl sample. Figures 14 and 15 show the corresponding results from FRS. It can be seen that the first signs of strain localization appear between photographs 4 and 5, which were both taken before peak stress. An elongated shear zone, with no significant signs of dilation or contraction, is shown well before any crack initiation appears, at least at the length scale of

STRAIN LOCALIZATION IN GEOMATERIALS

Fig. 13. (a) Axial stress v. vertical strain curve for biaxial test MB04 performed on Beaucaire Marl, which was carried out with lubricated Teflon bands as an anti-friction device to ensure that the conditions at the ends of the sample were well controlled. Numbers on the figure correspond to points at which photographs were taken. (b) Displacement jump profiles calculated along the left crack in test MB04; upper figure shows the crack-parallel jump component and lower figure shows the crack-normal component. Adapted from Tillard (1992) with permission.

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Fig. 14. Incremental displacement d fields (top row) and incremental shear strain 1s fields (bottom row) between the photographs 1 and 2, 2 and 3, 3 and 4, 4 and 5 calculated for test MB04 on Beaucaire Marl. Adapted from Tillard (1992) with permission.

observation. The shear zone can also be found later in the area ahead the advancing crack tip (Fig. 15). The displacement jump across the crack is aligned along the crack long axis; this means that no normal opening of the crack has occurred during the basic process of propagation. Figure 13b displays the profile of the components of the displacement jump along the left crack for the increments C5C7 and C7C8, parallel to the crack (Fig. 13b, top right), and normal to the crack (Fig. 13b, bottom right), respectively. The profiles show clearly that in both increments the normal jump is negligible compared with the tangential jump and the latter decreases progressively from the extremity, where the crack was initiated, to the active tip (vanishing at the tip as expected).

Localization in concrete and siltite A contrasting behaviour with respect to the crack development mode has been observed by Benaija (1992) in concrete using the stereophotogrammetry method (Torrenti et al. 1991, 1994). Figure 16 shows that, during an unconfined uniaxial compression test performed on a parallelepiped-shaped specimen of concrete, localization was detected at

the peak load using stereophotogrammetry. Moreover, the crack jump analysis showed that the detailed localization mechanism was a set of approximately axial cracks, organized in a band through the specimen. The displacement jump was essentially horizontal (i.e. normal to the crack direction) on each of these small cracks. This observation was confirmed in the other tests (Benaija 1992). This pattern of cracking is very like that observed in rocks and other materials by a number of workers (see, e.g. Riedel 1929; Hancock 1964; Petit 1987). Failure of a siltite specimen was observed using the same method on unconfined parallelepipedshaped specimens. The siltite has a uniaxial compressive strength of about 70 MPa, and its mineralogical composition is approximately 1/3 calcium carbonate, 1/3 quartz and 1/3 clay. To polarize the deformation process into the plane perpendicular to the large face, which was being photographed, a cylindrical hole was drilled in the centre of the face (see Fig. 17c, right). Comparison with uniaxial tests performed on the same specimens without a hole showed no effect of the hole either on peak load or on the equivalent strain at peak stress.

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Fig. 15. Stereophotogrammetic analysis of test MB04. Top row shows, from left to right, incremental displacement d and incremental shear strain 1s fields between photographs 5 and 7 (left pair) and 7 and 8 (right pair). Bottom row shows, from left to right, relative displacement along cracks, plotted as normal dn and the tangential dt component of this displacement for the same increments 5– 7 (left) and 7–8 (right). Sizes of symbols are proportional to the value of the scalar quantity being plotted. Adapted from Tillard-Ngan et al. 1993, with permission of Taylor & Francis Group.

Macroscopic cracks were initiated near the hole before the peak stress of the specimen was reached. Figure 17a shows the stress–strain response of the specimen, with numbers indicating the photographs, numbered sequentially. The top (axial stress, axial strain) curve of Figure 17a shows a continuous increase of the stress magnitude to photograph 7, and the lower curve of Figure 17a shows that the values of lateral strains in directions 2 and 3 (direction 1 being axial) remain very close over the same period. From photographs 7 to 11 the axial stress remains approximately constant whereas the strains in directions 2 and 3 diverge abruptly. After photograph 11 the specimen loses its strength and the axial stress drops suddenly. Figure 18 shows the loading steps 1–7, 7–11 and 11– 12, corresponding to the three phases described above. In increment 7– 11, the displacement jump (middle image of bottom row) shows a dominant shear component as well as a normal component. That is, ahead of the crack there is a concentration of shear strain. The siltite specimen fails completely

in increment 11– 12, the cracks crossing the specimen. The right part of the specimen is no longer constrained by the loading platen and elastic unloading can be observed: it should be noted that the length of the displacement vectors at the upper end of the right block in increment 11– 12 (postpeak loading) is approximately equal to the length of the displacement vectors at the bottom of the specimen during the pre-peak loading stage, increment 1–7. This indicates that the majority of the strain accumulated during loading is recovered elastically in the blocks after rupture; that is, here is a large elastic and recoverable, or brittle, element to the deformation of the siltite. This is in marked contrast to the Beaucaire Marl.

Localization in Vosges sandstone Axisymmetric triaxial tests were performed on dry Vosges Sandstone cylindrical specimens (diameter 40 mm, height either 80 or 40 mm), with a confining pressure varying between 0 and 60 MPa

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Fig. 16. Unconfined compression test of a parallelepiped-shaped concrete specimen. (a) Load–displacement curve; (b) displacement jump over the observed cracks during the load increment including the peak load; (c) incremental displacement fields before the peak, at peak, and after the peak. Adapted from Torrenti et al. (1994) with permission of Taylor & Francis Group.

(Be´suelle 1999; Be´suelle et al. 2000). A very smooth, lubricated, enlarged platen was used to reduce friction between the end of the specimen and the platen. In axisymmetric triaxial tests the applied axial stress value can either be higher than the confining pressure, producing a compression

test, or lower, producing an extension test. Both types of test were performed and all specimens reached failure, with development of localized deformation bands in all the specimens (Fig. 19). In the uniaxial compression tests a combination of axial splitting and inclined failure surfaces was

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Fig. 17. Results from an unconfined compression test on a siltite specimen. (a) Axial stress– axial strain curve with photograph numbers plotted. (b) Lateral strain–axial strain curve. (c) Photograph of the deformed specimen showing the central hole.

observed. Figure 20 shows the deviatoric stress and the volumetric strain, each plotted against axial strain in compression tests with a slenderness ratio (H/D) ¼ 2 over the complete range of confining pressure used. Be´suelle et al. (2000) reported that in axisymmetric triaxial compression tests performed under confining pressures between 10 and 30 MPa one or two parallel shear bands developed. In contrast, conjugate shear bands occurred from 40 to 60 MPa, the distance between the bands decreasing with increasing confining pressure, whereas the number of shear bands increased as confining pressure increased. The angle between a band and the axial stress direction (in compression tests this is the direction of the most compressive principal stress) increased with the confining pressure and became greater than p/4 at 50 MPa. The pattern of failure surfaces observed was similar in the extension tests. Between 40 and 60 MPa confining pressure there was one shear band at a small angle to the lateral stress direction (in extension tests this is the direction of the most compressive principal stress). Between 20 and 30 MPa, the failure surface was parallel to the direction of the most compressive stress. However, the localization pattern is sensitive to the test’s end conditions. A subsequent study (Be´suelle et al. 2003) was carried out on a similar Vosges Sandstone specimen tested in compression at a confining pressure of 5 MPa and with lower quality end lubrication using platens with several

small holes. The specimen was loaded and subjected to X-ray analysis and showed a distribution of localization similar to that for sand in Figure 10. Two specimens tested in axisymmetric compression tests (30 and 50 MPa confining pressure) were observed, unloaded, using the X-ray CT apparatus (Be´suelle et al. 2000). This technique provides a spatial view of the bulk density distribution inside the specimen. Figure 19b shows a reconstruction of specimens tested at 30 MPa and at 50 MPa in a plane parallel to the specimen axis. The denser zones within the specimen are brighter in the figure. It should be noted that the samples were not originally perfectly homogeneous, as the upper half of the left image is darker, indicating a less dense pre-test material. The shear bands typically develop in the less dense portions. At 30 MPa confining pressure, the shear band is darker than the surrounding material, which shows that the porosity is greater (and density lower) inside the band than outside it. At 50 MPa, the band has become brighter than the surrounding material, showing that the porosity is lower (and density higher) inside the band than outside it. A dilative volumetric strain is associated with the shear band at 30 MPa and a compactive volumetric strain is associated with the shear band at 50 MPa, showing that, at least for these conditions, the transition from dilation to compaction occurs between 30 and 50 MPa confining pressure. Be´suelle (2001) has discussed, from a theoretical viewpoint, the transition between compacting and dilating shear bands.

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Fig. 18. Unconfined compression test of a siltite specimen; pre-peak (left column) peak (middle column) and post-peak behaviour (right column). Incremental displacement field (top row), incremental shear strain field (middle row) and incremental relative displacement jump on crack (bottom row). Two macrocracks are propagating from the central hole. The displacement jump is, on average, parallel to the crack direction. Elastic unloading of the broken right piece of the specimen is observed in the last increment.

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Fig. 19. (a) Sketches of observed shear band patterns with changing confining pressure for compression tests (top two rows) with two slenderness ratio values H/D ¼ 2 and H/D ¼ 1, respectively, and for extension tests (bottom row). Value of confining pressure (in MPa) is shown next to each sketch. The angle of the shear bands with respect to the major principal stress direction increases with the confining pressure and bands also become more numerous and closer together with increases in the confining pressure. (b) X-ray CT reconstructions of specimens tested at 30 MPa (left) and 50 MPa (right) confining pressure. The dilating shear band is inclined and darker, and the compacting shear band is inclined and brighter than the average grey level of the specimen. Adapted from Be´suelle et al. (2000) with permission of Elsevier.

Observation by optical and electronic microscopy and a measurement of the surface porosity on thin slices of the previous specimens shows a similar result (Be´suelle et al. 2000). However, the internal structure of the localization appeared to be more complex than was suggested by X-ray CT. Microscopy shows a central part where grains were crushed and where the porosity was higher than in the material outside the bands. In addition, crack density was high at the band margins. At 30 MPa the surface porosity in these marginal areas was intermediate between the surface porosity in the central part of the band and that outside the band. At 50 MPa, the surface porosity on band margins was lower than that either outside the band or in the central part of the band. These results suggest that two steps occur during the strain localization process. A first step of localization is associated with a specific volumetric behaviour of the band, which itself depends on the confining pressure. A second step of localization takes place inside the bands and is always associated with a dilative and cataclastic deformation (a gouge layer). This second step probably occurs at larger strains, beyond the peak stress. These stages in the localization process have been observed quite widely in laboratory

experiments aimed at reproducing fault gouge development in natural rock (Mandl et al. 1997; Bartlett et al. 1981; Logan et al. 1992). Very similar features, but at a much wider range of length scales, have been observed in nature (Aydin 1978; Uehara & Shimamoto 2007; Wibberley et al. 2007). This accumulation of work gives confidence that these observations of localization are not restricted to any small subset of geomaterial or deformation conditions. The work reviewed here is focused on granular and less consolidated materials. It also provides very closely controlled experimental conditions, and therefore helps to determine how and why localization develops.

Conclusion Geomaterials may vary widely with respect to their physical and mechanical properties and can include granular or cohesive, ductile or brittle and finegrained or coarse materials. Experimental results have shown that the conditions for application of shear band analysis for prediction of onset of strain localization are met fairly well for granular materials. However, in a large number of cases the failure mode reported experimentally for rocks and other brittle, cohesive, materials consists of

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deviatoric stress (MPa)

(a)

120

80

60 MPa 50 MPa 40 MPa 30 MPa 20 MPa 10 MPa 0.1 MPa

40

axial strain

0 0.000

volumetric strain

(b)

0.004

0.008

0.012

0.007

0.000 axial strain

–0.007 Fig. 20. Deviatoric stress (a) and volumetric strain (b) v. axial strain in compression tests on Vosges Sandstone specimens at different confining pressures. Key to symbols representing the different confining pressures in given in the top left corner. Reprinted from Be´suelle et al. (2000) with permission of Elsevier.

one or more approximately planar failure surfaces. This differs from the behaviour of granular materials in that a granular material’s shear band thickness does not vanish, even at large strains. One situation arising from length-scale problems must also be considered: the ratio of the relevant physical system dimension to the shear band thickness can tend to infinity in many naturally occurring structures such as fault systems, even in granular materials. Large relative displacements can take place between the two sides of these shear bands in such a way that, at the field scale, the situation is no longer very different from that described above for cohesive materials. In addition, some cases of micro-scale observation and other cases of macro-scale observation suggest that, at some stage of its development, localization in rock takes the form of a concentrated band of local deformation. The final failure surface

(or surfaces) seems to be the ultimate outcome of the concentration of micro-damage developing first inside a non-zero thickness zone and then evolving into a macrocrack. In some cases, a gouge layer of granular material (residual particles of crushed grains) is formed inside the failure surfaces. This series of observations provides support for efforts to develop more comprehensive theoretical and numerical tools to model the localization phenomena in the shear band analysis framework.

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International Congress on Soil Mechanics and Foundation Engineering, San Francisco, 549–552. L ANIER , J. 1989. Recent trends in laboratory testing. In: D ARVE , F. (ed.) Constitutive Equations and Modelling. Elsevier, Amsterdam, 7– 26. L OGAN , J. M., D ENGO , C. A., H IGGS , N. G. & W ANG , Z. Z. 1992. Fabrics of experimental fault zones: their development and relationship to mechanical behaviour. In: E VANS , B. & W ONG , T.-F. (eds) Fault Mechanics and Transport Properties of Rock. Academic Press, London, 33–67. M ANDL , G., D E J ONG , L. N. J. & M ALTHA , A. 1997. Shear zones in granular materials—an experimental study of their structure and mechanical genesis. Rock Mechanics, 9, 95–144. M ENE´ NDEZ , B., Z HU , W. L. & W ONG , T. F. 1996. Micromechanics of brittle faulting and cataclastic flow in Berea Sandstone. Journal of Structural Geology, 18, 1– 16. M OKNI , M. 1992. Relations entre de´formations en masse et de´formations localise´es dans les mate´riaux granulaires. PhD thesis, Universite´ Joseph Fourier, Grenoble. M OKNI , M. & D ESRUES , J. 1999. Strain localization measurements in undrained plane-strain biaxial tests on Hostun rf sand. Mechanics of Cohesive-Frictional Materials, 4, 419– 441. O RD , A., V ARDOULAKIS , I. & K AJEWSKI , R. 1991. Shear band formation in Gosford Sandstone. International Journal of Rock Mechanics and Mining Sciences & Geomechanical Abstracts, 28, 397–409. P ATTERSON , M. S. 1978. Experimental Rock Deformation—the Brittle Field. Springer, Berlin. P ETIT , J. P. 1987. Criteria for the sense of movement on fault surfaces in brittle rocks. Journal of Structural Geology, 9, 597–608. R AYNAUD , S. 1995. Fracturation et dissolution sous contrainte des roches. PhD thesis, University of Montpellier. R ICE , J. R. 1976. The localization of plastic deformation. In: K OITER , W. T. (ed.) Theoretical and Applied Mechanics. North-Holland, Amsterdam, 207–220. R IEDEL , W. 1929. Zur mechanik geologischer brucherscheinungen. Zentralblatt fu¨r Mineralogie, Geologie und Palaeontologie, 1929B, 354 –368. R OGER , V. 2000. Etude expe´rimentale et the´oretique de la localization des de´formations dans les mate´riaux granulaires en condition isochore. Universite´ Joseph Fourier, Grenoble. R OGER , V., D ESRUES , J. & V IGGIANI , C. 1998. Experiments on strain localization in dense sand under isochoric conditions. In: O KA , F. (ed.) Localisation and Bifurcation Theory for Soils and Rocks. Balkema, Rotterdam, 239– 248. R UDNICKI , J. W. & R ICE , J. R. 1975. Conditions for localization of deformation in pressure sensitive dilatant materials. Journal of the Mechanics and Physics of Solids, 23, 371– 394. S CARPELLI , G. & M UIR W OOD , D. 1982. Experimental observations of shear band patterns in direct shear tests. In: V ERMEER , P. A. & L UGER , H. J. (eds) Proceedings of the IUTAM Conference on Deformation and Failure of Granular Media, Delft. Balkema, Rotterdam, 473–484. S COTT , R. F. 1987. Failure. Ge´otechnique, 37, 423–466.

STRAIN LOCALIZATION IN GEOMATERIALS T ATSUOKA , F., S AKAMOTO , M., K AWAMURA , T. & F UKISHIMA , S. 1986. Strength and deformation characteristics of sand in plane strain compression at extremely low pressures. Soils and Foundations, 26, 65– 84. T ATSUOKA , F., N AKAMURA , T., H UANG , C. C. & T ANI , K. 1990. Strength anisotropy and shear band direction in plane stress test of sand. Soils and Foundations, 30, 35– 54. T ILLARD , D. 1992. Etude de la rupture dans les ge´omateriaux cohisifs. Application a` la marne de Beaucaire. PhD thesis, Universite´ Joseph Fourier, Grenoble. T ILLARD -N GAN , D., D ESRUES , J., R AYNAUD , S. & M AZEROLLE , F. 1993. Strain localization in Beaucaire marl. In: A NAGNOSTOPOULOS , A. (ed.) Geotechnical Engineering of Hard Soils–Soft Rocks. Balkema, Rotterdam, 1679–1686. T ORRENTI , J. M., D ESRUES , J., B ENAIJA , E. H. & B OULAY , C. 1991. Stereophotogrammetry and localization in concrete under compression. Journal of Engineering Mechanics, 117, 1455–1465. T ORRENTI , J. M., D ESRUES , J., B ENAIJA , E. H. & B OULAY , C. 1994. Applications of stereophotogrammetry on concrete compression test. In: S ILVA G OMES , J. F. (ed.) Recent Advances in Experimental Mechanics. Balkema, Rotterdam, 63–68. T ULLIS , J. & Y UND , R. A. 1977. Experimental deformation of dry Westerly Granite. Journal of Geophysical Research, 82, 5705– 5718. U EHARA , S. & S HIMAMOTO , T. 2007. Permeability of fault rocks from the Median Tectonic Line in Ohshika-mura, Nagano, Japan as studied by pressure-cycling tests. In: L EWIS , H. & C OUPLES , G. D. (eds) The Relationship between Damage and Localization. Geological Society, London, Special Publications, 289, 143– 160. V ARDOULAKIS , I. 1979. Bifurcation analysis of the triaxial test on sand samples. Acta Mechanica, 32, 35– 54. V ARDOULAKIS , I. 1996a. Deformation of water-saturated sand: I. Uniform undrained deformation and shear banding. Ge´otechnique, 46, 441–456.

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V ARDOULAKIS , I. 1996b. Deformation in water-saturated sand: II. Effect of pore water flow and shear banding. Ge´otechnique, 46, 457–472. V ARDOULAKIS , I. & G RAF , B. 1985. Calibration of constitutive models for granular materials using data from biaxial experiments. Ge´otechnique, 46, 299– 317. V ARDOULAKIS , I. & S ULEM , J. 1995. Bifurcation Analysis in Geomechanics. Blackie, Glasgow. V ARDOULAKIS , I., G OLDSCHEIDER , M. & G UDEHUS , Q. G. 1978. Formation of shear bands in sand bodies as a bifurcation problem. International Journal for Numerical and Analytical Methods in Geomechanics, 2, 99–128. V IGGIANI , G., L ENOIR , N., B E´ SUELLE , P. D. M., D ESRUES , J. & K RETZSCHMER , M. 2004. X-ray micro tomography for studying localized deformation in fine-grained geomaterials under triaxial compression. Comptes rendus Me´canique, Acade´mie des Sciences, 332, 819–826. W AWERSIK , W. R. & F AIRHURST , C. 1970. A study of brittle failure in laboratory compression experiments. International Journal of Rock Mechanics and Mining Sciences, 7, 561–575. W IBBERLEY , C. A. J., P ETIT , J.-P. & R IVES , T. 2007. The mechanics of fault distribution and localization in high-porosity sands, Provence, France. In: L EWIS , H. & C OUPLES , G. D. (eds) The Relationship between Damage and Localization. Geological Society, London, Special Publications, 289, 19– 46. W ONG , T.-F. 1982. Micromechanics of faulting in Westerly Granite. International Journal of Rock Mechanics and Mining Sciences & Geomechanics Abstracts, 19, 49– 64. W ONG , T.-F., D AVID , C. & Z HU , W. L. 1997. The transition from brittle faulting to cataclastic flow in porous sandstones: mechanical deformation. Journal of Geophysical Research, 102, 3009–3025. Y OSHIDA , T., T ATSUOKA , F., S IDDIQUE , M. S. A. & K AMEGAL , Y. 1994. Shear banding in sand observed in plane strain compression. In: V ARDOULAKIS , I., C HAMBON , R. & D ESRUES , J. (eds) Localisation and Bifurcation Theory for Soils and Rocks. Balkema, Rotterdam, 165– 179.

The progression from damage to localization of displacement observed in laboratory testing of porous rocks J. M. LOGAN1,2,3 1

Department of Geological Sciences, University of Oregon, Eugene, OR 97403, USA

2

Department of Geosciences, Texas A&M University, College Station, TX 77843, USA 3

Present address: PO Box 1776, Bandon, OR 97411 (e-mail: [email protected]) Abstract: Laboratory experiments suggest that fault zones form in porous rocks through the extension and coalescence of fractures of predictable geometries. These fractures form in an array of Riedel fractures in R1, R2, P and Y orientations. Displacement along closely spaced fractures leads to the formation of comminuted fault gouge. Localization of displacement within the fault gouge progresses from distributed shearing to comminution and compaction of the fault rock material culminating in fractures in the Riedel orientations. Colour boundaries within the simulated gouge zones show the change in accommodation of displacement to fractures in the Y orientation as shear strain progresses. Clay– quartz mixtures demonstrate that the clay inhibits localization and the achievement of steady-state sliding as well as stick-slip. A reduction in the coefficient of friction does not occur until about 30% of the clay is present and continues to decrease until about 70% is reached. Localization of slip appears as a necessary condition for steady-state sliding as well as unstable behaviour. Field studies show the implication of grainsize reduction in the localization process by porosity decrease inhibiting fluid flow normal to the fault zone. The pervasive Y fractures, however, facilitate fluid migration parallel to some faults.

The progression from damage associated with fracture and fault formation to the localization of slip within fault zones, and associated changes in lithological properties and mechanical behaviour, have been the subject of laboratory investigations for over 30 years. Much of the work has been associated with triaxial testing of porous rocks, particularly sedimentary lithologies, under confining and pore pressures up to 100 MPa and temperatures to 1000 8C. Common lithologies have included sandstone and limestone, although material from natural fault zones and mixtures of quartz and phyllosilicates have also been used. Simulated fault zones may be created by distributing quartz or other disaggregated material between two saw-cut surfaces and deforming them in triaxial compression. Sequential tests at various displacements, shear strains, confining pressures, and temperatures have been accompanied by observational studies of the deformed specimens. Although triaxial tests have an inherent displacement limitation, pre-conditioning of the simulated fault rock material has helped to overcome this problem (Shimamoto 1985; Chester & Logan 1990). Torsion experiments have also attempted to overcome this limitation and have extended our understanding of the localization process (Tullis & Weeks 1986; Blanpied & Tullis 1998).

Fracture geometry and shear zones Much of the discussion concerning fault localization revolves around the geometry of fractures associated with faults and especially within the fault gouge, or fault rock. Field studies have focused primarily on fracture arrays associated with strike-slip faults, such as the early work by Tchalenko (1970). Analogue model studies using clay were preformed at least as long ago as the seminal work by Riedel (1929) in which Riedel shears were defined, whereas those involving shearing of unconsolidated sand have documented other elements of the fracture array (Fig. 1; Mandl et al. 1977; Naylor et al. 1986). Laboratory studies in triaxial apparatus of the behaviour of simulated fault gouge recognized for the first time a similar array of fractures to those found in field and model studies in experiments at confining pressures of 50 MPa and higher (Logan et al. 1981). Subsequent laboratory tests have enhanced our understanding of the localization processes.

The progression from an array of fractures to a coalesced zone of shear Do fractures form as an array of discrete features that subsequently coalesce into a zone along

From: LEWIS , H. & COUPLES , G. D. (eds) The Relationship between Damage and Localization. Geological Society, London, Special Publications, 289, 75– 87. DOI: 10.1144/SP289.5 0305-8719/07/$15.00 # The Geological Society of London 2007.

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Fig. 1. Schematic drawing of the orientations of fractures formed in association with fault shear zones. R1 and R2 fractures are commonly termed Riedel fractures after Riedel (1929). Other fracture orientations found in the array are X, P and Y, as shown, with the documented displacements along each. The overall sense of shear is shown by the bold arrows in the country rock bounding the shear zone. The fractures occur not simply as single ones, but as parallel features in a given orientation spanning the width and length of a shear zone.

which displacement is localized, or does a single fracture propagate, and with displacement, produce a damage halo around it? This question is still a major topic of laboratory and field

investigation, with both possibilities having supporting evidence. Some insight into the process is offered by analogue rock models undergoing strike-slip displacement along a saw cut in a forcing block of sandstone which was bounded by layers of Indiana limestone (Bartlett et al. 1981). The prismatic specimens were deformed dry in triaxial compression at room temperature and confining pressures to 100 MPa, with the saw cut at 458 to the loading axis. The change from distributed fractures to localization is associated with increasing shear displacement as shown in deformation maps constructed from thin sections. The thin sections show that the R and P shears develop concurrently, are initially short, and are not connected within the thin section (Fig. 2). Increasing displacement results in the R1 shears propagating along curved paths asymptotic to the Y shear orientation and the shear width, and average linear fracture density tending to increase. The increasing fracture density produces a through-going interconnected, gouge-filled fault zone, which develops as R shears are lengthened, new R shears are created, and R and P shears are connected.

Fig. 2. Right-lateral shear zone map of 0.5 cm Indiana limestone veneers, deformed at 50 MPa confining pressure in triaxial compression. Specimens were deformed to 0.14 cm, 0.26 cm, and 0.54 cm in right-lateral shear displacements. Continuous lines indicate fractures along which there is separation and dashed lines show traces of gouge. Black areas indicate material lost as a result of extreme comminution of the limestone. Arrows show sense of displacement. Typical orientations of R1, P and Y fractures are shown. (Modified from Bartlett et al. 1981.)

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Continued displacement appears to produce comminution of the rock within the through-going fault zone as the overall displacement is localized there. A halo of the original fractures forms a damage zone along which minor displacements may continue, but they remain a relic of the early stages of localization. Studies at increasing confining pressures show similar features, except that less displacement is necessary at the higher pressures for the zones to develop to equivalent stages.

Fracture sequence of within a non-clay fault zone Field studies have long demonstrated that faults are generally composed of relatively intact ‘country rock’ separated by a zone of fault gouge or fault rock. The thickness, composition and character of the latter may vary along both strike and dip, but the fault displacement is localized within this zone. A primary question has been whether the displacement is generally homogeneous within the zone or whether it is localized along one or more discrete horizons. Petrographic observations from triaxial experiments performed under 40 MPa confining pressure, at room temperature and a constant displacement rate of 1024 cm s21 but with increasing displacements provide some insight (Fig. 3). Simulated fault gouge of calcite was distributed along a saw-cut to a thickness of about 1 mm. The saw-cut was oriented at 358 to the maximum compressive stress. Homogeneous shearing was initially observed and no fractures are found until just before the

Fig. 3. Differential stress v. displacement curve averaged from five direct shear triaxial experiments. Prismatic specimens contained a saw-cut along which calcite gouge about 1 mm thick was distributed. The experiments were performed room dry, at 23 8C, a confining pressure of 40 MPa and constant displacement rate of 1024 cm s21. The shaded portion of the curve shows the spread of the data in that region. Stages of the evolution of the fracture array are indicated. (Modified from Logan et al. 1992).

Fig. 4. Photomicrograph of simulated fault gouge of quartz about 1 mm thick deformed in left-lateral shear and a confining pressure of 100 MPa at 100 8C in triaxial compression. The continuous R1 fractures that curve to become asymptotic to the Y fractures close to the interface with the sandstone forcing blocks should be noted. The R1 fractures also cut early formed R2 fractures (Higgs 1981).

ultimate strength where Riedel fractures are observed, but without any observable shear displacement. At about 4 mm of displacement, the grain size of the calcite has been markedly reduced and R1 fractures extend almost completely across the gouge layer. By about 7 mm, R1 fractures change orientation at the edges of the zone to parallel the boundaries, and thus take on a Y orientation (Fig. 4). P fractures are found at this stage also. At 17 mm displacement, Y fractures are the dominant fabric element. The stress– strain curve approaches steady-state behaviour with further displacement and no new fracture orientations develop. It at this point, however, that the sliding becomes unstable with cyclic stick-slip stress drops. Y fractures tend to be concentrated at the country rock interfaces, although a few are found in the central region of the gouge. Similar experiments were carried out to evaluate the influence of absolute displacement v. shear strain. These involved simulated gouge layers of differing thicknesses displaced to increments of displacement. They show that it is the magnitude of the shear strain, not the displacement, that correlates with the fabric development. Thicker gouge layers develop similar fabrics to thinner ones but require larger displacements to attain these conditions.

Partitioning of displacement and mechanical behaviour The preceding description has focused on the geometry of the fracture array and sequence of development accompanying the change from damage to localization along a fault zone. The question

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Fig. 7. Photomicrograph of simulated fault zone deformed in right-lateral shear showing boundary between white and black calcite. It should be noted that the boundary was initially deformed by homogeneous shearing and subsequently offset by R1 shear fractures. The gouge layer is about 1 mm thick.

Fig. 5. Photograph of cylindrical triaxial specimen, sliced to show the simulated gouge layer with white and black calcite. The interface between the two colours provides an internal marker of the displacement within the gouge layer. The gouge layer is oriented at 358 to the potential loading axis. The specimen is about 5 cm in diameter.

remains as to how the overall displacement of a shear zone is accommodated within the fracture array. Investigation into this question was difficult

Fig. 6. Differential stress v. shear displacement curves for triaxial experiments performed at confining pressures of 30 and 35 MPa. The curves are averages of two tests carried out at each confining pressure. The test specimens contained a layer of simulated gouge composed of black and white calcite as shown in Figure 4.

as internal markers within even simulated gouge layers were not available. However, experiments with normal white calcite and calcite with a black coating, although with similar mechanical properties, allowed the use of colour boundaries within a simulated fault zone to map the progressive changes in displacement accommodation. Simulated gouge layers using alternating sections of black and white calcite with their respective interface normal to the overall displacement were placed along saw-cuts oriented at 358 to the load axis (Fig. 5). The specimens were deformed room dry, at a temperature of 25 8C, a constant displacement of 1024 cm s21, and confining pressures to 200 MPa. After precompaction for about 5 h at 150 MPa confining pressure, the specimens had a density of about 1.57 g cm23 and porosity of about 42%. The specimens were then sheared at a confining pressure of 30 and 35 MPa (Fig. 6). The onset of stick-slip behaviour set the maximum displacement investigated. Thin sections of each specimen were then evaluated (Fig. 7). The Riedel array developed as expected, but as the fractures offset the colour interface, the separation could be used to document the relative displacement on each fracture and the contribution of each fracture type to the overall displacement (Fig. 8). Measurements were possible along R1, P, and Y fractures, but the displacements along R2 fractures were too small to measure. Additionally, the Riedel fractures bounded regions of homogeneous shearing on a much finer scale, so that this contribution to the displacement could also be determined (Fig. 8). By performing a sequence of experiments that were stopped at increasing overall displacements, changes in the contribution of the various fracture types and that of homogeneous shearing could be evaluated.

24.2 25.4 1.95 1.97 34.7 38.2 29.1 26.7

*Density change from precompacted value of 1.57 g cm23.

36.2 35.1 35 30 3029 3036

14.5 15.0

30 3069

3.6

42.6

49.4

8.0

1.93

22.9

During strain-hardening; before ultimate strength Strain-softening, after ultimate strength Steady-state sliding Steady-state sliding at onset of stick-slip 17.1 1.84 0.0 92.0 8.0 2.3 30

Density (g cm23) Displacement along Y fractures (% of total) Homogeneous shearing (% of total) Displacement along R1 shears (% of total) Total displacement (mm) Confining pressure (MPa) Experiment number

Table 1. Displacement Partitioning as a Function of Shear Strain at 30 – 35 MPa Confining Pressures

Table 1 summarizes the observations. It should be noted that at the lowest total displacement of 2.3 mm, homogeneous shearing of the calcite layer accommodates almost all of the displacement with only minor amounts of displacement along R1 fractures. As the total displacement increases, displacement along the R1 shears increases to about equal to that contributed by homogeneous shearing, with minor accommodation along Y fractures. The contribution of displacement along Y fractures steadily increases as the overall displacement of the simulated fault increases, so that at the onset of stick-slip the y fractures are accommodating relatively more of the displacement than the other contributing elements. Measurements of densities of the simulated gouge after each experiment were then made (Fig. 9) and compared with those at the start of the experiment (Table 1). As shearing of the gouge material progresses, the density increases with increasing displacement. This presumably is due to some rigid-body rotation of grains and comminution of some grains along the fractures. The other noticeable feature is that cyclic stress drops that characterize stick-slip do not occur until a significant amount of the overall displacement is accommodated along Y fractures. That is, although shear displacement does occur along R1 fractures, cyclic stress drops are not observed. The preceding summary of experimental results is schematically shown in Figure 10. It appears that homogeneous shearing accommodates the very earliest displacement of the zone, but remains relatively passive once the major R1, P, and Y fractures are formed. The relative displacement along these fractures increases until most is along Y fractures, which appears to coincide with some critical level of density change in the simulated gouge material. This also marks the onset of unstable, stick-slip sliding.

Density change* (%)

Fig. 8. Sketch of specimen with a simulated gouge zone of black and white calcite deformed at 35 MPa confining pressure and a total displacement of 14.5 mm. Typical orientations of R1, P and Y fractures are marked. The displacements of the black– white boundary are projected onto a line paralleling the contact with the forcing block, where the displacements for each fracture type and homogeneous shearing are then summed, with the results tabulated as shown in Table 1. (Modified from Logan et al. 1992).

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3121

Sliding mode when displacement stopped (see Fig. 5)

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Fig. 9. Density v. shear displacement of a typical specimen deformed at 30 MPa confining pressure. The increase in density accompanies dominantly homogeneous shearing of the simulated gouge, but remains relatively constant as localization of the displacement takes place.

Localization in polymineralic fault zones The above discussion has reviewed the observations on laboratory studies involving monomineralic simulated gouge material. However, commonly

Fig. 10. Schematic sketch illustrating the progression from distributed shearing in (a), as shown by the dark arrows, to localization of the displacement on Y fractures in (d) and (e). The principal fractures, R1, R2, P and Y are labelled, with the boundaries of the fault gouge delineated by the dotted lines. The fractures are shown with observed relative displacements along them. The evolution progresses with increasing shear strain from (a) to (e) with the principal elements shown at each stage. Numerous fractures in each orientation are common.

fault rock materials are polymineralic, and in particular those containing significant amounts of phyllosilicates or serpentine minerals have received attention addressing issues of potentially high pore pressures, low heat flow and possible low friction along fault zones (Sibson 1977; Wang & Mao 1979; Lachenbruch & Sass 1980; Dengo & Logan 1981; Morrow et al. 1992; Moore et al. 1997). Samples recovered from the US Geological Survey (USGS) Dry Lake Valley No. 1 Well, drilled into the San Andreas Fault, were deformed in triaxial compression. The samples contained about 30% quartz, 15% feldspar, with about 16% chlorite and 33% montmorillonite. The samples were disaggregated by ball milling and crushed material was distributed along a saw-cut surface to a thickness of 0.4 mm. They were then deformed at either 258 or 300 8C, room dry, and a confining pressure of 150 MPa. Although similar fabric arrays were found in all experiments to those previously described, there were notable differences, especially at 300 8C. As shown in Figure 11, Y fractures were not confined to the interface with the forcing blocks but developed throughout the shear zone offsetting the R fractures. Apparently, displacement shifted from one Y fracture to another to achieve the lowest frictional resistance. In addition, there appear to be Riedel and P fractures that have developed as a result of displacement between two adjacent Y fractures. They are confined between the Y fractures, producing a number of generations of Riedel fractures referred to as ‘Riedels within Riedels’.

Fig. 11. Photomicrograph of simulated gouge formed by disaggregating core material from the US Geological Survey Dry Lake Valley No. 1 Well drilled into the San Andreas Fault. The specimen was deformed in triaxial compression room dry, at a confining pressure of 150 MPa and a temperature of 300 8C. The simulated gouge is about 1 mm thick and was deformed in right-lateral shear. The pervasive development of the Y fractures, which cut earlier formed R1 shears, should be noted.

LOCALIZATION OF DISPLACEMENT

Fig. 12. Photograph of a similar experiment to that shown in Figure 10, displaying the pervasive elongation of quartz grains by very fine microfracturing. The specimen was deformed in right-lateral shear with elongated grains in the P orientation and along Y fractures.

Additionally, elongated stringers of quartz and feldspar developed in the P orientation are abundant (Fig. 12). These features are not found in experiments on monomineralic quartz. They are produced by extremely fine cataclastic deformation of the grains, which is hypothesized to be enhanced by fluids driven out of the phyllosilicates by the elevated temperatures. Studies on mixtures of quartz and montmorillonite under wet conditions were undertaken to look at their frictional properties (Logan & Rauenzahn 1987). Of significance here is the influence of the clay on localization of displacement. Triaxial friction experiments were carried out at confining pressures to 50 MPa and involved precompacting the specimens at 70 MPa to enhance localization. Velocity stepping sequences were run where sliding velocity was maintained for a pre-set period of time and then either increased or decreased a given amount. The sequences lasted for a total of 0.25, 5 and 40 h, respectively, under load. A series of velocities was predetermined along both increasing and decreasing paths, but always returning to previous velocities to determine if steady-state sliding had been reached. This was inferred when the same mechanical behaviour was recorded as in the previous exposure to that velocity. The gouge material was used containing increasing amounts of montmorillonite of 5, 15, 25, 50, 75 and 100% with corresponding decreases in the amount of quartz. The quartz grains had diameters ranging from 0.5 to 1.0 mm, with montmorillonite generally less than 0.01 mm in diameter. The gouge material was distributed along saw-cuts oriented at 358 to the loading axis. Two sets of experiments were run with simulated gouge layers

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Fig. 13. Photomicrograph of gouge zone of 100% quartz with an initial grain size of 0.5–1.0 mm. The specimen was preconditioned at 70 MPa and then deformed at 50 MPa confining pressure. The specimen was deformed in right-lateral shear and shows a well-developed R1 fracture. The gouge is about 0.85 mm thick.

of a thickness of either 0.3 or 1 mm. The specimens were saturated for 20 h in distilled water to ensure uniform moisture distribution but were vented to the atmosphere during the experiments. In experiments with 100% quartz and at all three velocity sequences, the specimens reached a steady-state sliding and after some stable sliding, cyclic stick-slip sliding occurred. The coefficient of friction in these experiments ranged from 0.49 to 0.51. The resulting fabric is similar to that found in previous studies (Fig. 13). R1 and P fractures are present with grains reduced in diameter along them. Displacement eventually localizes along one or more Y surfaces, leading to stick-slip sliding. At the other end of the compositional spectrum, 100% montmorillonite also achieved steady-state sliding but only at the intermediate and longer periods of displacement cycles (Fig. 14). In all cases, however, only stable sliding was observed. The coefficient of friction ranges from 0.08 to 0.14. The resulting fabric shows no evidence of grain-size reduction or fracture arrays (Fig. 15). The clay platelets do, however, appear to have rotated to form alignments in the R1, P, and Y orientations. These form continuous bands within the material, although there is no optical evidence that these are fractures. The R1 and P features are often long and frequently appear to curve to parallel the boundaries of the zone at the boundaries. The platelets appear to also rotate to Y orientations first at the edges of the simulated gouge, forming boundary zones there, with the zones appearing to migrate inward towards the centre. Impurities within the shear zone also rotate to parallel the

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Fig. 14. Typical coefficient of friction v. driving displacement rate curves for simulated gouges of different compositions. The upper curves are for gouges of 95% quartz and 5% montmorillonite, the lower for 100% montmorillonite and the central ones for compositions of 50–85% quartz. Curves for short, medium, and long displacement cycles are shown with the displacement paths marked by arrows. Experiments with 100% montmorillonite and the long displacement cycle with 95% quartz achieved steady-state sliding whereas the other compositions and cycles failed to do so.

clay platelets. There was no evidence of either grain-size reduction or recrystallization of the clays. Additions of montmorillonite to the quartz produce some significant changes in both mechanical behaviour and deformation within the simulated gouges. The addition of 5% montmorillonite to the quartz stabilized the sliding mode, although there is no recognizable difference in the resulting fabric. The coefficient of friction was within the

Fig. 15. Photomicrograph showing a portion of the simulated gouge zone composed of 100% montmorillonite. The specimen has undergone left-lateral shear and is about 0.85 mm thick. It should be noted that the clay platelets have rotated to R1, P and Y orientations. Larger dark fragments are impurities.

Fig. 16. Coefficient of friction v. per cent quartz for experiments with montmorillonite– quartz simulated gouge.

same range as for experiments with 100% quartz although the average was slightly higher (Fig. 16). The coefficient of friction remains generally unchanged from either 100% quartz or 100% montmorillonite, if either composes 70 –75% of the total composition (Fig. 16). Mixtures containing between 85% and 35% quartz did not reach steady-state sliding as they continue to strain-harden with displacement (Fig. 14), nor did they show any isolated stress drops or cyclic stick-slip. Between these end-members the value decreases monotonically as more clay is added to the material. Of interest are the resulting fabrics and their implications for localization of the overall displacement. Clay platelets generally surround the quartz grains and thus distribute the stress concentrations as they accommodate the accumulating displacement and minimize microfracturing (Fig. 17). Eventually, the quartz grains rotate to the P orientation, bringing them in greater contact and resulting in more microfracturing in the R and Y directions, although there are generally very few throughgoing fractures. Grain-size reduction of the quartz, although present, is much less than in the absence of the clay, with many grains remaining about the initial size. Overall slip does eventually localize along Y fractures but the specimens still show displacement-hardening. There are clearly fewer fractures in the Y orientation than in the case of 100% quartz and they form close to the interface with the forcing blocks. Thus, the presence of the clays stabilized the sliding mode from stick-slip behaviour, but more significantly, it delayed the localization process as shearing was initially distributed throughout the shear zone. Other mixtures where there is wide disparity in mechanical strength would be expected to behave in a similar fashion. This does not argue that localization does not eventually occur, but rather requires larger overall

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Fig. 17. Photomicrographs of simulated gouge composed of (a) 75% quartz and 25% montmorillonite and (b) 50% of each. Both specimens have undergone right–lateral shear under similar conditions to those for Figure 12. The lack of microfractures in the quartz grains and absence of through-going fractures of any orientation should be noted. The deformation is distributed without significant evidence of localization.

displacement under the same boundary conditions to reach a critical shear strain.

The mechanics of localization Laboratory experiments on rocks and simulated fault gouge, analogue models and finite-element studies have contributed to a synergistic view of the possible mechanical and kinematic factors that influence localization of displacement within the fault rock. Once a zone of generally continuous fractures forms so that shear is roughly confined to it, fracturing and grain-size reduction occur in a generally homogeneous fashion within it to accommodate the displacement. This is generally associated with a decrease in bulk density of the material. Evidence from ring-shear analogue models on granular materials (Mandl et al. 1977) and finite-element studies (Logan et al. 1992), suggest that as the stress level increases the principal stresses rotate within the simulated gouge zone. At some point, fractures will form in one or both of the R1 and R2 orientations in response to Mohr – Coulomb failure. The formation of the fractures is

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also facilitated as the energy necessary to continue to fracture individual grains throughout the zones increases exponentially to the point where localization requires less energy (Prasher 1987). The fractures initially form at high angles to the shear zone boundaries but will rotate as the stress field continues to do so, reaching a limit at about 458 to the boundaries. In the laboratory experiments, rotation of R1 fractures to become parallel to the interface supports this argument, as well as apparent rotation of early formed fractures to an R2 position. This rotation of orientation is clearly a function of increasing shear strain within the zone. Displacement accommodation along the R fractures appears limited because of the relatively rigid boundaries of the shear zone. This kinematic constraint leads to the formation of P and Y fractures. Because the P fractures propagate to intersect the simulated fault zone boundaries, they also are limited in the amount of displacement they can accommodate. Eventually, the majority of the displacement is localized along fractures in the Y orientation. Additional lengthening of R1 and P fractures, however, may continue to occur, which is in agreement with observations made in rotary shear experiments (Weeks & Tullis 1985; Tullis & Weeks 1986; Scruggs & Tullis 1998). The displacement, although localized, does not necessarily appear to be confined to one Y fracture, as shown from the experiments on samples from the USGS well (Fig. 11). Multiple sites of displacement may occur so that it is possible that heterogeneous states of stress continue to exist within the fault rock. Some domains between Y fractures may still be strain-hardening while others may be strainsoftening. The result may be development of Riedel fractures between adjacent Y fractures. Experiments similar to those described here demonstrate that whether the boundaries of the fault rock are a saw-cut or an irregular fracture the same localization process appears to occur. In these cases the Y fractures form to bridge the irregular fracture interface, forming planar features (Logan et al. 1992). The initial porosity of the zone, the width and overall displacement that determine the shear strain, and the mechanical properties of the deformed material all influence the speed and extent to which localization develops. Thus, thicker gouge material requires larger displacement for localization to occur. As triaxial tests and even direct shear experiments have limited displacement capabilities, care should be taken in comparing their fabrics with those of thinner fault rock zones. Although similar localization fabric elements have been observed in a wide range of lithologies from pure quartz to calcite (Friedman & Higgs 1981; Wang 1989), mixtures involving clays (Logan & Rauenzahn 1987; Moore et al. 1989),

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conditions leading to the localization of displacement. The early stages of localization found in the analogue experiments (Fig. 2) are seen in a quarry exposure where R1 fractures are seen to curve into Y orientation at the edge of the fault zone (Fig. 18). The common fault geometry inferred is a zone of fault rock or gouge surrounded by a halo of fractured rock that forms a damage zone. This is in turn surrounded by relatively undeformed country rock (Fig. 19). The laboratory experiments on localization as described indicate that the grain-size decrease accompanying the localization of displacement reduces porosity and permeability normal to the zone. The presence of fractures especially in the Y orientation may increase permeability parallel to the fault rock boundaries. Samples studied from the Oil Creek Sandstone in Oklahoma support such an interpretation where porosity measurements perpendicular to the fault zone show higher values on either side of the zone than in the gouge material (Fig. 20). The

Fig. 18. Photograph of a shear zone developed in Carrara marble in a quarry in Italy. The deformation is still distributed along R1 fractures that in places have curved to Y orientation. The zone is about 1.5 m wide in the lower portion.

granite (Blanpied & Tullis 1998) and to halite (Shimamoto 1986; Shimamoto & Logan 1986; Chester & Logan 1990), the amount of strain varies with the simulated gouge composition. Generally, weaker materials such as halite and calcite achieve steady-state fabrics with less shear strain than the stronger materials. The lower mechanical strength leads to increased fracturing and faster compaction, which promotes the localization process and changes in mechanical response.

Relevance of laboratory observations Although a number of experiments have resulted in some understanding of the processes that lead to localization of slip in zones of distributed shear under laboratory conditions, their relevance to natural conditions depends upon field observations. Field studies (Engelder 1974; Dengo 1982; Rutter et al. 1986; Chester & Logan 1987, 1989; Sammis et al. 1987; Chester et al. 1993; Chester & Chester 1998; Solum et al. 2003) and examination of faults exposed in mines (Wallace & Morris 1986) have developed a strong correlation between experimental observations and natural

Fig. 19. Schematic diagram of geometry, showing the fault rock or gouge within a halo of fractured material surrounded by relatively undeformed country rock. The nominal fault width is shown. The fracture halo is not necessarily symmetrical about the fault gouge, as is shown.

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Fig. 20. A profile of porosity measurements made across a fault zone in the Oil Creek Sandstone in Oklahoma. The porosity drops significantly within the gouge zones, but is slightly higher in the fractured zone between the two gouge layers.

fault shows two gouge zones separated by a highly fractured, but not intensely deformed region. The gouge zones show a greater development of Riedel, P and Y shears, which have resulted in porosity reductions within them. The intervening fracture zone shows porosity higher than that of the

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Fig. 22. Photograph of a fault zone from the same area as Figure 20; here, although the fluid has moved up along the fault from depth, fracturing in the hanging wall has permitted fluids to migrate about 2 m away from the fault. In contrast, permeability barriers on the footwall side of the fault zone have restricted fluid migration there.

gouge material, but lower than that of the general fracture halo surrounding the fault. The enhancement of fluid flow within a fault is shown by some exposures in the Lilstock area of the Somerset coast, UK. The presence of Y fractures has enhanced flow parallel to a fault, and comminution of the fault gouge material has restricted the fluid to the fault zone (Fig. 21). Fluid inclusion studies show temperatures of homogenization of about 100 8C, indicating that the fluids came from some depth and not from the surrounding country rock. In the same region, another fault clearly shows the enhancement of fluid flow within the fault (Fig. 22), but fracturing within the hanging wall has facilitated fluid migration outward from the fault. In contrast, the comminution of very fine grains along the footwall has formed a barrier to fluid migration there. Petrographic and fluid inclusion studies show three clear episodes of fluid migration. Fluids migrated from depth upward along Y fractures, precipitating calcite that sealed the fault. Subsequent movement on the fault produced Riedel fractures and eventual migration of a second cycle of fluids. This was followed by sealing and a third cycle of fracture, fluid flow and healing.

Conclusions

Fig. 21. Photograph of a fault zone in the area of Lilstock on the Somerset coast, UK. The fault cuts Jurassic limestones and shales and was mineralized by carbonate-rich fluids. The restriction of mineralization to the fault zone, should be noted.

The laboratory experiments suggest a localization process that is similar both at the initiation of a fault and once fault rock or gouge is formed. When the latter is sufficiently continuous major displacement is localized there. In both cases individual fractures extend and rotate, with those parallel to the overall trace of the fault accommodating

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most of the displacement. In the case of fault gouge, however, homogeneous shearing precedes the localization. Experiments on gouge of different colours clearly document this progression in monomineralic material. Experiments on quartz mixed with montmorillonite show that as little as 5% clay when mixed with quartz stabilizes the sliding mode, but the coefficient of friction is essentially unchanged until about 30% clay is added. Strain is localized within the clay and inhibits the localization process so that steady-state sliding was not attained. The coefficient of friction decreases monotonically until the percentage of clay reaches about 70%. Steady-state sliding is not achieved in both longand short-term experiments and the coefficient does not decrease as the percentage of clay is increased to 100%. Field studies show very similar features to the laboratory studies. Additionally, they demonstrate that the grain comminution that accompanies the localization reduces porosity and creates permeability barriers to fluid flow normal to the fault zones. In contrast, the Y fractures that accompany the localization enhance fluid migration parallel to the fault boundaries. Multiple cycles of fracturing, fluid migration and healing have been documented. The author wishes to acknowledge the research contributions of W. Bartlett, N. Higgs, C. Dengo, K. Rauenzahn, Z. Z. Wang, and C. Decker. Experimental assistance of J. Magouirk was invaluable. Thoughtful reviews by R. Gabrielsen, T. Murray, G. Couples and H. Lewis significantly improved the paper. The work has been supported in part by USGS grants 14-08-001-19293, 14-08-001-21181, 14-08-001-G-820, 14-08-001-G-981, and 14-08-001-G-1509, and NSF grants EAR-8200726 and EAR-8513651.

References B ARTLETT , W. L., F RIEDMAN , M. & L OGAN , J. M. 1981. Experimental folding and faulting of rocks under confining pressure; Part IX. Wrench faults in limestones. Tectonophysics, 79, 255–277. B LANPIED , M. L. & T ULLIS , T. E. 1998. Effect of shear heating on frictional properties of granite. Journal of Geophysical Research, 103, 489– 512. C HESTER , F. M. & C HESTER , J. S. 1998. Ultracataclastic structure and friction processes of the Punchbowl fault, San Andreas system, California. Tectonophysics, 295, 199– 221. C HESTER , F. M. & L OGAN , J. M. 1987. Composite planar fabric of gouge from the Punchbowl Fault, California. Journal of Structural Geology, 9, 621–634. C HESTER , F. M. & L OGAN , J. M. 1989. Implications for mechanical properties of brittle faults from observations of the Punchbowl Fault Zone, California. Pure and Applied Geophysics, 24, 89– 106.

C HESTER , F. M. & L OGAN , J. M. 1990. Frictional faulting in polycrystalline halite: correlation of microstructure, mechanisms of slip and constitutive behavior. In: DUBA , A. G., DURHAN , W. B., HANDIN , J. W. & WANG , H. F. (eds) The Brittle-Ductile Transition in Rocks. Geophysical Monograph, American Geophysical Union, 56, 49–56. C HESTER , F. M., E VANS , J. P. & B IEGEL , R. L. 1993. Internal structure and weakening mechanisms of the San Andreas fault. Journal of Geophysical Research, 98, 771–786. D ENGO , C. A. 1982. Structural analysis of the Polochic Fault Zone in Western Guatemala, Central America. PhD thesis, Texas A&M University, College Station. D ENGO , C. A. & L OGAN , J. M. 1981. Implications of the mechanical behaviour of serpentinite to seismogenic faulting. Journal of Geophysical Research, 86, 10771–10782. E NGELDER , J. T. 1974. Cataclasis and the generation of fault gouge. Geological Society of America Bulletin, 85, 1515– 1522. F RIEDMAN , M. & H IGGS , N. G. 1981. Calcite fabrics in experimental shear zones. In: CARTER , N. L., FRIEDMAN , M., LOGAN , J. M. & STEARNS , D. W. (eds) Mechanical Behavior of Crustal Rocks, Geophysical Monograph, American Geophysical Union, 24, 11–28. H IGGS , N. G. 1981. Mechanical properties of ultrafine quartz, chlorite and bentonite in environments appropriate to upper-crustal earthquakes. PhD thesis, Texas A&M University, College Station. L ACHENBRUCH , A. H. & S ASS , J. H. 1980. Heat flow and energetics of the San Andreas fault zone. Journal of Geophysical Research, 85, 6185–6222. L OGAN , J. M. & R AUENZAHN , K. A. 1987. Velocitydependent behaviour of mixed quartz– montmorillonite gouge. Tectonophysics, 144, 87– 108. L OGAN , J. M., H IGGS , N. & F RIEDMAN , M. 1981. Laboratory studies of natural fault gouge from the U.S.G.S. No. 1 well in the San Andreas fault zone. In: C ARTER , N. L., F RIEDMAN , M., L OGAN , J. M. & S TEARNS , D. W. (eds) Mechanical Behavior of Crustal Rocks, Geophysical Monograph, American Geophysical Union, 24, 121– 134. L OGAN , J. M., D ENGO , C. A., H IGGS , N. G. & W ANG , Z. Z. 1992. Fabrics of experimental fault zones: their development and relationship to mechanical behavior. In: E VANS , B. & W ONG , T.-F. (eds) Fault Mechanics and Transport Properties of Rock. Academic Press, London, 33– 67. M ANDL , G., DE J ONG , L. N. J. & M ALTHA , A. 1977. Shear zones in granular materials—an experimental study of their structure and mechanical genesis. Rock Mechanics, 9, 95–144. M OORE , D. E., S UMMERS , R. & B YERLEE , J. 1989. Sliding behaviour and deformation textures of heated illite gouge. Journal of Structural Geology, 11, 329–342. M OORE , D. E., L OCKNER , D. A., S HENGLI , M., S UMMERS , R. & B YERLEE , J. D. 1997. Strengths of serpentinite gouges at elevated temperatures. Journal of Geophysical Research, 102, 14787–17801. M ORROW , C., R ADNEY , B. & B YERLEE , J. 1992. Frictional strength and the effective pressure law of

LOCALIZATION OF DISPLACEMENT montmorillonite and illite clays. In: E VANS , B. & W ONG , T.-F. (eds) Fault Mechanics and Transport Properties of Rock. Academic Press, London, 69– 88. N AYLOR , M. A., M ANDL , G. & S IJPENSTEIN , C. J. K. 1986. Fault geometries in basement-induced wrench faulting under different initial stress states. Journal of Structural Geology, 8, 737–752. P RASHER , C. L. 1987. Crushing and Grinding Handbook. Wiley, New York. R IEDEL , W. 1929. Zur mechanik geologischer Brucherscheinungen, Zentralblatt fu¨r Mineralozie, Geologie und Palaeontologie, 1929B, 354–368. R UTTER , E. H., M ADDOCK , R. H., H ALL , S. H. & W HITE , S. H. 1986. Comparative microstructures of natural and experimentally produced clay bearing fault gouges. Pure and Applied Geophysics, 124, 3–29. S AMMIS , C. G., K ING , G. & B IEGEL , R. 1987. Kinematics of gouge deformation. Pure and Applied Geophysics, 125, 777– 812. S CRUGGS , V. J. & T ULLIS , T. E. 1998. Correlation between velocity dependence of friction and strain localization in large displacement experiments on feldspar, muscovite and biotite gouge. Tectonophysics, 295, 15–40. S HIMAMOTO , T. 1985. Confining-pressure reduction experiment: a new method for measuring frictional strength over a wide range of normal stress. International Journal of Rock Mechanics, Mineral Science & Geomechanical Abstracts, 22, 227– 236. S HIMAMOTO , T. 1986. A transition between frictional slip and ductile flow for halite undergoing large shearing deformation at room temperature. Science, 231, 711–714.

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S HIMAMOTO , T. & L OGAN , J. M. 1986. Velocitydependent behaviour of simulated halite shear zones: an analog for silicates. In: DAS , S., BOATWRIGHT , J. & SCHOLZ , C. H. (eds) Earthquake Source Mechanics. Geophysical Monograph, American Geophysical Union, 37, 49–64. S IBSON , R. H. 1977. Fault rocks and fault mechanisms, Journal of the Geological Society, London, 133, 191– 213. S OLUM , J. G., VAN DER P LUIJM , B. A., P EACOR , D. R. & W ARR , L. N. 2003. Influence of phyllosilicate mineral assemblages, fabrics, and fluids on the behaviour of the Punchbowl fault, southern California. Journal of Geophysical Research, 108, paper number ETG 5-1-11. T CHALENKO , J. S. 1970. Similarities between shear zones of different magnitude. Geological Society of America Bulletin, 81, 1625–1640. T ULLIS , T. E. & W EEKS , J. D. 1986. Constitutive behaviour and stability of frictional sliding of granite. Pure and Applied Geophysics, 124, 383–414. W ALLACE , R. E. & M ORRIS , H. T. 1986. Characteristics of faults and shear zones in deep mines. Pure and Applied Geophysics, 124, 107– 125. W ANG , C. & M AO , N. 1979. Shearing of saturated clays in rock joints at high confining pressure. Geophysical Research Letters, 6, 825 –828. W ANG , Z. Z. 1989. Frictional sliding and the fabric developed in experimental shear zones. PhD thesis, Texas A&M University, College Station. W EEKS , J. D. & T ULLIS , T. E. 1985. Frictional sliding of dolomite: a variation in constitutive behavior. Journal of Geophysical Research, 90, 7821– 7826.

Microscale damage evolution in compacting sandstone A. A. DIGIOVANNI1, J. T. FREDRICH2, D. J. HOLCOMB & W. A. OLSSON Sandia National Laboratories, PO Box 5800, Albuquerque, NM 87185-0750, USA 1

Present address: Hughes Christensen, The Woodlands, TX 77380, USA

2

Present address: BP America, Houston, TX 77079, USA (e-mail: [email protected]) Abstract: Recent field, laboratory, and theoretical studies suggest that under certain stress conditions, compaction of porous rock may be accommodated by narrow zones of localized compressive deformation oriented perpendicular to the maximum compressive stress. Triaxial compression experiments were performed on Castlegate sandstone, an analogue reservoir sandstone, that included acoustic emission detection and location. Initially, acoustic emissions were concentrated in horizontal bands that initiated at the sample ends (perpendicular to the maximum compressive stress) but, with continued loading, progressed axially towards the sample centre. High-resolution field-emission SEM was performed to elucidate the micromechanics of compaction. The microscopy revealed that compaction of this weakly cemented sandstone proceeded in two phases: an initial stage of porosity decrease accomplished by breakage of grain contacts and grain rotation, and a second stage of further porosity reduction accommodated by intense grain breakage and rotation. Quantitative stereological measurements corroborated the decrease in the intergrain spacing and the increase in grain boundary contacts that the microstructural observations suggest occurred during the first stage of compaction. The microstructural data show that a five-fold increase in the surface area per unit volume resulted from the extensive microfracturing that occurred during the second stage of compaction.

Hydrocarbon production, deformation, and fluid transport in reservoirs are linked strongly. Oil or gas extraction reduces pore pressure and causes an increase in the effective stress (e.g. Teufel et al. 1991). For very porous or weakly consolidated formations, the increase in effective stress may be sufficient to cause inelastic deformation of the reservoir rock (e.g. Goldsmith 1989; Jones & Leddra 1989; Rhett 1997; Schutjens et al. 1997; Fossum & Fredrich 1998, 2000a). The consequences of reservoir compaction can be severe, and include surface subsidence, casing damage, and other production problems (e.g. de Waal & Smits 1988; Smits et al. 1988; Ruddy et al. 1989; Elf Aquitaine 1991; Myer et al. 1996; Patillo et al. 1997; Fredrich et al. 2000). Small-scale faulting and deformation of porous sandstones has been studied extensively in the field (Aydin 1978; Aydin & Johnson 1983; Underhill & Woodcock 1987; Antonellini et al. 1994; Antonellini & Aydin 1995; Mollema & Antonellini 1996) and in the laboratory (Dunn et al. 1973; Brace 1978; Zhang et al. 1990; Ord et al. 1991; Scott & Nielsen 1991; Bernabe´ et al. 1992; Wong et al. 1992; Myer et al. 1996; Mair et al. 2000). Inelastic deformation under compressive stresses is generally considered to occur in two modes: shear localization, or distributed grain-scale deformation. In the field, shear localization often results in a complex network of deformation bands, characterized by layers of granulated deformation,

separating regions of comparatively undamaged host rock (Aydin 1978; Aydin & Johnson 1983). Although the faulting can develop eye or ramp structures (Antonellini et al. 1994), deformation primarily occurs in well-defined bands, which form in planes of high shear stress. Bands can range in length from centimetres to kilometres with shear offsets from tenths of metres to tens of metres and thickness on the scale of millimetres (Aydin 1978; Underhill & Woodcock 1987; Antonellini et al. 1994). Experimental studies show that bands may contain significant intragranular fracture and grain crushing or, conversely, grain integrity may be preserved with a grain fabric that is altered compared with the undisturbed rock (Aydin 1978; Underhill & Woodcock 1987; Antonellini et al. 1994). Critical parameters affecting localization and subsequently, the formation of bands, include porosity (Scott & Nielsen 1991; Wong et al. 1997) and secondary cementing phases (Bruno & Nelson 1991; David et al. 1998). Additionally, other microstructural parameters such as phase distribution (Bernabe´ et al. 1992) and grain size (Zhang et al. 1990; Shah & Wong 1996) can affect the deformation character. Recently, field investigations by Mollema & Antonellini (1996) have shown that in addition to the more commonly observed shear-dominated deformation banding, highly porous rocks may also

From: LEWIS , H. & COUPLES , G. D. (eds) The Relationship between Damage and Localization. Geological Society, London, Special Publications, 289, 89– 103. DOI: 10.1144/SP289.6 0305-8719/07/$15.00 # The Geological Society of London 2007.

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develop thin tabular zones known as compaction bands. These bands, observed in the field in rocks with porosity greater than 20–25%, are similar to shear localization phenomena in their macroscopic appearance, but are distinct in that there is no shear offset across the band. Faulting in pressuresensitive, dilatant materials can be described as a bifurcation of the macroscopic constitutive stress– strain relations governing deformation within the material (Rudnicki & Rice 1975), and recent theoretical and laboratory investigations determined that compaction band development could be accounted for with the traditional bifurcation analyses (Olsson 1999; Issen & Rudnicki 2000). Clear trends have been established in the mechanical response of rock as it undergoes the transition from elastic to inelastic loading for both hydrostatic and deviatoric load paths. As the applied mean stress is increased, the elastic behaviour is limited by a transition or yield point whereby further increases in the applied mean stress activate inelastic deformation mechanisms in the material. Cap plasticity models (e.g. DiMaggio & Sandler 1971) attempt to capture this behaviour using a pressuredependent shear yield and/or shear limit-state envelope with a hardening or hardening–softening elliptical end cap that defines pore collapse. The dominant microscale damage mechanisms in highly porous sedimentary rocks are grain rearrangement and pore collapse involving Hertzian-like fractures at grain-to-grain contacts (e.g. Mene´ndez et al. 1996). The microstructure, including porosity, grain size, and cementation, exerts fundamental control over the initiation and development of damage driven by the externally applied stresses. This investigation sought to identify the microstructural changes accompanying the formation of localized compaction in laboratory triaxial compression experiments performed on an analogue reservoir rock. Acoustic emission (AE) detection and location was used to identify band formation both spatially and temporally, and its progression with increasing strain. Sub-cores were removed from a deformed sample showing two regions of AE behaviour: one with minimal and diffusely spaced events, and another where the events were localized and more frequent. Microscopy was performed with specific emphasis on the grain-scale microfracture and deformation of the primary quartz phase as well as the secondary phyllosilicate and feldspar phases. Detailed stereological measurements were performed to quantify the microstructural evolution.

Laboratory experiments The rock used in our study is from the Castlegate Formation, and is a weak, high-porosity sandstone that is used commonly as an analogue reservoir rock.

The Castlegate rock is a fine- to medium-grained (c. 0.2 mm grain size) sublitharenite with quartz being the dominant phase (70 –80%) and a clay content that ranges from 5 to 10% (TerraTek, Inc.). Other minor phases include feldspar, siderite, and lithic fragments such as chert. The bulk porosity measured using gas porosimetry is 28% and the intrinsic permeability measured under tens of megapascals effective confining pressure is (0.2–0.4)  10212 m2. The microstructure of the pristine material, and its relation to the mechanical behaviour, is described below in more detail.

Experimental procedures Axisymmetric compression experiments were conducted on cylindrical test specimens prepared with their axes parallel to bedding and a diameter of 50.8 mm and length of 127.0 mm. Specimens were first wrapped in a thin foil of copper and 12 piezoelectric (PZT) transducers were then mounted on the copper jacket. The sample was then coated with a thick film of polyurethane to prevent the confining fluid from penetrating the specimens, and after curing, loaded in conventional triaxial compression using a servo-controlled triaxial system with the pore pressure drained to atmosphere. Confining pressure was measured with a conventional strain gauge transducer, and measurements of force on an external load cell were used to calculate axial stress. Axial and radial strains were measured using linear variable differential transformers that were mounted axially and radially at the middle of the sample. The sample of interest was loaded hydrostatically to a confining pressure of 80 MPa, and then loaded triaxially under displacement rate control to yield a strain rate of 1025 s21. AE arrival time and waveforms were recorded from the 12 PZT transducers and used later to determine event location within the sample. The criteria for location included that the event was recorded by at least four of the 12 transducers. The location error was c. +3 mm. Olsson & Holcomb (2000) have described the experimental programme in more detail.

Stress – strain behaviour The material behaviour during the hydrostatic loading was essentially isotropic, as indicated by the axial and radial strain data shown in Figure 1a. There was no pronounced inflection in the pressure –volume behaviour during the hydrostatic loading to 80 MPa, and the initial yield surface associated with the onset of inelastic compaction was intersected during the triaxial loading segment (Fig. 1b). After reaching a peak stress difference of c. 150 MPa, the sample experienced a small stress drop, and then deformed at an

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Fig. 1. Differential stress v. axial, radial, and volumetric strain. Behaviour observed in experiment CG6 during (a) hydrostatic and (b) triaxial compression loading. It should be noted that load–unload loops were performed at several stages for measurement of incremental elastic moduli.

approximately constant differential stress of 135 MPa before being unloaded at a total (elastic plus inelastic) axial strain of c. 5%. The deformation measured during the triaxial segment by the radial strain transducer that was located at the specimen mid-point tracks almost identically for both loading and unloading. This indicates that the radial deformation at that location up to the point of unloading was dominantly elastic.

Acoustic emission (AE) data The AE data were analysed in time and space. A random 40 000 of 10  106 total events were

captured and located. Three-dimensional renderings of the AE locations during the hydrostatic and deviatoric loading are shown in Figure 2. During the hydrostatic loading a small number of AE events were recorded (Fig. 2) that were distributed throughout the sample volume, with a slight concentration at the sample ends. The initial part of the triaxial segment was relatively quiet; but when the total axial stress increased to c. 150 MPa, an escalation in AE rate was associated with the development of two distinct bands of AE that were located initially at the sample ends (Fig. 2). The initiation of AE at the sample ends arose from the elastic modulus

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3

2

4

5

250 Total axial stress (MPa)

3

4

5

200 2

150 100 1 50 0 0.00

0.01

0.02 0.03 0.04 Axial strain

0.05

0.06

Fig. 2. (Top) Three-dimensional volume renderings of AE events during (1) the hydrostatic segment, and (2) during the triaxial loading segment to just prior to peak stress, (3) through peak stress and over the duration of the small stress drop, and (4 and 5) during the remainder of the triaxial loading segment. The extent of the five time segments is indicated on the stress– strain plot (bottom). Total axial stress v. axial strain during the hydrostat and triaxial loading are indicated. It should be noted that load –unload loops were performed at several stages for measurement of incremental elastic moduli. The diameter and length of the cylindrical sample are 50.8 mm and 127.0 mm, respectively, and the linear voxel dimension is 1 mm.

difference between the end cap and sample and resultant frictional constraint (Olsson & Holcomb 2000). As the stress reached the peak value, the AE bands moved away from the sample ends, and propagated towards the sample centre throughout the portion of the triaxial segment during which the stress maintained a roughly constant value (Fig. 2).

Macroscopic observations Visual observations of the sample made after removal of the jacketing material suggested the existence of three distinct zones of deformation. Both the upper and lower thirds of the sample exhibited macroscopic signs of deformation, including multiple inclined diffuse shear bands and a pervasive lightening in colour. These regions experienced a uniform diametrical increase of a few per cent, whereas the central region retained its original diameter. The colour change was

presumably associated with grain crushing and granulation, as these regions were very fragile and prone to surficial disintegration during even gentle handling. In contrast, the centre portion of the sample maintained its colour, retained its integrity, and appeared largely unchanged from its initial state. Post-test measurements of compressional (P) wave velocity corroborated the visual observations and suggested the existence of different deformation regimes within the sample. The P-wave velocity as measured diametrically in the upper and lower regions of the sample was reduced to 1.6 – 1.8 km s21, and the velocity gradually increased to near its original value (c. 2.8– 3.0 km s21) as the sample centre was approached (Fig. 3).

Microscopy To substantiate the occurrence of non-uniform or localized compaction and to elucidate the

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60 0 degrees 90 degrees

Axial Position (mm)

40

20

0

–20

–40

–60 1.2

1.4

1.6

1.8

2.0

2.2

2.4

2.6

2.8

3.0

3.2

P-Wave Velocity (km/s) Fig. 3. P-wave velocity measured at orthogonal points along the axis of the deformed specimen. The zero point on the vertical axis corresponds to the geometric centre of the deformed sample. The region closest to the centre experienced only diffuse AE activity and shows P-wave velocities of 2.8 km s21, the value measured for the pristine Castlegate sandstone (CG0).

micromechanics of the compaction process, we conducted microscopy on the deformed sample using light optical and scanning electron microscopy (SEM). After testing, the deformed sample was vacuum impregnated with a low-viscosity epoxy that was doped with a red dye (rhodamine-B). Following curing, several 25.4 mm diameter cylindrical sub-cores were diamond-cored from different locations within the large sample and then used to prepare polished (to 0.05 mm) thin sections for optical microscopy and thick sections for SEM. The sub-cores were taken horizontally from locations in the upper, central, and lower thirds of the sample, and orientation was maintained throughout the preparation of the polished sections. Observations made on sections prepared from the upper and lower portions of the sample were similar so that we do not discriminate between them here and these samples are denoted simply as CG6-SD. Specimens sub-cored from the central region of the sample are referred to as CG6-UD. A pristine sample was similarly impregnated, sub-cored, and sectioned for direct comparison with the deformed sample. This sample is referred to as CG0.

Observations Optical microscopy revealed the gross features of the deformation but could not resolve the details of the grain-scale cracking and fragmentation. To elucidate the fine-scale deformation, high-resolution

field-emission SEM was performed on carboncoated polished thick sections using a Hitachi S-4500 scanning electron microscope operated at 25 kV using the backscattered imaging mode.

Pristine sample A scanning electron micrograph of the undeformed Castlegate sandstone (CG0) is shown in Figure 4a. The dominant mineral phase is quartz and other detrital phases include potassium feldspar and chlorite (and/or possibly muscovite). Authigenic clays are present as pore lining and sometimes as pore filling material (Fig. 4a and b; see also Fig. 10), and include kaolinite and illite. There are areas of agglomerated clays, and partially leached grains occur regularly that contain significant microporosity (an example from the deformed sample can be seen in Fig. 6a). In general, clay permeates the grain framework, coating most surfaces, and acting to minimally cement grain contacts. A small amount of cementing silica overgrowth on quartz grains is also occasionally present. Highmagnification microscopy reveals the lamellar structure of many of the phyllosilicate phases as shown in Figure 4b and c, and in Figures 10 and 11. A significant aspect of the microstructure is the high number of point and tangential contacts between grains; the original texture of the sand has not been affected significantly by diagenesis, and a pre-compaction texture has been largely

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Fig. 4. (a) Backscattered scanning electron micrograph of pristine Castlegate sandstone (CG0). Quartz is the dominant phase, and the light grain at the centre of the left edge is potassium feldspar. The darker phases partially coating some quartz grains, and occasionally partially filling pores, are clays. The image is 1.00 mm across. (b) The brighter central grain is probably iron-rich detrital chlorite (or possibly biotite). Authigenic clays coat this grain and the surrounding quartz grains and cement them weakly. Energy-dispersive spectrometry was used to identify kaolinite and illite as the dominant clay phases. Illite can be seen extending from the grain surface in the lower left part of the image. The image is 0.20 mm across. (c) A detrital cluster of chlorite (or possibly muscovite) partially fills in a pore space in which authigenic clays are also present. The image is 0.15 mm across.

maintained. The small amount of pore-filling clays in combination with the occasional local enhancement of porosity by dissolution results in a pore network that is extremely well connected and with pore sizes that occasionally exceed the size of the largest grains by a factor of two or more.

Deformed sample As discussed above, the central region of the sample was associated with only diffuse AE activity during the hydrostatic and triaxial loading segments

(Fig. 2) and showed no macroscopic signs of deformation, whereas the outer zones experienced intense localized AE activity and compaction. The significant porosity reduction is revealed dramatically with optical microscopy (Fig. 5). The highly porous nature of the untested material is evident in Figure 5a, and the progressive loss of porosity is shown in Figure 5b and c. Although the macroscopic reduction of porosity is shown in the optical thin sections, the details are more clearly revealed through SEM. The sample region with minimal AE activity shows a

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Fig. 5. Optical thin sections of epoxy-impregnated Castlegate sandstone revealing the loss of macroscopic porosity as a result of compaction. (a) The pristine rock, CG0. (b) The initial stage of deformation with minimal diffused AE, CG6-UD. (c) The region of intense AE and grain comminution, CG6-SD. Scale bar represents 200 mm in all micrographs. Maximum compressive stress was horizontal in (b) and (c).

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Fig. 6. Backscattered images from (a) the central region of sample CG6 (CG6-UD), which appeared largely undeformed to the naked eye, and (b) the lower region of CG6 (CG6-SD), which exhibited obvious signs of deformation and through which the band of high AE activity propagated. In CG6-UD, grains are intact but closer together than in CG0, and edge (point) grain contacts have increased (decreased). In CG6-SD, porosity is further reduced and grains are intensely comminuted. The maximum compressive stress was horizontal. Higher magnification images from the areas outlined in white are shown in Figure 9. The images are 1.00 mm across.

microstructure that is altered subtly. The quartz and potassium feldspar framework grains have retained their integrity and are largely unfractured; only occasionally have point contacts resulted in fractures (Fig. 7). More significantly, the grain structure is more compact in that grains appear to be closer together with less pore space between neighbouring grains. In addition, there are fewer point or tangential contacts between grains; the dominant contacts are now line or edge. These observations suggest that a phase of porosity reduction occurred through breakage of grain contacts and subsequent rotation of grains to yield a more compact packing.

As noted above, the sample showed deformation that was obvious to the naked eye in the regions through which the bands of intense AE activity had migrated during the constant stress portion of the experiment. The microstructure is dramatically different in these regions of the sample (CG6-SD, Fig. 6b) as compared with both the central region of the sample with diffuse AE as well as the pristine rock; the original fabric of the sandstone is completely destroyed in the regions through which the AE bands propagated. The porosity is noticeably reduced both in terms of its bulk volume and the size of the remaining pores.

MICROSCALE DAMAGE EVOLUTION

Fig. 7. Backscattered electron image from sample CG6-UD, which experienced modest grain rearrangement and shows an example of Hertzian-like contact fracture between quartz grains. Maximum compressive stress was horizontal. The image is 0.33 mm across.

Within the compaction zone, grain fragmentation is profound, with few undamaged grains; many grains experienced severe contact fractures (Fig. 8) and were crushed extensively (Fig. 9). The large reduction in porosity was accommodated through both fragmentation and subsequent rotation of the crushed grains. Notably, some grains that were surrounded by sufficiently large amounts of clay or second phases remained intact. To some extent, deformation within the detrital chlorite and clay phases is commensurate with the observations of the framework grains in the two

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deformation regions. With the advent of grain rearrangement and slight porosity reduction, some of the intergranular phyllosilicate phases have been moderately transformed. High-resolution microscopy reveals that the nature of the local grain deformations is reflected in the compression of the layered phases between the harder phases such as quartz. For example, the layered grain in CG6-UD suggests the possibility of relative offsets between the layers (Fig. 10a). Similarly, the upper layers of the particle in Figure 10b appear to have experienced a moderate amount of buckling as a result of the small quartz grain in the top right part of the figure. (However, it should be noted that some detrital particles appear slightly deformed even in the pristine material.) In contrast, in the region of localized compaction (CG6-SD), many of the less rigid clay and detrital phases are deformed more markedly. The layered phases dramatically reveal local deformation because of their regular structure. During compaction, the lamellae can buckle severely to accommodate local rearrangement of the quartz grains (Fig. 11). For example, one region of the particle on the left of Figure 11a shows tightly packed layers with a small amount of in-plane offset. Further right in the same particle, the lamellae have splayed apart and are bent at nearly 908 to accommodate the encroachment of surrounding grains. A simple estimate of linear distortion revealed a c. 20% shortening of the layers where the greatest amount of buckling took place. However, not all phyllosilicates are severely deformed within the compaction region; there is a greater variation in the level of deformation experienced by the smaller particles; some particles remain relatively undeformed, whereas others are deformed more severely (Fig. 11b).

Quantitative stereology

Fig. 8. Backscattered electron image from sample CG6-SD, which experienced severe grain fracture, rearrangement, and consolidation marked by intense AE activity. Two grains have possibly impinged at a point contact, and the severe damage has obscured the original grain boundaries. Maximum compressive stress was horizontal. The image is 0.33 mm across.

Stereological techniques (Underwood 1970) were applied to the micrographs from both deformation regions to quantify the microstructural evolution of the compaction process. The mean phase intercept length, L3, and the number of interceptions of objects per unit line length, NL, were measured for the CG0 and CG6-UD samples. Measurements of the interception points were made in directions both parallel and perpendicular to the maximum compressive stress for the deformed samples, and in two perpendicular directions for the unoriented pristine sample. Measurements of L3 were not made on sample CG6-SD because of the complex microstructure (see Fig. 6b) associated with the second stage of compaction that was accommodated by grain fragmentation and crushing and associated with the intense AE activity.

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Fig. 9. High-magnification backscattered images showing intense grain-scale microcracking in CG6-SD (see Fig. 5, bottom, for detail area). (a) The grain on the right is quartz and is only slightly damaged. The lighter grain in the upper part of the image is potassium feldspar and is intensely comminuted. Fine microcracking parallel to the strong cleavage in the feldspar is apparent, as is subsequent rotation of the cleaved grain fragments. (b) Intense microcracking developed in two quartz grains that may have originally had a point contact. The maximum compressive stress was horizontal. The images are 0.20 mm across.

The method was implemented as follows. To record the number of interception points, original greyscale digital images at 1024  816 were quartered (to 513  409) and overlain with an array of either 13 or 17 evenly spaced test lines parallel to the long axis of the image. The long axes of the images were aligned with the maximum compression direction. Another array of 34 randomly distributed line segments of length l ¼ 75 mm was used for the CG0 and CG6-UD specimens to calculate L3. The quarter image fields were 398.4 mm in

height and 499.7 mm in width (corresponding exactly to the length of the evenly spaced lines). Three 1024  816 images (yielding 3  4 ¼ 12 quarter images) were used for each sample, so that an area equal to 2.4 mm2 was covered for each sample. The array of regularly spaced lines was also used to measure the number of intercepts of the object of interest per unit test line length, PL, here defined as boundaries separating individual grains (PLaa) and boundaries separating the pore and

Fig. 10. Backscattered images of phyllosilicate phases in sample CG6-UD. (a) The large bright grain is probably detrital chlorite or muscovite. The geometry suggests a small amount of deformation; some layers appear to be slightly offset and to protrude into the pore space. The image is 0.12 mm across. (b) Layers appear bent, twisted, and shortened in several places, and some of the grain fragments appear to push against the central particle (probably chlorite). Authigenic clays are found in the grain boundary regions of the surrounding quartz particles in both images and also interspersed with the detrital grains at the centre of both images. The image is 0.38 mm across. Maximum compressive stress was horizontal for both images.

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Fig. 11. Backscattered electron images of severely deformed phyllosilicate particles from CG6-SD. Commensurate with the intense grain damage in this sample is rearrangement of phyllosilicate particles into and around the contracting pore spaces. (a) The phyllosilicate (chlorite or muscovite) has deformed around an impinging quartz grain (bottom) and into the remaining pore space. The image is 0.12 mm across. (b) Clays (or possibly chlorite) deform around quartz fragments (right) to accommodate the reduction in overall volume, whereas some particles experience less deformation (left). The image is 0.15 mm across. Maximum compressive stress was horizontal.

grain phases (PLab) for the CGO and CG6-UD samples. Tangential intercepts were counted as 12. It should be noted, however, that for the CG6-SD sample no distinction was made between pore or grain boundaries or fracture surfaces in determining PL, as differentiating between them in heavily comminuted grains was impossible. Microporosity contained in the leached quartz grains and the clays was also not measured in our analysis. Consequently, the measurements reported here were intended to characterize exclusively the macroporosity changes with deformation. From the number of intercept points per line, the number of particles per unit length, NL(a), for the grain phase is given by:

and can be related to a mean grain diameter for a particular assumed grain shape as described by Underwood (1970). The calculations of NL and L3 were used to calculate the volume fraction for the phase of interest, VV, by the relation

1 NLðaÞ ¼ ð2PLaa þ PLab Þ: 2

SV ¼ 2PL :

ð1Þ

Because a boundary cannot exist between pore phases, NL(a) for the pore phase is given by 1 NLðaÞ ¼ PLab 2

ð4Þ

and the mean free path, l, between grains using l ¼ ½1  VVðaÞ =NLðaÞ :

ð5Þ

For an isotropic system, the surface area per unit volume SV is ð6Þ

In addition, for an axisymmetric structure, the surface area per unit volume is determined from orthogonal measurements of the number of intercepts per unit test line length, PL, where one set of test lines is coincident with the axis of symmetry

ð2Þ SV ¼ 1:571ðPL Þ ? þ0:429ðPL Þk;

The array of random line segments was used to determine the mean phase intercept length L3. For each line segment of length, l, the number of end points falling in the phase of interest, p, and the number of intersections of line segments with the phase boundary, P, were counted. The mean phase intercept length is given by L3 ¼ l p=P

VVðaÞ ¼ NLðaÞ L3

ð3Þ

ð7Þ

where ? and k indicate the measurements perpendicular and parallel to the axis of symmetry that is assumed to coincide with the maximum compressive stress direction. The results for the grain and pore phase are shown in Tables 1 and 2. The data clearly support the qualitative microscopy observations discussed above. Specifically, the particle size in the central region of the test specimen (sample CG6-UD) is

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Table 1. Stereological measurements of the grain phase Sample CG0† CG6-UD

k ?

PL(ab) mm21

PL(aa) mm21

NL(a) mm21

L3 mm

l mm

10.7 + 1.5 11.9 + 1.8 12.8 + 3.0

1.8 + 0.7 3.0 + 1.4 2.8 + 1.1

7.1 + 1.2 8.9 + 2.1 9.1 + 1.9

76.4 + 4.7 76.5 + 22.6

68.0 + 22.7 41.4 + 14.5

† Anisotropy was not anticipated in this sample and orthogonal measurements for a random orientation were identical within the limits of standard deviation.

Table 2. Stereological measurements of the pore phase Sample CG0 CG6-UD CG6-SD‡

k ? k ?

PL(ab) mm21

NL(a) mm21

L3 mm

VV%

SV mm21

11.2 + 1.6 10.1 + 2.2 10.3 + 2.4 45.4 + 10.6 52.8 + 11.0

5.6 + 0.8 5.0 + 1.1 5.2 + 1.2

37.8 + 8.9 32.1 + 10.7

21.0 + 5.5 15.4 + 3.0

22.4 + 3.2 20.6 + 4.5 102.4 + 21.5

‡ Because of the nature of damage in this sample, no differentiation was made between grain boundaries and fracture surfaces. SV was calculated using Eq. (6) for CG0 and Eq. (7) for the UD and SD samples.

not reduced in comparison with the pristine material CG0 (identical L3), and there is a 40% reduction in the inter-grain spacing for sample CG6-UD compared with the pristine sample CG0. This corresponds to a 27% reduction in the macroporosity or, equivalently, a net reduction in the volume fraction of macroporosity from 21 to 15%. (It should be noted also that the data for CG0 imply that the microporosity accounted for c. 7% of the total bulk porosity of 28%.) Furthermore, the increase in line and edge contacts in favour of point contacts is demonstrated by the measurements of grain boundary intercepts, PLaa, which increased by 67% in the parallel direction and by 75% in the perpendicular direction for the CG6-UD sample compared with the pristine sample CG0. Indirectly, this is also demonstrated by the 8% reduction of pore surface area per unit volume, SV, for CG6-UD. CG0, shown in Table 2. The measurements made within the heavily compacted region show a moderate increase in the number of intercepts orthogonal to the maximum compression direction (i.e. normal to the band direction). This indicates that microfracturing occurred preferentially out of the plane of the compaction localization. Finally, SV for CG6-SD increased more than five-fold after compaction, as a result of the substantial grain comminution shown in Figures 6, 8 and 9.

Discussion The microstructural observations confirm the existence of fundamentally distinct zones of deformation

within the test specimen. In conjunction with the macroscopic stress –strain and AE data, the microscopy analyses suggest that compaction of this weakly cemented sandstone under the triaxial load path followed during the test was accommodated in two distinct stages. The first stage of compaction was associated with breakage of the minimal cement bonding the framework grains and subsequent rotation of the intact framework grains to yield a more compact grain packing. This stage reduced the macroporosity by an estimated 27%. In the second stage, compaction became localized and was associated with intense grain comminution and continued rotation of the grain fragments, resulting in substantial additional reduction in porosity. The AE data reveal that the second stage of compaction propagated as localized bands of deformation through the test specimen. The mechanical data indicate that the sample deformed at an approximately constant deviatoric stress during the stage in which the localized compaction zones propagated through the sample. After the zone of localized compaction has passed completely through the sample, strain-hardening begins (Olsson & Holcomb 2000) and, presumably, further loading eventually leads to shear localization. Olsson (1999) suggested that the occurrence of localized compaction bands can be understood within the framework of the bifurcation theory of localization proposed originally by Rudnicki & Rice (1975). Issen & Rudnicki (2000) analysed this in detail and outlined the conditions under which compaction bands may form.

MICROSCALE DAMAGE EVOLUTION

Our microscopy observations have some implications for the micromechanical sources of AE during the inelastic deformation and failure of porous rock. In particular, the observations suggest that the processes involved in the first stage of compaction are reasonably inefficient generators of AE. As discussed above, the first stage of compaction was associated with breakage of grain bonds and subsequent rotation, but was marked by only limited diffuse AE activity. In contrast, the second stage of compaction, which was accommodated by intense grain-scale microcracking, was associated with intense AE activity. Lockner (1993) has discussed previously the complexities related to the interpretation of micromechanical sources of AE during the failure of low-porosity rock. The secondary clay phases that weakly cement grains and partially fill pores were deformed to some extent by the sliding and rotation of the harder quartz and feldspar phases in the initial deformation stage. Localized compaction showed continued deformation of the layered phases, often with spreading or separation of the mineral layers. It is clear that the softer phases distort to fill the changing fabric of the pore space during compaction. Moreover, within the region of localized compaction, the microstructure is characterized by significantly reduced macroporosity interspersed with fractured quartz grains and the more easily deformed phyllosilicate phases. The implication for permeability is a loss of the largest pathways, creating a more tortuous flow path with reduced throat sizes. Stereological measurements imply that only minimal anisotropy (,10%) develops in the grain structure during the first stage of compaction. In the second stage of deformation, SV increases by approximately a factor of five, which can be attributed to intense intragranular microcracking. Although grain boundaries were counted equally with fracture surfaces in this stage, the former constitute only a fraction of PL measurements prior to grain comminution (Table 1). The second stage of compaction produces about a 16% anisotropy in crack surface area per unit volume, which develops preferentially subparallel to the maximum compressive stress. The micromechanical compaction behaviour that is observed for the Castlegate sandstone is in marked contrast to previous observations for Berea sandstone by Mene´ndez et al. (1996). Those workers performed a suite of compaction tests under different load paths including hydrostatic compression and triaxial compression. They observed, however, that the initiation of inelastic yield (compaction) under both hydrostatic and triaxial loading conditions was associated with the onset of brittle microcracking at grain contacts

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(see Mene´ndez et al. 1996, fig. 5) and a substantial increase in SV. In contrast, our study of Castlegate sandstone shows clearly an initial stage of increased line and edge contacts and a reduction in SV with compaction that is accommodated by the breakage of grain contacts and subsequent rotation of intact grains with no grain fragmentation. The primary differences between the two sandstones are their fabric and cementation. The Berea sandstone has undergone a diagenesis that has resulted in a more indurated fabric with welldeveloped grain boundaries and significantly more cementation. In contrast, the Castlegate sandstone has maintained a pre-compacted fabric throughout its diagenesis, the grain contacts are often point or tangential, and the cementation is minimal. Thus, the Castlegate sandstone fabric is sufficiently weak that a large amount of porosity reduction can be accomplished through grain rearrangement alone without significant grain breakage and fragmentation. The different micromechanical behaviour observed for Castlegate sandstone v. Berea sandstone has significant implications for the evolution of fluid transport properties during deformation. Specifically, the pronounced grain crushing that is associated with the onset of inelastic compaction in the Berea sandstone reduces permeability dramatically by both reducing the local hydraulic conductance of the pores and increasing the tortuosity of the flow paths (Zhu & Wong 1997). Our microstructural analyses suggest that the permeability evolution during the initial stage of compaction in the Castlegate sandstone will differ from that observed for Berea sandstone, in that the dominant microstructural change relates to grain rearrangement as opposed to grain fragmentation. Therefore, we expect the connectivity of the flow paths to be approximately maintained during the initial stage of compaction. The second stage of compaction, which occurs as a localized band of deformation, is expected to result in significant spatial anisotropy of the fluid transport properties.

Conclusion Detailed microscopy studies were performed to elucidate the micromechanics of compaction during triaxial compression experiments performed on Castlegate sandstone. The microscopy revealed that compaction of this weakly cemented sandstone proceeded in two phases: an initial stage of porosity decrease that was accomplished by breakage of grain contacts and subsequent grain rotation, and a second stage of further volume reduction that was accommodated by intense grain breakage and rotation. The second stage of compaction was revealed through AE location to have occurred as

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a localized compaction band that spread through the sample in a direction perpendicular to the maximum compressive stress. Depending upon the generality of the deformation behaviour observed here (i.e. in terms of the microstructural characteristics, stress states, and load paths that favour such behaviour), researchers may need to consider whether this deformation mode of localized compaction needs to be considered in numerical simulations of complex geosystems. Applications involving compacting materials such as basin evolution or coupled reservoir –geomechanics modelling may need to include a more complex constitutive model that allows for localized compaction behaviour (e.g. Issen & Rudnicki 2000) rather than the uniform deformation that is assumed by traditional cap plasticity models (e.g. see Fossum & Fredrich 2000b). Barnichon & Charlier (1999) described recently the incorporation of the classic Rudnicki & Rice (1975) constitutive theory for shear bifurcation into finite-element models of sedimentary basin evolution. The authors would like to thank J. C. Lorenz for comments that improved the manuscript. We gratefully acknowledge support from the US Department of Energy (DOE) Office of Basic Energy Sciences, Chemical Sciences, Geosciences and Biosciences Division. This work was performed at Sandia National Laboratories funded by the US DOE under Contract No. DE-AC04-AL85000. Sandia is a multiprogramme laboratory operated by Sandia Corporation, a Lockheed Martin Company, for the US Department of Energy.

References A NTONELLINI , M. A. & A YDIN , A. 1995. Effect of faulting on fluid flow in porous sandstones: geometry and spatial distribution. AAPG Bulletin, 79, 642–671. A NTONELLINI , M. A., A YDIN , A. & P OLLARD , D. D. 1994. Microstructure of deformation bands in porous sandstones at Arches National Park, Grand County, Utah. Journal of Structural Geology, 16, 941–959. A YDIN , A. 1978. Small faults formed as deformation bands in sandstone. Pure and Applied Geophysics, 116, 913–930. A YDIN , A. & J OHNSON , A. M. 1983. Analysis of faulting in porous sandstone. Journal of Structural Geology, 5, 19–31. B ARNICHON , J.-D. & C HARLIER , R. 1999. Finite element modeling of shear banding in sedimentary basins undergoing tectonic compression, Paper presented at 2nd Euroconference on Rock Physics and Rock Mechanics, 14–18 November, Edinburgh, Scotland. B ERNABE´ , Y., F RYER , D. T. & H AYES , J. A. 1992. The effect of cement on the strength of granular rocks. Geophysical Research Letters, 19, 1511–1514. B RACE , W. F. 1978. Volume changes during fracture and frictional sliding: a review. Pure and Applied Geophysics, 116, 603–614.

B RUNO , M. S. & N ELSON , R. B. 1991. Microstructural analysis of the inelastic behaviour of sedimentary rock. Mechanics of Materials, 12, 95–118. D AVID , C., M ENE´ NDEZ , B. & B ERNABE´ , Y. 1998. The mechanical behaviour of synthetic sandstone with varying brittle cement content. International Journal of Rock Mechanics and Mining Sciences, 35, 759– 770. DE W AAL , J. A. & S MITS , R. M. M. 1988. Prediction of reservoir compaction and surface subsidence: field application of a new model. SPE Formation Evaluation, 347 –356. D I M AGGIO , F. L. & S ANDLER , I. S. 1971. Material model for granular soils. Journal of Engineering Mechanics, ASCE, 97, 935 –950. D UNN , D. E., F OUNTAIN , L. J. & J ACKSON , R. E. 1973. Porosity dependence and mechanisms of brittle fracture in sandstones. Journal of Geophysical Research, 78, 2403– 2417. E LF Aquitaine 1991. The role of rock mechanics in oil and gas exploration. Nature, 350, supplement, 8– 10. F OSSUM , A. F. & F REDRICH , J. T. 1998. Estimation of constitutive parameters for the Belridge Diatomite, South Belridge Diatomite field, SAND98-1407. Sandia National Laboratories, Albuquerque, NM. F OSSUM , A. F. & F REDRICH , J. T. 2000a. Constitutive Models for the Etchigoin Sands, Belridge Diatomite, and Overburden Formation at the Lost Hills Oil Field, California, SAND2000-0827. Sandia National Laboratories, Albuquerque, NM. F OSSUM , A. F. & F REDRICH , J. T. 2000b. Cap plasticity models and compactive and dilatant pre-failure deformation. In: G IRARD , J., L IEBMAN , M., B REEDS , C. & D OE , T. (eds) Proceedings of the 4th North American Rock Mechanics Symposium. Balkema, Rotterdam, 69–76. F REDRICH , J. T., A RGUELLO , J. G., D EITRICK , G. L. & DE R OUFFIGNAC , E. P. 2000. Geomechanical modeling of reservoir compaction, surface subsidence, and casing damage at the Belridge diatomite field. SPE Reservoir Evaluation & Engineering, 3, 348– 359. G OLDSMITH , A. S. 1989. Permeability decline and compressibility in sandstone reservoir rocks. In: M AURY , V. & F OURMAINTRAUX , D. (eds) Rock at Great Depth. Balkema, Rotterdam, 923– 928. I SSEN , K. A. & R UDNICKI , J. W. 2000. Conditions for compaction bands in porous rock. Journal of Geophysical Research, 105, 21529–21536. J ONES , M. E. & L EDDRA , M. J. 1989. Compaction and flow of porous rocks at depth. In: M AURY , V. & F OURMAINTRAUX , D. (eds) Rock at Great Depth. Balkema, Rotterdam, 891–898. L OCKNER , D. A. 1993. The role of acoustic emission in the study of rock fracture. International Journal of Rock Mechanics and Mining Sciences & Geomechanics Abstracts, 30, 883–899. M AIR , K., M AIN , I. & E LPHICK , S. 2000. Sequential growth of deformation bands in the laboratory. Journal of Structural Geology, 22, 25– 42. M ENE´ NDEZ , B., Z HU , W. & W ONG , T.-F. 1996. Micromechanics of brittle faulting and cataclastic flow in Berea sandstone. Journal of Structural Geology, 18, 1– 16. M OLLEMA , P. N. & A NTONELLINI , M. A. 1996. Compaction bands: a structural analog for anti-mode

MICROSCALE DAMAGE EVOLUTION 1 cracks in aeolian sandstone. Tectonophysics, 267, 209–228. M YER , L., J ACOBSEN , J., H ORSMAN , J. ET AL . 1996. Use of visualization techniques in analysis of well failures in diatomite reservoirs. Leading Edge. March, 185–189. O LSSON , W. A. 1999. Theoretical and experimental investigation of compaction bands in porous rocks. Journal of Geophysical Research, 104, 7219–7228. O LSSON , W. A. & H OLCOMB , D. J. 2000. Compaction localization in porous rock. Geophysical Research Letters, 27, 3537–3540. O RD , A., V ARDOULAKIS , I. & K AJEWSKI , R. 1991. Shear band formation in Gosford sandstone. International Journal of Rock Mechanics and Mining Sciences & Geomechanics Abstracts, 28, 397–409. P ATILLO , P. D., K RISTIANSEN , T. G., S UND , G. V. & K JELSTADII , R. M. 1997. Reservoir compaction and seafloor subsidence at Valhall. In: Eurock— Rock Mechanics in Petroleum Engineering, 1. Society of Petroleum Engineers, SPE/ISRM 47274, 377–386. R HETT , D. W. 1997. Ekofisk revisited: a new model of Ekofisk Reservoir Geomechanical Behavior. In: Eurock—Rock Mechanics in Petroleum Engineering, 1. Society of Petroleum Engineers, SPE/ISRM 47273, 367–376. R UDDY , I., A NDERSEN , M. A., P ATTILLO , P. D., B ISHLAWI , M. & F OGED , N. 1989. Rock compressibility, compaction, and subsidence in a high-porosity chalk reservoir: a case-study of Valhall field. Journal of Petroleum Technology, 41, 741– 746. R UDNICKI , J. W. & R ICE , J. R. 1975. Conditions for the localization of deformation in pressure sensitive dilatant materials. Journal of the Mechanics and Physics of Solids, 23, 371–394. S CHUTJENS , P. M. T. M., B LANTON , T. L., M ARTIN , J. W., L EHR , B. C. & B AAIJENS , M. N. 1997. Depletion induced compaction of an overpressured reservoir sandstone. In: Eurock—Rock Mechanics in Petroleum Engineering, 2. Society of Petroleum Engineers, SPE/ISRM 47277, 53– 62.

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S COTT , T. & N IELSEN , K. 1991. The effects of porosity on the brittle– ductile transition in sandstones. Journal of Geophysical Research, 96, 405– 414. S HAH , K. R. & W ONG , T.-F. 1996. Grain fracturing and comminution in porous materials. In: A UBERTIN , M., H ASSANI , F. & M ITRI , H. (eds) Rock Mechanics, Tools and Techniques, Proceedings of the 2nd North American Rock Mechanics Symposium. Balkema, Rotterdam, 859–866. S MITS , R. M. M., DE W AAL , J. A. & VAN K OOTEN , J. F. C. 1988. Prediction of abrupt reservoir compaction and surface subsidence caused by pore collapse in carbonates. SPE Formation Evaluation, 340– 346. T EUFEL , L. W., R HETT , D. W. & F ARRELL , H. E. 1991. Effect of reservoir depletion and pore pressure drawdown on in situ stress and deformation in the Ekofisk Field, North Sea. In: ROEGIERS , J.-C. (ed.) Proceedings of the 32nd US Symposium on Rock Mechanics. Balkema, Rotterdam, 63–72. U NDERHILL , J. R. & W OODCOCK , N. H. 1987. Faulting mechanisms in high-porosity sandstone; New Red Sandstone, Arran, Scotland. In: J ONES , M. E. & P RESTON , R. M. F. (eds) Deformation of Sediments and Sedimentary Rocks. Geological Society, London, Special Publications, 29, 91– 105. U NDERWOOD , E. E. 1970. Quantitative Stereology. Addison Wesley, Reading, MA. W ONG , T.-F., S ZETO , H. & Z HANG , J. 1992. Effect of loading path and porosity on the failure mode of porous rocks. Applied Mechanics Review, 45, 281– 293. W ONG , T.-F., D AVID , C. & Z HU , W. 1997. The transition from brittle faulting to cataclastic flow in porous sandstones: mechanical deformation. Journal of Geophysical Research, 102, 3009– 3025. Z HANG , J., W ONG , T.-F. & D AVIS , D. M. 1990. Micromechanics of pressure-induced grain crushing in porous rocks. Journal of Geophysical Research, 95, 341–352. Z HU , W. & W ONG , T.-F. 1997. The transition from brittle faulting to cataclastic flow: permeability evolution. Journal of Geophysical Research, 102, 3027–3041.

Influence of grain size and geothermal gradient on the ductile-to-brittle transition in arenaceous sedimentary rocks: implications for fault structure and fluid flow Q. J. FISHER, S. D. HARRIS, M. CASEY & R. J. KNIPE Rock Deformation Research Ltd, School of Earth and Environment, University of Leeds, Leeds, LS2 9JT, UK (e-mail: [email protected]) Abstract: To investigate the interaction between the rheology of arenaceous sedimentary rocks (sand and sandstone) and stress conditions during burial we have coupled published results from deformation experiments with a simple quartz cementation model. The model provides valuable insights into controls on sandstone deformation consistent with observations from nature. A transitional zone exists in subsiding sedimentary basins, here referred to as the ductile-to-brittle transition (DBT), above which faults in normally pressured arenites will tend to form fluid flow barriers, and below which they will tend to form conduits. The DBT depth in sandstone is dependent upon geothermal gradient, burial rate and grain size. Low geothermal gradients, rapid sedimentation rates and coarse grain sizes favour a deep DBT and vice versa. Fine-grained arenites may only deform in a brittle manner for most natural burial rates and geothermal gradients, explaining why they do not usually contain thick deformation band zones. Coarser-grained arenites may deform in a brittle–ductile or ductile manner, which is why they often contain thick deformation band zones and occasionally experience pervasive porosity collapse. Sandstones within high geothermal gradient areas may deform to produce fluid flow conduits at shallow depths when porosities in the sequence as a whole are high; this possibly favours fault-related mineralization.

The desire to predict subsurface fluid flow (e.g. Zoback & Byerlee 1976), reservoir compaction (e.g. Hettema et al. 2000), borehole stability (e.g. Moos et al. 2001) and seismicity (Griggs & Handin 1960) has led to numerous studies related to the deformation of arenites (here the term arenite is used to cover a range of lithologies from poorly lithified sands to well-lithified sandstones). Deformation experiments have provided a wealth of information on the links between porosity, permeability, grain size, stress conditions and failure mode (e.g. Griggs & Handin 1960; Scott & Nielson 1991; Wong et al. 1997). Rather surprisingly, however, only a few studies have explored the implications of these relationships for faulting and fluid flow in natural environments. In particular, little attention has been focused on linking deformation mechanisms in arenites to the rheology and stress conditions that are actually encountered in the subsurface. Such omissions are partly responsible for the commonly held, but unproven, belief that fault movement is always associated with a transient permeability increase. There have been some notable exceptions. For example, Scott & Nielson (1991) noted that sandstone could ‘exhibit ductile to brittle behaviour with increasing depth’ as a result of the evolution of stress and porosity during burial.

The primary aim of this paper is to explore the links between the rheology of arenites and the stress conditions that occur within subsiding sedimentary basins. The work is based on the premise that rheology and stress conditions encountered in the subsurface are commonly related. For example, effective stress often increases during burial within a subsiding basin (Engelder 1993) and diagenetic processes decrease the porosity of arenites (e.g. Ramm 1992; Ramm & Bjørlykke 1994) and increase their strength (e.g. Zhang et al. 1990; Dyke & Dobereiner 1991; Wong et al. 1997). Consequently, certain combinations of rheology and stress are far more likely to be encountered in the subsurface than others. For example, shallowly buried arenites in the North Sea are exposed to low effective stress conditions (,20 MPa) and are poorly lithified (e.g. Bjørlykke & Høeg 1997). On the other hand, those buried to 5 km are often exposed to far higher effectives stresses (e.g. 60 MPa) but tend to be stronger because of mesodiagenetic alteration (e.g. Bjørlykke & Høeg 1997). It should also be noted that many combinations of stress and rheology are extremely unlikely to occur outside of the laboratory. For example, Hirth & Tullis (1989) presented data showing that the Oughtibridge ganister begins to experience pervasive grain fracturing

From: LEWIS , H. & COUPLES , G. D. (eds) The Relationship between Damage and Localization. Geological Society, London, Special Publications, 289, 105–121. DOI: 10.1144/SP289.7 0305-8719/07/$15.00 # The Geological Society of London 2007.

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at 1200 MPa. However, short of a meteorite impact, it is difficult to envisage any natural circumstances where a rock with a porosity of 8% would ever become subject to such huge stresses. It should also be emphasized that stress in itself has a huge influence on rheology. For example, a porous rock may behave in a brittle manner under low effective stress conditions but in a ductile manner under higher effective stresses (e.g. Scott & Nielson 1991). A secondary aim of the paper is to investigate the implications that the relationship between the rheological and stress evolution of arenites has for fault-related fluid flow and fault zone structure. In particular, we explore whether the relationship between stress and rheological evolution of arenites may explain: (1) why some faults acts as conduits to fluid flow and others act as barriers; (2) why thick zones of deformation bands are often observed in medium- to coarse-grained sandstones but are rarely, if ever, observed in very fine-grained sandstones. To further investigate the relationship between stress conditions and rheology, we use the simple model for quartz cementation published by Walderhaug (1996) to estimate how porosity varies as a function of burial depth for a variety of geothermal gradients, burial rates and grain sizes. Assuming hydrostatic conditions, we then use the burial depth to calculate the overburden stress (Fisher et al. 1999). A robust yield or failure envelope for sand or sandstone, which is similar to that developed for the theory of critical-state soil mechanics, is then used to investigate likely deformation pathways. The paper begins, however, by summarizing some observations on the deformation behaviour of sandy material from oil reservoirs; these provide a way to validate the results from the model. We then present a general background to the rheology and deformation behaviour of arenites. Finally, we present some results from our model and explore their implications for the deformation and fluid flow behaviour of faults within normally pressured sand and sandstone in subsiding basins. It should be emphasized that the model we present is extremely simple and at best can be viewed as semi-quantitative. The model does, however, provide results that are entirely consistent with petrographic observations that we have made from several hundred petroleum reservoirs throughout the world. In doing so, the model provides a ‘rule of thumb’ that allows the deformation behaviour of arenites to be predicted based on information such as grain size, as well as thermal and deformation history. The link between stress and rheological evolution also provides an explanation as to why some faults in arenites act as barriers to fluid

flow whereas others act as conduits. This is an advantage over existing models, which tend to estimate whether or not a fault is likely to act as a conduit for fluid flow based mainly on whether or not it is active or critically stressed (e.g. Barton et al. 1995).

Natural examples of deformed arenites Advances in the quality of seismic data and basin modelling software make it possible to estimate the burial and deformation history of petroleum reservoirs. Such studies allow constraints to be placed on the stress and temperature history that reservoir sandstones have experienced. Clearly, many assumptions need to be made, which result in uncertainties in estimating factors such as depth, stress and temperature at the time of faulting. In many cases, we believe that the errors in our estimates are relatively small. For example, many faults within the Middle Jurassic units of the North Sea have an extensional offset and do not appear to offset the Base Cretaceous Unconformity; there are also often observable increases in the sediment thickness in the hanging wall of these faults. We therefore assume that faulting occurred soon after sediment deposition, under low effective stress conditions, and the maximum stress resulted from the weight of the overburden. In other cases, errors in such estimates are larger. For example, the extensional faults within the Rotliegendes sequence of the southern North Sea clearly extend further into the overburden and there is usually no increase in sediment thickness in the hanging wall of such faults. We can therefore be reasonably certain that these formed during deeper burial than the Middle Jurassic faults of the northern North Sea, but we cannot be certain of the exact depth at which they formed. Despite these uncertainties, we believe that the seismic and basin modelling results from petroleum reservoirs allow us to gain far more insights into the geological history of these rocks than is usually possible in outcrop examples. Faults within core material recovered from petroleum reservoirs therefore provide a valuable resource for studies attempting to understand the controls on sandstone deformation (e.g. Fisher & Knipe 1998, 2001). Here we present four examples that highlight the general consistency between natural deformation and the results of rock mechanics experiments that are reviewed briefly in this paper. It should be noted that these examples are entirely consistent with the observations we have made of fault rock microstructures and petrophysical properties from over 300 petroleum reservoirs, within many

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of the world’s major hydrocarbon-producing provinces (e.g. Caspian Sea, Gulf of Mexico, North Sea, Norwegian Continental Shelf, offshore Brazil, etc.).

Faults within poorly lithified sands formed during shallow burial The Middle Jurassic reservoirs of the northern North Sea experienced a major period of extensional faulting during Middle Oxfordian to Early Kimmeridgian times and did not experience significant fault development or reactivation during deeper burial (Ziegler 1990; Thomson & Underhill 1993). In other words, the major phase of deformation probably occurred under mean effective stresses of ,5 MPa. We have examined fault rocks from over 40 Middle Jurassic reservoirs from the northern North Sea that have experienced burial to depths of between 2 and 5 km. All of these reservoirs are at present at their maximum burial depth. Faults within clean sandstones (,15% clay) usually classify as disaggregation zones (Fisher & Knipe 1998, 2001). In hand specimen, these are usually very subtle, isolated, features (Fig. 1) whose microstructure (grain size and porosity) is usually indistinguishable from that of the host sandstone (Fig. 2a and b). Indeed, the microstructure of the disaggregation zones is so similar to that of the surrounding sandstone that it is not obvious why the faults are actually distinguishable in core. Impure sandstones (15 –40% clay) contain thin (,1 mm wide) phyllosilicate-framework fault rocks that have reduced poroperm properties compared with their hosts (Fisher & Knipe 1998, 2001). The poroperm reductions in these fault rocks result from the mixing and compaction of the poorly sorted sediment; grain fracturing is not usually apparent (Fig. 3a and b). On the hand-specimen scale, the phyllosilicate-framework fault rocks are usually isolated features (Fig. 4a) but can occur in clusters (Fig. 4b). The clusters are not, however, the tightly packed, subparallel features as seen in the cataclastic faults from the Rotliegendes sequence described below. Instead, larger spacings tend to exist between individual faults.

Faults formed in poorly lithified sediments during deeper burial The southern North Sea experienced regional subsidence while the Rotliegendes, Zechstein and Triassic sequences were deposited. Accelerated rifting during the Jurassic resulted in the development of extensional faults. Extension ended in the Late

Fig. 1. Hand specimen containing a disaggregation zone developed in a clean Middle Jurassic sandstone from the North Sea. It should be noted that it is an isolated feature that is difficult to distinguish from its surrounding host sediment. The fault formed during shallow burial and did not undergo grain fracturing.

Jurassic to Early Cretaceous when regional uplift culminated with the formation of the Cimmerian unconformity. The area experienced compression and fault reactivation during the Late Cretaceous. Episodic inversion and subsidence occurred throughout the Tertiary. We have examined faults from over 30 Rotliegendes reservoirs from the southern North Sea. These are at present at burial depths of 2–3.5 km. Basin modelling and vitrinite reflectance results indicate, however, that they have been buried to depths of .4.5 km (Archard et al. 1998; Rock Deformation Research Limited, unpubl. data). Cataclastic faults are the most common type of fault rock present within Rotliegendes reservoirs (Fisher & Knipe 2001); they occur as isolated features (Fig. 5a) or in dense clusters (Fig. 5b). The latter are identical to the zones of deformation bands described by Aydin (1978). The extent of grain fracturing varies significantly. The isolated

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Fig. 2. BSE images showing the microstructure of (a) an undeformed sandstone and (b) the disaggregation zone that it contains. It should be noted that the microstructure of the undeformed sediment and disaggregation zone are indistinguishable.

features tend to have a broad grain-size distribution as a result of the presence of large, mostly unfractured, grains as well as small fragments produced by cataclasis (Fig. 6a). The clustered faults commonly contain discrete slip surfaces that have experienced extensive grain-size reductions (Fig. 6b). Microstructural examination of many of the individual deformation bands within these clusters using cathodoluminescence (CL) reveals that individual quartz grains are often highly fractured. The fragments produced by grain fracturing have not been widely dispersed throughout the fault rocks, indicating that they have accommodated negligible strain (i.e. ,0.5 mm). On the other hand, the individual slip surfaces appear to have far larger offsets and the grain fragments produced by cataclasis have clearly separated from their parent grains. Microstructural analysis indicates that most cataclastic faults within the

Rotliegendes reservoirs formed before extensive mesodiagenetic alteration, such as quartz cementation, illite precipitation and grain-contact quartz dissolution (commonly referred to as ‘pressure solution’). In other words, the reservoirs were probably poorly lithified at the time of faulting.

Porosity collapse in sandstone reservoirs The late-stage mechanical compaction by grainfracture-assisted porosity collapse has only recently been recognized as an important process in the solidification of some reservoir sandstones (Fisher et al. 1999). The deformation process is essentially distributed cataclastic flow in which grain fracturing is distributed throughout the reservoir and not confined to localized faults such as within the deformation bands described above. The process was first recognized as being important within a deeply

Fig. 3. BSE images showing the microstructure of (a) an undeformed sandstone and (b) the phyllosilicateframework fault rock that it contains. The fault rock has a lower porosity than the undeformed sandstone but has not undergone grain fracturing.

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Fig. 4. Hand specimens containing (a) isolated and (b) clustered phyllosilicate-framework fault rocks developed in impure Middle Jurassic sandstone from the North Sea.

Fig. 5. Hand specimen of (a) an isolated cataclastic and (b) clusters of cataclastic faults developed in a Rotliegendes reservoir from the southern North Sea.

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Fig. 6. BSE images showing cataclastic faults from within a Rotliegendes reservoir with (a) minor grain fracturing and a broad grain-size distribution, and (b) intense grain fracturing and a narrow grain-size distribution.

buried Rotliegendes reservoir in which mesodiagenetic quartz cementation had been arrested by the presence of grain-coating clays (Fisher et al. 1999). Basin modelling and vitrinite reflectance data indicate that the Rotliegendes units in the area where porosity collapse has occurred could have been buried to depths of .4.5 km. More recently, porosity collapse has been identified as severely reducing reservoir quality in other areas, particularly those that have experienced rapid burial under low geothermal gradients (Rock Deformation Research Limited, unpubl. data). The average number of fractured grains, as well as the average number of fractures per grain, increases with grain size in all reservoirs that we have studied that have experienced porosity collapse (Fig. 7a and b). For example, in an outcrop example we have recently studied from the Caspian it was found that ,10% of grains were fractured within very fine-grained sandstones, whereas every grain was fractured within some of

the medium-grained sandstones from the same sequence. It is believed that these outcrop examples were buried to up to 2.5 km, under a geothermal gradient of ,20 8C km21 and then uplifted to the surface within 6 Ma.

Faulting of quartz-cemented sandstone The location of this reservoir is confidential; however, we present a brief review of the fault rocks that it contains as they contrast with the faults from the North Sea. The reservoir sandstone is composed almost entirely of quartz grains, and has a porosity of 5–15% (Fig. 8a). The main diagenetic process that affected the reservoir was extensive quartz cementation. Although we cannot reveal precise details of the burial and thermal history of this reservoir, the main thing to note is that it was buried to sufficiently high temperatures for sufficient lengths of time that quartz cementation became sufficiently pervasive to reduce the

Fig. 7. BSE images showing (a) fine- to medium-grained sandstone that has experienced extensive porosity collapse and (b) fine-grained sandstone in which porosity collapse has not been so intense.

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Fig. 8. BSE images showing (a) quartz-cemented sandstone from a South American oilfield and (b) a calcitecemented dilational fault that it contains.

porosity of the reservoir to as little as 5%. An important feature of these reservoirs that distinguishes them from the other examples discussed above is that the last generation of faulting occurred after quartz cementation. The resulting faults had a higher porosity than the surrounding undeformed reservoir and became cemented by calcite (Fig. 8b), possibly as a result of along-fault fluid flow.

In log– log space, the product of grain radius and porosity has an inverse linear correlation with the yield strength under hydrostatic conditions, p* (Fig. 10). A regression line through the data of Wong et al. (1997) provides the following relationship:

Deformation experiments

where p* is measured in MPa, f is the fractional porosity and R is the grain radius (mm). Wong et al. (1997) developed a yield –failure criterion for sandstone, similar to those used in critical-state soil mechanics, in which differential stress, q, is plotted against effective mean stress, p, each normalized to p* (Fig. 11a). Although boundaries between deformation modes are diffuse, a first approximation would be that dilatant brittle failure occurs when p/p* is ,0.25, brittle – ductile deformation occurs when p/p* is between 0.25 and 0.4, and distributed cataclastic flow occurs when p/p* is .0.4.

Deformation experiments show that most arenites deform in a non-linear, anelastic, manner at very low (e.g. ,5 MPa) mean effective stresses (Zhang et al. 1990). Deformation occurs in an elastic manner at higher effective stresses until yield, which is marked by a rapid increase in compressibility. Two end-member modes of deformation may be distinguished based on the post-yield macroscopic structure of samples (e.g. Griggs & Handin 1960). The first, localized or brittle deformation, results in the formation of discrete slip planes that accommodate most of the strain. The stress drop following yield indicates that this is a strain-softening process (Fig. 9). The second, distributed or ductile deformation, does not result in the formation of discrete slip surfaces; instead, strain is accommodated throughout the sample. Stress–strain curves indicate that initial yield results in strain hardening. A transitional regime, often referred to as the brittle-to-ductile transition, exists in which deformation occurs along multiple slip planes (Scott & Nielson 1991). Localized deformation leads to dilation, which results in a permeability increase if porosity is ,15% at the time of faulting, or a permeability decrease if the porosity is .15% (e.g. Zhu & Wong 1997). Distributed deformation leads to compaction and decreased permeability.

logð p Þ ¼ 3:9  1:1 logðfRÞ

ð1Þ

Model for the deformation of sand and sandstone Rheological evolution of sand and sandstone during burial Quartz cementation and grain-contact dissolution are the most important diagenetic processes responsible for the densification of sandstones during burial (e.g. Rittenhouse 1971a, b; McBride 1989; Dutton & Diggs 1992; Fisher et al. 2000). These processes are largely temperature controlled and do not become pervasive until .90 8C (Walderhaug 1996). Above c. 90 8C the extent of quartz cementation and grain-contact dissolution

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(i) Distributed deformation

σ1

Stress

σ3

σ1 σ3 (ii) Brittle-ductile transition σ1 σ3

(iii)Strain localisation

Strain Fig. 9. Diagram showing typical stress– strain curves obtained during sandstone deformation experiments. Included on the plot are sketches showing the typical form of the deformed samples following the experiments. Curve (i) is typical of rocks that deformed by distributed cataclastic flow. Curve (ii) is typical of rocks deformed at the brittle– ductile transition. Curve (iii) is typical of rock deformed by brittle failure. The diagram is based on Griggs & Handin (1960).

increase with time and temperature, resulting in decreased porosity and increased strength. Walderhaug (1996) presented a simple model to estimate the quartz cement content of sandstone based on the reactive quartz surface area and temperature history. The model assumes that, at a constant temperature, the volume of quartz cement, Vq (in cm3), that can be precipitated in 1 cm3 of sandstone can be related to the quartz surface area, A (in cm2), present and time t (in s), using the equation Vq ¼ MrAt=r

ð2Þ

where M is the molar mass of quartz (60.09), r is the rate of quartz precipitation (moles cm22 s21) and r is the density of quartz (2.65 g cm23). This was differentiated with respect to time to give dVq =dt ¼ MrA=r:

ð3Þ

Walderhaug (1996) also showed that the surface area of quartz, A, can be related to the surface area, A0, and the porosity, f0, at the start of quartz cementation and Vq by the equation A ¼ A0 ðf0  Vq Þ=f0 :

ð4Þ

This can be substituted into equation (3) to give the following ordinary differential equation for Vq: dVq =dt ¼ A0 ðf0  Vq ÞMr=ðf0 rÞ:

ð5Þ

The relationship between A0 and grain size presented by Walderhaug (1996) was A0 ¼ 3f V=R

ð6Þ

where A0 is equal to the cumulative surface area of spheres with a radius R and there is a total volume of detrital quartz f in a unit volume V. This estimate

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3.5

log p* = -1.1 log (Rø) + 3.9

R2 = 0.86

3

log p* (MPa)

2.5 2 1.5 1 2

log p* = -1.1 log(ϕRø) + 3.5

0.5

R2 = 0.86

0 0

0.5

1

1.5

2

2.5

3

3.5

log (φRø) Fig. 10. Log– log plot of p* against the product of grain radius and porosity. Included is the regression line through the data (black). The grey line and equation represent the minimum boundary to the data and were used to calculate p* values in the simulations presented. Data are from Wong et al. (1997).

of surface area, along with the quartz cementation rates calculated by Walderhaug (1994), were substituted into equation (5), which for a constant rate of sedimentation and geothermal gradient offers a simple analytical solution. We simulated the evolution of p* during burial for a variety of geothermal gradients, burial rates and grain sizes. At each depth we determined the vertical stress assuming lithostatic loading and hydrostatic pore fluid pressures. The lower bound to the data shown in Figure 10 was taken to calculate p* because rocks are likely to be weaker in nature than in the laboratory experiments as a (a) 0.8 0.7

(b) 0.8 0.7

ail

ure

n

ittl ef

0.3

0.6 0.5 0.4 0.3

Br

q/p*

0.5

q/p*

Duc t defo ile rma tio

0.6 0.4

result of scale (e.g. Singh & Huck 1973) and time (e.g. Lawn 1993) effects as well as the presence of an active fluid (Baud et al. 2000). The yield – failure curve of Wong et al. (1997) was approximated to an elliptical form (Fig. 11b) and then used to calculate the value of the second principal stress plane that was required for yield –failure based on the value of p* derived from the quartz cementation model of Walderhaug (1996). Two values are obtained: a lower value representing a normal fault regime and a higher value representing a thrust regime; in this paper we concentrate only on the former. These results were then used to

0.2

0.2

0.1

0.1

0 0

0.2

0.4

0.6 p/p*

0.8

1

0

0

0.2

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0.8

Fig. 11. (a) Yield– failure envelope developed by Wong et al. (1997); (b) the elliptical form for the yield–failure envelope used in the present paper.

1

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calculate p/p* at yield– failure to provide information on the likely failure mode. It should be emphasized that, because the rate of quartz cementation increases exponentially with temperature, the model predictions are not significantly affected by the exact values of p/p* chosen as cut-offs between brittle, brittle– ductile and ductile behaviour. For example, if a p/p* value of 0.5 was used to distinguish between brittle and brittle– ductile deformation, for an arenite with a grain size of 0.5 mm buried under a geothermal gradient of 30 8C km21 at a rate of 0.05 km Ma21, the model would predict that a change from brittle– ductile to ductile behaviour would occur at a burial depth of 4.3 km, as opposed to 4.6 km when assuming a p/p* cut-off of 0.25.

Model results Evolution of deformation of arenites during burial We first investigate how p/p*, porosity and quartz cement content vary during burial. The burial of p/p*

(a) 0.0

0.2

0.4

sand with an initial porosity of 30%, at a rate of 0.05 km Ma21, under a geothermal gradient of 30 8C km21 was simulated; five grain sizes were used: 0.1 mm, 0.2 mm, 0.3 mm, 0.5 mm and 1 mm. The results all show an increase in p/p* at failure until a maximum value after which p/p* begins to fall (Fig. 12). The magnitude of p/p* and the depth at which the maximum value occurs vary significantly as a function of grain size. For example, the results suggest that sand with a grain size of ,0.2 mm yields a p/p* of ,0.25 for the burial rate and geothermal gradient used in the simulations. In other words, deformation occurs by brittle faulting throughout burial. Sand with a grain size of 0.3 mm reaches values of p/p* of .0.25 at depths between 2.3 and 3.8 km. At depths of .3.8 km the ratio of p/p* falls below 0.25, suggesting that the failure mode returns to brittle dilational faulting; this change in deformation behaviour is known as the ductile-to-brittle transition (DBT; Fisher et al. 2003). Values of p/p* at yield increase above 0.4 for sands with a grain size .0.5 mm, suggesting that they would be susceptible to cataclastic flow. Porosity

(b) 0.6

0.8

1.0

0.0

0.1

0.2

0.3

0.4

0

1

0.1 mm 0.2 mm 0.3 mm 0.5 mm 1 mm

Burial depth (km)

2

3

4

5

6

Fig. 12. Plots of (a) p*/p and (b) porosity against burial depth for sandstone with an initial porosity of 30%, buried at a rate of 0.05 km Ma21, under a geothermal gradient of 30 8C km21, and with a grain-size of 0.1– 1 mm.

DUCTILE –BRITTLE TRANSITION IN SANDSTONES

0.00 0 2

0.05

p/p* 0.10 0.15

0.20

0.25

10 °C/km 20 °C/km 30 °C/km 40 °C/km

Burial depth (km)

4 6 8 10 12 14 16

Fig. 13. Plots of p/p* against depth for a sandstone with a grain size of 0.1 mm, buried at 0.1 km Ma21, under geothermal gradients of 10–40 8C km21.

The porosity of sandstone is an important control on whether brittle faulting increases or decreases permeability (Zhu & Wong 1997). We have therefore plotted porosity evolution during burial so that it can be compared with p/p* values (Fig. 12b). The results from the simulations indicate that arenites have a porosity of .15% when they deform by brittle faulting at shallow depths. However, arenites tend to have porosities of ,15% at depths deeper than the DBT. A fine-grained sandstone (,0.1 mm) buried under these conditions did not attain p/p* values of .0.1 in these simulations. We therefore ran a series of simulations at increased burial rates (0.1 km Ma21) and a variety of geothermal gradients (10–40 8C km21) to investigate whether higher p/p* values were obtained. The results indicate that p/p* does not reach 0.25, even for burial under a geothermal gradient as low as 10 8C km21 (Fig. 13).

Controls on the depth of the base of the ductile-to-brittle transition To investigate the controls on the depth of the DBT we simulated the burial of an arenite with a grain size of 0.3 mm for burial rates of 0.05–1 km Ma21 and geothermal gradients of 10 –80 8C km21. The

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change in p/p* as a function of depth, temperature and porosity for each simulation was noted. The effects of geothermal gradient and burial rate on the depth of the DBT are shown in Figure 14a. For burial under geothermal gradients of 60 and 80 8C km21 at rates of 0.1 and 0.05 km Ma21, the value of p/p* remained ,0.25, suggesting that similar rocks experiencing this burial – geothermal history would remain susceptible to only brittle failure throughout burial. The simulations suggest that increases in geothermal gradient are associated with significant reductions in the depth of the DBT. For example, the DBT occurs at c. 15 km under a geothermal gradient of 10 8C km21, whereas it occurs at c. 2 km under a geothermal gradient of 40 8C km21. There is a general decrease in the depth of the DBT with decreasing burial rate, although this effect is small in comparison with that caused by variations in geothermal gradient. It should be noted that, although these large depths quoted above (i.e. 15 km) are normally associated with crystal plastic behaviour, the very low geothermal gradients in the situation modelled would suppress this process. It appears that the DBT in arenites deposited under high geothermal gradients occurs at shallow depth where porosities are high (Fig. 14b). Arenites deposited under low geothermal gradients continue to behave in a ductile manner during deep burial when quartz cement contents are high and porosities are low. The temperature at the DBT increases from 130 8C for burial rates of 0.05 km Ma21 to 200 8C km21 for burial rates of 1 km Ma21 for geothermal gradients of 10 – 40 8C km21 (Fig. 14c). The temperature at the DBT is relatively insensitive to the geothermal gradients over this range.

Discussion Comparison between nature, laboratory experiments and simulations The model presented within the present paper makes many assumptions and simplifications, which limit the extent to which it can be applied in a quantitative manner. For example, the model of Walderhaug (1996) is a simplification in that it assumes that silica is supplied by local processes and that the amount of quartz cement is controlled by its precipitation kinetics. The model also does not take into account factors such as stress conditions, water chemistry, pH, etc. Despite its simplicity, the quartz cementation model of Walderhaug (1996) has provided accurate estimates of the amount of quartz cement in reservoirs

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(a)

Geothermal gradient (oC/km) 0

10

20

30

40

50

Burial depth (km)

0 5 10 15

0.05 km/M a 0.1 km/Ma 0.5 km/Ma 1 km/Ma

20 25

(b)

25

Porosity (%)

20 15 10 5 0 0

10

0

10 20 30 40 Geothermal gradient (oC/km)

20 30 40 o Geothermal gradient ( C/km)

50

Temperature(oC)

(c) 250 200 150 100 50 0 50

Fig. 14. Plots of (a) depth of the DBT, (b) porosity and (c) temperature at the DBT against geothermal gradient for a medium-grained sandstone with an initial porosity of 30% buried at rates of 0.05–1 km Ma21.

DUCTILE –BRITTLE TRANSITION IN SANDSTONES

with a wide range of geological ages, buried at varied rates under a wide range of geothermal gradients (Lander & Walderhaug 1999; Walderhaug et al. 2000). We also assume spherical grains, with uniform grain size, and that quartz cementation is the only process responsible for altering the rheology of arenites during burial. Large uncertainties exist as to how the relationship between p*, porosity and grain size established in the laboratory can be applied to nature, because of differences in scale, strain rate, etc. In addition, the effects of overpressure, which will be particularly significant in rapidly buried subsiding basins, have not been considered. Our combined quartz cementation –rock deformation model does appear to explain many of the deformation phenomena that we observe in nature, despite these shortcomings. The most obvious analogies between the results from the model and nature can be made by comparing the chronology of deformation, particularly its relation to quartz cementation. The earliest faults to form were those in the Middle Jurassic reservoir (Figs 1–4) and are similar to those formed in the laboratory by shear localization. The deformation bands within the Rotliegendes units (Figs 5 and 6) formed during deeper burial than for the Middle Jurassic examples but prior to quartz cementation. The clusters of deformation bands within the Rotliegendes samples have a similar structure to the deformation features formed in the laboratory at the brittle –ductile transition (e.g. Scott & Nielson 1991). Distributed grain fracturing in the Rotliegendes units also occurs before pervasive quartz cementation, but only within reservoirs that have experienced the deepest burial (Fisher et al. 1999). This distributed grain fracturing (Fig. 7) seems analogous to rocks deformed in the laboratory by cataclastic flow. The calcite-cemented faults in the quartz-cemented sandstone (Fig. 8) indicate that faulting occurred in a brittle manner, resulting in an increase in both porosity and permeability. These analogies indicate that: (1) faults in Middle Jurassic reservoirs and those within the quartz-cemented sandstones formed when p/p* was ,0.25; (2) deformation bands in the Rotliegendes units formed when p/p* was 0.25– 0.4; (3) reservoirs experience significant porosity collapse at p/p* values of .0.4. Thus, the natural examples have the same order of deformation as predicted by the simulations; that is, an increase in p/p* with increased burial followed by a decrease in p/p* after the onset of quartz cementation. The basic reason for the behaviour predicted by the simulations stems from the temperature dependence of quartz cementation. The temperature

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is too low for pervasive quartz cementation during initial burial. The sands therefore become exposed to ever-increasing vertical stresses without a significant increase in strength. Such a situation favours ductile behaviour. Once quartz cementation becomes pervasive there is a rapid increase in strength, allowing the rocks to support higher differential stresses favouring brittle failure. The model has also been successfully applied to predict phenomena outside the areas in which it was calibrated. In particular, this model predicted that coarse-grained sands, buried rapidly under a low geothermal gradient, would be particularly susceptible to porosity collapse (Fisher et al. 1999). Recent work on such sediments in a range of sedimentary basins has since proved this to be the case (Rock Deformation Research Limited, unpubl. data). We therefore believe that the model provides a useful tool to make qualitative predictions on the controls on the deformational behaviour of arenites during burial. The remainder of this paper will focus on examining the implications of these results.

Grain-size dependence on deformation style: implications for fault zone structure The results from the simulations clearly predict a grain-size dependence on deformation style. The observation that porosity collapse within sandstone reservoirs is far more pervasive in coarser-grained horizons clearly supports this prediction. Additionally, the simulations suggest that very high burial rates and low geothermal gradients are needed for fine-grained sandstones to deform in a brittle – ductile manner during burial in a normal faulting regime. Models of deformation band development commonly suggest that initial strain-hardening causes a widening of the zones of deformation bands until strain becomes localized along a single slip plane (e.g. Aydin 1978; Mair et al. 2000). It is probably valid to assume that this behaviour is indicative of rocks deforming at the brittle –ductile transition. The results from the simulations indicate that increased grain size is associated with an increased tendency to deform in a ductile or brittle– ductile manner. Hence, the simulations may explain the observations that fault damage zones are far thinner in fine-grained rocks than in coarser sandstone (Antonellini & Aydin 1995; Knott et al. 1996; Beach et al. 1999).

Implications for fault-related fluid flow The results from the simulations indicate three distinct stages in the deformation behaviour of

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(a) 0.1

0.2

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p/p* < 0.25,φ > 15%, brittle faulting, fault permeability is the same or lower than host, localised faulting without thick zones of deformation bands

σ1 1

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Deformation mode, fault permeability, fault zone structure

Porosity 0.0 0

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σ1 1

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Deformation mode, fault permeability, fault zone structure

Porosity

(b)

σ3

2

σ1

of deformation bands

σ1

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σ3

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σ3 fault permeability is lower than host, thick zones

3

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(C)

Deformation mode, fault permeability, fault zone structure

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brittle-ductile deformation 2

σ1 3

σ3

p/p* > 0.4, ductile deformation, grain-fracturing and porosity collapse distributed throughout sediment

4

σ1 5

6 0.0

brittle-ductile deformation p/p* < 0.25,φ < 15%, brittle faulting, fault permeability is the same or higher than host, localised faulting without thick zones of deformation bands

σ3 0.2

0.4

0.6

0.8

1.0

p/p*

Fig. 15. Plot of porosity (black) and p/p* (grey) against burial depth for (a) fine-grained (R ¼ 0.05 mm), (b) medium-grained (R ¼ 0.125 mm) and (c) coarse-grained (R ¼ 0.25 mm) arenites with an initial porosity of 30%, buried at a rate of 0.05 km Ma21, under a geothermal gradient of 30 8C km21. Predicted modes of deformation, based on porosity and p/p*, and their consequences for fault permeability and fault zone architecture are shown.

DUCTILE –BRITTLE TRANSITION IN SANDSTONES

many arenites. The first stage occurs after initial sediment deposition, where p/p* values are ,0.25 and deformation is likely to occur by brittle faulting. At extremely low stresses this is likely to result in the formation of disaggregation zones, which do not significantly affect fluid flow (Fisher & Knipe 1998, 2001). Under higher effective stresses, it is likely that arenites will deform to produce dilational cataclasites. The porosity of the arenite at this stage is ,15% and therefore the cataclasites will have reduced permeability (see Zhu & Wong 1997). The second stage occurs during deeper burial, when p/p* values reach .0.25. Here deformation will occur in a brittle–ductile or ductile manner, resulting in decreased permeability. The last stage occurs following the onset of quartz cementation and p/p* values fall back below 0.25. The porosity of the arenite will be ,15% for most burial rates and geothermal gradients, resulting in the formation of dilational faults with increased permeability. In other words, the results indicate that the faulting of porous sandstones often leads to a permeability reduction until below the DBT when they deform to produce conduits for fluid flow. It has long been recognized that some faults can act as conduits (e.g. Link 1952; Ramsay 1980; Sibson 1981; Thrasher et al. 1996) whereas others restrict fluid flow (e.g. Smith 1966; Watts 1987; Knipe et al. 1997; Gibson 1998; Fisher & Knipe 1998, 2001). It is commonly argued that the reason for this seemingly paradoxical behaviour is that active or critically stressed faults behave as conduits whereas inactive faults, or those that are not critically stressed, act as barriers to fluid flow (e.g. Sibson 1990; Barton et al. 1995; Wiprut & Zoback 2000). The model of fault permeability described in this paper explains the variability in the fluid flow behaviour without needing to invoke dramatic changes in fault permeability with activity. We view this as both an alternative and an improvement to many existing models because fault rocks in nature often do not show evidence of fault-related rock–fluid reaction, which we interpret to suggest that they have not acted as significant conduits for fluid flow despite once being active (e.g. Fisher & Knipe 1998, 2001; Fisher et al. 2003). It seems likely that the depth of the DBT will vary between sedimentary basins depending on geothermal gradient and burial rate. It is also likely that the DBT will be somewhat blurred in any single basin because the rate of change of rheology with depth is dependent on factors such as sediment composition and grain size. In other words, some horizons will reach the DBT at different depths to others. A diagram summarizing the effects of burial depth and grain size on fault zone permeability and architecture is provided in Figure 15.

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For significant fluid flow to occur along faults it is necessary that they have significantly higher permeability than the surrounding rock mass and that significant amounts of fluid are present. In other words, faults are most likely to act as conduits for significant fluid flow if they form when there is still a significant amount of fluid within the rock (Bjørlykke & Høeg 1997). The simulation results presented here indicate that the condition that favours fluid flow conduits in porous sediments is a very steep geothermal gradient. It seems that early cementation by quartz as a result of the high geothermal gradients allows the rocks to behave in a brittle manner very early. Sands deposited under low geothermal gradients become buried under very high confining stresses while they are still weakly cemented, a condition that favours ductile behaviour. Such arguments indicate the possibility of a link between fault mineralization and high geothermal gradients. In should be noted that observations that we have made from petroleum reservoirs also show that fault-related conduits often form by the deformation of arenites during uplift. In soil mechanics terminology, such sediments would be regarded as heavily overconsolidated. As our model deals solely with faulting in subsiding basins this situation was not discussed in this paper.

Conclusions (1) Sandstone strength increases in a non-linear manner during burial, as a result of diagenetic reactions such as quartz cementation. Effective stress also tends to increase during burial. The interaction between stress and rheology is controlled by factors such as grain size, burial rate and geothermal gradient. Linking a simple model for quartz cementation with a yield –failure envelope allows the relationships between these variables to be explored. (2) The model indicates that the failure mode of medium- and coarse-grained arenites buried under geothermal gradients of between 10 and 40 8C km21 will vary significantly during burial. Initially, they will deform by brittle faulting. The resulting fault rocks will have permeabilities the same as or lower than that of their host. At greater depths, sandstone will become prone to deformation by brittle –ductile and ductile processes, resulting in decreased permeability. Still deeper, sandstone will reach the DBT, after which it will be prone to brittle deformation, resulting in increased permeability. The tendency of a fault to act as a conduit or barrier is therefore dependent on the interaction of rheology and stress.

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(3) Fine-grained arenites tend to become quartz cemented before they reach effective stresses that are sufficiently high for brittle deformation, possibly explaining why thick zones of deformation bands are far more commonly observed in coarsegrained arenites than in finer-grained sandstones. (4) The depth of the DBT varies significantly, depending particularly on geothermal gradient and to a lesser extent on burial rate. The DBT occurs at far greater depths in basins with a low geothermal gradient than those where the gradient is steeper. (5) The porosity at the DBT is higher in basins with high geothermal gradients, possibly making them more prone to fault-related mineralization. The authors would particularly like to thank their sponsors (Amerada Hess, BG, BP, ChevronTexaco, ConocoPhilips, Encana, Kerr–McGee, Petrobras, Norskhydro, Shell, Statoil, Total) for providing financial support, core material, supporting data and stimulating discussion over the last 12 years.

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in Hydrocarbon Reservoirs. Geological Society, London, Special Publications, 147, 117– 134. F ISHER , Q. J. & K NIPE , R. J. 2001. The permeability of faults within siliciclastic petroleum reservoirs of the North Sea and Norwegian Continental Shelf. Marine and Petroleum Geology, 18, 1063–1081. F ISHER , Q. J., C ASEY , M., C LENNELL , M. B. & K NIPE , R. J. 1999. Mechanical compaction of deeply buried sandstones. Marine and Petroleum Geology, 16, 605–618. F ISHER , Q. J., K NIPE , R. J. & W ORDEN , R. 2000. The microstructure of deformed and undeformed sandstones from the North Sea: its implications for the origin of quartz cement. In: W ORDEN , R. H. & M ORAD , S. (eds) Quartz Cementation in Sandstones. International Association of Sedimentology, Special Publications, 29, 129–146. F ISHER , Q. J., C ASEY , M., H ARRIS , S. D. & K NIPE , R. J. 2003. The fluid flow properties of faults in sandstone: the importance of temperature history. Geology, 31, 965–968. G IBSON , R. G. 1998. Physical character and fluid flow properties of sandstone-derived fault zones. In: C OWARD , M. P., D ALTABAN , T. S. & J OHNSON , H. (eds) Structural Geology in Reservoir Characterization. Geological Society, London, Special Publications, 127, 83–98. G RIGGS , D. T. & H ANDIN , J. W. 1960. Observations on fracture and a hypothesis of earthquakes. In: GRIGGS , D. T. & HANDIN , J. (eds) Rock Deformation. Geological Society of America, Memoirs, 79, 347– 364. H ETTEMA , M. H. H., S CHUTJENS , P. M. T. M., V ERBROOM , B. J. M. & G USSINKLO , H. J. 2000. Production-induced compaction of a sandstone reservoir: the strong influence of stress path. SPE Reservoir Evaluation and Engineering, 3, 342– 347. H IRTH , G. & T ULLIS , J. 1989. The effects of pressure and porosity on the micromechanics of the brittle–ductile transition in quartzite. Journal of Geophysical Research, 94, 17825–17838. K NIPE , R. J., F ISHER , Q. J. ET AL . 1997. Fault seal analysis: successful methodologies, application and future directions. In: M OLLER -P EDERSON , P. & K OESTLER , A. G. (eds) Hydrocarbon Seals: Importance for Exploration and Production. Norwegian Petroleum Society, Special Publications, 7, 15–40. K NOTT , S. D., B EACH , A., B ROCKBANK , P. J., B ROWN , J. L., M C C ALLUM , J. E. & W ELBON , A. W. 1996. Spatial and mechanical controls on normal fault populations. Journal of Structural Geology, 18, 359–372. L ANDER , R. H. & W ALDERHAUG , O. 1999. Porosity prediction through simulation of sandstone compaction and quartz cementation. AAPG Bulletin, 83, 433–449. L AWN , B. 1993. Fracture of Brittle Solids, 2nd edn. Cambridge University Press, Cambridge. L INK , W. K. 1952. Significance of oil and gas seeps in world oil exploration. AAPG Bulletin, 36, 1505–1539. M AIR , K., M AIN , I. & E LPHICK , S. 2000. Sequential growth of deformation bands in the laboratory. Journal of Structural Geology, 22, 25– 42. M C B RIDE , E. F. 1989. Quartz cement in sandstone. A review. Earth-Science Reviews, 26, 69– 112.

DUCTILE –BRITTLE TRANSITION IN SANDSTONES M OOS , D., Z OBACK , M. D. & B AILEY , L. 2001. Feasibility study of the stability of openhole multilaterals, Cool Inlet, Alaska. SPE Drilling and Completion, 16, 140–145. R AMM , M. 1992. Porosity–depth trends in reservoir sandstones: theoretical models related to Jurassic sandstones, offshore Norway. Marine and Petroleum Geology, 9, 553–567. R AMM , M. & B JØRLYKKE , K. 1994. Porosity/depth trends in reservoir sandstones: assessing the quantitative effects of varying pore pressure, temperature history and mineralogy, Norwegian Shelf data. Clay Minerals, 29, 475–490. R AMSAY , J. 1980. The crack-seal mechanism of rock deformation. Nature, 284, 135 –139. R ITTENHOUSE , G. 1971a. Pore space reduction by solution and cementation. AAPG Bulletin, 55, 80– 91. RITTENHOUSE, G. 1971b. Mechanical compaction of sands containing different percentages of ductile grains: a theoretical approach. AAPG Bulletin, 55, 92–96. S COTT , T. E. & N IELSON , K. C. 1991. The effects of porosity on the brittle– ductile transition in sandstones. Journal of Geophysical Research, 96, 405–414. S IBSON , R. H. 1981. Fluid flow accompanying faulting: field evidence and models. In: S IMPSON , D. W. & R ICHARDS , P. G. (eds) Earthquake Prediction: an International Review. Maurice Ewing Series, American Geophysical Union, 4, 593–603. S IBSON , R. H. 1990. Conditions for fault-valve behaviour. In: K NIPE , R. J. & R UTTER , E. H. (eds) Deformation mechanisms, Rheology and Tectonics. Geological Society, London, Special Publications, 54, 15– 28. S INGH , M. M. & H UCK , P. J. 1973. Large scale triaxial tests on rocks. In: HARDY , H. R. & STEFANKO , R. (eds) New Horizons in Rock Mechanics, Proceedings of the 14th Symposium on Rock Mechanics, Pennsylvania State University. American Society of Civil Engineers, New York, 35–60. S MITH , D. A. 1966. Theoretical consideration of sealing and non-sealing faults. AAPG Bulletin, 50, 363–374. T HOMSON , K. & U NDERHILL , J. R. 1993. Controls on the development and evolution of structural styles in the Inner Moray Firth, UK North Sea. In: P ARKER , J. R. (ed.) Petroleum Geology of Northwest Europe, Proceedings of the 4th Conference. Geological Society, London, 1167–1178. T HRASHER , J., H OVLAND , M., F LEET , A. J., D U¨ PPENBECKER , S. & H AY , S. J. 1996. Understanding geology and the key to using seepage in

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exploration: the spectrum of seepage styles. In: S CHUMACHER , D. & A BRAMS , M. A. (eds) Hydrocarbon Migration and its Near-Surface Expression. AAPG Memoirs, 66, 223 –242. W ALDERHAUG , O. 1994. Precipitation rates for quartz cement in sandstones determined by fluid-inclusion microthermometry and temperature– history modelling. Journal of Sedimentary Research, A64, 324– 333. W ALDERHAUG , O. 1996. Kinetic modeling of quartz cementation and porosity loss in deeply buried sandstone reservoirs. AAPG Bulletin, 80, 731– 745. W ALDERHAUG , O., L ANDER , R. H., B JØRKUM , P. A., OELKERS, E. H., B JØRLYKKE , K. & N ADEAU , P. H. 2000. Modelling quartz cementation and porosity in reservoir sandstones—examples from the Norwegian continental shelf. In: WORDEN , R. & MORAD , S. (eds) Quartz Cementation in Sandstones. International Association of Sedimentologists, Special Publications, 29, 39–49. W ATTS , N. L. 1987. Theoretical aspects of cap-rock and fault seals for single and two phase hydrocarbon columns. Marine and Petroleum Geology, 4, 274– 307. W IPRUT , D. & Z OBACK , M. D. 2000. Fault reactivation and fluid flow along a previously dormant normal fault in the northern North Sea. Geology, 28, 595– 598. W ONG , T.-F., D AVID , C. & Z HU , W. 1997. The transition from brittle faulting to cataclastic flow in porous sandstones: mechanical deformation. Journal of Geophysical Research, 102, 3009– 3025. Z HANG , J., W ONG , T.-F. & D AVIS , D. M. 1990. Micromechanics of pressure-induced grain crushing in porous rocks. Journal of Geophysical Research, 95, 341– 352. Z HU , W. & W ONG , T.-F. 1997. The transition from brittle faulting to cataclastic flow: permeability evolution. Journal of Geophysical Research, 102, 3027– 3041. Z IEGLER , P. A. 1990. Tectonic and palaeogeographic development of the North Sea rift system. In: B LUNDELL , D. J. & G IBBS , A. D. (eds) Tectonic Evolution of the North Sea Rifts. Oxford Science Publications, Oxford, 1 –36. Z OBACK , M. D. & B YERLEE , J. D. 1976. Effect of high-pressure deformation on permeability of Ottawa Sand. AAPG Bulletin, 60, 1531– 1542.

Fracture aperture, length and pattern geometry development under biaxial loading: a numerical study with applications to natural, cross-jointed systems J. E. OLSON Petroleum and Geosystems Engineering Department, University of Texas at Austin, 1 University Station C0300, Austin, TX 78746, USA (e-mail: [email protected]) Abstract: Fracture mechanics modelling of fracture pattern development was used to analyse pattern geometry and population statistics for natural opening-mode fractures. Orthogonal fracture network geometries were generated under biaxial extension loading conditions from a slightly anisotropic initial strain state. Fracture statistics were analysed by grouping all fracture orientations into one population for these unique orthogonal pattern geometries. Fracture aperture distributions resembled negative exponential curve shapes, consistent with published observations for stratabound fractures in sedimentary rock. Fracture length distributions had a strongly power-law shape, and showed that longer fractures grew first and reached their fullest extent before shorter fractures began propagating. The power-law shape of the length distribution was first established by the growth of the longest fractures in the population, followed by the later growth of shorter fractures that extended the power-law shape to smaller sizes. The shortest fracture length at which the power-law distribution was truncated varied with the magnitude of the applied strain. Other variations in fracture pattern results were tied to mechanical layer thickness and subcritical crack growth propagation properties of the fractured media.

The characterization of natural, opening-mode fractures is a challenging task, particularly in the subsurface, where only limited information is available. Discovering the important processes responsible for fracture pattern development and the fundamental systematics of fracture attribute populations is essential for fracture analysis. Important fracture attributes include fracture orientation, planarity, spacing, aperture (opening displacement), and length. It is well established from mechanics that opening-mode fractures form perpendicular to the local least compressive principal stress (Lawn & Wilshaw 1975; Pollard & Aydin 1988). For conditions favouring vertical fractures (such as when the maximum compressive stress is vertical), this implies that an isolated opening-mode fracture (free from any mechanical interaction with other nearby fractures) propagates perpendicular to the remote least compressive horizontal stress, shmin. Based on both empirical and theoretical analysis, average fracture spacing is often noted to be controlled by mechanical layer thickness (e.g. Ladeira & Price 1981; Narr & Suppe 1991; Bai & Pollard 2000), with the added distinction that the amount of clustering in the pattern may vary as a function of mechanical crack interaction and the degree of pattern development (Delaney et al. 1986; Rives et al. 1992; Olson 1993; Wu & Pollard 1995; Gillespie et al. 2001; Olson 2004). The degree of planarity of a fracture (how straight it propagates) has been interpreted to be primarily controlled by

the stress state and the influence of other nearby fractures (Pollard et al. 1982; Olson & Pollard 1989; Bai & Gross 1999). For nominally horizontal, bedding plane outcrops, it can be said that crack path curving is enhanced by close crack-to-crack proximity and by a low in-plane differential stress (shmax 2 shmin). Straighter crack paths result when fractures are far apart and when the in-plane differential stress is high. Fracture size attributes such as aperture and length are often analysed with respect to their population statistics. Many researchers have proposed that the cumulative frequency of fracture aperture populations can be described as power-law functions (e.g. Sanderson et al. 1994; Clark et al. 1995; Marrett et al. 1999; Ortega & Marrett 2000). Gillespie et al. (1999) found a distinction between stratabound (fracture height contained by mechanical layer thickness) and non-stratabound opening-mode fractures, observing power-law aperture distributions for non-stratabound fractures and non-power-law distributions (normal, log-normal or negative exponential) for stratabound fractures. Ortega & Marrett (2000) also observed an influence of mechanical layer thickness on aperture distributions, but interpreted all distributions as powerlaw, with different fitting exponents depending on whether fracture length was less than or greater than mechanical layer thickness. Gillespie et al. (2001) also observed length distributions to be influenced by fracture height containment,

From: LEWIS , H. & COUPLES , G. D. (eds) The Relationship between Damage and Localization. Geological Society, London, Special Publications, 289, 123–142. DOI: 10.1144/SP289.8 0305-8719/07/$15.00 # The Geological Society of London 2007.

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observing lognormal distributions for stratabound and power-law distributions for non-stratabound fractures. Ortega & Marrett (2000) interpreted their fracture length distributions as power-law, but emphasized that length determination is not straightforward for natural fractures, particularly when they can be made up of en echelon segments that may or may not appear linked depending on the scale of observation. Although Ackermann et al. (2001) studied experimentally generated normal faults, not opening mode fractures, it is worth noting that they found the nature of fault length data to change depending on the magnitude of the strain, with power-law length distributions at low strain and negative exponential length distributions at higher strain. They also found mechanical layer thickness to be important in controlling fault length and displacement development. Most of the work on attributes discussed above was limited to singling out a particular set of nominally parallel fractures, even when multiple sets were present, because multiple fracture orientations are typically interpreted to be of different age and thus caused by different deformation events (Pollard & Aydin 1988). However, there are special cases where multiple fracture orientations can result from a single deformational event. Polygonal fracture patterns, such as those found in mudcracks and columnar jointing of volcanic rock, have nominally random orientations of fractures that can be linked to a single causative event, usually an isotropic tension caused by desiccation or cooling (Lachenbruch 1962; DeGraff & Aydin 1993). Orthogonal patterns may also be generated from a single deformational event, and are very commonly found in outcrops of thin sedimentary units. The focus of this study is patterns that are typically referred to as cross-jointed or ladder patterns (Pollard & Aydin 1988; Rawnsley et al. 1992; Gross 1993; Rives et al. 1994). An important attribute of cross-jointed fracture patterns is that there is typically an older, through-going fracture set that is very straight and regularly spaced (Fig. 1; Gross 1993; Bai et al. 2002). Later cross-joints typically propagate perpendicular to the through-going set and abut into them, although there can be a wide variation in the planarity and orientation of these later fractures. As described by Bai et al. (2002), there are multiple mechanisms that can be responsible for the generation of orthogonal fracture sets. The mechanism of interest in this study has been termed stress release, where the tensile stresses caused by a remotely applied extensional strain can be completely relieved by crack propagation perpendicular to that strain (Olson 1993; Wu & Pollard 1995). This stress release builds over time as the fracture pattern develops, and when the initial fracture set

becomes well-developed (through-going and regularly spaced), the local principal stress directions can change, causing later fractures to grow in a different (typically orthogonal) direction. Bai & Pollard (2000) illustrated how the reduction in tensile stress in one direction, induced by fracture propagation, can locally cause the principal stress directions to flip by 908, resulting in a subsequent flip in fracture propagation direction. They identified values of stress anisotropy and the ratio of fracture spacing to mechanical layer thickness for which cross-jointing is expected to occur. In addition to the stress analysis work exemplified by the study by Bai & Pollard (2000), several studies have been performed where fracture propagation was explicitly modelled. Olson (1997) simulated cross-joint development using an isotropic initial state followed by monotonic uniaxial extension, generating an early set of nominally parallel, through-going fractures between which later cross-joints developed, but the earlier fractures varied considerably in their orientation and were not as universally through-going as often observed in the field. Similar simulations run with biaxial extension loading from an isotropic initial state (Tuckwell et al. 2003; Rijken 2005; Olson et al. 2007) resulted in more polygonal fracture geometries. Olson et al. (2007) also showed that an initial straight fracture propagation followed by cross-jointing can be achieved under biaxial extension given a small initial strain anisotropy, but if that initial anisotropy is too large, it can prevent the principal stress direction change induced by stress release (Bai et al. 2002), suppressing the formation of cross-joints. This study further analyses the systematics of fracture patterns that grow under biaxial extension, conditions that are conducive to orthogonal pattern development. A process-oriented approach is adopted, using mechanics-based numerical experiments to examine the development of pattern geometry and fracture attribute distributions through time. The effects of mechanical layer thickness and subcritical index (stress corrosion cracking) are investigated.

The fracture mechanics model Natural, opening-mode fractures have been postulated to initiate from rock imperfections such as microcracks, grain boundary defects, eccentrically shaped pores, fossil fragments and flute casts (Engelder 1987; Pollard & Aydin 1988; Engelder & Lacazette 1990). These mechanical imperfections or flaws can range in size from tens of microns to a few centimeters. The numerical modelling presented here mimicked this initial flawed

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Fig. 1. Bedding plane exposures of cross-jointed natural fracture patterns showing planar, through-going joint sets (trend indicated with white arrow) with younger cross-joint development. (a) Jurassic limestone, Bristol Channel, UK. Pattern orientations vary from bed to bed. Simplest cross-joint pattern is in bed in foreground. (b) Jurassic sandstone, Oil Mountain, Wyoming, USA. (c) Triassic sandstone, Comb Ridge, Utah, USA. (d) Miocene dolomite beds (light coloured), Santa Barbara, California, USA. (e) Triassic sandstone, Comb Ridge, Utah, USA. (f) Jurassic sandstone, Oil Mountain, Wyoming, USA.

state of rock by randomly locating vertical, randomly oriented flaws of 0.2 m in length, all centred on a given horizontal plane within a semiinfinite elastic body. An additional restriction on the location of the initial flaws was that their centers could not be closer than two flaw lengths,

preventing initial flaw overlap or intersection. Although the 0.2 m flaw length is an order of magnitude or more larger than suggested by natural observations, smaller initial flaws were considered computationally impractical, given that the boundary element size of the numerical simulation

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scales with the initial flaws (each flaw is subdivided into two 0.1 m crack elements, to always have two independent crack tips). Fracture propagation was modelled by adding additional boundary elements of constant size (0.1 m length) at the crack tip, so the longer the fracture, the greater the number of elements it contained. For the 20 m by 20 m horizontal dimensions of the modelled fractured region, it would take a fracture with 200 elements of 0.1 m length to completely span the region from side to side. Reducing the initial flaw size by an order of magnitude (1 cm) would result in an order of magnitude more elements to populate fractures of a given length (e.g. 2000 elements for a 20 m fracture). Given the practical code limitation of about 10 000 single precision elements for a computer with 2 Gigabytes of memory, using the smaller flaw or element size would restrict the model to the growth of very few longer fractures (only five or 10). As the goal of the simulations was to generate realistic outcrop-scale fracture patterns containing hundreds of individual fractures, the larger initial flaw size was considered essential. In addition, previous simulation studies (Olson 1993; 2004) have demonstrated that fractures are self-organizing after propagation is initiated, such that the final fracture patterns generated are largely independent of the initial flaw distributions and are more dependent on mechanical boundary conditions and material properties. However, the main point of concern remains that the initial flaw length will influence the strain at which fracture propagation is initiated (the stress at the crack tip scales with the square root of fracture length). The numerical fracture model integrates the effects of mechanical crack interaction, fracture containment within prescribed mechanical layers (3D effects), subcritical propagation velocities, and displacement-driven loading conditions in a linear elastic media. The fractured region (Fig. 2) is an imaginary layer within a semi-infinite, rectangular parallelepiped (vertical dimension infinite). The fractures are prescribed to fully extend across the imaginary layer from the top to bottom but cannot propagate vertically across the layer boundaries. Thus, the fractures are all centred on the same horizontal plane with respect to z, have a constant height equal to the mechanical layer thickness, and can only propagate laterally, as described by Olson (2004). The fractures are assumed to be sharp-tipped at top and bottom (as if the layer boundaries were welded), and thus the size of their stress perturbation scales with their shortest dimension (typically the height for layer-contained or strata-bound fractures) (Olson 2003). This 3D aspect of the mechanics problem is accounted for by incorporating a 3D correction factor (Olson

2004) in the 2D, plane-strain, displacement discontinuity solution (Crouch 1976). This geometric conceptualization is chosen to correspond to the propagation of natural fractures within thin sedimentary units, where the fracture height is constrained by bedding planes between differing lithologies, and the fracture length is typically much greater than the fracture height. A mechanical effect that is not treated in this model is the influence of the mechanical properties of the layers adjacent to the fractured layer (Helgeson & Aydin 1991; Fischer et al. 1995), which can be critical in determining whether fractures will be contained within a particular mechanical layer or not. However, once fracture containment is presumed, the layering heterogeneity is considered to be a second-order effect on the mechanics of stratabound fractures (Bai et al. 2000). The model is implemented, as mentioned above, by subdividing fractures into straight, equal-length boundary elements of constant displacement discontinuity (opening and/or shearing) (Crouch 1976). The normal and shearing displacement discontinuities (Dn and Ds, respectively) of the numerical elements are referenced to the x–y plane and represent the opening (aperture) and shear offset of a fracture at a particular location along its length. Crack propagation direction and velocity depend upon the opening- and shearing-mode stress intensity factors, KI and KII, which can be computed as a function of the displacement discontinuities at the crack tip element as given by (Olson 1991) ! pffiffiffiffi pE pffiffiffiffiffiffi Dn KI ¼ 0:806 4ð1  n2 Þ Da ! pffiffiffiffi pE pffiffiffiffiffiffi Ds KII ¼ 0:806 4ð1  n2 Þ Da

ð1Þ

where E is Young’s modulus, n is Poisson’s ratio, and Da is the crack element length. The primary mechanism for propagation is stress corrosion cracking (subcritical crack growth), where propagation occurs at a stress intensity factor below fracture toughness (KIc) but above some minimum threshold stress intensity factor (K*I  KIc/10; Segall 1984), typically related to the hydrolytic weakening of intergranular bonds caused by pore waters (Anderson & Grew 1977; Atkinson & Meredith 1987). Given subcritical crack growth conditions (K*I  KI , KIc), propagation velocity, V, can be written as (Atkinson 1984; Swanson 1984; Olson 1993)  n Ktip V¼A KIc

ð2Þ

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z

y

x

Fig. 2. Idealization of a horizontal, fractured layer used for numerical propagation modelling. The body is homogeneous and isotropic, finite in extent in the horizontal x and y directions and infinite in vertical z direction. Grey x–y planes represent boundaries (bedding planes) defining the extent of the mechanical layer. Planes oriented in the x – z and y –z direction depict vertical fractures contained within the specified mechanical layer.

where n is the subcritical index and A is a proportionality constant. Ktip is based on maximum circumferential stress theory (Erdogan & Sih 1963) and can be written as       u u u Ktip ¼ cos2 KI cos 3KII sin ð3Þ 2 2 2 where u is the counter-clockwise positive angle defining propagation direction, found by solving KI sin u þ 3KII ð3 cos u  1Þ ¼ 0:

ð4Þ

Straight (in-plane) crack propagation occurs when KI . K*I and KII ¼ 0, which corresponds to a propagation angle of u ¼ 08 and Ktip ¼ KI. The maximum kink in crack path direction occurs when KI ¼ 0 and KII = 0, resulting in an angle

of u ¼ +70.58 for positive or negative KII, respectively. The displacement discontinuities for a given boundary element, Dn and Ds, are calculated as a function of the local stress boundary conditions on the element (Fig. 3a), the remote stress boundary conditions resolved onto the element (Fig. 3b), and stresses caused by the mechanical interaction with other nearby elements (Crouch 1976). The local and remote stress boundary conditions can be summarized with the driving stress term from fracture mechanics (Pollard & Segall 1987), which can be defined for mode I opening as   DsI ¼ snlocal  snremote

ð5Þ

where slocal is the locally applied normal traction n acting on the element (typically an internal fluid

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(a)

(b)

σnremote

σnlocal

σsremote σslocal

Fig. 3. (a) Diagram of the normal and shear stresses acting ‘internally’ on a boundary element that represents a fracture segment. The displacement discontinuity elements as prescribed by Crouch (1976) are analogous to dislocation elements, such that a single element represents both fracture surfaces. The local normal stress is typically provided by a fluid pressure in open natural fractures in the subsurface, and the shear stress comes from friction, which acts only if the fracture surfaces are in contact (i.e. the fracture is closed). For an open fracture at the Earth’s surface, the local stresses would be zero, representing traction-free boundaries. (b) Diagram showing the action of the remotely applied stresses resolved on a boundary element representing a fracture segment. These resolved . sremote (compression is stresses are acting from a distance act against the internally applied stresses of (a). If slocal n n j . jslocal j the element will slip (shear). positive), the fracture element will open. If jsremote s s

pressure in nature) and sremote is the resolution of n the remote boundary stresses acting perpendicular to the element. For mode II (in-plane) shearing, the driving stress can be defined as    sremote DsII ¼ slocal s s

ð6Þ

where slocal is the locally applied shear traction s acting on the element (typically a frictional resistance when the crack surfaces are in contact) and is the resolution of the remote boundary sremote s stresses acting parallel to the element. Mechanical interaction with other fracture elements adds additional resolved normal and shear stresses (Crouch 1976). For the modelling presented here, initial conditions were divided into stress and displacement components. The initial stress conditions were set to give both opening- and shearing-mode driving stresses equal to zero. The condition of DsII ¼ 0 was accomplished by having an isotropic remote horizontal stress state (sxx ¼ syy), which resolved no shear stress on any vertical fracture element regardless of its strike in the horizontal plane, comparable with the non-tectonic reference state of McGarr (1988). The condition of DsI ¼ 0 was achieved with an internal fluid pressure in the fracture elements that just balanced the remote horizontal compressive stress (Segall 1984; Olson 1993). Presuming a mechanism of natural hydraulic fracturing, Engelder & Lacazette (1990) proposed that, under static conditions, the fluid pressure in the fracture should be equal to the pore pressure in the formation (Pp). Consequently, these initial

stress conditions could be used to represent any depth in the subsurface where the pore pressure was equal to the horizontal in situ stress (Pp ¼ ¼ sremote ). Prescribing the exact loading or sremote xx yy burial path leading to this initial state is beyond the scope of this paper, but there are various models in the literature that could be manipulated to reach this point (Prats 1981; Zoback & Healy 1984; Blanton & Olson 1999). In addition, a zero shear-stress condition was imposed on the vertical boundaries (perimeter) of the solid to allow it to freely expand or contract in response to the lateral normal displacement loading (making these boundaries principal stress planes). The displacement boundary conditions were applied to the perimeter of the fractured body (Fig. 2) to simulate the effects of horizontal tectonic strains. An initial shortening strain in the x-direction of 1xx ¼ 23.6  1025 (1yy ¼ 0) was assumed to be the cumulative effect of horizontal tectonics up to the beginning of the simulations, during which no crack growth had occurred other than the creation of the starter flaws. The value of the initial 1xx was chosen to be large enough to align initial crack propagation in the x-direction (Olson et al. 2007), avoiding random polygonal fracture pattern development such as modelled by Tuckwell et al. (2003), but small enough to prevent the purely straight crack propagation indicative of large horizontal stress anisotropy (Olson & Pollard 1989; Olson et al. 2007). Fracture propagation was induced by the application of additional horizontal tectonic displacements constituting a biaxial extension of 1xx ¼ 1yy ¼ 9.1  1025 applied at a strain rate of 1.2  10219 s21 over a time period of

129

5.0

1.6E-04

4.0

1.2E-04

ε D = (ε yy − ε xx )/ε yy

3.0

ε yy

8.0E-05

ε xx

2.0

4.0E-05

1.0

Strain

Strain Ratio

DIAGENESIS AND FRACTURE MECHANICS

0.0E+00

0.0

–4.0E-05 0

5

10

15

20

25

Time (Ma)

Fig. 4. The strain history used for all numerical simulations, showing normal strains in x and y and the strain anisotropy, 1D, over a period of 24 Ma. The initial strain conditions were 1yy ¼ 0 and 1xx ¼ 23.6  1025, implying an infinite strain anisotropy. With the application of biaxial extension, the strain anisotropy is reduced. Uniaxial conditions occur when 1D ¼ 1 and isotropy occurs when 1D ¼ 0.

c. 24 Ma (Fig. 4). The total extension was small in magnitude but is comparable with that indicated by the observation of joints in the field (Segall 1984). The strain rate was chosen to represent a relatively stable sedimentary basin, not an active tectonic margin. Based on geodetic and geological data, crustal strain rates have been estimated to range from 1  10213 to 1  10220 s21 (Kreemer & Holt 2001; Kreemer et al. 2002). The horizontal extension that drove crack propagation was achieved in the numerical model by imposing 10 discrete increments of horizontal normal displacement at regular time intervals to the vertical edges of the semi-infinite solid, representing a constant strain rate deformation in a step-wise fashion. The tectonic (remote) strain anisotropy was quantified with the strain ratio, 1D, defined as 1D ¼ ð1yy  1xx Þ=1yy :

ð7Þ

1D . 1 indicates loading that is more anisotropic than uniaxial conditions (1D ¼ 1.0), whereas 1D , 1 indicates loading that is less anisotropic than uniaxial conditions. The limit of 1D ¼ 0.0 represents isotropic conditions. Thus, the larger the

value of 1D, the stronger the horizontal tectonic strain anisotropy, and the stronger the elasticinduced, tectonic stress anisotropy to align cracks for straight propagation (Olson & Pollard 1989).

Numerical modelling results Simulations were performed with 200 initial flaws, mechanical layer thicknesses of 2 and 4 m, and subcritical index values of n ¼ 20, 40 and 80. The other mechanical properties used in the simulations are indicated in Table 1. The subcritical index Table 1. Mechanical properties used in numerical simulations Property Fracture toughness, KIc Threshold stress intensity factor, K*I Young’s modulus, E Poisson’s ratio, n

Value 1

1.5 MPa m2 1 0.15 MPa m2 20 GPa 0.2

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values were based on the experimental range of values typical for sandstone, which average around n ¼ 50 (Holder et al. 2001; Rijken et al. 2002). The horizontal dimensions of the semiinfinite body were 22 m  22 m, and the body perimeter was divided into 2 m long boundary elements. Fracture propagation was limited to a 20 m  20 m sub-region centred within that body to avoid unwanted numerical edge effects caused by fracture interaction with these large edge elements (see Olson 1993).

with changing mechanical layer thickness. A stratabound fracture can be approximated as an elliptical fracture with a height (minor axis) equal to t (mechanical layer thickness) and a length (major axis) of L. The solutions for the minimum and maximum stress intensity factors around an elliptical fracture are (Kanninen & Popelar 1985)

Orthogonal trace pattern geometry

acting at the ends of the major axis (the lateral tips), and

All simulations had the identical strain history (Fig. 4): the strain state was strongly anisotropic (1D . 1) until about 9 Ma, while 1xx was contractional; it was uniaxial (1D ¼ 1) for the duration of one loading step, while 1xx ¼ 0 (from 9 to 11 Ma); and it became weakly anisotropic (1D , 1) after 11 Ma, when 1xx became extensional (positive) and no longer hindered cross-fracture growth. The results (Fig. 5) showed a progression in fracture pattern geometry from a clearly cross-jointed pattern, with the through-going fractures aligned with the initial x-directed shortening strain (Fig. 5a), to a more polygonal pattern with a slight preferential fracture trend in the x-direction (Fig. 5f). Increasing layer thickness and increasing subcritical index had similar effects on the fracture pattern: they both tended to increase fracture spacing and increase the tendency of crack paths to curve. The increase in fracture spacing with mechanical layer thickness was a direct result of the increasing size of the stress shadow around the fracture (area of stress release), which suppressed later fracture growth in the vicinity of early formed fractures and scaled in size with the fracture height (equal to mechanical layer thickness) (Pollard & Segall 1987; Bai & Pollard 2000; Olson 2004). This effect was clearly demonstrated by the difference between the patterns with a 2 m layer thickness (Fig. 5a, c and e) and those with a 4 m layer thickness (Fig. 5b, d and f ). Fracture paths were less straight for the thicker mechanical layer (t ¼ 4 m) than for the thinner one (t ¼ 2 m), all other factors being equal. This was caused by the higher crack-induced stresses in the thicker layer case that increased the degree of mechanical interaction between cracks. The magnitude of crack-induced stresses for a mode I crack should scale roughly with the magnitude of the stress intensity factor, KI, at the tip (Lawn & Wilshaw 1975). This scaling is precise in the near-tip region (at distances from the crack tip that are much less than the fracture dimensions) and only approximate further away from the crack, but it is sufficiently accurate to allow better understanding of the change in crack interaction

ðKI Þmin ¼

DsI

ðKI Þmax ¼

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pt2 =2L EiðkÞ

pffiffiffiffiffiffiffiffiffiffi DsI pt=2 EiðkÞ

ð8Þ

ð9Þ

acting at the ends of the minor axis (the top and bottom of the fracture). Ei(k) is the complete elliptic integral of the second kind, and k 2 ¼ [(L/2)22 (t/2)2]/(L/2)2. Based on the numerators of equations (8) and (9) (the variation of Ei(k) with t is weak) and assuming a given length and constant driving stress, both (KI)min and (KI)max increase strongly with increasing mechanical layer thickness, t. Given that the remote differential stress (caused in these simulations by the remote strain anisotropy) will not vary with layer thickness, the overall effective mechanical interaction between cracks is enhanced by the higher crack-induced stresses in thicker mechanical layers, and this mechanical interaction leads to more crack path curving. The effect of increasing subcritical index on the fracture pattern spacing and fracture path curving was attributed to fracture propagation timing. Fracture growth and strain anisotropy v. time for the 2 m layer cases (Fig. 6) showed that fracture propagation (indicated by the increasing number of crack elements) began later for higher subcritical index cases. This delay in crack propagation with increasing subcritical index was an outgrowth of subcritical propagation velocity law (equation (1)). Because (KI/KIC) , 1 for subcritical crack growth, the higher the subcritical exponent, n, the lower the propagation velocity for cracks of a given stress intensity factor. In the numerical model, crack growth occurred only when enough time had elapsed to add an entire boundary element (0.1m long in this case). Consequently, there was a characteristic time delay proportional to the propagation velocity before crack length increased. For instance, the n ¼ 20 case had no recordable growth until about 6.2 Ma into the simulation. For n ¼ 40, it took 11.5 Ma for growth to start, and for n ¼ 80 it took 17.8 Ma.

DIAGENESIS AND FRACTURE MECHANICS

Fig. 5. Final map-view trace patterns of fracture simulations for layer thicknesses of t ¼ 2 m and t ¼ 4 m and subcritical indices of n ¼ 20, 40 and 80. Small tick marks within patterns are unpropagated flaws. (Scales are in metres.)

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4000

6.0

5.0

3200

εD = (εyy– εxx )/εyy

Strain Ratio

4.0

2800 2400 2000

3.0

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Fig. 6. Fracture growth (as measured by boundary elements added) and strain ratio, 1D, v. time for the 2 m thick layer cases with n ¼ 20 (Fig. 5a), n ¼ 40 (Fig. 5c) and n ¼ 80 (Fig. 5e). Increasing subcritical index delays the initiation of fracture propagation and decreases the total amount of resultant growth as measured by elements added (equivalent to cumulative fracture length). The grey region at the bottom of the plot represents strain ratios that are less anisotropic than uniaxial loading.

The increased delay before fracture propagation began for the higher subcritical index material affected fracture spacing in a couple of ways. First, at any given point in the strain history, the flaw population had a range of propagation velocities that was a function of stress intensity factor according to equation (1). Only the fastest growing cracks accumulated enough length to add crack elements. Given the finite time span of the simulations, the lower the subcritical index of the material, the more the total fracture length accumulated (see the number of elements added for various subcritical indices shown in Fig. 6), and consequently the lower the average fracture spacing. The minimum average spacing between the through-going, x-parallel fractures was about 2.7 m for a subcritical index of n ¼ 20 (Fig. 5a). Increasing the subcritical index to n ¼ 40 increased average fracture spacing to about 3.1 ms (Fig. 5c), and at n ¼ 80, average fracture spacing increased to 4.4 m (Fig. 5e). Similar fracture spacing dependence on subcritical index was obtained for a layer thickness of 4 m (Fig. 5b, d and f ), with

values of 3.7, 4.4, and 7.3 m for subcritical indices of n ¼ 20, n ¼ 40, and n ¼ 80. Another subcritical index related factor that influenced the spacing between the more throughgoing, x-parallel fractures was the point in the strain history at which cross-fractures started forming. Once cross-fractures began to form, it was more difficult for x-parallel fractures to increase in length without intersecting other fractures. For the patterns generated in this study, crossjointing always occurred after the total x-strain became extensional (positive), or in other words, when 1D , 1. For the 2 m thick layer with n ¼ 20, x-parallel propagation of through-going fractures occurred between 6.2 and 10.7 Ma (Fig. 7a and b), with 1502 crack elements added while 1D  1.00 (Fig. 6). Cross-fracturing began at 11.3 Ma (Fig. 7c), with predominantly diagonal trends to the pre-existing, x-parallel fractures. At 13 Ma (Fig. 7d), cross-fracturing continued, but predominantly orthogonal to the through-going fractures. Some subsequent in-filling to the crossfractures also appeared in the form of younger and

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(b)

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Fig. 7. Fracture growth reconstruction for the fracture pattern of Figure 5a, showing the change in fracture growth direction with time. Bold lines indicate fractures added since previous time increment. Fracture orientation began to change after 10.7 Ma, when 1D became ,1. (Scales are in metres.)

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1

0.1 Cumulative Frequency

Power-Law

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Negative Exponential 0.01

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Aperture(m) Fig. 8. A semi-log plot of the cumulative frequency distribution of apertures for the case of Figure 5a. Labelled lines indicate best-fit power-law, log-normal, negative exponential and normal distribution curves for the given data (diamond markers).

shorter x-parallel fractures, which continued between 16.2 and 22.8 Ma (Fig. 7e). Additional y-parallel cross-fractures and some diagonally oriented fractures occurred at this late time as well. For the 2 m thick layer with n ¼ 40, fracture propagation did not initiate until about 11.5 Ma (Fig. 6), under remote conditions that were already more isotropic than uniaxial strain (1D , 1). This reduced anisotropy made it easier for cross-fractures to form as a result of the stress relief caused by the early x-parallel fracturing. The x-parallel fractures were still fairly straight, but the spacing was broader when cross-jointing began than for the n ¼ 20 case (one less throughgoing fracture propagated). The crack growth reconstruction for the n ¼ 40 case (not shown) indicated that cross-jointing began at 19.7 Ma (1D ¼ 0.50), which, compared with the n ¼ 20 case, was later in the strain history but earlier in

the fracture propagation history (only 1231 elements of straight crack growth had occurred prior to cross-jointing with n ¼ 40, compared with 1502 for n ¼ 20). The difference between the amount of parallel fracture infilling prior to crossjointing for n ¼ 20 v. n ¼ 40 was even more significant for the case of a 4 m thick layer (Fig. 5b v. Fig. 5d). For n ¼ 80 with the 2 m thick layer, propagation initiation was delayed until almost 17 Ma. The initial fracture propagation followed fairly non-planar paths, indicating strong mechanical interaction between nearby fractures (Olson & Pollard 1989). Mechanical interaction was stronger for this case of delayed fracture initiation for two reasons: (1) the strain anisotropy had become weak (1D ¼ 0.57) by 17 Ma, so the remote differential stress that could have aligned cracks into straight paths was weak when cracks started growing; (2)

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1 t = 2 m, n = 20 t = 2 m, n = 40 t = 2 m, n = 80

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Fig. 9. Aperture distributions (cumulative frequency plots) for the simulations of Figure 5: (a) 2 m thick layer with n ¼ 20, 40 and 80; (b) 4 m thick layer with n ¼ 20, 40 and 80. Near-vertical line on left side of plots represents the apertures of flaws that did not propagate. Distribution shapes vary according to subcritical index and layer thickness. The best negative exponential distributions are the cases of n ¼ 20.

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(a)

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Fig. 10. Length distributions for the simulations of Figure 5: (a) mechanical layer thickness of 2 m; (b) mechanical layer thickness of 4 m. All distributions exhibit a power-law shape for lengths greater than 2 m. Near-vertical trend on left-hand side of plot represents flaws that did not propagate. Curves show that a higher percentage of flaws propagated for lower subcritical indices and for thinner mechanical layer thickness.

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137

1 25.0 Ma 10.2 Ma 16.2 Ma

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Length (m) Fig. 11. Fracture length distribution through time for the 2 m thick layer case with n ¼ 20 (Fig. 5a). Vertical trend for 2.2 Ma indicates the length of the flaws, 0.2 m. The power-law trend is established by 10.6 Ma, and propagates to shorter lengths as time progresses.

the magnitude of overall strain at the time of crack initiation was higher than for the lower subcritical index cases (Fig. 4), meaning larger stress intensity factors when cracks started growing, and larger crack-induced stresses. The n ¼ 80 case for the 4 m thick layer shows similar results but with even less x-parallel fracture set development.

Fracture aperture development Based on field data, Gillespie et al. (1999) proposed that aperture distributions for stratabound veins followed normal, log-normal or negative exponential distributions, whereas non-stratabound vein apertures followed power-law distributions. Data from the n ¼ 20 case for a 2 m thick layer were plotted with best-fit curves for various distribution shapes (Fig. 8). The data did not show power-law character but were more similar to the negative exponential shape, consistent with the findings of Gillespie et al. (1999) for stratabound fractures. Looking at the aperture data from all of the simulations (Fig. 9), many exhibited similarities to a negative exponential shape. In addition to the distribution shape, there were several systematic variations in aperture worth

noting (Fig. 9). The 4 m thick layer cases had peak apertures almost twice those from the 2 m thick cases, demonstrating the aperture dependence on layer thickness in the simulations. There was also a systematic variation with subcritical index, where the n ¼ 80 cases had the greatest number of larger aperture fractures and the fewest smaller aperture fractures (the shallowest slope in the semi-log plot and the lowest ‘y-intercept’), whereas the n ¼ 20 cases had the fewest larger aperture fractures and the most smaller aperture fractures. Thus, as exemplified in the tracemap results (Fig. 5), fewer fractures grow in the higher subcritical index cases, but those fractures that do grow have larger apertures because they have to accommodate the same amount of overall strain and they have fewer nearby fractures that would interfere with their opening via mechanical interaction. (These generalizations exclude the fracture apertures that are less than 0.001 m, which plot as an almost vertical line along the cumulative frequency axis. They represent the flaws that did not grow, and were not considered in the analysis of distributions as they do not reflect the self-organizing growth and mechanical interaction process.)

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Fracture length analysis Whereas the aperture distribution shapes varied depending on subcritical index values, the length distributions for these same simulations were all strongly power-law (Fig. 10). Regardless of layer thickness or subcritical index, the longer end of the length distributions (greater than 1–2 m in length) plotted as a straight line on a log–log plot with a power-law exponent of c. 22. The main difference between length populations for the different cases was that the higher the subcritical index and the greater the layer thickness, the larger the portion of the initial flaw population that did not grow. For the 4 m thick layer, only 15% of the fractures exceeded the initial flaw length (0.2 m) for the n ¼ 80 case, whereas the n ¼ 20 case had almost 50% of the fractures growing beyond 0.2 m. Decreasing the layer thickness from 4 to 2 m for n ¼ 20 increased the percentage of fractures growing beyond 0.2 m from 50% to 80%. To understand how the power-law length distributions developed, it is interesting to look at the distribution shape through time, using the example of the 2 m thick layer with n ¼ 20 (Fig. 11). As shown in Figure 6, fracture growth did not start until about 6.2 Ma, so the length distribution at 2.2 Ma simply reflects the flaw population. There was a small amount of growth indicated by 8.4 Ma, including the appearance of the longest fracture in the simulation (open square plotted at 20 m), which completely spanned the breadth of the fractured region in the x-direction. Although other early fractures appear to span the entire fractured body (Fig. 7a), most are made up of overlapping en echelon segments, similar to fractures that have been documented in the field (Segall & Pollard 1983; Vermilye & Scholz 1995). The sequence of population distributions (Fig. 11) shows the growth of successively shorter fractures through time, filling in the power-law trend from longest to shortest. This sequence suggests that once a fracture started to grow, it quickly grew to the fullest extent possible, upon which its propagation was arrested. The maximum length of younger fractures was dictated by the probability of intersection with earlier propagated fractures. As time progressed and the fracture pattern filled in, the maximum likely length for a newly propagating fracture diminished. The shorter fractures, thus, were not those whose growth was terminated early in the deformation history, they were the ones that started growing the latest.

Discussion There have been several recent papers describing how cross-jointed patterns might develop (e.g.

Gross 1993; Rives et al. 1994; Bai et al. 2002); this study adds the perspective of controlled numerical experiments in fracture generation, along the lines of the study by Tuckwell et al. (2003) on polygonal fracture patterns. The modelled patterns (Fig. 5) demonstrated the potential influence of rock with different subcritical indices and varying mechanical layer thickness, indicating that cross-jointed patterns are more likely to form in thinner-bedded units and for low to intermediate subcritical index (n , 80). The results also showed how a small initial strain anisotropy can guide early fracture propagation direction to establish the through-going set, whereas initial isotropy gives a more polygonal pattern (Tuckwell et al. 2003; Olson et al. 2007), and a stronger anisotropy suppresses cross-jointing (Bai et al. 2002; Olson et al. 2003). Consequently, there is a fairly narrow range of boundary conditions from which crossjointed patterns will evolve, as delineated by Bai et al. (2002) based on stress analysis. The distribution analysis of the numerical fracture patterns presented here is somewhat unusual compared with what is typically done with field data. The analysis in this study lumped fractures of multiple orientations together into one population. Typically, analysed outcrops either have only one fracture orientation or the population is segregated into parallel sets. For instance, the numerical pattern in Figure 5a (as well as the natural patterns in Fig. 1) could be argued to consist of an early, regularly spaced, systematic set of parallel fractures, and a non-systematic assemblage (the cross-joints and later generations) that is probably much younger. Based on this interpretation of systematic v. non-systematic, the cross-joints would justifably be excluded from detailed analysis because they would be presumed to be caused by a different event from that which caused the systematic set. However, knowing a priori that the simulated fractures in this study were all generated as a result of one deformational event, it seems clear that all orientations should be analysed as members of one population. The corollary is that similar natural fracture sets should be treated likewise. The scaling of the numerical results, particularly for the lengths, validate this approach in that systematic results were obtained. The development of the fracture pattern through time as displayed in Figure 7 depicts a deformation process of sequential infilling (Gross 1993) in response to monotonically increasing strain, not a series of separate fracture events of varying orientation. The length distributions generated with the studied loading conditions were clearly power-law in character, but this power-law result does not indicate a universal characteristic for all fracturing processes. Simulations performed with the same

DIAGENESIS AND FRACTURE MECHANICS

numerical model under strongly anisotropic conditions have produced single orientation fracture sets with negative exponential length distributions (Olson 2004). The reason for the difference in distribution type for the cross-jointed v. parallel fracture patterns may be a difference in propagation arrest mechanisms. For the cross-jointed sets (natural and numerical), virtually every fracture abuts into a pre-existing one, suggesting that length is determined by how likely a fracture is to intersect pre-existing ones, and those intersections are typically at right angles. For a population of subparallel, straight fractures, pattern development takes on a different character, with propagation arrest caused by the indirect growth interference of mechanical interaction between non-intersecting, overlapping en echelon fracture tips (Pollard et al. 1982; Olson & Pollard 1989). A key characteristic for straight fracture growth, given a nonintersecting mechanism for propagation arrest, is that the application of additional strain can result in the lengthening of previously arrested fractures. This was demonstrated in growth experiments for parallel normal faults by Ackermann et al. (2001), where additional strain increased the lengths of all active faults, including the longest ones. For the cross-jointed patterns of this study, additional strain does not lengthen pre-existing fractures but initiates new ones. Thus, the numerical results suggest there is not a universal population scaling law for opening mode fracture lengths, but the scaling varies depending on the nature of the fracture interactions that limit length. The length development sequence of Figure 11 also brings up an interesting question with regard to how population distributions change as strain accumulates. The power-law length distributions for each snapshot in time were truncated at the shorter end, but became less truncated with further pattern development caused by increasing applied strain. In field data, the change in slope of a power-law distribution toward smaller values is usually attributed to a sampling limitation termed truncation bias (Ortega & Marrett 2000). If the numerical models are a valid proxy for natural fracture pattern development (at least in cross-jointed patterns), truncation bias in natural fracture populations could have a genetic cause related to the amount of strain experienced at a given locality. This is analogous to the concept applied to fracture spacing analysis by Rives et al. (1992) and Wu & Pollard (1995), among others, where fracture spacing distributions are interpreted to change through time as a function of how well-developed or ‘saturated’ a particular pattern becomes. Finally, projecting the power-law distribution of Figure 11 back toward a cumulative frequency of one can be used to estimate the expected abundance

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of shorter fracture lengths in the pattern if the loading were to continue past 25 Ma. The predicted value of the shortest fracture is about 1.3 m, assuming all flaws eventually propagated and the powerlaw length distribution was maintained. Clearly, field observations do not support an absence of fractures below 1.3 m, given the common observations of microfractures down to the grain scale (Kranz 1983; Laubach 1997). The documented progression of length distribution through time suggests that part of the solution to this discrepancy is to increase the number of initial flaws; that is, as fracture pattern development involves the infilling of the pattern with the propagation of new fractures, there need to be sufficient initiation sites or flaws for that progression to continue. In defence of the flaw number used for this study (200), the outcrop patterns depicted in Figure 1 do not suggest that further fracture propagation would be required beyond what is exemplified in Figure 5. Another question that needs to be addressed with respect to flaws is their initial lengths. To increase the initial number of flaws per unit area for a simulation, flaw length would need to be reduced to keep flaws from interacting strongly or amalgamating. The use of shorter, more realistic sized flaws will have a tendency to postpone fracture propagation because of the reduced stress intensity factor for shorter fractures given the same loading. Using equations (8) and (9) and assuming a penny-shaped crack, with t ¼ L ¼ r (where r is the radius of the penny-shape), the stress pffiffi intensity factor can be shown to scale as KIa r . The results from increasing the subcritical index to n ¼ 80 (Fig. 5e and f ) indicated that delaying propagation initiation under the prescribed loading scenario had the effect of increasing crack path curving and increasing fracture spacing (reducing the amount of propagation). Therefore, if the results of this study were to be applied to a situation where the starter flaws were smaller, it is expected that to obtain a cross-jointed pattern with very straight through-going fractures, a larger initial strain anisotropy would be required to suppress crack path curving and to postpone cross-joint initiation until the through-going fractures became more closely spaced.

Conclusions Multiple fracture orientations can be a consequence of multiple loading cycles with different principal stress –strain directions, such as could be caused by different tectonic events. If the earlier fracture sets are sealed with cement, later fractures may propagate across them rather than abut into them, as described by Gillespie et al. (2001) for

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Carboniferous limestones of The Burren, Ireland, where younger joints cross-cut earlier calcite-filled veins. This paper examines something different: the generation of multiple fracture orientations (in the form of a cross-jointed pattern) caused by a single loading cycle. The loading could be a result of uplift and erosion, differential expansion and contraction, or a small strain tectonic event. However, making the connection between the loading character (boundary conditions) and a particular type of geological event was not the purpose of this paper. The goal here was to associate a particular set of mechanical boundary conditions with a fracture pattern. Although this association is not necessarily unique (i.e. there may be multiple sequences of loading that could give similar-looking fracture patterns), and the investigation here was certainly not exhaustive, it does provide some interpretative guidelines for what type of mechanical boundary conditions or loading history are implied by fracture patterns observed in the field. This study showed how layer thickness and subcritical crack index exerted fundamental controls on fracture spacing, fracture timing, and the planarity of fracture paths. Fracture pattern geometry was also strongly influenced by strain anisotropy, which changed in magnitude with time given the conditions of a small initial anisotropy followed by biaxial extension. The trends observed from the simulations presented here show strong similarities to observed field relations, despite the simplifying assumptions required to make numerical modelling feasible, indicating that further investigations, particularly where some limiting assumptions could be relaxed, could provide additional advancement in the understanding of fracture processes. This study was supported by a grant from the Chemical Sciences, Geosciences and Biosciences Division, Office of Basic Energy Sciences, Office of Science, US Department of Energy program ‘Predicting fracture porosity evolution in sandstone’, and by the industrial associates of the Fracture Research and Application Consortium.

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V ERMILYE , J. M. & S CHOLZ , C. H. 1995. Relation between vein length and aperture. Journal of Structural Geology, 17, 423 –434. W U , H. & P OLLARD , D. D. 1995. An experimental study of the relationship between joint spacing and layer thickness. Journal of Structural Geology, 17, 887–905. Z OBACK , M. D. & H EALY , J. H. 1984. Friction, faulting, and in situ stress. Annales Geophysicae, 2, 689–698.

Permeability of fault rocks from the Median Tectonic Line in Ohshika-mura, Nagano, Japan as studied by pressure-cycling tests S. UEHARA1 & T. SHIMAMOTO2 1

Department of Urban and Environmental Engineering, Kyoto University, Kyoto 606-8501, Japan (e-mail: [email protected]) 2

Department of Geology and Mineralogy, Graduate School of Science, Kyoto University, Kyoto 606-8502, Japan

Abstract: We describe laboratory measurements of the permeability of fault rocks and their host protoliths obtained from the Median Tectonic Line (MTL), the largest strike-slip fault in Japan. The measurements are made using a gas-medium apparatus that simulates in situ conditions. Samples of fault gouge, cataclastic mylonite, and protoliths were collected from the Kitagawa and Ankoh outcrops of the MTL and adjacent areas in Ohshika-mura, Nagano Prefecture, central Japan. Permeabilities of these samples were measured at room temperature under dry conditions, with nitrogen as the pore fluid. Most samples from the incohesive fault zone have a permeability ranging between 10213 and 10217 m2 (100–0.01 mD). These permeabilities are greater than those of cemented cataclasites and mylonites by more than two orders of magnitude at all effective pressures (Pe) up to 180 MPa. Clayey fault gouge material has a permeability as low as 10219 m2 (0.1 mD) at high effective pressures, but such impermeable fault gouge does not constitute a continuous zone on the two outcrops we studied. Permeability of the incohesive fault rocks exhibits large hysteresis upon Pe cycling, compared with cataclasite and mylonite, because those cemented, cohesive fault rocks undergo much less inelastic deformation during the pressure cycling.

Fault zones exhibit a wide and seemingly unpredictable range of fluid transport properties, in comparison with their host rocks, because of the complicated and variable microstructure patterns that formed by a large number of possible deformation and alteration processes (e.g. Caine et al. 1996; Haneberg et al. 1999). Therefore, for full understanding of fluid circulation in the crust, it is important to cover the transport characteristics of a wide spectrum of fault zones. The permeability structure of fault zones may also strongly affect fault mechanics; if it is impermeable enough, there is a possibility that long-term fluid pressure build-up in a fault zone (e.g. Sleep & Blanpied 1992) or thermal pressurization of pore fluid by fault frictional heating (Mase & Smith 1987) may weaken a fault zone dramatically during a large earthquake. Caine et al. (1996) proposed a conceptual permeability structure model for fault zones consisting of ‘protolith’, ‘damage zone’ and ‘fault core’. The damage zone, consisting of fault breccia and fractured protolith, is relatively permeable, whereas the fault core (or fault gouge) may be impermeable. Such a fault-zone architecture could cause permeability anisotropy at a large scale; that is, easy fluid flow parallel to a fault zone through permeable damage zones and suppressed fluid flow across an impermeable fault core. Several permeability

measurement studies have reported that samples from fault core zones were less permeable than those obtained from other parts of the same fault zone (Evans et al. 1997; Seront et al. 1998). These results support the model of Caine et al. (1996). They also suggest the possibility of fault strength reduction in the event of pore-fluid pressure increase caused by factors such as thermal pressurization. Another complexity arises from the long tectonic history of large-scale faults, which have wide fault zones consisting of mylonites, cataclasites and incohesive fault rocks. The Median Tectonic Line (MTL) is the largest strike-slip fault in Japan, with a length of about 1200 km, and is a typical example of such long-lived, large-scale faults. The study by Wibberley & Shimamoto (2003), reporting on the permeability structure of the MTL at the Tsukide outcrop in the western part of Mie Prefecture, is perhaps the only work that revealed complex internal and permeability structures of such a large-scale fault. To develop this work further, we have worked on two wellknown outcrops of the MTL in Ohshika-mura, Kami-ina-gun, Nagano Prefecture, central Japan. This is the area where mylonitic rocks were described first in Japan, and very detailed work on structural and textural analyses of fault rocks has been conducted there (Hara et al. 1980;

From: LEWIS , H. & COUPLES , G. D. (eds) The Relationship between Damage and Localization. Geological Society, London, Special Publications, 289, 143–160. DOI: 10.1144/SP289.9 0305-8719/07/$15.00 # The Geological Society of London 2007.

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Takagi 1986; Michibayashi 1993; Michibayashi & Masuda 1993). This paper will present another example of complex internal structure of large-scale faults and, in comparison with the study by Wibberley & Shimamoto (2003), will elucidate heterogeneity in permeability structures along the fault.

Internal structure of the Median Tectonic Line in Ohshika-mura Geological setting The MTL extends from Kyushu, at the southwestern end of Japan, to the Kanto area NE of Tokyo, and forms the boundary between two contrasting metamorphic belts; the Ryoke belt to the NW (low P–high T type) and the Sambagawa belt to the SE (high P– low T type) (Fig. 1). In the study area, the Ryoke belt consists of the Hiji tonalite and metamorphic rocks, whereas the Sambagawa belt consists mainly of pelitic and mafic schists (Takagi 1986). The southeastern margin of the Ryoke belt constitutes a mega-shear zone with a width over 1 km, consisting of mylonites, cataclasites and incohesive fault rocks towards the MTL, whereas only incohesive fault rocks are developed on the Sambagawa side; the MTL fault zone is asymmetrical (Hara et al. 1980; Takagi 1986). Mylonitic deformation goes back to at least Cretaceous time (e.g. Ichikawa 1980).

The mylonite zone in Ohshika-mura is several hundreds of metres to 1000 m wide and exhibits a typical protomylonite–mylonite –ultramylonite sequence towards the MTL; it has been subdivided into several zones based on the size of deformed quartz grains (Hara et al. 1980; Takagi 1986; Michibayashi 1993; Michibayashi & Masuda 1993). The foliation pattern in the mylonite zone and the asymmetric microstructures in the ultramylonites clearly indicate that the sense of motion was sinistral during mylonitization. The mylonite zone was overprinted by cataclastic deformation during the exhumation process, forming a cohesive and structureless cataclasite zone of about 100 m width in Ohshika-mura (Takagi 1986). This cataclasite zone is terminated by the latest incohesive fault zone of about 5–30 m width along the present MTL. The timing and sense of fault motion during cataclastic deformation is not clear. The present MTL in Shikoku is the fastest-moving active fault in Japan, with an average dextral displacement rate of 5 –10 mm a21 (Okada 1980). However, the MTL is not regarded as an active fault in the eastern Kii Peninsula, central Japan, including Ohshika-mura and the Kanto area (The Research Group for Active Faults of Japan 1992). Total displacement along the MTL responsible for the formation of incohesive fault rocks has not yet been determined. The sense of fault motion at a fault gouge in Ohshika-mura was strike-slip (Tanaka et al. 1996).

Fig. 1. Location of the study area and regional setting of the Median Tectonic Line and metamorphic belts in central Japan.

PERMEABILITY STRUCTURE OF THE MTL FAULT

The MTL strikes approximately north –south and dips almost vertically in Ohshika-mura. Internal and permeability structures of MTL fault zones have been studied at outcrops at Kitagawa and Anko and their vicinities (Figs 2 and 3). These outcrops (Matsushima 1994; Tanaka et al. 1996) are parts of the MTL Museum in Ohshika-mura and have been preserved for geological excursions.

Internal structure of the MTL at the Kitagawa outcrop and its vicinity The MTL fault zone has been studied along Nishi Kobanazawa (a small mountain gorge) and at the Kitagawa outcrop (Fig. 2a). The mylonite zone is about 1 km wide and exhibits grain-size refinement of deformed quartz. The mylonitic foliation strikes around 0 –308E and dips vertically or steeply to the west. The Ryoke cataclasite is about 150 m wide along this gorge and is derived from mylonites of tonalites, and pelitic and silicic metamorphic rocks. The cataclasite adjacent to the incohesive fault zone is partially brecciated (left or west side of the MTL in Fig. 2b and c), and is partially cemented with carbonates (Fig. 4a). The fault zone at the Kitagawa outcrop consists of a black sandy foliated gouge along the MTL (the present material boundary between the Ryoke and Sambagawa belts), clayey gouge, white sandy gouge, partially brecciated Ryoke cataclasites, and fault breccia of Sambagawa schists (Fig. 2b and c). Here, ‘sandy’ means that this fault gouge is coarse in comparison with the clayey gouge, and has a typical grain size of 0.1–0.25 mm. The foliation in the black gouge strikes around north –south and dips almost vertically. Shear bands indicating sinistral slip were reported by Tanaka et al. (1996). A 10– 20 mm thick clayey gouge was found within the black gouge, but unlike in the MTL fault zone in western Mie Prefecture (Wibberley & Shimamoto 2003), this gouge zone is not continuous. This clayey gouge zone is likely to have been destroyed by the latest MTL movement. A zone of relatively coarse white gouge, 50 –150 mm in width, formed next to the black gouge. Mafic and pelitic Sambagawa schists are brecciated and their boundary with the black gouge zone is unclear because of the mixing of materials. X-ray diffraction (XRD) data reveal that the black gouge consists of quartz, mica minerals and other clay minerals (smectite, kaolinite); this is consistent with the results of Tanaka et al. (1996). Thin-section observation showed that irregularshaped needle-like minerals, identified as carbonates by XRD analysis, surround the aggregates of clay minerals (Fig. 4b). The carbonate minerals appear to be more abundant in clasts than in the

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matrix (Fig. 4c). Optical microscopic observation and XRD analysis also reveal that the white gouge consists of fine-grained quartz (,15 mm) and fine mica and carbonates (,30 mm), and that the breccia of Sambagawa schists contains white mica and chlorite. The samples for permeability measurements were collected at the sites marked by open circles in Figure 2b from the black gouge (KG01A, KG01B, KG01C), white gouge (KG02), gouge derived from pelitic schist (KG03) and clayey gouge (KG04) at the Kitagawa outcrop.

Internal structure of the MTL at the Ankoh outcrop and its vicinity There is another spectacular outcrop of the MTL along one side of Aoki River, about 1 km north of Jizo Pass (Fig. 3), and this outcrop is also preserved by the Oshika MTL Museum as an observation spot. The MTL fault zone is about 20 m wide here and consists of two black sandy gouge zones, separated by partially brecciated Ryoke cataclasites of about 6 m width (Fig. 3b and c). Black foliated cataclasite zones are formed within this cataclasite body. The western end of the outcrop consists of partially brecciated cataclasites derived from Ryoke mylonites of about 5 m width (left side of Fig. 3b and c). This cataclasites are lightly stained brownish red and do not have foliation. The black sandy gouge zone on the west is about 2 m wide and the one near the MTL (the material boundary between Ryoke and Sambagawa rocks) is about 1 m wide. These zones consist of incohesive foliated fault gouge with foliation striking N15 –308E and nearly vertical dips. A clayey gouge zone of a few tens of millimetres thickness was found in the black gouge, but the clayey gouge zone is not distinct or continuous. Fault motion along the MTL forming the gouge zones does not appear to be very large, although the exact displacement has not yet been determined. Green foliated cataclasites of about 2 m width have formed near the MTL (Fig. 3b). This foliated rock contains many clasts typical of cataclasites and is clearly different from brecciated Sambagawa schists. This rock may also be primarily of Ryoke origin, although chemical and mineralogical analysis is needed to confirm this. Dense veins of calcite formed in this rock nearly parallel to the fault plane (Fig. 4d). A fault breccia zone of 15 –20 m width derived from Sambagawa mafic and pelitic schists is developed on the east side of the MTL (part of this zone is seen at the right side of Fig. 3b and c). The MTL fault zone has also been studied along a small stream to the SW of the Ankoh outcrop

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Fig. 2. (a) Distribution of fault rocks in the Kitagawa area, and (b) a sketch and (c) a photograph of the Kitagawa outcrop of the Median Tectonic Line, showing sample locations (W).

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Fig. 3. (a) Distribution of fault rocks in the Ankoh area, and (b) a sketch and (c) a photograph of the Ankoh outcrop of the Median Tectonic Line, showing sample locations (W).

(Fig. 3a), where cataclasites of at least 100 m width are exposed well on the Ryoke side. This cataclasite is derived from Ryoke mylonites of tonalite and contains mylonite fragments of various sizes. The cataclasite zone is locally highly sheared to form foliated cataclasite with strikes of around N10– 208E and with almost vertical dips. Some foliated cataclasite is rich in phyllosilicates and has a schist-like appearance. The cataclasite is locally fractured and brecciated to form incohesive fault breccia.

The samples for permeability measurements were collected at the sites marked by open circles in Figure 3b from the black gouge (AKO01B, AKO02A), clayey gouge (AKO01A), Ryoke cataclasites (AKO08) and green foliated cataclasites (AKO05) at the Ankoh outcrop. The composition of clay minerals, typical grain size and the microstructure of the black foliated gouge are similar to those at the Kitagawa outcrop. Tanaka et al. (1996) reported shear bands indicating both sinistral and dextral slip at this site.

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Fig. 4. Optical photomicrographs taken under crossed polars of rocks collected from (a)–(c) the Kitagawa outcrop and its vicinity, and from (d) the Ankoh outcrop. (a) Ryoke cemented cataclasite adjacent to the black sandy gouge. (b), (c) Black sandy gouge. (d) Green foliated cataclasite next to the black gouge zone. White arrows in (a) and (b) indicate calcite or dolomite veins. In (c), heterogeneities are observed such as a clast (dashed lines) and matrix microstructure. In (d), we can observe a carbonate vein in the centre and small carbonate scattered clasts.

A simplified internal structure of the MTL in Ohshika-mura We now propose a simplified fault-zone model for the MTL in Ohshika-mura as shown in Figure 5, based on our data in Figures 2 and 4 and on previous studies (Takagi 1986; Matsushima 1994; Tanaka et al. 1996). The fault zone consists of an incohesive fault zone along the MTL (about 5–30 m wide), the Ryoke cataclasite zone (100 –150 m wide) and the Ryoke mylonite zone (about 1 km wide). The incohesive fault zone is composed of black sandy gouge (about 2 m to several metres wide) containing discontinuous clayey gouge (10–20 mm thick), white sandy gouge (up to about 150 mm wide with varying thickness), fault breccia of Sambagawa schists (about 5–30 m wide) and partially brecciated Ryoke cataclasites (a few metres to about 15 m wide). The fault rock

distribution indicates that, as the depth decreased, and the mode of deformation changed from plastic flow (mylonite) to cataclastic flow, and then to brittle deformation, the width of shear zone decreased. The permeability structure of the MTL will be examined based mainly on this simplified internal structure.

Sample preparation and experimental conditions For permeability measurements, we collected samples of fault rocks from the Ankoh and Kitagawa outcrops and their vicinities (Table 1). We collected samples of incohesive rocks by pushing a stainless steel tube of 25 mm diameter, or a copper tube of 20 mm diameter (with a wall thickness of 0.5 mm), into the outcrop. Specimens of

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Fig. 5. A schematic model of the distribution of fault rocks along the Median Tectonic Line around the study area.

cohesive rocks were made by laboratory coring from blocks using a 20 mm diamond core. The length of specimens ranges from 5 to 50 mm. Cores were taken parallel to their foliation and/or the MTL fault plane, with a few additional cores perpendicular to foliation. Specimens were dried at a temperature of 80 8C for several days to eliminate any pore water prior to permeability measurements. Then specimens were jacketed in polyolefin heat-shrink tubing and installed in the pressure vessel (Fig. 6). We used the experimental gas-medium apparatus at Kyoto University (e.g. Wibberley & Shimamoto 2003) to measure permeability. This apparatus is illustrated schematically in Figure 6. It has a servo-controlled piston (A in Fig. 6) to control pore pressure. This piston is very important for the pore-pressure oscillation method, which is the permeability measurement method used in this study, as described in detail below. The experiments reported here have all been conducted under isotropic (hydrostatic) stress conditions, using nitrogen as the pore fluid, and at room temperature. Although there are some disadvantages in using nitrogen as a pore fluid instead of water (e.g., the possibly dependence of permeability on the type of pore fluid, as discussed in Faulkner & Rutter (2000)), there are also some advantages. In particular, the experiment can be completed dramatically faster with nitrogen (in general), because it has a much lower viscosity. With the pore pressure, Pp, fixed at 20 MPa, the confining pressure, Pc, was incrementally increased

from 30 MPa to a peak value, after which it was decreased incrementally to the initial value. Permeability was measured at each stage in the confining pressure cycle. The peak value of confining pressure was either 110 MPa or 200 MPa. Permeability was measured using the porepressure oscillation method (Kranz et al. 1990; Fischer & Paterson 1992). In this method, pore pressure on the upstream side of the specimen is oscillated sinusoidally by means of the servocontrolled piston, and pore pressure on the downstream side is recorded. The time period of downstream pore-pressure oscillation is equal to that of the upstream oscillation, but the amplitude and the phase of the two oscillations differ (an example of our record is shown in Fig. 7). From the amplitude ratio and phase difference, permeability is calculated using the equations of Kranz et al. (1990) and Fischer & Paterson (1992). The amplitude of upstream pore-pressure oscillation was c. 2 MPa in all measurements. The frequencies of the oscillations at some data points are given in Table 2. One of the advantages of this method, as compared with other methods for permeability measurement, is that it is relatively easy to measure the amplitude and phase of the corresponding downstream pore-pressure oscillation, even if there are any external pore-pressure changes (e.g. those caused by gas leakage or temperature changes), because the time period of the response is known. Error bars (+2s) are added to some of the data points shown in Figures 8 and 9. The error bars are

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Table 1. Summary of laboratory-derived permeability data for all the fault zone rocks in this study Fault rock type

Sample no.

Direction

Locality

Permeabilty (10218 m2)

Specimen size (mm) Diameter

Length

Loading path Pe c. 30 MPa

Ryoke host rock (tonalite)

Fractured mylonite (from clast in cataclasite) Ryoke cemented cataclasite Green foliated cataclasite Sambagawa host rock (pelitic schist) Black sandy gouge

Clayey gouge White sandy gouge Pelitic-schist derived douge

x

Kitagawa

20.5

7.1

1.5

Pe 80 – 90 MPa

Pe 170 – 180 MPa

Pe c. 30 MPa

Peak Pe (MPa)

6.4  1021

–

9.4  1021

78

21

21

SR1d WOB10x WOB10z WOB12x WOB12z WOBN12 WOB13 AKS11

– x z x z x x x

Kitagawa Kitagawa Kitagawa Kitagawa Kitagawa Kitagawa Kitagawa Ankoh

20.5 20.5 19.2 20.5 19.2 19.3 19.1 24.7

12.4 11.7 11.2 6 6 16.2 17.3 25.9

1.7 1.8  1021 1.2  1021 2.8 – – 6.2  1022 –

3.7  10 4.7  1022 1.3  1021 7.3  1021 9.5  1022 2.0  1021 1.1  1022 2.9  1022

– – – 3.9  1021 6.8  1022 8.4  1022 ,3.9  1022* 1.6  1022

7.9  10 1.1  1021 3.3  1022 – – – 2.5  1022 –

81.8 80 81 170 175 183 181 181

AKO08 AKO05 EOB12

– x x

Ankoh Ankoh Kitagawa

19.4 20.5 19.1

18.7 6.5 9.9

3.8 1.2  1021 1.9  10

8.3  1022 3.1  1022 7.8  1021

,8.3  1022* ,3.1  1022* 6.3  1022

2.8  1021 2.1  1022 –

180 180 182

KG01A KG01B KG01C AKO01B AKO02A KG04 AKO01A KG02 KG03

x x x x x x x x x

Kitagawa Kitagawa Kitagawa Ankoh Ankoh Kitagawa Ankoh Kitagawa Kitagawa

25 20.1 19.6 24.3 24.7 19.5 19.75 25.7 25

19 16 14.05 39.8 25.85 16 15.8 21.4 48

6.1  103 – 5.4  104 2.6  104 9.6  103 9.0  1022 2.0  103 1.9  105 –

2.3  102 1.1  102 6.0  102 8.0  102 1.9  102 4.7 2.5  102 2.5  103 3.4  10

– – – 3.4  10 2.4 1.4  1021 6.6 – 3.4

2.5  102 1.3  102 6.8  102 1.4  102 7.6 3.1  1021 8.7  10 2.8  103 –

87 90 90.4 168 171 181 167 92.1 171

Direction: x, parallel to the MTL fault plane; z, perpendicular to the MTL fault plane; – , unknown. *Below the measurement range.

S. UEHARA & T. SHIMAMOTO

Tonalitic mylonite

SR1b

Unloading path

PERMEABILITY STRUCTURE OF THE MTL FAULT

151

Fig. 6. Experimental setup used for this study.

larger at low confining pressure (high permeability) because only a high-frequency sinusoidal oscillation is suitable for measurements of high permeability, and the oscillation tends to become disrupted at higher frequencies. At lower permeability, the error bars are much smaller.

Results of permeability measurements Figures 8–10 show permeability as a function of effective pressure, Pe (Pe ¼ Pc – Pp), where Pc and Pp are confining pressure and pore pressure, respectively. Permeability tends to decrease when Pe increases, as noted in many previous studies on laboratory permeability variation with pressure (e.g. Brace et al. 1968; Morrow et al. 1986; Bernabe 1987; Huenges & Will 1989). The amplitude ratio of pore-pressure oscillation at the downstream reservoir to that at the upstream reservoir and the phase lag at some data points are given in Table 2. For some experiments, it was theoretically impossible to calculate permeability from the observed pairs of the amplitude ratio and phase lag. In this case, if the observed phase lag is close to the lower limit expected theoretically from that amplitude ratio, we used the theoretical value of phase lag for the calculation of

permeability instead of the observed value (Table 2). In general, samples from the incohesive fault zone have permeabilities ranging from 10213 to 10217 m2 (100 –0.01 mD), more than two orders of magnitude greater than those of cemented cataclasites and mylonites at a given effective pressure. However, material from the clayey fault gouge zone has permeability as low as 10219 m2 (0.1 mD) at high effective pressures. The results for two mylonites (WOB10 and WOB12) indicate that permeability anisotropy exists. The permeability parallel to the foliation is greater than that perpendicular to the foliation (Fig. 9a). The ratio of anisotropy ranges from a factor of about three to nearly an order of magnitude. Many samples show a logarithmic decrease in permeability with increasing Pe, as found in some of the previous laboratory studies on fault rock permeabilities (e.g. Evans et al. 1997; Seront et al. 1998). Similar relationships in other rock types led David et al. (1994) to propose a negative exponential relationship between permeability, k, and Pe:

k ¼ k0 exp½gðPe  P0 Þ

ð1Þ

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Fig. 7. (a) Sketch of pore-pressure measurement setup. (b) An example of oscillated upstream pore pressure and (c) subsequent oscillation in the downstream pore pressure for permeability measurement by the pore pressure oscillation method, as recorded on a strip-chart recorder (the frequency of oscillation is 0.01 Hz).

where k0 is the permeability at Pe ¼ P0, a reference pressure, and g is the pressure sensitivity coefficient of permeability. Table 3 shows g and k0 of the measured samples when P0 is 0 MPa. k0 of incohesive rocks is around 10215 –10212 m2 (1–1000 mD), whereas k0 of cohesive rocks is around 10219 –10216 m2 (0.0001–0.1 mD). Variation of g from 0.04 to 0.07 indicates that, with Pe increase to 100 MPa on the confining path, the permeability of most incohesive rocks decreases by 2–3 orders of magnitude. On the other hand, the permeability of most of the cohesive samples decreases by less than two orders of magnitude over the same increase in pressure (Fig. 11). The incohesive fault rocks exhibit much larger hysteresis under Pe cycling than do the cohesive samples. In the case of the incohesive rocks, g values of the loading paths (0.039 –0.073) are generally larger than those of the unloading paths above Pe ¼ 30 MPa (0–0.017). On the other hand, g values below Pe ¼ 30 MPa (0.012–0.097) tend

to be close to those of the loading paths. The differences in g between the loading and unloading portions are larger for the incohesive rocks than for the cohesive rocks (0.013 –0.08 for the loading paths, and 0.015–0.044 for the loading paths). Some results on incohesive rock show that the slope of the curve slightly decreases as Pe increases at high Pe, and the measurement data deviate from an exponential relationship.

Permeability structure of the Median Tectonic Line To compare the permeability values of various fault rocks, and to evaluate the permeability structure of the MTL fault zone, the permeability values of samples at Pe ¼ 80 MPa from the loading (e.g. increasing) part of the pressure cycle are plotted as a function of distance from the MTL for the Kitagawa and Ankoh areas (Fig. 12). The value of

Table 2. Frequency of oscillation, amplitude ratio and phase lag at some data points Peak Pe (MPa)

At Pe c. 30 MPa on confining path Frequency of Amplitude ratio oscillation (Hz) (Adown / Aup)

At Pe c. 30 MPa on unconfining path

Frequency of Amplitude ratio Phase lag Frequency of Amplitude ratio Phase lag oscillation (Hz) (Adown/Aup) (d deg.) oscillation (Hz) (Adown/Aup) (d deg.)

87 90 90.4 92.1 171 181 167 168 171

2.2–3 0.6–1* 3 2 0.25 1.5–3 0.6 1.5 1

0.23–0.35 0.08–0.14 0.89–0.93 0.94–0.99‡ 0.036 0.12–0.2 0.018 0.038 0.033

72 – 86 87 – 115 22 – 28† 10.8– 19.8 109 – 116 78 – 105 89 – 95 90 88 – 93

0.25– 0.5 0.1– 0.2 0.5– 1.5 2 0.01 0.005 0.1 0.1 0.02

0.11– 0.056 0.057– 0.12 0.053– 0.13 0.11 0.02 0.011– 0.012 0.0071 0.022 0.013

104– 133 86– 107 82– 110 84– 97 99– 112 92– 101 101– 112 105– 115 89†

0.5 0.2–0.4 0.5–1.5 1.5– 2 0.02 0.005– 0.01 0.2 0.5 0.08

0.072 0.037 –0.071 0.057 –0.17 0.2 – 0.25 0.016 0.0082– 0.027 0.01 0.017 –0.018 0.01

80 81 170 175 182 183 181 78 81.8 180 181 180

0.003 0.005 0.05 0.002* 0.2 0.003* 0.01 0.1 0.08 0.003 0.001* 0.08

0.0095–0.01 0.003–0.0039 0.017 0.019 0.015–0.016 0.0086 0.0027–0.003 0.0041 0.0032 0.012 0.005 0.0043

89† 91† 89† 101 92 116 – 138 100 – 107 90† 99 91 – 94 89† 90†

0.001 0.001 0.025 0.001 0.002 0.001 0.001k 0.05 0.01 0.001} 0.001 0.002}

0.0071– 0.0081 0.0019 0.0059 0.019 0.0055– 0.0059 0.0085 0.0016– 0.0017 0.0034 0.0056 0.009 0.0012 0.004

90† 92– 106 90† 90 – 117† 90 104 97– 153 91 96 89† 102 95

0.002 0.001 – – 0.01* 0.002§ 0.005 0.08 0.04 0.001 0.001§ 0.005

0.0085 0.0048

93 92 62 – 95 76– 85† 110 – 127 107 – 146 97 – 102 116 – 119 89†

90 – 93 95† – – – – 0.0079– 0.0087 90 0.014 89† 0.0017– 0.0026 109 – 123 0.0032 92 0.003 –0.0031 90 0.0062 90† 0.005 90† 0.0051 90†

PERMEABILITY STRUCTURE OF THE MTL FAULT

Incohesive rocks KG01A KG01B KG01C KG02 KG03 KG04 AKO01A AKO01B AKO02A Cohesive rocks WOB10x WOB10z WOB12x WOB12z EOB12 WOBN12 WOB13 SR1b SR1d AKO05 AKS11 AKO08

Phase lag (d deg.)

At peak Pe

*At Pe ¼ 50 MPa. † Phase lag used for the calculation of permeability is the lowest value demanded theoretically from the measured amplitude ratio. ‡ Amplitude ratio used for the calculation of permeability is the range of the value demanded theoretically from the measured phase lag. § At Pe ¼ 10 MPa. k At Pe ¼ 150 MPa on the confining path. } At Pe ¼ 110 MPa on the confining path.

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Fig. 8. Permeability data for incohesive fault rock samples taken from (a) the Kitagawa outcrop, and (b) the Ankoh outcrop, as a function of effective pressure Pe (Pe ¼ Pc 2 Pp). The sampling locations are shown in Figures 2b and 3b. Error bars (+2s) are added to the data points for KG02, KG03 and KG04.

80 MPa represents a depth of several kilometres. Rock exposed at the surface probably went through an unloading path. However the permeability on the unloading path generally depends on the maximum previous effective pressure, and the value of this pressure is still unknown in this area. In addition, we included two kinds of maximum Pe in the experiments, and it is difficult to compare all of the data if we use the data for the unloading path. Therefore we chose the data from the loading path for the plot in Figure 12. The results show that host rocks, Ryoke mylonite and cemented cataclasite are more than three orders of magnitude less permeable than fault gouge at Pe ¼ 80 MPa. The permeability of mylonite is equal to or less than that of tonalite. The

mylonite specimens WOB10 and WOB12 show anisotropy in permeability; that is, the permeability in the direction parallel to the fault plane tends to be larger than that perpendicular to the fault plane. Clayey gouge is less permeable than sandy gouge, but it does not have good structural continuity.

Discussion Summary of internal and permeability structures of the MTL fault zones The MTL fault zones at the Kitagawa and Ankoh outcrops and in their vicinities in Ohshika-mura consist of a mylonite zone (about 1000 m wide),

Fig. 9. Permeability data for (a) cohesive Ryoke fault rocks and (b) Ryoke and Sambagawa host rocks from the Kitagawa area as a function of effective pressure Pe. The sampling locations are shown in Figure 2a, except for that of SR1 (tonalite), which was taken from about 3.4 km west of the Median Tectonic Line. The permeability of WOB13 could not be measured when Pe was 180 MPa because the amplitude of the downstream pore pressure could not be detected.

PERMEABILITY STRUCTURE OF THE MTL FAULT

155

Fig. 10. Permeability data for host rocks and cohesive fault rocks from the Ankoh area as a function of effective pressure Pe. The sampling locations are shown in Figure 3. The permeability of AKO05 and AKO08 could not be measured when Pe was .100 MPa because the amplitude of the downstream pore pressure was too small to be measured above background noise. Therefore, after Pe was increased to 180 MPa and decreased to 100 MPa, the measurements were continued.

cataclasites (about 100 –150 m wide), and incohesive fault rocks (about 5–30 m wide), as schematically shown in Figure 5. The present MTL as the material boundary between the Ryoke and Sambagawa belts is occupied by black sandy gouge of 1–2 m width, containing discontinuous clayey fault gouge of a few tens of millimetres width. Mylonites and cataclasites are recognized only on the Ryoke side, and fault breccia of a few tens of metres thickness is the only fault rock on the Sambagawa side. Thus the MTL fault zone is asymmetrical. Measured permeability at an effective pressure, Pe, of 80 MPa (Fig. 12) clearly shows that the distributions of permeability at the Kitagawa and Ankoh outcrops (and in their vicinities) are also similar to each other. The permeability measurements indicate that the permeability of the incohesive fault rocks is several orders of magnitude greater than those of the cohesive fault rocks (mylonites and cataclasites) and the host rocks of both the Ryoke and Sambagawa metamorphic rocks. The clayey fault gouge has a permeability value between these two, but the clayey gouge does not form a continuous zone at the Kitagawa and Ankoh outcrops. Thus, the incohesive fault zone should act as the primary fluid conduit at least at depths shallower than several kilometres. The pressure sensitivity coefficient of permeability, g in equation (1), is greater for the incohesive fault rocks than for the cohesive fault rocks and host rocks (Table 3), so that the difference in permeability between the

two groups decreases with increasing Pe. Taking the centre of the variation of permeability values, the difference between the permeability of the incohesive fault rocks, excluding clayey fault gouge, and those of the cohesive fault rocks and host rocks is about four orders of magnitude at Pe ¼ 80 MPa (Fig. 12). This contrast in permeability increases to about five orders of magnitude at Pe ¼ 30 MPa and decreases to about three orders at Pe ¼ 150 MPa. This change in permeability structure with depth must be taken into account in calculating large-scale fluid flow in fault zones. Real permeability structures of fault zones must be more complex than that in Figure 12 for at least two reasons. Permeability values in this figure are taken from pressure-increasing paths in Figures 8–10. However, the incohesive fault rocks (Fig. 8) exhibit much greater hysteresis upon pressure cycling than the cohesive fault rocks and host rocks (Figs 9 and 10) because porous incohesive fault rocks compact permanently at elevated pressures. Thus, the permeability of the incohesive rocks depends on their history. For instance, when the fault gouge was compacted at depth and then was uplifted with little disturbance, its permeability should be less than that shown in Figure 12. On other hand, the permeability of the cohesive fault rocks and host rocks increases dramatically with fracturing (Uehara & Shimamoto 2004). Thus a fault zone comprising those rocks can be a fluid conduit when fractured during seismic fault

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Table 3. Effective pressure sensitivity coefficient for the permeability Loading Peak Pe (MPa)

k0 (10216 m2)

g (MPa21)

R

87 90 90.4 92.1 171 181 167 168 171

3  102 7  10 4  103 1  104 2  10 3  10 9  10 1  103 4  102

0.058 0.048 0.073 0.070 0.039 0.062 0.045 0.051 0.057

0.983 0.997 0.995 0.992 0.968 0.974 0.982 0.984 0.997

80 81 170 175 182 183 181 78 81.8 180 181 180

5  1023 8  1023 5  1022 6  1023 7  1021 7  1023 2  1023 7  1022 5  1022 5  1023 2  1023 7  1021

0.031 0.054 0.018 0.016 0.042 0.013 0.029 0.037 0.033 0.032 0.017 0.080

0.991 0.984 0.922 0.766 0.970 0.953 0.956 0.900 0.998 0.946 0.828 0.995

Unloading (.30 MPa)

g (MPa21)

R

4  10 1  1021 1  1022 1 4 2  1021

0.0078 0.0082 0.0043 0.0054 0.0090 0.015 0.021 0.018 0.018

0.591 0.855 0.833 0.817 0.903 0.759 0.917 0.829 0.743

2  1023 9  1024 – – 3  1022 1  1023 4  1024 2  1022 3  1022 1  1023 1  1023 2  1022

0.021 0.027 – – 0.025 0.0032 0.016 0.015 0.027 0.019 0.0097 0.044

0.969 0.943 – – 0.926 0.754 0.975 0.912 0.913 0.598 0.858 0.868

k0 (10216 m2) 4 2 8

Unloading (.30 MPa

k0 (10216 m2)

g (MPa21)

R

2 1 7 3  10 8  1022 3  1023 7  1021 1 8  1022

20.0012 0.0034 0.0020 0.0019 0.0048 0.0048 0.017 0.0098 0.0079

0.148 0.815 0.657 – – 0.920 0.886 0.937 0.893

k0 (10216 m2) 9 3 1  10 5  10 2  1021 4  1022 2 3  10 6

k ¼ k0 exp[2g (Pe 2 Po)], Po ¼ 0 MPa. R, correlation coefficient; those for ‘unloading (,30 MPa)’ and other several data are not shown, because there are only two or three data points.

g (MPa21) 0.042 0.024 0.012 0.021 0.025 0.089 0.035 0.097 0.14

S. UEHARA & T. SHIMAMOTO

Incohesive rocks KG01A KG01B KG01C KG02 KG03 KG04 AKO01A AKO01B AKO02A Cohesive rocks WOB10x WOB10z WOB12x WOB12z EOB12 WOBN12 WOB13 SR1b SR1d AKO05 AKS11 AKO08

Unloading

PERMEABILITY STRUCTURE OF THE MTL FAULT

157

Fig. 11. Examples of fitting the function k ¼ k0 exp [2g (Pe 2 P0)] (P0 ¼ 0 MPa) to permeability measurement data for (a) incohesive rock specimens, and (b) cohesive rock specimens. Values of k0 and g used here are given in Table 3.

motion, but it can be a seal for fluid flow when cemented nearly completely. Thus the permeability structure of a fault zone can change with time (see the discussions by Uehara & Shimamoto, 2004).

Along-fault variation in permeability structure of the MTL fault zone The permeability of the clayey fault gouge at the Kitagawa outcrop decreases from about 10214 m2

Fig. 12. Distribution of fault rocks and permeability data for fault rocks from the Kitagawa area and the Ankoh area at Pe ¼ 80 MPa on the Pc confining path (*) as a function of distance from the Median Tectonic Line (MTL). ‘WOB10X’ and ‘WOB12X’ indicate permeability values in the direction parallel to the fault plane, and ‘WOB10Z’ and ‘WOB12Z’ are the values perpendicular to the fault plane.

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(10 mD) to about 10219 m2 (0.1 mD) with an increase in Pe from 10 to 180 MPa (Fig. 8a). This permeability is almost as low as that for the clayey fault gouges of the Nojima fault (Mizoguchi et al. 2000) and the MTL at the Tsukide outcrop in Mie Prefecture (Wibberley & Shimamoto 2003). However, the clayey fault gouge along the MTL in Ohshika-mura does not form a continuous zone at the Kitagawa and Ankoh outcrops, which is a different situation from that for fault gouges or fault cores at other locations (Caine et al. 1996, and references cited above). The difference between Tsukide and Ohshimamura is probably due to the difference in recent MTL activity. The MTL in Mie Prefecture, in the eastern part of the Kii Peninsula, and the MTL in Nagano Prefecture, central Japan, are not regarded as active faults in Japan (The Research Group for Active Faults of Japan 1992). However, the MTL in Wakayama Prefecture, in the western part of the Kii Peninsula, is regarded as very active, and the Tsukide outcrop is not far from this active portion. The fault gouge zone is several metres wide and the clayey fault gouge is continuous and straight (Wibberley & Shimamoto 2003). Despite the reasons for variation in fault zone structures, the difference in permeability between Tsukide and Ohshika-mura represents along-fault variation in permeability structure and has importance implication for seismogenic fault motion. If frictional heating by fault slip causes thermal expansion of pore fluid, and if the fault gouge zone adjacent to the slip plane is so impermeable and wide that the escape rate of the fluid from the gouge zone is less than the expansion rate, the thermally induced pore-pressure rise may reduce fault strength dramatically (Larchenbruch 1980; Mase & Smith 1987; Noda & Shimamoto 2005). Such thermal pressurization is not effective for a permeable fault gouge or core such as those in Ohshika-mura, and along-fault variation in permeability can be a significant source of heterogeneous mechanical behaviour along a fault.

Limitations of using laboratory data Permeability was measured in this study under limited conditions, and there are many factors that must be considered for the prediction of a more realistic permeability structure for a fault zone. Temperature and the existence of water might influence the permeability strongly, because they are related to the mechanism of rock deformation. In addition, the interaction of pore fluids with the pore surfaces (Faulkner & Rutter 2000) can lead to the dissolution and deposition of minerals, thus altering the mechanical and flow properties. Furthermore, the permeability of the fault zone might evolve with

continued fault activity. For example, the permeability of cataclasite measured in this study might differ enormously from that which might be observed just after crushing by fault motion. The role of differential stress, and other aspects of the geohistory, have not been addressed by our experimental work. Including those effects in laboratory studies with gas apparatus will be a challenge for the near future.

Upscaling It must be remembered that the permeability measured in this study is at the scale of the core sample, and larger-scale structures such as outcrop-scale fractures are not considered. In the case of cohesive rocks, the sample permeability is low. Therefore the fractures in them might have an important role as fluid conduits, and the constructed permeability architecture developed in this study is that of the ‘matrix permeability’, which might be the lowest limit of the actual permeability applying to a larger scale. To estimate the permeability of rock containing fractures, it is important to study the characteristics of the distribution of fractures (e.g. the size distribution, anisotropy of directions, and density (e.g. Haneberg et al. 1999)), and those of the permeability of a single fracture (e.g. its dependence on size, roughness, stress and temperature (e.g. Witherspoon et al. 1979; Barton et al. 1985; Raven & Gale 1985)). In the case of the fault breccia, we observed a block –matrix structure whose scale is larger than the specimen size. To estimate the flow properties of this block –matrix structure, we would have to take into account the permeability of the matrix parts and the block parts, and their structures and distributions. It would also, perhaps, be necessary to consider how the permeability might vary as a function of fluid –geomechanical interactions.

Deformation mechanism of incohesive rocks The results of permeability measurements show that the pore structure of incohesive rock relating to the permeability deforms inelastically during the loading– unloading cycle, as compared with cohesive rocks. Such inelastic deformation is probably caused by the rearrangement of grains and possibly by grain breakage. Because of such inelasticity of the pore structure, the permeability of incohesive rock during deconfinement is substantially influenced by the peak value of Pe to which the samples have been subjected (e.g. Bolton et al. 1998; Zhang & Cox 2000). Large changes in permeability were measured where Pe is less than 30 MPa during the deconfining portion of the test procedure in several samples of

PERMEABILITY STRUCTURE OF THE MTL FAULT

incohesive rocks. This suggests that, at low Pe (,30 MPa), the permeability changes of some incohesive rocks are affected strongly by elastic deformation. However, the effects vanish or are very small at high Pe (.30 MPa). In the case of typical overconsolidated soil, the slope of the effective pressure Pe –permeability curve is small at low Pe along the loading path, and at some value of Pe the slope increases. This tendency reflects the transition of the Pe –volumetric strain relationship from a small elastic deformation to a large plastic deformation. This Pe at the transition is called the overconsolidation pressure in soil mechanics, and is the highest effective pressure to which the material had previously been subjected (i.e. the palaeopressure). Our results for incohesive rock did not show such a change in the slope of the curve. One of the possible explanations for these results is that the palaeopressure may be lower than the 10–30 MPa effective pressure used at the start of our tests. Faulkner & Rutter (2000) showed similar results and discussed other possible explanations. Another possibility is that disturbance of incohesive specimens during the sampling process could effectively reset any memory of maximum previous effective stress.

159

mylonites display moderate hysteresis, suggesting the closing and opening of fractures upon pressure cycling. The clayey fault gouge zone material has permeability as low as 10219 m2 (0.1 mD) at high effective pressures, but such an impermeable fault gouge is very local and does not constitute a continuous zone. The MTL fault core in Ohshika-mura is much more permeable than that for other faults and for the MTL Tsukide outcrop in Mie Prefecture. The permeability structure is likely to vary along the MTL and this can cause heterogeneous seismic behaviour of the MTL because of the differences in the effectiveness of the thermal pressurization mechanism. The authors would like to express their sincere thanks to C. A. J. Wibberley, G. Couples, B. Crawford, O. Katz and an anonymous reviewer for critical reading of the manuscript and many suggestions to improve it; A. Tsutsumi, C. A. J. Wibberley, T. Hirose and K. Mizoguchi for useful discussions and collaboration in laboratory; and K. Kawamoto and F. Hara for valuable information in the field work and for providing the experimental samples. This work was partially supported by a Grant-in-Aid for Scientific Research, JSPS (12440136, 16340129), by the CAPROCKS project (Ref. 64/02/) and by the Kyoto University Active Geosphere Project for the 21st Century.

Conclusions The following fault-rock zonation has been mapped along the Median Tectonic Line (MTL) at the Kitagawa and Ankoh outcrops and in their vicinity in Ohshika-mura, Kami-ina-gun, Nagano Prefecture: (1) along the main MTL fault, incohesive fault-rock distributions (about 5 –30 m wide), consisting of a clayey gouge zone 1 –2 cm thick, a black sandy gouge in a zone ,2 m thick and brecciated Sambagawa rocks, were observed; (2) further from the MTL and into the Ryoke belt, a cataclasite zone (locally foliated under shear and with a schist-like appearance; 100 –150 m wide) and a mylonite zone (about 1 km wide) are observed. The fault-parallel permeability of the incohesive fault rocks is greater by a few to several orders of magnitude than those of the cohesive fault rocks (cataclasites and mylonites) and the Ryoke and Sambagawa host rocks; the incohesive fault zone near the MTL (the material boundary between the Ryoke and Sambagawa belts) should act as a fluid conduit. The incohesive fault rocks exhibit large hysteresis upon pressure cycling and samples tend to remember the low permeability values at the greatest pressure, suggesting nearly permanent compaction of the fault gouge. Thus their history is important for evaluating the permeability of the incohesive fault rocks. In contrast, the cataclasites and

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F AULKNER , D. R. & R UTTER , E. H. 2000. Comparisons of water and argon permeability in natural claybearing fault gouge under high pressure at 20 8C. Journal of Geophysical Research, 105, 16415– 16426. F ISCHER , G. J. & P ATERSON , M. S. 1992. Measurement of permeability and storage capacity in rocks during deformation at high temperature and pressure. In: E VANS , B. & W ONG , T.-F. (eds) Fault Mechanics and Transport Properties of Rocks. Academic Press, London, 213–252. H ANEBERG , W., M OZLEY , P., M OORE , J. & G OODWIN , L. (eds) 1999. Fault and Subsurface Fluid Flow in the Shallow Crust. Geophysical Monograph, American Geophysical Union, 113. H ARA , I., S HYOJI , K., S AKURAI , Y., Y OKOYAMA , S. & H IDE , K. 1980. Origin of the Median Tectonic Line and its initial shape. In: I CHIKAWA , K. (ed.) Median Tectonic Line of Southwest Japan. Memoirs of the Geological Society of Japan, 18, 27– 49. H UENGES , E. & W ILL , G. 1989. Permeability, bulk modulus and complex resistivity in crystalline rocks. In: BRIDGEWATER , D. (ed.) Fluid Movements— Element Transport and the Composition of the Deep Crust. Kluwer, Dordrecht, 361– 375. I CHIKAWA , K. (ed.) 1980. Median Tectonic Line of Southwest Japan. Memoirs of the Geological Society of Japan, 18. K RANZ , R. L., S ALTZMAN , J. S. & B LACIC , J. D. 1990. Hydraulic diffusivity measurements on laboratory rock samples using an oscillating pore pressure method. International Journal of Rock Mechanics and Mining Sciences & Geomechanics Abstracts, 27, 345– 352. L ARCHENBRUCH , A. H. 1980. Frictional heating, fluid pressure, and the resistance to fault motion. Journal of Geophysical Research, 85, 6097– 6112. M ASE , C. W. & S MITH , L. 1987. Effects of frictional heating on the thermal, hydrological, and mechanical response of a fault. Journal of Geophysical Research, 92, 6249–6272. M ATSUSHIMA , N. 1994. New interpretation of Median Tectonic Line of the Akaishi Mountain Lands, in Japan. Bulletin of Iida City Museum, 4, 113–124. M ICHIBAYASHI , K. 1993. Syntectonic development of a strain-independent steady state grain size during mylonitization. Tectonophysics, 222, 151–164. M ICHIBAYASHI , K. & M ASUDA , T. 1993. Shearing during progressive retrogression in granitoids: abrupt grain size reduction of quartz at the plastic– brittle transition for feldspar. Journal of Structural Geology, 15, 1427– 1432. M IZOGUCHI , K., H IROSE , T. & S HIMAMOTO , T. 2000. Internal and permeability structures of Nojima Fault; southwest Japan. EOS Transactions, American Geophysical Union, 2000 Fall Meeting, 81, 1145. M ORROW , C. A., Z HANG , B.-C. & B YERLEE , J. D. 1986. Effective pressure law for permeability of Westerly

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Insights into the faulting process from numerical simulations of rock-layer bending G. D. COUPLES1,2, H. LEWIS1,2, P. OLDEN1,2, G. H. WORKMAN3 & N. G. HIGGS4 1

Institute of Petroleum Engineering, Heriot – Watt University, Edinburgh EH14 4AS, UK (e-mail: [email protected]) 2

ECOSSE (Edinburgh Collaborative of Subsurface Science and Engineering), a part of the Edinburgh Research Partnership in Engineering and Mathematics 3

Applied Mechanics Inc., 3431 Bayou Court, Longboat Key, FL 34228, USA

4

Higgs– Palmer Technologies LLC, Remington Tower, Suite 707, 5810 East Skelly Drive, Tulsa, OK 74135, USA

Abstract: An elastic– plastic material model, with strain-hardening or -softening, and volumetric strains, implemented within a general-purpose finite-element system (SAVFEMTM), is shown to reproduce the stress –strain relationships and localized to de-localized (brittle to ductile) changes in strain response that have long been observed in typical laboratory experiments on common porous rocks. Based on that validation of the implementation, SAVFEMTM is then used to create numerical simulations that reproduce the patterns of localized shear zones, and their growth history, that occur in experimental (physical) models of fold–fault systems in layered rocks. These simulations involve a progressive evolution of the mechanical state, illustrating a geometrically dominated type of localization behaviour. Part of the deformation simulated here represents a crestal graben system. Analysis of the evolving mechanical state in the system of simulated faults poses challenges to some longstanding ideas concerning the way that faults operate, suggesting the need for a new fault-process paradigm.

One prime motivation for geomechanical research is to provide a physically based explanation of the causes of deformation features. This enhanced physical understanding can then be used to predict the initiation and evolution of a particular deformation feature, or sets of features. That is how it should work, but significant difficulties can arise as a result of a mismatch between the observations and the conceptual knowledge of deformation processes, and the technological capabilities needed to make robust predictions. In this paper, we present the results of a modelling programme that demonstrates a significant convergence between theory and practical capabilities, allowing us to critically assess a longstanding concept of fault formation. Our assessment addresses both the relationship between fault formation and the existing stress state, and the potential modification of that stress state by the fault initiation and development. Faults or shear zones are among the more important types of localized deformation, at least for a range of subsurface extraction and storage problems. This is due to the way that these features affect fluid flow along and across the zones, which can be represented as fault zone sealing potential, how they alter rock strength, and to the role that faults play in altering the shape of a deforming

system. There have been numerous studies that have attempted to explain the formation and evolution of faults and fault-like features in terms of primary mechanical concepts. In one of the seminal contributions to this topic, Anderson (1942) related the development of fault sets (representing a bulk strain state) to the notion of a putative causal stress state. Anderson’s assumed stress states were simple and highly idealized, and there was little or no evident consideration of the stress state being a consequence of deformation events. Hubbert (1951) was perhaps the pioneer in terms of relating faulting to a non-uniform mechanical state that was created in response to a loading process. A companion paper, by Hafner (1951), amplified Hubbert’s view of the interrelationship between loading and the resulting stress state by introducing (into the geoscience literature) an approach based on solving a boundary value problem to predict the development of fault arrays. Sanford (1959), using both experimental methods and simple elastic calculations, reinforced the idea that fault systems develop progressively in an evolving, and self-controlling (we now would say emergent) mechanical system. Anderson, Hubbert, Hafner, Sanford, and many who have followed, have explicitly or implicitly

From: LEWIS , H. & COUPLES , G. D. (eds) The Relationship between Damage and Localization. Geological Society, London, Special Publications, 289, 161–186. DOI: 10.1144/SP289.10 0305-8719/07/$15.00 # The Geological Society of London 2007.

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treated faults as macroscopic shear fractures writ large. That is, significant faults have usually been considered to be discrete pseudo-planar breaks that formed at an angle of c. 25–308 to the local direction of the maximum compressive principal stress. The faults then accumulated finite slip as a result of subsequent frictional motion that occurred within a stress state that was essentially the same as the one within which faulting was initiated. In this conceptual framework, curved faults were explained as being the consequence of yielding in a mechanical state where the stress trajectories themselves were initially curved as a result of the spatial arrangement of the loads. The apparent utility of boundary value problems in predicting faulting helped to spawn additional studies along similar lines. By applying the principle of superimposition that is inherent to the assumption of linear elasticity, almost any conceivable pattern of statically admissible loads can be applied to the boundaries of a model region (Sanford 1959; Couples 1977). In all but the most trivial of loading arrangements, spatial variations in stress orientations (curving trajectories), in magnitudes of all the stress components, and of quantities such as mean stress, are ubiquitous (Fig. 1).

Thus, the idea of spatially complex, but ordered, stress states has long been established as a foundation underpinning our explanation of the generation and evolution of realistic fault systems. Although it is true that an elastic, continuum model cannot remain strictly applicable after faulting is initiated, because the fault’s existence violates the assumption of a continuum, it is interesting that the fault-pattern predictions derived from the elastic solutions seem to be plausible as at least first-order explanations for realistic deformation systems. A range of modelling studies (Hubbert 1951; Sanford 1959; Serra 1973; Couples 1978; Couples & Stearns 1978) reinforce the idea that it is possible to associate the development of fault systems with a spatially heterogeneous mechanical state whose basic characteristics seem to be similar to the states predicted by simple elastic models. (See Stearns et al. (1981) for an expanded review of the papers mentioned here.) The studies noted above used an elastic continuum approach, precluding examination of how multiple, active faults interact during faulting and therefore also precluding an assessment of how the resulting mechanical state might differ from that predicted using the elastic continuum static

Stress Trajectories

Maximum principal stress Minimum principal stress Potential Faults

Below criterion

Exceeds criterion

Fig. 1. Elastic solution of boundary value problem using the method of Hafner (1951). This example illustrates a case with an end-load, with basal shear, and a standard state, leading to curved stress trajectories (upper image) and thus curved surfaces of predicted shear fractures or faults (lower image). The lower image depicts a boundary between states that exceed the Mohr–Coulomb failure criterion (continuous lines) and those states that are not at failure (dashed lines). This 2D model is 10 km high and 40 km long.

NUMERICAL SIMULATIONS OF FAULTING

stress state. A separate group of researchers also used elastic models to investigate fracturing and faulting (e.g. Eshelby 1957; Wiltschko 1979; Segall & Pollard 1980; Aydin & Schultz 1990; Crider & Pollard 1998; Maerten et al. 2002; Healy et al. 2006; de Joussineau & Petit 2007). In some of this work, faults or fractures were represented by explicit discontinuities using methods based on elastic dislocation theory and effective medium theory. Often these models were used to study how load rearrangements in the fault-tip region can perturb the stress state local to the fault tip. By adopting a line of reasoning similar to that suggested by Hafner, these models allow us to make a prediction of curved, or at least non-planar, fault propagation along a direction that is not parallel to the original fault orientation. Again, using the principle of superimposition (which is inherent to linear elasticity), more complicated models can be developed to examine how two or more faults (discontinuities) can interact. Studies using these methods help us to understand how one fault might inhibit the growth of another fault, or how one fault might make an adjacent fault more liable to grow, or how additional faults might form in the region of altered stress state. Similar reasoning is used to explain how joint systems might evolve when there is the potential for interactions (e.g. Healy et al. 2006; Olson 2007). Current implementations of this type of approach use boundary-element methods to calculate stress states for complex systems containing many faults (e.g. Willemse et al. 1996). Within those portions of such models that are not disrupted by discontinuities, the stresses are readily determined because of the assumption of elasticity; however, the discontinuities that are associated with the defined boundaries introduce the potential for non-linearity in the total state. Such studies reveal that multiple-fault systems can generate local complexity in the form of inhomogeneous stress patterns that could affect the growth of new minor faults (or joints), especially in sites such as the region between fault or joint tips. However, there is little support from this work for the idea that fault interactions could cause entire fault arrays to develop differently from the way predicted from the continuum models, or indeed, differently from the concept posed by Anderson. Thus, a key idea, that faults are the product of a stress state but that they do not significantly alter it, has been a functional paradigm for several decades. Developments in numerical simulation technology now provide us with an opportunity to reassess the faulting and stress-state evolution story. Do fault systems function according to an Andersonian view? Or do faults and stresses jointly evolve to create deformation systems, in which case our

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ideas concerning the role of stress may need to be updated? Examining this question of how the creation or growth of faults might change the subsequent development of entire mechanical systems requires us to move beyond the theory of elasticity. In practice, this requirement demands that we adopt the framework of plasticity theory, where yielding, and post-yield behaviours, are the primary factors that characterize material responses. The potential value of plasticity was first highlighted (in structural geology) by Ode´ (1960), who suggested that faults might be thought of as slip lines in a plastic medium. Ode´’s application of this theory to a geological problem was based on a robust line of work in soil mechanics, where slip lines represented a way to predict complex deformation responses in an age before numerical simulations were practical (e.g. Prager 1953; de Jong 1959; Abbott 1966). For example, slip lines (in two dimensions, or slip surfaces in three dimensions) can be derived via analytical (and graphical) methods for the case of materials ‘at yield’. They represent lines (planes) along which displacement discontinuities can occur because of the special conditions applying to a material at yield. In current terms, we would see the concept of slip lines within the context of localization and bifurcations (e.g. Rudnicki & Rice 1975). One important branch of current geomaterial plasticity is concerned with developing ways to represent material instabilities, or bifurcations, in a simulation environment (e.g. Chambon et al. 1994; Desrues & Chambon 2002; Collin et al. 2006). Numerical models focused on this issue are designed so that the stress state, prior to yielding, is absolutely uniform, and a research goal is to capture the effects of feedback caused by internal load rearrangements, themselves resulting from rock hardening or softening after yielding begins at an initiation point. One of the outcomes of that work is the discovery that it is necessary to introduce a new parameter to define the thickness of localization bands. Research in this subject is closely linked with experimental investigations, and the combination of these approaches is leading to insights into how localization processes take place (e.g. Desrues et al. 2007), and how to represent them in a practical way in numerical models. Here, we illustrate that localization can occur in simulations of plastic deformation without necessarily needing to adopt higher-order approaches that require specification of the band-thickness parameter. In one case, where we simulate the deformation of a laboratory rock cylinder, our approach leads to the development of simulated shear bands that are slightly too thick, but otherwise are good approximations of the experimental results. In our other simulations, localization occurs because of

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Package of layers undergoing flexure

Rigid blocks that are translating and rotating along a fault surface Fig. 2. Illustration of a fold–fault system in which the folding (flexure) of a layer, or sequence of layers, is causally associated with loading imposed by fault-associated translations and rotations operating transversely to the layer from beneath. This loading arrangement is an example of forced folding (see Stearns 1978). This image shows part of the experimental design used in the models described by Couples et al. (1994).

geometric requirements associated with the evolving model configuration. These latter simulations involve a fold–fault system (Fig. 2) that has been the subject of numerous theoretical (Hafner 1951; Hubbert 1951; Sanford 1959; Couples 1977; Gangi et al. 1977) and experimental (Sanford 1959; Friedman et al. 1976; Weinberg 1979; Couples et al. 1994) studies over a long time period. In the experimental investigations of this system, initial yield does not lead to a complete through-going failure. Instead, localization phenomena (i.e. plastic yielding) occur by means of entire systems of fault-like zones that progressively develop and evolve. By analysing how these fault-like features interact, and how the instantaneous stress state is dependent on their development, we can gain new insights into the sorts of questions that have been studied since Anderson (1942) proposed his explanation for faults. The paper begins with a section that summarizes a few necessary geomechanical concepts and the knowledge-base of experimentally derived rock deformational behaviours that are captured by the material descriptions implemented in the numerical poro-plastic material model. Next we describe how the poro-plastic conceptual model for rock mechanical responses has been implemented into a numerical simulation environment. More detailed material for these topics is contained in Appendices A and B.

The numerical implementation is validated by means of comparing calculated results with observations obtained via laboratory testing of rock cylinders. We then apply the new simulation capability to a model configuration that involves the bending of a package of rock layers (e.g. similar to Fig. 2). During that bending deformation, the model’s layers develop arrays of localized strain features, which we suggest are approximations to fault arrays. The simulated patterns of strain agree well with those observed to occur in a comparable suite of physical models. Analysis of the deformation sequence allows us to suggest a new faultprocess paradigm that differs considerably from the way that faults were understood to operate.

Poro-plastic rock deformation and its numerical implementation It has been known for some time (Handin & Hager 1957; Griggs & Handin 1960; Handin et al. 1963) that most of the common sedimentary rock types exhibit a progressive change in their macroscopic deformation behaviour as a function of increasing the effective overburden or confining pressure. This knowledge is grounded in the laboratory, but outcrop observations reinforce the story. In the rock mechanics laboratory, the confining pressure

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Fig. 3. Representative experimental results for the deformation of porous rocks. The samples depicted here are of Berea Sandstone, a fine-grained, well-sorted, quartz-rich rock with about 18% porosity. The four cylinders were deformed under different confining pressures: from left to right, the effective confining pressures increase from 50 MPa to 75 MPa to 90 MPa to 120 MPa. The macroscopically observable responses change as a function of the confining pressure. The examples on the left exhibit strongly localized shears that are widely spaced and appear as distinct features. The examples on the right are also deformed by shears, but the slip surfaces are more closely spaced and individually have smaller offsets. Throughout this suite of samples, the deformation mechanism remains constant: cataclasis, with grain breakages and rearrangements. The reflective surfaces of the images are due to the plastic jacketing material that is part of the experimental protocol.

is typically arranged to be the least compressive stress, s3. At low (relative) effective confining pressure, a given rock will deform by means of widely spaced, distinct shear zones (Fig. 3). When the confining pressure is greater, the shear zones become more closely spaced, and often are slightly ‘less sharp’. Macroscopically, the cylindrical rock specimens often develop a barrel shape during deformation at the higher confining pressures; this is sometimes called a ductile response. Generally, there is no change in the deformation mechanism that is operative across the range of experimental conditions that produces the macroscopic changes noted here, but there can be significant changes in the distribution and intensity of deformation. In the cases illustrated in Figure 3, all deformation occurs by means of cataclasis, with grain breakage and movement of separate particles occurring within distinct bands of deformation. The movement associated with these bands leads to the macroscopic strain that is observed for the whole specimen. The experimentally recorded stress –strain behaviours of typical sedimentary rocks are similarly dependent on the confining pressure. At low confining pressure, the yield point is typically well-defined (Fig. 4), whereas at higher confining pressure the yield point may be less obvious. Sometimes, the yield point is identified with the beginning of nonlinearity in the stress –strain response. Post-yield characteristics also differ depending on the conditions: cases at low confining pressure are more likely to be typified by a strain- (or work-)softening

(a)

5 Yield point difficult to identify 4 3 Δσ 2 Yield point is clearly determined 1 ε

(b) Coulomb τ failure criterion

1

σ 2

3

4

5

Fig. 4. Illustration of typical experimental measurements for the deformation of porous rocks. (a) Range of differential-stress–axial-strain plots observed for a single rock type deformed under increasing (effective) stress (confining pressure), as indicated by curves labelled 1 –5. (b) Typical Mohr circle plots of the data shown in (a). (Note the change in yield criterion line associated with change in stress– strain response that occurs between case 3 and case 4.)

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Pm

Dilatant Volumetric Strain Curve 5 Isotropic Compression

Curve 1

Compactant Volumetric Strain ΔV Vo

Fig. 5. Illustration of the way that volumetric strain (change of volume divided by original volume) develops as a function of increased confining pressure. Here, the plot uses Pm, the mean stress, which serves as a proxy for increased confining pressure. In a standard experimental protocol, the sample is initially loaded by an isotropic stress up to the designed magnitude of the confining pressure; differential stress is then generated by means of axial shortening. The volumetric strain follows the dashed line during the isotropic loading, and then diverges once a state of differential stress is created. Dilation occurs when the material experiences a volume increase compared with the volume expected under isotropic confinement. Compaction is the case where the sample volume decreases more than would be expected by the confinement. It should be noted how these data show a change from dilational to compactional responses as mean stress (confining pressure) becomes larger. The change from dilation to compaction occurs roughly in association with the stress –strain change that is represented by curve 3 (see Fig. 4).

response, whereas the higher confining pressure conditions often lead to strain-hardening behaviours. An important characteristic of the post-yield region is the volumetric strain that occurs as the rocks continue to yield. It is often observed that the low confining pressure case produces a volume increase, or dilation (Fig. 5), whereas higherpressure cases tend to result in a volume loss, or compaction, as yielding continues. Poro-plasticity is a theoretical approach (e.g. Schofield & Wroth 1968; Atkinson & Bransby 1978; Jones & Addis 1986; Muir Wood 1990; Coussy 1995; Cuss et al. 2003) that integrates, or explains, the main macroscopic geomechanical characteristics that have been outlined above. A summary of poro-plasticity is provided in Appendix A. To use poro-plasticity concepts in a practical way, the theoretical representation needs to be incorporated within a geomechanical simulation tool. We describe in general terms in Appendix B how the theory has been implemented within SAVFEMTM (Structural Analysis via Finite Element Method), which is a general-purpose simulation environment created and marketed by Applied Mechanics Inc. (AMI). Some of the strengths of SAVFEMTM include its treatment of frictional sliding, which

makes it suitable for addressing a range of problems of interest in structural geology, and its treatment of elastic–plastic material responses. The current SAVFEMTM state-of-the-art in this regard is a material type called a ‘general power-law material’. Appendix B describes the general characteristics of the SAVFEMTM simulation system, along with the key features of this advanced material type, and how it relates to the poro-plastic concepts outlined in Appendix A.

Calibration of material properties to laboratory data Using SAVFEMTM, we have loaded a cylindrical model (Fig. 6) that represents a rock sample like those typically examined in the laboratory (e.g. Fig. 3). Simulations using the cylindrical model reproduce the outcomes of laboratory experiments (one example being included here), and so validate that the numerical implementation is operating as intended, and that the input material parameters result in calculated deformation behaviour (yielding, and post-yield responses) that is the same as that which occurs in the laboratory tests. A key step in creating this model (and achieving the result shown) is to acknowledge that the ends of the experimental rock cylinder are in contact with steel pistons that are used to impose an axial shortening onto the cylindrical specimen. Rock materials and steel have different properties, so there is a small amount of movement that occurs at the rock –steel interface during an experiment. It is standard laboratory practice to lubricate that interface, giving it a low coefficient of friction, so we need to replicate that aspect of the experimental configuration in our model. In the cylinder models described here, we include a frictional surface along the cylinder ends to separate the rock material from the simulated loading system, and to allow the simulation outcome to replicate the (possibly significant) mechanical state that develops at that interface. SAVFEMTM represents such frictional surfaces by means of friction–gap elements. These elements allow adjacent parts of a model to slide past one another (or to separate) according to a set of rules. The elements of a discontinuity are represented by initially co-located grid points that belong to adjacent elements of the finite-element mesh that represents the continuum rock materials. For the frictional aspect of their behaviour, these grid points (or gap elements) remain locked if there is a compressive normal force (stress), and the shear force (stress) is below a value determined by the coefficient of friction. If the shear force versus the normal force ratio exceeds the sliding criterion,

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Fig. 6. Numerical simulation of the experimental deformation of a rock cylinder deformed under a confining pressure of 50 MPa. (a) Differential-stress v. axial-strain plot derived from calculated output of the model. This curve reproduces the experimental curve used to create the material properties. (b) Deformed finite-element mesh, showing localized zones of shear strain. (c) Cut-away view showing magnitude of effective plastic strain (see Appendix B, equation (B11), for definition), which in this case is dominated by the shear strain components. Dark blue indicates effective plastic strains of zero (i.e. elastic strains only). Lighter blue and green indicate increasing magnitude of equivalent plastic strain. (Compare this model result with the actual experimental results shown in Fig. 3.)

the element is allowed to slip, with the amount of slip determined by an iterative procedure: slip reduces the shear force by allowing the adjacent material to strain, but redistribution of load also occurs in the surrounding solid materials, thus complicating the non-linearity associated with such friction elements. If the normal force becomes tensile, the gap elements can separate. Significant amounts of sliding are made possible by defining some number of additional grid points (sometimes referred to as nodes) adjacent to the point initially defined at the location of a pair of continuum mesh grid points, such that these extra grid points become used in the calculation if sliding repositions the gap-element grid point into proximity with them. In effect, this design allows the user to tell the system that one part of the model can move relative to another part according to a frictional law. In the case of the cylinder ends, we specify the actual boundary conditions (axial displacement) on grid points that are outboard of the frictional surfaces, to better replicate actual laboratory conditions. Here we summarize one example to illustrate the validation process. The rock type used for this case is the well-studied Berea Sandstone (Coyner 1984; Bernabe´ & Brace 1990; Hart & Wang 1995; Katz et al. 2000; Wong et al. 2001). The results presented

here are at the lower end of the confining-pressure range represented by the four rock samples of Figure 3. The simulation run begins by imposing an isotropic stress state of 50 MPa. Then, the top of the numerical model, which is external to the rock cylinder, is displaced downwards in a series of steps to replicate the movement of the loading piston in the experimental setup. The grid points immediately below the cylinder (and separated from it by another frictional surface) remain fixed, representing a stationary steel piston in the laboratory apparatus. Within the mesh representing the rock cylinder, we fix the central grid point on the basal surface, but allow the other points of that surface to move laterally if frictional resistance (m ¼ 0.05) is overcome. Just as in the experimentally deformed cylinders, the model generates a reaction stress that increases approximately linearly as the axial shortening increases (Fig. 6a). By following the evolution of the deformation sequence, we extract values for the differential stress in the cylinder (axial stress minus confining pressure) and plot these against the axial strain. A sharp yield point appears as the material reaches its elastic limit. In this instance, the conditions are such that the material behaves, after yielding, with an initial work-softening style of response, and

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then enters a pseudo-steady-state where strain continues to accumulate while the stress state remains nearly constant. As post-yield strain accumulates in this model, it does so in localized bands (Fig. 6b). The spatial pattern of these bands is very much like the pattern observed in the actual experiment (see Fig. 3). From any viewpoint, such bands usually occur in conjugate pairs, which means that they have a pseudo-radial symmetry similar to the pattern usually observed in real experimental cases (e.g. Mair et al. 2002; Be´suelle et al. 2006). It is important to note that the development of plastic strain in this model does not depend on manual initiation by means of (arbitrarily) weakening some element in the model, which is a so-called seeding method. Instead, yielding occurs when one or more locations build up sufficient stress so that the yield criterion is met at that site. We assume that initial yielding at one or more sites is due to miniscule variations in stress state related to numerical round-off issues, or to tiny variations related to heterogeneities introduced by nonuniform sliding on the frictional ends. Once that initial yielding takes place, the rest of the model reorganizes the stress distribution, and these changes lead to further yielding, and the process continues. Although nature does not suffer numerical round-off difficulties, we think that it is reasonable to think of these modelling ‘artefacts’ as being analogous to the minor variations in material properties or loading that we can expect to be present in real rocks, as a consequence of grain-packing variations, cementation differences, imperfect bed shapes, etc. Our model cylinder is a continuum. When the strain localizes into prominent bands, as seen in this example (and in our other models, which are designed to reproduce tectonic processes), it does so as a continuum within the context of the theory that we assume to apply. How does this numerical response relate to very sharp localized shears and discontinuities (fractures) seen in real rocks? Do our localized bands have a sensible relationship with faults? Of course, natural localization phenomena can range from continuous (but intense) zones of strain, as seen in this model, to discrete features where the strain can be expressed as a distinct discontinuity. Does the model serve to represent this suite of deformation types in an appropriate fashion? One way to approach this question is to answer another question: What is the strain distribution in the bands that develop in this cylinder model? Perhaps the easiest way to answer this latter question is to consider the components of strain. For ease of illustration, we have taken a 2D slice through the model, along a diameter of the cylinder.

The localization bands expressed in this view have both volumetric and distortional strain components, with the distortional strain components being dominant. The distribution of plastic strains, and the pattern of localized deformation, is compatible with observational data on natural and experimental shear zones. On the basis of this and other similar results, we conclude that the localization characteristics of the material model in SAVFEMTM are suitably representing fault-like deformations.

Application to a geometrically complex problem There are different ways to use geomechanical simulation tools. In one approach, the goal is to create a uniform state and observe the development of material instabilities, and how these instabilities evolve into finite band-like features. In that usage, the intention is to assess how well the software implementation replicates the behaviour represented by the poro-plastic localization ideas. Because of the highly non-linear aspect of poroplasticity, the solution of the equations can only be accomplished numerically. Typically, the solution approach involves a form of finite-element modelling with a criterion for localization along with the usual material-property and yielding parameters. As a result of the nature of the discretization process and how this determines the spatial gradients permitted in the calculated mechanical state, it is necessary to introduce higher-order theories to simulate localization features such as a shear zone. After introducing these additional complexities, this class of geomechanical simulation codes is capable of replicating material instabilities that occur during a localization process (Chambon et al. 2004). The approach we are using here is based on the implementation in SAVFEMTM that does not include the higher-order (i.e. second gradient) approach or a specific localization criterion. Instead, the models that we are concerned with are dominated by their geometry. By this we mean that localization takes place if there is reinforcing feedback between the hardening or softening behaviour that occurs at yield, and the load redistribution process that occurs as deformation proceeds and loading is progressively changed. The cylindrical models described above illustrate this type of localization caused by the way that the deformation state ‘sees’ the finite boundaries of the sample and develops spatial heterogeneities of the deformation state inside the cylinder. This feedback process will become more obvious in the example described next.

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Reference physical rock model Couples et al. (1994) described a suite of physical, rock-layer models deformed under confining pressure in the rock mechanics laboratory. These rock models were designed to explore the processes of flexural deformation occurring in a layered system when it is subjected to an offset (displacement discontinuity) resulting from motions associated with fault-bounded blocks below. Such a situation could arise where a steeply dipping fault propagates upwards towards the base of the layered system. The natural examples that inspired the experimental design are the basement-cored uplifts that are well exposed in Wyoming, USA. Here, fault blocks of crystalline basement rocks are rotated and translated along typically steep curved faults (e.g. similar to the image in Fig. 2), causing deformations of the cover sequence (see Stearns 1978, and references therein). Erslev (1986), Brown (1988) and Narr & Suppe (1994) have provided some contrasting views of the interpretations and mechanics associated with these uplifts. It is not our purpose here to pursue that debate; rather, we simply wish to note the historical context that led to the preceding theoretical and experimental work on flexures causally associated with subjacent faulting. The layered rocks in the sedimentary cover succession in Wyoming exhibit a range of responses to the uplift and rotation of the basement blocks below. Where there is a ductile rock unit at the base of the cover sequence, the remainder of the overlying cover usually does not fault significantly; instead, it deforms into an approximately continuous fold. Such exposed folds (without major faulting) can have structural amplitudes in excess of 2 km, and so may represent analogues for large subsurface hydrocarbon reservoirs. Although major faulting does not completely separate the layered rock sequence, smaller (but still large) faults do occur. It is these arrays of faults within the layered sequence that we address here. The physical models described by Couples et al. (1994), representing an advanced stage in a continuing series of rock-layer models of forcedfolding (Friedman et al. 1976, 1980; Weinberg 1979), reproduce the primary behaviour of the interpreted natural system when deformed under confining pressure in a large-sample apparatus (Fig. 7). The suite of physical models gives insights into the layer-parallel motions that take place within the flexure, and the interactions that occur between the layered sequence and the rigid forcing blocks (Couples & Lewis 1998). Additional physical models that evaluate the effects of layer subdivisions, and further contrasts in material properties, continue that investigation sequence (Couples &

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Lewis 2000). However, the simpler suite, reported by Couples et al. (1994), is the basis for the numerical simulations that are presented below. The layer-package assembly used by Couples et al. (1994) in their experiments is composed of three distinct mechanical units. The thicknesses and mechanical behaviours of these thin units were selected to promote a system response like that derived from interpretations of the outcrop examples in Wyoming. At the base of the layerpackage is a ductile unit. In nature, this is composed of Cambrian shales and mudrocks, whereas in the experimental models, it is composed of lead sheet stock (like that used for roofing applications in the building trades). The middle unit of the three-layer assembly is, in nature, a package of layered carbonates, and in the experiments is a single layer of dolostone. In nature, the upper unit is a composite sequence of mixed siliciclastic rocks, evaporites and carbonates that, overall, behaves in a semiductile fashion. The experimental material used to represent this sequence is a single layer of porous limestone, which has a semi-ductile behaviour mode at the 50 MPa confining-pressure conditions used in the experiments. The layer contacts in the rock models are slightly lubricated to allow them to slip as frictional surfaces. Experimental procedures require that another layer of lead be placed above the limestone layer (to avoid jacket punctures). This last layer has no specific natural counterpart, but it is believed to have no significant mechanical impact on the experimental results. The deformation pattern that evolves in these rock models is highly reproducible over a series of experiments (Fig. 7: see also Couples et al. 1994). Here we will consider only the damage that is induced in the dolostone and limestone layers of the models. These patterns of damage will later be used for comparison with the numerical simulation results. Taking the limestone layer (the top, thick layer) first, a prominent graben system is developed in the crestal region of the antiform. The fault system propagates from the top towards the base of the layer as the degree of flexure increases. The fault system also widens, and individual fault surfaces increase their total displacement, while additional faults are initiated and then join the growth scheme. There is a sense that the faults grow in a self-similar fashion. A comparable, but slightly less prominent, system of deformation affects the synform position of the limestone layer. This ‘upside-down’ graben system is almost a mirror image of that which affects the crestal region, except that it is smaller. Between the two extensional areas, curved reverse faults cross the steep limb of the flexure (which we will call the forelimb). These features are oriented at high angles to the layer at the base of

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Fig. 7. High-pressure experimentally deformed models composed of rock layers that develop a flexural deformation style in response to a localized offset imposed along their base. The left images are line drawings that represent a composite interpretation of the damage distribution based on visual inspection of the model exterior, the slabbed surface of the model (see right images), and thin sections. The right images are photographs of these two models made after epoxy impregnation and cutting along the medial plane. The epoxy can be identified by its transluscent character in the images. The upper set of images is of model 46, and the lower set is of model 40 (see Couples et al. (1994) for a complete description of the experimental process and details of the models). (Note in particular the development of the crestal graben system in both models.) Reproduced from Couples (1986) with permission.

the limestone, but become subparallel to the layer boundary at the top of the unit. The dolostone, which behaves in more brittle manner at these conditions in standard rockcylinder tests, develops a separate pattern of deformation from that observed in the limestone layer. The fractures that are created in the dolostone are very localized. Most fractures penetrate across the whole of the dolostone layer, but some models show partly penetrating fractures. The fractures open in a wedge shape, indicating greatest along-layer extension at the upper arc of the crestal antiform, and along the lower arc of the synform. The dolostone layer has distinct hinges, with nearly straight limbs between them. The fractures are more numerous in the hinge areas. It should be noted that individual fractures do not typically cross the limestone –dolostone interface. It is worth mentioning the style of deformation that affects the lead layer at the base of the package. This lead experiences a high degree of ductile flow during the experiments. Major thickness changes occur, and the lead actually pinches off into separated upper and lower sections at high

structural relief, when the steel forcing blocks below the layer-package contact the base of the dolostone. Similar deformation is observed and interpreted in the natural examples (Couples 1986; Couples et al. 1994). As anticipated, it is difficult to numerically simulate this extreme flow of the lower lead layer.

Simulation results The layer-packages of the experimental rock models are just over 2 cm in thickness. Our 2D model of the package of layers (including the lowest highly ductile unit) is 23 mm high and 100 mm long. A mesh of 9300 finite elements uses general quadrilateral elements to discretize the model region, with mesh densification in the area that will become folded and faulted (Fig. 8). The layer contacts are welded together in this initial model; that is, no friction elements occur along the layer contacts, so no discontinuities are possible. The displacement boundary condition at the base of the model reproduces the rotation–translation of the main forcing blocks of steel in the experimental models, and is

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Fig. 8. Geometric aspects of the simulation model. (a) Illustration of finite-element mesh, showing the complete model prior to loading, and giving the dimensions. (b) Similar diagram showing the complete model after deformation. (c) An enlargement of the central region of the model, indicating the main layers, and the mesh densification in the area that will be deformed. (d) An enlargement of the deformed mesh in the central region of the model.

similar to that interpreted for the movement of basement blocks in natural prototypes. We start the simulation of the block uplift after letting the model equilibrate to an isotropic pre-stress state that replicates the confining pressure of the experimental conditions (50 MPa). At the natural scale, each layer of our model would represent a composite mechanical unit composed of numerous individual rock layers with bedding-plane contacts. We have not attempted any form of formal upscaling to create ‘equivalent’

materials to represent the behaviours of such real-world units. Instead, what we illustrate here are models where each layer is composed of material that is initially isotropic and homogeneous (as in the experiments themselves). The material model used for the ‘limestone’ and ‘dolostone’ layers is based on the Berea Sandstone model described above, with minor adjustments to that material type made to reflect differences in elastic modulus, uniaxial compressive yield strength, cohesion and angle of internal friction. The lead is

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Fig. 9. Plot of effective plastic strain (see Appendix B) through the central region of the simulation model with welded layer contacts. Yellow indicates non-permanent (elastic) strains. Colour scale shows plastic strains up to about 0.5. It should be noted that the strain is localized in distinct bands, and that there are regions with generally higher strain values. Also, it should be noted that the strain pattern changes at the boundaries between material types. The right side of the image gives the names used to refer to the component layers of the model.

simulated by a simple elastic –plastic material type that has properties that are characteristic of rock salt. When this SAVFEMTM model is simulated through a series of load steps, it calculates the stress and strain evolution of the entire system at each step. The simulation runs to approximately half of the (relative) structural relief achieved by the experimental models and some of the natural examples. This end point is the result of high distortions of elements in the lowest layer, which has the ‘salt’ rheology and represents the lead in the experiments, or shale in nature. The extreme strains in

these highly deformed elements create numerical problems that lead to an early abort. This is not a long-term problem, as the lowest unit can be re-meshed and the run can then be restarted. In spite of the problem with the lowest layer, the model runs to a sufficient state of deformation that the distribution of damage in the upper layers is very obvious, and its pattern of evolution is clear. The end point of this model run results in a state of deformation that is nearly identical to that observed in equivalent experimental models. Here we use a plot of effective plastic strain (a scalar

Fig. 10. Plot of the magnitude of maximum compressive principal stress (s1) across the region depicted in Figure 9. Scale bar shows magnitudes ranging from slightly below zero (i.e. tensile stress, dark red, with small regions of more tensile stress shown in white) to compressive stresses of 220 MPa (dark blue; higher values are indicated in white). This image also shows the deformed finite-element mesh. A noteworthy feature is the lateral flow of the lower ductile unit, which is comparable with the flow of the equivalent experimental material as reported by Couples et al. (1994) and Couples & Lewis (1998, 2000).

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Fig. 11. Plot of the magnitude of minimum compressive principal stress (s3) across the region depicted in Figure 9. Scale bar shows magnitudes ranging from slightly below zero (i.e. tensile stress, dark red, with small regions of more tensile stress shown in white) to compressive stresses of 110 MPa (dark blue; higher values are indicated in white). This image also shows the deformed finite-element mesh. (See also caption to Fig. 10.)

quantity that can be readily contoured; see Appendix B, equation (B11)) to discuss the results. The high-strain regions (Fig. 9) are direct analogues of the localization features created in the simulations of the rock cylinders. The strain plots show that there is a graben system at the crest of the antiform in the thick limestone-equivalent layer, and the same upside-down graben system in its synform position, as is seen in the experimentally deformed models (Fig. 7). The curving reverse faults that disrupt the forelimb are there, too. The calculation produces swarms of sharp strain features in the hinges of the dolostone-equivalent layer, with the correct pattern of layer-parallel

elongations: these are also seen in Figure 7. Load steps that lead up to the final calculated state that is shown in Figure 9 (not included here for space reasons) indicate that the graben systems develop progressively, just as they did in the experimental models. In short, the numerical results reproduce all of the characteristics of the rock-model experiment, and the main features observed in comparable natural examples. To examine the way that permanent (plastic) strain affects the resulting stress state, we wish to consider three regions that have generally elevated magnitudes of equivalent plastic strain (although it is important to appreciate that undeformed and

Fig. 12. Plot of effective plastic strain (see Appendix B) through the central region of the simulation model with frictional layer contacts. Yellow indicates non-permanent (elastic) strains. Colour scale shows plastic strains up to about 0.5. It should be noted that the strain is localized in distinct bands, and that there are regions with generally higher strain values; also, the strain pattern changes significantly at the boundaries between material types (along slip surfaces). The right side of the image gives the names used to refer to the component layers of the model.

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Fig. 13. Plot of the magnitude of maximum compressive principal stress (s1) across the region depicted in Figure 12. Scale bar shows magnitudes ranging from slightly below zero (i.e. tensile stress, dark red, with small regions of more tensile stress shown in white) to compressive stresses of 220 MPa (dark blue; higher values are indicated in white). This image also shows the deformed finite-element mesh. The sharp stress discontinuities along the slip surfaces should be noted.

little-deformed sub-regions exist within them). In reference to Figure 9, we wish to focus on the thicker layer (labelled semi-ductile) that is equivalent to the limestone layer of the experiments. The three regions consist of the crestal graben area, the forelimb area and the synformal area. If we examine the distribution of maximum compressive principal stress magnitudes (Fig. 10), we see that the three regions of increased plastic strain are each associated with a region of anomalous stress values. In the crestal graben area, there is, in general, a reduction in the maximum stress values (compared with the less-strained region to the

right side of the image, which is characterized by stresses close to the imposed confining pressure), but there are small areas with increased stresses that are interspersed within the area of reduced stresses. The synformal area shows a similar character, with generally low stresses and occasional high values. In contrast, the forelimb area is dominated by generally elevated stresses (but again, there are small interspersed regions that have lower stresses). Clearly, the deformation has caused extreme heterogeneity in the stress field. The pattern of least-compressive principal stress magnitudes (Fig. 11) is also affected by the three

Fig. 14. Plot of the magnitude of minimum compressive principal stress (s3) across the region depicted in Figure 12. Scale bar shows magnitudes ranging from slightly below zero (i.e. tensile stress, dark red, with small regions of more tensile stress shown in white) to compressive stresses of 110 MPa (dark blue; higher values are indicated in white). (See also caption to Fig. 13.)

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regions defined by the plastic strain distribution. However, the boundaries of the three regions are not as distinctly obvious in this plot, with almost the entire region of the double flexure exhibiting a reduction in the least stress magnitude, as compared with the confining pressure. The only site of increased s3 is in the shallow portion of the semi-ductile layer on the left side, above the synformal area, and where the forelimb shears are terminating. If we now consider the stiff, or brittle, layer, we can see that the principal stresses are affected here, too. Because of the greater stiffness (higher Young’s modulus), the material of the brittle layer develops larger stresses for the same strain magnitude, as compared with the material that forms the semi-ductile layer. Within the brittle layer, the s1 magnitudes are larger than 220 MPa underneath the crestal graben area (the white area in Fig. 10). Even the s3 values are increased at that location, particularly in the lower part of the layer. This model is characterized by heterogeneity in its stresses, with large-magnitude changes occurring over short spatial scales. Although this simulation result is impressive in terms of its reproduction of the outcomes from the experimental work, an even better match can be obtained if we acknowledge that frictional slip can occur between the rock layers in the physical models. To assess the role of such interlayer slip, we have constructed a separate numerical model that is identical with the one described above, except that the interfaces between the layers are represented by friction –gap elements with a coefficient of friction equal to 0.3. This new model, which allows for layer-parallel slip, develops a pattern of deformation that is very similar to the case described above, but there are differences (Fig. 12). The most important difference is that the localized strain features are very distinctly limited to the individual layers, with sharp terminations at the layer interfaces. The maximum (Fig. 13) and minimum (Fig. 14) principal stresses once again show a similar relationship to the regions of elevated equivalent plastic strain. The comments made in relation to the previous model still hold, but it should be noted that the existence of frictional surfaces at the layer contacts allows very pronounced stress differences to develop adjacent to those slipping interfaces. Stress magnitudes in these local spots are also slightly larger, compared with the no-slip case. The stress-trajectory patterns in this model with frictional layer contacts reveal additional aspects of the same story (Fig. 15). The maximum principal stress direction and magnitude varies through the model region. If we focus attention on the upper

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thick layer, s1 is dominantly vertical (layer-normal) in the crestal region (and also in its synformal equivalent), but displays local variations within the fault zones (see further discussion on this observation below). Although the antiformal and synformal locations have generally layer-normal s1 orientations, the stress trajectories of these regions are part of larger patterns that show curving stress trajectories, with local regions of layer-parallel s1, and large areas with trajectories that are not aligned with the boundaries of the model. The resulting pattern of stress trajectories is very similar to the pattern predicted by elastic solutions to a similar boundary-value problem (e.g. Hafner 1951; Couples 1977), and to the stress patterns inferred from petrofabric studies of the experimental models (Couples et al. 1994). In the lower, brittle layer of this image, there are similarities to that pattern, but the stresses vary much more markedly near the high-strain zones. Also, this layer is thinner, so the stress variations that depend on position in the layer are telescoped together and complicate the visual display. The magnitudes of the horizontal (x-direction; Fig. 16) and vertical (y-direction; Fig. 17) stress components also show considerable spatial variability. Similarly to the principal stress plots, one can see ‘boundaries’ between regions where the magnitudes of the stress components change. These positions of stress-magnitude change are associated with the regions of plastic strain noted above, and they are also the places where the patterns of stress trajectories change. It is very obvious that there is a strong interaction between the strains that accumulate to make permanent deformation, and the resulting stress state. In detail, the materials develop localized zones of high strain (and altered stresses) with intervening regions of less strained material (which have less unusual stresses). It should be noted that in those portions of the model where the materials need to elongate in response to the imposed loading, the horizontal stress component becomes small, or even tensile (as may the principal stresses). In a continuum approximation, which is what we have adopted here, it is necessary to permit the materials to develop a small tensile stress if we wish to allow them to undergo elongations. In nature, such elongations might be expressed by the opening of fractures or joints, but it is not possible to replicate huge numbers of such discontinuities within a continuum-dominated model system. Thus, our decision to allow a small tensile response (via the parameters used to define the material characteristics; see Appendix B) is a practical one that is governed by the limitations of the continuum approach. It is worth noting, however, that the tensile stresses as calculated are probably too large (too tensile).

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Fig. 15. Principal stress trajectory plot for the limestone-equivalent and dolostone-equivalent layers of the model shown in Figure 12. The blue (s3) and grey (s1) lines represent compressive stresses, and the red and pink lines represent their tensile stress equivalents, respectively. Line orientations show the direction of the principal stresses at each plotting point. Line lengths are related to the stress magnitude, but a non-linear scale is used (refer to Figs 13 and 14 for principal stress magnitude information). It should be noted that the stress trajectory patterns change at the boundaries between the material types, and also at the edges of the strained regions discussed in the text. The pattern depicted here is similar to that calculated by Hafner (1951) and Couples (1977) using elasticity theory.

Fig. 16. Horizontal stress component magnitudes for the model shown in Figures 12–15 (with frictional interfaces). Only the semi-ductile and brittle layers are shown in this case. This image uses a different display tool, and leaves off the finite-element mesh to allow details to be more easily seen. The large (to .300 MPa) and very large (600 MPa) stress values in the inner arcs of the antiform in the semi-ductile and brittle layers, respectively, should be noted.

Fig. 17. Vertical stress component magnitudes for the model shown in Figures 12–15 (with frictional interfaces). Only the semi-ductile and brittle layers are shown in this case. This image uses a different display tool, and leaves off the finite-element mesh to allow details to be more easily seen. The very low stress values around the synform region, which is protected from the boundary confining pressure by a self-organized arch that transfers the loads, should be noted.

The reason for this statement is that the opening of pre-existing discontinuities is likely to be associated with a low stiffness. In effect, the elastic modulus (or, better, its plastic equivalent) would be lower

in crack dilation, so the ratio of stress and strain during elongation would be different from that in the intact material, which means that the tensile stresses as calculated (using the same modulus as

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for intact material) are overestimated. If the dilation modulus was, say, an order of magnitude lower, then the ‘actual’ tensile stress would be an order of magnitude lower. The distribution of the vertical stress components also warrants some comment. Once away from the region of the flexure, the vertical stress magnitude is approximately constant and equal to the assigned confining pressure. Thus, it is clear that the effect of the bending is spatially limited. However, the region where the vertical stress is affected occupies a considerable area that extends beyond the points where the curvature of the layers becomes indistinct. Within the disturbed region, the stress magnitudes are both lower and higher than the confining pressure. The synform region reveals a significant unloaded zone associated with the auto-development of a stress arch (curved stress trajectories seen in Fig. 15). Within the crestal graben, there are places with greater, and smaller, values of vertical stress, as compared with the confining pressure.

Discussion The fidelity of these numerical models, relative to the pattern of deformation damage observed in the experimental rock models, is a significant step forward in geomechanical simulation capability. The shear zones that develop in the simulations do not require a seed, an anomalous, typically weak, cell, in order to initiate or grow. The progressive development of the organized arrays of shears is simply a consequence of the interaction between the material behaviour and the mechanical state that evolves in those portions of the flexure, which itself is a result of the imposed boundary displacements. In other words, the bending deformation that affects the layered materials is accomplished by the materials developing the pattern of strains that we see in the both the physical (rock) and numerical models. In current terminology, we will say that the self-organized patterns represent an emergent phenomenon. The detailed correspondence between these complex physical and numerical models, including the pattern and timing of induced shear zones, indicates that the numerical simulation approach that has been implemented in SAVFEMTM is a robust representation of the mechanical processes operating in the rocks that form the experimental models. Given that there is such a good correspondence between the numerical simulations and the experimental rock models, it seems reasonable to query the numerical models to discover additional information about the deformation process that we cannot readily obtain by other means. Probably the most important data of this sort concern the

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stress state. In our simulations, the stresses do not need to be inferred based upon some proxy measure such as petrofabric studies (see Friedman et al. 1976; Couples et al. 1994), and we can determine how the stresses change during progressive deformation. The evolution of stress is a difficult inference to make in experimental models, and requires the assembly of many separate results that we must assume to represent stages within a single process. If we ignore local variations, the stress distribution in our poro-plastic simulation models (Figs 9–17) is similar to the stress pattern calculated by elastic models of similar systems involving bending (e.g. Hafner 1951; Sanford 1959; Couples 1977; Gangi et al. 1977; Friedman et al. 1980). The similarity between poro-plastic (e.g. Fig. 15) and elastic results indicates that, for this deformation system (flexure of layers in bending), the introduction of anelastic behaviour does not lead to a wholesale alteration of the mechanical response. Indeed, the elastic predictions represent a good first-order approximation of the real, and realistic, stress distribution. This result explains why the early studies of flexure based on elasticity have proven to be so valuable. The reason for this similarity is because rocks generate large stresses in resisting loads that would cause shape changes. Even very small distortions, as are initially imposed during a stage where elasticity may be an appropriate description, already reveal the selforganized pattern of load-carrying forces or stresses that a material develops to resist the imposed changes. Plasticity alters that distribution of internal forces (which we express as a stress state), as we have seen in the results described here. Those alterations are due to changes in the materials that occur as a consequence of their deformation, leading to both lesser and greater capacity to carry loads. If localization occurs, the load redistributions are more pronounced. In spite of these materialproperty changes, in the case examined here, plastic yielding does not completely change the overall pattern of loads through the whole system. However, plastic responses do have an impact on the details of the stress and strain states in portions of the system, especially those associated with localizations. Let us focus on the crestalgraben array that contributes to the development of the antiform in the simulation model. Here, we see that the stress trajectories (in detail) do not form a simple smooth pattern (Fig. 18). Instead, there are direction changes indicating local reorientations of up to 908. Most of the extreme direction changes occur near the ends of individual shears, but complexity also is noted where shears intersect, and elsewhere. Along with direction changes, the magnitudes of the stress components also change significantly. The s1 magnitudes are generally

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Fig. 18. Enlarged view of the crestal-graben system developed in the model shown in Figure 9. (a) Plot of effective plastic strain, shown in grey-scale. (b) Trajectories of maximum-compressive principal stress (s1), with the length of each symbol indicating the magnitude of s1. This image shows the stress-state alteration that is caused by the non-linear faulting process. The blank-appearing areas have very low stress values, and their trajectories are very short (below the resolution of the graphics). (Note particularly the variety of orientations through this deformed region.)

reduced (but are occasionally increased; see Fig. 13) as a consequence of the deformation, and the s3 magnitudes (Fig. 14) are even smaller, and may become tensile. Where s1 is nearly vertical, its magnitude is generally much smaller than would be estimated from a simple density consideration (in a gravitational case), but it can also be larger. Based on related models that include the effects of gravity, and that show similar variations to those shown here, the vertical stress component is not well approximated by the often-assumed value of sv ¼ r.g.z where r is density, g is the gravitational acceleration, and z is depth. How can that be? The changed sv is a consequence of load redistribution. The total load is being partitioned, as an emergent response, by means of a series of self-generated load-bearing arches and relays. The places in the model where s1 is nearly horizontal represent the tops of these arches, or the bottom of an ‘anti-arch’ below the crestal graben. There are multiple tiers of such arches and relays that develop, serving as ‘stress guides’ through the system. All of the loads are carried in the simulation results, including, in models not shown here, those loads due to gravity. The finite-element technique ensures that the model is fully balanced, with static equilibrium being ensured, and spatial patterns of the stresses indicate how the deformation has been organized (or here self-organized) to carry the loads in a system that contains new materials whose current properties are the result of earlier stages of the deformation. Any variation from an expected stress state indicates a potential need to reassess the thinking associated with that

particular expectation. In other words, we suggest that common preconceptions about the way that deformation will occur could be incorrect. If we pursue that line of thought, we reach a position where we need to re-examine the thought processes that have led to our current working model for the causes of faults. As the main example, let us look at the behaviour of the crestalgraben fault array in our simulation models. The geomechanical processes operating here differ considerably from the classical idea in which faults operate as planes that experience sliding in a causative stress field that is both constant, certainly constant in orientation, and spatially homogeneous. In stark contrast to that classical view, our models indicate that fault systems can develop in a selforganizing fashion such that the stress state in the region of faulting is evolving, and dependent on the history of deformation, on the current loading, and on the altered material properties that have been created by earlier deformation stages. Stresses on and around the faults, including normal and shear components and principal stresses, do not fit the classical perception of fault causation. If we want to make predictions about the properties of any rocks affected by faults (such as predicting fault-seal characteristics at the scale of the resulting deformation heterogeneities; see Couples 2005), we need to acknowledge that the Andersonian idea about the constancy of stress is not adequate for the task. What we have learned is that an elastic prediction can be a reasonable approximation for the deformation state at a large scale, but that it is not

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possible to use this overall state to make predictions about deformation at smaller scales (down-scaling) when the small-scale processes involve localization behaviour, or responses that are not captured by the assumed material model. For example, opening of joints is not well represented by an elastic model tuned to a compressive loading modulus, and we need to invent ways to interpret the results in order to derive predictions about phenomena that are not explicitly included in a model. If the result for this fold–fault system is applicable in a wide range of deformation systems, by which we mean if there is a strong feedback process that governs the way that the system evolves, then there are important implications for how we go about making useful predictions, and how we need to build and operate our models, across many deformation settings. There is a considerable difference between associating a calculated simple stress state (as occurs in elastic models) with a prediction about the occurrence of faults, and then analysing the details of the deformation processes that occur during the faulting based on assuming that the simple state applies throughout. For decades, we have made a mistake in assuming that the large-scale association could be extrapolated down to the process scale.

Conclusions The current state-of-the-art in geomechanical simulation technology is based on a continuum approach that relies on poro-plasticity theory. The theoretical basis of plasticity has itself been progressively updated by an active community of specialists to its current form, to account for laboratory observations about rock behaviours, including characteristics such as strain-hardening or softening and localization. Using a simulation tool based on poroplasticity has allowed us to replicate the patterns of faulting that allow large-scale bending deformations to take place. Frictional slip between the layers in the flexure is once again shown to be an important factor. These results set the stage for further simulation work that can add to our understanding of the partitioning of deformation processes that occurs in multi-layer systems. Our results also shed new light on the mechanics of fault systems. Fault arrays do not evolve within a static stress regime. Instead, stresses are the consequence of the continuing deformation processes. Predictions about fault sealing, or other properties of fault systems, need to acknowledge that localization phenomena such as faults lead to self-organized deformation states that cannot be predicted based on static ideas. To understand the operative deformation processes, and thus to

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be able to make practical predictions, we need to reconsider some of the time-worn ideas that we have depended on for many decades. Our work on the problem of flexural deformation has progressively advanced over the course of several decades. Certain key figures have had a major influence in terms of crystallizing or changing our ideas. D. Stearns deserves a special mention in this regard, as do M. Fahy and D. Weinberg. The remaining original faculty and students of the Center for Tectonophysics (Texas A&M University) also played a major role in helping us to develop our ideas about the flexural (bending) deformation mode, and how subsidiary deformation types (such as the fault systems described here) relate to the whole process. A more recent impetus for this work has come from the sponsoring companies (Amerada Hess, BP, BG, ConocoPhillips, Department of Trade and Industry, Kerr McGee, Shell, Statoil, Total) of the BMFFFS (Behaviour and Modelling of Fault/Fracture/Fluids Systems) Project that was brokered by the Industry Technology Facilitator, and that involved researchers from the University of Leeds (Rock Deformation Research Group), the University of Kent, and Texas A&M University. We specifically acknowledge these colleagues for helpful discussions about the mechanics of faulting. We particularly wish to acknowledge extremely helpful comments by J. Rudnicki based on an earlier draft of this paper. The manuscript has also benefited from a helpful review by D. Healy.

Appendix A: Brief overview of poro-plastic rock deformation This summary is included to assist readers in gaining a basic understanding or poro-plastic concepts in geomechanics. To begin, we assume that the material is isotropic (for simplicity in explaining the basic concepts). The conditions of yielding can be represented by a curve in P –Q space (Fig. A1). The actual state of stress at some point in a material plots as a single point whose coordinates are the scalar invariants ‘Peff’ (mean effective stress), and ‘Q’, which is interpretable as a ‘differential stress’: 1 Peff ¼ I1  Pp ¼ skk  Pp 3

ðA1Þ

where Pp is the pore pressure, and the double index implies summation

Q¼

pffiffiffiffiffi J2 ¼

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  ffi 1 2 ðs1 þ s22 þ s23 Þ 2

ðA2Þ

where sn are the deviatoric stresses; for example, s1 ¼ s1  I1 , etc:

ðA3Þ

It can be helpful to think of P and Q in terms of their representation in the more familiar Mohr–Coulomb (s –t)

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space: P is the coordinate of the centre of the Mohr circle, and Q is related to its radius. In poro-plasticity, if the stress point lies underneath the yield curve, the material is ‘not-yielding’. When the stress condition changes, and the state point touches (lies on) the curve, the material is ‘at yield’. As noted in the main text, rocks can become ‘stronger’ or ‘weaker’ after initial yielding, which we call strain- (or work-)softening or -hardening, respectively. How are those changes represented in P –Q space? In brief, these changes mean that yielding causes the yield curve to ‘move’ and become a different curve; that is, the deformed rock becomes a new material. (Another way of representing the hardening– softening responses is to depict yielding in a higher-order space; see, for example, Couples 2005; fig. 7.) The ‘new’ rock typically also has a changed porosity, as a result of the volumetric strains that occur, as we noted above. Additional rock properties such as permeability and acoustic velocity are also changed. Poro-plasticity theory identifies two additional material parameters, m and b, which allow these changes in the yield curve and strain state to be expressed in terms of the differential equations that govern the material model. The parameter m is the slope of the yield surface in P– Q space, so m can be positive or negative, depending on whether yielding occurs to the left of right of the peak of the curve (Fig. 5). The parameter b is the slope of the plastic potential, in strain space. The plastic potential is used to predict the strains that result from plastic deformation (i.e. yielding). The shape of the plastic potential is similar to that of the yield curve, and the axes of the strain space, representing volumetric and distortional strains, are ‘parallel’ to the axes P and Q. It is worth noting that the actual plastic potential is not exactly like the yield curve in real rock materials. In the hypothetical case where the curves are identical in shape, the term ‘associated’ flow is used. Unfortunately, this approach over-predicts the volumetric strains, so the non-associated flow rule (and thus the parameter b) was invented to account for the observed behaviour of real rocks. As a guideline, both m and b are positive for lowerporosity rock undergoing dilatant deformation, and both are typically negative for high-porosity material undergoing compaction. Localization (Rudnicki & Rice 1975; Issen & Rudnicki 2000) is taken to mean a concentration of deformation (increased values of incremental volumetric and/ or distortional strain) into a ‘band’, which is a planar zone of small thickness. Within a volume of material that has experienced plastic deformation with localization, there can be multiple bands, possibly with different orientations, as is obvious from outcrop examination and experimental studies of deformed rocks. Predicting the formation of such a localization band (or set of them) is based on finding one or more solutions to an eigenvalue problem that involves an ‘instantaneous’ modulus Lijkl (such that dsij ¼ Lijkld1kl, where sij is the stress and 1ij

is the strain) and the unit vector normal to the (putative) band. Based on a suitable expression for the constitutive relationship (see Appendix B), the theory identifies the material parameters m and b that lead to the development of shear bands, dilation bands, or compaction bands (in an otherwise continuum model, so these bands represent discontinuities in the strain rate, not explicit displacement discontinuities). There is a continuing theme of research activity focused on further development of this theory (e.g. Olsson 1999; Chateau & Dormieux 2002; Lade 2002), and testing it by laboratory experimentation (e.g. Mene´ndez et al. 1996; Wong & Baud 1999; Mair et al. 2002; Desrues & Viggiani 2004). For our purpose in this paper, it is sufficient for the moment to note that there is a fairly robust theoretical basis that ‘explains’ (or at least agrees with) observations concerning the development of sets of shear zones in deforming porous rocks.

Appendix B: Numerical representation of material properties To use poro-plasticity theory in a practical way, it needs to be incorporated within a geomechanical simulation tool. In this section, we describe in general terms how the theory has been implemented within SAVFEMTM (Structural Analysis via Finite Element Method), which is a general-purpose simulation environment created by Applied Mechanics Inc. (AMI). Some of the strengths of SAVFEMTM include its treatment of frictional sliding, which makes it suitable for addressing a range of problems of interest in structural geology, and its treatment of elastic– plastic material responses. SAVFEMTM was adopted in the 1980s as a key geomechanics simulation tool by the Amoco Production Company research laboratory in Tulsa, Oklahoma. Scientists working there, along with other colleagues, collaborated with AMI to develop better material representations that were suitable for simulating the deformations of geomaterials. The current SAVFEMTM state-of-the-art in this regard is a material type called a ‘general power-law material’. The following describes the general characteristics of the SAVFEMTM system, along with the key features of this material type, and how it relates to the poro-plastic concepts outlined in Appendix A. The notation used in this Appendix follows the SAVFEMTM manual. In common with many finite-element approaches, SAVFEMTM uses the matrix-displacement method of structural analysis (Zienkiewicz & Taylor 1989). The continuum to be modelled is represented by a mathematical model composed of a finite number of discrete regions (elements). Each element is assumed to respond to deformation in such a way that the displacement pattern inside the element is expressible (by interpolation, using shape functions) as a simple function of the displacements of a finite set of defining grid points that usually lie along the edges or corners of the element. Using classical continuum mechanics theory, and expressions that define how

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Critical state indicative of constant-volume deformation (curve 3?)

Q Stress state indicative of dilational yielding (curve 1)

Stress state indicative of compactional yielding (curve 5)

Non-yielding state

Peff

Alternate interpretation of critical state (intersection of two criteria)

Fig. A1. Yield surface in P– Q space. Stress-state points are indicated for the post-yield parts of three curves illustrated in Figure 4. The ‘non-yielding’ point shows the pre-yield condition for curve 3 (see Fig. 4). The right-hand image reminds us that our understanding of the transition between dilational and compaction yielding is uncertain. For example, the smooth curve depicted in the main image could, in fact, be a misinterpretation of the intersection of two separate yield conditions (see Issen 2002, for a discussion of this topic).

one of the fundamental variables of the problem is what makes the analysis non-linear) Equation (B2) can be rewritten as

materials behave, the displacements of the points are related to point strains (via partial derivatives of the interpolation functions), and to point stresses via the constitutive relationships. Stresses and strains can be integrated over the element to form an element stiffness matrix relationship between the grid-point forces and displacements. Assembly of all element matrices into a global matrix relationship allows SAVFEMTM to solve the set of equations, after assuming that there is a minimization of global energy. For example, in a linear (elastic), smalldisplacement analysis, the global equations of static equilibrium can be written in matrix form:

where fRg is the vector sum of the internal and external applied forces; that is, fRg represents a residual (local equilibrium error) vector between the calculated state and the ‘true’ state. Based on the principle of virtual work (Zienkiewicz & Taylor 1989), we can show that

½KfDg ¼ fFg

dfRg ¼ ½KTS dfDg þ dfFI g  dfFg

ðB1Þ

where [K] is the global stiffness matrix, fDg is the displacement vector, and fFg is the applied force vector (square brackets denote a matrix, and curly brackets denote a vector). The user-defined constraints (e.g. boundary conditions) allow the equations to be solved via one of several methods (e.g. Bathe 1996; Carey & Oden 1984). The solution determines the displacement of every grid point, and hence allows the stresses and strains to be derived as noted above. Inelastic, or non-linear, situations introduce additional complexity because the stiffnesses (relationships between forces and displacements) change during the deformation. In a small-displacement, plastic analysis, SAVFEMTM employs an incremental tangent stiffness method to determine the solution to the time-independent (i.e. not ratedependent) equilibrium equations, which are now in the form ½KðfDgÞ fDg ¼ fFg

ðB2Þ

where [K(fDg)] is the stiffness matrix, written so as to indicate its dependence on the current deformation state represented by the displacement vector (this dependence on

fRg ¼ ½KðfDgÞ fDg  fFg ffi f0g

ðB3Þ

ðB4Þ

where [KTS] is the global incremental tangent stiffness matrix, which can be interpreted as the ‘instantaneous’ stiffness at the current deformation state (i.e. the slope of the curve), and dfFIg ¼ the global incremental force vector. If the applied forces are held constant (i.e. dfFg ¼ f0g), then the Newton–Raphson iteration method can be used to develop a solution procedure: dfDgiþ1 ¼ ½KTS 1 i dfRgi

for i ¼ 1, . . . , N

ðB5Þ

where dfRg0 ¼ dfFIg 2 dfFg and N is the number of iterations. Thus, we can determine a set of incremental displacements that, when summed, approximate the non-linear response of the system. Using the Newton– Raphson approach guarantees convergence if the load-step size is small enough. In practice, the full Newton– Raphson approach is computationally very expensive, so modifications of this scheme have been implemented. Additional grid points (i.e. smaller element sizes) reduce the model errors inherent in the numerical solution of such a nonlinear problem, but these additional grid points increase

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the size of the matrix and hence add to the solution cost. A challenge in using numerical simulations is to learn how to identify an acceptable balance between numerical accuracy and solution efficiency. Additional considerations arise in the case of largedisplacement, plastic analyses. The relationship between strain and displacement becomes highly non-linear if the equations continue to reference the initial geometric locations of the grid points (which is what happens in a Lagrangian, or ‘material’, coordinate system). However, that non-linearity is eliminated if the equations are formulated so that the strain–displacement relationship refers to the current (deformed) locations: this is an Eulerian, or spatial, coordinate system. The formulation of the largedisplacement solution method, as used for the models included in this paper, is Eulerian. It is based on an adaptation of the scheme suggested by McMeeking & Rice (1975), which is, in turn, based on the variational principle of Hill (1959) for incremental deformations. Following the McMeeking & Rice approach, we can derive ½½KS þ ½KL fDg ¼ fFg

ðB6Þ

where [K]S is the small-displacement stiffness matrix (as before), [K]L is the large-displacement (geometric) stiffness matrix, fDg ¼ fVg tR (where fVg is velocity, and tR is some reference time), and fFg is applied forces at time tR, which depends on loading-rate changes and changes of internal reactions. If we let [KTS] ¼ [K]S þ [K]L, then we have an expression similar to that in equation (B2). Thus, the iterative solution scheme as outlined above can be used to find the displacements that result from a problem involving both geometric and material non-linearities. The solution methods described above are formulated on the basis of forces and displacements. In contrast, our understanding of material behaviour is represented by a relationship between strain and stress. The links between these representations of deformation rely on the definitions of strain and stress from continuum mechanics, and the way in which these state variables are related to spatial locations by means of the shape functions that are assumed in the finite-element formulation (e.g. fdeijg ¼ [B] fdDg, where [B] is the strain– displacement relationship). Thus, the stiffness matrices are determined by equations that are linked to the material responses. Prior to yielding (i.e. in elastic conditions), or when unloading occurs, SAVFEMTM treats the material as an elastic solid, represented in standard fashion as fSij g ¼ ½DfeEij g

ðB7Þ

where fSijg is the stress ‘vector’ (with stress components: SXX, SYY, SZZ, SXY, SYZ, SZX) (these are Cauchy stresses), feEij g is the elastic strain ‘vector’ (with strain components: eEXX, eEYY, eEZZ, eEXY, eEYZ, eEZX) (if the analysis involves additional deformation contributions, such as strains arising from thermal or pore-pressure effects, the

preceding equation will be rewritten as fSijg ¼ [D] feijg, with feijg ¼ (feEij g þ feTij g þ fePP ij g) (these are logarithmic strains), and [D] is the deformation matrix (for historical reasons, we retain the continuum mechanics use of the letter ‘D’ for this matrix, even though this has the potential to cause confusion as the finite-element method uses the letter ‘D’ in the definition of the displacement vector fDg; the difference is indicated by curly and square brackets), with D1 D2 D ½D ¼ 2 0 0 0

D2 D1 D2 0 0 0

D2 D2 D2 0 0 0

0 0 0 0 0 0 G 0 0 G 0 0

0 0 0 0 0 G

where E is the elastic (Young’s) modulus, n is Poisson’s ratio, G is the shear modulus (¼E/2(1 þ n)), B is the bulk modulus (¼E/3(1 2 2n)), D1 ¼ B þ 4G/3 and D2 ¼ B 2 2G/3. The governing equations are solved by an incremental approach fdSij g ¼ ½DTS fdeij g  fSIij g

ðB8Þ

where fSIijg is the initial stress components (for that increment), and [DTS] is the material tangent stiffness stress– strain matrix. Elastic responses represent the (usually) major component of the behaviour of most materials, even those that go on to deform in an inelastic (plastic) fashion. (See comments on this issue in the Discussion section of the main text.) However, quantification of plastic deformation requires that we introduce additional equations defining material behaviours, especially a definition of the conditions of yielding (cessation of elastic response), a flow rule (how strain changes during plastic deformation), and a hardening rule (which specifies how the stiffnesses change as a function of plastic strain). Initially, SAVFEMTM represented the deformation behaviour of geomaterials using a method based on the classical Drucker–Prager (Drucker & Prager 1952) yield function: FDP ¼ a1 I1 þ ðJ2 Þ1=2

ðB9Þ

where FDP is the Drucker– Prager yield function, J2 is the second stress invariant (e.g. ((SXX 2 SYY)2 þ (SYY 2 SZZ)2 þ (SZZ 2 SXX)2)/6 þ S2XY þ S2YZ þ S2ZX) (see also equation (2)), and a1 is the Drucker– Prager constant (a is the equivalent of m described in Appendix A). The classic von Mises yield criterion, applicable to metals, where the isotropic component of stress (e.g. mean stress) does not affect the yield condition, is SeVM ¼ (3J2)1/2, with SeVM sometimes being called the von Mises effective stress. To allow both approaches to be selected, SAVFEMTM has modified the Drucker–Prager

NUMERICAL SIMULATIONS OF FAULTING criterion by multiplying both sides by the square root of three, leading to SeDP ¼ aDP I1 þ SeVM

ðB10Þ

where SeDP is the SAVFEM Drucker–Prager yield function (¼31/2 FDP), and aDP is the SAVFEM Drucker–Prager constant (¼31/2 a1). When the material undergoes yielding, plastic strains develop. If new load increments cause additional yielding, then the strain accumulates. This ‘flow’, or change of shape, is calculated by means of a flow rule. In SAVFEMTM, the approach uses the form of the Prandtl– Reuss equations fdePij g ¼ fdFP =dSij gF deP

ðB11Þ

where fdePijg is the incremental plastic strains, FP is the yield surface for plastic flow (¼SeDP ¼ aPF I1 þ SeVM), aPF is the Drucker–Prager constant for flow (if this constant equals the constant for yielding, i.e. aPF ¼ aPY, the flow rule is called an associated flow rule; for the case where aPF = aPY, the flow rule is called non-associated), and deP is a simple non-negative constant of proportionality; that is, the effective plastic strain, which is 1P ¼ [(de PXX2 þ de PYY2 þ de PZZ2 þ (de PXY2 þ de PYZ2 þ de PZX2]2/2 [3(1 þ 2a2PF)]1/2. Plasticity theory includes two popular ways in which changes to the yield surface (i.e. the stresses that cause yielding to occur) can be related to the changing magnitude of plastic strain (a measure of the amount of yielding). One method is called isotropic hardening, where the origin of the yield surface remains fixed (precisely, it is fixed in the pi-plane, a surface that is normal to the line of S1 ¼ S2 ¼ S3), but the deviatoric stresses (e.g. as represented by J2) are changeable. The other method is called kinematic hardening, and in this case, the origin of the yield surface translates across the pi-plane. SAVFEMTM allows a mixed hardening approach that combines isotropic and kinematic hardening, allowing

(3J2)1/2 = SeVM

(a)

183

the user to represent a range of material behaviours. In the context of these comments on hardening rules, the parameter FP in the preceding equations must be understood as referenced to the appropriate stress–space origin. The simplest implementation of a mean-stressdependent Drucker– Prager material is illustrated in Figure B1a. In this case, the yield surface is a simple, linear function of mean stress, and the stress–strain behaviour of the material has both an elastic aspect (determined by Young’s modulus and Poisson’s ratio, for example), and a plastic response whose tangent modulus is constant. SAVFEMTM also permits the definition of a multi-linear Drucker– Prager material type (Fig. B1b), for which the segments of the yield surface can have positive or negative slopes. Each linear segment of the yield surface is associated with its own aPY (and aPF of each segment is determined by multiplying the appropriate aPY by a constant specified by the user; e.g. aPF ¼ xPF aPY). To better represent the complexities of the deformation responses of geomaterials, SAVFEMTM has implemented a more complex material type called a general power-law material. The implementation of this material type is similar to that as outlined above for the Drucker–Prager material type, but it represents a generalization that is based on the three-parameter strength criterion of Menetrey & William (1993). An important difference between the general power-law material and the multi-linear Drucker– Prager material is the ability to specify each segment of the yield surface as having a non-linear shape, leading to a smoother transition between segments and a better fidelity with laboratory data and first-principles constitutive theories. The cross-section of the yield function can also be changed, from a circle, in Drucker–Prager, to a triangular form, by using the eccentricity parameter, e. The generalized power-law material yield function is Fi ¼ ai1 ðbe ÞP þ ai2 be rðe; uÞ þ ai3 b1  ai4 ¼ 0 ðB12Þ where Fi is the yield function, for i ¼ Y, and plastic potential, for i ¼ F; aij are user-defined constants,

(b)

SeVM (I13,SeVM3)

(I14,SeVM4) αPY

(I12,SeVM2) αPY2

SeDP

I1

(I11,SeVM2)

Fig. B1. Examples of yield surfaces in SAVFEMTM. (a) Typical linear Drucker–Prager material with single slope aDP defining SeDP as a function of the mean stress (I1). It should be noted that these images are drawn using the engineering sign convention, where compressive stresses are negative. (b) Multi-linear Drucker– Prager material type. Coordinates of the ends of each segment of the yield criterion are shown. Each segment has its own aPY as defined by the user (only one shown for clarity).

I1

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i ¼ Y or F, and j ¼ 1, . . . , 4, aij ¼ f(1P); 1P is the effective plastic strain (defined previously); be ¼ seVM/sC; seVM is the von Mises effective stress; sC is the uniaxial compressive yield stress; b1 ¼ I1/sC; r(e,u) is the elliptic function; e is eccentricity (0.5 , e  1.0) (yield surface is circular in pi-plane when e ¼ 1.0); u is the polar angle ¼ f(sij) ¼ (cos21 (33/2 J3/2J3/2 2 ))/3; P is the power-law exponent (1  P  1 ); J2 is the second deviatoric stress invariant ¼ (I21 þ 3I2)/3; J3 is the third deviatoric stress invariant ¼ (2I31 þ 9I1I2 þ 27I3)/27; I1 is the first stress invariant ¼ sxx þ syy þ szz; I2 is the second stress invariant ¼ t2xy þ t2yz þ t2zx 2 (sxxsyy þ syyszz þ szzsxx); I3 is the third stress invariant ¼ sxxsyyszz þ 2txytyztzx 2 (sxxt2yz þ syyt2zx þ szzt2xy). A unique feature of the general power-law material description is the use of the normalized plastic potential gradient vector (fdFF/dsijgN) within the definition of the plastic strain increments. For example, in a uniaxial response case, d1xx ¼ dsxx/E þ 0.816497 d1P for a von Mises material using the general power-law material description, whereas the same expression for a simple Drucker–Prager material description gives d1xx ¼ dsxx/E þ d1P (note the unit multiplier for the effective plastic strain term). This difference, as a result of normalization, is important in determining the multi-linear strain hardening behaviour of this material type.

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Improved seismic identification of inter-fault damage via a linked geomechanics – seismic approach S. A. HALL1,2, H. LEWIS1 & X. MACLE3 1

Institute of Petroleum Engineering, Heriot – Watt University, Edinburgh, EH14 4AS, UK 2

Present address: Laboratoire 3S-R, UJF-INPG-CNRS, Grenoble, France (e-mail: [email protected]) 3

Present address: 3 Alle´e de la Bellardie`re, 95220 Herblay, France

Abstract: Predicting deformation-driven permeability changes in the subsurface requires knowledge of the character and distribution of dilatant and compactant rock damage. Seismic reflection data can be used to gain insight into aspects of the deformation such as the geometry of seismically resolvable faults and bulk material property distributions. However, interpretations of material properties from seismic data are non-unique. This paper addresses the use of established seismic techniques to identify the signatures of fault-associated open fractures, modified and improved by a new linked geomechanics– seismic approach. The paper also addresses how each of stress state and open fractures affect seismic anisotropy. The geomechanics–seismic approach is demonstrated using a model of a North Sea hydrocarbon field in which a series of potential fracture arrays are assumed and the fracture apertures are modified to reflect the geomechanically generated stress states. Seismic anisotropy predictions based on these modified fracture distributions are then compared with a pre-existing seismic anisotropy interpretation to determine the best match. Using geomechanical simulation to support a seismic anisotropy-based method produces a higher-confidence result and can lead to better prediction of altered permeabilities in faulted regions. Because of the geomechanical focus of this Special Publication, the background for seismic identification of faults and inter-fault damage is also outlined, including a review of current seismic practice.

Faults and their associated damage produce changes in rock body geometry and material properties. Both of these elements can have a major effect on reservoir flow connectivity through changed flow-unit juxtaposition (geometry) and permeabilities (as a result of dilatant or compactant deformation). This is a geometrically complex problem with significant spatial variability, as highlighted by the local variations that can be seen in the damage around faults in outcrop (e.g. Fig. 1) and also in observed local perturbations of stress trajectories in numerical simulations (e.g. Maerten et al. 2002) or in photo-elastic deformation experiments (e.g. de Joussineau et al. 2003). Thus, to adequately understand and predict subsurface fluid flow, knowledge of the 3D, and sometimes 4D, distribution of faults and their dilatant or compactant damage zones is necessary. Here we consider the characterization of both the damage contained within the immediate fault zone and the typically less intense damage in the fault-bounded blocks. We refer to the latter as inter-fault damage. We are most concerned here with dilatant damage, such as open fractures that increase bulk permeability, as the seismic methods that we employ are more sensitive to such damage.

Extensively reported surface observations of semiordered and semi-parallel fracture sets support the proposition that such sets can be open under certain deformation histories and in situ stress states. However, it is difficult to identify, quantify or characterize such fracturing in the subsurface, by observation or by inference. The primary data typically used for assessing both the subsurface 3D geometry, such as faulting and folding, and material property distributions (e.g. inter-fault damage) are from seismic reflection surveys. Such seismic reflection surveys are used first to generate images of geological structures and second to provide insight into the rock and fluid property distributions through seismic traveltime and reflectivity values. Therefore, from seismic data sensu largo it is possible to identify the geometric consequences of faulting and also to infer some of the associated changes in seismic properties (e.g. wave propagation velocities) owing to the fault-associated damage. However, such traditional seismic methods are not directly suited to determining the type and distribution of deformation (e.g. small faults, deformation bands or open v. shear fractures) below the seismic wavelength, which can be several tens of metres.

From: LEWIS , H. & COUPLES , G. D. (eds) The Relationship between Damage and Localization. Geological Society, London, Special Publications, 289, 187–207. DOI: 10.1144/SP289.11 0305-8719/07/$15.00 # The Geological Society of London 2007.

188 S. A. HALL ET AL. Fig. 1. Photomosaic taken from the cliff of a wave-cut platform at Nash Point, South Wales, UK Field of view is 50–100 m. Fracture arrays in the left and left–centre are relatively uniform in density and direction changes are gradual. As a fault zone (right–centre) is approached fracture densities become more variable and fracture trends change abruptly. Fractures near and within the fault zone appear to be curvilinear but closer observation (inset) shows that they consist of connected linear segments of the ‘regional’ fracture sets.

SEISMIC– GEOMECHANICS DAMAGE PREDICTION

This is where seismic anisotropy methods come to the fore, as they have the potential for better assessment and characterization of bulk material property variations and, significantly, their directionality. Seismic anisotropy analysis for reservoir characterization is a relatively new area but one with increasing popularity because of the potential for identifying directionality that may relate, for example, to fracturing. Existing seismic anisotropy results (e.g. Hall 2000; Gray et al. 2002; Hall et al. 2002; Hall & Kendall 2003) show patterns similar to those seen in outcrop, in simulations and in laboratory experiments (see Fig. 2). However, as with most seismic data, the interpretation of seismic anisotropy observations is in general nonunique, so a wide range of stress and strain fields, deformation types and deformation distribution models could explain a given set of observations. A good example is the potential occurrence of spatially varying stress state and open fractures. The common interpretation of seismic anisotropies calculated from seismic survey data considers these two possibilities as individual and separate causes of anisotropy. However, if open fractures are present, it is very likely that the stress field and fractures are both acting, and interacting, to produce the observed seismic anisotropy signatures

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via some combination of fault-associated damage, perturbations to stress fields around the faults and fault-block-scale rotation of stress directions. Our main objective in this work is to provide new tools that improve seismic anisotropy interpretation for characterization of fault-associated damage. To achieve this we introduce geomechanical modelling into our method so that we can incorporate the effects of variable stress, strain and permanent deformation states into our seismic anisotropy predictions. These geomechanically informed seismic predictions can then be compared with observations derived from seismic survey data. We consider, as an initial step, the simplest case where geomechanically generated stress fields are used to calculate zones of dilatant damage and open fracture apertures and thus to forward model the seismic anisotropy signatures. This approach also permits us to investigate the influence and interaction of stress state and open fracturing on seismic wave propagation, and so address how these factors can influence the observable seismic anisotropy. Finally, as seismic anisotropy techniques are more sensitive to dilatant (semi-ordered) sets of deformation features we focus our attention on the characterization of damage in the form of open fracturing.

Fig. 2. Map of AVOA magnitude and orientations for part of a North Sea hydrocarbon field calculated from seismic survey data (after Hall & Kendall 2003). The AVOA is displayed with fault traces (dashed lines) determined from a coherency analysis. AVOA magnitude (colour) is defined by the difference in the two principal directions (f ¼ 0 and f ¼ 908 in equation (3)) normalized by their r.m.s. average value. Colour bar gives the normalized AVOA magnitude. Arrows are oriented in the direction of the most positive AVOA gradients. Calculated AVOA shows an anisotropy magnitude that is typically greater than one. Anisotropy direction overall tends to parallel faults, being most marked for the NW– SE-oriented faults and in the far eastern section. This correlation supports an association between faulting and anisotropy development, potentially including damage caused by the faulting process, or local stress reorientations because of the faulting, or the two factors interacting.

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Because of the geomechanical focus of this Special Publication we first present an overview of the seismic methods available for identifying and characterizing faults and their consequences, including damage. In this overview we focus on seismic anisotropy methods that can allow better characterization of inter-fault damage. This section is followed by a review of relevant theory, modelling and practical implementation for seismic anisotropy methods in the context of characterizing in situ inter-fault damage. The review provides the basis for the presentation of a nonlinear–elastic anisotropic geomechanics –seismic (GM–SEIS) modelling approach. The roles played by both fault-related stress-state variations and open fractures in generating seismic anisotropy distributions are assessed using linked seismic and geomechanical models of a faulted area of a North Sea hydrocarbon reservoir. In this example, we generate a small number of geomechanical simulations with different mechanical loading and assumed (potentially open) fracture arrays. The resultant stress fields are used to generate a series of forward-modelled seismic anisotropy interpretations that are compared with seismic anisotropy results for the area from Hall & Kendall (2003) as shown in Figure 2.

Fault and fracture detection using seismic data Geophysical methods provide the only practical means to characterize the subsurface on a reservoirwide scale. Of these, the seismic method is the most commonly used and provides information that is directly related to the mechanical properties of the rocks, albeit without a direct correlation to physical rock properties. The basic system of surface seismic data acquisition uses sound waves generated at the surface; for example, by a controlled explosion. Multiple receivers, also at the surface and at some distance (offset) from the source, record the energy that is reflected back to the surface from subsurface interfaces of property contrasts. Analysis of these reflections can provide insight into various important aspects of the subsurface such as 3D geometry and material property distributions. The relevant approaches for using such seismic data to detect and map faults and inter-fault damage are outlined in the following subsections. In the context of the seismic method, faults are taken to be larger features that may be seen to offset seismic reflectors (e.g. lithological boundaries). Fractures and other types of fault-associated damage are considered to be features of a length scale much smaller than a seismic wavelength, which is typically many tens of metres depending

on the frequency range of the source used. Such features cannot be directly imaged by seismic methods. However, the influence of inhomogeneities will be averaged over a seismic wavelength to give a bulk effect. In the context of seismic anisotropy analysis, if there is a common alignment of inhomogeneities there will be an effective anisotropic effect on the propagating seismic wave. Thus the seismic properties (e.g. velocities or reflectivities, as described below) will vary depending on the direction of propagation. In such a categorization there is obviously a ‘grey area’ at which point the fractures or faults have a spacing of a similar magnitude to the seismic wavelength. In such situations the inhomogeneities will act more as scatterers of the seismic energy and cannot be considered in terms of an effective anisotropy. Nor will they be seen as offset seismic reflections. In this work we consider the two ends of this scale in the knowledge that there will be a scale where neither method is applicable. Utilizing seismic data containing different source frequencies or the enhanced techniques discussed below may yield information on this intermediate scale.

Approaches for fault detection and mapping using seismic data Direct imaging of fault planes or fault zones from surface seismic data may sometimes be possible but more generally the position and character of fault planes is inferred from the offset of seismic reflections (see Fig. 3). Such offsets can be tracked spatially (Fig. 4a) and across a number of reflectors to produce a 3D interpretation of the fault plane or zone. Additionally, reflection horizons picked from seismic data can be analysed for curvature or dip-azimuth (e.g. Roberts 2001) to highlight the likely locations of faults that offset the reflecting event (e.g. Fig. 4b). A number of approaches are available for enhancing seismic data to identify and map faults. Of these some form of image coherency measure is foremost, as described by Bahorich & Farmer (1995). Such methods assess the local waveform similarity in three dimensions. Segments of seismic traces that intersect fault planes will show a different seismic character from that of adjacent segments, leading to distinct loss of trace-to-trace coherence. Therefore faults appear as lines of low coherence, which can be used to produce fault maps (e.g. Fig. 4c). On the basis of such faultdefining attributes, fault planes can either be interpreted in the traditional sense or ‘extracted’ using more novel approaches such as neural networks (e.g. Meldahl et al. 2001) or ‘ant tracking’ (Pedersen et al. 2002). Other fault-detection methods that can be used in the right situations include cross-well seismic,

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Fig. 3. Example of a vertical reflection section through a seismic data cube showing how complex subsurface structures can be imaged and how offset reflections highlight large-scale faults. Colour bar shows the relative magnitude of the seismic reflection strength. Black lines are representative fault interpretations.

time-lapse seismic and microseismic monitoring. The cross-well seismic method uses seismic source arrays in one subsurface borehole and receivers in an adjacent one; analysis of the propagation across a fault using such arrays could be used to determine some fault properties. Time-lapse seismic monitoring allows changes in subsurface properties, such as fluid distributions, to be assessed during production. Therefore if a fault acts as a boundary to flow, time-lapse images can be used to infer the existence of the fault and also to gain some indication of the across-fault flow properties (e.g. Sonneland et al. 2000). Microseismic monitoring involves ‘listening’ for small earthquakes that occur along and around fault planes. Mapping of microseismic event clouds can allow the identification of active fault zones (e.g. Maxwell et al. 1998) and analysis of the moment tensor of the events provides insight into the nature of the movement on the fault plane (e.g. Nolen-Hoeksema & Ruff 1999).

Seismic anisotropy attributes Standard seismic methods can be used, as described above, to determine the structural elements in the subsurface. However, there is also significant information to be gained about the subsurface rocks and fluids from seismic attributes such as traveltimes and amplitudes. With the appropriate data it is possible to assess the directionality of these properties and to investigate the anisotropic nature of the rocks. For example, where open fractures have subvertical alignment this can produce azimuthal anisotropy, which might lead to observable differences between seismic data acquired along different

azimuths as outlined in Figure 5. The following two subsections describe the key methods of seismic anisotropy analysis and in the subsequent section the theory underlying these phenomena is discussed. Traveltime anisotropy attributes: SWS and NMOA. In an anisotropic medium there will be three seismic body waves, one compressional and two transverse (with quasi-orthogonal polarizations), all with velocities that can vary with the direction of propagation. From these wave velocities two key forms of measurable traveltime characteristics in seismic data can be used to assess anisotropy. The better known is shear-wave splitting (SWS), or birefringence, and another approach is normal-moveout anisotropy (NMOA). SWS is a phenomenon associated with the transverse nature of shear waves such that, on entering an anisotropic medium, they could be polarized into two orthogonal propagation modes. In the simplest cases of layering or a single alignment of open fracturing (commonly referred to as transversely isotropic with a vertical or a horizontal symmetry axis; VTI or HTI, respectively) there will be a faster propagating shear wave polarized parallel to the fracture planes or layers and a slower mode polarized perpendicular (e.g. Crampin & Lovell 1991; Potters et al. 1999; see Fig. 5). Thus the faster mode will have a reduced traveltime through the medium and arrives ahead of the slower mode. The lag time between the two modes is referred to as the shear-wave splitting and can be used to determine the degree of anisotropy. Additionally, the polarizations of these two modes provide insight into the orientation of the principal axes of the anisotropy, which, in

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Fig. 4. Examples of seismic attributes for part of a North Sea hydrocarbon field that can be used to identify and map faults or fractures in the subsurface. (a) Time horizon (a reflection surface): horizon picked from the seismic data with fault traces identified where the horizon is offset abruptly as in Figure 3. Horizon colour represents two-way traveltime. Picked fault traces are shown by black lines; circle size represents the magnitude of fault offset. (b) Surface curvature map: time horizons may be displayed to show abrupt dip changes. Here blue indicates low dip change, yellow indicates high dip change. (c) Seismic coherency map: analysis of the trace-to-trace similarity along a reflection event (shown here), or throughout a volume, highlights discontinuities. (d) AVOA anisotropy magnitude and orientation. Such an attribute relates to bulk rock property directionality and thus reflects present-day stress state and pre-existing damage, including any fracturing. Discontinuities in AVOA magnitude or orientation could be used to identify faults but the interpretation is less straightforward; the vectors can correspond to the mapped fault traces but in other places do not and may be (sub-)parallel or (sub-)perpendicular to the larger faults.

the simple HTI fracture case, will correspond to the fracture-parallel and fracture-normal directions. Normal moveout (NMO) is a phenomenon involving the hyperbolic increase in recorded traveltime, t, from source to reflector to receiver with increasing offset. This NMO results from the velocity variation with depth and the approximately layered nature of the Earth and is defined by the

standard equation (e.g. Taner & Koehler 1969) in terms of the source–receiver offset, x, the wave propagation velocity, v, and the zero-offset traveltime, t0,

t2 ðxÞ ¼ t0 þ

x2 : v2

ð1Þ

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Fig. 5. Schematic representation of the influence of aligned fractures on the seismic properties of a medium.

When there is an azimuthal anisotropy of the seismic velocities (e.g. Fig. 5) the NMO will vary with azimuth (NMOA) as defined by an azimuthdependent velocity. For HTI media this variation with azimuth can generally be considered to be elliptical (e.g. Grechka & Tsvankin 1998). Thus traveltime variations can be measured and analysed using equation (1), and an azimuth-dependent velocity function, to provide insight into subsurface anisotropy. NMO(A) analysis is usually, but not exclusively, carried out with P (longitudinal) wave data. Reflectivity attributes: AVO and AVOA. Amplitude variation with offset (AVO) analysis considers the dependence of the reflected wave amplitude on the incidence angle. Exact reflection solutions to this problem have been provided by Aki & Richards (1980). However, the reflectivity is commonly approximated, for the purposes of data analysis; for example, for the P –P reflection amplitude, Rpp, between isotropic media (e.g. Chapman 1976; Shuey 1985), Rpp ¼

  1 Dvp Dr þ r 2 vp  2 ! 1 Dvp 2vs Dm þ sin2 i  vp m 2 vp 1 Dvp 2 sin i tan2 i þ 2 vp

for an incidence angle, i, P-wave and S-wave velocities, vp and vs, shear modulus, m, and density, r. The overbar denotes an average of the properties either side of the reflecting interface and D indicates the difference. The AVO trends in seismic data are well recognized as a useful seismic indicator of lithology and pore fluid properties (e.g. Chapman 1976; Ostrander 1984; Shuey 1985; Rutherford & Williams 1989; Castagna & Swan 1997). Significant insight into the character of the subsurface can be obtained from surface seismic data through the interpretation of AVO attributes (e.g. the intercept and gradient, the first and second terms in equation (2)). Where azimuthal anisotropy exists, the velocities of the media bounding an interface will have an azimuthal dependence. Therefore the amplitude of waves reflected from such an interface will vary not only with the angle of incidence but also with the azimuth (see Fig. 5). This is known as amplitude variation with offset and azimuth (AVOA) and can be used to characterize the azimuthal anisotropy, as first demonstrated by Lynn & Thomsen (1990). The AVOA of an interface between HTI media can be described by (e.g. Ru¨ger 1998; Vavrycˇuk & Psˇencˇ´ık 1998)

ð2Þ Rpp ði; fÞ ¼ A þ ðB þ D cos 2fÞ sin2 i þ ðC þ E cos 2f þ F cos 4fÞ sin2 i tan2 i

ð3Þ

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where A, B and C are the same terms as in the isotropic equation (2) above and D, E and F depend on the anisotropy (e.g. Ru¨ger 1998). Reflection amplitude methods have an advantage over traveltime or velocity methods because the azimuthal variations of thinner layers can be detected with good vertical resolution and characterization of azimuthal attributes local to the reflector of interest. Traveltime or velocity analysis requires a sufficient thickness to accumulate an observable effect and will thus also be more influenced by the propagation through overlying layers. Figures 2 and 4d provide examples of AVOA attributes and indicate the spatial variation in anisotropy magnitude and orientation that may be determined from such analyses.

Dilatational fault damage and seismic anisotropy Seismic anisotropy: what is it? This section is a brief review of the large field of seismic anisotropy that is the basis for the detection methods discussed above and also is important for the subsequent discussions. More in-depth reviews have been given by Crampin (1981), MacBeth (2002) or Thomsen (2002), for example. Anisotropy is defined as the directional variation of physical properties at a point within a medium, and, in the case of seismic anisotropy, this refers to variation in seismic velocity with direction. If an elastic medium is isotropic only two independent elastic constants, the Lame´ parameters, l and m, and the density of the medium, r, are necessary to fully describe the seismic velocities, vp ¼ [(ll þ 2m)/r]1/2 and vs ¼ (m/r)1/2. However, if a preferred alignment of small inhomogeneities (e.g. fractures or any other small, low aspect ratio inhomogeneity) exists, this can give rise to material anisotropy such that the elastic properties at a point depend on the direction of propagation. For the general 3D case, the 3  3 stress and strain tensors, sij and 1ij respectively, can be related by the stiffness, cijkl, sij ¼ s0ij þ cijkl 1kl þ cijklmn 1kl 1mn þ   

ð4Þ

where s0ij is a pre-stress (state of stress before the seismic wavefront arrives), cijkl is the 3  3  3  3 (linear) stiffness tensor, which relates tensorial stress and strain, and cijklmn is the higher-order elasticity tensor that describes nonlinear–elastic behaviour (i, j, k, l, m, n take the values 1, 2 or 3). Below, compliance will also be used, which is given by sijkl ¼ c21 ijkl . The stiffness

and compliance tensors will reflect the physical symmetry. For example, the simplest symmetry class, which includes the two principally cited cases of layering or vertical fractures, is hexagonal. The presence of more than one set of anisotropic features, such as layering plus open fractures, two or more open fracture directions or aligned minerals or vugs, can produce media with lower symmetry.

Factors causing or influencing seismic anisotropy The factors that cause and/or influence seismic anisotropy have generally been split into three main categories: (1) sedimentary or mineral fabrics; (2) deformation history; (3) pre-stress. The first of these is self-explanatory and includes features such as bedding and lamination, crossbedding and depositional-flow induced grain or mineral alignments. The second factor, deformation history, essentially encompasses all the permanent deformation experienced by the rocks to the present; these deformation features will vary spatially (e.g. Fig. 1). The third involves the in situ stress field. We feel that a better categorization considers (1) and (2) as before, but with (3) divided into pre-strain and pre-stress. The ‘pre’ element is the subsurface stress and strain field immediately before the passage of the seismic wave. These two factors could be considered as the same thing in a medium that has only seen elastic (recoverable) deformation, but they have been separated here to distinguish two different contributions, as described below. Pre-strain essentially describes the instantaneous non-linear–elastic response of the medium to loading. The response to loading could be described by load-dependent elastic moduli (i.e. cijkl ¼ f (1ij) in equation (4) or higher-order elasticities (i.e. cijklmn also in equation (4)). If there is a pre-existing material anisotropy, the changes in elastic moduli will vary with direction. The most obvious example of relevance here is differential aperture change of open fractures dependent on their alignment relative to the orientation of the applied loading. Such changes in fracture apertures can produce an observed anisotropy signature for the loaded state that differs from that of the unloaded (or differently loaded) state. This phenomenon could also ‘induce’ anisotropy in an originally isotropic system through the differential closure of randomly aligned low-aspect-ratio inhomogeneities (including fractures) under non-hydrostatic loading. This effect is often known as stress-induced anisotropy but here it is attributed to strain, as the effect is a direct consequence of closure (strain) of open

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apertures. In the literature this topic often comes under the heading of acousto-elasticity (e.g. Hughes & Kelly 1953; Thurston & Brugger 1964; Nur 1971; Zatsepin & Crampin 1997; Tod 2002). Pre-stress, the subsurface stress state before arrival of the seismic wavefront, refers to the pre-stress term in equation (4). Seismic anisotropy could result purely from the superposition of the stress caused by the propagating waves on an anisotropic pre-stress, as described by Biot (1940). Such effects cannot be described by a stress- or straindependent elasticity tensor cijkl, as the pre-strain could be considered to do, but rather depends on the differential pre-stress appearing explicitly in the velocity equations for the shear waves. However, for pre-stress to be a significant issue requires differential stresses of a similar magnitude to the shear modulus of the medium, which in most rocks is very unlikely. This factor can therefore generally be neglected. A good discussion of prestrain and pre-stress in the context of elastic wave propagation has been given by Rasolofosaon (1998).

Modelling the seismic properties of damaged rock The two main approaches for modelling wave propagation in media with aligned inhomogeneities are continuum and discrete methods. Discrete methods model wave propagation, in elastic media containing parallel open fractures, by solving the reflection–transmission problem at each fracture (e.g. Pyrak-Nolte et al. 1990). In continuum modelling of the elasticity of fractured media a material containing fractures is represented by an effective homogeneous medium of equivalent elastic constants and is thus both more flexible and less computationally intensive. The main assumption in this effective medium approach is that, for a time-variant applied stress (i.e. a seismic wave), the scale of the inhomogeneities, here fractures, must be much smaller than the (seismic) wavelength. This assumption is valid where fractures have very small apertures, the situation most prevalent in the subsurface scenarios of interest. The elasticity of a systematically fractured medium can therefore be easily characterized with reasonable accuracy using effective medium methods. Furthermore, different types of inhomogeneities with different alignments and point-to-point spatial variations can be modelled with ease. Approaches for modelling the effective elasticity of fractured media are described in the literature from a wide range of research disciplines. These approaches are briefly reviewed below before providing an optimal, generalized approach. Some

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aspects of this subject have been previously reviewed by Kachanov (1992) and Crampin (1993).

Effective medium approaches The work of Eshelby (1957), which describes the perturbation to the elastic field in and around an ellipsoidal inclusion, forms the basis for most approaches that describe the effective elasticity of media containing ellipsoidal inclusions (e.g. Kuster & Toksoz 1974; Tandon & Weng 1984), cracks (e.g. Budiansky & O’Connell 1976; Hoenig 1978; Hudson 1980, 1981) or fractures (e.g. Frazer 1990; Hudson & Liu 1999). In the context of effective medium modelling for seismic methods, inclusions are generally ellipsoidal inhomogeneities, which may contain any type of material of high or low velocity, cracks are microscopic-scale features, and fractures are infinitely long, thin, parallel features. A crack may provide an approximation to a fracture in the low-aspect-ratio limit and both are considered to contain weak material. Fractures can be further divided into three broad types (after Hudson & Liu 1999): (1) a plane of open voids with a large area of the fracture surfaces in contact (e.g. Hudson et al. 1996b); (2) a plane of welds such that the majority of the fracture is open (e.g. Hudson et al. 1997); (3) two parallel, unconnected planes. The fractures considered in this work are taken to be no more than a few metres long, and thus are intermediate in scale between cracks and fractures as described above. Based on an investigation into the bases of the different effective medium modelling approaches Hall (2000) described a general, unified framework for modelling the elasticity of fractured media; the Effective Flat Fracture Compliance (EFFEC) approach. This approach is initially based on the general approach of Schoenberg & Sayers (1995) but, to allow forward modelling and data interpretation, it is necessary to make assumptions about the nature of the aligned fracturing. Parallels are therefore drawn to a number of other fracture theories to extend the generalized approach to represent different fracture systems such as penny-shaped microcracks or long, parallel fractures with distributions of contacts between the faces. Such a representation is consistent with that adopted by other workers in the field in the past few years (e.g. Hudson & Liu 1999; MacBeth 2002) because of its versatility in representing a number of different fracture types and multiple fracture alignments.

Model basis The mathematical model for the damaged medium considered here is based on the assumption that the

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inhomogeneities (e.g. fractures) are thin, low-volume and open with parallel, poorly bonded faces, which may have points of contact along their length, such that ‘the solution [of the stress-strain problem] depends only on the tractions imposed on the plane of the crack by the incident field’ (Hudson 1980). Thus, in the presence of an average applied stress, s¯ij, there will potentially be a difference in the displacements of the two sides of the fracture to produce a displacement discontinuity, [ui], that is proportional to the tractions, ti, on the fracture surface, S. Therefore for a single fracture with normal nj, ti ¼ si nj

ð5Þ

and so the displacement discontinuity may be described as

½ui dS / sij nj :

ðmÞ ðmÞ ðmÞ

þ Zil nk n j

ðmÞ ðmÞ ðmÞ

þ Z jl nk ni :

The sum of all the displacement discontinuities in a volume, V, produces an additional strain in the medium such that the total strain is given by (Schoenberg & Sayers 1995) 1 X 2V r

ð

ð½ui nj þ ½uj ni ÞdS: ð7Þ

Sf

Here sijklb is the compliance of the unfractured background medium (which may have arbitrary anisotropy) and the second part of the equation represents the additional strain as a result of the presence of r displacement discontinuities. This additional strain may be related to the applied stress, which is assumed to be continuous across the medium, through the 3  3 fracture system compliance tensor, Zij, of Schoenberg & Sayers (1995): ð

½ui dS ¼ Zip s pq nq :

ð8Þ

Sr

Combining equations (7) and (8) allows an effective compliance tensor, sijkl, of a rock, with an unfractured background containing m (differently) aligned fracture sets each contributing an additional compliance s (m) ijklf , to be defined (following Schoenberg & Sayers 1995): sijkl ¼ sijklb þ

X m

ðmÞ

sijklf

ð10Þ

This representation describes the excess compliance owing to sets of the most general form of low-aspect-ratio inhomogeneities. However, if the fractures are rotationally invariant about their normal axis and have no significant preferential slip direction, then the fracture system compliance tensor may be simplified. For such fractures Zij is reduced to three non-zero elements, which are given by just two independent terms that are the normal and tangential fracture compliances, ZN and ZT (see Fig. 5), given by ð11Þ

ð6Þ

S

1X V r

1 ðmÞ ðmÞ ðmÞ ðmÞ ðmÞ ðmÞ sijklf ¼ Sm ½Zik nl n j þ Z jk nl ni 4

Zij ¼ ZN ni nj þ ZT ðdij  ni nj Þ

ð

1ij ¼ sijklb skl þ

where

ð9Þ

(Schoenberg & Sayers 1995), where dij is the Kronecker delta. One of the strengths of the above approach for modelling the effective elasticity of damaged media is its generality, such that few assumptions need be made about the nature of aligned inhomogeneities or voids. However, this can also be a drawback. The general approach is thus extended by drawing parallels to other theories to create a hybrid scheme for describing the elasticity of fractures and fractured media through the use of the ZN and ZT terms. A number of methods to quantify the additional fracture compliance are provided in the Appendix.

Influence of a pre-strain and directional non-linear elasticity The representation of fractures described above is strictly a linear elastic approach and so the (anisotropic) seismic velocities cannot vary under loading. However, one of the key components of the open fracture representation is the aperture (given explicitly or in terms of the aspect ratio), which is widely recognized to change under loading. Therefore the theory should be modified to address such effects. The effect on elastic moduli and seismic velocities of changes in fracture aperture as a result of loading will be non-linear, which could be represented using the higher-order elasticity coefficients in equation (4). The instantaneous elastic moduli for a given load could then be determined from the tangent, or possibly secant, modulus of the set of non-linear stress – strain curves. However, this is not a straightforward task and it raises many issues relating to the

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relationship between the relatively slow geomechanical loading and the instantaneous passage of a seismic wave. Therefore a more conceptual approach is needed. If a medium containing an open crack, fracture or weak inclusion is loaded, the strain will be primarily taken up by a shape change of the crack, fracture or inclusion; this problem has been considered by various researchers (e.g. Walsh 1965; Nur 1971; Schwartz et al. 1994; Zatsepin & Crampin 1997; Hudson 2000; Tod 2002). If the open feature is thin then it is reasonable to assume that the strain is predominantly taken up by normal displacements; that is, only strain components acting parallel to the crack-normal will cause a change in aperture. Where fracture length is constant the change in aperture, and aspect ratio, will be directly proportional to the strain. Therefore for a non-propagating fracture with normal ni aligned at some angle f to the x-axis, as a result of a plane strain 1ij in the horizontal xy-plane, the change in aspect ratio is given by Dd ¼ d0 ð1xx cos2 f þ 21xy cos f sin f þ 1yy sin2 fÞ:

ð12Þ

Tod (2002) provided a similar equation (based on the work of Eshelby (1957)) in terms of changes in stress, sij (and fluid pressure), which may be rewritten (assuming no change in fluid pressure) as Dd ¼

ðlb þ 2mb Þ ðsxx cos2 f pmb ðlb þ mb Þ þ 2sxy cos f sin f þ syy sin2 fÞ ð13Þ

where lb and mb are the background matrix Lame´ parameters. Clearly, as the normal (shortening) strain increases some apertures will be closed. Therefore, an associated change in the crack density is included through an assumed exponential density reduction with reducing aperture, such that when the crack aspect ratio goes to zero the crack density correspondingly reduces to zero. In the situation described by the equations given above, even an initially isotropic medium containing a random distribution of cracks will become anisotropic under non-hydrostatic loading. The symmetry of such a system will match that of the applied loading and the degree of anisotropy will be determined by the integral of crack closure, and the associated (modified) additional compliance, over all angles. In the presence of a single preferential alignment of cracks (and a linear elastic matrix) certain loads will induce a general closure

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of the similarly aligned cracks and thus a change in the degree of anisotropy, but the medium symmetry will be unchanged. If all cracks are fully closed the medium will approach an isotropic state. Non-hydrostatic loading of a system with more than one alignment of initially open cracks will lead to differential crack closure and therefore a change in both the magnitudes and the symmetries of the anisotropies. Thus it is clear that there are many combinations of applied loading and preexisting medium symmetries that may exist in the subsurface and could be invoked to explain observations of seismic anisotropy (such as in Fig. 2). The following section explores this concept in more detail through a modelled example.

Linked geomechanics and seismic anisotropy modelling In this section linked seismic and geomechanical models of part of a faulted North Sea hydrocarbon reservoir are used to investigate the influence of present-day stress state and existing open fracture arrays on seismic anisotropy signatures. The reservoir has a number of faults that have been mapped from offset seismic reflectors and coherency methods and there is a published seismic anisotropy interpretation for the area (Hall & Kendall 2003). In this demonstration we generate a number of geomechanical models of several fault blocks. The setup of the models varies only in the load applied to the model boundaries. We use the resultant simulated stress fields to predict the adjusted fracture apertures for a number of possible fracture arrays. These arrays of adjusted fracture apertures are the starting point for modelling of effective medium elasticities, which are used to predict seismic anisotropy signatures. For each scenario, we compare the predicted seismic anisotropy maps with the existing seismic anisotropy calculated for this part of the reservoir. This method is shown in Figure 6.

Model setup Structural model and geomechanical loading. The area used for this demonstration is outlined in Figure 2 and consists of several fault blocks on the plunging nose of a larger elongate dome structure. We create a series of 2D plan-view geomechanical simulations of these fault blocks with a slightly simplified fault pattern and for an area that is extended to outside the published AVOA study limits (Fig. 7) to produce a viable geomechanical model that is sufficiently simple for our purposes but also represents the basic characteristics of the mechanical system. Our purpose, in

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Geomechanical Simulation

Seismic Anisotropy

Fault block geometry, rock properties

Load fault block model using simple far -field displacements

Calculate stress & strain fields, dilation & compaction zones

Generate fractures

Assume fracture arrays a. Random b. Regional (conjugate) c. Fault- parallel d. Combinations

Equivalent medium elastic properties

Seismic Anisotropy calculations

Comparison with real reservoir

Fig. 6. Flow diagram showing the basic components of the linked geomechanics– seismic (GM–SEIS) method. Continuous-line arrows represent the sequence of steps. Dashed arrow represents a potential refinement.

Fig. 7. Diagram showing the fault blocks and mesh used in the geomechanical modelling plus the correspondence between the areas used in the geomechanical simulation and the seismic anisotropy study of Hall & Kendall (2003). Numbered x- and y-axes outline the area used for the seismic anisotropy study (Hall & Kendall 2003; Fig. 2) and the faults previously interpreted from the seismic survey are shown as fine black lines. The faults common to the geomechanical model and the seismic interpretation are shown as thick black lines; dashed lines show extensions of the seismically interpreted faults to create the complete fault blocks needed in the geomechanical model. The dotted faults (top right) were interpreted from the seismic data but were excluded from the geomechanical model for simplicity. The result is a geomechanical model with six fault blocks.

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geomechanical terms, is to induce a series of stress and strain fields within the fault blocks that are consistent with a series of far-field maximum principal strain directions that rotate from north– south to east–west. This is done by applying a larger shortening displacement in the assumed maximum principal strain direction and a normal traction in the orthogonal direction. The normal traction is chosen to achieve displacement magnitudes with a ratio of approximately 2:1 and bulk contractional strains of approximately 2–5%. Another model set uses either uniform inward displacement or uniform normal traction on all boundaries to represent a uniform far-field state. In this paper we present only geomechanical models with north– south and east –west maximum shortening. A simple rock material law (linear elastic and isotropic) is used to provide a demonstration of the method; geomechanical models using much more advanced material laws have been used but they require more complicated approaches for the linked geomechanics –seismic (GMSEIS) transform and are not reported here. The linear elastic material properties used represent a weak limestone and faults are treated either as zones of weak elastic materials or as frictional interfaces. Examples of the simulated stress fields are shown in Figure 8. Fracture array models. The initial seismic modelling step requires a series of assumed fracture arrays (see Fig. 9). The first array represents a medium with a random distribution of fractures in the horizontal plane and thus has no preloading anisotropy, although a non-linear response to loading and associated differential fracture closure can induce anisotropy. The second array is that most commonly invoked in (fractureinduced) seismic anisotropy modelling and interpretation; that is, a single, regionally extensive, set of parallel fractures. The remaining arrays are assemblages made up from combinations of different sets of regional fractures and/or fault-associated fractures, including one array involving only fault-parallel fractures. In each case all fractures are taken to be vertical. One of the chosen fracture assemblies resembles fractures observed by the authors on the coast of South Wales (Fig. 1) and those described by Guiton et al. (2003). Fault-associated fracturing is assumed to consist of fractures paralleling the fault traces, with equal number-density on either side of each fault and an exponential decrease in fracture number-density away from the mapped fault trace. Where faults abut or intersect, the fracture direction parallels the vector sum of the fault orientations. The number-density distribution is consistent with published data (e.g. Harris et al.

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2003) although such symmetry may not be the normal case.

Method Seismic attributes and geomechanics to seismic transform. For each modelled scenario, the fractures in the chosen array have a uniform initial aperture. Adjusted apertures (and so aspect ratios) are determined using the calculated stress tensors from the geomechanical simulation and equation (13). Crack densities are also adjusted as discussed. It is assumed that the influence of the material anisotropy and non-linear elasticity on the geomechanical model is small. Thus for each predefined fracture assemblage, with adjusted apertures and crack densities according to the given geomechanics-derived stress fields, the effective elasticity tensor is determined for each grid point using the EFFEC approach. For this initial demonstration simple penny-shaped voids that have loaddependent aspect ratios and crack densities are considered. We then derive the seismic anisotropy (NMOA) values and maps, which would result from the loading, using the approach outlined by Grechka et al. (2000). The modelled anisotropy magnitude (defined by the normalized difference between the two principal directions) and orientations are then compared visually with the anisotropy shown in Figure 2. Ultimately this comparison will involve full calculation of the predicted AVOA instead of the NMOA. The NMOA is taken to be good enough for comparison with the data-derived AVOA in this simple example, with the assumption that the anisotropy directions will be equivalent and the relative magnitudes and spatial distributions are comparable. Additionally, a statistical approach could be included to allow use of the GM –SEIS method to constrain and/or update the geomechanical models such that the most likely fracture distribution and loading can be determined. The approach presented here is highly simplified. For example, at this early stage only 2D geomechanical models have been used when in reality the problem is often fundamentally 3D. However, a wide range of geomechanical models of fold and fault scenarios, some of them in three dimensions, show the same patterns of variation in stress magnitudes and directions from an assumed uniform regional value as is seen in this work. It therefore seems likely that the variation of seismic anisotropies produced here can occur widely, although significant geometric changes in the third dimension will require very careful application. Our model does not include any rock property changes in the geomechanical models to reflect

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Fig. 8. Plots of maximum stress magnitudes and trajectories (or directions) for north–south-directed (top) and east–west-directed (bottom) loading as calculated in the geomechanical simulations. Stress magnitudes in MPa as shown on the colour bar. (a) and (c) show magnitudes of maximum principal stress for the model shortened using

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201

Fig. 9. Examples of fracture array patterns used in the GM–SEIS method in map view with all fractures presumed to be vertical. (a) The base case; this represents a distribution of low-aspect-ratio, small inhomogeneities with no preferential orientation in the horizontal plane. (b) – (d) Simplifications of commonly occurring fracture arrays: (b) a single set of parallel fractures that are transversely isotropic with a horizontal symmetry axis (HTI); (c) two sets of parallel fractures; (d) as in (c) with an additional north– south-oriented fracture set. (a)–(d) have uniform fracture density. (e) Fault-associated and fault-parallel fractures with an exponential reduction in fracture intensity away from the fault trace.

the occurrence of open fractures or material anisotropy and we have made an assumption that the effect of the non-linear elasticity is small. We also investigate the interaction of stress state and open fractures from the seismic viewpoint but, as the geomechanical models do not represent open fractures, their interaction in a geomechanical sense is absent. These assumptions and simplifications, although not valid in most cases, represent the firstorder behaviour of the system. Thus we are able to both demonstrate the use of linked geomechanical and seismic approaches in the identification of (dilatant) damaged rock and investigate the loading and open fracture causes of seismic anisotropy. Future work will progressively add greater realism; for example, by introducing open fractures only in the dilatant regions that develop in the geomechanical simulations and orienting them to be consistent with the local principal stress directions. These fractures would influence both the next stage of the simulated deformation and the seismic model. However, further technical and conceptual advances are necessary in geomechanics, in seismic methods and in their integration to achieve this level of sophistication. This is the topic of continuing research.

Results and discussion Figure 10 shows the modelled seismic velocity anisotropy (NMOA) maps for the base-case of a pre-loading isotropic distribution of fractures. Figure 11 gives the seismic anisotropy maps for a series of assumed fracture arrays and a north– south-directed maximum bulk shortening direction; the other shortening cases are omitted for brevity, as the initially isotropic examples (Fig. 10) indicate that a north– south direction provides a better first match to the observed seismic anisotropies. This series of results illustrates: (1) the differences in anisotropy as a result of loading of an initially isotropic medium with randomly oriented fracture damage (Fig. 10) versus that also involving nonrandom open fracture distributions (Fig. 11); (2) the differences in the predicted seismic anisotropy when the same applied shortening is combined with different fracture arrays; (3) the consequences, in terms of anisotropy, of regional versus fault-associated pre-existing fractures. The observations and associated inferences can be summarized as follows. (1) The geomechanical models with faults, represented either as weak elastic materials or as

Fig. 8. (Continued) north–south- and east– west-directed displacement, respectively. Both models were also subjected to a normal traction in the orthogonal direction. (b) and (d) show the principal stress trajectories (or directions) for the north– south- and east–west-directed shortening, respectively. Lines in (b) and (d) are oriented in the directions of the maximum and minimum principal stresses; blue and grey denote compressive principal stresses, and red and pink denote tensile principal stresses. It should be noted that maximum principal stress magnitudes and directions vary considerable across each model, with significant variations in most fault blocks and rapid, discontinuous, changes across faults. Unsurprisingly, the north– south shortened models have a dominant north–south-directed maximum principal stress direction and the east–west shortened models have a dominant east– west-directed maximum principal stress direction. However, in both cases, there are many deviations from the dominant direction, particularly in the central blocks.

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Fig. 10. Predicted NMOA anisotropy magnitudes and orientations in an initially isotropic but non-linear-elastic model. Anisotropy magnitude is represented by colour as given on the colour bar. Line orientation represents the direction of the maximum NMOA anisotropy. The upper plots represent north–south-directed loading and the lower plots east– west-directed loading. Magnitudes are calculated from the differences in the principal axes of the NMO ellipse normalized by their r.m.s. average value at each point.

frictional interfaces (not shown), develop significant variations in the stress and strain magnitudes and principal stress and strain trajectories (directions) when compared with models with the same far-field conditions but no faults. These variations occur across the fault blocks and not only adjacent to the faults (Fig. 9). In these situations the assumption of a uniform ‘regional’ stress orientation and magnitude will be inaccurate and could lead to erroneous interpretations. (2) Variations in the stress magnitudes and principal stress trajectories have a significant impact on the seismic anisotropy distribution in all the cases, producing magnification of the anisotropy value in some areas and reduction in others. The anisotropy directions also vary significantly. (3) The seismic anisotropy vectors for the initially isotropic fracture distribution case (Fig. 10) have the same orientations as the input

stress fields, but the magnitude and spatial variation of the seismic anisotropy differs. These differences occur because additional factors, such as the mean stress, are involved in the calculation of the nonlinear changes in fracture aperture and density that are used as input to the seismic anisotropy modelling. In these models the north–south-directed loading case produces a considerably better match of simulated anisotropy to seismic survey derived anisotropy than does the east –west loading, although neither case is a high-quality match. (4) The paired geomechanical and seismic simulations of the GM–SEIS approach produce significantly different calculated (synthetic) seismic anisotropy when either the loading direction or the fracture array is held constant and the other element is changed. The interaction of these two elements is critical to understanding and interpreting the seismic anisotropy observations derived from

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Fig. 11. Predicted NMOA anisotropy magnitudes and orientations for different fracture arrays with a North– South-directed loading. The fracture models are: (a) two sets of parallel fractures with a 608 angular separation where their bisector is oriented NW –SE; (b) three sets of parallel fractures, the central set being oriented NW– SE and the other sets being 608 clockwise and 608 anticlockwise; (c) fault-associated fracturing as outlined in Figure 9e; (d) a combination of (a) and (c).

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these models. As the GM–SEIS approach does not yet incorporate fractures into the geomechanics portion we have probably underestimated this effect. (5) None of the ‘regional’ or the faultassociated-only fracture arrays explain the seismic anisotropy calculated from the seismic survey data when they are assigned uniform fracture apertures. However, adjusting the fracture apertures and densities to reflect the local stress state provides spatial variations in the seismic anisotropy that are more like the patterns calculated from the seismic survey data. Despite this improvement the distribution of high anisotropy values associated with the faults in the real case is not reproduced in either the regional-only or the fault-associated-only fracture models. (6) North –south-directed loading of a model containing both fault-associated and regional conjugate fracture assemblages appears to best match the real case when the regional fractures are aligned with their angle-bisector oriented NW–SE, an orientation that is consistent with the local structure. Although none of the models exactly match the observed seismic anisotropy, it can be concluded that, for this example, a regional fracture array model is not appropriate and two of the most commonly invoked models, loading-induced anisotropy (Fig. 10) and a single set of aligned fractures, are also poor matches. This holds even when the local variations in stress magnitudes and directions are included. Such a conclusion is clearly case-specific. However, as a general conclusion, it has been demonstrated that, if feasible, it is prudent to assess the loading and fracture interactions and resulting seismic signatures for each possible case. This, however, would involve a large amount of work and we suggest that significant insight can be gained from comparison of geomechanical modelling and seismic anisotropy data for appropriate ‘vignettes’, such as presented here, to better interpret seismic anisotropy observations directly over a larger area.

Conclusion This work has provided an outline of the approaches available for using seismic data to identify faulting and fault-associated dilatant damage and has highlighted the potential for improving such methods through integration with geomechanics and linked geomechanical–seismic modelling. ‘Standard’ seismic imaging approaches can provide insight into the geometric characteristics of the subsurface, from which larger-scale faults and discontinuities can be identified or inferred. Furthermore, other standard approaches may provide some insight

into rock and fluid property spatial distribution and variations. However, to assess the nature of (inter-fault) damage, it is necessary to use the directional properties of the seismic wave propagation (i.e. seismic anisotropy). As such, the basis of seismic anisotropy, in relation to dilatant damage in the form of cracks or fractures, has been outlined and the methods used for its identification in three dimensions in the subsurface have been discussed. Unfortunately, there are inherent ambiguities in the interpretation of seismic anisotropy data for damage characterization. Therefore an approach has been suggested that will help constrain the ambiguities through a novel integration of seismic anisotropy and geomechanical modelling methods. Implementing this method requires some level of integration of the underlying seismic and geomechanical theories. As an initial demonstration we have presented a simple approach based on nonlinear elasticity to predict changes in open fracture apertures with loading. An application of this approach, for a North Sea hydrocarbon reservoir, has indicated that commonly used simple assumptions about damage, fracturing and the influence of loading are insufficient, as they miss the complex interactions and local variations that will very probably exist and dominate the system behaviour. Use of this integrated approach, and its future development, will allow better understanding of subsurface damage and stress states based on geomechanical and geological principles and supported by direct observation data from seismic (anisotropy) analysis. Future developments in this area require further analysis of the relationship between the instantaneous (elastic) seismic response and the (elastoplastic) geomechanical evolution. Also, statistical approaches are being developed to assess the goodness-of-fit between the seismic attribute fields predicted via the GM–SEIS modelling and those derived from the seismic survey data. Such an approach will lead to a maximum likelihood model, and to decreased uncertainties, for the fault damage distribution and current-day stress state. As part of this process it is also necessary to ‘drive’ the geomechanical modelling with better constrained mechanical loads in the form of normal tractions or displacements produced by improved analysis of the structural evolution of the rock mass. This work has been carried out in part through the sponsorship of the Robust Fracture Identification (RFI) project at Heriot–Watt University that was brokered by the ITF (Industry Technology Facilitator). The sponsors of the project are acknowledged for their financial support and permission to publish: AmeradaHess, BG Group, BP,

SEISMIC– GEOMECHANICS DAMAGE PREDICTION ConocoPhilips, Kerr– McGee, Shell, StatOil, Total and the UK DTI. S.A.H would also like to acknowledge M. Kendall of the University of Leeds for many useful discussions. X. M. thanks ChevronTexaco for their support of his MSc at Heriot–Watt University. We would also like to thank two reviewers for their critiques, which were very useful in refining various aspects of the paper.

Appendix: Characterization of the additional compliances, ZN and ZT

½ui dS ¼

S

a3 U ij s jk nk m

ðA1Þ

after Hudson et al. (1996a). Comparing equations (A1) and (8), the fracture compliance is given as Zij ¼

hc U ij m

ðA2Þ

where hc is the crack density, defined as Na 3/V, where N is the number of cracks, V the total volume of the medium and a the radius of the cracks. For rotationally invariant fractures U ij can be reduced to two terms, U 11 and U 33 . From equation (A2) these terms relate directly to ZN and ZT, respectively. Hudson (1981) and Crampin (1984) gave U 11 and U 33 for the case of a single penny-shaped crack, containing a weak isotropic material (with Lame´ parameters lf and mf) in an elastic medium (with Lame´ parameters lb and mb): U 11 ¼

4 ðlb þ 2mb Þ 1 ; 3 ðlb þ mb Þ 1 þ K

U 33 ¼

16 ðlb þ 2mb Þ 1 3 ð3lb þ 4mb Þ 1 þ M

ðA3Þ

where  K¼

4 k f þ mf 3 pdmb

 ðlb þ 2mb Þ ; ðlb þ mb Þ

plane with large areas where the faces are in contact with a small distribution of open voids. Such a case was presented by Hudson et al. (1996b) to describe the compliance of a single slip surface (i.e. a fracture or fault plane) where a single fracture is represented as a plane of ellipsoidal voids. In this case, the additional compliance terms, zN and zT, are (adapted from solutions derived by Hudson et al. 1996b)    3 hf l b þ mb ; U 11 1 þ h2f U 11 p mb lb þ 2mb    3 h p 3lb þ 4mb zT ¼ f U 33 1 þ h2f U 33 mb 4 lb þ 2mb

zN ¼

The displacement discontinuity in equation (6) is related to the remotely applied stress through the average fracture parameter U ij (Hudson 1980, 1981, 1986), which is similar to Zij in equation (8), such that ð

205

ðA4Þ

4mf ðlb þ 2mb Þ M¼ pdmb ð3lb þ 4mb Þ kf is the bulk modulus of the crack-fill such that kf ¼ lf þ 23mf and the aspect ratio, d, is defined as c/a (halfcrack thickness/crack radius). Using equations (A2)– (A4), ZN and ZT can now be determined for the case of penny-shaped cracks but a more realistic model of a fracture is perhaps one involving a plane of ellipsoidal voids. This is analogous to a fracture

ðA5Þ

¯ 11 and U ¯ 33 are the same terms as before (equations where U (A3)). The term hf is the ‘crack density’ of the voids within a single fracture plane and the second parts of equations (A5), in braces, describe the interaction of these in-plane voids. This representation does not provide the effective elasticity of a fractured medium but Schoenberg & Douma (1988), using Backus (1962) style averaging, showed that the displacement discontinuity caused by the total fracture system is defined by the displacement discontinuity of a single fracture divided by the average spacing of the fractures. This is based on the assumption that the thickness of the fractures is smaller than the spacing between each fracture. Hudson & Liu (1999) employed this method to determine the overall fracture system compliance tensor from the compliance of a single fracture described by equations (A5). For an inter-fracture spacing H the fracture system compliances are, therefore (after Hudson & Liu, 1999)    3 hf l b þ mb U 11 1 þ h2f U 11 p Hmb lb þ 2mb  rtmb 4 3ptmb ¼ k f þ mf þ H 4a 3 !)1   3 l b þ mb 4r2  1  pffiffiffiffi lb þ 2mb 3 p    3 h p 3lb þ 4mb ZT ¼ f U 33 1 þ h2f U 33 Hmb 4 lb þ 2mb    rtmb 3ptmb 3lb þ 4mb ¼ mf þ H 16a lb þ 2mb !)1 3 4r2 :  1  pffiffiffiffi 3 p

ZN ¼

ðA6Þ

In deriving the second set of definitions it is assumed that the fracture planes are made up of a sparse distribution of low-aspect-ratio ellipsoidal voids, with maximum width 2c, mean fracture half-thickness t ¼ 4c/3 and relative open area on the fracture r ¼ N/Vpa 2 (Hudson et al. 1996b).

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The converse image of a fracture to that given above is a plane of isolated welds between two loosely bonded planes. Hudson et al. (1997) applied a similar approach to that of Hudson et al. (1996b) to provide the compliance of a single fracture plane with a sparse distribution of welded areas between the faces. For an aligned set of fractures described in this way the fracture system compliances are  r 12  1 rw 4ðlb þ mb Þ w 1þ2 ; Ha pmb lb þ 2mb p  r 12  1 rw 8ðlb þ mb Þ w ZT ¼ 1þ2 Ha pmb lb þ 2mb p

ZN ¼

ðA7Þ

where rw ¼ (Nw/V)pa 2 is the relative area of weld between the fracture faces for a number-density of welds, Nw/V, of average radius, a, on a single fracture surface.

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Localization processes in a coupled hydrogeomechanically sensitive fractured system M. A. REYNOLDS1,2, G. D. COUPLES1, H. LEWIS1 & G. E. PICKUP1 1

Institute of Petroleum Engineering and ECOSSE Partnership, Heriot –Watt University, Edinburgh EH14 4AS, UK 2

Present address: Total E&P UK Limited, Geoscience Research Centre, Aberdeen AB12 3FG, UK (e-mail: [email protected])

Abstract: Simulation of coupled dynamic fluid flow and geomechanical loading of fractured systems shows that complex behaviours can result, even for geometrically simple fracture systems and simple loading. Using a bi-directionally coupled simulation tool, HYDRO– DDA, we examine how fluids and discontinuum processes interact in a fractured, porous rock layer that is being flexed. The changes in fracture aperture, and hence the equivalent permeabilities of this system, exhibit marked localization or delocalization responses in spite of the geometrical and mechanical simplicity of this model. Typically the linked flow-deformation behaviour develops markedly non-linear responses. In some cases the permeability varies by more than three orders of magnitude for minor changes in input variables. Upscaling methods that are suitable representations of this permeability variation are developed. These non-linear behaviours develop in a porous material, which would be expected to suppress non-linear effects. If the range of behaviours seen in this geometrically simple coupled system is typical of other, potentially more complex, fully coupled systems, then the results obtained here can be used to explain the spatially and temporally high variability of the permeability characteristics of fractured systems.

Many, if not most, rocks have been fractured or jointed at some point in their history, altering the flow properties of the rock mass. These changes in flow properties as a result of the open fractures can be very significant in real-life problems. Examples of the interest in fracture-related flow include groundwater supply (e.g. National Research Council 1996; Shapiro 2001) and waste disposal (e.g. Chapman 1994; Chaplow 1996; Gburek & Folmar 1999; McKeown et al. 1999), mining (e.g. Lewis & Couples 1999; Gleeson et al. 2001; Tripp & Vearncombe 2004), geothermal resources (e.g. Drury & Jessop 1982; Griesser & Rybach 1989; Slevinsky 2002; Flores-Armenta et al. 2005; Geiger et al. 2005), and the hydrocarbon industry (e.g. Aarseth et al. 1997; Odling et al. 1999; Economides & Nolte 2000; Geiger et al. 2004; Bourbiaux et al. 2005). Although open fractures might be the only flow path in some fractured masses involving certain rock types (such as crystalline lithologies), the work reported here deals with porous rocks where open fractures and matrix porosity both contribute to the flow and fluid storage systems. Thus, our study addresses situations comparable with fractured reservoir types II and III (Nelson 2001). When a rock mass contains an abundance of open or openable fractures, it rapidly becomes impractical to treat each fracture individually, and

a coarser length-scale representation is needed if the system is to be represented as an equivalent porous medium. The net flow effects of the fractures and the matrix can then be represented in the coarser-scale model by an equivalent flow property that describes their combined flow performance, averaged or homogenized across an appropriate volume. There are several approaches to determining this equivalent permeability value, all using some form of permeability upscaling. Some of the approaches are simple; for example, fractures are assigned to one or more parallel sets that are stochastically distributed through a medium (e.g. Schoenberg 1991). This approach is similar to effective medium theory, where the upscaled property is calculated analytically based on essential simplifying assumptions. In another approach, a detailed flow model is constructed in which both the fracture-bounded matrix blocks and the fractures themselves are represented (e.g. Long et al. 1985; Cacas et al. 1990; Basquet et al. 2005). This type of approach is sometimes referred to as a discrete fracture network model, for which the matrix may, or may not, play a role in the simulated flow process. Complicated, and realistic, fracture patterns can be evaluated by using models and simulation methods that explicitly represent the fractures (e.g. Geiger et al. 2004; Karimi-Fard et al. 2004; Ma & Couples 2004; Prevost et al. 2005), but all of these

From: LEWIS , H. & COUPLES , G. D. (eds) The Relationship between Damage and Localization. Geological Society, London, Special Publications, 289, 209–225. DOI: 10.1144/SP289.12 0305-8719/07/$15.00 # The Geological Society of London 2007.

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approaches are based on the assumption that the fracture pattern is fixed, with no potential for feedback in terms of geomechanical effects. Many modelling methods assign the same aperture to all fractures of a given set (i.e. fractures with the same orientation). Even so, fracture concentrations can lead to increased flow in a local region and the upscaled, equivalent permeability needs to be capable of representing these spatial variations such that the coarse-scale flow simulation still captures the heterogeneity of the flow field. This statement demands that the cell sizes of the coarse-scale model are sufficient, and that they lead to an equivalent system response. As real fracture patterns can exhibit many length scales, it is not clear a priori that one can arbitrarily choose the coarse cells without reference to the details of the way that the system behaves. Here, we show (in a fashion similar to that previously noted by Zhang & Sanderson (2002) that the effects associated with coupling between fractures and geomechanics can lead to self-organized outcomes that are not stationary, so there may not be a unique or simple relationship between the fractured system and its upscaled flow representation. A similar localization of the flow response could develop in response to a region of nominally similar fractures developing some larger apertures. It is typical in upscaling to assume that the model configuration is fixed, but aperture changes are a likely consequence of interactions within coupled systems. Although upscaling might still be used successfully here, especially if the fracture length is small relative to the size of the upscaled region, only upscaling methods based on explicit representations could be used. This interaction between fracture-pattern characteristics (which can be affected by geomechanical interactions) and flow is the focus of this work. One of the common causes of fracture aperture variation (from a notional starting uniform aperture) is the interaction of the fluid and the mechanical systems. This topic is properly part of hydrogeomechanics, which focuses on the relationship between the deformation state of the rock mass and its contained fluid system, both in the matrix pore system and in any fractures or other voids. Open fractures, typically idealized as planar-sided, of uniform aperture and of limited extent, respond to changes in the mechanical state of the rock mass. In this system the aperture of any open fracture of arbitrary orientation is a function of the state of stress in the solid phase and the magnitude of the fluid pressure. In addition, fractures that are aligned along the direction of the maximum effective compressive stress (seff 1 ) will be more open than are those fractures aligned perpendicular to seff 1 . High pore-fluid pressures enhance this effect by leading to a

smaller minimum effective compressive stress (seff 3 ). In the fractured systems considered in this paper, variations in both the effective stress state and the geometric characteristics of the fracture networks emerge or develop, even though both the mechanical and the fluid pressure elements of the system are very simple. It is this emergence of complexity from simplicity that is the major outcome of the study. Our modelling programme is designed to enhance our understanding of the coupling that exists between pore-fluid pressures and geomechanics. There are similarities between our work and the extensive results reported by Zhang & Sanderson (2002). However, there are important differences: one is that our fracture-bounded matrix blocks contribute to the fluid flow system, whereas Zhang & Sanderson’s models treated the matrix material as impermeable. Another difference is related to the way that we represent the mechanical system: we use DDA for that purpose (Rouainia et al. 2006) whereas Zhang & Sanderson used UDEC (Cundall & Hart 1983; see also http:// www.itascacg.com/udec.html). Although emergent complexity seems to be a common outcome of studies that examine coupled systems, it is not at all clear yet whether the outcomes are dominated by the specifics of the models, or by the methods used for simulation and the representations of the physics of the system implemented via those methods. A suite of direct comparisons, based on the same models but using different methods, seems warranted. The results described in this paper are based on a conceptual situation consisting of a pre-fractured layer that is subjected to mechanical loading leading to the occurrence of a monoclinal flexure, plus a simultaneous fluid-pressure differential acting from bottom to top. The models that we have created here are mainly intended to explore the behavioural outcomes of coupled, fractured systems, and were chosen to best illustrate these. With minor modifications these models could be used to model a layer at the base of a sealing sequence (or caprock) being deformed above a rejuvenated fault (Lewis et al. 2002). They could also be modified to represent the behaviour of a layer within a fractured reservoir. We first describe the basics of the simulation system, following that with a description of the simple model and its general behaviour. Then we focus on quantifying the variations of effective flow properties as a function of changes in the input, along with making an analysis of the degree of localization that emerges or disappears as the boundary conditions change. We introduce permeability upscaling techniques to assess the development of flow localization and delocalization. Finally, we

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examine the implications of the observed responses for more general cases and draw conclusions about issues that need to be considered in other upscaling procedures involving fractured systems.

The HYDRO – DDA system HYDRO–DDA (Couples et al. 2001, 2003; Rouainia et al. 2001, 2006) is a 2D simulation system formed by coupling a fluid flow simulation tool (HYDRO) with a geomechanics simulation tool (DDA: Discontinuous Deformation Analysis; for a sample of discussions of the DDA family of methods, see Shi 1988; Lin 1995; Bicanic 2001; Jing et al. 2001; Ohnishi et al. 2002; Wu et al. 2004). HYDRO is based on Darcy’s law, and is derived from hydrogeological principles, so the formulation uses hydraulic head as the representation of fluid potential energy (Rouainia et al. 2001, 2006), When we impose a top-surface fluid ‘pressure’, specified as a head of water given in metres, the absolute pressure is taken to be that for a column of fresh water of that height above the model’s top surface. For reference, a 1 m head of fresh water equates to a fluid pressure of 10 kPa (0.01 MPa), which is about 1.5 p.s.i. The fluid pressure (head) imposed on the base of the model is determined from the top-boundary pressure and the assigned fluid-potential gradient. HYDRO uses a linear finite-element approach (see, e.g. Istok 1990; Smith & Griffiths 1998) to capture the detailed geometry of the fractures and blocks systems (Fig. 1a). DDA represents the mechanical interactions of blocks that are bounded by frictional discontinuities (i.e. fractures; Fig. 1b).

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DDA was originally developed for use in civil engineering to address problems involving significant block motion, with a particular focus on the relative movements between blocks along their frictional contacts. DDA blocks are defined in a model as arbitrarily shaped polygons, so this approach is suited to represent the complexities of geomechanical interactions in fractured systems where the fracture pattern is irregular and block shapes may be complex. The HYDRO– DDA system couples these two simulation tools via a feedback mechanism that represents a staggered coupling arrangement (Rouainia et al. 2001, 2006; Reynolds 2004). A finite-element mesh of triangles is generated for use in HYDRO, capturing the configuration of the DDA blocks, with mesh refinements along the fracture traces. For the models described in this paper, the mesh refinement creates elements c. 0.5 mm in size along and near the fracture traces. Steady-state fluid pressures, calculated by HYDRO, are provided to DDA where they alter the geomechanical behaviour by changing the conditions for slip along the frictional block interfaces. DDA then calculates a new geometry, accounting for block movements and the stress changes within blocks, and the updated fracture apertures are used to calculate a new permeability field that is then used by HYDRO to update the flow calculation. These calculations continue in an iteration cycle until the observed pressure (represented as hydraulic head) changes, measured from one iteration to the next, are below some convergence criterion. The calculation of the permeability is deliberately kept simple in the models described here so

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Fig. 1. Example of a model design illustrating the two components of the HYDRO– DDA simulation environment. (a) Simplified, unstructured finite-element mesh of triangles used by HYDRO. The mesh is considerably more refined along the margins of the DDA blocks. (b) Suite of DDA blocks showing the complex shapes that can be designed. In this model, the block boundaries represent fractures. Using HYDRO–DDA it is possible to design both simple and complex fracture network models. Modified from Reynolds (2004) with permission.

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that we can minimize any concern about the system response being an artefact of the method used to determine the new permeability. Therefore HYDRO elements representing the matrix blocks keep the original 72 mD (milliDarcy) permeability and do not account for realistic, but small, poroelastic changes as a result of variations of effective stress inside the blocks. Elements of the HYDRO mesh that lie completely within an open fracture are assigned a maximum permeability of 5.8  105 mD. Elements of the HYDRO mesh that cross a matrix–fracture boundary (of the DDA system) are assigned an interpolated permeability based on their relative proportions of matrix and fracture components, expressed as an average ‘porosity’ (assuming that a fracture has 100% porosity, with the matrix porosity being constant at 5%). The porosity of any finite element then determines its permeability, and the porosity– permeability relationship is assumed to follow a cubic law (e.g. Zimmerman & Yeo 2000) anchored by the 72 mD and c. 5.8  105 mD values at the end-point porosities. These simplifications ensure a moderate range of permeability values, and a smoother transition from matrix to fracture permeability for the finite-element mesh. More extreme transitions in permeability around open fractures may well occur, but their use here might lead to numerical challenges in the HYDRO calculation, and thus (possibly) affect the whole simulation system because of oscillations between iterations (unless we implemented a higher-order scheme in HYDRO). Our choices thus reflect a conservative approach.

The model and its behaviour Deformations leading to finite strains in general are typically highly non-linear, and this is particularly true of deformations involving movement along discontinuities. The term ‘non-linear’ as used here encompasses both the idea of irreversible responses, including hysteresis, and the potential for large variations in the amount of a response as a function of small, similar, but different, changes in input. The simulation of such potentially non-linear geomechanical processes needs to be designed and executed in such a way that the simulation neither precludes the ‘natural’ responses of the system nor induces responses that are artefacts (Couples 2005). The implementation of DDA used in the work reported here (see Rouainia et al. 2006) is not ideal in this respect, as the imposition of boundary conditions is not very flexible, requiring some care in designing the model geometry and its loading. There are many geological settings where the effective flow properties of a bent, fractured layer

are of interest. We focus on a layer that is being slightly deformed in a flexural fashion, while also being subjected to a significant fluid-energy gradient from bottom to top. These relatively large fluid-energy differentials are chosen so that the responses cannot be confused with numerical round-off artefacts. Such a large fluid gradient might be expected in a sealing interval. However, we have also kept the matrix permeability at a level that is too high for a seal and more representative of a reservoir interval. There are two reasons for making this latter choice. The first is to ensure that the effects of the combined fracture and matrix system are tested. The second is that matrix permeability is expected to suppress open fracture-related effects, so any significantly nonlinear responses that develop in our models do so in spite of the dampening matrix effects. Even small deformations of rock layers can induce significant mechanical effects (Lewis et al. 2002, 2007). Such deformation can also change the flow properties of the layer. Here, we examine how the effective vertical permeability of a prefractured layer is affected by a simple flexural deformation, taking the form of a cantilever, and pose the question: does the combination of slight structural deformation and a large fluid-pressure gradient cause a pre-fractured layer to experience a large change in its flow properties? If there is a significant effect in our simple model, we can reason that the cause is not related to extremes of fracture patterns or loading, or to fractureonly behaviours. Our 2D model is 5 m thick and 35 m long and is in the vertical plane (Fig. 2). The starting configuration of the model is symmetrical (but the loading is not). The central part of the model contains two preexisting open fracture sets in a conjugate pattern about the vertical axis. Fractures dip to the left and the right, and are equally spaced. Each of the polygons defined by the open fractures is represented by a single DDA block. The fracture surfaces obey a linear friction law, and each is assigned the same static (¼sliding) friction coefficient. The friction angle is 308, except for special cases discussed below. The unfractured part of the layer is composed of multiple DDA blocks that are welded together and have no block-to-block frictional behaviour. These welded blocks have the matrix porosity and permeability values of 5% and 72 mD, respectively. One of the observations made from a series of continuum and discontinuum geomechanical models is that symmetry in loading can suppress behaviours that would appear in less symmetrical but otherwise similar models (see, e.g. Rouainia et al. 2006). We have designed our geometrically symmetrical models to embody a very simple, but

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Fig. 2. Schematic diagram illustrating the model type examined in this paper. The image shows DDA polygons and the applied boundary conditions used to investigate the effect of varying mechanical and hydraulic loads on a fracture network. (Note fixed mechanical boundary condition on right end, and downward-directed mechanical load at the top left edge.) The white polygons represent the fracture-bounded blocks in the area of interest. The grey region is composed of welded DDA blocks. All DDA matrix blocks are porous and permeable, allowing an investigation of the impact of fluid flow through fractured systems with a matrix component. Hydraulic boundary conditions consist of a constant head imposed on top and base, with an interpolated head boundary condition on the ends (not shown on right end for simplicity). The lower boundary always has the higher imposed head, so the fluid moves upwards. Modified from Reynolds (2004) with permission.

asymmetrical, variation in applied mechanical loading such that the fractured rock layer is being bent into a subtle monocline, resulting in a variation in reaction across the layer thickness and leading to stress differences from bottom to top, and along the layer. We anticipate that fractures will have a greater or lesser tendency to slip or open depending on their position in the layer (laterally and vertically). If movement on one fracture influences the potential for movement on others, then there is potential for interesting non-linear responses. By adding the effects of fluid pressure, whose spatial pattern may also be affected by flow variations caused by fracture movement, the system should prove instructive. There are many possible asymmetrical loading configurations that can cause a layer to flex into a monocline. We have chosen a cantilever (Fig. 2). This arrangement causes the top of the layer to elongate (over the fractured region) as a consequence of the slight rotations induced. In the version of HYDRO –DDA used for these models, the external mechanical confining pressure is related to the magnitude of the fluid pressure applied to that boundary, leading to an undeformed effective stress state that is near zero for all models.

When the model is simulated using HYDRO – DDA (Fig. 3), the pre-fractured layer undergoes a minor flexural bending, and each of the fracturebounded blocks moves slightly, with both translational and rotational motions. The flow system changes as a result of the deformation, with the fluid-pressure pattern within the model becoming spatially complex and so contributing to changes in effective stresses across the model region. Changed effective stresses then influence the deformation, with the consequent mechanical response also leading to changed stresses. Thus, the system illustrates bi-directional coupling between the fluid mechanics and the geomechanics. The flow system exhibits strongly localized flow as a consequence of the movement of the fractures. The mechanical system exhibits localization too, represented at one scale by movements on each of the fractures themselves, and, at a larger scale, by patterns of fracture movements. Table 1 describes the boundary conditions and fracture densities used in this suite of models. In addition to the spatially complex pattern of fluid pressures produced as a consequence of the fracture movements, the total fluid flux across the fractured region also shows the effects of the

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Fig. 3. Simulation results from two representative cantilever models selected out of a suite of similar models. The models shown here vary only in the magnitude of mechanical load applied to the top left edge, using values of 5 MPa (a) and 20 MPa (b). (Also see Table 1.) A head of 11 m is applied to the base of each model and a head of 1 m to the top. The top image of each pair of outputs shows the Darcy velocity (flux) by means of scaled arrows. The arrow scale is chosen to highlight the flow in the fractures, but flow also occurs through the matrix. The lower image of each pair shows the head distribution: grey-scale of the resulting head distributions varies linearly from 11 m (dark grey) to 1 m (white). It should be noted that in (a) the highest flux is on the right side but arrows are visible along most fractures, whereas in (b) the highest flux is at the left side.

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Table 1. HYDRO–DDA cantilever model mechanical, fluid boundary conditions and fracture densities for each figure Figure number

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Where suites of models have been produced, the full range of head difference or mechanical load values used in the suite is shown and not only those illustrated in the figures. Matrix porosity and permeability are 5% and 72 mD, respectively, in all suites listed here.

fracture aperture changes. As we know the overall pressure gradient, it is a simple matter to divide the total flux across the central fractured region by the imposed pressure gradient, to determine an equivalent permeability that accounts for the effects of both matrix and fracture-pattern contributions. Figure 3 illustrates the lowest and highest mechanical load results of a series of cantilever models for which the value of the mechanical load is varied, but the fluid potential is constant at 10 m head difference from bottom to top, with a 1 m head applied to the top of the model. The effective permeabilities are back-calculated to be 301.1 mD, 134.3 mD and 157.0 mD for the mechanical load cases of 20 MPa, 15 MPa and 5 MPa, respectively. The 10 MPa model (with 10 m head difference) produces significantly different effective permeability and is discussed below. Reynolds (2004) showed many examples similar to those described here. It is clear that for the set of models presented in this paper (Fig. 3), and those of Reynolds (2004), different mechanical loads in otherwise identical models lead to different equivalent permeabilities because of the way that the fractures move within each model, and that the relationship between applied mechanical loads and equivalent permeabilities is neither linear nor monotonically increasing. This result could be a consequence of a numerical problem resulting from the way the problem is solved; it could reflect the true behaviour of the system; or it could be a combination of both factors. However, higher effective permeabilities do correspond to more-opened fractures and lower permeabilities with less fracture opening, but the opening and closing of fractures is not simply related to the loading conditions. Sometimes, the non-linear dependences are startling. We discovered a very large, anomalous, non-linear relationship between the calculated

equivalent permeability of the fractured region and a particular set of imposed boundary conditions. Using the coarse fracture density cantilever model described above, and applying the same mechanical load in each simulation run, we find that there is a spike in the fluid flux (and hence the permeability) that occurs over a narrow range (about 3 m head) of imposed hydraulic gradients (Fig. 4). The equivalent permeability in this spike is c. 4.0  104 mD, about 1000 times greater than the values at higher and lower head differences, and is apparently due to a through-going set of wide open fractures. It appears that the combination of mechanical and hydraulic loads is balanced in this pattern to allow such open channels across the model, leading to the high fluid flux. This pattern of block movement is specific to this model configuration. However, a number other examples of such anomalous flow behaviour have been found in different model configurations investigated in other studies, showing that the sharp spike depicted in Figure 4 is not unique. This anomalous result, and its close correspondence to obviously wide-open fractures, gives confidence to the conclusion that the less dramatic non-linearities are a true reflection of the system behaviour. The narrowness of the parameter dependence of the known spikes means that it is very difficult to plan a programme of study aimed at finding them all: the parameter space is very large and the spikes are very narrow with sharp ‘shoulders’. The existence of such spikes raises concerns about whether it is safe to make predictions about the flow effects of fractured systems based on interpolations between a few model results. The large variation in the output, expressed as the equivalent permeability, as a function of a small change in the input is a very puzzling characteristic of many of our HYDRO– DDA models. It is suggestive of the spikes representing bifurcations in

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Fig. 4. Plot of net Darcy velocity of the fractured region of the cantilever models plotted against head difference, showing anomalous response over a narrow range of head gradient. All models are identical except for the difference in the hydraulic gradient. Model boundary conditions are listed in Table 1. Modified from Reynolds (2004) with permission.

non-linear, coupled systems resulting from some interaction between the fractures, matrix blocks and fluid flow through the system. We are hesitant to make that claim pending further work to investigate whether there are numerical or procedural artefacts that might be a more appropriate explanation. However, the range of model designs where such spikes have been observed is sufficiently varied that we feel the evidence is accumulating in favour of an important phenomenon operating in these complex coupled systems. Fracture spacing is expected to be important for flow in fractured systems. If we halve the spacing of both fracture sets in our model, the resulting flow system is less visually heterogeneous (Fig. 5) and the equivalent permeability of the fractured region is higher. This result is entirely as expected, but, because of the non-linear results for other parameters, and based on results for parameter values reported by Reynolds (2004), we would caution against making a global generalization about how

fracture spacing affects flow. We suggest that a better generalization is that there are effects that are related to the ratio of fracture spacing to the spacing of mechanical boundaries such as bedding planes or larger fracture elements.

Upscaling in non-linear coupled systems We have shown above that our models typically exhibit non-linear responses, some of which can be very large, to changes in the controlling parameters, either mechanical load or head gradient. In this section, we describe results that illustrate further dependences, assessing the outcomes in terms of how the system responses (effective permeabilities) could be used in upscaled models. As noted above, anomalous flow rates are associated with DDA block boundaries that are propped open in a stable configuration leading to well-connected flow pathways. Similar behaviour

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Fig. 5. Comparison showing the impact of fracture spacing. Left example has the same fracture spacing as the models shown in Figure 3. Right example has a fracture spacing half that of the model shown in Figure 3. Differences in degree of localization of flow should be noted. Fluid flow in the left case is more sensitive to variations in fracture aperture than that revealed by the right model. The model boundary conditions are listed in Table 1.

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is revealed in many of the simulations of Zhang & Sanderson (2002), where there is no matrix permeability. Here, because of the matrix permeability, even when the fracture network does not create a complete through-going pathway (because part of a fracture has closed), the remaining open fracture segments can individually act as local conduits, and can locally enhance the flux with flow also occurring through the matrix connections. It is very rare for the DDA blocks to stabilize in a configuration where a block– block boundary, a fracture, has a constant aperture along its length; most fractures vary in aperture. Thus, provided there is a head gradient, open fracture portions lead to increased values of fluid flux in very local regions of all models (Figs 3 and 5). By imposing an analysis cell onto portions of the fractured region, we can calculate the local equivalent permeability for an arbitrarily chosen area (Fig. 6). The size of the cell can be changed to provide upscaled effective permeabilities over larger or smaller areas. This then allows us to assess the suitability (or otherwise) of representing the combined flow effects of matrix þ fractures, and their geomechanical interactions, in an upscaled fashion using a particular cell size.

When this procedure is applied to a typical simulation result, choosing a range of cell sizes that become progressively smaller, the outcome is not surprising (Fig. 6). As the cell size is reduced, the calculated permeabilities show more variance from cell to cell. Because the matrix also contributes to the flow, it is not unusual to have highpermeability cells (related to open fractures in that cell) failing to make a continuous band across the model. We examined the possibility that the cell location could influence the outcomes by using the same cell sizes, but offsetting their locations relative to the boundary of the model and the edge of the fractured region. The results (not shown) reveal patterns that are like those in Figure 6, so we discount the role of cell location as a significant factor. The spatial heterogeneity also remains if the fracture density is increased. We have illustrated how fracture density, and the size of the mechanical load, can affect the way that fluid flow is localized through their influence on the geomechanical response of the fractured mass. Remaining parameters, such as absolute pressure (presented as the role of effective stress), the applied pressure differential, and the frictional character of the block interfaces, are discussed below.

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Fig. 6. Calculated equivalent permeability determined for rectangular cell-shaped sub-regions of a single model where the size of the sub-regions used is varied. Upper image shows the whole of the fractured region used in the calculations. The lower four images are subdivided into progressively smaller evaluation cells. Upscaled equivalent permeability for each cell is given by means of grey-scale ranging from 0 mD (white) to 8000 mD (black). Model has a 20 MPa mechanical load, and a 10 m hydraulic gradient (2 m m21). Model boundary conditions are listed in Table 1.

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The variability in these results will be illustrated using a simple descriptor, the coefficient of variation, Cv. Cv is defined as the standard deviation divided by the arithmetic mean (Jensen et al. 2000). When the Cv is ,0.5, the data are considered to be homogeneous; when 0.5 , Cv , 1.0, the data are said to be heterogeneous; and when 1.0 , Cv, the data are said to be very heterogeneous. In our context, heterogeneous results indicate strongly localized responses in the model at that scale of observation. Using Cv as a measure, we can more easily describe results from a large suite of models, without the need to include numerous plots that may not easily make our points in a clear fashion. For each model we calculate Cv for the distribution of equivalent permeabilities determined in each set of the four cell sizes depicted in Figure 6. If Cv is low (,0.5), we describe that result as having a homogeneous flow pattern, without

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localization. If Cv is large, we can say that there is strongly localized flow (and localized deformation). This latter case (high Cv) tends towards the situation we illustrated with the sharp permeability spike, whereas the case with low Cv is interpreted as a delocalized response. Of course, the large cell size calculations produce a small number of data points, so a localized v. delocalized designation is not robust. However, our purpose is to highlight heterogeneity in the system responses, and large values of Cv (even though they are based on small numbers of cells) provide a strong indication of heterogeneity for the coarse-cell cases. The first sensitivity that we examine is effective stress. Unsurprisingly, the effective stresses vary from location to location because of the asymmetrical loading and the resulting system response. The stress state in an unfractured layer would vary as a result of flexure; with the addition of

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1.8 2.0 m/m hydraulic gradient

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0.6 0.05 m/m hydraulic gradient

0.2

0

10

20

30

Absolute Hydraulic Head (m) Fig. 8. Plot showing range of Cv values plotted against absolute hydraulic head for different hydraulic gradients. Shaded regions indicate the outcomes from four hydraulic gradients. It should be noted that models homogenize (lower Cv values and small range) at lower fluid pressures when the hydraulic gradient is lowest.

the block–block interactions, the resulting spatial variation is stress state can be complex. It is therefore not a straightforward matter to choose a ‘representative’ stress value at some location to use as a reference. Instead, we will use the term ‘effective stress’ in a general fashion, and say that higher absolute fluid pressures are creating lower ‘effective stresses’. If we adopt a representative cantilever model, we can explore the role of absolute fluid pressure by running a series of simulations where the only imposed change is the fluid pressure value that is imposed on the top of the model; the bottom fluid pressure, which differs by an amount determined by the imposed gradient, changes with it. For these comparison simulations, the imposed hydraulic gradient is kept constant, the mechanical load is the same, and the configuration of the pre-fractured region is identical in each model. When the Cv of the effective permeability is plotted against the absolute hydraulic head imposed on the model’s top (Fig. 7), the results

show that models with higher effective stress (those with lower imposed hydraulic heads) exhibit higher Cv or more flow localization. As the flow localization is related to patterns of connected open fractures, we can also say that these models develop more deformation localization. The models with lower effective stress (higher imposed hydraulic heads) are distinctly more homogeneous. We interpret this to mean that the lower effective stresses are suppressing the non-linear aspect of the geomechanical system, itself associated with the frictional character of the block contacts. We will separately examine the frictional effects below. The next sensitivity is the hydraulic (pressure) gradient. We adopt the same representative cantilever model and impose the previously used fluid pressure as a boundary condition on the top of the model, but we adjust the fluid pressure on the lower boundary to give different gradients. The mechanical load is fixed, and the pre-fractured region is identical in each model.

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Fig. 9. Illustration showing the variation in ranges of Cv as a function of the coefficient of friction (expressed as the angle of friction). Models with low hydraulic gradient have more homogeneous flux values when they have higher-friction interfaces, whereas cases with high hydraulic gradient become more heterogeneous as friction increases. The data distributions for the 308 cases are from Figure 8. (See text for explanation of these results.)

The Cv distribution of the results (Fig. 8) indicates that smaller hydraulic gradients lead to delocalization (homogeneity) at a lower magnitude of absolute fluid pressure. The results for the largegradient (2 m m21; 0.02 MPa m21) cases indicate that localization always occurs (high heterogeneity). The difference between a model with a moderate gradient (0.125 m m21; 0.001 MPa m21), and a model with a low gradient (0.05 m m21; 0.005 MPa m21), is small: the 5 m thickness of the layer leads to absolute values of bottomboundary fluid pressure of 0.625 m and 0.25 m, respectively, higher than the imposed top-surface pressure for any simulation run. It seems surprising that these small differences, which lead to small differences in effective stress and hence mechanical response, can have that much effect on the presence or absence of localization. The final sensitivity examined here is the frictional properties of the block contacts. All of the

models discussed thus far use an angle of friction of 308. This is replaced by a value of 608. In a simply responding system an increased friction angle would be expected to reduce block– block movements, and so also diminish the system behaviours that are related to movement on the discontinuities. The results (Fig. 9) seem, at first, to be contradictory. The increased angle of friction causes the models with high hydraulic gradient to develop even more heterogeneity in their flow pattern; the models with low hydraulic gradient become less heterogeneous with less localization. One explanation for this difference is that, yet again, effective stresses are playing a key role. For the low-gradient models, the variation in effective stresses as a result of only fluid pressures is very small. If the mechanical state does not generate much spatial variability in terms of stresses, then it could be that the stress state on each block contact is more or less the

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same as for all block contacts. In other words, the blocks all slip only slightly, with less variability in slip from block to block, and hence smallermismatches of block edges and fracture apertures arise, than occurred in the 308 friction cases. This situation would lead to greater homogeneity of the flow field. Carrying this same logic to the high-gradient cases requires us to suggest that the interplay between the spatial distribution of fluid pressures and the (presumably) higher mechanical loads (compared with the 308 friction case) works so as to cause block-to-block movements to be strongly localized. This is in direct contrast to the way that we explained the operation for the low-gradient case. How can this be? As we argued that high friction tends to make the model act more like a continuum, this explanation could be applied to layers without pre-existing fractures. In practice, this means that we are posing the possibility that the ‘fractures’ in the high-friction case are only slightly effective as pre-existing discontinuities, and that the ones that do actually move are aligned along directions where a new fracture would have formed from a continuum. In the situation of a true continuum, the substantial spatial variability of the fluid pressure (associated with the high gradient) could be contributing to the tendency of materials to deform locally, via some form of positive feedback. Although the explanations offered concerning the effect of friction are plausible, the real explanation may well be related to modelling artefacts or some other phenomenon. Additional work is needed. If we interpret the present results as they stand, or if the additional work supports the argument, then the implication is that localization in a fully coupled system can be caused by several processes. These processes can interact to reinforce, or to suppress, the localization, and the combinations of parameters that lead to localization may occupy isolated regions in a multi-dimensional parameter space. Simple rules-of-thumb may not apply to such systems, and it may be very risky to attempt to upscale them to a single value. Probabilistic methods to describe the upscaled values may be more appropriate.

Discussion This study has suggested that the interplay between dynamic fluid flow and a geomechanically active fractured medium can lead to complexity in the response of the coupled system. The complexity in this case is represented by spatial variability in the way that the pre-fractured medium deforms, and hence in the magnitude and spatial pattern of fluid flow through the medium. Combinations

of input parameters encourage either localization or delocalization of the patterns of fracture movements, leading to spatial variations in the patterns of equivalent permeabilities that are determined for sub-regions of the model. The resulting complexity emerges even for our models that have simple loads, and simple configurations. Fluid flow through the matrix does not suppress the importance of fractures in terms of allowing complexity (expressed as localization) to happen. We noted above that the scale of observation affects the magnitude of the calculated effective permeabilities, and their spatial connectivity (with small observation regions, there is not necessarily a through-going pattern of high permeability). In typical fluid flow upscaling practice as used for hydrocarbon reservoirs, it is assumed that the solid elements of the model are static. In that case, the overall connectivity would not change as the fluid system was altered. Unconnected highpermeability cells could be eliminated from the model with essentially no discernible impact on the result (e.g. Zhang et al. 2007), except for special cases such as cells located near a well. However, for coupled systems where the solid framework geometry is changing, it is conceivable, and even likely, that the instantaneous local regions of high permeability could shift about, and the simulation forecast would be very misleading if these effects were not included. The results shown here are certainly related, in part, to the specifics of the model design. However, other model configurations that have been investigated by our group in industrially sponsored projects have also exhibited complex behaviours, so the outcomes illustrated here are not unique. Additional work is needed to identify the generalities that govern the way that such coupled systems operate. We took as our example a pre-fractured layer subjected to both a fluid-energy gradient and a slight deformation that contained elements expected in subsurface seals and reservoirs. The results suggest that it can be difficult to make a simple prediction concerning the net flow effects. Upscaling is very problematic and may not even be appropriate. Fractures may well not have constant apertures because of the spatial variation in the mechanical deformation of the rock mass. Although we did not run simulations where the fluid pressures changed (in a temporal sense), we can use the existing model results to infer that the effective permeability of the layer could be altered significantly as a consequence of such changes in the fluid system (as might happen during fluid extraction, for example). There are regions of the parameter space where the response appears to be smooth as a function of changes in

HYDROGEOMECHANICALLY SENSITIVE FRACTURES

some parameters, and thus potentially predictable. However, within that parameter space, there are small regions where the response departs considerably from the trend. We are reasonably confident that these unusual behaviours are due to the physics and not to the simulation method. If so, they represent a significant physical uncertainty that is not captured by simple rules-of-thumb. More work is needed to develop a robust understanding of the process interactions that appear to be operative in these coupled systems. We may lack a fully robust explanation of the underlying causes of the spatial complexity observed in our models, and by inference we also lack a similar temporal complexity as a result of the way that fluid systems and deformation can progressively evolve owing to natural or man-made reasons. But it is still possible to examine some of the consequences of such complexity. For this purpose, we assume that a typical fractured rock mass could well develop local regions of higher equivalent permeability, as a result of the interaction between the pore fluids and the fractured mass, in a fashion like that observed in these models. If we assume that the sizes of the local high-permeability regions of such a medium can be larger than those simulated in the present study (perhaps associated with larger fracture spacing, etc.), then we can consider some of the prediction problems that might arise in assessing the characteristics of such a fractured system. In a subsurface situation, anomalously high permeabilities are recognized via the flow performance. Typically, this is in a well that is extracting or injecting fluids. If there are matrix-material samples, and if that matrix has low permeability but the well produces fluid at a high rate, then we usually infer that a fracture system is allowing that high flow. In fact, this type of anomalous flow is the hallmark of what is known as a fractured reservoir (Nelson 2001). The usual interpretation that is attached to anomalously high flow rates is that the local region around the well is more fractured than other regions. That interpretation is then used to generate a model for the distribution of fractures. Such models usually depict patterns of high and low fracture intensities, with the patterns controlled by whatever assumption was selected by the modeller. The fracture-distribution model is then used to forecast reservoir performance. The process model identified here provides an alternative means of interpreting variations in flow rates in fractured reservoirs. If we acknowledge that fracture and fluid coupling can create regions of anomalous flow performance, even in systems with no spatial variation in fracture density or characteristics, then there is an additional way to

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explain the observations. At present, we do not know enough about how key parameters affect the response of such coupled systems, so it is not yet possible to set out the rules that might help to distinguish between different causative situations. However, as we know that the interaction between fluids and fractures can lead to significant permeability variations in space and time, it might be advisable to include these ideas within an analysis of the uncertainties associated with fracturedreservoir performance predictions.

Conclusions We have examined a simple model representing the bi-directional coupling between the deformation of a fractured rock layer and the flow of pore fluids through it. The nominal conditions are those of a fractured caprock seal, or of a flow barrier within an overpressured reservoir. The equivalent permeability of the system is sensitive to aspects such as the effective stress and fracture spacing. Small changes in the characteristics of the system can produce remarkably large changes in the permeability: up to a factor of 1000 in the cases examined here. Consideration of the spatial variability of the equivalent permeability of the fractured region reveals considerable heterogeneity. The highly heterogeneous cases are caused by localization of the deformation, such that individual fractures and groups of fractures become propped open by the set of mechanical and fluid loads. The results of this study provide a phenomenological alternative that can equally well explain anomalous flow performance in fractured reservoirs. This paper represents a portion of the work contained in a PhD thesis that was submitted to Heriot–Watt University by the senior author. The HYDRO–DDA simulation system was developed during EPSRC-sponsored LINK projects GR/M61009/01 and GR/M62532/01 based at the University of Glasgow and Heriot– Watt University. We thank G. Tuckwell, R. Fletcher and particularly T. Manzocchi for helpful reviews of this manuscript, although we are responsible for the scientific positions expressed herein. The senior author expresses particular thanks to R. Nelson and P. Corbett for their insights and feedback conveyed during the PhD viva.

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Proximity to a critical point: evidence from, and implications for, hydrocarbon reservoirs K. J. HEFFER Institute of Petroleum Engineering, Heriot – Watt University, Edinburgh, EH14 4AS, UK (e-mail: [email protected]) Present address: Kayness, Nightingale Avenue, West Horsley, Leatherhead, KT24 6PB, UK Abstract: Current models of lithospheric deformation that involve the concept of proximity to a critical point in its statistical mechanical sense are reviewed in the context of implications for fluid flow in hydrocarbon reservoirs. The data from hydrocarbon fields that support the applicability of this concept are listed. In particular, spatial and temporal correlations of fluctuations in oilfield production and injection rates are assessed. The long-range spatial characteristics of these correlations provide strong support for the reservoirs being close to a point of criticality, at least during their development lifetimes. The implications for reservoir simulation modelling, data acquisition and future research to further elucidate this area are outlined.

Faults and fractures can often have a large influence on fluid flow in oilfield reservoirs, in primary depletion and in secondary or tertiary recovery schemes. There is a tendency within the hydrocarbon industry to compartmentalize the category of ‘naturally fractured’ reservoirs (in which the fractures are obviously open and conductive from core, image log or well test analyses), and often to neglect fractures in other reservoirs where their influence is less obvious. Additionally, conventional reservoir engineering treats the properties of fractures as static in time, and often homogeneous in space. The postulate of this paper, supported by both theoretical considerations and empirical field data, is that fracture properties are in fact dynamic and heterogeneous, and have a significant influence on fluid flow in most reservoirs. Central to the theoretical considerations, from recent advances in understanding of the mechanical behaviour of the lithosphere, is the concept that much of it is close to a critical point.

Criticality First, an explanation of what is meant by a ‘critical point’ is needed. This has received copious descriptions (e.g. Binney et al. 1992) in the field of critical phenomena associated with the physics of continuous phase transitions in thermodynamic equilibrium (e.g. liquid –gas, ferromagnetic –paramagnetic, etc. transitions). Critical points are characterized by: infinite correlation lengths in fluctuations of the ‘order’ parameter (density, magnetization etc.); large susceptibility to external fields; power-law relationships; contributions from all length or time

scales to the process, such that no range of length scales can be treated independently of others.

Self-organized criticality It is necessary to tune a system in thermodynamical equilibrium to a critical point by adjustment of some control parameter (e.g. temperature). A mystery for some considerable time was therefore the appearance of power-law characteristics in the noise of systems which were far from equilibrium. So-called ‘1/f ’, ‘flicker’ or ‘pink’ noise lacks a characteristic time scale, and is therefore fractal or power-law in its power spectrum. One explanation of spontaneous evolution to a critical point, without the necessity for tuning, and of 1/f noise, was proposed by Per Bak with the model of self-organized criticality (SOC). The model has been popularized, and many expositions have been made that appeal to it in various physical systems (e.g. Bak 1997); prominent among the proposed applications is deformation of the Earth’s lithosphere (e.g. Sornette et al. 1990; Crampin 1994, 1999, 2000; Hooge et al. 1994; Bak 1997; Sornette 2000; Leary 2002). The crucial characteristics of a system that self-organizes to criticality, in the context of deformation, are as follows. (1) A threshold: in the case of deformation the threshold is that of mechanical failure; that is, fracture or reactivation of fracture, manifest as an earthquake. (2) Slow driving in comparison with the speed of failure: the build-up of stress between episodes of earthquakes is on a much longer time scale than the earthquakes themselves.

From: LEWIS , H. & COUPLES , G. D. (eds) The Relationship between Damage and Localization. Geological Society, London, Special Publications, 289, 227–239. DOI: 10.1144/SP289.13 0305-8719/07/$15.00 # The Geological Society of London 2007.

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(3) Many degrees of freedom: this implies that the system can take on many combinations of states, and equates to the existence of many configurations of stress state and points of failure on the spatial distribution of faults and fractures. (4) Avalanche mechanism: one failure (earthquake) can trigger another failure anywhere within the region of perturbed stress state around the fault, possibly at some distant location. Eventually such a system attains a state in which there are percolating paths (or zones) of incipient failure. Each fracture at failure threshold on this path can be termed, rather confusingly, ‘critically stressed’ (Barton et al. 1998; Barton 2000). For failure in shear, the resolved normal and shear stresses on the fracture plane are such as to put the fracture on the threshold for shear slip. The correlation length of stress diverges as the critical point is approached, until through-going failure occurs and redistributes the stress to a state of low spatial correlation (e.g. Weatherley et al. 2001). For any domain that has undergone sufficient deformation, displacements are concentrated on a few through-going fractures that allow the rock to fail, and along which most of the transport of energy (strain, hydraulic, thermal, chemical, etc.) takes place. If the lithosphere is close to criticality, an important implication is metastability: it requires very little perturbation to cause further failure. Even regions remote from the large faults that have been the focus for the majority of displacement can be critically stressed (Grasso & Sornette 1998, fig. 3). The degree of sensitivity is remarkable: the stress change sufficient to trigger after-shocks following a major earthquake has been interpreted as of the order of a bar or less (a few p.s.i.) (e.g. Stein 1999). If the point of perturbation happens to be close to an isolated region of critically stressed points, surrounded by stable rock, then the disturbance will not propagate very far; on the other hand, if a point close to a ‘main thoroughfare’ is disturbed, a major earthquake can result. This randomness has important, negative implications for the predictability of earthquakes. Grasso & Sornette (1998) and Sornette (2000) noted that, compared with the metastable regions, ‘the complementary finite fraction of the crust is relatively stable and resistant to perturbation’, and that ‘not all human perturbations will lead to induced seismicity and that some regions will be very stable against external disturbances’. It is recommended that the reader views a simulation of SOC to gain an impression of the spatial patterns of avalanche and of threshold regions (e.g. Maslov 2004). Results of SOC models (see also fig. 2 of Grasso & Sornette (1998), taken from Sornette

et al. (1994)) illustrate that, as well as critical areas, there is also a significant proportion of the domain that is not at a point close to failure. One can speculate that, if such figures were more finegrained, they would show fractal-like detail. Areas below threshold at the coarse scale might show, at the fine scale, a sparse proportion of sub-areas at threshold. Therefore, one can imagine that wherever one might perturb such a system, it is never spatially ‘very far’ from a point of metastability. There is currently insufficient precision in both theory and field measurements to be able to quantify ‘very far’, especially given that mechanical failures are along highly anisotropic paths. However, a perturbation that itself influences a spatially extensive region, say with fluid pressure change, is more likely to encompass some critically stressed set of points within it. Additionally, it should be noted that hydrocarbon reservoirs, by their very nature, are predominantly located in salient structures, which are likely to provide foci for local pathways of contemporary failure. Nevertheless, even if SOC is a valid model for the deformation of the lithosphere, there will be subcritical regions in many hydrocarbon reservoirs.

Self-organized sub-criticality Furthermore, the model of SOC for deformation is not universally accepted (Jensen 1998; Jaume´ & Sykes 1999; Main & Al-Kindy 2002). A variant model involves the relaxation of stress in a broad region around large earthquakes. The result is that the lithosphere self-organizes to a sub-critical state, in which the correlation length remains finite, although still large; power laws still hold within finite domains. The approach back to criticality by stress regeneration is slow, characterized in its later stages by accelerating seismicity. This model is referred to as self-organized sub-criticality (SOSC) or intermittent criticality. Al-Kindy & Main (2003), through analysis of global earthquake data, supported the view that it is a better model for crustal deformation. Despite the current imprecision about the degree to which the lithosphere is close to criticality, the magnitude and spatial extent of perturbations that are introduced to the stress state by oilfield developments implies that the majority of reservoirs are likely to be mechanically metastable, and it will take very little perturbation to cause further deformation events. Metastability implies that ‘as solid as rock’ carries the opposite meaning from that normally construed; and that the rock framework of a reservoir is as dynamic as the fluid within it. The field evidence that this is indeed the case is examined below.

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Conductivity of critically stressed fractures That the properties of critically stressed regions control the deformation of a body of rock implies that they also influence the behaviour of fluid contained in the pore space. This is not only through poroelastic effects changing pore volumes: there is another characteristic of critical stressing that is potentially of even more importance to fluid flow in a body of rock. Recently there has been good direct field evidence gathered from individual wells (Barton et al. 1998; Barton & Zoback 2000, 2002) that those faults or fractures that are critically stressed in the modern-day stress state are predominantly conductive; those fractures that are stable in the ambient stress state are predominantly nonconductive. Whatever the tectonic history that generated the natural fractures in a reservoir, there is therefore a potential overprint on their conductivity distribution controlled by the modern-day stress state. Fracture conductivities can be affected by not only pre-production in situ stresses, but also by the significant stress changes induced by field development (e.g. Segall & Fitzgerald 1998); evidence that this occurs is presented below. Influence of diagenesis. Critical stressing is not a universal criterion for fault and fracture conductivity: Stowell et al. (2001) and Laubach et al. (2004), in studies of wells in tight gas sands of the western USA, promoted the antithesis that diagenetic characteristics of the fractures provide the main control and that fractures retain conductivity despite large normal effective stresses. Dyke (1996) and Laubach (1988) also made the case that cement bridges created by partial mineralization of fractures during periods when they are open can produce low sensitivity of their permeabilities to normal stresses. Dyke (1996) also cited the maintenance of permeability in many (fractured) hydrocarbon reservoirs despite large decreases in reservoir pressure as support for the insensitivity of fracture apertures to stress changes. Rationalization. Can these opposing views be synthesized? Where and under what conditions might we expect one control on fracture conductivity to dominate over the other? The following scenarios are possible. (1) A sedimentary basin that is on the boundary between geographical regions that have different modern-day stress regimes might have a stress state that is only weakly anisotropic and over its duration has not caused much inelastic strain; alternatively or additionally, the stress state might only recently have switched, such that the rocks have not yet attained a new metastability. This would

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be equivalent to the whole basin being in a sub-critical state. (2) Conversely, the basin is indeed generally in a new state of self-organized criticality under the new stress state, but the sampled areas are remote from the paths of rupture threshold. It has already been noted that the concept of SOC does not imply that stresses are critical everywhere; only in specific locations where the stress state and some fractures are configured critically will stress state play a large role in control of fracture conductivity. The main sensitivity to stress of the conductivity of even partially mineralized fractures may be due to shear failure. It is possibly pertinent that the angles between the strikes of conductive fractures and direction of modern-day maximum horizontal stress (Shmax) in the examples given by Stowell et al. (2001) are on average close to either zero or 908: neither situation involves large shear stress on the fracture planes, and indeed the fractures are reported as lacking evidence of shear failure, leaving conductivity to be controlled more by pre-existing diagenetic effects. It should be noted that the criterion for fracture conductivity in some studies appears to be the openness of fractures in core from visual inspection, rather than independent measurement of fluid flow, especially on an inter-well or larger scale. Also, all studies of influence on fracture conductivity of stresses, which can show considerable local heterogeneity, need to include full and precise determinations of stress state tensors, as demonstrated, for example, by Barton et al. (1998).

Support for criticality from hydrocarbon fields: review of previous work Several researchers have examined aspects of oilfield data that support the concept of a significant proportion of the lithosphere being close to a critical point (e.g. Crampin 1994, 1999, 2000; Grasso & Sornette 1998; Crampin & Chastin 2000; Leary 2002). Those previously examined issues are briefly described below; the subsequent section will then examine some new indicators.

Induced (micro-)seismicity Segall (1989), Grasso & Sornette (1998), Rutledge et al. (1998) and Adushkin et al. (2000), amongst many others, have reviewed cases of induced seismicity, particularly in hydrocarbon reservoirs, in various tectonic regimes. Grasso & Sornette (1998) found that both pore pressure changes and mass (fluid) transfers leading to incremental effective deviatoric stresses (difference between

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maximum and minimum effective stresses) of less than 1 MPa were sufficient to trigger earthquakes with magnitudes ranging up to 7.0 in otherwise aseismic areas. Once triggered, stress variations of the order of 0.01 MPa were enough to sustain seismic activity. Grasso & Sornette also cited cases of mining-induced seismicity, which continues for much shorter periods following perturbation, as examples of regions that are sub-critical. Grasso (1992) made the point that known associations between seismic instabilities and hydrocarbon recovery occur in only 10% of hydrocarbon basins. However, apart from the apparent incongruity of seismic instabilities being specific to particular basins, hydrocarbon recovery is termed ‘aseismic’ for a lower threshold in detectability of earthquakes of magnitude greater than three. If the data on numbers of earthquakes and pressure depletion from hydrocarbon fields provided by Grasso & Sornette (1998, fig. 7) are normalized, albeit with considerable uncertainties, to compensate for the different lower limits of detectable magnitudes, then they collapse onto a broad exponential trend of cumulative number of earthquakes with magnitude greater than one v. cumulative pressure drop (Fig. 1 and Appendix). Only data from fields with pressure depletion, in contrast to secondary recovery schemes, are included in this plot; consequently, effective stresses are monotonically increasing in a manner that is comparatively uniform across cases, which may

partially explain the degree of consistency in the data. Figure 1 indicates that a pressure depletion in excess of some 30 MPa (4500 p.s.i.) is required to induce earthquakes of magnitude greater than three, in accord with the data from the Lacq field. This trend implies that even though induced seismicity fails to be detected by regional seismic networks with a threshold in magnitude of order three, a more sensitive local network installed purposely over or within an hydrocarbon field can detect lower magnitude microseismic events; the published literature provides no cases that contradict this. Although these projections, based on a small number of cases, need to be treated with caution, they seem more consistent with the view that stress states across the majority of reservoirs are predominantly a little below criticality, such that perturbations induce only minor seismic events on limited paths of criticality without extensive propagation; the likelihood of a more extensive failure path being triggered increases as pressure depletions increase in magnitude and therefore also in spatial influence.

Shear-wave splitting Shear seismic waves passing through fractured rock are polarized, or split, into two components, with essentially orthogonal vibrations. Observations of splitting of shear waves have implied a narrow range of fracture densities straddling the boundary

Normalised cumulative number of earth quakes of magnitude >1 with cumulative pressure depletion in various hydrocarbon fields

C u mulative number of earth quakes

10000

M>0

M>1 M>2

1000

M>3

Lacq Ekofisk

100

Assen Fashing Imogene

10

1 0

20

40

60

80

Cumulative pressure drop, MPa

Fig. 1. Cumulative number of earthquakes observed in various hydrocarbon fields with pressure depletion. Data adapted from Grasso & Sornette (1998) and normalized to a magnitude detection threshold of M . 1 using a Gutenberg–Richter relationship (see Appendix). Straight line tentatively depicting approximate trend through normalized data for M . 1 was fitted by eye. The projected parallel lines corresponding to cumulative frequencies of other magnitudes (M . 0, .2, .3) are shown by dotted lines.

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between essentially intact and heavily fractured rock (Crampin 1994). Splitting has been observed in most rocks in the crust, and the polarization of the faster component in most rocks, including most reservoir rocks, is aligned approximately in the direction of the maximum horizontal stress. Crampin interpreted that this is because almost all rocks contain grain-boundary microcracks or low aspect-ratio pores that are aligned, like hydraulic fractures, perpendicular to the direction of minimum horizontal stress, and sensitive to its changes. These distributions of aligned microcracks are referred to as extensive-dilatancy anisotropy. The narrow range of observed fracture densities, at the micro-scale, is consistent with larger rock volumes being close to critical, and therefore with the concept of self-organized criticality. Shearwave seismic data therefore appear to be an ideal tool with which to investigate more quantitatively the degree to which stress state locally approaches criticality and its time dependence during the life of a reservoir.

Direct measurements of stress state The stress states that have been directly measured in the field (mainly via hydraulic fractures or minifractures; or characteristics of wellbore breakout) are often consistent with control by frictional failure (Townend & Zoback 2000; Zoback et al. 2002). Zoback & Zinke (2002), building on the interpretations by Teufel & Rhett (e.g. 1991), showed that the maintenance of permeability in the Ekofisk and Valhall fields, evident from well production rates during pressure depletion, can be explained only by the effective stress state ‘tracking’ the frictional failure envelope for normal faults, which remain conductive.

1/k scaling of fracture porosity Well logs of porosity in fractured crystalline rock, where only fractures contribute to porosity, yield insight into the spatial correlation structure of strain associated with fracturing. In fact, observations by many researchers (e.g. Hewett 1981; Bean & McCloskey 1993; Bean 1996; Holliger 1996; Dolan et al. 1998; Leary, 1998, 2002; Marsan & Bean 1999; Leary & Al-Kindy 2002) have indicated that fluctuations in general heterogeneities in well logs through different lithologies show ‘1/f ’ scaling. Examples of 1/f noise are most often associated with temporal frequencies; it will be referred to here as 1/k noise to emphasise that it pertains in this context to spatial frequency, or wavenumber, k. (The wavenumber, or spatial frequency, k, is related to the wavelength, l, of an oscillation by k ¼ 2p/l). In terms of spatial

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wavenumber, k, the spectral densities of the heterogeneities show power-law behaviour S(k)  1/k b, where b  1.0–1.6. (The spectral density, S(k), describes the ‘amplitude’ of an oscillation with a particular wavenumber.) 1/k scaling in well logs has been interpreted by Crampin (1994, 2000), Leary (1998, 2002) and Crampin & Chastin (2000) as symptomatic of the involvement of SOC in structural deformation. Power-law relationships can arise in several ways other than SOC (Sornette 2000, chapter 14); in this case there are some empirical findings that make this association of a power law with deformation at a critical point more secure. (1) Investigations on a lattice model of the relationship between the effective stiffness modulus and the density of failed bonds yield power laws at the critical point of failure (e.g. Chakrabarti & Benguigui 1997). It can be shown from these that effective stiffness scales with length of the finite-size domain as a power law with exponents of c. 3–4. In other words, larger volumes of rock are, on average, less stiff than smaller, for deformation near the critical point. Such scaling of effective modulus, when used to modify the spectral density function for fluctuations in strain arising from elastic theory, gives rise to values of b  1–2, in line with findings of scaling from well logs. (2) If variations in surface depth from seismic surveys are taken as representative of the vertical components of displacement during deformation, then we can infer corresponding fluctuations of strain. Spectral power densities of displacement fluctuations in two dimensions empirically have negative exponents of 4 –5, implying that spectral power densities of strain fluctuations in two dimensions would have negative exponents of 2–3. These again equate to a b value in the range 1–2. The situation is not entirely clearcut: 1/k behaviour is found in sedimentary rocks as much as in crystalline ones (Leary & Al-Kindy 2002). Although the origin of scaling has been attributed to universal scaling of fractures along the borehole (Leary 1991; Holliger 1996), Bean (1996) showed that scaling in the lithology distribution can also be taken as a contributing cause. The similarity of the fractal dimensions of porosity in unfractured rock to those of rock whose porosity is associated mainly with fractures is possibly due to tectonic or deformational influences on diagenetic processes (compaction, dissolution, cementation, pressure solution) which overwrite the statistics of porosity derived from the original depositional process (Dolan et al. 1998). The influence of tectonism on deposition (e.g. in controlling avulsions of fluvial systems or the accommodation available for sedimentation) is also probably significant.

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Support for criticality from hydrocarbon fields: new indicators Directionality of injected fluids In support of the control of fracture conductivity by modern-day stress, at least under conditions pertaining to fluid injection schemes in oil reservoirs, Heffer & Lean (1993) showed the existence of a statistical correlation between the directionality of inter-well fluid flow indicated by tracers, pressure response, interference tests, etc. and the local azimuth of maximum horizontal stress (Shmax) in over 80 hydrocarbon fields. Crucially, this correlation holds equally well for the subset of reservoirs that are deemed not to be ‘naturally fractured’ in a conventional sense. The most convincing explanation of this correlation, supported by coupled modelling (Heffer & Koutsabeloulis 1993), is that the natural fractures or faults that do exist in the reservoir, although possibly not very conductive before field development, are activated, mainly in shear, by the changes to stress state caused by pressure or temperature changes (e.g. Chen & Teufel 2001), and become conductive in patterns that are strongly anisotropic relative to modern-day in situ stress orientations. The strong sensitivity, or susceptibility, of fracture conductivity to changing stress state is consistent with the concept of the reservoir being close (in a sense relevant to the scale of perturbations from oilfield operations) to a critical point. The modelling by Zhang & Sanderson (2002) of the fluid flow in a granular medium as the stress state approaches a critical point also indicates progressive development of fractures striking subparallel to Shmax, seeding from pre-existing larger fractures of a different strike.

Spatial and temporal correlations in production and injection rates Another analysis of oilfield data that is difficult to explain other than as the manifestation of changing permeabilities owing to dynamic geomechanical conditions in reservoirs is that of the correlations in well production-rate fluctuations (Heffer et al. 1997). The technique calculates the correlation between injection or production rates at a pair of wells, considered as a pair of time series. The results from applications to several fields indicate that rate correlations between over half a million well pairs have the following general properties: (1) highest positive correlations for well pairs aligned along a direction close to the local orientation of maximum horizontal principal stress; (2) on average negative correlations for well pairs aligned subparallel to the local orientation of minimum horizontal principal stress;

(3) many of the high correlations are at long range; (4) orientational trends that are similar to faulting trends appear in the correlations. These properties are best explained by the interpretation that the rate fluctuations in a field are (at least partially) due to geomechanical changes in the reservoir causing strain and therefore permeability changes. The appearance of long ranges in the correlations is suggestive that the system is close to a critical point. One way of further analysing the rate correlations is to find the principal components or eigensolutions of the matrix of correlations between all individual wells in the field (both injectors and producers). This matrix has þ1 in all of its diagonal locations, and values between 21 and þ1 in off-diagonal locations. The principal components of this correlation matrix are independent linear combinations of rate fluctuations from each of the wells; the eigenvalue corresponding to each principal component indicates the variance associated with its combination of rates; those principal components with the highest eigenvalues are the independent modes that ‘explain’ most of the variance in the rate fluctuation data. If the rate fluctuations are mainly caused by correlated strain changes in the reservoir, then it should be possible to interpolate the values of these eigenvectors between wells with theoretical strain correlation functions, to give a geomechanically sensible result. Making the assumption that the eigenvectors of the rate correlations are each proportional to variations in total volumetric strain across the field, these can be interpolated spatially using an isotropic long-range function for the spatial correlation of strain, equivalent to c. r 20.1, slightly less correlated than the 1/k scaling that has been observed in porosity logs (see ‘1/k scaling of fracture porosity’ above). Fitting this interpolation function to the data was effected with a minimum variance or kriging scheme. Figure 2 shows the results. There are some hints of lineations in Figure 2. However, lineations were even more obvious when the values of eigenvectors were interpolated utilizing anisotropic correlation functions for displacement fields (Heffer et al. 1999; Daly 2001), which can then be converted to the full strain tensor. A minimum variance filter method, similar to kriging, was employed, which produced an a posteriori estimate of the displacement vectors across the gridded domain, based on the a priori long-range, anisotropic, spatial covariances, and the ‘observations’ of total strain (individual components of each eigenvector) at each well. The arbitrary locations of wells together with anisotropic spatial correlation functions allowed a unique solution to the filter matrix inversion. With the assumption that the eigenvector values are a reflection of local values of total strain the filter is

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Fig. 2. Map of values at each well of eigenvector 1 from the rate temporal correlation matrix, interpolated areally using an isotropic correlation function c. r 20.1 with a minimum variance fitting technique.

mathematically linear. Of course, other equivalences for the eigenvector values might be assumed (e.g. maximum principal strain, or shear strain), but those would generally render the filter non-linear and computationally more difficult. Once the displacement vectors have been calculated across the grid, other geomechanical variables are readily derived. Figure 3 shows the interpolated maps of maximum horizontal principal strain corresponding to the five principal components with the highest eigenvalues. The interpolation has again been made on the basis that the principal component value corresponds to the local volumetric strain. The corresponding maps of horizontal displacement vectors are shown in the right-hand column of Figure 3. From the values of the eigenvalues given in the second column of Figure 3, the first two principal components are fairly equal in the amount of variance that they encapsulate, whereas the last two are only about half as important. Remarkably, not only do all of the maps of maximum principal strain show spatial coherence, the pattern for principal component 1 correlates well with the faulting pattern across the field, as shown in Figure 4. It also has highs around the periphery of the field, in accordance with the long-range correlation in rates previously noted. The interpreted displacements

follow a general inward direction towards central points within the field, especially for the first three principal components. The tentative mechanical interpretation is that compaction associated with depressurization causes lateral inward displacements, with extensional ‘splitting’ and associated shear, along particular fault trends and at the periphery of the field, where pressure gradients are highest. Principal components 2–5 show less correlation with the fault traces, but within themselves do display characteristic trends, some of which may reflect movement of individual fault blocks. This finding raises the possibility that independent principal components of a statistical nature are related to different geomechanical modes of deformation, although, because of the non-linearities involved, the physical modes are unlikely to be completely independent. Principal component analysis has been used for similar purposes in the science of climatology (e.g. Wikle 2002), where it is referred to as ‘empirical orthogonal function’ analysis. As Wikle (2002) implied, it is not the only tool available for isolating modes from correlated multivariable time series. Furthermore, whereas principal component analysis decomposes the total variance into contributions from orthogonal, or independent, modes, another method that might be applicable is factor analysis (e.g. Lawley & Maxwell 1971),

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Fig. 3. Interpolated maps of maximum horizontal principal strain corresponding to the five principal components with the highest eigenvalues. Scale of each panel is c. 43 km north–south and 49 km east–west.

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Fig. 4. Map of fault traces on top surface of reservoir superimposed upon the map of maximum horizontal principal strain for the first principal component from the rate correlation matrix. (Note the strong correlation of the trends in high strain with the faulting trends, and the location of high strains towards the periphery of the field.) Scale of panel is c. 43 km north– south and 49 km east– west.

which attempts to fit the whole covariance matrix with individual factors. It is speculated that these other methods of analysis might give better insight into the physical modes that lie behind the correlated fluctuations. The principal components described here should be viewed as a preliminary decomposition; one, however, that reveals a strong influence of fault trends on rate fluctuations. The characteristics of rate correlations that are consistent with proximity to a geomechanical critical point are: (1) their long-range spatial correlation; the influences on this of the field boundary and the unique length scale of the field-wide pressure distribution are as yet uncertain; (2) their relationship to faulting structure; (3) their relationship to modern-day stress state (Heffer et al. 1997). Further work is necessary to quantify the degree to which these data fit models of SOC or SOSC.

Implications for reservoir modelling Fracture modelling tools The problem of identifying and predicting the influence of faults and fractures on fluid flow in the reservoir can be divided into three parts. (1) What is the natural configuration of structural features, possibly deriving from several

tectonic episodes in the geological history of the region? (2) What is the distribution of hydraulic conductivities of natural faults and fractures prior to field development? Here the influences of diagenetic and stress controls both need to be considered. (3) What will be the influence of the changes in stress state, induced by the field development processes, in changing the conductivities of the faults and fractures? In all of these three stages the scale invariance, long-range correlations, strong fluctuations and anisotropy that are associated with the concept of self-organized criticality in deformation can be important. However, particular length scales introduced by heterogeneities associated with mechanical stratigraphy, diagenetic variations, structure from previous tectonic episodes, and the development pattern itself will interrupt the spatial scaling; these will affect the geomechanical solution and therefore fracture fluid flow, as well as having a direct influence on the matrix fluid flow. The contrast in mechanical properties at a boundary between beds, or between regions of different diagenetic histories, can focus stresses, such that there is a local increase in fracture density. Differential compaction can cause sufficient deformation to lead

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to fracturing. Stylolites, compaction features effected by pressure solution mechanisms, can also provide the focus for extensional strains with associated tension gashes. Moreover, stylolites can themselves show different characteristics depending upon, for example, the clay or shale content of the carbonate sequence, yielding another lithology control on fracturing. Geomechanical models, such as finite-element models, can be important in providing realistic constraints to mechanical behaviour, even though they must use effective properties rather than explicitly handle the stochastic, discontinuous and scale-ranging implications of critical point concepts. In particular, geomechanical modelling can allow prediction of the localized trends of rupture threshold along which fracture conductivity is likely to be stress-sensitive. In simulating the development stage especially, it is obviously important that geomechanical modelling be coupled with fluid flow simulation. There is sufficient field evidence presented in the two sections above to imply that the current neglect in most reservoir simulators of geomechanical changes, particularly their association with time-dependent permeabilities, means that there is a failure to capture a strong bias in directionality of fluid flow; other commercially important aspects of geomechanical behaviour, such as influences on vertical sweep efficiencies, or changing injectivities, are also being lost to modelling by this neglect.

Implications of 1/k scaling of heterogeneities for stochastic modelling The spatial correlation (strictly, generalized autocovariance function, GACV) that is equivalent to 1/k spectral densities (in 1D samples) is c. log(r), where r is the lag distance (Greenhall 1999). 1/k b spectral densities imply a GACV varying with distance, r, as c. r (b21). These covariance functions are obviously long range in nature, although they have a sharp drop-off at small lag distance. As b increases, so does the spatial correlation of the strain: combined with the orientational variations defined in equation (1), lineations of increasing scale are modelled. These forms of correlation imply: (1) a partial loss of predictability from well data even in the immediate surroundings; (2) a long-range correlation that has much more widespread influence than ‘usual’ variograms with finite ranges. The fact that 1/k scaling of heterogeneities has at least some origin in nearcritical geomechanical fluctuations implies that strong anisotropies in correlation are also to be expected.

Data acquisition The issues considered above place importance upon acquiring the following types of data from reservoirs. (1) Stress state variations across a reservoir to assess their likely influence on (a) the natural fracture conductivities, and (b) changes in fracture or fault conductivities resulting from stress changes induced by the proposed development process. Field measurements of stress state are vital for calibrating geomechanical models that interpolate or extrapolate structural variations in space or time. (2) Shear wave surveys, particularly timelapse repeat surveys, to validate and calibrate models of geomechanical and fracture behaviour. (3) Microseismic emissions that can provide direct evidence of spatial and temporal patterns of inelastic geomechanical changes associated with fluid flow.

Future work Items of future work that can improve understanding and allow more commercial value to be obtained in these topics are suggested as follows. (1) More comprehensive statistical analysis of rate fluctuations at wells, seeking a robust technique for extracting information on important structural features; this may include extensions to different frequencies of data collection, ultimately in real time. (2) Seeking the nature of signals of coupled geomechanical –flow processes, both theoretically and by potential detection in field data. (3) Improving the basis for modelling coupled geomechanical –flow processes so that it is sufficiently flexible to allow production and other field data to be adequately history-matched. Production data contain vital information on (changing) fracture or fault conductivities that cannot be practically measured in other ways: analysis of production data is as likely, if not more likely, to expose the length scales truly of most importance to reservoir fluid flow, than are extrapolations of very local fracture description data to other regions, and to other stress state conditions. Because geomechanical changes to rock have a direct influence on seismic wave characteristics, as well as an indirect influence through being associated with changes to the permeability field, which alters fluid distributions, it is important to incorporate coupled modelling into analyses of 4D seismic surveys, either P wave or S wave; this places even more emphasis on the need for a practical, flexible modelling tool. (4) Gaining a better measure of the degree to which hydrocarbon reservoirs approach a critical state, both before and during development.

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(5) Understanding the implications of timedependent permeabilities and porosities for standard reservoir engineering tools such as well test analyses.

Conclusions There are several strands of evidence from hydrocarbon fields that support the concept that crustal deformation self-organizes to a state close to geomechanical criticality. Many hydrocarbon field operations will maintain the stress state, or move it even closer to criticality. New evidence considered in this paper comes from spatial and temporal correlations in rate fluctuations that are long-range in nature and related to both stress state and faulting structure, as predicted by SOC models. The spatial patterns derived from rate fluctuations are consistent with localized linear trends at rupture threshold along which fracture conductivity is stress-sensitive and highly connected; away from these trends the stress state may be sub-critical, and other factors, particularly cementation, may control fracture conductivities. There are important implications for modelling reservoir heterogeneities and fluid flow, particularly in using long-range, anisotropic correlation functions for simulating heterogeneities and coupled geomechanical– flow models that allow for timedependent changes in fracture permeabilities when simulating reservoir development. Field data that can quantify these concepts and improve predictive capability from such models include precise stress state measurements, shear-wave surveys and recording of microseismic emissions. The author is grateful to P. Schutjens for reviewing an early draft of this paper and making very helpful comments. Discussions with P. Leary clarified some aspects of 1/k scaling.

Appendix Data from depleted hydrocarbon fields provided by Grasso & Sornette (1998, fig. 7) were normalized utilizing a Gutenberg –Richter law relating numbers of earthquakes and their magnitudes (e.g. Main & Al-Kindy 2002): N.1 ¼ N.M  10bðM1Þ where N.M is the cumulative number of earthquakes with magnitude greater than M; and b was given a nominal value of 0.94 for all the fields (Leonard et al. 2001). The results are shown in Figure 1. Uncertainties are involved in the b values chosen; also, a unique relationship may not be expected because of different configurations of pressure fields, poroelastic constants, stress heterogeneities, etc.

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Index Page numbers in italics refer to Figures; page numbers is bold refer to Tables.

acoustic emission 7 technique 90, 91– 92 amplitude variation with offset anisotropy (AVOA) 193 Ankoh (Japan) 145–147 see also Median Tectonic Line arenite see sandstone axial splitting 61 axisymmetric triaxial compression in sandstone 90, 95 experimental testing lab methods 90 lab results acoustic emission 91– 92 macroscopic observations 92 stress/strain behaviour 90– 91 microscopic methods 92– 93 microscopic results deformed 94–97 pristine 93– 94 stereology 97–100 results discussed 100–101 axisymmetric triaxial testing 49, 54, 57, 60, 61, 65– 69 Base Cretaceous Unconformity, North Sea 106 Bassin du Sud-Est (France) 20, 21 deformation bands distribution 20, 22, 23– 24 and fault evolution 30 growth 29–30 faults distribution 24–25, 29 orientation 27–28, 29 thickness 25–27 modelling of fluid pressure rise and deformation Coulomb plasticity softening 36–37 mechanical context 33–36 model description 40–43 model evolution 31– 33 model results 37–39 Beaucaire Marl 48 strain testing 62– 64 bending see poro-plastic deformation Berea Sandstone, deformation 165, 167 biaxial testing 50, 52– 54, 62 bifurcation theory 12, 13–15, 16 boundary values, and faulting 162 boundary-element method 163 brittle failure regime 10 damage v. localization 15– 16 brittle fracture, types 61 brittle materials 47, 59–61 brittle/ductile controls on transition 115 terminology 3 bulk density, effect in fault zones 83

Carboniferous Limestone, jointing 140 Castlegate Formation (analogue reservoir) 90, 95 experimental testing lab methods 90 lab results acoustic emission 91–92 macroscopic observations 92 stress/strain behaviour 90–91 microscopic methods 92–93 microscopic results deformed 94– 97 pristine 93–94 stereology 97– 100 results discussed 100 –101 cataclasis 3 cataclastic fault zones 19 cataclastic flow 61 cavitation 55 cement bridges 229 cementation effect on faulting 110–111 modelling sandstone behaviour method 111–114 results 114–115 clays effect in compaction experiments 93–94, 97 in gouge 82 see also montmorillonite coefficient of friction, and displacement 81, 82 cohesive bodies and strain testing ductile v. brittle 59– 61 lab tests 61–62 Beaucaire Marl 62–64 Red Vosges sandstone 65– 69 siltite 64–65 micro to macro cracking 61 compaction, v. dilation 166 compaction bands 12, 15, 49, 61, 89– 90 compaction studies in reservoir sandstone 90, 95 experimental testing lab methods 90 lab results acoustic emission 91–92 macroscopic observations 92 stress/strain behaviour 90–91 microscopic methods 92–93 microscopic results deformed 94– 97 pristine 93–94 stereology 97– 100 results discussed 100 –101 conductivity in fractures 229 sensitivity to stress 229, 232 contractancy 59

242 crack interactions 9– 10 non-interacting 10 propagation modes 12–13 crack tip 12–13 creep 3 Cretaceous sands of Provence deformation band fault network evolution 30 growth and propagation 29– 30 modelling Coulomb plasticity softening 36–37 mechanical context 33– 36 model description 40–43 model evolution 31–33 model results 37– 39 pore fluid pressure and compaction 30– 33 single v. multiple strand interpretation distribution 29 orientation 29 observations distribution 24– 25 orientation 27– 28 thickness 25– 27 critical point and criticality characterized 227 relation to failure 228 self-organized criticality (SOC) 227– 228 self-organized sub-criticality (SOSC) 228 lithosphere studies 1/k scaling 231 directionality 232 induced seismicity 229 –230 permeability and production rate 232 –235 shear wave splitting 231 stress state 231 cross-joint development early studies 124 exposures 125 modelling methods 124– 129 results 130 results discussed 138–139 summary 139–140 crystal plastic mechanisms 3 damage defined 1, 2 localization compared 3, 15– 16 macroscopic variables 8 –9 microscopic concepts 9– 11 parameter (D) 8 –9 strain organization 7 deformation impact on geomaterials 4 style 2 deformation bands see shear bands also compaction bands deformation bands, and strain hardening 20 deformation bands, in Provence fault network evolution 30 growth and propagation 29– 30

INDEX modelling Coulomb plasticity softening 36– 37 mechanical context 33–36 model description 40– 43 model evolution 31– 33 model results 37–39 pore fluid pressure and compaction 30–33 single v. multiple strand interpretation distribution 29 orientation 29 observations distribution 24–25 orientation 27–28 thickness 25–27 deformation in sandstones experiments 111 modelling 111– 112 methods 112 –114 results 114–115 modelling and experiments compared 115–117 fluid flow effects 117– 119 grain size effects 117 summary 119– 120 North Sea reservoirs cementation effects 110– 111 deep 107 –108 porosity collapse 108–110 shallow 107 see also compaction studies density, effect on strain localization 52, 59 diagenesis, relation to conductivity 229 dilatency, effect in strain testing 50, 55 dilation, v. compaction 166 distributed grain-scale deformation 89 dolostone deformation experiments 170 model deformation 171–172 drained testing 49–50, 56, 57 ductile deformation, damage parameter (D) 7, 8 –9 ductile materials 47, 59–61 ductile/brittle controls on transition115 terminology 3 earthquakes see seismicity effective elastic modulus 8 effective field 9 effective matrix 9 effective media theory (EMT) 9 effective stress, impact on modelling fracture-related flow 219– 220 Ekofisk field (North Sea) 231 elastic models, and faulting 162– 163 elastic v. inelastic behaviour 90 elastic wave, velocity change 7 electrical conductivity 7 en echelon cracks 12 energy theory 12 extension bands 15 extension fracture 61 extensive-dilatancy anisotropy 231

INDEX 1/f noise 227, 231 failure frictional 231 in relation to critical point 228 false relief stereophotogrammetry (FRS) 48 fault zones displacement accommodation 78– 79 experimental studies monomineralic zones 78– 79 polymineralic zones 80–83 field observations 77 fracture arrays 75–77 fracture development sequence 77 fracture halo 84 mechanics of localization 83–84 permeability structure model 143 see also gouge faulting and boundary values 162 and boundary-element method 163 and elastic models 162 –163 and localization 163–164 and plasticity 163 and poro-plastic deformation calibration of properties 166–168 introduction 164–166 numerical representation 180–184 simulation methods 169–170 simulation results 170– 177 simulation results discussed 177–179 theory 179– 180 and stress state 161–162 faults bifurcation theory applied 13–15 and fluid flow, effects of rheology and stress 106, 117–119 and fractures damage zones 187, 188 detection by seismic methods approaches 190–191 attributes 191– 194 introduction 189– 190 theory 190 initiation of 7 flicker 227 flow see fracture-related flow fluid flow directionality 232 and faults, effects of rheology and stress 106, 117–119 in fractures 229 impact of faults 19 see also reservoir modelling fluid transport, across faults, modelling 143 fold– fault systems 164 forced folding 169 fracture, and seismic anisotropy 189 fracture arrays, and faults, Nash Point 188 fracture geometry 75– 76 associated shear zones 76 displacement accommodation 78– 79 with increasing displacement 76–77 fracture halo 84

fracture mechanics 12–13 fracture sequence, non-clay fault zone 77 fracture-related flow simulation HYDRO-DDA system DDA tool 212– 213 HYDRO tool 211– 212 model behaviour 213– 216 model discussed 222 –223 upscaling sensitivities 216 –219 effective stress 219– 220 friction 221–222 hydraulic gradient 220– 221 introduction 209 upscaling 210 fractures 11– 12 attributes 123 aperture 123 length 123– 124 modelling 137, 138, 139 orientation and stress 123 planarity 123 modelling 137 spacing 123 modelling 130, 132 conductivity of 229 mechanics of localization 83– 84 modelling tools 235 multiple orientations 124 numerical modelling methods 124–129 results aperture 137 geometry 130– 137 length 138 results discussed 138 –139 summary 139–140 reservoir engineering approach to 227 France see Bassin du Sud-Est; Beaucaire Marl; Hostun RF sand; Red Vosges Sandstone friction and displacement 81, 82 impact on modelling fracture-related flow 221 –222 frictional failure 231 frictional surfaces in modelling 166–167 geomechanical models HYDRO-DDA system DDA tool 212 –213 HYDRO tool 211– 212 model behaviour 213–216 model discussed 222–223 upscaling sensitivities 216–219 effective stress 219 –220 friction 221–222 hydraulic gradient 220 –221 linked to seismic anisotropy model equations 205– 206 model method 199– 201 model results 201– 204 model set-up 197–199 summary 204

243

244 gouge in faults 84 experimental studies monomineralic zones 78–79 polymineralic zones 80–83 grain contact dissolution 111–112 grain crushing 61 grain size, relation to deformation 117 granular bodies and strain testing equipment 50–51 Hostun sand test 51, 52 density 52 mean stress level 52–56 patterns of localization 56–58 slenderness ratio 52 summary 59 volumetric strains 58–59 Hostun RF Sand (France) 48 localization patterns 56, 60 strain testing 51 hydraulic gradient, impact on modelling fracture-related flow 220–221 hydrocarbon reservoirs see reservoir rocks HYDRO-DDA system DDA tool 212– 213 HYDRO tool 211–212 model behaviour 213– 216 model discussed 222– 223 upscaling sensitivities 216– 219 effective stress 219– 220 friction 221 –222 hydraulic gradient 220– 221 Japan see Median Tectonic Line Jurassic, reservoirs of North Sea 107 1/k in modelling 236 noise 231 Kitagawa (Japan) 145, 146, 147 see also Median Tectonic Line limestone deformation experiments 169– 170 model deformation 171– 172 model results 174 lithosphere at criticality 228 critical point studies historical researches 229– 231 new data 232– 235 localization (strain localization) 2, 3, 48 application of bifurcation theory 13– 15, 16 cohesive body studies 48– 49 ductile v. brittle 59– 61 lab tests 48–49 Beaucaire marl 62–64 micro to macro cracking 61 Red Vosges sandstone 65– 69 siltite 64–65 experiments on Berea Sandstone 168 and faulting 163– 164

INDEX granular body studies 48– 49 equipment 50–51 Hostun sand test 51, 52 density 52 mean stress level 52–56 patterns of localization 56–58 slenderness ratio 52 summary 59 volumetric strains 58–59 mechanics of 83–84 phenomena fractures 11– 12 localization bands 12 shear band evolution 49– 50 strain organization 7 techniques used 48–49 test types 49 Median Tectonic Line 144–145 internal structure field observations 145–147 modelling 148 permeability studies 143 laboratory set-up 151 methods 148– 149 results 150, 151–152 structure 152–154 along-fault variation 157– 158 data limitations 158 deformation mechanisms 158 –159 upscaling 158 zones 154– 157 upscaling 158 deformation mechanisms 158–159 microcracks and microcracking 10, 16, 231 microscopy see SEM Miocene, strike slip events 20 modelling cementation sandstone behaviour method 111– 114 results 114–115 modelling cross joints methods 124–129 results 130 results discussed 138 –139 summary 139–140 modelling deformation band evolution Coulomb plasticity softening 36– 37 mechanical context 33– 36 model evolution 31–33 model results 37– 39 model description 40– 43 modelling faults and poro-plastic deformation calibration of properties 166– 168 introduction 164– 166 numerical representation 180– 184 simulation methods 169 –170 simulation results 170–177 plastic strain plots 172, 173 stress plots 172, 173, 174, 176 simulation results discussed 177– 179 theory 179–180 modelling fractures numerical modelling methods 124–129

INDEX results aperture 137 geometry 130–137 length 138 results discussed 138– 139 summary 139 –140 modelling fracture-related flow HYDRO-DDA system DDA tool 212– 213 HYDRO tool 211–212 model behaviour 213– 216 model discussed 222– 223 upscaling sensitivities 216– 219 effective stress 219–220 friction 221– 222 hydraulic gradient 220–221 modelling seismic properties equations 205–206 method 199– 201 results 201–204 set-up 197– 199 summary 204 montmorillonite, effect in gouge of 80, 81, 82 Nagano (Japan) see Median Tectonic Line Nash Point, fracture arrays 188 noise in systems 227, 231 normal-move-out anisotropy (NMOA) 191 North Sea reservoir sandstones fault timing 106 deep burial 107– 108 shallow burial 107 modelling fracture arrays and seismic signatures model description 205– 206 model method 199 –201 model results 201–204 model set-up 197– 199 summary 204 oilfields see reservoir rocks Oligocene, rifting events 20 orthogonal fractures 124 modelling 130 Oughtibridge gannister 105 –106 overconsolidation ratio 59, 61 P fractures 77, 78, 79, 80, 81, 82, 83 P shears 76, 85 permeability 7 calculation from HYDRO-DDA 211– 212 modelling 143 and production rate 232–235 reduction at faults 19, 20 relation to frictional failure 231 permeability structure studies laboratory set-up 151 methods 148– 149 results 150, 151–152 Median Tectonic Line 152–154 along-fault variation 157– 158 data limitations 158 deformation mechanisms 158 –159 upscaling 158 zones 154 –157

petroleum reservoirs see reservoir rocks pink noise 227 plane strain biaxial testing 49, 50, 51, 52–54, 62 simulation 172, 173 plasticity, and faulting 163 polygonal fractures 124 modelling 130– 138 poro-plastic deformation calibration of properties 166–168 introduction 164–166 numerical representation 180– 184 simulation methods 169–170 simulation results 170–177 plastic strain plots 172, 173 stress plots 172, 173, 174, 176 simulation results discussed 177 –179 theory 179–180 porosity collapse in North Sea reservoirs 108–110 fracture 231 reduction at faults 19, 20 pressure cycling tests for permeability structure laboratory set-up 151 methods 148–149 results 150, 151– 152 Median Tectonic Line studies 152– 154 along-fault variation 157–158 data limitations 158 zones 154– 157 pressure solution 3 Provence (France) see Bassin du Sud-Est quartz cement effect on faulting 110–111 modelling sandstone behaviour method 111–114 results 114–115 R fractures 77, 78, 79, 80, 81, 82, 83 R shears 76, 85 Red Vosges Sandstone 48 strain testing 65–69 reservoir modelling 1/k scaling 236 data acquisition 236 fracture modelling tools 235–236 future work 236 –237 reservoir rocks analogue rock see Castlegate Formation approach to fractures 227 critical point studies 1/k scaling 231 directionality 232 induced seismicity 229– 230 permeability and production rate 232– 235 shear wave splitting 230–231 stress state 231 extraction and effective stress 89 fault timing 106 deep burial 107–108 shallow burial 107 metastability 228

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246

INDEX

reservoir rocks (Continued) modelling fracture arrays and seismic signatures model description 205–206 model method 199– 201 model results 201–204 model set-up 197– 199 summary 204 rheology and stress, implications for faulting and fluid flow 105 –106 Riedel fractures 77, 85 experimental study monomineralic zones 78–79 polymineralic zones 80–83 Riedel shears 75, 76, 85 Rotliegendes, south North Sea 106, 107– 108 sands in experimental studies Cretaceous of Provence see deformation band study Hostun RF Sand (France) 48 localization patterns 56, 60 strain testing 51 sandstone deformation behaviour in North Sea reservoirs cementation effects 110–111 deep 107– 108 porosity collapse 108– 110 shallow 107 deformation experiments 111 deformation modelling 111– 112 methods 112– 114 results 114– 115 deformation modelling and experiments compared 115–117 fluid flow effects 117–119 grain size effects 117 summary 119–120 effects of rheology and stress, implications for fluid flow 105– 106 strain testing in Red Vosges Sandstone 48, 65– 69 see also reservoir rocks sealing, on fault zones 19 seismic anisotropy defined 194 factors causing 194–195 linked to geomechanical models model equations 205 –206 model method 199– 201 model results 201–204 model set-up 197– 199 summary 204 methods of fracture study approaches 190–191 attributes reflectivity 193 –194 travel time 191–193 introduction 189 –190 theory 190 modelling approaches 195 basis 195–196 seismicity in reservoirs induced 229–230 shear wave splitting 230– 231

SEM microscopy in compaction analysis, methods 92– 93 results deformed 97– 100 pristine 93–94 stereology 97– 100 results discussed 100 –101 shear bands 12 evolution 49– 50 experimental testing Beaucaire Marl 48, 62–64 Hostun Sand 48, 51, 56, 60 Red Vosges Sandstone 48, 65–69 factors affecting formation 47–48 strain rate 13 under compression 15 shear fracture 61 shear localization, compressive stress 89 shear zones in Berea Sandsone 165 displacement experiments 76– 77 and fractures 76 shear-wave splitting (SWS) 191, 230–231 siltite, strain testing 64– 65, 67, 68 slenderness ratio, effect on strain localization 52 slip lines and planes 163 stereology, quantitative 97– 98 method 98– 99 results 99–100 stereophotogrammetry 48, 52, 53, 64, 65, 68 strain in geomaterials 47–48 cohesive body studies ductile v. brittle 59–61 lab tests Beaucaire marl 62– 64 Red Vosges sandstone 65–69 siltite 64– 65 micro to macro cracking 61 granular body studies equipment 50–51 Hostun sand test 51, 52 factors affecting results density 52 mean stress level 52– 56 slenderness ratio 52 patterns of localization 56–58 summary 59 volumetric strains 58–59 materials studies 48 shear bands evolution 49–50 techniques 48–49 test types 49 strain hardening 20, 166 stress compressive 89 effective, and hydrocarbon extraction 89 simulation 172, 173, 174, 176 stress field and fractures 123, 124 stress intensity factor 12 stress and rheology, implications for faulting and fluid flow 105– 106 stress state calculation by boundary-element method 163 and faulting 161–162 stress/strain behaviour 90–91

INDEX stylolites 236 subcritical index and effect on fractures modelling methods 129– 130 results 130, 132, 134

introduction 76– 77 monomineralic zones 78–79 polymineralic zones 80– 83 triaxial strain testing 49, 54, 57, 60, 61

temperature, effect on cementation 111 –112 thickness of layers and effect on fractures modelling methods 129– 130 results 130, 131, 134, 135, 136 tomodensitometry 48– 49 triaxial compression experiments experimental testing with Castlegate Formation 90, 95 lab methods 90 lab results acoustic emission 91– 92 macroscopic observations 92 stress/strain behaviour 90– 91 microscopic methods 92– 93 microscopic results deformed 94–97 pristine 93– 94 stereology 97–100 results discussed 100–101

undrained strain testing 49– 50, 55–56 upscaling in modelling 210 HYDRO-DDA system 216– 219 sensitivities effective stress 219 –220 friction 221–222 hydraulic gradient 220 –221 Valhall field (North Sea) 231 volumetric strain variations 7 X-ray computed tomography 48–49 Y fractures 77, 78, 79, 80, 81, 82, 83 yield point 165 post-yield strain 167, 168 Young’s modulus 8

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The many kinds of porous geomaterials (rocks, soils, concrete, etc.) exhibit a range of responses when undergoing inelastic deformation. In doing so they commonly develop well-ordered fabric elements, forming fractures, shear bands and compaction bands, so creating the planar fabrics that are regarded as localization. Because these induced localization fabrics alter the bulk material properties (such as permeability, acoustic characteristics and strength), it is important to understand how and why localization occurs, and how it relates to its setting. The concept of damage (in several uses) describes both the precursor to localization and the context within which it occurs. A key theme is that geomaterials display a strong material evolution during deformation, revealing a close linkage between the damage and localization processes. This volume assembles perspectives from a number of disciplines, including soil mechanics, rock mechanics, structural geology, seismic anisotropy and reservoir engineering. The papers range from theoretical to observational, and include contributions showing how the deformed geomaterials’ emergent bulk characteristics, like permeability and seismic anisotropy, can be predicted. This book will be of interest to a wide range of geoscientists and engineers who deal with characterization of deformed materials.