To date, more than 7300 in situ stress orientations have been compiled as part of the World Stress Map project. Of these, over 4400 are considered reliable tectonic stress indicators, recording horizontal stress orientations to within <±25°. Remarkably good correlation is observed between stress orientations deduced from in situ stress measurements and geologic observations made in the upper 1–2 km, well bore breakouts extending to 4–5 km depth and earthquake focal mechanisms to depths of ∼20 km. Regionally uniform stress orientations and relative magnitudes permit definition of broad-scale regional stress patterns often extending 20–200 times the approximately 20–25 km thickness of the upper brittle lithosphere. The “first-order” midplate stress fields are believed to be largely the result of compressional forces applied at plate boundaries, primarily ridge push and continental collision. The orientation of the intraplate stress field is thus largely controlled by the geometry of the plate boundaries. There is no evidence of large lateral stress gradients (as evidenced by lateral variations in stress regime) which would be expected across large plates if simple resistive or driving basal drag tractions (parallel or antiparallel to absolute motion) controlled the intraplate stress field. Intraplate areas of active extension are generally associated with regions of high topography: western U.S. Cordillera, high Andes, Tibetan plateau, western Indian Ocean plateau. Buoyancy stresses related to crustal thickening and/or lithospheric thinning in these regions dominate the intraplate compressional stress field due to plate-driving forces. These buoyancy forces are just one of several categories of “second-order” stresses, or local perturbations, that can be identified once the first-order stress patterns are recognized. These second-order stress fields can often be associated with specific geologic or tectonic features, for example, lithospheric flexure, lateral strength contrasts, as well as the lateral density contrasts which give rise to buoyancy forces. These second-order stress patterns typically have wavelengths ranging from 5 to 10+ times the thickness of the brittle upper lithosphere. A two-dimensional analysis of the amount of rotation of regional horizontal stress orientations due to a superimposed local stress constrains the ratio of the magnitude of the horizontal regional stress differences to the local uniaxial stress. For a detectable rotation of 15°, the local horizontal uniaxial stress must be at least twice the magnitude of the regional horizontal stress differences. Examples of local rotations of SHmax orientations include a 75°–85° rotation on the northeastern Canadian continental shelf possibly related to margin-normal extension derived from sediment-loading flexural stresses, a 50°–60° rotation within the East African rift relative to western Africa due to extensional buoyancy forces caused by lithospheric thinning, and an approximately 90° rotation along the northern margin of the Paleozoic Amazonas rift in central Brazil. In this final example, this rotation is hypothesized as being due to deviatoric compression oriented normal to the rift axis resulting from local lithospheric support of a dense mass in the lower crust beneath the rift (“rift pillow”). Estimates of the magnitudes of first-order (plate boundary force-derived) regional stress differences computed from modeling the source of observed local stress rotations magnitudes can be compared with regional stress differences based on the frictional strength of the crust (i.e., “Byerlee's law”) assuming hydrostatic pore pressure. The examples given here are too few to provide a definitive evaluation of the direct applicability of Byerlee's law to the upper brittle part of the lithosphere, particularly in view of uncertainties such as pore pressure and relative magnitude of the intermediate principal stresses. Nonetheless, the observed rotations all indicate that the magnitude of the local horizontal uniaxial stresses must be 1–2.5+ times the magnitude of the regional first-order horizontal stress differences and suggest that careful evaluation of such local rotations may be a powerful technique for constraining the in situ magnitude stress differences in the upper, brittle part of the lithosphere.

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First‐ and second‐order patterns of stress in the lithosphere: The World Stress Map Project

The validity of the cubic law for laminar flow of fluids through open fractures consisting of parallel planar plates has been established by others over a wide range of conditions with apertures ranging down to a minimum of 0.2 µm. The law may be given in simplified form by Q/Δh = C(2b)3, where Q is the flow rate, Δh is the difference in hydraulic head, C is a constant that depends on the flow geometry and fluid properties, and 2b is the fracture aperture. The validity of this law for flow in a closed fracture where the surfaces are in contact and the aperture is being decreased under stress has been investigated at room temperature by using homogeneous samples of granite, basalt, and marble. Tension fractures were artificially induced, and the laboratory setup used radial as well as straight flow geometries. Apertures ranged from 250 down to 4µm, which was the minimum size that could be attained under a normal stress of 20 MPa. The cubic law was found to be valid whether the fracture surfaces were held open or were being closed under stress, and the results are not dependent on rock type. Permeability was uniquely defined by fracture aperture and was independent of the stress history used in these investigations. The effects of deviations from the ideal parallel plate concept only cause an apparent reduction in flow and may be incorporated into the cubic law by replacing C by C/ƒ. The factor ƒ varied from 1.04 to 1.65 in these investigations. The model of a fracture that is being closed under normal stress is visualized as being controlled by the strength of the asperities that are in contact. These contact areas are able to withstand significant stresses while maintaining space for fluids to continue to flow as the fracture aperture decreases. The controlling factor is the magnitude of the aperture, and since flow depends on (2b)3, a slight change in aperture evidently can easily dominate any other change in the geometry of the flow field. Thus one does not see any noticeable shift in the correlations of our experimental results in passing from a condition where the fracture surfaces were held open to one where the surfaces were being closed under stress.

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VALIDITY OF CUBIC LAW FOR FLUID FLOW IN A DEFORMABLE ROCK FRACTURE - eScholarship

A review of techniques, advances and outstanding issues in numerical modelling for rock mechanics and rock engineering

Fracture coalescence in rock-type materials under uniaxial and biaxial compression

The purpose of this review is to discuss the development and the state of the art in dynamic testing techniques and dynamic mechanical behaviour of rock materials. The review begins by briefly introducing the history of rock dynamics and explaining the significance of studying these issues. Loading techniques commonly used for both intermediate and high strain rate tests and measurement techniques for dynamic stress and deformation are critically assessed in Sects. 2 and 3. In Sect. 4, methods of dynamic testing and estimation to obtain stress–strain curves at high strain rate are summarized, followed by an in-depth description of various dynamic mechanical properties (e.g. uniaxial and triaxial compressive strength, tensile strength, shear strength and fracture toughness) and corresponding fracture behaviour. Some influencing rock structural features (i.e. microstructure, size and shape) and testing conditions (i.e. confining pressure, temperature and water saturation) are considered, ending with some popular semi-empirical rate-dependent equations for the enhancement of dynamic mechanical properties. Section 5 discusses physical mechanisms of strain rate effects. Section 6 describes phenomenological and mechanically based rate-dependent constitutive models established from the knowledge of the stress–strain behaviour and physical mechanisms. Section 7 presents dynamic fracture criteria for quasi-brittle materials. Finally, a brief summary and some aspects of prospective research are presented.

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A Review of Dynamic Experimental Techniques and Mechanical Behaviour of Rock Materials

Introduction. Estimating in-situ stresses. Methods of in-situ stress measurement. Hydraulic methods. Relief methods. Jacking methods. Strain recovery methods. Borehole breakout methods. Case studies and comparison between different methods. Monitoring of stress change. The state of stress in the earth's crust: from local measurements to the world stress map. Using stresses in rock engineering, geology and geophysics. Appendices.

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Rock stress and its measurement

Nearly 1500 stress orientation determinations are now available for Europe. The data come from earthquake focal mechanisms, overcoring measurements, well bore breakouts, hydraulic fracturing measurements, and young fault slip studies and sample the stress field from the surface to seismogenic depths. Three distinct regional patterns of maximum compressive horizontal stress (SHmax) orientation can be defined from these data: a consistent NW to NNW SHmax stress orientation in western Europe; a WNW-ESE SHmax orientation in Scandinavia, similar to western Europe but with a larger variability of SHmax orientations; and a consistent E-W SHmax orientation and N-S extension in the Aegean Sea and western Anatolia. The different stress fields can be attributed to plate-driving forces acting on the boundaries of the Eurasian plate, locally modified by lithospheric properties in different regions. On average, the orientation of maximum stress in western Europe is subparallel to the direction of relative plate motion between Africa and Europe and is rotated 17° clockwise from the direction of absolute plate motion. The uniformly oriented stress field in western Europe coincides with thin to medium lithospheric thickness (approximately 50–90 km) and high heat flow values (>80 m W/m2). In western Europe a predominance of strike-slip focal mechanisms implies that the intermediate principal stress is vertical. The more irregular horizontal stress orientations in Scandinavia coincide with thick continental lithosphere (110–170 km) and low heat flow (<50 m W/m2). The cold thick lithosphere in this region may result in lower mean stresses associated with far-field tectonic forces and allow the stress field to be more easily perturbed by local effects such as deglaciation flexure and topography. The stress field of the Aegean Sea and western Anatolia is consistent with N-S extension in a back arc setting behind the Hellenic trench subduction zone. The stress field is influenced in places by regional geologic structures, e.g., in the Western Alps, where SHmax directions show a slight tendency toward a radial stress pattern. Not all major geologic structures, however, appear to affect the SHmax orientation, e.g., in the vicinity of the Rhine rift system horizontal stress orientations are continuous.

Regional patterns of tectonic stress in Europe

Foreword Preface Dedication Acknowledgement List of Permissions 1. Introduction 1.1 Stresses in a Body 1.2 Importance of Rock Stress 1.3 History of Interest in Rock Stress Part I - Definition and Terminology 2. Stress Definition 2.1 Stress Tensor 2.2 Principal Stresses 2.3 Mohr Circle of Stress 2.4 Visualizing Stress 3. Rock Fracture Criteria 3.1 Phenomenological Theories 3.2 Mechanistic Failure Theories 3.3 Fracture Mechanics 3.4 Nonlinear Fracture Mechanics 4. Rock Stress Terminology 4.1 Gravity Stress 4.2 Tectonic Stress 4.3 Residual Stress 4.4 Structural Stress 5. Crustal Stress Models 5.1 Lithostatic Stress 5.2 Biaxial State of Stress 5.3 Tectonic Stress Field 5.4 Effective Stress 5.5 Laboratory Stress Profiles Part II - Measuring Stress 6. Physics of Stress Measurements 6.1 Mechanical Methods 6.2 Strain Gages 6.3 Diffraction Methods 6.4 Optical Methods 6.5 Ultrasonic Wave Speed 6.6 Micromagnetic Method 7. Measuring Crustal Stress - Borehole Methods 7.1 Classification of Measurement Techniques 7.2 Hydraulic Fracturing 7.3 Borehole Breakouts 8. Measuring Crustal Stress - Core-Based Methods 8.1 Anelastic Strain Recovery 8.1.1 Rheological Basis 8.1.2 Relaxation Apparatus 8.2 Kaiser Effect Part III - Interpreting Stress Data 9. Local Stress Data 9.1 Continental Deep Drilling Site KTB, Germany 9.2 Nuclear Waste Site Olkiluoto, Finland 9.3 San Andreas Fault Observatory at Depth, USA 10. Generic Stress Data 10.1 Magnitude-Depth Profiles 10.2 Orientation Maps and Smoothing 10.3 Stress State-Scale Relations 10.4 Best-Estimate Stress Model 11. Global Stress 11.1 European Stress 11.2 World Stress Map 11.3 Plate Tectonic Interpretation Epilogue Stress References Index Stress Movies Content on DVD-ROM DVD-ROM included inside back cover

Stress Field of the Earth's Crust

A series of uniaxial compression tests were performed on gypsum specimens with preexisting fractures to study the failure mechanism of fractures and rock bridges in fractured rock masses. The coalescence mechanism of two parallel and offset fractures was investigated by monitoring the process of fracture initiation and propagation with a video camera. The tests showed that two inclined parallel fractures can coalesce by shear failure and/or tensile failure under a uniaxial load. The coalescence path and mechanism mainly depend on the relative position of the two fractures. For instance, when the two fractures are coplanar or slightly offset, coalescence is generated by shear failure; when they are overlapping in the loading direction, coalescence is generated by mixed shear and tensile failure. Two types of preexisting fractures, one without initial surface contact and hence frictionless and another with surface contact and friction, were used to study the influence of fracture contact conditions on the coalescence path and load. It was found that coalescence of fractures with surface contact and friction requires loads as much as 35% higher than that for coalescence of fractures without contact and friction. A stress analysis was conducted in this study to explain the different coalescence mechanisms. The analytical work indicated that different fracture geometries produce significantly different stress fields in the rock bridge area and hence result in different failure modes.

Coalescence of fractures under shear stresses in experiments

This book presents some fundamental concepts behind the basic theories and tools of discrete element methods (DEM), its historical development, and its wide scope of applications in geology, geophysics and rock engineering. Unlike almost all books available on the general subject of DEM, this book includes coverage of both explicit and implicit DEM approaches, namely the Distinct Element Methods and Discontinuous Deformation Analysis (DDA) for both rigid and deformable blocks and particle systems, and also the Discrete Fracture Network (DFN) approach for fluid flow and solute transport simulations. The latter is actually also a discrete approach of importance for rock mechanics and rock engineering. In addition, brief introductions to some alternative approaches are also provided, such as percolation theory and Cosserat micromechanics equivalence to particle systems, which often appear hand-in-hand with the DEM in the literature. Fundamentals of the particle mechanics approach using DEM for granular media is also presented. Presents the fundamental concepts of the discrete models for fractured rocks, including constitutive models of rock fractures and rock masses for stress, deformation and fluid flow Provides a comprehensive presentation on discrete element methods, including distinct elements, discontinuous deformation analysis, discrete fracture networks, particle mechanics and Cosserat representation of granular media Features constitutive models of rock fractures and fracture system characterization methods detaiing their significant impacts on the performance and uncertainty of the DEM models

Ove Stephansson

Papers

Rock stress and its measurement

Regional patterns of tectonic stress in Europe

Stress Field of the Earth's Crust

Coalescence of fractures under shear stresses in experiments

Fundamentals of Discrete Element Methods for Rock Engineering : Theory and Applications