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Table Of Integrals Series And Products

This essential reference for graduate students and researchers provides a unified treatment of earthquakes and faulting as two aspects of brittle tectonics at different timescales. The intimate connection between the two is manifested in their scaling laws and populations, which evolve from fracture growth and interactions between fractures. The connection between faults and the seismicity generated is governed by the rate and state dependent friction laws - producing distinctive seismic styles of faulting and a gamut of earthquake phenomena including aftershocks, afterslip, earthquake triggering, and slow slip events. The third edition of this classic treatise presents a wealth of new topics and new observations. These include slow earthquake phenomena; friction of phyllosilicates, and at high sliding velocities; fault structures; relative roles of strong and seismogenic versus weak and creeping faults; dynamic triggering of earthquakes; oceanic earthquakes; megathrust earthquakes in subduction zones; deep earthquakes; and new observations of earthquake precursory phenomena.

The Mechanics of Earthquakes and Faulting

Geological atlas of western and central Europe

Scaling in fracture systems has become an active field of research in the last 25 years motivated by practical applications in hazardous waste disposal, hy- drocarbon reservoir management, and earthquake haz- ard assessment. Relevant publications are therefore spread widely through the literature. Although it is rec- ognized that some fracture systems are best described by scale-limited laws (lognormal, exponential), it is now recognized that power laws and fractal geometry provide widely applicable descriptive tools for fracture system characterization. A key argument for power law and fractal scaling is the absence of characteristic length scales in the fracture growth process. All power law and fractal characteristics in nature must have upper and lower bounds. This topic has been largely neglected, but recent studies emphasize the importance of layering on all scales in limiting the scaling characteristics of natural fracture systems. The determination of power law expo- nents and fractal dimensions from observations, al- though outwardly simple, is problematic, and uncritical use of analysis techniques has resulted in inaccurate and even meaningless exponents. We review these tech- niques and suggest guidelines for the accurate and ob- jective estimation of exponents and fractal dimensions. Syntheses of length, displacement, aperture power law exponents, and fractal dimensions are found, after crit- ical appraisal of published studies, to show a wide vari- ation, frequently spanning the theoretically possible range. Extrapolations from one dimension to two and from two dimensions to three are found to be nontrivial, and simple laws must be used with caution. Directions for future research include improved techniques for gathering data sets over great scale ranges and more rigorous application of existing analysis methods. More data are needed on joints and veins to illuminate the differences between different fracture modes. The phys- ical causes of power law scaling and variation in expo- nents and fractal dimensions are still poorly understood.

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Scaling of fracture systems in geological media

Propagation of Magma-Filled Cracks

Rock Fractures in Geological Processes: Properties of some crustal materials

Aims The way active faults transport crustal fluids is important in many fields of earth sciences, including petroleum geology, geothermal research, volcanology, seismology, and hydrogeology. There is increasing evidence that active, or potentially active, faults largely control fluid flow in solid rocks. In order to understand the flow of ground water and other crustal fluids in a region, it is necessary to know the general permeability structure of the fault zones in that region. This follows because the general hydromechanical structure of a fault zone, in relation to that of the host rock, determines whether the fault zone acts as a conduit that transports fluids or as a barrier to fluid flow. Here the focus is on the effects of faults on ground-water flow, but the results are, with suitable modifications, applicable to the transport of other crustal fluids. The principal aims of this chapter are to: Explain how fluids migrate to fault zones, in particular active fault zones. Present a general model on fluid transport along a fault zone. Explain the effects of fluid pressure on the apertures of fractures in the damage zone. Provide a simple model of fluid transport along the fault damage zone. Provide a simple model of fluid transport along the fault core. Discuss the general implications of the results for fluid transport by faults.

Rock Fractures in Geological Processes: Fluid transport in faults

Aims When rock is subject to higher stresses than it can sustain it fails. There are many modes of failure of rocks and other solids, but all belong primarily to one of two broad classes, namely ductile failure and brittle failure. However, some criteria apply both to strictly brittle rocks as well as to materials, such as granular media, that would normally be regarded as ductile. In fact, there is a gradation from strictly brittle (brittle fracture) to strictly ductile (plastic flow) in rocks and many other solids. In this chapter we discuss some of the main types of failure, focusing on the principal criteria for brittle failure, that is, failure through extension fractures and shear fractures. Brittle failure results in fractures which originate from stress concentrations (Chapter 6). The main aims of this chapter are to: Define the concepts of strength and failure. Explain the concept of a Coulomb material. Define and explain the Coulomb failure criterion. Explain and discuss some empirical failure criteria. Discuss the basic elements of the Griffith theory. Present the Griffith criterion for failure of rocks. Failure and strength Failure may be defined as the stress condition at which the solid either starts to flow or break. This stress condition thus defines the maximum stress or stress difference that the solid can sustain, that is, its load-bearing capacity. When the failure of the solid is through fracture formation, the failure is brittle . When the failure of the solid is through plastic flow, the failure is ductile .

Rock Fractures in Geological Processes: Theories of brittle failure of rocks

Dykes are extension fractures and form when the magmatic overpressure is high enough to rupture (break) the host rock. Their formation is entirely analogous to that of many joints and human-made hydraulic fractures, such as are used to increase permeability in reservoirs. When generating their paths, dykes use existing weaknesses (e.g., cooling joints) in the host rock. The maximum depth of large tension fractures below the surface of a rift zone, however, is mostly less than a few hundred metres. If the fractures extend to greater depths, they must change into closed normal faults which are generally not used as magma paths. There are thus no large tension fractures or wide-open faults at great depths ready to be filled with magma to form a dyke. While magma flow in dykes, as in other fluid-driven fractures, is at any point in various directions dyke segmentation may indicate the overall large-scale flow direction. Thus, dykes composed of large-spaced disconnected segments in lateral sections are primarily formed in vertical magma flow at segmentation depth whereas those composed of large-spaced disconnected segments in vertical sections are primarily formed in lateral magma flow. The far-field displacement and stress fields of segmented dykes are similar to those generated by single, continuous dykes of similar dimensions, particularly when the distances between the nearby tips of the segments become small in comparison with segment lengths. Most dykes become arrested and never supply magma to eruptions. Feeder-dykes normally reach the surface only along parts of their lengths (strike-dimensions). The volumetric flow or effusion rate of magma through a feeder-dyke or volcanic fissure depends on the aperture (opening) of the dyke or fissure in the 3rd power. All these theoretical and observational results are here applied to the dyke emplacements associated with the eruptions of Eyjafjallajökull 2010 and Bardarbunga-Holuhraun 2014. The results make it possible to (1) explain, broadly, the propagation-paths of the associated dykes, (2) the arrest and deflection (into sills) of many dyke segments, (3) the dimensions of the dykes, in particular (4) the dyke thicknesses, (5) the volumetric flow or effusion rates of the volcanic fissures, and (6) the location of the magma sources of the dykes.

Dyke-path formation in relation to the eruptions of Eyjafjallajökull 2010 and Bardarbunga-Holuhraun 2014

Based on eruptive volumes, eruptions can be classified as follows: small if the volumes are from less than 0.001 km3 to 0.1 km3, moderate if the volumes are from 0.1 to 10 km3, and large if the volumes are from 10 km3 to 1000 km3 or larger. The largest known explosive and effusive eruptions have eruptive volumes of 4000-5000 km3. The physics of small to moderate eruptions is reasonably well understood. For a typical mafic magma chamber in a crust that behaves as elastic, about 0.1% of the magma leaves the chamber (erupted and injected as a dyke) during rupture and eruption. Similarly, for a typical felsic magma chamber, the eruptive/injected volume during rupture and eruption is about 4%. To provide small to moderate eruptions, chamber volumes of the order of several tens to several hundred cubic kilometres would be needed. Shallow crustal chambers of these sizes are common, and deep-crustal and upper-mantle reservoirs of thousands of cubic kilometres exist. Thus, elastic and poro-elastic chambers of typical volumes can account for small to moderate eruptive volumes. When the eruptions become large, with volumes of tens or hundreds of cubic kilometres or more, an ordinary poro-elastic mechanism can no longer explain the eruptive volumes. The required sizes of the magma chambers and reservoirs to explain such volumes are simply too large to be plausible. Here I propose that the mechanics of large eruptions is fundamentally different from that of small to moderate eruptions. More specifically, I suggest that all large eruptions derive their magmas from chambers and reservoirs whose total cavity-volumes are mechanically reduced very much during the eruption. There are two mechanisms by which chamber/reservoir cavity-volumes can be reduced rapidly so as to squeeze out much of, or all, their magmas. One is piston-like caldera collapse. The other is graben subsidence. During large slip on the ring-faults/graben-faults the associated chamber/reservoir shrinks in volume, thereby maintaining the excess magmatic pressure much longer than is possible in the ordinary poro-elastic mechanism. Here the physics of caldera subsidence and graben subsidence is regarded as basically the same. The geometric difference in the surface expression is simply a reflection of the horizontal cross-sectional shape of the underlying magma body. In this new mechanism, the large eruption is the consequence—not the cause—of the caldera/graben subsidence. Thus, once the conditions for large-scale subsidence of a caldera/graben during an unrest period are established, then the likelihood of large to very large eruptions can be assessed and used in reliable forecasting.

Agust Gudmundsson

Papers

Rock Fractures in Geological Processes: Properties of some crustal materials

Rock Fractures in Geological Processes: Fluid transport in faults

Rock Fractures in Geological Processes: Theories of brittle failure of rocks

Dyke-path formation in relation to the eruptions of Eyjafjallajökull 2010 and Bardarbunga-Holuhraun 2014

The physics of large eruptions