Many of the Earth's most fascinating natural processes are related to rock fractures. Volcanic eruptions, tectonic earthquakes, geysers, large landslides and the formation and development of mid-ocean ridges all depend on fracture formation and propagation. Rock fractures are also of fundamental importance in more applied fields such as those related to fluid-filled reservoirs, deep crustal drilling, tunnelling, road construction, dams, geological and geophysical mapping and field geology and geophysics.

There has been great progress in understanding fracture initiation and propagation over the past decades. The results of this progress are summarised in many papers, textbooks and monographs within the fields of fracture mechanics and materials science. Much of this improved knowledge of fracture development is of great relevance for understanding and modelling geological processes that relate to rock fractures.

The purpose of this book is to offer a modern treatment of rock fractures for earth scientists and engineers. The book is primarily aimed at, first, undergraduate or beginning graduate students in geology, geophysics and geochemistry and, second, scientists, engineers and other professionals who deal with rock fractures in their work. The book has been designed so that it can be used (1) for an independent study, (2) as a textbook for a course in rock fractures in geological processes and (3) as a supplementary text for courses in structural geology, seismology, volcanology, hydrogeology, geothermics, hazard studies, engineering geology, rock mechanics and petroleum geology.

Aims

Rock fractures occur in a variety of geological processes and range in size from plate boundaries at the scale of hundreds of kilometres to microcracks in crystals at the scale of a fraction of a millimetre. This chapter provides a definition of a rock fracture as well as of some of the mechanical concepts used in the analysis of fractures. Some of these definitions are preliminary and more accurate ones will be provided in later chapters. The primary aims of this chapter are to:

• Provide a definition of a rock fracture.

• Indicate some of the many earth-science topics and processes where fractures play a fundamental role.

• Provide definitions of stress, strain, constitutive equations, and material behaviour.

• Explain the one-dimensional Hooke's law.

• Summarise some basic fracture-related definitions in structural geology.

• Discuss the basic information needed to solve fracture problems.

• Define and explain boundary conditions, rock properties, and rock-failure criteria.

• Explain accuracy, significant figures, and rounding of numbers.

• Explain the basic units and prefixes used.

Rock fractures

A rock fracture is a mechanical break or discontinuity that separates a rock body into two or more parts (Fig. 1.1). The continuity or cohesion of the rock body is lost across a fracture. A fracture forms in response to stress. More specifically, the rock breaks and forms a fracture when the applied stress reaches a certain limit, namely the rock strength. The stress associated with the fracture formation may be normal or shear or both.

Scientists and engineers use the International System of Units (in French: Système Internationale d'Unités – hence the acronym SI units), which has seven base units, relying on simple physical effects, and many derived units, only some of which (such as force and stress) are listed below. Also provided below are the dimensions of some of the quantities, where L denotes length, M mass, T time, and F force. More details are given by Huntley (1967), Gottfried (1979), Emiliani (1995), Benenson et al. (2002), Woan (2003), and Deeson (2007). The reference list for all the appendices is at the end of Appendix E.

Aims

Faults are measured in a very similar way as extension fractures. The main aims of this chapter are to:

• Give field examples of typical faults.

• Illustrate how the most important field measurements of faults are made.

• Provide typical field data on dip-slip faults.

• Provide field data on strike-slip faults.

• Explain and discuss oblique-slip faults.

Dip-slip faults

Dip-slip faults are those where the displacement is parallel with the dip of the fault (Fig. 12.1). They include normal faults, reverse faults, and thrust faults (Figs. 8.3 and 8.8). The block above the fault plane is the hanging wall, the block below the fault plane the footwall (Fig. 12.1). Depending on the sense of slip and the fault dip, the main types of dip-slip faults are normal faults, reverse faults, thrusts, and overthrusts. More specifically, when the hanging wall moves down relative to the footwall, the fault is a normal fault. When the hanging wall moves up relative to the footwall, the fault is a reverse fault.

When the dip of a fault is less than 45° it is referred to as a low-angle fault. A low-angle normal fault is just referred to as such, but sometimes as a lag fault. However, low-angle reverse faults have special names. In general, if the dip of the reverse fault is less than 45° (for some the angle should be 30°), it is referred to as a thrust or a thrust fault.

Aims

How do small extension fractures link to form larger fractures? More specifically, what main factors determine the path of an extension fracture? These topics are of great importance in understanding the development and maintenance of permeability in crustal rocks and reservoirs. The topic is also fundamental in understanding how volcanoes work. This follows because most volcanic eruptions are supplied with magma through dykes and inclined sheets, both of which are extension fractures. For assessing volcanic hazards and risks, we need to understand how dykes and sheets propagate to the surface, resulting in an eruption, or, alternatively, become arrested within layers or at layer contacts or other discontinuities. The main aims of this chapter are to present results on the:

• General development of tension fractures.

• General propagation and path-selection of hydrofractures.

• Conditions for hydrofracture deflection and arrest.

• Apertures of hydrofractures subject to constant fluid overpressure.

• Apertures of hydrofractures subject to varying fluid overpressure.

• Surface deformation related to arrested hydrofractures.

Development of tension fractures

Tension fractures develop in a similar way to hydrofractures. Most of the conclusions regarding hydrofracture propagation, deflection, and arrest, presented below, apply, with modifications, to tension fractures as well. The main differences are, first, that large tension fractures are restricted to shallow crustal depths, usually the uppermost 0.5–1 km of the crust (Chapters 7 and 8). Second, tension fractures are driven by external tensile stress, whereas hydrofractures are driven by internal fluid overpressure.

Aims

How faults transport crustal fluids is important in many fields of earth sciences, such as petroleum geology, geothermal research, volcanology, seismology, and hydrogeology. In order to understand the permeability evolution and maintenance of a fault zone, its internal hydromechanical structure and associated local stresses and mechanical properties must be known. The internal structure and, therefore, the local stresses in turn depend to a large degree on how the fault zone develops through time. The principal aims of this chapter are to present the current understanding of:

• How faults initiate and grow.

• The formation and development of the fault core.

• The formation and development of the fault damage zone.

• Local stresses within fault zones and their effects on permeability.

• The evolution of fault slip.

Initiation of faults

General

Faults, like any other tectonic fractures, initiate from existing weaknesses in the rock. Most weaknesses are joints and contacts/interfaces of various types. The basic theory of fracture initiation, including that of faults, is due to Griffith (1924). Earlier, the Griffith theory was discussed in connection with rock failure (Chapter 7), fracture mechanics (Chapter 10), and the maximum depth of tension fractures (Chapter 8). Here we use Griffith theory to explain the initiation of shear fractures from existing joints and other weaknesses in the host rock.

Aims

Fractures form when the stress in the rock becomes so high that the rock breaks. Stress is thus of fundamental importance for understanding rock fractures. Stress has already been defined (Chapter 1). Here the focus is on those aspects of stress analysis that are most useful for understanding rock fractures. The primary aims of this chapter are to:

• Explain some of the concepts from stress analysis, particularly those relevant to rock fractures.

• Clarify the difference between a stress vector (traction vector) and a stress tensor and the relationship between these concepts.

• Define the principal stresses and principal stress axes and planes.

• Explain the stress ellipsoid.

• Discuss stresses on an arbitrary plane and Mohr's circles of stress.

• Define mean stress and deviatoric stress.

• Indicate and explain the use of some special stress states.

• Define stress fields and stress trajectories and explain their use in stress analysis.

Some basic definitions

Stress at a point is a tensor of the second rank, whereas the stress on a plane is a vector, that is, the stress vector or traction (Chapter 1). The magnitude of a stress, given as, say, 10 MPa, is a scalar. Before we discuss the concept of stress in more detail, it is necessary to recall some basic definitions that are assumed to be known. This section, of necessity, must be brief. More details on tensors, vectors, and scalars are provided in some of the books listed at the end of this chapter.

Aims

How hydrofractures transport crustal fluids is important in many fields of earth sciences, including petroleum geology, geothermal research, volcanology, seismology, and hydro-geology. In the previous chapters the focus has been on the transport of ground water, geothermal water, and gas and oil. Here the main examples come from two fields: man-made hydraulic fractures, as are used to increase the permeability of reservoirs (petroleum and geothermal); and volcanology, that is, the transport of magma through vertical (dykes), inclined (sheets), and horizontal (sills) intrusions. It should be stressed, however, that the basic principles of fluid transport in hydrofractures are the same irrespective of the type of fluid. Thus, the results obtained here apply equally well, with suitable modifications, to hydrofracture transport of ground water, oil, and gas. The principal aims of this chapter are to explain and discuss:

• The source of the driving pressure (overpressure) in hydrofractures, including the effects of buoyancy.

• The initiation, propagation, and fluid transport in man-made (artificial) hydraulic fractures.

• Fluid transport in magma-driven fractures, primarily dykes and inclined sheets.

• Fluid transport in fractures driven by hydrothermal fluids, that is, fractures that subsequently become mineral veins.

Aims

This chapter gives an overview of the concepts of displacement and strain. In textbooks on structural geology, strain is normally treated in great detail. The reason is that strain, particularly finite strain, is of fundamental importance for understanding the kinematics of ductile deformation, such as occurs during folding. Here, however, the focus is on those aspects of strain that are useful when analysing stress-strain relations for brittle rocks, which to a first approximation behave as linearly elastic and are subject to small (usually less than 1%) strain. The main aims of this chapter are to:

• Explain the difference between displacement and strain.

• Provide examples of the use of each concept in geology.

• Explain the difference between deformation and strain.

• Discuss general aspects of infinitesimal strain.

• Provide some basic formulas for measuring strain.

• Discuss strain rates and how they are measured.

Basic definitions

Displacement refers to the change in position of a particle. When the displacement is very small, it is referred to as infinitesimal; when large, it is referred to as finite. Strain is also related to the displacement of particles from their original position to a new position. Strain and displacement are thus closely related. There is, however, a significant difference. Strain always involves changes in the internal configuration of the body. During strain, the distances between the particles that constitute the strained body change.

Aims

Rock strengths, such as tensile and shear strength, are one measure of its resistance to brittle failure. Another measure of rock resistance to brittle failure is the energy absorbed by the rock during fracture propagation. The critical energy absorbed is referred to as material toughness. A third measure of resistance to fracture is the stress intensity at the fracture tip that must be reached if the fracture is to propagate. The critical stress intensity is referred to as fracture toughness. Toughness is a basic concept in fracture mechanics that provides a physical framework for understanding many processes associated with rock fractures. The main aims of this chapter are to:

• Explain the concept of material toughness.

• Explain the concept of fracture toughness.

• Show how material and fracture toughness are related.

• Relate the concept of toughness to the initiation of a fracture.

• Introduce the concepts of process zone, fault core, and damage zone.

• Show how to calculate the toughness associated with extension fractures.

• Show how to calculate the toughness associated with faults.

Toughness

The toughness of a rock is a measure of its resistance to fracture. In other words, for a tough rock, large amounts of energy are needed to cause failure; much energy is absorbed by a tough rock as it fractures. This concept and the related concept of fracture toughness derive from materials science (Hull and Clyne, 1996; Chawla, 1998; Kobayashi, 2004) and fracture mechanics (Atkins and Mai, 1985; Broberg, 1999; Anderson, 2005).

Aims

What measurements can we make in the field so as to be able to use the analytical framework developed in earlier chapters? In particular, how do we analyse extension fractures in the field? The main aims of this chapter are to:

• Give field examples of the main types of extension fractures.

• Illustrate how the most important field measurements of extension fractures are made.

• Provide typical field data on tension fractures.

• Provide typical field data on joints.

• Provide typical field data on mineral veins.

• Provide typical field data on dykes.

• Provide data on man-made hydraulic fractures.

• Illustrate the connection between extension fractures and faults.

• Outline the general methods of field measurements of fractures.

Types of extension fractures

Tension fractures are formed by absolute tension, so that they are mostly confined to the shallow parts of the crust (Chapter 8). Hydrofractures are partly or entirely generated by an internal fluid overpressure and can form at any crustal depth if there is high fluid pressure. Tension fractures include many large tension fractures in rift zones, as well as many joints. Hydrofractures are fluid-driven rock fractures that include many joints, mineral veins, and dykes. Man-made fractures injected under fluid overpressure into reservoir rocks to increase their permeabilities, commonly referred to as hydraulic fractures, are also hydrofractures. In many hydrofractures, for example fractures formed by gas, oil, and ground water, the fluid may disappear after the fracture has formed. Other hydrofractures, such as dykes, are driven open by fluids that solidify in the fracture once it is formed.

Aims

Brittle rocks fail through fracture. Sometimes a single fracture develops during failure, sometimes several or many fractures. How and when rocks fail under loading has been studied extensively in laboratory experiments. Much of the theoretical background derives from studies of small rock specimens subject to various types of loading. The main scientific fields dealing with experimental rock deformation are rock mechanics and rock physics. The principal results are very thoroughly treated in many textbooks and monographs, so that only a brief overview will be given here. The main aims of this chapter are to:

• Describe the general behaviour of rock under loading.

• Define dilatancy.

• Discuss the various types of rock strength.

• Define the secant modulus and the tangent modulus.

• Describe the main stages leading to brittle failure.

• Describe strain hardening, strain softening, and the brittle–ductile transition.

• Define the main factors that affect the depth of the brittle–ductile transition.

• Define and describe the main mechanisms of ductile deformation in rocks.

Behaviour of rock under loading

At the Earth's surface, most solid rocks subject to short-term loading behave elastically. When the load on a rock exceeds a certain critical value, the rock fails through the formation of a fracture. Surface rocks thus generally behave as brittle solids. A well-known type of brittle failure of a solid rock at the surface is the fracturing that occurs when the load of a geological hammer is applied, as an impact, to the rock.

Aims

To understand fluid transport in the crust in general, and in rock fractures such as faults and hydrofractures in particular, some of the basic principles of fluid transport in porous and fractured media must be given. These principles include Darcy's law for flow in porous media and the cubic law for flow in fractures. Only the basic definitions and formulas that are needed in this and subsequent chapters are given. There are many excellent books on hydrogeology and fluid flow in porous media where the details can be found. The main aims of this chapter are to:

• Explain Darcy's law for fluid flow in porous media.

• Explain the cubic law for fluid flow in fractures.

• Present the model for fluid flow in a single set of fractures.

• Present models for fluid flow in orthogonal sets of fractures.

• Present some basic general results on fracture-related permeability.

Fluid transport in porous and fractured rocks

Fluid flow in crustal rocks is through pores, through fractures, or, most commonly, through both pores and fractures. How fluids are transported is of great academic and applied interest; understanding all kinds of fluid reservoirs depends on knowing and, preferably, being able to predict their permeabilities and fluid-transport or hydromechanical properties. Fluid flow in the crust is primarily through porous or fractured media, or a mixture of both (see Appendix E.2 for the physical properties of some crustal fluids).

Aims

Depending on the relative displacement across the fracture plane, all tectonic fractures are of two main mechanical types: extension fractures and shear fractures. In an extension fracture, the sense of displacement is perpendicular to, and away from, the fracture plane. In a shear fracture the sense of displacement is parallel with the fracture plane. To be able to distinguish between these two main types theoretically and in the field is of fundamental importance for understanding a variety of fracture-related problems and processes. These two types will be referred to many times in subsequent chapters. This brief chapter focuses on learning to recognise them in the field with reference to the basic theories for their formation discussed in earlier chapters. The main aims of this chapter are to:

• Explain the mechanical difference between extension fractures and shear fractures.

• Discuss the two main types of extension fractures, namely: (a) tension fractures, which are formed by absolute tension and usually close to the Earth's surface, and (b) fluid-driven fractures or hydrofractures, which are generated by fluid pressure and can form at any crustal depth.

• Show that shear fractures comprise all faults as well as some fractures that are normally classified as joints in the field.

• Provide field examples of the main mechanical types of fractures.

Aims

All brittle deformation is related to stress concentration. The deformation, such as the formation of fractures, occurs because the local stresses in certain parts of the crust or rock bodies are raised above the average (nominal) stress in the surrounding parts. Understanding stress raisers and concentrations is of fundamental importance for assessing likely sites for rock-fracture initiation and propagation; in fact, the Griffith theory of fracture (Chapter 7) is partly based on the idea of stress concentrations around small elliptical flaws in brittle materials. The main aims of this chapter are to:

• Define the concepts of a stress concentration and a stress raiser.

• Provide the basic equations for calculating stress concentrations around elliptical and circular holes and cavities.

• Explain stress concentrations in terms of the atomic structure of solids.

• Provide an analogy between stress concentrations and fluid flow around obstacles.

• Use the equations for stresses around elliptical cavities to explain stress concentrations around fluid-filled reservoirs and magma chambers.

• Use the equations for stresses around circular holes to explain stress measurements using hydraulic fractures injected from drill holes.

Basic definitions

Stress concentration occurs when a part of a loaded material has properties that differ from those of the surrounding parts. The easiest way to visualise stress concentration is to consider stresses in a body with a hole or a notch. A hole is a two-dimensional cavity.

Aims

How do we model fractures observed in the field? For example how do we determine the tensile stress that opened the fracture in Fig. 8.2 or the shear stress responsible for the vertical displacement on the normal fault in Fig. 8.1? Similarly, how can we estimate the magma overpressure that generated the dyke in Fig. 8.1? To answer these questions we need mathematical models of rock fractures. Such models are normally referred to as crack models, even though the words ‘fracture’ and ‘crack’ are often used as synonyms. The conceptual and analytical models provided in this chapter are fundamental for the remainder of the book. The main aims of this chapter are to:

• Explain the three main geometric crack models used for rock fractures, namely through-the-thickness cracks, part-through cracks, and interior cracks.

• Clarify further the concepts of strike dimension, dip dimension, and controlling dimension and how these are used for rock fractures.

• Explain the three main modes of displacement of the fracture surfaces or walls, namely mode I, mode II, and mode III, and for which types of rock fractures each mode is most appropriate.

• Explain what is meant by mixed-mode (hybrid) cracks and for which types of rock fractures these are applicable.

• Provide basic formulas and numerical examples on the use of the geometric models and crack-displacement modes to estimate the driving stresses and pressures associated with rock-fracture formation and development.

Aims

In this chapter we discuss the fundamental relationship between stress and strain, Hooke's law. This law describes approximately the stress–strain behaviour of many solid materials before failure, such as many metals, ceramics, and rocks. Elastic behaviour implies that the deformation is recoverable: a body that deforms elastically when loaded reverts to its original shape immediately when the load is removed. Most solid rocks behave as approximately elastic at low temperatures and pressures (that is, at shallow depths), up to strains of about 1%. Such strains are common in the crust before failure, hence the importance of Hooke's law for understanding processes leading to rock fractures. The main aims of this chapter are to:

• Explain the one-dimensional Hooke's law, as well as its extension to three dimensions.

• Discuss the elastic constants, the relations among them, and their physical meaning.

• Show how to estimate the vertical and horizontal stress in the Earth's crust.

• Provide information on the general state of stress in the Earth's crust.

• Discuss and explain the use of some reference states of stress in the crust.

• Explain the concept of elastic strain energy.

One-dimensional Hooke's law

Consider the comparatively isolated part of rock (a part of a basaltic dyke) in Fig. 4.1. This rock segment or ‘bar’ may serve as an illustration of the effects of the one-dimensional Hooke's law.