Synopsis

Chapter 5 summarizes the basic propagation processes that are encountered when studying radiation or edge diffraction. Three problems of progressive difficulty are studied. We begin by calculating the transient, antiplane radiation excited by a line source at the surface of a half-space. The Cagniard–deHoop method is used to invert the integral transforms. We then return to considering how plane waves and a knowledge of their interactions can be used to construct more general wavefields. We calculate the time harmonic, inplane radiation, from a two-dimensional center of compression buried in a half-space. Plane-wave spectral techniques are used and the resulting integrals are approximated by the method of steepest descents. This method is discussed in detail. Lastly, we extend our knowledge of plane-wave interactions by calculating the diffraction of a time harmonic, plane, antiplane shear wave by a semi-infinite slit or crack. This problem is solved exactly by using the Wiener–Hopf method and approximately by using matched asymptotic expansions. An Appendix describing the reduction of the diffraction integral to Fresnel integrals is included.

Antiplane Radiation into a Half-Space

We consider an elastic half-space. The x1 coordinate stretches along its surface and the positive x2 coordinate extends into the interior. At the origin a line load is applied to an otherwise traction-free surface. The line load is a tangentially acting force very localized in x1 and directed from −∞ to ∞ in the x3 direction.

The physical properties of the Earth's deep interior are unavoidably obscure. Direct human observation is limited to mines in the top 3 km or less, while deep drilling has provided core samples and instrument access to maximum depths of around only 12 km (Kozlovsky, 1987). Deeper physical evidence, down to several hundred kilometres, is obtained from xenoliths and other volcanic products ejected onto the surface from the mid-crust to upper mantle, but this still represents a very small portion of the 6370 km radius of the Earth. The bulk of our knowledge of the Earth's interior composition and physical properties has been deduced by indirect means. Major internal boundaries and structural features (Figure 2.1) have been delineated using seismic methods and supported by density models deduced from gravity readings (e.g. Ringwood, 1969, 1979; Brown and Mussett, 1993).

One conclusion from velocity modelling is that the interior of the Earth is significantly hotter than the surface. The internal heat is derived both from primordial sources related to the formation of the globe and from secondary processes generating heat internally.

Geothermal data provide critical constraints for models of global accretion, core–mantle segregation and crustal evolution. Temperature history is a controlling factor in the maturation of organic compounds into fossil fuels. Geothermal energy is a relatively clean, renewable means of heat and electricity production. Yet, in spite of the number and variety of applications of geothermal research, the means of measuring, assessing and interpreting basic geothermal data are poorly understood by all but a small number of specialists, even among those whose job it is to manipulate such data.

This book aims to bring together many aspects of basic geothermal research in an accessible way. Although some concepts require mathematical treatment, these will be kept to a minimum and, where possible, practical examples will be used as illustrations.

Synopsis

Chapter 1 provides the background, both the model equations and some of the mathematical transformations, needed to understand linear elastic waves. Only the basic equations are summarized, without derivation. Both Fourier and Laplace transforms and their inverses are introduced and important sign conventions settled. The Poisson summation formula is also introduced and used to distinguish between a propagating wave and a vibration of a bounded body.

A linear wave carries information at a particular velocity, the group velocity, which is characteristic of the propagation structure or environment. It is this transmitting of information that gives linear waves their special importance. In order to introduce this aspect of wave propagation, propagation in one-dimensional periodic structures is discussed. Such structures are dispersive and therefore transmit information at a speed different from the wavespeed of their individual components.

Model Equations

The equations of linear elasticity consist of (1) the conservation of linear and angular momentum, and (2) a constitutive relation relating force and deformation. In the linear approximation density ρ is constant. The conservation of mechanical energy follows from (1) and (2). The most important feature of the model is that the force exerted across a surface, oriented by the unit normal nj, by one part of a material on the other is given by the traction ti = τjinj, where τji is the stress tensor.

Synopsis

In Chapter 4 we discuss the formulation of integral representations of solutions to rather general problems in elastic-wave propagation. Two constructions are used: the reciprocity identity and the Green's tensor for a full space. For these representations for an infinite domain to be derived, the principle of limiting absorption is introduced. This is needed for time-harmonic problems because the disturbance, in a sense, has been going on forever, resulting in no initial wavefront being present. Moreover, we establish a uniqueness result, indicating as we do so both the role of the principle of limiting absorption and that of specifying an edge condition. The chapter closes with an example that uses these ideas to develop an integral representation for the scattering of an acoustic wave by an elastic inclusion.

Introduction

In Chapter 2 we moved away from discussing plane waves to an introduction of plane-wave spectral representations in Section 2.3. This allowed us to discuss more general wavefields and to understand their propagation characteristics in terms of those of plane waves. We continue with this general theme, but construct, in this chapter, both far more general representations and ones in physical space rather than in wavenumber space. Though we make limited use of it in the chapters that follow, this material is very important because it is the basis for formulating elastic-wave problems in a form suitable to be analyzed numerically.

Heat flow data have been used extensively by geoscientists to provide an indication of temperature variations within the Earth. However, there are relatively few experts available with the necessary experience to provide a critique of the central constraints or to obtain new data for second-order models. Terminology relating to thermal conductivity, diffusivity, vitrinite reflectance, xenolith geochemistry, and so on, is widely recognised but poorly understood, particularly by industry professionals.

As an example, the present state of geothermal modelling and interpretation in the petroleum industry is a peculiar one. On the one hand, there are numerous highly sophisticated software applications on the market for solving the complex equations governing the thermal evolution of sedimentary basins and the maturation of organic material.

By this point, the task of recreating the thermal history of a prospective petroleum exploration province probably seems fraught with uncertainty and complicated mathematics. Not only do we require estimates of such tenuous parameters as the thickness, temperature and thermal properties of the lithosphere, crust and sediments, but we must also interpret often-ambiguous geological evidence for stretching factor, extension rate, age, subsidence and magmatic underplating. Even then, all of these parameters intelligently and carefully chosen only provide a history of heat flow to be superimposed over a background value, which is, itself, often a matter of some conjecture.

With good-quality data, however, most of these complications can be largely overcome. The discussions in Chapter 7 focussed on forward modelling of heat flow history – estimating what the heat flow history of a basin should have been based on its general tectonic age and style. The alternative is to reverse model the thermal history – use physical data from the basin to deduce what the heat flow history must have been. This chapter details the steps for such a procedure.

We have looked at how heat generation and thermal gradient are measured or otherwise approximated within rocks. The last remaining parameter required to define steady-state heat flow is thermal conductivity. Simply put, thermal conductivity, λ, is a measure of how easily heat is transmitted through a material. It is a tensor operator that relates the heat flow vector to the thermal gradient vector within a body, and it is an inherent physical property. Many rocks are anisotropic, with conductivity dependent upon the direction of heat flow, but geothermal problems usually involve only the vertical component.

Thermal conductivity must be estimated over the same section that thermal gradient and heat generation are known. Where temperature is denned at discrete depths (e.g. Horner-corrected bottom-hole temperatures), an average thermal conductivity is required between each of those depths. Where a continuous temperature log is available, we require a continuous conductivity log. Heat flow remains undefined in any section where there is a gap in the thermal conductivity record.

The usual objective of thermal modelling is to identify the time and depth of hydrocarbon generation. Hydrocarbons are by-products of the metamorphism of organic material (kerogen) within sediments – a gradual process involving the expulsion of volatiles, gases, liquids and oils during the chemical alteration of buried organic matter. The thermal maturity of a rock is a measure of the degree to which this metamorphism has progressed.

Before we can make predictions about the timing of petroleum generation, we need to know two things. We need to understand the precise response of organic detritus to changes in thermal conditions, and we also need to know the thermal history of potential petroleum source beds. This chapter addresses those two requirements.

The Generation of Hydrocarbons from Organic Matter

The chemistry of converting organic matter into hydrocarbons is a whole separate field of expertise and is far too complex to investigate in great detail here. But a few words are necessary to illustrate the importance of temperature on the organic metamorphic process.

Wave propagation and scattering are among the most fundamental processes that we use to comprehend the world around us. While these processes are often very complex, one way to begin to understand them is to study linear wave propagation. This is a book describing such propagation.

I use the equations of linear elasticity to form a context for my description of wave propagation. However, the reader's knowledge of elasticity need not be very great, and experience with a related field theory, such as fluid mechanics or electromagnetic theory, is sufficient to understand what is written here. In many places I treat only the antiplane shear problem because I do not believe that the extra work needed to do the analogous inplane problem adds anything of significance to understanding the underlying wave processes. Nevertheless, where an inplane elastic problem introduces a unique feature, such as the presence of a nondispersive surface wave, that problem is treated.

This is also a book describing the parts of applied mathematics that describe the propagation and scattering of linear elastic waves. It assumes that the reader has a good background in calculus, differential equations, and complex analysis. By this I mean that the reader should have studied most of the topics in Courant and John, Introduction to Calculus and Analysis, Vols. 1 and 2 (1989) and in Boyce and DiPrima, Elementary Differential Equations and Boundary Value Problems (1992).