# Chapters

## Chapter 1: Some more or less basic notions

**Chapter DOI:** 10.2516/ifpen/2014002.c001

#### 1.1 Petroacoustics and Geoacoustics: Definition, etymology and particular branches of Acoustics

#### 1.2 Mechanics of continuous

#### 1.3 Physics in real media - Hierarchical structure of Geological media and continuum mechanics in such media

### Abstract

The first chapter deals with what we call some more or less basic notions that will be used in the following chapters. Some notions described in this chapter are well know and/or straightforward and can be found in any classical textbook on Continuum Mechanics or on Acoustics. Some other notions are unfortunately not commonly appreciated and need to be introduced for studying physics in geological media. The chapter is divided into three sections. First we introduce Petroacoustics, or more commonly Rock acoustics, and Geoacoustics, that is to say acoustics of geological media, as particular branches of Acoustics (section 1.1). Then we give the basics of classical Mechanics in Continuous Media, including the description of stress, strain and elastic wave propagation, together with the main deviations from the ideal homogeneous isotropic linearly elastic behaviour, that is to say heterogeneity, dispersion, attenuation, anisotropy, and nonlinearity possibly with the presence of hysteresis (section 1.2). Last, because natural media are all but continuous media at many scales, we describe in section 1.3 the way to adapt the previous descriptions to the case of discontinuous media with hierarchal structure, such as geological media, with the introduction of fundamental notions such as Representative Elementary Volume and Continuum Representation in such media. These are precisely the less obvious notions that are referred to in the title of this chapter.

## Chapter 2: Laboratory measurements

**Chapter DOI:** 10.2516/ifpen/2014002.c002

#### 2.1 General remarks

#### 2.2 Measurement of rock acoustic properties

#### 2.3 Experiment on wave propagation and analog modelling

### Abstract

In Chapter 2, we describe the most common techniques for performing acoustic experiments on rocks in the laboratory. The chapter is divided into three sections. First we discuss the reliability of petroacoustic measurements, we introduce the main petrophysical parameters (porosity, permeability), and we emphasize various experimental cautions (damage, saturation process.) (section 2.1). Then we introduce the two main types of experiments performed in petroacoustic laboratories, characterized by contrasted aims. The first type experiment, described in section 2.2, aims to measure the acoustic properties of geological materials. In this case it is important that the measured sample is representative of the studied geological formation. Another important aspect is the physical state of the rock sample. Obviously altered and/or damaged samples must be avoided. Finally the pressure and temperature state have to be as close as possible to the in-situ condition. Section 2.3 deals with the second type of experiments in rocks, aiming to better understand physical phenomena involved in elastic wave propagation, or to study wave propagation on scaled-down physical models in the laboratory. In this case, temperature and pressure condition, have less importance, unless these parameters are precisely in the central parameters of the study. The chosen materials, possibly artificial materials (such as sintered glass beads), can be chosen according to the purpose of the physical study.

## Chapter 3: Elastic waves in isotropic, homogeneous rocks

**Chapter DOI:** 10.2516/ifpen/2014002.c003

#### 3.1 General remarks

#### 3.2 Waves and lithology (mineralogy and porosity)

#### 3.3 Waves and permeability

#### 3.4 Waves and saturating fluids (modulus and viscosity)

#### 3.5 Waves and stress

#### 3.6 Waves and temperature (and phase change)

### Abstract

Chapter 3 addresses the dependence of the acoustic parameters (mainly velocity and attenuation) of geomaterials on their lithologic nature (mineralogy, porosity) and on physical parameters (fluid saturation, pressure, and temperature). All these relationships are obviously at the height of applications of petroacoustics to the interpretation of seismic data in a broad sense (*i.e.*, seismological data, applied seismic data, acoustic logs data.). As a matter of fact, it is from the quantitative knowledge of these relationships that we can hope to extract information such as porosity or saturation state of underground formations.

## Chapter 4: Elastic anisotropy

**Chapter DOI:** 10.2516/ifpen/2014002.c004

#### 4.1 A brief history of seismic anisotropy

#### 4.2 Anisotropy and Curie's symmetry principle

#### 4.3 Seismic anisotropy, the classical theory

#### 4.4 The main symptoms of seismic anisotropy

#### 4.5 Simplification of the formalism

#### 4.6 The main rock models incorporating elastic anisotropy

#### 4.7 Anisotropic viscoelasticity

#### 4.8 Seismic anisotropy for earth subsurface exploration and exploitation ... what is it for?

### Abstract

In Chapter 4 we discuss elastic anisotropy under different points of view but, as in the other chapters, always more or less in relation with experimental aspects. The chapter is divided into seven sections. In the first section (4.1), we summarize the history of seismic anisotropy. Section 4.2 introduces the symmetry principles in physical phenomena, due to the great scientist Pierre Curie, and the way they can simplify the description of elastic anisotropy. In the next section 4.3 we introduce the classical theory of static and dynamic elasticity in anisotropic media, and we describe and illustrate the main manifestations of elastic anisotropy in rock (*i.e.* directional dependence of the elastic wave velocities, shear-wave splitting of shear-wave birefringence, and the fact that the seismic rays are generally not normal to the wavefronts). Because rocks generally exhibit moderate to weak anisotropy strength it is possible to use perturbation theories to simplify the exact theoretical derivation as described in the next section (4.4). This is followed by a description of the main causes of elastic anisotropy and the corresponding rock physics models (section 4.5). In addition to elastic anisotropy, experimental studies have unambiguously other robust results, namely porous nature (poroelasticity), frequency dependence (viscoelasticity), or the dependence on stress-strain level (nonlinearity) which lead to use more sophisticated models as pointed out in the next part (section 4.6). The last section (4.7) explains how elastic anisotropy alters the seismic response and necessitates the adaptation of existing seismic processing tools to take into account the anisotropic case. Conversely it also explains how seismic response can be analyzed in order to characterize the studied rocks.

## Chapter 5: Frequency/wavelength dependence (impact of fluids and heterogeneities) *(Chapter under construction)*

### Abstract

The dependence of the mechanical properties of geological media with respect to frequency, or equivalently with wavelength, is illustrated by countless examples at various scales and is discussed in Chapter 5. This chapter also describes and details the main causes of this dependence. The chapter is divided into five sections. We start (section 5.1) by distinguishing the geometry-induced, or extrinsic, frequency/wavelength dependence from the intrinsic one, due to the property of the rock itself. The rest of the chapter is focused on intrinsic frequency/wavelength dependence. Next we describe the main causes of intrinsic frequency/wavelength dependence in rocks, which can be summarized in two words, namely fluids and heterogeneities. In the third section we describe the frequency/wavelength dependence due to the presence of fluid. It is essentially an anelastic mechanism (see Chapter 1 section 1.2.3.5), where the energy dissipation (conversion of wave energy to heat) is due to the viscosity of the saturating fluid. In contrast, the frequency/wavelength dependence due to the presence of heterogeneities described in section 4 is not due to energy dissipation but, rather, to energy redistribution from the first arriving coherent waves to the later chaotic arrivals, or codas, the total wave-field energy being conserved. Finally, instead of specifying the physical mechanisms involved in the frequency/wavelength dependence, an alternative way is to phenomenologically describe the mechanical behaviour of rock as done in the last section, by studying the empirical relation between the applied stress and the resulting strain. We shall see that, among the large class of phenomenological models, the sub-class of linear viscoelastic models can closely mimic the behaviour of a broad class of dissipative processes, resulting from rapid and small-amplitude variations in strain due to waves that propagate in rocks.

## Chapter 6: Poroelasticity applied to petroacoustics *(Chapter under construction)*

### Abstract

Chapter 6 deals with the poroelastic description of rock behaviour. In other words the chapter describes the elasticity of rocks considered as porous media. The chapter is divided into four sections. First we introduce the general field of Poromechanics, that is to say Mechanics in porous media, including the sub-fields of Poroelasticity and Poroacoustics, that is to say, respectively, Elasticity and Acoustics of porous media (section 6.1). Then we give the basics of the classical theory of poroelasticity, including the description of the stresses and the strains in porous media, of the static couplings (i.e., change of fluid pressure or mass due to applied stress, or change of porous frame volume due to fluid pressure or mass change]) and of the dynamic couplings (i.e., viscous and inertial couplings). The section ends with wave propagation (section 6.2), emphasizing the influence of the presence of macroscopic mechanical discontinuities, that is to say interfaces, and of fluid transfer through these interfaces on the observed wavefields. The next section (section 6.3) describes the various sophistications of the initial model imposed by experimental reality, mainly the necessity of integrating viscoelasticity [mainly due to the presence of compliant features (e.g., cracks, micro fractures)] and/or anisotropy into the poroelastic model. This leads to a new classification of wave propagation regimes in fluid-saturated porous media distinguishing four regimes represented in a (crack density)- (interface permeability) diagram [characterizing the fluid exchange through the macroscopic mechanical discontinuities (or interfaces)]. The last section explains how poroelastic signature of rocks can be used to characterize fluid substitution in different context of underground exploitation (section 6.4).

## Chapter 7: Nonlinear elasticity *(Chapter under construction)*

### Abstract

The perfect linear relation between stress and strain is often a convenient simplification in most real media, but does not reflect experimental reality. In fact, nonlinear elasticity is a pervasive characteristic of rocks, mainly due to the presence of compliant porosity (e.g., cracks, microfractures), but not only, and is addressed in Chapter 7. The chapter is divided into six parts. First we introduce the multiple aspects of nonlinear science and briefly introduce the history of nonlinear elasticity (section 7.1). Then we give the basics of nonlinear elasticity. This include the description of stresses in the presence of finite deformations, that is to say Cauchy stress relative to the present configuration and Piola-Kirchhoff stress relative to the reference configuration.

The classical third order nonlinear elasticity (implying expansion of the elastic deformation energy to the third power of the strain components) is detailed in the static case and in the dynamic case, especially wave propagation (section 7.2). Section 3 describes the main experimental manifestations of nonlinear elasticity, namely the stress-dependence of the velocities/moduli, the generation of harmonic frequency not present in the source frequency spectrum, and wave-to-wave interaction (section 7.3). Then we detail the two main fields of nonlinear elasticity in rocks (section 7.4), namely nonlinear acoustics (i.e., the study of wave of finite amplitude) and acoustoelasticity (i.e., the study of perturbative waves in statically pre-stressed media). In the next section we introduce the most used sophistications of the nonlinear elastic model, namely the higher order nonlinear models and nonlinear hysteretic models of Preisach type. Associated to Kelvin's description in eigenstresses and eigenstrains, the last approach demonstrates that there seems to be no limit in the sophistication of the models with media exhibiting simultaneously dispersion/attenuation, anisotropy, and nonlinearity possibly with the presence of hysteresis (section 7.5). In the last section (section 7.6) we illustrate how the multiple ramifications of nonlinear response of rocks may affect various areas of research in Geosciences. These include Rock mechanics, and more generally speaking material science, where the nonlinear response of material may be used for characterization purposes, and Seismology, where the spectral distorsion of seismic waves has to be considered. The characterization of material property change by monitoring the nonlinear response may be valuable (e.g., changes due to fluid saturation, to stress variations or to damage induced by fatigue.).

## Chapter 8: Applications to seismic interpretation *(Chapter under construction)*

### Abstract

Finally, in Chapter 8, we describe some case histories showing practical applications of each of the theories introduced in the previous Chapters. The Chapter is divided into four sections. In the first part (section 8.1) we deal with fracture characterization from the analysis of seismic anisotropy. Section 8.2 illustrates the application of Poroelasticity theory to seismic monitoring of subsurface exploitation with Hydrocarbon Reservoir monitoring and CO_{2} geological storage. This will be followed in the section 8.3 by the exploitation of the scattered seismic wavefields for the characterization of heterogeneity in the subsurface. The last section (8.4) illustrates by field examples how the principle of nonlinear elasticity can be exploited for inverting the stress state in the subsurface.