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Acoustic properties of rocks relate alternating stresses of varying frequency and elastic strains. In solids there are longitudinal and transversal waves, whose propagation is described by the wave equation. Longitudinal velocity Vp correlates with density and mean atomic weight of rocks. For rocks with a similar mean atomic weight there is a linear dependence of acoustic impedance Z vs. Vp. As a function of porosity, Vp may be estimated from modified Hashin-Shtrikman bounds. For sands and cemented sandstones the models of Dvorkin and Nur are applicable. Propagation velocities of elastic waves in rocks decrease with increasing temperature and increase with increasing pressure. To describe viscoelastic behavior of rocks, the concept of complex elastic moduli is used. Inner friction in rocks depends on temperature, pressure, porosity and pore saturation. Absorption coefficient and quality factor of rocks are frequency dependent. Rocks possess elastic intrinsic and extrinsic anisotropies. Anisotropy of elastic waves in minerals may be represented using pole diagrams. Focus Box 7.1: Models of sandstones after Dvorkin & Nur. Focus Box 7.2: Christoffel matrix and elastic wave velocities.
Darcy’s law connects the gradient of pressure and flow velocity in rocks via their permeability κ. Depending on flow velocity, pressure and matrix properties, there are law corrections after Forchheimer, Brinkman and Klinkenberg. Rock permeability depends on ambient pressure, temperature and deviatoric stresses. In sedimentary rocks the permeability is affected by quartz content, gravel fractions and grain sorting. There are two different permeability models for rocks: the capillary model of pores and the fracture model. For granular rocks the Kozeny–Carman equation is applicable, in which hydraulic radius and degree of tortuosity are involved. The effect of pore pressure can be described using the effective pressure transfer coefficient. Permeability and relative pore saturation are connected via the van Genuchten equation. Focus Box 5.1: Darcy flow in ducts of various geometry.
From stress-strain curves the tangential, average and secant elastic moduli can be estimated. Elastic moduli of solids have an atomistic background. Strain tensor is defined in matrix form, the elements of which represent relative deformations in respect of coordinate system axis and planes. The Poisson’s ratio ν in anisotropic rocks varies depending on the symmetry of spacing. The Poisson ratio depends on porosity, geometry of porous space, and their saturation. Hooke’s law establishes the linear relationship between the elements of stress and strain matrices. Taylor’s and Sack’s “homogenization” models are used to calculate effective elastic moduli. The averaging procedure after Voigt, Reuss, upper-lower bounds of Hashin–Shtrikman, direct and self-consistent methods and statistical continuum approach are used for calculations of elastic constants. Elastic moduli of rocks depend on pressure, temperature and porosity. Plasticity and viscous behavior may effectively be described by a combination of standard bodies: elastic springs, viscous dashpots, Saint-Venant friction and rupture elements. Their combinations connected in parallel and sequence may describe ductility and progressive failure. Friction in rocks depends on strain rate and the state of sliding contact after the Dietrich–Ruina law. Focus Box 4.1: Poisson’s ratio and crystal anisotropy.
Scales of rock heterogeneity are nano≪micro ≪ meso ≪ macro. In order to estimate effective physical parameters by using an averaging procedure over a certain representative reference volume, estimations of effective physical parameters may be done by taking the Reuss or the Voigt mean values or using the upper and lower Hashin–Shtrikman bounds. The fracture number per unit length of a rock is correlated with the rock quality or RQD index. Autorun analysis of rock fractures in scanlines yields a consistent criterion of rock quality. The stress tensor is defined in a symmetric matrix. In diagonal form, the elements of the stress matrix are principal normal stresses. The Mohr’s circle in 2D and 3D is used to define normal and tangential stress in an arbitrarily oriented plane. The strength of rocks depends on modes of deformation according to the failure criteria after Mohr-Coulomb, Griffith, and Hoek and Brown. The ISRM standard and four point bending tests of rock strength are designed to determine yield strength. Fracturing and failure modes in rocks are considered according to tensile fracture models after Griffith, Dugdale, Barenblatt and critical tip opening displacement. Focus Box 3.1: Percolation models in rocks.
Electrical conductivity and resistance obey Ohm’s law. Specific resistance may be measured in two- or four-electrode schemes. Mechanisms of electrical conductivity in rocks are ionic, electronic, anionic and protonic. Some mantle minerals, i.e. olivine, possess polaron conductivity. Conduction bands and density of states are considered for some minerals. Effective conductivity in heterogeneous rocks can be estimated from Wiener or Hashin–Shtrikman bounds, effective medium approximation (EMA) and resistor network models. The electrical conductivity of mineral aggregates can be effectively described by brick and percolation models. Diluted electrolytes and Kohlrausch’s law of independent movement of ions are considered in fluid-bearing rocks, whose electric conductivity obeys Archie’s law. Formation factor and cementation exponent are analyzed for sedimentary rocks. The relationship between rock conductivity and pore saturation is described by the Waxman–Smith model. Focus Box 8.1: Calculations of density of states (Fermi gas model). Focus Box 8.2: Reciprocal lattice and band gaps. Focus Box 8.3: Olivine structure.
Poroelastic problems are considered for drained and undrained rock conditions. The effective pressure coefficient a for bulk modulus is introduced to describe the effective pressure. The Skempton’s coefficient characterizes the relationship between pore pressure change and change in ambient pressure. The Gassman equation relates the bulk modulus of rock matrix, fluid and undrained rock with porosity and drained bulk modulus. Pore pressure follows the diffusion type equation: fluid transport and diffusion. Stress in rocks at depth may be estimated from hydraulic fracturing conditions. To calculate effective bulk and shear moduli of porous rocks, the differential Mori-Tanaka scheme is used. The effective bulk and shear moduli of a medium containing a family of identical-shaped pores can be expressed via the pore compressibility P and shear Q compliances. Focus Box 6.1: Elliptical coordinates and elliptical-shaped pores.
Displacement current and bounded electrical charges are responsible for rock polarization. Dielectric constant: Debye, Cole–Cole or Havriliak–Negami models. Three types of bulk polarization: polarization due to dipoles, ionic polarization, and electronic polarization. Interface polarization on grain boundaries is explained by the Maxwell–Wagner effect. At grain boundaries in contact with electrolyte fluid, electrical double layer is described by the Gouy–Chapman–Stern model. Reaction mechanisms between SiO2 and H2O may take place. Increased electrolyte concentration results in increased surface energy. Electrical flux density and dielectric losses. Dielectric properties of rocks. Energy transport, transmission and reflection coefficients of minerals for electromagnetic waves in rocks. Induced polarization of rocks. Mixing models for dielectric constant: effective medium approach, Wiener and Hashin–Shtrikman bounds, Lorenz equation and percolation model. Factors causing induced polarization in rocks and the Pelton model. Focus Box 9.1: Maxwell’s equations. Focus Box 9.2: Lorenz field and Clausius–Mossotti equation. Focus Box 9.3: Redox reactions, Warburg impedance, Nyquist plots.
Dia, para- and ferromagnetism of rocks and minerals correspond to the wide range of magnetic susceptibility. Atomistic models of dia- and paramagnetism are considered. The Langevin function describes magnetic saturation of paramagnetic particles, whose magnetic susceptibility depends on temperature according to the Curie–Weiß law. Ferromagnetism, antiferromagnetism, ferrimagnetism and canted antiferromagnetism are considered. Ferromagnetic minerals are characterized by magnetic domains whose boundaries experience Barkhausen jumps during magnetization-demagnetization. Magnetic domains are separated by Bloch walls. Koenigsberg’s ratio, i.e. the ratio of induced and remanent magnetizations, depends on the shape demagnetization factor a The concept of locking temperature based on the magnetization relaxation time is used to reconstruct paleomagnetic fields, i.e. in the case of magnetic stripes of mid-oceanic ridge basalts. Principles of chemical, pressure and detrital-remanent magnetization. Focus Box 10.1: Magnetic field of a small dipole. Focus Box 10.2: Brillouin function. Focus Box 10.3: Electron shells, orbitals and orbital hybridization. Focus Box 10.4: Extended Weiss model.
There are geogenic and cosmogenic contributions to the radioactivity of rocks. Radioactivity arises from the relationship between the atomic mass number, the number of protons and neutrons, and the atomic ordinal number in a radioactive element. Radioactive nuclei decay according to the exponential law. There are three natural decay series: uranium-radium, uranium-actinium, and thorium. The 87Sr/86Sr ratio in rocks and seawater is used for paleo-tectonic reconstructions. The measurement of gamma spectra is an important component of radiometry. Measurement of natural radioactivity using a gamma-spectrometer is considered. Fossil tracks of α-particles in rocks and minerals may be used to measure the time since the rock sample formed. Muon energetic spectra are used to estimate underground cavities. Radioactive emanations are connected with the gas radon, and their efficiency depends on rock porosity, saturation and mineral grain size. There are several methods for dating rocks using radioactive and stable isotope ratios: 87Sr/86Sr, ∝87Rb/86Sr and K-Ar. The Oklo reactor is a unique natural nuclear reactor. Focus Box 12.1: Binding energy Eb. Focus Box 12.2: Excitation and loss energy.
Two definitions of rock density are bulk density and solid matrix or skeleton density. Density depends on porosity, which is classified into open and closed porosity. Types of pores are defined by their interconnectivity, connectivity with surface, and geometry. Due to pore volume decrease with depth, sedimentary rocks possess three types of compressibility: bulk, matrix and pore volume compressibility. Density of rocks varies by density of constituting minerals and total porosity, and may be calculated as an arithmetic mean. Numerical modelling of rock densities may be done using Monte-Carlo simulations or on the basis of the software package Perple_X ’07. The capillary effect of fluid phase in pores depends on the wetting angle and specific surface energy. Drainage and imbibition are two processes that displace and saturate fluid in pores. Mercury porosimetry is a method to estimate pore volume by the intrusion of a nonwetting fluid. Gas-porosimetry relies on the known absorption of gas molecules on the pore surface, which is governed by the Gibbs equation. Kinetics of absorption/desorption processes are described by the Brunauer–Emmett–Teller (BET) equation. Focus Box 2.1: Models of porosity. Focus Box 2.2: Grain and atom arrangements in 3D. Focus Box 2.3: BET isotherm.
There are two main sources of silicate rocks in the solar system: chondrules and calcium-aluminum rich inclusions. After the stage of collisional sticking and coagulation of dust grains into rather large planetesimal bodies, the runaway and subsequent oligarchic growth resulted in the formation of four terrestrial planets. Sinking of metallic iron alloys and rising of light silicate rocks led to the shell structure of the Earth. The global circulation of material within the Earth’s mantle in the form of convection and plate tectonics is the principal driving mechanism of the global rock cycle. Geophysical methods are used to study the fine structure of the Earth’s shells, exploiting knowledge of the physical properties of rocks. Rocks are composed of mineral grains, so their classification is based on texture, structure, formation mechanisms, and fine and micro-structures of pore space and grain boundaries. Grain size or granular analysis may be performed, aiming to differentiate sedimentary rocks. This analysis uses median and sorting, and other statistical moments of grain size distributions. Focus Box 1.1: Basics of statistics: cumulative and probability density distribution functions, normal distribution.
Conduction of thermal energy according to Fourier’s law is the principal mechanism of heat transport in rocks, which is due to movement of electrons (electron conduction) and by lattice atoms (phonon or lattice conduction). Heat capacity of minerals at low temperature is mostly due to lattice contributions. At high temperatures, electron heat capacity and thermal conductivity are significant. Pressure dependence of thermal conductivity is described by the Bridgman equation. Pressure derivative is scaled to the Bridgman parameter. For thermal conductivity of cubic crystals above Debye temperature, Slack’s formula is used. The Wiedemann–Franz law relates thermal conductivity (?) and electrical conductivity. Increased concentration of vacancies reduces thermal conductivity, but it increases with tilt angle of grain boundaries. To measure thermal conductivity, Forbes, Ångström, Kohlrausch, and flash diffusivity methods are used. Phase transition and melting/crystallization affect heat capacity and thermal conductivity. Geothermal energy is connected with the properties of fluid-saturated rocks. Focus Box 11.1: Phonons and Debye temperature. Focus Box 11.2: Grüneisen parameter.
Rock physics encompasses practically all aspects of solid and fluid state physics. This book provides a unified presentation of the underlying physical principles of rock physics, covering elements of mineral physics, petrology and rock mechanics. After a short introduction on rocks and minerals, the subsequent chapters cover rock density, porosity, stress and strain relationships, permeability, poroelasticity, acoustics, conductivity, polarizability, magnetism, thermal properties and natural radioactivity. Each chapter includes problem sets and focus boxes with in-depth explanations of the physical and mathematical aspects of underlying processes. The book is also supplemented by online MATLAB exercises to help students apply their knowledge to numerically solve rock physics problems. Covering laboratory and field-based measurement methods, as well as theoretical models, this textbook is ideal for upper-level undergraduate and graduate courses in rock physics. It will also make a useful reference for researchers and professional scientists working in geoscience and petroleum engineering.
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