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In this section, we use the symbol s to represent the frequency, or Fourier transform variable, following the terminology of Bracewell (1965). When x refers to time, the symbol f is often used for frequency instead of s; when x refers to space, the symbol k is often used for spatial frequency instead of s. Do not confuse this with the use of s when defining the Laplace transform (Section 1.3) or S when defining the analytic signal (Section 1.2).
Spheres are often used as idealized representations of grains in unconsolidated and poorly consolidated sands. They provide a means of quantifying geometric relations, such as the porosity and the coordination number, as functions of packing and sorting. Using spheres also allows an analytical treatment of mechanical grain interactions under stress. Recent research also started to reveal how the properties of the granular medium change as particle shapes differ from perfect spheres.
Nur et al. (1991, 1995) and other workers have championed the simple, if not obvious, idea that the P and S velocities of rocks should trend between the velocities of the mineral grains in the limit of low porosity and the values for a mineral–pore-fluid suspension in the limit of high porosity.
If we wish to predict the effective dielectric permittivity ε of a mixture of phases theoretically, we generally need to specify: (1) the volume fractions of the various phases, (2) the dielectric permittivity of the various phases, and (3) the geometric details of how the phases are arranged relative to each other.
Biot (1956) derived theoretical formulas for predicting the frequency-dependent seismic velocities of saturated rocks in terms of the dry-rock properties. His formulation incorporates some, but not all, of the mechanisms of viscous and inertial interaction between the pore fluid and the mineral matrix of the rock.
Responding to the latest developments in rock physics research, this popular reference book has been thoroughly updated while retaining its comprehensive coverage of the fundamental theory, concepts, and laboratory results. It brings together the vast literature from the field to address the relationships between geophysical observations and the underlying physical properties of Earth materials - including water, hydrocarbons, gases, minerals, rocks, ice, magma and methane hydrates. This third edition includes expanded coverage of topics such as effective medium models, viscoelasticity, attenuation, anisotropy, electrical-elastic cross relations, and highlights applications in unconventional reservoirs. Appendices have been enhanced with new materials and properties, while worked examples (supplemented by online datasets and MATLAB® codes) enable readers to implement the workflows and models in practice. This significantly revised edition will continue to be the go-to reference for students and researchers interested in rock physics, near-surface geophysics, seismology, and professionals in the oil and gas industries.
If we wish to predict theoretically the effective elastic moduli of a mixture of grains and pores, we generally need to specify: (1) the volume fractions of the various phases, (2) the elastic moduli of the various phases, and (3) the geometric details of how the phases are arranged relative to each other. If we specify only the volume fractions and the constituent moduli, the best we can do is predict the upper and lower bounds (shown schematically in Figure 4.1.1).