Preface 1. Basic tools 2. Elasticity and Hooke's law 3. Seismic wave propagation 4. Effective media 5. Granular media 6. Fluid effects on wave propagation 7. Empirical relations 8. Flow and diffusion 9. Electrical properties Appendices.

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The Rock Physics Handbook: Tools for Seismic Analysis of Porous Media

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.

The Rock Physics Handbook

Acoustic metamaterials can manipulate and control sound waves in ways that are not possible in conventional materials. Metamaterials with zero, or even negative, refractive index for sound offer new possibilities for acoustic imaging and for the control of sound at subwavelength scales. The combination of transformation acoustics theory and highly anisotropic acoustic metamaterials enables precise control over the deformation of sound fields, which can be used, for example, to hide or cloak objects from incident acoustic energy. Active acoustic metamaterials use external control to create effective material properties that are not possible with passive structures and have led to the development of dynamically reconfigurable, loss-compensating and parity–time-symmetric materials for sound manipulation. Challenges remain, including the development of efficient techniques for fabricating large-scale metamaterial structures and converting laboratory experiments into useful devices. In this Review, we outline the designs and properties of materials with unusual acoustic parameters (for example, negative refractive index), discuss examples of extreme manipulation of sound and, finally, provide an overview of future directions in the field. Acoustic metamaterials can be used manipulate sound waves with a high degree of control. Their applications include acoustic imaging and cloaking. This Review outlines the designs and properties of these materials, discussing transformation acoustics theory, anisotropic materials and active acoustic metamaterials.

Controlling sound with acoustic metamaterials

This paper studies a macroeconomic model in which financial experts borrow from less productive agents. We pursue four sets of results: (i) The economy is prone to instability and occasionally enters volatile episodes. As volatility spikes agents precautionary motive increases depressing prices even further. Log-linear approximations fail to capture these non-linear effects that can cause economies to be significantly depressed for long periods of time. (ii) Endogenous risk during volatile episodes increases asset price correlations. (iii) Financial experts impose a negative externality on each other and on the labor sector by not maintaining adequate capital cushion, and funding structure. (iv) While risk sharing within the financial sector (through securitization and derivatives contracts) reduces many inefficiencies, it can also amlify systemic risks.

/pdf/a-macroeconomic-model-with-a-financial-sector-1s2akgr5uv.pdf

A macroeconomic model with a financial sector

Research into ultrasonic NDE techniques for the inspection of multilayered structures relies strongly on the use of modeling tools which calculate dispersion curves and reflection and transmission spectra. These predictions are essential to enable the best inspection strategies to be identified and their sensitivities to be evaluated. General purpose multilayer modeling tools may be developed from a number of matrix formulations which have evolved in the latter half of this century and there is now a formidable number of publications on the subject. This paper presents a review of the main developments of the matrix techniques, and their use in response and modal models, with emphasis on ultrasonics applications. >

Matrix techniques for modeling ultrasonic waves in multilayered media

A model for an imperfectly bonded interface between two elastic media is proposed. Displacement across this surface is not required to be continuous. The displacement discontinuity, or slip, is taken to be linearly related to the stress traction which is continuous across the interface. For isotropic interface behavior, there are two complex frequency dependent interface compliances, ηN and ηT, where the component of the slip normal to the interface is given by ηN times the normal stress and the component tangential to the interface is given by ηT times the shear stress and is in the same direction. Reflection and transmission coefficients for harmonic plane waves incident at arbitrary angles upon a plane linear slip interface are computed in terms of the interface compliances. These coefficients are frequency dependent even when the compliances are real and frequency independent. Examples of the effects of buried slip interfaces on reflection coefficient spectra and on Love‐wave dispersion relations are ...

https://www.math.purdue.edu/~santos/research/fracture_references/jasa_shoenberg_80.pdf

Elastic wave behavior across linear slip interfaces

A simple method for including the effects of geologically realistic fractures on the seismic propagation through fractured rocks can be obtained by writing the effective compliance tensor of the fractured rock as the sum of the compliance tensor of the unfractured background rock and the compliance tensors for each set of parallel fractures or aligned fractures. The compliance tensor of each fracture set is derivable from a second rank fracture compliance tensor. For a rotationally symmetric set of fractures, the fracture compliance tensor depends on only two fracture compliances, one controlling fracture compliance normal, the other, tangential, to the plane of the fractures. The stiffness tensor, which is more useful in the consideration of elastic wave propagation through rocks, can then be obtained by inversion. The components of the excess fracture compliance tensor represent the maximum amount of information that can be obtained from seismic data. If the background rock is isotropic and the normal a...

Seismic anisotropy of fractured rock

A model of parallel slip interfaces simulates the behaviour of a fracture system composed of large, closely spaced, aligned joints. The model admits any fracture system anisotropy: triclinic (the most general), monoclinic, orthorhombic or transversely isotropic, and this is specified by the form of the 3 × 3 fracture system compliance matrix. The fracture system may be embedded in an anisotropic elastic background with no restrictions on the type of anisotropy. To compute the long wavelength equivalent moduli of the fractured medium requires at most the inversion of two 3 × 3 matrices. When the fractures are assumed on average to have rotational symmetry (transversely isotropic fracture system behaviour) and the background is assumed isotropic, the resulting equivalent medium is transversely isotropic and the effect of the additional compliance of the fracture system may be specified by two parameters (in addition to the two isotropic parameters of the isotropic background). Dilute systems of flat aligned microcracks in an isotropic background yield an equivalent medium of the same form as that of the isotropic medium with large joints, i.e. there are two additional parameters due to the presence of the microcracks which play roles in the stress-strain relations of the equivalent medium identical to those played by the parameters due to the presence of large joints. Thus, knowledge of the total of four parameters describing the anisotropy of such a fractured medium tells nothing of the size or concentration of the aligned fractures but does contain information as to the overall excess compliance due to the fracture system and its orientation. As the aligned microcracks, which were assumed to be ellipsoidal, with very small aspect ratio are allowed to become non-fiat, i.e. have a growing aspect ratio, the moduli of the equivalent medium begin to diverge from the standard form of the moduli for flat cracks. The divergence is faster for higher crack densities but only becomes significant for microcracks of aspect ratios approaching 0.3.

https://www.math.purdue.edu/~santos/research/fracture_references/schoenberg_Douma_GP_1988.pdf

Elastic wave propagation in media with parallel fractures and aligned cracks

Matrix algebra and group theory combine to offer a formalism for the simple calculation of the elastic, anisotropic, homogeneous medium which is equivalent, in the long-wavelength limit, to a heterogeneous distribution of fine layers, each layer itself an elastic anisotropic medium. The properties of each anisotropic constituent in a set of fine layers map to an element of a commutative group. A reverse mapping returns the material properties of the constituent. Adding group elements gives the group element for the homogeneous medium equivalent to a heterogeneous set of layers. Addition of an inverse element--that is subtraction--provides the means to remove a set of layers from an anisotropic medium; then, if the remaining layer is a stable anisotropic medium, a valid decomposition of the original medium into anisotropic constituents is obtained. Within the group structure, eight subgroups corresponding to eight types of elastic symmetry systems may be identified, immediately yielding the symmetry of the equivalent medium, given the symmetry of its constituents. Sets of parallel fractures or aligned microcracks are also represented as group elements, allowing fractures and anisotropic rocks to be manipulated in a consistent and uniform manner. These group elements depend on at most six fracture parameters and are independent of the properties of the material in which the fractures are embedded. Multiple sets of fractures are easily taken into account by rotating (back in model space) a rock to a coordinate system appropriate to each fracture system, and then adding the appropriate fracture system group element.

A calculus for finely layered anisotropic media

Vertical fractures and horizontal fine layering combine to form a long-wavelength equivalent orthorhombic medium. Such media constitute a subset of the set of all orthorhombic media. Orthorhombic elastic symmetry is the lowest symmetry for which the slowness surface (the solution of the Christoffel equation) is bicubic rather than sextic. Various properties of orthorhombic media, such as the number and location of conical points and longitudinal directions, may be derived from the slowness surface or, because of its bicubic character, the squared slowness surface, which is a cubic surface. From the occurrence and angular orientation of some of these distinctive features, conclusions can be drawn with respect to the properties of the medium and to the parameters of the assumed underlying causes of the anisotropy. The estimation of these more subtle properties gains greater importance with the proliferation of multiazimuthal seismic surveys and the ability to drill along ever-more complicated 3-D well trajectories.

Michael Schoenberg

Papers

Elastic wave behavior across linear slip interfaces

Seismic anisotropy of fractured rock

Elastic wave propagation in media with parallel fractures and aligned cracks

A calculus for finely layered anisotropic media

Orthorhombic media: Modeling elastic wave behavior in a vertically fractured earth