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

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Methods of Numerical Integration

Amorphous materials as diverse as foams, emulsions, colloidal suspensions and granular media can jam into a rigid, disordered state where they withstand finite shear stresses before yielding. Here we review the current understanding of the transition to jamming and the nature of the jammed state for disordered packings of particles that act through repulsive contact interactions and are at zero temperature and zero shear stress. We first discuss the breakdown of affine assumptions that underlies the rich mechanics near jamming. We then extensively discuss jamming of frictionless soft spheres. At the jamming point, these systems are marginally stable (isostatic) in the sense of constraint counting, and many geometric and mechanical properties scale with distance to this jamming point. Finally, we discuss current explorations of jamming of frictional and non-spherical (ellipsoidal) particles. Both friction and asphericity tune the contact number at jamming away from the isostatic limit, but in opposite directions. This allows one to disentangle the distance to jamming and the distance to isostaticity. The picture that emerges is that most quantities are governed by the contact number and scale with the distance to isostaticity, while the contact number itself scales with the distance to jamming.

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Jamming of soft particles: geometry, mechanics, scaling and isostaticity.

Foreword 1. Introduction 2. Interactions at the grain level 3. The granular solid: statics and elasticity 4. The granular solid: plasticity 5. Granular gases 6. The granular liquid 7. Immersed granular media 8. Erosion and sediment transport 9. Geomorphology References Index.

Granular Media: Between Fluid and Solid

We consider a random aggregate of identical frictionless elastic spheres that has first been subjected to an isotropic compression and then sheared. We assume that the average strain provides a good description of how stress is built up in the initial isotropic compression. However, when calculating the increment in the displacement between a typical pair of contaction particles due to the shearing, we employ force equilibrium for the particles of the pair, assuming that the average strain provides a good approximation for their interactions with their neighbors. The incorporation of these additional degrees of freedom in the displacement of a typical pair relaxes the system, leading to a decrease in the effective moduli of the aggregate. The introduction of simple models for the statistics of the ordinary and conditional averages contributes an additional decrease in moduli. The resulting value of the shear modulus is in far better agreement with that measured in numerical simulations.

https://arxiv.org/pdf/cond-mat/0502647.pdf

Fluctuations and the effective moduli of an isotropic, random aggregate of identical, frictionless spheres

Physical experiments on wave propagation on granular material control two macroscopic parameters that are assumed to characterize the response of a given assembly: the isotropic pressure p0 and the solid volume fraction . Here, by means of numerical simulation, we investigate the effect of the coordination number (the average number of contacts per particle) and the fluctuation of the number of contacts per particle Z'. We adopt a numerical protocol to create several initial packings characterized by the same volume fraction and, for a given isotropic pressure, we are able to obtain different coordination numbers. That is, pressure and coordination number, for a given volume fraction, are independent. The result is that packings with fixed volume fraction and isotropic pressure exhibit a different elastic response. We attribute this behavior only to the coordination number as we find, surprisingly, that is linked to Z'.

http://iopscience.iop.org/article/10.1209/0295-5075/81/34006/pdf

Characterizing the shear and bulk moduli of an idealized granular material

In this paper, we propose a theoretical model to study the elastic response of a granular material idealized as a random aggregate of identical, elastic, frictional spheres that has been isotropically compressed. When we consider that contacting particles move according to the average deformation, the effective shear modulus is over-predicted with respect to numerical simulation, while the effective bulk modulus is almost captured. We improve upon this simple approach by relaxing the aggregate. The kinematics of a pair of contacting particles is then given by the average deformation and fluctuations in both translations and rotations. We determine analytical expressions for these fluctuations by means of force and moment equilibrium applied to each particle of the pair. In order to derive the incremental stress associated with an incremental deformation, we introduce conditional averages of the fluctuations that are functions of the statistical geometry of the packing. This brings the theoretical predictions of the effective moduli close to those measured in numerical simulations. The variability of the average number of particle contacts per particle is seen to play an important role in the statistical description of the aggregate.

The Initial Response of an Idealized Granular Material

We study an ideal granular aggregate consisting of elastic spherical particles, isotropic in stress and anisotropic in the contact network. Because of the contact anisotropy, a confining pressure applied at zero deviatoric stress, produces shear strain as well as volume strain. Our goal is to predict the coordination number k, the average number of contacts per particle, and the magnitude of the contact anisotropy ɛ, from knowledge of the elastic moduli of the aggregate. We do this through a theoretical model based upon the well known effective medium theory. However, rather than focusing on the moduli, we consider their ratios over the moduli of an equivalent isotropic state. We observe good agreement between numerical simulation and theory.

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Luigi La Ragione

Papers

Fluctuations and the effective moduli of an isotropic, random aggregate of identical, frictionless spheres

Characterizing the shear and bulk moduli of an idealized granular material

The Initial Response of an Idealized Granular Material

Contact anisotropy and coordination number for a granular assembly: a comparison of distinct-element-method simulations and theory.

Particle spin in anisotropic granular materials