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

The recognition that minimizing an integral function through variational methods (as in the last chapters) leads to the second-order differential equations of Euler-Lagrange for the minimizing function made it natural for mathematicians of the eighteenth century to ask for an integral quantity whose minimization would result in Newton’s equations of motion. With such a quantity, a new principle through which the universe acts would be obtained. The belief that “something” should be minimized was in fact a long-standing conviction of natural philosophers who felt that God had constructed the universe to operate in the most efficient manner—but how that efficiency was to be assessed was subject to interpretation. However, Fermat (1657) had already invoked such a principle successfully in declaring that light travels through a medium along the path of least time of transit. Indeed, it was by recognizing that the brachistochrone should give the least time of transit for light in an appropriate medium that Johann Bernoulli “proved” that it should be a cycloid in 1697. (See Problem 1.1.) And it was Johann Bernoulli who in 1717 suggested that static equilibrium might be characterized through requiring that the work done by the external forces during a small displacement from equilibrium should vanish. This “principle of virtual work” marked a departure from other minimizing principles in that it incorporated stationarity—even local stationarity—(tacitly) in its formulation. Efforts were made by Leibniz, by Euler, and most notably, by Lagrange to define a principle of least action (kinetic energy), but it was not until the last century that a truly satisfactory principle emerged, namely, Hamilton’s principle of stationary action (c. 1835) which was foreshadowed by Poisson (1809) and polished by Jacobi (1848) and his successors into an enduring landmark of human intellect, one, moreover, which has survived transition to both relativity and quantum mechanics. (See [L], [Fu] and Problems 8.11 8.12.)

Variational Principles in Mechanics

1.Introduction to Composite Materials 2. Macrochemical Behavior of a Lamina 3.Micromechanical Behavior of a Lamina 4.Macromechanical Behavior of a Laminate 5.Bending, Buckling, and Vibration of Laminated Plates 6.Other Analysis and Behavior Topics 7.Introduction to Design of Composite Structures Appendix A.Matrices and Tensors Appendix B.Maxima and Minima of Functions of a Single Variable Appendix C.Typical Stress-Strain Curves Appendix D.Governing Equations for Beam Equilibrium and Plate Equilibrium, Buckling, and Vibration Index

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Mechanics of composite materials

Today, surface-wave analysis is widely adopted for building near-surface S-wave velocity models. The surface-wave method is under continuous and rapid evolution, also thanks to the lively scientific debate among different disciplines, and interest in the technique has increased significantly during the last decade. A comprehensive review of the literature in the main scientific journals provides historical perspective, methodological issues, applications, and most-promising recent approaches. Higher modes in the inversion and retrieval of lateral variations are dealt with in great detail, and the current scientific debate on these topics is reported. A best-practices guideline is also outlined.

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Surface-wave analysis for building near-surface velocity models — Established approaches and new perspectives

The simplest effective model of a formation containing a single fracture system is transversely isotropic with a horizontal symmetry axis (HTI) Reflection seismic signatures in HTI media, such as NMO velocity and amplitude variation with offset (AVO) gradient, can be conveniently described by the Thomsen‐type anisotropic parameters e(V), δ(V), and γ(V) Here, we use the linear slip theory of Schoenberg and co‐workers and the models developed by Hudson and Thomsen for pennyshaped cracks to relate the anisotropic parameters to the physical properties of the fracture network and to devise fracture characterization procedures based on surface seismic measurements Concise expressions for e(V), δ(V), and γ(V) linearized in the crack density, show a substantial difference between the values of the anisotropic parameters for isolated fluid‐filled and dry (gas‐filled) penny‐sh aped cracks While the dry‐crack model is close to elliptical with e(V)≈δ(V), for thin fluid‐filled cracks e(V);≈0 and the absolute value

https://www.math.purdue.edu/~santos/research/9junio/Bakulin_HTI_GPY_2000.pdf

Estimation of fracture parameters from reflection seismic data—Part I: HTI model due to a single fracture set

We present a way to image through complex overburden. The method uses surface shots with downhole receivers placed below the most complex part of the troublesome overburden. No knowledge of the velocity model between shots and receivers is required. The method uses time-reversal logic to create a new downward-continued data set with virtual sources (VS's) at the geophone locations. Time reversal focuses energy that passes through the overburden into useful primary energy for the VS. In contrast to physical acoustics, our time reversal is done on a computer, utilizing conventional acquisition with surface shots and downhole geophones. With this approach, we can image below extremely complex (realistic) overburden — in fact, the more complex the better. We recast the data to those with sources where we actually know and can control the waveform that has a downward-radiation pattern that may also be controlled, and is reproducible for 4D even if the near-surface changes or the shooting geometry is altered sl...

The virtual source method: Theory and case study

Existing geophysical and geological data indicate that orthorhombic media with a horizontal symmetry plane should be rather common for naturally fractured reservoirs. Here, we consider two orthorhombic models: one that contains parallel vertical fractures embedded in a transversely isotropic background with a vertical symmetry axis (VTI medium) and the other formed by two orthogonal sets of rotationally invariant vertical fractures in a purely isotropic host rock. Using the linear‐slip theory, we obtain simple analytic expressions for the anisotropic coefficients of effective orthorhombic media. Under the assumptions of weak anisotropy of the background medium (for the first model) and small compliances of the fractures, all effective anisotropic parameters reduce to the sum of the background values and the parameters associated with each fracture set. For the model with a single fracture system, this result allows us to eliminate the influence of the VTI background by evaluating the differences between t...

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Estimation of fracture parameters from reflection seismic data—Part III: Fractured models with monoclinic symmetry

We propose an alternative solution that does not require knowledge of the near-surface velocity model. The price to pay is placing geophones in the Earth below the most complex near-surface part while keeping sources at the surface. Receivers may sit in horizontal or slanted wells, which may be producers/injectors or dedicated sidetracks. Utilizing time reversal logic, we convert surfaceto-downhole data into a new dataset with downhole Virtual Sources (VS) located at geophone positions. The resulting VS dataset with both downhole sources and receivers can be conventionally imaged requiring only the bottom portion of the velocity model below the receivers that is more simple to obtain. To illustrate the technique, we show application to one synthetic data set and one field case study.

Virtual Source: New Method For Imaging And 4D Below Complex Overburden

We develop a rock physics model based on nonlinear elasticity that describes the dependence of the effective stiffness tensor as a function of a 3D stress field in intrinsically anisotropic formations. This model predicts the seismic velocity of both P- and S-waves in any direction for an arbitrary 3D stress state. Therefore, the model overcomes the limitations of existing empirical velocity-stress models that link P-wave velocity in isotropic rocks to uniaxial or hydrostatic stress. To validate this model, we analyze ultrasonic velocity measurements on stressed anisotropic samples of shale and sandstone. With only three nonlinear constants, we are able to predict the stress dependence of all five elastic medium parameters comprising the transversely isotropic stiffness tensor. We also show that the horizontal stress affects vertical S-wave velocity with the same order of magnitude as vertical stress does. We develop a weakanisotropy approximation that directly links commonly measured anisotropic Thomsen parameters to the principal stresses. Each Thomsen parameter is simply a sum of corresponding background intrinsic anisotropy and stressinduced contribution. The stress-induced part is controlled by the difference between horizontal and vertical stresses and coefficients depending on nonlinear constants. Thus, isotropic rock stays isotropic under varying but hydrostatic load, whereas transversely isotropic rock retains the same values of dimensionless Thomsen parameters. Only unequal horizontal and vertical stresses alter anisotropy. Since Thomsen parameters conveniently describe seismic signatures, such as normal-moveout velocities and amplitudevariation-with-offset gradients, this approximation is suitable for designing new methods for the estimation of 3D subsurface stress from multicomponent seismic data.

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Andrey Bakulin

Papers

Estimation of fracture parameters from reflection seismic data—Part I: HTI model due to a single fracture set

The virtual source method: Theory and case study

Estimation of fracture parameters from reflection seismic data—Part III: Fractured models with monoclinic symmetry

Virtual Source: New Method For Imaging And 4D Below Complex Overburden

Nonlinear rock physics model for estimation of 3D subsurface stress in anisotropic formations: Theory and laboratory verification