# The Rock Physics Handbook, First Edition

Gary Mavko, Tapan Mukerji, and Jack Dvorkin

*Cambridge University Press, 1998*

The Rock Physics Handbook is published by Cambridge University Press.

Second Edition, 2009

Over the last three decades enormous strides have been made to understand the connections between physical properties of rocks and geophysical observables -- the science now known as Rock Physics. Scientists have discovered an increasing amount of order in relations that once appeared disappointingly scattered, such as seismic wave velocity versus porosity, porosity versus permeabiity, velocity versus fluid saturation and lithology. The Rock Physics Handbook conveniently brings together the theoretical and empirical relations that form the foundations of rock physics, with particular emphasis on seismic properties. It also includes commonly used models and relations for electrical and dielectric rock properties.

76 stand-alone articles concisely summarize a wide range of topics, including wave propagation, AVO-AVOZ, effective media, poroelasticity, pore fluid flow and diffusion. The book contains overviews of dispersion mechanisms, fluid substitution, and Vp-Vs relations. Useful empirical results on reservoir rocks and sediments, granular media, tables of mineral data, and an atlas of reservoir rock properties complete the text.

This distillation of an otherwise scattered and eclectic mass of knowledge is presented in a form that can be immediately applied to solve real problems. Geophysics professionals, researchers and students as well as petroleum engineers, well log analysists, and environmental geoscientists will value the Rock Physics Handbook as a resource for practical problem solving.

When we wrote the Rock Physics Handbook, we focused on collecting useful equations -- those relations that we derive once every two years and then forget, or find ourselves searching for in piles of articles, or somewhere in that shelf of books, or on scraps of paper taped to the side of the filing cabinet. Our approach was to present results, with a few of the key assumptions and limitations, and almost never any derivations. Our intention was to create a reference, and not a textbook. We assume that the reader will be generally aware of the various topics, and if not, we provide a few references to the more complete descriptions in books and journals. The Rock Physics Handbook is presented as 76 stand-alone articles. We wanted the user to be able to go directly to the topic of interest and to find all of the necessary information within a few pages, without the need to refer to previous chapters, as in a conventional textbook. As a result, there is an occasional redundancy in the explanatory text.

We wish to thank the faculty, students, and industrial affiliates of the Stanford Rock Physics and Borehole Geophysics (SRB) project for many valuable comments and insights. We found particularly useful discussions with Zhijing Wang, Thierry Cadoret, Ivar Brevik, Sue Raikes, Sverre Strandenes, Mike Batzle, and Jim Berryman. Li Teng contributed the chapter on anisotropic AVOZ, and Ran Bachrach contributed to the chapter on dielectric rock properties. Ranie Lynds helped with the graphics and did a marvelous job of proofing and editing. Thanks to Barbara Mavko for many useful comments on content and style. And as always, we are indebted to Amos Nur whose work, past and present, has helped to make the field of Rock Physics what it is today.

We hope you find this handbook useful.

### Table of Contents

- Part 1:
**Basic Tools**- 1.1 The Fourier Transform
- 1.2 The Hilbert Transform and Analytic Signal
- 1.3 Statistics and Linear Regression
- 1.4 Coordinate Transformations

- Part 2:
**Elasticity and Hooke's Law**- 2.1 Elastic Moduli -- Isotropic Form of Hooke's Law
- 2.2 Anisotropic Form of Hooke's Law
- 2.3 Thomsen's Notation for Weak Elastic Anisotropy
- 2.4 Stress-Induced Anisotropy in Rocks
- 2.5 Strain Components and Equations of Motion in Cylindrical and Spherical Coordinate Systems
- 2.6 Deformation of Inclusions and Cavities in Elastic Solids
- 2.7 Deformation of a Circular Hole -- Borehole Stresses
- 2.8 Mohr's Circles

- Part 3:
**Seismic Wave Propagation**- 3.1 Seismic Velocities
- 3.2 Phase, Group, and Energy Velocities
- 3.3 Impedance, Reflectivity, and Transmissivity
- 3.4 Reflectivity and AVO
- 3.5 AVOZ in Anisotropic Environments
- 3.6 Viscoelasticity and Q
- 3.7 Kramers-Kronig Relations Between Velocity Dispersion and Q
- 3.8 Waves in Layered Media: Full Waveform Synthetic Seismograms
- 3.9 Waves in Layered Media: Stratigraphic Filtering and Velocity Dispersion
- 3.10 Waves in Layered Media: Frequency Dependent Anisotropy and Dispersion
- 3.11 Scale-dependent Seismic Velocities in Heterogeneous Media
- 3.12 Scattering Attenuation
- 3.13 Waves in Cylindrical Rods -- The Resonant Bar

- Part 4:
**Effective Media**- 4.1 Hashin-Shtrikman Bounds
- 4.2 Voigt and Reuss Bounds
- 4.3 Wood's Formula
- 4.4 Hill Average Moduli Estimate
- 4.5 Composite with Uniform Shear Modulus
- 4.6 Rock and Pore Compressibilities and Some Pitfalls
- 4.7 Kuster Toksoz Formulation for Effective Moduli
- 4.8 Self-Consistent Approximations of Effective Moduli
- 4.9 Differential Effective Medium Model
- 4.10 Hudson's Model for Cracked Media
- 4.11 Eshelby-Cheng Model for Cracked Anisotropic Media
- 4.12 Elastic Constants in Finely Layered Media -- Backus Average

- Part 5:
**Granular Media**- 5.1 Packing of Spheres -- Geometric Relations
- 5.2 Random Spherical Grain Packings -- Contact Models and Effective Moduli
- 5.3 Ordered Spherical Grain Packings -- Effective Moduli

- Part 6:
**Fluid Effects on Wave Propagation**- 6.1 Biot's Velocity Relations
- 6.2 Geertsma-Smit Approximations of Biot's Relations
- 6.3 Gassmann's Relations
- 6.4 BAM -- Marion's Bounding Average Method
- 6.5 Fluid Substitution in Anisotropic Rocks: Brown and Korringa's Relations
- 6.6 Generalized Gassmann's Equations for Composite Porous Media
- 6.7 Mavko-Jizba Squirt Relations
- 6.8 Extension of Mavko-Jizba Squirt Relations for All Frequencies
- 6.9 BISQ
- 6.10 Anisotropic Squirt
- 6.11 Common Features of Fluid-Related Velocity Dispersion Mechanisms
- 6.12 Partial and Multi-Phase Saturations
- 6.13 Partial Saturation: White and Dutta-Ode Model for Velocity Dispersion and Attenuation
- 6.14 Waves in Pure Viscous Fluid
- 6.15 Physical Properties of Gases and Fluids

- Part 7:
**Empirical Relations**- 7.1 Velocity-Porosity Models: Critical Porosity and Nur's Modified Voigt Average
- 7.2 Velocity-Porosity Models: Geertsma's Empirical Relations for Compressibility
- 7.3 Velocity-Porosity Models: Wyllie's Time Average Equation
- 7.4 Velocity-Porosity Models: Raymer-Hunt-Gardner Relations
- 7.5 Velocity-Porosity-Clay Models: Han's Empirical Relations for Shaly Sandstones
- 7.6 Velocity-Porosity-Clay Models: Tosaya's Empirical Relations for Shaly Sandstones
- 7.7 Velocity-Porosity-Clay Models: Castagna's Empirical Relations for Velocities
- 7.8 Vp-Vs Relations
- 7.9 Velocity-Density Relations

- Part 8:
**Flow and Diffusion**- 8.1 Darcy's Law
- 8.2 Kozeny-Carman Relation for Flow
- 8.3 Viscous Flow
- 8.4 Capillary Forces
- 8.5 Diffusion and Filtration -- Special Cases

- Part 9:
**Electrical Properties**- 9.1 Electrical Properties: Bounds and Effective Medium Models
- 9.2 Electrical Properties: Velocity Dispersion and Attenuation
- 9.3 Electrical Properties: Empirical Relations
- 9.4 Electrical Conductivity in Porous Rocks

- Part 10:
**Appendices**- 10.1 Typical Rock Properties
- 10.2 Conversions
- 10.3 Moduli and Density of Common Minerals

- Index

### Errata for the 1st Edition (1998) of the Rock Physics Handbook

Thanks to careful readers, we have identified the following errors. Please let us know if you find any more errors.

- Page 219: Physical Properties of Gases and Fluids

The equation for Bo should have ...+T+17.8 within the brackets and not ...+T+1.78. This correction also applies to the worked example on page 219. - Pages 244-249: Empirical Relations

The section headings for "Greenberg-Castagna" (page 246), "Williams" (page 248), and "Xu and White" (page 249) should be two levels higher. They are not meant to be sub-sections within "Krief's Relation" but are separate sections by themselves. - Page 71: Viscoelasticity and Q

The third displayed equation for 1/Q from the bottom of the page should be an approximate equality and not an exact equality. - Page 155: Cemented Sand Model

Equation for C_tau, should have +10^(-4) as in Geophysics paper Dvorkin & Nur 1996, not -10^(-4). - Reference to J.G. Berryman, 1995 "A Handbook of Physical Constants", AGU - the citation should read "Rock Physics & Phase Relations: A Handbook of Physical Constants".
- Section 7.1, top of page 222: It says: "In the suspension domain, POR > POR_crit, the effective bulk and shear moduli can be estimated quite accurately using by the Reuss (iso-stress) average". Either the word "using" or the word "by" in this sentence needs to be removed.
- Section 7.8, page 245: It says that Beta = POR/POR_crit for 0<=POR<=POR_crit and that Beta = 1 for POR < POR_crit. The last inequality is incorrect. It should be "... for POR > POR_crit".
- Equation S=(3/2)(1-Phi)/d in the middle of page 261 should be S=6(1-Phi)/d
- Page 157: Typo in the denominator of the Geff calculation. (3/6)*(9*2+8*6)/(2+2*3) = 2.625. Then Geff = 3.97
- page 217, equation at the very bottom: 1/b = 0.306 - 7.6/M is right. The one in RPH and Batzle-Wang equation (16) giving b = 0.306 - 7.6/M is a typo.
- NOT A TYPO! Page 215, third line from the bottom: 1820S^2 should NOT be 820S^2 RPH is correct (1820 S^2). 820 S^2 in Batzle & Wang Geophysics paper is typo.

- Page 70 eqn.2 last term should be 2*mu*epsilon_ij.