This page intentionally left blank The Rock Physics Handbook, Second Edition Tools for Seismic Analysis of Porous Media The science of rock physics addresses the relationships between geophysical observations and the underlying physical properties of rocks, such as composition, porosity, and pore fluid content The Rock Physics Handbook distills a vast quantity of background theory and laboratory results into a series of concise, self-contained chapters, which can be quickly accessed by those seeking practical solutions to problems in geophysical data interpretation In addition to the wide range of topics presented in the First Edition (including wave propagation, effective media, elasticity, electrical properties, and pore fluid flow and diffusion), this Second Edition also presents major new chapters on granular material and velocity–porosity–clay models for clastic sediments Other new and expanded topics include anisotropic seismic signatures, nonlinear elasticity, wave propagation in thin layers, borehole waves, models for fractured media, poroelastic models, attenuation models, and cross-property relations between seismic and electrical parameters This new edition also provides an enhanced set of appendices with key empirical results, data tables, and an atlas of reservoir rock properties expanded to include carbonates, clays, and gas hydrates Supported by a website hosting MATLAB routines for implementing the various rock physics formulas presented in the book, the Second Edition of The Rock Physics Handbook is a vital resource for advanced students and university faculty, as well as in-house geophysicists and engineers working in the petroleum industry It will also be of interest to practitioners of environmental geophysics, geomechanics, and energy resources engineering interested in quantitative subsurface characterization and modeling of sediment properties Gary Mavko received his Ph.D in Geophysics from Stanford University in 1977 where he is now Professor (Research) of Geophysics Professor Mavko co-directs the Stanford Rock Physics and Borehole Geophysics Project (SRB), a group of approximately 25 researchers working on problems related to wave propagation in earth materials Professor Mavko is also a co-author of Quantitative Seismic Interpretation (Cambridge University Press, 2005), and has been an invited instructor for numerous industry courses on rock physics for seismic reservoir characterization He received the Honorary Membership award from the Society of Exploration Geophysicists (SEG) in 2001, and was the SEG Distinguished Lecturer in 2006 Tapan Mukerji received his Ph.D in Geophysics from Stanford University in 1995 and is now an Associate Professor (Research) in Energy Resources Engineering and a member of the Stanford Rock Physics Project at Stanford University Professor Mukerji co-directs the Stanford Center for Reservoir Forecasting (SCRF) focusing on problems related to uncertainty and data integration for reservoir modeling His research interests include wave propagation and statistical rock physics, and he specializes in applied rock physics and geostatistical methods for seismic reservoir characterization, fracture detection, time-lapse monitoring, and shallow subsurface environmental applications Professor Mukerji is also a co-author of Quantitative Seismic Interpretation, and has taught numerous industry courses He received the Karcher award from the Society of Exploration Geophysicists in 2000 Jack Dvorkin received his Ph.D in Continuum Mechanics in 1980 from Moscow University in the USSR He has worked in the Petroleum Industry in the USSR and USA, and is currently a Senior Research Scientist with the Stanford Rock Physics Project at Stanford University Dr Dvorkin has been an invited instructor for numerous industry courses throughout the world, on rock physics and quantitative seismic interpretation He is a member of American Geophysical Union, Society of Exploration Geophysicists, American Association of Petroleum Geologists, and the Society of Petroleum Engineers The Rock Physics Handbook, Second Edition Tools for Seismic Analysis of Porous Media Gary Mavko Stanford University, USA Tapan Mukerji Stanford University, USA Jack Dvorkin Stanford University, USA CAMBRIDGE UNIVERSITY PRESS Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo, Delhi, Dubai, Tokyo Cambridge University Press The Edinburgh Building, Cambridge CB2 8RU, UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521861366 © G Mavko, T Mukerji, and J Dvorkin 2009 This publication is in copyright Subject to statutory exception and to the provision of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press First published in print format 2009 ISBN-13 978-0-511-65062-8 eBook (NetLibrary) ISBN-13 978-0-521-86136-6 Hardback Cambridge University Press has no responsibility for the persistence or accuracy of urls for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate Contents Preface page xi Basic tools 1.1 1.2 1.3 1.4 The Fourier transform The Hilbert transform and analytic signal Statistics and probability Coordinate transformations 18 Elasticity and Hooke’s law 21 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 21 23 35 39 40 43 47 2.9 2.10 2.11 2.12 Elastic moduli: isotropic form of Hooke’s law Anisotropic form of Hooke’s law Thomsen’s notation for weak elastic anisotropy Tsvankin’s extended Thomsen parameters for orthorhombic media Third-order nonlinear elasticity Effective stress properties of rocks Stress-induced anisotropy in rocks Strain components and equations of motion in cylindrical and spherical coordinate systems Deformation of inclusions and cavities in elastic solids Deformation of a circular hole: borehole stresses Mohr’s circles Static and dynamic moduli Seismic wave propagation 81 3.1 3.2 Seismic velocities Phase, group, and energy velocities 81 83 v 54 56 68 74 76 vi Contents 3.3 3.4 3.5 3.6 3.7 3.8 3.9 3.10 3.11 3.12 3.13 3.14 3.15 3.16 NMO in isotropic and anisotropic media Impedance, reflectivity, and transmissivity Reflectivity and amplitude variations with offset (AVO) in isotropic media Plane-wave reflectivity in anisotropic media Elastic impedance Viscoelasticity and Q Kramers–Kronig relations between velocity dispersion and Q Waves in layered media: full-waveform synthetic seismograms Waves in layered media: stratigraphic filtering and velocity dispersion Waves in layered media: frequency-dependent anisotropy, dispersion, and attenuation Scale-dependent seismic velocities in heterogeneous media Scattering attenuation Waves in cylindrical rods: the resonant bar Waves in boreholes 138 146 150 155 160 Effective elastic media: bounds and mixing laws 169 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 4.10 4.11 4.12 4.13 4.14 4.15 4.16 4.17 Hashin–Shtrikman–Walpole bounds Voigt and Reuss bounds Wood’s formula Voigt–Reuss–Hill average moduli estimate Composite with uniform shear modulus Rock and pore compressibilities and some pitfalls Kuster and Toksoăz formulation for effective moduli Self-consistent approximations of effective moduli Differential effective medium model Hudson’s model for cracked media Eshelby–Cheng model for cracked anisotropic media T-matrix inclusion models for effective moduli Elastic constants in finely layered media: Backus average Elastic constants in finely layered media: general layer anisotropy Poroelastic Backus average Seismic response to fractures Bound-filling models 169 174 175 177 178 179 183 185 190 194 203 205 210 215 216 219 224 Granular media 229 5.1 5.2 Packing and sorting of spheres Thomas–Stieber model for sand–shale systems 229 237 86 93 96 105 115 121 127 129 134 vii Contents 5.3 5.4 5.5 Particle size and sorting Random spherical grain packings: contact models and effective moduli Ordered spherical grain packings: effective moduli 245 264 Fluid effects on wave propagation 266 6.1 6.2 6.3 6.4 266 272 273 6.20 6.21 Biot’s velocity relations Geertsma–Smit approximations of Biot’s relations Gassmann’s relations: isotropic form Brown and Korringa’s generalized Gassmann equations for mixed mineralogy Fluid substitution in anisotropic rocks Generalized Gassmann’s equations for composite porous media Generalized Gassmann equations for solid pore-filling material Fluid substitution in thinly laminated reservoirs BAM: Marion’s bounding average method Mavko–Jizba squirt relations Extension of Mavko–Jizba squirt relations for all frequencies Biot–squirt model Chapman et al squirt model Anisotropic squirt Common features of fluid-related velocity dispersion mechanisms Dvorkin–Mavko attenuation model Partial and multiphase saturations Partial saturation: White and Dutta–Ode´ model for velocity dispersion and attenuation Velocity dispersion, attenuation, and dynamic permeability in heterogeneous poroelastic media Waves in a pure viscous fluid Physical properties of gases and fluids 331 338 339 Empirical relations 347 7.1 Velocity–porosity models: critical porosity and Nur’s modified Voigt average Velocity–porosity models: Geertsma’s empirical relations for compressibility Velocity–porosity models: Wyllie’s time-average equation Velocity–porosity models: Raymer–Hunt–Gardner relations 6.5 6.6 6.7 6.8 6.9 6.10 6.11 6.12 6.13 6.14 6.15 6.16 6.17 6.18 6.19 7.2 7.3 7.4 242 282 284 287 290 292 295 297 298 302 304 306 310 315 320 326 347 350 350 353 viii Contents 7.5 7.8 7.9 7.10 7.11 7.12 7.13 7.14 Velocity–porosity–clay models: Han’s empirical relations for shaley sandstones Velocity–porosity–clay models: Tosaya’s empirical relations for shaley sandstones Velocity–porosity–clay models: Castagna’s empirical relations for velocities VP–VS–density models: Brocher’s compilation VP–VS relations Velocity–density relations Eaton and Bowers pore-pressure relations Kan and Swan pore-pressure relations Attenuation and quality factor relations Velocity–porosity–strength relations 358 359 363 380 383 383 384 386 Flow and diffusion 389 8.1 8.2 8.3 8.4 8.5 8.6 8.7 Darcy’s law Viscous flow Capillary forces Kozeny–Carman relation for flow Permeability relations with Swi Permeability of fractured formations Diffusion and filtration: special cases 389 394 396 401 407 410 411 Electrical properties 414 9.1 9.2 9.3 9.4 9.5 Bounds and effective medium models Velocity dispersion and attenuation Empirical relations Electrical conductivity in porous rocks Cross-property bounds and relations between elastic and electrical parameters 414 418 421 424 429 Appendices 437 Typical rock properties Conversions Physical constants Moduli and density of common minerals Velocities and moduli of ice and methane hydrate 437 452 456 457 457 7.6 7.7 A.1 A.2 A.3 A.4 A.5 355 357 269 6.1 Biot’s velocity relations 1=2 z ¼ ðo=or Þ 1=2 oa fl ¼ where ber( ) and bei( ) are real and imaginary parts of the Kelvin function, respectively, Jn( ) is a Bessel function of order n, and a is the pore-size parameter The pore-size parameter a depends on both the dimensions and the shape of the pore space Stoll (1974) found that values between 16 and 17 of the mean grain diameter gave good agreement with experimental data from several investigators For spherical grains, Hovem and Ingram (1979) obtained a ¼ fd/[3(1 – f)], where d is the grain diameter The velocity dispersion curve for fast P-waves can be closely approximated by a standard linear solid viscoelastic model when k/a2 ! (see Sections 3.8 and 6.12) However, for most consolidated crustal rocks, k/a2 is usually less than At very low frequencies, F(z) ! 1pand z), the ffiffiffi at very high frequenciesp(large asymptotic values are Tzị ! ỵ iị= and Fzị ! k=4ị1 ỵ iị= The reference frequency, fc, which determines the low-frequency range, f ( fc, and the high-frequency range, f )fc, is given by fc ¼ 2pfl k One interpretation of this relation is that it is the frequency where viscous forces acting on the pore fluid approximately equal the inertial forces acting on it In the high-frequency limit, the fluid motion is dominated by inertial effects, and in the lowfrequency limit, the fluid motion is dominated by viscous effects As mentioned above, Biot’s theory predicts the existence of a slow, highly attenuated P-wave in addition to the usual fast P- and S-waves The slow P-wave has been observed in the laboratory, and it is sometimes invoked to explain diffusional loss mechanisms Slow S-wave In the Biot theory the only loss mechanism is the average motion of the fluid with respect to the solid frame, ignoring viscous losses within the pore fluid The fluid strain-rate term is not incorporated into the constitutive equations As a result, the Biot relaxation term includes only a part of the drag force involving permeability, but does not account for the dissipation due to shear drag within the fluid Incorporation of the viscous term is achieved by volume averaging of the pore-scale constitutive relations of the solid and fluid constituents (de al Cruz and Spanos, 1985; Sahay et al., 2001; Spanos, 2002) This gives rise to two propagating shear processes, corresponding to in-phase and out-of-phase shear motion of the phases, with fast and slow S-wave velocities VSI and VSII, respectively The slow S-wave has the characteristics of a rapidly decaying viscous wave in a Newtonian fluid The slow S-wave 270 Fluid effects on wave propagation may play a role in attenuation of fast P- and S-waves by drawing energy from fast waves due to mode conversion at interfaces and discontinuities Incorporation of the fluid strain-rate term introduces an additional relaxation frequency, the saturated shear frame relaxation frequency (Sahay, 2008), given by ob ¼ mfr =, which is typically above Biot’s peak relaxation frequency Expressions for the complex fast and slow S-wave velocities derived from the extended theory are given by Sahay (2008) as VSI;SII ẳ Tặ p!1=2 T 4Á !' & mfr o gỵ g 1ỵ df mf i d f ms ỵ ioi =oị ob ỵ ioi =oị mf mfr ẳ io df n ỵ ioi =oị Tẳ n ẳ =fl ¼ kinematic shear viscosity of pore fluid g ¼ À mfr =m0 mfr ; m0 ¼ shear moduli of dry rock frame and solid mineral, respectively ms ¼ ị0 = ẳ solid mass fraction mf ẳ fl = ¼ fluid mass fraction df ¼ a À mf oi ¼ df oc ¼ df 2pfc ¼ a À mf fl k The frequency oi interpreted as the Biot critical frequency scaled by tortuosity, is the peak frequency associated with Biot relaxation Asymptotic approximations for o ( oi and oi ( o < ob are derived by Sahay (2008): rffiffiffiffiffiffi mfr i o > > 1À o ( oi d f mf > > oi < VSI % rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi& !' > > À Á ms mfr i oi mf o > > g a1 ỵ ỵ oi ( o < ob : ð1 À mf =aÞ o a ob mf a VSII % rffiffiffiffiffi n oa > > o Ài > > < oi oi o ( oi rffiffiffiffiffi > > n > > : o1=2 ð1 À iÞ 2a oi ( o < ob 271 6.1 Biot’s velocity relations The slow S-wave mode in the regime above the Biot relaxation frequency is a diffusive viscous wave with a phase velocity that has a square-root frequency dependence, and a diffusion constant given by the kinematic shear viscosity scaled by the tortuosity Below the Biot relaxation frequency the viscous drag of the fluid on the solid frame dominates over inertial effects The slow S-wave is highly attenuating with a linear and a quadratic frequency dependence of its attenuation and phase velocity, respectively (Sahay, 2008) Uses Biot’s theory can be used for the following purposes: estimating saturated-rock velocities from dry-rock velocities; estimating frequency dependence of velocities; and estimating reservoir compaction caused by pumping using the quasi-static limit of Biot’s poroelasticity theory Assumptions and limitations The use of Biot’s equations presented in this section requires the following considerations: the rock is isotropic; all minerals making up the rock have the same bulk and shear moduli; the fluid-bearing rock is completely saturated; and the pore fluid is Newtonian Caution For most crustal rocks the amount of squirt dispersion (which is not included in Biot’s formulation) is comparable to or greater than Biot’s dispersion, and thus using Biot’s theory alone will lead to poor predictions of high-frequency saturated velocities Exceptions include very-high-permeability materials such as ocean sediments and glass beads, materials at very high effective pressure, or near open boundaries, such as at a borehole or at the surfaces of a laboratory sample The recommended procedure is to use the Mavko–Jizba squirt theory (Section 6.10) first to estimate the high-frequency wet-frame moduli and then to substitute them into Biot’s equations The wavelength, even in the high-frequency limit, is much larger than the grain or pore scale Extensions Biot’s theory has been extended to anisotropic media (Biot, 1962) 272 Fluid effects on wave propagation 6.2 Geertsma–Smit approximations of Biot’s relations Synopsis Biot’s theoretical formulas predict the frequency-dependent velocities of saturated rocks in terms of the dry-rock properties (see also Biot’s relations, Section 6.1) Lowand middle-frequency approximations (Geertsma and Smit, 1961) of his relations may be expressed as VP2 ¼ 4 VP1 ỵ VP0 fc =f ị2 ỵ V ðf =f Þ2 VP1 P0 c where VP is the frequency-dependent P-wave velocity of saturated rock, VP0 is the Biot– Gassmann low-frequency limiting P-wave velocity, VP1 is the Biot high-frequency limiting P-wave velocity, f is the frequency, and fc is Biot’s reference frequency, which determines the low-frequency range, f ( fc, and the high-frequency range, f ) fc, given by fc ¼ 2pfl k where f is porosity, rfl is fluid density, is the viscosity of the pore fluid, and k is the absolute permeabililty of the rock Uses The Geertsma–Smit approximations can be used for the following: estimating saturated-rock velocities from dry-rock velocities; and estimating the frequency dependence of velocities Assumptions and limitations The use of the Geertsma–Smit approximations presented in this section requires the following considerations: mathematical approximations are valid at moderate-to-low seismic frequencies, so that f < fc This generally means moderate-to-low permeabilities, but it is in this range of permeabilities that squirt dispersion may dominate the Biot effect; the rock is isotropic; all minerals making up the rock have the same bulk and shear moduli; and fluid-bearing rock is completely saturated Caution For most crustal rocks the amount of squirt dispersion (not included in Biot’s theory) is comparable to or greater than Biot’s dispersion, and thus using Biot’s 273 6.3 Gassmann’s relations: isotropic form theory alone will lead to poor predictions of high-frequency saturated velocities Exceptions include very high-permeability materials such as ocean sediments and glass beads, or materials at very high effective pressure The recommended procedure is to use the Mavko–Jizba squirt theory (Section 6.10), first to estimate the high-frequency wet-frame moduli and then to substitute them into the Biot or Geertsma–Smit equations 6.3 Gassmann’s relations: isotropic form Synopsis One of the most important problems in the rock physics analysis of logs, cores, and seismic data is using seismic velocities in rocks saturated with one fluid to predict those of rocks saturated with a second fluid, or equivalently, predicting saturated-rock velocities from dry-rock velocities, and vice versa This is the fluid substitution problem Generally, when a rock is loaded under an increment of compression, such as from a passing seismic wave, an increment of pore-pressure change is induced, which resists the compression and therefore stiffens the rock The low-frequency Gassmann–Biot (Gassmann, 1951; Biot, 1956) theory predicts the resulting increase in effective bulk modulus, Ksat, of the saturated rock using the following equation: Kdry Ksat Kfl ; ẳ ỵ K0 Ksat K0 Kdry K0 Kfl ị msat ẳ mdry where Kdry is the effective bulk modulus of dry rock, Ksat is the effective bulk modulus of the rock with pore fluid, K0 is the bulk modulus of mineral material making up rock, Kfl is the effective bulk modulus of pore fluid, f is the porosity, mdry is the effective shear modulus of dry rock, and msat is the effective shear modulus of rock with pore fluid Gassmann’s equation assumes a homogeneous mineral modulus and statistical isotropy of the pore space but is free of assumptions about the pore geometry Most importantly, it is valid only at sufficiently low frequencies such that the induced pore pressures are equilibrated throughout the pore space (i.e., there is sufficient time for the pore fluid to flow and eliminate wave-induced pore-pressure gradients) This limitation to low frequencies explains why Gassmann’s relation works best for very low-frequency in-situ seismic data (