This article was originally published in Treatise on Geophysics, Second Edition, published by Elsevier, and the attached copy is provided by Elsevier for the author's benefit and for the benefit of the author's institution, for non-commercial research and educational use including without limitation use in instruction at your institution, sending it to specific colleagues who you know, and providing a copy to your institution’s administrator. All other uses, reproduction and distribution, including without limitation commercial reprints, selling or licensing copies or access, or posting on open internet sites, your personal or institution’s website or repository, are prohibited. For exceptions, permission may be sought for such use through Elsevier's permissions site at: http://www.elsevier.com/locate/permissionusematerial Schmitt D.R Geophysical Properties of the Near Surface Earth: Seismic Properties. In: Gerald Schubert (editor-in-chief) Treatise on Geophysics, 2nd edition, Vol 11. Oxford: Elsevier; 2015. p. 43-87. Author's personal copy 11.03 Geophysical Properties of the Near Surface Earth: Seismic Properties DR Schmitt, University of Alberta, Edmonton, AB, Canada ã 2015 Elsevier B.V. All rights reserved. 11.03.1 Introduction 44 11.03.2 Basic Theory 45 11.03.2.1 Hooke’s Constitutive Relationship and Moduli 45 11.03.3 Mineral Building Blocks 47 11.03.3.1 Elastic Properties of Minerals 47 11.03.3.2 Bounds on Isotropic Mixtures of Anisotropic Minerals 48 11.03.3.3 Isothermal Versus Adiabatic Moduli 49 11.03.3.4 Effects of Pressure and Temperature on Mineral Moduli 51 11.03.3.5 Mineral Densities 51 11.03.4 Fluid Properties 53 11.03.4.1 Phase Relations for Fluids 54 11.03.4.2 Equations of State for Fluids 54 11.03.4.2.1 Ideal gas law 56 11.03.4.2.2 Adiabatic and isothermal fluid moduli 56 11.03.4.2.3 The van der Waals model 57 11.03.4.2.4 The Peng–Robinson EOS 57 11.03.4.2.5 Correlative EOS models 58 11.03.4.2.6 Determining K from equations of state 58 f 11.03.4.3 Mixtures and Solutions 59 11.03.4.3.1 Frozen mixtures 59 11.03.4.3.2 Miscible fluid mixtures 60 11.03.5 The Rock Frame 65 11.03.5.1 Essential Characteristics 65 11.03.5.2 The Pore-Free Solid Portion 66 11.03.5.3 Influence of Porosity 67 11.03.5.4 Influence of Crack-Like Porosity 69 11.03.5.5 Pressure Dependence in Granular Materials 72 11.03.5.6 Implications of Pressure Dependence 73 11.03.5.6.1 Stress-induced anisotropy (acoustoelastic effect) 73 11.03.5.6.2 Influence of pore pressure 74 11.03.6 Seismic Waves in Fluid-Saturated Rocks 74 11.03.6.1 Gassmann’s Equation 75 11.03.6.2 Frequency-Dependent Models 76 11.03.6.2.1 Global flow (biot) model 77 11.03.6.2.2 Local flow (squirt) models 78 11.03.7 Empirical Relations and Data Compilations 78 11.03.8 The Road Ahead 81 Acknowledgments 81 References 81 Glossary Aspect ratio x (dimensionless) In the context of crack-like Adiabat An adiabatic path in P–V–T space. porosity, this refers to the aperture width of the crack to its Adiabatic A thermodynamic process in which no heat is length. allowed to transfer into or out of the system. The local Bulk modulus K (Pa) Also called the incompressibility. A measure of the resistance of a material to deformation for a compression and rarefaction and corresponding increase given change in pressure. and decrease of both pressure and temperature of a material À1 as compressional wave passes are assumed to be an adiabatic Compliances Sij (Pa ) The elastic mechanical parameters process. that generally relate stresses to strains. À1 Anisotropy The condition in which the physical properties Compressibility (Pa ) Inverse of the bulk modulus. of a material will depend on direction. Treatise on Geophysics, Second Edition http://dx.doi.org/10.1016/B978-0-444-53802-4.00190-1 43 Treatise on Geophysics, 2nd edition, (2015), vol. 11, pp. 43-87 Author's personal copy 44 Geophysical Properties of the Near Surface Earth: Seismic Properties Cricondenbar For fluid mixtures. The greatest pressure at Poisson’s ratio n (dimensionless) The negative of the ratio which both liquid and vapor phases can coexist. Above the between the radial and the axial strains induced by an axial cricondenbar, the mixture must be either a liquid or a stress. supercritical fluid phase. Polycrystal A material that is a mixture of mineral crystals Critical point For pure fluids, a point in P–V–T space at and that, often, is assumed to be free of pores. The properties which the liquid–vapor phase line terminates. The fluid will of the polycrystal are then taken to be representative of those be in the supercritical state for pressures and temperatures for the solid portion of the rock. above the critical pressure Pc and temperature Tc. At the Pseudocritical point For fluid mixtures, a point in P–V–T critical point the fluid will have the critical specific volume space where the bubble and dew lines meet. This point Vc or equivalently the critical density rc ¼M/Vc, where M is depends on the composition of the mixture and occurs at the chemical molecular weight. the pseudocritical pressure PPC and temperature TPC. Cricondentherm For fluid mixtures. The greatest Saturated The condition where the pore space of the rock is temperature at which both liquid phase and vapor phase can filled with fluids. still coexist. Above this temperature, the fluid will be either Saturation The fraction of the pore space that is filled with a vapor or supercritical fluid phase. given fluid. If only one fluid fills the pore volume, it will have À Density r (kg m 3) Mass per unit volume. a saturation of 1. If the pore volume is equally filled with two Equation of state A theoretical or empirical function or set different fluids, they each will have a saturation of 0.5. of functions that describes the material’s specific volume as a Shear modulus m (Pa) The elastic mechanical parameter function of pressure and temperature. relating shear stress to shear strain. Hooke’s law The mathematical relationship between stress Stiffness Cij (Pa) The elastic mechanical parameters that and strain via the elastic stiffnesses or conversely the strains generally relate strains to stresses. and the stresses via the elastic compliances. Strain eij or gij (dimensionless) Measures of the Isentropic A thermodynamic process in which the entropy deformation of a material. of the system remains constant. A reversible adiabatic Stress sij or tij (Pa) The ratio of an applied force to the area process is also isentropic. over which it is applied. Normal stresses sij are directed Isochor A thermodynamic path in P–V–T space in perpendicularly to the surface. Shear stresses tij are directed which the specific volume Vm or the density r remains along the plane of the surface. constant. Supercritical The condition for a fluid encountered in P–V–T Isotherm A thermodynamic path in P–V–T space in space in which it is no longer considered a liquid or a vapor which the temperature T remains constant. These are (gas) but a fluid with the characteristics of both. For single- often the conditions employed in conventional component fluids, the supercritical phase exists above the measurements of fluid properties particularly in the critical point at the critical pressure Pc and temperature Tc. petroleum industry. Young’s modulus E (Pa) Also often referred to as the Lame´ parameters l and m (Pa). The two elastic parameters modulus of elasticity. The elastic mechanical parameter relating stresses to strains in the Lame´ mathematical relating the linear axial strain induced to the applied axial formulation of Hooke’s law. normal stress. 11.03.1 Introduction computational power grows, the differences between inversion and advanced prestack migration in imaging will become less Geophysicists measure the spatial and temporal variations in distinct. electromagnetic, magnetic, and gravitational potentials and Velocity, as it is used in the geophysical community for seismic wave fields in order to make inferences regarding the wave speed, would certainly first come to a geophysicist’s internal structure of the Earth in terms of, respectively, its mind as a seismic property. It is also the seismic property that electrical resistivity (See Chapters 2.25, 11.04, 11.08, and is most often used to infer lithology. Liberally, compressional 11.10), its magnetism (See Chapters 2.24, 5.08, 11.05, wave velocities that can exist in crustal materials can range 11.11), its density (See Chapter 3.03, 11.05, 11.12), and its from a few hundred meters per second in air-saturated uncon- À solidated sediments to upward of 8 km s 1 for high-grade elasticity (See Chapters 1.26 and 2.12). In seismology the most basic observation is that of a seismic wave’s travel time metamorphosed rocks at the top of the mantle. Typically from its source to the point of measurement. Seismologists then, within a given geologic context, the velocities themselves continue to develop increasingly sophisticated analyses to con- or additional parameters derived from them such as the vert this basic observation into seismic velocities from which compressional/shear wave speed ratio VP/VS, Poisson’s ratio n, or the seismic parameter ’ ¼ V2 À 4V2/3 are useful indicators the Earth’s structure may be deduced. This holds true for the P S simplest 1-D seismic refraction analysis to the most compli- of lithology.