Elastic waves and rock properties
Seismic waves induce elastic deformation along the propagation path in the subsurface. The term elastic refers to the type of deformation that vanishes upon removal of the stress which has caused it. To study seismic amplitudes and thus investigate their use in exploration seismology, it is imperative that we review wave propagation in elastic solids. This gives us the opportunity to appreciate the underlying assumptions in estimating acoustic impedance and AVO attributes.
Elastic wave theory
To facilitate the forthcoming discussion on the link between elastic waves and rock properties, first, we summarize the definitions of elastic wave theory that should always be remembered.
- Stress is force per unit area. Imagine a particle represented by an infinitesimally small volume around a point within a solid body with dimensions (dx, dy, dz) as depicted in Figure L-1. The stress acting upon one of the surfaces, say dy − dz, can in general be at some arbitrary direction. It can, however, be decomposed into three components — one which is normal to the surface, and two which are tangential to the surface. The normal component of the stress is called the normal stress and the tangential components are called the shear stress. A normal stress component is tensional if it is positive and compressional if it is negative. Fluids cannot support shear stress. In a fluid medium, only one independent stress component exists — the hydrostatic pressure.
- Strain is deformation measured as the fractional change in dimension or volume induced by stress. Strain is a dimensionless quantity. The stress field away from the typical seismic source is so small that it does not cause any permanent deformation on rock particles along the propagation path. Hence, the strain induced by seismic waves is very small, usually around 10−6. Consider two points, P and Q, within a solid body as indicated in Figure L-2. Subject to a stress field, the solid is deformed in some manner and the particles at points P and Q are displaced to new locations P′ and Q′. Consider deformations of specific types illustrated in Figure L-3. The simplest deformation is the extension in one direction as a result of a tensional stress (Figure L-3a). The fractional change in length in a given direction is defined as the principal strain component. A positive strain refers to an extension and a negative strain refers to a contraction. Other types of deformation are caused by shearing (Figure L-3b), rotation (Figure L-3c), and a combination of the two (Figure L-3d). These angular deformations are called shear strains since they result in a shearing of the volume around a point within a solid body (Figure L-3b).
Figure L-3 Deformations caused by stress acting on one surface of the volume in Figure L-1: (a) linear deformation that results in extension of the side AB in the x direction by an amount BB′; (b) shearing only; (c) rotation only; (d) combined angular deformation (α) and rotation (β). See text for details.
- Elastic deformation is a deformation in solid bodies that vanishes once the stress is released.
- Hooke’s law for elastic deformations states that the strain at any point is directly proportional to the stresses applied at that point.
- Elastic moduli are material constants that describe stress-strain relations:
- Bulk modulus is the ratio of hydrostatic stress to volumetric strain; hence, it is a measure of incompressibility.
- Modulus of rigidity is the ratio of shear stress to shear strain; hence, it is a measure of resistance to shear stress.
- Young’s modulus is the ratio of the longitudinal stress to the longitudinal strain associated with a cylindrical rod that is subjected to a longitudinal extension in the axial direction. Since strain is a dimensionless quantity, Young’s modulus has the dimensions of stress.
- Poisson’s ratio is the ratio of the lateral contraction to longitudinal extension associated with a cylindrical rod that is subjected to a longitudinal extension in the axial direction. Since strain is a dimensionless quantity, Poisson’s ratio is a pure number.
- Seismic waves are elastic waves that propagate in the earth.
- P-waves (or equivalently, compressional waves, longitudinal waves, or dilatational waves) are waves with particle motion in the direction of wave propagation.
- S-waves (or equivalently, shear waves, transverse waves, or rotational waves) are waves with particle motion in the direction perpendicular to the direction of wave propagation.
- Reflection is the wavefield phenomenon associated with the fraction of incident wave energy that is returned from an interface that separates two layers with different elastic moduli.
- Refraction is the the wavefield phenomenon associated with the fraction of incident wave energy that is transmitted into the next layer.
- Diffraction is the wavefield phenomenon associated with energy that propagates outward from a sharp discontinuity in the subsurface.
In exploration seismology, we are primarily interested in compressional and shear waves that travel through the interior of solid layers, and thus are characterized as body waves. Whereas in earthquake seismology, we also make use of Love and Rayleigh waves which travel along layer boundaries, and thus are characterized as surface waves.
Both body waves and surface waves are different forms of elastic waves, each associated with a specific type of particle motion. In the case of compressional waves, the particle motion induced by a compressional stress is in the direction of wave propagation. The compressional stress causes a change in the particle dimension or volume. The more the rock resists to the compressional stress, the higher the compressional wave velocity. In the case of shear waves, the particle motion induced by a shear stress is in the direction perpendicular to the direction of wave propagation. The shear stress does not cause a change in the particle dimension or volume; instead, it changes particle shape. The more the rock resists shear stress, the higher the shear wave velocity. Under the assumption that both wave types are elastic, whatever the change induced by the wave motion — the elastic deformation in particle shape, dimension or volume, vanishes once the wave motion on the particle vanishes and is propagated onto the neighboring particle.
Figure 11.0-1 outlines the interrelationships between the various elastic parameters. Starting with Young’s modulus — the ratio of principal stress to principal strain, and Poisson’s ratio — the ratio of shear strain to principal strain, Lamé’s constants — λ and μ are defined. The Lamé constant μ is indeed the modulus of rigidity and the Lamé constant λ = κ − (2/3)μ, where κ is the bulk modulus (see the mathematical foundation of elastic wave propagation for more information). Then, the two wave velocities — compressional (P-waves) and shear (S-waves), are derived in terms of the Lamé constants, or the bulk modulus and modulus of rigidity, and density.
P- and S-wave velocities
From the definitions of the P- and S-wave velocities in Figure 11.0-1, note that both are inversely proportional to density ρ. At first thought, this means that the lower the rock density the higher the wave velocity. A good example is halite which has low density (1.8 gr/cm3) and high P-wave velocity (4500 m/s). In most cases, however, the higher the density the higher the velocity (Figure 11.0-2). This is because an increase in density usually is accompanied by an increase in the ability of the rock to resist compressional and shear stresses. So an increase in density usually implies an increase in bulk modulus and modulus of rigidity. Returning to the expressions for the P- and S-wave velocities in Figure 11.0-1, note that the greater the bulk modulus or the modulus of rigidity, the higher the velocity. Based on field and laboratory measurements, Gardner  established an empirical relationship between density ρ and P-wave velocity α. Known as Gardner’s formula for density, this relationship given by ρ = cα0.25, where c is a constant that depends on the rock type, is useful to estimate density from velocity when the former is unknown. With the exception of anhydrites, most rock types — sandstones, shales, and carbonates, tend to obey Gardner’s equation for density.
Figure 11.0-2 Variations in P-wave velocity with various rock types with different densities .
Figure 11.0-3 Crossplot of P-wave slowness versus S-wave slowness based on laboratory measurements with various rock types .
Figure 3.0-3 Velocity range for rocks of different lithologic compositions at different depths of burial. Adapted from ; courtesy American Association of Petroleum Geologists.
In introduction to velocity analysis and statics corrections, a brief review of the results of some of the key laboratory experiments on seismic velocities was made. For a given lithologic composition, seismic velocities in rocks are influenced by porosity, pore shape, pore pressure, pore fluid saturation, confining pressure, and temperature. It is generally accepted that confining pressure, and thus the depth of burial, has the most profound effect on seismic velocities (Figure 3.0-3). For instance, the P-wave velocities for clastics can vary from 2 km/s at the surface up to 5 km/s and for carbonates from 3 km/s at the surface up to 6 km/s at depths greater than 5 km.
Because of the large variations in P-wave velocities caused by all these factors, P-wave velocity alone is not adequate to infer the lithology, unambiguously. The ambiguity in lithologic identification can be resolved to some extent if we have the additional knowledge of S-wave velocities. Here, we examine the ratio of the S-wave velocity to the P-wave velocity, β/α, which only depends on Poisson’s ratio σ (Figure 11.0-1). In some instances, we refer to the inverse ratio α/β. The higher the Poisson’s ratio, the higher the velocity ratio α/β. This relationship is supported by the physical meaning of Poisson’s ratio — the ratio of shear strain to principle strain. A way to describe the physical meaning of Poisson’s ratio is to consider a metal rod that is subject to an extensional strain. As the rod is stretched, its length increases while its thickness decreases. Hence, the less rigid the rock, the higher the Poisson’s ratio. This is exactly what is implied by the expression in Figure 11.0-1 that relates the modulus of rigidity μ to Poisson’s ratio σ. Unconsolidated sediments or fluid-saturated reservoir rocks have low rigidity, hence high Poisson’s ratio and high velocity ratio α/β. Here is the first encounter with a direct hydrocarbon indicator — the P- to S-wave velocity ratio. Ostrander  was the first to publish the link between a change in Poisson’s ratio and change in reflection amplitude as a function of offset.
Figure 11.0-4 (a) Crossplot of P-wave velocity versus S-wave velocity derived from full-waveform sonic logs using rock samples with different lithologies — SS: sandstone, SH: shale and LS: limestone; (b) crossplot of the velocity ratio versus the P-wave velocity using the same sample points as in (a) .
Figure 11.0-5 Variation of the velocity ratio with respect to (a) shale content and (b) clay content .
Aside from the direct measurement of S-wave velocities down the borehole, there are three indirect ways to estimate the S-wave velocities. The first approach is to perform prestack amplitude inversion to estimate the P- and S-wave reflectivities and thus compute the corresponding acoustic impedances (analysis of amplitude variation with offset). The second approach is to record multicomponent seismic data and estimate the S-wave velocities from the P-to-S converted-wave component (4-C seismic method). The third approach is to generate and record S-waves themselves.
Figure 11.0-3 shows a plot of the S-wave slowness (inverse of the S-wave velocity) versus the P-wave slowness (inverse of the P-wave velocity) based on laboratory measurements . Figure 11.0-4a shows a plot of the P-wave velocity to S-wave velocity based on full-waveform sonic logs from a producing oil field . The key observation made from these results is that a lithologic composition may be associated with a reasonably distinctive velocity ratio α/β. The shale and limestone samples fall on a linear trend that corresponds to a velocity ratio of 1.9, whereas the dolomite samples have a velocity ratio of 1.8. The sandstone samples have a range of velocity ratio of 1.6 to 1.7. Lithologic distinction sometimes is more successful with a crossplot of P-wave to S-wave velocity ratio versus the P-wave velocity itself . This is illustrated in Figure 11.0-4b which shows the same sample points as in Figure 11.0-4a.
Effect of shale and clay content on the velocity ratio α/β is an important factor in lithologic identification. Field and laboratory data from sandstone cores indicate that the velocity ratio α/β increases with increasing shale and clay content as a result of a decrease in S-wave velocity (Figure 11.0-5).
Finally, effect of porosity on the velocity ratio α/β is generally dictated by the pore shape. For limestones with their pores in the form of microcracks, the velocity ratio increases as the percent porosity increases . For sandstones with their rounded pores, the velocity ratio does not increase as much with increasing porosity . The difference between the rounded pores and microcracks lies in the fact that it is easier to collapse a rock with microcracks, hence lower modulus of rigidity.
- Gardner et al. (1974), Gardner, G. H. F., Gardner, L. W., and Gregory, A. R., 1974, Formation velocity and density — The diagnostic basis for stratigraphic traps: Geophysics, 39, 770–780.
- Pickett, 1963, Pickett, G. R., 1963, Acoustic character logs and their applications in formation evaluation: J. Can. Petr. Tech., 15, 659–667.
- Sheriff, 1976, Sheriff, R. E., 1976, Inferring stratigraphy from seismic data: Am. Assn. Petr. Geol. Bull., 60, 528–542.
- Ostrander (1984), Ostrander, W. J., 1984, Plane-wave reflection coefficients for gas sands at nonnormal angles of incidence: Geophysics, 49, 1637–1648.
- Miller and Stewart, 1999, Miller, S. L. M. and Stewart, R. R., 1999, Effects of lithology, porosity and shaliness on P- and S-wave velocities from sonic logs: SEG Continuing Education Class Notes.
- Eastwood and Castagna, 1983, Eastwood, R. L. and Castagna, J. P., 1983, Basis for interpretation of vp/vS ratios in complex lithologies: 24th Ann. Meeting of Logging Symp., Soc. Prof. Well Log Analysts.
- Seismic resolution
- Analysis of amplitude variation with offset
- Acoustic impedance estimation
- Vertical seismic profiling
- 4-D seismic method
- 4-C seismic method
- Seismic anisotropy
- Mathematical foundation of elastic wave propagation