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# Compressional and shear velocities

(Redirected from Elastic wave velocities)

Elastic waves are comprised of compressional (or P-waves) and shear (or S-waves). In compressional waves, the particle motion is in the direction of propagation. In shear waves, the particle motion is perpendicular to the direction of propagation. Understanding the velocity of these waves provide valuable information about the rocks and fluids through which they propagate.

## Elastic wave velocity

Stress strain relationships in rocks considered only the static elastic deformation of materials. By adding the dynamic behavior, we arrive at how elastic waves propagate through materials. If a body is changing its speed as well as deforming, there will be an unbalanced force because of the acceleration described through Newton’s Second Law: ....................(1)

where ρ is density, a is acceleration, u is displacement, and t is time. Combining this with Hook’s Law (SeeStress strain relationships in rocks) gives the general wave equation. For a plane wave in the xx direction, this can be written as ....................(2)

However, if the material is being deformed, we will have strains associated with the change of displacement with position. In turn, these strains can be related to the stresses through the appropriate modulus, M: ....................(3)

For constant elastic components, this simplifies to ....................(4)

### Compressional velocity

The solution to this equation gives the compressional velocity ....................(5)

### Shear velocity

Similarly, for shear motion ....................(6)

and we get the shear velocity: ....................(7)

## Cracked rock

For some cracked rocks, different methods of calculating velocities and the effects of pore fluids are preferable. Numerous theories have been developed to describe the effects of crack-like pores. Most view cracks as ellipsoids with their aspect ratio, α, defined as the ratio of the semiminor to semimajor axes. Eshelby examined the elastic deformation of such elliptical inclusions, and these results were then applied to the compressibility of rocks by Walsh. In concept, long, narrow cracks are compliant and can be very effective at reducing the rock moduli at low crack porosities. The primary controlling factor for these elliptical fractures is the aspect ratio, α, defined as the ratio of the ellipse semiminor (a) to semimajor (b) axes: ....................(1)

The smaller the value of α, the softer the crack and cracked rock, resulting in lower velocities and stronger pressure dependence.

### Assumptions

Numerous assumptions are made in the derivation and application of cracked media models, such as the following:

1. Porous material is isotropic, elastic, monomineralic, and homogeneous
2. Fracture population is dilute, and few, or only first-order, mechanical interactions occur among fractures
3. Fractures can be described by simple shapes
4. Pore-fluid system is closed, and there is no chemical interaction between fluids and rock frame (however, shear modulus need not remain constant).

Some of these assumptions may be dropped, depending on the model involved. For example, Hudson specifically includes the effect of anisotropic crack distributions.

### Kuster-Tokuz model

One particularly useful result was derived by Kuster and Toksoz. Using scattering theory, they derived the general relation of bulk and shear moduli of the cracked rock (K*, μ*) to the crack porosity (c), aspect ratio (αm), mineral (K0, μ0), and inclusion or crack moduli (K′, μ′) (Cheng and Toksoz). ....................(2) ....................(3)

Here, T1 and T2 are scalar functions of K0, μ0, K′, and μ′, and correspond to Tiijj and Tijij in Kuster and Toksoz. This formulation allows the effects of several populations (several values of m ) of cracks to be summed. The general limitation is that the porosity for any particular aspect ratio cannot exceed the value of the aspect ratio itself.

The results of the Kuster-Toksoz model are shown in Fig. 1. Numerous important features should be noted. Velocities drop rapidly for long, narrow cracks (small α), with even small crack porosities. For such soft cracks, the increase in velocity is dramatic. At a shape close to spherical (α above about 0.5), the pores are stiff, and the change in density dominates. Thus, with α s close to unity, going from dry to water-saturated actually decreases the velocity. Notice also that for small aspect ratios, the shear velocity increases with water saturation. This requires a changing shear modulus with saturation, in direct violation of a primary assumption of Gassmann’s relations. This changing shear modulus is one reason why Gassmann’s relations may not work well in fractured rocks.

### Comparison between Gassmann's and Kuster-Tokuz technique

An example of a rock modeled by both Gassmann’s and Kuster-Toksoz techniques is shown in Fig. 2. For this limestone, Gassmann’s relations substantially underestimate the effect of liquid saturation. The Kuster-Toksoz prediction for oil saturation is close to the experimental observed values. However, the success of this model is not quite as spectacular as it seems, because an arbitrary population of fractures and aspect ratios (αms) can be included to force such a good fit. The actual population of cracks in rocks remains unknown.

The expressions in Eqs. 2 and 3 are complicated and difficult to apply. The linear relation of normalized velocities to crack aspect ratio and porosity suggests that a simplified form can be derived to give a first-order approximation. ....................(4)

## Anisotropy

In reality, most rocks are anisotropic to some degree. Some dominant lithologies, such as shales, are by definition anisotropic (otherwise, they are mudstones). In addition, many ubiquitous sedimentary features such as bedding will lead to anisotropy on a larger scale. In-situ stresses are anisotropic (Fig. 3), resulting in an anisotropy in rock properties.

Anisotropy in transport properties such as permeability is a common concern in describing reservoir flow. Fractured reservoirs typically have a preferred fracture and flow direction, and these directions often can be ascertained from oriented borehole or surface seismic data.

### Wave motion in anisotropic rocks

An interesting aspect of anisotropy is the phenomenon of shear-wave splitting. Elastic anisotropy means that the stiffness or effective moduli in one direction will be different from that in another. For shear waves, their particle motion will be approximately normal to the direction of propagation. The velocity will depend on the orientation of the particle motion. The shear wave will then "split" into two shear waves with orthogonal particle motion, each traveling with the velocity determined by the stiffness in that direction. An example of this is shown in Fig. 4 from Sondergeld and Rai. The recorded waveform can be seen as two distinct shear waves traveling at their own velocities. Note that when these distinct waves are examined in isolation, their velocity is independent of direction. A single input wave has been split into two waves. This is similar to the image splitting in optics when light travels through an anisotropic medium. On the other hand, because compressional waves have particle motion only along the direction of propagation, they have no splitting.

Although the split shear waves may travel each with a constant velocity, the amplitude within each will be strongly dependent on angle. The energy of the initial single shear wave is partitioned as vector components in each of the principal directions. This amplitude dependence on angle is shown in Fig. 5, also from Sondergeld and Rai. Figs. 3 and 4 demonstrate that measurement of seismic shear waves at the surface will be useful in delineating in-situ anisotropy directions. This anisotropy can then be related to factors such as oriented fractures and in-situ stress directions.

### Anisotropic parameter

A typical homogeneous but bedded sedimentary unit would have a horizontal plane of symmetry as well as a vertical symmetry axis of rotation. This situation is commonly referred to as Vertical Transverse Isotropy (VTI), although the term "Polar Anisotropy" has also been suggested (Thomsen). For "weak" anisotropy (Thomsen), the dependence of velocities as a function of angle (θ) from the symmetry axis can be written as ....................(1) ....................(2) ....................(3)

where Vp(θ) is the compressional velocity and VS-(θ) and VS

|(θ) are the shear velocities with particle polarizations perpendicular and parallel to the symmetry plane (e.g., bedding), respectively.

The Thomsen anisotropic parameter ε can be defined as ....................(4)

where VP0 is the compressional velocity parallel to the axis of symmetry, and VP90 is the velocity perpendicular to this axis. The parameter γ can be defined as ....................(5)

where VS0 is the shear velocity parallel to axis of symmetry, and VS|90 is the velocity perpendicular to this axis.

The anisotropic parameter δ is more difficult to characterize, and is the primary component modifying the compressional moveout velocity from the isotropic case. To describe it, we must refer back to stiffness defined in the generalized Hooke’s law given in Eq. 6. ....................(6) ....................(7)

The advantage of these formulations is that they can be extracted from observed shear-wave splitting or extracted from normal moveout (NMO) corrections during seismic processing. Thus, they provide a valuable tool to describe the anisotropic character of reservoirs from remote measurements.

## Relationships

Relationships have been developed between the elastic wave velocities and:

## Nomenclature

 K = bulk modulus, GPa or MPa Ko = mineral bulk modulus, GPa or MPa M = molecular weight, g/mo MO = reference oil molecular weight, g/mole t = time, s T1, T2 = Kunster-Toksoz coefficients Vrock = rock velocity, m/s Vp = compressional velocity, m/s Vs = shear velocity, m/s x = directional component, m y = directional component, m z = directional component, m εij = strain components, fractional εshear = shear strain, fractional μ = shear modulus, GPa or MPa λ = wavelength, MPa−1 ρ = density, kg/m3 or g/cm3 σij = stress components, GPa or MPa Vp = compressional velocity, m/s