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Variation of seismic velocity depending on either the direction of travel (for P- or S- waves) or the direction of polarization (for S-waves). Velocity anisotropy (or coefficient of anisotropy) is sometimes taken as the fractional difference between the maximum and minimum velocities in different directions, , often expressed as a percentage, sometimes as the ratio of maximum and minimum velocities, ; the numerical value usually makes clear which is meant. P-wave anisotropy is usually meant, unless S-wave anisotropy is specified, but anisotropy of P-waves usually implies anisotropy for S-waves and vice-versa. Anisotropy may be analyzed in terms of the symmetry of the medium:
- The general elasticity tensor (stiffness or its inverse compliance) is a 3X3X3X3 tensor relating stress and strain. Among its 81 components, it contains up to 21 independent constants, the number depending on the symmetry (see symmetry systems). Because of symmetries, this tensor may be written as a 6X6 matrix. In the general case, for each direction of plane-wave propagation there is one "quasi-longitudinal" wave and two "quasi-shear" waves (with two characteristic angles of polarization, and two different velocities, leading to "shear-wave splitting".
- In isotropic media (the simplest case) there are only two independent constants among 12 nonzero elements of this matrix. These may be taken as the longitudinal modulus M and the shear modulus . A non-zero linear combination, also appears in the matrix. Other elastic parameters, such as the bulk modulus may also be computed from these, see E-5. In the isotropic case, the longitudinal wave is exactly longitudinal (polarization vector exactly parallel to wave vector), and the two shear waves degenerate to one (with a velocity independent of polarization direction, which may lie anywhere within the plane exactly perpendicular to the wave vector, hence no shear-wave splitting).
- Polar anisotropy (transverse isotropy) is the simplest geophysical case of anisotropy. It has elastic properties that are independent of the azimuth about a polar axis (hence the name) of symmetry, which is usually vertical. It is associated with, for example, unfractured shales or thinly-bedded sequences (see Physical Causes of Anisotropy, below). An example is given in Figure A-14. Polar anisotropy has five independent constants among 12 nonzero elements of the stiffness (or compliance) matrix. However, the anisotropic seismic behaviour is governed directly by certain combinations of these five stiffnesses; for body waves they may be taken as the Thomsen anisotropic parameters.
- Azimuthal anisotropy is a general term describing all lower symmetries, which all have azimuthal variation of elastic properties. In exploration geophysics, it is usually caused by aligned fractures (see Physical Causes of Anisotropy).
- The simplest plausible case of azimuthal anisotropy, in exploration geophysics, is that of Orthorhombic anisotropy (more properly: "orthotropic"). It has the symmetry of a brick, with nine independent constants among 12 nonzero elements of the stiffness (or compliance) matrix. In exploration geophysics, it is usually caused by a single set of aligned fractures in an otherwise polar anisotropic medium, or perhaps two such sets, orthogonal to each other (see Physical Causes of Anisotropy). Despite the complexity, it is commonly feasible to analyze modern wide-azimuth (WAZ) datasets in terms of orthorhombic anisotropy .
- The most realistic case of anisotropy is that of monoclinic anisotropy. It has twelve independent constants among 18 nonzero elements of the stiffness (or compliance) matrix.
Physical Causes of Anisotropy
Anisotropy is always the large-scale expression of small-scale structure with preferred orientation. This means that the measurement of seismic anisotropy will vary with the wavelength used to measure it.
- At the scale of a single crystal (giga-Hertz frequencies), the small-scale structure is the arrangement of atomic cells.
- At the scale of a core (mega-Hertz frequencies), the small-scale structure is the micro-geometry of the arrangement of grains and pores. In some rocks, the crystals are randomly oriented, so that although each is intrinsically anisotropic, the random averaging means that the rock itself is isotropic (this might describe some sandstones). In other rocks (e.g. shales), there is a preferred orientation of some crystals (e.g. clays, which often have the shapes of platelets, and commonly lie horizontally), usually established by the direction of gravity during sedimentation and lithification. (The complementary pore space, occupied by fluid or kerogen, is then also flat-lying.) By itself, this usually leads to polar anisotropy.
Also at the core scale, there may be other physical causes in special circumstances. For example, if a salt body has flowed into place (rather than precipitated into place), the flow process can include re-crystallization, with preferential orientation of salt grains, related to the flow. (Tectonic flow in the upper mantle may include re-crystallization of olivine and other minerals, with preferred orientations related to that flow.)
At this core scale, there may also be micro-fractures, related to the state of stress, or a previous state of stress. It is sometimes thought that the stress itself (if the 3 principle stresses are not equal) makes for anisotropy . However, this direct stress-anisotropy is purely elastic, and so disappears once the stress is removed. By contrast, if the finite stress changes the micro-structure of the rock, e.g. by creating micro-fractures, this is called indirect stress-anisotropy, and may not be reversible, with the removal of the stress. (This could happen, for example, if the fractures were open for a long time, and minerals precipitated out of the fluid, on the crack-faces, so that the cracks do not close when the stress is removed.) In the lab, it is possible to distinguish these cases; normally one finds that the direct effect of stress is much smaller than the indirect effect.
If stress-aligned micro-fractures are present, they may be preferentially aligned with flat faces perpendicular to the direction of least compressive stress. There may be a second set, preferentially aligned with flat faces perpendicular to the direction of intermediate compressive stress. Rarely, there is a third set, preferentially aligned with flat faces perpendicular to the direction of most compressive stress. If (as is common in sedimentary basins), the maximum stress is oriented vertically, then the first two sets mentioned here are vertical, and orientated in two orthogonal horizontal directions.
- At the logging scale (kilo-Hertz frequencies), the small-scale structure may also include the layering which results from the sedimentary process, with layer-thicknesses small compared to the sonic wavelengths. By itself, this usually leads to polar anisotropy. The thin-layering need not not be periodic, although if the statistics of the layering are not stationary, then the sequence is effectively inhomogeneous, as well as anisotropic. There may also be stress-induced anisotropy (both direct and indirect), caused by the stress concentrations near the borehole, which accompany the creation of the borehole wall. This will cause lower symmetry of anisotropy.
- At the reservoir scale, (seismic frequencies), the small-scale structure may also include the layering which results from the sedimentary process, with layer-thicknesses small compared to the seismic wavelength. By itself, this usually leads to polar anisotropy. (In the sub-crustal lithosphere, there may be azimuthal anisotropy at this scale, caused by sheeted-dike emplacement at spreading ridges.)
There may also be both direct and indirect (or crack-related) stress-induced anisotropy; in field data, it may not be possible to distinguish between these causes. Hence, the observed anisotropy orientations might indicate directions of either the current stress-state, or a previous stress-state, or both. If the fractures yield monoclinic anisotropy, this is an indication of a complex geologic history, which places its complex character on the current anisotropy.
If aligned cracks are present, their shapes may be limited, top and bottom, by fracture-resistant beds. Since there is no corresponding limitation in the horizontal direction, they may be "ribbon-shaped" joints, rather than "penny-shaped" micro-cracks. Such joints usually dominate the hydraulic anisotropy, although they may or may not dominate the seismic anisotropy.
- Crampin, Stuart (1981). "A review of wave motion in anisotropic and cracked elastic-media". Wave Motion 3 (4): 343–391. doi:10.1016/0165-2125(81)90026-3. ISSN 01652125.
- Thomsen, Leon (1986). "Weak elastic anisotropy". GEOPHYSICS 51 (10): 1954–1966. doi:10.1190/1.1442051.
- Alkhalifah, Tariq; Tsvankin, Ilya (1995). "Velocity analysis for transversely isotropic media". GEOPHYSICS 60 (5): 1550–1566. doi:10.1190/1.1443888.
- Thomsen, Leon (2002). Understanding seismic anisotropy in exploration and exploitation: SEG-EAEG Distinguished Instructor Series #5. SEG and EAGE. doi:10.1190/1.9781560801986. ISBN 978-1-56080-198-6.
- Tsvankin, Ilya (1997). "Anisotropic parameters and P‐wave velocity for orthorhombic media". GEOPHYSICS 62 (4): 1292–1309. doi:10.1190/1.1444231.
- Backus, George E. (1962). "Long-wave elastic anisotropy produced by horizontal layering". Journal of Geophysical Research 67 (11): 4427–4440. doi:10.1029/JZ067i011p04427.