Seismic anisotropy is the directional dependence of the velocity of seismic waves in a medium (rock) within the Earth.
Description A material is said to be anisotropic if the value of one or more of its properties varies with direction. Anisotropy differs from the property called heterogeneity in that anisotropy is the variation in values with direction at a point while heterogeneity is the variation in values between two or more points. Seismic anisotropy can be defined as the dependence of seismic velocity on direction or upon angle. General anisotropy is described by a 4th order elasticity tensor with 21 independent elements. However, in practice observational studies are unable to distinguish all 21 elements, and anisotropy is usually simplified. In the simplest form, there are two main types of anisotropy, both of them are called transverse isotropy (it is called transverse isotropy because there is isotropy in either the horizontal or vertical plane) or polar anisotropy. The difference between them is in their axis of symmetry, which is an axis of rotational invariance such that if we rotate the formation about the axis, the material is still indistinguishable from what it was before. The symmetry axis is usually associated with regional stress or gravity.
Vertical transverse isotropy (VTI), transverse isotropy with a vertical axis of symmetry, is associated with layering and shale and is found where gravity is the dominant factor. Horizontal transverse isotropy (HTI), transverse isotropy with a horizontal axis of symmetry, is associated with cracks and fractures and is found where regional stress is the dominant factor. The transverse anisotropic matrix has the same form as the isotropic matrix, except that it has five non-zero values distributed among 12 non-zero elements. Transverse isotropy is sometimes called transverse anisotropy or anisotropy with hexagonal symmetry. In many cases the axis of symmetry will be neither horizontal nor vertical, in which case it is often called "tilted".
History of the recognition of anisotropy Anisotropy was first recognised in the 19th century following the theory of Elastic wave propagation. George Green (1838) and Lord Kelvin (1856) took anisotropy into account in their articles on wave propagation. Anisotropy entered seismology in the late 19th century and was introduced by Maurycy Rudzki. From 1898 till his death in 1916, Rudzki attempted to advance the theory of anisotropy, he attempted to determine the wavefront of a transversely isotropic medium (TI) in 1898 and in 1912 and 1913 he wrote on surface waves in transversely isotropic half space and on Fermat's principle in anisotropic media respectively. With all these, the advancement of anisotropy was still slow and in the first 30 years (1920–1950) of exploration seismology only a few papers were written on the subject. More work was done by several scientists such as Helbig (1956) who observed while doing seismic work on Devonian schists that velocities along the foliation were about 20% higher than those across the foliation. However the appreciation of anisotropy increased with the proposition of a new model for the generation of anisotropy in an originally isotropic background and a new exploration concept by Crampin (1987). One of the main points by Crampin was that the polarization of three component shear waves carries unique information about the internal structure of the rock through which they pass, and that shear wave splitting may contain information about the distribution of crack orientations. With these new developments and the acquisition of better and new types of data such as three component 3D seismic data, which clearly show the effects of shear wave splitting, and wide azimuth 3D data which show the effects of azimuthal anisotropy, and the availability of more powerful computers, anisotropy began to have great impact in exploration seismology in the past three decades.
Concept of seismic anisotropy Since the understanding of seismic anisotropy is closely tied to the shear wave splitting, this section begins with a discussion of shear wave splitting. Shear waves have been observed to split into two or more fixed polarizations which can propagate in the particular ray direction when entering an anisotropic medium. These split phases propagate with different polarizations and velocities. Crampin (1984) amongst others gives evidence that many rocks are anisotropic for shear wave propagation. In addition, shear wave splitting is almost routinely observed in three-component VSPs. Such shear wave splitting can be directly analyzed only on three component geophones recording either in the subsurface, or within the effective shear window at the free surface if there are no near surface low-velocity layers. Observation of these shear waves show that measuring the orientation and polarization of the first arrival and the delay between these split shear waves reveal the orientation of cracks and the crack density . This is particularly important in reservoir characterization. In a linearly elastic material, which can be described by Hooke's law as one in which each component of stress is dependent on every component of strain, the following relationship exists:
σ i j = C i j k l e k l i , j , k , l = 1 , 2 , 3 {\displaystyle \sigma _{ij}=C_{ijkl}e_{kl}\quad i,j,k,l=1,2,3}
where σ is the stress, C is the elastic moduli or stiffness constant, and e is the strain. The elastic modulus matrix for an anisotropic case is
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