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:
where is the stress, is the elastic moduli or stiffness constant, and is the strain.
The elastic modulus matrix for an anisotropic case is
The above is the elastic modulus for a vertical transverse isotropic medium (VTI), which is the usual case. The elastic modulus for a horizontal transverse isotropic medium (HTI) is:
For an anisotropic medium, the directional dependence of the three phase velocities can be written by applying the elastic moduli in the wave equation is; The direction dependent wave speeds for elastic waves through the material can be found by using the Christoffel equation and are given byG. Mavko, T. Mukerji, J. Dvorkin. The Rock Physics Handbook. Cambridge University Press 2003 (paperback).
where is the angle between the axis of symmetry and the wave propagation direction, is mass density and the are elements of the elastic stiffness matrix. The Thomsen parameters are used to simplify these expressions and make them easier to understand.
Seismic anisotropy has been observed to be weak, and Thomsen (1986) rewrote the velocities above in terms of their deviation from the vertical velocities as follows;
where
are the P and S wave velocities in the direction of the axis of symmetry () (in geophysics, this is usually, but not always, the vertical direction). Note that may be further linearized, but this does not lead to further simplification.
The approximate expressions for the wave velocities are simple enough to be physically interpreted, and sufficiently accurate for most geophysical applications. These expressions are also useful in some contexts where the anisotropy is not weak.
The Thomsen parameters are anisotropic and are three non-dimensional combinations which reduce to zero in isotropic cases, and are defined as
Origin of anisotropy Anisotropy has been reported to occur in the Earth's three main layers: the crust, mantle and the core.
The origin of seismic anisotropy is non-unique, a range of phenomena may cause Earth materials to display seismic anisotropy. The anisotropy may be strongly dependent on wavelength if it is due to the average properties of aligned or partially aligned heterogeneity. A solid has intrinsic anisotropy when it is homogeneously and sinuously anisotropic down to the smallest particle size, which may be due to crystalline anisotropy. Relevant crystallographic anisotropy can be found in the upper mantle. When an otherwise isotropic rock contains a distribution of dry or liquid-filled cracks which have preferred orientation it is named crack induced anisotropy. The presence of aligned cracks, open or filled with some different material, is an important mechanism at shallow depth, in the crust. It is well known that the small-scale, or microstructural, factors include (e.g. Kern & Wenk 1985; Mainprice et al. 2003): (1) crystal lattice preferred orientation (LPO) of constituent mineral phases; (2) variations in spatial distribution of grains and minerals; (3) grain morphology and (4) aligned fractures, cracks and pores, and the nature of their infilling material (e.g. clays, hydrocarbons, water, etc.). Because of the overall microstructural control on seismic anisotropy, it follows that anisotropy can be diagnostic for specific rock types. Here, we consider whether seismic anisotropy can be used as an indicator of specific sedimentary lithologies within the Earth's crust.
In sedimentary rocks, anisotropy develops during and after deposition. For anisotropy to develop, there needs to be some degree of homogeneity or uniformity from point to point in the deposited clastics. During deposition, anisotropy is caused by the periodic layering associated with changes in sediment type which produces materials of different grain size, and also by the directionality of the transporting medium which tends to order the grains under gravity by grain sorting. Fracturing and some diagenetic processes such as compaction and dewatering of clays, and alteration etc. are post depositional processes that can cause anisotropy.
The importance of anisotropy in hydrocarbon exploration and production In the past two decades, the seismic anisotropy has dramatically been gaining attention from academic and industry, due to advances in anisotropy parameter estimation, the transition from post stack imaging to pre stack depth migration, and the wider offset and azimuthal coverage of 3D surveys. Currently, many seismic processing and inversion methods utilize anisotropic models, thus providing a significant enhancement over the seismic imaging quality and resolution. The integration of anisotropy velocity model with seismic imaging has reduced uncertainty on internal and bounding-fault positions, thus greatly reduce the risk of investment decision based heavily on seismic interpretation.
In addition, the establishment of correlation between anisotropy parameters, fracture orientation, and density, lead to practical reservoir characterization techniques. The acquisition of such information, fracture spatial distribution and density, the drainage area of each producing well can be dramatically increased if taking the fractures into account during the drilling decision process. The increased drainage area per well will result in fewer wells, greatly reducing the drilling cost of exploration and production (E&P) projects.
The application of the anisotropy in petroleum exploration and production Among several applications of seismic anisotropy, the following are the most important: anisotropic parameter estimation, prestack depth anisotropy migration, and fracture characterization based on anisotropy velocity models.
Anisotropy parameter estimation The anisotropy parameter is most fundamental to all other anisotropy application in E&P area. In the early days of seismic petroleum exploration, the geophysicists were already aware of the anisotropy-induced distortion in P-wave imaging (the major of petroleum exploration seismic surveys). Although the anisotropy-induced distortion is less significant since the poststack processing of narrow-azimuth data is not sensitive to velocity. The advancement of seismic anisotropy is largely contributed by the Thomsen's work on anisotropy notation and also by the discovery of the P-wave time-process parameter . These fundamental works enable to parametrize the transverse isotropic (TI) models with only three parameters, while there are five full independent stiff tensor element in transverse isotropic (VTI or HTI) models. This simplification made the measurement of seismic anisotropy a plausible approach.