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Sandstone bar Scanning electron photomicrographs A laminated solid Random media Source-receiver configuration Seismograms Seismograms Seismograms Seismograms Seismograms Seismograms Seismograms Seismograms Seismograms Snapshots of wave propagation Equivalent medium theory A 1D random medium VTI model A snapshot of wave propagation through a VTI medium A snapshot of wave propagation through a VTI HTI model A snapshot of wave propagation through a HTI medium A snapshot of wave propagation through a HTI medium An orthorhombic model A snapshot of wave propagation through a orthorhombic medium A snapshot of wave propagation through a orthorhombic medium A snapshot of wave propagation through a orthorhombic medium A snapshot of wave propagation through a orthorhombic medium S-wavefront Monoclinic medium Snapshots of wave propagation through a monoclinic medium Snapshots of wave propagation through a monoclinic medium Snapshots of wave propagation through a monoclinic medium Snapshots of wave propagation through a monoclinic medium Snapshots of wave propagation through a monoclinic medium Snapshots of wave propagation through a monoclinic medium Snapshots of wave propagation through a monoclinic medium S-wavefronts in an monoclinic medium S-wavefronts in an monoclinic medium Shear-wave splitting Shear-wave splitting Alford rotation Fast and slow shear wave directions Wave splitting Synthetic zero-offset VSP data Source signature Zero-offset VSP data Shear sonic experiment Sonic shear-splitting data A sonic tool Slowness and velocity surfaces Wavefront surface Wavefronts of the qP-wave in four homogeneous VTI media Wavefronts of the qSV-wave in four homogeneous VTI media Weak approximation Amplitude as a function of source-obersevation distance Linear attenuation models Drift values Drift curve Drift curves The drift phenomena Positive drift Random media with an exponential autocorrelation function Negative drift

Images in: Chapter 12

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Sandstone bar

Images - Chapter 12 Crossbedding in a Devonian fluvial sandstone bar, from the Catskill delta, New York, USA.

Like all substances, rock formations are made of atoms, and they contain gaps or empty spaces (see Figure 2-3). This feature is especially true for sedimentary rocks, which comprise most petroleum reservoirs (see Appendix A). Except in chapters 1 and 2, our discussion in this book has so far totally disregarded the atom scale (microscopic scale) by assuming that most rock formations can be described as isotropic piecewise-continuous regions, separated by interfaces in which the medium parameters are discontinuous. In other words, we have so far described the subsurface as an isotropic heterogeneous medium in which elastic properties can vary from one point to another
(i.e., a heterogeneous-medium assumption), but for any given point of the medium, these elastic properties cannot vary with direction (i.e., the isotropic-medium assumption). The word point here represents a particle (representitive of a volume) at a macroscopic scale whose size is of the order of a quarter of a seismic wavelength, about 6 m or more.

Evidences of heterogeneities much smaller than the particle are abundant. The crossbedding in this figure and the photomicrograph of limestone in the next figure are just a few of many evidences of heterogeneities at scales much smaller than those of particles. Therefore, a model of the earth which ignores small-scale heterogeneities is bound to be inadequate for describing some rock formations. However, the laws of continuous mechanics that we are using today to study seismic wave propagation and to analyze seismic data are valid only at the particle scale. So the dilemma that petroleum seismologists face is how to process and interpret seismic data at the particle scale while taking into account some of the behaviors of rock formations at a scale much smaller than that of particles. One way of addressing this dilemma is to consider that rock formations can behave as an anisotropic medium at the particle scale; i.e., elastic properties at a given point of the medium can vary with direction (i.e., the anisotropic-medium assumption) in addition to the fact that these properties can vary from one point to another (i.e., the heterogeneous-medium assumption).

Scanning electron photomicrographs

Images - Chapter 12 Scanning electron photomicrographs showing (a) aligned and (b) randomly oriented grains. In the left image, the alignment is apparent because the solution effect has dissolved intergranular cement. In the right image, layers are aligned within each kaolinite booklet, but booklets are randomly oriented. These are samples of Bassien limestone from the Mukta field, offshore Bombay, India.

The two limestones in this figure provide good illustrations of anisotropic behavior at the particle scale. The measurements of velocity for the limestone with randomly packed grains in (a) do not show any noticeable velocity variations with direction, whereas the measurements of velocity for the limestone with aligned grains in (b) show significant velocity variations with direction. Therefore, the limestone in (a) can be treated as isotropic at the particle scale, whereas the limestone in (b) would be best treated as anisotropic. Otherwise we would not capture information about the grain alignments.

A laminated solid

Images - Chapter 12 White and Angona computed various velocities in a laminated solid as a function of the proportion of two materials (A and B) making up horizontal layers.
This figure shows compressional velocities for horizontal travel (dashed curve) and vertical travel (solid curve), as a function of the fractional amount of the higher-speed component (material A); VA is the P-wave velocity of material A, and VB is the P-wave velocity of material B.

Early contributions to this topic go back to the 19th century. Laboratory and field experiments in the 1950s detected velocity anisotropy when vertically and horizontally traveling waves were found to have different velocities. However, for most of the 70 or so years of petroleum exploration in the 20th century, petroleum seismologists have ignored anisotropy in their models of the earth, and their theoretical and practical developments assumed that waves propagate equally fast in all directions. There are good reasons for this omission. Seismic data through the 20th century were essentially dominated by the P-wave, for which the anisotropic effect is often small, with directional velocity differences of only 3 to 5 percent. When compared to errors due to the assumptions which were included in our models through the 20th century, such as the plane-layer approximation imposed by normal moveout (NMO) and stack processing or the two-dimensional (2D) earth model approximation imposed by acquisition geometry and limited computational resources, 5 percent anisotropy effect is in this context negligible. However, with recent advances in seismic acquisition and processing that we discussed in previous chapters, the reasons for ignoring anisotropy are no longer valid. Moreover, some petroleum seismologists now believe that getting a grip on anisotropic behavior of rock formations can mean the difference between success and failure in reservoir evaluation and development. For instance, if an amplitude versus offset (AVO) study does not take into account the anisotropic behavior of a shale cap rock, the underlying gas-bearing sandstone may be overlooked because the predicted AVO curve (for an oil sand overlain by isotropic shale) would not fit the observed AVO response from the actual survey.

Random media

Images - Chapter 12 Random media with an exponential autocorrelation function f(x,z) = exp{-√(x2/a2 + z2/b2)}, where a and b are the autocorrelation lengths. We can describe media in which the inhomogeneities are isotropic or elongated in a direction parallel to either of the two Cartesian directions. For example, (a) represents a random medium with a = 1 m and b = 1 m. (b) Same as Figure 1a with a = 5 m and b = 5 m. (c) Same as (a), with a = 5 m and b = 1 m. (d) Same as (a), with a = 10 m and b = 1 m. (e) Same as (a), with a = 1 m and b = 20 m. (f) Same as (a), with a = 1 and b = ∞ m.

Source-receiver configuration

Images - Chapter 12 The source-receiver configuration used to generate the seismograms described in this paper. The '*' represents the source position, and the 'o' represents receiver positions. The receivers are distributed along a quarter of a circle so that the incident angle θ varies between 0 and 90 degrees and the receivers are equally distant from the source point.

Seismograms

Images - Chapter 12 Horizontal- (a) and vertical- (b) component seismograms corresponding to a homogeneous medium. (c) The total magnitude. The parameters of the homogeneous medium are given in Table 12-2. The source characteristics are also given in Table 12-2. Notice that the source is explosive, and therefore it radiates only P-waves.

Seismograms

Images - Chapter 12 Horizontal- (a) and vertical- (b) component seismograms corresponding to a 1D random medium. The solid line represents traveltimes predicted by the effective medium theory. The fast velocity is 3509 m/s, and the slow velocity is 3405 m/s.

Seismograms

Images - Chapter 12 Horizontal- (a) and vertical- (b) component seismograms from a 1D random medium identical to the one used in the previous figure, corresponding to a vertical point force and a horizontal point force, respectively, instead of explosive source. We have recorded the S-wave arrivals. We can see that the P-wave and S-wave arrivals combined describe a transversely isotropic elastic medium. For example, the first S-wave arrival at θ = 0ο is at the same time as the S-wave arrival at θ = 90ο whereas the first P-wave arrival at θ = 0ο is later than the first P-wave arrival at θ = 90ο.

Seismograms

Images - Chapter 12 Horizontal- (a) and vertical- (b) component seismograms corresponding to a random medium with ro = 0 (zero aspect ratio).

Seismograms

Images - Chapter 12 Horizontal- (a) and vertical- (b) component seismograms corresponding to a random medium with ro = 0.05 (zero aspect ratio).

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