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Cracks, broadly defined as compliant discontinuities, are a major cause of elastic anisotropy. However, few models are available for quantifying crack properties relevant to anisotropy. We developed a rock-physics model to quantify crack angular distribution and normal-to-tangential compliance ratio from pressure-dependent acoustic velocities measured in the laboratory. Our model uses a rectangular function of variable width and amplitude to extract the maximum dip angle of cracks, which is a direct quantification of crack alignment relative to the bedding plane. We tested the model on an organic-rich shale data set and confirmed that crack alignment and compliance ratio strongly impact Thomsen anisotropy parameters, thus demonstrating the model as a useful tool for better understanding how cracks affect elastic anisotropy.

Aligned fractures/cracks in rocks are a primary source of elastic anisotropy. In an azimuthally anisotropic formation surrounding a borehole, shear-waves (S-waves) split into fast and slow waves that propagate along the borehole and are recorded by a borehole logging tool. However, when the formation has conjugate fractures with orthogonal strike directions, the azimuthal anisotropy vanishes. Hence, azimuthal anisotropy measurements may not be adequate to detect orthogonal fracture sets. We have developed a method for obtaining azimuthal and radial S-wave anisotropy parameters simultaneously from 4C array waveforms. The method uses a velocity tomogram around the borehole. The azimuthal and radial anisotropy were determined by integrating a shear velocity radiation profile along the radial direction at different azimuthal angles. Results indicate that this approach is reliable for estimating anisotropy properties in aligned crack systems. The advantage of this interpretation method is shown in multiple conjugate crack systems. Field data processing examples are used to verify the application of the proposed technique. Comparison of results against those obtained with a conventional procedure shows that the new method can not only provide estimates of azimuthal anisotropy, but also of the radial anisotropy parameter, which is important in fracture network evaluation.

Rock physics provides a crucial link between seismic and reservoir properties, but it requires knowledge of the elastic properties of rock components. Whereas the elastic properties of most rock components are known, the anisotropic elastic properties of clay are not. Scanning electron microscopy studies of clay in shales indicate that individual clay platelets vary in orientation but are aligned locally. We present a simple model of the elastic properties of a region (domain) of locally aligned clay platelets that accounts for the volume fraction, aspect ratio, and elastic-stiffness tensor of clay platelets, as well as the effective elastic properties of the interplatelet medium. Variations in clay anisotropy are quantified by examining the effects of varying model parameters upon the effective transverse-isotropic (TI) elastic-stiffness tensor of a domain. Statistics of these distributions and correlations between stiffnesses and anisotropy parameters enable the most probable sets of stiffnesses to be identified for rock physics calculations. The mean of these distributions is on the order of twice the mode for in-plane stiffnesses ([Formula: see text], [Formula: see text], [Formula: see text]), but it is of the same order as the mode for out-of-plane stiffnesses ([Formula: see text], [Formula: see text], [Formula: see text]). Despite random sampling, well-defined relations emerge, consistent with similar shale relations reported in the literature. Expressing these relations in terms of [Formula: see text] for a single domain of aligned clay platelets facilitates their general application. In the limit that the volume fraction approaches unity, the elastic stiffnesses thus derived reproduce those of the clay mineral assumed as platelets. Given the elastic-stiffness tensor of a single domain of aligned clay platelets, the effective TI elastic-stiffness tensor of clay is obtained by integrating over the clay-platelet orientation-distribution function.

Mudstones are often anisotropic, which complicates depth conversion in seismic exploration and monitoring subsurface reservoirs during injection or production. In addition, the physical and mechanical properties of mudstones are highly sensitive to their water content. The elastic anisotropy of mudstones is not well understood because of their complex nature and the lack of laboratory experiments performed on well-preserved samples. Triaxial deformation tests were performed on mudstone core plugs to investigate the impact of water saturation on the elastic anisotropy of the Whitby Mudstone (United Kingdom). The mechanical and physical properties of the Whitby Mudstone were estimated from stress-strain and ultrasonic wave velocity data obtained on core plugs with different water saturations under isotropic and anisotropic stress conditions. The Whitby Mudstone has extremely high intrinsic elastic anisotropy (0.3–0.4) due to its composition and lamination. This elastic anisotropy increases with decreasing water content. There are three competing mechanisms that play a key role in the anisotropy increase due to dehydration such as (1) density contrast in the pore space (i.e., the presence of purely brine or a mixture of brine and air in the pore space), (2) formation of dehydration fractures, and (3) frame stiffening. Increasing the mean effective stress leads to a decrease in Thomsen’s anisotropy parameters [Formula: see text] and [Formula: see text] because of the closure of defects, such as natural and dehydration fractures, and the formation of stress-induced fractures. The relationship between the wavefront anellipticity factor [Formula: see text] and the mean effective stress is nonmonotonic and can be related to the onset of inelastic deformation.

A borehole existing in any geologic formation concentrates the far-field tectonic and overburden stresses amplifying the magnitudes of certain stress components near the borehole. It is important to understand the magnitudes and patterns of this stress concentration because these lead to damage and can even collapse the borehole if sufficiently strong. The solution of the stress distributed near a borehole can be complicated considering the elastic anisotropy of rocks. We have developed programs (ASCIB3D) in MATLAB and Python to model the stress distribution around an inclined borehole in an arbitrarily oriented anisotropic medium. The program is built on the Lekhnitskij-Amadei solution. The input orientation of the far-field stresses and the elastic stiffness matrix of the medium into the program are geology angles instead of the rotation angles shown in previous studies, making the code more convenient for users. The sign convention for the inverse function, which is ignored in previous studies, is discussed in detail. The results indicate that the program ASCIB3D is a useful tool for modeling the stress distributed around an inclined borehole in the anisotropic formation and analyzing the effect of anisotropy and borehole inclination on stress distribution. The inclination and azimuth of the borehole and the anisotropy of the rocks affect the orientation and strength of the stress concentration.

Radiation patterns of earthquakes contain important information on tectonic strain responsible for seismic events. However, elastic anisotropy may significantly impact these patterns. We systematically investigate and visualize the effect of anisotropy on the radiation patterns of microseismic events. For visualization, we use a vertical-transverse-isotropic (VTI) medium. We distinguish between two different effects: the anisotropy in the source and the anisotropy on the propagation path. Source anisotropy mathematically comes from the matrix multiplication of the anisotropic stiffness tensor with the source strain expressed by the potency tensor. We analyze this effect using the corresponding radiation pattern and the moment tensor decomposition. Propagation anisotropy mathematically comes from the deviation between the polarization and the propagation direction of a quasi P-wave in an anisotropic medium. We investigate both effects separately by either assuming the source to be anisotropic and the propagation to be isotropic or vice versa. We find that both effects have a significant impact on the radiation pattern of a pure-slip source. Finally, we develop an alternative visualization of source mechanisms by plotting beach balls proportional to their potency tensors. For this, we multiply the potency tensor with an isotropic elasticity tensor having the equivalent shear modulus [Formula: see text] and [Formula: see text]. In this way, we visualize the tectonic deformation in the source, independently of the rock anisotropy.

The Bakken Formation consists of three members: The Upper Bakken and Lower Bakken are dark marine shales with high organic content, whereas the Middle Bakken consists of mixed carbonates and clastics and is the main reservoir unit, despite having low porosity and permeability. Dipole S-wave data acquired in a lateral well in the Middle Bakken Formation revealed this formation to be anisotropic. Backus upscaling of logs acquired in a nearby vertical pilot well in the same layers sampled by the lateral well gave estimates of the anisotropy that were too small to explain the S-wave anisotropy measured in the lateral well. The observed anisotropy was interpreted in terms of bedding-parallel compliant discontinuities such as microcracks and low-aspect-ratio pores. The presence of bedding-parallel microcracks and low-aspect-ratio pores may contribute to the permeability of the tight Middle Bakken reservoir, and the sensitivity of P- and S-wave velocities to the presence of microcracks and low aspect ratio pores suggested the use of sonic and seismic measurements for identifying the productive zones in the low-permeability Middle Bakken reservoir.

Velocity anisotropy and attenuation in weakly anisotropic and weakly attenuating structures can be treated uniformly using weak anisotropy-attenuation (WAA) parameters. The WAA parameters are constructed in a way analogous to weak anisotropy (WA) parameters designed for weak elastic anisotropy. The WAA parameters generalize WA parameters by incorporating attenuation effects. They can be represented alternatively by one set of complex values or by two sets of real values. Assuming high-frequency waves and using the first-order perturbation theory, all basic wave quantities such as the slowness vector, the polarization vector, propagation velocity, attenuation, and the quality factor are linear functions of WAA parameters. Numerical modeling shows that perturbation equations have different accuracy for different wave quantities. The propagation velocity usually is calculated with high accuracy. However, the attenuation and quality factor can be reproduced with appreciably lower accuracy. This happens mostly when the strength of velocity anisotropy is higher than 10% and attenuation is moderate or weak [Formula: see text]. In this case, the errors of the attenuation or [Formula: see text]-factor can attain values comparable to the strength of anisotropy or even higher. A simple modification of the equations by including some higher-order perturbations improves accuracy by three to four times.

Recent experimental studies of ultrasonic velocity anisotropy in kerogen‐rich shales indicate that these rocks are characterized by a very strong anisotropic response related to a very fine, bedding‐parallel lamination of organic matter and preferred orientation of clay particles in the rock matrix (Vernik and Nur, 1992). This intrinsic anisotropy is further enhanced in thermally mature shales by bedding‐parallel microcracks caused by the processes of hydrocarbon generation (Vernik, paper in preparation). However, the potential of recognizing mature source‐rocks in situ using downhole sonic, crosshole seismic, or VSP data clearly depends on our ability to discriminate between these two major causes of elastic anisotropy, remove the effect of the intrinsic anisotropy, and estimate crack density and crack porosity from velocity measurements.

Tube‐wave speed in the presence of a weakly anisotropic formation can be expressed in terms of an effective shear modulus for an equivalent isotropic formation. When combined with expressions for the speeds of the SH‐ and quasi‐SV‐waves along the borehole axis, a simple inversion procedure can be obtained to determine three of the five elasticities of a transversely isotropic (TI) formation tilted at some known angle with respect to the borehole axis. Subsequently, a fourth combination of elastic moduli can be estimated from the expression for the qP‐wave speed along the borehole axis. The possibility of determining all five elasticities of a TI formation based on an assumed correlation between two anisotropy parameters is discussed.

Berryman shows elegantly that the “inequality [Formula: see text] is true for any horizontally stratified, homogeneous material whose constituent layers are isotropic…” However, the final clause of this sentence “…, i.e., any homogeneous, transversely isotropic material,” is, if taken at face value, misleading. It is clear from the proof in the section “A fundamental inequality” that this statement is only shown to hold for lamellated media with isotropic lamellae, and that Berryman chooses arbitrarily and without any warning the phrase homogeneous, transversely isotropic to stand as a synonym for what Backus (1962) painstakingly describes as “smoothed, transversely isotropic, long‐wave equivalent (STILWE).” In view of the fact that even within the context of exploration seismics transverse isotropy can be due to causes other than horizontal stratification with isotropic constituents (e.g., schists can be intrinsically anisotropic, anisotropy might be due to preferential orientation of sandgrains or joints), I believe this choice to be unfortunate. It leads the unsuspecting reader to assume a wider applicability of the fundamental inequality than Berryman really intends to claim, and thus makes it unnecessarily difficult to understand this significant contribution.

Most bulk elastic media are weakly anisotropic. The equations governing weak anisotropy are much simpler than those governing strong anisotropy, and they are much easier to grasp intuitively. These equations indicate that a certain anisotropic parameter (denoted δ) controls most anisotropic phenomena of importance in exploration geophysics, some of which are nonnegligible even when the anisotropy is weak. The critical parameter δ is an awkward combination of elastic parameters, a combination which is totally independent of horizontal velocity and which may be either positive or negative in natural contexts.

We used the ultrasonic transmission method to measure P-, SH-, and SV-wave velocities for Chelmsford granite, Chicopee shale, and Berea sandstone in different directions up to 1 000 bars confining pressure. The velocity measurements indicate these three rocks are elastically anisotropic. The stiffness constants, dynamic Young’s moduli, dynamic Poisson’s ratios, and dynamic bulk moduli of the three rocks were also calculated. The elastic constants, together with velocity measurements, suggest that: (1) elastic anisotropy is due to the combined effects of pores or cracks and mineral grain orientation, and (2) elastic anisotropy decreases with increasing confining pressure. The residual anisotropy at higher confining pressure is due to mineral grain orientation.

Berryman and Levin made an assumption about constancy or limited variations of Poisson’s ratio in the thin layers, in their analyses of elastic anisotropy in thin‐layered media. Berryman states (p. 913): “Rare cases can occur with large variations in Poisson’s ratio.” However, on p. 911 Berryman does point out (with reference to Benzing) that range of variations of the parameter γ = VS/VP from 0.45 to 0.65 is typical of rocks. That corresponds to a range of variations of Poisson’s ratio of 0.373 to 0.134 (i.e., almost three times as much).

I regret that I do not read Russian. If I did, there would have been a much greater chance of my being familiar with the previous work of Lyakhovitskiy and Nevskiy on elastic anisotropy.

In a paper whose importance seems to have escaped notice, Thomsen (1) derived equations that give the moveout velocities of P, SV, and SH-waves when solids are weakly transversely isotropic and (2) tabulated experimentally determined elastic constants for a large number of rocks, crystals, and a few other solids. For rocks, one of the constants, delta, differed from zero by as much as 0.73 and −0.27. Delta is the fraction by which P-wave moveout velocity deviates from the vertical velocity [Thomsen’s equation (27a)]. Although some deltas indicated deviations from the vertical velocity smaller than 1 or 2 percent, most were larger and positive. Until the publication of Thomsen’s data, most of us concerned with elastic waves traveling in earth sections that act as transversely isotropic solids because the sections consist of thin beds had assumed the individual beds were isotropic solids, all with the same Poisson’s ratios. That assumption results in a zero value for delta and a moveout velocity equal to the vertical velocity. The validity of the assumption is now doubtful.