Abstract “Where are you going and what do you wish?” These questions are woven throughout all of our experiences, from earliest days, and shape our future. As some might remember, this question is a citation from the chronicle of Winken, Blinken and Nod. Together, we discuss where we are going and what we wish to accomplish. It is appropriate to state a reasonable working model of the earth, and then to review briefly the state of the art in regards to anisotropy, and finally to comment upon probable future developments. The focus herein is confined to the field of hydrocarbon exploration and development, and the applications of anisotropy in this field. The scale of interest, unless otherwise specified, is the seismic scale: frequencies between 5 to 100 hz, distances on the order of 10s of meters to kms.

Abstract In April 1856 Lord Kelvin (then still William Thomson) read a paper before the Royal Society, in which he outlined a description of the elastic tensor entirely based on the inherent symmetry of the material itself, i.e., without reference to any external coordinate system. The key concepts were the “perfect concurrence” (parallelity) and the orthogonality of stresses and strains, i.e., concepts known for vectors. This can be seen as a mapping of the space of tensors of rank 2 on a six-dimensional vector space. At the time of the presentation, the terms ‘tensor’ and ‘vector’ were not yet known, so that Kelvin had to develop his own terminology. For this mapping to be meaningful, it must preserve the norm , which Kelvin deduced from the invariance of the elastic energy. This requires that the tensor components are written as components of six-vectors in a way that differs from the conventional ‘contracted notation’. In this notation, the elastic tensor becomes a second rank tensor in six-dimensional space (with components that differ from the elements of the conventional 6x6 matrix). The (mutually orthogonal) eigenvectors of the elastic tensor provide a reference system in which the elastic tensor is diagonalized. The decomposition of the 21 independent components of the elastic tensor into the six eigenvalues (eigenstiffnesses) and the 15 independent parameters of the eigenstrains provide the basis for the coordinate-free description of the elastic tensor: the six eigenstiffnesses encode the properties related to the magnitude of stresses and strains, twelve parameters of the eigenstrains encode the geometric relations of stresses and strains (among others the symmetry properties of the elastic tensor), and three of the parameters of the eigenstrains describe the orientation of the system of eigenstrains with respect to the (arbitrary) external reference system. Though published in the Philosophical Transactions of 1856 and reprinted as an appendix to the entry «Elasticity» in the sixth edition of the Encyclopaedia Britannica (1878, vol. 7), the article seems to have made no impression on Kelvin’s contemporaries. The only reference known is a critical discussion in Todhunter and Pearson’s History of the Theory of Elasticity … of 1893, which shows that the writer was aware of but did not understand – Kelvin’s ideas. It has taken over hundred years for science to emulate Kelvin. The concepts proposed by Kelvin (by now independently proposed by several contemporary authors) allow to handle many problems of the theory of elasticity more concisely (and more elegantly) than with the conventional formalism.

Abstract A norm-preserving mapping of hierarchically symmetric tensors of rank 4 in three dimensions onto 21-dimensional vectors is proposed. It permits the definition of a metric of such tensors, of which the elastic tensors are a subset that satisfies the condition of stability. This metric allows to define the optimum approximation of an arbitrary tensor by one satisfying given symmetry conditions. The deviation of the original tensor from the symmetric approximation is defined as the angle between the two representing vectors. This concept allows to speak of one and the same medium as “deviating from transverse isotropy by a% , from orthorhombic symmetry by b%” (where b ≤ a) . Vectors corresponding to higher symmetry belong to subspaces of the 21-space, and the optimum approximation with given symmetries is the projection of the original vector on the different subspaces. This simple picture is complicated by the equivalence of a tensor to all tensors that can be obtained by a rotation of the 3D reference system. This equivalence is resolved by selecting from the equivalence set the member with the largest projection (and the smallest deviation). Normalization, i.e., the decomposition of the 21D vectors into magnitude and direction corresponds to a decomposition into overall stiffness and anisotropy. The normalized vectors have their end points on the surface of the unit sphere of the appropriate sub-space. The stability condition constrains the elastic vectors to a fraction 2 -k of the surface of the unit sphere, where k is the number of distinct ‘eigenstiffnesses’ of the symmetry system (i.e., two for isotropy, three for cubic symmetry, four for hexagonal and trigonal symmetry, and six for orthorhombic and lower symmetries). Together with a finite ‘threshold of distinctness’ the finite size of the region of stability leads to an estimate of the number of distinct anisotropic media that might exist. With realistic assumptions for this threshold, the numbers are modest for isotropic and cubic media, barely manageable for hexagonal (transversely isotropic) media, and of the order of 10 5 for orthorhombic media. With the same threshold, the total number of triclinic media is of the order 10 12 .

Abstract Physical modelling experiments have been conducted to study how the degree of anisotropic influence on spatial resolution varies with the orientation of the symmetry axis in transversely isotropic materials. Blocks of Phenolite laminated plastic comprising sheets of paper set in phenolic resin were used to simulate media such as shales having vertical and horizontal axis of symmetries respectively. Cylindrical holes of various sizes were drilled into the bottoms of both models. The hole sizes ranged from 0.83 to 1.17 and from 0.86 to 1.71 times the expected Fresnel-zone diameters in the horizontally layered and vertically fractured Phenolites respectively. These holes while serving as reflectors simulate a range of geological features, e.g. pinnacle reefs, intrusions, seals, and pipes etc. A three-dimensional physical modelling facility was used in collecting the experimental data. The spatial extents of the reflector boundaries are estimated by marking the half-amplitude points of the horizontal reflector events with respect to the amplitude at the centre of the reflectors. The degree of horizontal resolution is determined by comparing the seismically estimated reflector dimension with the true spatial dimension of the reflector. The resolving potentials of pressure and shear body waves in both media were compared using the collected data. Results obtained indicate that anisotropy can impose varying effects on the spatial resolving power of seismic waves. The degree of these effects depends on the curvature of the wavefront which changes with the orientation of the symmetry axis. Essentially, the lateral extent of discontinuities will be better imaged or resolved in a vertically fractured medium than in a similar medium that is horizontally layered.

Abstract Two numerical inversions were designed to calculate the 21 independent stiffnesses that define, in general, an anisotropic medium from either group- or phase-velocity data. The accuracy, robustness and computational complexity of the two inversion procedures — group velocity to stiffnesses and phase velocity to stiffnesses — were then compared. The group-velocity inversion overcomes the difficulty of calculating group velocity in a prescribed direction and can calculate group velocities accurately even in directions near shear-wave singularities. Although phase velocities are easier to calculate than group velocities, the group-velocity inversion performed better in laboratory tests because it is easier to acquire measurements of group velocities in many different directions. The group-velocity inversion for the general anisotropic tensor yielded the best results in its inversion of the 99 velocity data-points from a sphere of phenolic. The smallest velocity error (7.7 m/s) and the least average statistical uncertainty in the stiffnesses (0.04 GPa) came out of this application of the algorithm. The superior performance of this inversion is attributed to the freedom that the experimenter has to make as many measurements in whatever directions as are desirable, without having to cut the sample so that plane waves may be generated. In contrast to the group-velocity method, the phase-velocity inversion was the simplest, most robust and most accurate method in the theory and numerical testing but did not perform as well when faced with laboratory data. The drawback to the application of this method appears to be the limitation of inverting only plane-wave velocities, which limits the number of measurements that can be made. The model velocities fit the observed velocities with a statistical error of 18 m/s resulting in an average uncertainty in the stiffnesses of 0.16 GPa, statistical errors that are substantially higher than those for the group-velocity inversion. Despite this higher statistical error due to fewer measurements, the errors from the phase-velocity inversion are within the uncertainty estimates for the laboratory measurements.

Abstract The nonlinear (NL) elastic behavior of rocks evidenced by the stress-induced variations of the seismic velocities and related to the presence of mechanical defects (e.g. cracks) is now well established. Another classical result is the anisotropic behavior of rocks due to the spatial order exhibited by the orientation distribution function of heterogeneities (e.g. aligned microfractures, preferred orientation of grains). Both linear and NL elastic properties of the rock are affected by this order. We propose a global method for simultaneously characterizing anisotropy and nonlinearity in anisotropic media of arbitrary symmetry, such as rocks, based on a vectorial mapping of the elastic tensors associated with a suitable metric. The method provides the tools to (a) quantify linear elasticity and NL elasticity; (b) compare linear anisotropy and NL anisotropy; and (c) replace the considered medium of low symmetry by a simpler medium of higher symmetry, i.e. isotropic, transversely isotropic or orthotropic. The analysis of experimental data shows that rocks exhibit strikingly strong nonlinearity, orders of magnitude larger than most “intact” homogeneous materials (i.e. media without mechanical defects, such as crystals). In all the considered materials NL anisotropy is larger than linear anisotropy, but a material exhibiting a larger NL anisotropy does not necessarily exhibit larger linear anisotropy. For instance NL anisotropy can be comparable to larger in rocks than in their constituent minerals, whereas the linear anisotropy is weaker in the former than in the latter.

Abstract Synthetic sandstones have been manufactured under controlled stress conditions. The velocities of P- and S-waves were measured in two orthogonal directions, permitting studies of effects of cementation, compaction, unloading and reloading on velocities. Assuming that the samples and the experimental procedures are representative for earth materials and earth stress changes, these effects may be related to processes like lithification, burial / fluid withdrawal, uplift / coring, and burial after uplift / core testing under stress. The results show that unloading leads to largely reduced velocities, and also to a velocity anisotropy associated with the initial stress anisotropy. Strong stress dependency is prominent in materials which have been previously unloaded, indicating that laboratory measured stress dependency may often be a result of the unloading while coring.

Abstract An inversion method ispresented which provides a means to solve for the elastic moduliof anisotropic materials with symmetry as complicated as triclinic. Anisotropic elastic moduli were determinedfora layered glass plate—epoxy composite from oblique angle time—of—flight measurements using an immersion system. Both the glass and the epoxy behaveas isotropic materials in bulk form. However, once assembled to produce a heterogeneous structure(composite material) and probed with sufficiently long wavelengths, the heterogeneous stack appeared homogeneous but with anisotropic properties. With the measured properties of the constitutive bulk glass and epoxy and knowledge of their ratio in the layered stack,forward calculations are made to predict equivalent elastic moduli of the composite. Of theequivalent media theories available, a thickness—weighted averaging model as well as a theory which models the thin compliant epoxy bonds as a set of fractures are considered. Anisotropic elastic moduli predicted using these two theories are then compared to the measured elastic moduli of the glass—epoxy stack. The results show that epoxy does not behavethe same in an adhesive bond as it does in bulk. We therefore can not recommend using epoxy in the construction of physical models. However, in spite of the indeterminate behavior of epoxy, the equivalent media theories of Backus and of Schoenberg worked fairly well.

Abstract The ultrasonic anisotropic elastic properties of shales were measured as a function of effective pressure. Due to the low permeability of shales special equipment and techniques were required to control the over burden pressure and the sample pore pressure. An initial study looked at a preserved shale specimen of Jurassic age. Here measurements of compressional and shear wave velocities were taken on a set of three plugs cut from the shale and oriented at three angles (0, 90 and 45 degrees) with respect to the visual axis of symmetry. These measurements were taken at pressures up to 110 MPa and with the pore pressure drained to atmosphere. Next, an inversion procedure was created to reconstruct elastic stiffnessessusing the measured velocities. This inversion procedure involved taking an overdetermined set of measurements (minimum of 9 measurements for 3 sample plugs)and then minimizing a cost function based on the mis-fit between the predicted phase slownes surface and the laboratory measured slownesses. The final rms error for the reconstruction using the three samples was low, less than 3 μ /m. With a different block of the same shale, an additional experiment used a series of five plugs cut at five different angles relative to the axis of symmetry. The objective of this experiment was to make a additional test on the estimation of the elastic stiffnesses, and in particular for C 13, using a greatly overdetermined set data. The five samples gave a set of 15 measurements with which to estimate the 5 elastic parameters. The resultant reconstruction of elastic constants gave a low rms error of approximately 4 μs /m. In addition, the elastic stiffnesses for the separate blocks of shale were in dose agreement, suggesting that data sets collected on this shale may be representative. The reconstructed elastic stiffnesses for the experiment using the three samples were then analyzed in terms of their anisotropy and other properties as a function of effective stress. Up to 26 percent compressional wave anisotropy (Thompson’sparameter, ε) and up to 48 percent shear wave anisotropy (Thompson’s parameter,γ) was observed and it was found that both εγ and decreased as a function of increasing effective stress. In addition the anellipticity (deviation of the slowness surfaces from ellipses) was found to be positive and also decreased as a function of increasing effective stress. This decrease in anisotropy and anellipticity with increasing effective stress was accompanied by a corresponding decrease in porosity of the sample, from 10 1/2 to 8 1/2 percent. The decrease in overall anisotropy of the shales with increasing effective stress was seen to be consistent with theoretical modeling of shale properties where the shale anisotropy and anellipticity were predicted to decrease as a function of decreasing fluid filled porosity.

Abstract The failure of brittle rocks during compression is preceded by the formation, growth and coalescence of microcracks. Elastic wave velocities are reduced in the presence of open cracks and may therefore be used to monitor the progressive damage of the rock. In general, these microcracks are not randomly oriented and the rock displays a stress–induced seismic anisotropy which can be expressed in terms of a second–rank and fourth–rank crack density tensor. For open dry cracks the contribution of the fourth–rank crack density tensor to the elastic wave velocities is small.The results are compared with recent measurements of the ultrasonic compressional and shear wave velocities for propagation parallel and perpendicular to an increasing axial stress applied at constant confining stress to Berea sandstone. Inversion of the velocity measurements indicates that the microcracks propagate parallel to the maximum compressive stress, in agreement with current rock mechanics theory. A good fit to the data is obtained using only the second–rank crack density tensor even though, at high confining stress, the cracks are expected to be in partial contact along their length; This is consistent with the model of elastic wave propagation in a medium containing partially contacting fractures published by White (1983).

Abstract A self-consistent scheme has recently been implemented to study wave propagation through a variety of materials with microstructure such as, for example, anisotropic matrix containing a random array of aligned spheroidal inclusions, and its limiting case of penny-shaped cracks. Dispersion and attenuation curves calculated using this method show a characteristic resonance behaviour as a function of frequency that is independent of the wave type and the angle of incidence in almost all cases. Also, there is a frequency dependence of the azimuthal behaviour of the speed and attenuation coefficients in some cases. At low frequencies, very similar results may be found by employing the method of smoothing or the quasicrystalline approximation. But only the self-consistent method has been developed to yield wave speeds and attenuation for finite frequencies.