GeoScienceWorld
Volume

Advances in Anisotropy:

Selected Theory, Modeling, and Case Studies

Edited by Julie A. Hood

Abstract

Seventeen papers based on talks and posters presented at 7IWSA are included in this book:

In “General constitutive relations for layered media,” Schoenberg provides a combination of matrix algebra and group theory to offer a formalism for evaluating of the properties of the unique homogeneous anisotropic medium that is statically equivalent to a stationary distribution of homogeneous, but generally anisotropic, fine layers. The method is valid for systems representing fluid permeability, heat conductivity, and electrical properties such as electrical conductivity, dielectric permittivity, and magnetic permeability. Additionally, the formalism extends to higher dimensional elastic layered systems so that bi-anisotropic electromagnetic and piezoelectric layered systems can be studied.

“Upscaling: Elastic anisotropy from ultrasonic laboratory measurements to borehole seismic surveys,” by Hornby, applies the equivalent media theory of Backus (1962) to resolve discrepancies in anisotropic parameters for transversely isotropic media derived from ultrasonic laboratory measurements and seismic-scale VSP surveys. Hornby argues that the inversion of laboratory experiments generally leads to larger values of the δ parameter of Thomsen (1986), as well as smaller values for the normalized anisotropy parameter η proposed by Alkalifah and Tsvankin (1994). The difference is attributed to the effect of fine-scale layering on the larger wavelength VSP measurements. Hornby tests this hypothesis by the upscaling of elastic constants determined from in-situ depth-continuous borehole measurements. Although borehole measurements cannot resolve the complete set of elastic constants, remaining parameters are estimated from representative laboratory measurements of borehole samples. The resulting values for the anisotropic parameters δ and η are comparable to those determined from the VSP survey with remaining differences attributed to the effect of unresolved finer-scale layering.

In “Can we separate the effects of anisotropy and structure from surface seismic data?” Kuehnel and Li offer a practical method of correcting traveltimes for reflections in transversely isotropic media. They present a method of decomposing the traveltime equation for a reflected wave in a transversely isotropic layer in terms of dip and anisotropy, based on the traveltime of individual wavetype using an iterative procedure applied to a ray-tracing code.

In “P-wave traveltime anomalies below a dipping anisotropic thrust sheet,” Leslie and Lawton raise the awareness of the complications that can arise from this common subsurface configuration and begin to quantify the anisotropic effect in depth migration. Results from physical modeling and numerical ray tracing show that isotropic depth migration and velocity analysis produce significant artifacts in the synthetic data from such a model. Therefore, it is suggested that care be taken when using the isotropic assumption in depth migration even with the presence of mild anisotropy.

  1. Page 1
    Abstract

    Seventeen papers based on talks and posters presented at 7IWSA are included in this book:

    In “General constitutive relations for layered media,” Schoenberg provides a combination of matrix algebra and group theory to offer a formalism for evaluating of the properties of the unique homogeneous anisotropic medium that is statically equivalent to a stationary distribution of homogeneous, but generally anisotropic, fine layers. The method is valid for systems representing fluid permeability, heat conductivity, and electrical properties such as electrical conductivity, dielectric permittivity, and magnetic permeability. Additionally, the formalism extends to higher dimensional elastic layered systems so that bi-anisotropic electromagnetic and piezoelectric layered systems can be studied.

    “Upscaling: Elastic anisotropy from ultrasonic laboratory measurements to borehole seismic surveys,” by Hornby, applies the equivalent media theory of Backus (1962) to resolve discrepancies in anisotropic parameters for transversely isotropic media derived from ultrasonic laboratory measurements and seismic-scale VSP surveys. Hornby argues that the inversion of laboratory experiments generally leads to larger values of the δ parameter of Thomsen (1986), as well as smaller values for the normalized anisotropy parameter η proposed by Alkalifah and Tsvankin (1994). The difference is attributed to the effect of fine-scale layering on the larger wavelength VSP measurements. Hornby tests this hypothesis by the upscaling of elastic constants determined from in-situ depth-continuous borehole measurements. Although borehole measurements cannot resolve the complete set of elastic constants, remaining parameters are estimated from representative laboratory measurements of borehole samples. The resulting values for the anisotropic parameters δ and η are comparable to

  2. Page 5
    Abstract

    Matrix algebra and group theory combine to offer a formalism for the evaluation of the properties of the unique homogeneous anisotropic medium that is equivalent statically, or in the long wavelength limit, to a stationary distribution of homogeneous, but, in general, anisotropic, fine layers. The method was originally developed to evaluate the elastic moduli of finely layered regions, and is generalized to include any properties that are specified by a linear set of constitutive relations. The simple lossless situation of 3 × 3 real, symmetric, positive definite matrices relating two vector fields includes the cases of fluid permeability, heat conductivity, and electrical properties, such as electrical conductivity, dielectric permittivity, and magnetic permeability. Examples of higher dimensional cases are the above mentioned elastic layered systems (6 × 6), bi-anisotropic electromagnetic layered systems (6 × 6), and piezoelectric layered systems (9 × 9). In all cases, the properties of a set of fine layers of a given anisotropic constituent map to an element of a commutative group. A reverse mapping returns the standard material properties of the equivalent medium. The addition of group elements yields the group element for the homogeneous medium equivalent to the combined set of layers. Addition of an inverse element, i.e., subtraction, provides the means to remove a set of layers from an anisotropic medium; then, if the remaining medium is stable, a valid decomposition of the original medium into anisotropic constituents is obtained. Within the group structure, subgroups, corresponding to types of symmetry systems, may be identified, immediately giving the symmetry of the equivalent medium, given the symmetry of its constituents. Sets of layers are also represented as group elements whose moduli and overall thickness approach zero at the same rate, so that, in the limit, the set of layers becomes a set of interfaces. This allows such interfaces to be manipulated in a consistent and uniform manner.

  3. Page 21
    Abstract

    Comparisons of “intrinsic” anisotropy of shales measured on whole core at ultrasonic frequencies in the laboratory with seismic scale anisotropy estimated using walkaways VSP surveys reveal systematic differences that are attributed to fine layer effects. In a case study from the North Sea intrinsic shale anisotropy was determined both in the laboratory using samples of preserved core at ultrasonic frequencies and in situ using walkaway borehole seismic data at seismic frequencies. These observations were linked using depth continuous measurements of bedding from borehole logs, including the DSI* (Dipole Shear Sonic Imager) borehole sonic tool. A combination of the ultrasonic core measurements, the corssed-dipole borehole sonic measurements and a shale indicator (gamma-ray) resulted in an estimation of elastic stiffnesses as a function of depth. The result was a set of layers, at the DSI* sonic vertical acquisition interval of 0.5 ft (.15 m), with individual layer properties that are transversely anisotropic with a vertical axis of symmetry (TIV). Upscaling of the depth continous TIV parameters to long-wavelength (seismic) properties was then achieved using equivalent medium theory (Backus averaging). The equivalent medium results did not account for all the differences between the ultrasonic core results and the seismic frequency walkaway VSP results; these differences were attributed to the presence of layering on a finer scale than resolved by the borehole sonic survey. To test this theory, synthetic data were generated at a fine 0.5 ft (.15 m) scale and tests were made of anisotropy introduced by the layering using both the fine scale and a smoothed 3 ft (.91 m) response comparable to the DSP tool resolution. One conclusion is that the effects of the fine layering on the seismic scale anisotropy are underestimated using data acquired at the resolution of the current borehole sonic tools.

  4. Page 47
    Abstract

    Approximate traveltime equations for reflections from a dipping interface embedded in a weakly TI (transversely isotropic) medium are derived and formulated as four (dip and anisotropy-independent, and dip and anisotropy-dependent) terms. The accuracy of the traveltime equation and the contribution of each term may then be evaluated and corrected. The traveltime equation can be separated into these four terms for 15–20% anisotropy (Thomsen parameter ε = 0.15–0.2) and 20 degrees dip and incidence angles up to 30 degrees. The four terms may be ranked in order of magnitude as: isotropic dip term (zero-order), isotropic dip-residual term (first order), dip-independent anisotropic residual term (second order), and the dip- and anisotropy-dependent residual term (third order). A separation algorithm is then constructed based on this ordered relationship for estimating the dip and anisotropic residual terms from the traveltime of any given wavetype. Testing of different anisotropy models shows that this separation algorithm is valid for 15% anisotropy (ε = 0.15) and 20 degrees dip or up to 30 degrees of incidence angle.

  5. Page 77
    Abstract

    A two-dimensional physical model of a simple thrust fault was constructed which incorporates a variably dipping layer of phenolic laminate. This material possesses orthorhombic anisotropy, but was cut in a symmetry plane so that it approximates a transversely isotropic medium. It is weakly anisotropic with Thomsen parameters δ= 0.081, ε = 0.150 and γ = 0.035. A zero-offset ultrasonic survey was acquired across the model and the traveltimes from a flat reflector below the thrust yielded a time-structure anomaly with a magnitude of 100 ms. Anisotropic raytracing was undertaken through the model and numerical traveltimes agreed closely with the physical modeling results and showed that about 50 ms of the time-structure anomaly was caused by differences in the average isotropic properties of the model, and 50 ms was due to anisotropy in the dipping phenolic layer. The study indicates that velocity anisotropy, if unaccounted for, may cause significant errors in depth imaging in fold and thrust belts, and underlines the need for anisotropic depth migration algorithms.

  6. Page 85
    Abstract

    A method for studying temporal variations of wave propagation properties in the earth is specifically useful for studying crustal dynamics, but also for studying reservoir geophysics. Medium anisotropy causes three phases to propagate along almost the same path. This allows a determination of the relative difference of shear-wave velocities in a fashion, which is remarkably insensitive to potential errors. Even instrumental timing errors should not affect the results. If there are nearly identical doublet sources, which often occur naturally, one can determine differences of with extreme accuracy. This allows resolving small temporal variations of wave propagation properties, which cannot usually be detected within the larger measurement uncertainty of . An application to data from a hydraulic fracturing experiment in a deep drilling borehole (KTB) showed that such temporal variations indeed exist, even at substantial depth levels in the crust. The relative difference of split shear-wave velocities decreased by about 2% during a 12-hour interval in the experiment. This can be explained only in terms of changing effective elastic properties of the medium. The method can be used with artificial and natural sources.

  7. Page 99
    Abstract

    Schoenberg and Muir (1989) show how to generalize the layer-averaging method of Backus (1962) to the more general concept of a “layer group,” in which layers are not only averaged to find an equivalent homogeneous medium, but also added, subtracted, and negated. Dellinger and Muir (1993) show how to recast the Dix equations (Dix, 1955) into paraxial layer-group form.

    Backus averaging is a low-frequency approximation; the Dix equations are a high-frequency paraxial (small ray parameter) approximation. Layer matrices, which extrapolate a monochromatic wavefield across a layer, are exact for all frequencies and angles of propagation. In this paper I show how by suitable manipulation of layer matrices it is possible to exactly calculate the ensemble-average wavefield emerging from a statistically defined layer stack. This result is used to define an equivalent-medium system based on layer matrices.

    If the layer probabilities are independent, the theory is particularly simple and directly leads to an obvious layer-group formulation. The method also works for layers defined via Markov chains, although it has not yet been proven whether there is a (noncommutative) layer-group formulation for this case.

  8. Page 109
    Abstract

    Dipping parallel fractures in an elastic background medium that is assumed to be hexagonally symmetric with a vertical axis of rotational symmetry, commonly referred to as transversely isotropic (TI), combine to render the medium’s overall long wavelength elastic symmetry to be monoclinic. Four independent fracture compliances specify the elastic behavior of the most general fracture system described by the linear slip assumption consistent with the monoclinicity of the equivalent medium. In addition, two angles specify the orientation of the fracture planes, one is the horizontal strike direction and the other is the dip angle of the fracture planes. The mirror plane of symmetry of the monoclinic medium equivalent to such dipping fractures embedded in a TI background is perpendicular to the strike direction. Assuming that strike direction is known, a total of five fracture parameters along with five independent TI background compliances determine thirteen nonzero components of the equivalent medium’s compliance matrix. Thus, in addition to the standard inequalities which arise from the physical constraints that the background TI compliance matrix be positive definite and the fracture compliance matrix be nonnegative definite, the thirteen elastic compliances of such a monoclinic medium must satisfy three constraint equations. When a compliance matrix determined from measurements on an anisotropic medium displays monoclinic symmetry, and if these three hard constraints on its components are satisfied, then expressing the assumed TI symmetry conditions of the background in terms of the four fracture compliances and the dip angle provides a set of eight equations that are linear in the four fracture compliances, but nonlinear in the dip angle. The compliance matrix of the TI background can then be found by simple matrix subtraction. The solution is a valid decomposition of the monoclinic medium into a set of dipping fractures in a particular TI background medium. The existence of such a decomposition rigorously implies only that the anisotropic behavior is the same as that of a medium composed of those dipping fractures in that TI background medium; physically, the existence of the decomposition strongly suggests that the dipping fractures are real and are the cause of the observed monoclinicity.

  9. Page 127
    Abstract

    Crustal S-wave anisotropy has been suggested as a possible tool for monitoring stress and/or fluid changes associated with the earthquake cycle. We analyzed 363 microearthquakes occurring between 1981 and 1995 beneath Southern California Seismic Network station BWC, located approximately 25 km south of the Landers, California earthquake hypocenter(M1= 7.3; June 28, 1992) and 5 km west of the Joshua Tree earthquake (Mw= 6.1; April 23, 1992). S-wave seismograms exhibit a strong initial polarization alignment of approximately N4.5°E (in the seismometer coordinate system), followed by a 30-degree eastward rotation beginning around the time of the Landers earthquake. However, P-wave particle motions, similar earthquake analysis, and further investigation of the station service record indicate that this apparent rotation is an instrumental artifact due to a reorientation of the horizontal seismometers during post-Landers station maintenance.

    After using P-waves to calibrate horizontal seismometer azimuths both before and after the reorientation, the data are consistent with a stable fast horizontal anisotropic axis of 16.5 ± 14° E of N throughout the study period. Clear examples of S-wave splitting are observed, corresponding to a minimum inferred shallow (≲10 km) S-wave anisotropy of approximately 2%. S-wave splitting time difference estimates from similar earthquakes suggest a subsample (approximately 10%) decrease in anisotropy during the last month of the Joshua Tree aftershock-Landers foreshock interval. This trend was subsequently unaffected by the Landers mainshock. However, more detailed hypocenter estimates will be required to adequately separate effects due to hypocenter differences from the apparent temporal change in anisotropy.

  10. Page 143
    Abstract

    A method of determining the stress-dependent elastic properties of TI shales using only single test specimens is presented, which limits the effect of inherent material variability from specimen to specimen. The wave-speed measurements involve the use of an effective point source. Recognizing that the ultrasound traveltimes refer to anisotropic group velocities does not lead to an intractable problem, but rather to one that can be solved numerically in a systematic way.

    The optimum strategy for wave-speed inversion is found, and demonstrated with Monte Carlo simulated data sets drawn from a Cretaceous shale model, and to laboratory data obtained (by the authors) on Heather shale.

  11. Page 159
    Abstract

    If the shear-wave AVO gradients can be reliably extracted from prestack shear-wave amplitudes, they can supplement the fracture-detection algorithms based on the normal-incidence shear-wave reflection coefficients and the time difference of split shear waves.

    Approximate reflection coefficients in the two vertical symmetry planes of azimuthally anisotropic media explicitly describe the influence of medium parameters on the AVO gradients and the higher-angle terms of the shear-wave reflection response. The anisotropy in the subsurface is hereby characterized by the shear-wave splitting parameter γ and anisotropy parameters similar to Thomsen’s (1986) coefficients. The important result of this study is that the AVO gradients of shear waves propagating in the vertical symmetry planes of fractured media are sensitive to the shear-wave splitting parameter and to a parameter combination (ε(V)−δ(V). The later term carries information about the fracture filling and is important for time processing in anisotropic media. Analytic insight developed in this study naturally leads to an inversion algorithm for the anisotropy parameters of the fractured medium.

  12. Page 187
    Abstract

    Laboratory ultrasonic transmission experiments were performed on three dolomitic limestones (K43, K72, and C66) to investigate the presence of preferentially oriented planes of weakness. The directional difference in P-wave velocities is 0.6% to 12%, but the amplitude anisotropy is significantly greater at 30% to 94%, indicating it is more sensitive to anisotropy. Tests on both dry and saturated cores reveal that the addition of water in samples K43 and C66 decreases the attenuation ratio between the signal in the direction of the maximum saturated amplitude and 90 degrees, Q−1excess, which is consistent with the stiffening of compliant interfaces. Insight into the microstructure controlling attenuation is obtained by thin section analyses, which indicates the maximum amplitudes in the saturated specimens coincide with the orientation of one or more fractures in the cores. To test if the presence of fractures can explain the observed attenuation, the measured phase velocity and attenuation ratio Q−1excess in a direction parallel and perpendicular to the inferred plane of weakness are compared to values obtained from a nonwelded interface model for isotropic media. Two of the following three parameters are required to match theory with observations: a fracture stiffness; and/or a viscosity; and/or a time delay correction for dry cores. Only for sample K43 can relative compliance be estimated because its structure is simple and viscous loss mechanisms do not contribute significantly to the attenuation. The time delay correction required for the dry sample K43 is due to a combination of bedding anisotropy and porous mineral grains.

  13. Page 205
    Abstract

    Laboratory studies on acoustic wave propagation in synthetic rocks with parallel cracks show that the attenuation is strongly dependent on the incidence angle. The scattering which causes this attenuation also distorts the waveforms and changes the frequency content of the signal. The angular dependence of the attenuation therefore depends somewhat on the feature used as a measure of the signal strength.

    The study shows that under the conditions studied here, the attenuation anisotropy is much more visible than the velocity anisotropy. Most of the measurements used a 100 kHz excitation pulse, which corresponds to a wavelength-to-crack-diameter ratio in the range 1.7–3.0. In the dry sample, the attenuation anisotropy matches fairly well theoretical expectations, and is qualitatively similar to the velocity anisotropy. However, the attenuation anisotropy is much more visible: for the P-wave, the velocity ratio Vmin/Vmax = 0.80, while the amplitude ratio Amin/Amax = 0.01. At higher frequencies the P-wave velocity anisotropy vanishes, while the attenuation anisotropy remains high and clearly visible. The effect of water saturation is a significant reduction in the velocity anisotropy for the P-wave, while the attenuation anisotropy remains high. The S-waves are less affected by saturation.

  14. Page 217
    Abstract

    The goal of the study is to estimate theoretically the possible range of permeability and seismic velocity changes in low-porous fluid-saturated cracked rocks for different fluid composition and pore pressure under stress and temperature conditions typical for the upper 5–10 km of the crust. Clayey limestones saturated with oils of different composition and low-porosity granites saturated with brines were treated as models of reservoir and hot dry rocks.

    Macroscopic cracks are described as non-interacting parallel fluid-filled inclusions of an increased permeability compared with the background permeability of a host rock, and percolation effects are not considered in the study. As physical characteristics of pore fluids are PT dependent, variations of external stress and temperature can result in a drastic change of the macroscopic physical parameters of a rock. The model is applied to consider rock under the undrained regime, when there is no evacuation of fluid from crack volume. The effective-medium concept is used to model numerically variations of crack geometry (rock porosity), effective permeability and VS for different values of pore pressure (10–100 MPa) and external uniaxial compression applied to a rock mass at T|50–300°C.

    The results of the study show that variations of crack geometry and crack concentration due to the action of external stress are more sensitive to a composition of a saturating fluid than to initial pore pressure and/or temperature. Macroscopic properties of a rock depend almost linearly on fluid density and are less sensitive to temperature variations. Under external uniaxial stress, the behavior of both permeability and shear velocities is strongly anisotropic until stress does not exceed the critical value when all pre-existing cracks are closed. At the crustal depth of 5–10 km and high values of compression, only cracks saturated with high-density fluids can remain open. Permeability of fluid-saturated low-porosity rocks decreases with an increase of fluid temperature; shear velocities change simultaneously, but in the opposite direction.

  15. Page 239
    Abstract

    By a comparison of both equations and wave fronts, we show that effective models of stratified fluid-solid media can be considered as special cases of the Biot model. Explicit relationships which transform all Biot parameters to the effective media domain are presented. In the case of isotropic or transversely isotropic elastic layers this is just a special case of the transversely isotropic Biot model. However, for the lower symmetry cases more complex Biot models are necessary, and are presented.

    A model of alternating layers of solids with fluids can be used as a first approximation in representing fractured and porous media. However, use of traditional long-wave equivalent effective media models is not valid. Although this medium is not a strict porous medium (thus enabling the use of Biot theory formulations based on continuous pore space), neither are formulations based on monophase elasticity appropriate; even though normal displacements and stresses are continuous across boundaries for both phase, the constitutive equations representing this model are actually two-phase.

    The effective media model is a transitional model between a completely two-phase model in which all stresses and displacements are different in both phases, unlike in a monophase model where the stresses and displacements are coincident. This transitional character can be seen in the expression of the wave propagation, in particular as a second longitudinal wave propagating along the layers yet absent across lamination, and generating a triangular-shaped wave front. A relationship established between models confirms the longitudinal nature of this front.

    Additionally, two new features of wave propagation in the anisotropic Biot model are presented. Double loops on the shear wave front with four cusps in one quadrant are now observed. Additionally, the transition from slow longitudinal to shear mode occurs indicating that at orthogonal directions on the same front polarization changes from pure longitudinal to pure shear. These features can be used as potential indicators for the presence of anisotropic porous reservoirs.

  16. Page 255
    Abstract

    Fermat’s Principle states that for two points A and B in a velocity field, the ray path will be the trajectory between A and B along which the travel time is stationary. For many cases in isotropic media Fermat’s Principle seems intuitively obvious. For example, in a homogeneous, isotropic medium the least time travel path between two points is a straight line. In anisotropic media the results of Fermat’s Principle are less obvious and it is useful to have a rigorous proof. In this “tutorial style” paper we present a proof of Fermat’s Principle for anisotropic elastic media. The proof involves relationships between the slowness and wave surfaces. The slowness surfaces are defined by the determinant (S), or equivalently the eigenvalues (Gm), of the Kelvin-Christoffel matrix. The proof is given for both cases.

  17. Page 271
    Abstract

    A derivation of Fermat’s principle for general elastic anisotropic media is presented. It is shown that Fermat’s principle breaks down at the cusps of the wave surface. Applications of Fermat’s principle should therefore be restricted to rays associated with convex slowness surfaces.

  18. Page 289
    Abstract

    Elastic waves can, in principle, be classified according to their propagation velocity, e.g., using the values of the slowness or the group velocity surfaces along a given direction. The investigation is restricted to media with a distinct outer (P) sheet of the wave surface. We show that there are media with the same velocity distribution but drastically different polarization behavior. Such media are kinematically identical but dynamically different. Therefore, classification according wave velocity alone is not sufficient, and the identification of the wave type should be based on both velocity and the polarization distribution.

    For transversely isotropic symmetry there are at most two and, for orthorhombic symmetry, at most four media that have the same velocity distribution. There is always one with a polarization distribution topologically similar to that of isotropy, which is called “normal polarization.” All the other media are said to possess “anomalous polarization.” Note that this does not imply that such media can be found, but only that they are not forbidden by the laws of physics.

    In orthorhombic media there is a different set of four media closely related to the above-mentioned set. The two sets share all velocities along the axes, but the velocity distribution off the axes is different in the second set. All members of this set possess anomalous polarization.

Purchase Chapters

Recommended Reading