Quartic moveout coefficient for a dipping azimuthally anisotropic layer

Geophysics ◽  
2004 ◽  
Vol 69 (3) ◽  
pp. 699-707 ◽  
Author(s):  
Andrés Pech ◽  
Ilya Tsvankin
Keyword(s):  
Reservoir Characterization ◽  
Fractured Reservoirs ◽  
Symmetry Plane ◽  
Exact Expression ◽  
Naturally Fractured Reservoirs ◽  
P Wave ◽  
Orthorhombic Symmetry ◽  
Fracture Density ◽  
Weak Anisotropy ◽  
Velocity Anisotropy

Interpretation and inversion of azimuthally varying nonhyperbolic reflection moveout requires accounting for both velocity anisotropy and subsurface structure. Here, our previously derived exact expression for the quartic moveout coefficient A4 is applied to P‐wave reflections from a dipping interface overlaid by a medium of orthorhombic symmetry. The weak‐anisotropy approximaton for the coefficient A4 in a homogeneous orthorhombic layer is controlled by the anellipticity parameters η(1), η(2), and η(3), which are responsible for time processing of P‐wave data. If the dip plane of the reflector coincides with the vertical symmetry plane [x1, x3], A4 on the dip line is proportional to the in‐plane anellipticity parameter η(2) and always changes sign for a dip of 30○. The quartic coefficient on the strike line is a function of all three η–parameters, but for mild dips it is mostly governed by η(1)—the parameter defined in the incidence plane [x2, x3]. Whereas the magnitude of the dip line A4 typically becomes small for dips exceeding 45○, the nonhyperbolic moveout on the strike line may remain significant even for subvertical reflectors. The character of the azimuthal variation of A4 depends on reflector dip and is quite sensitive to the signs and relative magnitudes of η(1), η(2), and η(3). The analytic results and numerical modeling show that the azimuthal pattern of the quartic coefficient can contain multiple lobes, with one or two azimuths of vanishing A4 between the dip and strike directions. The strong influence of the anellipticity parameters on the azimuthally varying coefficient A4 suggests that nonhyperbolic moveout recorded in wide‐azimuth surveys can help to constrain the anisotropic velocity field. Since for typical orthorhombic models that describe naturally fractured reservoirs the parameters η(1,2,3) are closely related to the fracture density and infill, the results of azimuthal nonhyperbolic moveout analysis can also be used in reservoir characterization.

Estimation of fracture parameters from reflection seismic data—Part II: Fractured models with orthorhombic symmetry

Geophysics ◽  
2000 ◽  
Vol 65 (6) ◽  
pp. 1803-1817 ◽  
Author(s):  
Andrey Bakulin ◽  
Vladimir Grechka ◽  
Ilya Tsvankin
Keyword(s):  
Symmetry Axis ◽  
Fractured Reservoirs ◽  
Symmetry Plane ◽  
P Wave ◽  
Single Fracture ◽  
Orthorhombic Symmetry ◽  
Weak Anisotropy ◽  
Horizontal Symmetry ◽  
Vertical Symmetry Plane ◽  
Vertical Fractures

Existing geophysical and geological data indicate that orthorhombic media with a horizontal symmetry plane should be rather common for naturally fractured reservoirs. Here, we consider two orthorhombic models: one that contains parallel vertical fractures embedded in a transversely isotropic background with a vertical symmetry axis (VTI medium) and the other formed by two orthogonal sets of rotationally invariant vertical fractures in a purely isotropic host rock. Using the linear‐slip theory, we obtain simple analytic expressions for the anisotropic coefficients of effective orthorhombic media. Under the assumptions of weak anisotropy of the background medium (for the first model) and small compliances of the fractures, all effective anisotropic parameters reduce to the sum of the background values and the parameters associated with each fracture set. For the model with a single fracture system, this result allows us to eliminate the influence of the VTI background by evaluating the differences between the anisotropic parameters defined in the vertical symmetry planes. Subsequently, the fracture weaknesses, which carry information about the density and content of the fracture network, can be estimated in the same way as for fracture‐induced transverse isotropy with a horizontal symmetry axis (HTI media) examined in our previous paper (part I). The parameter estimation procedure can be based on the azimuthally dependent reflection traveltimes and prestack amplitudes of P-waves alone if an estimate of the ratio of the P- and S-wave vertical velocities is available. It is beneficial, however, to combine P-wave data with the vertical traveltimes, NMO velocities, or AVO gradients of mode‐converted (PS) waves. In each vertical symmetry plane of the model with two orthogonal fracture sets, the anisotropic parameters are largely governed by the weaknesses of the fractures orthogonal to this plane. For weak anisotropy, the fracture sets are essentially decoupled, and their parameters can be estimated by means of two independently performed HTI inversions. The input data for this model must include the vertical velocities (or reflector depth) to resolve the anisotropic coefficients in each vertical symmetry plane rather than just their differences. We also discuss several criteria that can be used to distinguish between the models with one and two fracture sets. For example, the semimajor axis of the P-wave NMO ellipse and the polarization direction of the vertically traveling fast shear wave are always parallel to each other for a single system of fractures, but they may become orthogonal in the medium with two fracture sets.

3-D description of normal moveout in anisotropic inhomogeneous media

Geophysics ◽  
1998 ◽  
Vol 63 (3) ◽  
pp. 1079-1092 ◽  
Author(s):  
Vladimir Grechka ◽  
Ilya Tsvankin
Keyword(s):  
Fractured Reservoirs ◽  
Symmetry Plane ◽  
Transversely Isotropic ◽  
Inhomogeneous Media ◽  
P Wave ◽  
Orthorhombic Symmetry ◽  
Azimuthal Variation ◽  
Anisotropic Models ◽  
Transversely Isotropic Media ◽  
Isotropic Media

We present a new equation for normal‐moveout (NMO) velocity that describes azimuthally dependent reflection traveltimes of pure modes from both horizontal and dipping reflectors in arbitrary anisotropic inhomogeneous media. With the exception of anomalous areas such as those where common‐midpoint (CMP) reflection time decreases with offset, the azimuthal variation of NMO velocity represents an ellipse in the horizontal plane, with the orientation of the axes determined by the properties of the medium and the direction of the reflector normal. In general, a minimum of three azimuthal measurements is necessary to reconstruct the best‐fit ellipse and obtain NMO velocity in all azimuthal directions. This result provides a simple way to correct for the azimuthal variation in stacking velocity often observed in 3-D surveys. Even more importantly, analytic expressions for the parameters of the NMO ellipse can be used in the inversion of moveout data for the anisotropic coefficients of the medium. For homogeneous transversely isotropic media with a vertical axis of symmetry (VTI media), our equation for azimuthally dependent NMO velocity from dipping reflectors becomes a relatively simple function of phase velocity and its derivatives. We show that the zero‐dip NMO velocity Vnmo(0) and the anisotropic coefficient η are sufficient to describe the P-wave NMO velocity for any orientation of the CMP line with respect to the dip plane of the reflector. Using our formalism, Vnmo(0) and η (the only parameters needed for time processing) can be found from the dip‐dependent NMO velocity at any azimuth or, alternatively, from the azimuthally dependent NMO for a single dipping reflector. We also apply this theory to more complicated azimuthally anisotropic models with the orthorhombic symmetry used to describe fractured reservoirs. For reflections from horizontal interfaces in orthorhombic media, the axes of the normal moveout ellipse are aligned with the vertical symmetry planes. Therefore, azimuthal P-wave moveout measurements can be inverted for the orientation of the symmetry planes (typically determined by the fracture direction) and the NMO velocities within them. If the vertical velocity is known, symmetry‐plane NMO velocities make it possible to estimate two anisotropic parameters equivalent to Thomsen’s coefficient δ for transversely isotropic media.

Anisotropic parameters and P‐wave velocity for orthorhombic media

Geophysics ◽  
1997 ◽  
Vol 62 (4) ◽  
pp. 1292-1309 ◽  
Author(s):  
Ilya Tsvankin
Keyword(s):  
Fractured Reservoirs ◽  
Transversely Isotropic ◽  
Azimuthal Velocity ◽  
Analytic Solutions ◽  
Seismic Inversion ◽  
P Wave ◽  
Weak Anisotropy ◽  
P Waves ◽  
Velocity Anisotropy ◽  
Wide Range

Although orthorhombic (or orthotropic) symmetry is believed to be common for fractured reservoirs, the difficulties in dealing with nine independent elastic constants have precluded this model from being used in seismology. A notation introduced in this work is designed to help make seismic inversion and processing for orthorhombic media more practical by simplifying the description of a wide range of seismic signatures. Taking advantage of the fact that the Christoffel equation has the same form in the symmetry planes of orthorhombic and transversely isotropic (TI) media, we can replace the stiffness coefficients by two vertical (P and S) velocities and seven dimensionless parameters that represent an extension of Thomsen's anisotropy coefficients to orthorhombic models. By design, this notation provides a uniform description of anisotropic media with both orthorhombic and TI symmetry. The dimensionless anisotropic parameters introduced here preserve all attractive features of Thomsen notation in treating wave propagation and performing 2-D processing in the symmetry planes of orthorhombic media. The new notation has proved useful in describing seismic signatures outside the symmetry planes as well, especially for P‐waves. Linearization of P‐wave phase velocity in the anisotropic coefficients leads to a concise weak‐anisotropy approximation that provides good accuracy even for models with pronounced polar and azimuthal velocity variations. This approximation can be used efficiently to build analytic solutions for various seismic signatures. One of the most important advantages of the new notation is the reduction in the number of parameters responsible for P‐wave velocities and traveltimes. All kinematic signatures of P‐waves in orthorhombic media depend on just the vertical velocity [Formula: see text] and five anisotropic parameters, with [Formula: see text] serving as a scaling coefficient in homogeneous media. This conclusion, which holds even for orthorhombic models with strong velocity anisotropy, provides an analytic basis for application of P‐wave traveltime inversion and data processing algorithms in orthorhombic media.

Estimation of fracture parameters from reflection seismic data—Part III: Fractured models with monoclinic symmetry

Geophysics ◽  
2000 ◽  
Vol 65 (6) ◽  
pp. 1818-1830 ◽  
Author(s):  
Andrey Bakulin ◽  
Vladimir Grechka ◽  
Ilya Tsvankin
Keyword(s):  
Shear Wave ◽  
Fractured Reservoirs ◽  
Shear Wave Splitting ◽  
Naturally Fractured Reservoirs ◽  
P Wave ◽  
S Wave ◽  
Monoclinic Symmetry ◽  
Weak Anisotropy ◽  
Wave Splitting ◽  
Single Set

Geophysical and geological data acquired over naturally fractured reservoirs often reveal the presence of multiple vertical fracture sets. Here, we discuss modeling and inversion of the effective anisotropic parameters of two types of fractured media with monoclinic symmetry. The first model is formed by two different nonorthogonal sets of rotationally invariant vertical fractures in an isotropic host rock; the other contains a single set of fractures with microcorrugated faces. In monoclinic media with two fracture sets, the shear‐wave polarizations at vertical incidence and the orientation of the NMO ellipses of pure modes in a horizontal layer are controlled by the fracture azimuths as well as by their compliances. While the S-wave polarization directions depend only on the tangential compliances, the axes of the P-wave NMO ellipse are also influenced by the normal compliances and therefore have a different orientation. This yields an apparent discrepancy between the principal anisotropy directions obtained using P and S data that does not exist in orthorhombic media. By first using the weak‐anisotropy approximation for the effective anisotropic parameters and then inverting the exact equations, we devise a complete fracture characterization procedure based on the vertical velocities of the P- and two split S-waves (or converted PS-waves) and their NMO ellipses from a horizontal reflector. Our algorithm yields the azimuths and compliances of both fracture systems as well as the P- and S-wave velocities in the isotropic background medium. In the model with a single set of microcorrugated fractures, monoclinic symmetry stems from the coupling between the normal and tangential (to the fracture faces) slips, or jumps in displacement. We demonstrate that for this model the shear‐wave splitting coefficient at vertical incidence varies with the fluid content of the fractures. Although conventional fracture models that ignore microcorrugation predict no such dependence, our conclusions are supported by experimental observations showing that shear‐wave splitting for dry cracks may be substantially greater than that for fluid‐filled ones.

scholarly journals Seismic AVOA Inversion for Weak Anisotropy Parameters and Fracture Density in a Monoclinic Medium

2020 ◽  
Vol 10 (15) ◽  
pp. 5136 ◽  
Author(s):  
Zijian Ge ◽  
Shulin Pan ◽  
Jingye Li
Keyword(s):  
Fractured Rock ◽  
Synthetic Data ◽  
Transversely Isotropic ◽  
P Wave ◽  
Inversion Method ◽  
Fracture Density ◽  
Weak Anisotropy ◽  
Amplitude Versus Offset ◽  
Porosity And Permeability ◽  
Transverse Isotropic

In shale gas development, fracture density is an important lithologic parameter to properly characterize reservoir reconstruction, establish a fracturing scheme, and calculate porosity and permeability. The traditional methods usually assume that the fracture reservoir is one set of aligned vertical fractures, embedded in an isotropic background, and estimate some alternative parameters associated with fracture density. Thus, the low accuracy caused by this simplified model, and the intrinsic errors caused by the indirect substitution, affect the estimation of fracture density. In this paper, the fractured rock of monoclinic symmetry assumes two non-orthogonal vertical fracture sets, embedded in a transversely isotropic background. Firstly, assuming that the fracture radius, width, and orientation are known, a new form of P-wave reflection coefficient, in terms of weak anisotropy (WA) parameters and fracture density, was obtained by substituting the stiffness coefficients of vertical transverse isotropic (VTI) background, normal, and tangential fracture compliances. Then, a linear amplitude versus offset and azimuth (AVOA) inversion method, of WA parameters and fracture density, was constructed by using Bayesian theory. Tests on synthetic data showed that WA parameters, and fracture density, are stably estimated in the case of seismic data containing a moderate noise, which can provide a reliable tool in fracture prediction.

Plane-wave attenuation anisotropy in orthorhombic media

Geophysics ◽  
2007 ◽  
Vol 72 (1) ◽  
pp. D9-D19 ◽  
Author(s):  
Yaping Zhu ◽  
Ilya Tsvankin
Keyword(s):  
Attenuation Coefficient ◽  
Wave Attenuation ◽  
Fractured Reservoirs ◽  
The Body ◽  
P Wave ◽  
Attenuation Coefficients ◽  
Velocity Anisotropy ◽  
Homogeneous Wave ◽  
Anisotropy Parameters ◽  
Attenuation Anisotropy

Orthorhombic models are often used in the interpretation of azimuthally varying seismic signatures recorded over fractured reservoirs. Here, we develop an analytic framework for describing the attenuation coefficients in orthorhombic media with orthorhombic attenuation (i.e., the symmetry of both the real and imaginary parts of the stiffness tensor is identical) under the assumption of homogeneous wave propagation. The analogous form of the Christoffel equation in the symmetry planes of orthorhombic and VTI (transversely isotropic with a vertical symmetry axis) media helps to obtain the symmetry-plane attenuation coefficients by adapting the existing VTI equations. To take full advantage of this equivalence with transverse isotropy, we introduce a parameter set similar to the VTI attenuation-anisotropy parameters [Formula: see text], [Formula: see text], and [Formula: see text]. This notation, based on the same principle as Tsvankin’s velocity-anisotropy parameters for orthorhombic media, leads to concise linearized equations for thesymmetry-plane attenuation coefficients of all three modes (P, [Formula: see text], and [Formula: see text]).The attenuation-anisotropy parameters also allow us to simplify the P-wave attenuation coefficient [Formula: see text] outside the symmetry planes under the assumptions of small attenuation and weak velocity and attenuation anisotropy. The approximate coefficient [Formula: see text] has the same form as the linearized P-wave phase-velocity function, with the velocity parameters [Formula: see text] and [Formula: see text] replaced by the attenuation parameters [Formula: see text] and [Formula: see text]. The exact attenuation coefficient, however, also depends on the velocity-anisotropy parameters, while the body-wave velocities are almost unperturbed by the presence of attenuation. The reduction in the number of parameters responsible for the P-wave attenuation and the simple approximation for the coefficient [Formula: see text] provide a basis for inverting P-wave attenuation measurements from orthorhombic media. The attenuation processing must be preceded by anisotropic velocity analysis that can be performed (in the absence of pronounced velocity dispersion) using existing algorithms for nonattenuative media.

scholarly journals Physical and numerical modeling of tuning and diffraction in azimuthally anisotropic media

Geophysics ◽  
2000 ◽  
Vol 65 (5) ◽  
pp. 1613-1621 ◽  
Author(s):  
Richard L. Gibson ◽  
Stephen Theophanis ◽  
M. Nafi Toksöz
Keyword(s):  
Numerical Modeling ◽  
Fractured Reservoirs ◽  
Homogeneous Model ◽  
Azimuthal Anisotropy ◽  
P Wave ◽  
Orthorhombic Symmetry ◽  
Fractured Reservoir ◽  
Sh Wave ◽  
Wave Data ◽  
Ray Methods

Fractured reservoirs are an important target for exploration and production geophysics, and the azimuthal anisotropy often associated with these reservoirs can strongly influence seismic wave propagation. We created a physical model of a fractured reservoir to simulate some of these propagation effects. The reservoir is represented by a phenolite disk that is thin with respect to the elastic wavelengths in the experiment, creating model dimensions that are representative of realistic reservoirs. Phenolite is strongly anisotropic with orthorhombic symmetry, which suggests that azimuthal amplitude versus offset (AVO) effects should be obvious in data. We acquired both SH- and P-wave data in common‐offset gathers with a near offset and a far offset and found that although the SH-wave data show clear azimuthal variations in AVO, the P-wave signals show no apparent changes with azimuth. We then applied numerical modeling to analyze the data. Because ray methods cannot model diffractions from the disk edge, we first used a ray‐Born technique to simulate variations in waveforms associated with such scattering. The synthetic seismograms reproduced variations in the SH-wave waveforms accurately, though the amplitude contrast between acquisition azimuths was overestimated. Assuming a laterally homogeneous model, we then applied ray methods to simulate tuning effects in SH- and P-wave data and confirmed that in spite of the large contrasts in elastic properties, the tuning of the P-wave reflections from the thin disk changed so there was negligible contrast in AVO with azimuth. Models of field scale reservoirs showed that the same effects could be expected for field applications.

Seismic critical-angle reflectometry: A method to characterize azimuthal anisotropy?

Geophysics ◽  
2007 ◽  
Vol 72 (3) ◽  
pp. D41-D50 ◽  
Author(s):  
Martin Landrø ◽  
Ilya Tsvankin
Keyword(s):  
Critical Angle ◽  
Transverse Isotropy ◽  
Symmetry Plane ◽  
Azimuthal Anisotropy ◽  
P Wave ◽  
Head Wave ◽  
Orthorhombic Symmetry ◽  
Amplitude Curve ◽  
Diagnostic Features ◽  
Azimuthal Variation

Existing anisotropic parameter-estimation algorithms that operate with long-offset data are based on the inversion of either nonhyperbolic moveout or wide-angle amplitude-variation-with-offset (AVO) response. We show that valuable information about anisotropic reservoirs can also be obtained from the critical angle of reflected waves. To explain the behavior of the critical angle, we develop weak-anisotropy approximations for vertical transverse isotropy and then use Tsvankin’s notation to extend them to azimuthally anisotropic models of orthorhombic symmetry. The P-wave critical-angle reflection in orthorhombic media is particularly sensitive to the parameters [Formula: see text] and [Formula: see text] responsible for the symmetry-plane horizontal velocity in the high-velocity layer. The azimuthal variation of the critical angle for typical orthorhombic models can reach [Formula: see text], which translates into substantial changes in the critical offset of the reflected P-wave. The main diagnostic features of the critical-angle reflection employed in our method include the rapid amplitude increase at the critical angle and the subsequent separation of the head wave. Analysis of exact synthetic seismograms, generated with the reflectivity method, confirms that the azimuthal variation of the critical offset is detectable on wide-azimuth, long-spread data and can be qualitatively described by our linearized equations. Estimation of the critical offset from the amplitude curve of the reflected wave, however, is not straightforward. Additional complications may be caused by the overburden noise train and by the influence of errors in the overburden velocity model on the computation of the critical angle. Still, critical-angle reflectometry should help to constrain the dominant fracture directions and can be combined with other methods to reduce the uncertainty in the estimated anisotropy parameters.

Sign in / Sign up

Export Citation Format

Share Document