Theory of traveltime shifts around compacting reservoirs: 3D solutions for heterogeneous anisotropic media

Geophysics ◽  
2009 ◽  
Vol 74 (1) ◽  
pp. D25-D36 ◽  
Author(s):  
Rodrigo Felício Fuck ◽  
Andrey Bakulin ◽  
Ilya Tsvankin
Keyword(s):  
Isotropic Medium ◽  
Transversely Isotropic ◽  
Anisotropic Media ◽  
Time Lapse ◽  
Theory Of Elasticity ◽  
Closed Form Expression ◽  
Reservoir Compaction ◽  
Velocity Changes ◽  
Zero Offset ◽  
Induced Velocity

Time-lapse traveltime shifts of reflection events recorded above hydrocarbon reservoirs can be used to monitor production-related compaction and pore-pressure changes. Existing methodology, however, is limited to zero-offset rays and cannot be applied to traveltime shifts measured on prestack seismic data. We give an analytic 3D description of stress-related traveltime shifts for rays propagating along arbitrary trajectories in heterogeneous anisotropic media. The nonlinear theory of elasticity helps to express the velocity changes in and around the reservoir through the excess stresses associated with reservoir compaction. Because this stress-induced velocity field is both heterogeneous and anisotropic, it should be studied using prestack traveltimes or amplitudes. Then we obtain the traveltime shifts by first-order perturbation of traveltimes that accounts not only for the velocity changes but also for 3D deformation of reflectors. The resulting closed-form expression can be used efficiently for numerical modeling of traveltime shifts and, ultimately, for reconstructing the stress distribution around compacting reservoirs. The analytic results are applied to a 2D model of a compacting rectangular reservoir embedded in an initially homogeneous and isotropic medium. The computed velocity changes around the reservoir are caused primarily by deviatoric stresses and produce a transversely isotropic medium with a variable orientation of the symmetry axis and substantial values of the Thomsen parameters [Formula: see text] and [Formula: see text]. The offset dependence of the traveltime shifts should play a crucial role in estimating the anisotropy parameters and compaction-related deviatoric stress components.

Estimation of layer thickness and velocity changes using 4D prestack seismic data

Geophysics ◽  
2006 ◽  
Vol 71 (6) ◽  
pp. S219-S234 ◽  
Author(s):  
Thomas Røste ◽  
Alexey Stovas ◽  
Martin Landrø
Keyword(s):  
Layer Thickness ◽  
Seismic Data ◽  
Single Layer ◽  
Time Lapse ◽  
Seismic Velocities ◽  
Seismic Line ◽  
Velocity Increase ◽  
Reservoir Compaction ◽  
Velocity Changes ◽  
Zero Offset

In some hydrocarbon reservoirs, severe compaction of the reservoir rocks is observed. This compaction is caused by production and is often associated with stretching and arching of the overburden rocks. Time-lapse seismic data can be used to monitor these processes. Since compaction and stretching cause changes in layer thickness as well as seismic velocities, it is crucial to develop methods to distinguish between the two effects. We introduce a new method based on detailed analysis of time-lapse prestack seismic data. The equations are derived assuming that the entire model consists of only one single layer with no vertical velocity variations. The method incorporates lateral variations in (relative) velocity changes by utilizing zero-offset and offset-dependent time shifts. To test the method, we design a 2D synthetic model that undergoes severe reservoir compaction as well as stretching of the overburden rocks. Finally, we utilize the method to analyze a real 2D prestack time-lapse seismic line from the Valhall field, acquired in 1992 and 2002. For a horizon at a depth of around [Formula: see text], which is near the top reservoir horizon, a subsidence of [Formula: see text] and a velocity decrease of [Formula: see text] for the sequence from the sea surface to the top reservoir horizon are estimated. By assuming that the base of the reservoir remains constant in depth, a reservoir compaction of 3.6% (corresponding to a subsidence of the top reservoir horizon of [Formula: see text]) and a corresponding reservoir velocity increase of 6.7% (corresponding to a velocity increase of [Formula: see text]) are estimated.

scholarly journals Transformation to zero offset in transversely isotropic media

Geophysics ◽  
1996 ◽  
Vol 61 (4) ◽  
pp. 947-963 ◽  
Author(s):  
Tariq Alkhalifah
Keyword(s):  
Ray Tracing ◽  
Impulse Response ◽  
Homogeneous Medium ◽  
Anisotropy Parameter ◽  
Transversely Isotropic ◽  
Anisotropic Media ◽  
Amplitude Distribution ◽  
Isotropic Media ◽  
Zero Offset ◽  
Vertical Inhomogeneity

Nearly all dip‐moveout correction (DMO) implementations to date assume isotropic homogeneous media. Usually, this has been acceptable considering the tremendous cost savings of homogeneous isotropic DMO and considering the difficulty of obtaining the anisotropy parameters required for effective implementation. In the presence of typical anisotropy, however, ignoring the anisotropy can yield inadequate results. Since anisotropy may introduce large deviations from hyperbolic moveout, accurate transformation to zero‐offset in anisotropic media should address such nonhyperbolic moveout behavior of reflections. Artley and Hale’s v(z) ray‐tracing‐based DMO, developed for isotropic media, provides an attractive approach to treating such problems. By using a ray‐tracing procedure crafted for anisotropic media, I modify some aspects of their DMO so that it can work for v(z) anisotropic media. DMO impulse responses in typical transversely isotropic (TI) models (such as those associated with shales) deviate substantially from the familiar elliptical shape associated with responses in homogeneous isotropic media (to the extent that triplications arise even where the medium is homogeneous). Such deviations can exceed those caused by vertical inhomogeneity, thus emphasizing the importance of taking anisotropy into account in DMO processing. For isotropic or elliptically anisotropic media, the impulse response is an ellipse; but as the key anisotropy parameter η varies, the shape of the response differs substantially from elliptical. For typical η > 0, the impulse response in TI media tends to broaden compared to the response in an isotropic homogeneous medium, a behavior opposite to that encountered in typical v(z) isotropic media, where the response tends to be squeezed. Furthermore, the amplitude distribution along the DMO operator differs significantly from that for isotropic media. Application of this anisotropic DMO to data from offshore Africa resulted in a considerably better alignment of reflections from horizontal and dipping reflectors in common‐midpoint gather than that obtained using an isotropic DMO. Even the presence of vertical inhomogeneity in this medium could not eliminate the importance of considering the shale‐induced anisotropy.

Imaging reflections in elliptically anisotropic media

Geophysics ◽  
1988 ◽  
Vol 53 (12) ◽  
pp. 1616-1618 ◽  
Author(s):  
Joe Dellinger ◽  
Francis Muir
Keyword(s):  
Point Source ◽  
Isotropic Medium ◽  
Short Note ◽  
Transversely Isotropic ◽  
Anisotropic Media ◽  
Virtual Image ◽  
Transversely Isotropic Medium ◽  
Sh Waves ◽  
Special Case

In an isotropic medium, waves reflected from a mirror form a virtual image of their source. This property of planar reflectors is generally not true in the presence of anisotropy. In their short note, Blair and Korringa (1987) show that for the special case of SH waves from a point source in a transversely isotropic medium, an aberration‐free image is formed for any orientation of the mirror. While their proof is mathematical, we show the same result in an intuitive, pictorial fashion and in the process discover that although the image is indeed aberration free, it is still distorted.

Dip moveout in anisotropic media

Geophysics ◽  
1990 ◽  
Vol 55 (7) ◽  
pp. 863-867 ◽  
Author(s):  
N. F. Uren ◽  
G. H. F. Gardner ◽  
J. A. McDonald
Keyword(s):  
Transversely Isotropic ◽  
Anisotropic Media ◽  
Point Scatterer ◽  
Velocity Function ◽  
Processing Step ◽  
Axis Of Symmetry ◽  
Zero Offset ◽  
Seismic Lines ◽  
Seismic Measurements ◽  
Simple Numerical Model

When a common‐midpoint gather is collected above a dipping reflector, the point at which reflection occurs moves updip as the source‐receiver offset increases. Stacking velocity is constant, but it is a function of dip. Hence a stacked trace is not equivalent to a zero‐offset recorded trace. Dip moveout (DMO) is a processing step which converts traces to the equivalent of true zero‐offset records, making migration after stack (MAS) equivalent to migration before stack (MBS). The theory of velocity‐independent Gardner DMO is extended to triaxially elliptically anisotropic media in this paper. It is shown that the transformation is exact for homogeneous elliptically anisotropic media and that, after DMO, the stacking velocity is the horizontal component of the elliptically anisotropic velocity function. By taking three distinct seismic lines, three of the six constants of triaxial elliptical anisotropy may be determined. The remaining three cannot be obtained from surface seismic measurements. A simple numerical model of a point scatterer in a transversely isotropic medium with a tilted axis of symmetry is used to generate examples. The DMO process works when the velocity function is elliptical, but is not exact when the velocity function is nonelliptical.

Elastic waves in anisotropic media

1959 ◽  
Vol 253 (1275) ◽  
pp. 563-580 ◽  
Keyword(s):  
Point Source ◽  
Elastic Medium ◽  
Elastic Waves ◽  
Isotropic Medium ◽  
Transversely Isotropic ◽  
Anisotropic Media ◽  
Transversely Isotropic Medium ◽  
Anisotropic Elastic Medium ◽  
Anisotropic Elastic ◽  
Fourier Integrals

The displacements due to a radiating point source in an infinite anisotropic elastic medium are found in terms of Fourier integrals. The integrals are evaluated asymptotically, yielding explicit expressions for displacements at points far from the source. The relative amplitudes of waves from a point source are thus determined, and it is found that although in general the decay of wave amplitudes is proportional to the distance from the source, it is possible that in certain directions the decay is less than this. The method used in this paper is also shown to be an alternative way of deriving known results concerning the geometry of the propagation of disturbances. As an example, the radiation in a transversely isotropic medium from an isolated force varying harmonically with time is discussed.

The migrator’s equation for anisotropic media

Geophysics ◽  
1990 ◽  
Vol 55 (11) ◽  
pp. 1429-1434 ◽  
Author(s):  
N. F. Uren ◽  
G. H. F. Gardner ◽  
J. A. McDonald
Keyword(s):  
Physical Model ◽  
Transversely Isotropic ◽  
Anisotropic Media ◽  
Vertical Axis ◽  
Original Structure ◽  
Model Data ◽  
P Waves ◽  
Reflection Seismic ◽  
Isotropic Media ◽  
Zero Offset

The migrator’s equation, which gives the relationship between real and apparent dips on a reflector in zero‐offset reflection seismic sections, may be readily implemented in one step with a frequency‐domain migration algorithm for homogeneous media. Huygens’ principle is used to derive a similar relationship for anisotropic media where velocities are directionally dependent. The anisotropic form of the migrator’s equation is applicable to both elliptically and nonelliptically anisotropic media. Transversely isotropic media are used to demonstrate the performance of an f-k implementation of the migrator’s equation for anisotropic media. In such a medium SH-waves are elliptically anisotropic, while P-waves are nonelliptically anisotropic. Numerical model data and physical model data demonstrate the performance of the algorithm, in each case recovering the original structure. Isotropic and anisotropic migration of anisotropic physical model data are compared experimentally, where the anisotropic velocity function of the medium has a vertical axis of symmetry. Only when anisotropic migration is used is the original structure recovered.

Traveltime approximations for a layered transversely isotropic medium

Geophysics ◽  
2006 ◽  
Vol 71 (2) ◽  
pp. D23-D33 ◽  
Author(s):  
Bjørn Ursin ◽  
Alexey Stovas
Keyword(s):  
Taylor Series ◽  
Isotropic Medium ◽  
Continued Fraction ◽  
Transversely Isotropic ◽  
Horizontal Distance ◽  
Weak Anisotropy ◽  
Sh Waves ◽  
Numerical Studies ◽  
Anisotropic Layers ◽  
Zero Offset

We consider multiple transmitted, reflected, and converted qP-qSV-waves or multiple transmitted and reflected SH-waves in a horizontally layered medium that is transversely isotropic with a vertical symmetry axis (VTI). Traveltime and offset (horizontal distance) between a source and receiver, not necessarily in the same layer, are expressed as functions of horizontal slowness. These functions are given in terms of a Taylor series in slowness in exactly the same form as for a layered isotropic medium. The coefficients depend on the parameters of the anisotropic layers through which the wave has passed, and there is no weak anisotropy assumption. Using classical formulas, the traveltime or traveltime squared can then be expressed as a Taylor series in even powers of offset. These Taylor series give rise to a shifted hyperbola traveltime approximation and a new continued-fraction approximation, described by four parameters that match the Taylor series up to the sixth power in offset. Further approximations give several simplified continued-fraction approximations, all of which depend on three parameters: zero-offset traveltime, NMO velocity, and a heterogeneity coefficient. The approximations break down when there is a cusp in the group velocity for the qSV-wave. Numerical studies indicate that approximations of traveltime squared are generally better than those for traveltime. A new continued-fraction approximation that depends on three parameters is more accurate than the commonly used continued-fraction approximation and the shifted hyperbola.

Time delays from stress-induced velocity changes around fractures in a time-lapse DAS VSP

2018 ◽  
Author(s):  
Gary Binder ◽  
Aleksei Titov ◽  
Diana Tamayo ◽  
James Simmons ◽  
Ali Tura ◽  
...  
Keyword(s):  
Time Delays ◽  
Time Lapse ◽  
Velocity Changes ◽  
Induced Velocity

scholarly journals Theory of interval traveltime parameter estimation in layered anisotropic media

Geophysics ◽  
2016 ◽  
Vol 81 (5) ◽  
pp. C253-C263 ◽  
Author(s):  
Yanadet Sripanich ◽  
Sergey Fomel
Keyword(s):  
Parameter Estimation ◽  
Fourth Order ◽  
Taylor Expansion ◽  
Symmetry Plane ◽  
Transversely Isotropic ◽  
Anisotropic Media ◽  
Interval Parameter ◽  
Effective Parameters ◽  
Zero Offset ◽  
Vti Media

Moveout approximations for reflection traveltimes are typically based on a truncated Taylor expansion of traveltime squared around the zero offset. The fourth-order Taylor expansion involves normal moveout velocities and quartic coefficients. We have derived general expressions for layer-stripping second- and fourth-order parameters in horizontally layered anisotropic strata and specified them for two important cases: horizontally stacked aligned orthorhombic layers and azimuthally rotated orthorhombic layers. In the first of these cases, the formula involving the out-of-symmetry-plane quartic coefficients has a simple functional form and possesses some similarity to the previously known formulas corresponding to the 2D in-symmetry-plane counterparts in vertically transversely isotropic (VTI) media. The error of approximating effective parameters by using approximate VTI formulas can be significant in comparison with the exact formulas that we have derived. We have proposed a framework for deriving Dix-type inversion formulas for interval parameter estimation from traveltime expansion coefficients in the general case and in the specific case of aligned orthorhombic layers. The averaging formulas for calculation of effective parameters and the layer-stripping formulas for interval parameter estimation are readily applicable to 3D seismic reflection processing in layered anisotropic media.

Estimating SV‐wave stacking velocities for transversely isotropic solids

Geophysics ◽  
1991 ◽  
Vol 56 (10) ◽  
pp. 1596-1602 ◽  
Author(s):  
Patricia A. Berge
Keyword(s):  
Shear Wave ◽  
Isotropic Medium ◽  
Transversely Isotropic ◽  
Anisotropic Media ◽  
Vertical Axis ◽  
Transversely Isotropic Medium ◽  
Transversely Isotropic Media ◽  
Isotropic Media ◽  
Snell’S Law ◽  
Snell's Law

Conventional seismic experiments can record converted shear waves in anisotropic media, but the shear‐wave stacking velocities pose a problem when processing and interpreting the data. Methods used to find shear‐wave stacking velocities in isotropic media will not always provide good estimates in anisotropic media. Although isotropic methods often can be used to estimate shear‐wave stacking velocities in transversely isotropic media with vertical symmetry axes, the estimations fail for some transversely isotropic media even though the anisotropy is weak. For a given anisotropic medium, the shear‐wave stacking velocity can be estimated using isotropic methods if the isotropic Snell’s law approximates the anisotropic Snell’s law and if the shear wavefront is smooth enough near the vertical axis to be fit with an ellipse. Most of the 15 transversely isotropic media examined in this paper met these conditions for short reflection spreads and small ray angles. Any transversely isotropic medium will meet these conditions if the transverse isotropy is weak and caused by thin subhorizontal layering. For three of the media examined, the anisotropy was weak but the shear wave-fronts were not even approximately elliptical near the vertical axis. Thus, isotropic methods provided poor estimates of the shear‐wave stacking velocities. These results confirm that for any given transversely isotropic medium, it is possible to determine whether or not shear‐wave stacking velocities can be estimated using isotropic velocity analysis.

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