Finite‐difference modelling of S‐wave splitting in anisotropic media

We have implemented a 3D finite-difference scheme to simulate wave propagation in arbitrary anisotropic media. The anisotropic media up to orthorhombic symmetry were modelled using a standard staggered grid scheme and beyond (monoclinic and triclinic) using a rotated staggered grid scheme. The rationale of not using rotated staggered grid for all types of anisotropic media is that the rotated staggered grid schemes are more expensive than standard staggered grid schemes. For a 1D azimuthally anistropic medium, we show a comparison between the seismic data generated by our finite-difference code and by the reflectivity algorithm; they are in excellent agreement. We conducted a study on zero-offset shear-wave splitting using the finite-difference modelling algorithm using the rotated staggered grid scheme. Our S-wave splitting study is mainly focused on fractured media. On the scale of seismic wavelenghts, small aligned fractures behave as an equivalent anisotropic medium. We computed the equivalent elastic properties of the fractures and the background in which the fractures were embedded, using low-frequency equivalent media theories. Wave propagation was simulated for both rotationally invariant and corrugated fractures embedded in an isotropic background for one, or more than one, set of fluid-filled and dry fractures. S-wave splitting was studied for dipping fractures, two vertical non-orthogonal fractures and corrugated fractures. Our modelling results confirm that S-wave splitting can reveal the fracture infill in the case of dipping fractures. S-wave splitting has the potential to reveal the angle between the two vertical fractures. We also notice that in the case of vertical corrugated fractures, S-wave splitting is sensitive to the fracture infill.

[1]  K. Fischer,et al.  Shear wave anisotropy in the Mariana Subduction Zone , 1998 .

[2]  M. Schoenberg,et al.  Elastic wave propagation in media with parallel fractures and aligned cracks , 1988 .

[3]  L. Thomsen.  Weak elastic anisotropy , 1986 .

[4]  I. Tsvankin.  Reflection moveout and parameter estimation for horizontal transverse isotropy , 1997 .

[5]  M. Savage,et al.  Shear-wave splitting beneath western United States in relation to plate tectonics , 1995 .

[6]  L. Frazer,et al.  Computation of synthetic seismograms for stratified azimuthally anisotropic media , 1990 .

[7]  L. Frazer,et al.  Reflection/transmission coefficients and azimuthal anisotropy in marine seismic studies , 1991 .

[8]  Ladislav Halada,et al.  3D Fourth-Order Staggered-Grid Finite-Difference Schemes: Stability and Grid Dispersion , 2000 .

[9]  Erik H. Saenger,et al.  Finite-difference modeling of viscoelastic and anisotropic wave propagation using the rotated staggered grid , 2004 .

[10]  K. R. Kelly,et al.  SYNTHETIC SEISMOGRAMS: A FINITE ‐DIFFERENCE APPROACH , 1976 .

[11]  Enru Liu,et al.  Behaviour of shear waves in rocks with two sets of parallel cracks , 1993 .

[12]  Francisco J. Serón,et al.  A numerical laboratory for simulation and visualization of seismic wavefields1 , 1996 .

[13]  Heiner Igel,et al.  Anisotropic wave propagation through finite-difference grids , 1995 .

[14]  M. Schoenberg,et al.  Seismic anisotropy of fractured rock , 1995 .

[15]  J. Carcione,et al.  Seismic modelingSeismic modeling , 2002 .

[16]  G. McMechan,et al.  3-D viscoelastic anisotropic modeling of data from a multicomponent, multiazimuth seismic experiment in Northeast Texas , 1995 .

[17]  C. Sayers.  Misalignment of the orientation of fractures and the principal axes for P and S waves in rocks containing multiple non-orthogonal fracture sets , 1998 .

[18]  David C. Booth,et al.  The anisotropic reflectivity technique: theory , 1983 .

[19]  Fumihito Yamazaki,et al.  Shear wave polarization anisotropy in the upper mantle beneath Honshu, Japan , 1983 .

[20]  Ilya Tsvankin,et al.  Anisotropic parameters and P-wave velocity for orthorhombic media , 1997 .

[21]  Paul G. Silver,et al.  Implications for continental structure and evolution from seismic anisotropy , 1988, Nature.

[22]  A. Bakulin,et al.  Estimation of fracture parameters from reflection seismic data—Part I: HTI model due to a single fracture set , 2000 .

[23]  Arthur Frankel,et al.  Three-dimensional simulations of ground motions in the San Bernardino Valley, California, for hypothetical earthquakes on the San Andreas fault , 1993, Bulletin of the Seismological Society of America.

[24]  Jean Virieux,et al.  SH-wave propagation in heterogeneous media: velocity-stress finite-difference method , 1984 .

[25]  R. Russo,et al.  Trench-Parallel Flow Beneath the Nazca Plate from Seismic Anisotropy , 1994, Science.

[26]  Ilya Tsvankin,et al.  Using AVO for fracture detection: Analytic basis and practical solutions , 1997 .

[27]  Moshe Reshef,et al.  A nonreflecting boundary condition for discrete acoustic and elastic wave equations , 1985 .

[28]  Géza Seriani,et al.  LOW AND HIGH ORDER FINITE ELEMENT METHOD: EXPERIENCE IN SEISMIC MODELING , 1994 .

[29]  Mrinal K. Sen,et al.  Seismic anisotropy in the core–mantle transition zone , 1998 .

[30]  Michael Schoenberg,et al.  Finite-difference modeling of faults and fractures , 1995 .

[31]  K. Nihei,et al.  Three-dimensional elastic wave scattering by a layer containing vertical periodic fractures. , 2003, The Journal of the Acoustical Society of America.

[32]  Paul L. Stoffa,et al.  TRAVELTIME COMPUTATION IN TRANSVERSELY ISOTROPIC MEDIA , 1994 .

[33]  K. Marfurt.  Accuracy of finite-difference and finite-element modeling of the scalar and elastic wave equations , 1984 .

[34]  L. Neil Frazer,et al.  Seismic waves in stratified anisotropic media , 1984 .

[35]  J. Virieux.  P-SV wave propagation in heterogeneous media: Velocity‐stress finite‐difference method , 1986 .

[36]  S. Shapiro,et al.  Modeling the propagation of elastic waves using a modified finite-difference grid , 2000 .

[37]  V. Grechka,et al.  Characterization of dipping fractures in a transverselyisotropic background , 2004 .

[38]  J. Hudson,et al.  Diffraction of seismic waves by cracks with application to hydraulic fracturing , 1997 .

[39]  Susan E. Minkoff,et al.  Spatial Parallelism of a 3D Finite Difference Velocity-Stress Elastic Wave Propagation Code , 1999, SIAM J. Sci. Comput..

[40]  J. Hudson.  Overall properties of a cracked solid , 1980, Mathematical Proceedings of the Cambridge Philosophical Society.