Effective velocities in fractured media: a numerical study using the rotated staggered finite‐difference grid

The modelling of elastic waves in fractured media with an explicit finite-difference scheme causes instability problems on a staggered grid when the medium possesses high-contrast discontinuities (strong heterogeneities). For the present study we apply the rotated staggered grid. Using this modified grid it is possible to simulate the propagation of elastic waves in a 2D or 3D medium containing cracks, pores or free surfaces without hard-coded boundary conditions. Therefore it allows an efficient and precise numerical study of effective velocities in fractured structures. We model the propagation of plane waves through a set of different, randomly cracked media. In these numerical experiments we vary the wavelength of the plane waves, the crack porosity and the crack density. The synthetic results are compared with several static theories that predict the effective P- and S-wave velocities in fractured materials in the long wavelength limit. For randomly distributed and randomly orientated, rectilinear, non-intersecting, thin, dry cracks, the numerical simulations of velocities of P-, SV- and SH-waves are in excellent agreement with the results of the modified (or differential) self-consistent theory. On the other hand for intersecting cracks, the critical crack-density (porosity) concept must be taken into account. To describe the wave velocities in media with intersecting cracks, we propose introducing the critical crack-density concept into the modified self-consistent theory. Numerical simulations show that this new formulation predicts effective elastic properties accurately for such a case.

[1]  Mark Kachanov,et al.  Effective Elastic Properties of Cracked Solids: Critical Review of Some Basic Concepts , 1992 .

[2]  P. C. Robinson,et al.  Numerical calculations of critical densities for lines and planes , 1984 .

[3]  P. C. Robinson.  Connectivity of fracture systems-a percolation theory approach , 1983 .

[4]  Guido Kneib,et al.  Accurate and efficient seismic modeling in random media , 1993 .

[5]  Yehuda Ben-Zion,et al.  Wrinkle-like slip pulse on a fault between different , 1997 .

[6]  T. Yamashita,et al.  Multiple scattering of SH waves in 2-D elastic media with distributed cracks , 1995 .

[7]  Robert W. Clayton,et al.  Finite difference simulations of seismic scattering: Implications for the propagation of short‐period seismic waves in the crust and models of crustal heterogeneity , 1986 .

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

[9]  Tai Te Wu,et al.  The effect of inclusion shape on the elastic moduli of a two-phase material* , 1966 .

[10]  S. Shapiro,et al.  A Numerical Study of Effective Velocities In Fractured Media Using the Rotated Staggered Finite Difference Grid , 2000 .

[11]  Predicting the Overall Properties of Composite Materials with Small-scale Inclusions or Cracks , 1989 .

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

[13]  The elastic modulus of media containing strongly interacting antiplane cracks , 1995 .

[14]  J. Berryman.  Single‐scattering approximations for coefficients in Biot’s equations of poroelasticity , 1992 .

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

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

[17]  T. Becker,et al.  On the Elastic and Viscous Properties of Media Containing Strongly Interacting In-plane Cracks , 1998 .

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

[19]  James G. Berryman,et al.  Long‐wavelength propagation in composite elastic media I. Spherical inclusions , 1980 .

[20]  Y. Guéguen,et al.  Comment [on “The elastic modulus of media containing strongly interacting antiplane cracks” by Paul M. Davis and Leon Knopoff] , 1996 .

[21]  J. Hudson,et al.  Seismic properties of rocks with distributions of small cracks , 1990 .

[22]  C. Edward.  High-order (space And Time) Finite-difference Modeling of the Elastic Wave Equation , 1990 .

[23]  J. Hudson,et al.  Equivalent medium representation of fractured rock , 2000 .

[24]  J. Douma.  THE EFFECT OF THE ASPECT RATIO ON CRACK-INDUCED ANISOTROPY1 , 1988 .

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

[26]  B. Budiansky,et al.  Elastic moduli of a cracked solid , 1976 .

[27]  J. Berryman,et al.  Differential effective medium modeling of rock elastic moduli with critical porosity constraints , 1995 .

[28]  F. Henyey,et al.  Self‐consistent elastic moduli of a cracked solid , 1982 .

[29]  J. D. Eshelby.  The determination of the elastic field of an ellipsoidal inclusion, and related problems , 1957, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.

[30]  Chuen Hon Arthur Cheng.  Crack models for a transversely isotropic medium , 1993 .

[31]  Reply [to “Comment on ‘Seismic velocities in dry and saturated cracked solids’ by Richard J. O'Connell and Bernard Budiansky”] , 1976 .

[32]  Johan O. A. Robertsson,et al.  A numerical free-surface condition for elastic/viscoelastic finite-difference modeling in the presence of topography , 1996 .

[33]  L. Knopoff,et al.  Elastic moduli of two-component systems , 1978 .

[34]  Bernard Budiansky,et al.  Seismic velocities in dry and saturated cracked solids , 1974 .

[35]  M. Bouchon,et al.  Numerical simulation of the propagation of P waves in fractured media , 1999 .

[36]  Amos Nur,et al.  Effects of stress on velocity anisotropy in rocks with cracks , 1971 .

[37]  W. M. Bruner,et al.  Comment on ‘Seismic velocities in dry and saturated cracked solids’ by Richard J. O'Connell and Bernard Budiansky , 1976 .

[38]  A. Norris.  A differential scheme for the effective moduli of composites , 1985 .