Annual Report 2000 - WIT
Annual Report 2000 - WIT Annual Report 2000 - WIT
182 REFERENCES Bourbié, T., Coussy, O., and Zinszner, B., 1987, Acoustics of porous media: Editions Technip. Harjes, H.-P., Bram, K., Dürbaum, H.-J., Gebrande, H., Hirschmann, G., Janik, M., Klöckner, M., Lüschen, E., Rabbel, W., Simon, M., Thomas, R., Tormann, J., and Wenzel, F., 1997, Origin and nature of crustal reflections: results from integrated seismic measurements at the KTB superdeep drill hole: Journal of Geophysical Research, 102, 18267–18288. Holliger, K., 1996, Upper-crustal seismic velocity heterogeneity as derived from a variety of p-wave logs: Geophys J. Int., 125, 813–829. Holliger, K., 1997, Seismic scattering in the upper crystalline crust based on the evidence from sonic logs: Geophys J. Int., 128, 65–72. Jia, Y., and Harjes, H.-P., 1997, Seismische q-werte als ausdruck von intrinsischer dämpfung und streudämpfung der kristallinen kruste um die ktb-lokation: ICDP- Kolloquium, Bochum. Jones, A. G., and Holliger, K., 1997, Spectral analyses of the ktb sonic an density logs using robust nonparametric methods: Journal of Geophysical Research, 102, 18,391–18,403. Kneib, G., 1995, The statistical nature of the upper continental crystalline crust derived from in situ seismic measurements: Geophys J. Int., 122, 594–616. Li, X.-P., and Richwalski, S., 1996, Seismic attenuation and velocities of p- and s- waves in the german ktb area: J. Appl. Geophys., 36, 67–76. Müller, T. M., Shapiro, S. A., and Sick, C., 2000, A weak fluctuation approximation of primary waves in random media: Waves Random Media, submitted. Pujol, J., Lüschen, E., and Hu, Y., 1998, Seismic wave attenuation in methamorphic rocks from vsp data recorded in germany's continental super-deep borehole: Geophysics, 63, 354–365. Sato, H., and Fehler, M., 1998, Wave propagation and scattering in the heterogenous earth: AIP-press. Shapiro, S. A., and Hubral, P., 1999, Elastic waves in random media: Springer. Shapiro, S. A., Schwarz, R., and Gold, N., 1996, The effect of random isotropic inhomogeneities on the phase velocity of seismic waves: Geophys. J. Int., 127, 783–794. Wu, R.-S., Xu, Z., and Li, X.-P., 1994, Heterogeneity spectrum and scale-anisotropy in the upper crust revealed by the german continental deep-drilling (ktb) holes: Geophysical Research Letters, 21, 911–914.
Wave Inversion Technology, Report No. 4, pages 183-191 Effective velocities in fractured media: Intersecting and parallel cracks E.H. Saenger and S.A. Shapiro 1 keywords: finite differences, effective velocities, fractured media ABSTRACT This paper is concerned with a numerical study of effective velocities in two types of fractured media. We apply the so-called rotated staggered finite difference grid technique. 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 hardcoded 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 crack density. The synthetic results are compared with several theories that predict the effective P- and S-wave velocities in fractured materials. For randomly distributed and randomly oriented rectilinear intersecting thin dry cracks the numerical simulations of velocities of P-, SV- and SH-waves are in excellent agreement with the results of a new critical crack density (CCD) formulation. On the other hand for randomly distributed rectilinear parallel thin dry cracks three different classical theories are compared with our numerical results. INTRODUCTION The problem of effective elastic properties of fractured solids is of considerable interest for geophysics, for material science, and for solid mechanics. In this paper we consider the problem of a fractured medium in two dimensions. With this work some broad generalizations can be elucidated that will help solving problems with more complicated geometries. Strong scattering caused by many dry cracks can be treated only by numerical techniques because an analytical solution of the wave equation is not available. So-called boundary integral methods are well suited to handle such discrete scatterers in a homogeneous embedding. They allow the study of SV-waves (Davis and Knopoff, 1995; Murai et al., 1995), SH-waves (Dahm and Becker, 1998) and P-waves (Kelner et al., 1 email: saenger@geophysik.fu-berlin.de, shapiro@geophysik.fu-berlin.de 183
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182<br />
REFERENCES<br />
Bourbié, T., Coussy, O., and Zinszner, B., 1987, Acoustics of porous media: Editions<br />
Technip.<br />
Harjes, H.-P., Bram, K., Dürbaum, H.-J., Gebrande, H., Hirschmann, G., Janik, M.,<br />
Klöckner, M., Lüschen, E., Rabbel, W., Simon, M., Thomas, R., Tormann, J., and<br />
Wenzel, F., 1997, Origin and nature of crustal reflections: results from integrated<br />
seismic measurements at the KTB superdeep drill hole: Journal of Geophysical<br />
Research, 102, 18267–18288.<br />
Holliger, K., 1996, Upper-crustal seismic velocity heterogeneity as derived from a<br />
variety of p-wave logs: Geophys J. Int., 125, 813–829.<br />
Holliger, K., 1997, Seismic scattering in the upper crystalline crust based on the evidence<br />
from sonic logs: Geophys J. Int., 128, 65–72.<br />
Jia, Y., and Harjes, H.-P., 1997, Seismische q-werte als ausdruck von intrinsischer<br />
dämpfung und streudämpfung der kristallinen kruste um die ktb-lokation: ICDP-<br />
Kolloquium, Bochum.<br />
Jones, A. G., and Holliger, K., 1997, Spectral analyses of the ktb sonic an density<br />
logs using robust nonparametric methods: Journal of Geophysical Research, 102,<br />
18,391–18,403.<br />
Kneib, G., 1995, The statistical nature of the upper continental crystalline crust derived<br />
from in situ seismic measurements: Geophys J. Int., 122, 594–616.<br />
Li, X.-P., and Richwalski, S., 1996, Seismic attenuation and velocities of p- and s-<br />
waves in the german ktb area: J. Appl. Geophys., 36, 67–76.<br />
Müller, T. M., Shapiro, S. A., and Sick, C., <strong>2000</strong>, A weak fluctuation approximation<br />
of primary waves in random media: Waves Random Media, submitted.<br />
Pujol, J., Lüschen, E., and Hu, Y., 1998, Seismic wave attenuation in methamorphic<br />
rocks from vsp data recorded in germany's continental super-deep borehole: Geophysics,<br />
63, 354–365.<br />
Sato, H., and Fehler, M., 1998, Wave propagation and scattering in the heterogenous<br />
earth: AIP-press.<br />
Shapiro, S. A., and Hubral, P., 1999, Elastic waves in random media: Springer.<br />
Shapiro, S. A., Schwarz, R., and Gold, N., 1996, The effect of random isotropic inhomogeneities<br />
on the phase velocity of seismic waves: Geophys. J. Int., 127, 783–794.<br />
Wu, R.-S., Xu, Z., and Li, X.-P., 1994, Heterogeneity spectrum and scale-anisotropy in<br />
the upper crust revealed by the german continental deep-drilling (ktb) holes: Geophysical<br />
Research Letters, 21, 911–914.
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