Abstract
A finite-difference approach of aP-SV modeling scheme is applied to compute seismic wave propagation in heterogeneous isotropic media, including fluid-filled boreholes. The discrete formulation of the equation of motion requires the definition of the material parameters at the grid points of the numerical mesh. The grid spacing is chosen as coarse as possible with respect to the accurate representation of the shortest wavelength. If we assume frequencies lower than 250 Hz then the grid spacing is usually chosen in the range of a few meters. One encounters difficulties because of the large-scale difference between the grid spacing and the size of the borehole, usually several centimeters.
These difficulties can be overcome by a grid refinement technique. This technique provides the construction of grids with varying grid spacing. The grid spacing in the vicinity of the borehole is chosen such that the borehole is properly represented. An example demonstrates the accuracy of this technique by comparisons with other methods. Unlike many analytical methods, the FD method can handle complex subsurface geometries. Further numerical examples of walk-awayVSP configurations show tube wave propagation within fluid-filled boreholes of realistic diameters.
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References
Bouchon, M., andSchmitt, D. P. (1989),Full-wave Acoustic Logging in an Irregular Borehole, Geophysics54, 758–765.
Bouchon, M. (1993),A Numerical Simulation of the Acoustic and Elastic Wave Fields by a Source on a Fluid-filled Borehole Embedded in a Layered Medium, Geophysics58, 475–481.
Cheng, C. H., andToksöz, M. N. (1981),Elastic Wave Propagation in a Fluid-filled Borehole and Synthetic Acoustic Logs, Geophysics46, 1042–1053.
Cheng, C. H., andToksöz, M. N. (1984),Generation, propagation and analysis of tube waves in a borehole. InVertical Seismic Profiling, Part B: Advanced Concepts (eds. Toksöz, M. N., and Stewart, R. R.) (Geophysical Press, London-Amsterdam 1984) pp. 276–287.
Crase, E. (1990),High-order (Space and Time) Finite-difference Modeling of the Elastic Wave Equation, Paper presented at the 60th SEG Annual Meeting, Expanded Abstracts, 987–991.
Dong, W., Bouchon, M., andToksöz, M. N. (1995),Borehole Seismic-source Radiation in Layered Isotropic and Anisotropic Media: Boundary Element Modeling, Geophysics60, 735–747.
Falk, J., Tessmer, E., andGajewski, D. (1995),Sesimic Modelling by the Finite-difference Method with Locally Varying Time Steps, Paper presented at the 57th EAEG Annual Meeting, Expanded Abstracts, D013.
Fornberg, B. (1988),Generation of Finite Difference Formulas on Arbitrarily Spaced Grids, Mathematics of Computation51, 699–706.
Jastram, C. (1993),Seismische Modellierung mit finiten Differenzen höherer Ordnung auf einem Gitter mit vertikal variierendem Gitterabstand, Ph.D. Thesis, Institute for Geophysics, University of Hamburg, Germany.
Jastram, C., andBehle, A. (1991),Elastic Modeling by Finite-difference and the Rapid Expansion Method (REM), Paper presented at the 61st SEG Annual Meeting, Expanded Abstracts, 1573–1576.
Jastram, C., andBehle, A. (1992),Acoustic Modelling on a Grid of Vertically Varying Spacing, Geophysical Prospecting40, 157–169.
Jastram, C., andTessmer, E. (1994),Elastic Modelling on a Grid with Vertically Varying Spacing, Geophysical Prospecting42, 357–370.
Kurkijan, A. L. (1986),Theoretical Far-field Radiation from Low-frequency Horizontal Acoustic Point Force in a Vertical Borehole, Geophysics51, 930–939.
Levander, A. R. (1988),Fourth-order Finite-difference P-SV Seismograms, Geophysics53, 1425–1436.
Moczo, P. (1989),Finite-difference Technique for SH Waves in 2-D Media Using Irregular Grids-Application to the Seismic Response Problem, Geophys. J. Int.99, 321–329.
Paillet, F. L., andWhite, J. E. (1982),Acoustic Modes of Propagation in the Borehole and their Relationship to Rock Properties, Geophysics47, 1215–1228.
Petrashen, G. I, Kashtan, B. M., andKiselev, Yu. V. (1994),The Quantitive Investigation of Nonstationary Interference Wave Fields in Layered-homogeneous Elastic Media with Plane-parallel Interfaces. I. Statements of the problems and Rational Methods of their Solution (in Russian), Zapiski naychnikh seminarov POMI 214, St. Petersburg, 7–186.
Schmitt, D. P., andBouchon, N. (1985),Full-wave Acoustic Logging: Synthetic Micro-seismograms and Frequency-wave Number Analysis, Geophysics50, 1756–1778.
Stephen, R. A., Cardo-Casas, F., andCheng, C. H. (1985),Finite-difference Synthetic Acoustic Logs, Geophysics50, 1588–1609.
Sun, R., andMcMechan, G. A. (1988),Finite-difference Modeling of Borehole Resonances, Energy Sources10, 55–75.
Tubman, K. M., Cheng, C. H., andToksöz, M. N. (1984),Synthetic Full Waveform Acoustic Logs in Cased Boreholes, Geophysics49, 1051–1059.
Virieux, J. (1986),P-SV Wave Propagation in Heterogeneous Media: Velocity-stress Finite-difference Method, Geophysics51, 889–901.
Yoon, K.-H., andMcMechan, G. A. (1992),3-D Finite-difference Modeling of Elastic Waves in Borehole Environments, Geophysics57, 793–804.
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Falk, J., Tessmer, E. & Gajewski, D. Tube wave modeling by the finite-difference method with varying grid spacing. PAGEOPH 148, 77–93 (1996). https://doi.org/10.1007/BF00882055
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DOI: https://doi.org/10.1007/BF00882055