Hostname: page-component-cd9895bd7-q99xh Total loading time: 0 Render date: 2024-12-27T09:32:58.168Z Has data issue: false hasContentIssue false
  • English
  • Français

The Parameter Averaging Technique in Finite-Difference Modeling of Elastic Waves in Combined Structures with Solid, Fluid and Porous Subregions

Published online by Cambridge University Press:  20 August 2015

Wei Guan  and
Hengshan Hu
Wei Guan*
Affiliation:
Department of Astronautics and Mechanics, Harbin Institute of Technology, Postbox 344, 92 West Dazhi Street, Harbin 150001, China
Hengshan Hu*
Affiliation:
Department of Astronautics and Mechanics, Harbin Institute of Technology, Postbox 344, 92 West Dazhi Street, Harbin 150001, China
*
Corresponding author.Email:[email protected]
Email:[email protected]
Get access

Abstract

To finite-difference model elastic wave propagation in a combined structure with solid, fluid and porous subregions, a set of modified Biot’s equations are used, which can be reduced to the governing equations in solids, fluids as well as fluid-saturated porous media. Based on the modified Biot’s equations, the field quantities are finite-difference discretized into unified forms in the whole structure, including those on any interface between the solid, fluid and porous subregions. For the discrete equations on interfaces, however, the harmonic mean of shear modulus and the arithmetic mean of the other parameters on both sides of the interfaces are used. These parameter averaging equations are validated by deriving from the continuity conditions on the interfaces. As an example of using the parameter averaging technique, a 2-D finite-difference scheme with a velocity-stress staggered grid in cylindrical coordinates is implemented to simulate the acoustic logs in porous formations. The finite-difference simulations of the acoustic logging in a homogeneous formation agree well with those obtained by the analytical method. The acoustic logs with mud cakes clinging to the borehole well are simulated for investigating the effect of mud cake on the acoustic logs. The acoustic logs with a varying radius borehole embedded in a horizontally stratified formation are also simulated by using the proposed finite-difference scheme.

Keywords

76M2065M0676S0535L0586A15Finite-differencewave equationporous mediumacoustic loggingnumerical simulation
Type
Research Article
Information
Communications in Computational Physics , Volume 10 , Issue 3 , September 2011 , pp. 695 - 715
Copyright
Copyright © Global Science Press Limited 2011

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Alford, R. M., Kelly, K. R., and Boore, D. M., Accuracy of finite-difference modeling of the acoustic wave equation, Geophysics., 39 (1974), 834–842.Google Scholar
[2]Berryman, J. G., and Pride, S. R., Dispersion of waves in porous cylinders with patchy saturation: formulation and torsional waves, J. Acoust. Soc. Am., 117 (2005), 1785–1795.Google Scholar
[3]Biot, M. A., Theory of propagation of elastic waves in a fluid-saturated porous solid, I-low-frequency range, J. Acoust. Soc. Am., 28 (1956), 168–178.Google Scholar
[4]Biot, M. A., Theory of propagation of elastic waves in a fluid-saturated porous solid, II-higher-frequency range, J. Acoust. Soc. Am., 28 (1956), 178–191.Google Scholar
[5]Biot, M. A., Mechanics of deformation and acoustic propagation in porous media, J. Appl. Phys., 33 (1962), 1482–1498.Google Scholar
[6]Biot, M. A., and Willis, D. G., The elastic coefficients of the theory of consolidation, J. Appl. Mech., 24 (1957), 594–601.Google Scholar
[7]Boutin, C., Bonnet, G., and Bard, P. Y., Green functions and associated sources in infinite and stratified poroelastic media, Geophys. J. R. Astr. Soc., 90 (1987), 521–550.Google Scholar
[8]Chew, W. C., and Weedon, W. H., A3-D perfectly matched medium from modified Maxwell’s equations with stretched coordinates, Microw. Opt. Technol. Lett., 7 (1994), 599–604.CrossRefGoogle Scholar
[9]Dai, N., Vafidis, A., and Kanasewich, E. R., Wave propagation in heterogeneous, porous media: a velocity-stress, finite-difference method, Geophysics., 60 (1995), 327–340.Google Scholar
[10]Deresiewicz, H., and Skalak, R., On the uniqueness in dynamic poroelasticity, Bull. Seism. Soc. Am., 53 (1963), 783–788.Google Scholar
[11]Dong, H., Kaynia, A. M., Madshus, C., and Hovern, J. M., Sound propagation over layered poro-elastic ground using a finite-difference model, J. Acoust. Soc. Am., 108 (2000), 494–502.CrossRefGoogle ScholarPubMed
[12]Gassmann, F., Über die elastizität poröser medien, Viertel. Naturforsch. Ges. Zürich., 96 (1951), 1–23.Google Scholar
[13]Guan, W., Hu, H., and He, X., Finite-difference modeling of the monopole acoustic logging in a horizontally stratified porous formation, J. Acoust. Soc. Am., 125 (2009), 1942–1950.Google Scholar
[14]Ge, D., and Yan, Y., Finite-Difference Time-Domain Method for Electromagnetic Waves, 2nd edn, XiDian University Press, Xi’an China, 2005.Google Scholar
[15]Johnson, D. L., Koplik, J., and Dashen, R., Theory of dynamic permeability and tortuosity in fluid-saturated porous media, J. Fluid. Mech., 176 (1987), 379–402.CrossRefGoogle Scholar
[16]Masson, Y. J., Pride, S. R., and Nihei, K. T., Finite difference modeling of Biot’s poroelastic equations at seismic frequencies, J. Geophys. Res., 111 (2006), B10305.Google Scholar
[17]Mittet, R., Free-surface boundary conditions for elastic staggered-grid modeling schemes, Geophysics., 67 (2002), 1616–1623.Google Scholar
[18]Norris, A. N., Radiation froma point source and scattering theory in a fluid-saturated porous solid, J. Acoust. Soc. Am., 77 (1985), 2012–2023.Google Scholar
[19]Plona, J., Observation of a second bulk compressional wave in a porous medium at ultrasonic frequency, Appl. Phys. Lett., 36 (1980), 259–261.Google Scholar
[20]Randall, C. J., Multipole borehole acoustic waveforms: synthetic logs with beds and borehole washouts, Geophysics., 56 (1991), 1757–1769.Google Scholar
[21]Roden, J. A., and Gedney, S. D., Convolution PML (CPML):an efficient FDTD implementation of the CFS-PML for arbitrary media, Micro. Opt. Tech. Lett., 27 (2000), 334–339.3.0.CO;2-A>CrossRefGoogle Scholar
[22]Rosenbaum, J. H., Synthetic microseismograms: logging in porous formations, Geophysics., 39 (1974), 14–32.CrossRefGoogle Scholar
[23]Schmitt, D. P., Effects of radial layering when logging in saturated porous formations, J. Acoust. Soc. Am., 84 (1988), 2200–2214.Google Scholar
[24]Song, R., Ma, J., and Wang, K., The application of the nonsplitting perfectly matched layer in numerical modeling of wave propagation in poroelastic media, Appl. Geophys., 2 (2005), 216–222.Google Scholar
[25]Song, R., Parallel Computation of Acoustic Field in Non-Axisymmetric Cased Holes & Theory and Method Studies of Acoustic Cementing Quality Evaluation, Ph.D. Thesis, Jilin University, China, 2008.Google Scholar
[26]Tsang, L., and Rader, D., Numerical evaluation of the transient acoustic waveform due to a point source in a fluid-filled borehole, Geophysics., 44 (1979), 1706–1720.Google Scholar
[27]Vernik, L., Predicting lithology and transport properties from acoustic velocities based on petrophysical classification of siliclastics, Geophysics., 59 (1994), 420–427.Google Scholar
[28]Wang, T., and Tang, X. M., Finite-difference modeling of elastic wave propagation: a nonsplitting perfectly matched layer approach, Geophysics., 68 (2003), 1749–1755.Google Scholar
[29]Zhang, J., Quadrangle-gride velocity-stress finite difference method for poroelastic wave equations, Geophys. J. Int., 139 (1999), 171–182.Google Scholar
[30]Zeng, Y. Q., He, J. Q., and Liu, Q. H., The application of the PML in numerical modeling of wave propagation in poroelastic media, Geophysics., 66 (2001), 1258–1266.Google Scholar
[31]Zhu, X., and McMechan, G. A., Numerical simulation of seismic responses of poroelastic reservoirs using Biot theory, Geophysics., 56 (1991), 328–339.Google Scholar