Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-29T14:25:50.397Z Has data issue: false hasContentIssue false
  • English
  • Français

Wave Propagation Across Acoustic/Biot’s Media: A Finite-Difference Method

Published online by Cambridge University Press:  03 June 2015

Guillaume Chiavassa  and
Bruno Lombard
Guillaume Chiavassa*
Affiliation:
Centrale Marseille, Laboratoire de Mecanique Modelisation et Procedes Propres, UMR 7340 CNRS, Technopole de Chateau-Gombert, 38 rue Frederic Joliot-Curie, 13451 Marseille, France
Bruno Lombard*
Affiliation:
Laboratoire de Mecanique et d’Acoustique, UPR 7051 CNRS, 31 chemin Joseph Aiguier, 13402 Marseille, France
*
Corresponding author.Email:[email protected]
Email:[email protected]
Get access

Abstract

Numerical methods are developed to simulate the wave propagation in heterogeneous 2D fluid/poroelastic media. Wave propagation is described by the usual acoustics equations (in the fluid medium) and by the low-frequency Biot’s equations (in the porous medium). Interface conditions are introduced to model various hydraulic contacts between the two media: open pores, sealed pores, and imperfect pores. Well-posedness of the initial-boundary value problem is proven. Cartesian grid numerical methods previously developed in porous heterogeneous media are adapted to the present context: a fourth-order ADER scheme with Strang splitting for time- marching; a space-time mesh-refinement to capture the slow compressional wave predicted by Biot’s theory; and an immersed interface method to discretize the interface conditions and to introduce a subcell resolution. Numerical experiments and comparisons with exact solutions are proposed for the three types of interface conditions, demonstrating the accuracy of the approach.

Keywords

35L0535L5065N0665N8574F10Biot’s modelporoelastic wavesjump conditionsimperfect hydraulic contacthigh-order finite differencesimmersed interface method
Type
Research Article
Information
Communications in Computational Physics , Volume 13 , Issue 4 , April 2013 , pp. 985 - 1012
Copyright
Copyright © Global Science Press Limited 2013

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]Berger, M. J., LeVeque, R. J., Adaptive mesh refinement using wave-propagation algorithms for hyperbolic systems, SIAM J. Numer. Anal., 35-6 (1998), 2298–2316.Google Scholar
[2]Berger, M. J., Oliger, J., Adaptative mesh refinement for hyperbolic partial differential equations, J. Comput. Phys., 53 (1984), 484–512.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., 28-2 (1956), 168–178.Google Scholar
[4]Biot, M. A., Theory of propagation of elastic waves in a fluid-saturated porous solid. II: High-frequency range, J. Acoust. Soc. Am., 28-2 (1956), 179–191.Google Scholar
[5]Blanc, E., Chiavassa, G., Lombard, B., Biot-JKD model: simulation of 1D transient poroelastic waves with fractional derivatives, submitted to J. Comput. Phys. (2012).Google Scholar
[6]Bourbié, T., Coussy, O., Zinszner, B., Acoustics of Porous Media, Gulf Publishing Company (1987).Google Scholar
[7]Carcione, J. M., Quiroga-Goode, G., Some aspects of the physics and numerical modeling of Biot compressional waves, J. Comput. Acoust., 3 (1995), 261–280.Google Scholar
[8]Carcione, J. M., Wave Fields in Real Media: Wave Propagation in Anisotropic, Anelastic, Porous and Electromagnetic Media, Elsevier (2007).Google Scholar
[9]Carcione, J. M., Morency, C., Santos, J. E., Computational poroelasticity - a review, Geophysics, 75-5 (2010), 229–243.Google Scholar
[10]Chekroun, M., le Marrec, L., Lombard, B., Piraux, J., Abraham, O., Comparisons between multiple scattering methods and direct numerical simulations for elastic wave propagation in concrete, Springer Proceedings in Physics, 128 (2009), 317–327.Google Scholar
[11]Chiavassa, G., Lombard, B., Time domain numerical modeling of wave propagation in 2D heterogeneous porous media, J. Comput. Phys., 230 (2011), 5288–5309.Google Scholar
[12]Diaz, J., Ezziani, A., Analytical solution for wave propagation in heterogeneous acoustic/porous media. Part 1: the 2D case, Comm. Comput. Phys., 7 (2010), 171–194.Google Scholar
[13]Edelman, I., On the existence of the low-frequency surface waves in a porous medium, C. R. Mecanique, 332 (2004), 43–49.Google Scholar
[14]Ezziani, A., Modelisation de la propagation d’ondes dans les milieux viscoelastiques et poroelastiques, PhD thesis, University Paris-Dauphine, (2005).Google Scholar
[15]Feng, S., Johnson, D. L., High-frequency acoustic properties of a fluid/porous solid interface. I. New surface mode, J. Acoust. Soc. Am., 74-3 (1983), 906–914.Google Scholar
[16]Feng, S., Johnson, D. L., High-frequency acoustic properties of a fluid/porous solid interface. II. The 2D reflection Green’s function, J. Acoust. Soc. Am., 74-3 (1983), 915–924.CrossRefGoogle Scholar
[17]Gubaidullin, A. A., Kuchugurina, O. Y., Seulders, D. M. J., Wisse, C. J., Frequency-dependent acoustic properties of a fluid/porous solid interface, J. Acoust. Soc. Am, 116-3 (2004), 1474–1480.Google Scholar
[18]Gurevich, B., Kelder, O., Smeulders, D. M. J., Validation of the slow compressional wave in porous media: comparison of experiments and numerical simulations, Trans. Porous Media, 36 (1999), 149–160.Google Scholar
[19]Gurevich, B., Schoenberg, M., Interface conditions for Biot s equations of poroelasticity, J. Acoust. Soc. Am., 105-5 (1999), 2585–2589.Google Scholar
[20]Gustafsson, B., The convergence rate for difference approximations to mixed initial boundary value problems, Math. Comput., 29-130 (1975), 396–406.Google Scholar
[21]Haire, T. J., Langton, C. M., Biot theory: a review of its application to ultrasound propagation through cancellous bone, Bone 24-4 (1999), 291–295.Google Scholar
[22]Kaser, M., Dumbser, M., la Puente, J. de, Igel, H., An arbitrary high order Galerkin method for elastic waves on unstructured meshes III: viscoelastic attenuation, Geophys. J. Int., 168-1 (2007), 224–242.Google Scholar
[23]Lemoine, G. I., LeVeque, R. J., Ou, M. J. Y., Poroelastic wave propagation using CLAWPACK, Proceedings of Waves Conference (2011), 173–176.Google Scholar
[24]LeVeque, R. J., Finite Volume Methods for Hyperbolic Problems, Cambridge University Press (2002).CrossRefGoogle Scholar
[25]LeVeque, R. J., Zhang, C., The immersed interface method for wave equations with discontinuous coefficients, Wave Motion, 25 (1997), 237–263.Google Scholar
[26]Li, Z., LeVeque, R. J., The immersed interface method for elliptic equations with discontinuous coefficients and singular sources, SIAM J. Num. Anal., 31 (1994), 1019–1044.Google Scholar
[27]Lombard, B., Piraux, J., Numerical treatment of two-dimensional interfaces for acoustic and elastic waves, J. Comput. Phys., 195-1 (2004), 90–116.Google Scholar
[28]Lombard, B., Piraux, J., Numerical modeling of transient two-dimensional viscoelastic waves, J. Comput. Phys., 230 (2011), 6099–6114.Google Scholar
[29]Lu, J. F., Hanyga, A., Wave field simulation for heterogeneous porous media with singular memory drag force, J. Comput. Phys., 208 (2005), 651–674.CrossRefGoogle Scholar
[30]Luppé, F., Conoir, J. M., Robert, S., Coherent waves in a multiply scattering poro-elastic medium obeying Biot’s theory, Waves Random Complex Media, 18-2 (2008), 241–254.Google Scholar
[31]Martin, R., Komatitsch, D., Ezziani, A., An unsplit convolutional perfectly matched layer improved at grazing incidence for seismic wave propagation in poroelastic media, Geophysics, 73-4 (2008), 51–61.Google Scholar
[32]Morency, C., Tromp, J., Spectral-element simulations of wave propagation in porous media, Geophys. J. Int., 175 (2010), 301–345.Google Scholar
[33]Plona, T. J., Observation of a second bulk compressional wave in a porous medium at ultra-sonic frequencies, App. Phys. Lett., 36-4 (1980), 259–261.CrossRefGoogle Scholar
[34]la Puente, J. de, Dumbser, M., Kaser, M., Igel, H., Discontinuous Galerkin methods for wave propagation in poroelastic media, Geophysics, 73-5 (2008), 77–97.Google Scholar
[35]Rasolofosaon, P. N. J., Importance of hydraulic condition on the generation of second bulk compressional wave in porous media, Appl. Phys. Lett., 52-10 (1988), 780–782.Google Scholar
[36]Rosenbaum, J. H., Synthetic microseismograms: logging in porous formation, Geophysics, 39-1 (1974), 14–32.Google Scholar
[37]Sidler, R., Carcione, J. M., Holliger, K., Simulation of surface waves in porous media, Geophys. J. Int., 183-2 (2010), 820–832.Google Scholar
[38]Stoll, R. D., Bryan, G. M., Wave attenuation in saturated sediments, J. Acoust. Soc. Am., 47-5 (1970), 1440–1447.Google Scholar
[39]Wu, K., Xue, Q., Adler, L., Reflection and transmission of elastic waves from a fluid-saturated porous solid boundary, J. Acoust. Soc. Am., 87-6 (1990), 2349–2358.Google Scholar