Finite-difference modeling of surface waves in poroelastic media and stress mirror conditions
Zhang Yu1,2,3, Ping Ping4, and Zhang Shuang-Xi1,2,3
1. School of Geodesy and Geomatics, Wuhan University, Wuhan 430079, China.
2. Key Laboratory of Geospace Environment and Geodesy, Ministry of Education, Wuhan 430079, China.
3. Collaborative Innovation Center for Geospatial Technology, Wuhan University, Wuhan 430079, China.
4. State Key Laboratory of Geodesy and Earth's Dynamics, Institute of Geodesy and Geophysics, Chinese Academy of Sciences, Wuhan 430077, China.
Abstract During seismic wave propagation on a free surface, a strong material contrast boundary develops in response to interference by P- and S- waves to create a surface-wave phenomenon. To accurately determine the effects of this interface on surface-wave propagation, the boundary conditions must be accurately modeled. In this paper, we present a numerical approach based on the dynamic poroelasticity for a space–time-domain staggered-grid finite-difference simulation in porous media that contain a free-surface boundary. We propose a generalized stess mirror formulation of the free-surface boundary for solids and fluids in porous media for the grid mesh on which lays the free-surface plane. Its analog is that used for elastic media, which is suitable for precise and stable Rayleigh-type surface-wave modeling. The results of our analysis of first kind of Rayleigh (R1) waves obtained by this model demonstrate that the discretization of the mesh in a similar way to that for elastic media can realize stable numerical solutions with acceptable precision. We present numerical examples demonstrating the efficiency and accuracy of our proposed method.
This work was sponsed by National Natural Science Foundation of China (NSFC, Grant No. 41304077), the Natural Basic Research Program of China (the “973 Project,” Grant No. 2013CB733303), and Postdoctoral Science Foundation of China (Grant No. 2014T70740).
Cite this article:
. Finite-difference modeling of surface waves in poroelastic media and stress mirror conditions[J]. APPLIED GEOPHYSICS, 2017, 14(1): 105-114.
[1]
Biot, M. A., 1962, Mechanics of deformation and acoustic propagation in porous media: Journal of Applied Physics, 33, 1482-1498.
[2]
Blanc, E., Chiavassa, G., and Lombard, B., 2014, Wave simulation in 2D heterogeneous transversely isotropic porous media with fractional attenuation: A Cartesian grid approach: Journal of Computational Physics, 275, 118−142.
[3]
Bohlen, T., and Saenger, E. H., 2006, Accuracy of heterogeneous staggered-grid finite-difference modeling of Rayleigh waves: Geophysics, 71(4), T109−T115.
[4]
Bouklas, N., Landis, C. M., and Huang, R., 2015, A nonlinear, transient finite element method for coupled solvent diffusion and large deformation of hydrogels: Journal of the Mechanics and Physics of Solids, 79, 21−43.
[5]
Carcione, J. M., 2007, Wave fields in real media: wave propagation in anisotropic, anelastic, porous, and electromagnetic media, 2nd ed: Elsiever, Amsterdam.
[6]
Carcione, J. M., and Helle, H. B., 1999, Numerical solution of the poroviscoelastic wave equation on a staggered mesh: Journal of Computational Physics, 154, 520-527.
[7]
Carcione, J. M., and Quiroga-Goode, G., 1995, Some aspects of the physics and numerical modelling of Biot compressional waves: Journal of Computational Acoustics, 3(4), 261-280.
[8]
Itza, R., Iturraran-Viveros, U., and Parra, J. O., 2016, Optimal implicit 2-D finite differences to model wave propagation in poroelastic media: Geophysical Journal International, 206, 1111−1125.
[9]
Kristek, J., Moczo, P., and Archuleta, R. J., 2002, Efficient methods to simulate planar free surface in the 3D 4th-order staggered-grid finite-difference schemes: Studia Geophysica et Geodaetica, 46, 355-381.
[10]
Levander, A. R., 1988, Fourth-order finite-difference P-SV seismograms: Geophysics, 53, 1425-1436.
[11]
Liu, X., Yin, X., and Li, H., 2014, Optimal variable-grid finite-difference modeling for porous media: Journal of Geophysics and Engineering, 11, 065011.
[12]
Liu, Y., and Sen, M., 2009, A new time-space domain high-order finite difference method for the acoustic wave equation: Journal of Computational Physics, 228, 8779−8806.
[13]
Liu, Y., and Sen, M. K., 2011, Scalar wave equation modeling with time- space domain dispersion-relation-based staggered-grid finite-difference schemes: Bulletin of the Seismological Society of America, 101, 141−159.
[14]
Luo, Y., Xia, J., Miller, R. D., Xu, Y., Liu, J., and Liu, Q., 2008, Rayleigh-wave dispersive energy imaging using a high-resolution linear radon transform: Pure and Applied Geophysics, 165, 903−922.
[15]
Masson, Y. J., and Pride, S. R., 2007, Poroelastic finite-difference modeling of seismic attenuation and dispersion due to mesoscopic-scale heterogeneity: Journal of Geophysical Research, 112, B03204.
[16]
Masson, Y. J., Pride S. R., and Nihei, K. T., 2006, Finite difference modeling of Biot’s poroelastic equations at seismic frequencies: Journal of Geophysical Research, 111, B10305.
[17]
Mittet, R., 2002, Free-surface boundary conditions for elastic staggered-grid modeling schemes: Geophysics, 67, 1616-1623.
[18]
Moczo, P., Kristek, J., Vavry?uk, V., Archuleta, R. J., and Halada, L., 2002, 3D heterogeneous staggered-grid finite-difference modeling of seismic motion with volume harmonic and arithmetic averaging of elastic moduli and densities: Bulletin of the Seismological Society of America, 92, 3042-3066.
[19]
Ouml;zdenvar, T., and McMechan, G. A., 1997, Algorithms for staggered-grid computations for poroelastic, elastic, acoustic, and scalar wave equations: Geophysical Prospecting, 45, 403-420.
[20]
Plona, T. J., 1980, Observation of a second bulk compressional wave in a porous medium at ultrasonic frequencies: Applied Physics Letters, 36, 259−261.
[21]
Pride, S. R., and Berryman, J. G., 1998, Connecting theory to experiment in poroelasticity: Journal of the Mechanics and Physics of Solids, 46, 719-747.
[22]
Pride, S. R., Gangi, A. F., and Morgan, F. D., 1992, Deriving the equations of motion for porous isotropic media: Journal of Acoustical Society of America, 92, 3278-3290.
[23]
Robertsson, J. O. A., 1996, A numerical free-surface condition for elastic/viscoelastic finite-difference modeling in the presence of topography: Geophysics, 61, 1921-1934.
Xu, Y., 2015, Surface technology and subsurface image, Chinese University of Geosciences Press, Wuhan.
[26]
Xu, Y., Xia, J., and Miller, R. D., 2007, Numerical investigation of implementation of air-earth boundary by acoustic-elastic boundary approach: Geophysics, 72, SM147-SM153.
[27]
Yang, Q., and Mao, W., 2017, Simulation of seismic wave propagation in 2-D poroelastic media using weighted-averaging finite difference stencils in the frequency-space domain: Geophysical Journal International, 208, 148−161.
[28]
Zeng, C., Xia, J., Miller, R. D., and Tsoflias, G. P., 2012, An improved vacuum formulation for 2D finite-difference modeling of Rayleigh waves including surface topography and internal discontinuities: Geophysics, 77, T1−T9.
[29]
Zhang, Y., Ping, P., Deng, P., Zhou, H., and Zhang, S., 2015, A free surface formulation for finite difference modeling of surface wave in porous media: 85th Annual International Meeting, SEG expanded abstract, 2406−2411.
[30]
Zhang, Y., Xu, Y., and Xia, J., 2011, Analysis of dispersion and attenuation of surface waves in poroelastic media in the exploration seismic frequency band: Geophysical Journal International, 182, 870−888.
[31]
Zhang, Y., Xu, Y., and Xia, J., 2012, Wave fields and spectra of Rayleigh waves in poroelastic media in the exploration seismic frequency band: Advances in Water Resources, 49, 62-71.
2017, Vol. 14 
