Modeling of Surface Waves in a Fluid Saturated Poro-Elastic Medium under Initial Stress Using Time-Space Domain Higher Order Finite Difference Method

Affiliation(s)

Birla Institute of Technology, Ranchi, India.

RVS College of Engineering & Technology, Jamshedpur, India.

Birla Institute of Technology, Ranchi, India.

RVS College of Engineering & Technology, Jamshedpur, India.

ABSTRACT

In this present context, mathematical modeling of the propagation of surface waves in a fluid saturated poro-elastic medium under the influence of initial stress has been considered using time dependent higher order finite difference method (FDM). We have proved that the accuracy of this finite-difference scheme is 2*M** *when we use 2nd order time domain finite-difference and 2*M*-th order space domain finite-difference. It also has been shown that the dispersion curves of Love waves are less dispersed for higher order FDM than of lower order FDM. The effect of initial stress, porosity and anisotropy of the layer in the propagation of Love waves has been studied here. The numerical results have been shown graphically. As a particular case, the phase velocity in a non porous elastic solid layer derived in this paper is in perfect agreement with that of Liu *et al*. (2009).

In this present context, mathematical modeling of the propagation of surface waves in a fluid saturated poro-elastic medium under the influence of initial stress has been considered using time dependent higher order finite difference method (FDM). We have proved that the accuracy of this finite-difference scheme is 2

Cite this paper

A. Ghorai and R. Tiwary, "Modeling of Surface Waves in a Fluid Saturated Poro-Elastic Medium under Initial Stress Using Time-Space Domain Higher Order Finite Difference Method,"*Applied Mathematics*, Vol. 4 No. 3, 2013, pp. 469-476. doi: 10.4236/am.2013.43070.

A. Ghorai and R. Tiwary, "Modeling of Surface Waves in a Fluid Saturated Poro-Elastic Medium under Initial Stress Using Time-Space Domain Higher Order Finite Difference Method,"

References

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[2] M. A. Biot, “Theory of Propagation of Elastic Solid in a Fluid-Saturated Porous Solid II. Low-Frequence Range,” Journal of the Acoustical Society of America, Vol. 28, 1956, pp. 179-191. doi:10.1121/1.1908241

[3] M. A. Biot, “Surface Instability in Finite Anisotropic Elasticity under Initial Stress,” Proceedings of the Royal Society A, Vol. 273, No. 1354, 1963, pp. 329-339. doi:10.1098/rspa.1963.0092

[4] A. M. Abd-Alla, S. R. Mahmoud and M. I. R. Helmi, “Effect of Initial Stress and Magnetic Field on Propagation of Shear Wave in Non-Homogeneous Anisotropic Medium under Gravity Field,” The Open Applied Mathematics Journal, Vol. 3, No. 1, 2009, pp. 58-65. doi:10.2174/1874114200903010058

[5] S. Gupta, A. Chattopadhyay and D. K. Majhi, “Effect of Initial Stress on Propagation of Love Waves in an Anisotropic Porous Layer,” Journal of Solid Mechanics, Vol. 2, No.1, 2010, pp. 50-62.

[6] S. Gupta, A. Chattopadhyay, S. K. Vishwakarma and D. K. Majhi, “Influence of Rigid Boundary and Initial Stress on the Propagation of Love Wave,” Applied Mathematics, Vol. 2, No. 5, 2011, pp. 586-594. doi:10.4236/am.2011.25078

[7] J. Virieux, “SH-Wave Propagation in Heterogeneous Media: Velocity Stress Finite Difference Method,” Geophysics, Vol. 49, No. 11, 1984, pp. 1933-1957. doi:10.1190/1.1441605

[8] J. Virieux, “P-SV Wave Propagation in Heterogeneous Media: Velocity Stress Finite Difference Method,” Geophysics, Vol. 51, No. 4, 1986, pp. 889-901. doi:10.1190/1.1442147

[9] A. Levander, “Fourth-Order Finite Difference P-SV Seismograms,” Geophysics, Vol. 53, No. 11, 1998, pp. 1425-1436. doi:10.1190/1.1442422

[10] K. Hayashi and D. R. Burns, “Variable Grid Finite-Difference Modeling Including Surface Topography,” 69th Annual International Meeting, SEG, Expanded Abstracts, Vol. 18, 1999, pp. 523-527.

[11] R. W. Graves, “Simulating of Seismic Wave Propagation in 3D Elastic Media Using Staggered-Grid Finite Differences,” Bulletin of the Seismological Society of America, Vol. 86, No. 4, 1996, pp. 1091-1106.

[12] J. Kristek and P. Moczo, “Seismic Wave Propagation in Visco-Elastic Media with Material Discontinuities: A 3D Forth-Order Staggered-Grid Finite Difference Modeling,” Bulletin of the Seismological Society of America, Vol. 93, No. 5, 2003, pp. 2273-2280. doi:10.1785/0120030023

[13] E. H. Saenger and T. Bohlen, “Finite Difference Modeling of Viscoelastic and Anisotropic Wave Propagation Using the Rotated Staggered Grid,” Geophysics, Vol. 69, No. 2, 2004, pp. 583-591. doi:10.1190/1.1707078

[14] T. Bohlen and E. H. Saenger, “Accuracy of Heterogeneous Staggered-Grid Finite-Difference Modeling of Rayleigh Waves,” Geophysics, Vol. 71, No. 4, 2006, pp. T109-T115. doi:10.1190/1.2213051

[15] J. Kristek and P. Moczo, “On the Accuracy of the Finite-Difference Schemes: The 1D Elastic Problem,” Bulletin of the Seismological Society of America, Vol. 96, No. 6, 2006, pp. 2398-2414. doi:10.1785/0120060031

[16] E. Tessmer, “Seismic Finite Difference Modeling with Spatially Variable Time Steps,” Geophysics, Vol. 65, No. 4, 2000, pp. 1290-1293. doi:10.1190/1.1444820

[17] B. Finkelstein and R. Kastner, “Finite Difference Time Domain Dispersion Reduction Schemes,” Journal of Computational Physics, Vol. 221, No. 1, 2007, pp. 422-438. doi:10.1016/j.jcp.2006.06.016

[18] Y. Liu and M. K. Sen, “Advanced Finite-Difference Method for Seismic Modeling,” Geohorizons, Vol. 14, No. 2, 2009, pp. 5-16.

[19] Y. Liu and M. K. Sen, “A Practical Implicit Finite Difference Method: Examples from Seismic Modeling,” Journal of Geophysics and Engineering, Vol. 6, No. 3, 2009, pp. 231-249. doi:10.1088/1742-2132/6/3/003

[20] Y. Liu and M. K. Sen, “A New Time-Space Domain High-Order Finite Difference Method for the Acoustic Wave Equation,” Journal of Computational Physics, Vol. 228, No. 23, 2009, pp. 8779-8806. doi:10.1016/j.jcp.2009.08.027

[21] M. A. Dublain, “The Application of High-Order Differencing to the Scalar Wave Equation,” Geophysics, Vol. 51, No. 1, 1986, pp. 54-66. doi:10.1190/1.1442040

[22] Y. Liu and M. K. Sen, “Acoustic VTI Modeling with a Time-Space Domain Dispersion-Relation-Based Finite-Difference Scheme,” Geophysics, Vol. 75, No. 3, 2010, pp. A11-A17. doi:10.1190/1.3374477

[23] Y. Liu and W. Xiucheng, “Finite Difference Numerical Modeling with Even Order Accuracy in Two Phase Anisotropic Media,” Applied Geophysics, Vol. 5, No. 2, 2008, pp. 107-114. doi:10.1007/s11770-008-0014-6

[24] X. Zhu and G. A. McMechan, “Finite difference Modeling of the seismic Response of fluid Saturated, Porous, Elastic Solid Using Biot Theory,” Geophysics, Vol. 56, No. 4, 1991, pp. 424-435.

[25] Y. Liu and M. K. Sen, “Scalar Wave Equation Modeling with Time-Space Domain Dispersion-Relation-Based Staggered-Grid Finite-Difference Schemes,” Bulletin of the Seismological Society of America, Vol. 101, No. 1, 2011, pp. 141-159. doi:10.1785/0120100041

[1] M. A. Biot, “Theory of Propagation of Elastic Solid in a Fluid-Saturated Porous Solid 1. Low-Frequence Range,” Journal of the Acoustical Society of America, Vol. 28, 1956, pp. 168-178. doi:10.1121/1.1908239

[2] M. A. Biot, “Theory of Propagation of Elastic Solid in a Fluid-Saturated Porous Solid II. Low-Frequence Range,” Journal of the Acoustical Society of America, Vol. 28, 1956, pp. 179-191. doi:10.1121/1.1908241

[3] M. A. Biot, “Surface Instability in Finite Anisotropic Elasticity under Initial Stress,” Proceedings of the Royal Society A, Vol. 273, No. 1354, 1963, pp. 329-339. doi:10.1098/rspa.1963.0092

[4] A. M. Abd-Alla, S. R. Mahmoud and M. I. R. Helmi, “Effect of Initial Stress and Magnetic Field on Propagation of Shear Wave in Non-Homogeneous Anisotropic Medium under Gravity Field,” The Open Applied Mathematics Journal, Vol. 3, No. 1, 2009, pp. 58-65. doi:10.2174/1874114200903010058

[5] S. Gupta, A. Chattopadhyay and D. K. Majhi, “Effect of Initial Stress on Propagation of Love Waves in an Anisotropic Porous Layer,” Journal of Solid Mechanics, Vol. 2, No.1, 2010, pp. 50-62.

[6] S. Gupta, A. Chattopadhyay, S. K. Vishwakarma and D. K. Majhi, “Influence of Rigid Boundary and Initial Stress on the Propagation of Love Wave,” Applied Mathematics, Vol. 2, No. 5, 2011, pp. 586-594. doi:10.4236/am.2011.25078

[7] J. Virieux, “SH-Wave Propagation in Heterogeneous Media: Velocity Stress Finite Difference Method,” Geophysics, Vol. 49, No. 11, 1984, pp. 1933-1957. doi:10.1190/1.1441605

[8] J. Virieux, “P-SV Wave Propagation in Heterogeneous Media: Velocity Stress Finite Difference Method,” Geophysics, Vol. 51, No. 4, 1986, pp. 889-901. doi:10.1190/1.1442147

[9] A. Levander, “Fourth-Order Finite Difference P-SV Seismograms,” Geophysics, Vol. 53, No. 11, 1998, pp. 1425-1436. doi:10.1190/1.1442422

[10] K. Hayashi and D. R. Burns, “Variable Grid Finite-Difference Modeling Including Surface Topography,” 69th Annual International Meeting, SEG, Expanded Abstracts, Vol. 18, 1999, pp. 523-527.

[11] R. W. Graves, “Simulating of Seismic Wave Propagation in 3D Elastic Media Using Staggered-Grid Finite Differences,” Bulletin of the Seismological Society of America, Vol. 86, No. 4, 1996, pp. 1091-1106.

[12] J. Kristek and P. Moczo, “Seismic Wave Propagation in Visco-Elastic Media with Material Discontinuities: A 3D Forth-Order Staggered-Grid Finite Difference Modeling,” Bulletin of the Seismological Society of America, Vol. 93, No. 5, 2003, pp. 2273-2280. doi:10.1785/0120030023

[13] E. H. Saenger and T. Bohlen, “Finite Difference Modeling of Viscoelastic and Anisotropic Wave Propagation Using the Rotated Staggered Grid,” Geophysics, Vol. 69, No. 2, 2004, pp. 583-591. doi:10.1190/1.1707078

[14] T. Bohlen and E. H. Saenger, “Accuracy of Heterogeneous Staggered-Grid Finite-Difference Modeling of Rayleigh Waves,” Geophysics, Vol. 71, No. 4, 2006, pp. T109-T115. doi:10.1190/1.2213051

[15] J. Kristek and P. Moczo, “On the Accuracy of the Finite-Difference Schemes: The 1D Elastic Problem,” Bulletin of the Seismological Society of America, Vol. 96, No. 6, 2006, pp. 2398-2414. doi:10.1785/0120060031

[16] E. Tessmer, “Seismic Finite Difference Modeling with Spatially Variable Time Steps,” Geophysics, Vol. 65, No. 4, 2000, pp. 1290-1293. doi:10.1190/1.1444820

[17] B. Finkelstein and R. Kastner, “Finite Difference Time Domain Dispersion Reduction Schemes,” Journal of Computational Physics, Vol. 221, No. 1, 2007, pp. 422-438. doi:10.1016/j.jcp.2006.06.016

[18] Y. Liu and M. K. Sen, “Advanced Finite-Difference Method for Seismic Modeling,” Geohorizons, Vol. 14, No. 2, 2009, pp. 5-16.

[19] Y. Liu and M. K. Sen, “A Practical Implicit Finite Difference Method: Examples from Seismic Modeling,” Journal of Geophysics and Engineering, Vol. 6, No. 3, 2009, pp. 231-249. doi:10.1088/1742-2132/6/3/003

[20] Y. Liu and M. K. Sen, “A New Time-Space Domain High-Order Finite Difference Method for the Acoustic Wave Equation,” Journal of Computational Physics, Vol. 228, No. 23, 2009, pp. 8779-8806. doi:10.1016/j.jcp.2009.08.027

[21] M. A. Dublain, “The Application of High-Order Differencing to the Scalar Wave Equation,” Geophysics, Vol. 51, No. 1, 1986, pp. 54-66. doi:10.1190/1.1442040

[22] Y. Liu and M. K. Sen, “Acoustic VTI Modeling with a Time-Space Domain Dispersion-Relation-Based Finite-Difference Scheme,” Geophysics, Vol. 75, No. 3, 2010, pp. A11-A17. doi:10.1190/1.3374477

[23] Y. Liu and W. Xiucheng, “Finite Difference Numerical Modeling with Even Order Accuracy in Two Phase Anisotropic Media,” Applied Geophysics, Vol. 5, No. 2, 2008, pp. 107-114. doi:10.1007/s11770-008-0014-6

[24] X. Zhu and G. A. McMechan, “Finite difference Modeling of the seismic Response of fluid Saturated, Porous, Elastic Solid Using Biot Theory,” Geophysics, Vol. 56, No. 4, 1991, pp. 424-435.

[25] Y. Liu and M. K. Sen, “Scalar Wave Equation Modeling with Time-Space Domain Dispersion-Relation-Based Staggered-Grid Finite-Difference Schemes,” Bulletin of the Seismological Society of America, Vol. 101, No. 1, 2011, pp. 141-159. doi:10.1785/0120100041