This study aimed to investigate shear wave (SH-wave) propagation through a multilayered cracked porous model with exponential heterogeneity. Moreover, the stability criterion of SH-wave propagation was analyzed by determining phase and group velocities of SH-wave. Haskell’s matrix method was used to determine complex dispersion relation for n − 1 media overlying an inhomogeneous porous half-space with fractures. Stability analysis was performed through the finite difference method. Moreover, the phase and group velocities were determined using the Courant number. Dispersion and damping equations were derived for n = 2 and 3. Classical Love wave equation was attained in each case using certain conditions. This equation validated the developed mathematical model. Stability analysis was conducted for reducing the errors and determining the conditions of convergence. The effects of the heterogeneity parameter, porosity, attenuation coefficient, Courant number, and discretization ratio on phase and group velocities were graphically observed. A two-dimensional plot was used to perform comparative analysis between the fractured porosity and isotropy. Cracked porous material, which has various applications in real world, was analyzed in this study. To the best of the authors’ knowledge, wave propagation in a multilayered system consisting of heterogeneous cracked porous material has not been examined previously. Thus, this study explores a new research area. The developed model can be successfully used to interpret seismic behavior during earthquakes. More complex structures can be developed on the basis of the considered model.
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Barenblatt GI, Zheltow IP, Kochina TN (1960) Basic concepts in the theory of seepage homogeneous liquids in fissured rocks. J Appl Math Mech 24:1286–1303. https://doi.org/10.1016/0021-8928(60)90107-6
Bednarik RG (2019) Rock metamorphosis by kinetic energy. Emerg Sci J 3:293–302. https://doi.org/10.28991/esj-2019-01192
Berryman JG, Wang HF (1995) The elastic coefficients of double-porosity models for fluid transport in jointed rock. J Geophys Res 100:34611–34627. https://doi.org/10.1029/95JB02161
Berryman JG, Wang HF (2000) Elastic wave propagation and attenuation in a double-porosity dual-permeability media. Int J Rock Mech Min Sci 37:63–78. https://doi.org/10.1016/S1365-1609(99)00092-1
Beskos DE (1989) Dynamics of saturated rocks. I: Equations of motion. J Eng Mech 115(5):982–995. https://doi.org/10.1061/(ASCE)0733-9399(1989)115:5(982)
Beskos DE, Vgenopoulou I, Providakis CP (1989) Dynamics of saturated rocks, II: body waves. J Eng Mech 115(5):996–1016. https://doi.org/10.1061/(ASCE)0733-9399(1989)115:5(996)
Biot MA (1956) Theory of propagation of elastic waves in a fluid-saturated porous solid. I. Low frequency range. II. Higher frequency range. J Acoust Soc Am 28(2):179–191. https://doi.org/10.1121/1.1908241
Biot MA (1962) Mechanics of deformation and acoustic propagation in porous media. J Appl Phys 33:1482–1498. https://doi.org/10.1063/1.1728759
Chattopadhyay A, Gupta S, Singh AK (2010) The dispersion of shear wave in multilayered magnetoelastic self-reinforced media. Int J Solids Struct 47(9):1317–1324. https://doi.org/10.1016/j.ijsolstr.2010.01.019
Dai ZJ, Kuang ZB (2006) Love waves in double porosity media. J sound vib 296(4-5):1000–1012. https://doi.org/10.1016/j.jsv.2006.03.029
Dong LJ, Hu QC, Tong XJ, Liu YF (2020) Velocity-free MS/AE source location method for three-dimensional hole-containing structures. Engineering, https://doi.org/10.1016/j.eng.2019.12.016
Emerman SH, Schmidt W, Stephen RA (1982) An implicit finite-difference formulation of the elastic wave equation. Geophysics 47:1521–1526. https://doi.org/10.1190/1.1441302
Gollob S, Kocur GK, Schumacher T, Mhamdi L, Vogel T (2017) A novel multi-segment path analysis based on a heterogeneous velocity model for the localization of acoustic emission sources in complex propagation media. Ultrasonics 74:48–61. https://doi.org/10.1016/j.ultras.2016.09.024
Gupta S, Bhengra N (2017) Implementation of finite difference approximation on the SH-wave propagation in a multilayered magnetoelastic orthotropic composite medium. Acta Mech 228(10):3421–3444. https://doi.org/10.1007/s00707-017-1884-6
Gupta S, Smita, Pramanik S (2017) SH-wave in a multilayered orthotropic crust under initial stress: a finite difference approach. Cogent Mathematics 4(1):1284294. https://doi.org/10.1080/23311835.2017.1284294
Haskell NA (1953) The dispersion of surface waves in multilayered media. Bull Seismol Soc Am 43:17–34. https://doi.org/10.1029/SP030p0086
Hokmabadi NN, Sarfarazi V, Moshrefifar M (2016) Investigation of separation non-persistent faults in fracture mechanism of rock bridge. Civil Engineering Journal 2 (7):348–357. https://doi.org/10.28991/cej-2016-00000039
Hu Q, Dong L (2019) Acoustic emission source location and experimental verification for two-dimensional irregular complex structures. IEEE Sens J, https://doi.org/10.1109/JSEN.2019.2954200
Kakar R, Kakar S (2012) Propagation of Love waves in a non-homogeneous elastic media. J Acad Ind Res 1(6):323–8
Kalyani VK, Sinha A, Chakraborty SK, Mahanti NC (2008) Finite difference modeling of seismic wave propagation in monoclinic media. Acta Geophys 56(4):1074. https://doi.org/10.2478/s11600-008-0049-3
Ke LL, Wang YS, Zhang ZM (2005) Propagation of Love waves in an inhomogeneous fluid saturated porous layered half-space with properties varying exponentially. J Eng Mech 131(12):1322–1328. https://doi.org/10.1061/(ASCE)0733-9399(2005)131:12(1322)
Ke LL, Wang YS, Zhang ZM (2006) Love waves in an inhomogeneous fluid saturated porous layered half-space with linearly varying properties. Soil Dyn Earthq Eng 26(6-7):574–581. https://doi.org/10.1016/j.soildyn.2006.01.010
Kelly KR, Ward RW, Treitel S, Alford RM (1976) Synthetic seismograms, a finite-difference approach. Geophysics 41:2–27. https://doi.org/10.1190/1.1440605
Moghaddam HN, Keyhani A, Aghayan I (2016) Modelling of crack propagation in layered structures using extended finite element method. Civil Engineering Journal 2(5):180–188. https://doi.org/10.28991/cej-2016-00000024
Pallavika VK, Chakraborty SK, Sinha A (2008) Finite difference modeling of SH-wave propagation in multilayered porous crust. Journal of Indian Geophysical Union 12(4):165– 172
Saha S, Chattopadhyay A, Singh AK (2018) Numerical modelling of SH-wave propagation in initially-stressed multilayered composite structures. Eng Computation, https://doi.org/10.1108/EC-05-2018-0207https://doi.org/10.1108/EC-05-2018-0207
Thompson WT (1950) Transmission of elastic waves through a stratified solid medium. J Appl Phys 21:89–93. https://doi.org/10.1063/1.1699629
Warren JE, Root PJ (1963) The behavior of naturally fractured reservoirs. Soc Pet Eng J 3:245–255. https://doi.org/10.2118/426-PA
Wilson RK, Aifantis EC (1982) On the theory of consolidation with double porosity. Int J Eng Sci 20:1009–1035. https://doi.org/10.1016/0020-7225(82)90036-2
Wilson RK, Aifantis EC (1984) A double porosity model for acoustic wave propagation in fractured-porous rock. Int J Eng Sci 22:1209–1217. https://doi.org/10.1016/0020-7225(84)90124-1
The authors are sincerely grateful to Indian Institute of Technology (Indian School of Mines), Dhanbad, India for providing great opportunity, guidance, best facilities and equipments.
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Gupta, S., Das, S. & Dutta, R. Finite difference modeling of shear wave propagation in multilayered fractured porous structures. Arab J Geosci 14, 224 (2021). https://doi.org/10.1007/s12517-020-06429-w
- Cracked porous system
- Haskell’s matrix method
- Finite difference method
- Phase and group velocities
- Courant number