Finite difference modeling of shear wave propagation in multilayered fractured porous structures

Abstract

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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Acknowledgments

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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Correspondence to Soumik Das.

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Appendix: Appendix

Appendix: Appendix

$$ \begin{array}{@{}rcl@{}} &&s_{1}^{(j)}=\bar{\zeta}_{12}^{(j)^{2}}+\omega^{2}\bar{\rho}_{22}^{(j)^{2}},~s_{2}^{(j)}=\bar{\zeta}_{13}^{(j)^{2}}+\omega^{2}\bar{\rho}_{33}^{(j)^{2}},\\ &&c_{sj}=\left[\frac{\bar{G}^{(j)}}{\left( \bar{\rho}_{11}^{(j)}+\frac{2\bar{\zeta}_{12}^{(j)^{2}}\bar{\rho}_{12}^{(j)}+\bar{\zeta}_{12}^{(j)^{2}}\bar{\rho}_{22}^{(j)}-\omega^{2}\bar{\rho}_{12}^{(j)^{2}}\bar{\rho}_{22}^{(j)}}{s_{1}^{(j)}}+\frac{2\bar{\zeta}_{13}^{(j)^{2}}\bar{\rho}_{13}^{(j)}+\bar{\zeta}_{13}^{(j)^{2}}\bar{\rho}_{33}^{(j)}-\omega^{2}\bar{\rho}_{13}^{(j)^{2}}\bar{\rho}_{33}^{(j)}}{s_{2}^{(j)}}\right)}\right]^{\frac{1}{2}},\\ &&\delta_{j}=\frac{\omega c_{sj}^{2}}{2\bar{G}^{(j)}}\left[\frac{\bar{\zeta}_{12}^{(j)}(\bar{\rho}_{12}^{(j)}+\bar{\rho}_{22}^{(j)})^{2}}{s_{1}^{(j)}}+\frac{\bar{\zeta}_{13}^{(j)}(\bar{\rho}_{13}^{(j)}+\bar{\rho}_{33}^{(j)})^{2}}{s_{2}^{(j)}}\right],~a_{1j}=\left( \frac{c}{c_{sj}}\right)^{2}-(1-\delta^{2}),~b_{1j}=2\left[\frac{\delta_{j}c^{2}}{c_{sj}^{2}}-\delta\right],\\ &&\bar{q}_{j}=\sqrt{\frac{\sqrt{a_{1j}^{2}+b_{1j}^{2}}+a_{1j}}{2}},~q_{j}^{\prime}=\sqrt{\frac{\sqrt{a_{1j}^{2}+b_{1j}^{2}}-a_{1j}}{2}},~j=1,2,3,\\ &&C_{1}=P_{2}+\frac{\alpha^{(1)}}{k_{1}({A_{1}^{2}}+{B_{1}^{2}})}(A_{1}P_{1}+B_{1}Q_{1}),~C_{2}=(\bar{q_{1}}^{2}-q_{1}^{\prime^{2}})(A_{1}P_{1}+B_{1}Q_{1})-2\bar{q_{1}}q_{1}^{\prime}(A_{1}Q_{1}-B_{1}P_{1}),\\ &&D_{1}=Q_{2}+\frac{\alpha^{(1)}}{k_{1}({A_{1}^{2}}+{B_{1}^{2}})}(A_{1}Q_{1}-B_{1}P_{1}),~D_{2}=(\bar{q_{1}}^{2}-q_{1}^{\prime^{2}})(A_{1}Q_{1}-B_{1}P_{1})+2\bar{q_{1}}q_{1}^{\prime}(A_{1}P_{1}+B_{1}Q_{1}),\\ &&C_{3}=\frac{\alpha^{(2)}}{k_{1}}+A_{2},~D_{3}=B_{2},~\bar{L_{11}}=\bar{L_{0}}-\frac{4\bar{G}^{(1)}e^{\alpha^{(1)}h_{1}}\bar{L_{1}}}{\bar{G}^{(2)}},~L_{11}^{\prime}=L_{0}^{\prime}-\frac{4\bar{G}^{(1)}e^{\alpha^{(1)}h_{1}}{L_{1}}^{\prime}}{\bar{G}^{(2)}},\\ &&\bar{L_{22}}=P_{3}(\bar{q_{2}}^{2}-q_{2}^{\prime^{2}})-2\bar{q_{2}}q_{2}^{\prime}Q_{3},~L_{22}^{\prime}=2\bar{q_{2}}q_{2}^{\prime}P_{3}+Q_{3}(\bar{q_{2}}^{2}-q_{2}^{'^{2}}),\\ &&\bar{L_{32}}=(A_{1}\bar{A_{2}}-B_{1}\bar{B_{2}})\left( A_{1}P_{2}-B_{1}Q_{2}+\frac{\alpha^{(1)}}{k_{1}}P_{1}\right)+ (A_{1}\bar{B_{2}}+B_{1}\bar{A_{2}})\left( A_{1}Q_{2}+B_{1}P_{2}+\frac{\alpha^{(1)}}{k_{1}}Q_{1}\right),\\ &&L_{32}^{\prime}=(A_{1}\bar{A_{2}}-B_{1}\bar{B_{2}})\left( A_{1}Q_{2}+B_{1}P_{2}+\frac{\alpha^{(1)}}{k_{1}}Q_{1}\right)- (A_{1}\bar{B_{2}}+B_{1}\bar{A_{2}})\left( A_{1}P_{2}-B_{1}Q_{2}+\frac{\alpha^{(1)}}{k_{1}}P_{1}\right),\\ &&\bar{L_{23}}=P_{1}(\bar{q_{1}}^{2}-q_{1}^{\prime^{2}})-2\bar{q_{1}}q_{1}^{\prime}Q_{1},~L_{23}^{\prime}=Q_{1}(\bar{q_{1}}^{2}-q_{1}^{\prime^{2}})+2\bar{q_{1}}q_{1}^{\prime}P_{1},\\ &&\bar{L_{33}}=(A_{1}\bar{A_{2}}-B_{1}\bar{B_{2}})\left( P_{4}\bar{A_{2}}-Q_{4}\bar{B_{2}}-\frac{\alpha^{(2)}}{k_{1}}P_{3}\right)+(A_{1}\bar{B_{2}}+B_{1}\bar{A_{2}})\left( P_{4}\bar{B_{2}}+Q_{4}\bar{A_{2}}-\frac{\alpha^{(2)}}{k_{1}}Q_{3}\right),\\ &&L_{33}^{\prime}=(A_{1}\bar{A_{2}}-B_{1}\bar{B_{2}})\left( P_{4}\bar{B_{2}}+Q_{4}\bar{A_{2}}-\frac{\alpha^{(2)}}{k_{1}}Q_{3}\right)-(A_{1}\bar{B_{2}}+B_{1}\bar{A_{2}})\left( P_{4}\bar{A_{2}}-Q_{4}\bar{B_{2}}-\frac{\alpha^{(2)}}{k_{1}}P_{3}\right),\\ &&\bar{L_{21}}=\bar{G}^{(2)}\bar{L_{2}}+\bar{G}^{(1)}e^{\alpha^{(1)}h_{1}}\bar{L_{3}}, L_{21}^{\prime}=\bar{G}^{(2)}L_{2}^{\prime}+\bar{G}^{(1)}e^{\alpha^{(1)}h_{1}}L_{3}^{\prime},~D^{\prime\prime}=\frac{\alpha^{(3)}}{k_{1}}+A_{3},~D^{\prime\prime\prime}=B_{3},\\ &&F_{1}=\cos(k_{1}{\Delta} x)\cosh(k_{1}\delta{\Delta} x)-1,~F_{2}=\cos(k_{1}{\Delta} z)\cosh(k_{1}\delta{\Delta} z)-1,~F_{3}=\sinh(\alpha{\Delta} z)\cos(k_{1}{\Delta} z)\sinh(k_{1}\delta{\Delta} z),\\ &&I_{1}=-4\bar{\rho_{11}}\sin\left( \frac{\omega{\Delta} t}{2}\right),~R_{s}=16\bar{\rho_{jj}}^{2}\sin^{2}\left( \frac{\omega{\Delta} t}{2}\right)+4\bar{\zeta_{1j}}^{2}({\Delta} t)^{2}\cos^{2}\left( \frac{\omega{\Delta} t}{2}\right),\\ &&I_{g}=\frac{1}{R_{s}}\left[\left\lbrace-4\bar{\zeta_{1j}}^{2}({\Delta} t)^{2}\cos^{2}\left( \frac{\omega{\Delta} t}{2}\right)+16\bar{\rho_{1j}}^{2}\sin^{2}\left( \frac{\omega{\Delta} t}{2}\right)\right\rbrace \times 4\bar{\rho_{jj}}\sin\left( \frac{\omega{\Delta} t}{2}\right)-32\bar{\zeta_{1j}}^{2}({\Delta} t)^{2}\bar{\rho_{1j}}\sin\left( \frac{\omega{\Delta} t}{2}\right)\cos^{2}\left( \frac{\omega{\Delta} t}{2}\right)\right],\\ &&I_{1}^{*}=-4\bar{\rho_{11}}\sin\left( \frac{\pi p c {\Delta} x}{c_{s0}\lambda}\right),~R_{s}^{*}=16\bar{\rho_{jj}}\sin^{2}\left( \frac{\pi p c {\Delta} x}{c_{s0}\lambda}\right)+4\bar{\zeta_{1j}}^{2}\left( \frac{p{\Delta} x}{c_{s0}}\right)^{2}\cos^{2}\left( \frac{\pi p c {\Delta} x}{c_{s0}\lambda}\right),\\ &&I_{g}^{*}=\frac{1}{R_{s}^{*}}\left[\left\{-4\bar{\zeta_{1j}}^{2}(\frac{p{\Delta} x}{c_{s0}})^{2}\cos^{2}\left( \frac{\pi p c {\Delta} x}{c_{s0}\lambda}\right)+16\bar{\rho_{1j}}^{2}\sin^{2}\left( \frac{\pi p c {\Delta} x}{c_{s0}\lambda}\right)\right\}\right.\\ &&\left.\times 4\bar{\rho_{jj}}\sin\left( \frac{\pi p c {\Delta} x}{c_{s0}\lambda}\right)-32\bar{\zeta_{1j}}^{2}(\frac{p{\Delta} x}{c_{s0}})^{2}\bar{\rho_{1j}}\sin\left( \frac{\pi p c {\Delta} x}{c_{s0}\lambda}\right)\cos^{2}\left( \frac{\pi p c {\Delta} x}{c_{s0}\lambda}\right)\right],~s=1,2,~g=3,5,~j=2,3\\ &&F_{1}^{*}=\cos\left( \frac{2\pi{\Delta} x}{\lambda}\right)\cosh\left( \frac{2\pi\delta{\Delta} x}{\lambda}\right)-1,~F_{2}^{*}=\cos\left( \frac{2\pi d_{1}{\Delta} x}{\lambda}\right)\cosh\left( \frac{2\pi d_{1}\delta{\Delta} x}{\lambda}\right)-1,\\ &&F_{3}^{*}=\sinh(\alpha d_{1}{\Delta} x)\cos\left( \frac{2\pi d_{1}{\Delta} x}{\lambda}\right)\sinh\left( \frac{2\pi d_{1}\delta{\Delta} x}{\lambda}\right). \end{array} $$

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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

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Keywords

  • Heterogeneity
  • Cracked porous system
  • Haskell’s matrix method
  • Finite difference method
  • Phase and group velocities
  • Courant number