# Shear Wave Splitting and Polarization in Anisotropic Fluid-Infiltrating Porous Media: A Numerical Study

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Mathematical Model

#### 2.1. Governing Equations

#### 2.2. Constitutive Laws

#### 2.2.1. Effective Stress Law for Anisotropic Elasticity

#### 2.2.2. Tensorial Nature of Biot’s Coefficient

#### 2.2.3. Darcy’s Law

## 3. Numerical Implementation

#### 3.1. Galerkin Form

#### 3.2. Matrix Form and Time Discretization

#### 3.2.1. Implicit Monolithic Schemes

Algorithm 1: Newton–Raphson Algorithm. |

Initialization: $\phantom{\rule{1.em}{0ex}}{\mathit{u}}_{n}={\left(\right)}^{{\mathit{u}}_{Sn}}T=0$ |

for $n=1:{n}_{end}$ |

$\phantom{\rule{1.em}{0ex}}{\mathit{F}}_{n+1}={\mathit{F}}_{n}+\theta \Delta \mathit{F},\phantom{\rule{1.em}{0ex}}{\mathit{u}}_{n+1}={\mathit{u}}_{n},$ |

for $i=1:{i}_{max}$ |

$\phantom{\rule{2.em}{0ex}}\Delta {\mathit{u}}_{n+1}^{i}={\mathit{u}}_{n+1}^{i}-{\mathit{u}}_{n}$ |

Compute matrix:$\phantom{\rule{0.166667em}{0ex}}{\mathit{M}}_{ii},\phantom{\rule{0.166667em}{0ex}}{\mathit{K}}_{ii}$ |

Compute residual: |

${\mathit{R}}^{i+1}=\left[\begin{array}{c}0\\ {\mathit{r}}_{{v}_{S}}\\ {\mathit{r}}_{{v}_{F}}\\ {\mathit{r}}_{{p}_{F}}\end{array}\right]={\mathit{F}}_{n+1}+\left[\begin{array}{c}0\\ {\mathit{a}}_{2}\\ {\mathit{a}}_{3}\\ {\mathit{a}}_{4}\end{array}\right]-\left[\begin{array}{cccc}\mathit{I}& 0& 0& 0\\ 0& {\mathit{M}}_{22}& 0& 0\\ 0& 0& {\mathit{M}}_{33}& 0\\ 0& 0& {\mathit{M}}_{43}& {\mathit{M}}_{44}\end{array}\right]\phantom{\rule{-0.166667em}{0ex}}\frac{\Delta {\mathit{u}}_{n+1}^{i}}{\Delta t}-\left[\begin{array}{cccc}0& -\mathit{I}& 0& 0\\ {\mathit{K}}_{21}& {\mathit{K}}_{22}& {\mathit{K}}_{23}& {\mathit{K}}_{24}\\ 0& {\mathit{K}}_{32}& {\mathit{K}}_{33}& {\mathit{K}}_{34}\\ 0& {\mathit{K}}_{42}& {\mathit{K}}_{43}& {\mathit{K}}_{44}\end{array}\right]\phantom{\rule{-0.166667em}{0ex}}\left[\begin{array}{c}\theta {\mathit{u}}_{n+1}^{i}+(1-\theta ){\mathit{u}}_{n}\end{array}\right]$ |

Check residual: |

if$\phantom{\rule{1.em}{0ex}}{\mathit{R}}^{i+1}<toll$ |

break |

end |

Compute Jacobi matrix: |

$\phantom{\rule{2.em}{0ex}}\mathit{J}=\mathit{M}/\Delta t+\theta \mathit{K}$ |

Compute Y-increment: |

$\phantom{\rule{2.em}{0ex}}d\mathit{u}={\mathit{J}}^{-1}{\mathit{R}}^{i+1}$ |

Update solution: |

$\phantom{\rule{2.em}{0ex}}{\mathit{u}}_{n+1}^{i+1}={\mathit{u}}_{n+1}^{i}+d\mathit{u}$ |

end |

end |

#### 3.2.2. Semi-Explicit/Implicit Splitting Scheme

Algorithm 2: Prediction/Correction Algorithm. |

Initialization: $\phantom{\rule{1.em}{0ex}}{\mathit{u}}_{Sn}=0,\phantom{\rule{1.em}{0ex}}{v}_{n}=0,\phantom{\rule{1.em}{0ex}}{\mathit{p}}_{n}=0$ |

for $n=1:{n}_{end}$ |

$\phantom{\rule{1.em}{0ex}}{\mathit{f}}_{n}={\mathit{f}}_{n}+\Delta \mathit{f},\phantom{\rule{1.em}{0ex}}{v}^{P}={v}_{n},\phantom{\rule{1.em}{0ex}}{\mathit{p}}^{P}={\mathit{p}}_{n},$ |

$\phantom{\rule{1.em}{0ex}}\Delta {\mathit{u}}_{S}=\Delta t{v}_{S}^{P},\phantom{\rule{1.em}{0ex}}{\mathit{u}}_{Sn}+\Delta {\mathit{u}}_{S}$ |

for $i=1:{i}_{max}$ |

Compute matrix:$\phantom{\rule{0.166667em}{0ex}}\mathit{M},\phantom{\rule{0.166667em}{0ex}}{\mathit{K}}_{ii},\phantom{\rule{0.166667em}{0ex}}{\overline{\mathit{K}}}_{ii}$ |

Compute prediction velocities: |

$\phantom{\rule{2.em}{0ex}}\left[\begin{array}{c}{v}_{S}^{*}\\ {v}_{F}^{*}\end{array}\right]={\left[\begin{array}{cc}\frac{{\mathit{M}}_{22}}{\Delta t}+{\mathit{K}}_{22}& -{\mathit{K}}_{23}\\ -{\mathit{K}}_{32}& \frac{{\mathit{M}}_{33}}{\Delta t}+{\mathit{K}}_{33}\end{array}\right]}^{-1}\phantom{\rule{-0.166667em}{0ex}}\left\{\begin{array}{c}\left[\begin{array}{cc}\frac{{\mathit{M}}_{22}}{\Delta t}& 0\\ 0& \frac{{\mathit{M}}_{33}}{\Delta t}\end{array}\right]\left[\begin{array}{c}{v}_{Sn}^{P}\\ {v}_{Fn}^{P}\end{array}\right]+\left[\begin{array}{cc}-{\mathit{K}}_{21}& {\mathit{K}}_{24}\\ 0& -{\overline{\mathit{K}}}_{34}\end{array}\right]\left[\begin{array}{c}{\mathit{u}}_{Sn}^{P}\\ {\mathit{p}}_{n}^{P}\end{array}\right]+\left[\begin{array}{c}{\mathit{a}}_{2}\\ {\mathit{a}}_{3}\end{array}\right]+\left[\begin{array}{c}{\mathit{f}}_{Sn}\\ 0\end{array}\right]\end{array}\right\}$ |

Compute pore fluid pressure: |

$\phantom{\rule{2.em}{0ex}}\left[\begin{array}{c}0\\ {\mathit{p}}_{n+1}^{*}\end{array}\right]=\left[\begin{array}{c}0\\ {\mathit{p}}_{n}^{P}\end{array}\right]+{\left[\begin{array}{cc}\mathit{I}& 0\\ 0& {\overline{\mathit{K}}}_{44}\end{array}\right]}^{-1}\phantom{\rule{-0.166667em}{0ex}}\left\{\begin{array}{c}\left[\begin{array}{cc}0& 0\\ -{\mathit{K}}_{42}& {\mathit{K}}_{43}\end{array}\right]\left[\begin{array}{c}{v}_{S}^{*}\\ {v}_{F}^{*}\end{array}\right]-\left[\begin{array}{c}0\\ {\overline{\mathit{f}}}_{Pn+1}\end{array}\right]\end{array}\right\}$ |

Compute velocities correction: |

$\phantom{\rule{2.em}{0ex}}\left[\begin{array}{c}{v}_{Sn+1}\\ {v}_{Fn+1}\end{array}\right]=\left[\begin{array}{c}{v}_{S}^{*}\\ {v}_{F}^{*}\end{array}\right]+{\left[\begin{array}{cc}\frac{{\mathit{M}}_{22}}{\Delta t}& 0\\ 0& \frac{{\mathit{M}}_{33}}{\Delta t}\end{array}\right]}^{-1}\phantom{\rule{-0.166667em}{0ex}}\left\{\begin{array}{c}\left[\begin{array}{cc}0& {\overline{\mathit{K}}}_{24}\\ 0& -{\mathit{K}}_{34}\end{array}\right]\left[\begin{array}{c}0\\ {\mathit{p}}_{n+1}-{\mathit{p}}_{n}^{P}\end{array}\right]+\left[\begin{array}{c}0\\ {\mathit{f}}_{Pn+1}\end{array}\right]\end{array}\right\}$ |

Compute solid displacements: |

$\phantom{\rule{2.em}{0ex}}{\mathit{u}}_{Sn+1}={\mathit{u}}_{Sn}+\frac{1}{2}\Delta t\left(\right)open="("\; close=")">{v}_{Sn+1}+{v}_{Sn}$ |

$\phantom{\rule{2.em}{0ex}}{\mathit{r}}_{i}={\mathit{u}}_{Sn+1}-{\mathit{u}}^{P}$ |

if$\phantom{\rule{1.em}{0ex}}{\mathit{r}}_{i}<toll$ |

break |

end |

Update prediction variables: |

$\phantom{\rule{2.em}{0ex}}{v}^{P}={v}_{n+1},\phantom{\rule{1.em}{0ex}}{\mathit{u}}_{S}^{P}={\mathit{u}}_{Sn+1},\phantom{\rule{1.em}{0ex}}\Delta {\mathit{u}}_{S}={\mathit{u}}_{Sn+1}-{\mathit{u}}_{Sn},$ |

end |

end |

## 4. Numerical Examples

#### 4.1. Benchmark Cases with Isotropic Elastic Materials

^{−3}s has been adopted, while for the semi-explicit scheme with quadratic elements, $dt$ = $2.5$ 10

^{−4}s has been assumed in order to satisfy the CFL stability condition.

#### 4.2. Dynamic Poroelastic Responses of Isotropic Porous Media

#### 4.3. Dynamic Poroelastic Responses with Transversely Isotropic Porous Media

#### 4.3.1. Effect of Different Rotation in a Transversely Isotropic Symmetry Axis of Soil Material

#### 4.3.2. Effect of Biot’s Effective Stress Coefficient Tensor on Wave Propagation

## 5. Discussion

- (i)
- the p-waves produce polarized vibrations along the direction of propagation (particles move along the wave’s direction of propagation) and subsequent compression and extension deformations along the same direction: they are visible along the vertical direction under the impulsive load (Figure 6 and Figure 9), even considering the anisotropic models (in this case they are coupled with the shear contribution, Figure 14 and Figure 18);
- (ii)
- p-waves are faster than s-waves: in all the models in fact the domain borders are reached in different times;
- (iii)
- (iv)
- the s-wave decouples into a wave polarized on the horizontal plane and into another one on the vertical plane: visible in the curves of effective shear stresses, Figure 15;

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

## References

- Biot, M.A. Theory of propagation of elastic waves in a fluid-saturated porous solid. II. Higher frequency range. J. Acoust. Soc. Am.
**1956**, 28, 179–191. [Google Scholar] [CrossRef] - Biot, M. Theory of elastic waves in a fluid-saturated porous solid. 1. Low frequency range. J. Acoust. Soc. Am.
**1956**, 28, 168–178. [Google Scholar] [CrossRef] - Berryman, J.G. Elastic wave propagation in fluid-saturated porous media. J. Acoust. Soc. Am.
**1981**, 69, 416–424. [Google Scholar] [CrossRef] - Paterson, M.; Wong, T. Experimental Rock Deformation—The Brittle Field; Springer Science & Business Media: Berlin, Germany, 2005. [Google Scholar]
- Na, S.; Sun, W. Wave propagation and strain localization in a fully saturated softening porous medium under the non-isothermal conditions. Int. J. Numer. Anal. Methods Geomech.
**2016**, 40, 1485–1510. [Google Scholar] [CrossRef] - Sharma, M. Propagation of seismic waves in patchy-saturated porous media: Double-porosity representation. Geophys. Prospect.
**2019**, 67, 2147–2160. [Google Scholar] [CrossRef] - Borja, R.; Sun, W. Estimating inelastic sediment deformation from local site response simulations. Acta Geotech.
**2007**, 2, 183–195. [Google Scholar] [CrossRef][Green Version] - Borja, R.; Sun, W. Coseismic sediment deformation during the 1989 Loma Prieta earthquake. J. Geophys. Res. Solid Earth
**2008**, 113, doi. [Google Scholar] [CrossRef][Green Version] - Sun, W. A unified method to predict diffuse and localized instabilities in sands. Geomech. Geoengin.
**2013**, 8, 65–75. [Google Scholar] [CrossRef][Green Version] - Na, S.; Sun, W.; Ingraham, M.; Yoon, H. Effects of spatial heterogeneity and material anisotropy on the fracture pattern and macroscopic effective toughness of Mancos Shale in Brazilian tests. J. Geophys. Res. Solid Earth
**2017**, 122, 6202–6230. [Google Scholar] [CrossRef][Green Version] - Thamarux, P.; Matsuoka, M.; Poovarodom, N.; Iwahashi, J. VS30 Seismic Microzoning Based on a Geomorphology Map: Experimental Case Study of Chiang Mai, Chiang Rai, and Lamphun, Thailand. ISPRS Int. J. Geo-Inf.
**2019**, 8, 309. [Google Scholar] [CrossRef][Green Version] - Crampin, S. Evaluation of anisotropy by shear-wave splitting. Geophysics
**1985**, 50, 142–152. [Google Scholar] [CrossRef] - Cardoso, L.; Cowin, S.C. Role of structural anisotropy of biological tissues in poroelastic wave propagation. Mech. Mater.
**2012**, 44, 174–188. [Google Scholar] [CrossRef][Green Version] - Aki, K.; Richards, P.G. Quantitative Seismology, 2nd ed.; University Science Books: Mill Valley, CA, USA, 2002. [Google Scholar]
- Vlastos, S.; Liu, E.; Main, I.; Schoenberg, M.; Narteau, C.; Li, X.; Maillot, B. Dual simulations of fluid flow and seismic wave propagation in a fractured network: Effects of pore pressure on seismic signature. Geophys. J. Int.
**2006**, 166, 825–838. [Google Scholar] [CrossRef][Green Version] - Boxberg, M.S.; Prévost, J.H.; Tromp, J. Wave propagation in porous media saturated with two fluids. Transp. Porous Media
**2015**, 107, 49–63. [Google Scholar] [CrossRef] - Crampin, S.; Peacock, S. A review of shear-wave splitting in the compliant crack-critical anisotropic Earth. Wave Motion
**2005**, 41, 59–77. [Google Scholar] [CrossRef] - Grechka, V.; Kachanov, M. Effective elasticity of fractured rocks: A snapshot of the work in progress. Geophysics
**2006**, 71, W45–W58. [Google Scholar] [CrossRef] - Grechka, V.; Vasconcelos, I.; Kachanov, M. The influence of crack shape on the effective elasticity of fractured rocks. Geophysics
**2006**, 71, D153–D160. [Google Scholar] [CrossRef] - Crampin, S.; McGonigle, R. The variation of delays in stress-induced anisotropic polarization anomalies. Geophys. J. Int.
**1981**, 64, 115–131. [Google Scholar] [CrossRef][Green Version] - Virieux, J. P-SV wave propagation in heterogeneous media: Velocity-stress finite-difference method. Geophysics
**1986**, 51, 889–901. [Google Scholar] [CrossRef] - Carcione, J.M. A generalization of the Fourier pseudospectral method. Geophysics
**2010**, 75, A53–A56. [Google Scholar] [CrossRef] - Prevost, J.H. Wave propagation in fluid-saturated porous media: An efficient finite element procedure. Int. J. Soil Dyn. Earthq. Eng.
**1985**, 4, 183–202. [Google Scholar] [CrossRef] - Sluys, L.; de Borst, R.; Mühlhaus, H. Wave propagation, localization and dispersion in a gradient-dependent medium. Int. J. Solids Struct.
**1993**, 30, 1153–1171. [Google Scholar] [CrossRef][Green Version] - Abellan, M.; De Borst, R. Wave propagation and localisation in a softening two-phase medium. Comput. Methods Appl. Mech. Eng.
**2006**, 195, 5011–5019. [Google Scholar] [CrossRef] - Cowin, S.C.; Doty, S.B. Tissue Mechanics; Springer Science & Business Media: Berlin, Germany, 2007. [Google Scholar]
- Ehlers, W. Porous Media: Theory, Experiments and Numerical Applications; Springer Science & Business Media: Berlin, Germany, 2002. [Google Scholar]
- De Marchi, N.; Salomoni, V.; Spiezia, N. Effects of Finite Strains in Fully Coupled 3D Geomechanical Simulations. Int. J. Geomech.
**2019**, 19, 04019008. [Google Scholar] [CrossRef] - White, J.; Borja, R. Stabilized low-order finite elements for coupled solid-deformation/fluid-diffusion and their application to fault zone transients. Comput. Methods Appl. Mech. Eng.
**2008**, 197, 4353–4366. [Google Scholar] [CrossRef] - Sun, W.; Ostien, J.; Salinger, A. A stabilized assumed deformation gradient finite element formulation for strongly coupled poromechanical simulations at finite strain. Int. J. Numer. Anal. Methods Geomech.
**2013**, 37, 2755–2788. [Google Scholar] [CrossRef][Green Version] - Sun, W. A stabilized finite element formulation for monolithic thermo-hydro-mechanical simulations at finite strain. Int. J. Numer. Methods Eng.
**2015**, 103, 798–839. [Google Scholar] [CrossRef][Green Version] - Wang, K.; Sun, W. A semi-implicit discrete-continuum coupling method for porous media based on the effective stress principle at finite strain. Comput. Methods Appl. Mech. Eng.
**2016**, 304, 546–583. [Google Scholar] [CrossRef][Green Version] - Wang, K.; Sun, W. A unified variational eigen-erosion framework for interacting brittle fractures and compaction bands in fluid-infiltrating porous media. Comput. Methods Appl. Mech. Eng.
**2017**, 318, 1–32. [Google Scholar] [CrossRef][Green Version] - Na, S.; Bryant, E.C.; Sun, W. A configurational force for adaptive re-meshing of gradient-enhanced poromechanics problems with history-dependent variables. Comput. Methods Appl. Mech. Eng.
**2019**, 357, 112572. [Google Scholar] [CrossRef][Green Version] - Carroll, M. An effective stress law for anisotropic elastic deformation. J. Geophys. Res. Solid Earth
**1979**, 84, 7510–7512. [Google Scholar] [CrossRef] - Nur, A.; Byerlee, J. An exact effective stress law for elastic deformation of rock with fluids. J. Geophys. Res.
**1971**, 76, 6414–6419. [Google Scholar] [CrossRef] - Sun, W.; Andrade, J.; Rudnicki, J.; Eichhubl, P. Connecting microstructural attributes and permeability from 3D tomographic images of in situ shear-enhanced compaction bands using multiscale computations. Geophys. Res. Lett.
**2011**, 38. [Google Scholar] [CrossRef][Green Version] - Cowin, S.C. Continuum Mechanics of Anisotropic Materials; Springer Science & Business Media: Berlin, Germany, 2013. [Google Scholar]
- Markert, B.; Heider, Y.; Ehlers, W. Comparison of monolithic and splitting solution schemes for dynamic porous media problems. Int. J. Numer. Methods Eng.
**2010**, 82, 1341–1383. [Google Scholar] [CrossRef] - Jansen, K.E.; Whiting, C.H.; Hulbert, G.M. A generalized-α method for integrating the filtered Navier–Stokes equations with a stabilized finite element method. Comput. Methods Appl. Mech. Eng.
**2000**, 190, 305–319. [Google Scholar] [CrossRef] - Huang, M.; Wu, S.; Zienkiewicz, O. Incompressible or nearly incompressible soil dynamic behaviour—A new staggered algorithm to circumvent restrictions of mixed formulation. Soil Dyn. Earthq. Eng.
**2001**, 21, 169–179. [Google Scholar] - Huang, M.; Yue, Z.Q.; Tham, L.; Zienkiewicz, O. On the stable finite element procedures for dynamic problems of saturated porous media. Int. J. Numer. Methods Eng.
**2004**, 61, 1421–1450. [Google Scholar] - De Boer, R.; Ehlers, W.; Liu, Z. One-dimensional transient wave propagation in fluid-saturated incompressible porous media. Arch. Appl. Mech.
**1993**, 63, 59–72. [Google Scholar] [CrossRef] - Yang, Z.; Li, Z. A numerical study on waves induced by wheel-rail contact. Int. J. Mech. Sci.
**2019**, 161, 105069. [Google Scholar] [CrossRef]

**Figure 6.**Deformed mesh (amplified by scale factor 500) and contours of norm of soil displacements $\left(\right)open="\parallel "\; close="\parallel ">{\mathit{u}}_{S}$ for benchmark (2).

**Figure 9.**Isotropic soil model with ${K}_{s}=5.2{10}^{7}Pa$: deformed meshes (amplified by scale factor 500) and contours of norm of soil displacements vector $\left(\right)open="\parallel "\; close="\parallel ">{\mathit{u}}_{S}$.

**Figure 13.**Cauchy pore pressure evolution in time: solid line for nodes E and G, marked line for nodes F and H.

**Figure 14.**Transversely isotropic soil model with $\alpha =90{}^{\circ}$: deformed meshes (amplified by scale factor 500) and contours of norm of soil displacements vector $\left(\right)open="\parallel "\; close="\parallel ">{\mathit{u}}_{S}$.

**Figure 15.**Transversely isotropic soil models. Effective Cauchy stress component calculated along the horizontal directions: solid line for model with $\alpha =0{}^{\circ}$, marked line for model with $\alpha =45{}^{\circ}$.

**Figure 18.**Transversely isotropic soil model with constitutive model of solid phase rotated by $\alpha =90{}^{\circ}$: deformed meshes (amplified by scale factor 500) and contours of norm of soil displacements vector $\left(\right)open="\parallel "\; close="\parallel ">{\mathit{u}}_{S}$.

**Figure 19.**Effect of Biot’s coefficient tensor: effective shear stress component calculated along the horizontal directions.

Parameter | Values | S.I. unit |
---|---|---|

E | 14.52 × 10${}^{6}$ | Pa |

$\nu $ | 0.30 | |

${n}_{0}^{{\scriptscriptstyle F}}$ | 0.33 | |

${k}^{{\scriptscriptstyle F}}$ | 10${}^{-2}$ | m/s |

${\rho}_{{\scriptscriptstyle S}}$ | 2000 | kg/m${}^{3}$ |

${\rho}_{{\scriptscriptstyle F}}$ | 1000 | kg/m${}^{3}$ |

Parameter | Values | S.I. Unit |
---|---|---|

E | 12.0 × 10${}^{6}$ | Pa |

$\nu $ | 0.25 | |

${n}_{0}^{{\scriptscriptstyle F}}$ | 0.33 | |

${k}^{{\scriptscriptstyle F}}$ | 10.0${}^{-2}$ | m/s |

${\rho}_{{\scriptscriptstyle S}}$ | 2000.0 | kg/m${}^{3}$ |

${\rho}_{{\scriptscriptstyle F}}$ | 1000.0 | kg/m${}^{3}$ |

${K}_{{\scriptscriptstyle F}}$ | 5.2 × 10${}^{9}$ | Pa |

Parameter | Values | S.I. Unit |
---|---|---|

${E}_{x},{E}_{y}$ | 9 × 10${}^{6}$ | Pa |

${E}_{z}$ | 15 × 10${}^{6}$ | Pa |

${\nu}_{xy},{\nu}_{yx}$ | 0.25 | |

${\nu}_{yz},{\nu}_{xz}$ | 0.21 | |

${\nu}_{zx},{\nu}_{zy}$ | 0.35 | |

${G}_{xy}=\frac{{E}_{x}}{2(1+{\nu}_{xy})}$ | 3.6 × 10${}^{6}$ | Pa |

${G}_{23},{G}_{31}$ | 6.0 × 10${}^{6}$ | Pa |

${n}_{0}^{{\scriptscriptstyle F}}$ | 0.33 | |

${k}_{x}^{{\scriptscriptstyle F}},{k}_{y}^{{\scriptscriptstyle F}}$ | 10${}^{-2}$ | m/s |

${k}_{z}^{{\scriptscriptstyle F}}$ | 10${}^{-4}$ | m/s |

${\rho}_{{\scriptscriptstyle S}}$ | 2000 | kg/m${}^{3}$ |

${\rho}_{{\scriptscriptstyle F}}$ | 1000 | kg/m${}^{3}$ |

${K}_{{\scriptscriptstyle m}}$ | 7.14 × 10${}^{6}$ | Pa |

${K}_{{\scriptscriptstyle S}}$ | 3.57 × 10${}^{9}$ | Pa |

${K}_{{\scriptscriptstyle F}}$ | 2.2 × 10${}^{9}$ | Pa |

Parameter | Values | S.I. Unit |
---|---|---|

${E}_{x},{E}_{y}$ | 1.8 × 10${}^{7}$ | Pa |

${E}_{z}$ | 3.0 × 10${}^{7}$ | Pa |

${\nu}_{xy},{\nu}_{yx}$ | 0.25 | |

${\nu}_{yz},{\nu}_{xz}$ | 0.21 | |

${\nu}_{zx},{\nu}_{zy}$ | 0.35 | |

${G}_{xy}$ | 7.2 × 10${}^{6}$ | Pa |

${G}_{yz},{G}_{zx}$ | 1.2 × 10${}^{7}$ | Pa |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

De Marchi, N.; Sun, W.; Salomoni, V.
Shear Wave Splitting and Polarization in Anisotropic Fluid-Infiltrating Porous Media: A Numerical Study. *Materials* **2020**, *13*, 4988.
https://doi.org/10.3390/ma13214988

**AMA Style**

De Marchi N, Sun W, Salomoni V.
Shear Wave Splitting and Polarization in Anisotropic Fluid-Infiltrating Porous Media: A Numerical Study. *Materials*. 2020; 13(21):4988.
https://doi.org/10.3390/ma13214988

**Chicago/Turabian Style**

De Marchi, Nico, WaiChing Sun, and Valentina Salomoni.
2020. "Shear Wave Splitting and Polarization in Anisotropic Fluid-Infiltrating Porous Media: A Numerical Study" *Materials* 13, no. 21: 4988.
https://doi.org/10.3390/ma13214988