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

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## 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: $\phantom{\rule{1.em}{0ex}}{\mathit{u}}_{n}={\left[{\mathit{u}}_{Sn}\phantom{\rule{0.166667em}{0ex}}{v}_{Sn}\phantom{\rule{0.166667em}{0ex}}{v}_{Fn}\phantom{\rule{0.166667em}{0ex}}{\mathit{p}}_{n}\right]}^{T}=0,\phantom{\rule{1.em}{0ex}}{\mathit{F}}_{n}={\left[0\phantom{\rule{0.166667em}{0ex}}{\mathit{f}}_{Sn}\phantom{\rule{0.166667em}{0ex}}{\mathit{f}}_{Fn}\phantom{\rule{0.166667em}{0ex}}{\mathit{q}}_{n}\right]}^{T}=0$ |

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: |

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

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

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

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**Figure 6.**Deformed mesh (amplified by scale factor 500) and contours of norm of soil displacements $\u2225{\mathit{u}}_{S}\u2225=\sqrt{{u}_{Sx}^{2}+{u}_{Sz}^{2}}$ 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 $\u2225{\mathit{u}}_{S}\u2225=\sqrt{{u}_{Sx}^{2}+{u}_{Sy}^{2}+{u}_{Sz}^{2}}$.

**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 $\u2225{\mathit{u}}_{S}\u2225=\sqrt{{u}_{Sx}^{2}+{u}_{Sy}^{2}+{u}_{Sz}^{2}}$.

**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 $\u2225{\mathit{u}}_{S}\u2225=\sqrt{{u}_{Sx}^{2}+{u}_{Sy}^{2}+{u}_{Sz}^{2}}$.

**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 |

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**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