Wave and current-induced dynamic response in a multilayered poroelastic seabed

Abstract

Waves and currents often coexist in ocean, which significantly changes the pore pressure and stress state in the seabed. Besides, stratification is a basic feature of a seabed due to natural deposition or artificial construction. An analytical solution to the dynamic response of a multilayered seabed to combined wave and current loading is proposed in this study. The seabed is modeled using Biot’s fully dynamic theory, where the effects of inertia and the compressibility of solids and fluids are included. Unlike previous investigations, stratification of seabed and non-linear interactions between waves and the current are considered in this study. The present solution is firstly validated against an existing analytical solution and a model test. Comprehensive parametric study is then conducted to study the influences of soil layering, current and waves on the dynamic response of a multilayered seabed. A seabed with layered soil properties (e.g. shear modulus, permeability) has substantially different pore pressure and stress state from a homogeneous seabed under combined wave and current loading. The current significantly influences the seabed response. An opposing current is beneficial to prevent both soil liquefaction and shear failure, and a following current is more likely to cause seabed instability. The present solution is a practical tool to evaluate seabed stability in an ocean environment with multiple layers, especially where waves and currents are prevailing.

Appendix

The solution matrices ${\mathbf{L}}_{11}^n$, ${\mathbf{L}}_{12}^n$, ${\mathbf{L}}_{21}^n$ and ${\mathbf{L}}_{22}^n$ in Eq. (17) can be given as follows:

$$ {\mathbf{L}}_{11}^n=\left(\begin{array}{ccc}-\mathrm{i}{k}_x& -\mathrm{i}{k}_x& -\mathrm{i}{b}^n\\ {}-{a}_1^n& -{a}_2^n& -{k}_x\\ {}-{\xi}_1^n{a}_1^n& -{\xi}_2^n{a}_2^n& -{\xi}_s^n{k}_x\end{array}\right) $$

(A1)

$$ {\mathbf{L}}_{12}^n=\left(\begin{array}{ccc}-\mathrm{i}{k}_x& -\mathrm{i}{k}_x& \mathrm{i}{b}^n\\ {}{a}_1^n& {a}_2^n& -{k}_x\\ {}{\xi}_1^n{a}_1^n& {\xi}_2^n{a}_2^n& -{\xi}_s^n{k}_x\end{array}\right) $$

(A2)

$$ {\mathbf{L}}_{21}^n=\left(\begin{array}{ccc}2{\mu}_n{k}_x\mathrm{i}{a}_1^n& 2{\mu}_n{k}_x\mathrm{i}{a}_2^n& 2{\mu}_n{\varOmega}_n\mathrm{i}\\ {}2{\mu}_n{k}_x^2-{H}_n{\left({k}_1^n\right)}^2& 2{\mu}_n{k}_x^2-{H}_n{\left({k}_2^n\right)}^2& 2{\mu}_n{k}_x{b}^n\\ {}{M}_n\left(1+{\xi}_1^n\right){\left({k}_1^n\right)}^2& {M}_n\left(1+{\xi}_2^n\right){\left({k}_2^n\right)}^2& 0\end{array}\right) $$

(A3)

$$ {\mathbf{L}}_{22}^n=\left(\begin{array}{ccc}-2{\mu}_n{k}_x\mathrm{i}{a}_1^n& -2{\mu}_n{k}_x\mathrm{i}{a}_2^n& 2{\mu}_n{\varOmega}_n\mathrm{i}\\ {}2{\mu}_n{k}_x^2-{H}_n{\left({k}_1^n\right)}^2& 2{\mu}_n{k}_x^2-{H}_n{\left({k}_2^n\right)}^2& -2{\mu}_n{k}_x{b}^n\\ {}{M}_n\left(1+{\xi}_1^n\right){\left({k}_1^n\right)}^2& {M}_n\left(1+{\xi}_2^n\right){\left({k}_2^n\right)}^2& 0\end{array}\right) $$

(A4)

where

$$ {a}_1^n=\sqrt{k_x^2-{\left({k}_1^n\right)}^2},{a}_2^n=\sqrt{k_x^2-{\left({k}_2^n\right)}^2},{b}^n=\sqrt{k_x^2-{\left({k}_s^n\right)}^2} $$

(A5)

$$ {H}_n={\lambda}_n+2{\mu}_n,{\varOmega}_n={k}_x^2-{\left({k}_s^n\right)}^2/2 $$

(A6)

$$ {\xi}_{1,2}^n=\frac{\left({\lambda}_n+{M}_n+2{\mu}_n\right){\left({k}_{1,2}^n\right)}^2-{\rho}^n{\omega}^2}{\rho_f^n{\omega}^2-{M}_n{\left({k}_{1,2}^n\right)}^2},{\xi}_s^n=-\frac{\rho_f^n}{\gamma_n} $$

(A7)

$$ {M}_n={K}_f^{\prime }/{\phi}_n,{\gamma}_n=\frac{\rho_f^n}{\phi_n}-\frac{\mathrm{i}{\rho}^ng}{\omega {k}_n} $$

(A8)

$$ {\left({k}_{1,2}^n\right)}^2=\frac{B_n\mp \sqrt{B_n^2-4{A}_n{C}_n}}{2{A}_n},{\left({k}_s^n\right)}^2=\frac{C_n}{\mu^n{\gamma}_n{\omega}^2} $$

(A9)

$$ {A}_n={M}_n\left({\lambda}_n+2{\mu}_n\right) $$

(A10)

$$ {B}_n=\left[{\gamma}_n\left({\lambda}_n+{M}_n^2+2{\mu}_n\right)+{M}_n\left({\rho}_n-2{\rho}_f^n\right)\right]{\omega}^2 $$

(A11)

$$ {C}_n=\left[{\rho}_n{\gamma}_n-{\left({\rho}_f^n\right)}^2\right]{\omega}^4 $$

(A12)