*2.2. Seabed Model*

It is widely known that saturated soil considered as a multi-phase material is formed by soil particles and the void of the skeleton. The pores of the solid phase are filled with the water and trapped air distributed through the body. Therefore, in order to simulate the interaction between the soil skeleton and pore water in a porous seabed, a seabed model is established based on the partially dynamic Biot's equation ( also known as "*u* − *p*" approximation) [45] with consideration of acceleration inertia term of flow and soil particles.

With the assumptions of a homogeneous and isotropic seabed and compressible pore fluid, the mass conservation equation of pore fluid can be expressed as [46]:

$$K\_s \nabla^2 p\_s - \gamma\_w n\_s \beta\_s \frac{\partial p\_s}{\partial t} + K\_s \rho\_f \frac{\partial^2 \varepsilon\_s}{\partial t^2} = \gamma\_w \frac{\partial \varepsilon\_s}{\partial t},\tag{9}$$

where *p<sup>s</sup>* is pore water pressure, *γ<sup>w</sup>* is the unit weight of water, *K<sup>s</sup>* is the soil permeability, *n<sup>s</sup>* is soil porosity; *β<sup>s</sup>* is the compressibility of pore fluid while *e<sup>s</sup>* is volume strain, which can be expressed as:

$$\beta\_s = \frac{1}{K\_w} + \frac{1 - S\_r}{P\_{wo}}, \text{ and } \epsilon\_s = \frac{\partial u\_s}{\partial x} + \frac{\partial w\_s}{\partial z}, \tag{10}$$

where *u<sup>s</sup>* and *w<sup>s</sup>* are the soil displacements in the *x*- and *z*- direction, respectively; *K<sup>w</sup>* is the bulk modulus of pore fluid (*K<sup>w</sup>* <sup>=</sup> 1.95 <sup>×</sup> <sup>10</sup>9N/m<sup>2</sup> [4]); *S<sup>r</sup>* is the degree of saturation and *Pwo* is absolute static water pressure, which defined as *Pwo* = *γwd*, in which *d* is the water depth.

Based on Newton's second law, the force equilibrium equation in a poro-elastic medium in horizontal and vertical directions can be given as:

$$\frac{\partial \sigma\_{\mathbf{x}}^{\prime}}{\partial \mathbf{x}} + \frac{\partial \tau\_{\mathbf{x}\mathbf{z}}}{\partial \mathbf{z}} = -\frac{\partial p\_{s}}{\partial \mathbf{x}} + \rho \frac{\partial^{2} u\_{s}}{\partial t^{2}},\tag{11}$$

$$\frac{\partial \mathbf{r}\_{\mathbf{x}\mathbf{z}}}{\partial \mathbf{x}} + \frac{\partial \sigma\_z'}{\partial \mathbf{z}} = -\frac{\partial p\_s}{\partial \mathbf{z}} + \rho \frac{\partial^2 w\_s}{\partial t^2},\tag{12}$$

where *σ* 0 *<sup>x</sup>* and *σ* 0 *<sup>z</sup>* are the effective normal stresses in horizontal and vertical direction respectively; *τxz* is shear stress component; *ρ* is the average density of a porous seabed and can be obtained by *ρ* = *ρ<sup>f</sup> n<sup>s</sup>* + *ρs*(1 − *ns*), in which *ρ<sup>f</sup>* is the fluid density while *ρ<sup>s</sup>* is the solid density.

Based on the pore-elastic theory, the effective normal stresses and shear stress can be expressed in term of soil displacements:

$$\sigma\_{\mathbf{x}}' = 2G \left[ \frac{\partial u\_s}{\partial \mathbf{x}} + \frac{\nu\_s}{1 - 2\nu\_s} \epsilon\_s \right],\tag{13}$$

$$\sigma\_z' = 2\mathcal{G} \left[ \frac{\partial w\_s}{\partial z} + \frac{\nu\_s}{1 - 2\nu\_s} \epsilon\_s \right] \,, \tag{14}$$

$$\pi\_{\rm xz} = G \left[ \frac{\partial u\_s}{\partial z} + \frac{\partial w\_s}{\partial x} \right],\tag{15}$$

where the shear modulus *G* is defined with Young's modulus (*E*) and the Poisson's ratio (*νs*) in the form of *E*/2(1 + *νs*).

Substituting (13)–(15) into (11)–(12), we have the governing equations for force balance as

$$\, \mathrm{G} \nabla^2 u\_s + \frac{\mathrm{G}}{1 - 2v\_s} \frac{\partial}{\partial \mathbf{x}} (\frac{\partial u\_s}{\partial \mathbf{x}} + \frac{\partial w\_s}{\partial \mathbf{z}}) = \frac{\partial p\_s}{\partial \mathbf{x}} + \rho\_s \frac{\partial^2 u\_s}{\partial t^2} \,. \tag{16}$$

$$G\nabla^2 w\_s + \frac{G}{1 - 2\nu\_s} \frac{\partial}{\partial z} (\frac{\partial u\_s}{\partial x} + \frac{\partial w\_s}{\partial z}) = \frac{\partial p\_s}{\partial z} + \rho\_s \frac{\partial^2 w\_s}{\partial t^2},\tag{17}$$
