## 2 The \(\mathbf{u}\)-\(\mathbf{p}\) Dynamic Formulation for Wave Motion in FSPM
### Expressions of \(\mathbf{u}\)-\(p\) Dynamic Formulation  
Based on the basic assumptions introduced above, the \(\mathbf{u}\)-\(p\) dynamic formulation for the wave motion in the FSPM can be expressed as follows [3]:  
\[\mathbf{L}^{\rm T}\mathbf{D}\mathbf{L}\mathbf{u}+\mathbf{L}^{\rm T}\mathbf{m}\mathbf{p}-\rho\ddot{\mathbf{u}}=0 \tag{1}\]  
\[\mathbf{m}^{\rm T}\mathbf{L}\dot{\mathbf{u}}+\frac{k_{f}}{\mu_{f}}\
abla^{\rm T}\
abla p- \rho_{f}\frac{k_{f}}{\mu_{f}}\mathbf{m}^{\rm T}\mathbf{L}\ddot{\mathbf{u}}-\frac{n}{E_{w} }\dot{\rho}=0 \tag{2}\]  
For the derivation of \(\mathbf{u}\)-\(p\) dynamic formulation, the following solid-fluid coupling constitutive relations are adopted from the elastic and isotropic assumption:  
\[\mathbf{\sigma}^{\prime}=\mathbf{D}\mathbf{e}+\mathbf{m}\mathbf{p} \tag{3}\]  
In Equations (1) and (2), the solid skeleton displacement \(\mathbf{u}\) and pore fluid pressure \(p\) are the basic variables which need to be solved. Equation (1) expresses the total momentum balance of the FSPM, and Equation (2) is the flow conservation equation of the pore fluid. \(\mathbf{u}\), \(\mathbf{u}\) and \(\ddot{\mathbf{u}}\) are the displacement, velocity and acceleration vectors of the solid skeleton, respectively. For two-dimensional wave motion problems, \(\mathbf{u}=\left\{ \begin{array}{ll}u_{x},&u_{y}\\ \end{array} \right\}^{\rm T}\), \(\dot{\mathbf{u}}=\left\{ \begin{array}{ll}\dot{u}_{x},&\dot{u}_{y}\\ \end{array} \right\}^{\rm T}\), \(\ddot{\mathbf{u}}=\left\{ \begin{array}{ll}\ddot{u}_{x},&\ddot{u}_{y}\\ \end{array} \right\}^{\rm T}\). \(\rho\) is the total mass density of the FSPM, \(\rho_{f}\) is the mass density of the pore fluid, \(n\) is the porosity, \(k_{f}\) is [PERSON]'s permeability coefficient, \(\mu_{f}\) and \(E_{w}\) are the viscosity coefficient and bulk modulus of the pore fluid, respectively. The gradient operator \(\
abla\) and differential operator \(\mathbf{L}\) are defined as \[\
abla^{\mathrm{T}}=\left[ \begin{array}{cc}\frac{\partial}{\partial x}&\frac{ \partial}{\partial y}\\ \end{array} \right]\), \(\mathbf{L}=\left[ \begin{array}{cc}\frac{\partial}{\partial x}&0\\ 0&\frac{\partial}{\partial y}\\ \frac{\partial}{\partial y}&\frac{\partial}{\partial x}\\ \end{array} \right]\)  
Vector \(\mathbf{m}\) can be expressed as  
\[\mathbf{m}^{\mathrm{T}}=\left[ \begin{array}{ccc}1&1&0\\ \end{array} \right]\]  
In Equation (3), \(\sigma^{\prime}\) and \(\mathbf{e}\) are the effective stress and strain vectors of the solid skeleton, respectively. For two-dimensional wave motion problems, \(\sigma^{\prime}=\left\{ \begin{array}{cc}\sigma^{\prime}x,&\sigma^{\prime}y,\tau^{\prime}xy\\ \end{array} \right\}^{\mathrm{T}}\), \(\mathbf{e}=\left\{ \begin{array}{cc}\varepsilon_{x},\varepsilon_{y},&\gamma xy \\ \end{array} \right\}^{\mathrm{T}}\). \(\mathbf{D}\) is the stiffness matrix of the solid skeleton and can be expressed as for the 2D case. \[\mathbf{D}=\left[ \begin{array}{ccc}\lambda+2G&\lambda&0\\ \lambda&\lambda+2G&0\\ 0&0&G\\ \end{array} \right]\]  
where \(\lambda\) and \(G\) are the Lame constant and shear modulus of the solid skeleton, respectively. They can be expressed as follows:  
\[\lambda=\frac{\
u E_{s}}{(1+\
u)(1-2\
u)},\ G=\frac{E_{s}}{2(1+\
u)} \tag{4}\]  
where \(E_{s}\) and \(\
u\) are the elastic modulus and Poisson ratio of the solid skeleton, respectively. In Equation (2), the term \(\rho_{f}\frac{k_{f}}{\mu_{f}}\mathbf{m}^{\mathrm{T}}L\bar{\mathbf{u}}\) can be neglected because previous research shows that it has minimum impact on the computation results [18].