# Scattering of Plane Waves by Cylindrical Cavity in Unsaturated Poroelastic Medium +
## 2 Model and Solution  
### Mathematical Models and Basic Equations  
A long cylindrical cavity of radius \(r=a\) in an infinite unsaturated porous medium is considered as shown in Figure 1. The hydraulic boundary conditions at the cavity surface are permeable. The dynamic equations for unsaturated porous media established by [PERSON] and [PERSON] [18] are applied to the model, which is expressed as:  
\[ \begin{array}{ll}n_{0}^{S}\rho_{0}^{S}\ddot{u}^{S}&=\left(M_{\rm{SS}}+n_{0}^{ S}\mu_{S}\right)\
abla\cdot\mathbf{u}^{S}+n_{0}^{S}\
abla\cdot\
abla\mathbf{u}^{S}+M_{\rm{ SW}}\
abla\
abla\cdot\mathbf{u}^{W}+M_{\rm{SN}}\
abla\
abla\cdot\mathbf{u}^{N}\\ &+\dot{\mu}^{W}\left(\dot{\mathbf{u}}^{W}-\dot{\mathbf{u}}^{S}\right)+\dot{\mu}^{N} \left(\dot{\mathbf{u}}^{N}-\dot{\mathbf{u}}^{S}\right)\\ n_{0}^{W}\rho_{0}^{W}\dot{\mathbf{u}}^{W}&=M_{\rm{SW}}\
abla\
abla\cdot\mathbf{u}^{S}+M _{\rm{WW}}\
abla\
abla\cdot\mathbf{u}^{W}+M_{\rm{WN}}\
abla\
abla\cdot\mathbf{u}^{N}- \dot{\mu}^{W}\left(\dot{\mathbf{u}}^{W}-\dot{\mathbf{u}}^{S}\right)\\ n_{0}^{N}\rho_{0}^{N}\ddot{\mathbf{u}}^{N}&=M_{\rm{SN}}\
abla\
abla\cdot\mathbf{u}^{S}+M _{\rm{WN}}\
abla\
abla\cdot\mathbf{u}^{W}+M_{\rm{NN}}\
abla\
abla\cdot\mathbf{u}^{N}- \dot{\mu}^{N}\left(\dot{\mathbf{u}}^{N}-\dot{\mathbf{u}}^{S}\right)\end{array}  \tag{1}\]  
where the superscript S is the solid component; W and N are the wetting and non-wetting fluids, respectively; \(\mu_{S}\) is the shear modulus of the unsaturated porous medium; and \(n_{0}^{\rm{g}}\)\(\rho_{0}^{\alpha}\) and \(\mathbf{u}^{\alpha}(\alpha=S,W,N)\) are the initial volume fraction, initial density, and displacements of individual components, respectively. \(M_{\text{SS}}\), \(M_{\text{WW}}\), \(M_{\text{NN}}\), \(M_{\text{SW}}\), \(M_{\text{SN}}\) and \(M_{\text{WN}}\) are elastic coefficients related to the bulk modulus \(K^{\alpha}(\alpha=S,W,N)\) of the various phases in unsaturated porous media, the bulk modulus of the solid skeleton \(\hat{K}\), the shear modulus of the solid skeleton \(\hat{G}\), the effective stress coefficient \(\alpha_{B}\), and [PERSON]'s ration \(v\). \(\hat{\mu}^{f}(f=W,N)\) is the soil water characteristic curve, which can be related to the intrinsic permeability of the porous medium \(k\) and the dynamic viscosity \(v^{f}\) of the \(f\)-fluid by the expressions given explicitly in [PERSON] and [PERSON] [18]. The V g model to express the soil-water characteristic curve proposed by [PERSON] [19] is applied here:  
\[p^{\text{N}}-p^{\text{W}}=\alpha^{-1}\Big{(}S_{\text{e}}{}^{-1/m}-1\Big{)}^{1-m} \tag{2}\]  
where \(\alpha\) and \(m\) are the [PERSON] model parameters depending on the properties of the three-phase model, and the effective degree of saturation, \(S_{\text{e}}\), given by:  
\[S_{\text{e}}= \begin{cases}0&S_{\text{r}}\leq S_{\text{rN}}\\ \frac{S_{\text{r}}-S_{\text{WW}}}{S_{\text{rN}}-S_{\text{rW}}}&S_{\text{rW}} \leq S_{\text{r}}\leq S_{\text{rN}}\\ 1&S_{\text{r}}\geq S_{\text{rN}}\end{cases}  \tag{3}\]  
where \(S_{\text{r}}\) represents the saturation of the unsaturated porous medium, \(S_{\text{rW}}\) represents the irreducible degree of saturation, and \(S_{\text{rN}}\) represents the air-entry degree of saturation. Based on the poroelastic model proposed by [PERSON] and [PERSON] [18], the stress tensor in an unsaturated porous medium is expressed by:  
\[\left. \begin{array}{l}\sigma^{S}=\big{(}M_{SS}\
abla\cdot\mathbf{u}^{S}+M_{SW} \
abla\cdot\mathbf{u}^{\text{W}}+M_{\text{SN}}\
abla\cdot\mathbf{u}^{\text{N}}\big{)} I+2n_{0}^{\text{S}}\mu_{\text{S}}\Big{[}\
abla\mathbf{u}^{\text{S}}+\big{(} \
abla\mathbf{u}^{\text{S}}\big{)}^{T}\Big{]}\\ \sigma^{W}=\big{(}M_{SW}\
abla\cdot\mathbf{u}^{S}+M_{WW}\
abla\cdot\mathbf{u}^{\text{ W}}+M_{\text{WN}}\
abla\cdot\mathbf{u}^{\text{N}}\big{)}\mathbf{I}\\ \sigma^{N}=\big{(}M_{SN}\
abla\cdot\mathbf{u}^{S}+M_{WN}\
abla\cdot\mathbf{u}^{\text{ W}}+M_{\text{NN}}\
abla\cdot\mathbf{u}^{\text{N}}\big{)}\mathbf{I}\end{array}