**The Biot–Rayleigh theory**

The BR dispersion equation [28] yields the complex wave number *k*,

$$\begin{vmatrix} a\_{11}k^2 + b\_{11} & a\_{12}k^2 + b\_{12} & a\_{13}k^2 + b\_{13} \\ a\_{21}k^2 + b\_{21} & a\_{22}k^2 + b\_{22} & a\_{23}k^2 + b\_{23} \\ a\_{31}k^2 + b\_{31} & a\_{32}k^2 + b\_{32} & a\_{33}k^2 + b\_{33} \end{vmatrix} = 0,\tag{A1}$$

where

$$a\_{11} = A + 2N + i(Q\_2\phi\_1 - Q\_1\phi\_2) \mathbf{x}\_1,\\ b\_{11} = -\rho\_{11}\omega^2 + i\omega(b\_1 + b\_2),$$

$$a\_{12} = Q\_1 + i(Q\_2\phi\_1 - Q\_1\phi\_2) \mathbf{x}\_2,\\ b\_{12} = -\rho\_{12}\omega^2 - i\omega b\_{1},$$

$$a\_{13} = Q\_2 + i(Q\_2\phi\_1 - Q\_1\phi\_2) \mathbf{x}\_3,\\ b\_{13} = -\rho\_{13}\omega^2 - i\omega b\_{2},$$

$$a\_{21} = Q\_1 - iR\_1\phi\_2 \mathbf{x}\_1,\\ b\_{21} = -\rho\_{12}\omega^2 - i\omega b\_{1},$$

$$a\_{22} = R\_1 - iR\_1\phi\_2 \mathbf{x}\_2,\\ b\_{22} = -\rho\_{22}\omega^2 + i\omega b\_{1},\\ \tag{A2}$$

$$a\_{23} = -iR\_1\phi\_2 \mathbf{x}\_3,\\ b\_{23} = 0,$$

$$a\_{31} = Q\_2 + iR\_2\phi\_1 \mathbf{x}\_1,\\ b\_{31} = -\rho\_{13}\omega^2 - i\omega b\_{2},$$

$$a\_{32} = iR\_2\phi\_1 \mathbf{x}\_2,\\ b\_{32} = 0,$$

$$a\_{33} = R\_2 + iR\_2\phi\_1 \mathbf{x}\_3,\\ b\_{33} = -\rho\_{33}\omega^2 + i\omega b\_{2}.$$

and

$$\mathbf{x}\_{1} = i(\phi\_{2}Q\_{1} - \phi\_{1}Q\_{2})/Z,\ \mathbf{x}\_{2} = i\phi\_{2}R\_{1}/Z,\ \mathbf{x}\_{3} = -i\phi\_{1}R\_{2}/Z,$$

$$Z = \frac{i\omega\eta\phi\_{1}^{2}\phi\_{2}\phi\_{20}R\_{0}^{2}}{3\kappa\_{10}} - \frac{\rho\_{f}\omega^{2}R\_{0}^{2}\phi\_{1}^{2}\phi\_{2}\phi\_{20}}{3\phi\_{10}} - \left(\phi\_{2}^{2}R\_{1} + \phi\_{1}^{2}R\_{2}\right). \tag{A.3}$$

where *ω* is the angular frequency; *φ*<sup>10</sup> and *φ*<sup>20</sup> are the local porosities of intergranular pores and microcracks, respectively; and *φ*<sup>1</sup> and *φ*<sup>2</sup> are the corresponding absolute porosities. *ρ<sup>f</sup>* , *η*, and *κ*<sup>10</sup> are the fluid density, the fluid viscosity, and rock permeability, respectively; *A*, *N*, *Q*1, *R*1, *Q*2, and *R*<sup>2</sup> are elastic parameters; *R*<sup>0</sup> is the radius of inclusion; *ρ*11, *ρ*12, *ρ*13, *ρ*22, and *ρ*<sup>33</sup> are density parameters; and *b*<sup>1</sup> and *b*<sup>2</sup> are Biot dissipative parameters. The expressions for all these quantities can be found in [28,29].
