Figure 1

Length scales in Eq. (2). The periodic medium is composed of identical unit cells $\Omega$. Internal free boundaries are denoted by $\Gamma$.Reuse & Permissions

Figure 2

The analytical solution for the dynamic permeability given by [2] for a porous medium of porosity $ϵ$ made of parallel-plane channels of aperture $2h$. $K^{\prime }$ and ${\omega }^{\prime }$ are the dimensionless permeability and frequency, $K^{\prime }=3K/\left(ϵh^2\right)$, and ${\omega }^{\prime }=h^2\omega /\left(3{\nu }_f\right)$. Solid line: $\text{Re}\left(K^{\prime }\right)$. Dashed line: $-\text{Im}\left(K^{\prime }\right)$.Reuse & Permissions

Figure 3

(Color online) The structured mesh ${\text{SCT}}_6$. The domain is represented by a regular array of cubic elements. Each of them is decomposed into six identical tetrahedra; one of them is shown in (a) and three of them in (b) where one of the symmetry planes is visible.Reuse & Permissions

Figure 4

(Color online) Model and real porous media discretized into ${128}^3$ elementary cubes. (a) Unimodal medium HP. (b) Bimodal medium with nonpercolating vugs NPV; ${ϵ}_p=0.24$ and ${ϵ}_v=0.08$. (c) Real sandstone of size $L=716.8\mu \text{m}$ composed of five components; gray cubes are filled by quartz, red by K-feldspar, green by biotite, blue by pyrite, and yellow by zircon, respectively. (d) Real vuggy carbonate; gray and red cubes are filled by calcite and pyrite, respectively.Reuse & Permissions

Figure 5

(Color online) The real (a) and imaginary (b) parts of the permeability $K_{xx,c}^{\prime }$ as functions of ${\omega }_c^{\prime }$ for the unimodal media with $ϵ=0.3$ and $l_c/L$ equal to $1/16(●)$, $3/32(◆)$, $1/8(■)$, $3/16(★)$, $1/4(◀)$, $3/8(\ast )$, and $1/2(▼)$. The dashed lines correspond to the permeabilities ${\overline{K}}_{xx,c}^{\prime }$ averaged over three independent realizations for each value of $l_c$; $(\times )$ is the corresponding static permeability.Reuse & Permissions

Figure 6

(Color online) The modulus of the dimensionless permeabilities $\left|K_c^{\prime }\right|$, $\left|K_{\Lambda }^{\prime }\right|$, and $\left|K_s^{\prime }\right|$ as functions of the frequencies (a) ${\omega }_c^{\prime }$, (b) ${\omega }_{\Lambda }^{\prime }$, and (c) ${\omega }_s^{\prime }$. Conventions are the same as in Fig. 5.Reuse & Permissions

Figure 7

(Color online) (a) The absolute permeability $K_{0,\Lambda }^{\prime }$ as a function of porosity $ϵ$ for unimodal media. Data are for $l_c/L=3/32$ $(▲)$, 3/16 $(●)$, and 3/8 $(■)$. The dashed line corresponds to the numerical fit (50). (b),(c),(d) The modulus of the permeability $\left|{\overline{K}}_{xx,c}^{\prime }\right|$ as a function of ${\omega }_c^{\prime }$. (b) Unimodal media. Data are for $ϵ=0.15$ (black dotted line), 0.2 (red dashed line), and 0.3 (blue solid line); $l_c/L=3/32$ $(△)$, 3/16 $(○)$, and 3/8 $(\square )$. (c) Bimodal media; data are for HP $(▲)$, NPV $(●)$, PV $(■)$, and HV $(★)$. (d) Real media: sandstone $(■)$ and carbonate $(▲)$; the solid line corresponds to $\text{Re}\left(K_{zz}\right)/l^2$ and the dashed line to $-\text{Im}\left(K_{zz}\right)/l^2$ with $l=5.6\mu \text{m}$.Reuse & Permissions

Figure 8

(Color online) The dynamic permeabilities (a) $\left|K_{zz,\Lambda }^{\prime }\right|$ and (b) $\left|K_s^{\prime }\right|$ as functions of the frequency ${\omega }_{\Lambda }^{\prime }$. Data are for unimodal media: $l_c=6a$ $(◆)$, $8a$ $(●)$, $12a$ $(▶)$, $16a$ $(■)$, and $24a$ $(★)$ with $ϵ=0.15$ (black), $ϵ=0.2$ (red), and $ϵ=0.3$ (blue); bimodal media: HP (blue $△$), HV (brown $◁$), NPV (red $○$), and PV (green $\square$); real media: vuggy carbonate $(+)$ and sandstone $(\times )$.Reuse & Permissions

Figure 9

(Color online) ${\beta }^{\prime }=-\beta E_s$ for media containing spherical cavities $(▽)$ or solid spheres $(\ast )$, for the reconstructed porous media and for the real media; the dash-dotted red lines are the analytical solutions (55) for spherical cavities $\left(ϵ⪡1\right)$ and (54) for solid spheres $\left(ϵ>1-\pi /6\right)$. The Poisson ratio is ${\nu }_s=0.275$ in all cases, except for the real media. Conventions are the same as in Fig. 8.Reuse & Permissions

Figure 10

(Color online) The compressional (a) and shear (b) wave celerities $c^{\prime }$ as functions of the frequency ${\omega }_{\Lambda }^{\prime }$ for unimodal media. Data are for $ϵ=0.15$ (black dotted line), 0.2 (red dashed line), and 0.3 (blue solid line); $l_c/L=3/32$ $(▲)$, 3/16 $(●)$, and 3/8 $(■)$.Reuse & Permissions

Figure 11

(Color online) Unimodal media. The penetration depths $h/\Lambda$ for the (a) compressional and (b) shear waves as functions of the frequency ${\omega }_{\Lambda }^{\prime }$. (c) The slow wave celerity $c_{\parallel }^{\prime }$ as a function of the frequency ${\omega }_{\Lambda }^{\prime }$. (d) The penetration depth $h/\Lambda$ for the slow wave as a function of ${\omega }_{\Lambda }^{\prime }$. Data are for $ϵ=0.15$ (black dotted line), 0.2 (red dashed line), and 0.3 (blue solid line); $l_c/L=3/32$ $(▲)$, 3/16 $(●)$, and 3/8 $(■)$.Reuse & Permissions

Figure 12

(Color online) Bimodal media. (a) Compressional and (b) shear wave celerities $c^{\prime }$ as functions of the frequency ${\omega }_{\Lambda }^{\prime }$. The penetration depths $h/\Lambda$ for the fast compressional (c) and shear (d) waves as functions of the frequency ${\omega }_{\Lambda }^{\prime }$. Data are for unimodal media HP $(△)$ and HV $(◁)$ and bimodal media NPV $(○)$ and PV $(\square )$.Reuse & Permissions

Figure 13

Real porous media: $L/a=128$. (a) The compressional (solid line) and the shear (dashed line) wave celerities $c^{\prime }$ as functions of the frequency ${\omega }_{\Lambda }^{\prime }$ for real sandstone $(■)$ and real carbonate $(▲)$. (b) The compressional (solid line) and shear (dashed line) wave penetration depths $h/\Lambda$ as functions of the frequency ${\omega }_{\Lambda }^{\prime }$ for the real sandstone $(■)$ and for the real carbonate $(▲)$. (c) The compressional (solid line) and the shear (dashed line) wave penetration depths compared to the wavelength $h/\lambda$ as functions of the frequency ${\omega }_{\Lambda }^{\prime }$ for the real sandstone $(■)$ and for the real carbonate $(▲)$.Reuse & Permissions

Figure 14

The slow wave celerity $c_{\parallel }^{\prime }$ (a) and its penetration depth compared to the wavelength $h/\lambda$ (b) as functions of the frequency ${\omega }_{\Lambda }^{\prime }$ for real sandstone $(■)$ and real carbonate $(▲)$; $L/a=128$.Reuse & Permissions

Figure 15

(Color online) (a) The slow wave celerity $c_{\parallel }^{\prime }$ and (b) its penetration depth compared to the wavelength $h/\lambda$ as functions of the frequency ${\omega }_{\Lambda }^{\prime }$; $L/a=128$. The dashed lines in (a) and (b) correspond to $0.1\sqrt{{\omega }^{\prime }}$ and to $0.1/\sqrt{{\omega }^{\prime }}$, respectively. Data are for sandstone $(\ast )$, real carbonate $(\star )$, unimodal media HP (blue $△$) and HV (brown $◁$), and bimodal media NPV (red $○$) and PV (green $\square$); unimodal reconstructed porous media with $l_c/a=6$ $(▲)$, $l_c/a=12$ $(●)$, and $l_c/a=24$ $(■)$ and with $ϵ=0.15$ (black), $ϵ=0.2$ (red) and $ϵ=0.3$ (blue).Reuse & Permissions

Figure 16

(Color online) (a) The fast compressional (black) and the shear (red) wave celerity $c^{\prime }$ as functions of the porosity $ϵ$ for the real sandstone $(\times )$ and for the real carbonate $(+)$, for unimodal media HP $(△)$ and HV $(◁)$, bimodal media NPV $(○)$ and PV $(\square )$, and unimodal media with $l_c/a=6$ $(▲)$, $l_c/a=12$ $(●)$, and $l_c/a=24$ $(■)$. The dash-dotted lines correspond to the fits (69). (b) The penetration depth $h/\Lambda$ for the compressional (open symbols) and the shear (solid symbols) waves as functions of ${\omega }_{\Lambda }^{\prime }$ for unimodal media HP (blue $△$) and HV (brown $◁$), and bimodal media NPV (red $○$) and PV (green $\square$); and unimodal reconstructed porous media with $l_c/a=6$ $(◇)$, $l_c/a=12$ $(▷)$, and $l_c/a=24$ $(\star )$ and with porosities $ϵ=0.15$ (black), $ϵ=0.2$ (red), and $ϵ=0.3$ (blue), real sandstone $(\ast )$ and real carbonate $(\times )$ (where green and magenta symbols are for the compressional and shear waves, respectively. The dashed line corresponds to $500/{\omega }^{\prime 2}$.Reuse & Permissions