Surface-Wave Anelasticity in Porous Media: Effects of Wave-Induced Mesoscopic Flow

Abstract

We study the anelastic properties (attenuation and velocity dispersion) of surface waves at an interface between a finite water layer and a porous medium described by Biot theory including the frequency-dependent effects due to mesoscopic flow. A closed-form dispersion equation is derived, based on potential functions and open and sealed boundary conditions (BC) at the interface. The analysis indicates the existence of high-order surface modes for both BCs and a slow true surface mode only for sealed BC. The formulation reduces to two particular cases in the absence of water and with infinite-thickness water layer, with the presence of pseudo-versions of Rayleigh and Stoneley waves. The mesoscopic flow affects the propagation of all the pseudo-surface waves, causing significant velocity dispersion and attenuation, whereas the effect of the BC is mainly evident at high frequencies, due to the presence of the slow Biot wave. The mesoscopic-flow peak moves to low frequencies as the thickness of the water layer increases. In all cases, the true surface wave resembles the slow P2 wave, and is hardly affected by the flow.

Appendices

Appendix A: Complex Coefficients of the Effective Biot Medium

From Pride et al. (2004) and Liu et al. (2009), the complex coefficients are

$$\begin{aligned} a_{11}^{*}= & {} a_{11}-\dfrac{{\textrm{i}}\omega a_{13}^2}{{\textrm{i}}\omega a_{33}-\gamma (\omega )}, \ \ \ \ \ a_{12}^{*}=a_{12}-a_{13}\dfrac{{\textrm{i}}\omega a_{23}+\gamma (\omega )}{{\textrm{i}}\omega a_{33}-\gamma (\omega )}, \nonumber \\ a_{22}^{*}= & {} a_{22}+a_{23}-(a_{23}+a_{33})\dfrac{{\textrm{i}}\omega a_{23}+\gamma (\omega )}{{\textrm{i}}\omega a_{33}-\gamma (\omega )}, \end{aligned}$$

(A1)

where $a_{ij}$ are the real double-porosity constants corresponding to the high-frequency response for which no internal fluid pressure relaxation can take place. These real coefficients are (Pride et al. 2004)

$$\begin{aligned} a_{11}= & {} 1/K^d, \ \ \ \ \ \ \ \ \ a_{22}=\dfrac{v_1\alpha _1}{K_1^{d}}\left( \dfrac{1}{B_1}-\dfrac{\alpha _1(1-Q_1)}{1-K_1^d/K_2^d}\right) ,\nonumber \\ a_{12}= & {} -\dfrac{v_1Q_1}{K_1^d}\alpha _1, \ \ \ a_{23}=-\dfrac{\alpha _1 \alpha _2 K_1^d/K_2^d}{(1-K_1^d/K_2^d)^2}\left( \dfrac{1}{K^d}-\dfrac{v_1}{K_1^d}-\dfrac{v_2}{K_2^d}\right) ,\nonumber \\ a_{13}= & {} -\dfrac{v_2Q_2}{K_2^d}\alpha _2, \ \ \ a_{33}=\dfrac{v_2\alpha _2}{K_2^{d}}\left( \dfrac{1}{B_2}-\dfrac{\alpha _2(1-Q_2)}{1-K_2^d/K_1^d}\right) , \end{aligned}$$

(A2)

where subscript $i=$ 1 or 2 denotes the phase 1 or 2, respectively, $v_i$ is the volume fraction, $K_i^d$ is the bulk modulus of the dry-rock frame, $K_\textrm{d}$ is the dry-rock bulk modulus of the composite,

$$\begin{aligned} Q_1=\dfrac{1}{v_1}\dfrac{1-K_2^d/K_\textrm{d}}{1-K_2^d/K_1^d}, \ \ \ \ \, Q_2=\dfrac{1}{v_2}\dfrac{1-K_1^d/K_\textrm{d}}{1-K_1^d/K_2^d}, \end{aligned}$$

(A3)

$B_i$ is the Skempton coefficient,

$$\begin{aligned} B_i=\dfrac{K_\textrm{s}-K_i^d}{K_\textrm{s}-K_i^d+\phi _i K_i^d(K_\textrm{s}/K_\textrm{f}-1)}, \end{aligned}$$

(A4)

where $K_\textrm{s}$ and $K_\textrm{f}$ are the bulk moduli of the grains and fluid, respectively, and $\phi _i$ is the porosity. Moreover, $\alpha _i$ is the Biot-Willis coefficient of phase i, given by

$$\begin{aligned} \alpha _i=(1-K_i^d/K_i^u)/B_i, \end{aligned}$$

(A5)

where $K_i^u$ is the Gassmann wet-rock bulk modulus (confining pressure change divided by dilatation for a sealed sample), and is given by

$$\begin{aligned} K_i^u=\dfrac{K_i^d}{1-B_i(1-K_i^d/K_\textrm{s})}. \end{aligned}$$

(A6)

Substituting equation (A6) into (A5), we obtain

$$\begin{aligned} \alpha _i=1-\dfrac{K_i^d}{K_\textrm{s}}. \end{aligned}$$

(A7)

The frequency-dependent internal transport coefficient $\gamma (\omega )$ is derived by Pride et al. (2004) as

$$\begin{aligned} \gamma (\omega )=\gamma _\textrm{m}\sqrt{1-{\textrm{i}}\dfrac{\omega }{\omega _\textrm{m}}}, \end{aligned}$$

(A8)

where $\gamma _\textrm{m}$ and $\omega _\textrm{m}$ are parameters dependent on the constituent properties and the mesoscopic geometry. When the embedded phase 2 is very permeable, $\gamma _\textrm{m}$ can be expressed by

$$\begin{aligned} \gamma _\textrm{m}=-\dfrac{\kappa _{1}K_1^d}{\eta L_1^2}\left[ \dfrac{a_{12}+B_0(a_{22}+a_{33})}{R_1-B_0/B_1}\right] \left[ (1+O(\kappa _1/\kappa _2)\right] , \end{aligned}$$

(A9)

where $\eta$ is the fluid viscosity, $\kappa _i$ is the permeability of phase i, $B_0$ is the static Skempton coefficient of the composite, given by

$$\begin{aligned} B_0=-\dfrac{a_{12}+a_{13}}{a_{22}+2a_{23}+a_{33}}, \end{aligned}$$

(A10)

$R_1$ is the ratio of the average confining pressure in phase 1 to the pressure applied to the external surface of the double-porosity composite, given by

$$\begin{aligned} R_1=Q_1+\dfrac{\alpha _1(1-Q_1)B_0}{1-K_1^d/K_2^d}-\dfrac{v_2}{v_1}\dfrac{\alpha _2(1-Q_2)B_0}{1-K_2^d/K_1^d}, \end{aligned}$$

(A11)

and $L_1$ represents the characteristic length of the fluid pressure gradient.

In terms of $\gamma _\textrm{m}$, $\omega _\textrm{m}$ is

$$\begin{aligned} \omega _\textrm{m}=\dfrac{\eta B_1 K_1^d}{\kappa _1\alpha _1}\left( \gamma _\textrm{m} \dfrac{V}{S}\right) ^2\left( 1+\sqrt{\dfrac{\kappa _1B_2K_2^d\alpha _1}{\kappa _2B_1K_1^d\alpha _2}} \right) ^2, \end{aligned}$$

(A12)

with V/S representing the volume-to-surface ratio, where S is the surface area of the interface between the two phases in each volume V of composite.

In a concentric sphere geometry (a composite (phase 1) of radius R contains a small sphere of radius a of phase 2), satisfying $\kappa _1/\kappa _2 \ll 1$,

$$\begin{aligned} L_1^2=\dfrac{9}{14}R^2\left[ 1-\dfrac{7}{6}\dfrac{a}{R}+O(a^3/R^3) \right] ,\ \ \ \ \ \dfrac{V}{S}=\dfrac{R^3}{3a^2}. \end{aligned}$$

(A13)

In this case, the volume fractions $v_1$ and $v_2$ are defined as

$$\begin{aligned} v_2=\left( \dfrac{a}{R}\right) ^3, \ \ \ \ v_1=1-v_2, \end{aligned}$$

(A14)

and the total porosity is

$$\begin{aligned} \phi =v_1\phi _1+v_2\phi _2. \end{aligned}$$

(A15)

Appendix B: Components of Matrix ${\textbf{M}}$

The elements of matrix ${\textbf{M}}$ are

$$\begin{aligned} M_{00}= & {} 0, \ \ M_{01}=2\xi _1, \ \ M_{02}=2\xi _2, \ \ M_{03}=-{\textrm{i}}(1+\xi _3^2), \end{aligned}$$

(B1)

$$\begin{aligned} M_{10}= & {} -(1-\textrm{exp}[{-2kH\xi _0}])\rho _0,\nonumber \\ M_{11}= & {} (A+Q+\nu _1(R+Q)+2N)(1/V_1^2)-2N/c^2,\nonumber \\ M_{12}= & {} (A+Q+\nu _2(R+Q)+2N)(1/V_2^2)-2N/c^2,\nonumber \\ M_{13}= & {} 2{\textrm{i}}N \xi _3 /c^2, \end{aligned}$$

(B2)

$$\begin{aligned} M_{20}= & {} \xi _0(1+\textrm{exp}[{-2kH\xi _0}]),\nonumber \\ M_{21}= & {} \xi _1(1-\phi +\phi \nu _1),\nonumber \\ M_{22}= & {} \xi _2(1-\phi +\phi \nu _2),\nonumber \\ M_{23}= & {} -{\textrm{i}}(1-\phi +\phi \nu _3),\ \end{aligned}$$

(B3)

$$\begin{aligned} M_{30}= & {} -(1-\textrm{exp}[{-2kH\xi _0}])\rho _0,\nonumber \\ M_{31}= & {} (Q+R\nu _1)/\phi /V_1^2+{\textrm{i}}(\nu _1-1)\dfrac{\xi _1}{c}Z_I\phi ,\nonumber \\ M_{32}= & {} (Q+R\nu _2)/\phi /V_2^2+{\textrm{i}}(\nu _2-1)\dfrac{\xi _2}{c}Z_I\phi ,\nonumber \\ M_{33}= & {} (\nu _3-1) \dfrac{1}{c} Z_I\phi . \end{aligned}$$

(B4)

Next, we compare the above equations with those of Feng and Johnson (1983a). By letting $H =+\infty$ and $P=A+2N$, and defining

$$\begin{aligned} D_0= & {} C_0, \ \ D_1=C_1, \ \ D_2=C_2, \ \ D_3=iC_3,\nonumber \\ \nu _1= & {} -G_{+}, \ \ \nu _2=-G_{-}, \ \ \nu _3=\dfrac{{{\tilde{\tau }}}-1}{{{\tilde{\tau }}}}, \ \ Z_I=T, \end{aligned}$$

(B5)

in the same manner as Feng and Johnson (1983a), the first equation in (30) becomes

$$\begin{aligned} 2\xi _1C_1+2\xi _2C_2+(1+\xi _3^2)C_3=0. \end{aligned}$$

(B6)

By substituting $\xi _j$ in equation (25) into (B6), we derive equation (C2) of Feng and Johnson (1983a).

The second equation in (30) becomes

$$\begin{aligned}{} & {} \rho _0 c^2C_0+ \bigg (\dfrac{\big [G_{+}(R+Q)-(P+Q)\big ]c^2}{V_1^2}+2N\bigg )C_1\nonumber \\{} & {} \quad +\bigg (\dfrac{\big [G_{-}(R+Q)-(P+Q)\big ]c^2}{V_2^2}+2N\bigg )C_2+2N\xi _3C_3=0, \end{aligned}$$

(B7)

which is Eq. (C1) of Feng and Johnson (1983a).

The third equation in (30) becomes

$$\begin{aligned} \xi _0C_0+\xi _1(1-\phi -\phi G_{+})C_1+\xi _2(1-\phi -\phi G_{-})C_2+\left( 1-\dfrac{\phi }{{{\tilde{\tau }}}}\right) C_3=0, \end{aligned}$$

(B8)

which is equation (C3) of Feng and Johnson (1983a), where they instead use $\alpha =\tau$.

The last equation in (30) becomes

$$\begin{aligned}{} & {} -\rho _0C_0+\left[ \dfrac{Q-RG_{+}}{\phi V_1^2}-{\textrm{i}}\dfrac{(G_{+}+1)\xi _1T\phi }{c}\right] C_1 \nonumber \\{} & {} \qquad +\left[ \dfrac{Q-RG_{-}}{\phi V_2^2}-{\textrm{i}}\dfrac{(G_{-}+1)\xi _2T\phi }{c}\right] C_2-{\textrm{i}}\dfrac{T\phi }{c{{\tilde{\tau }}}}C_3=0. \end{aligned}$$

(B9)

By multiplying $-\phi c^2$ on both sides, we derive equation (C4) of Feng and Johnson (1983a).

It is evident that the equations of Feng and Johnson (1983a) are special cases of our equations when $H=+\infty$. Moreover, our equations use frequency-dependent elastic coefficients associated with the mesoscopic fluid flow, and hence are more realistic for low-frequency surface-wave propagation.