On the Normal-Incidence Reflection Coefficient in Porous Media

Abstract

We compare the exact normal-incidence PP reflection coefficient [Geertsma–Smit expression] to approximations reported by several authors, based on open-pore boundary conditions at a plane interface between two porous media. The approximations correspond to low frequencies. Two of them are derived from the low-frequency Biot theory below the Biot characteristic frequency, but the results show significant differences much below the Biot frequency. Then, we extend the Geertsma–Smit equations by including the high-frequency viscodynamic operator (i.e., the full-frequency range Biot theory), showing that there are additional substantial differences at the high-frequency range. Use of this latter expression is required to honor the physics in the whole frequency range. We further generalize the Geertsma–Smit equations to the case of general boundary conditions other than the open-pore interface. At the seismic band, it is shown that the lossless (elastic) expression based on the Gassmann P-wave impedance is the reflection coefficient to use for practical applications. It is inferred that interpretations based on the frequency dependency of these approximations can be misleading, since this dependency does not provide a suitable description of the physics.

Appendix A: Approximations of the Normal-Incidence Reflection Coefficient

1.1 A.1 Bourbie et al. Equations

The normal-incidence PP reflection coefficient at an interface separating the incidence medium 1 and the transmission medium 2 reported by Bourbie et al. (1987) is

$$\begin{aligned} R = \frac{1-Z}{1+Z} \cdot (1 + \beta ) \exp ({\mathrm{i}}\tan ^{-1} \beta ), \end{aligned}$$

(16)

where

$$\begin{aligned} \beta = \frac{\sqrt{2} Z^2}{1-Z^2} \cdot \frac{(m_1 - m_2)^2 \phi _1 \phi _2 \sqrt{\omega /\omega _1^c}}{g_1 + g_2 Z \sqrt{\omega _2^c/\omega _1^c}} \end{aligned}$$

(17)

$$\begin{aligned} m_j = \frac{\alpha _j M_j}{E_{Gj}}, \quad g_j = \sqrt{\frac{(1- \alpha _j m_j) m_j \gamma _j}{\alpha _j}}, \quad \omega _j^c = \frac{\phi _j \eta _j}{\kappa _j \rho _{fj}} , \quad \gamma _j = \frac{\phi _j \rho _{fj}}{\rho _j} , \end{aligned}$$

(18)

$$\begin{aligned} Z = \sqrt{\frac{\rho _2 E_{G2}}{\rho _1 E_{G1}}}, \quad E_{Gj} = E_{mj} + \alpha _j^2 M_j , \quad E_{mj} = K_{mj} + 4 \mu _j/3, \end{aligned}$$

(19)

$$\begin{aligned} M_j^{-1} = (\alpha _j - \phi _j)/K_{sj}+ \phi _j/K_{fj}, \quad \alpha _j = 1 - K_{mj}/{K_{sj}}, \end{aligned}$$

(20)

$$\begin{aligned} \rho _j = (1 - \phi _j ) \rho _{sj} + \phi _j \rho _{fj}, \end{aligned}$$

(21)

where $\omega$ is the angular frequency, $K_m$ and $\mu$ are the dry-rock bulk and shear moduli, respectively, $\rho _s$ and $\rho$ are the solid and bulk densities, respectively, $\kappa$ is permeability, and ${\mathrm{i}}= \sqrt{-1}$.

1.2 A.2 Gurevich et al. Equations

Gurevich et al. (2004) approximations of the wavenumbers of the fast and slow waves are (their Eqs. 10 and 11),

$$\begin{aligned} l_1 \approx \omega \sqrt{\frac{\rho }{E_G}} , \quad l_2 \approx \sqrt{\frac{{\mathrm{i}}\omega \eta E_G}{\kappa M E_m}} , \end{aligned}$$

(22)

to be compared with the exact ones, obtained from the dispersion Eq. (5).

Gurevich et al. (2004, Eq. 22) PP reflection coefficient is

$$\begin{aligned} R = \frac{1- (1-X)Z}{1+(1+X)Z} , \end{aligned}$$

(23)

where

$$\begin{aligned} X = \frac{ \sqrt{- {\mathrm{i}}\omega \rho _1 E_{G1}} \cdot (m_1 - m_2)^2}{P_1 + P_2} , \end{aligned}$$

(24)

$$\begin{aligned} P_j = \sqrt{\frac{\eta _j M_j E_{mj}}{\kappa _j E_{Gj}}}. \end{aligned}$$

(25)

1.3 A.3 Silin and Goloshubin expression

The normal-incidence PP reflection coefficients at an interface separating the incidence medium 1 and the transmission medium 2 reported by Silin and Goloshubin (2010, their Eq. 88; see Eqs. 9, 15, 24, 35, 64, 75, 78, 84-87) is

$$\begin{aligned} R_k = r_0 + r_1 \sqrt{\epsilon _k}, \quad \epsilon = \frac{{\mathrm{i}}\omega \kappa \rho _f}{\eta }, \end{aligned}$$

(26)

where

$$\begin{aligned} r_0 = \frac{I_1 - I_2}{I_1 + I_2} , \quad r_1 = \frac{t - r}{I_1 + I_2} \cdot I_2 , \end{aligned}$$

(27)

$$\begin{aligned} I_j = \frac{E_{mj}}{v_{mj}}\sqrt{ \frac{u_j}{\xi _j}} = \sqrt{\rho _j E_{mj}} \sqrt{ \frac{u_j}{\xi _j}}, \quad u_j = \zeta _j^2 + \xi _j, \end{aligned}$$

(28)

$$\begin{aligned} \xi _j = \left[ \frac{\phi _j}{K_{fj}} + \frac{(K_{sj} - K_{mj}) (1 - \phi _j)}{K_{sj}^2} \right] K_{mj} , \quad \zeta _j= 1 - \frac{K_{mj} (1-\phi _j)^2}{K_{sj}}, \end{aligned}$$

(29)

$$\begin{aligned} r = \frac{A u_2}{D \zeta _2} , \quad t = \frac{A u_1}{D \zeta _1} , \end{aligned}$$

(30)

$$\begin{aligned} D = \frac{1}{\zeta _1 \zeta _2} \cdot \left( \frac{E_{m2} u_1 \sqrt{u_2}}{\sqrt{\bar{\epsilon }} v_{f2}} + \frac{E_{m1} u_2 \sqrt{u_1}}{v_{f1}} \right) , \end{aligned}$$

(31)

$$\begin{aligned} A = \frac{2 I_1 I_2}{I_1+I_2} \left( \frac{\zeta _1}{u_1} - \frac{\zeta _2}{u_2} \right) , \end{aligned}$$

(32)

$$\begin{aligned} v_{mj} = \sqrt{\frac{E_{mj}}{\rho _j}} , \quad v_{fj} = \sqrt{\frac{E_{mj}}{\rho _{fj}}} , \quad \bar{\epsilon }= \frac{\epsilon _2}{\epsilon _1} . \end{aligned}$$

(33)

Note that $I_j$ [Eq. (28)] are not Gassmann impedances $\left( \sqrt{\rho _j E_{mj}}\right)$ as in the other approximations. Since subindex k is undetermined in Silin and Goloshubin (2010) [see Eq. (26)], we consider two reflections coefficients, corresponding to $\epsilon$ of medium 1 ($k = 1$) and $\epsilon$ of medium 2 ($k = 2$).