Frequency-dependent elastic wave propagation through anisotropic media induced by saturated cracks

Abstract

Usually, there exist both pores and oriented cracks in real rocks, and the existence of oriented cracks leads to the anisotropy, alters the rock stiffness property, and affects the elastic wave velocities. Besides, the local flow (squirt flow) between pores and cracks is also considered an important factor influencing the elastic wave velocities. In this paper, the frequency-dependent model is derived based on the squirt flow model to describe the elastic wave characteristics in anisotropic media induced by cracks. Through the study of the elastic wave characteristics at different crack parameters, incidence angles, and frequencies, it is found that the increase of crack density decreases the rock stiffness and enhances the effect of squirt flow resulting in the decrease of wave velocities and the increase of the degree of anisotropy, and the change of aspect ratio only influences the dispersion range. Moreover, P-wave velocity is found to be greatly affected by the squirt flow in a direction perpendicular to crack plane, and as the polarization direction is always parallel to crack plane, the fast shear wave is not influenced by squirt flow. Besides, the effect of squirt flow decreases at high frequency, which leads to the increase of the rock stiffness and wave velocities. By applying the frequency-dependent model to a real case, the experimental data can be well-interpreted.

Appendix

Frequency-dependent crack strain

According to the fluid substitution equation in anisotropic media given by Brown and Korringa (1975), the relationship between the fluid pressure δP_f and the external stress δP is given as:

$$ \frac{\delta {P}_f}{\delta P}=\frac{S_{ij kl}^{dry}-{S}_{ij kl}^{sat}}{S_{ij\beta \beta}^{dry}-{S}_{ij\beta \beta}^m}=\frac{S_{\alpha \alpha \beta \beta}^{dry}-{S}_{\alpha \alpha \beta \beta}^{sat}}{S_{\alpha \alpha \beta \beta}^{dry}-{S}_{\alpha \alpha \beta \beta}^m} $$

(25)

The bulk modulus of anisotropic rock is defined as follows:

$$ {\displaystyle \begin{array}{l}{S}_{\alpha \alpha \beta \beta}^{sat}=1/{K}_0,\\ {}{S}_{\alpha \alpha \beta \beta}^{dry}=1/{K}_d,\\ {}{S}_{\alpha \alpha \beta \beta}^m=1/{K}_s,\end{array}} $$

(26)

where ${S}_{\alpha \alpha \beta \beta}^{sat}$, ${S}_{\alpha \alpha \beta \beta}^{dry}$, and ${S}_{\alpha \alpha \beta \beta}^m$ are the compliances of saturated anisotropic media, dry anisotropic media, and rock grain and K₀, K_d, and K_s are the stiffnesses of saturated anisotropic media, dry anisotropic media, and rock grain.

Therefore, the relationship between the fluid pressure δP_f and the external stress δP in anisotropic rock can be determined by Eqs. (25) and (26):

$$ \delta {P}_f=\frac{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{${K}_d$}\right.-\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{${K}_0$}\right.}{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{${K}_d$}\right.-\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{${K}_s$}\right.}\delta P $$

(27)

Due to small squirt flow quantity, it is not enough to change the pore pressure, thereby the fluid pressure δP_f and the pore pressure δP_p are approximately equal.

The fluid flow, induced by the small amplitude of the elastic waves, satisfies the mass conservation law (Tang and Chen 1989):

$$ iw{\int}_{Ac}W\cdot ndA+{q}^{\hbox{'}}=\frac{iw}{K_f}{V}_c\delta {P}_c $$

(28)

where A_c is the crack surface, V_c is the crack volume, δP_c is the crack pressure, and q^′ is the fluid flow rate in the connections of pores and cracks.

The squirt flow quantity q =  ∫ q^′dt, q^′is the fluid flow rate, in frequency domain, q = q^′/(−iw), thus, the crack strain can be obtained by Eq. (28):

$$ \left\langle {\varepsilon}_c\right\rangle =\frac{\int_{Ac}W\cdot ndA}{V_c}=\frac{q}{V_c}+\frac{\delta {P}_c}{K_f} $$

(29)

According to Tang’s model (2011), we can establish the relationship between the squirt flow quantity q and the crack pressure δP_c:

$$ q=\frac{4\upvarepsilon \left(1-v\right)V}{\mu_0}\left(\delta {P}_c-\delta P\right)f\left(\varsigma \right) $$

(30)

The squirt flow is defined as the following form:

$$ {S}_W=\frac{q}{\delta {P}_fV} $$

(31)

By combining Eqs. (27), (30), and (31), we can obtain the relationship between the crack pressure and squirt flow:

$$ \delta {P}_c=\left(\frac{\mu_0}{4\varepsilon \left(1-v\right)}\frac{S_w}{f\left(\varsigma \right)}+\frac{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{${K}_d$}\right.-\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{${K}_s$}\right.}{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{${K}_d$}\right.-\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{${K}_0$}\right.}\right){\delta P}_f $$

(32)

The frequency-dependent crack volume strain can be determined by using Eqs. (29), (31), and (32):

$$ \left\langle {\varepsilon}_c\right\rangle =\frac{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{${K}_d$}\right.-\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{${K}_0$}\right.}{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{${K}_d$}\right.-\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{${K}_s$}\right.}\left(\frac{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{${K}_d$}\right.-\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{${K}_s$}\right.}{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{${K}_d$}\right.-\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{${K}_0$}\right.}\frac{1}{K_f}+\left(1+\frac{3\gamma {\mu}_0}{2\left(1-v\right){K}_ff\left(\varsigma \right)}\right)\frac{S_W}{\phi_c}\right)\delta P $$

(33)