Stress-Associated Intrinsic and Scattering Attenuation from Laboratory Ultrasonic Measurements on Shales

Abstract

Seismic attenuation is sensitive to stress-induced subtle changes in the physical state of rocks. In this study, the stress- and frequency-associated attenuation is quantified through ultrasonic measurements on three differently oriented cylindrical shale samples under various axial stresses. As an improvement to the single-scattering model, the elastic Monte Carlo method is employed to investigate multiple-scattering attenuations by incorporating the boundary reflections and wave conversions. Our results show that, as the axial stress increases, the intrinsic attenuation decreases in all directions, while the scattering attenuation decreases slightly in the direction perpendicular to the bedding but increases largely and nonlinearly in other directions. These discrepancies result from different attenuation mechanisms. Both the intrinsic and scattering attenuation are found to be largest in the direction 45° to the bedding, but least in the perpendicular direction. The S-wave attenuation is larger and more sensitive to stress changes than P-wave attenuation due to its shorter wavelength. As expected from sandstone examples, the scattering attenuation in shales is significantly larger and more sensitive to stress changes than the intrinsic attenuation. The frequency dependence of scattering attenuation suggests that the peak frequency with the maximum scattering attenuation is independent of axial stresses, but varies in different directions of an individual rock with different heterogeneity and anisotropy scales. The peak frequency of S-coda is smaller and its peak scattering attenuation is larger than P-coda. In conclusion, the stress and frequency dependence of ultrasonic attenuations in shales differ largely in various directions, indicating significant anisotropy and heterogeneity.

Appendices

Appendix: Tests on Computational Accuracy and Stability

Since there is no analytic solution or Green function of the multiple isotropic scattering models for inhomogeneous 3-D medium, we use two analytic solutions for a 3-D infinite homogeneous model to investigate the accuracy of the Monte Carlo simulation. The first one is an approximate solution of the radiative transfer equation for a 3-D isotropic scattering medium (Paasschens 1997):

$$E\left( {r,t} \right) \approx \frac{{We^{{ - g_{0} vt}} }}{{4\pi vr^{2} }}\delta \left( {t - \frac{r}{v}} \right) + W\frac{{\left( {1 - r^{2} /\left( {vt} \right)^{2} } \right)^{1/8} }}{{4\pi vt/\left( {3g_{0} } \right)^{3/2} }}e^{{ - g_{0} vt}} M\left( {g_{0} vt\left( {1 - \frac{{r^{2} }}{{\left( {vt} \right)^{2} }}} \right)^{3/4} } \right)H\left( {t - \frac{r}{v}} \right),$$

(20)

where W is the source energy density, g ₀ is the scattering coefficient, v is the background velocity of the medium, t is the lapse time, r is the distance between the source and the receiver, δ is the Dirac function, H is the step function and

$$M\left( x \right) \equiv 8\left( {3x} \right)^{ - 3/2} \sum\limits_{n = 1}^{\infty } {\frac{{\varGamma \left( {\frac{3}{4}n + \frac{3}{2}} \right)}}{{\varGamma \left( {\frac{3}{4}n} \right)}}\frac{{x^{n} }}{n!}} \approx e^{x} \sqrt {1 + 2.026/x} , {\text{ where }}\varGamma \left( z \right) = \int_{0}^{\infty } {x^{z - 1} e^{ - x} {\text{d}}x} .$$

(21)

Here, Γ is the Gamma function. Except for the direct wave, the relative error of this approximation is in the order of 2–3% (Ugalde and Carcolé 2009). The second one is a complete integral solution to multiple isotropic scattered waves in a uniform random medium for an impulsive wave energy source (Zeng et al. 1991):

$$E\left( {r,t} \right) = \frac{{\delta \left( {t - \frac{r}{v}} \right)e^{{ - \left( {\eta_{i} + \eta_{s} } \right)vt}} }}{{4\pi vr^{2} }} + \sum\limits_{n = 1}^{2} {E_{n} \left( {r,t} \right) + \int_{ - \infty }^{ + \infty } {\frac{{e^{i\varOmega t} }}{2\pi }d\varOmega \cdot \int_{0}^{\infty } {\frac{{\left( {\frac{{\eta_{s} }}{k}} \right)^{3} \left[ {\tan^{ - 1} \left( {\frac{k}{{\left( {\eta_{i} + \eta_{s} } \right) + i\varOmega /v}}} \right)} \right]^{4} \sin \left( {kr} \right)}}{{2\pi^{2} vr\left[ {1 - \frac{{\eta_{s} }}{k}\tan^{ - 1} \left( {\frac{k}{{\left( {\eta_{i} + \eta_{s} } \right) + i\varOmega /v}}} \right)} \right]}}{\text{d}}k} } } ,$$

(22)

where E (r, t) represents the total wave energy at distance r at the lapse time t. η _i and η _s are the intrinsic and scattering coefficients, respectively. In Eq. (22), the first term represents the direct-wave energy and the second term indicates the nth-order scattered energy. The first- and second-order scattered wave energy terms are expressed as:

$$E_{1} \left( {r,t} \right) = \frac{{\eta_{s} H\left( {t - \frac{r}{v}} \right)e^{{ - \left( {\eta_{i} + \eta_{s} } \right)vt}} }}{4\pi rvt}\ln \frac{{1 + \frac{r}{vt}}}{{1 - \frac{r}{vt}}},$$

(23)

$$E_{2} \left( {r,t} \right) = \frac{{\eta_{s}^{2} H\left( {t - \frac{r}{v}} \right)e^{{ - \left( {\eta_{i} + \eta_{s} } \right)vt}} }}{16\pi } \cdot \left[ {\frac{{\pi^{2} }}{vt} - \frac{3}{r}\int_{0}^{{\frac{r}{vt}}} {\left( {\ln \frac{1 + \alpha }{1 - \alpha }} \right)^{2} {\text{d}}\alpha } } \right].$$

(24)

The third term in Eq. (22) is a sum of scattered wave energy for the orders higher than two. The integral with respect to k can be calculated using the method of discrete wavenumber sum (Bouchon 1979). It is observed from the integral solution that it is difficult to separate the intrinsic and scattering attenuation mathematically. However, the scattering effect becomes more dominant as the scattering order increases. It means that relatively accurate separation of intrinsic and scattering attenuation can be achieved by phenomenological study of coda waves.

Seismogram envelopes are synthesized for a 3-D infinite scattering medium characterized with the parameters: η _i  = 0.0 km⁻¹, η _s  = g ₀ = 0.01 km⁻¹ and v = 3.15 km/s. In this section, the normalized time τ, the normalized distance ρ and the normalized radius R _n of spherical volume at the receiver are used for comparison (Sato 1995):

$$\tau = g_{0} vt, \, \rho = g_{0} r, \, R_{\text{n}} = g_{0} r.$$

(25)

Figure 19 shows the normalized energy density at different normalized distances ρ = 0.8, 1.6 and 3.2. The average and plus or minus one standard deviation for 20 simulations are marked by the red solid lines and the black dotted lines, respectively. The number of particles used in this section is N = 10⁷. The normalized radius of spherical volume used to estimate the spatial density of energy particles at the receiver is R _n = 0.2. Except for ripples of the envelope at ρ = 3.2, the normalized energy density calculated using the Monte Carlo method is identical to that calculated using the analytic solution of Zeng et al. (1991) and Paasschens (1997) at different normalized distances. Ripples of the envelope in Zeng et al. (1991) are caused by the inaccurate calculation on the third term involving multiple integrals in Eq. (22). The ripples in the Monte Carlo solution scatter around the analytic solution, but do not show any apparent bias within one standard deviation. Increasing the number of particles will diminish the ripples. The broadness of the direct wave in the Monte Carlo solution is due to the use of a spherical volume instead of a point at the receiver.

Fig. 19

Normalized energy density at the normalized distance ρ = 0.8, 1.6 and 3.2. The normalized radius of spherical volume at the receiver is R _n = 0.2. The normalized energy density calculated with the Monte Carlo method is the average of 20 independent numerical simulations. The results (the red solid lines) are compared with the analytic solutions of Paasschens (1997) (the blue solid lines) and Zeng et al. (1991) (the green dashed lines). Standard deviations are indicated with the black dotted lines. The background velocity is V ₀ = 3.15 km/s. The intrinsic and scattering coefficient is η _i  = 0.00 km⁻¹ and η _i  = g ₀ = 0.01 km⁻¹, respectively. The parameters used in the Monte Carlo simulation are Δτ = 0.01 and N = 10⁷

Dependence of the normalized energy density on Δτ is also investigated. Normalized seismogram envelopes with ρ = 1.6 and R _n = 0.05 under numerical conditions Δτ = 0.01, 0.1, 0.2 and 0.4 are shown in Fig. 20. The average (the red solid lines) and standard deviation (the black dotted lines) are calculated by synthesizing 20 envelopes for each case. For comparison, the analytic solutions (the blue solid lines and green dash lines) are also plotted. As can be seen, the shapes of normalized energy density are identical if Δτ ≤ 0.1 which conforms with the decoupling principle expressed as Δτ ≪ 1.0 in the Monte Carlo simulation. As Δτ increases beyond 0.1, the normalized energy density calculated using the Monte Carlo method is slightly larger compared with the analytic solutions. Similar results are observed in Fig. 21 with R _n = 0.10 and Fig. 22 with R _n = 0.20. The mean free distance of the scattering medium is l ₀ = 1/g ₀ and thus the normalized mean free time is given by τ ₀ = g ₀ V ₀ l ₀/V ₀ = 1. It can be inferred that the upper limit of the time sample interval in the calculation for accurate results is about one-tenth of the mean free time of the scattering medium.

Fig. 20

Dependence of the normalized energy density on Δτ = 0.01, 0.1, 0.2 and 0.4 with ρ = 1.6 and R _n = 0.05. The normalized energy density calculated with the Monte Carlo method is the average of 20 independent numerical simulations. The results (the red solid lines) are compared with the analytic solutions of Paasschens (1997) (the blue solid lines) and Zeng et al. (1991) (the green dashed lines). Standard deviations are indicated with black dotted lines. The set of parameters used in the simulation are identical to those in Fig. 19

Fig. 21

Dependence of the normalized energy density on Δτ = 0.01, 0.1, 0.2 and 0.4 with ρ = 1.6 and R _n = 0.10. The normalized energy density calculated with the Monte Carlo method is the average of 20 independent numerical simulations. The results (the red solid lines) are compared with the analytic solutions of Paasschens (1997) (the blue solid lines) and Zeng et al. (1991) (the green dashed lines). Standard deviations are indicated with the black dotted lines. The set of parameters used in the simulation are identical to those in Fig. 19

Fig. 22

Dependence of the normalized energy density on Δτ = 0.01, 0.1, 0.2 and 0.6 with ρ = 1.6 and R _n = 0.20. The normalized energy density calculated with the Monte Carlo method is the average of 20 independent numerical simulations. The results (the red solid lines) are compared with the analytic solutions of Paasschens (1997) (the blue solid lines) and Zeng et al. (1991) (the green dashed lines). Standard deviations are indicated with the black dotted lines. The set of parameters used in the simulation are identical to those in Fig. 19

Figure 23 shows the dependence of the normalized energy density on the radius of spherical volume at the receiver. With the normalized radius R _n increasing from 0.03 to 0.2, envelopes of the energy density for the direct wave begin to broaden and decrease while the arrival time is still identifiable from the energy peak. Also, influence of the size of spherical volume on the coda portion is not too severe for the normalized radius of 0.05, 0.1 and 0.2. Note that numerical instability occurs in the case of R _n = 0.03 and 0.05, especially at large lapse time. The decay rate of coda portion for R _n = 0.05 is still trackable in the analytical solutions. Therefore, the radius of the spherical volume at the receiver should not be too small and its proper lower limit is about one-eighth of the source–receiver distance (normalized to be 1), that is 1/8 = 0.125.

Fig. 23

Dependence of the normalized energy density on the normalized radius R _n = 0.03, 0.05, 0.10 and 0.20 with ρ = 0.8 and Δτ = 0.01. The normalized energy density calculated with the Monte Carlo method is the average of 20 independent numerical simulations. The results (the red solid lines) are compared with the analytic solutions of Paasschens (1997) (the blue solid lines) and Zeng et al. (1991) (the green dashed lines). The set of parameters used in the simulation are identical to those in Fig. 19

Data and resources

The data for this research were from a laboratory ultrasonic measurement of 3-D cylindrical shale samples in the Key Laboratory of the Earth’s Deep Interior, Institute of Geology and Geophysics, Chinese Academy of Sciences, Beijing, China. It is not publicly available yet.