Fractal heterogeneities in sonic logs and low-frequency scattering attenuation

Attenuation in 3D fractal media

The energy spectrum, $E^{(1)}_{H,b}(k)$, of von Kármán's autocorrelation function $N_{H,b}(r)$ in equation 7 is real and even:

$${E^{(1)}_{H,b}(k)}~{\sigma^{-2}} = \int_{-\infty}^{+\infty}\hspace{-3mm}N_{H,b}(r)\,e^{ikr}dr\,,$$ (25)

$$= C^{(1)}_{H}~\frac{2b}{\left(1+b^2k^2\right)^{H+\frac{1}{2}}}\,.$$ (26)

Values of $S(k)$ defined by equation 22 are

$$S(0) = C^{(1)}_{H}\,b\,,$$ (27)

$$Re[S(2k_0)] = \frac{C^{(1)}_{H}\,b}{(1+4\,b^2k_0^2)^{H+\frac{1}{2}}}\,.$$ (28)

Coefficient $C^{(1)}_{H}$, defined by equation 8, is an increasing function of exponent $H$ and has to be calculated numerically, except for some specific values:

$$\begin{array}{lll} C^{(1)}_{H}\sim{\pi}/{\Gamma(H)}\rightarr... ...5}=1.3317\ldots\,, & C^{(1)}_{1}=\pi/2\,. \nonumber \end{array}$$

The dispersion relation of equation 21 solves for an explicit solution of attenuation and dispersion:

$$Re[k/k_0] = 1+\frac{\sigma^2}{2}\left(1+2k_0~Im[S(2k_0)]\right)\,,$$ (29)

$$Im[k/k_0] = \sigma^2k_0b~C^{(1)}_{H} \left[1-\frac{1}{(1+4\,b^2k_0^2)^{H+\frac{1}{2}}} \right].$$ (30)

When $H=0.5$, the derivation produces simple expressions as detailed in Appendix B. The use of $S^{\ast}(k)=S(-k)$ with the Kramers-Krönig relation can be used to determine the real part of $k$. In the context of the second-order approximation, scattering attenuation in a von Kármán isotropic medium is

$$\frac{1}{Q} = \frac{2\,Im[k]}{Re[k]} = 2~\sigma^2\,k_0b~C^{(1)}_{H} \left[1-\frac{1}{(1+4\,b^2k_0^2)^{H+\frac{1}{2}}}\right].$$ (31)

For $k_0b\,\ll\,1$, the scattering attenuation reduces to the Rayleigh diffusion regime:

$$\frac{1}{Q} \simeq 8\,\sigma^2\,C^{(1)}_{H}\left(H+\frac{1}{2}\right)\left(k_0b\right)^3.$$ (32)

Fractal heterogeneities in sonic logs and low-frequency scattering attenuation