Fractal heterogeneities in sonic logs and low-frequency scattering attenuation

Low-frequency waves in 3D isotropic heterogeneous media

A scalar wave $u(\mathbf{x},\omega)$ in a weakly inhomogeneous medium (Karal and Keller, 1964; Chernov, 1960; Tatarski, 1961) satisfies the Helmholtz wave equation

$$\Delta u(\mathbf{x},\omega) + k_0^2\left[1+f(\mathbf{x})\right]^2u(\mathbf{x},\omega) = 0\,,$$ (16)

where $f(\mathbf {x})$ is a small perturbation of the medium from homogeneity, and $k_0=\omega/c_0$. Phase velocity $c_0$ is the background velocity. Assuming a second-order stationary statistical distribution for fluctuations $f(\mathbf {x})$ and a zero expectation value $\langle f \rangle = 0$, spatial covariance of the velocity variations is defined by relation 1. Expectation $\langle u(\mathbf{x},\omega)\rangle$ of random plane-wave realizations is calculated (Karal and Keller, 1964) using a perturbation theory to the second order in $f(\mathbf {x})$ by

$$\left[\Delta+k_0^2(1+\sigma^2)\right]\langle u(\mathbf{x},\omega)... ...mega) \,\langle u(\mathbf{x}^{\prime},\omega)\rangle~d\mathbf{x}^{\prime}= 0\,,$$ (17)

where $G(\mathbf{x},\mathbf{x}^{\prime},\omega)$ is Green's function of the operator $\left[\Delta+k_0^2\right]$, and integration is performed over the 3D space. The dispersion relation for a plane wave propagating in the heterogeneous medium follows as

$$\frac{k^2}{k_0^2} = 1 + \sigma^2\left[1 - 4\,k_0^2 \int N(\mathbf{r})\,G(\mathbf{r},\omega)\,e^{i\,\mathbf{k}\cdot\mathbf{r}}d\mathbf{r}\right],$$ (18)

$$G(\mathbf{r},\omega) = \frac{e^{ik_0r}}{4\pi r}\,,$$ (19)

where $\mathbf{r} = \mathbf{x}^{\prime}-\mathbf{x}$ is the relative position, with an absolute value $r=\vert\mathbf{r}\vert$, and $G(\mathbf{r},\omega)$ is the 3D isotropic free-space Green's function with the outward radiation condition (Bleistein et al., 2001). The path of waves should be sufficiently long to significantly sample medium heterogeneities statistically (Gist, 1994). The Born approximation is present because of Green's function. Heterogeneities with the isotropic correlation function $N(r)$ produce an isotropic wave vector $\mathbf{k}$. Combining Green's function in equation 19 with the isotropic integral in equation 5 reduces the squared dispersion relation of equation 18 to

$$\frac{k^2}{k_0^2} = 1 + \sigma^2\left[1 - \frac{4\,k_0^2}{k}\int_{0}^{\infty}\hspace{-2mm} N(r)\,e^{ik_0r}\sin(kr)\,dr\right].$$ (20)

Second-order expansion $k/k_0=1+O(\sigma^2)$ in the solution constrains validity to the domain $k_0b\ll 1/\sigma$, where $b$ is the characteristic length scale of the heterogeneities. The second-order approximation for the 3D dispersion relation is finally

$$\frac{k}{k_0} = 1 + \frac{\sigma^2}{2} + i\,\sigma^2\,k_0\left[S(0) - S(2k_0)\right].$$ (21)

Quantity $S(k)$, introduced above, is related to the real and even function $E^{(1)}(k)$, defined by the isotropic integral of equation 4:

$$S(k) = \int_{0}^{+\infty}\hspace{-3mm}N(r)e^{ikr}dr\,,$$ (22)

$$E^{(1)}(k)~\sigma^{-2} = \int_{-\infty}^{+\infty}\hspace{-3mm}N(r)e^{ikr}dr = 2~\mbox{Re}[S(k)]\,,$$ (23)

$$2\,i\,\mbox{Im}[S(k)] = S(k) - S(-k).$$ (24)

Connection to the O'Doherty-Anstey formula is detailed in Appendix A.

Fractal heterogeneities in sonic logs and low-frequency scattering attenuation