Fractal heterogeneities in sonic logs and low-frequency scattering attenuation

Nonlinear parameter estimation on sonic well logs

We propose to use the synthesis of a random medium detailed in Table 2 for $s$=1 as a basis for the procedure to estimate heterogeneity parameters from sonic logs. We achieve optimization by using a weighted least-squares method in the spectral domain on the logarithm of the amplitude, with the model derived from equation 7:

$$\ln\vert F(k)\vert = \ln\vert F(0)\vert - p\ln\left[1+(kb)^2\right],$$ (13)

$$p = \frac{H}{2}+\frac{1}{4}\ \ \ \mbox{and}$$ (14)

$$\vert F(0)\vert = \sigma~\sqrt{2b~C^{(1)}_{H}}\,.$$ (15)

We estimate the three parameters, $b$, $H$, and $\sigma$, using a separable least-squares method (Golub and Pereira, 1973) for $\ln\vert F(0)\vert$ and the slope $p$, and a Gauss Newton optimization algorithm on the nonlinear parameter $b^2$. Parameter $\ln\vert F(0)\vert$ is included in the optimization algorithm because it is difficult to estimate directly from the zero-frequency component in the data. Standard deviation $\sigma$, extracted from relation 15, is confirmed by direct evaluation on the spatial signal. When applying the method, we first substract the signal expectation and use it as a scaling factor. We have tested the efficiency of the algorithm on synthetic fGn and fBm generated by the procedure in Table 2 with a discrete length of 4056 points. Three synthetic fractal signals and their parameter estimations are shown in Figure 2. Results of the validation tests are presented in Table 3.

cgaussM025,lligaussfM025,cgauss025,lligaussf025,cgauss05,lligaussf05
Figure 2. Synthetic signals generated as fGn with $H=-0.25$, $b=10$m, $\sigma=20~\%$ in (a); fBm with $H=0.25$, $b=5$m, $\sigma=30~\%$ in (c); and fBm with $H=0.5$, $b=5$m, $\sigma=30~\%$ in (e). Parameter estimations on the logarithm of the spectral amplitude are shown on the right (b,d,f).

signalA1,signalC2,llisignalfA1,llisignalfC2,rllisignalfA1L,rllisignalfC2L
Figure 3. Sonic log $V_P$ (a) from well N$^{\circ }$1 and $V_S$ (b) from well N$^{\circ }$3, scaled by their respective average value $V_0$. Parameter estimation on the logarithm of the spectral amplitude (c,d) shows the existence of different slopes for low, medium, and high frequencies. These tool artefacts are removed by restricting the estimation method to low frequency (e,f).

Parameters	$H$	$b$ (m)	$\sigma$ (%)
Generated fGn	-0.25	10.0	20
Recovered	-0.21	11.0	18
Generated fGn	-0.25	5.0	20
Recovered	-0.21	5.9	17
Generated fBm	0.25	10.0	30
Recovered	0.26	10.2	23
Generated fBm	0.50	5.0	40
Recovered	0.51	5.3	32
Generated fBm	0.75	3.0	40
Recovered	0.79	2.7	30

Table 3. Comparison of the stochastic medium parameters used to generate synthetics fGn and fBm and their recovery by the nonlinear estimation method.

Well log data come from a sandy channel reservoir with a clastic overburden, and the facies evolves from silty sandstone to mudstone, which is characteristic of alluvial deposition. Velocities $V_P$ and $V_S$ were both measured with a spatial sampling of $0.125$ m. Figure 3 shows the parameter estimation for two sonic logs. Comparison with the method applied to the synthetics in Figure 2 uncovers the existence of different slopes for different frequencies in Figures 3(c) and 3(d). We can reasonably delimit three domains, denoted (A) for low frequencies, (B) for medium frequencies, and (C) for very high frequencies. These domains can be identified by parameters $r_{S}$ and $r_{I}$, representing specific values of relative distance $r$, namely

$$\mbox{(A) for}\ \ r\geq r_{S}, \mbox{(B) for}\ \ r_{S}\geq r\geq r_{I}, \mbox{(C) for}\ \ r_{I}\geq r,$$

where 1m $<r_{S}<2$m and $r_{I}<1$m. The sharp break (B) in the medium frequencies followed by a white noise (C) at high frequencies is characteristic of the tool artefact. Data acquisition involves a convolution with a box-car window (Shiomi et al., 1997; Dolan et al., 1998). Application of the estimation method is thus restricted to relative distances $r>r_S$, and results are shown in Figures 3(e) and 3(f).

Results are summarized in Table 4 for the four different sonic logs $V_P$ and $V_S$. The ratio $\langle V_P \rangle / \langle V_S \rangle$ is almost constant for the four well logs and roughly equal to two. The updated estimation in the spatial wavelength domain (A) produces reasonable results in Table 4. Standard deviation $\sigma$ varies from 20 to 45 % and is larger for $V_S$ logs than for $V_P$ logs. Correlation length $b$ is about 5 m for both $V_P$ and $V_S$, except for the well N$^{\circ }$4, which is 2.5 m. Exponent $H$ for $V_P$ varies from 0.1 to 0.4 and for $V_S$ from 0.2 to 0.6. In Figure 4, comparison of the frequency content of one real sonic log with one realization of a synthetic fBm, generated using similar parameters, shows that the sonic-log data contain higher peaks for very large wavelengths. We detected in the different sonic well logs the recurrence of some particular spatial cycles at 2.5 m, 5 m, 10 m, and 20 m.

Well Log	b (m)	$H$	$\sigma$ (%)	$\langle{V}\rangle$ (m/s)
N$^{\circ }$1 $V_P$ full	0.79	1.32	17	2791
$r>1.5$ m	6.70	0.13	22
$V_S$ full	1.05	1.16	32	1218
$r>2.0$ m	5.92	0.21	35
N$^{\circ }$2 $V_P$ full	1.90	0.92	27	2842
$r>1.6$ m	5.34	0.38	29
$V_S$ full	2.84	0.83	45	1240
$r>1.5$ m	3.08	0.62	44
N$^{\circ }$3 $V_P$ full	1.34	1.16	20	2787
$r>1.9$ m	7.22	0.18	21
$V_S$ full	1.25	1.23	32	1216
$r>1.8$ m	5.01	0.32	36
N$^{\circ }$4 $V_P$ full	0.64	1.98	18	2745
$r>1.4$ m	2.58	0.39	38
$V_S$ full	0.57	2.26	32	1247
$r>1.3$ m	2.46	0.56	33

Table 4. Parameters estimation from four wells in a clastic overburden, for the full sonic logs, and for limited spatial frequency bandwidths using the indicated restriction on the relative distance $r$ to remove the tool artefacts. Relevant physical values are underlined.

rfsignalC2,fitfiltb025
Figure 4. Fourier spectrum of the scaled $V_S$ sonic log from well N$^{\circ }$ 3 (a). The shape of the low-frequency content is different from that of the Fourier spectrum of the fractional Brownian motion (b) synthesized with the von Kármán model using $H=0.25$, $b=5$m, and $\sigma=30~\%$.

Fractal heterogeneities in sonic logs and low-frequency scattering attenuation