Fractal heterogeneities in sonic logs and low-frequency scattering attenuation |
We propose to use the synthesis of a random medium
detailed in Table 2 for =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:
cgaussM025,lligaussfM025,cgauss025,lligaussf025,cgauss05,lligaussf05
Figure 2. Synthetic signals generated as fGn with , m, in (a); fBm with , m, in (c); and fBm with , m, in (e). Parameter estimations on the logarithm of the spectral amplitude are shown on the right (b,d,f). |
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signalA1,signalC2,llisignalfA1,llisignalfC2,rllisignalfA1L,rllisignalfC2L
Figure 3. Sonic log (a) from well N1 and (b) from well N3, scaled by their respective average value . 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). |
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Parameters | (m) | (%) | |
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 |
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 and were both measured with a spatial sampling of 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 3c and 3d.
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
and , representing specific values of relative distance , namely
Results are summarized in Table 4 for the four different sonic logs and . The ratio 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 varies from 20 to 45 % and is larger for logs than for logs. Correlation length is about 5 m for both and , except for the well N4, which is 2.5 m. Exponent for varies from 0.1 to 0.4 and for 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) | (%) | (m/s) | |
N1 full | 0.79 | 1.32 | 17 | 2791 |
m | 6.70 | 0.13 | 22 | |
full | 1.05 | 1.16 | 32 | 1218 |
m | 5.92 | 0.21 | 35 | |
N2 full | 1.90 | 0.92 | 27 | 2842 |
m | 5.34 | 0.38 | 29 | |
full | 2.84 | 0.83 | 45 | 1240 |
m | 3.08 | 0.62 | 44 | |
N3 full | 1.34 | 1.16 | 20 | 2787 |
m | 7.22 | 0.18 | 21 | |
full | 1.25 | 1.23 | 32 | 1216 |
m | 5.01 | 0.32 | 36 | |
N4 full | 0.64 | 1.98 | 18 | 2745 |
m | 2.58 | 0.39 | 38 | |
full | 0.57 | 2.26 | 32 | 1247 |
m | 2.46 | 0.56 | 33 |
rfsignalC2,fitfiltb025
Figure 4. Fourier spectrum of the scaled sonic log from well N 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 , m, and . |
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Fractal heterogeneities in sonic logs and low-frequency scattering attenuation |