Explanation of Acoustic Wave Propagation
Monopole compressional and shear
Compressional and shear waves (sometimes referred to as p- and s-waves) are excited in the formation, along with various modes in the borehole, by a monopole source operating at high frequencies (typically 10-20 kHz). They propagate as body waves in the formation and along the borehole. As they do so, they leak energy (refract) back into the borehole, creating headwaves in the borehole fluid. Compressional waves propagate along the borehole in the direction of the borehole axis with minute vibrations (or displacements) of the formation in the same direction. Shear waves propagate in the direction of the borehole axis with minute radial vibrations of the formation.
Monopole shear waves have a lower velocity (higher t), generally a larger amplitude, and a slightly lower frequency than the compressional waves. Shear waves have a larger refraction angle than the compressional waves. The mud speed is usually nearly constant, so that the refraction angle depends on the phase velocity of the body wave in the formation. As the shear t becomes large (soft formations), less shear energy is refracted back into the hole. If the shear t surpasses the mud slowness (typically 190 sec/ft), none of the shear waves will be detected by the receivers.
At low frequencies, perhaps a few kHz, where typical wavelengths in the mud are greater than the borehole size, monopole signals are dominated by the Stoneley wave, a dispersive mode of the borehole. Stoneley waves are guided waves associated with the solid-fluid boundary at the borehole wall, and their amplitude decays exponentially away from the boundary in both the fluid and formation. At extremely low frequencies, the slowness of this mode approaches that of the tube wave, while at higher frequencies, it approaches that of the Scholte (planar interface) wave. It is most easily excited using a low-frequency monopole source. For all frequencies, the Stoneley slowness is determined predominantly by the mud and to a lesser extent by the formation compressional and shear slownesses, formation permeability, and other variables.
In a dipole shear sonic tool, a directional (dipole) source and directional receivers are employed. The source is operated at low frequencies, usually below 4 kHz. Compressional and shear waves are excited along with a dispersive flexural mode of the borehole. The slowness of this mode has the same high-frequency limit as the Stoneley wave, but at low frequencies it approaches the formation shear slowness rather than the tube wave slowness.
The amplitudes of both the flexural and the shear wave are peaked in frequency, the flexural generally peaking higher. They fall off very rapidly toward low frequencies and more gradually toward high frequencies. The flexural mode dominates the response down to very low frequencies where the shear wavelength is several times the borehole diameter. At such low frequencies, the direct shear wave is the only appreciable feature on the waveform. However, the amplitude of the waves at these frequencies (below 1 kHz for a typical slow formation) is very low and noise is likely to be a problem. A practical frequency range is 1-4 kHz. In this range, the flexural mode dominates the signals, but travels at nearly the shear slowness. A continuous shear log then is obtained by measuring the flexural slowness at as low a frequency as is practical and applying a small correction.
In very fast formations, the dipole compressional signal is usually very weak and may not be visible. The flexural mode is very dispersive in fast formations, there being as much as a factor of two difference in slowness between low frequencies (shear slowness) and high frequencies (Scholte slowness, approximately the mud slowness). The flexural arrival is therefore quite long in duration and spreads rapidly as the transmitter receiver spacing is increased. Low-frequency components traveling near the shear slowness become well separated from the slower higher frequency components. Often the (nondispersive) shear headwave is detectable in fast formations.
In slow formations, the flexural mode is again dispersive, but to a much lesser extent. Typically, the ratio between the high- and low-frequency limiting values of the flexural slownesses is about 1.2 or less. The flexural arrival is shorter in time duration and the spectral content is concentrated at lower frequencies. As in the figure, a higher frequency compressional arrival is often visible in slow formations, and in large boreholes and very slow formations can become the largest amplitude event. A distinct shear headwave arrival cannot be detected in slow formations.
Slowness Time Coherence (STC) Analysis Acquisition Software
Slowness-Time Coherence examines each waveform set for coherent arrivals across the array. It does this by stepping a time window of fixed duration through a range of times across the waveforms and a range of slowness across the array. For each time and slowness step, the waveforms within the window are added or stacked and the corresponding stacked or coherent energy is computed. When the window moveout or slowness aligns with a particular component moveout across the array, the waveforms within the window add in phase, maximizing the coherent energy. Coherent arrivals are thus identified by maxima in the coherent energy.
The STC module is used to find and extract slowness (Dt) and other information about various coherent arrivals in the sonic waveforms. Then the STC computation performs a sequence of operations on a set of waveforms aimed at identifying coherent arrivals in the set and extracting their slownesses. The following steps taken are: Waveform filtering, Waveform stacking, Peak searching, and Labeling.
An additional step is needed to identify and separate the desired arrivals (flexural, compressional, shear, or Stoneley) from any others. This is done by the labeling algorithm part of the STC computation. The slowness, arrival time, and coherence of each arrival are examined and compared with the propagation characteristics expected of the compressional, shear or Stoneley waves for the given physical conditions. Classifying the arrivals in this manner gives a continuous log of wave-component slowness versus depth.
STC processing of high-frequency monopole waveforms generally results in compressional and shear slowness estimates in fast formations. Narrow band filtering is applied to low-frequency monopole (Stoneley) waveforms, since this mode is dispersive and we want to estimate slowness within a consistent band of frequencies. In slow formations, no shear slowness estimate is available from monopole waveforms.
Dipole labeling bias correction
In STC processing of dipole waveforms, a coherence peak corresponding to the dispersive flexural mode occurs at a slowness near that of the frequency of peak excitation after filtering. The estimate is therefore biased slower than the true shear, and must be corrected. The bias depends on the time signature of the source excitation, the filter characteristics, the borehole size and shear slowness. In slow formations, the correction is less than 10%, and usually much less. In fast formations, where the dispersion of the flexural mode is greater, a large correction is required only in large (>17 in.) boreholes. In a fast formation with a moderate hole size (<12 in.), very little or no bias is found.
Depth-derived borehole compensation
One way to obtain borehole compensation is to derive slowness (delta t) measurements from both upward and downward propagating waves. The effects of borehole size changes tend to have an opposite effect on the slownesses derived from each. The standard BHC tool accomplishes this by having a transmitter above and below the receivers. The Long Spaced Sonic (LSS) tool, though, simulates this with depth-derived borehole compensation. The DSI-2 employs the same depth-derived technique. Instead of having transmitters above and below the array, it constructs a pseudo-transmitter array from several tool positions as it moves up the hole. The pseudo-transmitter array looks like an array of transmitters with one receiver above. This approximates a single transmitter on top with a receiver array below.
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