Acoustic Mode

An acoustic mode is a specific pattern of elastic wave propagation in which acoustic energy travels through or along a medium in a characteristic waveform pattern, with its own velocity, spatial distribution, and frequency dependence. In the context of borehole acoustic logging, several distinct modes propagate simultaneously when a sonic tool fires its transmitter in a fluid-filled borehole: the compressional (P-wave) mode refracted along the borehole wall and arriving first at the receivers; the shear (S-wave) mode refracted along the borehole wall in fast formations; the Stoneley mode (a tube wave that travels along the fluid-rock interface at the borehole wall); and leaky P and pseudo-Rayleigh modes that carry energy at intermediate velocities. Each mode has a characteristic velocity, amplitude decay with distance, and frequency content that allow it to be identified and separated from other modes in the full acoustic waveform. Correct mode identification and extraction is the foundation of array sonic processing, which derives separate P-wave and S-wave velocities from a single borehole acoustic measurement.

Key Takeaways

The four primary acoustic modes in borehole acoustic logging are: the compressional (P) refracted headwave, traveling along the borehole wall at the formation P-wave velocity (typically 3,000 to 7,000 m/s); the shear (S) refracted headwave (in fast formations, Vs greater than Vmud), traveling at the formation shear velocity (typically 1,500 to 4,000 m/s) and arriving after the P-wave; the Stoneley wave (also called the tube wave), a low-frequency interface wave traveling at slightly below the borehole fluid velocity (800 to 1,400 m/s) with particle motion that involves both the fluid and the formation; and the pseudo-Rayleigh wave, a dispersive mode that propagates at near the shear velocity in fast formations and is generated by the interaction of P and S waves at the borehole wall. In slow formations (Vs less than Vmud), the shear headwave and pseudo-Rayleigh mode do not exist because the refraction condition is not satisfied, and the dipole acoustic mode (flexural wave) must be used to measure shear velocity instead.
The Stoneley wave is the mode most sensitive to formation permeability and borehole fluid mobility. The Stoneley wave is an interface wave: its energy is concentrated near the borehole wall, with both a fluid pressure component inside the borehole and a radial displacement component in the formation rock. When formation permeability is high, the Stoneley wave can drive fluid in and out of the formation pore space as it passes, causing additional attenuation (energy loss) and velocity dispersion (velocity change with frequency) of the Stoneley wave compared to an impermeable formation. Processing the Stoneley wave velocity and attenuation provides a continuous log of formation permeability that does not require laboratory calibration to specific formation types and works in both sandstone and carbonate formations. Stoneley wave permeability logs from array sonic tools are used in the WCSB to identify permeable fractures in tight Devonian carbonates and to prioritize perforation intervals in Viking and Cardium sandstones.
Dispersion is a key property of borehole acoustic modes: their velocity varies with frequency. The pseudo-Rayleigh mode is dispersive (it travels faster at high frequency, approaching the formation S-wave velocity, and slower at low frequency). The Stoneley mode is weakly dispersive in impermeable formations but strongly dispersive in permeable ones, because the frequency-dependent coupling of Stoneley energy into the formation pore fluid changes the effective velocity at each frequency. The flexural mode (used by dipole tools) is also dispersive: its low-frequency limit is the formation shear velocity, and it approaches the tube wave velocity at high frequency. Processing dispersive modes correctly requires frequency-domain analysis (computing the slowness at each frequency, the dispersion curve) rather than simple time-domain first-arrival picking.
Monopole and dipole source tools generate different mode spectra. A monopole source (a cylindrical transmitter that fires symmetrically in all directions, like a loudspeaker) efficiently generates P-wave, Stoneley, and pseudo-Rayleigh modes. It cannot efficiently generate the borehole flexural mode, which requires a dipole (push-pull) excitation that pushes the borehole fluid to one side and pulls it on the other, exciting a bending (flexural) wave. A dipole source generates mainly the flexural mode and is the standard method for measuring shear velocity in slow formations. Modern array sonic tools include both monopole and dipole sources (usually two orthogonal dipole directions for azimuthal anisotropy measurement), firing them in sequence and recording the complete waveform at each receiver for each source type.
Shear wave anisotropy measurement using crossed-dipole acoustic modes is a valuable formation evaluation technique in naturally fractured formations and in formations with strong stress-induced anisotropy. When two dipole sources oriented 90° apart (crossed dipoles) are fired sequentially and the waveforms at the receiver array are processed together (four-component crossed-dipole analysis), the processing can identify the fast and slow shear wave polarizations and their velocity difference, which is the shear wave splitting parameter. In a fractured formation, the fast shear wave is polarized parallel to the fracture strike (the fractures appear stiffer in that direction) and the slow shear wave is polarized perpendicular to the fractures. The splitting magnitude (velocity difference between fast and slow shear) is proportional to fracture density and aperture, providing a measure of the fracture intensity at each depth level without running an image log.

Separating Acoustic Modes in the Full Waveform

The full waveform recorded at each receiver in an array sonic tool is a superposition of all the acoustic modes arriving at that time. The challenge of processing is to unmix this composite waveform into its constituent mode components so that each mode can be separately analyzed for its velocity and amplitude. Several techniques are used for mode separation.

Time-frequency analysis (computing a spectrogram of the waveform) exploits the different frequency contents of the modes: the compressional wave arrives first at high frequency (typically 10 to 25 kHz); the Stoneley wave arrives last with predominantly lower frequency content (1 to 5 kHz); the pseudo-Rayleigh and shear waves arrive at intermediate times with intermediate frequencies. By selecting different time-frequency windows in the spectrogram, each mode can be approximately isolated.

Semblance processing across the receiver array exploits the different moveouts (differences in arrival time across receivers) of the modes: a P-wave moving at 5,000 m/s has a shorter moveout across a 60-cm receiver array than a Stoneley wave moving at 1,200 m/s. The slowness-time coherence (STC) panel plots the coherence of the waveforms across the receiver array as a function of assumed slowness and time window, producing peaks where the assumed slowness matches the actual mode velocity. Each peak in the STC corresponds to one acoustic mode, and the position of the peak gives the mode's slowness (interval transit time) directly. This STC processing is the standard method for first-arrival picking in array sonic tools and provides P-wave, S-wave, and Stoneley slownesses as distinct outputs.

Fast Facts

The theory of acoustic modes in fluid-filled boreholes was developed by several researchers in the 1950s and 1960s, with Biot (1956) providing the foundational theory of acoustic wave propagation in fluid-saturated porous media (the Biot theory underpins all quantitative relationships between acoustic velocity, porosity, and permeability). The Stoneley wave in boreholes was described by Stoneley (1924) in the original theoretical framework, though its practical application to permeability measurement did not develop until the work of Rosenbaum (1974) and Schmitt, Bouchon, and Bonnet (1988). The pseudo-Rayleigh and leaky-P modes were characterized by Cheng and Toksöz in the 1980s. Commercial dipole shear sonic tools (Schlumberger's DSI, Baker Hughes' XMAC) were introduced in the 1980s and 1990s, enabling shear velocity measurement in slow formations for the first time. Crossed-dipole acoustic mode analysis for shear anisotropy was commercialized in the mid-1990s and has since become a standard analysis in naturally fractured carbonate and deep shale gas plays where fracture orientation is a key completion parameter.

Acoustic Modes in Perforation and Completion Diagnostics

Acoustic mode analysis is not limited to pre-completion evaluation in open hole. Production logging with acoustic tools (run on wireline or coiled tubing through the production casing after completion) uses Stoneley wave transmission across perforations to diagnose whether individual perforations are open and communicating with the formation or are blocked by scale, sand, or collapsed casing. A perforation that is open communicates with the permeable formation, and the Stoneley wave driven by the acoustic tool couples energy into the formation fluid through the perforation, causing local attenuation of the Stoneley wave signal at that depth. A blocked perforation or non-communicating zone shows no Stoneley attenuation at the perforation depth.

This application of acoustic mode analysis (Stoneley wave reflections at perforations and formation features) provides a non-invasive completion efficiency diagnostic: by running an acoustic tool through the production casing before and after a fracturing or acidizing job, the operator can compare the Stoneley wave response at each perforation cluster and determine which clusters opened (showed new Stoneley attenuation) and which remained blocked. This approach was demonstrated in several WCSB horizontal wells and provides a valuable confirmation of hydraulic fracture entry at each stage without requiring a production log flowmeter survey.

An acoustic mode is also called a propagation mode, wave mode, or borehole mode in acoustic logging literature. Related terms include Stoneley wave (the low-frequency interface wave that propagates along the borehole wall and is sensitive to formation permeability; the acoustic mode whose attenuation and velocity provide a permeability indicator in full-waveform sonic processing), flexural wave (the bending wave mode excited by a dipole acoustic source in a borehole; its low-frequency phase velocity equals the formation shear velocity, enabling shear velocity measurement in slow formations where the shear headwave does not exist), dispersion curve (the plot of acoustic mode velocity versus frequency; used to identify and characterize dispersive borehole modes and to extract formation properties such as shear velocity and permeability from the frequency dependence of the mode), slowness-time coherence (STC, the semblance processing method used to identify and separate acoustic modes in full-waveform sonic data by finding the slowness value at which the waveforms across the receiver array are most coherent), and shear wave anisotropy (the difference in velocity between the fast and slow shear wave polarizations in a formation; measured by crossed-dipole acoustic mode analysis and used to characterize fracture orientation and density).

How Stoneley Wave Mode Analysis Identified a Previously Unknown Permeable Interval in a Nisku Well

An operator was evaluating a Devonian Nisku Formation well in the Kaybob area of west-central Alberta. A standard acoustic log (P-wave transit time only) had been run as part of the original wireline suite, and the porosity from the Wyllie equation showed a relatively uniform 6 to 9% dolomite porosity across the 22-metre Nisku interval. The neutron-density crossplot confirmed similar porosity with no obvious high-porosity zones. Initial production testing without acid stimulation showed very low gas rate (0.4 MMscf/d), consistent with a tight formation.

Before deciding whether to stimulate with acid fracturing, the operator ran a full-waveform array sonic log through the same interval to obtain Stoneley wave attenuation data. The Stoneley wave processing showed a striking anomaly at one specific 3-metre interval within the Nisku, centred at 3,242 metres depth: the Stoneley wave amplitude dropped by 65% through this interval (from 85 mV to 30 mV), indicating strong attenuation by a highly permeable zone. Above and below this interval, the Stoneley attenuation was minimal, indicating tight, low-permeability dolomite. The borehole image log run concurrently showed a cluster of open natural fractures at the same depth interval.