Normal Mode

Normal mode in petroleum well logging and acoustic measurement refers to a specific propagation mode of elastic wave energy in a fluid-filled borehole — a guided wave that travels along the borehole axis as a combination of fluid pressure waves and formation shear waves, with the energy dispersed across a characteristic frequency range and velocity that depends on borehole diameter, mud velocity, and formation shear velocity — with the fundamental (lowest frequency) normal mode being the Stoneley wave and higher modes being pseudo-Rayleigh modes used in acoustic logging to determine formation shear slowness in slow formations where refracted shear arrivals cannot be directly measured.

Key Takeaways

The Stoneley wave (also called the tube wave) is the fundamental normal mode of a fluid-filled borehole — it is a low-frequency, dispersive wave that propagates at a velocity below the mud compressional velocity, with its particle motion consisting of coupled pressure oscillation in the borehole fluid and radial motion of the borehole wall, and its phase velocity at low frequencies approaching the formation shear velocity — making Stoneley wave velocity a proxy for formation shear modulus (shear slowness) in both fast and slow formations.
Pseudo-Rayleigh modes (also called leaky modes) are higher-frequency normal modes that can only exist when the formation shear velocity exceeds the borehole fluid velocity — the pseudo-Rayleigh mode velocity is dispersive, falling between the formation shear velocity (at low frequency) and the borehole fluid velocity (at high frequency, where the cutoff frequency depends on borehole diameter and formation shear velocity).
In slow formations (where formation shear velocity is less than the borehole fluid velocity, common in unconsolidated sands, gas hydrates, and weak shales), pseudo-Rayleigh modes do not exist, and only the Stoneley wave can propagate as a guided normal mode — the dipole or quadrupole acoustic tools that excite flexural modes in the borehole are used to measure shear slowness in slow formations by analyzing the dispersive velocity of the borehole flexural mode rather than the Stoneley mode.
Stoneley wave permeability assessment exploits the sensitivity of Stoneley wave attenuation and velocity to formation permeability — in permeable formations, borehole fluid can flow in and out of the formation as the Stoneley wave passes, reducing Stoneley velocity and increasing attenuation compared to the impermeable case; the difference between modeled Stoneley response (for impermeable rock with the same elastic properties) and measured Stoneley response provides a qualitative permeability indicator, particularly useful in identifying permeable zones in hard rock formations where mud cake does not form.
Normal mode analysis from monopole acoustic log data involves separating the full waveform into its compressional, shear (in fast formations), Stoneley, and noise components through frequency-wavenumber (f-k) processing or semblance-based velocity analysis across multiple receiver array stations, with the Stoneley and pseudo-Rayleigh components arriving after the direct compressional and shear head waves and occupying distinct frequency bands that facilitate separation.

Fast Facts

The theoretical treatment of normal modes in fluid-filled boreholes was worked out by White and Zechman in 1968, building on Biot's 1952 theory of elastic wave propagation in porous media. The commercial application of Stoneley wave analysis for formation shear slowness and permeability assessment followed the development of long-spaced acoustic logging tools (Schlumberger's Long-Spaced Sonic Tool and the Digital Array Sonic in the 1980s) and full-waveform recording that captured the complete acoustic signal including the Stoneley arrivals after the compressional and shear head waves. Modern acoustic logging tools (Sonic Scanner, Isolation Scanner) record full waveforms at dozens of receivers in monopole and cross-dipole modes simultaneously, enabling simultaneous analysis of all propagating modes including the Stoneley normal mode, dipole flexural mode, and Stoneley permeability measurement in a single logging run.

What Is Normal Mode?

When an acoustic source fires in a fluid-filled borehole, several types of elastic waves propagate away from the source. Some waves radiate energy into the formation as head waves and body waves. Others are trapped in the borehole by the contrast between the slow borehole fluid and the faster formation — these trapped waves are guided along the borehole axis as normal modes, accumulating energy in the borehole with each reflection from the borehole wall.

The concept of a normal mode comes from the mathematics of wave propagation in bounded media: in any confined system, certain specific wave patterns can propagate without loss of energy to radiation — these are the normal modes of the system. For a cylindrical fluid-filled borehole, the normal modes are the Stoneley wave (monopole, azimuthally symmetric) at low frequencies and the pseudo-Rayleigh modes at higher frequencies, each with a characteristic dispersion relationship between wave velocity and frequency that depends on the borehole diameter and the elastic properties of both the borehole fluid and the surrounding formation.

Normal modes are important in well logging because their dispersion characteristics carry information about formation properties. By measuring how the velocity of normal mode waves varies with frequency (the dispersion curve), and comparing this to theoretical predictions for different formation shear velocities, the formation shear slowness can be determined even in slow formations where the conventional refracted shear wave cannot be detected. This makes normal mode analysis an essential tool for complete elastic property determination in formation evaluation programs.

Normal Modes in Acoustic Well Logging

Stoneley wave logging has evolved from a qualitative permeability indicator to a quantitative tool for both shear slowness measurement and permeability assessment. The Stoneley wave travels more slowly than both the compressional and shear head waves, arriving at acoustic receivers after the direct arrivals and occupying the low-frequency portion (0.5 to 5 kHz) of the recorded waveform. Its velocity is primarily controlled by the formation shear modulus at low frequencies — the low-frequency Stoneley velocity approaches the tube wave velocity (a function of the formation bulk and shear moduli and fluid compressibility), which can be inverted to give shear modulus and shear slowness.

Permeability assessment from Stoneley wave attenuation is based on Biot's theory of coupled fluid-solid wave propagation. In a permeable formation, the oscillating borehole pressure associated with the passing Stoneley wave drives fluid in and out of the formation through the permeable rock face, dissipating energy and reducing Stoneley wave amplitude. The degree of attenuation, relative to the theoretical impermeable response calculated from the measured formation acoustic velocities and density, is inversely related to the formation permeability — more permeable formations attenuate the Stoneley wave more. This Stoneley permeability approach works best in high-permeability formations (greater than 10 mD) and requires accurate compressional and shear velocity inputs to compute the baseline impermeable response accurately.

Dipole and cross-dipole shear logging, which replaced the need to use normal modes for shear velocity estimation in many applications, excites the borehole flexural mode — a different guided wave from the Stoneley normal mode but also dispersive — which can propagate in both fast and slow formations. The dipole tool measures shear slowness by fitting the flexural mode dispersion curve, analogous to the Stoneley dispersion analysis but using the antisymmetric (bending) mode of the borehole rather than the symmetric (breathing) Stoneley mode. Cross-dipole logging measures shear slowness in two orthogonal azimuths, detecting shear wave splitting caused by formation anisotropy — a capability that normal mode Stoneley analysis cannot provide.

Normal Mode Across International Jurisdictions

Canada (AER / WCSB): Normal mode acoustic analysis is used in WCSB tight gas and shale formations where formation shear velocity is a critical input to mechanical earth models for hydraulic fracture design. AER well completion submissions for Montney and Duvernay horizontal wells may include acoustic log data including full-waveform Stoneley analysis to support shale brittleness calculations (from compressional and shear velocity-derived Young's modulus and Poisson's ratio). Stoneley permeability assessment is used in hard carbonate formations of the Devonian reef complex (Leduc, Nisku) where conventional permeability log indicators are absent and Stoneley attenuation provides the only continuous permeability indicator between core plugs.

United States (API / BSEE): Normal mode acoustic logging is standard practice in Gulf of Mexico deepwater wells where the slow formation shear velocity of unconsolidated Miocene sands (often slower than borehole fluid, precluding conventional shear wave detection) requires dipole flexural mode or Stoneley analysis for shear slowness determination. Formation shear slowness from normal mode analysis provides the input for pore pressure prediction from acoustic velocity methods (Eaton's equation) used widely in deepwater drilling planning. BSEE well completion reports for deepwater Gulf of Mexico development wells routinely include full-waveform acoustic log suites covering both compressional and shear (from normal mode or dipole analysis) slowness.

Norway (Sodir / NORSOK): NCS logging programs include monopole acoustic full-waveform acquisition for Stoneley wave analysis in Jurassic Brent Group and Triassic sandstone formations where permeability heterogeneity between cemented and clean sand intervals is important for production allocation and waterflood design. Equinor's petrophysical standards specify acoustic full-waveform recording for NCS horizontal wells where shear velocity is needed for geomechanical modeling of wellbore stability during drilling and hydraulic fracture initiation. Sodir's mandatory log data submission requirements for NCS wells include acoustic slowness curves (compressional and shear from dipole or Stoneley analysis), submitted in DLIS format to the national Diskos database.

Middle East (Saudi Aramco): Saudi Aramco uses Stoneley wave normal mode analysis in Arab Formation carbonate wells to assess permeability distribution along horizontal well lengths, supplementing core plug permeability data with a continuous permeability indicator that identifies the most productive intervals for selective perforation and intelligent completion design. Aramco's acoustic logging program for Arabian carbonate reservoirs includes cross-dipole shear logging for anisotropy characterization and Stoneley normal mode analysis for permeability profiling — both measurements are acquired in a single pass with modern multi-component acoustic tools (Schlumberger Sonic Scanner, Baker Hughes XMAC Elite). The correlation between Stoneley attenuation-derived permeability and production log flow allocation in Arab D horizontal wells has been validated in multiple Aramco field studies.

Normal mode in borehole acoustics is also called a guided wave or borehole mode. The specific normal modes are: Stoneley wave (fundamental monopole mode), pseudo-Rayleigh wave (higher monopole modes in fast formations), and flexural mode (dipole excitation). Related terms include Stoneley wave, acoustic logging, shear slowness, full-waveform acoustic, dispersion, dipole sonic, and Biot theory. The term mode in this context refers to a specific pattern of wave propagation with a characteristic spatial structure and dispersion relationship, distinct from its use in statistics (the most common value) or other engineering contexts.