Dephasing
Dephasing in nuclear magnetic resonance (NMR) logging is the loss of phase coherence among the ensemble of hydrogen proton spins precessing about the static magnetic field after they have been tipped into the transverse plane by a 90-degree radiofrequency pulse — a process that causes the net transverse magnetization signal (which is the sum of all individual proton spin contributions) to decay to zero as the individual spins, initially precessing in synchrony, progressively fall out of phase with one another due either to spatial inhomogeneities in the static magnetic field or to intrinsic molecular-level interactions that are collectively called transverse relaxation; the two distinct dephasing mechanisms are (1) the free induction decay (FID), caused by magnetic field inhomogeneities that impose slightly different Larmor precession frequencies on protons at different spatial positions in the sensitive volume, and (2) true T2 transverse relaxation, caused by spin-spin interactions and molecular diffusion in magnetic field gradients that irreversibly randomize the phases of individual spins; the CPMG (Carr-Purcell-Meiboom-Gill) pulse sequence used in all commercial NMR logging tools corrects for FID dephasing (which is recoverable) by applying 180-degree refocusing pulses that rephase the magnetically spread-out spins, but cannot reverse the irreversible T2 relaxation processes, making the CPMG echo amplitude decay envelope the direct measurement of T2.
Key Takeaways
- Free induction decay (FID) from static field inhomogeneity is the dominant dephasing mechanism in the borehole NMR environment — the permanent magnets used in NMR logging tools to create the static polarizing field cannot produce a perfectly uniform field across the sensitive volume (the cylindrical shell of formation rock around the borehole where the NMR measurement occurs), and the spatial variation in field strength across this volume means that protons at different radial positions precess at slightly different Larmor frequencies (since the Larmor frequency is proportional to the local field strength); protons at a radius where the field is 0.1% higher than the nominal field precess 0.1% faster than the nominal frequency, accumulating a phase lead relative to the nominal protons at a rate of approximately 0.1% × gamma × B0 radians per second; even this small spread of precession frequencies causes the ensemble of proton phases to spread across 360 degrees (full dephasing) within approximately 1 millisecond for typical NMR tool field homogeneity specifications, making the FID decay time much shorter than the T2 relaxation times being measured (which range from 0.3 milliseconds for clay-bound water to 3,000 milliseconds for gas).
- CPMG echo refocusing reverses FID dephasing but not T2 dephasing — the fundamental insight of the CPMG sequence is that FID dephasing is reversible (the protons are dephased because they precess at different speeds, but they are still precessing coherently in the sense that each proton's phase is a deterministic function of its position in the field; applying a 180-degree refocusing pulse at time tau after the initial 90-degree pulse reverses the phase development of each proton, causing all protons to rephase and produce a measurable spin echo at time 2tau; applying repeated 180-degree pulses at intervals 2tau produces a series of spin echoes (the CPMG echo train) whose amplitudes decay only by the irreversible T2 processes (spin-spin interactions and diffusion) that cannot be reversed by the 180-degree pulse; the CPMG echo train amplitude at each echo time is therefore a direct measurement of the T2 relaxation that has occurred, free from the confounding effect of FID, which is the fundamental reason that T2 is the measurable quantity in borehole NMR rather than the more rapidly decaying free induction decay signal.
- Molecular diffusion in the NMR tool gradient field is a form of irreversible dephasing that contributes to T2 in NMR logging — the NMR tool's static magnetic field has a radial gradient (the field is not perfectly uniform, so there is a systematic field variation with radius) that is created by the tool design and cannot be eliminated; water molecules and light hydrocarbon molecules diffuse through this gradient field during the inter-echo interval (between the 180-degree refocusing pulses), acquiring random phase shifts that cannot be refocused by the CPMG 180-degree pulses because the molecules have moved to a different field position than they were at when the 180-degree pulse was applied; the diffusion contribution to the apparent T2 (called T2D, the diffusion-dominated T2) is proportional to gamma² × G² × D × tau², where G is the field gradient, D is the molecular diffusion coefficient, and tau is the inter-echo half-spacing; this diffusion-dephasing effect is exploited in gas identification because gas has a much higher diffusion coefficient than water or oil (D_gas approximately 10 to 100 × D_water at reservoir pressure), causing gas to have a much shorter apparent T2 than water in the same pore environment — the T2/T1 ratio technique exploits this diffusion dephasing of gas as its primary fluid discrimination mechanism.
- Spin-spin T2 relaxation is the intrinsic molecular-level dephasing that occurs even in a perfectly homogeneous magnetic field — individual proton spins interact with neighboring protons through their magnetic dipole fields, and these fluctuating local fields cause random phase perturbations that irreversibly dephase the spin ensemble; in bulk water far from any surface, spin-spin T2 is very long (several seconds) because the rapid Brownian motion of water molecules averages out the local field fluctuations before they can accumulate significant phase error; in the confined spaces of small rock pores, where protons are frequently near paramagnetic mineral surfaces (iron, manganese) that generate intense local field fluctuations, spin-spin T2 is dramatically shortened by surface relaxation to values as short as 0.3 to 5 milliseconds, encoding the pore size information into the T2 distribution that is the basis of NMR-derived pore size analysis and permeability estimation.
- T2* (T2-star) is the measured FID decay time that combines both the irreversible T2 and the reversible FID effects — in a free induction decay measurement (not corrected by CPMG refocusing), the signal decays with time constant T2* = 1/(1/T2 + 1/T2_FID), where T2_FID is the dephasing time from field inhomogeneity; in borehole NMR tools, T2_FID is approximately 0.1 to 1 millisecond while T2 ranges from 0.3 to 3,000 milliseconds, so T2* is dominated by T2_FID and contains no useful formation information; this is why all commercial borehole NMR tools use CPMG rather than FID — the FID decays too fast to contain the T2 information about pore size and fluid properties, making the CPMG refocusing a physical necessity rather than a design preference for borehole NMR measurements.
Fast Facts
The CPMG pulse sequence that corrects for magnetic field inhomogeneity dephasing was developed in three stages: Herbert Carr first proposed using a train of 180-degree pulses to refocus spins in 1954; Edward Purcell (with Carr) refined the approach; and Saul Meiboom and David Gill independently developed an improved version in 1958 that corrected a phase error in Carr's original sequence (the Meiboom-Gill modification), producing the stable echo trains that are the basis of all modern T2 measurements. The CPMG sequence is so fundamental to NMR relaxometry that it is used not only in borehole logging but in virtually every time-domain NMR T2 measurement in medicine, food science, materials testing, and petroleum engineering. Its adaptation to the harsh borehole environment — high temperature, mechanical vibration, limited power budget, and the requirement to operate at a specific sensitive volume radius dictated by the tool's magnet design — was the primary engineering challenge that delayed commercial borehole NMR logging until the late 1980s despite the NMR physics being understood since the 1950s.
What Is Dephasing?
In NMR, measuring the formation requires getting all the hydrogen protons precessing in unison — like a synchronized dance — and then watching them fall out of step with each other. The rate at which they fall out of step (dephase) contains the information about the formation: pore size, fluid type, and permeability.
But the problem is that the wellbore environment is imperfect. The magnetic field that aligns the protons is not perfectly uniform — it varies slightly across the sensitive volume because no permanent magnet can be machined with infinite precision. This field variation means protons in slightly stronger parts of the field precess slightly faster, while protons in weaker parts precess slightly slower. Within a millisecond, the once-synchronized ensemble has completely dephased due to this instrumental effect alone — obscuring the formation-related relaxation signal that the measurement is trying to detect.
The CPMG refocusing pulse is the elegant solution: apply a 180-degree flip that reverses the phase development and brings all the protons back into synchrony at the echo time, as if running a video of dephasing in reverse. This recovers the formation signal from the instrumental dephasing, revealing only the irreversible T2 relaxation caused by the formation's pore size and fluid properties. Understanding dephasing is therefore not an academic exercise — it is understanding why CPMG works, and why NMR logging tools must use it rather than simply measuring the free induction decay.
Dephasing Physics and NMR Log Interpretation
Inter-echo spacing (TE, typically 0.2 to 1.5 milliseconds in commercial borehole NMR tools) determines the trade-off between capturing short-T2 signals from clay-bound water and minimizing diffusion dephasing from long-echo spacings — shorter TE allows detection of rapidly relaxing T2 components (below 3 × TE, the components are significantly attenuated in the early echoes of the CPMG train) but provides more CPMG refocusing pulses per unit time, reducing the accumulation of diffusion-related dephasing; longer TE allows diffusion in the gradient field to accumulate more dephasing between pulses, enhancing the T2 shortening due to diffusion that is used for gas identification but simultaneously causing underestimation of water T2 in porous formations where diffusion at long TE artificially shortens the apparent T2 distribution; the standard TE of 0.2 to 0.6 ms used in most commercial tools balances these competing factors for typical reservoir conditions, but specialized acquisition modes with longer TE (1.0 to 2.0 ms) enhance the diffusion T2 shortening effect for gas identification purposes.
Signal-to-noise ratio (SNR) in the CPMG echo train limits the minimum detectable T2 relaxation component and therefore the minimum detectable porosity in any formation — each echo in the CPMG train contributes to the measured signal, and the total CPMG echo train at a given depth represents the sum of signals from all relaxing proton populations in the measurement volume; the noise on each echo is dominated by thermal noise in the receiver electronics (Johnson noise, proportional to the square root of temperature and bandwidth), and the total SNR scales with the square root of the number of echoes and the echo amplitude; in low-porosity formations (below 5% porosity), the echo amplitude is small and the SNR may be insufficient to resolve the T2 distribution reliably within a single tool pass, requiring stacking of multiple measurements (either by running slowly or by repeating the CPMG sequence multiple times at the same depth) to improve SNR before inversion.
Dephasing and NMR Logging Across International Jurisdictions
Canada (AER / WCSB): WCSB NMR logging applications that require understanding of dephasing effects are most common in Montney and Duvernay tight formation evaluation, where the short inter-echo spacing CPMG acquisition (TE 0.3 to 0.6 ms) is required to capture clay-bound water T2 components in the 0.5 to 3 millisecond range that would be missed with longer TE; understanding FID dephasing and CPMG correction is essential for evaluating whether apparent short-T2 signals in tight rock NMR logs represent genuine clay-bound water or instrumental artifacts from partial dephasing correction in the high-gradient NMR tool environment; oil sands NMR in the McMurray Formation uses very long TE CPMG to enhance diffusion dephasing and improve the signal contrast between bitumen (low diffusion, less dephasing) and water (high diffusion, more dephasing) that is exploited in the differential spectrum method for bitumen saturation determination.