Free-Induction Decay

Key Takeaways

• The physics of free-induction decay begins with the behavior of hydrogen proton spins in a magnetic field: each proton has a magnetic moment (an intrinsic quantum mechanical property) that in the presence of a static magnetic field B_0 aligns either parallel (low energy state) or antiparallel (high energy state) to the field, with a slight excess of protons in the parallel (lower-energy) state creating a net magnetization M_0 in the direction of the static field (the z-direction); when a 90-degree RF pulse at the Larmor frequency (omega_L = gamma * B_0, where gamma is the proton gyromagnetic ratio 267.52 MHz/T) is applied, it tips this net magnetization M_0 from the z-direction into the transverse (x-y) plane; after the RF pulse ends, the tipped magnetization M_xy precesses around B_0 at the Larmor frequency, inducing an oscillating voltage at the Larmor frequency in the receiver coil that constitutes the FID signal; simultaneously, two relaxation processes reduce the FID signal amplitude: T1 (longitudinal) relaxation gradually restores the z-component of magnetization to M_0 as protons return to the equilibrium parallel alignment with B_0 (with time constant T1 of approximately 0.1 to 10 seconds in porous rocks), and T2 (transverse) relaxation destroys the coherence of the transverse precessing magnetization as the individual proton spins dephase from each other due to spin-spin interactions (with time constant T2 of 0.1 ms to 3 seconds in reservoir rocks).
• Field inhomogeneity causes the observed FID to decay much faster than the intrinsic T2 relaxation of the formation because the static magnetic field in an NMR logging tool is not perfectly uniform across the sensitive volume (the shell or region around the tool's permanent magnet at the resonant frequency): in the region of field inhomogeneity, protons at slightly higher field precess faster than protons at slightly lower field, causing the proton population to quickly lose phase coherence and the FID signal to decay with a very short apparent time constant T2* (typically 0.1 to 10 ms for logging tools, versus the intrinsic T2 that may range from 1 ms to 3,000 ms in the same formation); because T2* is dominated by the tool's field inhomogeneity rather than the formation's intrinsic relaxation properties, the FID is not directly useful for petrophysical interpretation of pore size, fluid type, or porosity in oilfield NMR logging; the CPMG echo train (in which a series of 180-degree refocusing pulses at intervals of 2*tau restore the phase coherence of the proton spins and produce spin echoes whose peak amplitudes decay with the intrinsic T2 rather than the tool-dominated T2*) was adopted as the standard NMR logging measurement specifically to overcome the field inhomogeneity limitation of the simple FID measurement.
• In laboratory NMR spectrometers used for core plug analysis (in contrast to logging tools), the static field is highly homogeneous and T2* approaches the intrinsic T2, making the FID more useful for quantitative measurements of total hydrogen content (NMR porosity) and T2 relaxation times: high-field laboratory NMR spectrometers (at 400 MHz to 1 GHz, using superconducting magnets that produce fields 200 to 500 times stronger than oilfield logging tools) achieve field homogeneity of better than 1 part per billion across the sample, giving T2* values that approach T2 and making the FID itself a useful measurement for T2 spectrum analysis in small core plugs; the laboratory NMR measurements of T2 relaxation in core plugs (saturated with formation brine, formation oil, and gas at reservoir conditions) provide the calibration data that links the T2 relaxation time (measured by the logging tool's CPMG echo train) to petrophysical properties such as pore size distribution, irreducible water saturation, and free fluid index -- connecting the physics of the FID to the practical oilfield application.
• The free fluid index (FFI) and bound volume irreducible water (BVI) derived from NMR logging are calculated by partitioning the T2 distribution (the amplitude-weighted distribution of T2 relaxation times measured from the CPMG echo train) at a cutoff T2 value that separates pores large enough to contain mobile fluids (T2 above the cutoff, contributing to FFI) from pores small enough that capillary and surface forces bind the water immovably (T2 below the cutoff, contributing to BVI): the T2 cutoff is determined empirically by comparing the NMR T2 distribution of a core plug measured at laboratory conditions with the irreducible water saturation of the same plug measured by capillary pressure drainage, finding the T2 cutoff that equates the NMR-predicted irreducible saturation to the capillary pressure-measured value; typical T2 cutoff values are 33 ms for sandstones (pores smaller than approximately 10 microns are water-wet at irreducible conditions) and 92 ms for carbonates (pores smaller than approximately 50 microns are bound); the FID decay time constant T2* provides no information about the T2 distribution because it is dominated by field inhomogeneity rather than surface relaxation, which is why the CPMG echo train is required for petrophysical NMR and the FID alone is insufficient for oilfield applications.
• The relationship between T2 relaxation and pore surface area is the physical basis for using NMR measurements to estimate permeability from pore geometry: surface relaxation (the interaction of proton spins with the pore wall surface, which enhances T2 relaxation at the surface relative to the bulk fluid) causes the T2 of pore water to decrease with increasing surface-to-volume ratio of the pore (smaller pores with higher surface-to-volume ratio relax faster, shorter T2; larger pores with lower surface-to-volume ratio relax slower, longer T2); the relaxation time T2 for a pore of surface-to-volume ratio S/V is approximately 1/T2 = rho_2 * (S/V), where rho_2 is the surface relaxivity (a material property of the pore surface, typically 1 to 10 micrometers per second for water-wet sedimentary rocks); the empirical relationship between the T2 distribution (which reflects the pore size distribution through the surface relaxivity) and the permeability (which depends on the pore size distribution through the Kozeny-Carman relation) is the basis for the Timur-Coates and SDR (Schlumberger-Doll Research) permeability equations used to calculate permeability directly from the NMR log, without requiring core plugs for each well; the FID alone cannot provide the T2 distribution needed for these permeability calculations, which is why the CPMG-derived T2 spectrum is the key measurement in all oilfield NMR applications.