Differential Spectrum

A differential spectrum in petroleum well logging and formation evaluation is a display or mathematical transformation that shows the rate of change of a measured log value with respect to depth or another variable — computed as the first derivative of a continuous log curve — used to enhance detection of thin beds, bed boundaries, and subtle formation changes that may be difficult to identify in the raw log curve, and particularly applied in nuclear magnetic resonance (NMR) logging to display the T2 relaxation time distribution as a differential spectrum that reveals the pore size distribution and fluid typing of the formation.

Key Takeaways

In NMR logging, the T2 differential spectrum (also called the T2 distribution or T2 relaxation spectrum) is the primary output of the inversion process that converts raw multi-exponential NMR echo decay data into a distribution of relaxation times — each bin in the T2 distribution represents the fraction of pore fluid relaxing at that time constant, with short T2 components (less than 33 ms for most sandstones) corresponding to small pores (bound or capillary-held water) and long T2 components (greater than 100 ms) corresponding to large pores (moveable fluids including oil and gas).
The depth derivative of a density or sonic log — computed as dρ/dz or dΔt/dz — functions as a differential spectrum in the stratigraphic sense, enhancing bed boundary positions where the log value changes rapidly with depth; this depth differential log is used to precisely locate bed tops for correlation, calculate formation thickness in thin-bed intervals, and identify fractured intervals where micro-scale density or velocity variations cause short-wavelength log perturbations that are invisible in the smoothed raw curve.
In core analysis and petrography, a pore size differential spectrum (derived from mercury injection capillary pressure curves by computing dSHg/dlog(Pc)) shows the pore throat size distribution — the peaks in the differential curve indicate the dominant pore throat sizes that control mercury entry and fluid flow, with the height and position of peaks directly relating to the permeability and irreducible water saturation of the rock.
Differential spectra are sensitive to noise because differentiation amplifies high-frequency measurement noise — a smooth log curve with small random fluctuations becomes a noisy differential curve with large amplitude variations, requiring application of appropriate smoothing (Gaussian filter, running average) before differentiation to balance noise suppression against preservation of genuine formation boundaries that have length scales comparable to the smoothing window.
Spectral gamma ray logs display a differential spectrum in the radiometric sense — the energy spectrum of gamma ray counts detected at different energies is displayed as counts per energy bin, and the differential peaks at characteristic energies (1.46 MeV for potassium-40, 1.76 MeV for bismuth-214 from uranium decay series, 2.61 MeV for thallium-208 from thorium decay series) identify the uranium, thorium, and potassium contributions to total radioactivity for clay typing and lithology identification.

Fast Facts

The T2 differential spectrum from NMR logging is the foundation of NMR-based permeability estimation (Timur-Coates and SDR models) and free fluid index calculation. The 33-ms T2 cutoff used for sandstone reservoirs to separate bound fluid from free fluid was empirically established by comparing NMR T2 distributions with centrifuge-derived capillary pressure measurements on core samples, and has been validated across hundreds of sandstone core datasets worldwide. For carbonates, the T2 cutoff is typically higher (92 ms for many carbonates) due to the different pore geometry and surface relaxivity of carbonate minerals compared to quartz. The selection of the appropriate T2 cutoff is critical for NMR-based water saturation and producibility assessment — using the wrong cutoff can cause systematic errors in free fluid index of 5 to 20 porosity units that directly affect production rate predictions.

What Is a Differential Spectrum?

A spectrum in measurement science is a display of a property as a function of some variable — the amplitude of a signal as a function of frequency, the intensity of radiation as a function of energy, or the porosity of rock as a function of pore size. A differential spectrum is the derivative of that spectrum with respect to the variable — it shows not the value of the property itself but the rate at which it is changing. Differential spectra are widely used in analytical science because they highlight the positions and sharpness of features (peaks, edges, boundaries) that may be obscured in the raw spectrum by background levels or broad features.

In petroleum well logging, the concept of differential spectrum appears in multiple contexts. The most technically specific use is in NMR logging, where the T2 relaxation distribution — the fundamental output of NMR log inversion — is literally a differential spectrum of the measured magnetization decay, showing how the proton signal is distributed among relaxation time components. But the concept also applies to the depth derivative of conventional logs (density, sonic, resistivity, gamma ray), which enhances bed boundary detection by highlighting where log values are changing most rapidly with depth, and to mercury injection capillary pressure analysis, where the pore throat size distribution is derived as a differential of the cumulative mercury saturation curve.

Understanding what a differential spectrum represents — the local rate of change of a measured property — is essential for correctly interpreting these derived curves. A large amplitude in a differential spectrum does not mean the measured property is large at that point; it means the property is changing rapidly at that point. A flat region in the differential spectrum means the measured property is changing slowly — either because the formation is uniform or because the measurement is smoothly varying without sharp boundaries.

Differential Spectrum in NMR and Log Analysis

NMR T2 differential spectra are the key output used for formation evaluation from nuclear magnetic resonance logging. The raw NMR measurement is a series of echo amplitudes decaying in time according to the multi-exponential relaxation of pore fluids in the formation's magnetic field gradient. Each pore or pore size class contributes an exponential decay component with a characteristic relaxation time T2 that depends on the pore surface-to-volume ratio, the pore fluid type, and the formation temperature. The mathematical inversion of this multi-exponential decay to a T2 distribution (the differential spectrum) is an ill-posed problem solved by regularized least-squares or maximum entropy methods that find the simplest smooth distribution consistent with the measured echo amplitudes.

The resulting T2 distribution shows distinct populations for different fluid types and pore environments: clay-bound water (T2 less than 3 ms), capillary-bound water (3 to 33 ms in sandstone), moveable water and light oil (33 to 300 ms), viscous oil (overlapping the capillary-bound and moveable water regions, requiring diffusion editing to separate), and gas (short T2 from diffusion in the field gradient, overlapping bound fluid depending on gradient and sequence). The area under each portion of the T2 distribution is proportional to the volume of fluid in that pore environment, allowing partition of total NMR porosity into bound fluid volume and free fluid index without requiring resistivity-based water saturation.

Depth derivative differential logs are used in thin-bed analysis and stratigraphic interpretation. When a conventional log is differentiated with respect to depth, the resulting curve has positive excursions where the log increases downward (entry into a more radioactive, denser, or more resistive bed depending on the log type) and negative excursions where the log decreases. The zero-crossings of the differential log identify inflection points in the original log — which may correspond to bed mid-points rather than bed boundaries, depending on the tool resolution — while the peaks in the differential log amplitude identify the bed boundaries with sharp transitions. This technique is particularly useful for correlating formation tops between wells and for calculating net-to-gross from log responses in thinly laminated sequences.

Differential Spectrum Across International Jurisdictions

Canada (AER / WCSB): NMR T2 differential spectrum analysis is used in WCSB tight gas and oil sands reservoir characterization to differentiate producible from non-producible fluid volumes in formations where conventional resistivity-based water saturation is unreliable. AER core analysis requirements for Athabasca oil sands resource assessment accept NMR T2 distribution data as supplementary evidence for pore size distribution and producible bitumen estimation alongside conventional Dean-Stark extraction data. Depth derivative differential logs are used in WCSB sequence stratigraphy interpretation for identifying flooding surfaces, transgressive surfaces, and sequence boundaries in the thin-bedded Mannville and Viking clastic sequences.

United States (API / BSEE): NMR differential spectrum interpretation is standard practice in Gulf of Mexico deepwater formation evaluation, where complex hydrocarbon fluid systems (light oil, condensate, heavy oil, gas) require T2 distribution analysis supplemented by diffusion editing to separate fluid components within the T2 spectrum. SPE papers from Gulf of Mexico operators (Chevron, Shell, bp, ExxonMobil) document NMR T2 spectrum interpretation in Miocene turbidite sands, demonstrating that the T2 cutoff must be calibrated for the specific reservoir temperature, salinity, and fluid viscosity rather than applying universal sandstone cutoffs. BSEE offshore resource assessment submissions accept NMR-derived porosity and fluid typing as an alternative to or supplement of resistivity-based petrophysical evaluation.

Norway (Sodir / NORSOK): NCS reservoir characterization programs routinely include NMR logging and T2 differential spectrum analysis for North Sea sandstone and chalk reservoirs where formation water salinity variations and complex fluid systems require independent fluid typing confirmation beyond resistivity log interpretation. Equinor's petrophysical standards for NCS wells specify NMR log acquisition and T2 spectrum interpretation procedures for tight gas and condensate reservoir appraisal in Jurassic and Triassic formations. Norwegian research institutes (NORCE, previously IRIS) have published calibration studies comparing NMR T2 cutoffs measured in the laboratory on NCS core samples against production test data, providing region-specific T2 cutoff values for Brent Group and Statfjord Formation sands.

Middle East (Saudi Aramco): Saudi Aramco uses NMR T2 differential spectrum analysis extensively for Arab Formation carbonate reservoir evaluation, where the multi-modal T2 distributions (reflecting macro-pores, micro-pores, and vugs at very different T2 values) require careful interpretation to assign producible versus non-producible porosity fractions. Aramco's carbonate NMR calibration program has established T2 cutoffs and partitioning algorithms specific to Arab D and Arab C limestone and dolomite pore systems, calibrated against special core analysis data from Ghawar, Abqaiq, and other major fields. The distinction between productive large-pore (vuggy, moldic) carbonate porosity and non-productive microporosity in Arab Formation chalky carbonates is critical for waterflood design and is primarily assessed from NMR T2 distribution analysis.

Differential spectrum in NMR logging is also called the T2 distribution, relaxation time distribution, or NMR pore size distribution. The depth derivative of a log curve is also called the differential log, derivative log, or dip indicator log in stratigraphic contexts. Related terms include nuclear magnetic resonance (NMR), T2 relaxation, free fluid index, bound fluid volume, pore size distribution, mercury injection capillary pressure, and formation evaluation. The T2 cutoff is the threshold value in the T2 differential spectrum that separates bound fluid from free fluid volumes, determined empirically from core NMR measurements compared to centrifuge drainage experiments.