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This page provides an overview of the mathematical principles behind the petrophysical aspects of nuclear magnetic resonance (NMR) logging, including porosity, saturation, hydrocarbon identification, facies prediction, and permeability.
Extensive laboratory studies on NMR behavior and on the properties of fluid-saturated porous media have been conducted since the inception of NMR and throughout the development of NMR-logging tools. The results from these investigations have provided the petrophysical foundation for understanding the logging measurements and for developing interpretation models and applications.
Low-field, bench-top pulse-NMR spectrometers were developed concurrently with logging tools so that wellbore measurements could be duplicated on core samples in the laboratory. These instruments operate and record data in the same manner as NMR-logging tools. Because NMR analysis is nondestructive, NMR and conventional capillary-pressure measurements can be performed on the same samples, in both the saturated and partially saturated states. Low-field spectrometers provide the ability to make repeatable measurements of rock- and fluid-NMR properties. This ability, in turn, permitted correlation and calibration of laboratory and field measurements and also permitted direct transfer of interpretation models developed in the laboratory to logging data. Where core is unavailable for NMR-log calibration, new technology and methods now allow NMR petrophysical measurements on drill cuttings.
Laboratory NMR studies are routinely conducted for the following purposes:
- Verifying formation porosity
- Evaluating textural effects, such as microporosity, on NMR-log response
- Determining formation-specific models that enhance the accuracy of determining bulk-volume-irreducible (BVI) water, free-fluid index (FFI), and, ultimately, permeability
- Developing models to identify and quantify hydrocarbons, including residual oil
- Developing models to predict changes in pore size (facies)
Much of this work is summarized in Kenyon, Murphy, Woessner, and Dunn et al. The most recent laboratory studies have suggested that some established NMR core-log relationships should be further investigated to better account for data scatter. A related area of study not dealt with here is NMR imaging of fluid flow in core.
As in NMR logging, data quality is critical. To achieve the desired objectives, laboratory NMR studies should include a preplanning phase similar to that used in logging. (See Job planning for nuclear magnetic resonance (NMR) logging).
The basic petrophysical parameters of porosity, permeability, and producibility can be determined from either T1 or T2 echo-decay data. Until low-field spectrometers were developed, T1 was the preferred acquisition method in the laboratory, where time is not a concern. Because T1 measurement requires more time than T2, T2 became the primary acquisition mode in pulse-NMR logging because it allowed logging at speeds that were commercially viable. Fortunately, there is a correlation between T1 and T2; and T1 can be estimated from T2 data by extrapolating the T2 decay-obtained-polarization pulses of different lengths (i.e., using different values of TW; see Fig. 1).
Modern logging tools are capable of operating in either T1- or T2-acquisition modes. The logging mode is dictated by operational factors and job objectives and may, in fact, switch back and forth, as needed. (See Logging while drilling (LWD)).
NMR properties of fluids
T1 relaxation occurs when the precessing proton system transfers energy to its surroundings. T2 relaxation occurs through a similar transfer in energy and also through dephasing. Consequently, transverse relaxation, T2, is always faster than longitudinal relaxation, T1. The emphasis of the proton NMR techniques used in formation evaluation is on the NMR fluid response from the pore fluids where T2 ≤ T1. The NMR response in solids (i.e., the non-shale/clay component of the rock matrix) is very short compared to the pore-fluid signal, and is generally not measured by laboratory or logging devices.
NMR relaxation of fluids depends on whether the fluid is measured in bulk form as a wetting-pore fluid within a rock matrix or in a gradient magnetic field. Bulk relaxation is the intrinsic relaxation property of a fluid that is controlled by viscosity, chemical composition, temperature, and pressure: T2bulk≅T1bulk. Fluids contained within pores have different relaxation characteristics, namely those of surface relaxation.
Surface relaxation occurs at the fluid-solid interface between a wetting-pore fluid and the rock-pore walls (Fig. 2) and is different from the relaxation in either the solid or the fluid, individually. Surface relaxation dramatically decreases both T2 and T1 and is the dominant component contributing to T2. When a nonwetting fluid (e.g., oil) is also present in rock pores, the nonwetting fluid may continue to relax at its bulk-relaxation rate.
Surface relaxation is expressed by the following equations (Eqs.1 and 2):
where ρ2 = T2 surface relaxivity (i.e., T2 relaxing strength of the grain surfaces); ρ1 = T1 surface relaxivity (i.e., T1 relaxing strength of the grain surfaces); and (S/V) pore=ratio of pore surface to fluid volume.
Surface relaxivity varies with mineralogy; and, for simple pore shapes (S is the surface area of a pore and V is the volume of the same pore), S/V is a measure of pore size. In a brine-wet rock, T2 in smaller pores will be less than T2 in large pores; consequently, identical pore water in different rocks can have a wide range of relaxation times because of variations in surface relaxivity. Laboratory studies have demonstrated that in water-wet rocks, the surface and volume ratio (S/V) is also a measure of permeability.
Fluids controlled by surface relaxation exhibit T2 values that are not dependent on temperature and pressure. For this reason, laboratory NMR measurements made at room conditions are commonly used to calibrate formulas used to estimate petrophysical parameters such as permeability and bound water.
Diffusion-induced relaxation occurs when a significant gradient exists in the static magnetic field. Molecular diffusion in this gradient causes additional dephasing that contributes to increased T2 relaxation. In addition to the magnetic-field gradient, diffusion is also controlled by inter-echo spacing and fluid diffusivity, viscosity, molecular composition, temperature, and pressure.
Bulk-fluid processes and surface relaxation affect both T1 and T2, while diffusion only affects T2 relaxation. All three processes are independent and act in parallel according to the following equations (Eqs.3 and 4):
The relative importance of the three diffusion-relaxation mechanisms depends on (see Table 1):
- Fluid type in the pores (e.g., water, oil, or gas)
- Sizes of the pores
- Strength of the surface relaxation
- Wettability of the rock surface
Eq.5 states that the T2 decay associated with a single pore size in water-saturated rocks is proportional to the pore size. In fact, because reservoir rocks typically comprise a distribution of pore sizes and frequently contain more than one fluid type, a CMPG T2 spin-echo train actually consists of a distribution of T2 decays, rather than a single T2 decay. In these cases, the exponential decay is described by Kenyon et al. as follows:
where M(t) = measured magnetization at t; Mi(0) = initial magnetization from the ith component of relaxation; and T2i = decay constant of the i th component of transverse relaxation. The summation is over the entire sample (i.e., all pores and all different types of fluid).
Fig. 3 illustrates the multiexponential decay character of a porous medium containing pores of different sizes and a single wetting phase. Surface relaxation dominates when a short inter-echo spacing is used and the formation is only brine saturated. Under this condition, T2 is directly proportional to pore size. When all pores are assumed to have similar geometric shape, the largest pores (see Fig. 3, left column) have the lowest S/V and, thus, the longest T2. Medium-size pores have smaller S/V, yielding shorter T2 values. The smallest pores have the highest S/V and the shortest T2 values.
Fig. 3 – A 100% water-saturated pore (upper left) has a single T2 value (upper center) that depends on pore size and, thus, its spin-echo train exhibits a single exponential decay (upper right) that also depends on pore size. Multiple pores at 100% water saturation (bottom left) have multiple T2 values (bottom center) that depend on the pore sizes and, thus, their composite spin-echo train exhibits multiexponential decay (bottom right) that also depends on the pore sizes.
Eq.6 can also be expressed as follows: 
where (S/V)t is the surface-to-volume ratio of the ith pore. When t = 0, the following is true:
If the measured magnetization for 100% bulk water, M100%(0), is known, then M(0) and M0i can be calibrated to porosity by the following equation:
where ϕ = calibrated porosity of the formation, ϕi = calibrated porosity associated with all pores of the ith pore size (also known as the incremental porosity). Therefore, the T2 distribution (in the form of the amplitudes, M0i, associated with the time constants, T2i, is calibrated to the porosity distribution (i.e., the individual pores ϕi with the associated time constants T2i).
If the rock is water-wet and the pores are partially saturated (i.e., the pores contain oil and/or gas in addition to water), then the total signal comprises contributions from each component in the following equation:
where Moil = magnetization produced by oil protons in the pores, Mgas = magnetization produced by gas protons in the pores, T2oil = T2 of oil measured with a CMPG sequence, and T2gas = T2 of gas measured with a CMPG sequence.
In a brine-saturated rock, the T2 decay spectrum represents a pore-size distribution. However, when nonwetting fluids (e.g., oil or gas) are present, the T2 spectrum includes a bulk response from the nonwetting fluid, in addition to the pore-size response. Pores containing the nonwetting fluid either appear in the spectrum at a decay time that is faster than is normally associated with the pores, or do not appear at all if the surface layer is too thin. This behavior affects the appearance of the T2 spectrum and associated T2 distribution but not the total signal amplitude (i.e., porosity).
The raw data recorded by an NMR device are a series of spin-echo amplitudes (echo train) as a function of time, usually at fixed time increments (bins). These NMR measurements are statistical, and stacking is required to improve the precision of the log outputs. The precision depends on formation-fluid properties, activation (acquisition) type, and the number of pulses stacked. The data are mathematically inverted (mapped) by use of a best-fit curve to produce a distribution of T2 values as a function of relaxation time (Figs. 4 and 5). Initially, biexponential-fitting algorithms were used; however, T2 decay in fluid-saturated rocks is multiexponential, because of this, multiexponential models were developed and are commonly used for inverting the data. The inverse solution (T2 distribution) is a function of both the measured echo data and the chosen smoothness (regularization) for the inversion. However, because regularization is controlled in part by the S/N, the fit that is actually used is not unique; that is, there can be a number of differently shaped T2 distributions that fit the original echo-decay curve. In general, the area under the T2-distribution curve (porosity) and the general location in time of the high-porosity bins are robust; however, caution is advised when interpreting the fine details of the distribution. Because the spin echoes are measured over a very short time, an NMR tool travels no more than a few inches along the wellbore while recording the spin-echo train; thus, the recorded spin-echo data can be displayed on a log as a function of depth.
Fig. 5 - T2 distribution. Mathematical inversion is used to convert the spin-echo-decay data to a T2 distribution. This distribution is the most likely distribution of T2 values that produce the echo train. With proper calibration, the area under the T2-distribution curve is equal to the porosity. When the rock is water-saturated, this distribution will correlate with pore-size distribution. The presence of hydrocarbons will affect the T2 distribution depending on the hydrocarbon type, viscosity, and saturation.
The mathematical statement (Eq.9 and Eq.10) that T2 distribution observed in water-saturated rock represents the pore-size distribution and porosity of the rock has been confirmed using mercury-injection capillary pressure (MICP) methods (see Figs. 6 and 7).
NMR responds to pore-body size, and MICP responds to pore-throat size. In clastic rocks in which there is a good correlation between pore-body and pore-throat size, there is often good qualitative agreement between NMR and MICP data, as Figs. 6 and 7 illustrate. The NMR porosity, reflected in the T2 distribution, is a spectrum comprising rock matrix and fluid components (see Fig. 8).
Fig. 8 – NMR-porosity model. These figures illustrate the distribution of pore size and pore fluids used in NMR logging. The suggested NMR mnemonics are Halliburton’s. Because NMR tools respond to the invaded zone, a mud-filtrate component is added between movable water and oil. NMR-logging tools are sensitive to fluids but not to matrix materials and dry clay. NMR porosity is affected by HI, TW, and TE.
Although matrix minerals and dry clay may contain hydrogen atoms in the form of hydroxyl groups, the T1 relaxation times of these nuclei are too long to be polarized by a moving NMR-logging tool, and their T2 relaxation times are too short to be recorded. The fluid component is subdivided into bound and free subcomponents. The hydrogen nuclei of clay-bound water are adsorbed on the surfaces of clay grains. These hydrogen protons can be polarized by NMR-logging tools and recorded when a sufficiently short TE is used. Similarly, hydrogen protons in capillary-bound water and movable fluids (e.g., free water, mud filtrates, oil, and gas), are polarized and recorded by NMR-logging tools with appropriate values for TE and TW.
The porosity and pore-size information from NMR measurements can be used to estimate producible porosity (i.e., the movable fluids) and permeability and for hydrocarbon identification.
|M0||=||macroscopic magnetization, gauss/cm3|
|M0i||=||magnitude of the initial magnetization from the ith component, gauss/cm3|
|M100%||=||magnitude of the magnetization for 100% bulk water, gauss/cm3|
|Mi(0)||=||magnitude of the initial magnetization from the ith component of relaxation gauss/cm3|
|M(0)||=||magnitude of the initial magnetization, gauss/cm3|
|M(t)||=||measured magnetization at time t, gauss/cm3|
|(S/V)i||=||ratio of pore surface (S) to fluid volume (V), of the ith pore, 1/cm|
|(S/V)pore||=||ratio of pore surface (S) to fluid volume (V), 1/cm|
|T1||=||longitudinal relaxation time, seconds|
|T1bulk||=||pore-fluid bulk-T1 relaxation time, seconds|
|T1surface||=||pore-surface T1 relaxation time, seconds|
|T2||=||transverse relaxation time, seconds|
|T2bulk||=||pore-fluid bulk-T2 relaxation time, seconds|
|T2diffusion||=||pore-fluid T2 relaxation time in a magnetic field gradient, seconds|
|T2i||=||pore-fluid surface T2 relaxation time of the ith component, seconds|
|T2surface||=||pore-fluid surface T2 relaxation time, seconds|
|TE||=||CMPG interecho spacing, seconds|
|TW||=||polarization (wait) time, seconds|
|ρ1||=||T1 surface relaxivity, cm/sec|
|ρ2||=||T2 surface relaxivity, cm/sec|
|ϕi||=||calibrated porosity associated with all pores of the i th pore size|
|Moil||=||magnetization produced by oil protons in the pores|
|Mgas||=||magnetization produced by gas protons in the pores|
|T2oil||=||T2 of oil measured with a CMPG sequence|
|T2gas||=||T2 of gas measured with a CMPG sequence|
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