The T2 Distribution

Chapter 10, NMR Logging

Learning objectives

Explain why a rock's NMR decay is a sum of exponentials, one for each pore size
Read the T2 distribution as a pore-size spectrum through $T_2 = r/(3\,\rho_2)$
Compute the echo train forward from a pore-size distribution and see that it carries the same information
Explain why the surface relaxivity, a mineral property, shifts the whole spectrum, so that one cutoff cannot serve every rock

Many Pores, Many Decays

The single relaxation time of Section 10.1 is the simplest case. A real rock holds pores of many sizes at once, and the fluid in each relaxes at its own rate, so the echo train the tool records is a sum of exponentials:

M(t) = \sum_j a_j\,e^{-t/T_{2,j}},

where $a_j$ is the volume of fluid that relaxes with time $T_{2,j}$ . The T2 distribution is that set of amplitudes drawn against T2, a curve that says how much fluid relaxes at each time. At zero time the train is the sum of the amplitudes, which is the porosity. In PetroWiki's words, "In a brine-saturated rock, the T2 decay spectrum represents a pore-size distribution."

The T2 Axis Is a Pore-Size Axis

A proton relaxes when it meets a grain surface, at a rate that the surface relaxivity $\rho_2$ (a length per time, set by the mineral) multiplies by the pore's surface-to-volume ratio. For a spherical pore of radius $r$ that ratio is $3/r$ :

\frac{1}{T_2} = \rho_2\,\frac{S}{V} = \frac{3\rho_2}{r} \quad\Longrightarrow\quad T_2 = \frac{r}{3\rho_2}.

The relation holds when the fluid crosses the pore faster than the wall relaxes it, the fast-diffusion limit that Brownstein and Tarr (1979) analysed, and when surface relaxation is the only mechanism, which the figure assumes. T2 is then proportional to the pore radius: the short T2 on the left is small pores, the long T2 on the right is big ones, and reading the T2 distribution is reading the pore-size spectrum, on a scale that the relaxivity sets.

The figure builds a model sandstone to show it. Its porosity is that of Ogbon-1's upper oil sand, 23.0 %. The well has no pore-size data, so the pore sizes are the figure's own assumption: 30 % of the pore space in micropores of radius 1.0 μm and the rest in macropores of radius 10 μm, each group spread lognormally about its radius. At a surface relaxivity of 10 μm/s, the book's value for quartz, the sandstone surface (Section 10.1), the micropores relax at 33.3 ms and the macropores at 333 ms, and a T2 of 100 ms stands for a pore of radius 3.0 μm. Plates (a) and (b) show the same volumes against radius and against T2.

The Spectrum and the Decay Carry the Same Information

Plate (c) runs the measurement forward. Each pore size relaxes by its own exponential and the train is their sum, starting at the porosity, 23.0 %. Nothing is in the train that is not in the distribution, and an inversion of a measured train is how the tool estimates the distribution; the figure runs only the forward direction. It also shows why one exponential will not do. The log-mean T2 is

T_{2,\mathrm{lm}} = \exp\frac{\sum_j a_j \ln T_{2,j}}{\sum_j a_j},

the T2 that the SDR permeability equation of Section 9.5 takes, and here it is 167 ms. A single exponential at that T2 has 37 % of its start left at that time, and the real train has 42 %. The two curves part by as much as 12 % of the start, at 378 ms, with the train under the exponential early and over it late.

The Mineral Shifts the Whole Spectrum

The relaxivity is a property of the mineral, and it varies. Marschall, Gardner, Mardon and Coates (1995), correlating NMR with mercury-injection data on cores, found effective relaxivities of 6.4 to 25 μm/s for their sandstones, 1.0 to 3.2 μm/s for their limestones, and 4.6 and 5.4 μm/s for their two dolomites. Hold the pores fixed and lower the relaxivity from 10 μm/s to 5 μm/s, as for a dolomite, and every T2 becomes twice as long: the micropores at 66.7 ms, the macropores at 667 ms, the log-mean at 334 ms, with the porosity unchanged at 23.0 %. At 3 μm/s, as for a limestone, every T2 is 3.3 times longer.

The same pores sit at different T2 in different rocks, so no one T2 cutoff can be right for all of them. A line at 50 ms falls between the two peaks of the sandstone, at 33.3 ms and 333 ms, but across the micropore peak of the dolomite, at 66.7 ms. Setting the cutoff for each rock is the subject of Section 10.3. For the logging practice built on all this, see Kenyon (1997) and Coates, Xiao and Prammer (1999).

Where the Model Stops

The figure models surface relaxation alone. The brine's own bulk relaxation, and the diffusion in the tool's field gradient that Section 10.5 treats, shorten the longest T2 and are left out, so the figure's sliders stop where the largest macropores reach 2.0 s.

References

Kenyon, W. E. (1997). Petrophysical principles of applications of NMR logging. The Log Analyst, 38(2), 21-43.
Coates, G. R., Xiao, L., and Prammer, M. G. (1999). NMR Logging Principles and Applications. Halliburton Energy Services.
Brownstein, K. R. and Tarr, C. E. (1979). Importance of classical diffusion in NMR studies of water in biological cells. Physical Review A, 19(6), 2446-2453. doi:10.1103/PhysRevA.19.2446.
Marschall, D., Gardner, J. S., Mardon, D., and Coates, G. R. (1995). Method for correlating NMR relaxometry and mercury injection data. Society of Core Analysts conference paper SCA-9511.
PetroWiki. NMR petrophysics. Society of Petroleum Engineers, petrowiki.spe.org/NMR_petrophysics.