The T2 Distribution

Many Pores, Many Decays

The single relaxation time of the last section was a fiction of convenience. A real rock holds pores of many sizes at once, and the fluid in each relaxes at its own rate, so the measured decay is a sum of exponentials. Mathematically inverting that sum, untangling how much fluid decays at each rate, produces the T2 distribution: a curve of amplitude against T2 that says how much pore volume sits at each relaxation time.

The T2 Axis Is a Pore-Size Axis

What makes it powerful is surface relaxation. A proton relaxes when it touches a grain surface, so the rate scales with the surface-to-volume ratio of the pore:

for a spherical pore of radius $r$. T2 is therefore proportional to pore size: the short-T2 left of the distribution is small pores, the long-T2 right is big ones. Reading the T2 distribution is reading the pore-size spectrum directly, the very thing every permeability estimator was straining to infer.

Lithology Shifts the Whole Spectrum

The catch is $\rho_2$, the surface relaxivity, a mineral property. Clay-rich sandstones relax fast (high $\rho_2$), packing the pore-size spectrum at short T2; clean carbonates relax slowly (low $\rho_2$), stretching the same pores out to long T2. Slide the surface relaxivity and the whole distribution marches along the axis. That is why a single fixed T2 cutoff cannot serve every rock, and why the bound-fluid cutoff of the next section must be set per lithology.

References

• Kenyon, W. E. (1997). Petrophysical principles of applications of NMR logging. The Log Analyst, 38(2).
• Coates, G. R., Xiao, L., and Prammer, M. G. (1999). NMR Logging Principles and Applications. Halliburton Energy Services.