Permeability from NMR

What the Estimators Have Been Guessing

Every estimator so far has inferred the size of the throats from something else: the porosity, the irreducible water of Section 9.2, a crossplot. An NMR log is the one log built on pore size. A small pore holds its fluid against a short relaxation time, T2, and a large pore against a long one, so a distribution of T2 is a picture of the pore-size spectrum. How the tool measures it, and what the distribution means, is Chapter 10 (Sections 10.1 to 10.3). This section takes only the two best-known ways of turning such a distribution into a permeability.

Ogbon-1 has no NMR log. Its logs are the gamma ray, the SP, the density, the neutron, the sonic, the resistivity and the photoelectric factor, so there is no T2 curve to read. Figure 9.5 therefore builds a what-if spectrum from the two things the well does state: the porosity of the half foot read, from the density as Section 1.1 reads it, and the well's irreducible water saturation, 18 %. The irreducible water goes in a bound peak at 4 ms and the rest of the pore volume in a free peak at a T2 you set; each peak is a normal curve in the logarithm of T2, 0.3 decades wide, and the amplitudes add up to the porosity. The 4 ms, the width and the two-peak shape are the figure's own assumptions, not a measurement, and oil and water are not separated. At 5890 ft in the upper oil sand the rock is 23.2 % pore space, and a free peak at 300 ms is where the figure starts.

Timur-Coates: the Split

A T2 cutoff divides the spectrum. The fluid below it is taken as bound, the bulk volume irreducible, $BVI$; the fluid above it as free, the free-fluid index, $FFI$. For a sandstone with no core to calibrate against, the usual default cutoff is 33 ms, and for a carbonate 92 ms (Coates, Xiao and Prammer, 1999); the figure's cutoff slider moves it from 10 ms to 120 ms. The Timur-Coates permeability is the ratio of the two raised to the second power, times the porosity to the fourth:

with $\phi$ a fraction, $k$ in mD, and $C$ = 10 as the default. That is the common form with a prefactor of 10,000 mD and exponents 4 and 2 on a fractional porosity (US Patent 11,933,932 B1; US Patent 6,959,246 B2 calls 10,000, 4 and 2 the common values), and Coates, Xiao and Prammer (1999) describe $C$ as a constant that depends on the formation. Two readings of it are worth having. When the bound fluid is the irreducible water, $BVI = \phi S_{wi}$ (US Patent 6,959,246 B2), the ratio is $(1 - S_{wi})/S_{wi}$ and the transform is a statement about the porosity and the irreducible water, as Timur's is. And in practice the bound fluid is whatever lies below the cutoff, so the answer depends on where the cutoff is put.

With the what-if spectrum at 5890 ft, 4.2 % of the rock is bound and 19.0 % free, a ratio of 4.5, and Timur-Coates gives 595 mD. Had the cutoff fallen exactly on the irreducible water the ratio would be 4.6 and the answer 598 mD. Ogbon-1's own permeability at that depth, Timur's relation with the irreducible 18 %, is 425 mD, so Timur-Coates reads 1.40 times that with the default constant.

SDR: the Average

The SDR transform, from the Schlumberger-Doll Research work of Kenyon and co-authors (1988), uses no cutoff. It takes the logarithmic mean of the whole distribution, $T_{2gm}$, the geometric mean of T2 weighted by amplitude:

with $T_{2gm}$ in ms and $a$ in mD per ms squared. PetroWiki gives $a$ = 4.5 for a sandstone and 0.1 for a carbonate, and the figure takes the 4.5; US Patent 11,933,932 B1 gives 4.6 and 0.46, citing Singer (1997). The what-if spectrum has a log mean of 138 ms, so SDR gives 247 mD at 5890 ft, 0.58 times Ogbon-1's own. The two transforms read the same spectrum and differ by a factor of 2.4: neither constant has been calibrated to anything.

What Each One Sees

Move the cutoff from 33 ms to 100 ms and 1.1 % of the rock changes sides from free to bound: BVI rises from 4.2 % to 5.2 % and Timur-Coates falls from 595 mD to 339 mD, 1.76 times lower, for a spectrum that has not changed. SDR stays at 247 mD, because it never uses the cutoff.

Now shrink the pores instead: move the free peak from 300 ms to 100 ms. The log-mean T2 falls from 138 ms to 56.0 ms and SDR, which goes with its square, falls from 247 mD to 41 mD, 6.1 times lower. Timur-Coates falls 1.73 times, from 595 mD to 345 mD, and only because the lower tail of the free peak now crosses the 33 ms cutoff and BVI rises to 5.2 %. Timur-Coates sees how much fluid is on each side of a line; SDR sees how far along T2 the fluid sits.

The split fails when the peaks meet the cutoff. With the free peak at 40 ms and the cutoff at 33 ms, BVI is 11.6 % against the irreducible water of 4.2 %, the ratio is 1.00 and Timur-Coates reads 29 mD against 425 mD. The formula has not failed; its input has, because a ratio of free to bound fluid measures the irreducible water only while the bound and free fluids lie either side of the cutoff.

Calibrate, and Say to What

Both transforms carry a constant, and the sources are plain about it. PetroWiki says the model is "calibrated over a particular zone of interest and verified, wherever possible, by core or well/formation test data", and that calibration is a regression of core permeability on the transform. Ogbon-1 has no core, and its permeability is itself Timur's relation, built into the well from the porosity and the irreducible water, so the only calibration available is to that. Set at 5890 ft to give 425 mD, the constants are $C$ = 10.9 and $a$ = 7.8. On one spectrum both transforms then go as the porosity to the fourth power and read alike at every depth: 582 mD against 600 mD in the main sand at 6190 ft, and 7.6 mD against 5.1 mD in the tight limestone at 6080 ft. Timur's relation goes with the porosity to the power 4.4 and the transforms, with the ratio and the log mean the same in every bed of this figure, to the power 4, so one constant fits one porosity and drifts away from it.

The limestone carries a second warning. The 33 ms cutoff and both constants are a sandstone's; for a carbonate Coates, Xiao and Prammer give a cutoff of 92 ms, PetroWiki an SDR constant of 0.1 and US Patent 11,933,932 B1 0.46. In the tight limestone the figure says so, and its two numbers demonstrate the formulas, not the rock. Whatever the constants, the comparison here is between formulas. It says what each transform responds to, not how close either is to a rock.

Where Next

Section 9.6 runs a permeability down the whole of Ogbon-1. Chapter 10 builds what this section assumed: what an NMR tool measures (10.1), what the T2 distribution is (10.2), how the cutoff is chosen (10.3), and an NMR log on Ogbon-1 (10.6).

References

• Coates, G. R., Xiao, L. and Prammer, M. G. (1999). NMR Logging Principles and Applications. Halliburton Energy Services.
• Kenyon, W. E., Day, P. I., Straley, C. and Willemsen, J. F. (1988). A three-part study of NMR longitudinal relaxation properties of water-saturated sandstones. SPE Formation Evaluation, 3(3), 622-636. SPE-15643-PA.
• US Patent 11,933,932 B1. Method of estimating permeability using NMR diffusion measurements. https://patents.google.com/patent/US11933932
• US Patent 6,959,246 B2. Carbonate permeability. https://patents.google.com/patent/US6959246
• PetroWiki. Estimating permeability based on surface area and water saturation. https://petrowiki.spe.org/Estimating_permeability_based_on_surface_area_and_water_saturation
• PetroWiki. Permeability estimation with NMR logging. https://petrowiki.spe.org/Permeability_estimation_with_NMR_logging