Swirr from NMR

Summary

A nuclear magnetic resonance log measures a T2 distribution, in which small pores relax quickly and large pores slowly. The part of the distribution below a T2 cutoff is Bound volume irreducible, the bound fluid, and Irreducible water saturation is that volume divided by porosity. Use it where an NMR log and a cutoff calibrated to core exist; it is the only method here that is measured at the well, with no dependence on permeability or height.

Inputs and outputs

	Item	Units
Input	Total porosity	v/v
Input	T2 logarithmic mean	ms
Input	T2 distribution width	ln units
Input	T2 cutoff	ms
Output	Bound volume irreducible	v/v
Output	Free fluid index	v/v
Output	Irreducible water saturation	v/v

Equations

The NMR porosity splits at the T2 cutoff into the bound volume and the free fluid:

\[ \phit = \CpBVI + \CpFFI \qquad \CpBVI = \int_0^{\CpTtCut} \phit\,f(T_2)\,dT_2 \]

where \(f(T_2)\) is the normalized T2 distribution. Swirr is the bound fraction of the pore volume:

\[ \Swirr = \frac{\CpBVI}{\phit} \]

The calculator models the distribution as one lognormal mode with logarithmic mean \(T_{2,lm}\) and width \(\sigma_{T2}\), for which the bound fraction is the normal cumulative distribution function \(\Phi\):

\[ \frac{\CpBVI}{\phit} = \Phi\!\left(\frac{\ln \CpTtCut - \ln \CpTtLm}{\CpTtSig}\right) \]

Symbol	Variable	Units	Typical range
\(\phi_t\)	Total porosity	v/v	0 to 0.40
\(T_{2,lm}\)	T2 logarithmic mean	ms	5 to 1000
\(\sigma_{T2}\)	T2 distribution width	ln units	0.5 to 1.5
\(T_{2,cut}\)	T2 cutoff	ms	10 to 100
\(BVI\)	Bound volume irreducible	v/v	0.01 to 0.2
\(FFI\)	Free fluid index	v/v	0 to 0.3
\(S_{wirr}\)	Irreducible water saturation	v/v	0.05 to 0.5

Single-value calculator

Behavior

Swirr rises as the cutoff rises, because more of the distribution is counted as bound, and falls as the T2 log mean rises, because bigger pores relax more slowly. For a width of 1.0, a log mean of 100 ms gives Swirr of 0.011, 0.134 and 0.467 at cutoffs of 10, 33 and 92 ms; a log mean of 40 ms gives 0.083, 0.424 and 0.798, and a log mean of 250 ms gives 0.001, 0.021 and 0.159. Moving the cutoff from 33 to 92 ms multiplies Swirr by 3.5 for the log mean of 100 ms: the cutoff is the main uncertainty.

Parameter guidance

The cutoff is set by core NMR: a plug is measured fully saturated and again after centrifuging at a chosen capillary pressure, and the cutoff is the T2 at which the cumulative curves of the two states match. Commonly quoted typical values are about 33 ms for clastic rock and about 92 ms for carbonates, with clay-bound water often handled with a separate cutoff near 3 ms. Treat these as starting points to be checked; the cutoff varies with rock type, surface relaxivity and the pressure used for the centrifuge. A cutoff chosen at a different capillary pressure from the one that defines Swirr in the field gives an inconsistent Swirr. The T2 mean and width in the calculator stand in for a real distribution, and real distributions are often bimodal.

Worked example

A real log gives a T2 distribution by bins; here a small made-up distribution with a total porosity of 0.20:

import numpy as np
t2 = np.array([1, 3, 10, 30, 60, 100, 300, 1000])        # bin centres, ms
amp = np.array([0.004, 0.012, 0.030, 0.040, 0.030, 0.040, 0.034, 0.010])  # porosity per bin (v/v)
phit = amp.sum()
print(f"porosity from the distribution = {phit:.3f}")
for cut in (33, 92):
    bvi = amp[t2 < cut].sum()
    print(f"cutoff {cut:3d} ms: BVI = {bvi:.3f}, FFI = {phit - bvi:.3f}, Swirr = {bvi / phit:.3f}")

Output

porosity from the distribution = 0.200
cutoff  33 ms: BVI = 0.086, FFI = 0.114, Swirr = 0.430
cutoff  92 ms: BVI = 0.116, FFI = 0.084, Swirr = 0.580

Assumptions and limitations

The T2 distribution reflects pore size only. Hydrocarbon in the pores (especially gas or light oil) changes the T2 and the apparent porosity, and the measurement needs correction.
A single cutoff separates bound from free fluid. Mixed lithology, heavy iron-rich minerals and rough surfaces change the relaxivity, and so the cutoff.
The cutoff was calibrated at a capillary pressure that corresponds to irreducible saturation in the field.
Clay-bound water is counted as bound. The result is then on a total porosity basis.

QC checks

BVI plus FFI equals the NMR porosity, and the NMR porosity is close to the log porosity in water-bearing zones.
Swirr from NMR follows the trend of Buckles, permeability-based and RQI results in the same rock type.
The cutoff is the same as the one calibrated on core, and its source is documented.
BVI is not above the water saturation times porosity in hydrocarbon zones. Where it is, the cutoff is too long.

Going Deeper

NMR Swirr is the nearest thing to a direct log measurement of irreducible water, and so is a good reference for the other methods. Its weakness is that the bound/free division is a calibration to core at one capillary pressure, and not a property of the rock. Coates and co-authors describe the commonly used cutoffs; a distribution-based variant replaces the single cutoff with a pore-size-dependent weight. NMR permeability models, the Coates and SDR relations, use the same BVI and FFI as inputs, which connects this page to the Permeability step.

References

Coates, G.R., Xiao, L. and Prammer, M.G., 1999. NMR Logging Principles and Applications. Halliburton Energy Services, Houston, TX.

Python reference implementation

Python reference implementation

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