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Coates

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Summary

The Coates relation estimates Permeability from porosity and the ratio of free fluid to bound fluid, (1 - Irreducible water saturation) / Swirr. The same form applied to NMR, where the free-fluid and bound-fluid volumes are measured, is the Coates (free-fluid) permeability. Use it when effective porosity and a credible Swirr are available, or an NMR log.

Inputs and outputs

Item Units
Input Effective porosity v/v
Input Irreducible water saturation v/v
Input Permeability model multiplier P mD (equation dependent)
Input Permeability model porosity exponent Q dimensionless
Input Permeability model saturation exponent R dimensionless
Output Permeability mD
Output Log permeability log10(mD)

Equations

The free-fluid (Coates) form uses effective porosity and the ratio of free to bound water:

\[ k = \CpPermP\,\phie^{\CpPermQ}\left(\frac{1 - \Swirr}{\Swirr}\right)^{\CpPermR} \]

The widely quoted constants are \(P = 10^4\), \(Q = 4\) and \(R = 2\) (with \(k\) in mD), which is the same as

\[ \sqrt{k} = 100\,\phie^{2}\,\frac{1 - \Swirr}{\Swirr} \]

With NMR, the ratio is measured: \((1 - \Swirr)/\Swirr\) is replaced by FFI/BVI, and with the porosity in percent the equation is \(k = (\phi/C)^4\,(\mathrm{FFI}/\mathrm{BVI})^2\) with \(C\) = 10 for sandstones (see Swirr from NMR). The result is limited to the interval 0 to 10 000 mD in the calculator.

Symbol Variable Units Typical range
\(\phi_e\) Effective porosity v/v 0 to 0.35
\(S_{wirr}\) Irreducible water saturation v/v 0.05 to 0.5
\(P\) Permeability model multiplier P mD (equation dependent)
\(Q\) Permeability model porosity exponent Q dimensionless 2 to 6
\(R\) Permeability model saturation exponent R dimensionless 1 to 4
\(k\) Permeability mD 0.0001 to 10000
\(\log_{10}k\) Log permeability log10(mD) -4 to 4

Single-value calculator

Behavior

Permeability grows as the fourth power of porosity and as the square of the free-to-bound ratio. At a Swirr of 0.25 and constants of 10 000, 4 and 2, permeability is 9.0, 45.6, 144 and 352 mD at porosities of 0.10, 0.15, 0.20 and 0.25. The Swirr dependence is stronger at the low end than for the other models: at a porosity of 0.20, raising Swirr from 0.25 to 0.50 cuts permeability from 144 to 16 mD, a factor of 9, because the ratio goes from 3 to 1. The only reliable use of these equations is as a functional form to be calibrated to core: the constants below are starting values.

Parameter guidance

\(P\) is the calibration knob; start with the standard exponents of 4 and 2 and fit \(P\) to core for each rock type, and only free the exponents if the core data are plentiful. The Swirr must be on the effective porosity basis, because the free-fluid argument excludes clay-bound water, and it must come from a source that does not use permeability (see the Permeability step page). With NMR, the 10 in the NMR form is a typical value for sandstone, and carbonates are fitted separately; treat both as constants to be calibrated.

Worked example

Effective porosity 0.20 at Swirr 0.25 and 0.50, and the equivalent NMR-style form in percent porosity:

phie = 0.20
for swirr in (0.25, 0.50):
    ratio = (1 - swirr) / swirr
    k = 1e4 * phie ** 4 * ratio ** 2
    print(f"Swirr {swirr:.2f}: FFI/BVI = {ratio:.2f}, k = {k:.1f} mD")
# NMR form with porosity in percent: k = (phi / 10)^4 x (FFI/BVI)^2
k_nmr = (20.0 / 10.0) ** 4 * (0.75 / 0.25) ** 2
print(f"NMR form, phi = 20 pu, FFI/BVI = 3: k = {k_nmr:.1f} mD")

Output

Swirr 0.25: FFI/BVI = 3.00, k = 144.0 mD
Swirr 0.50: FFI/BVI = 1.00, k = 16.0 mD
NMR form, phi = 20 pu, FFI/BVI = 3: k = 144.0 mD

Assumptions and limitations

  • Permeability follows the product of porosity to a power and the free-to-bound fluid ratio. Both are surrogates of pore size.
  • Swirr is effective-porosity irreducible saturation and is accurate; the ratio is very sensitive to it at high Swirr.
  • The constants are for a clastic rock of the type used in the calibration.
  • For NMR, the cutoff that defines BVI is calibrated, and hydrocarbon effects on the NMR porosity have been corrected.

QC checks

  • Plot against core permeability on a log-log 1:1 plot. Scatter of a factor of 2 to 3 is normal.
  • Permeability increases with porosity and decreases with Swirr.
  • Check that no sample has a Swirr above about 0.5: the free-to-bound ratio is then below 1 and the result collapses.
  • With NMR, check that BVI plus FFI matches the porosity.

Going Deeper

The form was published by Coates and Dumanoir in 1974 as a log-based permeability estimate with an exponent tied to the cementation exponent, and then became the Coates free-fluid equation with fixed exponents in the producibility work of Coates and Denoo in the early 1980s. Its attraction is the use of the free-fluid ratio, which is exactly what an NMR log measures. The Coates-Dumanoir original is not reproduced here; the fixed-exponent form is the one in common use. A related NMR relation, the SDR equation, uses the log mean T2 instead of the free-fluid ratio.

References

  1. Coates, G.R. and Dumanoir, J.L., 1974. A new approach to improved log-derived permeability. The Log Analyst, 15(1), 17–31.

Python reference implementation

Python reference implementation

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