Timur

Summary

Timur's relation estimates Permeability from porosity and Irreducible water saturation, both in percent, with a refit of the Wyllie-Rose form to sandstone core. Use it for clean to moderately shaly sandstones with a credible Swirr and, as with every equation here, calibrate the multiplier to core.

Inputs and outputs

	Item	Units
Input	Total 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

Porosity and Swirr are entered in percent, which is why the factors of 100 appear:

\[ k = \CpPermP\,\frac{\left(100\,\phit\right)^{\CpPermQ}}{\left(100\,\Swirr\right)^{\CpPermR}} \]

with \(P = 0.136\), \(Q = 4.4\) and \(R = 2\) in the published form (k in mD). The result is limited to the interval 0 to 10 000 mD in the calculator.

Symbol	Variable	Units	Typical range
\(\phi_t\)	Total porosity	v/v	0 to 0.40
\(S_{wirr}\)	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

The behavior is almost the same as Wyllie-Rose, with the constants chosen for sandstone. At a Swirr of 0.25 permeability is 5.5, 32.5, 115.4 and 308 mD at porosities of 0.10, 0.15, 0.20 and 0.25; at a porosity of 0.20, Swirr of 0.15, 0.25 and 0.35 give 321, 115 and 59 mD. Doubling the Swirr divides permeability by 4, and a 10% error in Swirr is roughly a 20% error in permeability. 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

In percent units the multiplier looks very different from the Wyllie-Rose one, but with fractions the equation is k = 8581 x porosity^4.4 / Swirr^2, because 0.136 x 100^(4.4 - 2) = 0.136 x 100^2.4 = 8581; the worked example confirms it. Fit \(P\) to core first, and leave the exponents unless there are many cores in the rock type. Use effective or total porosity consistently with the fit; the published constants are for total (core) porosity as far as I know. Swirr must come from a method that does not use permeability, see the Permeability step page.

Worked example

Porosity 20% and Swirr 25%, and the conversion of the percent form to the fractional form:

phit, swirr = 0.20, 0.25
k = 0.136 * (100 * phit) ** 4.4 / (100 * swirr) ** 2.0
print(f"k = 0.136 x ({100 * phit:g})^4.4 / ({100 * swirr:g})^2 = {k:.1f} mD")
# the same equation with porosity and Swirr as fractions
p_frac = 0.136 * 100 ** (4.4 - 2.0)
print(f"fractional form: P = 0.136 x 100^2.4 = {p_frac:.0f}; k = {p_frac * phit ** 4.4 / swirr ** 2.0:.1f} mD")

Output

k = 0.136 x (20)^4.4 / (25)^2 = 115.4 mD
fractional form: P = 0.136 x 100^2.4 = 8581; k = 115.4 mD

Assumptions and limitations

The relation was derived for sandstones; carbonates and heavily clayey rock do not follow it.
Porosity and Swirr are in percent in the published form. A fraction in place of a percent gives a result that is wrong by orders of magnitude.
Swirr is the irreducible water saturation, not the log Sw.
The constants, calibrated on a data set of core samples, hold for the rock being evaluated.

QC checks

Check units: porosity and Swirr as percent in the equation (or with the converted multiplier).
Plot against core permeability on a log-log 1:1 plot: scatter of a factor of 2 to 3 is normal.
Permeability is smooth: it should not step where the Swirr method or rock type changes.
Compare with Wyllie-Rose and Tixier on the same inputs; a large difference means the constants need work.

Going Deeper

Timur published the relation in 1968 after fitting porosity, permeability and residual water saturation of sandstone cores from a number of fields (the count I remember is 155 samples). It is a statistical refit of the form that Wyllie and Rose proposed, and the near-equivalence of the two is a reminder that the sensible variation in the exponents between fits is within the scatter of the data. Its use with a Swirr from NMR or from a saturation-height model is the usual practice; its use with a log Sw is a common mistake.

References

Timur, A., 1968. An investigation of permeability, porosity, and residual water saturation relationships for sandstone reservoirs. The Log Analyst, 9(4), 8–17.

Python reference implementation

Python reference implementation

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