Tixier

Summary

The Tixier relation, in the form popularised by Morris and Biggs, gives the square root of Permeability as a multiple of porosity cubed over Irreducible water saturation. The multiplier is different for oil and gas. Use it as a quick check of other models or where the Swirr and porosity are good and core is scarce.

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

The square root of permeability is proportional to porosity cubed over irreducible water saturation:

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

with \(P = 250\) for a medium-gravity oil, \(P = 79\) for a dry gas, \(Q = 3\) and \(R = 2\) (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

At a Swirr of 0.25 and the oil multiplier, permeability is 1.0, 11.4, 64 and 244 mD at porosities of 0.10, 0.15, 0.20 and 0.25. Compare the 6.4 mD that the gas multiplier of 79 gives at a porosity of 0.20: the multiplier alone changes the answer by a factor of 10. At a porosity of 0.20 Swirr of 0.15, 0.25 and 0.35 give 178, 64 and 33 mD. Permeability grows as the sixth power of porosity, which makes this model steeper than the others, and so more sensitive to porosity error: a 10% error in porosity is a 77% 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

Pick the multiplier by fluid, as published: 250 for medium-gravity oil and 79 for gas. That the fluid appears in a constant of a relation based on irreducible water is the clearest sign that the multipliers are calibrations to data and not physics. The porosity is total porosity and the Swirr is the irreducible value from a method that does not use permeability; see Swirr. Fit \(P\) to core for the rock type, and fix the exponents at 3 and 2 unless the core data justify changing them.

Worked example

Porosity 0.20 and Swirr 0.25 with the oil and gas multipliers:

phit, swirr = 0.20, 0.25
for fluid, p in (('oil', 250.0), ('gas', 79.0)):
    root_k = p * phit ** 3 / swirr
    print(f"{fluid}: sqrt(k) = {p:g} x {phit:g}^3 / {swirr:g} = {root_k:.3f}, k = {root_k ** 2:.1f} mD")
print(f"porosity +10%: k x {1.1 ** 6:.2f}")

Output

oil: sqrt(k) = 250 x 0.2^3 / 0.25 = 8.000, k = 64.0 mD
gas: sqrt(k) = 79 x 0.2^3 / 0.25 = 2.528, k = 6.4 mD
porosity +10%: k x 1.77

Assumptions and limitations

Permeability depends on porosity and Swirr only, with the fluid captured by a multiplier.
The multiplier 250 or 79 suits the fluid and the rock of the original data. Another reservoir will have a different value.
Swirr is the irreducible water saturation, and the porosity is on the basis of the calibration.
The relation is for clean to moderately shaly sandstone.

QC checks

Plot against core permeability on a log-log 1:1 plot; scatter of a factor of 2 to 3 is normal.
Check the fluid: using the oil multiplier in a gas zone overstates permeability by about a factor of 10.
Permeability follows porosity to a high power, so check that porosity is not noisy or too high from gas effects.
Compare with Timur and Wyllie-Rose, which should be of the same size.

Going Deeper

Tixier proposed in 1949 that permeability could be found from the gradient of resistivity in the transition zone. The relation that is usually quoted as Tixier's, with a cube of porosity over Swirr, appeared later and is often credited with Morris and Biggs, who wrote it with separate multipliers for oil and gas. I could not fully trace which of the forms in circulation goes with which author, and the equation is now a template. It is used mainly as a sanity check of a better calibrated model.

References

Tixier, M.P., 1949. Evaluation of permeability from electric-log resistivity gradients. Oil and Gas Journal, 48(6), 113–133.
Morris, R.L. and Biggs, W.P., 1967. Using log-derived values of water saturation and porosity. Transactions of the SPWLA 8th Annual Logging Symposium, Paper X.

Python reference implementation

Python reference implementation

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