The Timur and Tixier Estimators

Permeability from Porosity and Irreducible Water

A log reads a porosity and, through Archie's equation, a water saturation. It does not read a permeability, which is a rate set by the throats between the pores (Section 9.1). The oldest workaround is to estimate it from two quantities the logs can supply: the porosity $\phi$ and an irreducible water saturation $S_{wi}$, the water a rock keeps however far it is drained (Section 1.3). Figure 9.2 sets two such relations beside Ogbon-1's permeability. Timur (1968) is a power law based on core from sandstone reservoirs, 155 samples from three oil fields according to PetroWiki; Tixier (1949) is quoted as a different power law of the same two quantities:

with $k$ in millidarcies. Timur's is written with $\phi$ and $S_{wi}$ in per cent; written with fractions its constant is 8581 in place of 0.136, because $100^{4.4}/100^{2} = 100^{2.4}$. PetroWiki gives both: "For b=4.4, the value of a is 0.136 if Φ and Swi are in percent and 8,581 if Φ and Swi are fractional values." Tixier's relation is quoted as $k^{1/2} = 250\,\phi^{3}/S_{wi}$ by Torres, Hurtado and Aldana, and as $\log k = 6\log\phi - 2\log S_{wi} + 2\log 250$ by Hurtado, Díaz and Torres, and neither says in what units. The book reads it with fractions, as the kernel does: in per cent it would give 29 billion millidarcies at 23 % porosity and 18 % water.

Two Exponents and One Water Term

Both relations fall as $1/S_{wi}^{2}$, so the irreducible water cancels from their ratio and moves both estimates together: at an irreducible water of 30 % in place of 18 % each is 0.36 times what it was, and doubling the irreducible water divides either by four. They differ in the porosity exponent, 4.4 for Timur and 6 for Tixier once it is squared out, so the ratio of Tixier's to Timur's goes as $7.28\,\phi^{1.6}$: 0.55 at 20 % porosity, and 1 at 28.9 %, the porosity at which the two lines cross. Below it Tixier's is the lower. At a porosity of 20 % and an irreducible water of 20 %, Timur gives 180 mD and Tixier 100 mD.

Ogbon-1's reservoir lies wholly below the crossing, at 8.0 to 27.0 % porosity, so Tixier reads under Timur at every half foot of it. In the main sand, at 6150 ft and 25.6 % porosity, Tixier gives 538 mD against Timur's 655 mD, 0.82 times; in the tight limestone, at 6080 ft and 8.5 %, it gives 0.7 mD against 5.1 mD, 0.14 times, a factor of seven. No single factor calibrates one relation to the other over that range. Each is an empirical power law and has to be checked against core in the rock it is used in.

Why the Irreducible Water

The irreducible water is in the formula as a stand-in for the rock's internal surface. The water that stays when a rock is drained is held by surface tension at the internal surfaces and by capillary pressure (Section 7.3), so a rock that keeps much of it has much surface for its pore volume and fine throats, and a rock that keeps little is coarser and flows more freely. PetroWiki's account of Timur's relation is plain that this is a substitution and not a derivation: "There is no theoretical basis for the substitution of Swi for specific surface area Σ, so although the form of Eq. 2 is similar to that of Eq. 1, it is strictly an empirical relationship."

The Irreducible Water Is Not the Log's Saturation

A log gives a water saturation, and it is tempting to put it in the formula. That is right only where the rock is at its irreducible water. Plate (b) of Figure 9.2 rings the estimate made that way. At 5890 ft in the upper oil sand the log reads 19 % water and Timur gives 387 mD against the 425 mD built, 0.91 times. At 6180 ft, near the end of the pay in the main sand, the log reads 41 %, the rock being on the drainage curve of Section 1.5 and above its irreducible 18 %, and the estimate is 117 mD against 603 mD, 0.19 times. At 6260 ft, in the water leg, the log reads 100 % and the estimate is 21 mD against 635 mD, 0.03 times: the pores are full of water because oil never drained them, not because the rock holds no more. Over the 277.0 ft of net pay the geometric mean falls from 101 mD to 66 mD. PetroWiki warns of the same error: "Thus, a log-based estimate of saturation immediately above the oil/water contact will overestimate Swi." The irreducible water has to come from core, from capillary pressure (Chapter 11), from NMR (Section 10.3), or from the same rock higher in the column.

Ogbon-1 Was Built With Timur's Relation

Ogbon-1's permeability is not a measurement. It was built with Timur's relation at the well's irreducible water, 18 %, in every reservoir half foot, and it stands in for the core the well does not have. The figure takes the porosity as an analyst would, from the density log with each bed's matrix, so Timur at 18 % returns the built permeability exactly: 425 mD at 5890 ft and a geometric mean of 101 mD over the net pay, where Tixier's relation gives 42 mD. That shows nothing about which relation is right. It is a property of a synthetic well, and a real well, whose permeability comes from core, would not close so well with either. The tight limestone is a carbonate, and Timur's paper is about sandstone reservoirs, yet the well was built with his relation in it.

The porosity enters with its exponent, so a log error is multiplied: one porosity unit of error at 5890 ft changes Timur's answer by 20 % and Tixier's by 29 %. Section 9.3 sets permeability against porosity alone, a trend fitted to core, and Section 9.6 reads the permeability of the whole well.

References

• Timur, A. (1968). An investigation of permeability, porosity, and residual water saturation relationships for sandstone reservoirs. The Log Analyst, 9(4). Also listed as Transactions of the SPWLA 9th Annual Logging Symposium, Paper J.
• Tixier, M. P. (1949). Evaluation of permeability from electric-log resistivity gradients. Oil and Gas Journal, June 1949, p. 113. As cited in US Patent 6,691,037, Log permeability model calibration using reservoir fluid flow measurements.
• PetroWiki. Estimating permeability based on surface area and water saturation. https://petrowiki.spe.org/Estimating_permeability_based_on_surface_area_and_water_saturation
• Torres, J., Hurtado, N. and Aldana, M. Comparación de tres técnicas distintas con datos reales de pozo, en la determinación de la permeabilidad. arXiv:1311.3704.
• Hurtado, N., Díaz, R. and Torres, J. Training hybrid neuro-fuzzy system to infer permeability in wells on Maracaibo Lake, Venezuela. arXiv:1409.1264.