The Indonesia Equation

A Sum of Square Roots

Section 8.2 wrote the clay's current as a conductance in parallel with the brine's. Poupon and Leveaux (1971) wrote it another way: they add the square roots of a shale term and a sand term and scale the sum by the saturation:

The second term is the sand's, the clean-rock conductance of Archie's equation under a root. The first is the shale's, from its volume $V_{sh}$ and its resistivity $R_{sh}$, read from a nearby shale. With no shale the first term is zero and what is left is Archie's equation. Solved for the saturation the equation is a closed form, not a quadratic, and PetroWiki notes that for that reason it always gives a result greater than zero.

The equation is empirical. Hari Kumar writes of the genesis of the equation "to meet the requirements of fresh water formations of Indonesia". PetroWiki says the formula was empirically modeled with field data in water-bearing shaly sands, that its detailed functionality for hydrocarbon-bearing sands is unsupported except by common sense and long-standing use, and that although it was initially modeled on Indonesian data it can be applied everywhere. Take it as a fitted form with a long record, not a derivation.

The Exponent

The odd part is the exponent. A shale volume between 0 and 1 raised to a power below 1 is larger than itself, and $1 - V_{sh}/2$ is below 1 for any shale, so $V_{sh}^{\,1-V_{sh}/2}$ is always more than $V_{sh}$. It is $V_{sh}$ times $V_{sh}^{-V_{sh}/2}$, a factor that rises from 1 with no shale to its greatest, 1.20 times, at a shale volume of $1/e$, 37 %, and falls back toward 1 at both ends: 1.20 times at 30 % shale, the figure's start, and 1.17 times at 60 %. Simandoux's equation divides its water term by $1 - V_{sh}$; this one does not. The sources checked give no derivation of the exponent: it is a fit.

A What-If Rock

Testing the equation needs a rock whose true saturation is known and which has clay current in it, and Ogbon-1 is not that rock: its resistivity was built as clean Archie in every bed, the shaly oil sand included (Section 7.1). So plates (a) and (b) use a what-if rock, the figure's own device and not the rock of Section 8.2: the shaly oil sand's mean porosity, 17.0 %, and mean built water saturation, 19.6 %, with clay added. Its resistivity is written forward from a law you choose, Poupon and Leveaux's or Simandoux's, and then each model reads that resistivity back. The law that wrote the rock reads it back exactly, which checks the arithmetic and says nothing about whether the law is right; the other models show how far a different law reads from it.

With 30 % shale of 2.20 ohm.m, Ogbon-1's brine of 0.040 ohm.m and the Indonesia law, the rock reads 21.8 ohm.m, where the same rock without clay reads 36.0 ohm.m. Clean Archie puts all of that on the water and reads 25.2 %, 5.6 points high. Simandoux reads 15.5 %, 4.1 points low, and Indonesia the 19.6 % it wrote. Switch to Simandoux's law and the rock reads 15.1 ohm.m; now Simandoux is exact and Indonesia reads 23.6 %, 4.0 points high. A gap of about four points either way is the model-choice uncertainty, and a log does not say which law a real rock follows, so the choice has to be calibrated against core. On Ogbon-1's own logs, with the well's shale resistivity, Indonesia reads above Simandoux at every one of the 772 clean half feet.

Fresh Water

Plate (b) moves the rock's water from Ogbon-1's brine to fresher water. Fresher water carries less current, so the clay's share grows: the shale term is 22 % of Indonesia's sum in the brine and 50 % at 0.50 ohm.m. The models part with it. At 0.50 ohm.m and 30 % shale the rock that obeys the Indonesia law reads 112 ohm.m; clean Archie reads 39.3 %, 19.7 points high, and Simandoux 6.3 %, 13.3 points low, where in the brine they read 25.2 % and 15.5 %. That is the water the equation was made for, and also the reason for caution about Ogbon-1: Hari Kumar's abstract says that the "Indonesian equation at low Rw regime demands truncation for the overshooting results of the model", and Ogbon-1's 0.040 ohm.m is a low Rw (the paper's own boundary between low and high was not read). PetroWiki, by contrast, says the model can be applied everywhere. The two views are an argument for calibrating, not an answer.

On Ogbon-1

Plate (c) runs the equation on Ogbon-1 itself, with the shale volume from the gamma ray (Section 3.2) and the deep log as $R_t$. The well has no clay conduction, so clean Archie returns the built saturation and a shaly-sand model reads below it: the correction removes water that was never there. At 5890 ft, where the gamma-ray shale volume is 4 %, Indonesia reads 18.4 % against the 18.9 % built; at 5980 ft, where it is 19 %, Indonesia reads 16.6 % against 19.5 % and Simandoux 13.4 %. Over the shaly oil sand as a whole, with a gamma-ray shale volume averaging 22 %, Indonesia averages 16.3 % and Simandoux 12.8 % against the 19.6 % built, and over the net pay of the whole well Indonesia averages 21.2 % against 23.0 %. A shaly-sand model does not fix Ogbon-1; it over-corrects it. The Indonesia model takes the effective porosity (PetroWiki), and Section 4.6 separates it from the total; Ogbon-1's density porosity has no shale term, so the two agree in its sands.

The next two models tie the clay's current to the clay itself, its cation exchange and its bound water, instead of fitting a shale term to field data: Waxman-Smits in Section 8.4 and the Dual-Water model in Section 8.5. Section 8.6 puts the models on the whole of Ogbon-1.

References

• Poupon, A. and Leveaux, J. (1971). Evaluation of water saturation in shaly formations. SPWLA 12th Annual Logging Symposium, Dallas, paper SPWLA-1971-O.
• Hari Kumar, K. C. (2010). On the application of Simandoux and Indonesian shaly sand resistivity interpretation models in low and high Rw regimes. 8th Biennial International Conference and Exposition on Petroleum Geophysics, Society of Petroleum Geophysicists, India. Oil and Natural Gas Corporation.
• PetroWiki. Water saturation determination. petrowiki.spe.org.