Why Clean Archie Fails

Clay Carries Current

Archie's equation (Section 7.1) rests on one assumption about the rock, which PetroWiki states in a sentence: "In the clean-sand model, the formation water is the only electrically conductive medium." In a clean sand that holds. In a shaly sand it does not, because clay conducts too: the cations held on the clay's surfaces carry current in parallel with the brine in the pores. Waxman and Smits derived an equation that links the electrical conductivity of a water-saturated shaly sand to the conductivity of its water and to the rock's cation-exchange capacity per unit pore volume (Section 8.4), and Worthington (1985) traces how the shaly-sand concepts evolved.

Two paths for the current in parallel add their conductances, so a shaly sand's conductivity is the sand's plus the clay's. Figure 8.1 writes that sum in the form Section 8.2 develops, the modified Simandoux form, here for $n = 2$:

The first term is clean Archie's conductance divided by $1 - V_{sh}$, the sand's path; the second is the clay's, in proportion to the shale volume and the water saturation and in inverse proportion to the shale's resistivity $R_{sh}$. The factor $1 - V_{sh}$ is what makes this the modified form of Simandoux's equation. With no shale the clay term vanishes and the sand's term is clean Archie's.

A What-If Rock, Because Ogbon-1 Cannot Show It

Ogbon-1 cannot show the failure. Its resistivity was built as clean Archie in every bed, the shaly oil sand (5946 to 6018 ft, built with a shale volume averaging 21.9 %, which its gamma ray, read as Section 7.6 reads it, gives as 21.8 %) included, so clean Archie reads that sand's real log as 19.6 % water, the saturation it was built with (Section 7.6). The figure therefore makes a what-if rock, labelled as one: the porosity of the shaly oil sand (17.0 %, its density porosity averaged over its half feet), the well's brine of 0.040 ohm.m and its shales' 2.20 ohm.m, and a shale volume and a water saturation you set, starting at 22 % and 20 %, near the shaly oil sand's own. The rock is made forward with the law above, so the water it holds is known, and clean Archie is held against it. What the figure shows is what the law implies, not a log: a real rock is not exactly Simandoux's.

The Over-Read

At the default the clay carries 35 % of the current, so the deep log reads 17.5 ohm.m where a clean sand with the same 20 % water would read 34.5 ohm.m, 1.97 times as much. Clean Archie takes all of that conductance for brine and reads 28.1 % water, 8.1 points too much (a point is a percentage point of saturation). The error grows with the shale: 14.2 points at 40 % shale, the book's cutoff (Section 13.1), and 21.9 points at 60 %. It has one direction. A parallel path can only add conductance, so at a given water saturation the resistivity can fall and never rise, and Archie's equation reads a lower resistivity as more water. PetroWiki puts it in one line: "In shaly rocks, Rt is suppressed and Archie Sw calculations are too high."

In this form two terms lower the resistivity, and plate (b) draws them apart. With the clay's path switched off, the dotted line, the sand's term still carries the factor $1/(1 - V_{sh})$ and clean Archie would read 22.6 %, 2.6 points too much; the other 5.4 points of the 8.1 points are the clay's own path.

The over-read matters because of the cutoff. In this rock, water of 39.1 % already reads as the book's 50 % water cutoff, so every bed that holds between 39.1 % and 50 % is pay that clean Archie throws away, a window of 10.9 points of saturation. A bed that holds 40 % reads 51.0 % and is rejected.

What Sets the Size

The clay's share is set by how much current the brine itself carries. A fresher brine carries less: at 0.120 ohm.m, three times the well's 0.040 ohm.m, the clay carries 62 % of the current and the over-read grows from 8.1 points to 16.6 points. A more conductive shale carries more: at 1.00 ohm.m the clay carries 54 % and the over-read is 13.5 points, at 8.00 ohm.m it carries 13 % and the over-read is 4.3 points. Ogbon-1's brine is a salty 0.040 ohm.m, the setting where clay matters least; the same shale in fresh water is the worse problem.

Ogbon-1, Made Shaly

Plate (c) runs the same law through each half foot of the shaly oil sand, with its own built porosity, saturation and shale volume, your brine and your shale's resistivity. Over its 72.0 ft of pay the real log gives 19.6 % with clean Archie, the built value, and the same half feet made as the what-if rock give 27.6 %. All 72.0 ft stay under the 50 % cutoff, so here the over-read costs saturation and no pay; the rock at 40 % water would lose it.

A shaly-sand model run on Ogbon-1's real log has the opposite problem. There is no clay conduction in that log, so the model takes out clay current that was never there and reads less water than the truth: Section 8.2's Simandoux equation, run on each half foot as Section 7.6 reads it, averages 12.8 % there against the true 19.6 %. Chapter 8 says so wherever it runs a model on the well. The well tests a model's arithmetic; it is never evidence that the model fixes a real log.

Closing the Gap

The remedy is to count the clay's current. Section 8.2 solves the parallel-conductance law above for the saturation (the Simandoux equation), Section 8.3 gives the Indonesia equation, Section 8.4 Waxman-Smits and Section 8.5 the dual-water model, and Section 8.6 runs them on Ogbon-1, where, for the reason above, they read low. Each of those sections builds its own what-if rock, with the clay conducting by the law that section teaches, and says what it is: they are not one rock.

References

• PetroWiki (SPE). Water saturation determination. petrowiki.spe.org/Water_saturation_determination.
• Waxman, M. H. and Smits, L. J. M. (1968). Electrical conductivities in oil-bearing shaly sands. SPE Journal, June 1968, 107-122. SPE-1863-A. doi:10.2118/1863-A.
• Simandoux, P. (1963). Dielectric measurements on porous media, application to the measurement of water saturations: study of the behaviour of argillaceous formations. Revue de l'Institut Français du Pétrole, 18 (Supplementary Issue), 193-215.
• Worthington, P. F. (1985). The evolution of shaly-sand concepts in reservoir evaluation. The Log Analyst, 26(1), 23-40.