The Simandoux Equation

Chapter 8, Shaly-Sand Saturation Models

Learning objectives

  • Write the modified Simandoux equation as two parallel paths for the current, a water path that goes as Sw2S_w^{2} and a shale path that goes as SwS_w, and solve its quadratic for the water saturation
  • Say which Simandoux equation the kernel and Figure 8.2 compute, the modified form and not the 1963 original, and where its two modifications come from
  • Test the model against a rock whose water saturation is known, and say what an error in the shale resistivity does to the answer
  • Say why the model reads less water than the truth on Ogbon-1, whose resistivity was built as clean Archie, and why that is an over-correction and not a fix

Two Paths for the Current

Section 8.1 left a rock in which clay carries current beside the brine, and clean Archie, which gives the whole of the current to the water, reading too much water. Simandoux's answer is to write the two paths down. The measured conductivity, 1/Rt1/R_t, is a water path, which carries current in proportion to the square of the water saturation, plus a shale path, which carries it in proportion to the shale volume and to the saturation:

1Rt=ϕmSw2a Rw (1−Vsh)+Vsh SwRsh.\frac{1}{R_t} = \frac{\phi^{m}S_w^{2}}{a\,R_w\,(1-V_{sh})} + \frac{V_{sh}\,S_w}{R_{sh}}.

The first term is the water path, clean Archie's own with a (1−Vsh)(1-V_{sh}) in its denominator, and the second is the clay: the more of it there is, and the lower the shale resistivity RshR_{sh}, the more current it takes.

Which Simandoux Equation

This is the modified Simandoux equation, here with the well's saturation exponent of 2 (the kernel takes any; at 2 its root is the quadratic's), and it is the one the kernel and Figure 8.2 compute; it is not the 1963 original. Reviews of the history say that Bardon and Pied (1969) modified Simandoux's equation by including the water saturation in its shale term, and that Schlumberger's form divides the water term by 1−Vsh1-V_{sh}; Metwally, Mabrouk and Mahmoud (2022) date that form to 1975, and Sam-Marcus et al. (2018) call the division a crude way to calibrate the equipment, with no physical basis. The equation here has both changes. The reviews differ on small points of the original's algebra, and the 1963 paper, a study of laboratory mixtures of sorted sand and natural clay (Sam-Marcus et al.), was not read for this book.

Put 22 % clay into the what-if rock, the upper oil sand's porosity of 23.0 % and19 % of the pore space water and the shale path carries 24 % of the current: therock reads 12.5 ohm.m, not the 20.9 ohm.m it reads clean. Archie, which gives allthe current to the water, reads 24.6 %, 5.6 saturation points too much water.Simandoux, given the shale's own 2.20 ohm.m, takes the shale path out and reads19.0 %, the saturation built. On Ogbon-1, at 5982 ft, in the shaly oil sand,oil-bearing, the shale volume is 22 %. The well's resistivity is clean Archie, whichreads the built 19.6 %; Simandoux takes out a clay current the well does not haveand reads 13.1 %, 6.5 saturation points below the truth. Over the well's clean rockSimandoux reads 280.0 ft of net pay and an average of 19.3 % water over the net pay,against the 277.0 ft and 23.0 % built.(a) Read over built010203040500.801.001.251.501.752.00Shale volume (%)Saturation read over builtArchieSimandoux(b) The two paths51020501000.0010.010.1110Water saturation (%)Conductivity (S/m)sumwatershalemeasured(c) Ogbon-102550751005800590060006100620063006400Water saturation (%)Depth (ft)Key: in (a) the teal line is Archie and the red-brown line Simandoux, each the water saturation readfrom the what-if rock over the one it was built with, against the clay in it, the dashed black line thesaturation built (the ratio 1) (Simandoux lies on it when the shale resistivity is the rock's own), anda dot your clay. In (b) the dashed grey curve is the clean rock, the teal curve the water path (slope2), the brown curve the shale path (slope 1), the black curve their sum, the dashed red-brown line theconductivity measured; the dots are where Archie and Simandoux read. In (c) the line is the saturationbuilt on Ogbon-1, which is also what clean Archie reads there, the dots the Simandoux reading, one every5 ft, and the dashed vertical line the book's 50 % cutoff; the shaded beds are the four reservoir zonesand the dotted horizontal line the depth read. Ogbon-1 was built as clean Archie, so Simandoux readsless water than the truth there.Archie reads (what-if) 24.6 % (built 19.0 %). Shale's share of the current 24 % (shale path over 1/Rt).Simandoux on Ogbon-1 here 13.1 % (built 19.6 %). Net pay on Ogbon-1, Simandoux 280.0 ft (Ogbon-1; built277.0 ft). Simandoux reads (what-if) 19.0 % (built 19.0 %).

A Quadratic in the Saturation

Written as A Sw2+B Sw−1/Rt=0A\,S_w^{2} + B\,S_w - 1/R_t = 0, with A=ϕm/(a Rw(1−Vsh))A = \phi^{m}/(a\,R_w(1-V_{sh})) and B=Vsh/RshB = V_{sh}/R_{sh}, the equation has one positive root:

Sw=−B+B2+4A/Rt2A.S_w = \frac{-B + \sqrt{B^{2} + 4A/R_t}}{2A}.

The tool measures the whole conductivity; Simandoux gives the shale path its share and the water the rest, and the root is the saturation at which the water path carries that rest. Plate (b) draws the bookkeeping: the water path has the slope 2 on logarithmic axes and the shale path the slope 1, their sum is the rock, and each equation reads the saturation where its curve meets the measured conductivity. With no clay BB is zero, and the root is Archie's.

A Test Where the Truth Is Known

Ogbon-1 cannot test a shaly-sand model (see below), so plates (a) and (b) use a what-if rock, this section's own and not Section 8.1's: the upper oil sand's porosity, 23.0 %, and a water saturation you set, 19 % to begin with, with clay added, its resistivity built forward from the equation above (the figure's own forward law; the kernel has only the inverse). Clean, the rock reads 20.9 ohm.m. Put in 22 % clay, the shale volume of the shaly oil sand, and the shale path carries 24 % of the current, so the rock reads 12.5 ohm.m. Archie reads that as 24.6 % water, 5.6 saturation points too much; Simandoux, given the shale's own 2.20 ohm.m, reads 19.0 %, the saturation the rock was built with. The error of clean Archie grows with the clay: 1.14 times the built saturation at 10 % clay, 1.40 times at 30 % and 1.70 times at 50 %, where the shale carries 14 %, 28 % and 31 % of the current.

The clay matters most where the current is small. With the same 22 % clay the shale carries 24 % of the current at 19 % water and 7 % at 80 %, because the water path grows as the square of the saturation and the shale path only as the saturation, and Archie reads 1.30 times the built saturation at 19 % water and 1.17 times at 80 %. The test is built with the equation it tests, so Simandoux returns the saturation exactly when it is given the right inputs; that shows the arithmetic and the sensitivity, not that real clay obeys the equation. Given a shale resistivity twice the rock's it reads 20.3 %, 1.3 saturation points too high, and given half it reads 16.6 %, 2.4 points too low.

On Ogbon-1: an Over-Correction

Ogbon-1's resistivity was built as clean Archie in every bed, the shaly oil sand included (Section 7.6): there is no clay current in it to take out. A shaly-sand model run on the well therefore reads less water than the truth, and plate (c) shows it, with each half foot read as Section 7.6 reads it, the density porosity and the gamma-ray shale volume. At 5982 ft the shale volume is 22 % and clean Archie reads the built 19.6 %; Simandoux reads 13.1 %. Over the shaly oil sand, 5946 to 6018 ft, it averages 12.8 % against the 19.6 % built, over the upper oil sand 17.4 % against 18.9 %, and it reads below the built saturation at every one of the well's 772 clean half feet. It counts 280.0 ft of net pay against the 277.0 ft built, averaging 19.3 % water over its own pay against 23.0 % over the built. That is not a fix for Ogbon-1; it is the model taking a current out of a well that has none. On a real shaly sand, where clay does carry current, the same step is in the direction the physics asks for.

What Simandoux Needs, and Where It Strains

It asks for two numbers beyond Archie's: the shale volume (Section 3.2) and the shale resistivity, read from a nearby shale, and the figure shows what an error in the second does. It is a total-shale model, which Hamada and Al-Awad describe as used whatever the distribution of the shale, so it makes no distinction between dispersed and laminated clay, and the factor on its water term has no firm physical basis. The sections that follow write the clay's current another way: the Indonesia equation (Section 8.3), Waxman-Smits (Section 8.4) and the dual-water model (Section 8.5), and Section 8.6 runs them on Ogbon-1.

References

  • 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.
  • Bardon, C. and Pied, B. (1969). Formation water saturation in shaly sands. SPWLA 10th Annual Logging Symposium, Houston, 25-28 May 1969.
  • Metwally, A. M., Mabrouk, W. M. and Mahmoud, A. I. (2022). A numerical approach to accurately estimate water resistivity (Rw) and saturation (Sw) in shaly sand formations. Contributions to Geophysics and Geodesy, 52(3), 423-441. doi:10.31577/congeo.2022.52.3.4.
  • Sam-Marcus, J., Enaworu, E., Rotimi, O. J. et al. (2018). A proposed solution to the determination of water saturation: using a modelled equation. Journal of Petroleum Exploration and Production Technology, 8, 1009-1015. doi:10.1007/s13202-018-0453-4.
  • Hamada, G. M. and Al-Awad, M. N. J. Petrophysical evaluation of low resistivity sandstone reservoirs. Society of Core Analysts, paper SCA1998-51.

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