The Simandoux Equation

Two Conductances in Parallel

Simandoux was the first widely used shaly-sand model and the most direct. It simply adds the clay path that clean Archie left out, writing the measured conductivity as a water path plus a shale path:

The first term is the brine in the pores, the same conductance clean Archie uses, slightly boosted by the $(1-V_{sh})$ that accounts for the sand being diluted by shale. The second term is the shale itself, conducting in proportion to how much of it there is and how conductive it is.

A Quadratic in Sw

Because $S_w$ appears squared in the water term and linearly in the shale term, the equation is a quadratic, $A S_w^2 + B S_w - 1/R_t = 0$, solved in closed form for its positive root. The bookkeeping is the whole idea: the tool gives the total conductance, Simandoux assigns the shale its share and lets the water explain the rest. When $V_{sh}\to 0$ the shale term vanishes and the equation collapses back to clean Archie, which is exactly what a good correction should do.

What Simandoux Needs, and Where It Strains

It asks for only two extra numbers: the shale volume $V_{sh}$ from the gamma ray, and the shale resistivity $R_{sh}$ read from a nearby shale bed. That simplicity made it the workhorse for decades. Its weakness is that the shale term is empirical, linear in $V_{sh}$ and tied to a single $R_{sh}$, so it can over-correct in very dirty sands and it does not distinguish dispersed clay from laminated shale. The next models address exactly those limits.

References

• Simandoux, P. (1963). Mesures dielectriques en milieu poreux, application a mesure des saturations en eau. Revue de l'Institut Francais du Petrole, 18.
• Asquith, G. and Krygowski, D. (2004). Basic Well Log Analysis, 2nd ed. AAPG Methods in Exploration 16.