Waxman-Smits

Conductance from the Clay Surface

Simandoux and Indonesia fit the shale empirically. Waxman-Smits derives it. Clay minerals carry a negative surface charge, and the cations that balance it, the cation exchange capacity, sit in a loose cloud at the surface and conduct. Counting those cations per unit pore volume gives $Q_v$, and giving them an equivalent conductance $B$ makes the wet-rock conductance a simple sum of the brine and the clay:

With hydrocarbon present the brine term carries $S_w^2$ and the clay term $S_w$, so the saturation comes from the same kind of quadratic as Simandoux, but now every piece has a physical meaning.

The Salinity Story

The model earns its keep by explaining when the shale matters. The clay conductance $B,Q_v$ is a property of the rock and barely changes; the brine conductance $C_w = 1/R_w$ is a property of the water and changes enormously. In saline water $C_w$ is huge and the clay is a rounding error, so clean Archie is nearly right. In fresh water $C_w$ collapses and the clay can carry half the current or more, so the correction becomes large. The curve makes it visible: the clay share of the conductance climbs from a few percent in brine to dominant in fresh water. That is the single most important fact about shaly-sand evaluation.

Strength and Cost

Because it is physical, Waxman-Smits travels well once $Q_v$ is known, and it is the reference model for careful work. Its cost is exactly that $Q_v$: it comes from core cation-exchange measurements or a calibrated $Q_v$-porosity relation, which not every well has. The original form is iterative; for $n^{*}=2$ it closes to the quadratic used here. Dual-Water, next, is a clever repackaging of the same physics that avoids measuring $Q_v$ directly.

References

• Waxman, M. H. and Smits, L. J. M. (1968). Electrical conductivities in oil-bearing shaly sands. SPE Journal, 8(2).
• Waxman, M. H. and Thomas, E. C. (1974). Electrical conductivities in shaly sands. Journal of Petroleum Technology, 26(2).