One of the saturation from resistivity models:
s_w = \left[ \frac{\sqrt{\frac{1}{R_t}}}{ \frac{V_{sh}^{\left( 1-0.5 \, V_{sh} \right)}}{\sqrt{R_{sh}}} + \sqrt{ \frac{\phi_e^m}{A \, R_w}}} \right]^{\, (2/n)}
where
1.5 ÷ 2.5, default value is 2
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Well & Reservoir Surveillance / Well logging / Reservoir Data Logs (RDL) / Formation Resistivity Log @model
Poupon, André, and J. Leveaux. "Evaluation Of Water Saturation In Shaly Formations." Paper presented at the SPWLA 12th Annual Logging Symposium, Dallas, Texas, May 1971.