Water Saturation LAMBDAs

Water saturation in a clean formation from the general Archie equation. Porosity, water resistivity and true resistivity fix the bulk volume of water once the tortuosity factor, cementation exponent and saturation exponent are chosen. Sw = ((a / φ^m) * (Rw / Rt))^(1/n). Porosity is a fraction, resistivities are ohm·m, and a, m and n are dimensionless. With Vsh = 0 the shaly-sand equations in this batch reduce toward this one, at the a and m those equations bake in.

Water saturation in a shaly sand from the Simandoux total-shale equation in the form Bassiouni publishes. The shale conducts, so Archie read on the same logs overstates Sw. This solution is the quadratic for saturation exponent 2, with tortuosity 0.8 and cementation exponent 2 already substituted, which is why the leading factor is 0.4 Rw / φe² and the term under the square root carries a 5. Sw = (0.4 Rw / φe²) * (−Vsh/Rsh + sqrt((Vsh/Rsh)² + 5 φe² / (Rw Rt))). Set Vsh to 0 and it is Archie at a = 0.8, m = 2, n = 2.

Water saturation from the Fertl and Hammack shaly-sand equation. The first term is the clean-sand Archie saturation at n = 2, written with a formation factor you supply. The second term subtracts the extra conductivity of the shale. Sw = sqrt(F Rw / Rt) − Vsh Rw / (0.4 φe Rsh). A large shale term can push the result through zero; the equation is returned as written, not clipped to 0–1.

Water saturation from a pulsed-neutron capture measurement. The tool reads a formation capture cross section; the matrix, water and hydrocarbon cross sections, with porosity, split that reading into water and hydrocarbon. Sw = ((Σlog − Σma) − φ (Σh − Σma)) / (φ (Σw − Σh)). Cross sections must share one unit, usually capture units. The result is a fraction.