PO.PP.F.Humble

Formation resistivity factor from Archie's equation. It is the ratio of the resistivity of a rock fully saturated with water to the resistivity of that water, expressed through porosity. F = a / φ^m. Porosity is a fraction. a and m come from a log-log plot of F against porosity, or from a named pair such as Humble.

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.

Sandstone permeability from the formation factor, the correlation Carothers published in 1968. It is a statistical fit, not a pore-scale derivation, and it applies to sandstones. The limestone fit is a different equation and is not this one. k = 7×10^8 / F^4.5, with k in millidarcies and F dimensionless.

Porosity from the density log in a clean formation. The bulk density sits between the matrix density and the fluid density, and the fraction of the way across is the porosity. φ = (ρma − ρb) / (ρma − ρf). All three densities share one unit, g/cm³ on a density log. Shale and gas corrections are not in this equation.

Equivalent formation-water resistivity from the static spontaneous potential, for water in which NaCl dominates. The SP equation is SSP = −K log10(Rmfe / Rwe), with K = 61 + 0.133 T and T in °F. Solving for the water resistivity gives Rwe = Rmfe · 10^(SSP / K). SSP is negative in the usual case of a fresher mud filtrate opposite a saltier formation water, and Rwe then comes out smaller than Rmfe. The result is the equivalent resistivity at formation temperature, which is what the NaCl form of the SP equation solves. The chart that turns Rwe into true Rw is not a closed form and is not in this lambda.

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.