Petrophysics LAMBDAs

The linear gamma-ray shale index. It places a reading between the clean-sand and shale picks on a 0 to 1 scale. It is the input the Larionov shale-volume equations expect, and it is the shale volume itself only when the index is taken as linear. IGR = (GR − GRclean) / (GRshale − GRclean). Readings outside the two picks fall outside 0–1; the index is not clipped.

Shale volume for unconsolidated Tertiary rocks from the Larionov equation. Young rocks need less shale volume than a linear index implies at the same gamma ray, because the curve bows down. Vsh = 0.083 (2^(3.7 IGR) − 1). IGR is the linear gamma-ray index, a fraction. At 0 and at 1 the result is 0 and about 1.

Shale volume for older, consolidated rocks from the Larionov equation. The curve sits above the Tertiary one: the same gamma-ray index means more shale. Vsh = 0.33 (2^(2 IGR) − 1). IGR is the linear gamma-ray index, a fraction.

Three shale indices from a spectral gamma ray log, returned as one row: thorium, potassium, then the uranium-free curve. Each index is the linear pick between the clean and shale responses of that curve. Thorium and potassium avoid uranium that is not in the clay, which is why the total gamma ray on its own misleads in a radioactive sand. I_Th = (Th_log − Th_min) / (Th_shale − Th_min), and the same shape for potassium and for the uranium-free gamma ray. The three numbers are not averaged and not reduced to a minimum; that choice is left to the sheet.

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.

Sonic porosity from the Raymer-Hunt-Gardner field transform. It does not use a fluid transit time. An empirical constant, about 0.67 in sand and about 0.60 in carbonate, scales the fractional increase of transit time over the matrix. φ = C (Δt − Δtma) / Δt. Transit times share one unit, µs/ft on a sonic log. C is an input because the published range is 0.624 to 0.70, not a single lithology switch.

Sonic porosity from the Wyllie time-average equation, divided by the compaction factor for an uncompacted sand. The time average overstates porosity when the adjacent shale is slower than 100 µs/ft. Dividing by Δtsh/100 brings it back. φ = ((Δt − Δtma) / (Δtf − Δtma)) / (Δtsh / 100). When the shale transit time is 100 µs/ft the divisor is 1 and the result is the compacted Wyllie porosity. Transit times are µs/ft.

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.

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.

Formation resistivity factor for sucrosic sandstones from the Humble equation. It is Archie's equation with a = 0.62 and m = 2.15, the pair Winsauer, Shearin, Masson and Williams fitted at Humble Oil. F = 0.62 / φ^2.15. Porosity is a fraction, not a percent.

The static spontaneous potential, which is the maximum deflection a thick clean bed can develop. It is fixed by the equivalent resistivities of the mud filtrate and the formation water and by temperature. SSP = −K log10(Rmfe / Rwe), with K = 61 + 0.133 T and T in °F. A fresher filtrate opposite saltier water gives a negative SSP, in millivolts.

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.

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.