The Multimineral Solver

Part 12, Chapter 12: Lithology and Mineralogy

Learning objectives

Write each log as a linear response to the mineral volume fractions
Assemble the response matrix with a sum-to-one closure
Invert the system to get all mineral volumes at once
Recognize the crossplots as special cases of this solver

One System Behind Every Plot

The crossplots each solve one special case by hand. The general tool is a linear system, because every log is a linear mixture of the volume fractions: each component contributes its own response in proportion to how much of it is present, and a closure says the volumes fill the rock,

\rho_b = \sum_i V_i\,\rho_i, \quad \phi_N = \sum_i V_i\,\phi_{N,i}, \quad U = \sum_i V_i\,U_i, \quad \sum_i V_i = 1.

Invert and Read

With the density, the neutron, and the photoelectric cross section plus the closure, four equations fix four unknowns, quartz, calcite, dolomite, and porosity. Inverting the response matrix returns every volume at once, the same answers the M-N, MID, and triangle plots read off geometrically, but for any minerals and any logs, and with no chart to squint at. At zero noise it recovers the input mixture exactly.

The Honest Limits

A determined system passes measurement error straight into the volumes, and a wrong response, an unmodeled mineral or a bad matrix value, bends the answer or drives a volume negative, the flag that the model is incomplete. Real analysis carries more logs than unknowns and solves by least squares, the extra equations averaging the noise down, and adds a gamma-ray row whenever clay is in play. The closer that follows runs exactly this solve down the Ogbon-1 well.

References

Mayer, C. and Sibbit, A. (1980). GLOBAL, a new approach to computer-processed log interpretation. SPE Annual Technical Conference, SPE 9341.
Quirein, J. et al. (1986). A coherent framework for developing and applying multiple formation evaluation models. SPWLA Annual Logging Symposium.