Porosity from Density

Part 4, Chapter 4: Density and Neutron Porosity

Learning objectives

Invert the density mixing law to get porosity
Choose the right matrix density for the rock at hand
Explain how the wrong matrix biases the porosity
Read a negative density porosity as a matrix-mismatch flag

Running the Mix Backward

The density log gives bulk density; porosity comes from running the mixing law backward. Solve $\rho_b = (1-\phi)\rho_{ma} + \phi\rho_f$ for the porosity and you get the density porosity

\phi_D = \frac{\rho_{ma} - \rho_b}{\rho_{ma} - \rho_f}.

It is the lever arm from the last section read as a number: how far RHOB sits from the matrix point, divided by the full matrix-to-fluid span. Give it a matrix density and a fluid density and every foot of bulk density becomes a porosity.

The Matrix You Assume

The catch is that $\rho_{ma}$ is assumed, not measured. The same RHOB read against a sandstone (2.65), a limestone (2.71), and a dolomite (2.87) gives three different porosities, and the heavier the matrix you assume, the higher the porosity you read. In the figure, one RHOB line crosses three matrix levers and splits into three answers. In a clean known sandstone you pick 2.65 and trust the number; in mixed or unknown lithology you cannot, which is exactly why the neutron log and the density-neutron crossplot come next, to pin down the lithology the density log alone cannot.

When Porosity Goes Negative

Push the bulk density above the matrix density and $\phi_D$ goes negative. A negative density porosity is impossible as a real pore volume, so it is a useful alarm: either the assumed matrix is too light (a sandstone matrix over a dolomite, say) or heavy accessory minerals such as pyrite are present. Read it as the log telling you the matrix assumption is wrong, not as a number to carry forward.

References

Asquith, G. and Krygowski, D. (2004). Basic Well Log Analysis, 2nd ed. AAPG Methods in Exploration 16.
Tiab, D. and Donaldson, E. (2015). Petrophysics, 4th ed. Gulf Professional Publishing.