Porosity from the Sonic

Part 5, Chapter 5: Sonic and Acoustic Logs

Learning objectives

Invert the sonic time average for porosity
Read sonic porosity as a point on the Wyllie line
Recall the matrix transit times
Explain how the wrong matrix biases the sonic porosity

The Time Average, Inverted

The sonic gives DT; porosity comes from running the time average backward. Solving $DT = (1-\phi)DT_{ma} + \phi DT_f$ for the porosity gives the Wyllie sonic porosity

\phi_S = \frac{DT - DT_{ma}}{DT_f - DT_{ma}}.

It is the same lever idea as the density, in time: how far the measured DT sits above the matrix transit time, divided by the full matrix-to-fluid span.

A Straight Line

Plotted against porosity, the Wyllie relation is a straight line from the matrix point (DT_ma at zero porosity) up to the fluid point (DT_f at unit porosity). Reading sonic porosity is just finding where the measured DT crosses that line. A quartz matrix starts the line at about 55.5 us/ft, a limestone at 47.6, a dolomite at 43.5.

The Matrix Again

And the same warning applies as for the density: the matrix transit time fixes where the line starts, so the assumed matrix controls the answer. For one measured DT, a faster matrix (a lower DT_ma) reads a higher porosity, because the lever arm above the matrix point is longer. Pick the matrix wrong and the sonic porosity is wrong, which is one more reason the porosity logs are read together rather than alone.

References

Wyllie, M. R. J., Gregory, A. R., and Gardner, L. W. (1956). Elastic wave velocities in heterogeneous and porous media. Geophysics, 21(1).
Asquith, G. and Krygowski, D. (2004). Basic Well Log Analysis, 2nd ed. AAPG Methods in Exploration 16.