The Sonic Log

Part 5, Chapter 5: Sonic and Acoustic Logs

Learning objectives

Define interval transit time DT as slowness and relate it to velocity
Write the sonic time average DT = (1-phi)DTma + phi*DTf
Recall the matrix and fluid transit times
Explain why porosity raises DT and lowers the velocity

Timing the Pulse

The sonic tool fires a sound pulse from a transmitter and times its arrival at receivers a fixed distance away. What it reports is the interval transit time DT, the time the pulse takes to cross one foot of formation, in microseconds per foot. DT is a slowness, the inverse of velocity: $V_p = \dfrac{10^6}{DT}$ . A fast, dense rock has a low DT (a quartz matrix is about 55.5 us/ft, a velocity near 18,000 ft/s); a slow, porous rock has a high DT.

The Time Average

Like the density log, the sonic obeys a mixing law, but in time rather than volume. The pulse crosses a foot of rock that is part solid matrix and part pore fluid, and it spends time in each: a fast stretch through the matrix and a slow stretch through the fluid. Adding them gives the Wyllie time average

DT = (1-\phi)\,DT_{ma} + \phi\,DT_f.

Matrix is acoustically fast; pore fluid is slow (about 189 us/ft for water, far slower for gas). So every bit of porosity stacks slow fluid time onto the transit time.

Slowness and Velocity

Because DT and velocity are inverses, the same fact reads two ways: porosity raises the transit time and lowers the velocity. That falling velocity is exactly the quantity seismic surveys measure from the surface, which is why the sonic log is the bridge between the well and the seismic image, the subject that closes this chapter. First, though, the next section runs the time average backward to read porosity.

References

Asquith, G. and Krygowski, D. (2004). Basic Well Log Analysis, 2nd ed. AAPG Methods in Exploration 16.
Ellis, D. and Singer, J. (2007). Well Logging for Earth Scientists, 2nd ed. Springer.