Reconciling Lab, Log, and Seismic

Three Bands, One Rock

Everything in this part converges here. A single rock, unchanged, hands three different velocities to the three instruments that measure it, and now we know why. On the illustrative dispersion curve the seismic survey, working at tens of hertz, catches the rock at its relaxed limit and reads about 3.142 km/s. The sonic log, running at tens of kilohertz, catches the same rock part-way up its dispersion step and reads about 3.221 km/s. The laboratory ultrasonic measurement, at 1 MHz, catches it near the unrelaxed limit and reads about 3.292 km/s. That is a spread of roughly 5 percent between the lab and the seismic survey, and none of the three is a mistake. Each instrument is telling the truth about the rock at its own frequency. The scale gap of Part 9.1 and the dispersion of Part 9.4 have combined into a single, unavoidable fact: there is no one velocity of the rock to reconcile them to.

Rules to Carry to the Wellsite

Since there is no single velocity, the discipline is to be explicit about which band you are working in and to move data between bands deliberately. Three rules follow. First, model at the band you predict for. If the product is a synthetic seismogram or an AVO model for a surface survey, you want the relaxed, low-frequency velocity, and that is where Gassmann lives; the fluid-substitution machinery of Part 4 is a seismic-band tool by construction, because it assumes the fully relaxed pore pressure of the low-frequency limit. Second, correct lab and log data downward before you trust them at seismic scale. An ultrasonic core velocity sits near the unrelaxed top of the step; carried straight into a seismic model it will be too fast, and it must be brought down by a dispersion correction first. The sonic log, part-way up the step, needs a smaller correction in the same direction. Third, read a mismatch as information. When a sonic log and a surface seismic velocity disagree by a few percent, dispersion is the first suspect, ahead of processing error or a bad tie, because a few percent is exactly the size of the step this physics predicts.

The Close of the Theory Arc

These rules are not fussy bookkeeping; they are what keeps a rock-physics model honest across the enormous frequency range the discipline actually spans. The lab measures the rock most directly but at the wrong frequency; the seismic survey works at the right frequency but sees the rock most coarsely; the sonic log sits in between on both counts. A model that ignores which band its inputs and outputs belong to will mis-tie by a few percent every time, and a few percent of velocity is the difference between a good depth and a dry hole. With this, the theoretical arc of the course closes. We began with velocity as a model output and a reflection as an ambiguous sum; we built the bounds, the fluid and contact and inclusion models, the empirical ties and the calibration discipline, the pressure and stress effects, and finally the scale-and-frequency reconciliation that tells you which number the models are even talking about. Part 10 puts the whole toolkit to work, taking these methods off the page and onto interpretation capstones where the reconciled numbers do their job.

References

• Batzle, M. L., Han, D.-H., & Hofmann, R. (2006). Fluid mobility and frequency-dependent seismic velocity: Direct measurements. Geophysics, 71(1), N1-N9.
• Mavko, G., Mukerji, T., & Dvorkin, J. (2009). The Rock Physics Handbook (2nd ed.). Cambridge University Press.