Scales and Frequencies
Learning objectives
- Compute wavelength from velocity and frequency, and compare it with the rock's own scales
- Place laboratory, log, and seismic measurements on one frequency axis, five orders of magnitude apart
- Explain velocity dispersion in saturated rocks: relaxed at low frequency, unrelaxed at high
- State the ground rule: never compare velocities across frequency bands without a model
Three Rulers
Ask three instruments for the velocity of the same sandstone and you have asked three different questions. An ultrasonic bench measures a core plug at around a megahertz: the wavelength is millimeters, and the wave feels individual laminae. A sonic log works near ten kilohertz: the wavelength is tens of centimeters, averaging the borehole wall. A seismic survey works at tens of hertz: at km/s and 50 Hz, m, and the wave averages the whole reservoir interval with its neighbors. Same rock, three rulers five orders of magnitude apart in frequency, each averaging an utterly different volume of rock.
Dispersion: the Fluid Has a Clock
Scale is not the only difference; the velocities themselves disagree. In a fluid-saturated rock, a passing wave squeezes some pores more than others. Give the pore fluid time and the pressure differences bleed away between pores: the rock responds relaxed, softer. Squeeze fast and the fluid is trapped where it sits: the rock responds unrelaxed, stiffer. So saturated-rock velocity typically rises with frequency, often by a few percent between the seismic band and the ultrasonic bench, and Gassmann, the theory at the heart of Part 4, lives explicitly at the relaxed, low-frequency end.
Slide across five orders of magnitude and watch both stories at once: the velocity climbing its dispersion step between the relaxed and unrelaxed limits, and the wavelength shrinking past the reservoir, the bed, the core plug, the grain. The curve's shape here is illustrative; Part 9 builds the mechanisms (Biot flow, squirt, patchy saturation) that set its height and its corner frequency for a real rock.
The Ground Rule
Hence a rule this course will not break: a velocity is not a number, it is a number at a frequency and at a scale. Comparing an ultrasonic core measurement against a sonic log, or a log against a stacking velocity, without asking what frequency and what averaging volume each represents, is a category error that has produced many phantom discrepancies. Reconciling the three rulers is a modeling task, and Part 9 does it properly. With the motivation now complete, the next section draws the map of the toolkit this course will build.
References
- Mavko, G., Mukerji, T., & Dvorkin, J. (2009). The Rock Physics Handbook (2nd ed.). Cambridge University Press.
- Simm, R., & Bacon, M. (2014). Seismic Amplitude: An Interpreter's Handbook. Cambridge University Press.