Velocity Dispersion

Why One Rock Has More Than One Velocity

Part 4.5 showed the seam where Gassmann stops being safe: Gassmann assumes the pore pressure has time to equalise everywhere during a wave cycle, and at high enough frequency it does not. That failure has a name and a shape. Velocity dispersion is the increase of velocity with frequency, and it runs between two limits. At low frequency the wave is slow and the fluid has all the time it needs to flow and equilibrate pressure; this is the relaxed limit, the Gassmann velocity. At high frequency the wave oscillates faster than the fluid can move, pressure cannot equalise, the trapped fluid stiffens the rock, and the velocity is higher; this is the unrelaxed limit. Between them the velocity climbs through a step. On the illustrative curve carried from Part 4.5, a rock reads about 3.142 km/s in the relaxed limit and about 3.292 km/s at 1 MHz, close under its unrelaxed limit of 3.299, a rise of roughly 5 percent, with the midpoint of the step near 3.221 km/s. The size of that step and where on the frequency axis it sits are what the mechanisms decide.

Two Mechanisms, Two Places on the Axis

There are two ways for fluid flow to disperse a velocity, and they live at different frequencies. The first is Biot flow: the whole body of pore fluid moves relative to the solid frame as the wave passes, and the transition is set by the balance of viscous drag against fluid inertia. For common rocks and brines the Biot transition typically sits high, above the sonic band and often near or beyond the ultrasonic. The second is squirt flow: at the pore scale the wave squeezes fluid out of thin, compliant cracks and grain-contact spaces into the rounder, stiffer pores nearby, a local shuttling over microns rather than a bulk motion. Squirt has its transition wherever the crack geometry and fluid viscosity place it, and for many rocks that falls between the log band and the laboratory band, right in the gap between how a well is logged and how a core plug is measured. In most rocks squirt is the stronger of the two, the one that does most of the dispersing across the bands we care about, because a rock has far more compliant crack space to squeeze than the Biot mechanism's bulk flow exploits.

Reading the Curve Honestly

The single number to keep is that a velocity is not a property of a rock; it is a property of a rock at a frequency. The illustrative step here is a clean logistic for teaching, and its specific velocities (3.142, 3.221, 3.292) are the curve's, not measurements. What is real, and what the mechanisms deliver, is the physics that fixes the two ends and roughly where the transition falls: the relaxed end is the Gassmann limit from Part 4.5, the unrelaxed end reflects the extra stiffness of fluid that cannot escape, and squirt usually controls the climb between the log and lab bands. The mechanism frequencies themselves are quoted qualitatively on purpose, because the exact transition depends on crack aspect ratios and viscosity that are rarely known to better than an order of magnitude. This dispersion is not a loose end; it is one half of a single phenomenon. A rock that changes velocity with frequency must also lose energy, and the loss peaks exactly where the velocity is climbing fastest. That pairing, dispersion and attenuation as two faces of the same physics, is the subject of the next section.

References

• Biot, M. A. (1956). Theory of propagation of elastic waves in a fluid-saturated porous solid. I. Low-frequency range. Journal of the Acoustical Society of America, 28(2), 168-178.
• Mavko, G., & Jizba, D. (1991). Estimating grain-scale fluid effects on velocity dispersion in rocks. Geophysics, 56(12), 1940-1949.