Velocity and Pressure

Part 8, Part 8: Pressure, Stress, and Time-Lapse

Learning objectives

Read the velocity-pressure curve as steep at first, then flattening, as microcracks close
Use the empirical form and anchor it: Vp near 2.95 km/s at 2 MPa rising to 3.68 at 20 MPa and 3.98 at 60 MPa
Separate the two effects: rapid crack closing early, then the gentle Hertz-Mindlin cube-root stiffening
Explain why velocity flattens with depth, so the same pressure step buys less and less velocity

Two Regimes in One Curve

Raise the effective pressure on a rock and its velocity climbs, but not evenly. At the lowest pressures the climb is steep; at high pressures it nearly stops. The reason is that a rock holds two kinds of compliance. Thin microcracks and loose grain contacts close first, and they are disproportionately soft, so squeezing them shut stiffens the rock quickly. Once they are closed, only the stiff, rounded pore space and the grain framework remain, and those respond to further pressure gently. The curve of velocity against effective pressure therefore rises fast and then bends over, and the bend marks the pressure at which the cracks have mostly shut.

An Empirical Form With Numbers

The standard way to capture this shape is an empirical exponential: $V_P = V_\infty - \Delta V\,e^{-P_{eff}/P^\ast}$ , where $V_\infty$ infty is the high-pressure velocity the rock approaches once the cracks are shut, $\Delta V$ is the total velocity the cracks are worth, and $P^\ast$ sets how fast they close. This is a descriptive form, not a law derived from grain mechanics, and it is labelled illustrative in the widget below. With $V_\infty = 4.0$ infty=4.0 km/s, $\Delta V = 1.2$ km/s, and $P^\ast = 15$ MPa, it reads $V_P = 2.95$ P=2.95 km/s at 2 MPa, 3.14 at 5, 3.38 at 10, 3.68 at 20, 3.92 at 40, and 3.98 at 60. The first 18 MPa, from 2 to 20, buys 0.73 km/s; the next 40 MPa, from 20 to 60, buys only 0.29. The cracks did most of their closing early, and the curve is flat long before the deepest pressure.

Why the Rigorous Part Agrees

The exponential describes the crack-closing part; the granular part has a firmer footing. Part 5 gave the Hertz-Mindlin pack, whose moduli grow as the effective pressure to the one-third power, $P_{eff}^{1/3}$ eff1/3. That cube-root law is itself a flattening curve: doubling the pressure multiplies the modulus by only $2^{1/3}$ , about 1.26, and velocity by less. So both pieces of the physics, the fast crack-closing early and the slow contact-stiffening throughout, point the same way, and the observed velocity flattens with depth. The practical consequence is sharp. A pressure change of a few MPa near the surface moves velocity a great deal; the same change at reservoir depth moves it little. Which is exactly why a small pore-pressure change during production can still be seen at depth if it lands where the curve is steep, and why, at a reservoir already near $V_\infty$ infty, you must look at the fluid instead. The next section reads this curve backwards to catch overpressure.

References

Eberhart-Phillips, D., Han, D.-H., & Zoback, M. D. (1989). Empirical relationships among seismic velocity, effective pressure, porosity, and clay content in sandstone. Geophysics, 54(1), 82-89.
Mavko, G., Mukerji, T., & Dvorkin, J. (2009). The Rock Physics Handbook (2nd ed.). Cambridge University Press.