Vs From Vp

Part 7, Part 7: Calibration, From Data to Model

Learning objectives

State the missing-log problem: shear velocity is often not recorded, yet AVO and fluid substitution need it
Use Castagna's mudrock line and the Greenberg-Castagna lithology lines to predict brine-rock Vs from Vp
Anchor the numbers: at Vp 3.5 km/s the mudrock line gives Vs about 1.85 and the sand line about 1.96 km/s
Follow the correct hydrocarbon workflow: predict Vs on the brine leg, then fluid-substitute, never apply a brine line to a gas sand

The Log That Is Not There

Two of the workflows this course has built lean on the shear velocity. AVO reads the way a reflection changes with angle, and that angular behavior is governed by the contrast in $V_p/V_s$ across the interface. Gassmann fluid substitution needs the shear modulus to carry the dry frame from one fluid to another. Yet the shear velocity is the log most often absent: many wells were logged for $V_p$ p and density alone, and older wells for $V_p$ p only. So a routine and unavoidable task is to predict $V_s$ s from what you do have, and the empirical backbones of the previous section are exactly the tool. Someone measured $V_p$ p and $V_s$ s together on many brine-saturated rocks and fitted the relationship; you evaluate that fit at your measured $V_p$ p and read off a shear velocity.

Mudrock and the Lithology Lines

The oldest and simplest is Castagna's mudrock line, $V_s = (V_p - 1.36)/1.16$ 1.36)/1.16, a single straight fit for water-wet clastics. Greenberg and Castagna refined it in 1992 into separate lines for pure lithologies: sandstone, limestone, dolomite, and shale each get their own regression, because a given $V_p$ p implies a different $V_s$ s depending on what the rock is made of. At $V_p = 3.0$ p=3.0 km/s the mudrock line gives $V_s = 1.41$ s=1.41, the sandstone line 1.56, and the limestone line 1.52. Step up to $V_p = 3.5$ p=3.5 and they read 1.85, 1.96, and 1.85; at $V_p = 4.0$ p=4.0 they read 2.28, 2.36, and 2.16. The lithology matters: a limestone and a sandstone at the same $V_p = 4.0$ p=4.0 differ by 0.2 km/s in predicted $V_s$ s, which is the difference between a right and a wrong AVO gradient. Choose the line for the rock you have, and where a zone is a mix, the full Greenberg-Castagna scheme averages the pure-lithology predictions by volume fraction.

Brine Lines and the Classic Error

There is one discipline these lines demand, and it is the source of a common and costly mistake. Every one of them was fitted to brine-saturated rock. They tell you the shear velocity a rock would have if its pores held water. A gas sand does not: gas lowers the P-wave velocity sharply while leaving the shear velocity almost untouched, so its measured $V_p/V_s$ is far lower than any brine line predicts. Feed the gas sand's low $V_p$ p straight into a brine line and it will hand back a shear velocity that is too low, poisoning the very AVO analysis you were trying to set up. The correct move respects the physics of Part 4. Predict $V_s$ s on the brine leg, either at a nearby wet interval or by first substituting the zone to brine, then run Gassmann forward to put the hydrocarbon back and let it lower $V_p$ p while shear stays fixed. The brine line supplies the frame's shear behavior; Gassmann supplies the fluid effect. With a shear velocity in hand for every zone, the one property still missing from many old wells is density, and that is the next section.

References

Castagna, J. P., Batzle, M. L., & Eastwood, R. L. (1985). Relationships between compressional-wave and shear-wave velocities in clastic silicate rocks. Geophysics, 50(4), 571-581.
Greenberg, M. L., & Castagna, J. P. (1992). Shear-wave velocity estimation in porous rocks: Theoretical formulation, preliminary verification and applications. Geophysical Prospecting, 40(2), 195-209.