Brocher

Summary

Brocher's regression predicts Shear velocity from Compressional velocity in km/s with a fourth-degree polynomial fitted to crustal rocks over a Vp range of 1.5 to 8 km/s. It is a broad, single-curve relation, from sediment to crystalline rock, not tied to one lithology. Use it for regional models, deep sections or velocity models where lithology is not known.

Inputs and outputs

Equations

Compressional velocity in km/s, from the slowness in µs/ft:

Brocher's regression of shear velocity on compressional velocity, with both in km/s, valid for \(1.5 \le \vsVp \le 8\) km/s:

The shear slowness and the velocity ratio are:

Single-value calculator

Behavior

The curve is close to the mudrock line over the middle of the range and diverges from it at both ends. At 90 µs/ft (Vp = 3.387 km/s) it gives 1.756 km/s against 1.747 for the mudrock line; at Vp = 3 km/s the two are within 0.001 km/s (1.413 and 1.414). At Vp = 5 km/s it gives 3.011 km/s (Vp/Vs = 1.66) where the mudrock line gives 3.138, and at 7 km/s it gives 3.998 against 4.862: the straight line overpredicts Vs in hard rock, and the polynomial bends to a Vp/Vs near 1.7. At the low end, Brocher gives 0.337 km/s at Vp = 1.5 against 0.121 for the mudrock line. The calculator reports whether Vp is inside the valid range, which is a slowness of 38.1 to 203.2 µs/ft.

Parameter guidance

There are no parameters to pick. Range. Do not use it outside Vp of 1.5 to 8 km/s: the polynomial is a fit to data in that range and can give nonphysical values outside it. In slowness the range is 38.1 to 203.2 µs/ft. Which rocks. The regression is for a general crustal rock population, mixed sedimentary and crystalline. It is not specific to clastic reservoir rocks and not tuned to water-saturated shale, so in a clastic well it is less accurate than the lithology-specific lines when the lithology is known. Units. Vp and Vs are in km/s, as for every line on this step. Calibration. As for the other regressions, a scale and shift on the shear slowness can be fitted to a measured shear log in an offset well.

Worked example

A slowness of 90 µs/ft, converted to km/s, with the polynomial evaluated term by term; the range check; and a comparison at Vp of 5 km/s with the mudrock line:

dt = 90.0                                  # us/ft
vp = 304.8 / dt                            # km/s
c = [0.7858, -1.2344, 0.7949, -0.1238, 0.0064]
terms = [ci * vp**i for i, ci in enumerate(c)]
vs = sum(terms)
print(f'Vp = 304.8/{dt:g} = {vp:.4f} km/s ({1e6 / dt:.0f} ft/s)')
print('terms:', ', '.join(f'{t:.4f}' for t in terms))
print(f'Vs = {vs:.4f} km/s = {vs * 3280.84:.0f} ft/s;  DTS = {304.8 / vs:.1f} us/ft;  Vp/Vs = {vp / vs:.3f}')
print(f'valid range of slowness: {304.8 / 8:.1f} to {304.8 / 1.5:.1f} us/ft')
vp = 5.0
b = sum(ci * vp**i for i, ci in enumerate(c))
print(f'at Vp = 5 km/s: Brocher {b:.3f} km/s, mudrock line {0.8621 * vp - 1.1724:.3f} km/s')

Output

Vp = 304.8/90 = 3.3867 km/s (11111 ft/s)
terms: 0.7858, -4.1805, 9.1171, -4.8088, 0.8419
Vs = 1.7555 km/s = 5760 ft/s;  DTS = 173.6 us/ft;  Vp/Vs = 1.929
valid range of slowness: 38.1 to 203.2 us/ft
at Vp = 5 km/s: Brocher 3.011 km/s, mudrock line 3.138 km/s

Assumptions and limitations

• The relation is an average over many rock types; a given rock can be off it by several tenths of a km/s.
• Vp is within 1.5 to 8 km/s, the range of the fit.
• Velocities are those of water-saturated rock at the frequency and confining stress of the data used by the fit; gas, very high temperature and shallow, unconsolidated sediment are outside it.
• The polynomial is a fit, not a physical model: it has no mineral or porosity terms and its shape between data points is not guaranteed.

QC checks

• Vp is inside 1.5 to 8 km/s everywhere. The flag in the calculator marks the intervals that are not.
• Vp/Vs of the modeled curve is between 1.6 and 2.2 in normal rocks, decreasing with increasing Vp above about 3 km/s.
• In clastic sections, compare with the mudrock line and the lithology-specific lines. Where they differ by more than 0.1 km/s, check against a measured shear log.
• The result does not show wiggles or a reversal in trend that are not in the sonic.

Going Deeper

Brocher's 2005 paper was written for crustal seismology, with relations for Vs, density and Poisson's ratio as functions of Vp built from a compilation of laboratory and field measurements, and the polynomial is a regression through those data: it is sometimes called a Vp-Vs relation for 'average' rock. Its value is as a smooth, single-valued relation that is acceptable from soft sediment to basement, which makes it suited to large-scale velocity models and to wells that pass through unknown lithology. In reservoir work it is a fallback behind the lithology-specific lines.

References

1. Brocher, T.M., 2005. Empirical relations between elastic wavespeeds and density in the Earth's crust. Bulletin of the Seismological Society of America, 95(6), 2081–2092.

Python reference implementation

Related methods

• Greenberg-Castagna
• Castagna
• Eskandari
• Carroll