Carroll

Summary

The Carroll relation is a power law of Shear velocity in Compressional velocity in km/s. It comes from a 1969 study of the acoustic properties of volcanic rock as far as I can establish, and the coefficients on this page were not checked against the paper. Treat the page as an example of a power-law Vp-Vs form, and as a bound for stiff rock rather than as a reservoir regression.

Inputs and outputs

Equations

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

The power-law regression, with both velocities in km/s, with the coefficients as used in practice (not verified against the publication):

The shear slowness and velocity ratio follow as before:

Because the exponent is below one, the ratio increases with \(\vsVp\), and the relation passes through the origin: zero velocity gives zero Vs.

Single-value calculator

Behavior

The power law predicts a much stiffer rock than the shale line in soft rock, and a softer one in hard rock. At 90 µs/ft (Vp = 3.387 km/s) it gives 2.052 km/s, against 1.747 km/s from the mudrock line, a shear slowness of 148.5 against 174.4 µs/ft. At 110 µs/ft (Vp = 2.771 km/s) the difference is larger: 1.741 against 1.216 km/s, or 175.1 against 250.6 µs/ft. The velocity ratio rises with Vp (1.50 at 2 km/s, 1.62 at 3, 1.70 at 4, 1.77 at 5 and 1.88 at 7 km/s) which is the opposite of the trend in clastic rocks, where Vp/Vs falls as the rock stiffens. A ratio of 1.5 at 2 km/s means a Poisson's ratio of 0.10, too low for a soft sediment. The relation crosses the mudrock line at about 4.1 km/s and lies below it at higher velocity, for example 2.823 against 3.138 km/s at 5 km/s, where the Brocher curve gives 3.011.

Parameter guidance

There are no parameters in the form shown. Where it applies. The form is for stiff, low-porosity rock of the kind it was derived on (volcanic rock, as I understand the source), and not for water-saturated shale or sand, where the power law gives Vs that is too fast. Do not use it in clastic reservoirs without comparing against measured shear. If you need a power law, fit one locally: take the logarithm of both velocities and regress \(\ln V_s\) on \(\ln V_p\), which gives the exponent as the slope and the coefficient as the exponential of the intercept. Units. km/s; a different unit changes the coefficient.

Worked example

A slowness of 90 µs/ft, with the unit conversion, followed by a table of Vp/Vs and Poisson's ratio against Vp, and the log-log fit that would reproduce the form from data:

import math
dt = 90.0
vp = 304.8 / dt
vs = 0.75609 * vp**0.81846
print(f'Vp = {vp:.4f} km/s ({1e6 / dt:.0f} ft/s);  Vs = {vs:.4f} km/s ({vs * 3280.84:.0f} ft/s);  DTS = {304.8 / vs:.1f} us/ft')
print(' Vp km/s   Vs km/s   Vp/Vs   Poisson   mudrock Vs')
for p in (2.0, 3.0, 4.0, 5.0, 6.0, 7.0):
    s = 0.75609 * p**0.81846
    r = p / s
    nu = (r * r - 2) / (2 * (r * r - 1))
    print(f'{p:8.1f} {s:9.3f} {r:7.3f} {nu:9.3f} {0.8621 * p - 1.1724:11.3f}')
# recover the form from synthetic points on the curve by a log-log least squares fit
ps = [2.0, 3.0, 4.0, 5.0, 6.0]
xs = [math.log(p) for p in ps]; ys = [math.log(0.75609 * p**0.81846) for p in ps]
n = len(xs); sx, sy = sum(xs), sum(ys)
b = (n * sum(x * y for x, y in zip(xs, ys)) - sx * sy) / (n * sum(x * x for x in xs) - sx * sx)
a = math.exp((sy - b * sx) / n)
print(f'log-log fit: Vs = {a:.5f} Vp^{b:.5f}')

Output

Vp = 3.3867 km/s (11111 ft/s);  Vs = 2.0520 km/s (6732 ft/s);  DTS = 148.5 us/ft
 Vp km/s   Vs km/s   Vp/Vs   Poisson   mudrock Vs
     2.0     1.333   1.500     0.100       0.552
     3.0     1.858   1.615     0.189       1.414
     4.0     2.351   1.701     0.236       2.276
     5.0     2.823   1.771     0.266       3.138
     6.0     3.277   1.831     0.287       4.000
     7.0     3.718   1.883     0.304       4.862
log-log fit: Vs = 0.75609 Vp^0.81846

Assumptions and limitations

• Stiff, low-porosity rock: the relation has Vp/Vs below 1.9 over the whole range up to 7 km/s, which is typical of hard rock and not of shale.
• A single power law through the origin describes the relation between Vp and Vs.
• The coefficients on this page are not confirmed against the paper.
• Rock is not strongly fractured, gas-bearing or altered, which all lower Vs relative to Vp.

QC checks

• Compare with a measured shear log, or with the mudrock line and the lithology lines: a difference of more than 0.2 km/s in clastics means the power law should not be used there.
• Vp/Vs of the result increases with Vp. If your measured data show the opposite, the relation does not describe the rock.
• Poisson's ratio of the result is at least 0.1 in the whole range used; lower values at low Vp mean the rock is softer than the model.

Going Deeper

Power laws between Vp and Vs are an old form in rock physics, since a straight line on a log-log plot is the simplest relation with no intercept that goes through the origin. The first compilations of Vp-Vs data for rocks used it before the lithology-specific lines of the 1980s and 1990s, which replaced it for clastics. For a clastic reservoir it is mostly of historical or comparative interest: it shows what a stiff-rock model would predict and so how much the choice of regression matters. A local log-log regression on measured shear data, in the same form, is a reasonable approach when the data are available.

References

1. Carroll, R.D., 1969. The determination of the acoustic parameters of volcanic rocks from compressional velocity measurements. International Journal of Rock Mechanics and Mining Sciences & Geomechanics Abstracts, 6(6), 557–579.

Python reference implementation

Related methods

• Greenberg-Castagna
• Castagna
• Brocher
• Eskandari