Greenberg-Castagna

Summary

Greenberg and Castagna's method predicts Shear velocity from Compressional velocity in km/s with a separate regression for each lithology, and combines them for a mixed lithology with a Voigt-Reuss-Hill average weighted by the lithology volumes. Use it when mineral or lithology volumes are available and the rock is brine-saturated clastic or carbonate.

Inputs and outputs

Equations

The compressional velocity in km/s comes from the slowness in µs/ft:

For each lithology the shear velocity is a polynomial in \(\vsVp\), brine-saturated, in km/s:

With lithology volume fractions \(X_i\) (here \(\vsXss\), \(\vsXls\), \(\vsXdol\), \(\vsXsh\)) normalized to sum to one, the Voigt (arithmetic) and Reuss (harmonic) averages of the four lines are:

and the estimate is their mean, the Hill average:

Single-value calculator

Behavior

The four lines are close together at high velocity and separate at low velocity. At 90 µs/ft (Vp = 3.387 km/s) they give 1.868 (sandstone), 1.781 (limestone), 1.897 (dolomite) and 1.739 km/s (shale), so the lithology choice is worth up to 0.16 km/s of Vs, or about 15 µs/ft of slowness. At 140 µs/ft (Vp = 2.177 km/s) the range is 0.808 (shale) to 1.192 km/s (dolomite). The default mix of 60% sandstone, 10% limestone and 30% shale gives a Voigt value of 1.8204, a Reuss value of 1.8185 and a Hill value of 1.8195 km/s: a shear slowness of 167.5 µs/ft and a Vp/Vs of 1.861. The Voigt and Reuss bounds are within 0.002 km/s here because the lines are close, so which average is used matters little for clastics. In the plot the solid curve is the mix and the dashed curves are the four pure lines, with Vp on the horizontal axis.

Parameter guidance

Lithology fractions come from the mineral inversion or the volume of shale and clean-matrix curves, and need not add to one: they are normalized. Use the carbonate and clastic fractions of the solid rock, not of the bulk volume including pores, since the lines already include the effect of porosity through Vp. Which lines. Quartz sandstone uses the sandstone line, and clay-rich rock the shale line. Calcite uses the limestone and dolomite the dolomite line. A shaly sand is a mix of the two. Calibration. Check the modeled slowness against measured shear slowness in an offset well, and apply a scale and a shift if there is a systematic difference. Units. The coefficients are for Vp and Vs in km/s. Convert from µs/ft with 304.8/DT; do not apply them to ft/s or µs/ft. Fluids. The lines are for water-saturated rock. In gas or light-oil zones Vp is lowered and Vs is not, so the regression underestimates Vs; see the step page for the treatment.

Worked example

A slowness of 90 µs/ft converted to velocity, then the four lines, and a mix of 60% sandstone, 10% limestone and 30% shale:

dt = 90.0                                 # us/ft
v_fts = 1e6 / dt                         # ft/s
vp = v_fts * 0.3048 / 1000               # km/s
print(f'Vp = 10^6/{dt:g} = {v_fts:.0f} ft/s = {vp:.4f} km/s  (check 304.8/DT = {304.8 / dt:.4f})')
lines = {
    'sandstone': 0.80416 * vp - 0.85588,
    'limestone': -0.05508 * vp**2 + 1.01677 * vp - 1.03049,
    'dolomite': 0.58321 * vp - 0.07775,
    'shale': 0.76969 * vp - 0.86735,
}
for k, vs in lines.items():
    print(f'{k:10s} Vs = {vs:.4f} km/s = {304.8 / vs:6.1f} us/ft')
x = {'sandstone': 0.6, 'limestone': 0.1, 'dolomite': 0.0, 'shale': 0.3}
voigt = sum(x[k] * lines[k] for k in x)
reuss = 1 / sum(x[k] / lines[k] for k in x)
hill = 0.5 * (voigt + reuss)
print(f'Voigt = {voigt:.4f}, Reuss = {reuss:.4f}, Hill = {hill:.4f} km/s')
print(f'shear slowness = 304.8/{hill:.4f} = {304.8 / hill:.1f} us/ft;  Vp/Vs = {vp / hill:.3f}')
# limiting case: a single lithology reproduces its own line
print(f'100% sandstone: {0.80416 * vp - 0.85588:.4f} km/s')

Output

Vp = 10^6/90 = 11111 ft/s = 3.3867 km/s  (check 304.8/DT = 3.3867)
sandstone  Vs = 1.8675 km/s =  163.2 us/ft
limestone  Vs = 1.7812 km/s =  171.1 us/ft
dolomite   Vs = 1.8974 km/s =  160.6 us/ft
shale      Vs = 1.7393 km/s =  175.2 us/ft
Voigt = 1.8204, Reuss = 1.8185, Hill = 1.8195 km/s
shear slowness = 304.8/1.8195 = 167.5 us/ft;  Vp/Vs = 1.861
100% sandstone: 1.8675 km/s

Assumptions and limitations

• The rock is water-saturated. The regressions were derived for brine-saturated rock; in gas or oil zones Vp is lower at the same Vs and the regression underestimates Vs.
• The lithology lines hold across the range of velocity of the data they were fitted to, mainly consolidated clastics and carbonates; they extrapolate poorly in unconsolidated sand and in very hard rock.
• Lithology fractions are correct. Wrong fractions give wrong Vs, though the four lines are close in many clastic rocks.
• A Voigt-Reuss-Hill average of the lithology lines is a reasonable representation of a mixture. It is an empirical choice, and not a rigorous effective-medium result.
• The sonic log is a clean compressional slowness, free of cycle skips and washouts.

QC checks

• Vp/Vs of the result is in the range 1.5 to 2.5, and Poisson's ratio between 0 and 0.45.
• The modeled shear slowness reproduces the measured shear log in at least one offset well within a few percent, with no depth trend in the residual.
• Modeled shear slowness is always larger than compressional slowness times 1.41 (so that Poisson's ratio is not negative).
• A change in modeled Vs follows a change in lithology fractions, not noise in the sonic.
• Voigt and Reuss mixes are close. A large difference points to a very large contrast between the lines being mixed, or a bad fraction.

Going Deeper

Greenberg and Castagna's paper also gives a general formulation, with the shear velocity of a rock found from its mineral content, porosity and fluid by an iterative scheme based on Gassmann's equations. The lithology lines on this page are the simplest part of that work, the empirical Vp-Vs relations for brine-saturated rock, and are the part most often used in log analysis. The Voigt-Reuss-Hill average comes from Hill's 1952 work on aggregates. For a single mineral component the average of the two bounds is the one commonly used; for a mix of similar lines, as here, it makes little difference which is used. The lines for different lithologies are based on different data sets and cross at some velocities, so with an inconsistent combination of fractions a lithology change can produce a small jump.

References

1. Greenberg, M.L. and Castagna, J.P., 1992. Shear-wave velocity estimation in porous rocks: theoretical formulation, preliminary verification and applications. Geophysical Prospecting, 40(2), 195–209.
2. Castagna, J.P., Batzle, M.L. and Kan, T.K., 1993. Rock physics: the link between rock properties and AVO response. In Offset-Dependent Reflectivity: Theory and Practice of AVO Analysis, 135–171. Society of Exploration Geophysicists.
3. Mavko, G., Mukerji, T. and Dvorkin, J., 2009. The Rock Physics Handbook: Tools for Seismic Analysis of Porous Media, 2nd edition. Cambridge University Press.
4. Hill, R., 1952. The elastic behaviour of a crystalline aggregate. Proceedings of the Physical Society, Section A, 65(5), 349.

Python reference implementation

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