Castagna

Summary

Castagna's mudrock line is a straight-line relation between compressional and shear velocity in water-saturated clastic silicate rock, Shear velocity = 0.8621 Compressional velocity in km/s - 1.1724 in km/s. Castagna and co-authors also published limestone and dolomite relations. Use the mudrock line for shale-dominated clastics when nothing better than the sonic is available.

Inputs and outputs

Equations

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

The mudrock line of Castagna and co-authors, in km/s, is often written as \(\vsVp = 1.16\,\vsVs + 1.36\). Solved for the shear velocity:

For carbonates, the limestone and dolomite relations (the same lines as in the Greenberg-Castagna page, to rounding) are:

The shear slowness, velocity ratio and dynamic Poisson's ratio follow:

Single-value calculator

Behavior

At 90 µs/ft (Vp = 3.387 km/s) the mudrock line gives Vs of 1.747 km/s, a shear slowness of 174.4 µs/ft, Vp/Vs of 1.938 and a Poisson's ratio of 0.319. The limestone line gives 1.781 and the dolomite line 1.897 km/s at the same Vp, which is a shear slowness of 171.1 and 160.6 µs/ft. The lines separate as velocity falls: at 140 µs/ft the mudrock, limestone and dolomite lines give 0.705, 0.922 and 1.192 km/s. The mudrock line, as a straight line with a large negative intercept, extrapolates to a very small Vs at low Vp and to a large Vp/Vs: at 140 µs/ft the ratio is 3.1. The plot has Vp on the horizontal axis. The calculator returns an empty result if the line gives a Vs of zero or less.

Parameter guidance

The mudrock line has no parameters to pick. It was derived for water-saturated clastic silicate rock, mainly shale and sandy shale, over a Vp range of roughly 1.5 to 5.5 km/s, so check that the data are in that range. For a sandstone, use the Greenberg-Castagna sandstone line, which is higher at a given Vp. For carbonate use the limestone or dolomite lines according to the mineral. Units. Use km/s; the intercept is meaningless in other units. For conversions, 304.8/DT in µs/ft gives km/s. In ft/s multiply by 0.3048 and divide by 1000, or multiply km/s by 3280.84 for ft/s. Calibration. A scale and shift on the shear slowness, fitted to offset wells, is the usual correction for regional differences.

Worked example

A slowness of 90 µs/ft, with conversion between slowness, ft/s and km/s, the three lines, and the check that the inverse form of the mudrock line returns the compressional velocity:

dt = 90.0                                    # us/ft
v_fts = 1e6 / dt
vp = v_fts * 0.3048 / 1000
print(f'DT {dt:g} us/ft -> {v_fts:.0f} ft/s -> Vp = {vp:.4f} km/s')
lines = {'mudrock': 0.8621 * vp - 1.1724,
         'limestone': -0.05508 * vp**2 + 1.01677 * vp - 1.03049,
         'dolomite': 0.58321 * vp - 0.07775}
for k, vs in lines.items():
    r = vp / vs
    nu = (r**2 - 2) / (2 * (r**2 - 1))
    print(f'{k:10s} Vs = {vs:.4f} km/s ({vs * 3280.84:6.0f} ft/s), DTS = {304.8 / vs:6.1f} us/ft, Vp/Vs = {r:.3f}, Poisson = {nu:.3f}')
print(f'inverse form: 1.16 x {lines["mudrock"]:.4f} + 1.36 = {1.16 * lines["mudrock"] + 1.36:.4f} km/s (compare Vp = {vp:.4f})')

Output

DT 90 us/ft -> 11111 ft/s -> Vp = 3.3867 km/s
mudrock    Vs = 1.7472 km/s (  5732 ft/s), DTS =  174.4 us/ft, Vp/Vs = 1.938, Poisson = 0.319
limestone  Vs = 1.7812 km/s (  5844 ft/s), DTS =  171.1 us/ft, Vp/Vs = 1.901, Poisson = 0.309
dolomite   Vs = 1.8974 km/s (  6225 ft/s), DTS =  160.6 us/ft, Vp/Vs = 1.785, Poisson = 0.271
inverse form: 1.16 x 1.7472 + 1.36 = 3.3868 km/s (compare Vp = 3.3867)

Assumptions and limitations

• The rock is water-saturated clastic silicate rock. In gas or oil the Vp is lowered at constant Vs, and the line underestimates Vs.
• A single straight line represents the Vp-Vs relation for all shale and shaly sand; the scatter around the line, of order 0.1 km/s or more, is not described.
• Vp is within the range used to derive the line (roughly 1.5 to 5.5 km/s). It is less accurate for loose, very soft sediment and for hard, low-porosity rock, where the line gives too large a Vs (see the plot at low and high Vp).
• The limestone and dolomite lines hold for clean carbonates of those minerals.
• Velocities are for the frequency of the sonic log; no correction for dispersion is applied.

QC checks

• The modeled Vs is smaller than Vp / 1.41 (so that Poisson's ratio is positive) and Vp/Vs is between 1.5 and 3 for shale.
• The modeled curve overlays the measured shear log in an offset well in shale, with no trend with depth.
• In clean sand and in carbonate, check that the right line is used; the mudrock line gives too low a Vs in a clean sandstone (about 0.12 km/s lower at Vp = 3.4 km/s than the Greenberg-Castagna sandstone line).
• Poisson's ratio of the modeled curve is physically plausible (0.1 to 0.4 for most rock).

Going Deeper

The mudrock line comes from Castagna, Batzle and Eastwood's 1985 analysis of laboratory and log data for water-saturated clastic silicate rocks, in which the relation between Vp and Vs in the shales and sandy shales was close to linear. It became the standard way to estimate Vs for AVO and seismic modeling when no shear log was available. Its limits are well known: sandstones lie above it, and gas-bearing sands move off it in the other direction, so AVO workers use it as a baseline for the 'wet' trend and read the departure as a fluid indicator. A single line also hides lithology effects, which is the motivation for the lithology-specific lines of Greenberg and Castagna.

References

1. Castagna, J.P., Batzle, M.L. and Eastwood, R.L., 1985. Relationships between compressional-wave and shear-wave velocities in clastic silicate rocks. Geophysics, 50(4), 571–581.
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.

Python reference implementation

Related methods

• Greenberg-Castagna
• Brocher
• Eskandari
• Carroll