Eskandari

Summary

The Eskandari relation is a second-degree polynomial regression of Shear velocity on Compressional velocity in km/s attributed to a regional study of a carbonate reservoir. The coefficients on this page could not be checked against the publication: treat the page as the structure of a locally fitted Vp-Vs regression, and calibrate or replace it with your own fit.

Inputs and outputs

Equations

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

The quadratic regression, with both velocities in km/s, in the form and with the coefficients as used in practice (not verified against the publication, see the review notes):

with the shear slowness and velocity ratio, as on the other pages:

A quadratic in \(\vsVp\) has a maximum at \(\vsVp = 1.6126 / (2 \times 0.11236) = 7.18\) km/s, where \(\vsVs = 3.48\) km/s. Beyond that point the relation predicts that Vs decreases as Vp increases, which is not physical, so the equation is limited to lower velocities.

Single-value calculator

Behavior

The curve is steeper than the mudrock line at low Vp and flatter at high Vp. At 90 µs/ft (Vp = 3.387 km/s) it gives 1.867 km/s, 0.12 km/s above the mudrock line (1.747), a shear slowness of 163.3 µs/ft. At Vp = 5 km/s it gives 2.948 km/s (Vp/Vs = 1.70), and at 7 km/s 3.477 km/s (Vp/Vs = 2.01), where the quadratic is close to its maximum and Vp/Vs rises again. At low velocity it gives small Vs and large Vp/Vs: 0.470 km/s at Vp = 2 km/s (ratio 4.3). Quadratic regressions are not meant to be used far from the data they were fitted to, and this one has no stated range in the form here.

Parameter guidance

There is nothing to pick in the form given, but the right way to use any such regression is to refit it. Local calibration. If measured shear logs exist in the field or an offset well in the same carbonate, fit a quadratic of Vs on Vp by least squares and compare its coefficients with those above; use the local fit where they differ. Range. Restrict use to the velocity range of the calibration data, normally consolidated carbonate with Vp above about 3.5 km/s, and keep Vp below 7 km/s because of the maximum. Lithology. The relation is for carbonates; for clastics use the lithology lines of the Greenberg-Castagna page or the mudrock line. Units. km/s, as above.

Worked example

A slowness of 90 µs/ft, with unit conversion, and a comparison with the mudrock line at several velocities. The comparison shows where the two give different results, which is the practical reason for a local calibration:

dt = 90.0
vp = 304.8 / dt
vs = -0.11236 * vp**2 + 1.6126 * vp - 2.3057
print(f'Vp = {vp:.4f} km/s ({1e6 / dt:.0f} ft/s)')
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'vertex of the parabola: Vp = {1.6126 / (2 * 0.11236):.2f} km/s')
print(' Vp km/s   DT us/ft   Eskandari-type   mudrock   difference')
for p in (3.0, 4.0, 5.0, 6.0, 7.0):
    e = -0.11236 * p**2 + 1.6126 * p - 2.3057
    m = 0.8621 * p - 1.1724
    print(f'{p:8.1f} {304.8 / p:10.1f} {e:15.3f} {m:9.3f} {e - m:12.3f}')

Output

Vp = 3.3867 km/s (11111 ft/s)
Vs = 1.8669 km/s = 6125 ft/s;  DTS = 163.3 us/ft;  Vp/Vs = 1.814
vertex of the parabola: Vp = 7.18 km/s
 Vp km/s   DT us/ft   Eskandari-type   mudrock   difference
     3.0      101.6           1.521     1.414        0.107
     4.0       76.2           2.347     2.276        0.071
     5.0       61.0           2.948     3.138       -0.190
     6.0       50.8           3.325     4.000       -0.675
     7.0       43.5           3.477     4.862       -1.385

Assumptions and limitations

• The relation is a regional regression for a carbonate reservoir and may not transfer to other fields, minerals or porosity ranges.
• Vp-Vs relations depend on pore type and fluid; carbonates with vugs and fractures scatter widely around any single curve.
• The coefficients on this page are not confirmed against the paper.
• The relation is monotonic only below Vp = 7.18 km/s.

QC checks

• Vp is inside the velocity range of the calibration data and below 7 km/s.
• Vp/Vs between 1.6 and 2.1 over the carbonate; higher values mean the relation is outside its range.
• The model is compared with the measured shear log in an offset well, and fitted again locally if the residual is biased.
• Compare with the limestone and dolomite lines of Greenberg and Castagna: the difference (0.03 to 0.09 km/s at 90 µs/ft and up to 0.23 km/s at 70 µs/ft) is a measure of uncertainty of the model.

Going Deeper

The study usually cited for this relation applied multiple regression and neural networks to predict shear wave velocity from wireline logs in an Iranian carbonate reservoir. The single-variable quadratic in Vp is the simplest form that came out of that kind of work; the multi-log variants add neutron porosity and density as predictors. The general point is that a Vp-Vs relation fitted to one carbonate reservoir is a local model. As in much petrophysical regression, the form (a quadratic) is incidental and the data it was fitted to are what matter, so a local fit to measured Vs is better than any imported set of coefficients.

References

1. Eskandari, H., Rezaee, M.R. and Mohammadnia, M., 2004. Application of multiple regression and artificial neural network techniques to predict shear wave velocity from wireline log data for a carbonate reservoir, South-West Iran. CSEG Recorder, 29(7), 42–48.

Python reference implementation

Related methods

• Greenberg-Castagna
• Castagna
• Brocher
• Carroll