Vclay from GR (Stieber)

Summary

The Stieber relation turns the Gamma ray index into Clay volume with a hyperbolic curve that lies below the straight line \(V_{cl} = I_{GR}\). It removes much of the clay volume that the linear index assigns to cleaner intervals, while still reaching 1 at the clay point. Use it when the linear index is known to overestimate clay volume and you only have a gamma ray.

Inputs and outputs

Equations

The gamma ray index scales the log between the clean and clay values and is clamped to the interval 0 to 1:

The Stieber relation then gives the clay volume:

Single-value calculator

Behavior

The dashed line is the linear index, which is what you get with no transform. The Stieber curve starts with a slope of 1/3 at \(\IGR = 0\) and steepens to a slope of 3 at \(\IGR = 1\), so most of the correction is in the cleaner half of the range. At \(\IGR = 0.5\) the clay volume is 0.25, half the linear value.

Parameter guidance

The method has no parameters of its own. The result depends entirely on the two gamma ray picks, Clean gamma ray and Clay gamma ray, which are shared by every gamma-ray-based clay volume method. How to pick them, and why they should be picked per zone, is covered on the Clay Volume page under shared parameter picking.

Because the curve is nonlinear, a small error in the picks changes the answer by different amounts at different gamma ray values. Near the clay point the slope is high, so an underestimated Clay gamma ray pushes clay volume to 1 quickly.

Worked example

A reading of 65 gAPI, with a clean pick of 20 gAPI and a clay pick of 120 gAPI:

gr, gr_clean, gr_clay = 65.0, 20.0, 120.0
igr = min(1.0, max(0.0, (gr - gr_clean) / (gr_clay - gr_clean)))
vcl = igr / (3.0 - 2.0 * igr)
print(f"IGR = ({gr:g} - {gr_clean:g}) / ({gr_clay:g} - {gr_clean:g}) = {igr:.3f}")
print(f"Vcl = {igr:.3f} / (3 - 2 x {igr:.3f}) = {vcl:.3f}")
print(f"Linear index would give {igr:.3f}, so Stieber is {igr - vcl:.3f} lower")

Output

IGR = (65 - 20) / (120 - 20) = 0.450
Vcl = 0.450 / (3 - 2 x 0.450) = 0.214
Linear index would give 0.450, so Stieber is 0.236 lower

Assumptions and limitations

• The gamma ray responds only to clay. Radioactive feldspars, micas, glauconite, uranium-rich organic matter and heavy-mineral sands all raise the gamma ray without adding clay, so clay volume is overestimated.
• A single, correct pair of clean and clay picks applies across the interval. If the clay mineralogy or the clean-sand response changes with depth, the picks must change too.
• The gamma ray is environmentally corrected and normalized across wells, as covered in Stage 1.
• The Stieber curve is a fixed shape. It does not adapt to the rock, so it may under-predict clay in formations where the linear index is already close to correct.

QC checks

• The result must be between 0 and 1, and equal to 0 at the clean pick and 1 at the clay pick.
• The Stieber value must never exceed the linear index at the same depth.
• Compare with another independent clay indicator, such as neutron-density, in a clean-looking and a shaly interval. If the gamma-ray-based clay volume is consistently higher, suspect a radioactive mineral.
• Check that a visibly clean sand reads close to zero, not a few percent.

Going Deeper

The relation belongs to a family of hyperbolic transforms of the form

where \(a = 1\) gives the linear index and \(a = 3\) gives the Stieber relation used here. The Stieber form can also be written as \(0.5\,\IGR / (1.5 - \IGR)\). It is one of several nonlinear gamma ray transforms, alongside the Larionov relations for younger and older rocks and the Clavier relation. They differ in how strongly they reduce the linear index, and none is derived from first principles. Each was fitted to a particular set of data, so the choice between them is an empirical calibration against core, an independent clay indicator, or local practice.

References

1. Stieber, S.J., 1970. Pulsed neutron capture log evaluation, Louisiana Gulf Coast. SPE 2961, SPE Annual Fall Meeting, Houston.

Python reference implementation

Related methods

• Vclay from GR (Linear Index)
• Vclay from GR (Larionov, Tertiary)
• Vclay from GR (Larionov, Older Rocks)
• Vclay from GR (Clavier)