Although a relationship between gamma ray log response and shale volume had been recognized since the introduction of gamma ray logging in the late 1930s and early 1940s, the formula for gamma ray index, and the equating of gamma ray index to shale volume apparently appeared in the late 1960s. Contemporaneously there appeared three similar, alternative, non-linear relationships in 1969, 1970, and 1971. These functions were based upon observations and empirical graphical functions. Subsequently, these graphical functions were fit using very dissimilar-looking formulas. Only the 1969 data set was published in support of the graphical functions. No attempt to link these functions with a single formula was ever made, and only vague verbal explanations have been offered for the non-linear functions. Further, the 1969 publication was in Russian, partly mistranslated, and the mistranslation never corrected. Consequently, two of the resulting formulas are misapplied. In this article I review the four standard non-linear functions (i.e., Larionov’s two, Stieber’s, Clavier’s), examine their similarities, and show that a single function would serve the same purpose as all four, thereby eliminating a source of confusion for formation evaluators. When these shale (or clay) volume versus gamma ray index transforms are inverted to functions of gamma ray index versus shale (or clay) fractional volume a remark-able property is revealed: the increment of radioactivity per unit shale volume decreases with increases in fractional shale volume. In other words, if one unit of shale per unit volume produces a gamma ray intensity of 10 API units we would think it strange if 10 units of shale per unit volume produced only, say, 60 API units of gamma radiation (instead of 100). Yet, this is the message contained in these functions. The cause for this phenomenon has been speculated upon, but only briefly and not often. To remedy this lack of speculation, I propose a physical model and give it mathematical form. This model is in-tended as a challenge to theoretical-minded petrophysicists to falsify it, make it better, or propose an alternative and more realistic model. I also provide (in Appendix C) a digital listing of all the published graphical data in the literature that support the introduction of the non-linear shale (and clay) fractional volume – gamma ray index transforms.
Sampling heuristics for active function learning
