Transfer Functions for Predicting Borehole Gamma-Ray Logs

Abstract

If the gamma-ray emission rate of rocks varies only with depth, the intensity recorded in a vertical borehole is mathematically predictable. The Fourier transforms of these two functions of depth relate multiplicatively via transfer functions expressible as infinite series whose terms depend on Bessel functions. The results encode readily as programs with imperceptible run times that sample the transfers regularly for use with digitized data. Separate transfers treat non-directional and directional detectors, both supposed of negligible size and located on the borehole axis. Predicting the log from a synthetic test bed confirms and illustrates the theory.

Appendices

Appendix: List of Variables

I omit symbols used only fleetingly, such as $c, h, L, \theta , \varpi$, or as summation indices, like n.

$a =$ :

Borehole radius, axis to casing

$b =$ :

Casing thickness

$e=$ :

$\exp (1)$

$i=$ :

$\sqrt{-1}$

$r =$ :

A radius, different in Propositions 1 and Lemma 5

s, t:

Sections 2 and 3 only, defined for Eq. 5

s :

Beyond Sect. 3, as introduced in Eq. 13

t(z):

Outside Sects. 2 and 3, defined in Eq. 1

$u =$ :

Spatial frequency

$w =$ :

Dummy variable of integration

$z =$ :

Distance down hole from surface (except from mid-sill in Sect. 3)

A :

Defined for Eq. 5, maximum value A₀

$I(z) =$ :

Measured intensity

$I_s$ :

Modified Bessel function

${\mathcal {I}}_s$ :

Introduced in Eq. 13

$J(u) =$ :

Transform of I(z)

K :

Defined for Eq. 5

$N =$ :

Emission rate, factor of K

${\mathbb {N}}$ :

Natural numbers $1,2,3,\ldots$

${\mathcal {P}}, {\mathcal {Q}}, {\mathcal {R}}$ :

Defined by Eq. 7

$R = s+t$ :

Sections 2 and 3 only (symbol reused in Lemma 5)

T(u):

Defined in Eq. 2

$T^{(1)}(u), T^{(2)}(u)$ :

Transfers for spherical and linear detector

V :

Rock volume with differential $\mathrm{d}V$

$\alpha =$ :

Parameter in Eqs. 1 and 2, outside Sect. 3 only

$\alpha , \beta , \gamma =$ :

Coordinates of emitting element, in Sect. 3 only

$\epsilon =$ :

Detector efficiency, factor of K

$\kappa , \lambda$ :

Defined by Eq. 6

$\mu , \nu , \sigma , \tau$ :

Defined for Eq. 5

$\phi , \chi , \psi$ :

Defined by Eqs. 6 and 15

$\varGamma$ :

Usual notation for the gamma function

$\digamma$ :

Defined by Eq. 6 (digamma)

$\varDelta$ :

Defined by Eq. 6

Errata

In Dickson and Craig (2012), for the denominator u in (A-3) read $\pi {u}$. The borehole diameters on page D163 should be 10.16 and 11.43 cm.