Determining elemental concentrations and formation matrix type from natural gamma ray spectral logs

Natural gamma radiation is detected by a scintillation detector in a well logging sonde and separated into at least six separate energy regions. The gamma ray counts in the higher four energy regions are used to derive borehole compensated elemental concentrations of potassium, uranium and thorium. The gamma ray counts lower two energy regions are used to identify formation lithology.

FIELD OF THE INVENTION 
The present invention is concerned with detecting the naturally occurring 
gamma radiation from the earth formations penetrated by a well borehole. 
More specifically, the invention concerns methods and apparatus for 
detecting gamma radiation and measuring the gamma ray count rate in 
defined energy ranges. The naturally emitted gamma rays enable 
determination of relative concentrations of potassium, uranium and thorium 
in the earth formations surrounding a well borehole. The penetration of 
the gamma ray photons is a function of earth formation matrix type. The 
gamma ray count rates in the selected energy windows are isolated to 
define a factor expressive of formation matrix type. 
BACKGROUND OF THE INVENTION 
Naturally occurring gamma radiation for K-U-T (potassium, uranium and 
thorium) elements yields gamma ray intensity vs. energy spectra in the 
vicinity of a well borehole observed by highly stable scintillation 
detectors in the well borehole. The radiation at the scintillation 
detector and its associated photomultiplier produces a pulse height 
spectrum proportional to the energy of gamma rays impinging on the 
scintillation crystal. The spectrum is divided into selected energy ranges 
or windows. Three windows are centered about selected gamma ray emission 
peaks for the naturally occurring gamma rays of the K-U-T elements. Gamma 
ray count rates in each of the three energy ranges are transmitted to the 
surface and processed by a technique known as spectrum stripping wherein 
standard calibration spectra, for each of the individual elements 
(obtained in standard boreholes) are applied to the unprocessed data 
(count rates) of the selected windows (energy ranges) to detect each of 
the three elements of interest. The "stripping constants" are derived from 
measurements of the standard gamma ray energy spectra in standard 
boreholes containing essentially only one of the three elements to enable 
the stripping constants to be applied to the measured spectrum in an 
unknown earth formation surrounding a borehole. The concentrations of the 
three elements of interest are determined after application of the 
stripping constants. After carrying out specified procedures, elemental 
concentrations of the K-U-T elements are obtained. A fourth window is used 
to compensate the K-U-T concentrations for borehole effects, as described 
in U.S. patent application filed May 21, 1981, Ser. No. 265,736 now U.S. 
Pat. No. 4,436,996. Fifth and sixth windows in the observed spectrum are 
processed to isolate a factor indicative of formation matrix type. 
The actual gamma ray count rate achieved at a scintillation detector in a 
well borehole is dependent on the Compton attenuation coefficient .eta.. 
Each photon has a point of origination somewhere in the adjacent earth 
formations in traveling toward the scintillation detector. The attenuation 
of the gamma ray photon flux along the path of travel is dependent on the 
thickness of the material, the density of the material and the formation 
matrix type of the material. The gamma ray photons travel along a path 
having a length which is statistically determined from the distributed 
emission sources, namely the K-U-T elements. The present invention 
provides a measurement of formation matrix type of an adjacent formation 
by utilization of the measured natural gamma ray spectrum observed at a 
scintillation crystal coupled with signal processing procedures as 
described below. 
Major attenuation factors of the gamma ray flux include pair production, 
Compton scattering, and photoelectric absorption. Below certain energy 
levels, pair production is negligible and, therefore, not significantly 
involved in the method described herein. 
The observed or measured gamma ray energy spectra are thus separated into 
six energy level windows. The location of the six energy windows in the 
observed gamma ray spectrum is important. There are three predominant 
energy peaks for the K-U-T elements, and windows are normally defined to 
observe the peaks. THe K-U-T peaks are 1.46, 1.76 and 2.61 MeV gamma 
radiation peaks for potassium (K.sup.40), uranium (Bi.sup.214) and thorium 
(Tl.sup.208), respectively. A fourth energy window is defined in the 
Compton dominated spectrum to compensate for borehole and formation 
density induced changes in the calculated K-U-T concentrations. The fifth 
window (which may or may not overlap the fourth window) is sufficiently 
high in energy range to be above the effects of photoelectric absorption 
so that the primary mode of photon attenuation is Compton scattering. The 
sixth window is defined at very low gamma ray energy levels where 
photoelectric absorption is of major importance and relative attenuation 
due to pair production or the Compton effect is minimized. 
The measured count rates in the fifth and sixth windows can be used to 
define a ratio which, after isolation of K-U-T elemental concentration 
effects, is primarily a function of the formation photoelectric absorption 
cross-section U, and the borehole parameters adjacent to the tool. 
BRIEF SUMMARY OF THE DISCLOSURE 
This disclosure sets forth methods and apparatus for determining formation 
matrix type of earth formations adjacent to a well borehole. This is 
accomplished through measuring the naturally occurring gamma ray energy 
spectrum attributable to K-U-T elements and through evaluating the 
spectrum for an indication of formation matrix type.

DETAILED DESCRIPTION OF THE INVENTION 
The present invention is concerned with formation matrix type measurements. 
Before formation matrix type valuations are discussed, certain preliminary 
measurements must be described. The preliminary measurements of a 
naturally occurring gamma ray spectrum defined in selected energy windows 
are made and elemental concentrations are determined. Gamma ray 
attenuation due to Compton scattering is determined and density is 
obtained. Then, photoelectric absorption attenuation is also determined 
leading to evaluation of formation matrix type. 
Logs of natural gamma ray activity in three energy ranges have been used to 
estimate the potassium (K), uranium (U) and thorium (Th) content of earth 
formations. These logs (commonly referred to K-U-T logs) were initially 
used to determine other important information about the earth formations 
penetrated by a well borehole such as: 
(1) the oxidation state of the bed at the time of deposition; 
(2) quantity of organic material in sedimentary layers which (together with 
Item 1) leads to source rock bed identification; 
(3) the depositional environment of the bed (i.e. continental vs. marine 
deposition); 
(4) water movement in downhole formations which, in turn, may indicate 
fractures, faulting or permeability; 
(5) water movement in the borehole region which may indicate channeling or 
water producing perforations; 
(6) more accurate shale content determinations for a particular bed and; 
(7) clay typing and marker bed identification. 
The two commercially available services, responsive to naturally occurring 
gamma radiation, each use scintillation type gamma ray detectors which are 
biased to record the gamma radiation in either three or five energy bands 
or windows. Count rate contributions from the decay of each of the 
elements of interest, or their daughter decay products are mathematically 
stripped or fitted from the composite count rates observed within the 
three or five energy windows. Elemental concentrations may then be 
computed from the stripped or fitted count rates. In the second type of 
commercial logging operation for this purpose currently available, the use 
of five energy windows provides an overdetermined set of relations which 
may be used to statistically enhance the count rate information from each 
of the energy windows. Elemental concentrations may then be computed from 
the stripped or fitted count rates. However, no basic change from a 
comparison of unknown spectra with standard gamma ray spectra taken under 
standard borehole conditions is contemplated in either of the commercially 
available techniques at present. The method of the present invention uses 
multiple (6) windows for entirely different reasons. This photoelectric 
method to determine lithology is passive, that is, lacking in a gamma ray 
source as exemplified in the publication "the Lithodensity Log," SPWLA. 
1979 European Symposium by F. Felder & C. BOyeldieu. 
Changes in borehole conditions can introduce errors in concentration 
calculations of the elements that can approach an order of magnitude. Such 
errors, although large, were tolerable in early applications of the K-U-T 
log for minerals exploration. However, as the applications of such logs 
become more sophisticated, errors of this magnitude became unacceptable. 
In the present invention, a K-U-T log enhanced with borehole compensation 
by utilizing the response measured in a fourth energy window. Fifth and 
sixth energy windows are used in formation matrix type determination. The 
fourth, or compensation, window monitors the shape of the gamma ray 
spectrum caused by variations in borehole conditions. The response of the 
compensation window is then used to correct the response of the count rate 
in the first three energy windows to some standard borehole and formation 
geometry. The fifth window measured count rate is used with the sixth 
window count rate to determine formation matrix type or lithology. 
Referring now to FIG. 1, a natural gamma ray spectrum is illustrated 
schematically in which the gamma ray intensity or count rate is plotted as 
a function of gamma ray energy over the energy range from 0-3 MeV. Six 
energy windows contemplated for use, according to the present invention, 
are illustrated superimposed on the gamma ray spectrum of FIG. 1. The 
fifth and sixth windows will be defined in more detail later. The energy 
bands or windows labelled window 1, window 2 and window 3 are chosen to 
include the 2.61, 1.76 and 1.46 MeV gamma radiations from the decay of 
thorium (Tl.sup.208), uranium (Bi.sup.214), and potassium (K.sup.40). The 
total count rate recorded in each window may be expressed as given in 
Equations (1)-(4). 
EQU C.sub.1 =C.sub.1,T (1) 
EQU C.sub.2 =C.sub.2,U +K.sub.2,1,T C.sub.1,T (2) 
EQU C.sub.3 =C.sub.3,K +K.sub.3,1,T C.sub.1,T +K.sub.3,2,U C.sub.2,U (3) 
EQU C.sub.4 =K.sub.4,3,K C.sub.3,K +K.sub.4,2,U C.sub.2,U +K.sub.4,1,T 
C.sub.1,T (4) 
wherein Equations (1)-(4): 
C.sub.j =the count rate measured in window j 
j=1, . . . , 4; 
C.sub.j,k =the count rate contribution in window j due only to activity 
from the element k(K=T,U,K); 
K.sub.i,j,k ="stripping" constants defined as: 
EQU K.sub.i,j,k =C.sub.i,k /C.sub.j,k (5) 
The constants K.sub.i,j,k are measured in "standard" borehole conditions 
surrounded by formations, each of which contains only K, only U, or only 
Th. The constants K.sub.i,j,k are, therefore, known calibration constants. 
For standard borehole conditions, equations (1), (2) , and (3) are solved 
for C.sub.1,T, C.sub.2,U,C.sub.3,K by 
EQU C.sub.1,T =C.sub.1 (6) 
EQU C.sub.2,U =C.sub.2 -(K.sub.2,1,T C.sub.1,T) (7) 
EQU C.sub.3,K =C.sub.3 -(K.sub.3,1,T C.sub.1)-K.sub.3,2,U (C.sub.2 -K.sub.2,1,T 
C.sub.1) (8) 
where all terms on the right hand side of equations (6), (7) and (8) are 
either measured quantities (C.sub.1, C.sub.2, C.sub.3) or the several 
known calibration constants (K.sub.i,j,k). The relationship between 
C.sub.j,k and the corresponding elemental concentrations M.sub.K,M.sub.U 
and M.sub.T will be discussed later. 
In non-standard borehole conditions, the stripping constants K.sub.i,j,k 
measured in "standard" borehole conditions do not hold true if there is 
deviation from the standard configuration. Examples of stripping constants 
are shown in Table I for a selected group of Compton dominated energy 
ranges, with the "nonstandard" stripping constants, calculated using Monte 
Carlo techniques, being denoted by primes: 
Tool Diameter: 35/8" (centralized) 
Standard Borehole: 10" F.W. (fresh water) Filled (with 38% porosity oil 
sand formation). 
Non-Standard Borehole: 10", 51/2" F.W.CSG (casing)+CMT (cement) (with 38% 
porosity oil sand formation). 
TABLE 1 
______________________________________ 
Window Stripping Constants 
# Energy Range 
Standard K.sub.i,j,k 
Nonstandard K'.sub.i,j,k 
______________________________________ 
1 2.0-3.0 MeV K.sub.2,1,T = 0.118 
K'.sub.2,1,T = 0.130 
2 1.6-2.0 MeV K.sub.3,1,T = 0.157 
K'.sub.3,1,T = 0.235 
3 1.1-1.6 MeV K.sub.4,1,T = 0.357 
K'.sub.4,1,T = 0.529 
4 0.5-1.1 MeV K.sub.3,2,U = 0.406 
K'.sub.3,2,U = 0.388 
5 .15-0.5 MeV K.sub.4,2,U = 0.647 
K'.sub.4,2,U = 0.951 
K.sub.4,3,K = 0.657 
K'.sub.4,3,K = 0.955 
K.sub.5,1,T = 0.864 
K'.sub.5,1,T = 1.029 
K.sub.5,2,U = 1.647 
K'.sub.5,2,U = 1.650 
______________________________________ 
CORRECTION OF STRIPPING CONSTANTS FOR NON-STANDARD BOREHOLE CONDITIONS 
USING THE RESPONSE OF THE FOURTH WINDOW COUNT C4 
There are three major parameters which affect the stripping constants as 
borehole and, to a lesser extent, formation conditions change. They are 
given by: 
(a) .eta..ident..SIGMA. .rho. .chi. where .rho. and .chi. are the 
densities and effective thickness, respectively, of each intervening 
material such as borehole fluid, casing, and rock matrix between the 
detector within the sonde and the source of radiation; 
(b) E.sub.k .ident.the primary gamma ray energy from element k (see FIG. 
1); 
(c) .DELTA.E.sub.i,j .ident.the difference in the midpoint of energy window 
j and energy window i (see FIG. 1). 
The stripping constants K.sub.i,j,k obtained from a standard borehole at 
standard conditions must be corrected with functions of .eta., E.sub.k and 
.DELTA.E.sub.i,j to obtain correct stripped count rates C.sub.1,T and 
C.sub.2,U, and C.sub.3,K in non-standard boreholes. This operation can be 
expressed mathematically by rewriting Equations (1), (2) and (3). 
EQU C.sub.1,T =C.sub.1 (9) 
EQU C.sub.2,U =C.sub.2 -[L(.eta.)G.sub.2,1,T (E.sub.T, 
.DELTA.E.sub.2,1)]K.sub.2,1,T C.sub.1 (10) 
EQU C.sub.3,K =C.sub.3 -[L(.eta.)G.sub.3,1,T 
(E.sub.T,.DELTA.E.sub.3,1)]K.sub.3,1,T C.sub.1 -([L(.eta.)G.sub.3,2,U 
(E.sub.U,.DELTA.E.sub.3,2)]K.sub.3,2,U (C.sub.2 -[L(.eta.)G.sub.2,1,T 
(E.sub.T .DELTA.E.sub.2,1)]K.sub.2,1,T C.sub.1)) (11) 
Again, the terms C.sub.1, C.sub.2, and C.sub.3 are measured values while 
the constants K.sub.i,j,k are known calibration constants measured under 
standard borehole conditions. From Equations (9), (10) and (11), the 
problem is to determine the unknown or remaining stripping function 
correction term L(.eta.) and G.sub.i,j,k (E.sub.k,.DELTA.E.sub.i,j) where 
i,j, and k denote the same quantities as those used with the stripping 
constants K.sub.i,j,k. 
Physically, the product [L(.eta.)G.sub.i,j,k 
(E.sub.k,.DELTA.E.sub.i,j)]K.sub.i,j,k is simply a stripping constant for 
a nonstandard borehole condition, K'.sub.i,j,k. Using the data in Table 1, 
the ratio 
EQU K'.sub.i,j,k /K.sub.i,j,k =L(.eta.)G.sub.i,j,k (E.sub.k,.DELTA.E.sub.i,j) 
is plotted as a function of .DELTA.E.sub.i,j for k=T (thorium), k=U 
(uranium), and k=K (potassium) in FIG. 2. 
From FIG. 2, it can be seen that the midpoint E.sub.k and widths of windows 
1 through 4 were selected such that L(.eta.)G.sub.4,j,k 
(E.sub.k,.DELTA.E.sub.4,j) is essentially constant (=1.47) for all values 
of j and k. This is important considering the fourth window (i=4) is used 
as a "compensation" window to monitor the shape of the observed spectrum 
to thereby adjust the stripping constants for varying borehole conditions. 
This means that regardless of the relative concentrations of the K-U-T 
elements within the formation, the effects of the borehole on the 
stripping constant for each element will be reflected consistently in the 
fourth window. Mathematically, the effect can be seen as follows. 
The count rate in window four, for any borehole condition is Equation (4) 
rewritten as: 
EQU C.sub.4 =L(.eta.)G.sub.4,1,T (E.sub.T,.DELTA.E.sub.4,1)K.sub.4,1,T 
C.sub.1,T +L(.eta.)G.sub.4,2,U (E.sub.U,.DELTA.E.sub.4,2)K.sub.4,2,U 
C.sub.2,U +L(.eta.)G.sub.4,3,K (E.sub.K,.DELTA.E.sub.4,3)K.sub.4,3,K 
C.sub.3,K (12) 
where the constants K.sub.4,j,k are stripping constant measured in the 
"standard" borehole conditions surrounded by formations each of which 
contains only K, only U, or only Th. 
But from FIG. 2, we have seen that windows have been selected such that: 
EQU L(.eta.)G.sub.4,j,k (E.sub.k,.DELTA.E.sub.4,j).apprxeq.Constant (13) 
for all values of j and element k for this particular non-standard 
borehole. Although L(.eta.) will vary with changing borehole conditions, 
G.sub.4,j,k (E.sub.k,.DELTA.E.sub.4,j) will remain constant since it is a 
function only of E.sub.k and .DELTA.E.sub.4,j which, of course, are fixed 
on defining the energy windows. We can then write equation (13) for any 
borehole condition as: 
EQU L(.eta.)G.sub.4,j,k (E.sub.k,.DELTA.E.sub.4,j)=L(.eta.)G (14) 
where G is a constant (1.47 in this instance). Substituting (14) into (12) 
yields 
EQU C.sub.4 =L(.eta.)G[K.sub.4,1,T C.sub.1,T +K.sub.4,2,U C.sub.2,U 
+K.sub.4,3,K C.sub.3,K ] (15) 
FIG. 2A shows additional ratios K'.sub.i,j,T /K.sub.i,j,T obtained from 
experimental data and Monte Carlo calculations using tool, borehole, and 
formation parameters such that L(.eta.)G varied over a suitable range from 
0.87 to 7.06. These data indicate that, within the limits of statistical 
accuracy, the products L(.eta.)G.sub.i,j,k (E.sub.k,.DELTA.E.sub.i,j) can 
be represented by an equation of the form: 
EQU L(.eta.)G.sub.i,j,k 
(E.sub.k,.DELTA.E.sub.i,j)=1+F(E.sub.k,.DELTA.E.sub.i,j)m(L(.eta.)G-1 (16) 
where F(E.sub.K,.DELTA.F.sub.i,j).ident.a function only of E.sub.k and 
.DELTA.E.sub.i,j ; 
##EQU1## 
Fitting Equation (16) to the data shown in FIG. 2A gives: 
EQU .eta..perspectiveto.0.8 
EQU F(E.sub.T,.DELTA.E.sub.2,1).perspectiveto.0.22 
EQU L(.eta.)G.sub.2,1,T (E.sub.T,.DELTA.E.sub.2,1)=1+0.22m(L(.eta.)G-1).sup.0.8 
and F(E.sub.T,.DELTA.E.sub.3,1)=0.50 (17) 
EQU L(.eta.)G.sub.3,1,T 
(E.sub.T,.DELTA.E.sub.2,1)=1.0+0.50m(L(.eta.)G-1).sup.0.8 (18) 
Likewise, using uranium as a source, it can be shown that: 
EQU F(E.sub.U,.DELTA.E.sub.3,2)=0.08 
or 
EQU L(.eta.)G.sub.3,2,U 
(E.sub.U,.DELTA.E.sub.3,2)=1.0+0.08M(L(.eta.)G-1).sup.0.8 (19) 
Substituting Equations (17), (18), and (19) into Equations (9), (10), and 
(11) yields: 
EQU C.sub.1,T =C.sub.1 (20) 
EQU C.sub.2,U =C.sub.2 -(1+0.22m(L(.eta.)G-1).sup.0.8)K.sub.2,1,T C.sub.1 (21) 
EQU C.sub.3,K =C.sub.3 -[(1+0.50m(L(.eta.)G-1).sup.0.8)K.sub.3,1,T C.sub.1 
]-[(1+0.08(L(.eta.)G-1).sup.0.8)K.sub.3,2,U (C.sub.2 
-(1+0.50(L(.eta.)G-1).sup.0.8)K.sub.2,1,T C.sub.1)] (22) 
These three equations, along with Equation (15), now give four equations 
and four unknowns namely C.sub.1,T ; C.sub.2,U 1; C.sub.3,K ; and 
[L(.eta.)G]; dependent on C.sub.1, C.sub.2, C.sub.3, and C.sub.4 (which 
are measured) and the constants K.sub.i,j,k which are known calibration 
constants. 
For borehole conditions normally encountered, 0.5&lt;L(.eta.)G&lt;1.5. For this 
range of L(.eta.)G, the expressions for L(.eta.)G.sub.i,j,k 
(E.sub.k,.DELTA.E.sub.i,j) can be approximated, with good accuracy, by 
rewriting Equations (17), (18) and (19) as: 
EQU L(.eta.)G.sub.2,1,T (E.sub.T,.DELTA.E.sub.2,1).apprch.1+0.27(L(.eta.)G-1) 
(17a) 
EQU L(.eta.)G.sub.3,1,T (E.sub.T,.DELTA.E.sub.3,1).apprch.1+0.57(L(.eta.)G-1) 
(18a) 
EQU L(.eta.)G.sub.3,2,U (E.sub.U,.DELTA.E.sub.3,2).apprch.1+0.08(L(.eta.)G-1) 
(19a) 
yielding in Equations (20), (21), and (22): 
EQU C.sub.1,T .apprch.C.sub.1 (20a) 
EQU C.sub.2,U .apprch.C.sub.2 -(1+0.27(L(.eta.)G-1))K.sub.2,1,T C.sub.1 (21a) 
EQU C.sub.3,K .apprch.C.sub.3 -[(1+0.57(L(.eta.)G-1))K.sub.3,1,T C.sub.1 
]-[(1+0.08(L(.eta.)G-1))K.sub.3,2,U (C.sub.2 
-(1+0.27(L(.eta.)G-1))K.sub.2,1,T C.sub.1)] (22a) 
Note that in standard borehole conditions L(.eta.)G.sub.i,j,k =1, and this 
is reflected in Equation (16). 
The solution of the four simultaneous equations (that is, (16) to (19) 
inclusive) is time consuming. There is, however, an iteration technique 
which could also be used to obtain approximate real time solutions in 
logging operations. A flow chart or logic flow of the solution is shown in 
FIG. 3. Of course, with a computer with sufficient speed, the analytical 
solution can be used rather than the iterative approach described 
hereinafter. 
Upon entry into the iteration technique scheme at logic block 31, the 
function L(.eta.)G is set initially equal to 1.0 as a first guess (i.e. a 
value corresponding to standard borehole conditions). Control is then 
transferred to logic block 32 where computations of C.sub.1,T, C.sub.2,U 
and C.sub.3,K are made using Equations (20), (21), and (22) respectively. 
Substituting these variables, together with the assumed value of L(.eta.)G 
into Equation (15) results in a calculation of the expected count rate 
C.sub.4 ' obtained from window 4 (C.sub.4 ' replaces C.sub.4 in the 
equation to denote a calculated rather than a measured value). Control is 
then transferred to logic block 33 where a comparison test is made to 
determine if the computed C.sub.4 ' is about equal to the observed fourth 
window count rate C.sub.4. If C.sub.4 and C.sub.4 ' are within some 
preselected value .DELTA.C.sub.4 of each other, then the iteration scheme 
is complete and control is transferred to logic block 35 where the correct 
value of L(.eta.)G is output. If the test at block 33 does not pass, then 
non-standard borehole conditions are being encountered as indicated at 
logic block 36 where a second test to determine if C.sub.4 &gt;C.sub.4 '. If 
C.sub.4 &lt;C.sub.4 ' then control is transferred to block 37 where (at block 
38) the function L(.eta.)G is increased. If C.sub.4 &lt;C.sub.4 ' then 
control is transferred to logic blocks 39 and 40 where the function 
L(.eta.)G is decreased. Exit from either logic block 38 or 40 is to loop 
back to block 32 where another iteration is begun with the updated value 
of the function L(.eta.)G. In this manner, the function L(.eta.)G, for the 
particular borehole conditions being encountered may be determined. 
The rate at which the solution converges can be seen with the following 
exemplary well: 
EXAMPLE=a fresh water filled 51/2" casing in a 10" borehole with annular 
cement: 
______________________________________ 
C.sub.1 = 1.7 
These observed count rates are in 
C.sub.2 = 1.14 
arbitrary units but the relative 
C.sub.3 = 2.87 
elemental proportions are typical of 
C.sub.4 = 3.80 
those observed in an average kaolinite 
having 13 ppm of Th, 2 ppm of U, and 
0.42% K; 
______________________________________ 
setting .DELTA.L(.eta.)G=1.0 for a standard 10" open borehole filled with 
fresh water for which stripping constants are: 
______________________________________ 
K.sub.2,1,T = 0.118 
K.sub.4,1,T = 0.357 
K.sub.3,1,T = 0.157 
K.sub.4,2,U = 0.647 
K.sub.3,2,U = 0.406 
K.sub.4,3,K = 0.657 
L(.eta.).sup.--G 
C.sub.1,T C.sub.2,U 
C.sub.3,K 
C.sub.4 ' 
C.sub.4 
______________________________________ 
1.00 1.7 0.939 2.222 2.674 
3.80 
1.10 1.7 0.934 2.197 2.921 
3.80 
1.20 1.7 0.928 2.171 3.160 
3.80 
1.30 1.7 0.923 2.146 3.398 
3.80 
1.40 1.7 0.918 2.122 3.633 
3.80 
1.50 1.7 0.912 2.096 3.832 
3.80 
______________________________________ 
Therefore, L(.eta.)G=1.47 as the iterations converge. 
In many applications, iteration on each subsequent data set may 
conveniently begin in block 31 assuming as an initial value the L(.eta.)G 
obtained as a result of convergence on prior data set(s). In intervals of 
constant or slowly changing borehole conditions, this will result in more 
rapid convergence than if standard conditions were initially assumed. 
CONVERSION OF C.sub.1,T and C.sub.2,U and C.sub.3,K TO ELEMENTAL 
CONCENTRATIONS 
The stripped count rates C.sub.i,k are converted to the corresponding 
elemental concentrations M.sub.k using the relationship: 
EQU M.sub.k =C.sub.i,k /B(.eta.)Q.sub.k (23) 
where 
Q.sub.k (k=T,U,K)=calibration constants measured with the tool in a 
standard borehole surrounded by one of three formations containing known 
cocentrations of only Th, only U, or only K; 
B(.eta.)=a term which normalizes the calibration constants Q.sub.k, which 
were measured using standard borehole conditions, to borehole which are 
non-standard. 
Monte Carlo calculations have shown that, to a good approximation, .eta. 
can be computed from L(.eta.)G (which is obtained in the previously 
described iteration process) using the equation: 
EQU .eta.=(L(.eta.)G-0.093)/0.0324 (24) 
Note that the standard borehole L(.eta.)G=1 and .eta.=28. Also, Monte Carlo 
calculations show that the function form of B(.eta.) can be approximated 
by: 
EQU B(.eta.)=6.91 exp (-0.01+0.001.eta..sup.2) (25) 
To summarize, M.sub.k is computed from the corresponding stripped count 
rate C.sub.i,k as follows: 
(a) L(.eta.) G is obtained from the previously described iteration 
technique or direct solution of the set of four equations; 
(b) .eta. is computed from Equation (24) using L(.eta.)G; 
(c) B(.eta.) is computed from Equation (25) using .eta.; and 
(d) M.sub.k is computed from Equation (23) using B(.eta.) and the 
appropriate stripped count rate C.sub.i,k. 
The improvement in accuracy of the resulting M.sub.k values in non-standard 
boreholes can be demonstrated by again using hypothetical Monte Carlo data 
computed in a standard and non-standard borehole. 
For a (non-standard) 10" borehole containing a 51/2" fresh water filled 
casing and a cement annulus, the borehole compensated stripped count rates 
are: 
EQU C.sub.1,T =1.7, C.sub.2,U =0.92, C.sub.3,K =2.11 (26) 
with an iterated value of L(.eta.)G=1.47. Using Equation (24), one obtains 
EQU .eta.=42.5 (27) 
Using Equation (25) and the result of (27) yields: 
EQU B(.eta.)=0.599 (28) 
For the purposes of this hypothetical demonstration, assume: 
EQU Q.sub.k =1 for K=T, U, and K (29) 
Substituting the values from Equations (29), (28) and (26) into Equation 
(23) yields: 
EQU M.sub.T =2.83, M.sub.U =1.54, M.sub.K =3.52 (30) 
Monte Carlo calculations using the standard borehole geometry with 
identical elemental concentrations yielded values of: 
EQU M.sub.T =2.80, M.sub.u =1.52 M.sub.k =3.65 (31) 
thereby indicating good agreement. 
APPLICATION OF BOREHOLE COMPENSATION TECHNIQUE TO GAMMA RAY SPECTRA 
ANALYZED BY THE METHOD OF LEAST SQUARES FITTING 
Assuming standard borehole conditions, N.sub.i, the total number of gamma 
ray counts in energy channel i having a midpoint in gamma radiation energy 
band E.sub.i, is given by: 
EQU N.sub.i =.SIGMA..sub.k W.sub.k N.sub.i,k (32) 
where 
N.sub.i,k =number of gamma ray counts in energy channel i from the spectrum 
of element k, measured in standard borehole conditions (the "library" 
spectrum): 
W.sub.k =a term proportional to the concentration of element k within the 
formation. 
The terms of interest W.sub.k (related to the elemental concentrations) are 
determined using the least square criterion: 
##EQU2## 
where N.sub.i =the number of gamma ray counts observed in energy channel 
i. 
When non-standard borehole conditions are encountered. the standard or 
"library" spectra counts N.sub.i,k must be modified by 
EQU N'.sub.i,k =L(.eta.)G(E.sub.i)N.sub.i,k (34) 
where 
N'.sub.i,k =the library spectra counts in channel i from element k for 
non-standard borehole conditions; 
G(E.sub.i)=a term which is a function of the gamma ray energy E.sub.i 
recorded in the mid-point of energy chanel i. Again, using the 
least-square criterion: 
EQU .SIGMA..sub.i (N.sub.i -N.sub.i ').sup.2 =MINIMUM VALUE (35) 
where 
EQU N.sub.i '=.SIGMA..sub.k W.sub.k N'.sub.i,k (36) 
The terms G(E.sub.i) can be computed or measured. Therefore, the set of 
equations generated by the least squared criterion can be solved for 
W.sub.k and L(.eta.). 
Elemental concentrations, M.sub.k, are then computed from: 
EQU M.sub.k =W.sub.k /B(.eta.)Q.sub.k ' (37) 
where 
Q.sub.k '=calibration constants measured with the tool in a standard 
borehole surrounded by one of three formations containing known 
concentrations only Th, only U, or only K. 
For Equation (37), B(.eta.) and .eta. are given by Equations (25) and (24), 
respectively. 
Definition of the Fifth and Sixth Energy Windows Shown in FIG. 1 
The gamma ray energy spectrum of FIG. 1 was separated into six energy 
windows. The upper four windows were discussed hereinabove. The fifth 
window is located such that gamma radiation which is susceptible to 
attenuation primarily only through Compton scattering is detected therein, 
and also preferably low enough in energy that discernable spectral 
differences are not apparent for different sources (K-U-T). It encompasses 
a higher energy range relative to that in the sixth window, which is 
designed to emphasize photoelectric absorption effects in formation matrix 
elements. The fifth window is, therefore, selected where the attenuation 
is primarily due to Compton scattering, while the sixth window is selected 
in an energy range where photoelectric absorption from formation matrix 
elements such as calcium, silicon, and magnesium is significant. The count 
rate relationships in windows 5 and 6 of FIG. 1 are given by Equations 
(38) and (39): 
EQU C.sub.5 =f(M.sub.T,M.sub.U,M.sub.K,.eta.) (38) 
EQU C.sub.6 =f'(M.sub.T,M.sub.U,M.sub.K,.eta.,P) (39) 
In these equations, the concentrations of the K-U-T elements are common 
factors in both relationships. In addition, the attenuation .eta. is 
common to both relationships. The photoelectric attenuation factor P is 
discussed below. 
The probability of gamma ray photoelectric absorption in a material is 
related both to the energy (E.gamma.) of the gamma ray and the atomic 
numbers (Z) of the elements present. If this probability is expressed as a 
microscopic absorption cross-section, .sigma., then for element i: 
##EQU3## 
For a given gamma energy, this can be reduced to: 
EQU .sigma..sub.i =KZ.sub.i.sup.4.6 (41) 
K is a constant for a given gamma ray energy. 
For a molecule j composed of n.sub.i atoms of element Z.sub.i, then the 
microscopic molecular cross section is given by: 
##EQU4## 
If N represents the number of molecules per unit volume of a molecular 
material, then the total macroscopic photoelectric absorption cross 
section per unit volume (U) for a material is given by 
##EQU5## 
For sand, dolomite, and limestone U values for approximately 30 KeV gamma 
rays are 4.80 cm.sup.-1, 8.97 cm.sup.-1, and 13.76 cm.sup.-1 respectively. 
Hence it can be seen that U is substantially different for these rock 
matrices at low gamma energies. 
For a composite earth formation containing porosity .phi., the composite 
U.sub.FM is given by: 
EQU U.sub.FM =(1-.phi.)U.sub.MA+.phi.U.sub.F, (44) 
where U.sub.MA and U.sub.F are the U values for the matrix and fluid 
respectively. 
If the path followed by low energy scattered gamma rays from the source in 
the formation to the detector in the tool intersects several materials, j, 
each having photoelectric absorption cross section U.sub.j, then the total 
photoelectric attenuation factor, P, prior to detection is given by: 
##EQU6## 
where x.sub.j is the average path length traveled by the gamma rays in 
traversing material j between the source and detector. 
A ratio R of the observed gamma ray count rates in windows 6 and 5 can be 
obtained by combining equations (38) and (39): 
##EQU7## 
In light borehole mud environments and relatively good borehole conditions, 
the borehole terms and formation fluid terms in (46) are small relative to 
the formation matrix terms (x.sub.FM U.sub.MA (1-.phi.)). In many 
applications, spectral effects due to the relative source concentrations 
(M.sub.TH,M.sub.U,M.sub.K) are similar in windows 5 and 6, and hence R is 
not strongly affected by source type. In these instances 
EQU U.sub.ma =f"(R) (47). 
Hence the ratio R can be used directly to indicate formation lithology. 
If borehole terms are not negligible, then 
EQU U.sub.ma =f'" (48) 
(R, mudweight, caliper); and predetermined computer implemented corrections 
to the observed ratio R can be used, based on mud weight information and 
borehole size information from the caliper. 
The sensitivity of observed natural gamma spectra at low energies to 
different formation lithology types can be seen in FIG. 5. This figure 
shows spectra in test pit sand and limestone formations having somewhat 
similar natural gamma source distributions. Observe that at low energies 
there is a markedly greater attenuation of the observed gamma rays in the 
limestone formation, which has higher U.sub.ma. From this figure it is 
apparent that a ratio R, using the windows shown in the figure, will be 
sensitive to lithology. The ratio also serves to normalize differences in 
source strength in the different formations. 
Referring now to FIG. 6, a flow chart of the process for deriving the 
measurement of formation matrix type (U.sub.ma) is shown in more detail. 
The computer 54 of FIG. 1 may be programmed according to the flow chart of 
FIG. 6 to derive the lithology indicator just discussed as follows. 
Control is transferred to the program of FIG. 6 periodically from the main 
control program of the computer 54 of FIG. 1. For example, on a time or 
depth basis, the main control program of computer 54 enters the subroutine 
program of FIG. 6 to compute the formation matrix type indicator as a 
function of borehole depth. At logic block 61 the routine determines the 
count rates in energy windows 5 and 6 of FIG. 5 and control is transferred 
to logic block 62 where the ratio R is determined as indicated. 
Control is then transferred to logic block 63 where a test is performed to 
determine if borehole terms are negligible. If the borehole terms are 
negligible, control is passed to logic block 64 where the function 
U.sub.ma is computed according to equation (47). If borehole terms are not 
negligible as indicated by the test at block 63, then control is 
transferred to logic block 64 where caliper and mudweight data from the 
memory of computer 54 of FIG. 1 are utilized to determine U.sub.ma 
according to equation (48). In either event control is then returned to 
the main control program of the computer 54 of FIG. 1 and the lithology 
indicative parameter U.sub.ma is displayed as a function of borehole 
depth. 
WELL LOGGING SYSTEM 
Referring now to FIG. 4, a well logging system in accordance with the 
concepts of the present invention is illustrated schematically. An uncased 
well borehole 41 penetrates earth formation 46. The borehole 41 contains a 
well bore fluid 42 to control pressure in subsurface formations. Suspended 
in the borehole 41 by an armored well logging cable 54 is a sonde 43 
containing instrumentation for measuring gamma ray spectral 
characteristics of the earth formations 46 penetrated by the borehole 41. 
The sonde 43 contains a low atomic number pressure housing, such as the 
one described in U.S. patent filed June 16, 1982, Ser. No. 388,844. 
Signals from the downhole sonde 43 are conducted to the surface on 
conductors of the cable 56 and supplied to a surface computer 54 which 
performs the hereinbefore described signal processing techniques in order 
to extract the elemental constituents of potassium, uranium and thorium 
present in the earth formations 46 which are then recorded as a function 
of borehole depth on the recorder 55. The well logging cable 56 passes 
over a sheave wheel 44 which is electrically or mechanically coupled (as 
indicated by a dotted line 45) to the computer 54 and recorder 55 in order 
to provide depth information about the downhole sonde 43 for the surface 
recording process. The surface computer 54 may be a model PDP-11 provided 
by Digital Equipment Corp. of Cambridge, Mass. and can be programmed in a 
high level language such as FORTRAN to perform the previously described 
computations and to drive the output displays. 
The downhole sonde 43 contains, near the lower end thereof, a gamma ray 
detecting system comprising a scintillation crystal 47 and a 
photomultiplier and amplifier package 48, and may include a gain 
stabilization circuit. Power for the operation of the downhole sonde 
system is supplied from a surface power supply 53 via conductors of the 
cable 56 to a downhole power supply 51 where it is converted to 
appropriate voltage levels and supplied to the downhole circuitry 
components of the system as indicated in FIG. 4. Gamma ray spectral 
signals are supplied from the photomultiplier system 48 to a pulse height 
analyzer 49 where they are separated into count rates in the six energy 
windows hereinbefore described. The pulse height analyzer provides the six 
output signals corresponding to the count rates in each of the energy 
windows herein described to a telemetry system 50 where the pulse height 
gamma ray spectral information is converted to an appropriate wave form 
for transmission to the surface via conductors of the well logging cable 
56. Downhole control circuits 51 provide timing pulses to the pulse height 
analyzer and telemetry system in order to synchronize the transmission at 
regular data intervals from the downhole sonde 43 to the surface 
equipment. These synchronization signals are also encoded in the telemetry 
system 50 and supplied to the surface computer 54. 
Thus, naturally occurring gamma rays from the earth's formations 46 are 
detected by the scintillation crystal 47, photomultiplier detector system 
48 in the downhole sonde 43, broken down into their energy constituents by 
the pulse height analyzer 49 and telemetered to the surface by the 
telemetry system 50 on conductors of the armored well logging cable 56. At 
the surface, the signals are processed in accordance with the hereinbefore 
described techniques in order to extract the radioactive elemental 
constituency of earth formations 46 penetrated by the borehole 41, and to 
discern formation matrix type. 
The foregoing description may make other alternative arrangements according 
to the concepts of the present invention apparent to those skilled in the 
art.